Probing Fermi-surface spin-textures via the nonlinear Shubnikov-de Haas effect

TL;DR

This work introduces the nonlinear Shubnikov-de Haas (NSdH) effect as a sensitive probe of spin-orbit–induced spin textures on the Fermi surface. Using a Keldysh-based Landau-level formalism, the authors derive a general second-order conductivity $\sigma_{ijl}$ and show that NSdH oscillations carry a distinctive phase and amplitude signature tied to the spin texture, enabling discrimination between linear and cubic Rashba couplings. Through concrete Ge-based 2D hole gas models, they demonstrate that linear versus cubic SOI produces opposite or exact antisymmetric phase relations among nonlinear conductivity components (e.g., $\sigma_{xxx}$ and $\sigma_{xyy}$), and that the NSdH response tracks the underlying spin texture as a function of chemical potential. They further provide symmetry-based arguments and practical expressions for nonlinear resistivity, showing how NSdH can be used to quantify cubic Rashba strength and to characterize SOI in materials relevant to topology, spintronics, and solid-state quantum information technologies.

Abstract

The coupling of spin and electronic degrees of freedom via the spin-orbit interaction (SOI) is an essential ingredient for many proposed future technologies. However, probing the strength and nature of SOI is a significant challenge, especially in heterostructures. Here, we consider the nonlinear Shubnikov-de Haas (NSdH) effect, a quantum oscillatory effect that occurs under conditions similar to those of the well-known SdH effect, but is second order in the applied electric field. We demonstrate that, unlike its linear counterpart, the NSdH effect is highly sensitive to the spin textures that arise from SOI. In particular, we show that the phase and beating of NSdH oscillations in nonlinear conductivities can clearly distinguish between different types of SOI. As a demonstration, we show how NSdH can distinguish between the linear and cubic Rashba couplings that are expected in germanium heterostructures. Our results establish the NSdH effect as a powerful and sensitive probe of SOI, offering a new framework for characterizing materials relevant to topology, spintronics, and solid-state quantum information technologies.

Figures (3)

• Figure 1: Phenomenology of NSdH effect. (a) Schematic of our setup. We assume a lock-in measurement in which a low-frequency ac current $I^\omega$ (corresponding to an ac electric field $E(\omega)$ in our theoretical framework) is injected, and the $2\omega$ components of the longitudinal and Hall voltages are read out. (b) Schematic illustrations of the in-plane spin textures on the Fermi surface at $B=0$. They correspond to (i,ii) single-Fermi-surface cases of ${ {\bm S} }_{ {\bm k} }^{(1)}$ with (i) $\alpha > 0, \, \beta_2 = 0$ and (ii) $\alpha < 0, \, \beta_2 = 0$, and (iii,iv) two-Fermi-surface cases for (iii) ${ {\bm S} }_{ {\bm k} }^{(1)}$ and (iv) ${ {\bm S} }_{ {\bm k} }^{(3)}$, respectively. (c,d) Linear (black lines) and nonlinear SdH oscillations (red: $\sigma_{xxx}$, blue: $\sigma_{xyy}$) for the model exhibiting a single Fermi surface with (c) $\alpha = 20$ meV nm and (d) $\alpha = -20$ meV nm. We set $m^* = \infty$, $\lambda = 20$ nm, $\beta_{1,2}= 0$ meV nm$^3$, $g_{x,y} = 1.244$, $g_z=10$, $(B_x, B_y) = (0~{\rm T}, \, 0.1~{\rm T})$, and $\mu = 0.6$ meV in both cases. We set the impurity-scattering parameter to $n_{\rm i} u^2 = 100$ meV$^2$ nm$^2$. (e,f) Density of states (DoS), and linear and nonlinear SdH oscillations for 2DHGs. We consider $\hbar^2 / (2m^*) = 620$ meV nm$^2$, $\lambda = 0$, and $g_z = 10$ with (e) $\alpha = 1.5$ meV nm, $\beta_{1,2} = 0$, $g_{x,y} = 1.244$, and (f) $\beta_1 = 190$ meV nm$^3$, $\alpha = \beta_{2} = 0$, $g_{x,y} = 0.207$, inspired by the Ge [110] and Ge [100] cases, respectively Scappucci2021XGLL2021DLBLKNLKL2024. We employ an in-plane magnetic field of $(B_x, B_y) = (0~{\rm T}, \, 5~{\rm T})$, and set the impurity parameter to $n_{\rm i} u^2 = 60$ meV$^2$ nm$^2$. Purple and gray dotted lines indicate the positions of the beating nodes in $\sigma_{xii}$ and $\sigma_{xx}$, respectively. We use a sufficiently large number of Landau levels, $N = 400$, in all panels (c)–(f).
• Figure 2: Spin texture dependence of NSdH effect. (a,b) Chemical-potential dependence of the nonlinear SdH oscillations for Ge [110] parameters (i.e. $\alpha = 1.5$ meV nm, $\beta_1 = 190$ meV nm$^3$, $\beta_2 = 23.75$ meV nm$^3$, and $g_{x,y} = 1.244$). We set (a) $\mu = 1.5$ meV and (b) $\mu = 9$ meV. The plot of $\sigma_{xyy}$ in (b) is magnified by a factor of five to make the phase difference relative to $\sigma_{xxx}$ visible. (c,d) Linear Rashba SOI dependence of the linear and nonlinear SdH oscillations for Ge [100] parameters (i.e. $\alpha = 0$ meV nm, $\beta_1 = 190$ meV nm$^3$, $\beta_2 = 23.75$ meV nm$^3$, and $g_{x,y} = 0.207$) at a common chemical potential $\mu = 1$ meV, with (c) $\alpha = 0$ and (d) $\alpha = 1.5$ meV nm. (e,f) $\alpha$ dependence of (e) the phase difference $\Delta \varphi$ between $\sigma_{xxx}$ and $\sigma_{xyy}$ and (f) the maximum value of $|\sigma_{xii}|$ for the Ge [100] parameters. For all panels in this figure, we assume a purely imaginary constant self-energy $\gamma = 0.07$ meV Zhang2024, and the other parameters are the same as those in Figs. \ref{['fig:1']}(e,f).
• Figure S1: The linear and nonlinear SdH oscillations in (a,b) Ge [110] parameters at $\mu=1.5$ meV and (c,d) Ge [100] parameters at $\mu=1$ meV in (a,c) self-consistent Born approximation (SCBA) with $n_{\rm i} u^2 = 60~{\rm meV^2~nm^2}$ and (b,d) imaginary constant self-energy of $i{\gamma} = 0.07i$ meV.

Theorems & Definitions (1)

• proof