Hybridization at Superconductor-Semiconductor Interfaces

Hybrid superconductor-semiconductor devices are currently one of the most promising platforms for realizing Majorana zero modes. Their topological properties are controlled by the band alignment of the two materials, as well as the electrostatic environment, which are currently not well understood. Here, we seek to fill in this gap and address the role of band bending and superconductor-semiconductor hybridization in such devices by analyzing a gated single Al-InAs interface using a self-consistent Schrödinger-Poisson approach. Our numerical analysis shows that the band bending leads to an interface quantum well, which localizes the charge in the system near the superconductor-semiconductor interface. We investigate the hybrid band structure and analyze its response to varying the gate voltage and thickness of the Al layer. This is done by studying the hybridization degrees of the individual subbands, which determine the induced pairing and effective $g$ factors. The numerical results are backed by approximate analytical expressions which further clarify key aspects of the band structure. We find that one can obtain states with strong superconductor-semiconductor hybridization at the Fermi energy, but this requires a fine balance of parameters, with the most important constraint being on the width of the Al layer. In fact, in the regime of interest, we find an almost periodic dependence of the hybridization degree on the Al width, with a period roughly equal to the thickness of an Al monolayer. This implies that disorder and shape irregularities, present in realistic devices, may play an important role for averaging out this sensitivity and, thus, may be necessary for stabilizing the topological phase.

Popular Summary

Interfaces between metals and semiconductors are ubiquitous in semiconducting electronics. Such interfaces have recently attracted renewed attention in the context of engineered topological superconducting nanowires, which harbor charge-neutral, zero-energy edge excitations, the so-called Majorana zero modes. Remarkably, these zero modes can, in principle, be harnessed for noise-immune quantum computing. Here, we analyze a gated single Al-InAs interface using numerical and analytical methods. The Al-InAs system is currently the most promising candidate for the creation of topological superconductors because of its extremely clean interfaces.

There is strong experimental evidence of Majorana zero modes in metal-semiconductor interfaces when spin-orbit coupling and a magnetic field are present. But these modes are accessible only below a critical temperature, at which the metal becomes superconducting. The systems are typically operated at temperatures well below 1 K. Many microscopic details of the interfaces and the degree of metal-semiconductor hybridization that control topological properties—including the nature of the chemical bounds at the interface, the degree of interaction between the electron’s charge and spin, and the electrostatic landscape—have remained unclear, however. Here, we resolve the energy spectrum and the electrostatic landscape of the Al-InAs interface. We show that strong hybridization is achievable but that it requires a fine balance of parameters and a controlled Al width. We also demonstrate that the degree of hybridization depends almost periodically on the Al width, with a period roughly equal to the thickness of the Al monolayer. This finding implies that disorder and shape irregularities, present in realistic devices, may average out this sensitivity and stabilize the topological phase.

We expect that our findings will help improve designs of the next generation of Majorana devices and shed light on engineering semiconductor-superconductor hybrids.

Figure 1

(a) Schematic of the hybrid device consisting of a layer of metal, semiconductor, and dielectric with translational invariance in the plane. The rightmost edge of the dielectric is in contact with a gate electrode which keeps it at a voltage $V_G$ with respect to the grounded metallic layer. This translates into a voltage difference between the grounded metallic layer and the lower edge of the semiconductor, which we denote $V_D$. (b) Band diagram for the metal and semiconductor region before contact. (c) Band diagram of the metal and semiconductor region after contact. (d) Self-consistent band edge profiles in the semiconductor region for several values of $V_D$. Solid lines indicate the results obtained via the self-consistent Schrödinger-Poisson method and dashed lines indicate the results obtained from the Thomas-Fermi approximation. (e) Self-consistent charge density profiles in the semiconductor region for several values of $V_D$. Solid lines indicate the results obtained via the self-consistent Schrödinger-Poisson method and dashed lines indicate the results obtained from the Thomas-Fermi approximation.

Figure 2

Hybrid band structure of Al-InAs obtained with parameters $L_1=5\mathrm{nm}$, $L_2=100\mathrm{nm}$, $\mathrm{\Phi }=0.1\mathrm{eV}$, and $V_D=-0.5\mathrm{V}$. (a) Large-scale enlargement of the band structure showing that the bands are predominately Al-like with a narrow region of band segments, which have a large InAs weight. (b) Enlargement showing that band segments with strong hybridization appear only for narrow ranges of $k$ values. (c) Enlargement at the Fermi level revealing that this specific choice of parameters does not lead to states with strong hybridization at the Fermi energy.

Figure 3

(a)–(f) Band structure dependence on Al thickness and $V_D$. Horizontal direction indicates changing Al thickness and vertical indicates changing $V_D$. The semiconductor weights at the Fermi level of crossing bands are displayed in the top right-hand corner of each panel.

Figure 4

Band structure dependence on $\mathrm{\Phi }$ obtained with parameters $L_1=4.75\mathrm{nm}$, $L_2=100\mathrm{nm}$, and $V_D=-0.5\mathrm{V}$. Increasing the value of $\mathrm{\Phi }$ leads to a deeper well at the superconductor-semiconductor interface, which gives rises to more InAs-like bands below the Fermi level. In panels (a)–(c) we show the band structure for $\mathrm{\Phi }=0.1\mathrm{eV}$, $\mathrm{\Phi }=0.3\mathrm{eV}$, and $\mathrm{\Phi }=0.5\mathrm{eV}$, respectively.

Figure 5

(a)–(f) InAs weights for $L_1=5\mathrm{nm}$ and $\mathrm{\Phi }=0.1\mathrm{eV}$. Solid purple line: Weights obtained numerically using the self-consistent Schrödinger-Poisson method for $L_2=100\mathrm{nm}$. Dotted blue line: Weights obtained numerically using the Schrödinger equation within the square-well model for $L_2^{\prime }=16\mathrm{nm}$. Dashed red line: Weights obtained within the square-well model using the approximate analytical expressions of Eqs. (14), (17), and (22) for $L_2^{\prime }=16\mathrm{nm}$. The energetically highest weakly modified Al level is the one with $n_{*}=27$. We observe that the numerical methods are in good agreement up to $n=30$. For $n>30$, the numerical results obtained using the square-well model show significant deviations from the ones calculated with the actual triangular potential. The analytical results follow the numerical ones and manage to capture the qualitative features of the weights. However, the approximate analytical approach is inadequate to describe bands with $n\ge 31$.

Figure 6

(a)–(f) InAs weights for $L_1=4.75\mathrm{nm}$ and $\mathrm{\Phi }=0.1\mathrm{eV}$. Solid purple line: Weights obtained numerically using the self-consistent Schrödinger-Poisson method for $L_2=100\mathrm{nm}$. Dotted blue line: Weights obtained numerically using the Schrödinger equation within the square-well model for $L_2^{\prime }=16\mathrm{nm}$. Dashed red line: Weights obtained within the square-well model using the approximate analytical expressions of Eqs. (14), (17), and (24) for $L_2^{\prime }=16\mathrm{nm}$. The energetically highest weakly modified Al level is the one with $n_{*}=26$. The two numerical methods yield quite similar results up to $n=28$. For $n>28$, the numerical results obtained using the square-well model show significant deviations from the ones calculated using the Schrödinger-Poisson method. The analytical results follow the numerical ones and manage to capture the qualitative features of the weights. The approximate analytical expression fail to describe weights corresponding to $n\ge 29$.

Figure 7

Panels (a) and (b) depict the two possible hybrid band structure scenarios for energies in the vicinity of the isolated InAs conduction band edge. The band structures shown are obtained via the self-consistent Schrödinger-Poisson method for $\mathrm{\Phi }=0.1\mathrm{eV}$ and $L_2=100\mathrm{nm}$. In (a) [(b)] we have $L_1=5\mathrm{nm}$ [$L_1=4.75\mathrm{nm}$] and we have additionally shown the relevant pure Al and InAs bands for $L_2^{\prime }=16\mathrm{nm}$. The dashed (cyan) ellipses illustrate band segments of strong superconductor-semiconductor hybridization which glue the pure Al and InAs bands together and generally appear for $k_{\ell }\le k\le k_h$; see Eq. (15). In (a) we encounter scenario I in which $k_{\ell }=0$ for $n=n_{*}+1$ and a band appears below the pure InAs bands due to the hybridization-induced downward bending of the $(n_{*}+1)$th pure Al band. This scenario is realized when the condition of Eq. (16) is satisfied. In contrast, in (b) this condition is not satisfied and a separated band does not appear below the pure InAs levels. However, gluing segments with $k_{\ell }>0$ appear, to connect the pure InAs and Al bands.

Figure 8

(a)–(d) InAs weights shown in the upper (lower) panels for $L_1=5\mathrm{nm}$ ($L_1=4.75\mathrm{nm}$) and $\mathrm{\Phi }=0.1\mathrm{eV}$. Solid purple line: Weights obtained numerically using the self-consistent Schrödinger-Poisson method for $L_2=100\mathrm{nm}$. Dashed red line: Weights obtained within the square-well model using $L_2^{\prime }=16\mathrm{nm}$. Dotted green line: Weights obtained using the extended square-well model which assumes an energy-dependent $L_2^{\prime }$ for bands above the Fermi level. Here we have used $L_2^{\prime }=40.8\mathrm{nm}$ ($L_2^{\prime }=91.6\mathrm{nm}$) for the left (right) panels. In both cases, the extended analytical model yields a significantly improved agreement with the Schrödinger-Poisson approach.

Figure 9

Hybrid band structure for an Al-InAs nanowire with a square $100\times 100{\mathrm{nm}}^2$ InAs cross section. The band structure is obtained after imposing confinement along one of the planar directions (e.g., $y$). The parameter values are the same as in Fig. 3. The confinement leads to a “splitting” of the original bands shown in Fig. 3, thus resulting into a finite number of channels which are shifted in energy but exhibit similar hybridization profiles. We remark that, for presentation purposes, we have not included the confinement channels originating from purely metallic bands. Note also that we have neglected the possible electrostatic effects near the boundaries of the confined dimension.

Figure 10

Self-consistent band edge profiles and band structures obtained using the methods described in Appendix pp2 with $L_1=4.75\mathrm{nm}$ and $V_D=-0.5\mathrm{V}$. Results for both $\mathrm{\Phi }=0.1\mathrm{eV}$ and $\mathrm{\Phi }=0.3\mathrm{eV}$ are shown. (a) Band edge profiles obtained for $\mathrm{\Phi }=0.1\mathrm{eV}$. (b) Band structures obtained for $\mathrm{\Phi }=0.1\mathrm{eV}$. The numbers in the lower left-hand corner are the weights at the Fermi energy of the bands marked by the black circle. (c) Band edge profiles obtained for $\mathrm{\Phi }=0.3\mathrm{eV}$. (d) Band structures obtained for $\mathrm{\Phi }=0.3\mathrm{eV}$. The numbers in the lower left-hand corner are the weights at the Fermi energy of the bands marked by the black circle.

Figure 11

Plot of the lhs (dashed blue lines) and rhs (solid gold lines) of the direct and inverted transcendental Eqs. (c1) and (c2) for $\mathrm{\Phi }=0.1\mathrm{eV}$ and $L_2^{\prime }=16\mathrm{nm}$. In (a) and (c) $L_1=5\mathrm{nm}$, and in (b) and (d) $L_1=4.75\mathrm{nm}$. The eigenenergies are obtained when the lhs and rhs lines cross. Depending on the energy regime, it is convenient to use the direct or inverted transcendental equation in order to obtain the eigenspectrum. The approximate analytical expressions for the eigenenergies are obtained via a Taylor expansion of the lhs or/and the rhs about zeros of the respective tangent or cotangent. These zeros are related to the pure Al and InAs levels and the values $E_{n,k}^{\mathrm{Al}}$ and $E_{n,k}^{\mathrm{InAs}}$, as well as $E_{n+1/2,k}^{\mathrm{Al}}$ and $E_{n+1/2,k}^{\mathrm{InAs}}$. Panel (a) is employed for inferring weakly modified Al-like bands and InAs-like bands within scenario I. Panel (b) is employed for inferring weakly modified Al-like bands within scenario II. Panel (c) is employed for inferring the Al-InAs gluing segments within scenario I. Panel (d) is employed for inferring modified InAs-like bands within scenario II.