Effective mass theory of the Rashba effect in heterostructures

Within the envelope function scheme, a theory of the Rashba spin-orbit parameter is developed for an arbitrary heterostructure with sharp interfaces. In the first order of perturbation theory, the zero-field spin splitting in a given subband is found to be determined solely by the properties of the wave function at the interfaces; moreover, our results identify the proper electric field responsible for this splitting. The present work gives clear-cut proof that discontinuities in the band-structure parameters in the conduction band are the origin of the Rashba effect. Comparing our results to the Raman scattering data of Jusserand et al. [Phys. Rev. B 51, R4707 (1995)], in its simplest form our present theory rules out the valence-band offset as the source of the Rashba spin splitting, contrary to common belief. The corrections found here give hope for the use of accurate measurements of the Rashba spin splitting in heterostructures as a means to deduce conduction band offsets.

Figure 1

Ground state wave function $\zeta \left(z\right)$ computed for two values of the electric field $E_z$ in the well; the potential used in the calculation is also shown for $E_z=261.8\mathrm{kV}∕\mathrm{cm}$. The calculation is done with the effect of nonparabolicity and anisotropy included for a symmetric QW of width $d=180\mathrm{\AA }$ at $k=0$ and $\theta =0$. As the field gets stronger, the electron is pushed against the left-hand interface, thus increasing its probability there and reducing it in the right-hand interface.

Figure 2

Rashba spin splitting ${\delta }_{\mathbf{k}}^{\left(1\right)}$ in the ground subband as a function of the wave vector $k$ along the [1 0] direction. Curve (1): Contribution of the conduction-band offset. Curve (2): Contribution of the parallel to the interface mass mismatch. Curve (3): Contribution of the transverse mass mismatch. Curve (4): Contribution of the SO coupling mismatch between well and barriers. The solid line curve in black is the total spin splitting and the black square is the experimental point from Ref. 18.

Figure 3

Rashba spin-orbit parameter ${\alpha }_{\mathbf{k}}^{\left(n\right)}$ for the lowest three subbands $n=1$, 2, 3. As the subband index $n$ increases, the Rashba parameter decreases because in higher subbands the average electric field is weaker.

Figure 4

Rashba spin orbit parameter ${\alpha }_{\mathbf{k}}^{\left(1\right)}$ in the ground subband as a function of the wave vector $\mathbf{k}$ along [1 0]. As the magnitude of the wave vector increases, ${\alpha }_{\mathbf{k}}^{\left(1\right)}$ decreases in magnitude because of nonparabolicity.

Figure 5

Rashba spin orbit parameter ${\alpha }_{\mathbf{k}}^{\left(1\right)}$ in the ground subband as a function of the wave vector for three electric field strengths $E_z$. As the electric field increases, the whole curve representing ${\alpha }_{\mathbf{k}}^{\left(1\right)}$ vs $k$ rises up to higher values.