Microscopic theory of current-driven domain-wall motion and spin-current-assisted creep in the ferromagnetic semiconductor (Ga,Mn)As

The measurement by Yamanouchi, Chiba, Matsukura, Dietl and Ohno (Phys. Rev. Lett. 96, 096601 (2006)) of current-driven domain-wall velocities in (Ga,Mn)As over five decades, up to 22 m/s, corroborated spin-transfer theory only after assuming that the spin-current polarization $P$ entering the torque formulas equals the thermodynamic spin polarization $P_s$, and left a fourfold microscopic problem: (i) the relation between $P$ and $P_s$ for the complex valence band; (ii) the microscopic origin and magnitude of the nonadiabatic spin torque, bounded only by an upper limit on the ratio $\tau_{\rm ex}/\tau_{\rm sf}$; (iii) a microscopic theory of the empirical spin-current-assisted creep law $\ln v_{\rm eff} = a(T) - b(T) j^{-\nu}$ with $\nu \approx 0.5$ and $b(T) \propto (T_C - T)^\gamma$, $\gamma \approx 2$; and (iv) the experimental disentangling of thermal activation, pinning, and torque contributions at low currents. Later work has measured and microscopically calculated the carrier spin-relaxation time and the nonadiabatic torque in (Ga,Mn)As, extracted the current spin polarization from domain-wall dynamics, and recast current-driven creep in a universal elastic-interface framework. The surviving open core is a material-specific microscopic theory that predicts, rather than fits, the spin-transfer parameters and their temperature dependence, and derives the current-assisted creep barrier and its near-Curie-temperature scaling in realistic pinning landscapes.

Background

(Ga,Mn)As is a dilute ferromagnetic semiconductor in which substitutional Mn spins are coupled antiferromagnetically to holes residing in the strongly spin-orbit-coupled heavy-hole valence band of GaAs; the p-d exchange between Mn moments and holes, with coupling constant $\beta$, mediates ferromagnetism with Curie temperatures of order 100 K in the layers used here. Grown under suitable strain, (Ga,Mn)As acquires perpendicular magnetic anisotropy, so up- and down-magnetized domains are separated by a domain wall (DW), a planar magnetic texture of nanometer width (about 17 nm in the source layers) that can be pushed through a lithographic track either by a magnetic field or by an electrical current of density $j$.

A current of spin-polarized holes exerts torques on the wall. In the adiabatic spin-transfer picture (carrier spins track the local magnetization and transfer angular momentum to the wall, as in Slonczewski's torque), the wall velocity grows as $A(j^2 - j_C^2)^{1/2}$ with a critical current density $j_C$ (Tatara and Kohno, 2004), or is linear in $j$ above a pinning threshold (Barnes and Maekawa, 2005); the efficiency $A = g\mu_B P/(2eM)$ and $j_C = 2eK\delta_W/(\pi\hbar P)$ are set by the spin-current polarization $P$ (with $g$ the hole g-factor, $\mu_B$ the Bohr magneton, $e$ the elementary charge, $M$ the magnetization, $K$ the anisotropy energy density, and $\delta_W$ the wall width). Because holes have a finite spin relaxation time $\tau_{\rm sf}$ (limited by spin-orbit scattering and by spin mixing from the noncollinear magnetization inside the wall), while their precession time in the Mn exchange (molecular) field is $\tau_{\rm ex} = \hbar g\mu_B/(|\beta|M)$ with $\beta$ the p-d exchange coupling above, Zhang and Li (2004) predicted an additional nonadiabatic torque, giving a subthreshold, linear-in-$j$ wall mobility $C = A\,\tau_{\rm ex}/(\tau_{\rm sf}\alpha_G)$ with Gilbert damping $\alpha_G$. In the later spin-transfer literature this nonadiabatic torque is conventionally parametrized by a dimensionless coefficient also denoted $\beta$; to avoid confusion with the p-d coupling, the nonadiabatic coefficient is written $\beta_{\rm STT}$ below.

In the pinned, thermally activated regime, an elastic interface such as a DW line creeps — that is, it advances by rare thermally activated hops over the frozen (quenched) disorder landscape — with a stretched-exponential Arrhenius velocity law, generically $\ln v = a - bF^{-\mu}$ for a driving force $F$, whose exponent $\mu$ reflects the universality class of the disorder (for a one-dimensional wall in short-range two-dimensional disorder, the quenched Edwards-Wilkinson class, the standard universality class of elastic-interface depinning in random-bond disorder, gives $\mu = 1/4$).

Yamanouchi et al. (2006) measured DW velocities by magneto-optical Kerr microscopy in (Ga,Mn)As tracks with perpendicular anisotropy, calibrating Joule heating with the device resistance, and found (a) above-threshold velocities consistent with spin-transfer theory, provided $P$ was identified with the thermodynamic spin polarization $P_s = 6k_BT_CM/[(S+1)p\beta M_S]$ of the p-d Zener mean-field model ($S$ the Mn spin, $p$ the hole concentration, $M_S$ the saturation magnetization), an identification uncertain because $p$ and the p-d coupling $\beta$ were poorly known and the multiband valence band makes the relation between $P$ and $P_s$ nontrivial; (b) an upper limit $\tau_{\rm ex}/\tau_{\rm sf} \approx 10^{-2}$ on the nonadiabaticity from the subthreshold slope, without a microscopic determination; (c) an empirical creep scaling $\ln v_{\rm eff} = a(T) - b(T) j^{-\nu}$ with $\nu = 0.5 \pm 0.1$, $a(T) \sim (T_C - T)$ and $b(T) \propto (T_C - T)^\gamma$ with $\gamma = 2 \pm 0.5$, where $v_{\rm eff}$ is the measured effective wall velocity, which they conjectured — but did not derive — to arise from critical magnetization fluctuations that locally depress $j_C$ near the Curie temperature $T_C$; and (d) an unresolved experimental decomposition of low-current motion into thermal activation, extrinsic pinning, and torque contributions. What is asked for is the microscopic physics behind this phenomenology.

Problem Statement

Provide a microscopic, quantitative account of electrically driven domain-wall motion in the ferromagnetic semiconductor (Ga,Mn)As that connects the measured velocity-current-temperature data of Yamanouchi et al. (2006) to the underlying spin-transfer physics. In the generality of that source experiment, this requires: (1) establishing the relation between the spin-current polarization $P$ entering spin-transfer formulas and the thermodynamic spin polarization $P_s$ of the complex heavy-hole valence band, including the propagation of the uncertainties in hole concentration $p$, p-d exchange coupling $\beta$, and band-structure effects; (2) determining the microscopic origin and magnitude of the nonadiabatic spin-transfer torque — equivalently, the carrier spin relaxation time $\tau_{\rm sf}$, or the nonadiabatic coefficient $\beta_{\rm STT}$ and its ratio to the Gilbert damping $\alpha_G$ in the notation of Zhang-Li-type theories — together with its temperature dependence, improving on the source's upper bound $\tau_{\rm ex}/\tau_{\rm sf} \lesssim 10^{-2}$; (3) deriving, from microscopic pinning physics and critical magnetization fluctuations, the observed subthreshold creep law $\ln v_{\rm eff} = a(T) - b(T) j^{-\nu}$ — its exponent $\nu$, and the prefactors $a(T)$ and $b(T)$ with $b(T) \propto (T_C - T)^\gamma$ near the Curie temperature — for realistic disorder landscapes; and (4) decomposing the low-current motion into its thermal-activation, extrinsic-pinning, and torque components across wide temperature and disorder regimes. These four items are jointly posed and together constitute one objective: a predictive theory, validated against experiment, of how spin-polarized hole currents move and pin domain walls in (Ga,Mn)As.

References

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