Majorana Kramers doublets in $d_{x^2-y^2}$-wave superconductors with Rashba spin-orbit coupling

In this work, we show that a quasi-one-dimensional $d_{x^2-y^2}$-wave superconductor with Rashba spin-orbit coupling is a DIII class, time-reversal invariant, topological superconductor (TS), which supports a Majorana Kramers Doublet (MKD) at each end of the TS. A MKD is a pair of Majorana end states (MESs) protected by time-reversal symmetry (TRS). An external magnetic field breaks TRS and drives the system from DIII to D class in which case a single MES appears at each end of the TS. We show that a MKD induces resonant Andreev reflection with zero-bias conductance peak of $4e^2/h$. Experimental realizations of the proposed model are discussed.

Figure 1

A quasi-one-dimensional $d_{x^2-y^2}$-wave superconductor with spin-orbit coupling. (a) Majorana Kramers doublets appear in the absence of an external magnetic field. (b) Single MESs appear in the presence of a magnetic field.

Figure 2

(a) Excitation energy versus chemical potential. The parameters of $H_{1\mathrm{D}}$ are: $L=2000a$, $t=12$, ${\Delta }_0=1$, and ${\alpha }_R=4$, where $L$ is the length of the wire, $a$ is the lattice spacing. Zero-energy modes exist in the region highlighted in red. (b) The sum of the amplitude of the ground-state wave functions, $|{\Psi }_1{|}^2+{\left|{\Psi }_2\right|}^2$, vs $x$, where $x$ is the site label. In the topologically nontrivial regime, e.g., $\mu ={\alpha }_R/2$, the ground-state wave functions are localized at the edge. (c) In the topologically trivial regime, e.g., $\mu =1.5{\alpha }_R$, the ground-state wave functions are predominantly in the bulk.

Figure 3

Excitation energy as a function of chemical potential. The parameters of $H_{1\mathrm{D}}$ are $t=12$, ${\Delta }_0=1$, ${\alpha }_R=4$, and $V_z=2$. Single MESs appear near the band bottom at $\mu \approx -2t$, indicated by the blue line. Double MESs appear near the band center at $\mu \approx 0$ indicated by the red line. The insert is an enlargement of the gray section that indicates that there are two states near-zero energy at $\mu \approx 0$.

Figure 4

a) Excitation energy vs chemical potential. The parameters in $H_{\text{q1D}}$ are $V_z=0$, $t=12$, ${\Delta }_0=1$, and ${\alpha }_R=4$. On-site Gaussian disorder with variance ${\Delta }_0^2$ is present. The width of the wire is $W=6a$ and the length is $L=1400a$. All the states shown are doubly degenerate. Zero-energy modes associated with MKDs appear in a wide range of chemical potential. The Kramers degenerate ground states are depicted in red. (b) The parameters in (b) are the same as in (a) except $V_z=2$. Nondegenerate zero energy modes appear even when TRS is broken. The ground state is depicted in blue.

Figure 5

(a) A normal lead is attached to the end of a TS. (b) ZBC from the normal lead to the TS versus chemical potential at $V_z=0$. The ZBC is quantized at $4e^2/h$ in the topologically nontrivial regime, due to the presence of double MESs. (c) ZBC vs chemical potential at $V_z=2{\Delta }_0$. The ZBC is quantized at $2e^2/h$ in the topologically nontrivial regime due to the presence of a single MES.

Figure 6

Excitation energy of $H_{1\mathrm{D}}\left(k\right)$ as a function of chemical potential in the presence of disorder. The parameters in this figure are the same as the parameters in Fig. 3 of the main text except the fact that on-site potential disorder with variant ${\Delta }_0^2$ is added to the Hamiltonian. This is in sharp contrast to the multichannel case when the system is in the D class in which case an even number of Majorana fermions are not protected against disorder.

Figure 7

(a) Energy spectrum of a wire with $d_{x^2-y^2}$-wave, $s$-wave, $p$-wave pairing and Rashba spin-orbit coupling. ${\Delta }_s={\Delta }_p=0.2{\Delta }_0$ is assumed. The parameters used are the same as the parameters in Fig. 4a of the main text. (b) A plot of the sum of the amplitudes of the ground state wave functions of the wire ${\left|\Psi \right|}^2=|{\Psi }_1{|}^2+{\left|{\Psi }_2\right|}^2$ in the topologically nontrivial regime. Here, ${\Psi }_1$ and ${\Psi }_2$ are the Kramers pair. It is evident that the wave functions are localized at the ends. (c) A plot of the sum of the amplitudes of the ground-state wave functions of the wire in the topologically trivial regime. The wave functions are predominately in the bulk.

Figure 8

(a) A systematic picture of CeCoIn${}_5$ modeled as a trilayers system on a substrate. The top and bottom layers break the mirror symmetry with respect to the $z$ axis locally. (b) The energy spectrum of a finite system with open boundary conditions. The parameters used are ${\Delta }_0=2,t=9{\Delta }_0,{\alpha }_R=2{\Delta }_0$, and $t_z=4.5{\Delta }_0$. (c) The ground-state wave functions in the topologically trivial regime at $\mu =-2{\alpha }_R$. (d) The ground-state wave functions in the topologically nontrivial regime at $\mu =-1.25{\alpha }_R$.

Figure 9

The bulk band structure for a system with periodic boundary conditions in the $x$ direction. There are no Majorana end states due to the periodic boundary conditions. The spectrum is fully gapped in the topologically nontrivial regime. The system can undergo a quantum phase transition from topologically nontrivial to topologically trivial phase or vise versa when the bulk gap is closed by tuning the chemical potential.

Figure 10

The sum of the amplitude of the two ground-state wave functions, ${\left|\psi \right|}^2=|{\psi }_1{|}^2+{\left|{\psi }_2\right|}^2$, of a system which consists of a quasi-one-dimensional superconducting wire placed on top of a two-dimensional $d_{x^2-y^2}$-wave superconductor. The parameters are given in the text. (a) The ground-state wave function on the quasi-one-dimensional wire when the wire is in the topologically nontrivial regime. It is evident that the ground-state wave function is localized at the ends of the wire. (b) The ground-state wave functions leak into the bulk of the background $d$-wave superconductor. The wave functions leak into the nodal directions in which $|k_x|=|k_y|$.