Majorana fermions in semiconductor nanowires

We study multiband semiconducting nanowires proximity coupled with an $s$-wave superconductor and calculate the topological phase diagram as a function of the chemical potential and magnetic field. The nontrivial topological state corresponds to a superconducting phase supporting an odd number of pairs of Majorana modes localized at the ends of the wire, whereas the nontopological state corresponds to a superconducting phase with no Majoranas or with an even number of pairs of Majorana modes. Our key finding is that multiband occupancy not only lifts the stringent constraint of one-dimensionality, but also allows having higher carrier density in the nanowire. Consequently, multiband nanowires are better suited for stabilizing the topological superconducting phase and for observing the Majorana physics. We present a detailed study of the parameter space for multiband semiconductor nanowires focusing on understanding the key experimental conditions required for the realization and detection of Majorana fermions in solid-state systems. We include various sources of disorder and characterize their effects on the stability of the topological phase. Finally, we calculate the local density of states as well as the differential tunneling conductance as functions of external parameters and predict the experimental signatures that would establish the existence of emergent Majorana zero-energy modes in solid-state systems.

Figure 1

(Top) Low-energy BdG spectrum of a SM with proximity-induced superconductivity. The effective SM-SC coupling is $\gamma =0.25$ meV and the chemical potential is $\mu =0$. The induced gap vanishes at $k=0$ in the presence of a Zeeman field $\Gamma =\sqrt{{\gamma }^2+{\mu }^2}=0.25$ meV. The full lines are obtained by solving Eq. (16), while the symbols represent the spectrum given by Eq. (17). (Bottom) Low-energy BdG spectrum for $\gamma =2$ meV, $\mu =0$, and three different values of the Zeeman field. The filled area corresponding to energies above ${\Delta }_0=1$ meV represents the SC continuum. Note that for energies $E_{\mathit{k}}<{\Delta }_0/2$, Eq. (17) represents a very good approximation of the BdG spectrum even in the strong-coupling regime.

Figure 2

Dependence of the minimum quasiparticle excitation gap in the BdG spectrum given by Eq. (16) on the Zeeman field $\Gamma$ for different SM-SC couplings. The chemical potential is $\mu =0$ and the Rashba coefficient is ${\alpha }_r=0.15$ eV Å for the curve represented by small (green) circles and ${\alpha }_r=0.1$ eV Å for the other three curves. The system becomes gapless at ${\Gamma }_c=\sqrt{{\gamma }^2+{\mu }^2}$. The superconducting state with $\Gamma <{\Gamma }_c$ is topologically trivial, while for $\Gamma >{\Gamma }_c$ one has a topological SC that supports Majorana bound states. Note that the optimal quasiparticle gap for the topological SC has a weak dependence on the SM-SC coupling but varies strongly with the strength of the spin-orbit coupling.

Figure 3

(Top) Nonuniform SM-SC coupling $\tilde{t}\left(y\right)={\tilde{t}}_0[1-\theta p\left(y\right)]$ with a smooth profile $p\left(y\right)$ and $\theta =[\tilde{t}\left(0\right)-\tilde{t}\left(L_y\right)]/\tilde{t}\left(0\right)=0.8$. (Bottom) Dependence of the induced gap on the nonuniformity of the coupling across the wire. The inhomogeneous proximity effect induces inter-subband pairing with ${\Delta }_{n_y{n}_y\pm 1}\approx {\Delta }_{n_y{n}_y}/2$ for $\theta >0.7$.

Figure 4

Spectrum of an infinite nonsuperconducting wire in the presence of a Zeeman field. The energies are obtained by diagonalizing the Hamiltonian ${\tilde{H}}_{\mathrm{nw}}+{\tilde{H}}_{\mathrm{Zeeman}}$ for $\mu =5E_{\alpha }$ and $\Gamma \approx 15E_{\alpha }$ and are measured relative to the chemical potential. Due to the presence of the Zeeman field, each $n_y$ band is split into two subbands marked $\{n_y-\}$ and $\{n_y+\}$. Note that, for this value of $\Gamma$, the subbands $\{1+\}$ and $\{2-\}$ are degenerate at $k_x=0$.

Figure 5

(Top) BdG energy spectrum for a finite superconducting wire obtained by numerical diagonalization of $H_{\mathrm{eff}}$. The parameters $\mu$ and $\Gamma$ are the same as in Fig. 4 and $\mathit{n}$ labels the eigenvalues of $H_{\mathrm{eff}}$ staring with the lowest energy and has the same sign as $E_{\mathit{n}}$. The in-gap states are Majorana zero-energy modes. (Bottom) The particle-component of the wave-function amplitude for the lowest energy states. The Majorana modes ($\mathit{n}=1$) are localized at the ends of the wire, while the finite energy states extend over the entire wire.

Figure 6

(Top) Low-energy spectra for weak SM-SC coupling ($\overline{\gamma }=0.25{\Delta }_0$) in the vicinity of the sweet spot ($\mu =14.5E_{\alpha }$, $\Gamma =15.3E_{\alpha }$). The yellow circles correspond to inhomogeneous coupling with $\theta =0.8$ and takes into account dynamical effects, the black squares are obtained by neglecting dynamical effects, that is, ${\left(Z^{1/2}\right)}_{n_y{n}_y^{\prime }}={\delta }_{n_y{n}_y^{\prime }}$, and the red diamonds are for a homogeneous coupling. (Bottom) Same as in the upper panel for $\overline{\gamma }={\Delta }_0$ near the sweet spot ($\mu =14.5E_{\alpha }$, $\Gamma =24.2E_{\alpha }$). Note that inclusion of dynamical effects through $Z^{1/2}$ renormalizes the energy scales, while inhomogeneous coupling play a critical role in establishing a finite gap near the sweet spot. The location of a given sweet spot depends on the coupling strength.

Figure 7

Sequence of low-energy spectra obtained for four different values of the Zeeman field separated by points with a vanishing minigap. The spectra with an odd number of pairs of zero-energy modes ($N$) characterize topological SC phases, while those with an even $N$ correspond to trivial SC phases. Note the overall decrease of the minigap with the Zeeman field. The system is characterized by the following parameters: $\mu =30E_{\alpha }$, $\overline{\gamma }=0.25{\Delta }_0$, and $\theta =0.8$.

Figure 8

Phase diagram of the multiband nanowire as function of the Zeeman field $\Gamma$ and the chemical potential $\mu$. The quasiparticle gap ${\Delta }_{\infty }^{*}$ of the effective low-energy Hamiltonian (26) vanishes at the phase boundaries. Superconducting phases characterized by an odd (even) number of pairs of zero-energy Majorana modes are topologically nontrivial (trivial). The coupling of the SM nanowire to the $s$-wave SC is characterized by $\overline{\gamma }=0.25{\Delta }_0$ and $\theta =0.8$. The inhomogeneous coupling induces off-diagonal pairing ${\Delta }_{n_y{n}_y^{\prime }}$, which removes the x points, creating regions with stable nontrivial (near the sweet spots) or trivial SC.

Figure 9

Phase diagram of the multiband nanowire at intermediate coupling. The SM-SC coupling is $\overline{\gamma }={\Delta }_0$, while the other parameters are the same as in Fig. 8. Note that the sweet spots of the topological phase characterized by $N=1$ are significantly expanded as compared with the weak-coupling case shown in Fig. 8. As a result, tuning the Zeeman field in the vicinity of $\Gamma =30E_{\alpha }$ allows for huge variations of the chemical potential without crossing a phase boundary, that is, without closing the gap.

Figure 10

Dependence of the gap on the Zeeman field for a fixed chemical potential. The top panel corresponds to weak SM-SC coupling with $\overline{\gamma }=0.25{\Delta }_0$ and $\theta =0.8$, while the bottom panel is obtained for an intermediate coupling with $\overline{\gamma }={\Delta }_0$ and $\theta =0.8$. Within the red (dark gray) regions superconductivity is topologically trivial, while in the yellow (light gray) regions the system is topologically nontrivial. These curves correspond to horizontal cuts in the phase diagrams shown in Figs. 8 and 9, respectively, through a sweet spot of the phase $N=1$. Note that in the limit $\theta \to 0$, that is, for uniform SM-SC coupling, the width of the lower field topologically nontrivial region shrinks to zero for both coupling strengths.

Figure 11

Dependence of the quasiparticle gap on the chemical potential for a fixed Zeeman field. The top panel corresponds to weak coupling, $\overline{\gamma }=0.25{\Delta }_0$, and $\Gamma =25E_{\alpha }$, while the lower panel is obtained at intermediate coupling, $\overline{\gamma }={\Delta }_0$, and $\Gamma =32E_{\alpha }$. The curves represent vertical cuts through the $N=1$ topological phase in the phase diagrams shown in Figs. 8 and 9. At intermediate coupling (bottom panel) the gap is finite over the entire chemical potential range, making the topological SC phase in a system with average chemical potential $\overline{\mu }\approx 60E_{\alpha }$ robust against variations of the chemical potential of the order $\delta \mu =5$ meV.

Figure 12

(Top) Spectrum of a system characterized by $\mu =54.5E_{\alpha }$, $\Gamma =35E_{\alpha }$, $\overline{\gamma }={\Delta }_0$, and $\theta =0.8$. The states with $\mathit{n}=\pm 1$ are Majorana zero modes, $\mathit{n}=2,3$ correspond to states localized near the ends of the wire, and states with $\mathit{n}⩾4$ are extended states. The minigap is ${\Delta }^{*}=E_2$ and the minimum quasiparticle (QP) gap is ${\Delta }_{\infty }^{*}=E_4$. The finite energy localized states can be viewed as precursors of the extra pair of Majorana bound states that obtains for $\Gamma >40E_{\alpha }$. (Bottom) Dependence of the quasiparticle (QP) gap ${\Delta }_{\infty }^{*}$ and minigap ${\Delta }^{*}$ on the Zeeman field for a system with the same parameters as in Fig. 10. Note that ${\Delta }^{*}={\Delta }_{\infty }^{*}$ in the vicinity of phase transition points. The quasiparticle gap ${\Delta }_{\infty }^{*}$ has the same values as in Fig. 10.

Figure 13

Phase diagram of the multiband nanowire at different SM-SC tunneling strengths $\gamma$ and interband pairing ${\Delta }_{12}={\Delta }_{23}={\Delta }^{\prime }$ calculated analytically for the effective three-band model. Panels (a)–(c) correspond to $\gamma =5,20,40$, respectively, and superconducting gaps $\Delta /E_{\alpha }=4$ and ${\Delta }^{\prime }/E_{\alpha }=0$. Panels (d)–(f) correspond to $\gamma =5,20,40$, respectively, and superconducting gaps $\Delta /E_{\alpha }=4$ and ${\Delta }^{\prime }/E_{\alpha }=1$. We used here $\delta E_{12}/E_{\alpha }=30$ and $\delta E_{13}/E_{\alpha }=80$.

Figure 14

Phase diagram of a multiband wire in a transverse external field. The strength of the SM-SC coupling is $\overline{\gamma }={\Delta }_0$, with uniform coupling ($\theta =0$, i.e., ${\Delta }^{\prime }=0$) for the top panels and nonuniform tunneling ($\theta \approx 0.4$, ${\Delta }^{\prime }=\Delta /4$) corresponding to the bottom panels. The amplitude of the transverse potential is $V_0=0$ for panels (a) and (d), $V_0=50E_{\alpha }$ for (b) and (e), and $V_0=100E_{\alpha }$ for (c) and (f). Note that the topologically nontrivial phase has a vanishing width at the sweet spots in the absence of inter-subband pairing (a)–(c) for all values of the external potential. For nonuniform coupling (e) and (f), the transverse potential reduces the width of the topological phase near the sweet spots.

Figure 15

Diagrammatic perturbation theory in the tunneling between SM and SC. Disorder averaging is performed at each order in tunneling $t$. The thick solid line represents disorder-averaged Green's function in the SC $\overline{G}(\mathit{p},\omega )$. The bottom diagram corresponds to irreducible contributions, as far as disorder averaging is concerned.

Figure 16

Broadening of the subbands in a nanowire with random edges. For $n_y>2$ the broadening becomes comparable with the interband gap in the presence of fluctuations $\Delta L_y$ representing a few percent of the width $L_y$; that is, the subbands lose their identity.

Figure 17

Profile of the variation of the nanowire width, $\Delta L_y$, as a function of position along the wire. The characteristic length scale for profile I (top panel) is about $10\%$ of the nanowire length $L_x$, while for profile II (bottom panel) it is approximately $5\%$. Each profile generates a series of particular disorder realizations characterized by different values of the maximum amplitude $|\Delta L_y{|}_{\max }$. Random edge profiles corresponding to a given characteristic length and having the same maximum amplitude have similar effect on the low-energy physics of the nanowire.

Figure 18

Spectrum of a weakly coupled nanowire ($\overline{\gamma }=0.25{\Delta }_0$) with random edges in the topological phase with $N=1$ (see Fig. 8). In the presence of disorder the minigap ${\Delta }^{*}$ decreases, but remains finite. The calculation was done using the random edge profiles shown in Fig. 17 for maximum amplitudes of $2\%$ and $10\%$. Longer-range disorder (top panel) has stronger effects than short-range disorder (bottom panel). Small amplitude variations of the nanowire width of the order $2\%$ generate a reduction of ${\Delta }^{*}$ up to $30\%$, but further increasing the amplitude has a weak effect at low energies.

Figure 19

Spectrum of a nanowire with random edges at intermediate coupling, $\overline{\gamma }={\Delta }_0$. The control parameters correspond to a point in the phase diagram inside the topological phase with $N=1$ (see Fig. 9). The random edges are given by the profiles in Fig. 17.

Figure 20

Induced carrier density as a function of distance from the impurity. The two curves, corresponding to different external sets of parameters for the nanowire, have been shifted for clarity. The upper curve corresponds to a system with a single occupied subband and is characterized by $\lambda \approx 45a$ (i.e., $63\%$ of the induced charge is in a disk of radius $45a$), while the lower curve is for a system with three occupied subbands and has $\lambda \approx 30a$. Note that the effective impurity potentials that generate these density profiles are given by Eq. (42) with the corresponding values of $\lambda$.

Figure 21

(Top) Comparison between the low-energy spectrum of a clean nanowire (yellow squares) and the spectrum of a nanowire with a charged impurity with $q=e$ (red circles). Note that the two spectra are almost on top of each other. (Bottom) Electron wave function amplitudes for two low-energy states (marked by arrows in the top panel) in the presence of the charged impurity. The Majorana zero modes ($\mathit{n}=\pm 1$, not shown) and the in-gap modes ($\mathit{n}=\pm 2,\pm 3$, see Fig. 12 and the corresponding discussion) are localized near the ends of the wire and are not affected by the presence of the impurity. The low-energy bulk states $\left|\mathit{n}\right|>4$ that are extended in a clean nanowire become localized near the impurity.

Figure 22

Specific disorder realizations in nanowires with charged impurities. The dots represent the locations of the impurities. Each impurity has a charge $q=\pm e$ and is positioned one lattice spacing away from the surface of the nanowire along the $z$ direction. The linear impurity densities $n_{\mathrm{imp}}$ are $1\mu {\text{m}}^{-1}$ (A), $2\mu {\text{m}}^{-1}$ (B), $4\mu {\text{m}}^{-1}$ (C), and $8\mu {\text{m}}^{-1}$ (D).

Figure 23

Low-energy spectra of a superconducting nanowire with charged disorder for two different sets of control parameters and four disorder realizations. The symbols correspond to the specific disorder realizations shown in Fig. 22. Note that (i) the minigap ${\Delta }^{*}$ is finite for impurity concentrations $n_{\mathrm{imp}}⩽4\mu {\text{m}}^{-1}$ and collapses for $n_{\mathrm{imp}}=8\mu {\text{m}}^{-1}$, and (ii) there is an overall tendency of the low-energy features to move at lower energies when increasing the impurity concentration, but the exact value of the minigap depends on the specific disorder realization, for example, in the lower panel ${\Delta }^{*}(n_{\mathrm{imp}}=4)>{\Delta }^{*}(n_{\mathrm{imp}}=2)$.

Figure 24

Dependence of the lowest three energies $E_{\mathit{n}}$ ($\mathit{n}=1,2,3$) on the Zeeman field for a disordered nanowire with $n_{\mathrm{imp}}=4\mu {\text{m}}^{-1}$ (top) and $n_{\mathrm{imp}}=7\mu {\text{m}}^{-1}$ (bottom). The lowest energy state is characterized by a gap ${\Delta }_{\mathit{1}}=E_{\mathit{1}}$ (red/dark gray region) that vanishes for $\Gamma >21.6E_{\alpha }$, that is, when the system enters the topological SC phase. The gap between the first and the second levels, ${\Delta }_{\mathit{2}}=E_{\mathit{2}}-E_{\mathit{1}}$ (yellow), becomes the minigap in the topological phase. The transition zone is characterized by a high density of low-energy modes and expands with increasing the impurity concentration.

Figure 25

Low-energy spectra of a nanowire with long-range disorder. The system in characterized by $\mu =54.5E_{\alpha }$ and $\gamma ={\Delta }_0$ and the impurity potential is given by Eq. (44) with with a profile $v_{\mathrm{imp}}$ as in Fig. 17 (bottom panel) and different amplitudes $U_0$. The parameters $U_0$ and $\Gamma$ are expressed in units of $E_{\alpha }$. For $\Gamma =29$ the minigap collapses for $U_0⩾60$, while for a Zeeman field $\Gamma =32$ it survives up to an amplitude $U_0\approx 100$.

Figure 26

Random coupling at the SM-SC interface. The coupling $\tilde{t}$ contains a random component $\Delta \tilde{t}$ that is constant within patches of length $l$, but takes a random value within each patch. The patch sizes are $\mathit{l}=20a$ (top) and $\mathit{l}=60a$ (middle). The strength of $\Delta \tilde{t}$ within the patches is color coded. Note that only part of the interface is shown, as the typical length of a nanowire used in the calculations is of the order ${10}^{4}a$. (Bottom) Dependence of the induced gap ${\Delta }_{11}$ on the position along the wire for $\Gamma =32E_{\alpha }$, $\mu =54.5E_{\alpha }$, $\overline{\gamma }={\Delta }_0$, $\theta =0.8$, and a random coupling with $\mathit{l}=60a$ and an amplitude ${\left(\Delta \tilde{t}\right)}_{\max }=0.25\tilde{t}$. The huge local variations of the coupling are strongly reduced by the integration over $y$ (see main text) and generate fluctuations of the order of $10\%$ in ${\Delta }_{11}\left(x\right)$.

Figure 27

Spectra of a nanowire with random SM-SC coupling. (Top) Nanowire with short-range fluctuations of the SM-SC coupling. The random coupling is considered within the patch model described in the text and corresponds to the distributions shown in Fig. 26. (Bottom) Nanowire with long-range SM-SC coupling fluctuations. The variations of $\tilde{t}$ along the wire have a profile as shown in the top panel of Fig. 17 and an amplitude $\Delta \tilde{t}$. Note that for $\Delta \tilde{t}=0.25\tilde{t}(y=0)$ the gap collapses.

Figure 28

Typical spectra for a trivial SC with $N=0$ (red squares), a topological SC with $N=1$ (orange diamonds), and a trivial SC with $N=2$ (black circles). The system is characterized by $\mu =54.5E_{\alpha }$, $\theta =0.8$, and $\overline{\gamma }={\Delta }_0$. The finite energy in-gap states (for $\Gamma =15E\alpha$ and $\Gamma =45E_{\alpha }$) together with the Majorana zero modes are localized near the ends of the wire (see also Fig. 12), while the rest of the states extend throughout the entire system. The corresponding LDOS is shown in Fig. 29.

Figure 29

LDOS for a clean nanowire in three different phases: trivial SC phase with $N=0$ ($\Gamma =15E_{\alpha }$, top), topological SC phase with $N=1$ ($\Gamma =25E_{\alpha }$, middle), and trivial SC phase with $N=2$ ($\Gamma =45E_{\alpha }$, bottom). The corresponding spectra are shown in Fig. 28. Notice the finite energy in-gap states localized near the ends of the wire (top and bottom) and the zero-energy Majorana modes (middle and bottom). The weight of the zero-energy modes in the $N=2$ phase (bottom) is twice the weight of the Majorana modes in the topological SC phase with $N=1$ (middle). However, the clearest distinction can be made between the $N=0$ and the $N=1$ phases. The LDOS is integrated over the transverse coordinates $y$ and $z$.

Figure 30

LDOS for a nanowire with charged impurities. The top picture corresponds to a trivial SC state with $N=0$, while the bottom picture is for a system with $N=1$. The linear impurity density is $n_{\mathrm{imp}}=7\hspace{0.5em}\mu \mathrm{m}$. Notice that all the low-energy states are strongly localized, but the clear-cut distinction between the two phases holds.

Figure 31

Energy spectrum and LDOS in the vicinity a topological phase transition. (Top) The spectra of a system with $\Gamma =21.5E_{\alpha }$ without disorder (blue diamonds) and in the presence of charged impurities ($n_{\mathrm{imp}}=0.7/\mu \text{m}$, orange squares). Note the absence of a gap. The corresponding LDOS are shown in the middle (clean system) and bottom (disordered wire) panels. The spectral weight is distributed over the entire energy range, including at positions near the ends of the wire.

Figure 32

Differential conductance for tunneling into the end of a superconducting nanowire. The curves correspond to different values of the Zeeman field ranging from $\Gamma =11E_{\alpha }$ (bottom) to $\Gamma =36E_{\alpha }$ (top) in steps of $E_{\alpha }$. The curves were shifted vertically for clarity. The trivial SC phase ($\Gamma <21E_{\alpha }$) is characterized by a gap that vanishes in the critical region ($\Gamma \approx 21E_{\alpha }$). The signature of the topological phase is the zero-energy peak resulting from tunneling into the Majorana mode. The differential conductance was calculated at a temperature $T\approx 50$ mK for a disordered wire with a linear density of charged impurities $n_{\mathrm{imp}}=7/\mu \text{m}$.