Intersubband-induced spin-orbit interaction in quantum wells

Recently, we have found an additional spin-orbit (SO) interaction in quantum wells with two subbands [Bernardes et al., Phys. Rev. Lett. 99, 076603 (2007)]. This new SO term is nonzero even in symmetric geometries, as it arises from the intersubband coupling between confined states of distinct parities, and its strength is comparable to that of the ordinary Rashba. Starting from the $8\times 8$ Kane model, here we present a detailed derivation of this new SO Hamiltonian and the corresponding SO coupling. In addition, within the self-consistent Hartree approximation, we calculate the strength of this new SO coupling for realistic symmetric modulation-doped wells with two subbands. We consider gated structures with either a constant areal electron density or a constant chemical potential. In the parameter range studied, both models give similar results. By considering the effects of an external applied bias, which breaks the structural inversion symmetry of the wells, we also calculate the strength of the resulting induced Rashba couplings within each subband. Interestingly, we find that for double wells the Rashba couplings for the first and second subbands interchange signs abruptly across the zero bias, while the intersubband SO coupling exhibits a resonant behavior near this symmetric configuration. For completeness we also determine the strength of the Dresselhaus couplings and find them essentially constant as function of the applied bias.

Figure 1

(Color online) (a) Schematic of the band structure of direct gap zincblend semiconductors near the $\Gamma$ point $\left(k=0\right)$. The label ${\Gamma }_6$ represents the $s$ states in the conduction band, while ${\Gamma }_7$ (split-off holes) and ${\Gamma }_8$ (heavy holes and light holes) represent the $p$ states in the valence bands. (b) Band offsets for a double quantum well of width $L_w$ with a central barrier of width $L_b$. The relationships among the several offset parameters are given in Eqs. (16, 17).

Figure 2

(Color online) Schematic view of our quantum-well system. The doping densities ${\rho }_a$, ${\rho }_b$ and the external gate voltages $V_a$ and $V_b$ can be used to control the degree of the structural inversion asymmetry.

Figure 3

(Color online) (a) Self-consistent potential energy $V_{\text{sc}}$ (thick solid line) and the corresponding wave functions ${\psi }_0$ and ${\psi }_1$ for the single well ${\text{Al}}_{0.48}{\text{In}}_{0.52}\text{As}/{\text{Ga}}_{0.47}{\text{In}}_{0.53}\text{As}$ with external gates $V_a=0\text{eV}$ and $V_b=1.2\text{eV}$ (see Fig. 2). The electronic Hartree potential $V_e$ (short dashed line), the external gate plus modulation doping contributions $V_g$ (long dashed line), and the corresponding force fields $F_e=-d{V}_e/dz$ and $F_g=-d{V}_g/dz$ are also shown in (b). The two levels in the well (solid lines) denote the energies of the first and second subband edges, while the dotted level indicates the chemical potential.

Figure 4

(Color online) Rashba $\alpha$, Dresselhaus $\beta$, and intersubband-induced $\eta$ SO coupling constants for the ${\text{Al}}_{0.48}{\text{In}}_{0.52}\text{As}/{\text{Ga}}_{0.47}{\text{In}}_{0.53}\text{As}$ quantum well as functions of the gate voltage $V_b$ (see Fig. 2). In (a) the total 2D electron density is kept constant at $n_T=20\times {10}^{11}{\text{cm}}^{-2}$ and in (b) the chemical potential is kept constant at $\mu =200\text{meV}$.

Figure 5

Several distinct contributions to the coupling constants $\eta$ (a), ${\alpha }_0$ (b) and ${\alpha }_1$ (c) for the single GaInAs quantum well shown in Fig. 4a ($n_T$-constant model) as functions of the external gate $V_b$ $\left(V_a=0\right)$. These contributions arise from the electron density (Hartree potential), the external gate (together with donor regions), and the structural well potential; these are denoted by the superscripts $e$, $g$ and $w$, respectively.

Figure 6

(Color online) Rashba $\alpha$, Dresselhaus $\beta$, and the intersubband-induced $\eta$ SO couplings for a InSb double quantum well as functions of the right gate voltage $V_b$. In (a) the total electron density is kept constant at $n_T=10\times {10}^{11}{\text{cm}}^{-2}$ and in (b) the chemical potential is kept constant at $\mu =100\text{meV}$ (relative to initial bottom well).

Figure 7

Rashba $\alpha$, intersubband-induced $\eta$, and Dresselhaus $\beta$ couplings vs $V_b$ about the symmetric configuration $V_b=V_a=0\text{eV}$ for the double InSb well in Fig. 6a.

Figure 8

(Color online) Ratios ${\alpha }_v/\eta$ and ${\beta }_v/\eta$ for the InSb double well in Fig. 6. The insets are blowups around $V_b=0$, the symmetric configuration.

Figure 9

Different contributions to the SO couplings $\eta$, ${\alpha }_0$, and ${\alpha }_1$ for the InSb double well in the constant areal density model [see Fig. 6a] as functions of the external gate $V_b$ $\left(V_a=0\right)$. In the subfigures we show the contributions to the coupling constants coming from the areal electronic density, indicated by the superscript $e$, and from the external $\text{gate}+\text{donor}$ regions $g$, and also from the structural potential, being $w$ for the well and $b$ for the central barrier.

Figure 10

(Color online) Top row: Wave functions ${\psi }_0$ and ${\psi }_1$ for the InSb double well (Sec. 5B1) with external biases $V_b=+0.3\text{eV}$, $V_b=0\text{eV}$, and $V_b=-0.3\text{eV}$ (first, second, and third columns, respectively). The electronic Hartree contribution $V_e\left(z\right)$ to the potential energy and the corresponding force field $F_e=-d{V}_e/dz$ are shown on the second row. The third row shows the quantities $F_e^{00}\left(z\right)={\psi }_0{F}_e{\psi }_0$ and $F_e^{11}\left(z\right)={\psi }_1{F}_e{\psi }_1$ and the bottom row shows $F_e^{01}\left(z\right)={\psi }_0{F}_e{\psi }_1$. Here $n_T=10\times {10}^{11}{\text{cm}}^{-2}$. The vertical dotted lines mark the center of the 20-nm-wide AlInSb barrier within the 50-nm-wide InSb well.

Figure 11

(a) Total electronic areal density $n_T=n_0+n_1$, (b) subband areal densities $n_0$ and $n_1$, (c) ratio ${\mathcal{E}}_{\text{SO}}/\Delta \mathcal{E}$ and (d) effective-mass ratios ${\overline{m}}_{\pm }=m_{\pm }^{\ast }/m^{\ast }$ [Eq. (53)], as functions of the external gate $V_b$ $\left(V_a=0\right)$ in the $\mu$-constant model $\left(\mu =100\text{meV}\right)$.

Figure 12

(Color online) Schematic representation of doping layers of width $w$ and densities ${\rho }_a$ and ${\rho }_b$ plus the external gates $V_a$ and $V_b$. By varying the external gates (we usually keep $V_a=0$ and vary $V_b$) we can alter the spatial symmetry of our quantum wells; see Fig. 2. The curves illustrate the calculated $\text{gate}+\text{modulation}$ doping potential $V_g\left(z\right)$ for $V_a=V_b$ (dashed line) and $V_a>V_b$ (solid line), both with ${\rho }_a>{\rho }_b$.