Effects of Gate-Induced Electric Fields on Semiconductor Majorana Nanowires

We study the effect of gate-induced electric fields on the properties of semiconductor-superconductor hybrid nanowires which represent a promising platform for realizing topological superconductivity and Majorana zero modes. Using a self-consistent Schrödinger-Poisson approach that describes the semiconductor and the superconductor on equal footing, we are able to access the strong tunneling regime and identify the impact of an applied gate voltage on the coupling between semiconductor and superconductor. We discuss how physical parameters such as the induced superconducting gap and Landé $g$ factor in the semiconductor are modified by redistributing the density of states across the interface upon application of an external gate voltage. Finally, we map out the topological phase diagram as a function of magnetic field and gate voltage for InAs/Al nanowires.

Popular Summary

Practical quantum computers will depend on the implementation of robust quantum bits (qubits) to store and process information. Majorana zero modes, electron-hole superpositions that provide exponential protection against external noise, are ideal states to realize reliable qubits. However, the systems supporting Majorana modes exhibit complex behavior that obscures desired experimental signatures and makes the design of Majorana-based qubits a challenge. Here, we develop a comprehensive quantitative theoretical treatment aimed at understanding such complex behavior in semiconductor-superconductor nanowires that support Majorana modes.

Our model treats the strong-coupling regime between the semiconductor and superconductor on an equal footing, and it shows the effects of both disorder and external electric fields. We show how physical parameters, such as the semiconductor’s induced gap and the Landé $g$ factor, are affected by the presence of external fields and disorder. We then obtain the topological phase diagram as a function of the experimental control parameters, magnetic field, and gate voltage, taking into account the presence of disorder.

By showing how the topological state can be tuned via external gates and how it is affected by the presence of disorder, our results provide useful guidance for the fabrication and tuning of devices aiming to realize Majorana-based topological qubits, the next step toward the realization of a topologically protected and, therefore, fault-tolerant quantum computer.

Figure 1

(a) SM-SC heterostructure based on hexagonal nanowire. 10-nm-thick Al layer (blue) is deposited on two facets of InAs (brown) hexagonal wire with a height of 60 nm. The back gate is shown schematically in gray. (b) Rectangular geometry of the wire that supports the same number of subbands. The back gate is emulated by a boundary condition at the bottom.

Figure 2

The electrostatic calculation uses Dirichlet boundary conditions at $z=0$ and $z=L_z^{\mathrm{SM}}$, i.e., the top and bottom of the semiconducting wire. At $z=0$, the boundary condition is given by the gate voltage $V_g$, while at the interface to the aluminum it is given by the band offset $W$ (see also Table 1). This leads to an accumulation layer at the interface.

Figure 3

(a) Electronic density in the cross-sectional cut of the nanowire for $V_g=0$ obtained by the Thomas-Fermi approximation. (b) Square modulus of eigenstates of the wire in the normal state at $B=0$ with energies $-0.096$, $-0.068$, $-0.052$, $-0.023$, $-0.021$, and $-0.006\mathrm{eV}$ for panels 1–6, respectively.

Figure 4

(a) Electrostatic potential profile $\varphi (z)$ and (b) electronic density $n(z)$ in the semiconducting part of the system obtained from a self-consistent Schrödinger-Poisson calculation for three representative values of the gate voltage.

Figure 5

Eigenstates of the Hamiltonian Eq. (4) at $k_x=0$ for $V_g=-0.15\mathrm{V}$, $V_g=0\mathrm{V}$, and $V_g=0.2\mathrm{V}$. Panels (a)–(c) show the electrostatic potential for reference purposes, with horizontal lines denoting the bottom of each band below the Fermi energy. The color scale indicates the weight in the superconductor, and in the semiconducting part, the intensity indicates the square modulus of the eigenstate. Panels (d)–(f) show the eigenfunctions explicitly with the same color coding.

Figure 6

Top: Band structure in the normal state at $V_g=-0.15\mathrm{V}$. Color indicates the weight of the state in the superconductor. Hybridization between states is seen by the changing color of the subbands. Bottom: Band structure of the system in the superconducting state. The induced gap in each subband depends on the hybridization to the superconductor. It can clearly be seen that bands with stronger hybridization (red) are characterized by a larger induced gap. Here we used the same parameters as in Figs. 5 and 5.

Figure 7

Characterization of the superconducting state at $B=0$, i.e., in the trivial $s$-wave superconducting phase, as a function of the gate voltage $V_g$. (a) Density of states, (b) induced gap (black) and weight in the superconductor (green), (c) Fermi energy, (d) Fermi momentum, (e) Fermi velocity, and (f) SC coherence length. The discontinuities correspond to transitions where bands are driven below the chemical potential and thus become occupied [for an illustration, consider the transition between Figs. 8 and 8]. In panel (b), the correspondence between the magnitude of the induced gap and the hybridization between SM and SC (as measured by the $W^{\mathrm{SC}}$) is clearly shown. In panels (c) and (d), all bands are shown in gray, while the occupied band closest to the Fermi energy is highlighted in black.

Figure 8

Illustration of typical band structures, and the two different scenarios for the topological phase transition from an even to an odd number of occupied spin subbands. Panels (a) and (c) show the situation without magnetic field, $B=0$, where a subband is slightly below [(a), one spin-degenerate Fermi point] or above the chemical potential [(c), no Fermi points]. Upon turning on a finite field that exceeds the distance from the bottom of these subbands to the chemical potential, the corresponding panel on the right [(b) and (d)] is obtained, where an odd number of spin-split subbands is occupied.

Figure 9

The evolution of the system parameters as a function of a magnetic field $B$. Estimates for the (a) gap, (b) absolute value of the effective Fermi level of the subband $|ϵ(k_x=0)|$ (where $k_x=0$ is the location of the band bottom), (c) Fermi momentum, and (d) Fermi velocity as a function of magnetic field. Colors correspond to different gate voltages: for blue lines, the gate voltage is more positive, leading to a situation as sketched in Figs. 8 and 8. Red lines, on the other hand, have a more strongly negative gate voltage, thus leading to the situation of Figs. 8 and 8. Gray color corresponds to the threshold value $V_g=-0.427\mathrm{V}$, at which the subband is characterized by a vanishing effective chemical potential.

Figure 10

Absolute value of $g$ factor (left-hand column) and ratio of the induced gap to the critical magnetic field $2\mathrm{\Delta }/({\mu }_B{B}_c)$ (right-hand column) as a function of gate voltage in the vicinity of threshold gate voltage values $V_g=-0.07$ (a),(b), $-0.15$ (c),(d), $-0.45$ (e),(f), $-0.6$ (g),(h), $-2.3$ (i),(j) at which the number of subbands change in the system. The red dashed line is the absolute value of bare semiconductor $g$ factor $|g_{\mathrm{SM}}^{\mathrm{bare}}|$.

Figure 11

Gate voltage dependence of (a) the absolute value of effective $g$ factor and (b) the weight in superconductor. Panel (c) relates $|g|$ to the weight in the superconductor $W^{\mathrm{SC}}$. Dots represent typical values for the highest-occupied band at a given gate voltage. The dashed horizontal values are the absolute values of the bare $g$ factor in the semiconductor (red) and superconductor (green). When the coupling to the superconductor is weak, as indicated by a small weight of the wave functions in the SC, the $g$ factor is close to the bare InAs value. The opposite limit occurs in the strongly hybridized regime at more negative gate voltages.

Figure 12

(a) Topological phase diagram and magnitude of the spectral gap over a range of gate voltages from zero to very strongly negative. Clearly, the gap at zero field is largest for negative gate voltages where hybridization with the Al shell is strongest. The boundaries of the topological phase are marked with solid black lines. Dependence of the (b) gap normalized to the bare Al gap $E_g/{\mathrm{\Delta }}_0$ and estimates for (c) Fermi velocity and (d) coherence length as a function of magnetic field for the gate voltages indicated by dashed lines in (a).

Figure 13

Phase diagram (a),(c),(e) and band structure (b),(d),(f) in the vicinity of threshold gate voltages, corresponding to the change in the number of subbands: $V_g=-0.15$ (a),(b); $V_g=-0.6$ (c),(d); $V_g=-2.3$ (e),(f). Black dots in the left-hand panels correspond to the numerically calculated phase boundary. Red and green curves in left-hand panels are the estimates obtained using the standard relation Eq. (9) and the bare SM’s $g$ factor for the red curves and the renormalized $g$ factor, Fig. 10, for the green ones.

Figure 14

(a) SM-SC heterostructure in the slab geometry in the presence of disorder on the surface of superconductor (not to scale). (b) Square modulus of the eigenstate in the presence of disorder highlighting the enhanced scattering in the superconductor.

Figure 15

(a) Weight in the superconductor and (b) induced gap as a function of the gate voltage for a range of disorder strengths $U_D=0–1\mathrm{eV}$ averaged over 25 disorder realizations. The presence of disorder increases the induced gap.

Figure 16

Topological phase phase diagram in the presence of disorder. Plotted are the positions of the local minima $B_c^{*}$ of critical magnetic fields in $B$, $V_g$ plane. Empty dots corresponds to separate disorder realizations, filled symbols are disorder averages with the corresponding error bars. Increasing disorder strength leads to larger critical fields.