Two-terminal charge tunneling: Disentangling Majorana zero modes from partially separated Andreev bound states in semiconductor-superconductor heterostructures

We show that a pair of overlapping Majorana bound states (MBSs) forming a partially separated Andreev bound state (ps-ABS) represents a generic low-energy feature in spin-orbit-coupled semiconductor-superconductor (SM-SC) hybrid nanowire in the presence of a Zeeman field. The ps-ABS interpolates continuously between the “garden variety” ABS, which consists of two MBSs sitting on top of each other, and the topologically protected Majorana zero modes (MZMs), which are separated by a distance given by the length of the wire. The really problematic ps-ABSs consist of component MBSs separated by a distance of the order of the characteristic Majorana decay length $\xi$, and have nearly zero energy in a significant range of control parameters, such as the Zeeman field and chemical potential, within the topologically trivial phase. Despite being topologically trivial, such ps-ABSs can generate signatures identical to MZMs in local charge tunneling experiments. In particular, the height of the zero-bias conductance peak (ZBCP) generated by ps-ABSs has the quantized value $2e^2/h$, and it can remain unchanged in an extended range of experimental parameters, such as Zeeman field and the tunnel barrier height. We illustrate the formation of such low-energy robust ps-ABSs in two experimentally relevant situations: a hybrid SM-SC system consisting of a proximitized nanowire coupled to a quantum dot and the SM-SC system in the presence of a spatially varying inhomogeneous potential. We then show that, unlike local measurements, a two-terminal experiment involving charge tunneling at both ends of the wire is capable of distinguishing between the generic ps-ABSs and the non-Abelian MZMs. While the MZMs localized at the opposite ends of the wire generate correlated differential conduction spectra, including correlations in energy splittings and critical Zeeman fields associated with the emergence of the ZBCPs, such correlations are absent if the ZBCPs are due to ps-ABSs emerging in the topologically trivial phase. Measuring such correlations is the clearest and most straightforward test of topological MZMs in SM-SC heterostructures that can be done in a currently accessible experimental setup.

Figure 1

Schematic representation of the SM-SC heterostructure in Ref. [21] (see also Refs. [49, 50]). Proximitized semiconductor nanowire coupled to a quantum dot, where the dot is represented by the segment of the wire that is not covered by the superconductor. (a) The induced paring is ${\mathrm{\Delta }}_{\mathrm{ind}}=0.25\mathrm{meV}$ in the proximitized region and vanishes in the dot region. (b) Effective potential $V_1\left(x\right)$ normalized by ${\mathrm{\Delta }}_{\mathrm{ind}}=0.25\mathrm{meV}$; the dot corresponds to a potential valley. (c) Effective potential $V_1\left(x\right)$ normalized by ${\mathrm{\Delta }}_{\mathrm{ind}}=0.25\mathrm{meV}$; the dot is modeled as a potential step. In this paper, we model the quantum dot as a potential step, while Refs. [49, 50] model the quantum dot as a potential valley. Which one of the profiles $V_1\left(x\right)$ or $V_2\left(x\right)$ better represents the potential characterizing the quantum dot in the quantum dot–nanowire–superconductor hybrid devices realized in the laboratory depends on the specific materials, the work function difference between the semiconductor and the superconductor, and on the voltage applied under the dot region.

Figure 2

Low-energy spectrum as function of the applied Zeeman field for a system with an effective potential profile $V_1\left(x\right)$ (top and middle panels) and $V_2\left(x\right)$ (bottom panel). The potential profiles, the system parameters ${\mathrm{\Delta }}_{\mathrm{ind}}$ and $\mu$ (measured relative to the bottom of the band), as well as the length of the wire are given in Fig. 1. The values of the Rashba spin-orbit coupling $\alpha$ indicated on the figure are in units of $\mathrm{meV}\AA$. We have considered a single chain with lattice constant $a=10\mathrm{nm}$, $t_x=12.7\mathrm{meV}$ (corresponding to an effective mass $m_{\mathrm{eff}}=0.03m_0$), $t_y={\alpha }_y=0$, and the Zeeman field oriented along the wire. The spatial profiles of the MBSs ${\chi }_A$ and ${\chi }_B$ associated with the low-energy modes labeled by roman numerals are shown in Figs. 3 and 4.

Figure 3

Spatial profiles of the Majorana bound states ${\chi }_A$ and ${\chi }_B$ [yellow (light gray) and red (gray), respectively] defined by Eqs. (2) and (3) for the low-energy modes labeled by roman numerals in Fig. 2. Mode (I) is a “true” Majorana mode characterized by two MBSs localized near the opposite ends of the wire (top panel). Modes (II) and (III) are “standard” ABSs consisting of two MBSs that are practically on top of each other (middle and lower panels). The values of the model parameters are given in Figs. 1 and 2.

Figure 4

Spatial profiles of the Majorana bound states ${\chi }_A$ and ${\chi }_B$ [yellow (light gray) and red (gray), respectively] defined by Eqs. (2) and (3) for the low-energy modes marked by roman numerals in the lower panel of Fig. 2. The trivial low-energy modes (IV) and (V) are partially separated ABSs localized near the left end of the system, which contains the quantum dot. Mode (VI) consists of two MZMs localized at the opposite ends of the wire. The values of the model parameters are given in Figs. 1 and 2.

Figure 5

Schematic representation of the experimental setup in Ref. [20]. Metallic leads are coupled to each end of a semiconductor nanowire proximity coupled to an $s$-wave superconductor. Potential gates create tunneling barriers and control the electrostatic energy of the heterostructure. A bias voltage $V_s$ is applied across the wire.

Figure 6

(a) Inhomogeneous potential profiles with characteristic widths ${\delta }_V=0.2\mu \mathrm{m}$ [red (dark gray)] and ${\delta }_V=2.0\mu \mathrm{m}$ [blue (gray)]. The maximum height/depth of the potential is $V_0$, which can be positive or negative. (b), (c) Zeeman field dependence of the low-energy spectra. (b) A short-range potential inhomogeneity with ${\delta }_V=0.2\mu \mathrm{m}$ and $V_0=-0.55\mathrm{meV}$ induces an ABS that crosses zero energy in the topologically trivial region. A robust MZM emerges in the topological regime [orange (gray)]. (c) A long-range inhomogeneity with ${\delta }_V=2\mu \mathrm{m}$ and $V_0=-1\mathrm{meV}$ generates four nearly zero-energy MBSs (a ps-ABS near each end) in the topologically trivial regime [yellow (light gray)] (see Fig. 7). The model parameters are $t_x=12.7\mathrm{meV}$, $t_y={\alpha }_y=0$ (single chain), $\alpha =250\mathrm{meV}\AA$, $\mu =-0.5\mathrm{meV}$ [panel (b)], $\mu =-0.75\mathrm{meV}$ [panel (c)], ${\mathrm{\Delta }}_{\mathrm{ind}}=0.25\mathrm{meV}$, and the length of the wire is $2\mu \mathrm{m}$.

Figure 7

Majorana wave functions ${\chi }_A,{\chi }_B$ defined in Eqs. (2) and (3) for the lowest-energy modes in the topologically trivial regime. (a) A localized potential inhomogeneity (${\delta }_V=0.2\mu \mathrm{m}$) generates two strongly overlapping MBSs, which correspond to a regular “garden variety” ABS. Such an Andreev bound state resonance can be easily destabilized from zero energy by a change in the Zeeman field (see Fig. 6). (b) By increasing the width of the inhomogeneity (${\delta }_V=0.6\mu \mathrm{m}$), the lowest-energy MBSs ${\chi }_A,{\chi }_B$ become spatially separated and another pair of MBSs starts to separate spatially. (c) Four weakly overlapping MBSs forming a pair of ps-ABSs in a system with long-range potential inhomogeneity (${\delta }_V=2\mu \mathrm{m}$). The green and red states at the two ends originate from the lowest-energy spin subband, while the blue and orange states originate from the first higher-energy spin subband. The values of the model parameters are given in Fig. 6.

Figure 8

Low-energy spectra in the presence of a nonhomogeneous potential. The Zeeman field is $\mathrm{\Gamma }=1.75\mathrm{\Delta }$ and the chemical potential (defined relative to the bottom of the top band when $V_0=0$) is $\mu =2\mathrm{\Delta }$. The values of the other parameters are given in Fig. 6. As the characteristic length scale ${\delta }_V$ increases, robust low-energy ps-ABSs emerge in the topologically trivial regime [yellow (light gray) regions in (c)].

Figure 9

(Top) Measurements of the local density of states (LDOS) as a function of bias potential and Zeeman splitting associated with a long-length-scale potential inhomogeneity with characteristic length ${\delta }_V=2.0\mu \text{m}$ (Fig. 6). For sufficiently long potential length scales zero-energy crossings in the LDOS spectrum exist in the topologically trivial regime [yellow (light gray)] due to a ps-ABS at each end. (Bottom) Logarithm of the energy splitting $\mathrm{\Delta }E\propto exp(-L/\xi )$ as a function of wire length $L$ showing exponential protection of the energy splitting in both the topologically trivial regime [yellow (light gray) dots, $\mathrm{\Gamma }=2.5]$ and the topological regime [orange (gray) dots, $\mathrm{\Gamma }=3.5]$. LDOS was taken at a temperature of $T\approx 20\mathrm{mK}$, with the values of the other parameters given in Fig. 6. In the trivial regime, the energy splitting is exponentially suppressed in the length of the wire only when the length scale of the potential fluctuation is given by the length of the wire $L$.

Figure 10

(a) Single-lead differential conductance plot of a $2\text{−}\mu \text{m}$ superconductor-semiconductor heterostructure with a normal lead at one end of the wire. The Zeeman field ranges from $\mathrm{\Gamma }=1.5\mathrm{\Delta }$ to $4\mathrm{\Delta }$ with profiles offset for clarity. (b) Two-lead differential conductance in the Coulomb blockade regime as a function of bias potential $V$ and Zeeman splitting $\mathrm{\Gamma }$ with long-length-scale potential fluctuations ${\delta }_V=2.0\mu \text{m}$. Peaks exist in both the topologically trivial [yellow (light gray)] and the topological [orange (gray)] regimes. These peaks are suppressed at zero-energy crossings in the spectrum. (c) Two-lead “teleportation” differential conductance as a function of bias potential at $\mathrm{\Gamma }=4\mathrm{\Delta }$, showing exponential suppression with length of the wire. In this figure, all conductance calculations were performed for a chain at a temperature $T\approx 20\mathrm{mK}$, with the values of the other parameters given in Fig. 6.

Figure 11

Schematic representation of the inhomogeneous semiconductor-superconductor device along with the two effective potential profiles considered in this study. The potential maxima at the ends of the wire represent the left and right tunnel barriers. The effective potential is given in units of the induced gap, ${\mathrm{\Delta }}_{\mathrm{ind}}=0.25\mathrm{meV}$, while the chemical potential can be controlled using back gates (here we show the case corresponding to $\mu =0$).

Figure 12

Low-energy spectrum as a function of the applied Zeeman field for the effective potential profile shown in the top panel of Fig. 11 and different values of the chemical potential. The lowest-energy modes (red lines) represent a pair of partially overlapping MBSs that, in general, are not localized near the ends of the wire. The model parameters are $t_x=12.7\mathrm{meV}$, $t_y={\alpha }_y=0$ (single chain), $\alpha =250\mathrm{meV}\AA$, and ${\mathrm{\Delta }}_{\mathrm{ind}}=0.25\mathrm{meV}$.

Figure 13

Low-energy spectrum as function of the applied Zeeman field for the effective potential profile shown in the bottom panel of Fig. 11 and different values of the chemical potential. The blue lines represent partially overlapping MBSs. Note that for certain values of the chemical potential, there are additional subgap low-energy modes. The model parameters are given in Fig. 12.

Figure 14

Differential conductance as function of the applied Zeeman field and bias potential (energy) for the parameters corresponding to the panels marked by (1) and (3) in Fig. 12 and (2) in Fig. 13. The traces labeled by $L$ and $R$ correspond to tunneling from the left and right leads, respectively. The traces shown in the top panels are correlated for Zeeman fields larger than about $0.5\mathrm{meV}$, those in the middle panels show correlations above $0.7\mathrm{meV}$, while the traces in the lower panels are uncorrelated.