Pseudospin-electric coupling for holes beyond the envelope-function approximation

In the envelope-function approximation, interband transitions produced by electric fields are neglected. However, electric fields may lead to a spatially local $\left(k\text{−}\mathrm{independent}\right)$ coupling of band (internal, pseudospin) degrees of freedom. Such a coupling exists between heavy-hole and light-hole (pseudo)spin states in III-V semiconductors, such as GaAs, or in group IV semiconductors (germanium, silicon,...) with broken inversion symmetry. Here, we calculate the electric-dipole (pseudospin-electric) coupling for holes in GaAs from first principles. We find a transition dipole of 0.5 debye, a significant fraction of that for the hydrogen-atom $1s\to 2p$ transition. In addition, we derive the Dresselhaus spin-orbit coupling that is generated by this transition dipole for heavy holes in an asymmetric quantum well. A quantitative microscopic description of this pseudospin-electric coupling may be important for understanding the origin of spin splitting in quantum wells, spin coherence/relaxation $(T_2^{*}/T_1)$ times, spin-electric coupling for cavity-QED, electric-dipole spin resonance, and spin nonconserving tunneling in double quantum dot systems.

Figure 1

GaAs band structure generated using the elk code [18, 19]. The band structure is given between high-symmetry points $L\to \mathrm{\Gamma }\to X$. The labels ${\mathrm{\Gamma }}_{j_1{j}_2},j_1\in \{6,7,8\},j_2\in \{c,v\}$ for each band at the $\mathrm{\Gamma }$ point indicate the representation of the basis states with the subscript $j_2=c$ ($v$) indicating conduction- (valence-)band states. See Sec. 3.3 of Ref. [1] for more details. Inset: schematic showing detail of the topmost valence bands and lowest conduction band. The labels CB, HH, LH, SO indicate the conduction band, heavy-hole band, light-hole band, and split-off band, respectively.

Figure 2

Magnitude of the cubic spin-orbit coupling coefficients ${\gamma }_1^R$ [in blue, see Eq. (35)], ${\gamma }_2^D$ [in green, see Eq. (37)], and ${\gamma }_1^D$ [in yellow, see Eq. (36)] as a function of electric field $E_z$. The material parameters used (for GaAs) are given in Table 2.

Figure 3

Spin splitting $\mathrm{\Delta }$ in GaAs from eigenvalues of Eq. (33) (in blue) as a function of $k_{\parallel }=k_x$ (${\mathit{k}}_{\parallel }=k_{\parallel }\widehat{\mathit{x}}$) for an electric field $E_z=-{10}^6\mathrm{V}/\mathrm{m}$. The top horizontal axis indicates the hole sheet density, $n_p=k_{\parallel }^2/\left(2\pi \right)$, associated with a Fermi wave vector, $k_F=k_{\parallel }$. The splitting is linear in $k_{\parallel }$ for $k_{\parallel }\lesssim 0.02{\mathrm{nm}}^{-1}$, meaning the linear Dresselhaus spin-orbit coupling dominates in this region. The splitting due to the spin-orbit coupling term with coefficient ${\gamma }_1^R$ (in yellow) dominates at large $k_{\parallel }$. The material parameters used (for GaAs) are given in Table 2.

Figure 4

Magnitude of the linear Dresselhaus spin-orbit coupling coefficient $\beta$ (black solid line) as a function of electric field $E_z$. The dotted lines give the different contributions to $\beta$: ${\beta }_{C_k}$ [in green, see Eq. (40)], ${\beta }_{b1}$ [in red, see Eq. (43)], ${\beta }_{\chi }$ [in blue, see Eq. (41)], ${\beta }_0$ [in yellow, see Eq. (42)], and ${\beta }_{b2}$ [in purple, see Eq. (44)]. The material parameters used (for GaAs) are listed in Table 2. The gray line indicates the value of $E_z$ for which ${\beta }_{C_k}+{\beta }_{b1}=0$.

Figure 5

Total spin splitting $\mathrm{\Delta }$ (blue line) and spin splitting assuming $\chi \to 0$ (yellow line). For this choice of electric field ($E_z=-9\times {10}^6\mathrm{V}/\mathrm{m}$), ${\beta }_{\chi }$ is the largest contribution to the spin splitting (see Fig. 4). The top horizontal axis indicates the hole sheet density, $n_p=k_{\parallel }^2/\left(2\pi \right)$, associated with a Fermi wave vector, $k_F=k_{\parallel }$. The material parameters used (for GaAs) are displayed in Table 2.

Figure 6

Heavy-hole spin splitting in GaAs computed by diagonalizing the effective $2\times 2$ Hamiltonian obtained from the Schrieffer-Wolff transformation, as described in the main text (in green), by diagonalizing the $4\times 4$ Hamiltonian in the lowest valence-band subband (in blue) and by diagonalizing the $8\times 8$ Hamiltonian in the two lowest valence-band subbands (in yellow). The top horizontal axis indicates the hole sheet density, $n_p=k_{\parallel }^2/\left(2\pi \right)$, associated with a Fermi wave vector, $k_F=k_{\parallel }$. All calculations were performed for a triangular well with electric field $E_z=-{10}^6\mathrm{V}/\mathrm{m}$ with material parameters used (for GaAs) given in Table 2.

Figure 7

Linear spin-orbit coupling coefficients. The magnitude of the Dresselhaus coefficient (in blue) $\beta$ is larger than the magnitude of the Rashba coefficient (in yellow) $\alpha$ for all values of $E_z$ consi- dered. The values for the material parameters (for GaAs) are listed in Table 2.

Figure 8

Cubic spin-orbit coupling coefficients $-{\gamma }_1^D$ (in yellow), $-{\gamma }_3^D$ (in green), and ${\gamma }_2^R$ (in blue) as a function of electric field $E_z$. $|{\gamma }_1^D|$ is orders of magnitude smaller than the dominant cubic spin-orbit coupling term characterized by ${\gamma }_1^R$ (see Sec. 4a), however it is larger (in magnitude) than both ${\gamma }_3^D$ and ${\gamma }_2^R$, for the values of $E_z$ considered here. The values for the material parameters (for GaAs) are listed in Table 2.