## 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.

### Keywords

- spintronics
- spin-orbit interaction
- Rashba term
- Dresselhaus term
- persistent spin helix
- semiconductor
- quantum well

## 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 *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

where the first two terms refer to the kinetic contributions, in which

where *in the absence* of SO interaction [24]. Here we have defined _{th} quantized energy level (wave function) and

with

with

and the intersubband SO field is

where

The Rashba and Dresselhaus SO coefficients,

and

with the Rashba coefficients

Note that the Rashba strength

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

## 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.,

### 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

The width

### 3.2 Relevant parameters

In our GaAs/

### 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

Now, we turn to the second regime, in which we consider a well of

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

From Figure 3(e), we also find that

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 *effective* double wells.

In Figure 2(c), we show the linear Dresselhaus couplings

Now, we are ready to determine the persistent-spin-helix (PSH) symmetry points of the two subbands, at which the Rashba

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

For the second regime, we show in Figure 5(b) the intersubband SO couplings for a well of *enough* wells, where

### 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

In Figure 6(c), we show the dependence of

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

## 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

### 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

In Figure 7(a) and (b), we show the gate dependence of intrasubband Rashba terms in our GaAs/*local* fields (i.e., electron Hartree plus structural well) and the *universal* external gate field. For a larger value of

Figure 8(c) shows the dependence of linear intrasubband Dresselhaus coupling

Now, we determine the PSH symmetry points, where the Rashba

Besides altering the PSH symmetry condition involving SO terms of the first harmonic (

In Figure 9(a) and (b), we show the gate dependence of the intersubband Rashba

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

## 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 *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/

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.