Tunneling Current Measurement Scheme to Detect Majorana Zero Mode Induced Crossed Andreev Reflection

TL;DR

This paper proposes a tunneling current measurement scheme using a metallic ring to detect Majorana zero modes via crossed Andreev reflection, observing flux-dependent current peaks.

Contribution

It introduces a novel experimental setup with a metallic ring to directly measure Majorana-induced crossed Andreev reflection through tunneling current.

Findings

Detection of flux-dependent positive and negative current peaks

Distinct wave vector propagation of electrons and holes enables measurement

Potential for improved Majorana zero mode detection methods

Abstract

We propose a scheme to detect the Majorana-zero-mode-induced crossed Andreev reflection by measuring tunneling current directly. In this scheme a metallic ring structure is utilized to separate electron and hole signals. Since tunneling electrons and holes have different propagating wave vectors, the conditions for them to be constructively coherent in the ring differ. We find that when the magnetic flux threading the ring varies, it is possible to observe adjacent positive and negative current peaks of almost equal amplitudes.

Equations

$${\varphi }_D\pm {\varphi }_M=2n\pi (n\in \mathbb{Z}),$$

$$H_M=i\frac{E_M}{2}{\gamma }_1{\gamma }_2.$$

$$H_T^{(i)}=\int dx\frac{t_i}{\sqrt{2}}{\gamma }_i[{\psi }_i^{†}(x)-{\psi }_i(x)]\delta (x),$$

$$S_{L}=\left(\begin{array}[]{ccc}-(a+b)&\sqrt{\epsilon}&\sqrt{\epsilon}\\ \sqrt{\epsilon}&a&b\\ \sqrt{\epsilon}&b&a\end{array}\right)\,,$$

$$S_{M}=\left(\begin{array}[]{cccccccc}r_{1}^{(e)}&t_{1}^{(e)}&w_{2}^{(e)}&w_{2}^{(e)}&\tilde{r}_{1}^{(e)}&\tilde{t}_{1}^{(e)}&\tilde{w}_{2}^{(e)}&\tilde{w}_{2}^{(e)}\\ t_{1}^{(e)}&r_{1}^{(e)}&w_{2}^{(e)}&w_{2}^{(e)}&\tilde{t}_{1}^{(e)}&\tilde{r}_{1}^{(e)}&\tilde{w}_{2}^{(e)}&\tilde{w}_{2}^{(e)}\\ w_{1}^{(e)}&w_{1}^{(e)}&r_{2}^{(e)}&t_{2}^{(e)}&\tilde{w}_{1}^{(e)}&\tilde{w}_{1}^{(e)}&\tilde{r}_{2}^{(e)}&\tilde{t}_{2}^{(e)}\\ w_{1}^{(e)}&w_{1}^{(e)}&t_{2}^{(e)}&r_{2}^{(e)}&\tilde{w}_{1}^{(e)}&\tilde{w}_{1}^{(e)}&\tilde{t}_{2}^{(e)}&\tilde{r}_{2}^{(e)}\\ r_{1}^{(h)}&t_{1}^{(h)}&w_{2}^{(h)}&w_{2}^{(h)}&\tilde{r}_{1}^{(h)}&\tilde{t}_{1}^{(h)}&\tilde{w}_{2}^{(h)}&\tilde{w}_{2}^{(h)}\\ t_{1}^{(h)}&r_{1}^{(h)}&w_{2}^{(h)}&w_{2}^{(h)}&\tilde{t}_{1}^{(h)}&\tilde{r}_{1}^{(h)}&\tilde{w}_{2}^{(h)}&\tilde{w}_{2}^{(h)}\\ w_{1}^{(h)}&w_{1}^{(h)}&r_{2}^{(h)}&t_{2}^{(h)}&\tilde{w}_{1}^{(h)}&\tilde{w}_{1}^{(h)}&\tilde{r}_{2}^{(h)}&\tilde{t}_{2}^{(h)}\\ w_{1}^{(h)}&w_{1}^{(h)}&t_{2}^{(h)}&r_{2}^{(h)}&\tilde{w}_{1}^{(h)}&\tilde{w}_{1}^{(h)}&\tilde{t}_{2}^{(h)}&\tilde{r}_{2}^{(h)}\end{array}\right)\ ,$$

$$j=j_{\text{(in)}}[P_e-(v_h/v_e)P_h],$$

Full text

Majorana zero modes (MZM) have been attracting the interest of the condensed matter physics community in recent years Alicea_Review due to their potential application in topological quantum computation Nayak_TQC_Review . Several proposals have been made to realize MZM in realistic systems Read_Green ; FuKane_MF_TIsurf ; Sau_MF_SOC_Zeeman_sWave . Among them, the last one Sau_MF_SOC_Zeeman_sWave is based on the structure of a strong spin-orbit coupled semiconductor nanowire in proximity to an s-wave superconductor. This proposal, due to its convenience, has spurred many experimental efforts immediately Mourik_Ex_MF_Semi_SC_Wire ; Das_Ex_ZBP ; Deng_Ex_ZBP ; Churchill_Ex_ZB_oscillation . More recently, several other different methods Nadj_Ex_MZM_Fe_Chain ; JPXu_Ex_MZM_Vortex_TSC ; QLHe_Ex_ChiralMF_QSH have been applied for the observation of MZM.

Several methods can be used to detect the existence of MZM Alicea_Review , and the tunneling spectroscopy is one of them. It is generally based on the MZM-induced Andreev reflection (AR) Nilsson_CAR ; Law_MF_Induced_AR . There are two types of AR: local AR (LAR) and crossed AR (CAR). MZM-induced LAR results in a zero-bias peak (ZBP) of the differential conductance in the tunneling process. Up until now, ZBP has been observed by many experimental groups Mourik_Ex_MF_Semi_SC_Wire ; Das_Ex_ZBP ; Deng_Ex_ZBP . However, the conclusion of the existence of MZM cannot be made unambiguously through the observation of ZBP only, for some other mechanisms may produce similar results. More experimental evidence is needed to obtain definitive results. In contrast to LAR, it is believed that MZM-induced CAR does not result in any charge tunneling signal, since the electron tunneling probability equals to that for the hole in CAR. Instead of the tunneling current, it is suggested that shot noise Blanter_Buttiker_ShotNoise_Review has special properties in MZM-induced CAR and can, in principle, be detected. Nonetheless, to the best of our knowledge, no experiment has ever been done till now to measure the shot noise in a CAR set-up.

In this work, we propose a new scheme to observe the tunneling current directly through the MZM-induced CAR process. It goes beyond the traditional belief that the shot noise measurement is necessary in the observation of MZM-induced CAR. In order to separate the electron and hole tunneling signals, we employ a metallic ring to hold tunneling electrons and holes. A magnetic flux is threaded through the ring, so that an electron/hole acquires an Aharonov-Bohm (AB) phase when traveling around the ring AB_Paper . The resonance condition for the electron/hole to be constructively coherent in the ring is

[TABLE]

where $\phi_{D}=kL$ ($k$ is the wave vector of the electron/hole, $L$ is the circumference of the ring) and $\phi_{M}=2\pi\Phi/\Phi_{0}$ ($\Phi$ is the magnetic flux threading the ring in units of the flux quantum $\Phi_{0}$) are the dynamical phase and the magnetic phase (AB phase) that an electron/hole acquires when traveling one complete circle around the ring, and $\pm$ denotes clockwise/counterclockwise circulations. Because under a finite bias, tunneling electrons (above the Fermi level) have slightly larger wave vector than tunneling holes (below the Fermi level), they are not constructively coherent simultaneously. If we couple an external lead weakly to the ring, only when the resonant condition is fulfilled, there is a remarkable tunneling signal. Therefore, one can expect to observe electron and hole tunneling peaks separately corresponding to different magnetic fluxes.

Suppose there exists a pair of MZM close in space with a coupling energy $E_{M}$. The pair can be the MZM located at the opposite ends of a topological superconducting wire Sau_MF_SOC_Zeeman_sWave . The Hamiltonian for the two coupled MZM is

[TABLE]

Here $\gamma_{1}$ and $\gamma_{2}$ are Majorana operators that satisfy $\gamma_{1}^{\dagger}=\gamma_{1}$, $\gamma_{2}^{\dagger}=\gamma_{2}$, $\gamma_{1}^{2}=\gamma_{2}^{2}=1$ and $\gamma_{1}\gamma_{2}+\gamma_{2}\gamma_{1}=0$. Then, the MZM $\gamma_{i}$ ($i=1,2$) is coupled to the $i$-th ring through a tunneling Hamiltonian

[TABLE]

where $\psi_{i}(x)$ and $\psi_{i}^{\dagger}(x)$ are electron annihilation and creation operators in the $i$-th ring, which satisfy the usual fermion commutation relations, and $t_{i}$ denotes the coupling strength.

Our proposed transport experiment set-up is shown in Fig. 1. We let one lead bridge a source reservoir to the left ring and another lead bridge the right ring to a drain reservoir. The coupling between a lead and a ring can be described by a tri-junction scattering matrix Tri-junction_Scattering

[TABLE]

where $\epsilon$ denotes the tunneling strength, $a=(\sqrt{1-2\epsilon}-1)/2$ and $b=(\sqrt{1-2\epsilon}+1)/2$. The lead and the two ring-arms consist of the three channels of this scattering matrix. First, electrons incident from the source flow along the left lead and tunnel into the left ring. Next, electrons in the left ring tunnel into the right ring (may be converted to holes at the same time) through MZM-induced CAR. This process is shown in Fig. 2 and can be described by a scattering matrix

[TABLE]

of which the eight channels are lower left electron, upper left electron, lower right electron, upper right electron, lower left hole, upper left hole, lower right hole and upper right hole. For the detailed form and a derivation of these scattering matrix elements, see the supplementary material Supplement_Material . The MZM-induced CAR dominates LAR when the electron incident energy $E$ and the MZM energy width $t_{i}^{2}/\hslash v_{F}$ ($v_{F}$ is the Fermi velocity) are much smaller than the MZM energy $E_{M}$ Nilsson_CAR . Lastly, electrons/holes in the right ring tunnel to the second lead, and outflow to the drain.

We require that the metallic rings are coupled weakly to both the leads and the MZM. In this case the rings are almost isolated, and the tunneling electrons/holes can stay in the ring for a relatively long time. Thus, the resonance condition Eq.(1) plays an important role in the transport process.

The function of the left ring is to single out the incident electron energy. Only when an electron is constructively coherent in the ring (i.e., fulfills the resonant condition), it has a substantial contribution to the incident flow of CAR. It is worth mentioning that two directly coupled metallic rings were investigated in previous works David_Two_Rings ; Avishai_Two_Rings ; Lei_Two_Rings , and two metallic rings coupled by a p-wave wire was studied in David_pWave_Rings . All these studies focused on equilibrium properties, especially persistent currents inside the rings.

We first study the probability, $P_{e}$ and $P_{h}$, for the tunneling electron/hole to flow out from the second lead when an electron is incident with a specific energy from the first lead. The probabilities are, of course, affected by the magnetic fluxes $\Phi_{M}^{(1)}$ and $\Phi_{M}^{(2)}$, and we prefer to express them as functions of the magnetic phases $\phi_{M}^{(1)}$ and $\phi_{M}^{(2)}$. To solve the problem, we need to combine the scattering process at the tri-junction of the first lead and the left ring, the motion of electron/hole in the left ring, the CAR between the left ring and the right ring, the motion of electron/hole in the right ring and the tunneling process between the right ring and the second lead. The details of the calculation for $P_{e}$ and $P_{h}$ are given in the supplementary material Supplement_Material .

The result of a numerical calculation is shown in Fig. 3, under the setting of a group of typical values of the relevant parameters. The magnetic fluxes are varied such that the magnetic phases scan from [math] to $2\pi$. (Any other situation is equivalent to the one in this range due to the Byers-Yang theorem Byers_Yang .) The electron incident energy $E$ and the MZM width (due to their couplings to the rings) $\Gamma_{i}=t_{i}^{2}/\hslash v_{F}$ are both set to be much smaller than the MZM coupling energy $E_{M}$, so that CAR dominates LAR. The circumferences of the two rings are chosen a little bit different in order to distinguish their resonance conditions.

From Fig. 3, it is clear that the electron/hole tunneling probability peaks when both the resonance conditions $\phi_{e/h}^{(1)}\pm\phi_{M}^{(1)}=2n\pi$ and $\phi_{e/h}^{(2)}\pm\phi_{M}^{(2)}=2n^{\prime}\pi$ are fulfilled. Under our setting, $\phi_{e}$ and $\phi_{h}$ are only slightly different. So the electron and hole tunneling peaks are very close to each other. Nonetheless, they can be clearly distinguished. The exact locations of the peaks are actually a little further apart than those the resonance condition predicts, due to the quantum mechanical level repulsion.

Next, we consider the tunneling current Andreev_Scattering_Current

[TABLE]

where $j_{\text{(in)}}$ represents the current carried by the incident electron, $P_{e/h}$ is the electron/hole tunneling probability and $v_{e/h}$ is the velocity. In our case since the incident energy $E$ is small (compared to $E_{F}$), $v_{e}\approx v_{h}\approx v_{F}$, and so $j\approx j_{\text{(in)}}(P_{e}-P_{h})$. In our model, this current is contributed solely from the MZM-induced CAR process. When $P_{e}=P_{h}$, the MZM-induced CAR process does not generate a current, which is the traditional belief and the case for a setup where the tunneling is measured directly across the topological superconducting wire.

Figure 4 shows the tunneling current in the second lead. From Fig. 4(a), there exist separate regions of positive and negative currents. The positive current corresponds to the situation in which the electron tunneling dominates, while the negative current corresponds to that in which the hole tunneling dominates. The boundaries between these regions are the lines on which $\phi_{M}^{(2)}$ fulfills $k_{F}L_{2}\pm\phi_{M}^{(2)}=2n^{\prime}\pi$. If we fix the magnetic flux threading the left ring, and vary the flux of the right ring, we see a negative current peak immediately followed by a positive one, or vice versa, as shown in Fig. 4(b). The separation of the adjacent opposite peaks is due to slightly different wave vectors of electrons and holes. Moreover, the almost equal amplitudes of the peaks imply the tunneling is through MZM, which couple in the same way to electrons as to holes.

The above discussions are restricted to the situation in which a single electron is incident with a specific energy. In realistic situations, we should connect the first lead to a source and the second lead to a drain, and apply a negative voltage bias between them. We should also ground the superconductor hosting the MZM. Electrons are incident from the source, and it depends on the flux threading the second ring whether it is electron or hole that dominates the flow to the drain. When electrons dominate, there is little current flowing through the superconductor to the ground. While if holes dominate, there is an electron current flowing through the superconductor to the ground. The external connection of our set-up and the overall physical phenomenon are analogous to those studied in DF_MF_Converter .

Assuming the temperature is well below the bias voltage (and also $E_{M}$), all the electron states within the range of the bias participate in the transport process, according to the Landauer-Büttiker formalism Landauer_Buttiker . This complication, however, does not change our established picture of the tunneling current. The reason is twofold. First, the resonance has a width, originating from the coupling of the ring to the lead and MZM. [This width also determines the phase resolution of a typical tunneling signal shown in Fig. 4(b)]. If the applied bias is made to be smaller than the resonance width, it is possible that all incident electrons are effectively in resonant states. Second, the positions of tunneling peaks (in the magnetic-phase vs tunneling-current diagram) may slightly vary for different incident electrons, but the overall effect is not that their tunneling currents are canceled, for the wave vectors of the holes are always smaller than those of the electrons, resulting in all the positive current peaks are separated from all the negative ones by a boundary set by the Fermi vector $k_{F}$. Therefore, the line shape of the tunneling current as shown in Fig. 4(b) can be qualitatively taken as a signature of MZM-induced CAR.

In our model we implicitly assume that the electrons are spinless, by considering that the electron spin, in an experimental set-up, can be fully polarized in the presence of a Zeeman magnetic field. We also implicitly assume that there is only a single channel in the metallic ring. Actually, we do not intend the rings to have multiple channels, for in that case electrons in different channels can have distinct wave vectors, such that in one magnetic-phase period (from [math] to $2\pi$) there can exist multiple resonant states, of which if they have overlaps, the tunneling currents may cancel.

Finally, we discuss the parameters used in our numerical calculations and the feasibility of an experimental verification. For the MZM, we believe Mourik_Ex_MF_Semi_SC_Wire ; Deng_Ex_ZBP that a superconducting gap $\Delta$ of 0.25 meV can be induced in the InSb nanowire by superconducting Nb. When the spin-orbit energy of InSb is 0.3 meV, the coherence length $\xi$ on the wire is estimated to be 185 nm. Then, in order to let the MZM coupling energy $E_{M}$ ($\approx\Delta e^{-l/\xi}$) to be around 0.01 meV, the length of the InSb nanowire $l$ should be about 595 nm. For the metallic rings, we want the size of the rings to be as large as possible. We follow GaAs_GaAlAs_Loop that electron gas at a GaAs-GaAlAs hetero-interface can have a Fermi wavelength $\lambda_{F}=42~{}\text{nm}$ and a Fermi velocity $v_{F}=2.6\times 10^{5}~{}\text{m/s}$, which lead to the Fermi vector $k_{F}=0.15~{}\text{nm}^{-1}$ and the Fermi energy $E_{F}=12.835~{}\text{meV}$. The mobility of the electron gas attains $1.14\times 10^{6}~{}\text{cm}^{2}/\text{Vs}$, which makes the elastic mean free path $11~{}\mu\text{m}$ GaAs_GaAlAs_review . Under this condition, we can safely stay in the phase coherent transport regime as we restrict the circumferences of the rings to be no more than 3 $\mu\text{m}$.

We conclude by noting that the persistent current Buttiker_Persistent_Current inside the right ring also changes sign from the tunneling electron dominated regime to the tunneling hole dominated one when the magnetic flux is adjusted. This change in sign of the persistent current can be detected by placing a high sensitivity magnetometer near the right ring. In summary, we have presented a method using the constructive coherence condition in a metallic ring to separate the electron and hole tunneling signals of the Majorana zero mode induced crossed Andreev reflection.