Probing Majorana zero modes by measuring transport through an interacting magnetic impurity

TL;DR

This paper investigates how transport measurements through an interacting magnetic impurity can reveal the presence of Majorana zero modes at the boundary of a topological superconductor, providing robust signatures even with strong interactions.

Contribution

It introduces an exactly solvable model showing that Majorana zero mode signatures in conductance are robust against interactions and proposes combined conductance and noise measurements to identify topological states.

Findings

Half-integer zero-bias conductance $G_0=e^2/2h$ due to MZM hybridization.

Robust MZM conductance signature persists at large interactions.

Vanishing Fano factor $F=0$ indicates topological regime.

Abstract

Motivated by recent experiments we consider transport across an interacting magnetic impurity coupled to the Majorana zero mode (MZM) observed at the boundary of a topological superconductor (SC). In the presence of a finite tunneling amplitude we observe hybridization of the MZM with the quantum dot, which is manifested by a half-integer zero-bias conductance $G_0=e^2/2h$ measured on the metallic contacts. The low-energy feature in the conductance drops abruptly by crossing the transition line from the topological to the non-topological superconducting regime. Differently from the in-gap Yu-Shiba-Rosinov-like bound states, which are strongly affected by the on-site impurity Coulomb repulsion, we show that the MZM signature in the conductance is robust and persists even at large values of the interaction. Interestingly, the topological regime is characterized by a vanishing Fano factor, $F=0$, induced by the MZM. Combined measurements of the conductance and the shot noise in the experimental set-up presented in the manuscript allow to detect the topological properties of the superconducting wire and to distinguish the low-energy contribution of a MZM from other possible sources of zero-bias anomaly. Despite being interacting the model is exactly solvable, which allows to have an exact characterization of the charge transport properties of the junction.

Equations

$$H=H_{\text{imp}}+H_C+H_K+H_{T,C}+H_{T,K},$$

$$H_{\text{imp}}=\frac{U}{4}{\Omega }_d-\frac{h}{2}\left(n_{\uparrow }^d-n_{\downarrow }^d\right)+\frac{\mu }{2}\left(n_{\uparrow }^d+n_{\downarrow }^d-1\right)$$

$$H_K=\sum _{j=1}^{\infty }[(-t{c}_j^{†}{c}_{j+1}+\Delta c_j{c}_{j+1}+\text{H.c.})-\mu c_j^{†}{c}_j]$$

$$H_{T,C}=V_c\sum _{\alpha =L,R}\left(c_{1\alpha }^{†}{d}_{\uparrow }+\text{H.c.}\right)$$

$$H_{T,K}=-i\sum _j{V}_j{\gamma }_j{\gamma }_{\uparrow }^d,$$

$${\xi }_{\uparrow }^{\eta }={\xi }_{\uparrow }^d(1-2n_{\downarrow }^d),{\gamma }_{\uparrow }^{\eta }={\gamma }_{\uparrow }^d,$$

$$H^{\ast }=$$

$$-i\frac{U}{4}{\gamma }_{\uparrow }^{\eta }{\xi }_{\uparrow }^{\eta }-\frac{(\mu +h)-i(\mu -h){\gamma }_{\uparrow }^{\eta }{\xi }_{\uparrow }^{\eta }}{4}{q}_{\downarrow }^d$$

$$+i\frac{V_c}{2}\sum _{\alpha =L,R}\left({\gamma }_{\uparrow }^{\eta }{\xi }_{1\alpha }-q_{\downarrow }^d{\xi }_{\uparrow }^{\eta }{\gamma }_{1\alpha }\right).$$

$$H^{\ast }$$

$$+\frac{1}{2}\sum _j\left({\vec{\eta }}_{\uparrow }^{†}\cdot {\hat{V}}_j\cdot {\vec{c}}_j+\text{H.c.}\right)-\frac{1}{2}{\vec{\eta }}_{\uparrow }^{†}\cdot \frac{U}{2}{\sigma }^z\cdot {\vec{\eta }}_{\uparrow },$$

$$\hat{V}_{j}=iV_{j}\left(\begin{array}[]{cc}1&1\\ 1&1\end{array}\right),$$

$$\hat{V}_{c}=\frac{V_{c}}{2}\left(\begin{array}[]{cc}(1+q^{d}_{\downarrow})&-(1-q^{d}_{\downarrow})\\ (1-q^{d}_{\downarrow})&-(1+q^{d}_{\downarrow})\end{array}\right).$$

$$J_Q=-i\frac{V_c}{4}\sum _{\alpha =L,R}\text{sign}(\alpha )\left[{\gamma }_{1\alpha }{\gamma }_{\uparrow }^{\eta }+q_{\downarrow }^d{\xi }_{1\alpha }{\xi }_{\uparrow }^{\eta }\right]$$

$$S_Q=\int d(t-t^{\prime })\frac{⟨\{\delta J_Q(t),\delta J_Q(t^{\prime })\}⟩}{2},$$

$$⟨J_Q⟩=$$

$${\hat{\mathbf{T}}}^{R/A}(ϵ)={\hat{V}}_c^{†}\cdot {\hat{\mathbf{G}}}_{{\eta }_{\uparrow }}^{R/A}(ϵ)\cdot {\hat{V}}_c,$$

$$⟨⟨J_Q⟩⟩=\sum _{n_{\downarrow }^f=0,1}p(n_{\downarrow }^f)⟨J_Q(n_{\downarrow }^f)⟩$$

$$G(\varphi )=\frac{\partial ⟨J_Q⟩}{\partial \varphi }\simeq \frac{e^2}{2h}\left[1-{\left(\frac{\varphi }{2{\Gamma }_{c}Z}\right)}^2\right],$$

$$S$$

$$\displaystyle\text{Tr}\Big{[}\left(\hat{\mathbf{T}}^{R}_{\eta_{\uparrow}}(\epsilon)+\hat{\mathbf{T}}^{A}_{\eta_{\uparrow}}(\epsilon)\right)\cdot\left(\hat{\mathbf{T}}^{R}_{\eta_{\uparrow}}(\epsilon)+\hat{\mathbf{T}}^{A}_{\eta_{\uparrow}}(\epsilon)\right)\Big{]},$$

$$⟨S_Q⟩=\sum _{n_{\downarrow }^f=0,1}p(n_{\downarrow }^f)S_Q(n_{\downarrow }^f).$$

$$⟨S_Q⟩\simeq \frac{e^3}{h}\frac{{\varphi }^3}{24{\left({\Gamma }_{c}Z\right)}^2}\left[1-\frac{3}{10}{\left(\frac{\varphi }{{\Gamma }_{c}Z}\right)}^2\right],$$

$$F=\frac{S_{Q}}{J_{B}}\Big{|}_{\phi=0}=q_{\text{e}}$$

$$J_B=\frac{e^2}{h}\varphi -⟨⟨J_Q⟩⟩.$$

$$F=\frac{{\left(2{\Gamma }_c/U\right)}^2}{1+{\left(2{\Gamma }_c/U\right)}^2}.$$

Full text

Introduction.

After the seminal paper by Kitaev Ref. Kitaev (2001) that predicted the existence of electronic collective modes reminiscent of the Majorana fermions speculated in 1937 by Ettore Majorana Majorana (2008), quasi one-dimensional systems, hosting two or more Majorana zero modes (MZMs), have attracted both experimental Das et al. (2012); Mourik et al. (2012); Albrecht et al. (2016); Deng et al. (2016); Lutchyn et al. (2018); Zhang et al. (2018); Gül et al. (2018); Nichele et al. (2017); He et al. (2017); Estrada Saldaña et al. (2019) and theoretical Oreg et al. (2010); Fu and Kane (2008); Ivanov (2001); Read and Green (2000) interest. Intrigued by exciting prospects in fault-tolerant quantum computation Nayak et al. (2008); Alicea et al. (2011); Alicea (2012), existing theoretical studies focused on zero-bias and current measurement across a junction of metallic leads and topological superconductors (SCs) Affleck and Giuliano (2013); Giuliano and Affleck (2019); Affleck and Giuliano (2014); Fidkowski et al. (2012); Chung et al. (2011); Peng et al. (2015); Chevallier et al. (2018), shot noise measurement Devillard et al. (2017); Giuliano et al. (2018), interferometer measurement Dessotti et al. (2016), persistent current in hybrid normal-superconducting rings Jacquod and Büttiker (2013); Pientka et al. (2013); Nava et al. (2016, 2017) and topological realization of the Kondo effect Béri and Cooper (2012); Eriksson et al. (2014a, b).

Recently, a new direction has emerged which explores the interplay between pure Majorana physics and electronic correlations Chen et al. (2018); Rahmani et al. (2015a, b); Affleck et al. (2017).

In this letter we fully characterize the electronic transport through a novel class of experimentally realizable systems Deng et al. (2016); Zhang et al. (2018) which have recently attracted great interest for their easily realization and control. The MZM, emerging at the endpoint of a one dimensional semi-infinite wire with strong spin-orbit interaction (i.e. InAs wire) deposited on top of a s-wave SC and exposed to an external magnetic field, is coupled to an interacting magnetic impurity that can be used as a spectrometer. By coupling the dot to two metallic fully-polarized contacts we can probe the properties of the MZM though measurement of the current and the shot noise across the junction.

Model Hamiltonian.

To model the junction displayed in Fig. 1 we consider the Hamiltonian

[TABLE]

where

[TABLE]

is the dot Hamiltonian, with $n^{d}_{\sigma}=d^{\dagger}_{\sigma}d_{\sigma}$ the number operator on the impurity site and $\Omega_{d}=(2n^{d}_{\uparrow}-1)(2n^{d}_{\downarrow}-1)$. In (2) $U$ denotes the on-site interaction, $\mu$ the gate potential and $h$ the Zeeman field applied on the dot level. The Hamiltonian of the semi-infinite Kitaev chain reads

[TABLE]

where $t$ is the hopping amplitude between nearest neighbor sites, $\Delta$ the p-wave superconducting pairing and $\mu$ the chemical potential of the wire. We notice that left $(L)$ and right $(R)$ metallic contacts are described by Hamiltonian (3) with $\Delta=0$ and different electrochemical potentials $\mu_{L}=-\mu_{R}=\phi/2$. In our model both the Kitaev and the metallic chains are described by spinless particles. This is a natural assumption if one consider that topological SCs are realized in one dimensional p-wave SCs characterized by strong spin-orbit coupling and large magnetic fields, and if we assume fully-polarized ferromagnetic contacts. In this regime the magnetic exchange between the impurity spin and the leads is suppressed and the low-energy physics is dominated by the coupling with the MZM Shindou et al. (2010); Žitko and Simon (2011); Lee et al. (2013); Cheng et al. (2014); Weymann and Wójcik (2017). The tunneling between the dot and the metallic contacts reads:

[TABLE]

where $V_{c}$ is the tunneling amplitude and $\alpha=L,R$. Finally, we consider the hybridization with the boundary site of the semi-infinite Kitaev chain:

[TABLE]

where the sum extends to the semi-infinite Kitaev chain and we have introduced the Majorana operators $\gamma=c+c^{\dagger}$ and $\xi=-i(c-c^{\dagger})$. The simple model in Eq. (5) allows to study exactly the effect of correlations on the non-local Majorana edge state tunnel-coupled to an interacting quantum dot.

The interacting model is exactly solvable because the $d_{\downarrow}$ electrons are localized and $n^{d}_{\downarrow}$ can be treated as a $\mathbb{Z}_{2}$ real number $(=0,1)$. This property makes the Hamiltonian (1) an effective quadratic model, where similarly to the Falicov-Kimball model (FKM) Falicov and Kimball (1969) the $\downarrow$ configuration is obtained by minimizing the ground-state energy of the $\uparrow$ degrees of freedom.

In the absence of metallic contacts, $V_{c}=0$, the equilibrium properties of the model in Eq. (1) has been already studied in Ref. Shankar and Maciejko (2019). It is convenient to perform the following gauge transformation:

[TABLE]

in terms of $\gamma^{\eta}_{\uparrow}$ and $\xi^{\eta}_{\uparrow}$ fermions the Hamiltonian (1) becomes com :

[TABLE]

To avoid irrelevant complications we consider the case $\mu=h=0$. Introducing the Dirac (complex) fermion $\eta_{\uparrow}=\gamma^{\eta}_{\uparrow}+i\xi^{\eta}_{\uparrow}$, the model Hamiltonian reads

[TABLE]

where in the Nambu representation $\vec{\psi}=(\psi,\,\,\psi^{\dagger})^{T}$, $\hat{V}_{j}$ is the hybridization matrix between the dot and the $j$-th site of the Kitaev chain:

[TABLE]

and $\hat{V}_{c}$ couples the metallic contacts to the dot

[TABLE]

To characterize the transport properties of the junction we compute the charge current, $J_{Q}=(J_{L}-J_{R})/2$ with $J_{\alpha}=-i[N_{\alpha},H]$, that in the new representation (6) reads:

[TABLE]

where $\text{sign}(L)=+1$ and $\text{sign}(R)=-1$, and the zero frequency limit of the $J_{Q}$ fluctuations

[TABLE]

where $\delta J_{Q}=J_{Q}-\langle J_{Q}\rangle$. In the following we study transport through the junction by performing calculations with Keldysh Green’s function technique Rammer (2007); Haug and Jauho (1996), which we compare with the scattering matrix approach Blanter and Büttiker (2000); Nazarov and Blanter (2009); Nilsson et al. (2008).

Probing MZMs with charge conductance and shot noise.

Experimental measurements of charge conductance at the boundary of topological materials reveal the emergence of low-energy MZMs Das et al. (2012); Albrecht et al. (2016); Deng et al. (2016); Lutchyn et al. (2018); Zhang et al. (2018); Gül et al. (2018); Nichele et al. (2017) and provide an experimental tool to detect topological transitions by studying surface states via STM Jäck et al. (2019); Li et al. (2018); Schindler et al. (2018); Drozdov et al. (2014); Murani et al. (2017).

In this letter we present a detailed characterization of the low-energy signatures observed in the charge conductance and shot noise measurements that allows to classify different regions of the Kitaev chain phase diagram. Despite being done on a toy model, the analysis may give physical insight for the understanding of the ongoing experiments where the effect of local on-site interaction cannot be neglected. We start by reporting the expression of the current flowing through the metallic contacts:

[TABLE]

where $f_{L}(\epsilon)=f(\epsilon-\phi/2)$, $f_{R}(\epsilon)=f(\epsilon+\phi/2)$, Tr is the trace in the $2\times 2$ Nambu space, $\hat{\mathbf{T}}^{R/A}(\epsilon)$ is the impurity transfer matrix

[TABLE]

and $\bar{\rho}(\epsilon)$ is the boundary density of states for the semi-infinite normal contacts (we refer to the supplemental material for more details sup ).

The resulting value of the current is obtained by averaging over the spin $\downarrow$ configurations:

[TABLE]

where in the absence of any gate potential or Zeeman field on the quantum dot $p(0)=p(1)=1/2$.

In the topological regime, the low-energy physics is governed by the in-gap states that emerge from the hybridization between the real and imaginary part of the spin up dot fermion and the MZM of the Kitaev chain. The coupling between $\gamma^{d}_{\uparrow}$ and $\gamma_{1}$ induces an energy splitting $\sim V$, while the quantum dot interaction generates an energy splitting $\sim U$ between $\gamma^{d}_{\uparrow}$ and $\xi^{d}_{\uparrow}$. The combined effect of the dot-Kitaev chain coupling and the interaction, on an odd number of MZMs, is to split two of them by a term $\sim f(U,V)$ that eventually, for $U$ strong enough, wash them out from the superconducting gap. Whereas, the third one is a topologically protected, and robust to the interaction, zero energy mode. In the trivial regime, we have an even number of MZMs, then no zero energy mode is preserved as any finite interaction induces a hybridization $\sim U$ between them.

These features can by easily detected resorting to the scattering matrix approach of Ref. Nilsson et al. (2008); Nazarov and Blanter (2009) that allows to interpret the transport properties of the system in terms of the scattering processes across the junction (a detailed description is given in the supplementary material sup ). In the trivial regime (left panel of Fig. 2), the presence of massive in-gap modes suppresses low-energy scattering processes so that the $L$ and $R$ contacts are disconnected in the large $U/V_{c}$ limit. On the contrary, in the topological regime (right panel of Fig. 2), the presence of the MZM keeps alive all the scattering processes at low-energy. The normal transmission (T), the Andreev reflection (A) and the crossed Andreev reflection (C) are equal to one fourth at any value of $U$ and $V$. As a consequence, the charge current, $J_{Q}$, that measures the charge imbalance between left and right lead, is $\propto A+T\sim 1/2$ and the zero-bias conductance is reduced from $e^{2}/h$ to $e^{2}/2h$, as already observed in previous studies Liu and Baranger (2011); Lee et al. (2013); López et al. (2014); Vernek et al. (2014); Galpin et al. (2014). Interestingly, the on-site local repulsion does not modify the result $e^{2}/2h$ while it affects the curvature of the low-bias conductance by renormalizing the MZM:

[TABLE]

where $\Gamma_{c}=2\pi\bar{\rho}(0)V^{2}_{c}$ is the hybridization with the metallic contacts, $\bar{\rho}(\omega)$ the boundary metallic density of states and $Z$ the quasiparticle renormalization factor. The latter quantity is shown in the color map 3, where we analyze the evolution of $Z$ in different regions of the phase diagram of the Kitaev chain. We stress that Eq. (16) is valid in the topological regime $|\mu|<2t$ where the SC posses a non-trivial topology and a MZM appears at the edge of the semi-infinite Kitaev chain. Differently, in the region $|\mu|>2t$, the MZM disappears and we enter in the Coulomb blockade regime where the zero-bias conductance is suppressed.

The topological transition is associated to a drastic variation of the conductance $G(\phi)$. Indeed, as shown in Fig. 4, by crossing the critical line, $\mu=2t$, we observe a jump from $G_{0}=e^{2}/2h$ in the topological region to $G_{0}\simeq 0$ in the trivial one, which allows to distinguish the two different phases. Moreover, we notice that in the non-topological region, for $\mu/t\simeq 2.5$, the conductance presents coherent in-gap peaks attributable to Andreev bound states induced by the impurity, reminiscent of Yu-Shiba-Rosinov states LUH (1965); Shiba (1968); Rusinov (1969).

The effect of the interaction on the $G(\phi)$ is shown in Fig. 5, where we report the evolution of the low-energy MZM and of the Yu-Shiba-Rosinov-like bound states as a function of $U/t$. Being non-topological, the latter features are strongly affected by the interaction, and indeed, as shown in Fig. 5, above a certain value of $U/t$ they enter in the continuum of Cooper-pairs excitations of the SC. On the other hand, the contribution to the zero-bias conductance $G_{0}$ of the MZM is robust and persist for any value of $U/t$. The interaction renormalizes the coupling (5) $V\to V\sqrt{Z}$ between the dot and the MZM by the quasiparticle renormalization factor $Z$, displayed in Fig. 3, and enhances the curvature of the conductance close to the zero-bias anomaly (16).

In order to have a complete characterization of the junction we compute the shot noise $S_{Q}$ that at zero temperature reads:

[TABLE]

for more details we refer the interested reader to the supplementary material sup . Analogously to the previous case, we perform the average over $\downarrow$ configurations

[TABLE]

A complete characterization of the low-energy transport properties is given in Fig. 6, where we plot the current $J_{Q}$, the corresponding charge-conductance $G(\phi)$ and its fluctuations $S_{Q}(\phi)$ as a function of the applied bias. We notice that dependently on the region of the Kitaev phase diagram 3 we predict different low-energy response. In particular behavior (I) and (II) denote the presence of a MZM, while (III) and (IV) characterize the non-topological regime. Differently from (I) and (IV), regions (II) and (III) present additional in-gap bound states distinguished by sharp peaks in $G(\phi)$ away from the zero-bias anomaly.

We observe that an additional signature of the MZM is given by the low-bias behavior of the shot noise $S_{Q}(\phi)$, which is shown in the bottom panel of Fig. 6. Indeed, in the topological regime, for small bias, the shot noise goes like:

[TABLE]

while it becomes linear in the non-topological region, $S_{Q}\propto\phi$ for $|\mu|>2t$. The evaluation of the shot noise allows to compute the Fano factor

[TABLE]

which determines the charge of the elementary carriers de Picciotto et al. (1997). In Eq. (20) we have introduced the backscattering current, defined as the deviation from the unitary transmission through the junction Sela et al. (2006):

[TABLE]

As a consequence of the small bias behavior of Eqs. (16) and (19), the topological regime $|\mu|<2t$ is characterized by a vanishing Fano factor $F=0$, independently from the value of the interaction $U/t$, as shown in the bottom panel of Fig. 7. On the other hand, in the non-topological region $F$ is a function of $U/t$ which becomes equal to $1$ in the non-interacting limit $U/t\to 0$. In particular for $|\mu|>2t$ we find:

[TABLE]

Therefore, experimental measurements of the shot noise give additional informations complementary to those attainable by studying the characteristic zero-bias conductance $e^{2}/2h$. Combined measurements of the conductance and the shot noise in the experimental set-up presented in Fig. 1 allow to detect the topological properties of the superconducting wire and to distinguish the low-energy contribution of a MZM from other possible sources of zero-bias anomaly. We argue that the predicted behavior of the conductance and the shot noise persists even for a more realistic model Hamiltonian, that presents a non-vanishing tunnel-coupling with the spin $\downarrow$ fermionic operator in the quantum dot Lee et al. (2013); Cheng et al. (2014); Weymann and Wójcik (2017); Górski et al. (2018). However, a detailed analysis of this problem is left to future investigations.

Conclusions.

The present results show that transport measurements give a detailed characterization of the topological phase diagram of real materials and reveal MZM in nano-wires. The presence of a MZM is outlined by a fractional zero-bias conductance $e^{2}/2h$ that, we have shown, is robust against the dot interaction. Additionally, for small values of the on-site repulsion, we find in-gap bound states that represent the only low-energy feature in the topologically trivial region of the phase diagram in Fig. 3. Furthermore, we find that the topological regime is characterized by a vanishing Fano factor induced by the tunnel-coupling with the MZM at the edge of the superconducting wire. Our analysis gives a complete characterization of charge transport measurements that can experimentally detect the presence of MZM on the edge of real materials and, indirectly, allows to reconstruct their topological phase diagram.

Acknowledgements.

This work has been supported by the European Union under H2020 Framework Programs, ERC Advanced Grant No. 692670 “FIRSTORM”. We are grateful to Michele Fabrizio, Domenico Giuliano and Roberto Raimondi for discussions and comments on the manuscript. We thank Shankar Ganesh and Joseph Maciejko for useful correspondence at the early stage of this work.