Quantized charge transport in chiral Majorana edge modes

TL;DR

This paper demonstrates that Majorana edge modes in nanoscopic Shiba islands exhibit universal, quantized charge conductance proportional to the Chern number and carry chiral supercurrents, advancing topological quantum device prospects.

Contribution

It reveals the universal electronic and transport properties of Majorana modes in Shiba islands, including quantized conductance and chiral supercurrents, linking them to topological invariants.

Findings

Majorana modes have quantized charge conductance proportional to Chern number.

Majorana modes carry chiral supercurrents reflecting the Chern number sign.

Shiba islands are promising for future topological quantum devices.

Abstract

Majorana fermions can be realized as quasiparticles in topological superconductors, with potential applications in topological quantum computing. Recently, lattices of magnetic adatoms deposited on the surface of s-wave superconductors - Shiba lattices - have been proposed as a new platform for topological superconductivity. These systems possess the great advantage that they are accessible via scanning-probe techniques, and thus enable the local manipulation and detection of Majorana modes. Realizing quantum bits on this basis will rely on the creation of nanoscopic Shiba lattices, so-called Shiba islands. Here, we demonstrate that the topological Majorana edge modes of such islands display universal electronic and transport properties. Most remarkably, these Majorana modes possess a quantized charge conductance that is proportional to the topological Chern number, C, and carry a supercurrent whose chirality reflects the sign of C. These results establish nanoscopic Shiba islands as promising components in future topology-based devices.

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Full text

Quantized charge transport in chiral Majorana edge modes

Stephan Rachel

Institut für Theoretische Physik, Technische Universität Dresden, 01062 Dresden, Germany

  

Eric Mascot

Department of Physics, University of Illinois at Chicago, Chicago, IL 60607, USA

  

Sagen Cocklin

Department of Physics, University of Illinois at Chicago, Chicago, IL 60607, USA

  

Matthias Vojta

Institut für Theoretische Physik, Technische Universität Dresden, 01062 Dresden, Germany

Center for Transport and Devices of Emergent Materials, Technische Universität Dresden, 01062 Dresden, Germany

  

Dirk K. Morr

Department of Physics, University of Illinois at Chicago, Chicago, IL 60607, USA

**Majorana fermions majorana37nc171 can be realized as quasiparticles in topological superconductors lutchyn-10prl077001 ; oreg-10prl177002 ; mourik-12s1003 , with potential applications in topological quantum computing moore-91npb362 ; read-00prb10267 ; ivanov01prl268 ; nayak-08rmp1083 . Recently, lattices of magnetic adatoms deposited on the surface of $s$-wave superconductors – Shiba lattices – have been proposed as a new platform for topological superconductivity li-16nc12297 ; rontynen-15prl236803 . These systems possess the great advantage that they are accessible via scanning-probe techniques, and thus enable the local manipulation and detection of Majorana modes. Realizing quantum bits on this basis will rely on the creation of nanoscopic Shiba lattices, so-called Shiba islands. Here, we demonstrate that the topological Majorana edge modes of such islands display universal electronic and transport properties. Most remarkably, these Majorana modes possess a quantized charge conductance that is proportional to the topological Chern number, ${\cal C}$, and carry a supercurrent whose chirality reflects the sign of ${\cal C}$. These results establish nanoscopic Shiba islands as promising components in future topology-based devices. **

Originally proposed as elementary particles with the peculiar property of being their own antiparticles majorana37nc171 , Majorana fermions have been argued to exist in several condensed-matter settings: in certain fractional quantum Hall states moore-91npb362 , in superfluid He-3 volovik88zetf123 , in fractionalized spin liquids kitaev06 , in chiral superconductors kallin-16rpp054502 , and in hybrid superconducting structures mourik-12s1003 ; nadj-perge-14s602 ; ruby-15prl197204 ; pawlak-16npj16035 . Besides being of fundamental interest as exotic particles, Majorana fermions hold great potential for realizing fault-tolerant topological quantum bits due to their non-Abelian braiding statistics moore-91npb362 ; read-00prb10267 ; ivanov01prl268 ; nayak-08rmp1083 . Realizing this potential, however, necessitates the ability to create, detect, and manipulate single Majorana fermions in nanoscopic topological superconductors.

First steps towards achieving this goal have been taken through the creation of Shiba chains – chains of magnetic Fe atoms – on the surface of Pb, an $s$-wave superconductor with strong Rashba spin-orbit interaction li-14prb235433 ; nadj-perge-14s602 ; ruby-15prl197204 ; pawlak-16npj16035 . These chains were shown to realize one-dimensional topological superconductivity, with Majorana zero modes being localized at the chain ends nadj-perge-14s602 ; ruby-15prl197204 ; pawlak-16npj16035 . Such surface systems can be investigated via scanning tunneling spectroscopy (STS), providing not only spectroscopic but also spatially resolved insight into the nature of Majorana modes. Most recently, the concept of Shiba chains was theoretically generalized to Shiba lattices li-16nc12297 ; rontynen-15prl236803 – two-dimensional lattices of magnetic adatoms placed on an $s$-wave superconductor – as a new platform for chiral superconductivity with dispersing Majorana edge modes. Such systems are characterized by a non-zero Chern number ${\cal C}$, similar to the integer quantum Hall effect, and display a universal thermal Hall conductivity $\kappa^{xy}/T={\cal C}\frac{\pi^{2}k_{B}^{2}}{3h}$ chiu-16rmp35005 . Whether the Majorana edge modes in Shiba lattices also exhibit universal charge transport properties, in analogy to the quantized charge Hall conductivity $\sigma^{xy}=\frac{e^{2}}{h}{\cal C}$ in quantum Hall systems  thouless-82prl405 , that would allow one to uniquely identify them, is presently unclear volovik88zetf123 ; chiu-16rmp35005 .

Motivated by the recent progress in the nanoscale design of artificial magnetic structures paris-experiment ; WiesendangerPC ; FrankePC and the relevance of nanosocpic topological superconductors for quantum computing, we investigate the electronic properties of Shiba islands, i.e., nanoscale islands of magnetic atoms placed on the surface of an $s$-wave superconductor. We identify key characteristic features of chiral Majorana modes localized along the edges of Shiba islands that can be probed via STS; these features provide direct insight into the topologically invariant Chern number of the system, allowing us to unambiguously distinguish topologically trivial from non-trivial phases. In particular, we find that (i) the differential tunneling conductance associated with the flow of charge from the STS tip into Majorana modes is quantized (apart from small finite-size effects) and proportional to $|{\cal C}|$, and (ii) Majorana modes are equally spaced in energy, and carry a supercurrent whose chirality reflects the sign of the Chern number ${\rm sgn}({\cal C})$. Moreover, we demonstrate that zero-energy Majorana modes – crucial ingredients for topological braiding operations – can be created by threading a $\pi$-flux through the island. These results represent crucial steps towards realizing topological quantum bits and designing complex topological phases.

Shiba-island model:

To investigate the spectroscopic and transport properties of topological superconductors, we consider a finite two-dimensional island of magnetic adatoms deposited on the surface of an $s$-wave superconductor, as schematically shown in Fig. 1(a). We take the magnetic moments of the adatoms to be aligned ferromagnetically, and describe them as classical moments, as spin-flip processes leading to Kondo screening are suppressed in a superconducting host. The system is then described by the Hamiltonian  li-16nc12297 [assuming a square-lattice structure, as realized e.g. in Pb(100)]

[TABLE]

where $c_{{\bf r},\sigma}^{\dagger}$ creates an electron at lattice site ${\bf r}$ with spin $\sigma$, and $\sigma^{i}$ are the spin Pauli matrices. $t$ and $\Delta_{s}$ are the hopping and pairing amplitudes of the superconductor, $\mu$ is the chemical potential, $\alpha$ denotes the Rashba spin-orbit coupling arising from the breaking of the inversion symmetry at the surface nadj-perge-14s602 , $J$ is the magnetic exchange coupling, $t_{\rm tip}$ is the amplitude for electron tunneling from the STS tip into the system, and $H_{\rm tip}$ describes the electronic structure of the tip [see supplemental information (SI) Sec. I]. Finally, $\{{\bf R}\}$ denotes the sites of the magnetic Shiba island, and we consider only the two-dimensional surface of the superconductor. For a Shiba lattice, i.e., when the surface is fully covered by magnetic adatoms li-16nc12297 , the system described by Eq. (1) possesses three topologically distinct phases with Chern numbers ${\cal C}=-1,0$, and $+2$. One can tune between these phases by varying the chemical potential, with phase transitions at $\mu_{c,\pm}^{(1)}=\pm\sqrt{J^{2}-\Delta^{2}}$ and $\mu_{c,\pm}^{(2)}=\pm 4t\mp\sqrt{J^{2}-\Delta^{2}}$. Provided the superconductor is sufficiently thin, such tuning of $\mu$ can be achieved via gating privat . We employ the non-equilibrium Keldysh Green’s function formalism [see SI Sec. I for details] to compute the local density of states (LDOS), the differential conductance, and charge currents in the Shiba islands.

Edge states:

We begin by considering a Shiba island of radius $R=15a_{0}$, and choose a set of parameters, $(\alpha,\Delta_{s},J)=(0.2,0.3,0.5)t$ to ensure that the superconducting coherence length, $\xi\approx 6a_{0}$ is smaller than $R$. In Fig. 1 (d,e,f), we plot the LDOS at the edge of the island for values of $\mu$ corresponding to the three different phases. The bulk-boundary correspondence qi-11rmp1057 implies that the island possesses low-energy edge states in the topological phases with ${\cal C}=+2,-1$ [see Figs. 1 (d) and (f)], which are absent in the ${\cal C}=0$ phase [Fig. 1 (e)]. Due to the finite size of the island, the topological edge modes are discrete levels, equally spaced in energy inside the superconducting gap. The spatially resolved LDOS of these low-energy mode reveals that most of their spectral weight is confined to the edge of the island [see Fig. 1(b,c)], thus confirming that these modes are indeed the putative edge modes associated with the topological phases. The distribution of the edge modes’ spectral weight along the island’s edge depends strongly on the shape of the underlying Fermi surface in the normal state: for a circular Fermi surface [${\cal C}=-1,\mu=-4t$, inset of Fig. 1(b)] the weight distribution is approximately uniform along the edge, while it is highly anisotropic for a nearly nested Fermi surface [${\cal C}=2,\mu=0$, inset of Fig. 1(c)]. With decreasing $\xi$ (see SI Sec. II), the topological modes become more closely confined to the edges of the island, and their spectral weight in the interior of the island decreases. Similar edge states are also found in irregularly shaped islands (see SI Sec. III), as their existence only relies on the topological character of the system’s ground state, and not its particular geometry.

Tunneling conductance:

To further explore the properties of the topological edge modes, we consider the differential tunneling conductance associated with the flow of charges between an STS tip and a site ${\bf r}$ in the sample. As the main difference between the topological non-trivial and trivial phases lies in the presence of topological low-energy edge modes in the former, and their absence in the latter, we plot in Fig. 2 the maximum differential conductance $G=\max_{\Delta V}\left[dI({\bf r})/dV\right]$ in a narrow voltage range, $\Delta V$, around zero voltage (see SI Sec. IV). In the topological phases, $\Delta V$ contains only a single edge mode (being two-fold degenerate in the ${\cal C}=2$ phase), with $G$ thus characterizing its conductance.

For a finite-sized Shiba lattice (see SI, Fig. S2), we find the conductance of the topological edge modes to be quantized (within $0.1\%$) and given by $G=|{\cal C}|\,2e^{2}/h$, with deviations from this quantized value becoming larger as one approaches the phase boundaries at $\mu_{c}^{(1,2)}$, see Fig. 2. This overall scale of $G$ is consistent with the observation that for a normal tip-superconductor junction, the quantum of conductance per fermionic degree of freedom is given by $2e^{2}/h$ due to Andreev scattering. In the trivial ${\cal C}=0$ phase, the conductance vanishes (for $t_{\rm tip}\rightarrow 0$) due to the absence of low-energy modes. The transition from $G\not=0$ to $G=0$ for the finite Shiba lattice occurs approximately at the same critical values of $\mu_{c}^{(1)}=\pm 0.4t,\mu_{c}^{(2)}=\pm 3.6t$ as in the thermodynamic limit. For the Shiba island with $R=15a_{0}$, $G$ is still (nearly) quantized in the ${\cal C}=-1$ phase, but deviations from the quantized value increase more rapidly in the ${\cal C}=+2$ phase as $\mu_{c}^{(1)}$ is approached, which we attribute to the finite size of the island (the same conclusion also holds for irregularly shaped islands, see SI Sec.III).

To further substantiate the topological character of the nanoscopic system, we adapt and modify a real-space Chern number formalism previously used for disordered topological insulators prodan1 and apply it to our Shiba islands. Numerical results for the modified Chern number, $\tilde{\cal C}$, are in the inset of Fig. 2, for details see SI Sec. V. While $\tilde{\cal C}$ does not reach integer values for the nanoscopic island (as topological invariants are strictly defined only in the thermodynamic limit), it clearly distinguishes between topologically trivial and non-trivial phases; moreover it correctly reflects ${\rm sgn}({\cal C})$ and also allows one to quantitatively distinguish between the ${\cal C}=-1$ and ${\cal C}=2$ phases.

Supercurrents:

The combination of magnetic moments and Rashba interaction leads to the emergence of superconducting chiral spin-triplet correlations, whose nature reflects the broken time-reversal symmetry (TRS) inside the Shiba island, and the preserved TRS outside (see SI Sec.VI). Such chiral spin-triplet correlations imply the existence of a supercurrent in the system, i.e., currents that flow even in the absence of any applied bias $V$. A plot of the supercurrents carried by the lowest-energy edge modes in the topologically non-trivial ${\cal C}=-1$ and ${\cal C}=2$ phases [Figs. 3(a,b)] reveals that they are not only confined to the edge of the island, but that their chirality (i.e., direction of flow) reflects the sign of the Chern number, ${\rm sgn}({\cal C})$. Recent advances in observing charge currents at small length scales Ji12 ; Rod16 ; Tet16 suggest that such supercurrents could be observed in the near future. This would imply that transport experiments could not only measure the magnitude of ${\cal C}$ via the differential conductance, but also its sign, providing unprecedented insight into the nature of topological phases.

Majorana character of edge states:

The realization of fault-tolerant topological quantum computing requires the creation of zero-energy Majorana modes with exotic braiding properties nayak-08rmp1083 . Such Majorana zero modes have been predicted to exist  kitaev01pu131 ; lutchyn-10prl077001 ; oreg-10prl177002 ; li-14prb235433 and subsequently been observed at the ends of the so-called Kitaev chains mourik-12s1003 ; nadj-perge-14s602 ; ruby-15prl197204 ; pawlak-16npj16035 . In contrast, the finite size of the Shiba islands considered above results in topological edge states that are located at discrete, and non-zero energies, $\pm E_{i}$. The symmetric states at $\pm E_{i}$ arise from the intrinsic particle-hole symmetry of the superconductor and can be considered a pair of Majorana fermions. To create an isolated zero-energy Majorana mode, it is necessary to spatially separate the two Majorana fermions forming a pair. This can be achieved by inserting a $\pi$-flux in the center of a Shiba island: this flux creates a half-quantum vortex alicea-12rpp076501 and binds one of the two Majorana fermions. As a result, the Majorana modes become spatially separated, with one mode being located at the island’s edge and one mode in the island’s center at the $\pi$-flux, as follows from a plot of the low-energy LDOS in Fig. 4. This spatial separation leads to an exponential suppression of the coupling between the Majorana modes with increasing radius of the island and a shift of their energies $\pm E_{i}$ to zero, as shown in the inset of Fig. 4, where we contrast the low-energy part of the LDOS in the absence (black) and presence of the $\pi$-flux (purple). We thus obtain two zero-energy Majorana modes in the limit $R\rightarrow\infty$, which can be thought of as analogue to the Majorana bound states at the ends of a Kitaev or Shiba chain.

Summary:

Identifying the characteristic electronic and transport properties of topological phases in nanoscopic Shiba islands is a key step in the quantum engineering of Majorana fermions. The robustness and universality of the results (see SI Secs. II and III) is of crucial importance for applications. Moreover, designing the magnetic structure or geometry of Shiba islands may enable creating more complex topological states and manipulating the spatial structure of Majorana modes. Both of these possibilities hold exciting new potential for realizing topological quantum computing.

I Methods

To investigate the spectroscopic and charge transport properties of nanoscopic topological superconductors, we employ the non-equilibrium Keldysh formalism. This approach allows us to simultaneously compute the local density of states, the current flowing between an STS tip and the system, and the associated differential conductance, as well as the supercurrents flowing inside the topological superconductor. Furthermore, we extended a formalism to compute the Chern number in real space, allowing to us characterize the topological phase of Shiba islands.

II Acknowledgements

The authors thank B. A. Bernevig, T. Neupert, A. Petrescu, E. Prodan, J. Röntynen, K. J. Franke, A. Kapitulnik, and R. Wiesendanger for discussions. This work was supported by the DFG through SFB 1143, SPP 1666, and GRK 1621 (S.R. and M.V.) and by the U. S. Department of Energy, Office of Science, Basic Energy Sciences, under Award No. DE-FG02-05ER46225 (E.M., S.C., and D.K.M.).