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.
- spin-orbit interaction
- Rashba term
- Dresselhaus term
- persistent spin helix
- quantum well
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 . Recently, the SO effects  have attracted renewed interest in diverse fields of condensed matter, including the persistent spin helix (PSH) [5, 6, 7, 8], topological insulators , 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 , 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 , 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  and crossed spin helices .
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 . 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 direction. We start from the 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]:
where the first two terms refer to the kinetic contributions, in which is the electron effective mass and is the in-plane electron momentum with and directions. The third term describes the confining electron potential with the subscript indicating that it is determined self-consistently within the (Poisson-Schrödinger) Hartree approximation. The confining potential contains the structural potential arising from the band offsets, the doping potential , the electron Hartree potential , and the external gate potential [13, 23, 24]. The last two terms, and , correspond to the Rashba and Dresselhaus SO interactions, respectively. Note that in Eq. (1), we have added the Dresselhaus term empirically [4, 13, 24], since to obtain one has to take into account the remote bands (e.g., -conduction band) . The Rashba term reads with determining the Rashba strength and the spin Pauli matrices. The constants and are determined by the bulk band parameters [13, 24]:
where is the fundamental bandgap, is the Kane parameter, and is the split-off gap, in the well layer. The parameters , , and 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 , with the bulk Dresselhaus parameter and . 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 and the corresponding eigenspinors , , of the well in the absence of SO interaction . Here we have defined (), , as the th quantized energy level (wave function) and as the electron spin component along the direction. We then can straightforwardly obtain an effective 2D model by projecting Eq. (1) onto the basis . The effective 2D model with two subbands in the coordinate system , under the basis set reads :
with , the identity matrix (in both spin and orbital subspaces), and the Pauli (“pseudospin”) matrices acting within the orbital subspace. The term describes the Rashba and Dresselhaus SO contributions in terms of intra- and intersubband SO fields and , respectively:
with the electron factor, the Bohr magneton, and . Explicitly, the intrasubband SO field is
and the intersubband SO field is
where is the angle between and the axis.
with the Rashba coefficients , and the Dresselhaus coefficients and . The coefficient (Eq. (6)) is the renormalized “linear” Dresselhaus coupling, due to the cubic correction , where is the th-subband Fermi wave number with the th-subband occupation.
Note that the Rashba strength (Eq. (8)) can be written in terms of several distinct “individual” contributions, i.e., , with the gate contribution, the doping contribution, the electron Hartree contribution, and the structural contribution. Similar for the intersubband Rashba term , the matrix element for each contribution is calculated between different subbands. For convenience, below we use and 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 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.  and Rigamonti and Proetto  at zero temperature. As this discontinuity vanishes for K, we believe this is a minor effect in our system when 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., ). By tuning , we alter electron occupations from two subbands to one subband. For each value of , within the self-consistent Hartree approximation, we determine the relevant SO strengths, i.e., the intrasubband , , and intersubband Rashba couplings, and similarly for the Dresselhaus term, the intrasubband and the intersubband . Finally, the case of the two regimes in between is discussed as well.
The quantum wells we consider are similar to the samples experimentally studied by Koralek et al. : the 001-grown GaAs wells of width sandwiched between 48-nm As barriers. Our structure contains only one delta-doping (Si) layer, sitting 17 nm away from the well interface, with donor concentration . Our prior simulated SO couplings  on these samples agree well with the data obtained via the transient-spin-grating technique , where there is no bias applied and all wells share the same areal electron density . Here we go beyond the experiment in  by using an external bias, which can vary the electron density and control the SO interaction. At zero bias we assume .
The width of quantum wells we consider ranges from 20 to 70 nm. Two distinct regimes of the SO interaction we found are marked off around nm. Without loss of generality, we focus on a well of nm for the first regime and a well of nm for the second one. For the -nm well, we choose and K, the same as experimental parameters in , not only ensuring the electron occupation of the second subband at zero bias  but also retaining this second subband occupation over a broad range of ’s. While for the 65-nm well, we choose relatively small values of and , i.e., and K, to exclude the electron occupation of a higher third subband.
3.2 Relevant parameters
In our GaAs/As wells, the values of relevant band parameters (see Eqs. (2) and (3)) are as follows: eV and eV [4, 29, 30]. The corresponding counterparts of and in As barriers are eV and eV [13, 24, 30]. As a consequence, we have the band offsets for conduction and valence bands, eV, eV, and eV [13, 24, 30]. For the Kane parameter, we choose nm [4, 29]. Then, we obtain and (Eqs. (2) and (3)). We treat the bulk Dresselhaus constant 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) eV. We use this value in our simulations, consistent with the one obtained in a recent study by Walser et al. .
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 nm. In Figure 2(b), we show the dependence of Rashba () and Dresselhaus coefficients of the two subbands on . We find that and have the same sign and they both decrease to zero near the symmetric configuration (at eV) (see arrow in Figure 2(b)) and further change their sign. For the confining electron potential and wave functions in this symmetric geometry of the well, see Figure 2(a). Clearly, our wells are asymmetric at because of the one-side delta doping. In contrast to the Rashba term, we find that the Dresselhaus couplings remain essentially constant with . The inset in Figure 2(b) shows the electron density of the two subbands with as a function of , and the second subband starts to be unpopulated around eV.
Now, we turn to the second regime, in which we consider a well of nm, as shown in Figure 2(c). As opposed to the first regime, we find that and 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 , the nonballistic spin field transistor operating with orthogonal spin quantization axes , and the transition from a topological insulator to Dirac semimetal .
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 and separately, i.e., , , and . For the structural contribution, , is dominated by the first term, while is dominated by the second term, due to the already discussed tendency of the subband wave functions of and to respectively localize on the right and left halves of the well (zero bias for instance). This leads to opposite signs between and . Similarly, for , the electron Hartree contributions of the two subbands have opposite signs as well since mostly has opposite slopes with respect to for and . However, for , 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 and depends on the interplay of the three constituents as a function of gate voltages.
From Figure 3(e), we also find that remains essentially constant with at lower voltages, precisely, as its constituent behaves except for a prefactor difference. In contrast, it is found that sensitively depends on in the whole parameter range studied here. Moreover, and become zero (see arrow in Figure 3(d) and left arrow in Figure 3(e)) across the symmetric configuration, corresponding to eV. Interestingly, we observe that can even attach zero at 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 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. , 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 in the second regime. We observe that the inequality , 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 , we find [see crossing (black dot) between and 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 and the renormalized linear Dresselhaus (due to cubic corrections) are equal in strength, i.e., , as shown in Figure 4. We observe that takes place at eV (see right arrow in Figure 4) for the first subband, while always holds over the range of 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 and 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 —thus tuning to match with . These symmetry points are also crucial for the persistent skyrmion lattice  as well as the nonballistic spin field effect transistors  operating with orthogonal spin quantization axes .
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 . For the latter, it may in general destroy the helix but has a negligible effect on the results for our wells here .
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 and its distinct constituents including , , and and Dresselhaus coupling for a well of nm. We find that both and exhibit basically the linear behavior with . Owing to the orthogonality between and , the gate plus doing contribution vanishes. As for the constituents and , the dominant variation with the gate is the latter as the electron density in wells changes with , even though the magnitude of is greater than .
For the second regime, we show in Figure 5(b) the intersubband SO couplings for a well of nm. In contrast to the first regime, here largely remains constant as increases. And a maximum of occurs at eV (symmetric configuration). This arises from the fact that the wave functions of the two subbands, and , are mostly separated in very asymmetric configurations (Figure 3(b) and (c)), which renders (depending on the overlap of and ) almost constant and relatively weak. Similar to , 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 and have vanishing overlap, both and 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 nm, below we discuss how the two regimes transit from one to the other. To explore this, we only focus on , which is mostly in contrast between the two regimes.
In Figure 6(c), we show the dependence of on . Clearly, for the 20- and 24-nm wells, changes almost linearly, showing the usual behavior of the Rashba coupling, as demonstrated in the first regime. However, for the wells of , and 70 nm, we observe the unusual behavior, which is referred to the second regime. When in the range of nm, the gate dependence of 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 and wave function profiles for the wells of 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.
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 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 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.
The main structure of our well is again similar to the samples experimentally studied by Koralek et al. : the 001-grown GaAs well of width nm sandwiched between 48-nm As barriers. To have double (triple) wells, we introduce one (two) additional As barrier layers of length nm embedded inside the above structure. We choose the total electron density . The temperature is held fixed at K. Note that we choose relatively low and , instead of the experimental values and K in , 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 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 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/wells, for several Al contents of the central barrier As. At (single well), we find the usual scenario, i.e., and have the same sign and both change almost linearly with . 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 with follows. In contrast, when 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 and , 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, first increases with , while peaks at some point depending on 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 , increases with more abruptly before it peaks, see black circles in Figure 7(b) for a set of values of . Note that in magnitude consistently increases with , since the local fields and the universal external gate field contribute to constructively.
Figure 8(c) shows the dependence of linear intrasubband Dresselhaus coupling on . As opposed to , we find that is weakly dependent on (meV Å) for all values of considered here. On the other hand, as 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 and becomes more close for a larger value of .
Now, we determine the PSH symmetry points, where the Rashba and the renormalized linear Dresselhaus are matched. The cubic Dresselhaus term depends on the electron occupations and therefore alters the condition of the PSH symmetry. Figure 8(e) and (f) show the gate dependence of , , and , for and , respectively. Clearly, we see the distinction between and because of the correction from , 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 (); see black point in Figure 8(e) and (f). To attain the PSH symmetry for the second subband, one needs an even wider well , where is weaker so that and have the feasibility of matched strength by tuning .
Besides altering the PSH symmetry condition involving SO terms of the first harmonic (/functions of ), the cubic Dresselhaus also has the third harmonic () contribution; see Eq. (6). This leads to the decay of the PSH due to the D’yakonov-Perel’ (DP) spin relaxation mechanism . Specifically, the third harmonic Dresselhaus coupling-induced spin relaxation rate at the PSH symmetry point is written as , with the momentum relaxation time. For GaAs wells, with eV , , and ps [13, 31], we find .
In Figure 9(a) and (b), we show the gate dependence of the intersubband Rashba and Dresselhaus couplings, which depend on the overlap of the wave functions of the two subbands; see Eqs. (8) and (9). For the intersubband Rashba coupling , we find that it remains essentially constant at , since and are well overlapped even at asymmetric configurations; see Figure 8(a) with eV. However, when 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, and tend to be separated. Therefore, exhibits the strongest at the symmetric configuration of the system (i.e., ). Note that at the symmetric configuration, and are perfectly overlapped. Furthermore, the separation of and is more distinct for a larger (same ), cf. Figure 8(b) and (c), which renders more sensitively dependent on near . As for the intersubband Dresselhaus term , on the one hand, it depends on the overlap of and and accordingly strongly depends on near , similar to . On the other hand, as opposed to , the term vanishes at and further changes sign across the symmetric configuration, similar to the intrasubband Rashba terms .
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 and are more overlapped in a triple well under the same external conditions (e.g., same ), 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 varies, as shown in Figure 10(c). Moreover, we find that the Dresselhaus term 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.
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 . We have considered a set of wells of the width 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 and can be controlled by the gate, can attain zero in certain asymmetric configurations, and 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 always holds over a broad range of 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 , 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/As well with one or two additional 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 studied here. As the Al content of the embedded barrier(s) varies, we find distinct scenarios of the electrical control of SO terms, e.g., linear or nonmonotonic dependence of on , same or opposite signs between and , and inert or abrupt change of with near the symmetric configuration. In addition, we find that the gate dependence of SO terms is more smooth and 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).
This work was supported by the National Natural Science Foundation of China (Grant No. 11874236), FAPESP, and Capes.