First-principles theory of dilute magnetic semiconductors

This review summarizes recent first-principles investigations of the electronic structure and magnetism of dilute magnetic semiconductors (DMSs), which are interesting for applications in spintronics. Details of the electronic structure of transition-metal-doped III-V and II-VI semiconductors are described, especially how the electronic structure couples to the magnetic properties of an impurity. In addition, the underlying mechanism of the ferromagnetism in DMSs is investigated from the electronic structure point of view in order to establish a unified picture that explains the chemical trend of the magnetism in DMSs. Recent efforts to fabricate high-$T_C$ DMSs require accurate materials design and reliable $T_C$ predictions for the DMSs. In this connection, a hybrid method (ab initio calculations of effective exchange interactions coupled to Monte Carlo simulations for the thermal properties) is discussed as a practical method for calculating the Curie temperature of DMSs. The calculated ordering temperatures for various DMS systems are discussed, and the usefulness of the method is demonstrated. Moreover, in order to include all the complexity in the fabrication process of DMSs into advanced materials design, spinodal decomposition in DMSs is simulated and we try to assess the effect of inhomogeneity in them. Finally, recent works on first-principles theory of transport properties of DMSs are reviewed. The discussion is mainly based on electronic structure theory within the local-density approximation to density-functional theory.

Figure 1

(Color online) Schematic illustrating the MRAM device. Each magnetic pillar consists of three layers; two magnetic layers (depending on the magnetization direction of these layers) are separated by a nonmagnetic layer, yielding a structure with a regular GMR functionality. The binary code is stored as the coupling between the two magnetic layers in a pillar and it is read via the magnetoresistive property of the pillar (see text).

Figure 2

(Color online) Spin-polarized light-emitting diode. Spin-polarized holes of the Mn-GaAs region recombine with electrons in the InGaAs region, with emission of circularly polarized light.

Figure 3

(Color online) Predicted ordering temperatures of Mn-doped DMS materials. The concentration of Mn atoms is 5%. From 50.

Figure 4

(Color online) Schematic of the electronic structure of $3d$ TM impurities in semiconductors at (a) the tetrahedral $\left(T_d\right)$ substitutional site and (b) the tetrahedral $\left(T_d\right)$ interstitial site.

Figure 5

Schematic explanation of the microscopic mechanism of the reduction of intra-atomic Coulomb repulsion $\left(U_0\right)$ of a $3d$ TM impurity $\left(3d^N\right)$ in semiconductors (82). (a) In the free $3d$ TM atoms the additionally attached electron $\left(e^{-}\right)$ leads to a large $U_0\sim 20\mathrm{eV}$ due to the strong localization at the $3d$ atomic orbital. (b) In the semiconductor, due to the repulsive Coulomb interaction and the strong $p\text{−}d$ hybridization, the additional electron is delocalized; 90% of the charge is located in extended impurity band states at the ligand sites and only 10% $\left(\Delta Q\right)$ is localized on the $3d$ TM atomic site. This is called the Haldane-Anderson mechanism and reduces the effective intra-atomic Coulomb interaction $\left(U_{\mathrm{eff}}\right)$ by almost two orders of magnitude.

Figure 6

Change in the doping dependence of the magnetic moment $\left(\Delta M\right)$ and number of $3d$ electrons $\left(\Delta \mathrm{Mn}3d\right)$ in (Zn, Mn)O. Upon hole (acceptor) doping by N, the number of $3d$ electrons with up-spin states $\left(\Delta \mathrm{Mn}3d^{\uparrow }\right)$ decreases; on the other hand, $\left(\Delta \mathrm{Mn}3d^{\downarrow }\right)$ with down-spin states increase. Thus the change in the total $3d$ TM charge, both up and down spin, upon hole doping is only 10% of the change in the total charge, and the remaining 90% of the total charge must exist in extended impurity band states at the ligand sites. In contrast, the change in the magnetic moment $(\Delta M=\Delta \mathrm{Mn}3d^{\uparrow }-\Delta \mathrm{Mn}3d^{\downarrow })$ is very large, since the exchange-correlation interaction is not well screened by multipole screening.

Figure 7

Schematic showing the $T_d$ symmetry-preserving breathing mode and other lower-symmetry JT distorted modes. The arrows indicate the direction of displacement of As atoms.

Figure 8

DOS for Mn-doped GaN for an ideal structure with a tetrahedral $T_d$ symmetry. From 131.

Figure 9

DOS for Mn-doped GaN in a JT-distorted structure with $D_{2d}$ symmetry. From 131.

Figure 10

(Color online) Calculated density of states of GaN-based (upper panels) and GaAs-based (lower panels) DMSs with $3d$ impurities ranging from Cr to Co. Dotted lines show total averaged DOS per unit cell and solid lines show partial DOS per atom of $3d$ components at TM sites. (Note the different scales.) A ferromagnetic configuration is assumed and the TM concentration is 5% for all cases.

Figure 11

(Color online) Calculated density of states of ZnO-based (upper panels) and ZnTe-based (lower panels) DMSs. Transition-metal impurities ranging from Cr to Co are considered. Dotted lines show total averaged DOS per unit cell and solid lines show partial DOS per atom of $3d$ components at TM sites. (Note the different scales.) A ferromagnetic configuration is assumed and the TM concentration is 5% for all cases.

Figure 12

Chemical trends of the energy difference $\left(\Delta E\right)$ between ferromagnetic (FM) and paramagnetic (DLM) states for GaN-, GaP-, GaAs-, and GaSb-based DMSs. From 175.

Figure 13

Chemical trends of the energy difference $\left(\Delta E\right)$ between FM and paramagnetic (DLM) states for ZnO-, ZnS-, ZnSe-, and ZnTe-based DMS. From 175.

Figure 14

Calculated $T_C^{\mathrm{MFA}}$ of (Ga,Mn)N, (Ga,Mn)P, (Ga,Mn)As, and (Ga,Mn)Sb as a function of Mn concentration. From 172.

Figure 15

$T_C^{\mathrm{MFA}}$ of (Ga,Mn)N (dashed line) and (Ga,Mn)As (solid line) calculated with additional donor (O for GaN and antisite As for GaAs) or acceptor doping (Mg for both compounds). From 168.

Figure 16

Calculated total DOS (solid lines) and partial density of $3d$ states at Mn site (dotted lines) in (Ga,Mn)N for various additional carrier concentrations. From 168.

Figure 17

(Color online) Calculated density of states in (a) (Ga,Mn)N, (b) (Ga,Mn)P, (c) (Ga,Mn)As, and (d) (Ga,Mn)Sb. Total DOS (dotted lines) and partial density of $3d$ states at Mn site (solid lines) are shown. The Mn concentration is 5% and a ferromagnetic configuration is assumed for all cases. Up spin (down spin) means majority (minority) spin in ferromagnetic state.

Figure 18

Schematic diagram of the electronic structure in the case of double exchange. (a) Schematic of the spin-polarized DOS of a transition-metal impurity in a wide-gap semiconductor. The $d$ level is located in the band gap for the spin-up states. The full lines show the spin-up and spin-down impurity bands for a small concentration, the dashed lines for a larger concentration. The gray areas indicate that more energy can be gained by band broadening, since states are transferred to lower energies. (b) Schematic representation of the hybridization-induced energy gain $\Delta E_{DX}$ for a diatomic system with unperturbed levels ${ϵ}_1={ϵ}_2$.

Figure 19

(Color online) Calculated exchange interactions between Cr impurities in (a) (Ga,Cr)N and (b) (Zn,Cr)Te.

Figure 20

Schematic diagram of the spin-polarized DOS in the case of $p\text{−}d$ exchange, when the majority-spin $d$ level lies below the valence $p$ band and the minority level above. The dashed curve represents the valence band after interaction with the magnetic impurities ($p\text{−}d$ mixing).

Figure 21

$T_C^{\mathrm{MFA}}$ values of (Ga,Mn)As calculated within the LDA (closed circles) and the $\mathrm{LDA}+U$ (open circles). The inset shows partial DOS of Mn $d$ states in 5% Mn-doped GaAs calculated by use of the LDA (solid lines) and the $\mathrm{LDA}+U$ (dotted lines). For the $\mathrm{LDA}+U$ calculations a value $U=4\mathrm{eV}$ is assumed. From 172.

Figure 22

(Color online) Calculated total and partial DOSs of (Ga,Mn)As by (a) the LDA and (b) the PSIC method. In the calculations, Mn concentrations are fixed to 5%. Light part indicates Mn 3d components and dark part indicates total density of states. (c) Experimental photoemission spectrum of (Ga,Mn)As (146). The experimental Mn concentration is 6.9%.

Figure 23

(Color online) Calculated total and partial DOSs of (Zn,Cr)Te by (a) the LDA and (b) the PSIC method. In the calculations, Cr concentrations are fixed to 5%. Light part indicates Cr 3d components and dark part indicates total density of states. (c) and (d) Experimental photoemission spectra of (Zn,Cr)Te (114) and the experimental Cr concentrations are 3% and 15%, respectively.

Figure 24

Schematic diagram of the electronic structure in the case of superexchange. (a) Schematic diagram of the DOSs of two antiparallel-aligned impurity systems 1 and 2 with equal concentrations $c∕2$. The full lines show the LDOS without hybridization between the antiparallel-aligned impurity systems; the dotted lines refers to the LDOS that includes the hybridization effects. (b) Schematic of the hybridization-induced energy gain $\Delta E$ of Eq. (21) for a diatomic system with unperturbed levels ${ϵ}_1<{ϵ}_2$. t is the hopping matrix element between the impurity states.

Figure 25

(Color online) Mean-field Curie temperature $T_C^{\mathrm{MFA}}$ and exchange coupling constants $J_{ij}$ for (Ga,Fe)N and (Ga,V)As. (a) $T_C^{\mathrm{MFA}}$ as a function of impurity concentration $c$. (b) $J_{ij}$ for the Fe-Fe coupling in (Ga,Fe)N for various concentrations as a function of distance. (c) $J_{ij}$ for the V-V interaction in (Ga,V)As. From 18.

Figure 26

(Color online) Curie temperature of (Ga,Mn)N in the mean-field approximation calculated using the LDA and the $\mathrm{LDA}+U$ method.

Figure 27

(Color online) Schematic diagram of the spin-polarized DOS of two V impurities in ferromagnetic (Ga,V)As. From 18.

Figure 28

(Color online) Spin-polarized local DOS of an impurity in a DMS material (dashed curves, left scale) as a function of $E-E_F$. Exchange coupling constant $J_{01}\left(E_F^{*}\right)$ for nearest-neighbor impurities (full curve, right scale) as a function of an arbitrary change in $E_F^{*}$. (a) 5% Mn in (Ga,Mn)N, (b) 5% Mn in (Ga,Mn)As, (c) 5% Fe in (Ga,Fe)N (LDA method), and (d) 5% Fe in (Ga,Fe)N ($\mathrm{LDA}+U$ with $U=4\mathrm{eV}$). From 18.

Figure 29

Exchange interactions in $({\mathrm{Ga}}_{0.95-y},{\mathrm{Mn}}_{0.05},{\mathrm{As}}_y)\mathrm{As}$ without $(y=0)$ and with As antisites $(y=0.01)$. The exponential decay of these interactions with distance along the dominating [110] direction is shown in the inset (see text).

Figure 30

Local density of states on Mn atoms in $({\mathrm{Ga}}_{0.95},{\mathrm{Mn}}_{0.05})\mathrm{As}$ alloy for majority and minority states. In the inset the host GaAs density of states is shown. For comparison, local densities of states on Mn atoms in $({\mathrm{Ga}}_{0.95},{\mathrm{Mn}}_{0.05})\mathrm{N}$ and on Cr atoms in $({\mathrm{Zn}}_{0.8},{\mathrm{Cr}}_{0.2})\mathrm{Te}$ are also shown.

Figure 31

Bloch spectral functions for majority and minority bands in $({\mathrm{Ga}}_{0.95},{\mathrm{Mn}}_{0.05})\mathrm{As}$ (lower panels). For comparison the host GaAs band structure (left upper panel) and the corresponding Bloch spectral function (right upper panel; a small broadening is used) are shown. In the lower figures, the vertical lines denote the Fermi energy.

Figure 32

Exchange interactions along the dominating [110] direction in ${\mathrm{Cu}}_{1-x}{\mathrm{Mn}}_x$ alloys for $x=0.005$, 0.02, and 0.05. Interactions are multiplied by the RKKY-like factor ${(d∕a)}^3$, where $d$ is the distance between two Mn atoms and $a$ is the lattice constant. Corresponding exchange interactions between two isolated impurities in a Cu host are shown in the inset.

Figure 33

Exchange interactions along the dominating [110] direction in semi-Heusler (NiMnSb) and Heusler $\left({\mathrm{Ni}}_2\mathrm{Mn}\mathrm{Sb}\right)$ alloys, illustrating the effect of half metallicity. Interactions are multiplied by the RKKY-like factor ${(d∕a)}^3$, where $d$ is the distance between two Mn atoms and $a$ is the lattice constant.

Figure 34

Exchange interactions in $({\mathrm{Ga}}_{0.95},{\mathrm{Mn}}_{0.05})\mathrm{As}$ along various directions in real space. A similar dependence for $({\mathrm{Ga}}_{0.95},{\mathrm{Mn}}_{0.05})\mathrm{N}$ is shown in the inset for comparison.

Figure 35

Exchange interactions for various DMS alloys as a function of the distance $d$ between magnetic impurities ($a$ denotes the lattice constant).

Figure 36

Concentration dependence of exchange interactions in $({\mathrm{Ga}}_{0.95},{\mathrm{Mn}}_{0.05})\mathrm{As}$ along the dominating [110] direction. In the inset we show the same dependence but here it is scaled by a RKKY-like factor ${(d∕a)}^3$.

Figure 37

The effect of electron correlations on exchange interactions in $({\mathrm{Ga}}_{0.95},{\mathrm{Mn}}_{0.05})\mathrm{As}$. (a) Corresponding exchange interactions evaluated in the framework of the LDA and $\mathrm{LDA}+U$ approaches are shown as functions of the distance between Mn atoms. (b) Along the dominating [110] direction the exchange interactions scaled by the RKKY-like factor ${(d∕a)}^3$ are shown. The first four $d∕a$ values in (b) correspond to the four maxima at $d∕a=0.71$, 1.41, 2.12, and 2.83 in (a).

Figure 38

The stability of the ferromagnetic state in $({\mathrm{Ga}}_{0.95-y},{\mathrm{Mn}}_{0.05},{\mathrm{As}}_y)\mathrm{As}$ as a function of As antisite concentration $y$: (a) total energy picture and (b) the lattice Fourier transform of exchange interactions between Mn atoms (see text). The order parameter $r$ characterizes the fraction of spins pointing in a direction opposite to that of the majority of the atoms: values $r=0$ and 0.5 correspond to the ferromagnetic and DLM states, respectively. Other values of $r$ characterize various possible uncompensated DLM states.

Figure 39

(Color online) Critical temperatures for a toy model (see text) as a function of concentration and range of the exchange interactions. Results from nearest-neighbor interactions are indexed with a 1 (black dots), while results from nearest and next-nearest interactions are indexed with a 2 (squares), etc. The calculated results from Monte Carlo (MC) simulations are normalized with the corresponding values from the virtual crystal approximation (VCA).

Figure 40

(Color online) Curie temperature in ${\mathrm{Ga}}_{1-x}{\mathrm{Mn}}_x\mathrm{As}$ as a function of $x$. The triangles denote the calculated values within the SC-LRPA. The other symbols are the measured values for as-grown (59) and annealed samples (58, 60). The experimental values observed by 138 and by 45 are also indicated.

Figure 41

(Color online) Curie temperature in GaMnAs as a function of the concentration of the magnetically active Mn. Note that the experimental data corresponding to a fixed nominal Mn concentration are plotted as a function of $x_{\mathrm{eff}}$. The values of $\gamma =n_h∕x_{\mathrm{eff}}$ are also shown.

Figure 42

(Color online) Critical temperatures of Mn-doped GaN as function of Mn concentration $x$. See text for details.

Figure 43

(Color online) Curie temperature in ZnCrTe as a function of Cr concentration. We compare the Curie temperature calculated within MFA-VCA SC-LRPA (19), and MC simulations.

Figure 44

(Color online) Ordering temperature calculated as a function of defect concentration by MFA and MCS for two different types of defect, zinc vacancies $\left(V_{\mathrm{Zn}}\right)$ and N substituting O $\left({\mathrm{N}}_{\mathrm{O}}\right)$, for ${\mathrm{Mn}}_{0.05}{\mathrm{Zn}}_{0.95}\mathrm{O}$-based systems.

Figure 45

(Color online) Calculated Curie temperatures with Monte Carlo simulations for ${\mathrm{Zn}}_{1-x}{\mathrm{Mn}}_x\mathrm{O}$ systems. Filled symbols correspond to N substitution, while empty symbols are for $V_{\mathrm{Zn}}$. In parentheses the concentration of the defect is given.

Figure 46

(Color online) Distributions of local magnetic fields for (a) Mn-doped GaAs, (b) Mn-doped GaN, and (c) Cr-doped ZnTe for different concentrations $x$ of the magnetic impurities.

Figure 47

(Color online) Total DOS (thin line) and PDOS of $3d$ states per impurity atom at Si sites (bold line) for four cases of a substitutional TM: (a) ${\mathrm{Si}}_{1-x}{\mathrm{V}}_x{\mathrm{O}}_2$, (b) ${\mathrm{Si}}_{1-x}{\mathrm{Cr}}_x{\mathrm{O}}_2$, (c) ${\mathrm{Si}}_{1-x}{\mathrm{Mn}}_x{\mathrm{O}}_2$, and (d) ${\mathrm{Si}}_{1-x}{\mathrm{Co}}_x{\mathrm{O}}_2$, at $x=0.05$.

Figure 48

Stability of the ferromagnetic states in $(\mathrm{Cu},\mathrm{TM})\mathrm{Al}{\mathrm{O}}_2$. The total energy difference per unit cell between ferromagnetic and paramagnetic states in V-, Cr-, Mn-, Fe-, Co-, and Ni-doped $\mathrm{Cu}\mathrm{Al}{\mathrm{O}}_2$ is plotted for 5%, 10%, 15%, 20%, and 25% concentration of TM atoms. A positive difference indicates that the ferromagnetic state is more stable than the paramagnetic state. From 112.

Figure 49

Total density of states per unit cell (solid line) and local density of $d$ states at TM site per atom (dashed line) in (a) Mn-, (b) Fe-, (c) Co-, and (d) Ni-doped $\mathrm{Cu}\mathrm{Al}{\mathrm{O}}_2$ in a ferromagnetic state. The TM atoms occupy Cu substitutional sites. The TM concentration is 5% for each case. The horizontal axis is the energy relative to the Fermi energy.

Figure 50

Calculated $T_C$ of $(\mathrm{Cu},\mathrm{Mn})\mathrm{Al}{\mathrm{O}}_2$, $(\mathrm{Cu},\mathrm{Fe})\mathrm{Al}{\mathrm{O}}_2$, $(\mathrm{Cu},\mathrm{Co})\mathrm{Al}{\mathrm{O}}_2$, and $(\mathrm{Cu},\mathrm{Ni})\mathrm{Al}{\mathrm{O}}_2$ using the MCS as a function of concentration.

Figure 51

Density of states as a function of energy relative to the Fermi energy for 20% Mn doped in (a) CoTiSb and (b) NiTiSn calculated by use of the KKR-CPA method. The solid and dotted lines indicate total and partial Mn $d$ densities of states.

Figure 52

Magnetic properties of Ni(Ti,Mn)Sn. (a) Calculated exchange coupling constants between Mn atoms in Ni(Ti, Mn)Sn as a function of distance between two Mn atoms. Mn 10%, 20%, and 30% cases are indicated by solid, dotted, and dashed lines. (b) Calculated Curie temperature of Mn-doped NiTiSn as a function of Mn concentration. The Curie temperature is calculated with the mean-field approximation (MFA, solid lines), random-phase approximation in the virtual crystal approximation (RPA-VCA, dotted lines), random-phase approximation (RPA, dashed lines), and Monte Carlo simulation (MCS, filled squares).

Figure 53

(Color online) Calculated mixing energy of (a) Mn in GaN, (b) Cr in GaN, (c) Mn in GaAs, and (d) Cr in ZnTe. From 172.

Figure 54

Effective pair interactions between (a) Mn in GaN, (b) Cr in GaN, (c) Mn in GaAs, and (d) Cr in ZnTe as function of distance normalized to the lattice constant. Negative interactions indicate that the pair interactions are attractive.

Figure 55

(Color online) Cr configuration in (Zn,Cr)Te (a) under 3D spinodal decomposition (dairiseki phase) after 100 Monte Carlo steps. (b) The 1D konbu phase obtained by layer-by-layer crystal growth based on 2D spinodal decomposition. Cr concentration is 5% for both phases. Cr sites are indicated by spheres and nearest Cr-Cr pairs are combined by bars. By controlling the dimensionality of the phase separation, we can fabricate either the dairiseki phase or the konbu phase.

Figure 56

$T_C^{\mathrm{RPA}}$ of (a) (Ga,Mn)N, (b) (Ga,Cr)N, (c) (Ga,Mn)As, and (d) (Zn,Cr)Te as a function of the number of Monte Carlo steps per impurity. As increasing the Monte Carlo steps, the phase separation develops. For higher concentrations. $T_C$ increases due to the increased percolation path by clustering. On the other hand, for lower concentrations, $T_C$ decreases and the system becomes superparamagnetic.

Figure 57

(Color online) Simulated results of the superparamagnetic blocking phenomenon in (Ga,Mn)N. Left-hand panels: Magnetization as function of temperature starting from parallel or antiparallel configurations of initial magnetization to the external field. Right-hand panels: Magnetization as a function of external field. The simulations are performed for homogeneous Mn distribution [(a) and (d)], dairiseki phase [(b) and (e)], and konbu phase [(c) and (f)]. The insets in the lower panels show snapshots of Mn distribution in (Ga,Mn)N for the respective phases. Temperature $\left(k_{B}T\right)$ and external field are scaled by the strength of the nearest-neighbor interaction $\left|J_{01}\right|$. From 172.

Figure 58

Residual resistivity of the $\left({\mathrm{Ga}}_{1-x}{\mathrm{Mn}}_x\right)\mathrm{As}$ alloy as a function of the Mn concentration $x$ calculated using the TB-LMTO method (204) from the total conductivity (○) and its coherent part (▼), and using the KKR method (130) (×). Experimental values for annealed thin films (60) are displayed as well (◇).

Figure 59

Effect of As antisites on the transport properties of (Ga,Mn)As. (a) Residual resistivity and (b) spin polarization of the conductivities as function of the As antisite concentration $y$ in the bulk $\left({\mathrm{Ga}}_{0.95-y}{\mathrm{Mn}}_{0.05}{\mathrm{As}}_y\right)$ alloy in the ferromagnetic (FM, ◆) and the uncompensated disordered-local-moment (DLM, ◻) state (200). Influence of As antisites and of magnetic order on transport properties of Mn-doped GaAs.

Figure 60

Effect of Mn interstitials on the transport properties of (Ga,Mn)As. (a) Residual resistivity and (b) spin polarization of the conductivities as functions of the Mn interstitial concentration $z$ in the bulk $\left({\mathrm{Ga}}_{0.95+z}{\mathrm{Mn}}_{0.05-z}\right)\mathrm{As}{\mathrm{Mn}}_z^i$ alloy in the ferromagnetic (FM, ◆) and the antiferromagnetic (AFM, ◻) state. From 200. Influence of Mn interstitials and of magnetic order on transport properties of Mn-doped GaAs.

Figure 61

Relation between Curie temperature and conductivity as calculated for three different models of the (Ga,Mn)As alloy with the nominal Mn content $x_{\mathrm{Mn}}^{\mathrm{tot}}=0.067$, assuming that compensating defects are As antisites (◆), Mn interstitials (◻), and both (○) (120). Experimental values for as-grown and annealed thin films are also shown (×).