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Temperature-dependent magnetization in diluted magnetic semiconductors

We calculate magnetization in magnetically doped semiconductors assuming a local exchange model of carrier-mediated ferromagnetic mechanism and using a number of complementary theoretical approaches. In general, we find that the results of our mean-field calculations, particularly the dynamical mean field theory results, give excellent qualitative agreement with the experimentally observed magnetization in systems with itinerant charge carriers, such as Ga_{1-x}Mn_xAs with 0.03<x<0.07, whereas our percolation-theory-based calculations agree well with the existing data in strongly insulating materials, such as Ge_{1-x}Mn_x. We comment on the issue of non-mean-field like magnetization curves and on the observed incomplete saturation magnetization values in diluted magnetic semiconductors from our theoretical perspective. In agreement with experimental observations, we find the carrier density to be the crucial parameter determining the magnetization behavior. Our calculated dependence of magnetization on external magnetic field is also in excellent agreement with the existing experimental data.


I. INTRODUCTION
Diluted magnetic semiconductors (sometimes also referred to as doped magnetic semiconductors -we will use the abbreviation "DMS" to denote both of these equivalent terminologies) have recently attracted a great deal of attention 1,2 for their potential in combining ferromagnetic and semiconductor properties in a single material. The prototypical DMS material is Ga 1−x Mn x As 3 (typically x ≈ 1 − 10%) with the Mn ions substitutionally (in the ideal situation) replacing Ga at the cation sites. Mn ions in Ga 1−x Mn x As serve a dual purpose, acting both as dopants (acceptors in this case) and as magnetic impurities, whose spins align at the ferromagnetic transition. For x ≈ 1 − 7%, Ga 1−x Mn x As is found to be ferromagnetic with the ferromagnetic transition temperature (or, equivalently, the Curie temperature) T c ≈ 10 − 100K. The optimum value of x, which corresponds to the highest value of reported value of T c , is around 5%. Other DMS materials of current interest include In 1−x Mn x As, 4 Ga 1−x Mn x P, 5 Ge 1−x Mn x , 6 and Ga 1−x Mn x Sb. 7 In spite of a great deal of recent experimental and theoretical activity, 1,2 there is not yet a consensus on the fundamental mechanism leading to ferromagnetism in these systems as well as on the definitive predictive theory quantitatively describing this ferromagnetic mechanism. It is therefore important to work out detailed and experimentally falsifiable consequences for various proposed theoretical models and ideas. In this context, it is unfortunate that much of the theoretical DMS literature, perhaps due to the considerable technological motivation in creating room-temperature ferromagnetic semiconductors for projected spintronics applications, has concentrated on the calculations of T c in various DMS materials. Such theoretical predictions of T c invariably involve tuning free parameters (e. g., the strength of the exchange coupling between carriers and local moments), whose values are often unknown a priori. This signifi-cantly reduces the practical importance of these predictions except perhaps in the broadest qualitative sense of identifying the crucial controlling parameters which determine and limit T c in DMS materials. On the other hand, the temperature dependence M (T ) of the spontaneous magnetization possesses many characteristics, such as concavity/convexity of the curve, value of saturation magnetization, critical behavior at the point of ferromagnetic transition etc., which cannot all be described by just tuning parameters of a given model. Thus study of M (T ) has a very high potential for elucidating the physics behind DMS ferromagnetism in real systems.
A particular issue of considerable significance in DMS ferromagnetism has been the non-trivial non-mean-fieldlike behavior of the spontaneous magnetization as a function of temperature. This was already apparent in the very first reported observation 4 of DMS ferromagnetism in a III 1−x Mn x V material, namely In 1−x Mn x As, where the experimental M (T ) curve exhibited an untypical outwardly concave shape strikingly different from the usual convex M (T ) behavior expected within the textbook Weiss mean-field theory 8 and seen routinely in conventional ferromagnetic materials. The In 1−x Mn x As (with x = 0.013) system studied in Ref. 4 was an insulating system (i. e. with resistivity increasing monotonically with decreasing temperature), and the insulating ferromagnetic DMS systems studied so far in the literature almost always exhibit qualitatively similar non-mean-fieldlike concave M (T ) behavior. 6 Two of us have recently shown 9 that such a manifestly non-mean-field-like concave M (T ) behavior in insulating DMS materials can be understood on the basis of a magnetic percolation transition of bound magnetic polarons in the strongly localized carrier system. Earlier numerical simulations 10,11 in the strongly localized regime had already indicated that the M (T ) behavior of DMS ferromagnets could have concave outward shapes as seen experimentally. Very recent numerical simulations 12 in the strongly localized regime have verified our polaron percolation picture of DMS ferromagnetism in the insulating regime.
Even in the "metallic" (i. e. with resistivity decreasing with T below the ferromagnetic transition temperature) DMS systems, 1,3,13,14,15,16,17,18,19,20 such as optimally doped Ga 0.95 Mn 0.05 As, where long range magnetic ordering of the Mn magnetic moments is created presumably by a dilute gas of delocalized holes mediating the magnetic interaction, the experimentally observed M (T ) often appears to be very different from the classic mean-field shape. 8 Although the "metallic" Ga 1−x Mn x As DMS system does not exhibit the manifestly concave temperature dependent magnetization seen in insulating DMS systems, the observed M (T ) in metallic DMS is often 1,3,13,14,15,16,17,18,19,20 almost linear in temperature for 0.5T c T < T c , with a temperaturedependent behavior intermediate between the concave M (T ) shapes of the localized theory 9 and the textbook convex magnetization curves. 8 We mention that very recent annealing experiments 13,14,15,16,17,18,19,20 in Ga 1−x Mn x As (with x = 0.05 − 0.10) demonstrate that M (T ) behavior (as well as the value of T c ) can be strongly affected by annealing, and in particular, "optimal" annealing (to be found empirically for each sample purely by trial and error since the precise role of annealing in improving the materials quality is unknown at this stage) may lead to reasonably mean-field-like convex M (T ) shape with a concomitant enhancement of T c . "Non-optimal" annealing, on the other hand, leads to suppression of T c and strongly non-mean-field-like (and non-universal) M (T ) behavior.
Finally, most of DMS magnetization measurements in the literature, particularly for the insulating DMS systems, show a saturation magnetization (for B = 0) considerably smaller in magnitude than that expected from the full ordering of all the magnetic ions, indicating that a large fraction (sometimes as much as 90%) of the magnetic ions do not contribute to the global DMS ferromagnetism.
Motivated by the desire for illuminating the physical mechanisms underlying DMS ferromagnetism, we theoretically consider in this paper the temperaturedependent magnetization M (T ) in three-dimensional DMS systems using a number of complementary theoretical approaches. The calculations we present here are based on the static (Weiss) molecular mean-field theory, 21 the dynamical mean-field theory (DMFT), 22 and the percolation theory. 23 Of particular interest is the important issue of correlations between magnetic and transport properties of various DMS materials since such correlations should (and do) exist in magnetic systems where the ferromagnetism is mediated by carriers leading to the global ordering of the dopant local moments. While earlier studies invariably concentrated on either localized-carrier ("insulating") and itinerantcarrier ("metallic") material, we consider both these regimes as well as the crossover between them. We provide detailed numerical results for our calculated magnetization M (T, B), where B is the applied external magnetic field and T the temperature, in various regimes of the system parameter space. These M (T, B) results should help ascertain the applicability of various theoretical models to specific experimental DMS materials of current interest. This paper is organized as follows. In Sec. II, we describe our model and our theoretical approaches based on the Weiss mean-field theory, DMFT, and the percolation theory, and present our numerical results for M (T, B) in each of the theoretical approaches providing brief discussion of our results in light of the existing experimental data in the literature. We conclude in Sec. III summarizing our qualitative findings and providing a critical perspective on what our theoretical results on magnetization imply for the microscopic mechanisms underlying DMS ferromagnetism with the particular emphasis on the correlations between transport characteristics ("metallic" or "insulating") and M (T ) behavior (convex, concave, or "linear").

A. The model
We assume in this work that the fundamental mechanism underlying ferromagnetism in DMS materials (e. g. is the carrier -local-moment (kinetic or p-d) exchange coupling, which eventually leads to a global ferromagnetic ordering of the impurity local moments (i. e. Mn) for T < T c overcoming any direct antiferromagnetic (super-exchange) interaction between the local moment spins themselves. This is certainly the prevalent viewpoint for DMS ferromagnetism, at least for Ga 1−x Mn x As system following the pioneering work of Ohno and his collaborators. It is natural to ask about the evidence for this belief in carrier-mediated DMS ferromagnetism induced in the Mn local moments. Firstprinciples band theory calculations indicate that there is strong p-d hybridization between the Mn d-levels and the valence band p-states of GaAs. This leads to strong kinetic exchange coupling between hole spins and Mn spins, which is the basis of our model. Experimentally, there are a number of compelling circumstantial indications of Ga 1−x Mn x As being a carrier-mediated ferromagnetic material. First, there is the strong correlation (made even stronger by the recent annealing experiments) between transport and magnetic properties. This, however, is not definitively conclusive since strongly insulating Ga 1−x Mn x As (and In 1−x Mn x As) samples are also found to be ferromagnetic (albeit with much lower T c values). Perhaps the most compelling evidence supporting the carrier-mediated ferromagnetic mechanism is the observed agreement (so far available only in Ga 1−x Mn x As DMS systems) between the magnetization measured di-rectly using a SQUID magnetometer and that inferred by analyzing the anomalous Hall effect data. Such an consistency between direct and "transport-inferred" magnetization strongly suggests a carrier-mediated exchange mechanism underlying DMS ferromagnetism. In addition, there are a number of experiments dealing with optical properties (again, available only for Ga 1−x Mn x As so far) which indicate very strong correlation between the magnetic band structure and the magnetization of the system, leading again to the condition that the global ferromagnetic ordering of the Mn local moments is most likely induced by the hole spin polarization in DMS systems. The related issue of whether these carriers are valence band holes or impurity band holes is more difficult to settle. Given the strong inherent disorder in DMS materials and rather strong exchange coupling between hole and Mn spins (leading to local binding of holes and Mn ions), it is natural to think that all of the physics (both localized and delocalized) are essentially impurity band physics. This viewpoint, which we have adopted in our theories presented in this paper, has received strong support from recent optical experiments. Also our DMFT results, as presented in Sec. II C, clearly demonstrate the important role of impurity band physics in DMS magnetic properties.
Since the most intensively studied DMS materials of current interest are Mn-doped III-V semiconduc- where the carriers are typically holes, we will other refer to the semiconductor carriers as "holes" in the rest of the article without any loss of generality. So our basic model is that of a finite density n i of magnetic dopants ("impurities") interacting through a local exchange coupling with a finite density n c of holes in the host semiconductor material with n c /n i ≪ 1. We assume that magnetic impurities under consideration enter substitutionally at the cation sites (e. g. Mn impurities at Ga sites). Recent experimental annealing studies of Ga 1−x Mn x As have shown 13,14,15,16,17,18,19 that lattice defects may be playing an important role in determining magnetic and transport properties of the samples, but we assume in our work that these defects enter our theory only in determining the basic parameters of the model, namely, the density of magnetically active dopants n i , the hole density n c , and perhaps the local effective exchange coupling J between the holes and the magnetic impurities, and do not include any defects into our model explicitly. Examples of such defects which enter our model through its parameters rather than directly are given by antisite defects in the host semiconductor material (i. e. As at Ga sites in GaAs) and Mn interstitials (i. e. Mn atoms at interstitials sites rather than cation substitutional sites), which may very well be important in providing substantial compensation in the semiconductor, leading to the experimental fact that the hole density n c is usually a small fraction of the magnetic dopant density instead of there being a one-to-one correspondence between dopants and holes. The local moment density n i in our model is not necessarily the total Mn concentration in the system, since the presence of Mn interstitial defects could lower the density of magnetically active Mn ions. In this work we use n i to denote local moment volume density and x to denote the fraction of Ga atoms replaced by Mn dopants.
Similarly, consistent with the spirit of our minimal model we also neglect all band structure effects in our theory, making the simplest approximation such as a single parabolic band with a single effective mass or a single simple tight-binding carrier band characterized by an effective band width parameter. This is not because realistic band structure effects are not of any importance in DMS ferromagnetism, in fact we believe that spinorbit coupling in Ga 1−x Mn x As valence band hole states may play a quantitative role in Ga 1−x Mn x As ferromagnetism, particularly in the relatively disorder-free "metallic" (x ≈ 0.05) systems where the holes are likely to be GaAs valence band hole states with strong spin-orbit coupling. Our reasons for neglecting spin-orbit coupling and other band structure effects in our theory are the following: (1) our interest in this paper is in theoretically exploring M (T, B) within a minimal model, which requires only n i , n c , and J, leaving out all non-essential complications; (2) if needed, band structure effects can be systematically included in the future by appropriately extending the model; (3) due to the inevitable presence of strong exchange coupling and strong disorder in DMS materials, a starting point based on perfect "realistic" valence band hole states may be inapplicable -in fact, we believe that much of the DMS physics is occurring in the impurity band of the host semiconductor with the itinerant and the localized carriers being the extended and the localized hole states in the impurity band of the system and not the valence band states, a view point strongly supported by several recent experimental results. 24,25,26,27 The Hamiltonian of exchange interaction between magnetic impurities and holes we use in this paper reads where J, which has units of energy, is the exchange coupling between an impurity spins S j located at R j and a hole spin density s(r), and a 3 0 is the unit cell volume needed for proper normalization. The impurity spin S in Eq. (1) is assumed to be completely classical in the theory whereas the carrier spin s is treated quantummechanically. This is justified because the impurity spin is large, e. g. S = 5/2 for Mn in Ga 1−x Mn x As.
Our model (except for the mean-field theory considerations of Sec. II B) omits the direct Mn−Mn antiferromagnetic exchange interaction assuming its effects to be either negligibly small or incorporated into the effective parameters of the model. Actually, in the parameter range of interest to us (x ≪ 1), where DMS ferromagnetism typically occurs, the magnetic impurities are separated from each other by non-magnetic atoms, and this antiferromagnetic interaction, which rapidly decays with the distance, should be negligible. We also ignore any specific hole-hole interaction effect in our theory. These approximations are nonessential and are done in the spirit of identifying the minimal DMS magnetic model of interest. Both of these effects, which may be of quantitative importance in some situations, can be included in the theory by adjusting the parameters of the model or perhaps at the cost of introducing more unknown parameters characterizing these interactions. The possible effects of including these interactions in our calculations will be discussed later in the paper.
With this introduction the full Hamiltonian is given by where the first term is the "band" Hamiltonian with m being the relevant effective mass, V (r) is the random potential arising from (non-magnetic) disorder, W (r − R i ) is the Coulomb potential due to a magnetic impurity located at R i , and the second term is the local exchange coupling between the moments of magnetic impurities and the carrier spins [i. e. precisely the H M defined through Eq. (1)], with σ αβ being the Pauli matrices. The last two terms in Eq. (2) are simply the Zeeman energies of the local moments and the holes respectively, g i and g c are the corresponding g-factors, µ B is the Bohr magneton, and B is the external magnetic field. The local exchange coupling can be ferromagnetic (J < 0) or antiferromagnetic (J > 0) without affecting DMS ferromagnetism [first principles band theory 28,29,30 suggests local antiferromagnetic coupling (J > 0) for the holes in Ga 1−x Mn x As]. Finally, we mention that the magnetic interaction Hamiltonian defined by Eq. (1) is sometimes referred to as the s-d (or s-f) exchange Hamiltonian 31 or the Zener model 32 in the literature although it was originally introduced 33 by Nabarro and Fröhlich in a slightly different context. The physics we are interested in is how the local exchange interaction defined in Eqs. (1) and (2) could lead to global ferromagnetic ordering of the impurity local moments below a Curie temperature T c . We mention that we choose S = 5/2 and s = 1/2 in all our numerical calculations below.
Unfortunately none of the parameters J, n c , n i of our model is directly experimentally measurable. This is why we have emphasized throughout this work qualitative behavior in temperature-dependent magnetization as a function of the system parameters. The carrier density n c is hard to measure even in metallic ferromagnetic materials because of the problems associated with anomalous Hall effect (and the situation is obviously worse in strongly insulting systems). The local moment density n i is unknown because only a fraction of the incorporated Mn atoms are magnetically active due to the invariable presence of Mn interstitials and other possible defects in the system. What is known about Ga 1−x Mn x As is that it is a heavily compensated system in the sense that the density of holes, is much less than the density of Mn, typically n c /n i ∼ 0.1. Much of this compensation most likely arises from various defects invariably present in low-temperature MBE grown Ga 1−x Mn x As. The two most "harmful" defects in this respect are Mn interstitials and As antisites. Both of these defects act as effective double donors, producing two electrons each. Thus the holes produced by the magnetically and electrically active "desirable" Mn 2+ ions (sitting at substitutional cation sites), with each substitutional Mn ion producing one hole, could be heavily compensated by the defects leading to the existing situation in Ga 1−x Mn x As where n c /n i ≪ 1. All aspects of Kondo physics are completely negligible for the current problem. Kondo physics is relevant only in the complementary regime of high carrier density (n c /n i ≫ 1), where a paramagnetic carrier ground state entails as the impurity spin is quenched by the free carriers. For itinerant delocalized carriers (i. e., metallic DMS), the indirect exchange coupling between the local moments (induced by carrier spin polarization) is precisely the RKKY interaction. The relevance (or perhaps even, the dominance) of RKKY physics (leading to a magnetic ground state) over the Kondo physics in the low carrier density limit of the Kondo lattice system has occasionally been mentioned in the Kondo-effect literature. 34 The crucial element of RKKY physics playing a key role in DMS ferromagnetism is the relatively low values of carrier density in these materials leading to k F 3 √ n i ≪ 1 (where k F is the Fermi wavevector associated with the carrier density n c and 3 √ n i is the characteristic interimpurity separation) so that the RKKY interaction is essentially always ferromagnetic, and the RKKY spinglass type behavior predominant in disordered magnetic metallic systems may not arise here since the interaction is mostly ferromagnetic avoiding effects of frustration.
In principle one could start from the general Hamiltonian (2), and try to develop a theory for carrier localization and magnetism on an equal footing. Such an ambitious attempt would be essentially futile, due to the enormous complexity of the problem, which would require both disorder and exchange interaction to be treated nonperturbatively. We therefore adopt the reasonable empirical approach of building into our basic model the metal-lic (itinerant carriers) or the insulating (localized carriers) nature of the system, and develop separate complementary theoretical approaches for the two situations in order to compare and contrast the nature of DMS ferromagnetism in metallic and insulating systems, both with the local exchange magnetic Hamiltonian (1) coupling the impurity moments to carrier spins.
We use complementary theoretical approaches (degenerate-and non-degenerate-carrier Weiss static mean-field theory, dynamical mean field theory (DMFT), and percolation theory) in this paper; two of which apply to the limiting cases of extended "metallic" system (degenerate-carrier mean-field theory and DMFT) and the other two to the strongly localized "insulating" system (non-degenerate-carrier mean-field theory and percolation theory). In principle, DMFT can interpolate smoothly between the systems with extended and localized carriers, but we have neglected localization effects in our DMFT calculations carried out so far. Our results for temperature-and magnetic-field-dependent magnetization M (T, B) exhibit qualitative behavior very similar to that seen in experiments which, given the minimal nature of our theoretical model, is all we can expect of the theory. In our view, at this early stage of the development of DMS materials and their physical understanding (and given the metastable and fragile complex nature of the DMS systems), where all the materials details (e. g. defects and disorder of many possible types) qualitatively affecting the system magnetization may not even be known at the present time, it is premature to demand (or impose, for example, by tuning the parameters of the theoretical model) quantitative agreement between theory and experiment. In this respect we strongly disagree with the statements asserting that DMS ferromagnetism is a well-understood problem based on a model of free valence band holes interacting with Mn local moments. The excellent qualitative agreement between our calculated magnetization results in different theoretical approaches and the experimental magnetization data in the existing literature provides useful insight into the possible magnetic mechanisms underlying "metallic" and "insulating" DMS systems.

B. The Static Mean-Field Theory
The basic idea underlying the static mean-field theory, as applied to ferromagnetism in DMS, is to represent action of all impurity/hole spins upon a given impurity/hole spin as an effective "mean field," whose value is determined by the average values of the spins acting upon this given spin. The resulting equations for the spins of impurities and holes are to be solved self-consistently, finally yielding the equilibrium magnetization at a given temperature.
The difference between the mean-field theory considered in this Section and the canonical Weiss mean-field model arises from the existence of two interacting species of spins, those of holes and of impurities in a DMS system. As a result, we have two effective fields -one determined by the average value of a hole spin and the other determined by the average value of an impurity spin. In the framework of Hamiltonian (2), the effective field acting upon holes has contributions coming from magnetic impurities and from the external magnetic field B, where the direction of the z axis is chosen to coincide with the direction of applied magnetic field B, or, in the case of B = 0, with the direction of spontaneous magnetization of impurities. The effective field acting upon impurities is a sum of contributions from holes and the external magnetic field. Relative simplicity of the static mean-field theory allows us to account also for possible direct antiferromagnetic interaction between magnetic impurities by adding the following term to the Hamiltonian which yields one more contribution to the effective field acting upon magnetic impurities: where z AF is the effective number of surrounding impurities a given impurity interacts with. The response of the impurity spin to this effective field B (i) eff is given by is the Brillouin function. The magnetic response of hole spins to effective field B (c) eff produced by impurities strongly depends on whether the hole gas is degenerate or not. The two complementary cases of nondegenerate and degenerate holes will be considered below in Secs. II B 1 and II B 2 respectively. We mention that the non-degenerate (degenerate) situation applies primarily to the insulating ("metallic") DMS systems.

Non-degenerate holes
The case of non-degenerate holes, when the hole spin distribution is not affected by the Pauli exclusion principle, corresponds to two physical situations. The first one occurs when the holes are localized with strong on-site repulsion, so there is only one hole at each localization center. This scenario is relevant to DMS with localized carriers, which will also be considered in Sec. II D using the percolation theory. The second situation takes place in a gas of delocalized holes when the temperature is higher than the Fermi energy.
When the Pauli exclusion principle plays no role in the spin distribution of electrons, the latter is determined by Boltzmann statistics, and the average hole spin, as determined by the effective mean-field B (c) eff is given by similarly to Eq. (6), with g c being the hole g-factor.
Combining Eqs. (3), (5), (6), and (8), we obtain the self-consistent equation for S z : where and J is assumed to be negative (antiferromagnetic interaction between impurities and holes). Similarly to the text-book mean-field theory, Eq. (9) has a non-trivial solution in the absence of the external magnetic field only if temperature T is below a certain value, which is the ferromagnetic transition temperature T c . Using the expansion for the Brillouin function we arrive at the following expression: In the absence of antiferromagnetic interaction between the impurities, T 1 = 0 so the ferromagnetic transition temperature equals T c0 . Equation (9) in the absence of the external magnetic field and at J AF = 0 reduces to the average hole spin is still given by Eqs. (8) and (3) with B = 0. The solution of Eq. (14) can be found numerically. The resulting magnetization curves for impurities and holes for several values of n c /n i are shown in Fig. 1. Note that specific values of J, n i , and n c are not of any relevance here -only the ratio n c /n i is the important tuning parameter in determining magnetization as a function of T /T c . The most important salient feature of Fig. 1 is the highly "non-mean-field-like" concave magnetization behavior for low values (≤ 0.2) of n c /n i . The reason for such behavior is that if we have one hole per many impurities, the effective field B (c) eff acting on holes is much stronger than its counterpart B (i) eff acting on impurities. As a result, the hole magnetization grows as for T T c0 and reaches unity at some temperature of the order of T c0 , while the impurity magnetization is still much less than unity. As the temperature is getting lower, the impurity magnetization grows as T n c n i sS (s + 1)(S + 1) and approaches unity only at Finite antiferromagnetic coupling J AF suppresses T c , as one can easily see from Eq. (30). In Fig. 2    In discussing the results shown in Fig. 2 we first note that the actual direct Mn-Mn antiferromagnetic coupling J AF is expected to be very small in Ga 1−x Mn x As for low values of x, i.e. for relatively large Mn-Mn separation, since the direct antiferromagnetic coupling falls off exponentially with inter-atomic distance. For larger values of x, however, effects of antiferromagnetic Mn-Mn coupling may very well be important in determining the DMS magnetic behavior. In general, finite J AF suppresses both T c and M (T ) as one would expect. In particular, the zero-temperature magnetization M (T → 0) could be strongly suppressed far below the saturation magnetization by the finite AF coupling, particularly for lower hole densities, An interesting feature of Fig. 2 is that at low carrier densities larger values of J AF may actually restore (albeit with strongly suppressed value of saturation magnetization) the convex M (T ) shape (e. g. the z AF |J AF |/J = 5.0 × 10 −3 curve for n c /n i = 0.1 in Fig. 2(b)). For higher hole densities, however, M (T ) becomes more concave for J AF = 0.
The magnetic susceptibility of the system is essentially that of impurities (since n c /n i ≪ 1 and S > s), and above the critical temperature is given by where T * ≡ T 1 + T 2 1 + T 2 c0 , and χ 0 = n i (g i µ B ) 2 S(S + 1)/3T is the paramagnetic susceptibility of bare impurities.
Finally, we show in Fig. 3

Degenerate hole gas
For metallic DMS systems where ferromagnetic T c is optimum (at least for Ga 1−x Mn x As), the carrier system is typically delocalized, with the Fermi energy substantially exceeding the ferromagnetic transition temperature. In such systems, the formalism of Sec. II B 1, developed for the non-degenerate hole system is not applicable anymore, and we have to use the generic expression for the hole polarization, where D(ε) is the hole density of states. If the effective field B (c) eff acting on carriers is weak, we can expand the density of states up to the first order in the effective field to obtain For very low temperatures, k B T ≪ E F (in general, this condition is satisfied in metallic Ga 1−x Mn x As systems where typically k B T c /E F < 0.1 noting that both T c and E F decrease with lowering the hole density n c ), we get Linearizing Eq. (6) and using Eqs. (3) and (5), we arrive at where and T 1 is given by Eq. (11).
In the absence of antiferromagnetic interaction between magnetic impurities, J AF = 0, the ferromagnetic transition temperature equals T c0 . As one can see from Eq. Before presenting our numerical results for the calculated temperature-dependent DMS magnetization for the degenerate-carrier mean-field theory, we mention that equation (20) for the ferromagnetic transition temperature in the Zener model was first derived in Ref. 35 more than forty years ago and has recently been rediscovered 36,37 in the context of DMS ferromagnetism.
In Fig. 4 we show our calculated impurity and carrier magnetization for the degenerate-carrier Weiss meanfield theory using the same parameters (typical for Ga 1−x Mn x As) as for the corresponding non-degenerate case shown in Fig. 1. In general, except for the lowest hole density with n i /n c = 0.05, the impurity magnetization is convex, but looks quite different from the classic text book Weiss form except perhaps at very high carrier densities when n i /n c ≈ 1. In particular, for the realistic value of n c /n i = 0.1 (which is thought to apply to many Ga 1−x Mn x As samples), M (T ) in Fig. 4 has the non-mean-field-like straight line shape at lower temperatures. For higher values of hole density, n c /n i =0.2 and 0.5, the magnetization is much closer to the classical Brillouin shape whereas for lower carrier densities (e. g., n c /n i = 0.05) even this metallic DMS system starts exhibiting the concave magnetization curve typical of the insulating DMS systems discussed in the last sub-section (II B 1) of this article.
The origin of the concave shape of the magnetization curve is exactly the same as for the non-degenerate case discussed in Sec. II B 1, namely, the magnetization of holes saturates at T T c0 , much earlier than magnetization of impurities, which, upon saturation of the hole magnetization grows as given by Eq. (14), which represents a concave curve provided n c /n i ≪ 1. Thus, the concave magnetization behavior may be generic to the low carrier density limit of DMS ferromagnets, independent of whether they are metallic or insulating although the highly concave magnetization curves of Fig. 1(a) are clearly much more typical of insulating DMS systems than the metallic ones. The carrier magnetization results presented in Fig. 4(b) are very similar to textbook convex Weiss magnetization behavior. We note that the degenerate-carrier mean-field theory results shown in Fig. 4(a) are qualitatively very similar to the experimentally measured temperature dependent magnetization data in metallic Ga 1−x Mn x As systems. In particular, recent annealing experiments, where annealing leads to better metallicity (e. g., higher conductivity) by virtue of increasing the hole density, show qualitative trends strikingly similar to the results of Fig. 4(a).
Next we include the direct antiferromagnetic coupling (4) between the local moments in our consideration. With J AF = 0, the critical temperature is given by Eq. (20). The calculated impurity magnetization in the presence of the antiferromagnetic coupling is shown in Fig. 5. Comparing with the non-degenerate-hole model we find that the degenerate hole case is more quantitatively sensitive to the antiferromagnetic coupling although the qualitative effect of a finite J AF in both metallic and insulating DMS materials is basically the same, namely suppression of the transition temperature and the magnetization.
Similar to the results shown for the insulating systems, we present in Fig. 5 calculated impurity magnetization for two values of the ratio n c /n i = 0.5 [ Fig. 5(a)] and 0.1 [ Fig. 5(b)].
In Fig. 6 the calculated external magnetic field and temperature dependence of the dopant magnetization is shown for fixed parameter values J, S, s, and for various density ratio n c /n i . In the main figure we show M (T, B) as a function of temperature for a magnetic field value, B = 2.6T (solid lines), and in the inset we show M (T, B) as a function of the external field for fixed temperatures. Our calculated M (T, B) behavior is roughly qualitatively similar to experimental observations in Ga 1−x Mn x As systems in the "metallic" regime.
The magnetic susceptibility of the system is essentially that of impurities, similarly to Sec. II B 1, Eq. (16), and above the critical temperature is given by where χ 0 = n i (g i µ B ) 2 S(S + 1)/3T is the paramagnetic susceptibility of bare impurities. The Weiss molecular mean-field theory results (for localized carriers in Sec. II B 1 and for delocalized carriers in Sec. II B 2) presented above for DMS magnetization qualitatively agree very well with the existing experimental data. In particular, the basic trend of our results shown in Figs. 1-6, that the spontaneous magne- tization is strongly suppressed (perhaps even into a very unusual outwardly concave shape) at low carrier densities and for more insulating systems whereas at higher carrier densities and for more metallic systems the magnetization has the usual convex textbook shape, is in excellent agreement with experiment. Our degenerate-carrier mean-field results (see Fig. 3(a)) also reproduce the almost linear magnetization curves seen in GaMnAs for intermediate carrier densities. But obviously one needs to go beyond the Weiss mean-field theory for a deeper (and more quantitative) understanding of DMS magnetization, if for no other reason than to validate (or to ascertain the regime of validity of) the simple static meanfield theory. At low carrier density the mean-field theory should work better since on the average there are many Mn local moments between any two holes, but typically the number of Mn atoms per hole is around 3-4 so the quantitative applicability of MFT is questionable. Note that the hole-hole interaction neglected in our calculations may be included in a crude approximate fashion by incorporating a Stoner enhancement of the carrier susceptibility, though in general the strength of this enhancement is unknown.
In the next two sections we go beyond the Weiss static mean-field theory and develop two more sophisticated approximation schemes to theoretically study DMS magnetization. These are DMFT (section II C) and the percolation theory (section II D).

C. The Dynamical Mean-Field Theory (DMFT)
In this section we use a recently developed nonperturbative method, the "dynamical mean-field theory" (DMFT), 38 to calculate magnetization for the minimal model (Eq. (2)) of dilute magnetic semiconductors. The DMFT has been recently applied to the DMS system to calculate the magnetic transition temperature and the optical conductivity 22 . DMFT is essentially a lattice quantum version of the Weiss mean field theory where the appropriate density of states (including impurity band formation) along with temporal fluctuations are incorporated within an effective local field theory.
We model the Ga 1−x Mn x As system as a lattice of sites, which are randomly nonmagnetic (with probability 1−x) or magnetic (with probability x), where x now indicates the relative concentration (i.e. per Ga site) of active Mn local moments in GaMnAs. The DMFT approximation amounts to assuming that the self energy is local or momentum independent, Σ(p, iω n ) → Σ(iω n ), and then all of the relevant physics may be determined from the local (momentum-integrated) Green function defined by where we have normalized the momentum integral to the volume of the unit cell a 3 0 , and µ is the chemical potential and h the external magnetic field. G loc is in general a matrix in spin and band indices and depends on whether one is considering a magnetic (a) or non-magnetic (b) site. Since G loc is a local function, it is the solution of a local problem specified by a mean-field function g 0 , which is related to the partition function Z loc = dS exp(−S loc ) with action on the a (magnetic) site and on the non-magnetic (b) site. The local Green function is calculated exactly as where the thermal average . . . is taken with respect to the orientation of the local spin S. The a-site mean-field function g a 0 can be written as g a 0αβ = a 0 + a 1 m · σ αβ with m the magnetization direction and a 1 vanishing in the paramagnetic state. It is specified by the condition that the local Green function computed from Z loc , namely δ ln Z loc /δg a 0 = (g a 0 − Σ) −1 is identical to the local Green function computed by performing the momentum integral using the same self energy.
The form of the dispersion given in full Hamiltonian Eq. (2) applies only near the band edges. It is necessary for the method to impose a momentum cutoff, arising physically from the carrier band-width. We impose the cutoff by assuming a semicircular density of states D(ǫ) = a 3 The parameter t is chosen to correctly reproduce the band edge density of states. Other choices of upper cutoff would lead to numerically similar results. This choice of cutoff corresponds to a Bethe lattice in infinite dimensions. Other (perhaps more realistic) choices for the density of states would give magnetization results qualitatively similar to our results since the band edge density of states has the correct physical behavior in our model. For this N (ǫ) the self consistent equation for g 0 obeys the equation where the angular brackets denote averages performed in the ensemble defined by the appropriate Z loc . Within this approximation the normalized magnetization M (T ) of the local moments is given by As the temperature is increased, the spins disorder and eventually the magnetic transition temperature is reached. Above this temperature, g 0 is spin-independent. By linearizing the equation in the magnetic part of g 0 with respect to a 1 we may obtain the ferromagnetic transition temperature T c . The details on the calculation of T c are given in Ref. 22. The critical temperature T c and magnetization of the local moment depend crucially on J/t, x, and carrier density n c . Note that in the DMFT calculation it is more convenient to express our results in terms of n, the relative concentration of active local moments rather than n i , the absolute impurity density. Before we show our calculated magnetization we describe the density of state (DOS) of dynamical meanfield calculations applying to simple semi-circle models. In Fig. 7 we show the DMFT density of states for majority spin near the band edge corresponding to the disordered spin state T > T c , and the inset shows the T = 0 ferromagnetic state. The evolution of the energy (ω) dependent DOS is shown as the carrier-spin coupling J is increased from zero; note that the method works equally well in the ordered T = 0 state and the disordered spin T > T c case, and predicts the formation of a spin polarized impurity band for J > t. For small J we see the expected band shift proportional to xJ. For J > J c an impurity band centered at ∼ −J and containing x states is seen to split off from the main band, where the critical coupling J c ∼ t. In the DMFT calculations we parametrized the exchange coupling J in terms of the band width parameter t, subsuming the unit cell volume a 3 0 implicitly. In Fig. 8 we show the normalized magnetization of the local moments as a function of temperature for different values of J =1.0, 1.5, 2.0t and x = 0.05, and for various hole densities, n c /n i . For the small coupling constant J = t the impurity band is not formed, but for J =1.5, 2.0t we have a spin polarized impurity band. In discussing our DMFT magnetization calculations we note that the DMFT results shown in Fig. 8 are qualitatively roughly similar to the delocalized static (degenerate) mean-field-theory results we obtained in Sec. II B 2. For example, the results of Fig. 8 approximately resemble those shown in Fig. 4(a) Fig. 8 are in much better agreement with the magnetization measurements in metallic GaMnAs systems in the sense that the linear behavior of M (T ) for lower T (with almost a kink just below T c as can be seen in Fig. 8) is much more pronounced (as in the experimental data) than our delocalized mean-field-theory results shown in Fig. 4(a). This is both gratifying and expected because DMFT is a substantial improvement on the Weiss MFT as it incorporates the physics of spinsplit impurity bands through the appropriate quantum self-energy corrections not included in the simple MFT of Sec. II B. In fact, for very low carrier densities we obtain outwardly concave M (T ) in our DMFT calculations (as we expect to do in the strongly nondegenerate limit), but the computational convergence in our DMFT numerical calculations is rather poor in this regime of unrealistically low (n c /n i ≤ 0.01) carrier densities, and therefore we refrain from showing these results. One can, however, detect very slight concavity in the lowest carrier density (n c /n i = 0.04) M (T ) results shown in Fig.  8. We point out that the critical magnetic properties in DMFT are the same as in the static MFT, and therefore all the DMFT critical exponents are equal to those in the Weiss MFT.

results of
In particular, it should be possible to obtain the M (T ) behavior for the localized carrier case (cf. Sec. II B 1) also from DMFT by incorporating impurity band localization in the DMFT formalism. Our current theory does not include localization, and the impurity band (or valence band) carriers in our DMFT calculations are all delocalized metallic carriers. First principles band theory calculations indicate that the actual exchange coupling in GaMnAs may be close to critical J c , and as such impurity band physics may be quite important for understanding DMS magnetization. To incorporate physics of localization one needs to include disorder effects (invariably present in real DMS systems) in the model. All our MFT calculations (both DMFT in Sec. II C and the static MFT of Sec. II B) are done in the virtual crystal approximation where effective field is averaged appropriately leaving out random disorder effects explicitly. In the next section we explicitly incorporate disorder in the theory by developing a percolation theory approach to DMS magnetization for the strongly localized insulating systems.
The calculated magnetization as a function of temperature is shown in Fig. 9 for various external magnetic field values. The inset shows the magnetization as a function of external field at T = T c . Our calculated M (T, B) behavior is roughly qualitatively similar to the mean-field results in section II B 2. At the critical temperature T c M (T c , B) shows a mean-field behavior, M (T c , B) ∝ B 1/3 . The DMFT results shown in Fig. 9 are qualitatively similar to DMS experimental results.

D. Magnetization in the percolation formalism
Our recently developed percolation theory 9 applies strictly in the regime of strongly localized holes where the dynamical mean-field theory for delocalized carriers described in Sec. II C has little validity. These two theories, mean-field theory and percolation theory, are therefore complementary. Interestingly, however, the perturbation theory and the dynamical mean-field theory are not mutually exclusive in spite of their regimes of validity being different, and in particular, a significant aspect of our percolation approach is its ability to reproduce qualitatively the mean-field theory results of the last section both for T c and M (T ).
The percolation theory assumes the same carriermediated ferromagnetism model of Sec. II B, but now the carriers are pinned down with the localization radius a B . The localized carriers are therefore taken to be in the impurity band and disorder, completely neglected in the mean-field theory of Sec. II B and II C, now plays a key role in the carrier localization. The mean-field theory and the perturbation theory are therefore complementary in the sense that one (the mean-field theory) completely neglects disorder, and the other (the percolation theory) includes disorder at a very fundamental level i. e. by starting from the picture of localized carriers in a strongly disordered system. The magnetic impurities in this case are assumed to be completely randomly distributed in the host semiconductor lattice in contrast to the mean-field case where the carrier states are free and the disorder is neglected.
As we have demonstrated in Ref. 9, the problem of ferromagnetic transition in a system of bound magnetic polarons can be rigorously reduced to the problem of overlapping spheres studied in the percolation theory. 23 The latter problem studies spheres of the same radius r randomly placed in space (three-dimensional in our case) with some concentration n. Overlapping spheres form "clusters;" as the sphere radius r becomes larger, more and more spheres join into clusters, the clusters coalesce, and finally at some critical value of the sphere radius an infinite cluster spanning the whole sample appears. This problem has only one dimensionless parameter, r 3 n, and therefore can be easily studied by means of Monte-Carlo simulations.
Each sphere of the overlapping spheres problem corresponds to a bound magnetic polaron, which is a complex formed by one localized hole and many magnetic impurities with their spins polarized by the exchange interaction with the hole spin. The concentration n of spheres is therefore equal to the concentration n c of localized holes. The expression for the effective polaron radius is not trivial and has been found in our earlier work. 9 The resulting formal relation between the physical parameters of the system under consideration and the only parameter of the overlapping spheres problem reads: 9 Here 0.64 ≈ 0.86 3 is the critical value of the parameter r 3 n at which the infinite cluster appears, and T c is the Curie temperature of the ferromagnetic system under consideration, derived in Ref. 9, B n c is shown schematically in Fig. 10.
Since the magnetic characteristics of the sample are mostly due to magnetic impurities rather than holes, the quantity of interest is the number of magnetic impurities in a cluster, which is proportional to the cluster volume. The magnetic properties of the system can be expressed in terms of the following quantities, which can be easily found: (i) concentration P(vn; r 3 n)dv of clusters with volume between v and v + dv, (ii) the fraction of volume P ∞ (r 3 n) taken by the infinite cluster, and (iii) the fraction P 0 (r 3 n) which does not belong to any sphere or cluster of spheres. Clearly, these quantities obey the relation Figure 11 shows the behavior of the three terms of the above equation as a function of temperature T , with r 3 n related to T /T c by Eq. (29) and a 3 B n c = 10 −3 . Having found these quantities, one can easily find the magnetic properties of the system.
The spins of non-connected clusters are not correlated and average out when we calculate the spontaneous magnetization of the whole sample. The only non-vanishing contribution comes from the infinite cluster. The total magnetic moment of the sample per unit volume is therefore Here and in the following equations we write the characteristics P ∞ , P 0 , P(vn) of the overlapping sphere problem as functions of the physical parameters T /T c and a 3 B n c instead of r 3 n. The relation between these parameters is given by Eq. (29). The temperature dependence of the spontaneous magnetization given by Eq. (31) is plotted on Fig. 12 for two experimentally viable values of a 3 B n c . As already mentioned before, such strongly concave M (T ) behavior is often observed in insulating DMS systems, see for example Refs. 4,6,39 The magnetic susceptibility of a sample has contributions coming from polaron clusters and free spins. Using the classical expression for the susceptibility of a free 3T and the fact that the total spin of a cluster with volume v equals vn i we arrive at The first term in brackets diverges as (T − T c ) γ with γ ≈ 1.7 at T → T c . The second term in brackets does not have any singularity at T → T c . For the temperature dependence of the susceptibility given by Eq. (32), see Fig. 13.
Using the classical magnetization relation where L(x) ≡ B ∞ (x) = cotan x − 1/x is the Langevin function, we obtain the expression for the magnetic moment per unit volume in finite magnetic field Figures 14 and 15 illustrate the dependence of the magnetization on the temperature and magnetic field respectively. Since the spins of polarons and their clusters are much large than those of three impurities, the magnetization curve of Fig. 15 has two characteristic scales. First, polarons and their clusters are polarized (inset of Fig. 15), then the free spins are polarized at much larger fields. We mention that our percolation-theory critical exponents are γ ≈ 1.7 and β ≈ 0.4 for the susceptibility (Fig. 13) and magnetization (Fig. 12), as compared with the mean-field results of γ = 1 and β = 0.5 and the best existing numerical estimates of γ ≈ 1.39 and β ≈ 0.37 for the three-dimensional Heisenberg model. 40

III. CONCLUSION
The temperature-dependent magnetization results presented in this work for different but interrelated theoretical models are qualitatively consistent with one another. In particular, the two theories for metallic DMS systems with itinerant carriers, namely the degeneratecarrier Weiss static mean-field theory (see Sec. II B 2) and the dynamical mean-field theory (see Sec. II C), both give onwardly convex M (T ) carriers, similar (but not identical) to that in the textbook molecular mean-field result, with the convexity being enhanced (suppressed) with increasing (decreasing) delocalized carrier density. In typical situations involving itinerant carriers in metallic DMS systems, we find temperature dependent magnetization curves which are almost linear at lower temperatures in excellent agreement with experimental observations in metallic GaMnAs systems. Such apparent "non-meanfield-like" magnetization behavior, found here both in our delocalized-carrier static and dynamic mean-field theories, arises from a combination of reasons: (1) the double Brillouin function [i. e. B(B(x))] form of the coupled field magnetization [see Eq. (9)]; (2) the low values of n c /n i , leading to the Mn moments feeling on the average a much reduced number of free carriers. As the carrier density is increased, for example by suitable annealing in recent experimental studies, such linear magnetization curves evolve toward the more conventional outwardly convex magnetization, as can be seen in our results and in recent annealing experiments. 15,16,17,18,19 In the case of localized carriers appropriate for insulating DMS systems (many of which are also found to be ferromagnetic with well-defined Curie temperatures, e. g. Ga 1−x Mn x As for x < 0.03, In 1−x Mn x As, 4 and Ge 1−x Mn x 6 ), our two complementary theories, namely the non-degenerate-carrier static mean-field theory (Sec. II B 1) and the percolation theory (Sec. II D), give qualitatively similar magnetization behavior. In particular, at low carrier densities the M (T ) curves in strongly insulating DMS systems exhibit strikingly "non-mean-field-like" outwardly concave magnetization behavior, as often observed in insulating DMS systems. Again, this unusual concave magnetization behavior arises from a combination of the strongly localized nature of the carrier system and the low values of the carrier density (i. e. n c /n i ≪ 1) in DMS materials. We emphasize, however, a very important feature of theoretical results presented in this work: the convex magnetization behavior in the strongly metallic case and the concave behavior in the insulating case are not a sharp dichotomy in DMS properties -these are really the two extremes of a continuum of possible magnetization behavior in DMS materials, as can be inferred from the results of Sec. II B. The typical DMS magnetization behavior should lie somewhere in between these two extremes of highly concave (low carrier density and strongly localized insulating DMS) and highly convex (high carrier density and strongly delocalized metallic DMS) magnetization, leading to the generic experimental observation of almost linear M (T ) behavior in ferromagnetic Ga 1−x Mn x As.
Another often-mentioned peculiar aspect of DMS magnetization, namely the low value of the saturation magnetization (compared with the number of Mn ions in the sample as given by the value of x in Ga 1−x Mn x As), is also apparent in our theoretical results presented in this work. In particular, the presence of direct antiferromagnetic coupling between the Mn moments may drastically suppress the saturation magnetization, as can be seen in the results presented in Figs. 2 and 5 of this paper. This absence of complete magnetization saturation in our meanfiled results is akin to having effective ferrimagnetism in the system. In addition, the calculated magnetization is may be much lower than the saturation value except at very low temperatures, see Fig. 12, which could also explain the lack of magnetization saturation. It is likely that in the real DMS materials an appreciable fraction of the Mn moments are magnetically inactive because they do not sit in cation substitutional sites, but rather at defect sites such as interstitials and are antiferromagnetically coupled to other Mn moments. Such magnetically inactive Mn atoms could be one of the reasons for the low values of saturation magnetization in Ga 1−x Mn x As. We also note that our percolation theory provides another possible explanation for the observed low values of saturation magnetization in low carrier density insulating DMS materials. Since the infinite cluster of percolating bound magnetic polarons triggering the long-range ferromagnetic ordering necessarily leaves out a large number of Mn moments (which are not parts of the infinite cluster except at T → 0), one naturally expects a very low saturation magnetization except perhaps at T ≪ T c .
A recent series of potentially important annealing experiments in metallic Ga 1−x Mn x As samples by several different groups may eventually shed considerable light on our understanding of the mechanisms underlying DMS ferromagnetism. These experiments, while differing somewhat on the details, all find that suitable low temperature "optimal" annealing may enhance the magnetic properties of Ga 1−x Mn x As by increasing T c and more importantly for our purpose, by enhancing M (T ) to more convex almost standard mean-field-like behavior. This enhancement of magnetization seems to correlate well with improvement in the metallicity of the annealed sample (e. g. higher conductivity) and with an increase in the hole density. Annealing may also be enhancing the magnetic properties by annealing away some of the Mn interstitials. Our theoretical results are in excellent agreement with these annealing experiments, since increasing n c /n i does lead to enhanced (and more convex) magnetization in our theory. In this context, it will be very helpful to have more detailed information on the M (T ) behavior in strongly localized insulating samples both for Ga 1−x Mn x As with x < 0.03 and for other insulating DMS materials (e. g. Ge 1−x Mn x ).
Our final comment addresses the role of disorder (essentially neglected in our work, except for the percolation theory part, as we treat it in the virtual crystal approximation, i. e., we assume that holes and local moments only feel the average effective field ), which is invariably strong (and not yet well understood) in DMS materials. In particular, even the "metallic" DMS systems are in effect very poor metals with mean-free paths which are at or below the Ioffe-Regel limit (with low-temperature mobilities of the order of a few cm 2 /V·s only), indicating the presence of very strong disorder. 41 In the presence of such strong disorder, various spin glass ground states may compete with ferromagnetic ground states. We believe that it is important to look for signatures of spin glass physics in low-temperature DMS magnetic properties. It may very well turn out that spin glass phases dominate the regime of parameter space (e. g. x < 1% or x > 10% in Ga 1−x Mn x As) where a ferromagnetic ground state does not seem to stabilize in DMS materials. This work is supported by US-ONR and DARPA.