Spin-dependent phenomena and device concepts explored in (Ga,Mn)As

Over the past two decades, the research of (Ga,Mn)As has led to a deeper understanding of relativistic spin-dependent phenomena in magnetic systems. It has also led to discoveries of new effects and demonstrations of unprecedented functionalities of experimental spintronic devices with general applicability to a wide range of materials. This is a review of the basic material properties that make (Ga,Mn)As a favorable test-bed system for spintronics research and a discussion of contributions of (Ga,Mn)As studies in the general context of the spin-dependent phenomena and device concepts. Special focus is on the spin-orbit coupling induced effects and the reviewed topics include the interaction of spin with electrical current, light, and heat.

Figure 1

Schematic illustration of the long-range Coulomb and the two short-range potentials each contributing $\sim 30\mathrm{meV}$ to the binding energy of the ${\mathrm{Mn}}_{\mathrm{Ga}}$ acceptor. From [174].

Figure 2

TBA-Anderson density of states of ${\mathrm{Ga}}_{0.9}{\mathrm{Mn}}_{0.1}\mathrm{As}$ and its orbital composition. The position of the Fermi energy is indicated by a vertical line. From [174].

Figure 3

Density of states for (Ga,Mn)As mixed crystals with various content of Mn obtained in the $\mathrm{G}\mathrm{G}\mathrm{A}+U$ theory. The lines represent the total DOS while the shaded areas show the partial density of Mn $d$ states. From [174].

Figure 4

(a) Temperature dependent remanent magnetization, resistivity, and temperature derivative of the resistivity of a nominally 11% Mn-doped (Ga,Mn)As. From [131]. (b) Magnetic contribution to the specific heat of a 2.6% Mn-doped (Ga,Mn)As. Adapted from [341]. Upper inset: Temperature derivative of resistance and a multiple of the specific heat plotted against temperature for Ni. From [120]. Lower inset: Schematic diagram of the spin-spin correlation function in low and large $k$-vector limits. From [85].

Figure 5

Field and temperature dependences of resistance in ${\mathrm{Ga}}_{0.95}{\mathrm{Mn}}_{0.05}\mathrm{As}/\mathrm{GaAs}$ (compressive strain, upper panel) and in tensile strained ${\mathrm{Ga}}_{0.957}{\mathrm{Mn}}_{0.043}\mathrm{As}/(\mathrm{In},\mathrm{Ga})\mathrm{As}$ (lower panel) for magnetic field perpendicular to the film plane. Starting from the top, subsequent curves at $B=0$ correspond to temperatures in K: 70, 60, 80, 50, 90, 40, 100, 30, 125, 20, 2, 5, 10, 150, 200, 300 (upper panel) and to 50, 60, 40, 70, 30, 80, 90, 20, 100, 2, 10, 5, 125, 150, 200, 300 (lower panel). From [177].

Figure 6

(a) Conductance fluctuations for three wires of different length $L$. For the shortest wire the amplitude of the conductance fluctuations is about $e^2/h$, expected for conductors with all spatial dimensions smaller or comparable to $L_{\mathrm{\Phi }}$. The inset shows an electron micrograph of a 20 nm wide wire with a potential probe separation of $\sim 100\mathrm{nm}$. (b) Conductance vs magnetic field of the 200 nm wire for different temperatures between 20 mK and 1 K. From [309].

Figure 7

(a) Electron micrograph of a (Ga,Mn)As ring sample with a diameter of $\sim 100\mathrm{nm}$. (b) Comparison of the magnetoconductance trace of the ring sample with the conductance of a wire of comparable length and 20 nm width. (c) The corresponding Fourier transform taken from the conductance of the ring and wire. The region where ABE oscillations are expected is highlighted. From [309].

Figure 8

(a) Magnetization $M$, temperature derivative of the resistivity normalized to the peak value $(d\rho /dT{)}^{*}$, and resistivity $\rho (T)$ of an optimized 20 nm thick epilayer with 7% nominal Mn doping. (b), (c) Frequency dependence of the damping factor and field dependence of the SWR frequencies of the same sample. (d)–(f) Same as (a)–(c) for a material differing by having only one of the synthesis parameters not optimized (epilayer thickness of 500 nm is too large). From [202].

Figure 9

(a) Optimal growth temperature $T_G$ as a function of the nominal Mn doping $x$. (b) Dependence of the Curie temperature $T_c$ on the annealing time for two different annealing temperatures $T_A$ in a 15 nm thick (Ga,Mn)As epilayer with 13% nominal Mn doping grown at optimal $T_G$. (c), (d) $\rho (T)$ and $(d\rho /dT{)}^{*}$ in the $x=13\%$ epilayer grown at optimal $T_G$ in the as-grown state, for optimal $T_A$ and annealing time 0.5 h, and for optimal $T_A$ and optimal annealing time of 8 h. (e), (f) Same as (c), (d) for a $x=13\%$ epilayer grown at 10° below the optimal $T_G$; $(d\rho /dT{)}^{*}$ is not plotted for the as-grown insulating and paramagnetic sample. From [202].

Figure 10

Curie temperature $T_c$ vs hole density $p$ normalized to $x_{\text{eff}}$ ($N_{\text{eff}}=4x_{\text{eff}}/a^3$) representing the concentration of Mn magnetic moments which contribute to the magnetic order. The squares correspond to samples from [319] prepared under optimized growth conditions where the hole density $p$ is obtained from high-field Hall measurements. The circles correspond to samples from [72], where $p$ is obtained from ion channeling measurements. The stars correspond to samples from [242] prepared under optimized growth conditions, where $p$ is obtained from ion channeling measurements. The line is the prediction of the microscopic calculation of [128].

Figure 11

(a) Temperature dependence of the conductivity $\sigma (T)$ of optimized (Ga,Mn)As epilayers with depicted nominal Mn doping. Dashed lines indicate the activated parts of $\sigma (T)$ of the insulating paramagnetic (Ga,Mn)As with 0.05% Mn doping, corresponding to the Mn acceptor level and the band gap, respectively. (b) Conductivity at 4 K as a function of the nominal Mn doping. The open symbol corresponds to a paramagnetic sample. (c) Sharp Curie point singularities in the temperature derivative of the resistivity in the series of optimized ferromagnetic (Ga,Mn)As epilayers with metallic conduction. (d)–(f) Hole density $p$, magnetization $M$ and corresponding Mn-moment density $N_{\mathrm{M}\mathrm{n}}$, and Curie temperature $T_c$ as a function of the nominal Mn doping in the series of optimized (Ga,Mn)As epilayers. From [202].

Figure 12

(a) Dependence of magnetic anisotropy constants on nominal Mn doping. (b) Dependence of the Gilbert damping constant $\alpha$ and the spin-stiffness constant $D$ on nominal Mn doping. Measurements were performed at 15 K. From [202].

Figure 13

Schematic comparison of Ohmic Mott (GMR) and Dirac (AMR) devices and tunneling Mott (TMR) and Dirac (TAMR) devices. At the bottom a Dirac device is shown based on the chemical potential anisotropy which has no immediate counterpart in Mott spintronics. The short thick arrows show the magnetization direction, and the long thin (thick) lines with arrows depict low (high) electrical current. The broken current lines illustrate stronger scattering. Nonrelativistic (relativistic) densities of states are schematically illustrated in the TMR (TAMR, chemical potential AMR) panels.

Figure 14

Schematic illustrations of the skew (Mott) scattering AHE and SHE (top panels) and the intrinsic AHE and SHE due to the anomalous transverse component of the spin-dependent velocity originating from the spin-orbit coupled band structure in a clean crystal (bottom panels). In the AHE, an electrical current driven through a ferromagnetic conductor $j_e^s$ is spin polarized and the spin-dependent transverse deflection of electrons produces a transverse voltage. In the SHE, an unpolarized electrical current $j_e$ is driven through a normal conductor and the spin-dependent transverse deflection of electrons produces a transverse spin current. Opposite spins accumulate at opposite edges but unlike the AHE the transverse voltage remains zero.

Figure 15

Measured longitudinal and transverse in-plane AMR curves at external fields smaller than the saturation field (0.1 and 0.25 T) and larger than the saturation field (0.7 T). The solid lines represent fits to the experimental data. From [159].

Figure 16

Left panel: Cross section (parallel to the $k_x,k_y$ plane) of the 3D radial spin texture belonging to the two heavy-hole bands of (Ga,Mn)As in a spherical approximation. Right top panel: Nonzero backscattering elements when neglecting the nonmagnetic part of the Mn-impurity potential. The corresponding AMR has a positive sign. The purely magnetic Mn impurity is illustrated by a dot with an arrow. Right bottom panel: Nonzero backscattering elements for the same strengths of the nonmagnetic and magnetic parts of the Mn-impurity potential. The corresponding AMR has a negative sign. The combined ionized acceptor and magnetic nature of the Mn impurity is illustrated by a dot with a negative sign and an arrow. (Electrical current $\mathbf{I}\parallel x$.) From [295].

Figure 17

(a) Device schematic showing the contact geometry and the crystallographic directions. (b) Hysteretic magnetoresistance curves acquired at 4.2 K with 1 mV bias by sweeping the magnetic field along the 0°, 50°, and 55° directions. Spin-valve-like features of varying widths and signs are clearly visible, delimited by two switching events labeled $H_{c1}$ and $H_{c2}$. The magnetoresistance is independent of the bias direction or amplitudes up to 1 meV. (c) TAMR along 30° for temperatures from 1.6 to 20 K, showing a change of sign of the signal. The curves are vertically offset for clarity. From [99].

Figure 18

Calculated (a) TMR values and (b) TAMR values represented as a function of the Fermi and spin splitting energy for a 6 nm (In,Ga)As barrier with a band offset of 450 meV. White lines represent the four bands at the center of the Brillouin zone. Gray lines indicate the Fermi energy for different hole concentrations. From [79].

Figure 19

(a) TMR measurements as a function of the magnetic field at 1 mV and 3 K for a $128\mu {\mathrm{m}}^2$ junction. (b) TMR measurements as a function of resistance area (RA) product at 3 K for four (un)annealed junctions. (c) TMR at 1 mV as a function of the temperature before and after annealing. (d) TAMR measurements as a function of the magnetic field at 1 mV and 3 K. From [79].

Figure 20

Top panel: Schematics of a capacitor with an ultrathin (3.5 nm) (Ga,Mn)As layer. Bottom panel: Experimental temperature dependence of the spontaneous moment for selected values of gate voltage. Temperatures at which the moment disappears define the Curie temperature $T_c$, as marked by arrows. From [256].

Figure 21

(a) Electron micrograph of the central part of a (Ga,Mn)As SET device. (b) Polar plot of the source-drain resistance $R_{\mathrm{sd}}$ at 1.6 K showing the strong anisotropy as a function of the magnetization direction. From [259].

Figure 22

(a) Resistance $R_S=V_S/I$ of the unstructured bar (see schematic diagram) vs up and down sweeps of in-plane magnetic field parallel and perpendicular to the current direction. (b) $R_S$ vs the angle between the current direction and an applied in-plane magnetic field of 5 T, at which $\mathbf{M}\parallel \mathbf{B}$. (c) Channel resistance $R_C$ vs gate voltage and down sweep of the magnetic field parallel to current. (d) $R_C$ vs gate voltage and the angle between the current direction and an applied in-plane magnetic field of 5 T. From [330].

Figure 23

(a) Two opposite transistor characteristics in a gate-voltage range $V$ ($V_{G0}$) to 1.04 V ($V_{G1}$) for two different magnetization orientations ${\mathbf{M}}_0$ and ${\mathbf{M}}_1$; corresponding Coulomb blockade oscillations in a larger range of $V_G=0.6$ to 1.15 V are shown in the inset. Switching between low-resistance (1) and high-resistance (0) states can be performed electrically or magnetically. (b) Hysteretic magnetoresistance at constant gate voltage $V_{G1}$ illustrating the nonvolatile memory effect in the magnetic mode. (c) Illustration of integrated transistor (electric mode) and permanent storage (magnetic mode) functions in a single nanoscale element. (d) The transistor characteristic for $\mathbf{M}={\mathbf{M}}_1$ is reminiscent of an $n$-type field-effect transistor and is inverted (reminiscent of a $p$-type field-effect transistor) for $\mathbf{M}={\mathbf{M}}_0$; the inversion can also be realized in the nonvolatile magnetic mode. From [327].

Figure 24

(a) Schematic showing the SET channel separated by ${\mathrm{AlO}}_x$ dielectric from the ferromagnetic (Ga,Mn)As back gate (BG). The SET comprises Al leads and island, and ${\mathrm{AlO}}_x$ tunnel barriers. (b) Coulomb oscillations for the SET on ${\mathrm{Ga}}_{0.97}{\mathrm{Mn}}_{0.03}\mathrm{As}$ for two different polar angles $\mathrm{\Phi }$ of the magnetization. (c) Magneto-Coulomb oscillations shown by the same SET by varying the angle of magnetization for two different gate voltages. (d) Magnetization vector with respect to (Ga,Mn)As crystal axes. (e) Schematic explaining the spin-gating phenomenon: reorientation of the magnetization from ${\mathbf{M}}_1$ to ${\mathbf{M}}_2$ causes a change in the chemical potential of the (Ga,Mn)As BG. This causes charge to flow onto the back gate from the reservoir (Res.). The net effect is to alter the charge on the back gate and therefore the SET conductance. The externally applied electrochemical potential on the gate ${\mu }_{ec}=q{V}_g$ is held constant. From [49].

Figure 25

(a) Layout of the device showing the $5\mu \mathrm{m}$ mesa and step for DW pinning in perpendicular magnetic anisotropy (Ga,Mn)As film. (b) $7\mu \mathrm{m}$ wide magneto-optical images with a $5\mu \mathrm{m}$ mesa in the center show that DW moves in the opposite direction to the current independent of the initial magnetization orientation, and that DW displacement is proportional to pulse duration (c). The lowest panel in (b) shows destruction of the ferromagnetic phase by Joule heating. From [333].

Figure 26

(a) Illustration of the steady-state nonequilibrium carrier spin polarization $\mathbf{s}$ and corresponding adiabatic STT (${\mathrm{STT}}_{\mathrm{AD}}$) acting on magnetization $\mathbf{m}$ in the ${\tau }_s\gg {\tau }_{\text{ex}}$ limit (left) and nonadiabatic STT (${\mathrm{STT}}_{\mathrm{NA}}$) in the ${\tau }_s\ll {\tau }_{\text{ex}}$ limit (right). (b) Schematic diagram of the predicted DW velocity as a function of the driving current in the presence of adiabatic and nonadiabatic STTs and $\beta /\alpha <1$ or $\beta /\alpha >1$, and of the predicted shift of the WB threshold current $j_{\mathrm{WB}}$ for two values of the in-plane magnetocrystalline constant $K_{u,1}<K_{u,2}$, controlled in situ by a piezostressor. (c) Measured DW velocity vs driving current density at piezovoltages $-200$ or $+200\mathrm{V}$, strengthening or weakening the [$1\overline{1}0$] in-plane easy axis, respectively. Open symbols correspond to the [$1\overline{1}0$]-oriented microbar with less internally stable Néel DW and filled symbols to the [110]-oriented microbar with more internally stable Bloch DW. The character of the measured data, including the shift of the WB threshold current, implies STTs with $\beta /\alpha <1$. (d) $\mathrm{\Delta }{v}_{\mathrm{DW}}=v_{\mathrm{DW}}(+200\mathrm{V})-v_{\mathrm{DW}}(-200\mathrm{V})$ vs current density illustrates the piezoelectric control of the DW mobility achieved starting from lower currents in the [$1\overline{1}0$]-oriented microbar with less internally stable DW. From [60].

Figure 27

(a) Atomic force micrograph of the studied sample with eight nonmagnetic metal contacts. (b) Diagram of device orientation with respect to crystallographic axes, with easy and hard magnetization axes marked with dashed and dot-dashed lines, respectively. Measured directions of the $H_{\text{eff}}$ field are shown for different current directions. Orientation of the effective SOT field with respect to the current direction for (c) Dresselhaus and (d) Rashba spin-orbit interactions. From [39].

Figure 28

(a), (b) Transverse anisotropic magnetoresistance $R_{xy}$ as a function of external field direction ${\phi }_H$ for $H=10\mathrm{mT}$ and current $I=\pm 0.7\mathrm{mA}$. The angles ${\phi }_H^{(i)}$ mark magnetization switchings. (c) Magnetization switches between the [010] and [$\overline{1}00$] directions when alternating $I=\pm 1\mathrm{mA}$ current pulses are applied with the current $\mathbf{I}\parallel [1\overline{1}0]$. The pulses have 100 ms duration and are shown schematically above the data curve. $R_{xy}$ is measured with $I=10\mu \mathrm{A}$. Adapted from [39].

Figure 29

Left panel: Rashba spin texture in equilibrium with zero net spin density. Right panel: Nonequilibrium redistribution of eigenstates in applied electric field resulting in a nonzero spin density due to broken inversion symmetry of the spin texture.

Figure 30

(a) Rashba and Dresselhaus spin textures. (b) For the case of a Rashba-like symmetry, the out-of-plane nonequilibrium carrier spin density that generates the intrinsic antidamping SOT has a maximum for $\mathbf{E}$ (anti)parallel to $\mathbf{M}$. In this configuration the equilibrium effective field $B_{\text{eff}}^{\text{eq}}$ and the additional field $\mathrm{\Delta }{B}_{\text{eff}}\perp \mathbf{M}$ due to the acceleration are perpendicular to each other causing all spins to tilt in the same out-of-plane direction. (c) For the case of a Rashba-like symmetry, the out-of-plane nonequilibrium carrier spin density is zero for $\mathbf{E}\perp \mathbf{M}$ since $B_{\text{eff}}^{\text{eq}}$ and $\mathrm{\Delta }{B}_{\text{eff}}$ are parallel to each other. (d) The analogous physical phenomena for zero magnetization induces a tilt of the spin out of the plane that has opposite sign for momenta pointing to the left or the right of the electric field, inducing in this way the intrinsic SHE. From [270], and [150].

Figure 31

Measured in-plane and out-of-plane SOT fields in (Ga.Mn)As. In-plane spin-orbit field and coefficients of the $\cos {\theta }_{\mathbf{M}-\mathbf{E}}$ and $\sin {\theta }_{\mathbf{M}-\mathbf{E}}$ fits to the angle dependence of the out-of-plane SOT field for our sample set. For the in-plane fields, a single sample in each microbar direction is shown (corresponding to the same samples that yield the out-of-plane data points). In the out-of-plane data, two samples are shown in each microbar direction. The symmetries expected for the antidamping SOT, on the basis of the theoretical model for the Dresselhaus term in the spin-orbit interaction, are shown by shading. All data are normalized to a current density of $10^5{\mathrm{A}\mathrm{cm}}^{-2}$. From [150].

Figure 32

(a) Infrared absorption of a series of optimized $(\mathrm{Ga},\mathrm{Mn})\mathrm{As}/\mathrm{GaAs}$ epilayers with nominal Mn doping $x=0.1\%–14\%$ plotted from the measured optical transmissions of the samples ($T$) and of the reference bare GaAs substrate ($T_0$). (b) Real part of the ac conductivity (lines) obtained from the measured complex conductivity in the terahertz range (points) and from fitting the complex conductivity in the infrared range to the measured transmissions. (c) Comparison of the infrared absorption in as-grown and annealed 4.5% doped sample. Inset: Comparison to GaAs:C samples with carbon doping densities $2\times 10^{19}$ and $2\times 10^{20}{\mathrm{cm}}^{-3}$. (d) Height of the (Ga,Mn)As midinfrared absorption peak as a function of Mn doping. (e) Position of the peak inferred from the transmission measurements and from the fitted ac conductivities. (f) Zero frequency conductivities obtained from dc transport measurements and from extrapolated optical ac conductivities measured in the terahertz range. From [131].

Figure 33

(a) ${\mathrm{Ga}}_{0.98}{\mathrm{Mn}}_{0.02}\mathrm{As}$ sample orientation with respect to the applied magnetic field and the four-step magnetization reversal process as consecutive 90° jumps (shown by dotted arrows) between the four easy directions (1)–(4). (b) Field dependences of the magnetic linear dichroism for different angles $\theta$ between the incident polarization and the [100] crystallographic direction, measured at a wavelength of $\lambda =815\mathrm{nm}$. (1)–(4) corresponds to the magnetization directions indicated in (a). $H_{12}$ and $H_{23}$ are the magnetic field values required for making jumps $(1)\to (2)$ and $(2)\to (3)$, respectively. (c) Spectra of the polar magneto-optical Kerr effect and magnetic linear dichroism (at $\theta =135^{\circ }$); (d) absorption spectrum at 5 K. (c), (d) Fabry-Pérot oscillations in the signal due to the finite buffer thickness have been removed numerically using a bandpass filter. From [138].

Figure 34

(a) Schematic diagram of the experimental setup for a detection of the magnetization precession induced in (Ga,Mn)As by an impact of the femtosecond laser pump pulse. Rotation of the polarization plane of reflected linearly polarized probe pulses is measured as a function of the time delay $\mathrm{\Delta }t$ between pump-and-probe pulses. The orientation of magnetization in the sample is described by the in-plane angle $\phi$ and the out-of-plane angle $\theta$. The external magnetic field $H_{\text{ext}}$ is applied in the sample plane at an angle ${\phi }_H$. (b) Dynamics of the magneto-optical signal induced by an impact of the pump pulse on the sample that was measured by probe pulses with different polarization orientations $\beta$. (c) Time evolution of the in-plane magnetization angle $\delta \phi (t)$, the out-of-plane angle $\delta \theta (t)$, and the magnitude $\delta M_s(t)/M_0$; the dotted line depicts the in-plane evolution of the easy-axis position around which the magnetization precesses. (d) Orientation of magnetization at different times after the impact of the pump pulse; the sample plane is represented by the vertical line and the equilibrium position of the easy axis is depicted by the gray spot. From [287].

Figure 35

Schematic illustration (top inset) of the optical spin-transfer torque induced by the rate $P$ of the photocarrier spin injection along the light propagation axis $\widehat{\mathbf{n}}$ (normal to the sample plane). The steady-state component of the nonequilibrium spin density $\mathbf{s}$ is oriented in the plane of the sample and perpendicular to the in-plane equilibrium magnetization vector. The (Ga,Mn)As sample is placed on a piezoelectric stressor (lower inset) which allows one to control the magnetic anisotropy in situ. Top panel: Precession of the magnetization induced in (Ga,Mn)As by ${\sigma }^{+}$ and ${\sigma }^{-}$ circularly polarized pump pulses. The points are the measured rotations of the polarization plane of the reflected linearly polarized probe pulse as a function of the time delay between pump-and-probe pulses. The experiment was performed on the (Ga,Mn)As sample attached to a piezostressor at applied bias $U=-150\mathrm{V}$ for which the ${\sigma }^{+}$ and ${\sigma }^{-}$ circularly polarized pump pulses produce signals with opposite sign corresponding to the opposite sign of the optical STT and no polarization-independent (${\sigma }^{+}+{\sigma }^{-}$) signal for this piezovoltage. Bottom panel: Same as the top panel for a piezovoltage $U=+150\mathrm{V}$. Here magnetization dynamics is excited by both the optical STT and a polarization-independent mechanism. Adapted from [201].

Figure 36

(a) Schematic illustration of the thermally excited precession of magnetization $\mathbf{M}(t)$ around the transient quasiequilibrium easy axis (EA). ${\mathbf{M}}_0$ is the magnetization vector aligned with in-plane equilibrium EA before the pump pulse. (b) Schematic illustration of optical SOT induced by the in-plane transverse component ${\mathbf{s}}_{\phi }$ of the nonequilibrium hole spin polarization. On the time scale of the magnetization precession, optical SOT causes an instantaneous tilt of the magnetization $\mathbf{M}(t_1)$ which allows one to clearly distinguish optical SOT from the considerably slower thermal excitation mechanism. The initial optical SOT induced tilt of magnetization can yield precession angles that are inaccessible in the thermally induced magnetization dynamics. (c) Definition of the coordinate system. (d) Time evolution of the magnetization vector measured in a (Ga,Mn)As material with nominal Mn doping $x=3\%$. The direction of the time increase is depicted by arrows. Magnetization tilt angles $\delta \phi$ and $\delta \theta$ are measured with respect to equilibrium EA. From [288].

Figure 37

Top eight panels: AHE and ANE loops at $T=10\mathrm{K}$ for different samples (left column) and at different temperatures for the 4% annealed sample (right column). In the left column, ANE data of 0.04*, 0.05*, and 0.07* samples were multiplied by $-1$ (* means that the sample was annealed). Bottom four panels: zero-field ANE coefficient. The solid lines are the best fits using Eq. (22) [or equivalently Eq. (25)], and the dashed curves are the best fits with $n=1$. Adapted from [230].

Figure 38

(a) Transverse AMT $S_{x,y}$ and transverse AMR $R_H$ in a 3.9% Mn-doped (Ga,Mn)As. (b) The relative orientation of $-\nabla T$, $\mathbf{M}$ and magnetic field $\mathbf{H}$. The four directions marked as I, II, III, and IV are easy directions of $\mathbf{M}$. (c) Angular dependence of the transverse AMT. (d) Comparison of $S_{xy}/S_{xx}$ and $R_H/R$, and sample magnetization $M$ measured by SQUID. Note that we use the terms transverse AMT and transverse AMR instead of the alternative planar Nernst effect and planar Hall effect ([231]) to clearly distinguish the fact that the effects shown here are the symmetric off-diagonal coefficients even in $\mathbf{M}$. From [231].

Figure 39

In magnetic tunnel junctions, thermal differences in the electron distributions and strong asymmetry in the spin-dependent tunneling channels are depicted. $T(E)$ is the transmission of the full tunnel junction, for which either the ferromagnetic electrodes can be a highly spin-polarized half-metal or the combination of the barrier and the ferromagnet exhibits half-metallic characteristics. The function $T(E)[-{\partial }_{E}f(E,\mu ,T)]$ is given in a darker color. The thick line marks the resulting value of the geometric center determining the Seebeck coefficient in the parallel magnetization $S_P$ and antiparallel magnetization $S_{AP}$ of the electrodes. Note that we use the term TMT instead of the alternative magneto-Seebeck effect to distinguish it clearly from the spin-Seebeck effect discussed in Sec. 3d4. Adapted from [311].

Figure 40

Thermovoltage in a $(\mathrm{Ga},\mathrm{Mn})\mathrm{As}/i-\mathrm{GaAs}/\mathrm{GaAs}:\mathrm{Si}$ tunnel junction as a function of the magnetization angle. 0 is along the [010] crystal axis. From [200].

Figure 41

Top panel: Measurement geometry of the spin-Seebeck effect. (a) Transverse voltage $V_y$ as a function of the applied field $B$ from the strip contact 0.3 mm above the scratch (star) with an applied $\mathrm{\Delta }{T}_x$ of 0.63 K. (b) Spatial dependence of the spin-Seebeck coefficient $S_{xy}$ before and after the scratch. The scratched region is indicated by the shaded region. (c) Temperature dependence of $S_{xy}$ after the scratch at various positions along the sample. Adapted from [119].