Theory of ferromagnetic (III,Mn)V semiconductors

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 postgrowth-treatment techniques have played a central role in the field, often pushing the limits of dilute Mn-moment 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. This review focuses on the theoretical understanding of the origins of ferromagnetism and basic structural, magnetic, magnetotransport, and magneto-optical 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 $173\mathrm{K}$ 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.

Figure 1

(Color online) Hall resistance versus external magnetic field for annealed ${\mathrm{Ga}}_{0.94}{\mathrm{Mn}}_{0.06}\mathrm{As}$ at $110\mathrm{K}$ (black, largest Hall signal), $130\mathrm{K}$ (red, medium Hall signal), and $140\mathrm{K}$ (green, smallest Hall signal). From 82.

Figure 2

(Color online) Top panel: False-color scanning electron microscope (SEM) picture (side view) of a double constriction showing some of the outer wires with the voltage leads. Insets: Relative magnetization for different parts and the resulting schematic magnetoresistance trace for sweep up (solid line) and sweep down (dashed line). Bottom panel: Measured magnetoresistance in a sample with tunnel barriers at the constrictions. From 258.

Figure 3

(Color online) Top panel: Substitutional ${\mathrm{Mn}}_{\mathrm{Ga}}$ and interstitial ${\mathrm{Mn}}_I$ in GaAs. Bottom panel: Two $e_g$ $3d$ orbitals and three $t_{2g}$ $3d$ orbitals of Mn.

Figure 4

(Color online) STM imaging of a ${\mathrm{Mn}}_{\mathrm{Ga}}$ impurity. Top panel: Energy-band diagram for the negative (left) and positive (right) bias. Bottom panel: ${\mathrm{Mn}}_{\mathrm{Ga}}$ impurity in the ionized $A^{-}\left(d^5\right)$ (left) and the neutral $A^0(d^5+\text{hole})$ (right) state. From 345.

Figure 5

(Color online) Electron-picture cartoon: Splitting of the isolated Mn acceptor level (top panel) and of the top of the valence band in the many-Mn-atom system (bottom panel) due to $p\text{−}d$ hybridization.

Figure 6

(Color online) The spin-resolved total DOS of (GaMn)As and (GaMn)N for a Mn concentration of 6.25%. Calculations are performed within LSDA and $\mathrm{LDA}+U$ approaches. For (GaMn)N, both zinc-blende and wurzite structures are presented. The spin-up (spin-down) DOS is shown above (below) the abscissa axis. The total DOS is given per chemical unit cell of the semiconductor. From 266.

Figure 7

(Color online) As in Fig. 6 but for the partial Mn $3d$ DOS. In $\mathrm{LDA}+U$ the $p\text{−}d$ hybridization in (Ga,Mn)As is weak for both majority-band and minority-band states.

Figure 8

(Color online) The potential-energy surface for Mn adsorption on GaAs(001), plotted in a plane normal to the surface and containing the As surface dimer. The minimum-energy adsorption site is the subsurface interstitial site labeled $i$; the corresponding surface geometry is shown [light gray for As, dark gray for Ga, yellow (white) for Mn]. Typical adsorption pathways funnel Mn adatoms to this interstitial site (heavy curves) or to a cave site $c$ (light curves). Inset: Binding energy of a Mn adatom centered on the As dimer; for comparison, results are also shown for a Ga adatom. When additional As is deposited, the metastable Mn site $s$ becomes more favorable and leads to partial incorporation of substitutional Mn. From 88.

Figure 9

(Color online) Theoretical LMTO formation energy $E\left({\mathrm{Mn}}_{\mathrm{Ga}}\right)$ of substitutional ${\mathrm{Mn}}_{\mathrm{Ga}}$ in ${\mathrm{Ga}}_{0.96}{\mathrm{Mn}}_{0.04}\mathrm{As}$ as a function of the concentration of various donors. From 210.

Figure 10

(Color online) Theoretical LMTO formation energy $E\left({\mathrm{Mn}}_I\right)$ of the interstitial Mn impurity in ${\mathrm{Ga}}_{0.96}{\mathrm{Mn}}_{0.04}\mathrm{As}$ a function of the concentration of various donors. From 210.

Figure 11

(Color online) Theoretical LMTO variations $\Delta E$ of the formation energies of the interstitial Mn impurity and ${\mathrm{As}}_{\mathrm{Ga}}$ antisite defect in ${\mathrm{Ga}}_{1-x}{\mathrm{Mn}}_x\mathrm{As}$ a function of the concentration $x$ of Mn atoms substituted in the Ga sublattice. Variations are measured from the values for the reference material with 4% of ${\mathrm{Mn}}_{\mathrm{Ga}}$ From 210.

Figure 12

(Color online) Main panel: Theoretical TBA equilibrium partial concentrations of substitutional ${\mathrm{Mn}}_{\mathrm{Ga}}$ (red upper line line) and interstitial ${\mathrm{Mn}}_I$ (blue lower line) impurities. Triangles and circles represent experimental ${\mathrm{Mn}}_{\mathrm{Ga}}$ and ${\mathrm{Mn}}_I$ partial concentrations, respectively. Inset: Formation energies of ${\mathrm{Mn}}_{\mathrm{Ga}}$ and ${\mathrm{Mn}}_I$ as a function of total Mn concentration. From 146.

Figure 13

(Color online) Theoretical lattice constant of (Ga, Mn)As as a function of the concentration of the impurities ${\mathrm{Mn}}_{\mathrm{Ga}}$ (circles), ${\mathrm{Mn}}_I$ (full triangles), and ${\mathrm{As}}_{\mathrm{Ga}}$ (empty triangles). From 208.

Figure 14

Relaxed lattice constants for $50\text{−}\mathrm{nm}$-thick (Ga,Mn)As films before annealing (squares) and after annealing (triangles) as a function of substitutional ${\mathrm{Mn}}_{\mathrm{Ga}}$ content. Inset: Change of the relaxed lattice constant as a function of the change in the fraction of interstitial Mn due to the out-diffusion of interstitial Mn during annealing (i.e., ${\mathrm{Mn}}_{I,i}-{\mathrm{Mn}}_{I,f}$). From 356.

Figure 15

(Color online) Ferromagnetic transition temperatures of (Ga,Mn)As calculated within the effective Hamiltonian and virtual crystal approximation: mean field [thick (black) line], Stoner enhancement of $T_c$ [thin (red) line], and $T_c$ after correcting for correlated Mn orientation fluctuations using a spin-wave approximation [(blue) symbols]. From 146.

Figure 16

(Color online) $T_c$ calculations within the microscopic TBA CPA model: $T_c$ vs hole density (left panel); $T_c$ vs number of holes per ${\mathrm{Mn}}_{\mathrm{Ga}}$ (right panel). The overall theoretical $T_c$ trend is highlighted in gray. From 146.

Figure 17

(Color online) LDA/CPA (red filled symbols) and $\mathrm{LDA}+U∕\mathrm{CPA}$ (blue open symbols) calculations of the Curie temperature as a function of Mn doping in uncompensated (one hole per Mn) (Ga,Mn)As and (Ga,Mn)N. From 266.

Figure 18

Comparison of ab initio Curie temperatures of ${\mathrm{Ga}}_{1-x}{\mathrm{Mn}}_x\mathrm{As}$ (diamonds) and ${\mathrm{Ga}}_{1-x}{\mathrm{Mn}}_x\mathrm{N}$ (squares) obtained with the spin-wave approach (solid line, filled symbols), the virtual-crystal-approximation and RPA (dashed line, filled symbols), the mean-field approximation (dotted line, filled symbols), and Monte Carlo simulations (dash-dotted line, open symbols). From 124.

Figure 19

(Color online) Temperature dependence of remanent magnetization and inverse paramagnetic susceptibility for ${\mathrm{Ga}}_{0.91}{\mathrm{Mn}}_{0.09}\mathrm{As}$ sample with $T_c=173\mathrm{K}$. Inset: Hysteresis loop for the same sample at $172\mathrm{K}$. From 334.

Figure 20

(Color online) Experimental $T_c∕x_{\mathrm{eff}}$ vs hole density relative to effective concentration of ${\mathrm{Mn}}_{\mathrm{Ga}}$ moments. Deviations from linear dependence on $x_{\mathrm{eff}}$ are seen only for high compensations $(1-p{a}_{\mathrm{lc}}^3∕4x_{\mathrm{eff}}=1-p∕N_{\mathrm{Mn}}^{\mathrm{eff}}>40\%)$ in agreement with theory. For weakly compensated samples $T_c$ shows no sign of saturation with increasing $x_{\mathrm{eff}}$. Theoretical (gray) $T_c$ trend from Fig. 16 is plotted for comparison. From 146.

Figure 21

(Color online) Cartoon of Zeeman coupling with an external magnetic field assuming $g>0$, in the electron and hole pictures. In the ferromagnetic state the valence band is spin split at zero magnetic field (solid lines). The majority bands in both electron and hole pictures move up in energy when the field is applied (dashed lines), resulting in a negative band contribution to the magnetization.

Figure 22

(Color online) Main panel: Mean-field total magnetization per Mn as a function of hole density relative to the local-Mn-moment density. Lower inset: Hole contribution to magnetization (see definition in the text) per Mn. Upper inset: Hole contribution to magnetization per hole. Results are obtained using the TBA CPA model. From 143.

Figure 23

(Color online) Integrated total and $d$- and $sp$-projected magnetizations per Mn as a function of hole density relative to the local Mn-moment density. See text for definition of $m_{\mathrm{TBA}}^{\mathrm{int}}$. From 143.

Figure 24

Magnetization of the hole liquid (squares) in ${\mathrm{Ga}}_{1-x}{\mathrm{Mn}}_x\mathrm{As}$ computed as a function of the hole concentration for the spin-splitting parameter $B_G=-30\mathrm{meV}$ (corresponding to $x=0.05$ at $T=0$). Crosses show the spin and orbital contributions to the hole magnetization. Results were obtained using the six-band Kohn-Luttinger parametrization of the valence band and the kinetic-exchange model. From 69.

Figure 25

(Color online) Mean-field kinetic-energy contribution to the hole magnetization per Mn as a function of hole density. Results were obtained using the six-band Kohn-Luttinger parametrization of the valence band and the kinetic-exchange model. From 143.

Figure 26

(Color online) SQUID magnetization per effective density of uncompensated ${\mathrm{Mn}}_{\mathrm{Ga}}$ local moments in as-grown (open symbols) and annealed (filled symbols) (Ga,Mn)As materials plotted as a function of hole density per effective density of uncompensated ${\mathrm{Mn}}_{\mathrm{Ga}}$ local moments. From 143.

Figure 27

Computed minimum magnetic field $H_{un}$ (divided by $M$) necessary to align magnetization $M$ along the hard axis for (a) compressive and (b) tensile biaxial strain in a ${\mathrm{Ga}}_{1-x}{\mathrm{Mn}}_x\mathrm{As}$ film for various values of the spin-splitting parameter $B_G$. (a) The easy axis is along the [001] direction and in the (001) plane at low and high hole concentrations for compressive strain, respectively. (b). The opposite behavior is observed for tensile strain. The symbol $\left[100\right]\to \left[001\right]$ means that the easy axis is along [100], so that $H_{un}$ is applied along [001] ($B_G=-30\mathrm{meV}$ corresponds to the saturation value of $M$ for ${\mathrm{Ga}}_{0.95}{\mathrm{Mn}}_{0.05}\mathrm{As}$). From 69

Figure 28

(Color online) Theoretical spin-stiffness parameter in ${\mathrm{Ga}}_{0.95}{\mathrm{Mn}}_{0.05}\mathrm{As}$ as a function of the hole density calculated using the Holstein-Primakoff representation of fluctuating Mn local spins and the KL kinetic-exchange description of hole bands.

Figure 29

(Color online) Theoretical Gilbert damping coefficient in (Ga,Mn)As. From 298.

Figure 30

Experimental peak-to-peak ferromagnetic resonance linewidth and corresponding Gilbert coefficient in as-grown (filled symbols) and annealed (open symbols) ${\mathrm{Ga}}_{0.92}{\mathrm{Mn}}_{0.08}\mathrm{As}$ samples measured as a function of temperature for [001] and [110] dc magnetic-field orientations (main plot) and as a function of the field angle at $4\mathrm{K}$ (inset). From 298.

Figure 31

Mean-field LSDA CPA Curie temperatures vs $T=0$ dc conductivities calculated for $\left({\mathrm{Ga}}_{0.95y}{\mathrm{Mn}}_{0.05}{\mathrm{As}}_y\right)\mathrm{As}$ alloys with varying As antisite content $y$ (full dots) and the experimental values obtained in as-grown and annealed (Ga,Mn)As thin films with the corresponding level of compensation (42) (open squares). Note that these are illustrative calculations since in the experiment the change in the unintentional impurity concentration upon annealing is mostly due to the out-diffusion of interstitial Mn. From 326.

Figure 32

Curie temperature vs room-temperature conductivity for (Ga,Mn)As films with $x=0.08$ (squares), $x=0.06$ (circles), and $x=0.05$ (triangles). Straight lines are to guide the eye. Data points correspond to samples annealed at different temperatures and times. Note that similar trends are obtained for Curie temperatures plotted vs low-temperature conductivities, whose values are systematically larger by $\sim 20–40\%$ compared to room-temperature conductivities. From 82.

Figure 33

(Color online) Schematic illustration of AMR configurations: electrical current $\mathbf{I}$ flows along the $\widehat{\mathbf{x}}$ direction and magnetization is aligned by an external magnetic field in either of the two in-plane directions $\mathbf{M}\Vert \mathbf{I}$ $(\mathbf{M}\Vert \widehat{\mathbf{x}})$ and $\mathbf{M}\perp \mathbf{I}$ $(\mathbf{M}\Vert \widehat{\mathbf{y}})$, or in the out-of-plane direction perpendicular to the current, $\mathbf{M}\perp \text{plane}$ $(\mathbf{M}\Vert \widehat{\mathbf{z}})$. The nonzero coefficient ${\mathit{AMR}}_{ip}$, defined as the relative difference in resistivities in the in-plane $\mathbf{M}\perp \mathbf{I}$ and $\mathbf{M}\Vert \mathbf{I}$ configurations, originates from spin-orbit coupling. In strained (Ga,Mn)As materials, ${\mathit{AMR}}_{ip}$ is in general not equal to ${\mathit{AMR}}_{op}$, defined as the relative difference in resistivities in the $\mathbf{M}\perp \text{plane}$ and $\mathbf{M}\Vert \mathbf{I}$ configurations. From 214.

Figure 34

Theoretical AMR effects in (Ga,Mn)As obtained from the KL kinetic-exchange model and Boltzmann transport theory. Curves correspond to total Mn doping $x_1$ and compensation due to As antisites or total Mn doping $x_2$ and compensation due to Mn interstitials. Main panel: Out-of-plane AMR coefficient as a function of strain for several Mn dopings. Lower inset: Out-of-plane and in-plane AMR coefficients as a function of strain. Upper inset: out-of-plane AMR coefficient as a function of the hole density. From 139.

Figure 35

(Color online) Upper and middle panels: Experimental field-induced changes in the resistance of ${\mathrm{Ga}}_{0.95}{\mathrm{Mn}}_{0.05}\mathrm{As}$ grown on a GaAs substrate under compressive strain for current along the [110] crystal direction (upper panel) and [100] crystal direction (middle panel) for three different orientations of the magnetization as indicated in Fig. 33. Lower panel: Experimental magnetoresistance curves in ${\mathrm{Ga}}_{0.957}{\mathrm{Mn}}_{0.043}\mathrm{As}$ grown on (In,Ga)As substrate under tensile strain (current along the [110] crystal direction). From 214.

Figure 36

Comparison between experimental and theoretical anomalous Hall conductivities. From 145.

Figure 37

(Color online) Theoretical Hall factors for $p=0.8{\mathrm{nm}}^{-3}$; $\hslash ∕\tau =50$ (dashed lines), and $150\mathrm{meV}$ (solid lines). Top panels: Only intraband transitions are taken into account. Bottom panels: Intraband and interband transitions are taken into account. Left panels: GaAs $(x=0)$; zero spin-orbit coupling (red upper lines) and ${\Delta }_{\mathrm{S}\mathrm{O}}=341\mathrm{meV}$ (blue lower lines). Right panels: (Ga,Mn)As with ${\mathrm{Mn}}_{\mathrm{Ga}}$ concentration 4% (green upper lines) and 8% (brown lower lines). ${\rho }_{xy}(B=0)=0$ in all panels except for the bottom left panel where ${\rho }_{xy}(B=0)\ne 0$ due to the anomalous Hall effect. From 146.

Figure 38

Experimental resistivity of ${\mathrm{Ga}}_{1-x}{\mathrm{Mn}}_x\mathrm{As}$ for $x=8\%$ vs temperature for various annealing times. From 252.

Figure 39

KL kinetic-exchange model resistivity, normalized with respect to the value above $T_c$, vs $T$ for a Mn doping of 5% and hole densities $p=0.2$ and $0.6{\mathrm{nm}}^{-3}$. From 192.

Figure 40

(Color online) Temperature dependence of the resistivity for a $200\text{−}\mathrm{nm}$-thick film of ${\mathrm{Ga}}_{1-x}{\mathrm{Mn}}_x\mathrm{As}$ with $x=0.053$ in the high-temperature paramagnetic region. Solid squares and open circles show experimental data and the fit using Eq. (42), respectively. From 241.

Figure 41

(Color online). Schematic diagrams of electron excitations across the band gap induced by circularly polarized light absorption in undoped (left) and $p$-type (right) DMSs with spin-split bands. Only the heavy-hole band is shown. For a given circular polarization of the absorbed light, optical selection rules allow transitions between the heavy-hole valence band and the conduction band with one spin orientation only and therefore make it possible to spectrally resolve the band splitting. The corresponding MCD signal can change sign in the doped system due to the Moss-Burnstein effect, as illustrated.

Figure 42

Top panel: Theoretical absorption edge for two circular polarizations in $p\text{−}(\mathrm{Ga},\mathrm{Mn})\mathrm{As}$ computed for spin-splitting parameter $B_G=-10\mathrm{meV}$ (corresponding to $T=0$ mean field from local Mn moments of density $x=1.7\%$) and hole concentration $3.5\times {10}^{20}{\mathrm{cm}}^{-3}$. Inset: The Fermi sea of the holes shown how it reverses the relative positions of the edges corresponding to ${\sigma }^{+}$ and ${\sigma }^{-}$ polarizations in agreement with experimental findings. Bottom panel: Spectral dependence of magnetic circular dichroism in the optical range in (Ga,Mn)As computed for hole concentration $3.5\times {10}^{20}{\mathrm{cm}}^{-3}$ and various spin-splitting parameters $B_G$. The magnitudes of MCD at given $B_G$ are normalized by its value at $1.78\mathrm{eV}$. From 69.

Figure 43

Experimental MCD spectra of (a) undoped semi-insulating GaAs substrate and (b), (c) epitaxial ${\mathrm{Ga}}_{1-x}{\mathrm{Mn}}_x\mathrm{As}$ films at $T=5\mathrm{K}$ and $B=1\mathrm{T}$. The spectrum of GaAs is magnified ten times because the signal is weaker than that of ${\mathrm{Ga}}_{1-x}{\mathrm{Mn}}_x\mathrm{As}$. From 12.

Figure 44

(Color online) Schematic diagram illustrating absorption due to electron excitations to the impurity band (left panel) and due to intra-valence-band excitations (right panel).

Figure 45

Impurity-band model calculations of $\sigma \left(\omega \right)$ vs $\omega$ for a $10\times 10$ periodic system with 26 spins $(x\sim 0.25)$, $J∕t=2.5$, $p=0.3$, and for different temperatures, as indicated. Inset: Drude weight $D$ vs temperature $T$ (8).

Figure 46

KL kinetic-exchange model calculations of infrared conductivity $\mathrm{Re}\left[\sigma \left(\omega \right)\right]$ and absorption coefficient $\alpha \left(\omega \right)$ for carrier densities from $p=0.2\text{to}0.8{\mathrm{nm}}^{-3}$ in the direction indicated by the arrow, for ${\mathrm{Ga}}_{0.95}{\mathrm{Mn}}_{0.05}\mathrm{As}$ (299). Disorder is treated within the first-order Born approximation. Dot-dashed line is the experimental absorption curve for a sample with 4% Mn doping. From 125.

Figure 47

Infrared absorption and conductivity of a metallic (Ga,Mn)As computed using the KL kinetic-exchange model and exact-diagonalization technique in a finite size disordered system. Here $p=0.33{\mathrm{nm}}^{-3}$ and $n_{\mathrm{Mn}}=1{\mathrm{nm}}^{-3}$ $(x\approx 4.5\%)$. From 346.

Figure 48

Top panel: Real part of the conductivity derived from measured transmission spectra for a paramagnetic $x=1.7\%$ sample (dashed line), a ferromagnetic $x=5.2\%$ sample (thick solid line), and the LT GaAs film (thin black line). Bottom panel: Temperature dependence of conductivity for $x=5.2\%$ sample on a logarithmic scale for $T>T_c$ and $T<T_c$. From 293.

Figure 49

(Color online) Left panel: A schematic qualitative diagram illustrating the requirements for ferromagnetic ordering in systems with coupled local and itinerant moments. The $y$ axis represents the strength of the exchange coupling relative to the itinerant-system Fermi energy and the $x$ axis the ratio between carrier and local-moment densities. In the weak-coupling regime the interaction between local moments is described by RKKY theory in which polarization of band electrons (holes) due to the interaction with the local moment at one site is propagated to neighboring sites. The mechanism corresponds to out-of-phase Friedel oscillations of carriers scattering off an attractive impurity potential for one carrier spin orientation (maximum density at the impurity site), and a repulsive potential for the opposite carrier spin polarization (minimum density at the impurity site). With increasing carrier density, the ferromagnetic RKKY state gradually approaches a regime of frustrated RKKY interactions. The RKKY picture does not apply to strong couplings. Here the tendency to order ferromagnetically is weakened by correlated (flip-flop) quantum fluctuations of the interacting local and itinerant moments, making the moment disappear eventually. This is the so-called Kondo singlet regime. Right panel: The robustness of the ferromagnetic state is viewed from the ferromagnetic low-temperature side. With increasing hole density, the ferromagnetically stiff mean-field state will start to suffer from the occurrence of lower-energy spin wave excitations until finally the energy of spin waves will become negative, signaling the instability of the ferromagnetic state. In the strong-coupling regime, hole spins will tend to align locally and instantaneously, in opposition to the fluctuating Mn moment orientations. Since the hole Fermi energy is relatively small in this regime, the kinetic-energy cost of these fluctuations is small, resulting in soft spin-wave excitations for the magnetic system. The long-range ferromagnetic order disappears eventually at temperatures much smaller than the mean-field $T_c$, while short-range order may still exist above the Curie temperature.