This review is a part of the Ferromagnetic Semiconductor Spintronic Web Project.
Abstract The body of research on (III,Mn)V diluted magnetic semiconductors (DMSs) initiated during the 1990s has concentrated on three major fronts: (i) the microscopic origins and fundamental physics of the ferromagnetism that occurs in these systems, (ii) the materials science of growth and defects, and (iii) the development of spintronic devices with new functionalities. This article reviews the current status of the ﬁeld, concentrating on the ﬁrst two, more mature research directions. From the fundamental point of view, (Ga,Mn)As and several other (III,Mn)V DMSs are now regarded as textbook examples of a rare class of robust ferromagnets with dilute magnetic moments coupled by delocalized charge carriers. Both local moments and itinerant holes are provided by Mn, which makes the systems particularly favorable for realizing this unusual ordered state. Advances in growth and postgrowthtreatment techniques have played a central role in the ﬁeld, often pushing the limits of dilute Mnmoment densities and the uniformity and purity of materials far beyond those allowed by equilibrium thermodynamics. In (III,Mn)V compounds, material quality and magnetic properties are intimately connected. The present review focuses on the theoretical understanding of the origins of ferromagnetism and basic structural, magnetic, magnetotransport, and magnetooptical characteristics of simple (III,Mn)V epilayers, with the main emphasis on (Ga,Mn)As. Conclusions are arrived at based on an extensive literature covering results of complementary ab initio and effective Hamiltonian computational techniques, and on comparisons between theory and experiment. The applicability of ferromagnetic semiconductors in microelectronic technologies requires increasing Curie temperatures from the current record of 173 K in (Ga,Mn)As epilayers to above room temperature. The issue of whether or not this is a realistic expectation for (III,Mn)V DMSs is a central question in the ﬁeld and motivates many of the analyses presented in this review. 
Semiconductor physics and magnetism are established subﬁelds of condensedmatter physics that continue to reveal a rich variety of unusual phenomena, often in new types of solidstate materials. The properties of semiconductors are extraordinarily sensitive to impurity atoms, defects, and charges on external gates. Magnetism is a collective electronic phenomenon with an ordered state that is often stable to exceptionally high temperatures. Magnetic order, when it is present, has a large impact on other material properties including transport and optical properties. In both semiconductor and magnetic cases, sophisticated and economically important technologies have been developed to exploit the unique electronic properties, mainly for information processing in the case of semiconductors and for information storage and retrieval in the case of magnetism.
The realization of materials that combine semiconducting behavior with robust magnetism has long been a dream of material physics. One strategy for creating systems that are simultaneously semiconducting and magnetic, initiated in the late 1970s Jaczynski et al. (1978); Gaj et al. (1978), is to introduce local moments into wellunderstood semiconductors. The result is a new class of materials now known as diluted magnetic semiconductors (DMSs). Over the past 15 years, building on a series of pioneering publications in the 1990s Munekata et al. (1989); Ohno et al. (1992); Munekata et al. (1993); Ohno et al. (1996b); Hayashi et al. (1997); Van Esch et al. (1997); Ohno (1998), it has been established that several (III,V) compound semiconductors become ferromagnetic when heavily doped with Mn, and that the ferromagnetic transition temperatures can be well above 100 K. In semiconductors like GaAs and InAs, Mn has been shown to act both as an acceptor and as a source of local moments. These (III,Mn)V materials are examples of ferromagnetic semiconductors, a phrase we reserve for magnetic systems in which ferromagnetism is due primarily to coupling between magnetic element moments that is mediated by conductionband electrons or valenceband holes. This deﬁnition implies that, in ferromagnetic semiconductors, magnetic properties can be inﬂuenced by the same assortment of engineering variables that are available for other more conventional semiconductor electronic properties. In the bestunderstood arsenide DMSs, semiconductor valenceband carriers participate in the magnetic order. The materials require participation of valenceband holes for the formation of a ferromagnetic state. Efforts to increase their critical temperatures further run into incompletely understood fundamental limits on the ratio of the magnetic transition temperature to the Fermi temperature of the freecarrier systems and are also affected by the role of disorder in these heavily doped materials. The tension between achieving high Curie temperatures and the desire for low, and therefore gateable, carrier densities is among the major issues in the study of these materials.
In this article we review the considerable theoretical progress that has been made in understanding the very broad range of properties that occur in (III,Mn)V ferromagnetic semiconductor epilayers in different regimes of Mn content and defect density. The main focus of this article is on the extensively studied (Ga,Mn)As ferromagnetic semiconductor, but we also make frequent comments on other (III,Mn)V DMSs. Comparisons to experimental data are made throughout the article. In Section I we review progress that has been achieved in the effort to realize useful DMS materials for spintronics (or magnetoelectronics). In Section II we discuss the properties of dilute Mn atoms in a (III,V) crystal, and the various mechanisms that can couple the orientations of distinct moments and lead to ferromagnetism. In Section III we discuss several different strategies that can be used to elevate material modeling from a qualitative to a more quantitative level. Sections IV–VII address a variety of different characteristics of (III,Mn)V layers, including their structural, magnetic, magnetotransport, and magnetooptical properties. Finally, in Section VIII we discuss the ferromagnetic ordering physics in (III,Mn)V DMSs in the broad context of magnetic interactions in systems with coupled local and itinerant moments, and then extrapolate from (III,Mn)V materials to comment on the effort to ﬁnd hightemperature ferromagnetism in other DMS materials. We conclude in Section IX with a brief summary.
To partially remedy omissions in the bibliography that originate from our incomplete coverage of this topic, we refer to an extended database of published work and preprints maintained at http://unix12.fzu.cz/ms. The structure of the database is similar to the structure of this review and we encourage the reader in need of a more detailed bibliography to use this resource.
A number of review articles on various aspects of the physics of DMSs have been published previously and may help the reader who seeks a broader scope than we are able to supply in this review. The extensive body of research on DMSs in the 1980s, focused mostly on (II,Mn)VI alloys, has been reviewed by Furdyna and Kossut (1988); Furdyna (1988), and Dietl (1994). Several extended papers cover the experimental properties of (III,Mn)V DMSs, particularly (Ga,Mn)As and (In,Mn)As, interpreted within the carriermediated ferromagnetism model Ohno (1999); Matsukura et al. (2002); MacDonald et al. (2005). Theoretical predictions based on this model for a number of properties of bulk DMSs and heterostructures have been reviewed by Dietl (2002); Lee et al. (2002); König et al. (2003), and Dietl (2003). A detailed description of widebandgap and oxide DMSs can be found in Pearton et al. (2003); Graf et al. (2003b); Fukumura et al. (2004, 2005); Liu et al. (2005). We also mention here several specialized theoretical reviews focusing on the predictions of densityfunctional ﬁrstprinciples calculations for (III,Mn)V DMSs Sanvito et al. (2002); Sato and KatayamaYoshida (2002), on Mndoped IIVI and IIIV DMSs in the lowcarrierdensity regime Bhatt et al. (2002), and on effects of disorder in (Ga,Mn)As Timm (2003).
IIIV materials are among the most widely used semiconductors. There is little doubt that ferromagnetism in these materials would enable a host of new microelectronics device applications if the following criteria were met: (i) the ferromagnetic transition temperature should safely exceed room temperature, (ii) the mobile charge carriers should respond strongly to changes in the ordered magnetic state, and (iii) the material should retain fundamental semiconductor characteristics, including sensitivity to doping and light, and electric ﬁelds produced by gate charges. For more than a decade these three key issues have been the focus of intense experimental and theoretical research into the material properties of Mndoped IIIV compounds. At ﬁrst sight, fundamental obstacles appear to make the simultaneous achievement of these objectives unlikely. Nevertheless, interest in this quest remains high because of the surprising progress that has been achieved. Highlights of this scientiﬁc endeavor are brieﬂy reviewed in this introductory section.
Under equilibrium growth conditions the incorporation of magnetic Mn ions into IIIAs semiconductor crystals is limited to approximately 0.1%. Beyond this doping level, surface segregation and phase separation occur. To circumvent the solubility problem a nonequilibrium, lowtemperature molecularbeamepitaxy (LT MBE) technique was applied and led to the ﬁrst successful growth of (In,Mn)As and (Ga,Mn)As DMS ternary alloys with more than 1% Mn. Since the ﬁrst report in 1992 of a ferromagnetic transition in ptype (In,Mn)As at a critical temperature T_{c} = 7.5 K Ohno et al. (1992), the story of critical temperature limits in (III,Mn)V DMSs has unfolded in different stages. Initial experiments in (In,Mn)As suggested an intimate relation between the ferromagnetic transition and carrier localization, reminiscent of the behavior of manganites (perovskite (La,A)MnO_{3} with A=Ca, Sr, or Ba) in which ferromagnetism arises from a Zener doubleexchange process associated with delectron hopping between Mn ions Coey et al. (1999). (We comment at greater depth on qualitative pictures of the ferromagnetic coupling in Sections 2.1 and 8.1.) This scenario was corroborated by a pioneering theoretical ab initio study of the (In,Mn)As ferromagnet Akai (1998) and the mechanism was also held responsible for mediating ferromagnetic MnMn coupling in some of the ﬁrst ferromagnetic (Ga,Mn)As samples with T_{c}’s close to 50 K Van Esch et al. (1997).
In 1998 the Tohoku University group announced a jump of T_{c} in ptype (Ga,Mn)As to 110 K Ohno (1998) and pointed out that the criticaltemperature value was consistent with the kineticexchange mechanism for ferromagnetic coupling, also ﬁrst proposed by Zener (see Section 2.1). In its simplest form, ferromagnetism in this picture follows Dietl et al. (1997) from RudermanKittelKasuyaYosida (RKKY) indirect coupling between Mn dshell moments mediated by induced spin polarization in a freehole itinerantcarrier system (see Sections 2.1, 5.1.2, and 8.1). Zener proposed this mechanism originally for transitionmetal ferromagnets for which the applicability of this picture is now known to be doubtful because of the itinerant character of transitionmetal d electrons. The model of Mn(d^{5}) local moments that are exchangecoupled to itinerant spband carriers does, however, provide a good description of Mndoped IVVI and IIVI DMSs Dietl (1994). The key difference between (III,Mn)V materials like (Ga,Mn)As and IVVI and IIVI DMSs is that Mn substituting for the trivalent cation (Ga) is simultaneously an acceptor and a source of magnetic moments. Theoretical criticaltemperature calculations based on the kineticexchange model predict roomtemperature ferromagnetism in (Ga,Mn)As with 10% Mn content. In spite of these optimistic predictions, the goal of breaking the 110 K record in (Ga,Mn)As remained elusive for nearly four years. Only recently has progress in MBE growth and in the development of post growth annealing techniques Hayashi et al. (2001); Yu et al. (2002); Edmonds et al. (2002a); Chiba et al. (2003a); Ku et al. (2003); Eid et al. (2005) made it possible to suppress extrinsic effects, pushing T_{c} in (Ga,Mn)As up to 173 K Wang et al. (2005a); Jungwirth et al. (2005b). T_{c} trends in current high quality (Ga,Mn)As epilayers are consistent with the Zener kineticexchange model Jungwirth et al. (2005b). The current T_{c} record should be broken if DMS material with a higher concentration of substitutional Mn ions can be grown.
Based on the few experimental and theoretical studies reported to date, (III,Mn)Sb DMSs are expected to fall into the same category as (Ga,Mn)As and (In,Mn)As DMSs. The kineticexchange model calculations predict T_{c}’s that are small compared to their arsenide counterparts Dietl et al. (2000); Jungwirth et al. (2002b). This difference, conﬁrmed by experiment Abe et al. (2000); Wojtowicz et al. (2003); Panguluri et al. (2004); Csontos et al. (2005), is caused by the weaker pd exchange and smaller magnetic susceptibility (smaller effective mass) of itinerant holes in the largerunitcell antimonides. Also consistent with the kineticexchange model is the remarkable observation of an increase of T_{c} in (In,Mn)Sb by 25% induced by the applied hydrostatic pressure Csontos et al. (2005).
Moving in the opposite direction in the periodic table toward (III,Mn)P and (III,Mn)N appears to be the natural route to highT_{c} ferromagnetic semiconductors. The kineticexchange model predicts T_{c}’s far above room temperature in these smallerlatticeconstant materials, in particular, in (Ga,Mn)N Dietl et al. (2000). Also, the solubility limit of Mn is much larger than in arsenides, making it possible in principle to grow highlyMndoped DMSs under or close to equilibrium conditions. However, the nature of magnetic interactions in Mndoped phosphides and nitrides is not completely understood either theoretically or experimentally Liu et al. (2005). As the valenceband edge moves closer to the Mn d level and the pd hybridization increases with increasing semiconductor gap width and decreasing lattice constant, charge ﬂuctuations of d states may become large Sanyal et al. (2003); Wierzbowska et al. (2004); Sandratskii et al. (2004). With increasing ionicity of the host crystal, the Mn impurity may also undergo a transition from a d^{5} divalent acceptor to a d^{4} trivalent neutral impurity Luo and Martin (2005); Kreissl et al. (1996); Schulthess et al. (2005). In either case, the picture of ferromagnetism based on the Zener kineticexchange model needs to be reconsidered in these materials.
Experimental critical temperatures close to 1000 K have been reported in some (Ga,Mn)N samples Sasaki et al. (2002a). It is still unclear, however, whether the hightemperature ferromagnetic phase should be attributed to a (Ga,Mn)N ternary alloy or to the presence of ferromagnetic metal precipitates embedded in the host GaN lattice. Reports of (Ga,Mn)N epilayers synthesized in cubic and hexagonal crystal structures, of ptype and ntype ferromagnetic (Ga,Mn)N, and of multiple ferromagnetic phases in one material all add to the complex phenomenology of these widegap DMSs Korotkov et al. (2001); Graf et al. (2002, 2003a); Arkun et al. (2004); Edmonds et al. (2004c); Hwang et al. (2005); Edmonds et al. (2005b); Sawicki et al. (2005a).
Uncertainties apply also to the interpretation of ferromagnetism seen in the (Ga,Mn)P samples studied to date, which have been prepared by postMBE ion implantation of Mn followed by rapid thermal Theodoropoulou et al. (2002); Poddar et al. (2005) or pulselasermelting annealing Scarpulla et al. (2005). Experiments in these materials have not yet established unambiguously the nature of magnetic interactions in the (III,Mn)P compounds. However, a comparative study of (Ga,Mn)P and (Ga,Mn)As prepared by the postMBE ion implantation and pulselasermelting annealing suggests carriermediated origin of ferromagnetism in the (Ga,Mn)P material Scarpulla et al. (2005).
Our current understanding of the material physics of (III,Mn)V DMS epilayers suggests that synthesis of a roomtemperature ferromagnetic semiconductor will require a level of doping and defect control comparable to what has now been achieved in highquality (Ga,Mn)As samples, Mn densities of order 10%, and may require the use of widergap IIIV alloys.
Finally, we note that efforts to enhance the Curie temperature in Mndoped (III,V) semiconductors have also led to material research in more complex semiconductor heterostructures with highly Mndoped monolayers (δdoped layers), showing promising results Nazmul et al. (2005); Kawakami et al. (2000); Chen et al. (2002); FernándezRossier and Sham (2002); Nazmul et al. (2003); Sanvito (2003); Myers et al. (2004); Vurgaftman and Meyer (2001).
Spintronic devices exploit the electron spin to manipulate the ﬂow of electrons and therefore require materials in which the charge and spin degrees of freedom of carriers are strongly coupled Wolf et al. (2001); De Boeck et al. (2002); Zutic et al. (2004). The most robust, and currently the most useful, spintronic devices rely on the collective behavior of many spins in ferromagnetic materials to amplify the coupling of external magnetic ﬁelds to electronic spins, a coupling that is very weak for individual electrons. The intrinsically large spinorbit interaction in IIISb and IIIAs valenceband states makes these hosts ideal candidates for exploring various spintronic functionalities. In (Ga,Mn)As DMS epilayers, for example, the measured anisotropic magnetoresistance (AMR) effect (the relative difference between longitudinal resistivities for different magnetization orientations) can reach ~ 10% Wang et al. (2002); Baxter et al. (2002); Jungwirth et al. (2002a); Tang et al. (2003); Jungwirth et al. (2003b); Matsukura et al. (2004); Wang et al. (2005c); Goennenwein et al. (2005).
A particularly strong manifestation of valenceband spinorbit coupling occurs in the antisymmetric offdiagonal element of the resistivity tensor. The anomalous Hall effect (AHE) shown in Fig. 1, which completely dominates the lowﬁeld Hall response in (Ga,Mn)As and some other IIIV DMSs, has become one of the key tools used to detect the paramagnetic/ferromagnetic transition Ohno et al. (1992); Ohno (1998). Its large value is due to the spinpolarization of holes and provides strong evidence for the participation of mobile charge carriers in the ordered magnetic state of these DMSs.

In metals, the current response to changes in the magnetic state is strongly enhanced in layered structures consisting of alternating ferromagnetic and nonmagnetic materials. The giantmagnetoresistance effect Baibich et al. (1988), which is widely exploited in current technology, for example, in ﬁeld sensors and magnetic randomaccess memories, reﬂects the large difference between resistivities in conﬁgurations with parallel and antiparallel polarizations of ferromagnetic layers in magnetic superlattices or trilayers like spin valves and magnetic tunnel junctions Gregg et al. (2002). The effect relies on transporting spin information between layers and therefore is sensitive to spincoherence times in the system. Despite strong spinorbit coupling, which reduces spin coherence in DMSs, functional spintronic trilayer devices can be built, as demonstrated by the measured large MR effects in (Ga,Mn)Asbased tunneling structures Tanaka and Higo (2001); Chiba et al. (2004a); Saito et al. (2005); Mattana et al. (2005). The coercivities of individual DMS layers can be tuned via exchange biasing to an antiferromagnet Eid et al. (2004) which is a standard technique used in metal giantmagnetoresistance devices Gregg et al. (2002).
DMS ferromagnets possess all properties that are exploited in conventional spintronics. They qualify as ferromagnetic semiconductors to the extent that their magnetic and other properties can be altered by usual semiconductor electronics engineering variables. The achievement of ferromagnetism in an ordinary IIIV semiconductor that includes several percent of Mn demonstrates on its own the sensitivity of magnetic properties to doping. Remarkably, doping proﬁles and, correspondingly, magnetic properties can be grossly changed, even after growth, by annealing. Early studies of (Ga,Mn)As indicated that annealing at temperatures above the growth temperature leads to a reduction of magnetically and electrically active Mn ions and, at high enough annealing temperatures, to the formation of MnAs clusters Van Esch et al. (1997). On the other hand, annealing at temperatures below the growth temperature can substantially improve magnetic and transport properties of the thin DMS layers due to the outdiffusion of charge and momentcompensating defects, now identiﬁed as interstitial Mn Yu et al. (2002); Edmonds et al. (2002a); Chiba et al. (2003a); Ku et al. (2003); Eid et al. (2005).
(In,Mn)Asbased ﬁeldeffect transistors were built to study electric ﬁeld control of ferromagnetism in DMSs. It has been demonstrated that changes in the carrier density and distribution in thinﬁlm DMS systems due to an applied bias voltage can reversibly induce the ferromagnetic/paramagnetic transition Ohno et al. (2000). Another remarkable effect observed in this magnetic transistor is electricﬁeldassisted magnetization reversal Chiba et al. (2003b). This novel functionality is based on the dependence of the hysteresis loop width on bias voltage, again through the modiﬁed chargedensity proﬁle in the ferromagnetic semiconductor thin ﬁlm.

Experiments in which ferromagnetism in a (III,Mn)V DMS system is turned on and off optically add to the list of functionalities that result from the realization of carrierinduced ferromagnetism in a semiconductor host material Munekata et al. (1997); Koshihara et al. (1997). The observed emission of circularly polarized light from a semiconductor heterostructure, in which electrons (holes) injected from one side of the structure recombine with spinpolarized holes (electrons) emitted from a DMS layer Fiederling et al. (1999); Ohno et al. (1999), is an example of phenomena that may lead to novel magnetooptics applications.
Tunneling anisotropic magnetoresistance (TAMR) is another novel spintronic effect observed in (Ga,Mn)As Gould et al. (2004); Brey et al. (2004); Rüster et al. (2005); Saito et al. (2005). TAMR, like AMR, arises from spinorbit coupling and reﬂects the dependence of the tunneling density of states of the ferromagnetic layer on the orientation of the magnetization with respect to the current direction or crystallographic axes.
The larger characteristic electronic length scales in DMSs compared to ferromagnetic metals make it possible to lithographically deﬁne lateral structures with independent magnetic areas coupled through depleted regions that act as tunnel barriers and magnetic weak links. The electrical response to magnetization reversals in these spintronic nanodevices can lead to MR effects with magnitudes of order 1000% Rüster et al. (2003), as shown in Fig. 2, and with a rich phenomenology Giddings et al. (2005). Wider lateral constrictions have been used to demonstrate controlled domainwall nucleation and propagation in DMS stripes Rüster et al. (2003); Honolka et al. (2005), a prerequisite for developing semiconductor logic gates based on magnetic domain manipulation Gate and Register (2002); Allwood et al. (2005). (Ga,Mn)As nanoconstrictions with lateral side gates have revealed a new effect, Coulomb blockade anisotropic magnetoresistance, which reﬂects the magnetization orientation dependence of the singleelectron charging energy Wunderlich et al. (2006). These spintronic singleelectron transistors offer a route to nonvolatile, lowﬁeld, and highly electrosensitive and magnetosensitive operation.
The magnetic dipoledipole interaction strength between two discrete moments separated by a lattice constant in a typical solid is only ~ 1 K, relegating direct magnetic interactions to a minor role in the physics of condensedmatter magnetic order. Relativistic effects that lead to spinorbit coupling terms in the Hamiltonian provide a more plausible source of phenomena that are potentially useful for spintronics. Although these terms are critical for speciﬁc properties like magnetic anisotropy, they are rarely, if ever, crucial for the onset of the magnetic order itself. Instead the universal ultimate origin of ferromagnetism is almost always the interplay between the electronic spin degree of freedom, the repulsive Coulomb interactions between electrons, and the fermionic quantum statistics of electrons. The Pauli exclusion principle correlates the spin and orbital parts of the manyelectron wave function by requiring the total wave function to be antisymmetric under particle exchange. Whenever groups of electrons share the same spin state, the orbital part of the manybody wave function is locally antisymmetric, lowering the probability of ﬁnding electrons close together and hence the interaction energy of the system. Because magnetic order is associated with the strong repulsive Coulomb interactions between electrons, it can persist to very high temperatures, often to temperatures comparable to those at which crystalline order occurs. Ferromagnetism can be as strong as chemical bonds. Very often the quantum ground state of a manyelectron system has nonzero local spin density, aligned either in the same direction in space at every point in the system as in simple ferromagnets, or in noncollinear, ferrimagnetic, or antiferromagnetic materials in conﬁgurations in which the spin direction varies spatially.
Although this statement on the origin of magnetic order has very general validity, its consequences for a system of nuclei with a particular spatial arrangement, are extraordinarily difficult to judge. Because ferromagnetism is a strongcoupling phenomenon, rigorous theoretical analyses are usually not possible. There is no useful universal theory of magnetism. Understanding magnetic order in a particular system or class of systems can be among the most challenging of solid state physics problems Ashcroft and Mermin (1976); Marder (1999). For most systems, it is necessary to proceed in a partially phenomenological way, by identifying the local spins that order, and determining the magnitude and sign of the exchange interactions that couple them by comparing the properties of simpliﬁed (often ‘spinonly’) model Hamiltonians with experimental observations.
One approach that is totally free of phenomenological parameters is densityfunctional theory (DFT), including its spindensityfunctional (SDF) generalizations in which energy functionals depend on charge and spin densities. Although DFT theory is exact in principle, its application requires that the formalism’s exchangecorrelation energy functional be approximated. Approximate forms for this functional can be partially phenomenological (making a pragmatic retreat from the ab initio aspiration of this approach) and are normally based in part on microscopic calculations of correlation effects in the electrongas model system. This is the case for the oftenused local(spin)density approximation (L(S)DA) von Barth and Hedin (1972). For many magnetic metals, in which correlations are somewhat similar to those in the electrongas model system, ab initio LSDA theory provides a practical and sufficiently accurate solution of the magnetic manybody problem Jones and Gunnarsson (1989). This is particularly true for the elemental transitionmetal ferromagnets Fe, Co, and Ni and their alloys Moruzzi and Marcus (1993); Marder (1999). In practice LSDA theory functions as a meanﬁeld theory in which the exchange energy at each point in space increases with the selfconsistently determined local spin density. With increasing computer power, LSDA theory has been applied to more complex materials, including DMSs.
As we discuss below, both phenomenological and DFT approaches provide valuable insight into (III,Mn)V ferromagnetism. Model Hamiltonian theories are likely to remain indispensable because, when applicable, they provide more transparent physical pictures of ferromagnetism and often enable predictions of thermodynamic, transport, and other properties that are sometimes (depending on material complexity), beyond the reach of ab initio theory techniques. Of particular importance for DMSs is the capability of model Hamiltonians to describe localized electronic levels coincident with an itinerantelectron band which, strictly speaking, is beyond the reach of the effectively oneparticle band theories of solids that emerge from LSDA theory Anderson (1961); Schrieffer and Wolff (1966). Interpreting experiments with model Hamiltonian approaches can, on the other hand, be misleading if the model is too simpliﬁed and important aspects of the physics are absent from the model. What is more, even simpliﬁed models usually leave complex manybody problems that cannot be completely solved.
Magnetism in (Ga,Mn)As and some other (III,Mn)V ferromagnets originates from Mn local moments. (As already pointed out, Mndoped phosphides and nitrides DMSs are less well understood; however, local Mn moments are likely to play an important role in these materials as well.) The dependence of the energy of the system on the relative orientation of Mn moments is generally referred to as an exchange interaction. This terminology is part of the ‘jargon’ of magnetism and recognizes that Fermi statistics is the ultimate origin. Several types of qualitative effect that lead to exchange interactions can be separately identiﬁed when addressing magnetic order in (III,Mn)V DMSs; the applicability of each and the relative importance of different effects may depend on the doping regime and on the host semiconductor material. In this section we ﬁrst introduce the terminology that is commonly used in the magnetism literature, by brieﬂy reviewing some of the effects that can lead to magnetic coupling, most of which have been recognized since near the dawn of quantum mechanics.
For spins carried by itinerant electrons, exchange interactions are often most simply viewed from a momentum space rather than a realspace point of view. Stoner’s itinerant exchange Ashcroft and Mermin (1976) favors spontaneous spin polarization of the entire electron gas because electrons are less likely to be close together and have strongly repulsive interactions when they are more likely to have the same spin. Because the band energy is minimized by double occupation of each Bloch state, the Stoner ferromagnetic instability occurs in systems with a large density of states at the Fermi energy. This helps to explain, for example, why ferromagnetism occurs in the late 3d transition elements. A large density of states makes it possible to gain exchange energy by moving electrons from one spin band to the other while keeping the kineticenergy cost sufficiently low. Since the key spins in many (III,Mn)V DMS materials are localized the Stoner mechanism does not drive ferromagnetism, although we will see in Section 5.1 that it still plays a minor supporting role.
In many systems, including (III,Mn)V DMSs, both the local nature of the moments and strong local Coulomb interactions that suppress charge (valence) ﬂuctuations play a key role and have to be included even in a minimal model. Many mechanisms have been identiﬁed that couple localized spins in a solid. The origin of Heisenberg’s direct exchange Ashcroft and Mermin (1976) between two local spins is the difference between the Coulomb energy of a symmetric orbital wave function (antisymmetric singlet spin wave function) state and an antisymmetric orbital wavefunction (symmetric triplet spin wavefunction) state. Kramer’s superexchange interaction Anderson (1950), applies to local moments that are separated by a nonmagnetic atom. In a crystal environment, an electron can be transferred from the nonmagnetic atom to an empty shell of the magnetic atom and interact, via direct exchange, with electrons forming its local moment. The nonmagnetic atom is polarized and is coupled via direct exchange with all its magnetic neighbors. Whether the resulting superexchange interaction between local moments is ferromagnetic or antiferromagnetic depends on the relative sign of the two directexchange interactions Goodenough (1958); Kanamori (1959). In (III,Mn)V materials, superexchange gives an antiferromagnetic contribution to the interaction between Mn moments located on neighboring cation sites.
Zener’s doubleexchange mechanism Zener (1951b) also assumes an intermediate nonmagnetic atom. In its usual form, this interaction occurs when the two isolated magnetic atoms have a different number of electrons in the magnetic shell and hopping through the intermediate nonmagnetic atom involves magneticshell electrons. Combined with the onshell Hund’s rule, double exchange couples magnetic moments ferromagnetically. Parallel spin alignment is favored because it increases the hopping probability and therefore decreases the kinetic energy of spinpolarized electrons. A version of double exchange, in which Mn acceptor states form an impurity band with mixed spd character, has often been referred to in the (III,Mn)V literature. In this picture electrical conduction and MnMn exchange coupling are both realized through hopping within an impurity band. The potential importance of double exchange is greater at lower Mn doping and in widergap (III,Mn)V materials.
Finally, we identify Zener’s kineticexchange Zener (1951a) or indirectexchange interaction. It arises in models with local, usually dshell or fshell, moments whose coupling is mediated by s or pband itinerant carriers. The local moments can have a ferromagnetic directexchange interaction with band electrons on the same site and/or an antiferromagnetic interaction due to hybridization between the local moment and band electrons on neighboring sites Bhattacharjee et al. (1983); Dietl (1994). Polarization of band electrons due to the interaction at one site is propagated to neighboring sites. When the coupling is weak (the band carrier polarization is weak, e.g., at temperatures near the Curie temperature), the effect is described by RKKY theory, which was originally applied to carriermediated indirect coupling between nuclear moments Fröhlich and Nabarro (1940); Ruderman and Kittel (1954); Bloembergen and Rowland (1955); Yosida (1957) and between local dshell moments in metals Zener (1951a); Kasuya (1956); Yosida (1957). The range of this interaction can be long and interactions between separate local moments can be either ferromagnetic or antiferromagnetic and tend to vary in space on the length scale of the itinerant band’s Fermi wavelength. Unlike the doubleexchange case, magnetic order in this case does not lead to a signiﬁcant change in the width of the conducting band. This type of mechanism certainly does play a role in (III,Mn)V ferromagnetism, likely dominating in the case of strongly metallic (Ga,Mn)As, (In,Mn)As, and Mndoped antimonides. There is no sharp distinction between impurityband doubleexchange and kineticexchange interactions; the former is simply a strongcoupling, narrowband limit of the latter.
The starting point for developing a useful predictive model of (III,Mn)V ferromagnetism is achieving a full understanding of the electronic state with a single Mn impurity in the host lattice. We need to fully understand the character of the isolated local moments before we can critically discuss how they are coupled. The character of the local moment need not be the same in all (III,Mn)V materials. The remaining subsections will focus on properties of a Mn impurity in GaAs and on the nature of ferromagnetic coupling in (Ga,Mn)As and related arsenide and antimonide DMSs. At the end of this section, we comment on how things might change in widergap hosts like GaP and GaN.
Among all (III,V) hosts, Mn impurity has been studied most extensively in GaAs. The elements in the (Ga,Mn)As compound have nominal atomic structures [Ar]3d^{10}4s^{2}p^{1} for Ga, [Ar]3d^{5}4s^{2} for Mn, and [Ar]3d^{10}4s^{2}p^{3} for As. This circumstance correctly suggests that the most stable and therefore most common position of Mn in the GaAs host lattice is on the Ga site where its two 4s electrons can participate in crystal bonding in much the same way as the two Ga 4s electrons. The substitutional Mn_{Ga}, and the less common interstitial Mn_{I}, positions are illustrated in Fig. 3. Because of the missing valence 4p electron, the Mn_{Ga} impurity acts as an acceptor. In the electrically neutral state, labeled as A^{0}(d^{5} + hole), Mn_{ Ga} has the character of a local moment with zero angular momentum and spin S = 5∕2 (Landé g factor g = 2) and a moderately bound hole. The local moment is formed by three occupied spd bonding states with dominant t_{2g} (3d_{xy}, 3d_{xz}, 3d_{yz}) character and by two occupied e_{g} (3d_{x2y2}, 3d_{z2}) orbitals that are split from the t_{2g} states by the tetrahedral crystal ﬁeld and do not strongly hybridize with the sp orbitals. All occupied d orbitals have the same spin orientation and together comprise the S = 5∕2 local moment. The weakly bound hole occupies one of the three antibonding spd levels with dominant As 4p character. The charge e ionized Mn_{Ga} acceptor center, labeled as A^{}(d^{5}), has just the S = 5∕2 local spin character.

Electron paramagnetic resonance (EPR) and ferromagnetic resonance (FMR) experiments conﬁrm the presence of the A^{}(d^{5}) center through the entire range of Mn concentrations in both bulk and epilayer (Ga,Mn)As Almeleh and Goldstein (1962); Szczytko et al. (1999b); Sasaki et al. (2002b). The S = 5∕2 local moment on Mn was detected through a resonance line centered at g = 2 and, in lowMndensity samples, through a sextet splitting of the line due to the hyperﬁne interaction with the I = 5∕2 ^{55}Mn nuclear spin. The neutral Mn_{Ga} centers are more elusive because of nearly full compensation by unintentional donor impurities at low Mn concentrations and because of the metalinsulator transition at high Mn concentrations. Nevertheless, a multitude of experimental techniques, including EPR Schneider et al. (1987), infrared (IR) spectroscopy Chapman and Hutchinson (1967); Linnarsson et al. (1997), and magnetization measurements Frey et al. (1988), have detected the A^{0}(d^{5} + textrmhole) center in (Ga,Mn)As. Strikingly direct evidence was given by an STM experiment Yakunin et al. (2004b,a); Kitchen et al. (2005), shown in Fig. 4, in which the state of a single impurity atom was switched between the ionized A^{}(d^{5}) and neutral A^{0}(d^{5} + hole) states by applying a bias voltage that corresponded to a binding energy E_{b} ≈ 0.1 eV.

The binding energy E_{b} = 112.4 meV inferred from IR spectroscopy Linnarsson et al. (1997); Chapman and Hutchinson (1967) is consistent with the above STM measurement and with inferences based on photoluminescence experiments Lee and Anderson (1964); Schairer and Schmidt (1974). These observations identify Mn as a moderately shallow acceptor in GaAs whose band gap is E_{g} = 1.52 eV. The binding energy, which governs the electrical behavior of the Mn impurity, has contributions from both Coulomb attraction between the hole and the A^{}(d^{5}) core (and a central cell correction) and spindependent pd hybridization. The latter effect is responsible for the exchange interaction on which this review centers. We now discuss this parameter in more detail.
The top of the GaAs valence band is dominated by 4p levels which are more heavily weighted on As than on Ga sites. Direct exchange between holes near the top of the band and localized Mn d electrons is weak since Mn_{Ga} and As belong to different sublattices. This fact allows pd hybridization to dominate, explaining the antiferromagnetic sign of this interaction Bhattacharjee et al. (1983) seen in experiment Okabayashi et al. (1998).
There is a simple physical picture of the pd exchange interaction which applies when interactions are treated in a meanﬁeld way, and therefore also applies as an interpretation of LSDA calculations. Given that the ﬁlled, say spindown, Mn dshell level is deep in the valence band and that the empty spinup d level is above the Fermi level and high in the conduction band, hybridization (level repulsion of likespin states) pushes the energy of spindown valenceband states up relative to the energy of spinup valenceband states. The resulting antiferromagnetic coupling between valenceband states and local Mn spins is illustrated schematically in Fig. 5. The same basic picture applies for itinerant valenceband states in a heavily doped metallic DMS and for the acceptor state of an isolated Mn_{Ga} impurity. Note that the cartoon band structure in Fig. 5 is plotted in the electron picture while the DMS literature usually refers to the antiferromagnetic pd coupling between holes and local Mn moments. We comment in detail on the equivalent notions of pd exchange in the physically direct electron picture and the computationally more convenient hole picture for these ptype DMSs in Section 5.2.

We have already mentioned in Section 2.1 the conceptional inadequacy of effective singleparticle theories, including the LSDA, in dealing with local moment levels coincident with itinerantelectron bands. Anderson suggested a manybody model Hamiltonian that circumvents this problem by including explicitly the Coulomb correlation integral of localized electron states in the Hamiltonian Anderson (1961); Haldane and Anderson (1976); Fleurov and Kikoin (1976). The problem is that the change in the effective potential when the number of occupied localized orbitals changes by one, the Hubbard constant U, can be comparable to or larger than other band parameters, invalidating any meanﬁeldlike approach. The consequences of this fact can be captured at a qualitative level in models that include the Hubbard U Krstajić et al. (2004). In these phenomenological models, the localized orbital part of the Hamiltonian generally has an additional parameter, the Hund’s rule constant J_{H}. This parameter captures the local directexchange physics, which favors spinpolarized openshell atomic states. For the case of the Mn(d^{5}) conﬁguration, J_{H} forces all ﬁve singly occupied d orbitals to align their spins in the ground state. Recently, considerable effort has been devoted to developing approaches that combine the local correlation effects induced by the U and J_{H} terms in phenomenological models, with SDF theory Anisimov et al. (1991); Perdew and Zunger (1981); Park et al. (2000); Sandratskii et al. (2004); Petit et al. (2006); Wierzbowska et al. (2004); Filippetti et al. (2005); Schulthess et al. (2005). We comment on these ab initio techniques in Section 3.1.
When hybridization Slater and Koster (1954); Harrison (1980) between the localmoment and bandelectron states is weak it can be treated perturbatively. The SchriefferWolff transformation of the Anderson Hamiltonian,
removes the hybridization (the last term in Eq. (1) and leads to a model in which the localmoment spin interacts with the valence band via a spinspin interaction only, ∑ _{k′k}j_{k′k}s_{d} ⋅ s_{k′k}, with the number of electrons in each band ﬁxed Schrieffer and Wolff (1966). Here we assume for simplicity a single localized orbital and a single itinerant band, and use k to represent band states and d to represent the localized impurity state, s labels spin, ε_{α} is the single particle energy, and n_{α} = c_{α}^{†}c_{ α} and c_{α} are standard second quantization operators. This procedure is normally useful only if the hybridization is relatively weak, in which case it is not usually a problem to forget that the canonical transformation should also be applied to operators representing observables. Strictly speaking, the SchriefferWolff transformation also leads to a spinindependent interaction Schrieffer and Wolff (1966) which is normally neglected in comparison with the stronger spinindependent longrange part of the Coulomb potential.Since valenceband states of interest in GaAs, even in heavily doped samples, are near the Brillouinzone center Γ, the single phenomenological constant extracted from experiment for this interaction should be thought of as its value when both initial and ﬁnal states are at the Γ point. The quantity ∑ _{k′k}j_{k′k}s_{d}s_{k′k} is then approximated by J_{0}s_{d} ⋅ s_{k=0}, where
 (2) 
V _{pd} represents the As porbital–Mn dorbital hybridization potential (neglecting again the multipleorbital nature of p and d levels for simplicity), ε_{d} < 0 is the singleparticle atomiclevel energy of the occupied Mn d state measured from the top of the valence band, and ε_{d} + U > 0 is the energy cost of adding a second electron to this orbital Schrieffer and Wolff (1966). When the hole density is large or holes are more strongly localized near Mn acceptors, the crystalmomentum dependence of this interaction parameter cannot be entirely neglected Timm and MacDonald (2005).
Since J_{0} originates from hopping between Mn and neighboring As atoms, the pd exchange potential J(R_{i} r) produced by Mn impurity at site R_{i} has a range of order one lattice constant, and
 (3) 
In Eq. (3) we assumed that the perfectcrystal Bloch function, ψ_{n,k}(r) = exp(ikr)u_{n,k}(r), is composed of a slowly varying envelope function and a periodic function u_{n,k}(r) with the normalizations 1∕V ∫ dr ψ_{n,k}^{*}(r)ψ_{ n′,k′}(r) = δ_{n,n′}δ_{k,k′} and 1∕Ω_{u.c.} ∫ _{u.c.}dr u_{n}^{*}(r)u_{ n′}(r) = δ_{n,n′}. Here V is the crystal volume and Ω_{u.c.} is the unitcell volume. In GaAs, Ω_{u.c.} = a_{lc}^{3}∕4=0.045 nm^{3} and the lattice constant a_{lc} = 0.565 nm. These wave functions can be obtained from k ⋅ p theory which treats the band Hamiltonian of the system perturbatively, expanding around the Γ point Dietl et al. (2001b); Abolfath et al. (2001a). J_{0} in Eq. (3) corresponds to the average value of J(R_{i}  r) experienced by the k = 0 Bloch state over the ith unit cell.
The slowly varying envelope function experiences an effective zerorange pd exchange potential, since
The value of the J_{pd} constant is often considered to be independent of the host semiconductor Dietl et al. (2000, 2001b); Jungwirth et al. (2002b). Indeed, the increase of ∣V _{pd}∣^{2} ~ a_{ lc}^{7} Harrison (1980) in Eq. (2) with decreasing lattice constant is partly compensated by smaller Ω_{u.c.} (~ a_{lc}^{3}), and the increase of ∣1∕ε_{d}∣ in a widergap host is partly compensated by the decrease of the term 1∕(ε_{d} + U). Although it may have similar values in many materials, J_{pd} will tend to be larger in largergap, smallerlatticeconstant hosts.
In a virtualcrystal meanﬁeld approximation, the pd exchange potential due to the Mn impurities in a Ga_{1x}Mn_{x}As DMS, xN_{0}Ω_{u.c.} ∑ _{Ru.c.}J(R_{u.c.}  r)〈S〉⋅ s, has the periodicity of the host crystal. (Here 〈S〉 is the meanﬁeld Mn spin.) The valenceband states in this approximation experience an effective singleparticle kineticexchange ﬁeld, h_{MF} = N_{Mn}J_{pd}〈S〉, where N_{Mn} = xN_{0} is the Mn_{Ga} density.
Finally, we discuss the relationship between the exchange constant J_{pd} and exchange constant, ɛ, used to provide a k ⋅p interpretation of spectroscopic studies of the neutral A^{0}(d^{5} + hole) center. We emphasize that the use of a k ⋅p approach assumes that the bound hole is spread over at least several lattice constants in each direction. The fact that it is possible to achieve a reasonably consistent interpretation of detailed spectroscopic data in this way is in itself strong support for the validity of this assumption. Coupling between the weakly bound hole moment J and the local spin S of the Mn_{Ga} core is expressed as ɛS ⋅ J, where J = j + L, j is the (atomic scale) total angular momentum operator of the band hole at the Γ point (j = 3∕2 or 1/2 for the As 4p orbitals forming the band states near k = 0), and L is the additional (hole binding radius scale) angular momentum acquired by the hole upon binding to the Mn_{Ga} impurity. The IR spectroscopy data Linnarsson et al. (1997) have been analyzed Bhattacharjee and ŕ la Guillaume (2000) within a spherical approximation, i.e., considering only the L = 0, slike bound state. (Note that a sizable anisotropic dlike component in the boundhole ground state has been identiﬁed in the analysis of the STM data Tang and Flatté (2004); Yakunin et al. (2004b).) Further simpliﬁcation is achieved by neglecting the admixture of the two j = 1∕2 (j_{z} = ą1∕2) Γpoint states which is justiﬁed by the large spinorbit splitting Δ_{SO} = 341 meV of these states from two heavyhole states (j = 3∕2, j_{z} = ą3∕2) and two lighthole states (j = 3∕2, j_{z} = ą1∕2).
Writing the groundstate wave function as ψ_{jz}(r) = F_{jz}(r)u_{jz}(r) with a spherically symmetric envelope function F_{jz}(r) and for j_{z} = ą3∕2,ą1∕2 (j = 3∕2), the expectation value of the exchange potential reads
where ∣f(R_{I})∣^{2} = 〈F_{ jz}^{*}(r)F_{ jz}(r)〉_{u.c.} is the mean value of the slowly varying envelope function squared within the unit cell containing the Mn_{Ga} impurity. Equation (5) implies that
 (6) 
i.e., the ratio between ɛ and J_{pd} is determined by the strength of the hole binding to the A^{}(d^{5}) Mn_{ Ga} core and is larger for more localized holes.
A combination of IR data and theoretical calculations has been used to analyze this in more detail Linnarsson et al. (1997); Bhattacharjee and ŕ la Guillaume (2000). First of all, the value of the g factor g = 2.77 of the neutral A^{0}(d^{5} + hole) Mn_{ Ga} complex obtained from IR spectroscopy measurements is in agreement with the theoretical value expected for the total angular momentum state F = S  J = 1 of the complex, conﬁrming the antiferromagnetic character of the pd coupling between the hole and the local Mn spin. The contribution of the pd potential to the binding energy is then given by ɛS ⋅ J = ɛ[F(F + 1)  S(S + 1)  J(J + 1)]∕2, which for the F = 1 ground state gives 21ɛ∕4. The IR spectroscopy measurement of the splitting 2ɛ between the F = 1 and F = 2 states gives ɛ ≈ 5 meV, i.e., the contribution to the binding energy from the pd interaction is approximately 26.25 meV. The remaining binding energy, 112.4  26.25 = 86.15 meV, is due to the central ﬁeld potential of the impurity. Bhattacharjee and Benoit Bhattacharjee and ŕ la Guillaume (2000) used the hydrogenicimpurity model with a screened Coulomb potential and a central cell correction, whose strength was tuned to reproduce the value 86.15 meV, to obtain a theoretical estimate for ∣f(R_{I})∣^{2} ≈ 0.35 nm^{3}. From Eq. (6) they obtained J_{ pd} ≈ 40 meV nm^{3}. Given the level of approximation used in the theoretical description of the A^{0}(d^{5} + hole) state, this value is in a reasonably good agreement with the exchange constant value J_{pd} = 54 ą 9 meV nm^{3} (N_{ 0}β = 1.2 ą 0.2 eV) inferred from photoemission data Okabayashi et al. (1998). We note here that photoemission spectroscopy Okabayashi et al. (1998, 1999, 2001, 2002); Rader et al. (2004); Hwang et al. (2005) has represented one of the key experimental tools to study the properties of Mn impurities in DMSs, in particular, the position of the Mn d level and the strength and sign of the pd coupling. An indirect measurement of the J_{pd} constant is performed by ﬁtting the photoemission data to a theoretical spectrum of an isolated MnAs_{4} cluster Okabayashi et al. (1998). This procedure is justiﬁed by the shortrange character of the pd exchange interaction.
Most of the singleMnimpurity spectroscopic studies mentioned above were performed in samples with doping levels x < 0.1% for which the Ga_{1x}Mn_{x}As random alloy can be grown under equilibrium conditions. In these materials Mn can be expected to occupy almost exclusively the lowenergy Gasubstitutional position. Ferromagnetism, however, is observed only for x > 1% which is well above the equilibrium Mn solubility limit in GaAs and therefore requires a nonequilibrium growth technique (in practice lowtemperature MBE) to avoid Mn precipitation. The price paid for this is the occurrence of a large number of metastable impurity states. The most important additional defects are interstitial Mn ions and As atoms on cation sites (antisite defects). Both act as donors and can have a severe impact on the electric and magnetic properties of DMS epilayers. More unintended defects form at higher Mn doping because of the tendency of the material, even under nonequilibrium growth conditions, toward selfcompensation.
Direct experimental evidence for Mn impurities occupying interstitial (Mn_{I}) rather than substitutional positions was uncovered by combined channeling Rutherford backscattering and particleinduced xray emission measurements Yu et al. (2002). This technique can distinguish between Mn_{I} and Mn_{Ga} by counting the relative numbers of exposed Mn atoms and the ones shadowed by latticesite host atoms at different channeling angles. In highly doped asgrown samples, the experiment identiﬁed nearly 20% of Mn as residing on interstitial positions. The metastable nature of these impurities is manifested by the substantial decrease in their density upon postgrowth annealing at temperatures very close to the growth temperatures Yu et al. (2002); Edmonds et al. (2002a); Chiba et al. (2003a); Ku et al. (2003); Stone et al. (2003). Detailed resistancemonitored annealing studies combined with Auger surface analysis established the outdiffusion of Mn_{I} impurities toward the free DMS epilayer surface during annealing Edmonds et al. (2004a). The characteristic energy barrier of this diffusion process is estimated to be 1.4 eV. (Note that a factor of 2 was omitted in the original estimate of this energy by Edmonds et al. (2004a).)
Isolated Mn_{I} spectroscopy data are not available in (Ga,Mn)As, underlying the importance of theoretical work on the electric and magnetic nature of this impurity. Density functional calculations Ernst et al. (2005) suggest, e.g., that minorityspin Mn_{I} d states form a weakly dispersive band at ~ 0.5 eV below Fermi energy, a feature that is absent in the theoretical Mn_{Ga} spectra. Ab initio totalenergy calculations Máca and Mašek (2002); Mašek and Máca (2003) showed that Mn can occupy two metastable interstitial positions, both with a comparable energy, one surrounded by four Ga atoms (see Fig. 3) and the other surrounded by four As atoms. The two Mn_{I} states have similar local magnetic moments and electronegativity.
Calculations have conﬁrmed that Mn_{I} acts as a double donor, as expected for a divalent metal atom occupying an interstitial position. Each interstitial Mn therefore compensates two substitutional Mn acceptors. It seems likely that because of the strong Coulombic attraction between positively charged Mn_{I} and negatively charged Mn_{Ga} defects, mobile interstitials pair up with substitutional Mn during growth, as illustrated in Fig. 3 Blinowski and Kacman (2003). The total spin of a Mn_{Ga}Mn_{I} pair inferred from ab initio calculations is much smaller than the local spin S = 5∕2 of the isolated Mn_{Ga} acceptor. This property, interpreted as a consequence of shortrange antiferromagnetic interactions between twolocal moment defects that have comparable local moments, has been conﬁrmed experimentally Edmonds et al. (2005a). According to theory, the strength of this magnetic coupling contribution to the Mn_{Ga}Mn_{I} binding energy is 26 meV Mašek and Máca (2003).
Ab initio calculations of the valenceband spin splitting indicate that J_{pd} coupling constants of interstitial and substitutional Mn are comparable Mašek and Máca (2003). This would suggest a negligible net pd coupling between the antiferromagnetically coupled Mn_{Ga}Mn_{I} pair and valenceband holes. We do note, however, that these LSDA calculations give J_{pd} ≈ 140 meV nm^{3}, which is more than twice as large as the experimental value, and so this conclusion must be regarded somewhat cautiously. It is generally accepted that this discrepancy reﬂects the general tendency of the DFT LSDA and similar theories to systematically underestimate the splitting between occupied Mn d states in the valenceband continuum and empty Mn d states. Equation (2) illustrates how this deﬁciency of SDF band theories translates into an overestimated strength of the pd exchange interaction. Apart from this quantitative inaccuracy, the relative strength of the pd interaction of Mn_{I} compared to Mn_{Ga} is still a somewhat controversial issue in the theoretical literature Blinowski and Kacman (2003); Mašek and Máca (2003), which has not yet been settled experimentally. This property and many others related to magnetism are sensitive to details of the SDF implementation to the way in which disorder is accounted for, and even to technical details associated with the way in which the BlochSchrödinger equation is solved numerically. For example, the positive charge of the Mn_{Ga}Mn_{I} (singleacceptor doubledonor) pair, which will tend to reduce its exchange coupling with valenceband holes due to Coulomb repulsion, is not included in all approaches.
In addition to direct hole and localmoment compensation effects of Mn_{I} defects on ferromagnetism in (Ga,Mn)As, structural changes they induce in the crystal are indirectly related to important magnetic properties, particularly to the various magnetic and transport anisotropies. Ab initio theory predicts that the separation of the four nearest As neighbors surrounding the Mn_{I} in a relaxed lattice is increased by 1.5% compared to the clean GaAs lattice Mašek et al. (2003). Because of Coulomb repulsion between Ga cations and Mn_{I} defects, an even larger lattice expansion (~ 2.5%) is found for the Mn_{I} in the four Ga tetrahedral positions. When grown on a GaAs substrate, this effect of interstitial Mn leads to a latticematching compressive strain in the (Ga,Mn)As thin layers that induces a large uniaxial magnetic anisotropy, as discussed in detail in Section 5.3.
Lowtemperature growth of GaAs is known to lead to the incorporation of high As antisite defect levels. This property is a combined consequence of the nonequilibrium growth conditions and As overpressure often used in the MBE process to assure the twodimensional (2D) growth mode. These doubledonor defects are likely to also be present the (Ga,Mn)As epilayers and may contribute to hole compensation.
Unlike Mn_{I} impurities, As antisites are stable up to ~450 ^{∘}C Bliss et al. (1992). This is well above the temperature at which Mn precipitation starts to dominate the properties of (Ga,Mn)As and therefore As antisites cannot be removed from the epilayer by a postgrowth annealing treatment. Experimental studies suggest that the degradation of (Ga,Mn)As magnetic properties due to hole compensation by As antisites can be reduced by using As_{2} dimers instead of As_{4} tetramers and by maintaining a strictly stoichiometric growth mode Campion et al. (2003b).
The following elements of the qualitative picture of ferromagnetism in (Ga,Mn)As emerge from the experimental data and theoretical interpretations discussed in this section (see also Dietl (2002)). The lowenergy degrees of freedom in (Ga,Mn)As materials are the orientations of Mn local moments and occupation numbers of acceptor levels near the top of the valence band. The number of local moments participating in the ordered state and the number of holes may differ from the number of Mn_{Ga} impurities in the IIIV host due to the presence of charge and momentcompensating defects. Hybridization between Mn d orbitals and valenceband orbitals, mainly on neighboring As sites, leads to an antiferromagnetic interaction between the spins that they carry.
At low concentrations of substitutional Mn, the average distance between Mn impurities (or between holes bound to Mn ions) r_{c} = (3∕4πN_{Mn})^{1∕3} is much larger than the size of the bound hole characterized approximately by the impurity effective Bohr radius a^{*} = εˉh∕m^{*}e^{2}. Here N_{ Mn} = 4x∕a_{lc}^{3} is the number of Mn impurities per unit volume, ε and a_{lc} are the semiconductor dielectric function and lattice constant, respectively, and m^{*} is the effective mass near the top of the valence band. For this very dilute insulating limit, a theoretical concept was introduced in the late 1970s in which a ferromagnetic exchange interaction between Mn local moments is mediated by thermally activated band carriers Pashitskii and Ryabchenko (1979). Experimentally, ferromagnetism in (Ga,Mn)As is observed when Mn doping reaches approximately 1% Ohno (1999); Campion et al. (2003a); Potashnik et al. (2002) and the system is near the Mott insulatortometal transition, i.e., r_{c} ≈ a^{*} Marder (1999). At these larger Mn concentrations, the localization length of impurityband states is extended to a degree that allows them to mediate ferromagnetic exchange interaction between Mn moments, even though the moments are dilute. Several approaches have been used to address ferromagnetism in DMSs near the metalinsulator transition (for a review see Dietl (2002); Timm (2003), including ﬁnitesize exact diagonalization studies of holehole and holeimpurity Coulomb interaction effects Timm et al. (2002); Yang and MacDonald (2003), and the picture of interacting bound magnetic polarons or holes hopping within the impurity band Inoue et al. (2000); Litvinov and Dugaev (2001); Durst et al. (2002); Chudnovskiy and Pfannkuche (2002); Kaminski and Das Sarma (2002); Berciu and Bhatt (2001); Bhatt et al. (2002); Alvarez et al. (2002). Because of the hopping nature of conduction and the mixed spd character of impurityband states, the regime is sometimes regarded as an example of doubleexchange ferromagnetism.
At even higher Mn concentrations, the impurity band gradually merges with the valence band Krstajić et al. (2004) and impurity states become delocalized. In these metallic (Ga,Mn)As ferromagnets, on which we focus in the following sections, the coupling between Mn local moments is mediated by the pd kineticexchange mechanism Dietl et al. (1997); Matsukura et al. (1998); Jungwirth et al. (1999); Dietl et al. (2000); Jain et al. (2001). A qualitatively similar picture applies for (In,Mn)As and Mndoped antimonides. In the metallic limit the inﬂuence of Coulomb and exchange disorder on perfectcrystal valenceband states can be treated perturbatively.
The crossover from impuritybandmediated to Bloch valencebandmediated interactions between Mn moments is a gradual one. In the middle of the crossover regime, it is not obvious which picture to use for a qualitative analysis, and quantitative calculations are not possible within either picture. Strongly localized impurityband states away from Fermi energy may play a role in spectroscopic properties Okabayashi et al. (2001), even when they play a weaker role in magnetic and transport properties. The crossover is controlled not only by the Mn density but (because of the importance of Coulomb interaction screening) also by the carrier density. There is a stark distinction between the compensation dependence predicted by the impurityband and Bloch valenceband pictures. When the impurityband picture applies, ferromagnetism does not occur in the absence of compensation Kaminski and Das Sarma (2003); Das Sarma et al. (2003); Scarpulla et al. (2005) because the impurity band is ﬁlled. Given this, we can conclude from experiment that the impurityband picture does not apply to optimally annealed (weakly compensated) samples which exhibit robust ferromagnetism.
The phenomenology of Mndoped phosphide or nitride DMSs is more complex, with many aspects that probably cannot be captured by the Zener kineticexchange model Dietl et al. (2002); Krstajić et al. (2004). Experimentally, the nature of the Mn impurity is very sensitive to the presence of other impurities or defects in the lattice Korotkov et al. (2001); Graf et al. (2002, 2003a); Arkun et al. (2004); Edmonds et al. (2004c); Hwang et al. (2005). This substantially complicates the development of a consistent ferromagnetism picture in these materials. Larger pd coupling in widegap DMSs (reported, e.g., in the photoemission experiment Hwang et al. (2005)), and stronger bonding of the hole to the Mn ion, might shift the metalinsulator transition to higher Mn densities. At typical dopings of several percent of Mn, the impurity band is still detached from the valence band Kronik et al. (2002), and ferromagnetic MnMn coupling is mediated by holes hopping within the impurity band. Recent experiments indicate that this scenario may apply to (Ga,Mn)P with 6% Mn doping Scarpulla et al. (2005). LSDA calculations suggest Sanyal et al. (2003); Wierzbowska et al. (2004); Sandratskii et al. (2004) that pd hybridization can be so strong that the admixture of Mn 3d spectral weight at the Fermi energy reaches a level at which the system effectively turns into a dband metal. To illustrate this trend we show in Fig. 6 the LSDA and LDA+U calculations of the spinsplit total density of states (DOS), and in Fig. 7 the results for the Mn d state projected DOS in (Ga,Mn)As and (Ga,Mn)N Sandratskii et al. (2004). Indeed, in the widergap (Ga,Mn)N the spectral weight of Mn d orbitals at the Fermi energy is large and is not suppressed even if strong onsite correlations are accounted for by introducing the phenomenological Hubbard parameter in the LDA+U method (see next section for details on combined Hubbard model and SDF techniques).


Another possible scenario for these more ionic IIIP and IIIN semiconductors, supported by EPR and optical absorption measurements and ab initio calculations Graf et al. (2002); Luo and Martin (2005); Kreissl et al. (1996); Schulthess et al. (2005), involves a transition of the substitutional Mn from a divalent (d^{5}) impurity to a trivalent (d^{4}) impurity. This strongly correlated d^{4} center, with four occupied d orbitals and a nondegenerate empty d level shifted deep into the host band gap, may form as a result of a spontaneous (JahnTeller) lowering of the cubic symmetry near the Mn site. If the energy difference between divalent and trivalent Mn impurity states is small, the DMS will have a mixed Mn valence which evokes the conventional doubleexchange mechanism. Systems with dominant d^{4} character of Mn impurities, reminiscent of a chargetransfer insulator, will inevitably require additional charge codoping to provide for ferromagnetic coupling between dilute Mn moments Schulthess et al. (2005).
Our focus is on the theory of (III,Mn)V ferromagnets and we therefore present in this section an overview of the different approaches that can be used to interpret the existing experimental literature and to predict the properties of materials that might be realized in the future. Since the electronic and magnetic properties of ferromagnetic semiconductors are extremely sensitive to defects that are difficult to control in real materials and may not be completely characterized, the ability to make reasonably reliable theoretical predictions that are informed by as many relevant considerations as possible can be valuable to discover useful new materials. For example, we would like to make conﬁdent predictions of the ferromagnetic transition temperature on a (III,Mn)V material as a function of the density of substitutional Mn, interstitial Mn, codopants, antisite defects, and any other defects whose importance might be appreciated in the future. This ability is developing, although there is no simple silver bullet that solves all difficulties for all host materials, and there may still be some considerations that are important for less studied host materials and are not yet part of the discussion. In this section we address in a general way the strengths and weaknesses of different theoretical approaches. The following sections of the review compare predictions made with different types of theoretical approaches with experimental data on a variety of important electronic and magnetic properties.
In SDF theory Hohenberg and Kohn (1964); Kohn and Sham (1965) manybody effects are buried in a complex exchangecorrelation energy functional. Once an approximation is made for this functional, predictions for electronic and magnetic properties depend only on the particular arrangement of atomic nuclei under consideration. In principle nuclear positions can be relaxed to make sure that the spatial distribution of nuclei is metastable and therefore realizable. The exchangecorrelation energy functional leads to a selfconsistently determined spindependent exchangecorrelation potential that appears in an effective independentparticle Hamiltonian. The main technical challenge in DFT applications is the development of numerically efficient methods that provide accurate solutions of singlebody Schrödinger equations (see review articles Jones and Gunnarsson (1989); Sanvito et al. (2002)). DFT is established as a ﬂexible and valuable tool for studying the microscopic origins of magnetism and for predicting electronic, magnetic, and groundstate structural properties in a wide variety of materials Jones and Gunnarsson (1989); Moruzzi and Marcus (1993). It has the advantage that it is a ﬁrstprinciples approach without any phenomenological parameters. DFT falls short of being a complete and general solution to the manyelectron problem only because the exact form of the exchangecorrelation energy functional is unknown. A simple and widely successful approximation is the LSDA von Barth and Hedin (1972).
The problem of solving LSDA equations with adequate accuracy remains a challenge even in perfectly ordered crystals. In DMSs the degrees of freedom that are important for ferromagnetism, the orientations of the Mn local moments, typically reside on approximately 1/40 of the atomic sites, which further complicates numerical implementation of the LSDA technique. Other length scales that are characteristic for the physics of interest in these materials, like the Fermi wavelengths of valenceband carriers, are also longer than the atomic length scale on which DFT interrogates matter. This property limits the number of independent magnetic degrees of freedom that can be included in a DFT simulation of DMS materials. The problem is exacerbated by the alloy disorder in (Ga,Mn)As. Even if all Mn atoms substitute for randomly chosen lattice sites, it is necessary to ﬁnd a way to average over microrealizations of the alloy.
Disorderaveraging coherentpotential approximation (CPA) Soven (1967); Velický et al. (1968) and supercell approaches have been used successfully in combination with DFT calculations to address those physical parameters of (III,Mn)V DMSs that are derived from totalenergy calculations, such as lattice constants, formation and binding energies of various defects, and type of magnetic order (see e.g. Park et al. (2000); van Schilfgaarde and Mryasov (2001); Sanvito et al. (2002); Máca and Mašek (2002); Erwin and Petukhov (2002); Sato et al. (2003); Edmonds et al. (2004a); Sandratskii et al. (2004); Wierzbowska et al. (2004); Mahadevan et al. (2004); Petit et al. (2006); Xu et al. (2005); Luo and Martin (2005)).
Supercell calculations have usually studied interactions between Mn moment orientations by comparing the energies of parallelspin and oppositespin orientation states in supercells that contain two Mn atoms. An effective spin Hamiltonian can be extracted from this approach if it is assumed that interactions are pairwise and of Heisenberg form. Even when these assumptions are valid, the interaction extracted from these calculations is the sum of interactions at a set of separations connected by the supercell lattice vector. If the MnMn spin interaction has a range larger than a couple of lattice constants, this poses a problem for the supercell approach. Longerrange interactions can, however, be estimated using a spinwave approach which allows spinorientation variations that are incommensurate with the supercell. Effective interactions extracted in this way lead to a classical Heisenberg model from which the critical temperature can usually be calculated without substantial further approximation Sandratskii et al. (2004); Xu et al. (2005). The CPA approach can estimate the energy cost of ﬂipping a single spin in the ferromagnetic ground state, which is proportional to the meanﬁeld approximation for the critical temperature of the effective Heisenberg model Sandratskii et al. (2004); Sato et al. (2003), and in this sense is limited in its predictive powers when meanﬁeld theory is not reliable. Alternatively, a more detailed picture of magnetic interactions is obtained by direct mapping of the CPA total energy to the Heisenberg Hamiltonian Kudrnovský et al. (2004); Liechtenstein et al. (1987); Bergqvist et al. (2004).
LSDA predictions for spectral properties, like the local DOS, are less reliable than predictions for totalenergyrelated properties. This is especially true for states above the Fermi energy, and is manifested by a notorious inaccuracy in predicting semiconductor band gaps. From a DFT point of view, this inconsistency arises from attempting to address the physics of quasiparticle excitations using groundstate DFT. In Mndoped DMSs, the LSDA also fails to account for strong correlations that suppress ﬂuctuations in the number of electrons in the d shell. One generally accepted consequence is that the energy splitting between the occupied and empty d states is underestimated in SDF theory, leading to an unrealistically large dstate local DOS near the top of the valence band and to an overestimate of the strength of the pd exchange.
LDA+U Anisimov et al. (1991) and selfinteractioncorrected (SIC) LSDA Perdew and Zunger (1981) schemes have been used to obtain more realistic energy spectra and help to establish theoretically microscopic origins of ferromagnetism in (III,Mn)V semiconductor alloys Park et al. (2000); Shick et al. (2004); Sandratskii et al. (2004); Petit et al. (2006); Filippetti et al. (2005); Wierzbowska et al. (2004); Schulthess et al. (2005). LDA+U schemes used in studies of (III,Mn)V DMSs combine SDF theory with the Hubbard description of strongly correlated localized orbitals. Additional parameters from the Hubbard model are added to the energy functional; they are obtained by ﬁtting to experiment or, in principle, can be calculated selfconsistently Anisimov et al. (1991). The SIC LSDA method is based on realizing that spurious selfinteractions present in the SDF lead to unphysically large energy penalties for occupying localized states. Subtracting these interactions of a particle with itself from the density functional suppresses the tendency of the LSDA to delocalize strongly correlated atomic orbitals.
A practical approach that circumvents some of the complexities of this strongly correlated manybody problem is based on the Anderson manybody Hamiltonian theory Anderson (1961); Haldane and Anderson (1976); Fleurov and Kikoin (1976); Krstajić et al. (2004) and a tightbindingapproximation (TBA) bandstructure theory Slater and Koster (1954); Harrison (1980). The TBA Hamiltonian includes the 8×8 sp^{3} term with secondneighborinteraction integrals describing the host semiconductor Talwar and Ting (1982) and terms describing hybridization with nonmagnetic impurities and Mn. Effective singleparticle TBA theory is obtained from the Anderson Hamiltonian by replacing the density operators in the Hubbard term in Eq. (1) with their mean values Mašek (1991). In the TBA model, local changes of the crystal potential at Mn and other impurity sites are represented by shifted atomic levels. The parameters chosen for the atomiclevel shifts and hopping amplitudes between atoms can be inferred from experiment in a manner which corrects for some of the limitations with LSDA theory. The parametrization, summarized by Talwar and Ting (1982) and Mašek (1991), provides the correct band gap for the host crystal and appropriate exchange splitting of Mn d states. In the calculations, the hole density can be varied independently of Mn doping by adding nonmagnetic donors (e.g., Si or Se in GaAs) or acceptors (e.g. C or Be in GaAs).
Although the TBA model is a semiphenomenological theory, it shares with ﬁrstprinciples theories the advantage of treating disorder microscopically. A disadvantage of the tightbinding model approach, which is often combined with the CPA, is that it normally neglects Coulomb interaction effects which inﬂuence the charge and spin densities over several lattice constants surrounding Mn ion positions. Curie temperatures, magnetizations, the lifetimes of Bloch quasiparticle states, the effects of doping and disorder on the strength of pd exchange coupling, and the effective MnMn magnetic interaction are among the problems that have been analyzed using this tool Blinowski and Kacman (2003); Jungwirth et al. (2003a); Timm and MacDonald (2005); Jungwirth et al. (2005b,a); Sankowski and Kacman (2005); Vurgaftman and Meyer (2001); Tang and Flatté (2004).
The highest critical temperatures in (Ga,Mn)As DMSs are achieved in optimally annealed samples and at Mn doping levels above 1.5% for which band holes are itinerant, as evidenced by metallic conductivities Campion et al. (2003a). In this regime, semiphenomenological models that are built on crystal Bloch states rather than localized basis states for the band quasiparticles might be expected to provide more useful insights into magnetic and magnetotransport properties. A practical approach to this type of modeling starts from recognizing that the length scales associated with holes in DMS compounds are still long enough that a k ⋅ p, envelope function description of semiconductor valence bands is appropriate. Since for many properties it is necessary to incorporate spinorbit coupling in a realistic way, six or eightband KohnLuttinger (KL) k ⋅ p Hamiltonians that include the spinorbit splitoff band are desirable Luttinger and Kohn (1955); Vurgaftman et al. (2001).
The kineticexchange effective Hamiltonian approach Zener (1951a); Bhattacharjee et al. (1983); Furdyna (1988); Dietl (1994) asserts the localized character of the ﬁve Mn_{Ga} d orbitals, forming a moment S = 5∕2, and describes hole states in the valence band using the KL Hamiltonian and assuming the pd exchange interaction between Mn_{Ga} and hole spins. As discussed in Section 2.2.1, the exchange interaction follows from hybridization between Mn d orbitals and valenceband p orbitals. The approach implicitly assumes that a canonical transformation has been performed which eliminated the hybridization Schrieffer and Wolff (1966); Timm (2003). The k ⋅ p approximation applies when all relevant wave vectors are near the Brillouinzone center and the model also assumes from the outset that states near the Fermi energy mainly have the character of the host semiconductor valence band, even in the neighborhood of a substitutional Mn. When these assumptions are valid it follows from symmetry considerations that the spindependent part of the effective coupling between Mn and band spins is an isotropic Heisenberg interaction characterized by a single parameter. If the KL Hamiltonian parameters are taken from the known values for the host IIIV compound Vurgaftman et al. (2001), the strength of this exchange interaction J_{pd} can be extracted from one set of data, for example, from spectroscopic studies of isolated Mn acceptors as explained in Section 2.2.1, and used to predict all other properties Dietl et al. (1997); Matsukura et al. (1998); Jungwirth et al. (1999); Dietl et al. (2000); König et al. (2000). Since the value of J_{pd} can be obtained from experiments in a paramagnetic state the approach uses no free parameters to model ferromagnetism in these systems. In the absence of an external magnetic ﬁeld the KL kineticexchange Hamiltonian has the general form
 (7) 
where ℋ_{holes} includes the k ⋅ p KL Hamiltonian and the interactions of holes with the random disorder potential and with other holes. The second term in Eq. (7) represents the pd exchange interaction between local Mn spins S_{I} and hole spins s_{i}.
The k ⋅ p approach has the advantage that it focuses on the magnetic degrees of freedom introduced by dilute moments, which can simplify analysis of the model’s properties. Disorder can be treated in the model by introducing Bornapproximation lifetimes for Bloch states or by more sophisticated, exactdiagonalization or Monte Carlo methods Jungwirth et al. (2002a); König et al. (2003). This approach makes it possible to use standard electrongas theory tools to account for holehole Coulomb interactions Jungwirth et al. (1999). The envelope function approximation is simply extended to model magnetic semiconductor heterostructures, like superlattices or quantum wells Lee et al. (2002); Kechrakos et al. (2005); Souma et al. (2005); Frustaglia et al. (2004); Brey and Guinea (2000); Lee et al. (2000). This strategy will fail, however, if the pd exchange is too strong and the Mn acceptor level is correspondingly too spatially localized or too deep in the gap. For example, Mndoped GaP and GaN compounds are likely less favorable for this approach than (III,Mn)As and (III,Mn)Sb compounds. Generally speaking, the advantages of a fully microscopic approach have increasing importance for more localized acceptors, and hence shorterrange MnMn interactions, while the advantages of the k ⋅ p approach are more clear when acceptors are more shallow and MnMn interactions have longer range.
There has also been theoretical work on (III,Mn)V DMS materials based on still simpler models in which holes are assumed to hop between Mn acceptor sites, where they interact with Mn moments via phenomenological exchange interactions Berciu and Bhatt (2001); Chudnovskiy and Pfannkuche (2002); Bhatt et al. (2002); Alvarez et al. (2002); Mayr et al. (2002); Fiete et al. (2003). Hamiltonians used in these studies have the form
 (8) 
or something similar, where ĉ_{iσ}^{†} creates a hole at site i with spin σ, the hole spin operator σ_{I} = ĉ_{Iα}^{†}σ_{ αβ}ĉ_{Iβ}, and σ_{αβ} are the Pauli matrices. The models apply at least qualitatively in the lowMndensity limit and are able to directly attack the complex and intriguing physics of these unusual insulating ferromagnets. Insulating ferromagnetism persists in this limit even when the carrier density is not strongly compensated Fiete et al. (2003). The dependence of ferromagnetism in this regime on the degree of compensation has not yet been systematically studied experimentally and seems certain to pose challenging theoretical questions. The models that have been used to study ferromagnetism in this regime can easily be adapted to include holes that are localized on ionized defects which may occur in addition to Mn acceptors.
Other related models assume that Mn acceptors are strongly compensated so that the density of localized holes is much smaller than the density of Mn ions, leading to a polaronic picture in which a single hole polarizes a cloud of Mn spins Kaminski and Das Sarma (2002); Durst et al. (2002). The freeparameter nature of these phenomenological models means that they have only qualitative predictive power. They are not appropriate for highT_{c} (Ga,Mn)As materials which are heavily doped by weakly compensated Mn acceptors and are metallic. On the other hand, the impurityband models may represent a useful approach to address experimental magnetic and transport properties of ferromagnetic (Ga,Mn)P where holes are more strongly localized Scarpulla et al. (2005); Kaminski and Das Sarma (2003); Das Sarma et al. (2003).
Experimental efforts to increase Mn doping in (Ga,Mn)As DMSs beyond the solubility limit of 0.1% have been assisted by modern ab initio theoretical studies of impurity formation energies and effects related to the growth kinetics Erwin and Petukhov (2002); Mašek et al. (2002); Mašek and Máca (2003); Mašek et al. (2004); Mahadevan and Zunger (2003); Edmonds et al. (2004a). Substitution of Ga by Mn is expected, based on these studies, to be enhanced when the Ga chemical potential is kept low relative to the Mn chemical potential, i.e., under Gapoor, Mnrich growth conditions Mahadevan and Zunger (2003). Calculations also suggest that one of the major drawbacks of bulk growth techniques is that they allow phaseseparated precipitates, such as MnAs, to attain their most stable freestanding lattice geometry, leading to relatively low formation energies for these unwanted phases Mahadevan and Zunger (2003).
In thinﬁlm epitaxy, competing phases are forced to adopt the crystal structure of the substrate, which can signiﬁcantly increase their formation energies Mahadevan and Zunger (2003). Nonequilibrium LT MBE has been a particularly successful growth technique which allows a synthesis of singlephase (Ga,Mn)As DMSs with Mn concentrations up to ~ 10%. If the growth temperature is precisely controlled, a 2D growth mode of uniform DMSs can be maintained, and at the same time a large number of Mn atoms is incorporated in Mn_{Ga} positions Foxon et al. (2004).
In (Ga,Mn)As DMSs a signiﬁcant fraction of Mn atoms is also incorporated in interstitial positions Yu et al. (2002). Adsorption pathways that can funnel Mn to interstitial sites have been identiﬁed theoretically (see Fig. 8) using ab initio calculations of the potentialenergy surface of Mn adsorbed on GaAs(001) Erwin and Petukhov (2002). Firstprinciples calculations have also conﬁrmed that interstitial Mn_{I} impurities are metastable in GaAs, showing that the three distinct positions they can occupy are two tetrahedral T (As_{4} or Ga_{4}) positions surrounded by four nearneighbor As or Ga atoms, and one hexagonal position with three Ga and three As nearest neighbors. Among the three interstitial sites the hexagonal position is clearly less favorable, especially so in an overall ptype (Ga,Mn)As material Mašek and Máca (2003). The typical energy barrier for Mn diffusion between interstitial sites is approximately 1 eV Mašek and Máca (2003); Edmonds et al. (2004a). On the other hand, diffusion of Mn between Ga substitutional positions involves a kickout mechanism of Mn_{I} + Ga_{Ga} → Mn_{Ga} + Ga_{I} for which the typical barrier is about 3 eV Erwin and Petukhov (2002). Interstitial Mn_{I} is therefore much more mobile than substitutional Mn_{Ga}.

As mentioned in Section 2.3.1, Mn_{I} donors are likely to form nearneighbor pairs with Mn_{Ga} acceptors in asgrown materials due to the strong Coulomb attraction. The net magnetic moment of such a pair is close to zero Blinowski and Kacman (2003); Mašek and Máca (2003); Edmonds et al. (2004a, 2005a). Although Mn_{I} can be removed by lowtemperature annealing, the number of substitutional Mn_{Ga} impurities will remain smaller than the total nominal Mn doping. The Mn_{Ga} doping efficiency is therefore one of the key parameters that may limit the maximum T_{c} that can be achieved in (Ga,Mn)As epilayers, as we discuss in detail in Section 5.1. Ab initio calculations of the formation energies can be used to estimate the dependence of Mn_{Ga} and Mn_{I} partial concentrations on total Mn doping in asgrown materials Mašek et al. (2004); Jungwirth et al. (2005b). Similarly, correlated doping effects can be studied for other defects that occur frequently in LT MBE (Ga,Mn)As materials, such as arsenic antisites As_{Ga} Mašek et al. (2002).
Correlations between acceptors (Mn_{Ga}) and donors (Mn_{I} or As_{Ga}) in IIIV semiconductors like GaAs are strong due to the nearly covalent nature of bonding in these crystals. The cohesion energy of covalent networks has a maximum if the Fermi energy E_{F } lies within a band gap. Whenever E_{F } is shifted to the valence band or conduction band the strength of bonds is reduced because of the occurrence of unﬁlled bonding states or occupied antibonding states, respectively.
In the case of weak doping, small changes in the impurity concentration can easily move E_{F } across the band gap, with a negligible inﬂuence on the energy spectrum. The dependence of the formation energy, i.e., the energy cost for incorporating a particular impurity in a crystal, on the number of electrically active impurities can then be represented by the corresponding change in E_{F } multiplied by the charge state of the impurity Mahadevan and Zunger (2003). In the case of strongly doped and mixed crystals, the redistribution of electronic states in the valence band due to impurities may play a more important role and should therefore be included in microscopic calculations.
In general, the formation energy as a function of impurity concentrations can be obtained from the compositiondependent cohesion energy W_{coh} of the crystal. Assuming a sample consisting of N unit cells of the impure (mixed) crystal, the formation energy E_{A} of an acceptor A replacing atom X is deﬁned as the reaction energy of the substitution process,
sample + A → sample with one extra acceptor + X.
The corresponding reaction energy is
where W_{tot}(x_{A},x_{D}) is the total energy of the doped crystal normalized to a unit cell, x_{A} and x_{D} are the acceptor and donor concentrations, and the last two terms represent the total energies of freestanding atoms X and A. With increasing size of the sample, N →∞, the ﬁrst term in Eq. (9) approaches the derivative of W_{tot}(x_{A},x_{D}) with respect to x_{A}, and atomic energies can be absorbed using the relation between the total and cohesion energies. As a result, the cohesion energy represents a generating functional for the formation energies, i.e.,
 (10) 
The formation energy of a donor D substituting for an atom Y has the same form with A ↔ D and X ↔ Y .
Having deﬁned E_{A} and E_{D}, we note that in the lowconcentration regime where they depend linearly on x_{A} and x_{D},
 (11) 
i.e., that the mutual inﬂuence of the two kinds of impurities is symmetric. K(x_{A},x_{D}) in Eq. (11) plays the role of a correlation energy characterizing the codoping process. For positive K(x_{A},x_{D}), the formation energy of one impurity increases in the presence of the other. In this case the material tends to be either n type or p type rather than a compensated semiconductor. On the other hand, negative correlation energy indicates that the presence of impurities of one kind makes the incorporation of the other dopants easier. In this case compensation is favored.
Doping correlations over a wide and continuous range of impurity concentrations have been studied using the CPA, combined with either the parametrized TBA model or the ab initio linearized muffintin orbital (LMTO) DFT method Mašek et al. (2004); Jungwirth et al. (2005b). Results of these calculations are summarized in Figs. 9 and 10. The zero of energy is set to correspond to the formation energy of a reference Ga_{0.96}Mn_{0.04}As system with all Mn atoms occupying substitutional Ga positions. Four representative examples are considered here. Se_{As} and Si_{Ga} are typical single donors in GaAs that occupy the anion and cation sublattice, respectively. The As antisite defects As_{Ga} and Mn_{I} interstitials are the most important native defects in (Ga,Mn)As, both acting as double donors. Figure 9 shows that the formation energy of Mn_{Ga} decreases with increasing the number of donors. The curves are grouped into pairs according to the charge state of the donors, with only a minor inﬂuence of the particular chemical origin of the defect. The dependence is almost linear for low concentrations and the slope of the function is roughly proportional to the charge state of the donor. All this indicates that the variation of the formation energy of Mn_{Ga} is mostly determined by the Fermilevel effect described above, and that the redistribution of the density of states induced by donor defects plays a minor role. Formation energies of the interstitial Mn_{I} in the tetrahedral T(As_{4}) position are shown in Fig. 10. For the donor Mn_{I} impurity, the formation energy increases with the density of other donors. This means that the creation of Mn_{I} is efficiently inhibited in the presence of As_{Ga}. Analogous results are obtained for As_{Ga} antisite defects.


In Fig. 11 we show the change of formation energies of As_{Ga} and Mn_{I} as a function of the number of Mn_{Ga}. In both cases, the formation energy is a decreasing function of the density of Mn_{Ga}. This selfcompensation tendency is an important mechanism controlling the properties of asgrown (Ga,Mn)As mixed crystals. It explains the observed charge compensation in asgrown materials and is responsible to a large degree for the lattice expansion of highly Mndoped (Ga,Mn)As DMSs, as we discuss in Section 4.2.

The formation energies can be used to theoretically estimate partial concentrations of substitutional Mn_{Ga}, x_{s}, and interstitial Mn_{I}, x_{i}, in asgrown (Ga,Mn)As materials. An assumption is made in these calculations, whose validity is tested by a comparison with experimental data, that the probabilities of Mn atoms to occupy substitutional or interstitial positions are determined by the respective formation energies E_{s} and E_{i}, even in the nonequilibrium LTMBEgrown materials.
The balanced distribution of Mn_{Ga} and Mn_{I} is reached when Mašek et al. (2002); Jungwirth et al. (2005b)
 (12) 
as also expected from the growth point of view. Partial concentrations x_{s,i} of Mn are obtained by solving Eq. (12) together with the condition 0 ≤ x_{s,i} ≤ x, x_{s} + x_{i} = x. In Fig. 12 we show results of TBA CPA calculations Jungwirth et al. (2005b); for x > 1.5% x_{s} ≈ 0.8x, and x_{i} ≈ 0.2x. LMTO CPA theory calculations give very similar predictions Mašek et al. (2004).
Linear relations between x_{s}, x_{i}, and x reﬂect the fact that the difference of the formation energies of Mn_{Ga} and Mn_{I} impurities (see inset of Fig. 12) can be, up to x = 10%, approximated by a linear function of x_{s} and x_{i},
Equation (13) indicates that, for x < 1.5%, Mn_{Ga} has a lower formation energy than Mn_{I} and Mn atoms tend to occupy substitutional positions. At and above x ≈ 1.5%, Δ(x_{s},x_{i}) approaches zero and both Mn_{Ga} and Mn_{I} are formed.The theoretical results are in very good agreement with experimental data, as shown in Fig. 12. The balance considerations, conﬁrmed experimentally in samples with Mn_{Ga} concentrations up to 6.8%, suggest that there is no fundamental physics barrier to increasing Mn_{Ga} concentration up to 10% and beyond. Very precise control over the growth temperature and stoichiometry is, however, required for maintaining the 2D growth mode of the uniform (Ga,Mn)As materials at these high doping levels.
Finally, we note that during growth the formation energies control incorporation of Mn atoms, assuming that the total amount of Mn in the material is related to a sufficiently high chemical potential in the Mn source. The annealing processes, on the other hand, do not depend on the formation energies but rather on the energy barriers surrounding individual metastable positions of Mn in the lattice. The barriers are larger for Mn_{Ga} Erwin and Petukhov (2002); Mašek and Máca (2003) so that postgrowth lowtemperature annealing can be used to remove Mn_{I} without changing the number of Mn_{Ga} signiﬁcantly.

Changes in the lattice constant of (Ga,Mn)As DMSs, relative to the lattice constant of undoped GaAs, are too small to signiﬁcantly suppress or enhance pd kinetic exchange or other magnetic coupling mechanisms. Direct effects of dopinginduced lattice distortion on the onset of ferromagnetism are therefore negligibly small. Nevertheless, variations in the lattice parameter provide a measure of impurity concentrations in the DMS material. The impurities do, of course, control ferromagnetism through their doping properties. Also, because thinﬁlm (Ga,Mn)As epilayers are not relaxed, latticeconstant mismatch between the DMS layer and substrate induces strains that in many cases determine magnetocrystalline and magnetotransport anisotropies, as we discuss in detail in Sections 5.3.1 and 6.2. Theoretical calculations Mašek et al. (2003) of the dependence of the lattice constant on the density of the most common impurities in DMSs represent therefore another piece to the mosaic of our understanding of ferromagnetism in these complex systems.
Considering the values of the atomic radii of Mn (R_{Mn} = 1.17 Ĺ) and Ga (R_{Ga} = 1.25 Ĺ), the substitutional Mn_{Ga} impurity may be expected to lead to only very small changes (reductions) in the lattice constant. This expectation is consistent with the calculated Zhao et al. (2002) lattice constant of a hypothetical zincblende MnAs crystal whose value is comparable to that of GaAs. On the other hand, As antisites produce an expansion of the GaAs lattice Liu et al. (1995); Staab et al. (2001) and a similar trend can be expected for interstitial Mn.
Modern densityfunctional techniques allow one to move beyond intuitive theoretical considerations and discuss the dependence of the lattice constant on impurity concentrations on a more quantitative level Mašek et al. (2003). The CPA is again a useful tool here for studying (Ga,Mn)As properties over a wide range of impurity concentrations. Some quantitative inaccuracies in theoretical results due to the limitations of the LMTO CPA approach (e.g., neglect of local lattice relaxations) have been corrected by using the fullpotential linearized augmentedplanewave supercell method Mašek and Máca (2005). Starting with an ideal (Ga,Mn)As mixed crystal with all Mn atoms occupying substitutional Ga positions, these calculations give the following Vegard’s law expression for the doping dependence of the lattice constant:
 (14) 
with the expansion coefficient a_{s} ranging from 0.05 to 0.02 depending on the method used in the calculation Mašek et al. (2003); Mašek and Máca (2005). As expected, a changes only weakly with the Mn_{Ga} density x_{s}.
A similar linear dependence is obtained for hypothetical crystals where As_{Ga} (or Mn_{I}) is the only impurity present in the material, as shown in Fig. 13. According to the more reliable fullpotential supercell calculations Mašek and Máca (2005), the compositiondependent lattice constant is found to obey
 (15) 
where x_{i} and y are the densities of Mn_{I} and As_{Ga}, respectively, and a_{0} is the lattice constant of pure GaAs.

Recently, several experimental works Potashnik et al. (2002); KuryliszynKudelska et al. (2004); Sadowski and Domagala (2004); Zhao et al. (2005) have studied the dependence of lattice constants in (Ga,Mn)As materials on disorder, based on a comparison between asgrown and annealed samples. Measurements conﬁrmed that both As_{Ga} and Mn_{I} defects lead to a signiﬁcant expansion of the lattice. In samples grown under Asrich conditions, which are expected to inhibit formation of Mn_{I} impurities, annealing has virtually no effect on the measured lattice constant. This is consistent with the stability of As_{Ga} defects up to temperatures that are far above the annealing temperatures. Mn_{I} impurities, on the other hand, can be efficiently removed by lowtemperature annealing. Consistently, annealing leads to a signiﬁcant reduction of the lattice constant in materials that contain a large number of these defects in the asgrown form, as shown in Fig. 14. On a quantitative level, experimental data suggest a stronger lattice expansion due to Mn_{Ga} and a weaker expansion due to As_{Ga} and Mn_{I}, compared to the theoretical predictions of Eq. (15). The quantitative disagreement can be attributed, in part, to the simpliﬁed description of the system within the theoretical model. Also, the presence of other lattice imperfections or inaccuracies in the determination of experimental doping values may have partly obscured the direct quantitative comparison between experiment and theory.

Curie temperatures in metallic (Ga,Mn)As have been studied theoretically starting from the k⋅p kineticexchange effective Hamiltonian Jungwirth et al. (1999); Dietl et al. (2000); Jungwirth et al. (2002b); Brey and GómezSantos (2003); Das Sarma et al. (2004); Jungwirth et al. (2005b) and from microscopic TBA or SDF bandstructure calculations Sandratskii and Bruno (2002); Sato et al. (2003); Sandratskii et al. (2004); Hilbert and Nolting (2005); Jungwirth et al. (2003a); Timm and MacDonald (2005); Jungwirth et al. (2005b); Xu et al. (2005); Bouzerar et al. (2005a,b); Bergqvist et al. (2004, 2005). For a more detailed description of these theoretical approaches see Sections 3.1–3.3. The advantage of the k⋅p kineticexchange model is that it uses the experimental value for the pd coupling constant J_{pd}, i.e., it correctly captures the strength of the magnetic interaction that has been established to play the key role in ferromagnetism in (Ga,Mn)As. The model also accounts for the strong spinorbit interaction present in the host valence band, which splits the three p bands into a heavyhole, lighthole, and splitoff bands with different dispersions. The spinorbit coupling was shown König et al. (2001a); Brey and GómezSantos (2003) to play an important role in suppressing magnetization ﬂuctuation effects and therefore in stabilizing the ferromagnetic state up to high temperatures. On the other hand, describing the potentially complex behavior of Mn_{Ga} in GaAs by a single parameter may oversimplify the problem. Calculations omit, for example, the suppression of T_{c} in lowholedensity (Ga,Mn)As materials due to the direct antiferromagnetic superexchange contribution to the coupling of nearneighbor Mn pairs. The whole model inevitably breaks down in DMS systems with holes strongly bound to Mn acceptors or with large charge ﬂuctuations on Mn_{Ga} d shells.
The advantage of microscopic approaches to Curietemperature calculations is that they make no assumption about the character of Mn_{Ga} impurities in GaAs and their magnetic coupling. They are therefore useful for studying material trends in T_{c} as a function of Mn doping or the density of other intentional or unintentional impurities and defects present in real systems. Because spinorbit interactions add to the numerical complexity of calculations that are already challenging, they have normally been neglected in this approach. Another shortcoming, discussed already in Section 3.1, of LSDA approaches is an overestimated strength of the pd exchange as compared to experiment. Within the meanﬁeld approximation, which considers thermodynamics of an isolated Mn moment in an effective ﬁeld and neglects correlated MnMn ﬂuctuations, microscopic calculations typically yield larger T_{c}’s than the effective Hamiltonian model, which uses the experimental value for J_{pd}. Stronger pd exchange and the omission of spinorbit coupling effects in the DFTs, however, also leads to a larger suppression of the Curie temperature due to ﬂuctuation effects. (A closer agreement on the character of the T_{c} versus Mndoping curves, calculated within the two formalisms, is obtained when the deﬁciencies of LSDA theories are partly eliminated by using, e.g., the LDA+U.) Despite the above weaknesses of semiphenomenological and microscopic calculations, a qualitatively consistent picture is clearly emerging from these complementary theoretical approaches that, as we discuss below, provides a useful framework for analyzing measured T_{c}’s.
Our review of theoretical T_{c} trends in (Ga,Mn)As starts with the results of the KL kineticexchange Hamiltonian and meanﬁeld approximation to set up a scale of expected Curie temperatures. These estimates, which are not accurate in all regimes, are simpliﬁed by assuming a homogeneous distribution of Mn_{Ga} ions and neglecting the role of other defects, apart from their potential contribution to hole or moment compensation Dietl et al. (2000, 1997); Jungwirth et al. (1999); König et al. (2003). For microscopic models this assumption is equivalent to the virtualcrystal approximation. Microscopic TBA calculations have shown very little effect of positional disorder on the strength of magnetic couplings in (Ga,Mn)As epilayers with metallic conductivities, partly justifying the virtualcrystal approach Jungwirth et al. (2003a). In addition, detailed theoretical studies conﬁrm the absence of any signiﬁcant magnetic frustration associated with the random positions of Mn_{Ga} moments in the lattice in more metallic ferromagnetic semiconductors Timm and MacDonald (2005); Fiete et al. (2005). In the very dilute limit, however, T_{c} becomes sensitive to the distribution of Mn moments in the lattice Bergqvist et al. (2004).
In the meanﬁeld approximation Jungwirth et al. (1999); Dietl et al. (2000), each local Mn_{Ga} moment is described by a Hamiltonian S_{I} ⋅ H_{MF} where S_{I} is the Mn_{Ga} local spin operator, H_{MF} = J_{pd}〈s〉, and 〈s〉 is the mean spin density of the valenceband holes (for the deﬁnition of the J_{pd} ﬁeld, see Section 2.2.1). H_{MF} is the effective ﬁeld experienced by the local moments due to spin polarization of the band holes, analogous to the nuclear Knight shift. Similarly h_{MF} = J_{pd}N_{Mn}〈S〉 is the effective magnetic ﬁeld experienced by the valenceband holes, which is proportional to the density and mean spin polarization of Mn_{Ga} local moments. The dependence of 〈S〉 on temperature and ﬁeld H_{MF} is given König et al. (2003) by the Brillouin function Ashcroft and Mermin (1976)
 (16) 
The Curie temperature is found by linearizing H_{MF} and B_{s} around 〈S〉 = 0:
Here χ_{f} is the itineranthole spin susceptibility given by
 (18) 
and e_{T } is the total energy per volume of the holes. Equations (16) and (17) give
 (19) 
The qualitative implications of this T_{c} equation can be understood within a model itineranthole system with a single spinsplit band and an effective mass m^{*}. The kinetic energy contribution e_{ k} to the total energy of the band holes gives a susceptibility
 (20) 
where k_{F } is the Fermi wave vector. Within this approximation T_{c} is proportional to the Mn_{Ga} local moment density, to the hole Fermi wave vector, i.e., to p^{1∕3} where p is the hole density, and to the hole effective mass m^{*}.
A more quantitative prediction for the Curie temperature is obtained by evaluating the itineranthole susceptibility using a realistic band Hamiltonian,
 (21) 
where ℋ_{KL} is the sixband KL Hamiltonian of the GaAs host band Vurgaftman et al. (2001) and s is the hole spin operator Dietl et al. (2000, 2001b); Abolfath et al. (2001a). The results, represented by the thick (black) line in Fig. 15, are consistent with the qualitative analysis based on the parabolic band model, i.e., T_{c} roughly follows the ~ xp^{1∕3} dependence. Based on these calculations, roomtemperature ferromagnetism in (Ga,Mn)As is expected for 10% Mn_{Ga} doping in weakly compensated samples Jungwirth et al. (2005b).
Holehole Coulomb interaction effects can be included in the lowest order of perturbation theory by adding the hole exchange contribution to the total energy Mahan (1981). The thin (red) line in Fig. 15 shows this Stoner T_{c} enhancement calculated numerically using the kineticexchange effective model with the sixband KL Hamiltonian. T_{c} stays roughly proportional to xp^{1∕3} even if holehole exchange interactions are included, and the enhancement of T_{c} due to interactions is of the order ~ 10–20% Jungwirth et al. (1999); Dietl et al. (2001b); Jungwirth et al. (2005b).

The meanﬁeld effective Hamiltonian analysis above neglects discreteness in random Mn_{Ga} positions in the lattice and magnetic coupling mechanisms additional to the kineticexchange contribution, particularly the nearneighbor superexchange. The former point can be expected to inﬂuence T_{c} at large hole densities, i.e., when the hole Fermi wavelength approaches interatomic distances. Of course, the entire phenomenological scheme fails on many fronts when the Fermi wavelength approaches atomic length scales since it is motivated by the assumption that all relevant length scales are long; the k⋅p band structure, the use of host material band parameters, and the neglect of momentum dependence in the J_{pd} parameter all become less reliable as the hole density increases to very large values. The approximations are apparently not fatal, however, even for x ~ 10% and any degree of compensation.
In the opposite limit of strongly compensated systems, where the overall magnitude of the holemediated exchange is weaker, antiferromagnetic superexchange can dominate the nearneighbor Mn_{Ga}Mn_{Ga} coupling Kudrnovský et al. (2004), leading to a reduced Curie temperature Jungwirth et al. (2005b); Sandratskii et al. (2004). We emphasize that the k⋅p kineticexchange model cannot be applied consistently when nearestneighbor interactions dominate, since it implicitly assumes that all length scales are longer than a lattice constant. We also note that net antiferromagnetic coupling of nearneighbor Mn_{Ga}Mn_{Ga} pairs is expected only in systems with large charge compensations. In weakly compensated (Ga,Mn)As the ferromagnetic contribution takes over Kudrnovský et al. (2004); Mahadevan and Zunger (2004); Dietl et al. (2001b).
In addition to the above effects related to random Mn distribution, Mn positional disorder can directly modify pd interactions when the coherence of Bloch states becomes signiﬁcantly disturbed. Microscopic theories, such as the TBA and CPA Jungwirth et al. (2005b) or ab initio approaches based on either the CPA or supercell band structures Sandratskii et al. (2004); Sato et al. (2003), capture these effects on an equal footing and can be used to estimate trends in the meanﬁeld T_{c} beyond the virtualcrystal approximation.

The meanﬁeld CPA Curie temperatures are obtained by evaluating the energy cost of ﬂipping one Mn_{Ga} moment with all other moments held ﬁxed in the ferromagnetic ground state. It can be evaluated for any given chemical composition Liechtenstein et al. (1987); Mašek (1991) and deﬁnes an effective exchange ﬁeld H_{eff} acting on the local moment. This energy change corresponds to H_{MF} in the kineticexchange model used in the previous section, i.e.,
 (22) 
Results based on microscopic TBA bandstructure calculations are shown in Fig. 16 as a function of hole density for several Mn_{Ga} localmoment concentrations Jungwirth et al. (2005b). (The hole density is varied independently of Mn_{Ga} doping in these calculations by adding nonmagnetic donors or acceptors.) Comparison with Fig. 15 identiﬁes the main physical origins of the deviations from the T_{c} ~ xp^{1∕3} trend. Black dots in the left panel of Fig. 16 correspond to a relatively low localMn_{Ga}moment concentration (x = 2%) and hole densities ranging up to p = 4N_{Mn}, and show the expected suppression of T_{c} at large p. The effect of superexchange in the opposite limit is clearly seen when inspecting, e.g., the x = 10% data for p < 1 nm^{3}. The meanﬁeld TBA CPA Curie temperature is largely suppressed here or even negative, meaning that the ferromagnetic state becomes unstable due to shortrange antiferromagnetic coupling. Note that the neglect of Coulomb interactions in these TBA CPA calculations likely leads to an overestimated strength of the antiferromagnetic superexchange. The inhomogeneity of the carrier distribution in the disordered mixed crystal may also contribute to the steep decrease of T_{c} with increasing compensation seen in Fig. 16.
Although the Curie temperatures in the left panel of Fig. 16 appear to depart from the T_{c} ~ xp^{1∕3} dependence, the linearity with x is almost fully recovered when T_{c} is plotted as a function of the number of holes per Mn_{Ga} local moment p∕N_{Mn} (see right panel of Fig. 16). Note that for compensations (1  p∕N_{Mn}) reaching 100% this property of the superexchange coupling is reminiscent of the behavior of (II,Mn)VI diluted magnetic semiconductors Furdyna (1988) in which Mn acts as an isovalent magnetic impurity. The dependence on p in (Ga,Mn)As is expected to become very weak, however, when approaching the uncompensated state. Similarly, prospects for substantial increases in T_{c} by nonmagnetic acceptor codoping of weakly compensated material appear to be quite limited.
In the left panel of Fig. 17 we show meanﬁeld CPA Curie temperatures in uncompensated (p∕N_{Mn} = 1) (Ga,Mn)As DMSs as a function of Mn doping calculated using LDA and LDA+U ab initio methods. The LDA+U calculations, which give more realistic values of the pd exchange coupling, conﬁrm the linear dependence of T_{c} on x, showing no signs of saturation even at the largest doping x = 10% considered in these calculations.

The potential inﬂuence of correlated Mnmoment ﬂuctuations (corrections to meanﬁeld theory) on ferromagnetic ordering in (Ga,Mn)As can be recognized by considering, within a simple parabolicband model, the RKKY oscillation effect, which occurs as a consequence of the 2k_{F } anomaly in the wavevectordependent susceptibility of the hole system Dietl et al. (1997); Brey and GómezSantos (2003). In this theory, which treats the hole system perturbatively around the paramagnetic state, the sign of the holemediated Mn_{Ga}Mn_{Ga} coupling varies as cos(2k_{F }d), where d is the distance between Mn_{Ga} moments, and its amplitude decays as d^{3}. Estimating the average Mn_{Ga}Mn_{Ga} separation in a (Ga,Mn)As random alloy as = 2(3∕4πN_{Mn})^{1∕3} for uncompensated (Ga,Mn)As systems and neglecting spinorbit coupling and band warping, cos(2k_{F } ) ≈1, which means that the role of the RKKY oscillations cannot be generally discarded. In realistic valence bands oscillations are suppressed due to nonparabolic and anisotropic dispersions of heavy and lighthole bands and due to strong spinorbit coupling Brey and GómezSantos (2003); König et al. (2001a).
More quantitatively, the range of reliability and corrections to the meanﬁeld approximation in (Ga,Mn)As can be estimated by accounting for the suppression of the Curie temperature using the quantum theory of longwavelength spin waves or using Monte Carlo simulations which treat Mn moments as classical variables. For weak pd exchange coupling, SN_{Mn}J_{pd}∕E_{F } ≪ 1, where E_{F } is the hole Fermi energy, the spin polarization of the hole system is small and the RKKY and spinwave approximations treat collective Mnmoment ﬂuctuations on a similar level. The advantage of spinwave theory is that it can be used to explore the robustness of ferromagnetic states over a wider range of pd couplings, including the more strongly exchangecoupled asgrown materials with large Mn density and large hole compensation (see Section 8.1 for a general discussion of magnetic interactions in the two couplingstrength limits).
Calculations in metallic systems have been performed starting from the k⋅p kineticexchange effective Hamiltonian Schliemann et al. (2001b); Brey and GómezSantos (2003); Jungwirth et al. (2005b) or from the SDF bandstructure calculations Xu et al. (2005); Hilbert and Nolting (2005); Bouzerar et al. (2005a); Bergqvist et al. (2004). (Monte Carlo studies of T_{c} in the regime near the metalinsulator transition have been reported by Mayr et al. (2002).) Within a noninteracting spinwave approximation, magnetization vanishes at the temperature where the number of excited spin waves equals the total spin of the ground state Jungwirth et al. (2002b, 2005b):
 (23) 
where k_{D} = (6π^{2}N_{ Mn})^{1∕3} is the Debye cutoff and D(T) = D_{ 0} 〈S〉(T)∕S is proportional to the zerotemperature spinwave stiffness parameter A (D_{0} = 2A∕SN_{Mn}) and the meanﬁeld temperaturedependent average spin on Mn, 〈S〉(T) König et al. (2001a); Schliemann et al. (2001a). (For a detailed discussion of the micromagnetic parameter A, see Section 5.3).
Comparing the spinstiffness results obtained using the k⋅p kineticexchange model with a simple parabolic band and with the more realistic spinorbitcoupled KL Hamiltonian, the spin stiffness is observed to always be much larger in the KL model König et al. (2000, 2001a). For (Ga,Mn)As, the parabolicband model underestimates D by a factor of ~10–30 for typical hole densities. This larger spin stiffness in spinorbitcoupled valence bands is due to the heavyhole–lighthole mixing. Crudely, the largemass heavyhole band dominates spin susceptibility and enables local (meanﬁeld) magnetic order at high temperatures, while the dispersive lighthole band dominates spin stiffness and enables longrange magnetic order. The analysis highlights that the multiband character of the semiconductor valence band plays an important role in the ferromagnetism of (Ga,Mn)As.
Critical temperature estimates based on Eq. (23), the KL kineticexchange model, and including also the Stoner enhancement of T_{c} are summarized in Fig. 15 (blue symbols). These T_{c} estimates indicate that T_{c} will remain roughly proportional to x even at large dopings. The suppression of T_{c} due to spin waves increases with increasing hole density relative to the local moment concentration, resulting in saturation of the critical temperature with increasing p at about 50% compensation.
The suppression of T_{c} due to correlated Mnmoment ﬂuctuations is also observed in LSDA calculations Xu et al. (2005); Hilbert and Nolting (2005); Bergqvist et al. (2004, 2005); Bouzerar et al. (2005a,b). The trend is illustrated in Fig. 18 where collective ﬂuctuations are accounted for using spinwave theory or the Monte Carlo approach Hilbert and Nolting (2005); Bergqvist et al. (2004); similar trends of suppressed meanﬁeld T_{c} due to collective Mnmoment ﬂuctuations have been predicted by spinwave theory using a more elaborate, selfconsistent randomphase approximation (RPA) technique on random lattice Bouzerar et al. (2005a). A larger suppression of the meanﬁeld T_{c} seen in ab initio calculations, compared to KL kineticexchange model results, can be attributed partly to the simpler, threefolddegenerate LSDA valenceband structure in theories that neglect spinorbit coupling. Also, the stronger pd exchange in the LSDA theories may result in a weaker spin stiffness of the magnetic system, as holes are more strongly bound to Mn acceptors and the holemediated MnMn coupling has a more shortrange character. The enhancement of ﬂuctuation effects in stronger pd coupled systems is clearly seen in Fig. 18 when comparing LSDA results for narrowergap (weaker pd exchange) (Ga,Mn)As and widergap (stronger pd exchange) (Ga,Mn)N.

Quantitative discrepancies between the KL kineticexchange model and LSDA results for the meanﬁeld T_{c} and for the suppression of ferromagnetism due to collective Mnmoment ﬂuctuations partly cancel out, leading to similar overall predictions for Curie temperatures in (Ga,Mn)As. Based on the T_{c} analysis alone it is therefore difficult to determine whether magnetic interactions have a short or longrange character in (Ga,Mn)As DMSs.
Theoretically, the localization of the hole around the Mn impurity and the range of magnetic MnMn interactions can be studied using microscopic TBA or ab initio calculations of the charge and moment distributions in the lattice or by mapping the total energy of the DMS crystal to the Heisenberg Hamiltonian van Schilfgaarde and Mryasov (2001); Sandratskii and Bruno (2002); Sato et al. (2003); Wierzbowska et al. (2004); Mahadevan and Zunger (2004); Kudrnovský et al. (2004); Hilbert and Nolting (2005); Schulthess et al. (2005); Timm and MacDonald (2005). Moving down the anion column in the periodic table from the nitride DMSs to antimonides, holes become more delocalized Mahadevan and Zunger (2004) and, consequently, MnMn interactions are more long range in these microscopic calculations. In (Ga,Mn)As the LSDA theory predicts shortrange magnetic coupling while the LDA+U or SICLSDA results suggest that holes which mediate the MnMn exchange interaction are more delocalized van Schilfgaarde and Mryasov (2001); Wierzbowska et al. (2004); Schulthess et al. (2005).
Combined theoretical and experimental studies of remanent magnetization, micromagnetic parameters, and magnetotransport coefficients, discussed in Sections 5.27, indicate that in highquality (Ga,Mn)As ferromagnets with metallic conductivities (conductivity increases with decreasing temperature) holes are sufficiently delocalized to make the kineticexchange model approach applicable. It is natural to expect that the freecarriermediated ferromagnetism picture will also apply in narrowergap antimonide DMSs, such as (In,Mn)Sb, with even larger conductivities due to the smaller hole effective mass Wojtowicz et al. (2003). The smaller effective mass and larger unitcell volume, as compared to arsenide DMSs, explain the smaller Curie temperatures in (III,Mn)Sb. This trend is illustrated in Table 1 by comparing the respective meanﬁeld KL kineticexchange model T_{c}’s. In Mndoped phosphides and nitrides the suppression of T_{c} due to effects beyond the meanﬁeld virtualcrystal approximation, seen in LSDA calculations and related to the shortrange nature of magnetic interactions, may prevail Scarpulla et al. (2005). We note also that LSDA Curietemperature studies mentioned above do not capture the possible transition of the Mn impurity state in wide gap IIIV semiconductors to the highly correlated trivalent (d^{4}) center with four strongly localized d electrons and an empty d state deep in the gap, in which case the holemediated ferromagnetism picture implied by these calculations is not applicable Luo and Martin (2005); Kreissl et al. (1996); Schulthess et al. (2005).

Experimentally, T_{c} trends have been most extensively studied in (Ga,Mn)As Ohno (1998); Potashnik et al. (2002); Edmonds et al. (2002a); Yu et al. (2003); Chiba et al. (2003a); Ku et al. (2003); Stone et al. (2003); Jungwirth et al. (2005b). In Fig. 19 we show the temperaturedependent magnetization and inverse susceptibility of the current record T_{c} material Wang et al. (2005a). The Brillouinfunction character of the magnetization curve conﬁrms that the meanﬁeld theory is appropriate in these highquality DMS materials with metallic conductivities. The Curie temperatures for a series of asgrown and annealed (Ga,Mn)As samples with experimentally characterized charge and moment compensations are plotted in Fig. 20 Jungwirth et al. (2005b). The concentration of uncompensated Mn moments in the plot is x_{eff} = x_{s}  x_{i}, where it is assumed that Mn_{I} donors present in the system are attracted to Mn_{Ga} acceptors and that these pairs couples antiferromagnetically, as discussed in Sections 2.3.1 and 4.1. (The consistency of this assumption is conﬁrmed by independent magnetization studies reviewed in Section 5.2.) The experimental T_{c}∕x_{eff} plotted against p∕N_{Mn}^{eff} in Fig. 20, where N_{ Mn}^{eff} = 4x_{ eff}∕a_{lc}^{3}, show a common T_{c} trend that is consistent with theoretical expectations Jungwirth et al. (2005b); Bouzerar et al. (2005a); Bergqvist et al. (2005). In particular, theory and experiment agree on the very weak dependence of T_{c}∕x_{eff} on p∕N_{Mn}^{eff} for low compensation and the relatively rapid fall of T_{c}∕x_{eff} with decreasing p∕N_{Mn}^{eff} for compensations of ~ 40% or larger. It should be noted that the maximum experimental x_{eff} is only 4.6% in the asgrown sample and 6.8% after annealing for a total Mn concentration x = 9%, hence the modest T_{c}’s observed so far. Achieving T_{c} values close to room temperature in (Ga,Mn)As, which is expected to occur for x_{eff} ≈ 10%, appears to be essentially a material growth issue, albeit a very challenging one Jungwirth et al. (2005b).


Only a few experimental studies of LTMBEgrown (III,Mn)Sb DMSs have been reported to date Abe et al. (2000); Wojtowicz et al. (2003). The Curie temperatures measured in these materials are lower than the T_{c}’s in Mndoped arsenides, consistent with kineticexchange model predictions in Table 1. The nature of ferromagnetism and therefore the interpretation of the experimental Curie temperatures observed in phosphide and nitride DMSs are not established yet, as already pointed out in Section 1.2.
In this section we focus on lowtemperature ferromagnetic moments in (Ga,Mn)As DMSs with metallic conductivities. Early experimental studies, reporting large apparent magnetization deﬁcits in (Ga,Mn)As Ohno and Matsukura (2001); Korzhavyi et al. (2002); Potashnik et al. (2002), motivated a theoretical search for possible intrinsic origins of frustrating magnetic interactions in this material. Using a wide spectrum of computational techniques, ranging from ab initio methods Korzhavyi et al. (2002); Mahadevan et al. (2004); Kudrnovský et al. (2004) and the microscopic TBA Timm and MacDonald (2005) to k⋅p kineticexchange models Schliemann and MacDonald (2002); Schliemann (2003); Brey and GómezSantos (2003); Zaránd and Jankó (2002); Fiete et al. (2005), theoretical studies have identiﬁed several mechanisms that can lead to noncollinear ground states. The observation that longwavelength spin waves with negative energies frequently occur within the parabolicband kineticexchange model illustrates that randomness in the distribution of Mn moments can result in an instability of the collinear ferromagnetic state Schliemann and MacDonald (2002). Frustration can be further enhanced when positional disorder is combined with anisotropies in MnMn interactions. The pd character of electronic states forming the magnetic moment leads to magnetic interaction anisotropies with respect to the crystallographic orientation of the vector connecting two Mn moments Mahadevan et al. (2004); Kudrnovský et al. (2004); Timm and MacDonald (2005); Brey and GómezSantos (2003). When spinorbit coupling is taken into account, magnetic interactions also become anisotropic with respect to the relative orientation of the MnMn connecting vector and the magnetic moment Zaránd and Jankó (2002); Fiete et al. (2005); Schliemann (2003); Timm and MacDonald (2005).
Some degree of noncollinearity is inevitable as a combined consequence of positional disorder and spinorbit coupling. Nevertheless, it was argued theoretically that a large suppression of ferromagnetic moment is not expected in metallic (Ga,Mn)As samples with Mn concentrations above ~ 1% Timm and MacDonald (2005). The minor role of noncollinearity is due largely to the longrange character of magnetic interactions, which tends to average out the frustrating effect of anisotropic coupling between randomly distributed Mn impurities Timm and MacDonald (2005); Zhou et al. (2004). Indeed, ab initio, microscopic TBA, and k⋅p kineticexchange model calculations of zerotemperature magnetic moments in (Ga,Mn)As ferromagnets which neglect effects that would lead to noncollinearity Dietl et al. (2001b); Wierzbowska et al. (2004); Schulthess et al. (2005); Jungwirth et al. (2005a) are consistent with experiments reported in a series of highquality (Ga,Mn)As ferromagnets Edmonds et al. (2005a); Wang et al. (2005b); Jungwirth et al. (2005a). This rules out any marked intrinsic frustrations in the ground state of these DMSs. Substantial magnetization suppression seen in many early (Ga,Mn)As samples can be attributed primarily to the role played in those samples by interstitial Mn atoms and other unintentional defects.
We start the discussion by identifying the key physical considerations that inﬂuence the groundstate magnetization of (Ga,Mn)As ferromagnets by focusing on a single Mn(d^{5}+hole) complex and approximating the total magnetization in the collinear state with a simple sum of individual (identical) Mn(d^{5}+hole)complex contributions. This crude model is used only to qualitatively clarify the connection between pd hybridization and antiferromagnetic kineticexchange coupling, the sign of the hole contribution to total moment per Mn, and the expected meanﬁeld contribution to magnetization per Mn from Mn local moments and from antiferromagnetically coupled holes. We also explain in this section that quantum ﬂuctuations around the meanﬁeld ground state are generically present because of the antiferromagnetic character of the pd kineticexchange interaction.
Magnetization at T = 0 is deﬁned thermodynamically by the dependence of the groundstate energy E on external magnetic ﬁeld B:
 (24) 
To avoid confusion that may result from using the hole picture to describe magnetization of carriers in ptype (Ga,Mn)As materials, we recall ﬁrst the relationship between magnetizations evaluated using the physically direct electron picture and magnetizations evaluated using the indirect but computationally more convenient hole picture. In meanﬁeld theory magnetization is related to the change of singleparticle energy with ﬁeld, summed over all occupied orbitals. Orbitals that decrease in energy with increasing ﬁeld make a positive contribution to the magnetization. For B ∥ + , the delectron spins are aligned along the (z) direction (down spins) and the majorityspinband electrons have spin up due to antiferromagnetic pd exchange coupling. Then, if the majority band moves up in energy with B and the minority band moves down, as illustrated in the left part of Fig. 21, the band kinetic energy increases with B and, according to Eq. (24), the corresponding contribution to the magnetization is negative. In the hole picture, we obtain the same respective sense of the shifts of the majorityhole and minorityhole bands, as illustrated in the right part of Fig. 21, and therefore the correct sign of the magnetization (negative in our case). The cartoon shows that in order to circumvent the potentially confusing notion of the spin of a hole in magnetization calculations, it is safer to start from the full Hamiltonian ℋ(B) in the physically direct picture of electron states, where the sign of the coupling of the electron spin to the ﬁeld and the exchange energy are unambiguously deﬁned. The electron picture → hole picture transformation (ℋ(B) →ℋ(B)) and the clearly deﬁned notion of majority and minority bands in either picture guarantees the sign consistency of the calculated magnetization. Note that the language used here neglects spinorbit interactions, which lead to singleparticle orbitals that do not have deﬁnite spin character. Although spinorbit interactions are important they can be neglected in most qualitative considerations Jungwirth et al. (2005a).

The electronelectron exchange energy has a negative sign and its magnitude increases monotonically when moving from the paramagnetic to the halfmetallic (empty minority band) state. This together with Eq. (24) implies that the magnetization contribution from the electronelectron exchange energy has the same sign as the contribution from the kinetic energy. Using the same arguments as above we see that in the electronelectron exchangeenergy case the sign of magnetization is also treated consistently by the electron picture → hole picture transformation.
The meanﬁeld groundstate wave function of the Mn(d^{5}+hole) complex is ∣S_{z} = S〉∣j_{z} = +j〉 and the magnetization per Mn is m_{MF} = (g_{S}S  g_{j}j)μ_{B}, where S and j are local delectron and hole moments and g_{S} and g_{j} are the respective Landé g factors. The ﬁve d electrons have zero total orbital angular momentum, i.e., g_{S} = 2, and for the spin j = 1∕2 hole (g_{j} = 2) we get m_{MF} = 4μ_{B}. However, hole states near the valenceband edge have p character, so more realistically we should consider g_{j}j = 4∕3 × 3∕2 = 2, which gives m_{MF} = 3μ_{B}. We show below that this basic picture of a suppressed m_{MF} due to holes applies also to highlyMndoped (Ga,Mn)As materials, although the magnitude of the meanﬁeld hole contribution is weaker because of the occupation of both majority and minority hole bands and, partly, because of spinorbit coupling effects.
The twospin S and j model can also be used to demonstrate the presence of quantum ﬂuctuations around the meanﬁeld ground state, which is a consequence of the antiferromagnetic sign of the S ⋅ j coupling Jungwirth et al. (2005a). In the limit of B → 0 the twospin Hamiltonian is given by
 (25) 
For antiferromagnetic coupling (J > 0), S_{tot} = S j, and the corresponding groundstate energy E_{AF} = (∣J∣∕2)[(S  j)(S  j + 1)  S(S + 1)  j(j + 1)] = ∣J∣(Sj + j) is lower than the meanﬁeld energy ∣J∣Sj. The meanﬁeld ground state is not exact here and quantum ﬂuctuation corrections to the magnetization will be nonzero in general. The difference between magnetizations of the exact and meanﬁeld states is obtained from the corresponding expectation values of the Zeeman Hamiltonian g_{S}μ_{B}BŜ_{z} + g_{j}μ_{B}Bĵ_{z}, and from Eq. (24) Jungwirth et al. (2005a):
 (26) 
When j = 1∕2 and g_{S} = g_{j} = 2 the quantum ﬂuctuation correction to the magnetization vanishes even though the meanﬁeld ground state is not exact. The correction remains relatively weak also in the case of j = 3∕2 and g_{j} = 4∕3, for which m_{QF} = 0.25μ_{B}.
As in the Mn(d^{5}+hole) complex, the magnetization of coupledMnmoment systems can be decomposed into meanﬁeld contributions from Mn local moments and valenceband holes and a quantum ﬂuctuation correction. At a meanﬁeld level, the TBA description of (Ga,Mn)As mixed crystals is particularly useful for explaining the complementary role of local and itinerant moments in this ptype magnetic semiconductor and we therefore start by reviewing this approach Jungwirth et al. (2005a). In Fig. 22 the microscopic TBA CPA magnetic moments per Mn, m_{TBA}, in (Ga,Mn)As ferromagnets are plotted as a function of p∕N_{Mn}. The value of m_{TBA} is obtained here using the electron picture by integrating over occupied states up to the Fermi energy. Spinorbit coupling is neglected in these TBA calculations and only the spinpolarization contribution to magnetization is considered in m_{TBA}, which simpliﬁes the qualitative discussion below.
A common way of microscopically separating contributions from local atomic and itinerant moments is by projecting occupied electron states onto Mn d and sp orbitals, respectively. In this decomposition, resulting local Mn moments are smaller than 5μ_{B} per Mn due to the admixture of d character in empty states near the valenceband edge. The effective kineticexchange model corresponds, however, to a different decomposition of contributions, in effect associating one spectral region with local Mn moments and a different spectral region with itineranthole moments. The kineticexchange model, in which local moments have S = 5∕2, is obtained from microscopic models, e.g. from the TBA and CPA, by expressing m_{TBA} as the difference between a contribution m_{TBA}^{int} calculated by integrating over all electronic states up to midgap, i.e., including the entire valence band, and a contribution corresponding to the integral from the Fermi energy to midgap. As long as the valenceband–conductionband gap is nonzero, the former contribution is independent of valenceband ﬁlling and equals the moment of an isolated Mn atom, 5μ_{B}. The latter term, plotted in the lower inset of Fig. 22, represents magnetization of itinerant holes.

The applicability of the kineticexchange model relies implicitly on the perturbative character of the microscopic pd hybridization. The level of pd hybridization over a typical doping range is illustrated in Fig. 23, which shows the orbital composition of m_{TBA}^{int}. The ﬁlled symbols are calculated including spectral weights from all spd orbitals while the halfopen and open symbols are obtained after projecting onto d and sp orbitals, respectively. If there were no hybridization, m_{TBA}^{int} projected on d orbitals would equal the total m_{TBA}^{int} and the sporbitalprojected m_{ TBA}^{int} would vanish. In TBA CPA calculations, the dorbitalprojected m_{TBA}^{int} is reduced by only 10% as compared to the total m_{TBA}^{int} and therefore the pd hybridization can be regarded as a weak perturbation. The nearly constant value of the dorbitalprojected m_{TBA}^{int} also suggests that the kineticexchange coupling parameter J_{pd} in the effective kineticexchange Hamiltonian is nearly independent of doping over the typical range of Mn and hole densities.

The decrease of m_{TBA} in Fig. 22 with increasing p∕N_{Mn} clearly demonstrates the antiferromagnetic pd exchange. The initial common slope for data corresponding to different Mn concentrations reﬂects the halfmetallic nature of the hole system (only majority hole band occupied) when spinorbit interactions are neglected. Here the hole contribution to magnetization per volume is proportional to p, i.e., magnetization per Mn is proportional to p∕N_{Mn}. The change in slope of m_{TBA} at larger hole densities, which now becomes Mndensity dependent, reﬂects the population of the minorityspin hole band and therefore the additional dependence of hole magnetization on exchange splitting between majority and minorityhole bands. Note that the maximum absolute value of the hole contribution to magnetization per hole (see upper inset of Fig. 22) observed in the halfmetallic state is 1μ_{B} in these TBA calculations, which assume j = 1∕2 and g_{j} = 2 holes.
Similar conclusions concerning the character of contributions to the magnetization of (Ga,Mn)As have been inferred from LDA+U and SIC LSDA supercell calculations Wierzbowska et al. (2004); Schulthess et al. (2005). (These microscopic calculations also neglect spinorbit coupling.) The halfmetallic LDA+U band structure in the case of zero charge compensation (p∕N_{Mn} = 1) results in a total magnetization per Mn of 4μ_{B} Wierzbowska et al. (2004), in agreement with the corresponding m_{TBA} values. In SIC LSDA calculations Schulthess et al. (2005), the system is not completely half metallic and, consistently, the total moment per Mn is larger than 4μ_{B}. The LDA+U and SIC LSDA local moments on Mn are 4.7μ_{B} and 4.5μ_{B}, respectively, in good agreement again with the dprojected m_{TBA}^{int} values. In both ab initio calculations the oppositely aligned moment on the As sublattice extends over the entire supercell, conﬁrming the delocalized character of holes and the antiferromagnetic sign of the pd exchange.
The KL kineticexchange model calculations Dietl et al. (2001b); Jungwirth et al. (2005a) have been used to reﬁne, quantitatively, predictions for the total magnetization based on the above microscopic theories. In particular, the number of minority holes at a given total hole density is underestimated in these TBA and ab initio approaches. This is caused in part by the quantitative value of the exchange spin splitting of the valence band which, e.g., in the TBA CPA calculations is a factor of 1.5–2 larger than the value inferred from experiment. Neglecting the spinorbit interaction also results in three majority bands that are degenerate at the Γ point, instead of only two bands (heavy hole and light hole) in the more realistic spinorbitcoupled band structure. (This deﬁciency is common to all calculations that neglect spinorbit coupling.) In addition to having more states available in the majority band, which leads to underestimating the minority hole density, these microscopic calculations also omit the reduction of the mean spin density in the majority band caused by the spinorbit coupling. The total magnetization values will be underestimated due to these effects. On the other hand, assuming only the spin contribution to the hole magnetization leads to an overestimated total magnetization, as already illustrated in Section 5.2.1.
In the kineticexchange effective model the T = 0 localmoment contribution to the magnetization per Mn is 5μ_{B}. As emphasized above, this is not in contradiction with the smaller delectron contribution to the magnetic moment in microscopic calculations. The kinetic band energy contribution to the meanﬁeld magnetization per Mn is obtained by numerically integrating over all occupied hole eigenstates of the Hamiltonian (21) and by ﬁnding the coefficient linear in B of this kineticenergy contribution to the total energy Dietl et al. (2001b); Jungwirth et al. (2005a). Results of such calculations are summarized in Fig. 24, which shows the spin and orbital contributions to the magnetization of holes, and in Fig. 25, showing hole moment per Mn m_{MF}^{kin} for several local Mnmoment and hole densities. Note that the decoupling of the hole magnetization into spin and orbital terms is partly ambiguous in the spinorbitcoupled valence bands and that only the total moment m_{MF}^{kin} has a clear physical meaning Dietl et al. (2001b); Jungwirth et al. (2005a). As expected holes give a negative contribution to magnetization, i.e., they suppress the total magnetic moment. The magnitude of the meanﬁeld magnetization per hole ∣m_{MF}^{kin}∣N_{ Mn}∕p is smaller than 2μ_{B} obtained in Section 5.2.1 for the isolated spinorbitcoupled hole bound to the Mn impurity. It is due to occupation of both majority and minority heavy and lighthole bands at these typical (Ga,Mn)As hole densities (see inset of Fig. 25). Data shown in Fig. 24 and in the main panel of Fig. 25 indicate a ~(0.2–0.4)μ_{B} suppression of the meanﬁeld moment per Mn due to the hole kineticenergy contribution to magnetization.


The hole exchange energy contribution to the total meanﬁeld magnetization was found to be negative and nearly independent of x and p in the typical doping range, and its magnitude is about a factor of 5 smaller than the magnitude of the term originating from the hole kinetic band energy Jungwirth et al. (2005a). Quantum ﬂuctuation corrections lead to a 1% suppression of the meanﬁeld moment per Mn Jungwirth et al. (2005a). (Details of these calculations, using the imaginarytime pathintegral formulation of quantum manybody theory combined with the HolsteinPrimakoff bosonic representation of the Mn local moments, can be found in König et al. (2001a) and Jungwirth et al. (2005a).) Combining all these considerations the T = 0 magnetization per Mn in the effective kineticexchange model has a positive meanﬁeld contribution equal to 5μ_{B} from Mn local moments and a negative contribution from band holes and quantum ﬂuctuations which suppress the moment per Mn by ~5–10 %.
The total magnetization per nominal Mn density and per effective density of uncompensated Mn_{Ga} local moments has been measured by a superconducting quantum interference device (SQUID) in a series of asgrown and annealed (Ga,Mn)As samples Jungwirth et al. (2005a). The characterization of these materials has already been discussed in the previous section (see Fig. 20 and the related text). Within experimental uncertainty, the SQUID magnetization was found to be independent of the magnetization orientation, in agreement with theoretical expectations Jungwirth et al. (2005a). The moment decreases with increasing nominal Mn concentration, and increases on annealing Potashnik et al. (2002); Jungwirth et al. (2005a). This is consistent with the anticipated formation of interstitial Mn for doping above ~2% Jungwirth et al. (2005b), given the antiferromagnetic coupling between Mn_{I} and Mn_{Ga} Blinowski and Kacman (2003); Edmonds et al. (2005a), and with breaking of this coupling by lowtemperature annealing Yu et al. (2002); Edmonds et al. (2004a). In agreement with the above theoretical calculations, the total magnetization per effective density of uncompensated Mn_{Ga}, m_{SQUID}^{eff}, falls within the range (4–5)μ_{B} for all samples studied (See Fig. 26). Furthermore, although there is appreciable scatter, it can be seen that samples with lower hole densities tend to show higher m_{SQUID}^{eff}, consistent with a negative contribution to magnetization from antiferromagnetically coupled band holes.
The local and dstateprojected contributions from Mn to the magnetic moment in (Ga,Mn)As have been probed experimentally by measuring the xray magnetic circular dichroism (XMCD) Edmonds et al. (2004b); Jungwirth et al. (2005a); Edmonds et al. (2005a). In agreement with the SQUID measurements and theoretical expectations, the XMCD data are independent, within experimental uncertainty, of the direction of magnetization Jungwirth et al. (2005a). The data are listed in Table 2 for two annealed samples with low and high Mn doping. In both cases, magnetic moments of around 4.5μ_{B} were obtained, showing a negligible dependence on the hole density. Similar results were found for samples with intermediate Mn doping Jungwirth et al. (2005a). Experimental XMCD results are in good agreement with corresponding TBA values indicated by halfopen symbols in Fig. 23 and with LDA+U and SIC LSDA Mn local moments Wierzbowska et al. (2004); Schulthess et al. (2005). (Note that these microscopic calculations account only for the spin angular momentum contribution to the local Mn 3d moment since spinorbitcoupling effects were neglected.)


A small set of parameters is often sufficient to phenomenologically describe the longwavelength properties of ferromagnets. The description, which usually captures all properties that are relevant for applications of magnetic materials, starts from the micromagnetic energy functional e[n] of the spatially dependent magnetization orientation n = M∕M Aharoni (2001). The micromagnetic energy functional treats longranged magnetic dipole interactions explicitly and uses a gradient expansion for other terms in the energy. The magnitude M of the magnetization is one of the micromagnetic parameters characterizing the material. Zerothorder terms in the energy functional are the magnetic anisotropy energy e_{ani}[n] and the Zeeman coupling to an external magnetic ﬁeld if present, μ_{0}H ⋅ nM. The leading gradient term e_{ex}[n] referred to as the exchange energy in micromagnetic theory, represents the reduction in magnetic condensation energy when the magnetization orientation is not spatially constant. Micromagnetic parameters used to characterize the magnetic anisotropy energy depend on the symmetry of the system. For example, in a ferromagnet, which possesses uniaxial anisotropy with the easy axis aligned along the z direction, e_{ani} = K_{u}n_{z}^{2}, where K_{ u} < 0 is the uniaxial anisotropy constant. As we explain below, the magnetic anisotropy of (Ga,Mn)As ferromagnets is a combination of the cubic term, the inplane uniaxial anisotropy, and the uniaxial term induced by the growthdirection latticematching strain which often dominates in (Ga,Mn)As epilayers. Large anisotropies and relatively small magnetic moments of these dilute magnetic systems rank (Ga,Mn)As DMSs among hard ferromagnets (magnetic hardness parameter κ ~∣K_{u}∕μ_{0}M^{2}∣^{1∕2} > 1) with outstanding micromagnetic properties, including frequently observed singledomainlike characteristics of ﬁeldinduced magnetization reversals Ohno (1998); Abolfath et al. (2001b); Dietl et al. (2001b); Potashnik et al. (2003); Wang et al. (2005d); Goennenwein et al. (2005).
Anisotropy of the exchange term in the micromagnetic functional is often neglected in which case it can be written as e_{ex} = A(∇n)^{2}, where A is the spinstiffness constant. The collective magnetization dynamics is described by the LandauLifshitzGilbert equation,
 (27) 
where μ_{0}H_{e} = ∂e∕∂M and α is the Gilbert damping micromagnetic parameter. In this section we review KL kineticexchange model calculations for micromagnetic parameters of metallic (Ga,Mn)As ferromagnets Dietl et al. (2001b); Abolfath et al. (2001a); Dietl et al. (2001a); König et al. (2001a); Brey and GómezSantos (2003); Sinova et al. (2004b) and predictions based on the microscopic values of these parameters for anisotropy ﬁelds, the characteristic size of domains, and the critical current in currentinduced magnetization switching Abolfath et al. (2001a); Dietl et al. (2001a); Sinova et al. (2004b).
Magnetocrystalline anisotropy, which is the dependence of the ferromagnet energy on the magnetization orientation with respect to the crystallographic axes, is a spinorbitcoupling effect often associated with localized electrons in magnetic d or f shells. Local Mn moments in (Ga,Mn)As, however, are treated in the KL kineticexchange model as pure spins S = 5∕2 with angular momentum L = 0, and therefore do not contribute to anisotropy. The physical origin of the anisotropy energy in this model is spinorbit coupling in the valence band Dietl (2001); Abolfath et al. (2001a). Even within the meanﬁeld approximation to the KL kineticexchange model, magnetic anisotropy has a rich phenomenology which has explained a number of experimental observations in (III,Mn)V DMSs, including easyaxis reorientations as a function of hole density, temperature, or strains in the lattice Ohno et al. (1996a); Liu et al. (2003); Sawicki et al. (2004, 2005b); Masmanidis et al. (2005); Liu et al. (2004b).
The remarkable tunability of the magnetic properties of (Ga,Mn)As DMSs through latticematching strains is an important byproduct of LT MBE growth of ferromagnetic ﬁlms with lattices locked to those of their substrates in the plane perpendicular to the growth axis. Xraydiffraction studies have established that the resulting strains are not relaxed by dislocations or other defects even for ~ 1μmthick epilayers Shen et al. (1999); Zhao et al. (2005). We mentioned in Section 4.2 that the lattice constant of relaxed (Ga,Mn)As is larger than the lattice constant of GaAs, especially if interstitial Mn_{I} or As_{Ga} antisites are present in the DMS crystal. (Ga,Mn)As grown on a GaAs substrate is therefore under compressive strain. Tensile strained (Ga,Mn)As DMSs have been produced using (Ga,In)As substrates Ohno et al. (1996a); Shono et al. (2000); Liu et al. (2003).
Strain in the [001] growth direction breaks the cubic symmetry of (Ga,Mn)As resulting in a combined cubic and uniaxial anisotropy form of the energy functional,
Here K_{c1} and K_{c2} are the two lowestorder cubic anisotropy constants Dietl et al. (2001b); Abolfath et al. (2001a). Strain must be included in the k⋅p description in order to evaluate the uniaxial anisotropy constant K_{u}. For small strains this is done by expressing the positional vector r^{′} in the strained lattice in terms of r in the unstrained lattice as r_{α}^{′} = r_{ α} + ∑ _{β}e_{αβ}r_{β}, and expanding the KL Hamiltonian in lowest order of the strain constants e_{αβ} Jones and March (1973); Chow and Koch (1999). In (Ga,Mn)As epilayers grown along the [001] direction, the strain constant e_{xx} = e_{yy} can be tuned from approximately 1% to +1%. For larger strain values, the uniaxial term dominates the total anisotropy energy Dietl et al. (2001b); Abolfath et al. (2001a). In Fig. 27 we show meanﬁeld KL kineticexchange model calculations of the uniaxial anisotropy ﬁeld μ_{0}H_{u} = ∣2K_{u}∕M∣ relative to the T = 0 magnetization M = gμ_{B}N_{Mn}S. μ_{0}H_{u} corresponds to the minimum external magnetic ﬁeld necessary to align the magnetization M along the hard axis. Figure (27) illustrates the dependence of the easyaxis orientation on the hole density and on the sign of e_{xx}. In particular, the easy axis is in plane for compressive strain (e_{xx} < 0) and out of plane for tensile strain (e_{xx} > 0), consistent with experiment Ohno et al. (1996a); Liu et al. (2003). The ability to manipulate the easy axisorientation from in plane to out of plane has many implications for fundamental research on DMS materials and is very attractive also from the point of view of potential applications in magnetic recording technologies.

(Ga,Mn)As epilayers have also a relatively strong inplane uniaxial anisotropy component Liu et al. (2003); Sawicki et al. (2004, 2005b); Tang et al. (2003). The inplane easy axis is consistently associated with particular crystallographic directions and can be rotated from the [110] direction to the [110] direction by lowtemperature annealing. Although the origin of the inplane uniaxial anisotropy has not been established, its dependence on hole concentration (varied by annealing) and temperature was modeled successfully within the KL kineticexchange model by assuming that it is associated with a small shear strain e_{xy} ≈ 0.05% Sawicki et al. (2005b).
Thermal and quantum magnetization ﬂuctuation effects in metallic (Ga,Mn)As DMSs have been described by HolsteinPrimakoff boson representation of Mn local spin operators. Assuming that ﬂuctuations around the meanﬁeld orientation are small, the relationship between spinraising and spinlowering operators and boson creation and annihilation operators is S^{+} = b and S^{} = b^{†} (S_{x} = (S^{+} + S^{})∕2, S_{ y} = (S^{+} S^{})∕2i) Auerbach (1994); König et al. (2000, 2001a). After integrating out the itineranthole degrees of freedom in the coherentstate pathintegral formalism of the manybody problem, the partition function
 (29) 
depends only on the bosonic degrees of freedom (represented by the complex numbers z and in the effective action S_{eff}). The independentspinwave theory is obtained by expanding S_{eff}[ z] up to quadratic order in z and , i.e., spin excitations are treated as noninteracting bosons. The spinstiffness parameter A is then calculated by ﬁtting the microscopic spinwave dispersion at long wavelength by the form
 (30) 
to match the conventions used for exchange and anisotropy constants in micromagnetic theory. Typical values of A in (Ga,Mn)As derived from the above manybody formalism and the KL kineticexchange description of hole bands are shown in Fig. 28. Values are consistent with the experimental spinstiffness parameter in Ga_{0.949}Mn_{0.051}As measured by ferromagnetic resonance Goennenwein et al. (2003).

The damping of smallconeangle magnetization precession in a ferromagnet is parametrized by the Gilbert coefficient. For small ﬂuctuations of the Mn magnetization orientation in (Ga,Mn)As around the easy axis, Eq. (27) can be used to derive an expression for the linear response of a magnetic system to weak transverse ﬁelds in terms of the phenomenological constants of micromagnetic theory. For zero external static magnetic ﬁeld and zero wave vector (uniform rotation), the corresponding inverse susceptibility reads
 (31) 
where = K_{u}∕(ˉhN_{Mn}S) and ω is the frequency of the external rf ﬁeld perturbation.
Microscopically, Gilbert damping in (Ga,Mn)As DMSs was attributed to the pd exchange coupling between local Mn moments and itinerant holes Sinova et al. (2004b). The elementary process for this damping mechanism is one in which a localmoment magnon is annihilated by exchange interaction with a band hole that suffers a spin ﬂip. This process cannot by itself change the total magnetic moment since the exchange Hamiltonian commutes with the total spin S + s. Net relaxation of the magnetization requires another independent process in which the itineranthole spin relaxes through spinorbit interactions. A fully microscopic theory of the kineticexchange contribution to the Gilbert coefficient was derived by comparing Eq. (31) with microscopic linear response theory and by identifying the Gilbert coefficient with the dissipative part of the quantummechanical susceptibility Sinova et al. (2004b); Tserkovnyak et al. (2004),
 (32) 
Here i = x,y and S_{i}(r,t) = M_{i}(r,t)∕(gμ_{B}) are the Mn transverse spindensity operators. As in the microscopic theory of the spin stiffness, the correlation function (32) for the uniform (k = 0) precession mode was evaluated using the the longwavelength noninteracting spinwave form of the partition function (29) Sinova et al. (2004b). Gilbert coefficients in (Ga,Mn)As DMSs obtained within this formalism agree quantitatively with experimental values of α in homogeneous (annealed) systems measured from the width of the ferromagnetic resonance curves, as shown in Figs. 29 and 30.


Theoretical values of magnetic anisotropy and spin stiffness have been combined to estimate the typical domain size in tensilestrained (Ga,Mn)As epilayers with outofplane easy axis Dietl et al. (2001a). The calculated lowtemperature width of a singledomain stripe of 1.1 μm compares favorably with the experimental value of 1.5 μm seen in the microHallprobe experiments Shono et al. (2000). Near T_{c} the discrepancy between theoretical and experimental domain sizes becomes large, however, and has been attributed to critical ﬂuctuation effects not included in the meanﬁeld theory.
The calculated Gilbert damping coefficient and magnetic anisotropy constants were used to predict critical currents for spintransfer magnetization switching Slonczewski (1996); Berger (1996) in (Ga,Mn)Asbased tunneling structures Sinova et al. (2004b). Spinpolarized perpendiculartoplane currents in magnetic multilayers with nonparallel spin conﬁgurations can transfer spin between magnetic layers and exert currentdependent torques Slonczewski (1996). The critical current for magnetization switching is obtained by adding the torque term to the LandauLifshitzGilbert equation (27) Slonczewski (1996); Sinova et al. (2004b). The critical currents ~ 10^{5} A cm^{2} obtained in these calculations and conﬁrmed in experiment Chiba et al. (2004b) are two orders of magnitude smaller than those observed typically in metals. The small moment densities explain a large part of the orders of magnitude reduction in the critical current. This ﬁnding suggests that DMS materials have the potential to be particularly useful for exploiting the currentinduced magnetization reversal effect in magnetic tunnel junctions.
Studies of the temperaturedependent resistivity, anisotropic magnetoresistance, and anomalous and ordinary Hall effects have been used to characterize DMS materials and to test different theoretical models describing these ferromagnets. In this section we review dc magnetotransport properties of (Ga,Mn)As focusing mainly on the metallic regime.
(Ga,Mn)As materials can exhibit insulating or metallic behavior depending on the doping and postgrowthannealing procedures. (Strictly speaking, a material is deﬁned as being metallic if its resistivity is ﬁnite in the limit T → 0, although for practical reasons this adjective is often used to describe a material whose resistivity decreases with increasing temperature over most or all of the range of temperatures studied in a particular series of experiments.) In optimally annealed samples with low density of unintentional defects, metallic behavior is observed for Mn doping larger than approximately 1.5% Campion et al. (2003a); Potashnik et al. (2002). In Section 2.4 we discussed this observation as a consequence of the Mott metalinsulator transition due to doping with substitutional Mn_{Ga} acceptors. We also introduced in Sections 2.4 and 3.4 some of the theoretical work that has qualitatively addressed magnetic and transport properties of DMS systems near the metalinsulator transition. To our knowledge, a systematic experimental analysis has not yet been performed that would allow a reliable assessment of the theory predictions in this complex and intriguing regime. On the other hand, the dc transport in metallic (Ga,Mn)As DMSs has a rich phenomenology which has been explored in a number of experimental works and microscopic understanding of many of these effects is now well established.
LSDA CPA bandstructure calculations combined with Kubo linear response theory were used to study correlations between the lowtemperature conductivity and density of various defects in the lattice, the hole density, and T_{c} in metallic (Ga,Mn)As Turek et al. (2004). The theory tends to overestimate the conductivity in lowcompensation materials but the overall range of values ~ 100–1000 Ω^{1}cm^{1} for typical material parameters is consistent with measured data. As illustrated in Figs. 31 and 32, the theory models capture at a qualitative level the correlation between large conductivities and high Curie temperatures seen in experiment Campion et al. (2003a); Potashnik et al. (2002); Edmonds et al. (2002a). Comparable absolute values of the T = 0 conductance and similar trends in the dependence of the conductance on the density of impurities were obtained using the KL kineticexchange model and the semiclassical Boltzmann description of the dc transport Jungwirth et al. (2002a); LópezSancho and Brey (2003); Hwang and Das Sarma (2005). We note, however, that there are differences in the detailed microscopic mechanisms limiting the conductivity in the two theoretical approaches. Scattering in the ab initio theory is dominated by the pd exchange potential on randomly distributed Mn atoms, and by local changes of the crystal potential on the impurity sites. Longrange Coulomb potentials produced by Mn_{Ga} acceptors and other charged defects are omitted in the CPA approach.


In the kineticexchange effective Hamiltonian model elasticscattering effects were included using the ﬁrstorder Born approximation Jungwirth et al. (2002a). The corresponding transportweighted scattering rate from the Mn_{Ga} impurity has contributions from both the pd exchange potential and the longrange Coulomb potential,
with scattering matrix elements Here ε_{host} is the host semiconductor dielectric constant, ∣z_{i}〉 is the multicomponent eigenspinor of the KL Hamiltonian (21) and E_{i,} is the corresponding eigenenergy, and the ThomasFermi screening wave vector q_{TF} = , where 𝒟(E_{F }) is the density of states at the Fermi energy Jungwirth et al. (2002a). (Analogous expressions apply to scattering rates due to other defects.)The relative strengths of scattering off the pd exchange and Coulomb potentials can be estimated by assuming a simple parabolicband model of the valence band characterized by the heavyhole effective mass m^{*} = 0.5m_{ e}. The pd kineticexchange contribution in this approximation is Γ_{pd} = N_{Mn}J_{pd}^{2}S^{2}m^{*}∕(4πˉh^{4}) and the scattering rate due to the screened Coulomb potential Γ_{C} is given by the standard BrooksHerring formula Brooks (1955). In (Ga,Mn)As with p = 0.4 nm^{3} and x = 5%, these estimates give ˉhΓ_{pd} ~ 20 meV and ˉhΓ_{C} ~ 150 meV. For lower compensation (p ≈ 1 nm^{3}), the screening of the Coulomb potential is more efficient, resulting in values of Γ_{C} below 100 meV, but still several times larger than Γ_{pd}. (Note that these elasticscattering rates are smaller, although by less than a factor of 10, than other characteristic energy scales such as the Fermi energy and the spinorbitcoupling strength in the GaAs valence band which partly establishes the consistency of this theoretical approach.)
The dominance of the Coulomb potential in Born approximation scattering rates for typical chemical compositions is conﬁrmed by calculations based on the sixband KL Hamiltonian Jungwirth et al. (2002a). Good agreement between T = 0 conductivity values obtained using the ab initio and kineticexchange model theories should therefore be taken with caution as it may originate, to some extent, from the stronger local pd exchange in the LSDA CPA theory, which partly compensates the neglect of longrange Coulomb potentials in this ab initio approach.
Boltzmann transport theory combined with the KL kineticexchange model of the (Ga,Mn)As band structure is a practical approach for studying magnetotransport effects that originate from the spinorbit coupling. In Section 5.3 we reviewed the predictions of the model for the role of spinorbit interaction in magnetic properties of (Ga,Mn)As. Here we focus on the anisotropic magnetoresistance (AMR) effect which is the transport analog of the magnetocrystalline anisotropy Wang et al. (2002); Baxter et al. (2002); Jungwirth et al. (2002a); Tang et al. (2003); Jungwirth et al. (2003b); Matsukura et al. (2004); Wang et al. (2005c); Goennenwein et al. (2005).
The AMR effect can be regarded as the ﬁrst spintronic functionality implemented in microelectronic devices. AMR magnetic sensors replaced simple horseshoe magnets in harddrive read heads in the early 1990s. With the introduction of giantmagnetoresistancebased devices in 1997, a new era was launched in the magnetic memory industry. In ferromagnetic metals, AMR has been known for well over a century. However, the role of the various mechanisms held responsible for the effect has not been fully clariﬁed despite renewed interest motivated by practical applications Jaoul et al. (1977); Malozemoff (1985). The difficulty in metal ferromagnets partly stems from the relatively weak spinorbit coupling compared to other relevant energy scales and the complex band structure. In (Ga,Mn)As ferromagnets with strongly spinorbitcoupled holes occupying the top of the valence band, modeling of the AMR effect can be accomplished on a semiquantitative level without using free parameters, as we review below.
(Ga,Mn)As epilayers with broken cubic symmetry due to growthdirection latticematching strains are usually characterized by two AMR coefficients,

Reminiscent of the magnetocrystalline anisotropy behavior, the theory predicts rotations as a function of the magnetization strain direction corresponding to the high (or low) resistance state. Calculations illustrating this effect are shown in Fig. 34 and the experimental demonstration in (Ga,Mn)As epilayers with compressive and tensile strains is presented in Fig. 35. In the top panel of Fig. 35, corresponding to (Ga,Mn)As material with compressive strain, M ⊥ plane (M ∥ẑ) is the highresistance conﬁguration, the intermediateresistance state is realized for inplane M ⊥ I (M ∥ŷ), and the lowresistance state is measured when M ∥ I (M ∥). In the sample with tensile strain, the inplane M ⊥ I and M ⊥ plane curves switch places, as seen in the bottom panel of Fig. 35. Consistent with these experimental observations the theoretical AMR_{ip} and AMR_{op} coefficients are negative and ∣AMR_{ip}∣ < ∣AMR_{op}∣ for compressive strain and ∣AMR_{ip}∣ > ∣AMR_{op}∣ for tensile strain.


The anomalous Hall effect (AHE) is another transport phenomenon originating from spinorbit coupling which has been used to study and characterize ferromagnetic ﬁlms for more than one hundred years Chien and Westgate (1980). The difficulties that have accompanied attempts at accurate microscopic description of the effect in metal ferromagnets are reminiscent of those in the AMR. The success of AHE modeling in (Ga,Mn)As materials Jungwirth et al. (2002c, 2003b); Edmonds et al. (2003); Dietl et al. (2003), reviewed in this section, has had implications also beyond the ﬁeld of DMSs. It helped to motivate a reexamination of the AHE in transition metals and and in a series of more complex ferromagnetic compounds, which has led to a signiﬁcant progress in resolving the microscopic mechanism responsible for the effect in these materials Yao et al. (2004); Fang et al. (2003); Lee et al. (2004); Dugaev et al. (2005); Kötzler and Gil (2005); Sinitsyn et al. (2005); Haldane (2004).
The Hall resistance R_{xy} ≡ ρ_{xy}∕d of a magnetic thin ﬁlm of thickness d is empirically observed to contain two distinct contributions, R_{xy} = R_{O}B + R_{A}M Chien and Westgate (1980). The ﬁrst contribution arises from the ordinary Hall effect (OHE) which is proportional to the applied magnetic ﬁeld B, and the second term is the AHE which may remain ﬁnite at B = 0 and depends instead on the magnetization. R_{O} and R_{A} are the OHE and AHE coefficients, respectively. To put the AHE studies in DMSs in a broader perspective, the following discussion offers a brief excursion through the history of AHE theory and an overview of microscopic mechanisms discussed in this context in the literature (a more detailed survey is given by, e.g., Sinova et al. (2004a)).
The ﬁrst detailed theoretical analysis of the AHE was given by Karplus and Luttinger Karplus and Luttinger (1954), where they considered the problem from a perturbative point of view (with respect to an applied electric ﬁeld) and obtained a contribution to the anomalous Hall conductivity in systems with spinorbitcoupled Bloch states given by
 (36) 
where f_{n,} is the Fermi occupation number of the Bloch state ∣u_{n,}〉. This contribution is purely a property of the perfect crystal and has become known in recent years as the intrinsic AHE. It leads to an AHE coefficient R_{A} proportional to ρ_{xx}^{2} and therefore can dominate in metallic ferromagnets that have a relatively large resistivity. The intrinsic AHE is related to Blochstate Berry phases in momentum space and depends nonperturbatively on the spinorbit interaction strength when degeneracies in momentum space are lifted by spinorbit coupling Sundaram and Niu (1999); Jungwirth et al. (2002c); Onoda and Nagaosa (2002); Burkov and Balents (2003); Haldane (2004); Dugaev et al. (2005). This point is particularly relevant for ferromagnetic semiconductors because all carriers that contribute to transport are located near particular points in the Brillouin zone, often highsymmetry points at which degeneracies occur.
Shortly after the seminal work of Karplus and Luttinger, Smit proposed a different interpretation of the AHE based on a picture of asymmetric spindependent skew scattering off an impurity potential involving spinorbit coupling Smit (1955); Nozieres and Lewiner (1973); LerouxHugon and Ghazali (1972). Analytically, the skew scattering appears in the secondorder Born approximation applied to the collision term of the Boltzmann transport equation. This mechanism gives a contribution to R_{A} ∝ ρ_{xx}, i.e., proportional to the density of scatterers, and dependent on the type and range of the impurity potential.
The AHE conductivity has a number of contributions in addition to the skew scattering and intrinsic contributions, which can originate from spinorbit coupling in either the disorder scattering or the Bloch bands. Among these, side jump scattering has been identiﬁed as an important contribution Berger (1970). Side jumps due to spinorbit coupling in Bloch bands appear as a ladder diagram vertex correction to the intrinsic anomalous Hall effect and their importance depends on the nature of that coupling Dugaev et al. (2005); Sinitsyn et al. (2005).
The ratio of intrinsic and skewscattering contributions to the AHE conductivity can be approximated, assuming a single spinorbitcoupled band and scattering off ionized impurities, by LerouxHugon and Ghazali (1972); Chazalviel (1975):
 (37) 
where N∕p is the ratio of the density of ionized impurities and the carrier density, r_{s} is the average distance between carriers in units of the Bohr radius, l is the mean free path, and c ~ 10, varying slightly with scattering length. For the shortrange scattering potential considered by Luttinger Luttinger (1958) and Nozieres and Lewiner Nozieres and Lewiner (1973), V () = V _{0}δ( ):
 (38) 
The estimate is a useful ﬁrst guess at which mechanism dominates in different materials but one must keep in mind the simplicity of the models used to derive these expressions.
In (Ga,Mn)As, Eq. (37) gives a ratio of the intrinsic to skewscattering contribution of the order of 50 and the intrinsic AHE is therefore likely to dominate Dietl et al. (2003). Consistently, the experimental R_{A} in metallic (Ga,Mn)As DMSs is proportional to ρ_{xx}^{2} Edmonds et al. (2002b). Microscopic calculations of the intrinsic lowtemperature AHE conductivity in (Ga,Mn)As were performed by taking into account the Berryphase anomalous velocity term in the semiclassical Boltzmann equation leading to Eq. (36) Jungwirth et al. (2002c), and by applying the fully quantummechanical Kubo formalism Jungwirth et al. (2003b). In both approaches, the KL kineticexchange model was used to obtain the hole band structure.
In the Kubo formula, the dc Hall conductivity for noninteracting quasiparticles at zero external magnetic ﬁeld is given by Onoda and Nagaosa (2002); Jungwirth et al. (2003b)
The real part of Eq. (39) in the limit of zero scattering rate (Γ → 0) can be written as Realizing that the momentum matrix elements 〈n′∣ _{α}∣n〉 = (m∕ˉh)〈n′∣∂ℋ()∕∂_{α}∣n〉, Eq. (40) can be shown to be equivalent to Eq. (36). The advantage of the Kubo formalism is that it makes it possible to include ﬁnite lifetime broadening of quasiparticles in the simulations. (See Section 6.2 for the discussion of quasiparticle scattering rates in (Ga,Mn)As. Whether or not lifetime broadening is included, the theoretical anomalous Hall conductivities are of order 10 Ω^{1} cm^{1} for typical (Ga,Mn)As DMS parameters. On a quantitative level, a nonzero Γ tends to enhance σ_{AH} at low Mn doping and suppresses σ_{AH} at high Mn concentrations, where quasiparticle broadening due to disorder becomes comparable to the strength of the kineticexchange ﬁeld.Systematic comparison between theoretical and experimental AHE data is shown in Fig. 36 Jungwirth et al. (2003b). Results are plotted versus nominal Mn concentration x while other parameters of the measured samples are listed in the legend. Experimental σ_{AH} values are indicated by ﬁlled squares and empty triangles correspond to theoretical data obtained for Γ = 0. Results shown in halfopen triangles were obtained by solving the Kubo formula for σ_{AH} with nonzero Γ due to scattering off Mn_{Ga} and As antisites, or Mn_{Ga} and Mn_{I} impurities. Calculations explain much of the measured lowtemperature AHE in metallic (Ga,Mn)As DMSs, especially so for the Mn_{I} compensation scenario. The largest quantitative discrepancy between theory and experiment is for the x = 8% material, which can be partly explained by a nonmeanﬁeldlike magnetic behavior of this speciﬁc, more disordered sample.

In DMSs the AHE has played a key role in establishing ferromagnetism and in providing evidence for holemediated coupling between Mn local moments Ohno et al. (1992); Ohno (1998). The dominance of the AHE in weakﬁeld measurements (see Fig. 1) allows the Hall resistance to serve also as a convenient proxy for magnetization. On the other hand, the same property can obscure Hall measurements of the hole density. If magnetization is not fully saturated at low ﬁelds, for example, then ρ_{AH} = σ_{AH}ρ_{xx}^{2} will increase with increasing external ﬁeld B through the dependence of σ_{AH} on M(B), and hole densities derived from the slope of ρ_{xy}(B) will be too low. Accurate determination of hole densities in DMSs is essential, however, and the Hall effect is arguably the most common and accurate nondestructive tool for measuring the level of doping in semiconductors. Hall experiments performed in high magnetic ﬁelds to guarantee magnetization saturation seem a practical way for separating AHE contributions, especially in samples showing weak longitudinal magnetoresistance Ohno (1999); Edmonds et al. (2002b). Holedensity measurements performed using this technique assume that the Hall factor, r_{H} = (ρ_{xy}(B)  ρ_{AH})∕(B∕ep), with ρ_{AH} = ρ_{xy}(B = 0), is close to 1 despite the multiband spinorbitcoupled nature of hole dispersion in (Ga,Mn)As ferromagnets. In the following paragraph we brieﬂy review a theoretical analysis of this assumption Jungwirth et al. (2005b).
Microscopic calculations in nonmagnetic ptype GaAs with hole densities p ~ 10^{17}–10^{20} cm^{3} have shown that r_{ H} can vary between 0.87 and 1.75, depending on doping, scattering mechanisms, and the details of the model used for the GaAs valence band Kim and Majerfeld (1995). An estimate of the inﬂuence on r_{H} from the spin splitting of the valence band and from the anomalous Hall term is based on the KL kineticexchange model description of the hole band structure. The Hall conductivity has been obtained by evaluating the Kubo formula at ﬁnite magnetic ﬁelds that includes both intraband and interband transitions. The approach captures the anomalous and ordinary Hall terms on equal footing Jungwirth et al. (2005a).

Many of the qualitative aspects of the numerical calculations, shown in Fig. 37, can be explained using a simple model of a conductor with two parabolic uncoupled bands. Note that the typical scattering rate in (Ga,Mn)As epilayers is ˉh∕τ ~ 100 meV and that the cyclotron energy at B = 5 T is ˉhω ~ 1 meV, i.e., the system is in the strongscattering limit, ωτ ≪ 1. In this limit, the twoband model gives resistivities
where the indices 1 and 2 correspond to the ﬁrst and second bands, respectively, the total density p = p_{1} + p_{2}, and the zeroﬁeld conductivity σ_{0} = e^{2}τp∕m^{*}. Equation (41) suggests that in the strongscattering limit the multiband nature of hole states in (Ga,Mn)As should not result in a strong longitudinal magnetoresistance. This observation is consistent with the measured weak dependence of ρ_{xx} on B for magnetic ﬁelds at which magnetization in the (Ga,Mn)As ferromagnet is saturated Edmonds et al. (2002b).The simple twoband model also suggests that the Hall factor r_{H} is larger than 1 in multiband systems with different dispersions of individual bands. Indeed, for uncoupled valence bands, i.e., when accounting for intraband transitions only, the numerical Hall factors in the top panels of Fig. 37 are larger than 1 and independent of τ as also suggested by Eq. (41). The suppression of r_{H} when spinorbit coupling is turned on, shown in the same graphs, results partly from depopulation of the angular momentum j = 1∕2 splitoff bands. In addition to this “twoband model”like effect, the interLandaulevel matrix elements are reduced due to spinorbit coupling since the spinor part of the eigenfunctions now varies with the Landaulevel index. In ferromagnetic Ga_{1x}Mn_{x}As bands are spin split and higher bands depopulated as x increases. In terms of r_{H}, this effect competes with the increase of the interLandaulevel matrix elements since spinors are now more closely aligned within a band due to the exchange ﬁeld produced by polarized Mn moments. Increasing x can therefore lead to either an increase or a decrease in r_{H} depending on other parameters, such as the hole density (compare top right panels of Fig. 37).
Interband transitions result in a more singlebandlike character of the system, i.e., r_{H} is reduced, and the slope of the ρ_{xy}(B) curve now depends more strongly on τ. Although the AHE and OHE contributions to ρ_{xy} cannot be simply decoupled, comparison of the numerical data in the four panels conﬁrms the usual assumption that the AHE produces a ﬁeldindependent offset proportional to magnetization and ρ_{xx}^{2}. Comparison also suggests that after subtracting ρ_{xy}(B = 0), r_{H} can be used to determine the hole density in (Ga,Mn)As with accuracy that is better than in nonmagnetic GaAs with comparable hole densities. For typical hole and Mn densities in experimental (Ga,Mn)As epilayers the error of the Hall measurement of p is estimated to be less than ą20% Jungwirth et al. (2005a).
Typical Fermi temperatures T_{F } = E_{F }∕k_{B} in ferromagnetic (Ga,Mn)As are much larger than the Curie temperature, relegating direct Fermi distribution effects of ﬁnite temperature to a minor role in transport. The carriermediated nature of ferromagnetism implies, however, strong indirect effects through the temperature dependence of the magnetization. A prime example is the AHE which from early studies of (III,Mn)V DMSs has served as a practical tool to accurately measure Curie temperatures Ohno et al. (1992). A rough estimate of T_{c} can also be inferred from the temperaturedependent longitudinal resistivity which exhibits a shoulder in more metallic (optimally annealed) samples and a peak in less metallic (asgrown) materials near the ferromagnetic transition Edmonds et al. (2002a); Potashnik et al. (2001); Van Esch et al. (1997); Matsukura et al. (1998); Hayashi et al. (1997). An example of this behavior is shown in Fig. 38 for a (Ga,Mn)As material with 8% nominal Mn doping Potashnik et al. (2001).
The shoulder in ρ_{xx}(T) has been qualitatively modeled using the meanﬁeld, KL kineticexchange Hamiltonian given by Eq. (21). Solutions to the Boltzmann equation LópezSancho and Brey (2003); Hwang and Das Sarma (2005) are shown in Fig. 39. The temperature dependence of the longitudinal conductivity follows in this theory from variations in parameters derived from the spinpolarized hole band structure (e.g., Fermi wave vector) and from variations in screening of impurity Coulomb potentials.

The peak in resistance near T_{c} has been discussed in terms of scattering effects beyond the lowestorder Born approximation and by using a network resistor model Timm et al. (2005). It has also been suggested that this transport anomaly in more highly resistive DMSs is a consequence of the change in localization length caused by the ferromagnetic transition Zaránd et al. (2005).

Above the Curie temperature, measurements of ρ_{xx} have been used to estimate the value of J_{pd}. Assuming scattering off the pd exchange potential at randomly distributed paramagnetic Mn_{Ga} impurities and parabolic hole bands, the corresponding contribution to the resistivity is approximated by
 (42) 
where χ_{⊥} and χ_{∥} are the transverse and longitudinal magnetic susceptibilities Dietl (1994); Omiya et al. (2000); Matsukura et al. (2002). This theory overestimates critical scattering, particularly near the Curie temperature where the susceptibility diverges. Far from the transition on the paramagnetic side, however, ﬁtting Eq. (42) to experimental magnetoresistance data gives an estimate of the J_{pd} Omiya et al. (2000) which is consistent with values inferred from spectroscopical measurements Okabayashi et al. (1998) (see Fig. 40).

Prospects for new technologies, based, for example, on materials in which the ferromagnetic transition can be controlled by light or on (III,Mn)V Faraday isolators monolithically integrated with existing semiconductor lasers, have motivated research in magnetooptical properties of DMSs Munekata et al. (1997); Koshihara et al. (1997); Sugano and Kojima (2000); Matsukura et al. (2002). Apart from these applied physics interests, ac probes have been used to study DMS materials with a wide range of experimental techniques. In Sections 2.2.1 and 5.2.2 we mentioned xray spectroscopies (corelevel photoemission and XMCD) used to characterize Mn 3d states and detect the sign and magnitude of the pd exchange coupling. Dispersions of hole bands in DMSs have been studied by angleresolved photoemission with ultraviolet excitation Okabayashi et al. (2002, 2001); Asklund et al. (2002) and infraredtoultraviolet spectroscopic ellipsometry Burch et al. (2004). Raman scattering induced by excitations in the visible range was used as an alternative means of estimating hole densities Seong et al. (2002); Limmer et al. (2002); Sapega et al. (2001). Spectroscopic studies of isolated Mn(d^{5}+hole) impurities in the infrared region provided key information on the valence of Mn in (Ga,Mn)As, as discussed in Section 2.2, and cyclotron resonance measurements were used to study highlyMndoped DMS materials in this frequency range Khodaparast et al. (2004); Sanders et al. (2003); Mitsumori et al. (2003). Microwave EPR and FMR experiments mentioned in Section 2.2 and 5.3 have been invaluable for understanding the nature of Mn in IIIV hosts at low and high dopings and for characterizing magnetocrystalline anisotropies and magnetization dynamics in ferromagnetic materials. In this section we review some studies of the magnetooptical responses Ando et al. (1998); Kimel et al. (2005); Matsukura et al. (2002); Beschoten et al. (1999); Szczytko et al. (1999a); Hrabovsky et al. (2002); Lang et al. (2005), particularly magnetic circular dichroism (MCD), in the visible range and infrared absorption Singley et al. (2002, 2003); Hirakawa et al. (2002, 2001); Nagai et al. (2001); Szczytko et al. (1999a); Burch et al. (2005).
Optical absorption due to electron excitations across the band gap is a standard characterization technique in semiconductors. In (Ga,Mn)As, absorption occurs in the visible range and the position of its edge on the frequency axis depends on the circular polarization of incident light. Analysis of this magnetooptical effect provides information on the pd exchangeinduced band splitting and on doping in the DMS material Dietl (2002); Matsukura et al. (2002).

The schematic diagram in Fig. 41 shows that for a given sign of the exchange coupling the order of absorption edges corresponding to two circular photon polarizations can reverse in ptype materials, compared to systems with a completely ﬁlled valence band. Calculations for (Ga,Mn)As that include this MossBurnstein effect were carried out using the meanﬁeld KL kineticexchange model Dietl et al. (2001b). The resulting absorptions α^{ą} of the σ^{ą} circularly polarized light, and MCD, deﬁned as Sugano and Kojima (2000)
 (43) 
are shown in Fig. 42. As suggested in the cartoon, the sign of the MCD signal in (Ga,Mn)As is opposite to the one obtained in bulk (II,Mn)VI DMSs where the sense of band splittings is the same as in (Ga,Mn)As but Mn substituting for the groupII element is an isovalent neutral impurity Dietl (1994). (The MossBurnstein sign change in MCD was also observed in codoped ptype (II,Mn)VI quantum well Haury et al. (1997).)

Experimentally, the incorporation of several percent of Mn in GaAs enhances MCD, as shown in Fig. 43 Ando et al. (1998); Matsukura et al. (2002), and the sign of the signal near bandgap frequencies is consistent with the above theory which assumes antiferromagnetic pd exchange coupling Ando et al. (1998); Beschoten et al. (1999); Szczytko et al. (1999a); Dietl et al. (2001b).

In (III,Mn)V DMSs, light absorption can occur at subbandgap frequencies due to excitations from the valence band to the Mn impurity band Hwang et al. (2002); Alvarez and Dagotto (2003); Craco et al. (2003) in more insulating materials and due to intravalenceband excitations Sinova et al. (2002); Yang et al. (2003); Sinova et al. (2003) in more metallic systems, as illustrated schematically in Fig. 44. The infrared absorption associated with substitutional Mn impurities is spectrally resolved from higherenergy excitations to donor levels of the most common unintentional defects, such as Mn_{I} interstitials and As_{Ga} antisites in (Ga,Mn)As, and therefore represents another valuable probe into intrinsic properties of these systems. Since infrared wavelengths are much larger than typical (submicron) DMS epilayer widths, the absorption is related to the real part of the conductivity by Sugano and Kojima (2000)
 (44) 
where Y and Y _{0} are the admittances of the substrate and free space, respectively.

Model Hamiltonians (8) (see Section 3.4) combined with dynamical meanﬁeldtheory or Monte Carlo simulations were used to study the role of the impurity band in infrared absorption Hwang et al. (2002); Alvarez and Dagotto (2003). In this theory, the impurity band forms when the strength of the model effective exchange interaction J is comparable to the width of the main band, characterized by the hopping parameter t. A nonDrude peak is observed in the frequencydependent conductivity, associated with transitions from the main band to the impurity band. The behavior is illustrated in Fig. 45 together with the predicted temperature dependence of the absorption spectra obtained using the Monte Carlo technique. As discussed in Section 3.4, these model calculations are expected to apply to systems with strong pd exchange coupling, like (Ga,Mn)P, and possibly also to (III,Mn)V DMSs, which are strongly compensated due to the presence of unintentional donor defects. (Impuritybandmediated ferromagnetism does not occur in uncompensated samples.)

Theoretical infrared absorption spectra calculated using the k ⋅ p model for (Ga,Mn)As DMSs with delocalized holes in the semiconductor valence band, plotted in Fig. 46, show similar nonDrude characteristics with a peak near the excitation energy of 220 meV. The underlying physics is qualitatively different, however, as the peak in these KL kineticexchange model calculations originates from heavyhole to lighthole intravalenceband transitions Sinova et al. (2002). These results were obtained by evaluating the Kubo formula for ac conductivity assuming noninteracting holes and modeling disorder within the ﬁrstorder Born approximation (see Eq. (33) in Section 6.1). In Fig. 47 we show theoretical predictions of exactdiagonalization studies based on the KL kineticexchange Hamiltonian but treating disorder effects exactly in a ﬁnitesize system. The results correct for the overestimated dc conductivity in the former model, which is a quantitative deﬁciency of the Born approximation as already mentioned in Section 6.1. At ﬁnite frequencies, the theoretical absorption in these metallic (Ga,Mn)As DMSs is almost insensitive to the way disorder is treated in the simulations, as seen from Figs. 46 and 47.


Experimental infrared absorption studies in ferromagnetic (Ga,Mn)As epilayers exhibit several common features summarized in Fig. 48 Singley et al. (2002, 2003); Hirakawa et al. (2002, 2001); Nagai et al. (2001); Szczytko et al. (1999a); Burch et al. (2005). Ferromagnetic materials (x = 5.2% curves in Fig. 48) show a nonDrude behavior in which the conductivity increases with increasing frequency in the interval between 0 and 220 meV, a broad absorption peak near 220–260 meV that becomes stronger as the sample is cooled, and a featureless absorption up to approximately 1 eV. As seen in Fig. 48, the peak is absent in the reference LTMBEgrown GaAs sample conﬁrming that the infrared absorption in ferromagnetic (Ga,Mn)As is related to changes induced by Mn impurities, in the band structure near the Fermi energy.

The presence of a ﬁnitefrequency peak in both impurityband and KL kineticexchange models for the infrared conductivity leads to an ambiguity in the interpretation of existing data, which have for the most part been taken in asgrown, presumably heavily compensated material. The metallic behavior of the x = 5.2% material below T_{c}, seen in the lower panel of Fig. 48, favors the intervalenceband absorption scenario. On the other hand, the large compensation likely present in asgrown lowT_{c} (Ga,Mn)As suggests that many holes may be strongly localized and that both absorption mechanisms may contribute to the measured absorption peak. Experiments in a series of samples interpolating between asgrown and optimally annealed limits, analogous to the resistancemonitored annealing studies Edmonds et al. (2002a, 2004a), should enable interpretation of infrared absorption spectra in (Ga,Mn)As DMSs. These studies will hint toward necessary reﬁnements of the simpliﬁed theories used so far, e.g., inclusion of the energy dependence of J_{pd} and a more quantitative theory of the impurityband model. The support for either scenario by these experiments has to be considered in conjunction with other available data in a selfconsistent picture, e.g., in an impurityband picture T_{c} is predicted to approach zero as the system reaches zero compensation, whereas the KL kineticexchange model has an opposite trend.
Systems with local moments coupled to itinerant electrons are common in condensedmatter physics and exhibit a wide variety of behaviors. Ferromagnets are far from the most common lowtemperature states. For that reason it is useful to ask how (III,Mn)V materials, and (Ga,Mn)As with its robust ferromagnetic order in particular, ﬁt in this larger context. This general qualitative analysis can help to identify some of the key factors that might limit the strength of ferromagnetic interactions in highly doped and strongly pdcoupled DMS ferromagnets for which meanﬁeld theory predicts the highest Curie temperatures.
An important class of materials that has been extensively studied is heavy fermions, in which felectron local moments are exchange coupled to band electrons Stewart (1984). Kondo lattice models, which are believed to qualitatively describe heavyfermion systems, assume that local moments exist at each lattice site. Models of DMS systems that make a virtualcrystal approximation (see Section 2.2.1and 5.1.1) place moments on all lattice sites and are therefore Kondo lattice models, often with speciﬁc details that attempt to capture some of the peculiarities of speciﬁc DMS materials. Theories of Kondo lattice model often start from a comparison of the RKKY (see Section 2.1) and Kondo temperature scales Doniach (1977); Degiorgi (1999); Tsunetsugu et al. (1997). The characteristic RKKY temperature refers to the strength of interactions between local moments mediated by a weakly disturbed carrier system and it is proportional to the meanﬁeld T_{c} given by Eq. (19). The Kondo scale refers to the temperature below which strong correlations are established between an isolated local moment and the carrier system with which it interacts. Standard scale estimate formulas Doniach (1977) applied to the case of DMS ferromagnets imply that the Kondo scale is larger than the RKKY scale when the meanﬁeld exchange coupling SN_{Mn}J_{pd} is larger than the Fermi energy E_{F } of the hole system, in other words, in the strongcoupling regime. The Kondo scale falls rapidly to small values at weaker coupling. (In heavyfermion materials the Kondo temperature scale is larger than the RKKY temperature scale.)

Optimally annealed metallic (Ga,Mn)As materials are on the weakcoupling side of this boundary, but (Ga,Mn)N and possibly (Ga,Mn)P may be starting to reach toward the strongcoupling limit (if the simple S = 5∕2 local moment model still applies in these materials). As the strongcoupling limit is approached, quantum ﬂuctuations (see discussion below Eq. (25)) will play a greater role, reducing the saturation moment per Mn and eventually driving down the ferromagnetic transition temperature. When the Kondo scale is much larger than the RKKY scale, the local moments are screened out by strongly correlated band electron spin ﬂuctuations and effectively disappear before they have the opportunity to couple.
On the weakcoupling side, RKKY interactions in Kondo lattice models tend to lead to ferromagnetism only when the number of itinerant electrons per moment is small, i.e., only when at least the nearneighbor RKKY interaction is ferromagnetic Tsunetsugu et al. (1997). One of the surprising features of (Ga,Mn)As is the property that ferromagnetism still occurs when the number of itinerant electrons per moment is ~ 1. As mentioned in Section 5.1.2, this property follows from the speciﬁc multiband electronic structure and spinorbit coupling at the top of the valence band. Nevertheless, frustrating antiferromagnetic RKKY interactions and exchange interactions that promote noncollinear magnetic states (see Section 5.2) will eventually become important for sufficiently large carrier densities. As this regime is approached from the ferromagnetic side, the transition temperature will be suppressed. These tendencies are summarized schematically in Fig. 49.
Similar considerations apply in assessing the robustness of ferromagnetism in the ordered state as characterized by the spinstiffness micromagnetic parameter Schliemann et al. (2001a); König et al. (2001b, 2003). We used this approach in Section 5.1.1 when analyzing the limitations of the meanﬁeld theory in (Ga,Mn)As. Starting from the ferromagnetic state, longwavelength spinorientation modulation will tend to lower the energy of some MnMn interactions for sufficiently high carrier densities. The spin stiffness will weaken as the frustrated magnetism regime is approached, until ﬁnally the energy of ferromagnetic spin waves will become negative, signaling the instability of this state Schliemann and MacDonald (2002). Similarly, for sufficiently strong coupling, the band system will be fully (nearly when spinorbit coupling is included) spin polarized and the cost of spinorientation spatial variation will be borne mainly by the kinetic energy of the hole system and will no longer increase with exchange coupling. As shown schematically in the lower panel of Fig. 49, hole spins in this regime are locally antiferromagnetically locked to the ﬂuctuating Mn moment orientations. For relatively small hole Fermi energies the kineticenergy cost of these ﬂuctuations is small, resulting in soft spinwave excitations of the magnetic system König et al. (2001b). In this regime longrange ferromagnetic order disappears at temperatures smaller than the meanﬁeld T_{c}, i.e., shortrange order still exists above the Curie temperature.
The white bottom left area in the panels of Fig. 49 qualitatively depicts the parameter range in which the ferromagnetic RKKY meanﬁeld state applies. Here T_{c} increases with the carrier density and local moment density, and (quadratically) with the strength of the pd exchange coupling. For a ﬁxed ratio of p∕N_{Mn}, increasing N_{Mn} corresponds to moving only slowly (as N_{Mn}^{1∕3}) upward in the diagrams. This may explain why (Ga,Mn)As materials with larger Mn doping and with similar hole compensations as in ferromagnetic systems with low localmoment density do not show any marked weakening of the ferromagnetic state. With p∕N_{Mn} still ﬁxed, attempts to increase T_{c} in (III,Mn)V DMSs by increasing the J_{pd} constant, in, e.g., ternary host alloys of Ga(As,P), might at some point reach the boundary of the soft spinwave (Kondo screened) state. Similarly, the Kondo lattice model allows only a limited space for enhancing the robustness of the ferromagnetic state by tweaking the carrier and localmoment densities independently. In this case moving horizontally from the boundary of the frustrated RKKY (ferromagnetically unstable) state is accompanied by approaching vertically the soft spinwave (Kondo screened) regime, and vice versa. Viewed from the opposite perspective, however, it is astonishing that a window in this parameter space has been found by the material research of (III,Mn)V compounds for robust DMS ferromagnets with Curie temperatures close to 200 K. The diagrams do not imply any general physical mechanism that limits T_{c} in these materials below room temperature.
Our remarks on the cartoons in Fig. 49 refer to the properties of Kondo lattice models that have been adjusted to reﬂect peculiarities of the zincblende semiconductor valence band. We have so far neglected the importance of disorder and Coulomb scattering in DMSs, and these can modify some parts of the simple qualitative picture depicted in Fig 49. This is particularly true in the verylowdensity isolated Mn limit, i.e., the very strong exchangecoupling limit. Because of Coulomb attraction between valenceband holes and the charged Mn ion which carries the local moment, a total angular momentum F = 1 isolated bound state is created, as explained in Section 2.2.1, instead of the strongly correlated Kondo singlet. The importance of Coulomb interactions and disorder is lessened by screening and Pauli exclusion principle effects when both the Mn density and carrier density are high.
In this section we narrow down the discussion of magnetism in localmoment systems to semiconducting compounds, focusing on the phenomenology of ferromagnetic DMSs other than (III,Mn)V materials. Almost any semiconducting or insulating compound that contains elements with partially ﬁlled d or f shells (local moments) will order magnetically at a sufficiently low temperature. Semiconductors and insulators with high density of magnetic moments usually order antiferromagnetically, however, although ferromagnetism does occur in some cases. A famous example of a ferromagnetic system that can be regarded as a doped semiconductor is provided by the manganite family (e.g., La_{1x}Sr_{x}MnO_{3}), whose ferromagnetism is favored by the doubleexchange mechanism and occurs over a wide range of transition temperatures from below 100 to nearly 400 K. The onset of magnetic order in these systems is accompanied by a very large increase in conductivity. For a review, see, for example, Coey et al. (1999). Other wellknown densemoment (of order one moment per atom) ferromagnetic semiconductors with strong exchange interaction between itinerant and local spins include Eu and Cr chalcogenides Kasuya and Yanase (1968); Mauger and Godart (1986); Baltzer et al. (1966); van Stapele (1982), such as rocksalt EuO and spinel CdCr_{2}Se_{4} with Curie temperatures 70 and 130 K, respectively.
DMS systems in which magnetic atoms are introduced as impurities have moments on only a small fraction of all atomic sites. The mechanisms that control magnetic order are therefore necessarily associated with the properties of these impurities. The coupling between moments will generally depend on locations of the dilute moments in the host lattice, on doping properties of the magnetic impurities, and on other dopants and defects present in the material. It seems plausible therefore that when DMS systems are ferromagnetic, their magnetic and magnetotransport properties will be more sensitive to material properties that can be engineered.
This review has concentrated on (Ga,Mn)As and related materials in which, as we have explained, substitutional Mn acts both as an acceptor and as a source of local moments. Ferromagnetism is carrier mediated and it has been demonstrated that it persists to surprisingly high temperatures. More may be achieved in the future by tweaking these materials. On the other hand, there is a vast array of alternate DMS materials that could be contemplated. Research to date has only scratched the surface of the volume of possibilities—we are truly still at the beginning of the road in studying diluted moment magnetism in semiconductors. Each system brings its new challenges. The interpretation of simple magnetic and transport characterization measurements is often not immediately obvious, in particular because of the possibility that the moments segregate into crystallites of one of a variety of available densemoment minority phases which are often thermodynamically more stable. In addition, magnetic properties depend in general not only on the dilute moment density, which normally is well controlled and variable, but also on the partitioning of local moments among many available sites in the host crystal, which is not always known and is usually much harder to control. The search for promising DMS materials would be simpliﬁed if ab initio DFT methods had reliable predictive power. Unfortunately this luxury appears to be absent in many cases because of extreme sensitivity of magnetic properties to details of the electronic structure and because of strongcorrelation effects that are often present in these systems. We mention brieﬂy in the following some other classes of diluted magnetic semiconductors that have been studied.
The class of DMS ferromagnets that is closest to (III,Mn)V materials comprises the (II,Mn)VI compounds codoped with groupV element acceptors. Examples include p(Zn,Mn)Te:N Ferrand et al. (2001) and p(Be,Mn)Te:N Hansen et al. (2001). These materials differ from (Ga,Mn)As mainly because the local moments and holes are provided by different types of impurities and can be controlled independently. Although the physics behind ferromagnetism seems to be very similar in the two classes of materials, the highest ferromagnetic transition temperatures that have been achieved are much smaller in the case of codoped (II,Mn)VI materials, ~ 2 K rather than ~ 200 K. The difference is explained partly by difficulty in achieving the same extremely high hole doping (~ 10^{21} cm^{3}) in (II,Mn)VI materials that has been achieved in (Ga,Mn)As and partly by a favorable interplay between electrostatic and magnetic effects in (III,Mn)V materials. In (III,Mn)V materials, unlike (II,Mn)VI materials, the Mn moment is charged and attracts holes. The tendency of holes to have a higher density near Mn sites tends to increase the effective strength of the pd exchange interaction. This effect is magniﬁed when two Mn moments are on neighboring cation sites. In (Ga,Mn)As the interaction between Mn moments on neighboring cation positions is ferromagnetic, compared to the strongly antiferromagnetic interactions seen in (II,Mn)VI materials. (For a discussion of the interplay between electrostatic and magnetic interactions see Śliwa and Dietl (2005).) In pdoped (II,Mn)VI materials, competition between antiferromagnetic nearneighbor interactions and the longerrange carriermediated ferromagnetic interactions suppresses the magnetic ordering temperature. This competition apparently does not occur in (III,Mn)V ferromagnets with large hole densities.
(Zn,Mn)O is an interesting IIVI counterpart of the nitride IIIV DMS Liu et al. (2005). With advances in oxide growth techniques, (Zn,Mn)O can be considered to be much like other (II,Mn)VI DMS materials and its investigation was originally motivated by theoretical work Dietl et al. (2000) that extrapolated from experience with (III,Mn)V DMS ferromagnets and predicted large T_{c}’s. Studies of this material have provided clear evidence of strong pd exchange but so far have led to inconsistent conclusions about the occurrence of longrange magnetic order Sharma et al. (2003); Fukumura et al. (2005); Petit et al. (2006); Lawes et al. (2005).
Tetrahedral DMS materials doped with transitionmetal atoms other than Mn have shown promising results. For instance, (Zn,Cr)Te is apparently homogeneous and has the required coupling between local moments and carriers Mac et al. (1996); Saito et al. (2003), and Curie temperatures as high as 300 K have been reported for this material. It may be, though, that the ferromagnetism is due to superexchange interactions rather than being carrier mediated since it occurs at very small ratios of the carrier density to the moment density Saito et al. (2002). Another interesting material with Cr moments is (Ga,Cr)N which exhibits ferromagnetism at ~ 900 K Liu et al. (2004a). The question still at issue in this material is the possible role of densemoment precipitates.
Traditional groups of ferromagnetic DMSs also include (IV,Mn)VI solid solutions with the rocksalt structure Story et al. (1986, 1992); Eggenkamp et al. (1993). Although the band structures of IVVI and IIIV semiconductors are quite different, these DMS ferromagnets (e.g., (Pb,Sn,Mn)Te) appear to have a carriermediated mechanism quite similar to that of (Ga,Mn)As. Holes with densities up to 10^{21} cm^{3} are supplied in these materials by cation vacancies, rather than by Mn substitution for divalent cations. The reported Curie temperatures in (Pb,Sn,Mn)Te are below 40 K Lazarczyk et al. (1997).
DMS ferromagnetism with Si or Ge as the host semiconductor is obviously attractive because of the greater compatibility of these materials with existing siliconbased technology. In Si, Mn impurities favor the interstitial position, which complicates the synthesis of a uniform DMS system. Mn in Ge, on the other hand, is a substitutional impurity and ferromagnetism has been reported in MBEgrown Ge_{x}Mn_{1x} thinﬁlm DMSs Park et al. (2002); Li et al. (2005). Careful studies of Ge_{x}Mn_{1x} Li et al. (2005) have demonstrated that slow lowtemperature growth is required to avoid the formation of thermodynamically stable densemoment ferromagnetic precipitates; it is likely that the hightemperature ferromagnetism sometimes found in these materials is due to precipitates. The latest studies Li et al. (2005) appear to indicate that true longrange order in Ge_{x}Mn_{1x} emerges only at low temperatures ~ 12 K and that weak coupling between remote moments is mediated by holes which are tightly bound to Mn acceptors. Further work is necessary to determine whether this picture of magnetism, reminiscent of the polaronic physics discussed in the context of (Ga,Mn)P or lowcarrierdensity (III,Mn)V systems Scarpulla et al. (2005); Kaminski and Das Sarma (2003), applies to Ge_{x}Mn_{1x} DMSs.
The possible presence of densemoment thermodynamically stable precipitates has also confused studies of oxidesemiconductor DMS systems. More consistent evidence of aboveroomtemperature ferromagnetism has been reported in Codoped TiO_{2} although the origin of ferromagnetism in this material is still under debate (see, e.g., Matsumoto et al. (2001) and recent reviews Prellier et al. (2003); Fukumura et al. (2005)). Mndoped indium tin oxide (ITO) is another promising candidate for a transparent ferromagnetic semiconductor which could be easily integrated into magnetooptical devices. Particularly encouraging is the observation of a large anomalous Hall effect showing that charge transport and magnetism are intimately connected in this oxide DMS Philip et al. (2004).
Other interesting related materials are the Mndoped IIIVV_{2} chalcopyrites surveyed theoretically in a ﬁrstprinciples calculation study by Erwin and Zutic (2004). Three of these compounds, CdGeP_{2}, ZnGeP_{2}, and ZnSnAs_{2}, have shown ferromagnetism experimentally. The origin of this ferromagnetic behavior has not yet been explored extensively.
Finally we mention recent observations of ferromagnetic order up to ≈ 20 K in a layered semiconductor, Sb_{2}Te_{3} doped with V or Cr Dyck et al. (2002, 2005). These highly anisotropic materials combine DMS behavior with strong thermoelectric effects. The character of the ferromagnetic coupling in these compounds is unclear at present.
This article is a review of theoretical progress that has been achieved in understanding ferromagnetism and related electronic properties in (III,Mn)V DMSs. The materials we have focused on have randomly located Mn(d^{5}) local moments which interact via approximately isotropic exchange interactions with itinerant carriers in the semiconductor valence band. (Ga,Mn)As is the most thoroughly studied and bestunderstood system in this class. Some (III,Mn)V materials may exhibit ﬂuctuations in the Mn valence between Mn(d^{5}) and Mn(d^{4}) conﬁgurations or have dominant Mn(d^{4}) character, possibly (Ga,Mn)N, for example. Magnetic and other properties of materials in the latter class will differ qualitatively from those of (Ga,Mn)As and this review makes no attempt to discuss the theory that would describe them. When we refer to (III,Mn)V ferromagnetism below, it should be understood that any materials that prove to be in the latter class are excluded.
Interest in DMS ferromagnetism is motivated by the vision that it should be possible to engineer systems that combine many of the technologically useful features of ferromagnetic and semiconducting materials. This goal has been achieved to an impressive degree in (III,Mn)V DMSs, and further progress can be anticipated in the future. The goal of hightemperature semiconductor ferromagnetism ﬂies in the face of fundamental physical limits, and the fact that so much progress has nevertheless been achieved is due to a serendipitous combination of attributes of (III,Mn)V materials. We have reserved the term “ferromagnetic semiconductor” for materials in which the coupling between local moments is mediated by carriers in the host semiconductor valence or conduction band. Then magnetic properties can be adjusted over a broad range simply by modifying the carrier system by doping, photodoping, gating, heterojunction bandstructure engineering, or any technique that can be used to alter other semiconductor electronic properties. Most of these tuning knobs have already been established in (III,Mn)V ferromagnets.
Progress that has been made in achieving (Ga,Mn)As ferromagnetism and in understanding its phenomenology has a few lessons. The analysis of any DMS should start with understanding the properties of isolated defects associated with the magnetic element. In the case of (Ga,Mn)As materials the desirable magnetic defect is substitutional Mn_{Ga}, because Mn then both introduces a local moment and acts as an acceptor. Holes doped in the system by Mn_{Ga} impurities provide the glue that couples the moments together. Understanding the role of other defects that are present in real materials is also crucial. Substitutional Mn_{I} is particularly important in (Ga,Mn)As because it reduces the number of free moments and reduces the density of the hole gas that mediates ferromagnetism. Learning how to remove defects that are detrimental to strong magnetic order is key to creating useful materials. These two steps have been largely achieved in (Ga,Mn)As. There is every reason to believe that, if the same progress can be made in other DMS materials, some will be even more magnetically robust.
We have reviewed in this article a number of theoretical approaches that shed light on what controls key properties of ferromagnetic (III,Mn)V semiconductors. Firstprinciples electronic structure calculations give a good overview of fundamental material trends across the series and explain many structural characteristics of these alloys. Semiphenomenological microscopic tightbinding models provide a convenient way to use experimental information to improve the quantitative accuracy of the description. Another phenomenological description that successfully models magnetic, magnetotransport, and magnetooptical properties is a singleparameter theory that adopts a k ⋅ p description for host semiconductor valence bands, and assumes that the exchange interaction between local moments and band electrons is short ranged and isotropic. The single exchange parameter that appears in this theory can be determined by ﬁtting to known properties of an isolated Mn local moment, leading to parameterfree predictions for ferromagnetism. Qualitative models which focus on what kind of physics can occur generically for randomly located local moments that are exchange coupled to either localized or itinerant band electrons also provide useful insights for interpreting experiments.
The most important properties of (III,Mn)V materials are their Curie temperature and ferromagnetic moment, which reﬂect both the strength of the coupling between Mn local spins and its range. The highest ferromagnetic transition temperatures in (Ga,Mn)As epilayers have so far been achieved with substitutional Mn_{Ga} fractions in the neighborhood of 5% by postgrowth annealing, which eliminates most interstitial Mn ions. Achieving T_{c} values close to room temperature in (Ga,Mn)As, which is expected to occur for 10% Mn_{Ga} doping, appears to be essentially a material growth issue, albeit a very challenging one. In optimally annealed samples, experimental and theoretical considerations indicate that MnMn exchange interactions are sufficiently long range to produce a magnetic state that is nearly collinear and insensitive to microrealization of the Mn_{Ga} spatial distribution. Magnetization and Curie temperatures in these systems are well described by meanﬁeld theory.
The magnetic and transport properties of highquality (Ga,Mn)As materials are those of a lowmomentdensity, lowcarrierdensity metallic ferromagnet, with a few special twists. Because of the strong spinorbit interactions in the valence band, metallic (Ga,Mn)As shows a large anomalous contribution to the Hall effect and the source of magnetic and transport anisotropies is more itinerant electrons, unlike the transitionmetal case in which anisotropies originate primarily in delectron spinorbit interactions. The small moment densities lead to a large magnetic hardness, reﬂected in a singledomainlike behavior of many (Ga,Mn)As thin ﬁlms. They also explain a large part of the orders of magnitude reduction in the current densities required for transport manipulation of the magnetic state through spinmomentumtransfer effects. The lowcarrier density of the itinerant holes responsible for magnetic coupling means that they are concentrated around a particular portion of the Brillouin zone in the valence band which has a large oscillator strength for optical transitions to the conduction band. This property opens up opportunities for optical manipulation of the magnetic state that do not exist in transitionmetal ferromagnets and have not yet been fully explored. The research reviewed here that is aimed at an understanding of the optical properties of (Ga,Mn)As ferromagnets is still incomplete, particularly for ideal annealed materials, and will be important in setting the groundwork for the exploration of new effects.
These conclusions do not necessarily apply to all (III,Mn)V ferromagnets. For example, material trends suggest that widerbandgap hosts would have stronger exchange scattering that would lower the conductivity, shorten the range of MnMn exchange interactions, and increase the importance of quantum ﬂuctuations in Mn and bandhole spin orientations. This could eventually lead to Curie temperatures signiﬁcantly below the meanﬁeld estimates. In the opposite limit, when the exchange interaction is weak enough to be treated perturbatively, sign variations in the RKKY MnMn interaction are expected to lead to frustration and weaken ferromagnetism at large carrier densities. One of the important miracles of (Ga,Mn)As ferromagnetism is that this effect is much weaker than would naively be expected because of the complex valenceband structure. In (Ga,Mn)As, incipient frustration that limits magnetic stiffness may be responsible for the weak dependence of the ferromagnetic transition temperature on carrier density. This property of (Ga,Mn)As suggests that little progress on the T_{c} front is likely to be gained by nonmagnetic acceptor codoping.
Disorder is an inevitable part of the physics of all DMS ferromagnets because of the random substitution of elements possessing moments for host semiconductor elements. Even in metallic, ideal annealed samples that have only substitutional Mn_{Ga} impurities, randomness in the Mn microstructure leads to both Coulomb and spindependent exchange potential scattering. For (Ga,Mn)As, Coulomb scattering dominates over exchange potential scattering, limiting the conductivity to ~ 100–1000 Ω^{1}cm^{1}. Frustration and disorder are certainly very important near the onset of ferromagnetism at low Mn density, where the network of exchange interactions that lead to longrange order is still tenuous. Studies of wellcharacterized materials with low Mn fraction near the metalinsulator transition are now possible because of progress in understanding the role of defects and are likely to exhibit complex interplay between the disorder and the Coulomb and exchange interactions.
We thank all colleagues who have given us the permission to show their results in this review. Among the many stimulating discussions we would like to especially acknowledge our interactions with experimentalists at the University of Nottingham. Work on the article was supported by by the Grant Agency of the Czech Republic through Grant No. 202/05/0575, by the Academy of Sciences of the Czech Republic through Institutional Support No. AV0Z10100521, by the Ministry of Education of the Czech Republic Center for Fundamental Research LC510, by the U.K. EPSRC through No. Grant GR/S81407/01, by the Welch Foundation, by the U.S. Department of Energy under Grant No. DEFG0302ER45958, and by the U.S. Office of Naval Research under Grant No. ONRN000140610122.
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