Control of Spin-Orbit Interaction in Conventional Semiconductor Quantum Wells

Abstract

In this chapter, we demonstrate, focusing on GaAs quantum wells (QWs), a full control of spin-orbit (SO) interaction including both the Rashba and Dresselhaus terms in conventional semiconductor QWs. We determine the SO interaction in GaAs from single to double and triple wells, involving the electron occupation of either one or two subbands. Both the intraband and interband SO coefficients are computed. Two distinct regimes, depending on the QW width, for the control of SO terms, are found. Furthermore, we determine the persistent-spin-helix (PSH) symmetry points, where the Rashba and the renormalized (due to cubic corrections) Dresselhaus couplings are matched. These PSH symmetry points, at which quantum transport is diffusive (2D) for charge while ballistic (1D) for spin, are important for longtime and long-distance coherent spin control that is the keystone in spintronic devices.

1. Introduction

The spin-orbit (SO) interaction is a relativistic effect coupling spatial and spin degree of freedom via an effective magnetic field, facilitating spin manipulation in semiconductor nanostructures [1, 2]. For instance, the proposal of Datta and Das for a spin field-effect transistor highlights the use of the SO interaction of Rashba [3]. Recently, the SO effects [4] have attracted renewed interest in diverse fields of condensed matter, including the persistent spin helix (PSH) [5, 6, 7, 8], topological insulators [9], and Majorana fermions [10, 11].

In zinc-blende-type crystals, such as GaAs, there are two dominant contributions to the SO interaction. The bulk inversion asymmetry leads to the Dresselhaus coupling [12], which in heterostructures contains both linear and cubic terms. The linear term mainly depends on the quantum-well confinement and the cubic one on the electron density [7, 13]. Additionally, the structural inversion asymmetry in heterostructures gives rise to the linear Rashba coupling [14], which can be electrically controlled by using an external bias [15, 16]. Extensive studies on the SO interaction have been focused on n-type GaAs/AlGaAs wells with only one-subband electron occupation [7, 13, 17]. Recently, quantum wells with two populated subbands have also drawn attention in both experiment [18, 19, 20] and theory [21, 22, 23, 24, 25], because of emerging new physical phenomena including the intersubband coupling-induced spin mixing [19] and crossed spin helices [25].

In this chapter, we report our recent results on the electric control of SO interactions in conventional semiconductor quantum wells. Firstly, we focus on the case of single GaAs wells and performed a detailed self-consistent calculation to determine how the SO coupling (both the magnitude and sign) changes as a function of the gate voltage $V_{\mathrm{g}}$ . The gate-altered electron occupations between the first two subbands have been taken into account, and both the intraband and interband SO coefficients are computed. We find two distinct regimes for the electric control of SO interactions. Secondly, we consider the case of multiple wells and determine the SO interaction in GaAs from single to double and triple wells. Furthermore, we determine the persistent-spin-helix (PSH) symmetry points, where the Rashba and the renormalized (due to cubic corrections) Dresselhaus couplings are matched. These PSH symmetry points, at which quantum transport is diffusive (2D) for charge while ballistic (1D) for spin, are important for longtime and long-distance coherent spin control that is the keystone in spintronic devices.

2. Model Hamiltonian

The quantum wells that we consider are grown along the $z\Vert \left[001\right]$ direction. We start from the $8\times 8$ Kane model, involving both conduction and valence bands, and use the folding down procedure [4, 24], to obtain an effective 3D Hamiltonian for electrons only [4, 13, 24]:

$${\mathcal{H}}^{3\mathcal{D}}=\frac{{\mathrm{\hslash }}^2{k}_{\parallel }^2}{2m^{\ast }}-\frac{{\mathrm{\hslash }}^2}{2m^{\ast }}\frac{{\partial }^2}{\partial z^2}+V_{\mathit{sc}}\left(z\right)+{\mathcal{H}}_{\mathcal{R}}+{\mathcal{H}}_{\mathcal{D}},$$ E1

where the first two terms refer to the kinetic contributions, in which $m^{\ast }$ is the electron effective mass and $k_{\parallel }=\sqrt{k_x^2+k_y^2}$ is the in-plane electron momentum with $x\Vert \left[100\right]$ and $y\Vert \left[010\right]$ directions. The third term $V_{\mathit{sc}}$ describes the confining electron potential with the subscript $\mathit{sc}$ indicating that it is determined self-consistently within the (Poisson-Schrödinger) Hartree approximation. The confining potential contains the structural potential $V_{\mathrm{w}}$ arising from the band offsets, the doping potential $V_{\mathrm{d}}$ , the electron Hartree potential $V_{\mathrm{e}}$ , and the external gate potential $V_{\mathrm{g}}$ [13, 23, 24]. The last two terms, ${\mathcal{H}}_{\mathcal{R}}$ and ${\mathcal{H}}_{\mathcal{D}}$ , correspond to the Rashba and Dresselhaus SO interactions, respectively. Note that in Eq. (1), we have added the Dresselhaus term ${\mathcal{H}}_{\mathcal{D}}$ empirically [4, 13, 24], since to obtain ${\mathcal{H}}_{\mathcal{D}}$ one has to take into account the remote bands (e.g., $\mathbf{p}$ -conduction band) [4]. The Rashba term reads ${\mathcal{H}}_{\mathcal{R}}=\eta \left(z\right)\left(k_x{\sigma }_y-k_y{\sigma }_x\right)$ with $\eta \left(z\right)={\eta }_{\mathrm{w}}{\partial }_z{V}_{\mathrm{w}}+{\eta }_{\mathrm{H}}{\partial }_z\left(V_{\mathrm{g}}+V_{\mathrm{d}}+V_{\mathrm{e}}\right)$ determining the Rashba strength and ${\sigma }_{x,y,z}$ the spin Pauli matrices. The constants ${\eta }_{\mathrm{w}}$ and ${\eta }_{\mathrm{H}}$ are determined by the bulk band parameters [13, 24]:

$${\eta }_{\mathrm{w}}=\frac{P^2}{3}\left(\frac{{\delta }_{\mathrm{v}}/{\delta }_{\mathrm{c}}}{E_g^2}-\frac{{\delta }_{\mathrm{\Delta }}/{\delta }_{\mathrm{c}}}{{\left(E_g+\mathrm{\Delta }\right)}^2}\right),$$ E2

$${\eta }_{\mathrm{H}}=-\frac{P^2}{3}\left(\frac{1}{E_g^2}-\frac{1}{{\left(E_g+\mathrm{\Delta }\right)}^2}\right),$$ E3

where $E_g$ is the fundamental bandgap, $P$ is the Kane parameter, and $\mathrm{\Delta }$ is the split-off gap, in the well layer. The parameters ${\delta }_c$ , ${\delta }_v$ , and ${\delta }_{\mathrm{\Delta }}$ stand for band offsets between well and barrier layers, of the conduction band, heavy hole (and light hole), and split-off hole, respectively. A schematic of the band offsets for a GaAs/AlGaAs well is shown in Figure 1. The Dresselhaus term is ${\mathcal{H}}_{\mathcal{D}}=\gamma \left[{\sigma }_x{k}_x\left(k_y^2-k_z^2\right)+\mathrm{c}.\mathrm{c}.\right]$ , with $\gamma$ the bulk Dresselhaus parameter and $k_z=-i{\partial }_z$ . From the 3D Hamiltonian (Eq. (1)), we are now ready to derive an effective 2D Hamiltonian for electrons. This 2D model is similar to the well-known Rashba model but now for wells with two subbands. We first self-consistently determine the spin-degenerate eigenvalues ${\mathcal{E}}_{{\mathbf{k}}_{\parallel }\nu }={\mathrm{ℰ}}_{\nu }+{\mathrm{\hslash }}^2{k}_{\parallel }^2/2m^{\ast }$ and the corresponding eigenspinors $\mid {\mathbf{k}}_{\parallel }\mathit{\nu \sigma }⟩=\mid {\mathbf{k}}_{\parallel }\nu ⟩\otimes \mid \sigma ⟩$ , $⟨\mathbf{r}|{\mathbf{k}}_{\parallel }\nu ⟩=exp\left(i{\mathbf{k}}_{\parallel }\cdot {\mathbf{r}}_{\parallel }\right){\psi }_{\nu }\left(z\right)$ , of the well in the absence of SO interaction [24]. Here we have defined ${\mathcal{E}}_{\nu }$ ( ${\psi }_{\nu }$ ), $\nu =1,2$ , as the $\nu$ _th quantized energy level (wave function) and $\sigma =\uparrow ,\downarrow$ as the electron spin component along the $z$ direction. We then can straightforwardly obtain an effective 2D model by projecting Eq. (1) onto the basis $\left\{|{\mathbf{k}}_{\parallel }\mathit{\nu \sigma }⟩\right\}$ . The effective 2D model with two subbands in the coordinate system $x_{+}\Vert \left(110\right)$ , $x_{-}\Vert \left(1\overline{1}0\right)$ under the basis set $\left\{|{\mathbf{k}}_{\parallel }1\uparrow ⟩,|{\mathbf{k}}_{\parallel }1\downarrow ⟩,|{\mathbf{k}}_{\parallel }2\uparrow ⟩,|{\mathbf{k}}_{\parallel }2\downarrow ⟩\right\}$ reads [26]:

$${\mathcal{H}}^{2D}=\left(\frac{{\mathrm{\hslash }}^2{k}_{\parallel }^2}{2m^{\ast }}+{\mathcal{E}}_{+}\right)\mathbf{1}\otimes \mathbf{1}-{\mathcal{E}}_{-}{\tau }_z\otimes \mathbf{1}+{\mathcal{H}}_{\mathcal{R}\mathcal{D}},$$ E4

with ${\mathcal{E}}_{\pm }=\left({\mathcal{E}}_2\pm {\mathcal{E}}_1\right)/2$ , $\mathbf{1}$ the $2\times 2$ identity matrix (in both spin and orbital subspaces), and ${\tau }_{x_{+},x_{-},z}$ the Pauli (“pseudospin”) matrices acting within the orbital subspace. The term ${\mathcal{H}}_{\mathcal{R}\mathcal{D}}$ describes the Rashba and Dresselhaus SO contributions in terms of intra- and intersubband SO fields ${\mathbf{B}}_{\mathrm{S}O}^{\nu }$ and ${\mathbf{B}}_{\mathrm{S}O}^{12}$ , respectively:

$${\mathcal{H}}_{\mathcal{R}\mathcal{D}}=\frac{1}{2}g{\mu }_B\sum _{\nu =1,2}\left[{\tau }_{\nu }\otimes \mathit{\sigma }\cdot {\mathrm{B}}_{\mathrm{S}O}^{\nu }+{\tau }_{x_{+}}\otimes \mathit{\sigma }\cdot {\mathbf{B}}_{\mathrm{S}O}^{12}\right],$$ E5

with $g$ the electron $g$ factor, ${\mu }_B$ the Bohr magneton, and ${\tau }_{1,2}=\left(\mathbf{1}\pm {\tau }_z\right)/2$ . Explicitly, the intrasubband SO field is

$$\begin{array}{ll}{\mathrm{B}}_{\mathrm{S}O}^{\nu }=\frac{2}{g{\mu }_B}k& \left\{\left[\left({\alpha }_{\nu }-{\beta }_{\nu ,\mathrm{e}\mathit{ff}}\right)sin\theta -{\beta }_{\nu ,3}sin3\theta \right]{\widehat{\mathbf{x}}}_{+}-\right.\\ & \left.\left[\left({\alpha }_{\nu }+{\beta }_{\nu ,\mathrm{e}\mathit{ff}}\right)cos\theta -{\beta }_{\nu ,3}cos3\theta \right]{\widehat{\mathbf{x}}}_{-}\right\},\end{array}$$ E6

and the intersubband SO field is

$${\mathbf{B}}_{\mathrm{S}O}^{12}=\frac{2}{g{\mu }_B}k\left[\left(\eta -\mathrm{\Gamma }\right)cos\theta {\widehat{\mathbf{x}}}_{+}-\left(\eta +\mathrm{\Gamma }\right)sin\theta {\widehat{\mathbf{x}}}_{-}\right],$$ E7

where $\theta$ is the angle between $\mathbf{k}$ and the $x_{+}$ axis.

Figure 1.

A schematic of the band offsets for GaAs/AlGaAs well, in which $E_g$ ( $E_g^b$ ) and $\mathrm{\Delta }$ ( ${\mathrm{\Delta }}^b$ ) are the fundamental bandgap and the split-off gap in the well (barrier), respectively. ${\delta }_c$ , ${\delta }_v$ , and ${\delta }_{\mathrm{\Delta }}$ represent the band offsets between well and barrier layers, of the conduction band, heavy hole (and light hole), and split-off hole, respectively.

The Rashba and Dresselhaus SO coefficients, ${\alpha }_{\nu }$ , ${\beta }_{\nu }$ , $\eta$ , and $\mathrm{\Gamma }$ , appearing in Eqs. (6) and (7) read

$${\eta }_{{\mathit{\nu \nu }}^{\text{'}}}=⟨{\psi }_{\nu }|{\eta }_{\mathrm{w}}{\partial }_z{V}_{\mathrm{w}}+{\eta }_{\mathrm{H}}{\partial }_z\left(V_{\mathrm{g}}+V_{\mathrm{d}}+V_{\mathrm{e}}\right)|{\psi }_{{\nu }^{\text{'}}}⟩,$$ E8

and

$${\mathrm{\Gamma }}_{{\mathit{\nu \nu }}^{\text{'}}}=\gamma ⟨{\psi }_{\nu }|k_z^2|{\psi }_{{\nu }^{\text{'}}}⟩,$$ E9

with the Rashba coefficients ${\alpha }_{\nu }\equiv {\eta }_{\mathit{\nu \nu }}$ , $\eta \equiv {\eta }_{12}$ and the Dresselhaus coefficients ${\beta }_{\nu }\equiv {\mathrm{\Gamma }}_{\mathit{\nu \nu }}$ and $\mathrm{\Gamma }\equiv {\mathrm{\Gamma }}_{12}$ . The coefficient ${\beta }_{\nu ,\mathrm{e}\mathit{ff}}={\beta }_{\nu }-{\beta }_{\nu ,3}$ (Eq. (6)) is the renormalized “linear” Dresselhaus coupling, due to the cubic correction ${\beta }_{\nu ,3}=\gamma k_{\nu }^2/4$ , where $k_{\nu }=\sqrt{2\pi n_{\nu }}$ is the $\nu$ th-subband Fermi wave number with $n_{\nu }$ the $\nu$ th-subband occupation.

Note that the Rashba strength ${\alpha }_{\nu }$ (Eq. (8)) can be written in terms of several distinct “individual” contributions, i.e., ${\alpha }_{\nu }={\alpha }_{\nu }^{\mathrm{g}}+{\alpha }_{\nu }^{\mathrm{d}}+{\alpha }_{\nu }^{\mathrm{e}}+{\alpha }_{\nu }^{\mathrm{w}}$ , with ${\alpha }_{\nu }^{\mathrm{g}}={\eta }_{\mathrm{H}}⟨{\psi }_{\nu }|{\partial }_z{V}_{\mathrm{g}}|{\psi }_{\nu }⟩$ the gate contribution, ${\alpha }_{\nu }^{\mathrm{d}}={\eta }_{\mathrm{H}}⟨{\psi }_{\nu }|{\partial }_z{V}_{\mathrm{d}}|{\psi }_{\nu }⟩$ the doping contribution, ${\alpha }_{\nu }^{\mathrm{e}}={\eta }_{\mathrm{H}}⟨{\psi }_{\nu }|{\partial }_z{V}_{\mathrm{e}}|{\psi }_{\nu }⟩$ the electron Hartree contribution, and ${\alpha }_{\nu }^{\mathrm{w}}={\eta }_{\mathrm{w}}⟨{\psi }_{\nu }|{\partial }_z{V}_{\mathrm{w}}|{\psi }_{\nu }⟩$ the structural contribution. Similar for the intersubband Rashba term $\eta ={\eta }^{\mathrm{g}}+{\eta }^{\mathrm{d}}+{\eta }^{\mathrm{e}}+{\eta }^{\mathrm{w}}$ , the matrix element for each contribution is calculated between different subbands. For convenience, below we use ${\alpha }_{\nu }^{\mathrm{g}+d}={\alpha }_{\nu }^{\mathrm{g}}+{\alpha }_{\nu }^{\mathrm{d}}$ and ${\eta }^{\mathrm{g}+d}={\eta }^{\mathrm{g}}+{\eta }^{\mathrm{d}}$ for the contribution from the gate plus doping potential. We should emphasize that all of the SO coupling contributions above depend on the total self-consistent potential $V_{\mathit{sc}}$ as our wave functions are calculated self-consistently.

It is worth noting that here we do not consider in our model the many-body effect-induced discontinuity of the electron density upon occupation of the second subband, as demonstrated by Goni et al. [27] and Rigamonti and Proetto [28] at zero temperature. As this discontinuity vanishes for $T>30$ K, we believe this is a minor effect in our system when $T=75$ K. However, it is conceivable that related features can manifest in the SO couplings at zero temperature. Additional work is needed to investigate this interesting possibility.

3. Two distinct regimes for the control of SO interaction

In this section, we first introduce the structure of our wells and relevant parameters adopted in our simulation. Then we discuss our calculated SO couplings for the two distinct regimes. In either regime, we focus on a well having a two-subband electron occupation at zero bias (i.e., $V_{\mathrm{g}}=0$ ). By tuning $V_{\mathrm{g}}$ , we alter electron occupations from two subbands to one subband. For each value of $V_{\mathrm{g}}$ , within the self-consistent Hartree approximation, we determine the relevant SO strengths, i.e., the intrasubband ${\alpha }_{\nu }$ , $\nu =1,2$ , and intersubband $\eta$ Rashba couplings, and similarly for the Dresselhaus term, the intrasubband ${\beta }_{\nu }$ and the intersubband $\mathrm{\Gamma }$ . Finally, the case of the two regimes in between is discussed as well.

3.1 System

The quantum wells we consider are similar to the samples experimentally studied by Koralek et al. [7]: the 001-grown GaAs wells of width $w$ sandwiched between 48-nm ${\mathrm{Al}}_{0.3}{\mathrm{Ga}}_{0.7}$ As barriers. Our structure contains only one delta-doping (Si) layer, sitting 17 nm away from the well interface, with donor concentration $n_d$ . Our prior simulated SO couplings [26] on these samples agree well with the data obtained via the transient-spin-grating technique [7], where there is no bias applied and all wells share the same areal electron density $n_e$ . Here we go beyond the experiment in [7] by using an external bias, which can vary the electron density and control the SO interaction. At zero bias we assume $n_e=n_d$ [7].

The width $w$ of quantum wells we consider ranges from 20 to 70 nm. Two distinct regimes of the SO interaction we found are marked off around $w=w_c=30–35$ nm. Without loss of generality, we focus on a well of $w=25$ nm for the first regime and a well of $w=65$ nm for the second one. For the $25$ -nm well, we choose $n_d=8.0\times {10}^{11}$ ${\mathrm{cm}}^{-2}$ and $T=75$ K, the same as experimental parameters in [7], not only ensuring the electron occupation of the second subband at zero bias [26] but also retaining this second subband occupation over a broad range of $V_{\mathrm{g}}$ ’s. While for the 65-nm well, we choose relatively small values of $n_d$ and $T$ , i.e., $n_d=4.0\times {10}^{11}$ ${\mathrm{cm}}^{-2}$ and $T=0.3$ K, to exclude the electron occupation of a higher third subband.

3.2 Relevant parameters

In our GaAs/ ${\mathrm{Al}}_{0.3}{\mathrm{Ga}}_{0.7}$ As wells, the values of relevant band parameters (see Eqs. (2) and (3)) are as follows: $E_g=1.519$ eV and $\mathrm{\Delta }=0.341$ eV [4, 29, 30]. The corresponding counterparts of $E_g$ and $\mathrm{\Delta }$ in ${\mathrm{Al}}_{0.3}{\mathrm{Ga}}_{0.7}$ As barriers are $E_g^b=1.951$ eV and ${\mathrm{\Delta }}^b=0.329$ eV [13, 24, 30]. As a consequence, we have the band offsets for conduction and valence bands, ${\delta }_c=\left(E_g^b-E_G\right)\times 60.4\%=0.261$ eV, ${\delta }_v=E_g^b-E_G-{\delta }_c=0.171$ eV, and ${\delta }_{\mathrm{\Delta }}={\delta }_v-\mathrm{\Delta }+{\mathrm{\Delta }}^b=0.159$ eV [13, 24, 30]. For the Kane parameter, we choose $P=1.0493$ nm [4, 29]. Then, we obtain ${\eta }_w=3.97$ ${\mathring{\mathrm{A}}}^2$ and ${\eta }_H=-5.30$ ${\mathring{\mathrm{A}}}^2$ (Eqs. (2) and (3)). We treat the bulk Dresselhaus constant $\gamma$ as an empirical parameter. We have recently done detailed calculations on a set of GaAs wells and have found via a realistic fitting procedure (theory and experiment) $\gamma \sim 11.0$ eV $\cdot {\mathring{\mathrm{A}}}^3$ [13]. We use this value in our simulations, consistent with the one obtained in a recent study by Walser et al. [31].

3.3 Numerical outcome: two distinct regimes

Below we discuss our self-consistent outcome for the SO couplings. We present our calculated intra- and intersubband SO couplings in the two distinct regimes. The behavior of the SO interaction in the first regime as a function of the gate voltage is usual. As a consequence, we mainly focus on the second regime, in which new features of the SO interaction emerge.

3.3.1 Intrasubband SO couplings: both Rashba and Dresselhaus terms

We consider in the first regime a well of $w=25$ nm. In Figure 2(b), we show the dependence of Rashba ${\alpha }_{\nu }$ ( $\nu =1,2$ ) and Dresselhaus ${\beta }_{\nu }$ coefficients of the two subbands on $V_{\mathrm{g}}$ . We find that ${\alpha }_1$ and ${\alpha }_2$ have the same sign and they both decrease to zero near the symmetric configuration (at $V_{\mathrm{g}}\sim 0.36$ eV) (see arrow in Figure 2(b)) and further change their sign. For the confining electron potential $V_{\mathit{sc}}$ and wave functions ${\psi }_{\nu }$ in this symmetric geometry of the well, see Figure 2(a). Clearly, our wells are asymmetric at $V_{\mathrm{g}}=0$ because of the one-side delta doping. In contrast to the Rashba term, we find that the Dresselhaus couplings ${\beta }_{\nu }=\gamma ⟨{\psi }_{\nu }|k_z^2|{\psi }_{\nu }⟩$ remain essentially constant with $V_{\mathrm{g}}$ . The inset in Figure 2(b) shows the electron density $n_{\nu }$ of the two subbands with $n_1+n_2=n_e$ as a function of $V_{\mathrm{g}}$ , and the second subband starts to be unpopulated around $V_{\mathrm{g}}\sim 0.5$ eV.

Figure 2.

(a) Self-consistent potential $V_{\mathit{sc}}$ and wave function profiles ${\psi }_{\nu }$ of a 25-nm well at $V_{\mathrm{g}}=0.36$ eV (symmetric configuration). The horizontal lines inside the well respectively indicate the two-subband energy levels and the Fermi level. Gate control of Rashba ${\alpha }_{\nu }$ and Dresselhaus ${\beta }_{\nu }$ SO couplings, for the $25$ -nm (b) and 65-nm wells (c). In (b), the inset shows electron density $n_{\nu }$ with $n_1+n_2=n_e$ as a function of $V_{\mathrm{g}}$ . In (c), the vertical dashed line at $V_{\mathrm{g}}\sim 0.26$ eV marks regions of one- and two-band occupations. The temperature is chosen at 75 K for the 25-nm well and 0.3 K for the 65-nm well, from Wang et al. [32].

Now, we turn to the second regime, in which we consider a well of $w=65$ nm, as shown in Figure 2(c). As opposed to the first regime, we find that ${\alpha }_1$ and ${\alpha }_2$ could have opposite signs. This is because electrons occupying the first and second subbands tend to be distributed on opposite sides for a wide well, as a result of the electron Hartree potential created “central barrier”; see Figure 3(a)–(c). We emphasize that the sign of SO couplings is crucial in diverse subjects of the field of spintronics, e.g., the persistent skyrmion lattice [25], the nonballistic spin field transistor operating with orthogonal spin quantization axes [33], and the transition from a topological insulator to Dirac semimetal [34].

Figure 3.

(a)–(c) Potential $V_{\mathit{sc}}$ and wave function profiles ${\psi }_{\nu }$ ( $\nu =1,2$ ) for a 65-nm well at three values of voltages, with $V_{\mathrm{g}}=0.18$ (a), 0 (b), and 0.24 eV (c). The horizontal lines inside the well indicate the two-subband energy levels and the Fermi level. Dependence of different contributions to the intrasubband Rashba strength of the first (d) and second (e) subbands on the gate voltage for the 65-nm well. In (d), the inset shows the electron density $n_{\nu }$ with $n_1+n_2=n_e$ as a function of gate voltage. The vertical dashed line in (d) (and inset) marks regions of one- and two-band occupations. The temperature is chosen at 0.3 K, from Wang et al. [32].

To see more details about the sign change of Rashba couplings, we show in Figure 3(d) and (e) the gate dependence of distinct contributions of ${\alpha }_1$ and ${\alpha }_2$ separately, i.e., ${\alpha }_{\nu }^{\mathrm{e}}$ , ${\alpha }_{\nu }^{\mathrm{w}}$ , and ${\alpha }_{\nu }^{\mathrm{g}+d}$ . For the structural contribution, ${\alpha }_{\nu }^{\mathrm{w}}\propto ⟨{\psi }_{\nu }|{\partial }_z{V}_{\mathrm{w}}|{\psi }_{\nu }⟩={\left|{\psi }_{\nu }\left(w/2\right)\right|}^2-{\left|{\psi }_{\nu }\left(-w/2\right)\right|}^2$ [26], ${\alpha }_1^{\mathrm{w}}$ is dominated by the first term, while ${\alpha }_2^{\mathrm{w}}$ is dominated by the second term, due to the already discussed tendency of the subband wave functions of ${\psi }_1$ and ${\psi }_2$ to respectively localize on the right and left halves of the well (zero bias for instance). This leads to opposite signs between ${\alpha }_1^{\mathrm{w}}$ and ${\alpha }_2^{\mathrm{w}}$ . Similarly, for ${\alpha }_{\nu }^{\mathrm{e}}\propto ⟨{\psi }_{\nu }|{\partial }_z{V}_{\mathrm{e}}|{\psi }_{\nu }⟩$ , the electron Hartree contributions of the two subbands have opposite signs as well since $V_{\mathrm{e}}$ mostly has opposite slopes with respect to $z$ for $z>0$ and $z<0$ [24]. However, for ${\alpha }_{\nu }^{\mathrm{g}+d}\propto ⟨{\psi }_{\nu }|{\partial }_z\left(V_{\mathrm{g}}+V_{\mathrm{d}}\right)|{\psi }_{\nu }⟩$ , the corresponding contributions of the two subbands have the same sign, due to the fact that the gate plus doping potential is linear across the well region. The relative sign of the total contribution to ${\alpha }_1$ and ${\alpha }_2$ depends on the interplay of the three constituents as a function of gate voltages.

From Figure 3(e), we also find that ${\alpha }_2$ remains essentially constant with $V_{\mathrm{g}}$ at lower voltages, precisely, as its constituent ${\alpha }_2^{\mathrm{w}}$ behaves except for a prefactor difference. In contrast, it is found that ${\alpha }_1$ sensitively depends on $V_{\mathrm{g}}$ in the whole parameter range studied here. Moreover, ${\alpha }_1$ and ${\alpha }_2$ become zero (see arrow in Figure 3(d) and left arrow in Figure 3(e)) across the symmetric configuration, corresponding to $V_{\mathrm{g}}\sim 0.18$ eV. Interestingly, we observe that ${\alpha }_2$ can even attach zero at $V_{\mathrm{g}}\sim 0.24$ eV (see right arrow in Figure 3(e)), at which the well is very asymmetric (Figure 3(c)).

Before moving into the Dresselhaus couplings in this second regime, it is worth noting that the electron densities of the two subbands exhibit the anticrossing-like behavior near the symmetric configuration (at $V_{\mathrm{g}}\sim 0.18$ eV), as shown in the inset of Figure 3(d). The feature of anticrossing of electron densities has been experimentally observed in double wells by Fletcher et al. [35], thus implying that our wells in the second regime do behave like effective double wells.

In Figure 2(c), we show the linear Dresselhaus couplings ${\beta }_{\nu }=\gamma ⟨{\psi }_{\nu }|k_z^2|{\psi }_{\nu }⟩$ in the second regime. We observe that the inequality ${\beta }_1<{\beta }_2$ , which is valid in the first regime, only holds in configurations near the symmetric geometry. However, if the well is very asymmetric, i.e., at $V_{\mathrm{g}}=0$ , we find ${\beta }_1>{\beta }_2$ [see crossing (black dot) between ${\beta }_1$ and ${\beta }_2$ in Figure 2(c)], in contrast to the first regime.

Now, we are ready to determine the persistent-spin-helix (PSH) symmetry points of the two subbands, at which the Rashba ${\alpha }_{\nu }$ and the renormalized linear Dresselhaus ${\beta }_{\nu ,\mathrm{e}\mathit{ff}}$ (due to cubic corrections) are equal in strength, i.e., ${\alpha }_{\nu }\approx {\beta }_{\nu ,\mathrm{e}\mathit{ff}}$ , as shown in Figure 4. We observe that ${\alpha }_1\approx {\beta }_{1,\mathrm{e}\mathit{ff}}$ takes place at $V_{\mathrm{g}}\sim 0.17$ eV (see right arrow in Figure 4) for the first subband, while ${\alpha }_2\approx -{\beta }_{2,\mathrm{e}\mathit{ff}}$ always holds over the range of $V_{\mathrm{g}}=0-100$ meV (see left arrow in Figure 4) for the second subband. This is possibly facilitating the locking of the PSH symmetry in practice. Note that in contrast to the well matched $\mid {\alpha }_2\mid$ and ${\beta }_{2,\mathrm{e}\mathit{ff}}$ here, a deviation between them can possibly occur for other wells with strong SO couplings (e.g., InAs-based wells). However, this deviation can always be possibly overcome by properly varying the electron density, which controls the cubic ${\beta }_{2,3}$ —thus tuning ${\beta }_{2,\mathrm{e}\mathit{ff}}$ to match with $\mid {\alpha }_2\mid$ . These symmetry points are also crucial for the persistent skyrmion lattice [25] as well as the nonballistic spin field effect transistors [5] operating with orthogonal spin quantization axes [33].

Figure 4.

The dependence of intrasubband Rashba ${\alpha }_{\nu }$ and renormalized Dresselhaus ${\beta }_{\nu ,\mathrm{e}\mathit{ff}}$ coefficients on $V_{\mathrm{g}}$ for the 65-nm well. The vertical dashed line marks regions of one- and two-band occupations. The temperature is held fixed at 0.3 K, from Wang et al. [32].

We emphasize that, for the PSH symmetry points that we determined above, the effect of the interband SO couplings (see Section 3.3.2) and of the random Rashba coupling [36, 37, 38, 39] has been ignored. For the former, it is only relevant near the crossing(s) of the two-subband branches, as discussed in [25]. For the latter, it may in general destroy the helix but has a negligible effect on the results for our wells here [26].

3.3.2 Intersubband SO couplings: both Rashba and Dresselhaus terms

Below we turn to the interband SO terms. Referring to the first regime, in Figure 5(a), we show the intersubband Rashba coupling $\eta$ and its distinct constituents including ${\eta }^{\mathrm{g}+d}$ , ${\eta }^{\mathrm{e}}$ , and ${\eta }^{\mathrm{w}}$ and Dresselhaus coupling $\mathrm{\Gamma }$ for a well of $w=25$ nm. We find that both $\eta$ and $\mathrm{\Gamma }$ exhibit basically the linear behavior with $V_{\mathrm{g}}$ . Owing to the orthogonality between ${\psi }_1$ and ${\psi }_2$ , the gate plus doing contribution ${\eta }^{\mathrm{g}+d}$ vanishes. As for the constituents ${\eta }^{\mathrm{w}}$ and ${\eta }^{\mathrm{e}}$ , the dominant variation with the gate is the latter as the electron density in wells changes with $V_{\mathrm{g}}$ , even though the magnitude of ${\eta }^{\mathrm{w}}$ is greater than ${\eta }^{\mathrm{e}}$ .

Figure 5.

Gate control of intersubband Rashba $\eta$ and Dresselhaus $\mathrm{\Gamma }$ strengths for a 25-nm well (a) and a 65-nm well (b) the temperature is chosen at 75 K for the 25-nm well and 0.3 K for the 65-nm well, from Wang et al. [32].

For the second regime, we show in Figure 5(b) the intersubband SO couplings for a well of $w=65$ nm. In contrast to the first regime, here $\eta$ largely remains constant as $V_{\mathrm{g}}$ increases. And a maximum of $\mid \eta \mid$ occurs at $V_{\mathrm{g}}\sim 0.18$ eV (symmetric configuration). This arises from the fact that the wave functions of the two subbands, ${\psi }_1$ and ${\psi }_2$ , are mostly separated in very asymmetric configurations (Figure 3(b) and (c)), which renders $\eta$ (depending on the overlap of ${\psi }_1$ and ${\psi }_2$ ) almost constant and relatively weak. Similar to $\eta$ , $\mathrm{\Gamma }$ also weakly depends on the gate except for configurations of the well near the symmetric geometry. Finally, we emphasize that, for wide and asymmetric enough wells, where ${\psi }_1$ and ${\psi }_2$ have vanishing overlap, both $\eta$ and $\mathrm{\Gamma }$ tend to be zero.

3.4 Two regimes in between for the control of SO couplings

Now, it is clear that the SO couplings show distinct behaviors for the two regimes. By analyzing SO couplings for a set of wells of $w=20-70$ nm, below we discuss how the two regimes transit from one to the other. To explore this, we only focus on ${\alpha }_2$ , which is mostly in contrast between the two regimes.

In Figure 6(c), we show the dependence of ${\alpha }_2$ on $V_{\mathrm{g}}$ . Clearly, for the 20- and 24-nm wells, ${\alpha }_2$ changes almost linearly, showing the usual behavior of the Rashba coupling, as demonstrated in the first regime. However, for the wells of $w=45,60$ , and 70 nm, we observe the unusual behavior, which is referred to the second regime. When $w$ in the range of $w=30-35$ nm, the gate dependence of ${\alpha }_2$ behaves in between the two regimes, i.e., there is neither the linear behavior in the first regime nor the sign of double changes in the second one. To gain more insight into this, we examine the confining potential $V_{\mathit{sc}}$ and wave function profiles ${\psi }_{\nu }$ for the wells of $w=30$ and 35 nm (Figure 6(a) and (b)), which are used to determine the SO coupling. It is clear that the envelope wave functions for the 30- and 35-nm wells look like a bridge connecting the two regimes, cf. Figure 2(a) for the first regime, Figure 3(a) for the second regime, and Figure 6(a) and (b) for the two regimes in between.

Figure 6.

Confining potential $V_{\mathit{sc}}$ and wave function profiles ${\psi }_{\nu }$ in GaAs wells at $V_{\mathrm{g}}\sim 0.36$ eV (symmetric geometry), for a 30-nm well (a) and a 35-nm well (b). the horizontal lines inside the wells indicate the two-subband energy levels and the Fermi level (c). the dependence of intrasubband Rashba ${\alpha }_2$ of the second subband on $V_{\mathrm{g}}$ , for a set of wells of w = 20,24,30,35,45,60, and 70 nm. The vertical dashed lines mark regions of one- and two-band occupations. The temperature is held fixed at 0.3 K, from Wang et al. [32].

To deplete the second subband occupation, it is clear that a wider well requires in general a larger value of gate voltage (see vertical dashed lines for wells of $w=20-45$ nm), as the confinement becomes weaker for a wider well. However, for the 60- and 70-nm wells (see vertical dashed lines for wells of $w=60-70$ nm), we find that the voltages needed to deplete the second subband occupation are even less than the one for the 24-nm well. This is attributed to the low electron density we choose for wide wells, the choice of which is to exclude the electron occupation of a higher third subband.

4. Control of SO interaction from single to double and triple wells

With the knowledge of the SO interaction in single wells (Section 3), below we consider the case of multiple wells and determine the electrical control of the SO interaction in GaAs from single to double and triple wells.

4.1 System

The main structure of our well is again similar to the samples experimentally studied by Koralek et al. [7]: the 001-grown GaAs well of width $w=26$ nm sandwiched between 48-nm ${\mathrm{Al}}_{0.3}{\mathrm{Ga}}_{0.7}$ As barriers. To have double (triple) wells, we introduce one (two) additional ${\mathrm{Al}}_x{\mathrm{Ga}}_{1-x}$ As barrier layers of length $b=3$ nm embedded inside the above structure. We choose the total electron density $n_e=6.0\times {10}^{11}$ ${\mathrm{cm}}^{-2}$ . The temperature is held fixed at $T=0.3$ K. Note that we choose relatively low $n_e$ and $T$ , instead of the experimental values $n_e=8.0\times {10}^{11}$ ${\mathrm{cm}}^{-2}$ and $T=75$ K in [7], on the one hand, to ensure the second subband electron occupation and on the other hand, to exclude the electron occupation of a higher third subband, for all values of $V_g$ s studied here.

4.2 SO coupling coefficients

To explore the SO features from single to double and triple wells, firstly, we focus on the case of having only one ${\mathrm{Al}}_x{\mathrm{Ga}}_{1-x}$ As additional barrier embedded at the center of the system.

In Figure 7(a) and (b), we show the gate dependence of intrasubband Rashba terms in our GaAs/ ${\mathrm{Al}}_{0.3}{\mathrm{Ga}}_{0.7}\mathrm{As}$ wells, for several Al contents $x$ of the central barrier ${\mathrm{Al}}_x{\mathrm{Ga}}_{1-x}$ As. At $x=0$ (single well), we find the usual scenario, i.e., ${\alpha }_1$ and ${\alpha }_2$ have the same sign and both change almost linearly with $V_g$ . In this case, electrons occupying the first and second subbands spread over the whole well region (Figure 8(a)) and “see” the same fields (i.e., slope of potentials) that determine Rashba couplings (Eq. (8)). Thus, the above behavior of ${\alpha }_{\nu }$ with $V_g$ follows. In contrast, when $x$ is away from zero (double well), electrons of the two subbands tend to localize on the left and right sides of the well, respectively, due to the effect of the central barrier (Figure 8(b) and (c)). This results in the opposite signs of ${\alpha }_1$ and ${\alpha }_2$ , as the local field on the left and right sides of the well is almost reversed; see potential profiles in wells shown in Figure 8(b) and (c). In addition, ${\alpha }_2$ first increases with $V_g$ , while peaks at some point depending on $x$ and further decreases, following from the compensated interplay of local fields (i.e., electron Hartree plus structural well) and the universal external gate field. For a larger value of $x$ , ${\alpha }_2$ increases with $V_g$ more abruptly before it peaks, see black circles in Figure 7(b) for a set of values of $x$ . Note that ${\alpha }_1$ in magnitude consistently increases with $V_g$ , since the local fields and the universal external gate field contribute to ${\alpha }_1$ constructively.

Figure 7.

Intrasubband Rashba ${\alpha }_1$ (a) and ${\alpha }_2$ (b) couplings as a function of $V_{\mathrm{g}}$ in GaAs/ ${\mathrm{Al}}_{0.3}{\mathrm{Ga}}_{0.7}\mathrm{As}$ wells for several Al contents $x$ of a central barrier ${\mathrm{Al}}_x{\mathrm{Ga}}_{1-x}\mathrm{As}$ . In (a), the inset shows the gate dependence of electron occupations $n_1$ , $n_2$ , and $n_e=n_1+n_2$ . In (b), the black circles indicate where ${\alpha }_2$ exhibits the maximum, from Wang et al. [40].

Figure 8.

(a)–(c) Self-consistent potential $V_{\mathit{sc}}$ and wave function profiles ${\psi }_{\nu }$ for a GaAs/ ${\mathrm{Al}}_{0.3}{\mathrm{Ga}}_{0.7}$ As well with ${\mathrm{Al}}_x{\mathrm{Ga}}_{1-x}\mathrm{As}$ as a central barrier, with $x=0$ (a), 0.1 (b), and 0.3 (c). We consider $V_g=0.1$ eV. The horizontal lines inside the well indicate the two-subband energy levels and the Fermi level. (d) Intrasubband Dresselhaus terms ${\beta }_{\nu }$ as a function of $V_{\mathrm{g}}$ , for several values of $x$ s with $x=\mathrm{0,0.1,0.2,0.3}$ . (e) and (f) Gate dependence of SO strengths ( ${\alpha }_{\nu }$ , ${\beta }_{\nu }$ , ${\beta }_{\nu ,\mathrm{e}\mathit{ff}}$ ) for $x=0$ and $x=0.3$ , respectively. The black points indicate the PSH symmetry points (i.e., ${\alpha }_{\nu }=\pm {\beta }_{\nu ,\mathrm{e}\mathit{ff}}$ ), from Wang et al. [40].

Figure 8(c) shows the dependence of linear intrasubband Dresselhaus coupling ${\beta }_{\nu }=\gamma ⟨k_z^2⟩$ on $V_g$ . As opposed to ${\alpha }_{\nu }$ , we find that ${\beta }_{\nu }$ is weakly dependent on $V_g$ ( $\pm \sim 0.5$ meV Å) for all values of $x$ considered here. On the other hand, as $x$ increases, the coupling between the left and right sides of the well is getting weaker, leading to a shrinking of the energy separation of the two subbands. Consequently, the strength of ${\beta }_1$ and ${\beta }_2$ becomes more close for a larger value of $x$ .

Now, we determine the PSH symmetry points, where the Rashba ${\alpha }_{\nu }$ and the renormalized linear Dresselhaus ${\beta }_{\nu ,\mathrm{e}\mathit{ff}}={\beta }_{\nu }-{\beta }_{\nu ,3}$ are matched. The cubic Dresselhaus term ${\beta }_{\nu ,3}=\gamma k_{\nu }^2/4=\mathit{\gamma \pi }{n}_{\nu }/2$ depends on the electron occupations and therefore alters the condition of the PSH symmetry. Figure 8(e) and (f) show the gate dependence of ${\alpha }_{\nu }$ , ${\beta }_{\nu }$ , and ${\beta }_{\nu ,\mathrm{e}\mathit{ff}}$ , for $x=0$ and $0.3$ , respectively. Clearly, we see the distinction between ${\beta }_{\nu }$ and ${\beta }_{\nu ,\mathrm{e}\mathit{ff}}$ because of the correction from ${\beta }_{\nu ,3}$ , in particular for the first subband, which has a higher electron occupation (see inset of Figure 7(a)). In the parameter range considered here, we only attain the PSH symmetry for the first subband ( ${\alpha }_1={\beta }_{1,\mathrm{e}\mathit{ff}}$ ); see black point in Figure 8(e) and (f). To attain the PSH symmetry for the second subband, one needs an even wider well [26], where ${\beta }_2$ is weaker so that ${\alpha }_2$ and ${\beta }_2$ have the feasibility of matched strength by tuning $V_g$ .

Besides altering the PSH symmetry condition involving SO terms of the first harmonic ( $sin$ / $cos$ functions of $\theta$ ), the cubic Dresselhaus ${\beta }_{3,\nu }$ also has the third harmonic ( $3\theta$ ) contribution; see Eq. (6). This leads to the decay of the PSH due to the D’yakonov-Perel’ (DP) spin relaxation mechanism [41]. Specifically, the third harmonic Dresselhaus coupling-induced spin relaxation rate at the PSH symmetry point is written as ${\mathrm{\Gamma }}_D^{\nu }={\gamma }^2{k}_{\nu }^6{\tau }_P/4{\mathrm{\hslash }}^2$ [42], with ${\tau }_P$ the momentum relaxation time. For GaAs wells, with $\gamma \sim 11.0$ eV ${\mathring{\mathrm{A}}}^3$ , $k_{\nu }\sim 0.2$ ${\mathrm{nm}}^{-1}$ , and ${\tau }_P\sim 1.0$ ps [13, 31], we find ${\mathrm{\Gamma }}_D^{\nu }\sim 0.07$ ${\mathrm{ns}}^{-1}$ .

In Figure 9(a) and (b), we show the gate dependence of the intersubband Rashba $\eta$ and Dresselhaus $\mathrm{\Gamma }$ couplings, which depend on the overlap of the wave functions of the two subbands; see Eqs. (8) and (9). For the intersubband Rashba coupling $\eta$ , we find that it remains essentially constant at $x=0$ , since ${\psi }_1$ and ${\psi }_2$ are well overlapped even at asymmetric configurations; see Figure 8(a) with $V_g=0.1$ eV. However, when $x$ is nonzero, electrons of the two subbands have the tendency of localizing on opposite sides of the well at asymmetric configurations (Figure 8(b) and (c)), namely, ${\psi }_1$ and ${\psi }_2$ tend to be separated. Therefore, $\eta$ exhibits the strongest at the symmetric configuration of the system (i.e., $V_g=0$ ). Note that at the symmetric configuration, ${\psi }_1$ and ${\psi }_2$ are perfectly overlapped. Furthermore, the separation of ${\psi }_1$ and ${\psi }_2$ is more distinct for a larger $x$ (same $V_g$ ), cf. Figure 8(b) and (c), which renders $\eta$ more sensitively dependent on $V_g$ near $V_g=0$ . As for the intersubband Dresselhaus term $\mathrm{\Gamma }=⟨1|k_z^2|2⟩$ , on the one hand, it depends on the overlap of ${\psi }_1$ and ${\psi }_2$ and accordingly strongly depends on $V_g$ near $V_g=0$ , similar to $\eta$ . On the other hand, as opposed to $\eta$ , the term $\mathrm{\Gamma }$ vanishes at $V_g=0$ and further changes sign across the symmetric configuration, similar to the intrasubband Rashba terms ${\alpha }_{\nu }$ .

Figure 9.

Intersubband Rashba $\eta$ (a) and Dresselhaus $\mathrm{\Gamma }$ (b) couplings as a function of $V_{\mathrm{g}}$ in GaAs/ ${\mathrm{Al}}_{0.3}{\mathrm{Ga}}_{0.7}\mathrm{As}$ wells, for several Al contents $x$ of the central barrier ${\mathrm{Al}}_x{\mathrm{Ga}}_{1-x}\mathrm{As}$ with $x=\mathrm{0,0.1,0.2,0.3,0.4}$ , from Wang et al. [40].

Finally, we consider the case of our system having two additional barriers, namely, a triple well. As compared to the double well case, the wave functions of the two subbands ${\psi }_1$ and ${\psi }_2$ are more overlapped in a triple well under the same external conditions (e.g., same $V_{\mathrm{g}}$ ), cf. Figure 10(a) and (b), due to the electron distribution in the middle of three wells. As a consequence, the relatively smooth change of SO terms even near the symmetric configuration follows when $V_g$ varies, as shown in Figure 10(c). Moreover, we find that the Dresselhaus term ${\beta }_{\nu }$ is stronger than that in our double well, cf. Figures 8(d) and 10(c). Note that the basic feature of the electrical control of the SO interaction in our triple well is similar to that in the double well, for the parameter range studied, under which the two-subband electron occupancy occurs.

Figure 10.

Confining potential and the two-band wave function for a GaAs/ ${\mathrm{Al}}_{0.3}{\mathrm{Ga}}_{0.7}$ As double well (a) and triple well (b) with embedded barrier ${\mathrm{Al}}_x{\mathrm{Ga}}_{1-x}$ As of $x=0.3$ , at $V_g=0.2$ eV. The horizontal lines inside the well indicate the two-subband energy levels and the Fermi level (c). gate control of intrasubband Rashba ( ${\alpha }_{\nu }$ ) and Dresselhaus ( ${\beta }_{\nu }$ ) and Intersubband Rashba ( $\eta$ ) and Dresselhaus ( $\mathrm{\Gamma }$ ) for the GaAs/ ${\mathrm{Al}}_{0.3}{\mathrm{Ga}}_{0.7}$ As triple well with the embedded barrier ${\mathrm{Al}}_x{\mathrm{Ga}}_{1-x}$ As of $x=0.3$ , from Wang et al. [40].

5. Conclusion

In this chapter, firstly, we consider two distinct regimes of the control of the SO interaction in conventional semiconductor quantum wells. Specifically, we have performed a detailed self-consistent calculation on realistic GaAs wells with gate-altered electron occupations from two subbands to one subband, thus determining how the SO coupling (both the magnitude and sign) changes as a function of the gate voltage $V_{\mathrm{g}}$ . We have considered a set of wells of the width $w$ in a wide range and found two distinct regimes. In the first regime, the behavior of the SO interaction is usual, e.g., a linear gate control of the Rashba terms. In contrast, in the second regime, there are emerging new features that one has to pay attention to when controlling SO couplings, e.g., the relative signs (same or opposite) of ${\alpha }_1$ and ${\alpha }_2$ can be controlled by the gate, ${\alpha }_2$ can attain zero in certain asymmetric configurations, and ${\alpha }_2$ remains essentially constant within a particular gate voltage range. In addition, we have determined the persistent-spin-helix symmetry points of the two subbands and found that the condition ${\alpha }_2=-{\beta }_{2,\mathrm{e}\mathit{ff}}$ always holds over a broad range of $V_{\mathrm{g}}$ s, thus possibly facilitating the locking of the symmetry point for the second subband in practice. Some of these features in the second regime can in principle be observed in proper double-well structures [24], as wide wells behave like “effective” double wells owing to the electron Hartree potential-induced central barrier. Moreover, the “symmetric configuration” mentioned throughout this work only refers to regions near the well (i.e., not far into the barriers). We cannot make our wells universally/fully symmetric by only tuning the gate voltage, due to the one-side doping in our system. This partial symmetric configuration is enough to render the intrasubband Rashba couplings to zero, since the envelope wave functions decay very quickly into the barriers. Our results should be timely and important for experiments controlling/tailoring the SO coupling universally, particularly for the unusual electrical control of the SO coupling in the second regime.

Secondly, we have investigated the full scenario of the electrical control of the SO interaction in a realistic GaAs/ ${\mathrm{Al}}_{0.3}{\mathrm{Ga}}_{0.7}$ As well with one or two additional ${\mathrm{Al}}_x{\mathrm{Ga}}_{1-x}$ As barriers embedded, in the course of the transition of our system from single to double and triple wells. We constantly consider the two-subband electron occupancy for all values of gate voltage $V_g$ studied here. As the Al content of the embedded barrier(s) $x$ varies, we find distinct scenarios of the electrical control of SO terms, e.g., linear or nonmonotonic dependence of ${\alpha }_2$ on $V_g$ , same or opposite signs between ${\alpha }_1$ and ${\alpha }_2$ , and inert or abrupt change of $\eta$ with $V_g$ near the symmetric configuration. In addition, we find that the gate dependence of SO terms is more smooth and ${\beta }_{\nu }$ is more stronger in our triple well, compared to the double well case. Moreover, we observe that the basic scenario of the electrical control of the SO interaction in our triple and double wells is similar, in the parameter range studied here. These results are expected to be important for a broad control of the SO interaction in semiconductor nanostructures.

As a final remark, in the case of three-subband electron occupancy which is not considered here, the electrical control of SO couplings is possibly distinct between double and triple wells because of a higher third subband occupation. More work is needed to investigate this interesting possibility (higher electron density).

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant No. 11874236), FAPESP, and Capes.