Improved effective equation for the Rashba spin-orbit coupling in semiconductor nanowires

Semiconductor Rashba nanowires are quasi-one-dimensional systems that have large spin-orbit (SO) coupling arising from a broken inversion symmetry due to an external electric field. There exist parametrized multiband models that can describe accurately this effect. However, simplified single band models are highly desirable to study geometries of recent experimental interest, since they may allow to incorporate the effects of the low dimensionality and the nanowire electrostatic environment at a reduced computational cost. Commonly used conduction band approximations, valid for bulk materials, greatly underestimate the SO coupling in zinc-blende crystal structures and overestimate it for wurtzite ones when applied to finite cross-section wires, where confinement effects turn out to play an important role. We demonstrate here that an effective equation for the linear Rashba SO coupling of the semiconductor conduction band can reproduce the behavior of more sophisticated eight-band $\mathrm{k}·\mathrm{p}$ model calculations. This is achieved by adjusting a single effective parameter that depends on the nanowire crystal structure and its chemical composition. We further compare our results to the Rashba coupling extracted from magnetoconductance measurements in several experiments on InAs and InSb nanowires, finding excellent agreement. This approach may be relevant in systems where Rashba coupling is known to play a major role, such as in spintronic devices or Majorana nanowires.

Figure 1

(a) Sketch of the type of systems studied in this work: an infinite semiconductor nanowire with hexagonal cross section of width $W_{\mathrm{wire}}$ (green) is placed over a dielectric substrate (purple) and may be covered by a metal (grey) on one or several of its facets. A back gate (black) allows to tune the chemical potential inside the wire. At its facets, the nanowire may develop a charge accumulation layer that we simulate with a surface charge density ${\rho }_{\mathrm{surf}}$. This charge layer may be present or absent at the interfaces with metals, depending on the chemical details of the heterojunction. All these elements contribute to create an electrostatic profile inside the wire that in turn influences the Rashba spin-orbit (SO) coupling of its energy bands. (b) Schematics of the eight lowest-energy bands of III-V semiconductors around the $\mathrm{\Gamma }$ point where these compounds exhibit a direct gap. These are grouped into four quasidegenerate pairs that comprise the conduction band, the light-hole, heavy-hole, and split-off valence bands. Their corresponding states, $|C_{\uparrow ,\downarrow }\rangle$, $|H{H}_{\uparrow ,\downarrow }\rangle$, $|L{H}_{\uparrow ,\downarrow }\rangle$, $|S{O}_{\uparrow ,\downarrow }\rangle$, serve as a truncated basis for the $\mathrm{k}·\mathrm{p}$ Kane model. ${\mathrm{\Delta }}_g$ is the semiconductor gap between conduction and valence bands, ${\mathrm{\Delta }}_{\mathrm{soff}}$ is the gap between valence and split-off bands, $P$ is the coupling between conduction and valence bands, and ${\gamma }_i$ are the intravalence band couplings, see Appendix pp1.

Figure 2

Rashba SO coupling modulus versus gate voltage for an InAs nanowire in an electrostatic environment like the one of Fig. 1. Different transverse subbands within the first conduction band of the wire are considered [(a)–(d)]. Dots correspond to ${\alpha }_R$ obtained from the eight-band (8B) $\mathrm{k}·\mathrm{p}$ Kane model of Appendix pp1 using Eq. (2). Solid lines correspond to ${\alpha }_R$ obtained from the effective conduction band (CB) approximation discussed in this work. In red (blue), the result for a zinc-blende (wurtzite) crystal using the improved Eq. (8) with the corresponding parameter $P_{\mathrm{fit}}$ displayed in Table 1. In green the result using the original Kane parameter $P$. In black the Rashba coupling obtained in the approximation ${\mathrm{\Delta }}_g\gg \left|e\varphi \left(\vec{r}\right)+E\right|$, i.e., using the simplified Eq. (7). Parameters are: $W_{\mathrm{wire}}=80$ nm, $W_{\mathrm{substrate}}=20$ nm, $W_{\mathrm{layer}}=10$ nm, ${\epsilon }_{\mathrm{wire}}={\epsilon }_{\mathrm{InAs}}=15.15$, ${\epsilon }_{\mathrm{substrate}}={\epsilon }_{{\mathrm{HfO}}_2}=25$, $V_{\mathrm{layer}}=0\mathrm{V}$ and ${\rho }_{\mathrm{surf}}=5\times {10}^{-3}\left(\frac{e}{{\text{nm}}^3}\right)$. The charge density of the wire ${\rho }_{\mathrm{mobile}}$ has been neglected for simplicity. The 8B model parameters are given in Tables 2 and 3.

Figure 3

Rashba coupling modulus versus gate voltage for the lowest energy subband within the conduction band approximation. (a) In green the results obtained using the improved Eq. (8) but with the original Kane parameter $P$. (b) In black, using the simplified Eq. (7). [(c) and (d)] In red and blue the results of the improved equation using the corresponding $P_{\mathrm{fit}}$ parameter displayed in Table 1 for a zinc-blende and a wurtzite crystal, respectively. Dots are used for full Schrödinger-Poisson simulations, solid lines for the Thomas-Fermi approximation and dotted lines for ${\rho }_{\text{mobile}}=0$, i.e., ignoring the mobile charge density. Same parameters as the ones of Fig. 2 (except ${\rho }_{\mathrm{mobile}}$). Temperature $T=10$ mK.

Figure 4

Fitting parameter $P_{\mathrm{fit}}$ of Eq. (8) as a function of the nanowire's width $W_{\mathrm{wire}}$ (curve with dots) for different semiconductors and crystal structures (a)–(d). This parameter is extracted by fitting Eq. (8) to 8B model results. The electrostatic environment corresponds to the one of Fig. 1 with parameters as in Fig. 2, but changing the dielectric permittivity of the wires to their corresponding ones, particularly ${\epsilon }_{\mathrm{InAs}}=15.15$, ${\epsilon }_{\mathrm{InSb}}=16.8$, ${\epsilon }_{\mathrm{GaAs}}=12.9,$ and ${\epsilon }_{\mathrm{GaSb}}=15.7$, extracted from Ref. [60]. For comparison, with a dashed line we show $P_{\mathrm{fit}}$ for a specific diameter, $W_{\mathrm{wire}}=80$ nm, which is the tabulated value given in Table 1, and with a solid line we show the original Kane parameter $P$.

Figure 5

Electrostatic environment modeling of some experimental setups (left), and corresponding effective Rashba couplings (right) obtained with magnetoconductance measurements (dots) and with conduction band (CB) numerical simulations (solid lines). In red we present results using the improved Eq. (8) for zinc-blende crystals, while in blue using the simplified Eq. (7). The shown experimental data corresponds to (a) Dhara et al. [62], (b) Liang et al. [25], (c) Takase et al. [27], (d) Scherübl et al. [26], and (e) Takase et al. [64]. The charge density of the wire is taken into account in the Thomas-Fermi approximation and the surface charge is ${\rho }_{\mathrm{surf}}=5\times {10}^{-3}\left(\frac{e}{{\text{nm}}^3}\right)$. The geometrical parameters used in the simulations can be found in the main text. The dielectric constants, extracted from Refs. [60, 76, 77], are ${\epsilon }_{\mathrm{InAs}}=15.15$, ${\epsilon }_{\mathrm{InSb}}=16.8$, ${\epsilon }_{{\mathrm{SiO}}_2}=3.9$, ${\epsilon }_{{\mathrm{HfO}}_2}=25$, ${\epsilon }_{{\mathrm{Al}}_2{\mathrm{O}}_3}=9,$ and ${\epsilon }_{\mathrm{PEO}}={10}^4$. In agreement with the experiments, in our simulations we fix the temperature to $T=1.7\mathrm{K}$ in (a), $T=4\mathrm{K}$ and the back gate to 0 in (b), $T=1.5\mathrm{K}$ in (c), $T=4.2\mathrm{K}$ and the back gate to 15 V in (d), and $T=1.7\mathrm{K}$ in (e).

Figure 6

(a) Modulus of the SO coupling as a function of the wire's width $W_{\mathrm{wire}}$ for a zinc-blende InAs nanowire in a homogeneous electric field $\vec{E}=(0,2,0)\mathrm{meV}\mathrm{nm}$. Different degrees of approximation are compared in this figure. In red we show the SO coupling found with the 8B model Hamiltonian of Eq. (A7), considered as the exact result in this work. The result to zeroth-order in the conduction band approximation is shown in green, to first order in cyan and to second order in pink. Dots mark the maximum of each curve. The dashed black curve corresponds to the improved SO Eq. (8), which considers a constant $P_{\mathrm{fit}}$ parameter fixed to $W_{\mathrm{wire}}=80$ nm. (b) Effective electron mass $m$ over bare electron mass $m_e$ versus $W_{\mathrm{wire}}$. In red the 8B model result and in green the value of the zeroth-order conduction band approximation.

Figure 7

Rashba SO coupling modulus versus gate voltage for different zinc-blende (ZB) (111) nanowires (rows) and different transverse modes (columns). Dots correspond to ${\alpha }_R$ obtained from the 8B $\mathrm{k}·\mathrm{p}$ Kane model of Appendix pp1 using Eq. (2). Solid lines correspond to ${\alpha }_R$ obtained from the improved Eq. (8) within the conduction band approximation discussed in Sec. 2b. The corresponding $P_{\mathrm{fit}}$ parameters are given in Table 1. Dashed lines correspond to ${\alpha }_R$ obtained in the approximation ${\mathrm{\Delta }}_g\gg \left|e\varphi \left(\vec{r}\right)+E\right|$, i.e., using the simplified Eq. (7). The simulations have been performed using the same electrostatic environment and parameters as in Fig. 2 of the main text, but changing the dielectric permittivity of the wires to their corresponding ones, ${\epsilon }_{\mathrm{InAs}}=15.15$, ${\epsilon }_{\mathrm{InSb}}=16.8$, ${\epsilon }_{\mathrm{GaAs}}=12.9$, or ${\epsilon }_{\mathrm{GaSb}}=15.7$. These values are extracted from Ref. [60].

Figure 8

Rashba SO coupling modulus versus gate voltage for a zinc-blende (111) InAs nanowire with different geometries and located in different electrostatic environments, sketched to the left of each sub-figure. Dots correspond to ${\alpha }_R$ of the lowest-energy subband obtained from the 8B model Hamiltonian using Eq. (2). Solid red curves correspond to the conduction band SO coupling obtained from the improved Eq. (8), by fitting $P_{\mathrm{fit}}$ in each case to the 8B model result. The resulting $P_{\mathrm{fit}}$ values are shown in each figure legend. Parameters: (a) same as in Fig. 2 of the main text; (b) $W_{\mathrm{wire}}=80$ nm, $W_{\mathrm{substrate}}=50$ nm, $W_{\mathrm{layer}}=10$ nm, ${\epsilon }_{\mathrm{wire}}={\epsilon }_{\mathrm{InAs}}=15.15$, ${\epsilon }_{\mathrm{substrate}}={\epsilon }_{{\mathrm{HfO}}_2}=25$, $V_{\mathrm{layer}}=0.2$ V and ${\rho }_{\mathrm{surf}}=0$; (c) $W_{\mathrm{wire}}=60$ nm, $W_{\mathrm{substrate}}=400$ nm, the separation between side gates and the nanowire is 70 nm, ${\epsilon }_{\mathrm{wire}}={\epsilon }_{\mathrm{InAs}}=15.15$, ${\epsilon }_{\mathrm{substrate}}={\epsilon }_{{\mathrm{SiO}}_2}=3.9$, $V_{\mathrm{bg}}=0$ V, $V_{\mathrm{L}}=0$ V and ${\rho }_{\mathrm{surf}}=5\times {10}^{-3}\left(\frac{e}{{\mathrm{nm}}^3}\right)$; (d) $W_{\mathrm{wire}}=40$ nm, $W_{\mathrm{substrate}}=40$ nm, ${\epsilon }_{\mathrm{wire}}={\epsilon }_{\mathrm{InAs}}=15.15$, ${\epsilon }_{\mathrm{substrate}}={\epsilon }_{{\mathrm{SiO}}_2}=3.9$ and ${\rho }_{\mathrm{surf}}=5\times {10}^{-3}\left(\frac{e}{{\mathrm{nm}}^3}\right)$. Notice that in this last case the nanowire has a square section instead of hexagonal. The charge density inside the wire ${\rho }_{\mathrm{mobile}}$ has been neglected for simplicity in (a)–(c), while it has been included in the Thomas-Fermi approximation in (d). The 8B model parameters are given in Table 2.

Figure 9

Conduction band effective mass $m$ versus gate voltage for different zinc-blende (ZB) (111) semiconductor nanowires (rows) and different transverse modes (columns). Dots correspond to $m$ obtained from the 8B $\mathrm{k}·\mathrm{p}$ Kane model of Appendix pp1 using Eq. (2). Solid lines correspond to $m$ obtained from the improved effective mass Eq. (G1). The corresponding $P_{\mathrm{mfit}}$ parameters are collected in Table 4. Dashed lines correspond to the zeroth-order simplified Eq. (6). The simulations have been performed using the same environment and parameters as in Fig. 7.