Determination of spin-orbit interaction in semiconductor nanostructures via nonlinear transport

We investigate nonlinear transport signatures stemming from linear and cubic spin-orbit interactions in one- and two-dimensional systems. The analytical zero-temperature response to external fields is complemented by a finite-temperature numerical analysis, establishing a way to distinguish between linear and cubic spin-orbit interactions. We also propose a protocol to determine the relevant material parameters from transport measurements attainable in realistic conditions, illustrated by values for Ge heterostructures. Our results establish a method for the fast benchmarking of spin-orbit properties in semiconductor nanostructures.

Figure 1

Band dispersions (left) and normalized first- and second-order conductivities (right), ${\sigma }^{\left(1\right)}$ and ${\sigma }^{\left(2\right)}$, respectively, for a 1D system with linear and cubic SOIs [see Eq. (1)] with (a) $\mathit{B}=5\widehat{\mathit{y}}$ T and (b) slightly rotated towards the SOI direction [$\mathit{B}=(5\widehat{\mathit{y}}+1\widehat{\mathit{z}})$ T]. Here, ${\sigma }^{\left(\ell \right)}$ for $\ell =1,2$ were numerically obtained for $T=0$ (solid lines) and $T=0.5$ K (dashed lines), for ${\hslash }^2/2m^{*}=310\mathrm{meV}{\mathrm{nm}}^2, \alpha =10\mathrm{meV}\mathrm{nm}, \beta =116\mathrm{meV}{\mathrm{nm}}^3$, and $g_y=2.07$ [95]. For comparison, the ratio between $|{\sigma }^{\left(2\right)}{|}_{\text{max}}$ in (a) and in (b), i.e., $|{\sigma }^{\left(2\right)}{|}_{\text{max,a}}/{\left|{\sigma }^{\left(2\right)}\right|}_{\text{max,b}}$, is approximately 0.04.

Figure 2

Normalized second-order longitudinal ${\sigma }_{xxx}^{\left(2\right)}$ and transversal ${\sigma }_{xyy}^{\left(2\right)}$ conductivities, as a function of Fermi energy $\mu$, for planar Ge with (a) only cubic SOI, $({\beta }_1,{\beta }_2)=(190,23.75)\mathrm{meV}{\mathrm{nm}}^3$, and (c) both linear ($\alpha =1.5\mathrm{meV}\mathrm{nm}$) and cubic SOI, with $B=5$ T aligned in the $y$ direction. Solid lines (circles) result from numerical evaluation of Eq. (9) at $T=0$ ($T>0$). The comparison between the numerical results for $T>0$ presented in (a) [(c)] and the $T=0$ analytical expressions of Eqs. (10, 11, 12) and Eqs. (13) and (14) is shown in (i) and (ii) [(iii) and (iv)], respectively. In (iii) we compare results including higher-order (HO) terms to the linear-order (LO) analytical results from Eqs. (10)–(12). (b) [(d)] Fermi contours for representative $\mu$'s marked in (a) [(c)]. Normalized ${\sigma }_{xxx}^{\left(2\right)}$ and ${\sigma }_{xyy}^{\left(2\right)}$ as function of $\mathit{B}=B\widehat{\mathit{y}}$, for two representative $\mu$'s in (e) Ge[100] and (f) Ge[110]. Colored solid and dashed lines represent results obtained from a numerical evaluation of Eq. (9) at $T=0$, while black solid lines stem from the corresponding analytics [see Eqs. (13) and (14)]. Assuming $\tau =10$ ps [102], we have (a) $|{\sigma }_{xxx}^{\left(2\right)}{|}_{\max }\approx 1.67$ nA m ${\mathrm{V}}^{-2}$, (c) $|{\sigma }_{xyy}^{\left(2\right)}{|}_{\max }\approx 8.97$ nA m ${\mathrm{V}}^{-2}$, (e) $|{\sigma }_{xxx}^{\left(2\right)}{|}_{\max }\approx 0.77$ nA m ${\mathrm{V}}^{-2}$, and (f) $|{\sigma }_{xxx}^{\left(2\right)}{|}_{\max }\approx 8.76$ nA m ${\mathrm{V}}^{-2}$.

Figure 3

Longitudinal ${\rho }_{xxx}^{\left(2\right)}$ (left panel) and transversal ${\rho }_{xyy}^{\left(2\right)}$ (right panel) resistivities, in units of $\mathrm{m}\mathrm{\Omega }\mathrm{mm}{\mathrm{A}}^{-1}$, as function of $\mu$, for planar Ge with (a) only cubic SOI, (${\beta }_1,{\beta }_2)=(190,23.75)\mathrm{meV}{\mathrm{nm}}^3$, and (b) both linear ($\alpha =1.5\mathrm{meV}\mathrm{nm}$) and cubic SOI, with $B=5$ T aligned in the $y$ direction. We assume $\tau =10$ ps [102]. Solid lines result from the numerical evaluation of Eq. (9) at $T=0$, while dashed black lines stem from the analytical results in Eqs. (10, 11, 12, 13, 14) and (17) and (18) for $T=0$. The values of $\mu$ at which ${\sigma }^{\left(2\right)}$ has discontinuities (due to the touching of the Fermi contours) are marked with vertical dashed lines as in Fig. 2.