Non-Abelian quantum order in spin-orbit-coupled semiconductors: Search for topological Majorana particles in solid-state systems

We show that an ordinary semiconducting thin film with spin-orbit coupling can, under appropriate circumstances, be in a quantum topologically ordered state supporting exotic Majorana excitations which follow non-Abelian statistics. The key to the quantum topological order is the coexistence of spin-orbit coupling with proximity-induced $s$-wave superconductivity and an externally induced Zeeman coupling of the spins. For the Zeeman coupling below a critical value, the system is a nontopological (proximity-induced) $s$-wave superconductor. However, for a range of Zeeman coupling above the critical value, the lowest energy excited state inside a vortex is a zero-energy Majorana fermion state. The system, thus, has entered into a non-Abelian $s$-wave superconducting state via a topological quantum phase transition (TQPT) tuned by the Zeeman coupling. In the topological phase, since the time-reversal symmetry is explicitly broken by the Zeeman term in the Hamiltonian, the edge of the film constitutes a chiral Majorana wire. Just like the $s$-wave superconductivity, the Zeeman coupling can also be proximity induced in the film by an adjacent magnetic insulator. We show this by an explicit model tight-binding calculation for both types of proximity effects in the heterostructure geometry. Here we show that the same TQPT can be accessed by varying the interface transparency between the film and the superconductor. For the transparency below (above) a critical value, the system is a topological (regular) $s$-wave superconductor. In the one-dimensional version of the same structure and for the Zeeman coupling above the critical value, there are localized Majorana zero-energy modes at the two ends of a semiconducting quantum nanowire. In this case, the Zeeman coupling can be induced more easily by an external magnetic field parallel to the wire, obviating the need for a magnetic insulator. We show that, despite the fact that the superconducting pair potential in the nanowire is explicitly $s$ wave, tunneling of electrons to the ends of the wire reveals a pronounced zero-bias peak. Such a peak is absent when the Zeeman coupling is below its critical value, i.e., the nanowire is in the nontopological $s$-wave superconducting state. We argue that the observation of this zero-bias tunneling peak in the semiconductor nanowire is possibly the simplest and clearest experiment proposed so far to unambiguously detect a Majorana fermion mode in a condensed-matter system.

Figure 1

(Color online) (a) The proposed heterostructure of a semiconductor (SM) sandwiched between an $s$-wave SC and a magnetic insulator (MI). In this geometry, the semiconducting film can support non-Abelian topological order. (b) Single-particle band structure in the semiconducting film with and without the Zeeman splitting induced by the MI. Without the Zeeman splitting, the two spin-orbit shifted bands touch at $k_x=k_y=\left|k\right|=0$ (red lines). Then, for any value of the chemical potential, the system has two Fermi surfaces. With a finite Zeeman splitting, the bands have an energy gap near $k_x=k_y=\left|k\right|=0$ (blue lines). If the chemical potential lies in the gap, the system just has one Fermi surface (indicated by the dotted circle).

Figure 2

(Color online) Complex roots $z=z_R+ız_I$ of the characteristic equation, Eq. (25), shown in the complex plane. Only solutions with $\text{Re}\left(z\right)=z_R>0$ are physically acceptable. In the non-Abelian phase $\left[C_0=\left({\Delta }^2+{\mu }^2-V_Z^2\right)<0\right]$, three roots with positive real parts and one root with negative real part for $\lambda =-1$ while there are only two roots on either side of the imaginary axis for $C_0>0$ for $\lambda =-1$. The roots in the $\lambda =1$ channel are the negative of the roots in the $\lambda =1$ channel.

Figure 3

(Color online) (a) Plots for the individual components of the four-component wave function $\Psi \left(r\right)={\left[u_{\uparrow }\left(r\right),u_{\downarrow }\left(r\right),v_{\downarrow }\left(r\right),-v_{\uparrow }\left(r\right)\right]}^T$ for the zero-energy Majorana state at the north pole. The components of $\Psi$ satisfy $u_{\sigma }=v_{\sigma }^{\ast }$ confirming the Majorana nature of the wave function. We also show the semiconductor heterostructure on the surface of a sphere with a vortex and an antivortex (with reduced superconducting amplitudes at the vortex cores) situated at the north and the south poles. (b) Numerical results for the vortex minigap $\Delta E$ (solid line) and bulk gap (dashed line) plotted against the spin-orbit-coupling strength $\alpha$ on the semiconductor. In these plots we have used $\Delta =0.5$, $\mu =0.0$, and $\eta =1.0$ in the units where $V_z=1$. The spin-orbit-coupling strength $\alpha =0.3$ in (a) and varies for (b). In these units, the size of the vortex core has been taken to be unity.

Figure 4

(Color online) Quasiparticle gap versus Zeeman coupling for various values of spin-orbit interaction $\alpha$. The strength of the spin-orbit coupling in the inset is such that $\alpha =0.3$ corresponds to $0.1\text{eV}\text{Å}$. The other parameters are taken to be $\Delta =0.5$ and $\mu =0.0$. The gap vanishes at the critical value $V_z=\sqrt{{\Delta }^2+{\mu }^2}$. The spin-orbit coupling has negligible effect below this critical point and the superconducting gap is of a conventional $s$-wave type. Above the critical value and in the absence of spin-orbit coupling, the superconducting gap is destroyed by the Zeeman coupling. Spin-orbit coupling opens up a gap in this phase leading to re-entrant superconductivity which is topological.

Figure 5

(Color online) Band structure for a semiconductor-ferromagnetic insulator heterostructure described by Eqs. (68, 69, 71, 73) with $N=10$ and ${\tilde{t}}_M=250\text{meV}$. The red points (the wide bands in the upper panel) represent semiconductor states while the magenta bands (narrow bands in the upper panel) are ferromagnetic insulator states. The lower panel shows the low-energy behavior around $\mathbf{k}=0$. Notice the spitting of the semiconductor band at $\mathbf{k}=0$ due to the effective Zeeman term induced by ferromagnetic proximity effect.

Figure 6

(Color online) Dependence of the amplitude of low-energy states on the distance from the semiconductor-MI interface for three different sets of parameters. In order of increasing penetration depth into the MI: lowest semiconductor band $\left(n=1\right)$ and ${\tilde{t}}_m=0.25\text{eV}$ (blue), second semiconductor band $\left(n=2\right)$ and ${\tilde{t}}_m=0.25\text{eV}$ (black), and $n=2$, ${\tilde{t}}_m=0.5\text{eV}$ (red). For a given set of parameters, a certain fraction of each low-energy wave function penetrates the insulator leading to ferromagnetic correlations in the semiconductor thin film, i.e., to an effective Zeeman field induced by proximity effect.

Figure 7

(Color online) Dependence of the normalized effective Zeeman splitting on the semiconductor thickness for different sets of parameters (dots). The continuous lines represent the normalized wave-function amplitude at the interface (the $N\text{th}$ layer) ${\left|{\psi }_n\left(N\right)\right|}^2/{\left|{\psi }_n\left(10\right)\right|}^2$. The reference gap is ${\tilde{\Gamma }}_0=2.955\text{meV}$. The wave-function amplitude in the absence of an applied bias, $V\left(z\right)=0$, is determined using Eq. (76) for $n=1$ (lower dark gray curve) and $n=2$ (upper dark gray curve), while for a linear bias $V\left(z\right)=V_{0}z/w$ with $V_0=\pm 1\text{eV}$ the amplitude is determined numerically (light gray curves).

Figure 8

(Color online) Dependence of the normalized Zeeman splitting on the interface transparency for a system with $N=12$ and different model parameters (dots). The straight lines are guide for the eyes. The reference interface coupling is ${\tilde{t}}_{m0}=250\text{meV}$. The rather small deviations from a linear dependence indicate that dynamical effects are negligible, i.e., neglecting the frequency dependence in Eq. (75) is a good approximation.

Figure 9

(Color online) Dependence of the effective Zeeman splitting on the ferromagnetic insulator gap for the first $(n=1$, lower set of dots) and second $(n=2$, upper set of dots), semiconductor bands. For $n=2$ an additional linear bias is applied. The semiconductor has $N=10$ layers and the interface coupling with the ferromagnetic insulator is ${\tilde{t}}_m=250\text{meV}$. The continuous lines are calculated using the effective low-energy theory described by Eq. (81). Notice the remarkable agreement between the low-energy theory and the exact results and the difference in energy scales between the insulating gap and the induced gap.

Figure 10

(Color online) BdG spectrum of the full Hamiltonian (68) for a semiconductor thin film with $N=10$ sandwiched between a MI ($\Gamma =0.2\text{eV}$, ${\tilde{t}}_m=250\text{meV}$, and $\tilde{\Gamma }=2.95\text{meV}$) and an $s$-wave SC ($\Delta =1.5\text{meV}$ and ${\tilde{t}}_s=130\text{meV}$). The red points (highly dispersive bands inside the MI and SC gaps) represent states that reside (mostly) within the semiconductor, magenta designates the MI bands (upper panel, dark gray bands with a 0.2 eV gap), while the SC states are blue (upper and lower panels, bands with a 3meV gap), while SC states are blue (upper and lower panels, bands with a 3meV gap). Due to the effective Zeeman filed, the semiconductor bands are split and only one crosses the chemical potential (see upper panel). As a result of the superconductor proximity effect, a small gap opens at the crossing points (see lower panel).

Figure 11

(Color online) Dependence of the low-energy spectrum on the coupling between the semiconductor and the SC: ${\tilde{t}}_s=0$ (red, maximum gap at $\mathbf{k}=0$), ${\tilde{t}}_s=130\text{meV}$ (black, intermediate gap at $\mathbf{k}=0$), and ${\tilde{t}}_s=190\text{meV}$ (yellow, small gap at $\mathbf{k}=0$). The semiconductor film has $N=10$ layers and is also coupled (at the opposite surface) to a ferromagnetic insulator $\left({\tilde{t}}_m=250\text{meV}\right)$. A proximity-effect-induced superconducting gap opens at finite $k$. As ${\tilde{t}}_s$ is increased, the minimal gap shifts to lower wave vectors and rest increases, then decreases and eventually vanishes at a critical coupling before opening again (see also Fig. 12).

Figure 12

(Color online) Upper panel: comparison between the solution of the effective low-energy theory given by Eq. (84) (lines) and the numerical solution of Hamiltonian (68) (dots) for two values of the chemical potential, $\mu =0$ (minimum gap at $k_x\approx 0.021/\mathrm{a}$ and $\mu =-1\text{meV}$ (minimum gap at $k_x\approx 0.007/\mathrm{a}$. Only positive energies are shown. The semiconductor film $\left(N=10\right)$ is coupled to both a ferromagnetic insulator $\left({\tilde{t}}_m=250\text{meV}\right)$ and an $s$-wave SC $\left({\tilde{t}}_s=200\text{meV}\right)$. Lower panel: dependence of the induced minimum gap on the SC interface transparency. The vanishing of the gap at a critical value ${\tilde{t}}_{\mathrm{SC}}\left(\mu \right)\left[{\tilde{t}}_{\mathrm{SC}}\right(0)\approx 188\mathrm{meV}$, ${\tilde{t}}_{\mathrm{SC}}(-1)\approx 148\mathrm{meV}$] reflects a quantum phase transition between a topologically nontrivial SC (at small values of ${\tilde{t}}_s$) and a trivial $s$-wave SC (at large couplings).

Figure 13

(Color online) Details of the low-energy spectrum for the case when the $n=2$ semiconductor band has the minimum in the vicinity of the chemical potential. The semiconductor film has $N=10$ layers and the couplings at the interfaces with the ferromagnetic insulator and the SC are $\left({\tilde{t}}_m=250\text{meV}\right)$ and $\left({\tilde{t}}_s=200\text{meV}\right)$, respectively. The low-energy physics in the vicinity of $k=0$ (upper panel) is described by the effective theory given by Eq. (84) with a coupling constant $g_s$ that includes the amplitude of the $n=2$ state at the interface. The $n=1$ semiconductor bands cross the chemical potential at larger values of $k$ (lower panel) and are gaped due to SC proximity effect. Note that the gap at large wave vectors does not vanish at a critical coupling.

Figure 14

(Color online) (a) Geometry to detect zero-energy Majorana fermions using STM spectroscopy on a semiconducting nanowire. The Zeeman splitting is induced by a parallel magnetic field while the chemical potential is controlled by an external gate (not shown). The Majorana fermion mode localized at the end of the nanowire gives rise to a zero-bias peak in the STM tunneling spectrum from the end. The tunneling spectrum from the bulk of the nanowire is gapped. (b) Band structure of the nanowire in the topological superconducting phase.

Figure 15

(Color online) (a) Conductance $dI/dV$ as a function of voltage ${\mu }_t=V$. For the plot we have taken $\eta ={\hslash }^2/2m^{\ast }$, where $m^{\ast }=0.04m_e$ for InAs, $\Delta =0.5\text{meV}$ for Nb, $\mu =0$, $\alpha =0.1\text{eV}\text{Å}$, and $T=100\text{mK}$. The different values of $V_Z$ in millielectron volt are given in the inset. The distance of the STM tip from the end of the nanowire has been taken as $a=100\text{nm}$. The plots for $V_Z>0.5\text{meV}$ (corresponding to $B_x\sim 0.5\text{T}$ for InAs with $g_{\text{InAs}}\sim 35$) are in the topological phase and show a zero-bias peak while the plots for $V_Z<0.5\text{meV}$ do not. (b) Position dependence of the zero-bias conductance in the topological phase. The amplitude of the zero-bias conductance is seen to be localized near the end of the wire and corresponds to the localized wave function of the Majorana mode at the end of the wire.