Rapid Transition of the Hole Rashba Effect from Strong Field Dependence to Saturation in Semiconductor Nanowires

The electric field manipulation of the Rashba spin-orbit coupling effects provides a route to electrically control spins, constituting the foundation of the field of semiconductor spintronics. In general, the strength of the Rashba effects depends linearly on the applied electric field and is significant only for heavy-atom materials with large intrinsic spin-orbit interaction under high electric fields. Here, we illustrate in 1D semiconductor nanowires an anomalous field dependence of the hole (but not electron) Rashba effect (HRE). (i) At low fields, the strength of the HRE exhibits a steep increase with the field so that even low fields can be used for device switching. (ii) At higher fields, the HRE undergoes a rapid transition to saturation with a giant strength even for light-atom materials such as Si (exceeding 100 meV Å). (iii) The nanowire-size dependence of the saturation HRE is rather weak for light-atom Si, so size fluctuations would have a limited effect; this is a key requirement for scalability of Rashba-field-based spintronic devices. These three features offer Si nanowires as a promising platform for the realization of scalable complementary metal-oxide-semiconductor compatible spintronic devices.

Figure 1

The strength of the electron and hole Rashba effects in 1D InAs nanowires. (a) Predicted Rashba parameter ${\alpha }_R$ as a function of electric field for both electrons (open circles) and holes (dots) in a $R=15\mathrm{nm}$ InAs nanowire, by performing atomistic pseudopotential method calculations. The vertical arrow separates the low-field region, where HRE depends strongly on field (an effect that can be used beneficially for applications in spintronic devices), and the intermediate field region where field saturation exists. The magnitude of the HRE at saturation is far larger than the ERE, and can reach giant values in nanowires even for light-atom materials such as Si. The black solid line represents a linear fit to electron ${\alpha }_R$. (b) Same as in panel (a) but as a function of nanowire radius $R$, upon application of an electric field $E_x=30\mathrm{kV}/\mathrm{cm}$. In the case of holes, two solid curves represent the best fit of the rise and fall parts to $12.9\times R^{2.3}$ and $715.9\times R^{-0.4}$, respectively. In the case of electrons, the black solid line is the best fit to the well-known band-gap dependence of the ERE Rashba parameter [13].

Figure 2

Energy separations of hole subbands in InAs nanowires. (a) Band structure of a $R=15\mathrm{nm}$ InAs nanowire under an electric field of $100\mathrm{kV}/\mathrm{cm}$. Inset: Schematic of an InAs nanowire under an electric field. (b) Energy separations of the first excited ($h1$) and second excited ($h2$) hole subbands relative to the ground hole subband ($h0$) at the zone center as a function of electric field $E_x$ for the $R=15\mathrm{nm}$ InAs nanowire. (c) Energy separations as a function of nanowire radius $R$ under an electric field $E_x=30\mathrm{kV}/\mathrm{cm}$.

Figure 3

Electrical response of the HRE in nanowires. The HRE ${\alpha }_R$ as a function of applied electric field for (a) GaAs and InAs nanowires and (b) Ge and Si nanowires with varying radius $R$. (c) Corresponding HRE coefficient $r_{ss}^h$ in the subsaturation region as a function of nanowire radius $R$ for InAs, GaAs, Ge, and Si nanowires. The black solid line represents the ERE coefficient $r_{41}^e$ of InAs nanowires for comparison. (d) Saturation HRE ${\alpha }_R^{\mathrm{sat}}$ decays as a $R^{-\lambda }$ power law toward its asymptotic zero value in bulk. The best fit indicates $\lambda =0.39$, 0.44, 0.61, and 0.22 for InAs, GaAs, Ge, and Si nanowire materials, respectively.