Open access peer-reviewed chapter

Control of Spin-Orbit Interaction in Conventional Semiconductor Quantum Wells

By Jiyong Fu, Wei Wang and Minghua Zhang

Submitted: May 28th 2018Reviewed: September 2nd 2018Published: December 20th 2018

DOI: 10.5772/intechopen.81260

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 . 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 Vg. 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 z001direction. We start from the 8×8Kane 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]:

H3D=2k22m22m2z2+Vscz+HR+HD,E1

where the first two terms refer to the kinetic contributions, in which mis the electron effective mass and k=kx2+ky2is the in-plane electron momentum with x100and y010directions. The third term Vscdescribes the confining electron potential with the subscript scindicating that it is determined self-consistently within the (Poisson-Schrödinger) Hartree approximation. The confining potential contains the structural potential Vwarising from the band offsets, the doping potential Vd, the electron Hartree potential Ve, and the external gate potential Vg[13, 23, 24]. The last two terms, HRand HD, correspond to the Rashba and Dresselhaus SO interactions, respectively. Note that in Eq. (1), we have added the Dresselhaus term HDempirically [4, 13, 24], since to obtain HDone has to take into account the remote bands (e.g., p-conduction band) . The Rashba term reads HR=ηzkxσykyσxwith ηz=ηwzVw+ηHzVg+Vd+Vedetermining the Rashba strength and σx,y,zthe spin Pauli matrices. The constants ηwand ηHare determined by the bulk band parameters [13, 24]:

ηw=P23δv/δcEg2δΔ/δcEg+Δ2,E2
ηH=P231Eg21Eg+Δ2,E3

where Egis the fundamental bandgap, Pis the Kane parameter, and Δis the split-off gap, in the well layer. The parameters δc, δv, and δΔstand 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 HD=γσxkxky2kz2+c.c., with γthe bulk Dresselhaus parameter and kz=iz. 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 Ekν=ν+2k2/2mand the corresponding eigenspinors kνσ=kνσ, rkν=expikrψνz, of the well in the absence of SO interaction . Here we have defined Eν(ψν), ν=1,2, as the νth quantized energy level (wave function) and σ=,as the electron spin component along the zdirection. We then can straightforwardly obtain an effective 2D model by projecting Eq. (1) onto the basis kνσ. The effective 2D model with two subbands in the coordinate system x+110, x11¯0under the basis set k1k1k2k2reads :

H2D=2k22m+E+11Eτz1+HRD,E4

with E±=E2±E1/2, 1the 2×2identity matrix (in both spin and orbital subspaces), and τx+,x,zthe Pauli (“pseudospin”) matrices acting within the orbital subspace. The term HRDdescribes the Rashba and Dresselhaus SO contributions in terms of intra- and intersubband SO fields BSOνand BSO12, respectively:

HRD=12gμBν=1,2τνσBSOν+τx+σBSO12,E5

with gthe electron gfactor, μBthe Bohr magneton, and τ1,2=1±τz/2. Explicitly, the intrasubband SO field is

BSOν=2gμBkανβν,effsinθβν,3sin3θx̂+αν+βν,effcosθβν,3cos3θx̂,E6

and the intersubband SO field is

BSO12=2gμBkηΓcosθx̂+η+Γsinθx̂,E7

where θis the angle between kand the x+axis. Figure 1.A schematic of the band offsets for GaAs/AlGaAs well, in which E g ( E g b ) and Δ ( Δ b ) are the fundamental bandgap and the split-off gap in the well (barrier), respectively. δ c , δ v , and δ Δ 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, αν, βν, η, and Γ, appearing in Eqs. (6) and (7) read

ηνν'=ψνηwzVw+ηHzVg+Vd+Veψν',E8

and

Γνν'=γψνkz2ψν',E9

with the Rashba coefficients ανηνν, ηη12and the Dresselhaus coefficients βνΓννand ΓΓ12. The coefficient βν,eff=βνβν,3(Eq. (6)) is the renormalized “linear” Dresselhaus coupling, due to the cubic correction βν,3=γkν2/4, where kν=2πnνis the νth-subband Fermi wave number with nνthe νth-subband occupation.

Note that the Rashba strength αν(Eq. (8)) can be written in terms of several distinct “individual” contributions, i.e., αν=ανg+ανd+ανe+ανw, with ανg=ηHψνzVgψνthe gate contribution, ανd=ηHψνzVdψνthe doping contribution, ανe=ηHψνzVeψνthe electron Hartree contribution, and ανw=ηwψνzVwψνthe structural contribution. Similar for the intersubband Rashba term η=ηg+ηd+ηe+ηw, the matrix element for each contribution is calculated between different subbands. For convenience, below we use ανg+d=ανg+ανdand ηg+d=ηg+ηdfor 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 Vscas 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 T>30K, we believe this is a minor effect in our system when T=75K. 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., Vg=0). By tuning Vg, we alter electron occupations from two subbands to one subband. For each value of Vg, within the self-consistent Hartree approximation, we determine the relevant SO strengths, i.e., the intrasubband αν, ν=1,2, and intersubband ηRashba couplings, and similarly for the Dresselhaus term, the intrasubband βνand the intersubband Γ. 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. : the 001-grown GaAs wells of width wsandwiched between 48-nm Al0.3Ga0.7As barriers. Our structure contains only one delta-doping (Si) layer, sitting 17 nm away from the well interface, with donor concentration nd. 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 ne. 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 ne=nd.

The width wof quantum wells we consider ranges from 20 to 70 nm. Two distinct regimes of the SO interaction we found are marked off around w=wc=3035nm. Without loss of generality, we focus on a well of w=25nm for the first regime and a well of w=65nm for the second one. For the 25-nm well, we choose nd=8.0×1011cm2and T=75K, 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 Vg’s. While for the 65-nm well, we choose relatively small values of ndand T, i.e., nd=4.0×1011cm2and T=0.3K, to exclude the electron occupation of a higher third subband.

3.2 Relevant parameters

In our GaAs/Al0.3Ga0.7As wells, the values of relevant band parameters (see Eqs. (2) and (3)) are as follows: Eg=1.519eV and Δ=0.341eV [4, 29, 30]. The corresponding counterparts of Egand Δin Al0.3Ga0.7As barriers are Egb=1.951eV and Δb=0.329eV [13, 24, 30]. As a consequence, we have the band offsets for conduction and valence bands, δc=EgbEG×60.4%=0.261eV, δv=EgbEGδc=0.171eV, and δΔ=δvΔ+Δb=0.159eV [13, 24, 30]. For the Kane parameter, we choose P=1.0493nm [4, 29]. Then, we obtain ηw=3.97Å2and ηH=5.30Å2(Eqs. (2) and (3)). We treat the bulk Dresselhaus constant γas 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) γ11.0eVÅ3. 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 w=25nm. In Figure 2(b), we show the dependence of Rashba αν(ν=1,2) and Dresselhaus βνcoefficients of the two subbands on Vg. We find that α1and α2have the same sign and they both decrease to zero near the symmetric configuration (at Vg0.36eV) (see arrow in Figure 2(b)) and further change their sign. For the confining electron potential Vscand wave functions ψνin this symmetric geometry of the well, see Figure 2(a). Clearly, our wells are asymmetric at Vg=0because of the one-side delta doping. In contrast to the Rashba term, we find that the Dresselhaus couplings βν=γψνkz2ψνremain essentially constant with Vg. The inset in Figure 2(b) shows the electron density nνof the two subbands with n1+n2=neas a function of Vg, and the second subband starts to be unpopulated around Vg0.5eV. Figure 2.(a) Self-consistent potential V sc and wave function profiles ψ ν of a 25-nm well at V 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 α ν and Dresselhaus β ν SO couplings, for the 25 -nm (b) and 65-nm wells (c). In (b), the inset shows electron density n ν with n 1 + n 2 = n e as a function of V g . In (c), the vertical dashed line at V g ∼ 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. .

Now, we turn to the second regime, in which we consider a well of w=65nm, as shown in Figure 2(c). As opposed to the first regime, we find that α1and α2could 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 . Figure 3.(a)–(c) Potential V sc and wave function profiles ψ ν ( ν = 1 , 2 ) for a 65-nm well at three values of voltages, with V 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 ν 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. .

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 α1and α2separately, i.e., ανe, ανw, and ανg+d. For the structural contribution, ανwψνzVwψν=ψνw/22ψνw/22, α1wis dominated by the first term, while α2wis dominated by the second term, due to the already discussed tendency of the subband wave functions of ψ1and ψ2to respectively localize on the right and left halves of the well (zero bias for instance). This leads to opposite signs between α1wand α2w. Similarly, for ανeψνzVeψν, the electron Hartree contributions of the two subbands have opposite signs as well since Vemostly has opposite slopes with respect to zfor z>0and z<0. However, for ανg+dψνzVg+Vdψν, 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 α1and α2depends on the interplay of the three constituents as a function of gate voltages.

From Figure 3(e), we also find that α2remains essentially constant with Vgat lower voltages, precisely, as its constituent α2wbehaves except for a prefactor difference. In contrast, it is found that α1sensitively depends on Vgin the whole parameter range studied here. Moreover, α1and α2become zero (see arrow in Figure 3(d) and left arrow in Figure 3(e)) across the symmetric configuration, corresponding to Vg0.18eV. Interestingly, we observe that α2can even attach zero at Vg0.24eV (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 Vg0.18eV), 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 βν=γψνkz2ψνin the second regime. We observe that the inequality β1<β2, 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 Vg=0, we find β1>β2[see crossing (black dot) between β1and β2in 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 ανand the renormalized linear Dresselhaus βν,eff(due to cubic corrections) are equal in strength, i.e., ανβν,eff, as shown in Figure 4. We observe that α1β1,efftakes place at Vg0.17eV (see right arrow in Figure 4) for the first subband, while α2β2,effalways holds over the range of Vg=0100meV (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 α2and β2,effhere, 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 β2,3—thus tuning β2,effto match with α2. 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 . Figure 4.The dependence of intrasubband Rashba α ν and renormalized Dresselhaus β ν , e ff coefficients on V 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. .

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 ηand its distinct constituents including ηg+d, ηe, and ηwand Dresselhaus coupling Γfor a well of w=25nm. We find that both ηand Γexhibit basically the linear behavior with Vg. Owing to the orthogonality between ψ1and ψ2, the gate plus doing contribution ηg+dvanishes. As for the constituents ηwand ηe, the dominant variation with the gate is the latter as the electron density in wells changes with Vg, even though the magnitude of ηwis greater than ηe. Figure 5.Gate control of intersubband Rashba η and Dresselhaus Γ 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. .

For the second regime, we show in Figure 5(b) the intersubband SO couplings for a well of w=65nm. In contrast to the first regime, here ηlargely remains constant as Vgincreases. And a maximum of ηoccurs at Vg0.18eV (symmetric configuration). This arises from the fact that the wave functions of the two subbands, ψ1and ψ2, are mostly separated in very asymmetric configurations (Figure 3(b) and (c)), which renders η(depending on the overlap of ψ1and ψ2) almost constant and relatively weak. Similar to η, Γalso 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 ψ1and ψ2have vanishing overlap, both ηand Γtend 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=2070nm, below we discuss how the two regimes transit from one to the other. To explore this, we only focus on α2, which is mostly in contrast between the two regimes.

In Figure 6(c), we show the dependence of α2on Vg. Clearly, for the 20- and 24-nm wells, α2changes 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 win the range of w=3035nm, the gate dependence of α2behaves 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 Vscand wave function profiles ψνfor the wells of w=30and 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 sc and wave function profiles ψ ν in GaAs wells at V g ∼ 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 α 2 of the second subband on V 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. .

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=2045nm), 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=6070nm), 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. : the 001-grown GaAs well of width w=26nm sandwiched between 48-nm Al0.3Ga0.7As barriers. To have double (triple) wells, we introduce one (two) additional AlxGa1xAs barrier layers of length b=3nm embedded inside the above structure. We choose the total electron density ne=6.0×1011cm2. The temperature is held fixed at T=0.3K. Note that we choose relatively low neand T, instead of the experimental values ne=8.0×1011cm2and T=75K 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 Vgs 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 AlxGa1xAs 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/Al0.3Ga0.7Aswells, for several Al contents xof the central barrier AlxGa1xAs. At x=0(single well), we find the usual scenario, i.e., α1and α2have the same sign and both change almost linearly with Vg. 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 ανwith Vgfollows. In contrast, when xis 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 α1and α2, 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, α2first increases with Vg, while peaks at some point depending on xand 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, α2increases with Vgmore abruptly before it peaks, see black circles in Figure 7(b) for a set of values of x. Note that α1in magnitude consistently increases with Vg, since the local fields and the universal external gate field contribute to α1constructively. Figure 7.Intrasubband Rashba α 1 (a) and α 2 (b) couplings as a function of V g in GaAs/ Al 0.3 Ga 0.7 As wells for several Al contents x of a central barrier Al x Ga 1 − x 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 α 2 exhibits the maximum, from Wang et al. . Figure 8.(a)–(c) Self-consistent potential V sc and wave function profiles ψ ν for a GaAs/ Al 0.3 Ga 0.7 As well with Al x Ga 1 − x 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 β ν as a function of V g , for several values of x s with x = 0,0.1,0.2,0.3 . (e) and (f) Gate dependence of SO strengths ( α ν , β ν , β ν , e ff ) for x = 0 and x = 0.3 , respectively. The black points indicate the PSH symmetry points (i.e., α ν = ± β ν , e ff ), from Wang et al. .

Figure 8(c) shows the dependence of linear intrasubband Dresselhaus coupling βν=γkz2on Vg. As opposed to αν, we find that βνis weakly dependent on Vg(±0.5meV Å) for all values of xconsidered here. On the other hand, as xincreases, 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 β1and β2becomes more close for a larger value of x.

Now, we determine the PSH symmetry points, where the Rashba ανand the renormalized linear Dresselhaus βν,eff=βνβν,3are matched. The cubic Dresselhaus term βν,3=γkν2/4=γπnν/2depends on the electron occupations and therefore alters the condition of the PSH symmetry. Figure 8(e) and (f) show the gate dependence of αν, βν, and βν,eff, for x=0and 0.3, respectively. Clearly, we see the distinction between βνand βν,effbecause of the correction from βν,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 (α1=β1,eff); 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 β2is weaker so that α2and β2have the feasibility of matched strength by tuning Vg.

Besides altering the PSH symmetry condition involving SO terms of the first harmonic (sin/cosfunctions of θ), the cubic Dresselhaus β3,νalso has the third harmonic (3θ) 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 ΓDν=γ2kν6τP/42, with τPthe momentum relaxation time. For GaAs wells, with γ11.0eV Å3, kν0.2nm1, and τP1.0ps [13, 31], we find ΓDν0.07ns1.

In Figure 9(a) and (b), we show the gate dependence of the intersubband Rashba ηand Dresselhaus Γcouplings, which depend on the overlap of the wave functions of the two subbands; see Eqs. (8) and (9). For the intersubband Rashba coupling η, we find that it remains essentially constant at x=0, since ψ1and ψ2are well overlapped even at asymmetric configurations; see Figure 8(a) with Vg=0.1eV. However, when xis 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, ψ1and ψ2tend to be separated. Therefore, ηexhibits the strongest at the symmetric configuration of the system (i.e., Vg=0). Note that at the symmetric configuration, ψ1and ψ2are perfectly overlapped. Furthermore, the separation of ψ1and ψ2is more distinct for a larger x(same Vg), cf. Figure 8(b) and (c), which renders ηmore sensitively dependent on Vgnear Vg=0. As for the intersubband Dresselhaus term Γ=1kz22, on the one hand, it depends on the overlap of ψ1and ψ2and accordingly strongly depends on Vgnear Vg=0, similar to η. On the other hand, as opposed to η, the term Γvanishes at Vg=0and further changes sign across the symmetric configuration, similar to the intrasubband Rashba terms αν. Figure 9.Intersubband Rashba η (a) and Dresselhaus Γ (b) couplings as a function of V g in GaAs/ Al 0.3 Ga 0.7 As wells, for several Al contents x of the central barrier Al x Ga 1 − x As with x = 0,0.1,0.2,0.3,0.4 , from Wang et al. .

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 ψ1and ψ2are more overlapped in a triple well under the same external conditions (e.g., same Vg), 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 Vgvaries, as shown in Figure 10(c). Moreover, we find that the Dresselhaus term βνis 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/ Al 0.3 Ga 0.7 As double well (a) and triple well (b) with embedded barrier Al x 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 ( α ν ) and Dresselhaus ( β ν ) and Intersubband Rashba ( η ) and Dresselhaus ( Γ ) for the GaAs/ Al 0.3 Ga 0.7 As triple well with the embedded barrier Al x Ga 1 − x As of x = 0.3 , from Wang et al. .

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 Vg. We have considered a set of wells of the width win 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 α1and α2can be controlled by the gate, α2can attain zero in certain asymmetric configurations, and α2remains 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 α2=β2,effalways holds over a broad range of Vgs, 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/Al0.3Ga0.7As well with one or two additional AlxGa1xAs 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 Vgstudied here. As the Al content of the embedded barrier(s) xvaries, we find distinct scenarios of the electrical control of SO terms, e.g., linear or nonmonotonic dependence of α2on Vg, same or opposite signs between α1and α2, and inert or abrupt change of ηwith Vgnear the symmetric configuration. In addition, we find that the gate dependence of SO terms is more smooth and βνis 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.

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Jiyong Fu, Wei Wang and Minghua Zhang (December 20th 2018). Control of Spin-Orbit Interaction in Conventional Semiconductor Quantum Wells, Functional Materials, Dipti Sahu, IntechOpen, DOI: 10.5772/intechopen.81260. Available from:

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