Metallization of a Rashba wire by a superconducting layer in the strong-proximity regime

Semiconducting quantum wires defined within two-dimensional electron gases and strongly coupled to thin superconducting layers have been extensively explored in recent experiments as promising platforms to host Majorana bound states. We study numerically such a geometry, consisting of a quasi-one-dimensional wire coupled to a disordered three-dimensional superconducting layer. We find that, in the strong-coupling limit of a sizable proximity-induced superconducting gap, all transverse subbands of the wire are significantly shifted in energy relative to the chemical potential of the wire. For the lowest subband, this band shift is comparable in magnitude to the spacing between quantized levels that arises due to the finite thickness of the superconductor (which typically is $\sim 500$ meV for a 10-nm-thick layer of aluminum); in higher subbands, the band shift is much larger. Additionally, we show that the width of the system, which is usually much larger than the thickness, and moderate disorder within the superconductor have almost no impact on the induced gap or band shift. We provide a detailed discussion of the ramifications of our results, arguing that a huge band shift and significant renormalization of semiconducting material parameters in the strong-coupling limit make it challenging to realize a topological phase in such a setup, as the strong coupling to the superconductor essentially metallizes the semiconductor. This metallization of the semiconductor can be tested experimentally through the measurement of the band shift.

Figure 1

A quantum wire is lithographically defined within a semiconducting 2DEG and coupled to a superconducting layer with width $W$ and thickness $d$, as studied in Refs. [56, 57].

Figure 2

(a) Excitation spectrum in the absence of tunneling ($t=0$) obtained numerically from Eq. (1). The two lowest subbands of the wire are distinguished by color, and black curves correspond to subbands of the superconductor. (b) If a weak tunnel coupling is turned on ($t/t_s=0.035$), the subbands of the wire undergo a substantial shift in energy ($\delta E_n\sim 10\mathrm{\Delta }$). Tight-binding parameters are fixed to $d/a=42$, $W/a=175$, ${\mu }_s/t_s=0.1$ [112], $\mathrm{\Delta }/t_s={10}^{-4}$, $t_w/t_s=5$, ${\mu }_w=0$, $\alpha /\left(a{t}_s\right)=0.05$, ${\mathrm{\Delta }}_Z=0$.

Figure 3

(a) Proximity-induced gap $E_g$ increases as a function of tunneling strength $t$, with a large gap being induced only for $t\sim t_s$. (b) The four lowest transverse subbands of the wire (distinguished by color) shift in energy as $t$ is increased. The energy of each subband is measured at $k=0$, as schematically indicated by gray dashed subbands, and the energy of occupied bands is negative. Inset: subband energy increases quadratically with index $n$ (evaluated at $t=0.25t_s$), in agreement with Eq. (5). (c) The effective mass $m^{*}$, which is obtained by fitting panel (b) to Eq. (5), increases as a function of $t$. In the limit of large $t$, the mass approaches that of the superconductor ($m^{*}/m_e\to 1$). (d), (e) The spin-orbit splitting $E_{\text{SO}}$ and $g$ factor $\left|g\right|$ of each subband are reduced as a function of $t$, also approaching their values in the superconductor ($E_{\text{SO}}\to 0$ and $\left|g\right|\to 2$) in the limit of large $t$. All parameters are the same as in Fig. 2 [with $\left|g\right|=15$ in panel (e)].

Figure 4

(a) The induced gap $E_g$ is the same for both widths $W/a=1$ (black) and $W/a=175$ (red) at all values of $t$. Inset: $E_g$ (blue) is independent of width $W$ over several orders of magnitude. (b) Energy of lowest subband at $k=0$, $E_1\left(0\right)$, is also the same for $W/a=1$ and 175 at all values of $t$. (c), (d) Excitation spectrum for $W/a=1$ and 175, respectively. The spectrum of the lowest wire subband $E_1\left(k\right)$ (blue) is virtually unchanged as the width $W$ is increased and does not couple to low-energy superconducting subbands (black) appearing for $W/a=175$. All parameters are the same as in Fig. 2.

Figure 5

Induced gap $E_g$ as a function of superconductor thickness $d$ in a disordered 2D tight-binding model (with length $L/a=3\times {10}^4$) with various disorder strengths ${\sigma }_{\mu \left(\mathrm{\Delta }\right)}$, plotted for (a) $t=0.05t_s$ and (b) $t=0.25t_s$. In both the weak- and strong-coupling limits, the gap is largely unaffected by disorder unless chemical potential fluctuations in the superconductor are extreme (${\sigma }_{\mu }=1000\mathrm{\Delta }={\mu }_s$). Tight-binding parameters are fixed to $d/a=42$, $W/a=1$, ${\mu }_s/t_s=0.1$, $\mathrm{\Delta }/t_s={10}^{-4}$, $t_w/t_s=5$, ${\mu }_w/\mathrm{\Delta }=1$, $\alpha =0$, ${\mathrm{\Delta }}_Z=0$.

Figure 6

(a) Induced gap $E_g$ for various strengths of tunneling fluctuations: ${\sigma }_t=0$ (black), ${\sigma }_t=0.2t$ (blue), ${\sigma }_t=t$ (green), and ${\sigma }_t=2t$ (red). The gap is largely unaffected unless fluctuations become comparable to tunneling strength $t$. (b) Strong surface disorder (green) slightly enhances the induced gap compared with the clean limit (black). Surface disorder is incorporated through chemical potential fluctuations with ${\sigma }_{\mu }=1000\mathrm{\Delta }$ on the five sites furthest from the interface and with ${\sigma }_{\mu }=10\mathrm{\Delta }$ on the remaining sites within the superconductor. Tight-binding parameters are the same as in Fig. 5.

Figure 7

(a) Proximity-induced gap $E_g$ and (b) energy of lowest subband at $k=0$, $E_1\left(0\right)$, for parameters corresponding to epitaxial Al/InAs. The tunneling strength $t=0.1t_s$ is chosen large enough such that a sizable gap $E_g$ is induced for all values of $d$. In the strong-coupling limit, the wire subband undergoes a huge band shift $\delta E_1=\left|E_1\left(0\right)\right|\sim 200$ meV. The tight-binding parameters are set to $t_s=117$ eV (corresponding to lattice spacing $a=0.175$ Å), ${\mu }_s=11.7$ eV, $\mathrm{\Delta }=250\mu \mathrm{eV}$, $t_w=50t_s$ (corresponding to $m^{*}=0.02m_e$), ${\mu }_w=0$, $\alpha =0.42$ eV Å (corresponding to $E_{\text{SO}}=250\mu \mathrm{eV}$), and ${\mathrm{\Delta }}_Z=0$.

Figure 8

Schematic illustration of the metallization of the Rashba wire. (a) Spectrum of the wire in the absence of tunneling. We assume the chemical potential to be tuned to the crossing point of the lowest subband and take the wire to have a small effective mass $m^{*}\sim 0.02m_e$ and large spin-orbit splitting $E_{\text{SO}}\sim 250\mu \mathrm{eV}$. (b) Renormalization of the spectrum in the strong-coupling limit. The lowest wire subband experiences a large band shift (taken from Fig. 7 to be $\delta E_1\sim -200$ meV), and renormalization of the effective mass ($m^{*}\sim 0.2m_e$) leads to the occupation of many wire subbands. The subbands of the wire also have a significantly reduced spin-orbit splitting $E_{\text{SO}}\sim 25\mu \mathrm{eV}$.

Figure 9

(a) The band shift is significantly reduced by increasing the thickness $d$. (b) When $d$ is increased, the crossover from weak coupling ($E_g\ll \mathrm{\Delta }$) to strong coupling ($E_g\sim \mathrm{\Delta }$) occurs at much smaller $t$. Tight-binding parameters are the same as in Fig. 7.

Figure 10

(a) Proximity-induced gap $E_g$, (b) energy of lowest subband at $k=0$, $E_1\left(0\right)$, (c) effective mass $m^{*}$, (d) spin-orbit splitting $E_{\text{SO}}$, and (e) $g$ factor plotted as a function of tunneling strength $t$. Tight-binding parameters are the same as those in Fig. 7, with $d=9.08$ nm (corresponding to $d/a=519$).

Figure 11

Energy of lowest subband at $k=0$, $E_1\left(0\right)$, plotted as a function of tunneling strength $t$ for various wire chemical potentials ${\mu }_w$. While the band shift $\delta E_1$ is dependent on ${\mu }_w$, the bottom of the band always approaches the same energy in the strong-coupling limit. This indicates that the spectrum is determined entirely by the superconductor in this limit. Tight-binding parameters are the same as in Fig. 7.

Figure 12

(a) Energy of four lowest transverse subbands at $k=0$, $E_n\left(0\right)$, plotted as a function of tunneling strength $t$. Parameters are the same as in Fig. 2, with ${\mu }_w=150\mathrm{\Delta }$. Comparing with Fig. 3 (which has ${\mu }_w=0$), we see that all four subbands approach the same energy in the limit of large $t$ regardless of ${\mu }_w$. (b) Effective mass $m^{*}$ for both ${\mu }_w=150\mathrm{\Delta }$ (black curve) and ${\mu }_w=0$ [red curve, corresponding to Fig. 3]. Because the band shifts are smaller when ${\mu }_w=150\mathrm{\Delta }$, the mass is renormalized more quickly as a function of $t$.