\textit{Ab initio} studies of influence of periodic-direction electric fields on spin lifetime and spin diffusion length and the validation of an \textit{ab initio} matrix-drift-diffusion model

TL;DR

The study extends an ab initio density-matrix framework to include electric-field drift along periodic directions via a Wannier-based covariant derivative, enabling first-principles predictions of how ${\bf E}$ alters spin lifetime $\tau_s$ and diffusion length $l_s$ in diverse materials. It demonstrates that DP-type spin relaxation and field-induced effective magnetic fields from Rashba SOC can significantly modify spin dynamics, with results diverging from standard drift-diffusion predictions in certain systems such as GaAs, GaN, and graphene–hBN. The work introduces an ab initio matrix-drift-diffusion (ab-mDD) model and RR-based reductions to approximate the full solution, and assesses their accuracy across EY and DP regimes; while some systems are well-described, others require the full microscopic approach, particularly when the equilibrium density matrix departs from Fermi-Dirac statistics. Overall, the framework highlights the necessity of microscopic, ab initio methods to reliably capture electric-field effects on spin transport and points to future extensions including field-dependent scattering.

Abstract

Recently, we developed an \textit{ab initio} approach of spin lifetime (τ_{s}) and spin diffusion length (l_{s}) in solids [Phys. Rev. Lett. 135, 046705 (2025)], based on a density-matrix master equation with quantum treatment of electron scattering processes. In this work, we extend the method to include the drift term due to an electric field along a periodic direction, implemented using a Wannier-representation-based covariant derivative. We employ this approach to investigate the electric-field effect on τ_{s} and l_{s} of monolayer WSe_{2}, bulk GaAs, bulk GaN, and graphene-h-BN heterostructure. We find that an electric field reduces τ_{s} of GaAs, due to the induced D'yakonov-Perel'-type spin relaxation. In GaN and graphene-h-BN, τ_{s} is significantly affected, partly because the electric field generates an effective magnetic field corresponding to the k-derivative of Rashba spin-orbit (magnetic) field. Our results show that l_{s} can be significantly enhanced or suppressed by a moderate downstream or upstream field respectively. While the standard drift-diffusion model performs well for WSe_{2}, it can introduce large errors of the electric-field-induced changes of l_{s} in GaAs, GaN and graphene-h-BN. Our proposed \textit{ab initio} matrix-drift-diffusion model improves results for GaAs and GaN, but still fails for graphene-h-BN. Thus, to accurately capture the influence of electric fields on l_{s} in realistic materials, it is necessary to go beyond the drift-diffusion model and adopt a microscopic \textit{ab initio} methodology. Moreover, in graphene-h-BN, we find that the field-induced changes of τ_{s} and l_{s} are not only governed by the drift term in the master equation, but are also significantly affected by the electric-field modification of the equilibrium density matrix away from Fermi-Dirac distribution function.

Figures (6)

• Figure 1: Calculated electric-field ($E_{x}$) dependent spin lifetimes ($\tau_{s}(E_{x})$) by different methods. $\tau_{s}(E_{x})$ of (a) holes of WSe$_{2}$ at 50 K, (b) electrons of GaAs at 300 K, and (c) electrons of GaN at 100 K. (d) $\tau_{s}\left(E_{x}=0\right)$ of GaN at different $B_{y}$ (external magnetic field along $y$). (e) and (f) are $\tau_{s}\left(E_{x}\right)$ of graphene-$h$-BN at 300 K with Fermi level $E_{F}$=0.1 eV. "Full" means that $\tau_{s}$ is calculated by the full ab initio approach solving full EVP Eq. \ref{['eq:sevp']}, with a self-consistently-computed field-dependent equilibrium density matrix $\rho^{\mathrm{eq}}$ (see Sec. \ref{['subsec:solve_rhoeq_E']}). "$n$-RR" means that $\tau_{s}(E_{x})$ is computed by Eq. \ref{['eq:RR_EVP_ls']} using $n$th-order RR method considering a certain number of spin "relevant" decay modes (see Sec. \ref{['subsec:RR']}). "ab-mDD" means our proposed ab initio matrix-drift-diffusion model (Sec. \ref{['subsec:ab-mDD']} and Eq. \ref{['eq:ab-mDD']}). "DP" corresponds to Eq. \ref{['eq:tausE_DP']}, which is a DP-like model derived from 1-RR. "Full, $\rho^{\mathrm{eq}}$=$f$" means that $\tau_{s}$ is calculated by solving full EVP Eq. \ref{['eq:sevp']}, but with $\rho^{\mathrm{eq}}$ being its zero-electric-field value - Fermi-Dirac function $f$.
• Figure 2: Calculated $\tau_{s}(E_{x})$ of GaAs, GaN and graphene-$h$-BN by solving full EVP and the ab-mDD model. "no $\Gamma^{E}$" or "no $\Omega^{t}$" means that the $\Gamma^{E}$ (Eq. \ref{['eq:GE']}) or $\Omega^{t}$ (Eq. \ref{['eq:Omegat']}) term of the model is not considered, respectively.
• Figure 3: (a) Schematic of the spin-diffusion setup. Spin are injected from a ferromagnet (FM) at $x<0$ through a transparent interface into the material at $x\ge0$, where they diffuse under zero or finite ${\bf E}$ along $x$ axis. (b) $E_{x}$ dependent spin diffusion length ($l_{s}(E_{x})$) of holes of monolayer WSe$_{2}$ at 50 K by different methods. "Model 1" uses the analytical formula Eq. \ref{['eq:model']} with $\eta$ being $\eta_{1}$=$e/(v_{Fx}^{2}\widetilde{m}_{x})$ (Eq. \ref{['eq:model1']}). "Full" means that $l_{s}$ is calculated by solving full EVP Eq. \ref{['eq:gevp']}. "$n$-RR" means that $l_{s}(E_{x})$ is computed by solving the reduced EVP Eq. \ref{['eq:RR_EVP_ls']} using $n$th-order RR method with basis functions $V^{R(L)}$ given by Eq. \ref{['eq:RR_ls']} (see Sec. \ref{['subsec:RR']}).
• Figure 4: $l_{s}(E_{x})$ of bulk GaAs at 300 K and bulk GaN at 100 K, computed by different methods. Panel (g) shows $l_{s}$ of GaN at $E_{x}$=-700 V/cm computed by RR methods of different orders ($n$) and with different types of basis functions $V^{R(L)}$, compared with results by solving full EVP. "Model 1, 2 and 3" refer to the model $l_{s}(E_{x})$ formula Eq. \ref{['eq:model']} with the parameter $\eta$ set as $\eta_{1}$, $\eta_{2}$=$\mu/D_{s}$ and $\eta_{3}$=$e/(k_{B}T)$ (Eqs. \ref{['eq:model1']}-\ref{['eq:model3']}), respectively. "no $v_{d}^{0}$" corresponds to the ab-mDD model without the $v_{d}^{0}$ term (Eq. \ref{['eq:vd0']}). $V^{R(L)}$ of $n$-RR, $n$-RR-B and $n$-RR-C are given in Eqs. \ref{['eq:RR_ls']}, \ref{['eq:RR-B']} and \ref{['eq:RR-C']}, respectively.
• Figure 5: $l_{s}(E_{x})$ of graphene-$h$-BN at 300 K, computed by different methods.