⇦Chinese Physics Letters, 2019, Vol. 36, No. 10, Article code 107102 Charge Transport Properties of the Majorana Zero Mode Induced Noncollinear Spin Selective Andreev Reflection * Xin Shang (尚欣), Hai-Wen Liu (刘海文)**, Ke Xia (夏钶)** Affiliations Department of Physics, Beijing Normal University, Beijing 100875 Received 7 September 2018, online 21 September 2019 *Supported by the National Natural Science Foundation of China under Grant Nos 11674028, 61774017, 11734004 and 21421003, and the National Key Research and Development Program of China under Grant No 2017YFA0303300.
**Corresponding author. Email: haiwen.liu@bnu.edu.cn; kexia@bnu.edu.cn Citation Text: Shang X, Liu H W and Xia K 2019 Chin. Phys. Lett. 36 107102    Abstract We study the charge transport properties of the spin-selective Andreev reflection (SSAR) effect between a spin polarized scanning tunneling microscope (STM) tip and a Majorana zero mode (MZM). Considering both the MZM and the excited states, we calculate the conductance and the shot noise power of the noncollinear SSAR using scattering theory. We find that the excited states give rise to inside peaks. Moreover, we numerically calculate the shot noise power and the Fano factor of the SSAR effect. Our calculation shows that the shot noise power and the Fano factor are related to the angle between the spin polarization direction of the STM tip and that of the MZM, which provide additional characteristics to detect the MZM via SSAR. DOI:10.1088/0256-307X/36/10/107102 PACS:71.10.Pm, 74.45.+c, 74.78.-w, 85.75.-d © 2019 Chinese Physics Society Article Text The Majorana zero mode (MZM) is a special type of Bogoliubov quasiparticle excitation with non-Abelian statistics, which forms the base of topological quantum computations,[1–7] and has received a large amount of research interest since being proposed. There are a number of methods that have been used to generate and detect the MZM in condensed matter systems,[5,7–9] these include a chiral p-wave superconductor,[10] the $\nu =5/2$ fractional quantum Hall system,[1,2] topological insulator (TI)/s-wave superconductor (SC) interfaces with the MZM in the vortex core,[11] proximity-induced superconductors for spin-orbit coupled nanowires,[6,12] and a triple-quantum dot system.[13] An electron with its spin direction aligned with the MZM will undergo an Andreev reflection, while an electron with opposite spin direction will not.[14–18] This allows us to detect the existence of the MZM using ferromagnetic STM.[14–18] However, previous studies of 2D SSAR have only the collinear spin transport, for example, the spin polarization of the lead being parallel or antiparallel to that of the MZM. The excited states, which have low energy above the MZM, were ignored.[18] To obtain more information concerning SSAR, we consider both the MZM and the excited states in the SSAR effect when the spin polarization direction of the STM tip and the MZM are noncollinear. In addition, compared to the conductance, the shot noise may shed more light on the underlying physical properties of the system.[19] In particular, the shot noise can be used to determine the charge and statistics of the quasiparticles relevant to transport, and to reveal information concerning the potential and internal energy scales of mesoscopic systems.[19] In this study, we calculate the shot noise power and the Fano factor of the SSAR to provide additional approaches to detect the MZM using SSAR.
cpl-36-10-107102-fig1.png
Fig. 1. The Majorana induced spin selective Andreev reflection. An MZM exists in the vortex core of a 3D TI, which is under an s-wave superconductor. An electron with the same spin as the MZM will undergo an Andreev reflection, whereas an electron with an opposite spin direction will not. The conductance and the shot noise power of this SSAR induced current may be related to the spin polarization of the STM tip.
As shown in Fig. 1, we consider a TI covered by a superconductor. Superconductivity is induced in the TI via the proximity effect, and a vortex state is formed in the surface of the TI under a magnetic field. At the center of the vortex core ($r=0$), the spin polarization of the MZM is parallel to the magnetic field. Let us construct the Hamiltonian of the vortex state in a topological superconductor (TS). This TS is modeled by a helical surface state with Rashba spin-orbit coupling and proximity-induced superconductivity.[18] The helical surface state is the surface state of a 3D TI in the $x$–$y$ plane. We can generalize the Hamiltonian in the $x$–$y$ plane to a spherical surface of radius $R$. The Hamiltonian of the vortex state can be written as $$ H_{\nu}=\left[\begin{matrix} H_{\rm 0e} & {\it \Delta} \\ {\it \Delta} ^{*} & H_{\rm 0h}\\ \end{matrix}\right],~~ \tag {1} $$ where $H_{\rm 0e(h)}$ is the single-electron (hole) Hamiltonian of a helical surface state, and ${\it \Delta}$ is the proximity-induced superconducting order parameter, and the superscript $*$ represents the conjugation. The single-electron Hamiltonian of a helical surface state[18] is $$\begin{align} H_{\rm 0e}=-\frac{\alpha}{R\hbar}\hat{L}\times \hat{\sigma}-\mu,~~ \tag {2} \end{align} $$ where $\alpha$ is the spin-orbit coupling strength, $\hat{\sigma}$ represents the Pauli matrices, $\hat{L}$ is the orbital angular momentum, and $\mu$ is the chemical potential. The single-hole Hamiltonian $H_{\rm 0h}$ is defined as $-\sigma_{y}H_{\rm 0e}\sigma _{y}^{*}$ with $\sigma_{y}^*$ being a conjugated Pauli matrix. The vortex state can be described as ${\it \Delta} ={\Delta}(\theta)e^{i\phi}$. Here the factor $e^{i\phi}$ describes a vortex with a winding number of 1 and ${\Delta}(\theta)={\it \Delta}_{0}\tanh (\frac{R\sin\theta}{\xi_{0}})$ with $\xi_{0}$ characterizing the size of the vortex core. For details of how to obtain zero mode, the first and second excited states, one can see the Supplementary Material. Next, let us consider the total Hamiltonian of a system with a vortex state coupling to a spin polarization STM tip. The Hamiltonian of the STM tip is $$ H_{L}=\left[{\begin{matrix} H_{L, e} & 0\\ 0 & H_{L, h}\\ \end{matrix}} \right],~~ \tag {3} $$ where $H_{L, e}$ ($H_{L, h}$ defined as $-\sigma_{y}H_{L, e}\sigma _{y}^{\ast}$) is the Hamiltonian of the electron (hole) on the STM tip. The Hamiltonian of the electron on the STM tip can be described as[20] $$\begin{align} H_{L, e}=\sum\nolimits_\sigma \hat{d}_{{L}, \sigma}^{+}(\varepsilon_{\sigma}-{\mu}_{L}+\hat{\sigma} \cdot \hat{M})\hat{d}_{{L}, \sigma},~~ \tag {4} \end{align} $$ where $\hat{d}_{{L}, \sigma}^{(+)}$ denotes the electron annihilation (creation) operator of the STM tip with $\sigma$ spin, ${\mu}_{L}$ indicates the chemical potential of the STM tip (set to zero), $\varepsilon_{\sigma}$ is the kinetic energy of the STM tip with $\sigma$ spin, $\hat{M}$ is the spin related potential, and $\hat{\sigma}$ are Pauli matrices. The coupling between the vortex states and the STM tip can be described using the following Hamiltonian[20] $$\begin{align} H_{\nu-{L}}=\,&\sum\nolimits_{n, \sigma} \Big\{2t_{n\sigma}\Big(\cos\frac{\theta}{2}\hat{d}_{{L}, \sigma}^{+}\\ &-\sigma \cdot \sin\frac{\theta}{2}\hat{d}_{{L}, \bar{\sigma}}^{+}\Big)\hat{c}_{n, \sigma}+{\rm c.c.}\Big\},~~ \tag {5} \end{align} $$ where $t_{n\sigma}$ is the coupling strength between the vortex and the STM tip. The retarded Green's function of the system can be obtained via Dyson's equation $$\begin{align} G^{\rm tot}=\frac{1}{{(G^{0, {\rm R}})}^{-1}-{\it \Sigma} ^{r}},~~ \tag {6} \end{align} $$ where the single-particle retarded Green's function $G^{0, {\rm R}}$ can be constructed with the wave functions $\hat{{\it \Psi}}_{m}$ and the eigenvalue $E_{m}$ of the vortex state. The self-energy ${\it \Sigma} ^{r}=H_{v-{L}}\lambda^{r}H_{{L}-v}$ with $\lambda^{r}=\sum\nolimits_m \frac{| \phi_{m}^{1}\rangle\langle\phi_{m}^{1}|}{E-E_{m}+i\delta}$ is the single-particle retarded Green's function of the STM tip, $E_{m}$ is the eigenvalue of $H_{L}$, $|\phi_{m}^{1}\rangle$ is the eigenfunction of $H_{L}$, and $\delta$ is a positive infinitesimal.[20] The $S$ matrix can be obtained via the Fisher–Lee formula[21] $$ S=\left[ {\begin{matrix} r_{\rm ee} & r_{\rm eh}\\ r_{\rm he} & r_{\rm hh}\\ \end{matrix}} \right]=-I+i{\it \Gamma} ^{1/2}\times G^{\rm tot}\times {\it \Gamma} ^{1/2},~~ \tag {7} $$ where ${\it \Gamma}$ is the broadening function defined as ${\it \Gamma} =i({\it \Sigma} ^{r}-{\it \Sigma} ^{r+})$, $r_{\rm ee(hh)}$ is a 2$\times 2$ matrix describing the probability of an electron (hole) being reflected as an electron (hole), while $r_{\rm eh(he)}$ is a 2$\times 2$ matrix describing the probability of an electron (hole) being reflected as a hole (electron) in spin space. The current $I_{\rm c}$ can be defined as $$\begin{alignat}{1} I_{\rm c}=\,&\frac{e}{h}\int_0^\infty [\langle a_{\rm e}^{+}(E)a_{\rm e}(E)\rangle-\langle b_{\rm e}^{+}(E)b_{\rm e}(E)\rangle\\ &-\langle a_{\rm h}^{+}(E)a_{\rm h}(E)\rangle+ \langle b_{\rm h}^{+}(E)b_{\rm h}(E)\rangle]dE,~~ \tag {8} \end{alignat} $$ where $a_{\rm e(h)}^{+(-)}(E)$ is the generate (annihilation) operator of an incoming electron (hole), $b_{\rm e(h)}^{+(-)}(E)$ is the generate (annihilation) operator of a outgoing electron (hole). The differential conductance can also be obtained.[22] The shot noise includes additional information concerning the fluctuation and can be calculated by[19] $$\begin{alignat}{1} \!\!\!\!\!P_{\rm s}(t-t')=\frac{1}{2}\langle\Delta I_{L}(t)\Delta I_{L} (t')+\Delta I_{L}(t')\Delta I_{L}(t)\rangle,~~ \tag {9} \end{alignat} $$ where $\Delta I_{L}(t)=I_{L}(t)-I_{L0}$, and $I_{L0}$ is the average of $I_{L}(t)$. The Fano factor[18] is defined as $$\begin{align} F=\frac{P_{\rm s}}{2eI}.~~ \tag {10} \end{align} $$ Both the shot noise power $P_{\rm s}$ and the Fano factor $F$ can be obtained from the $S$-matrix. Then, we calculate the charge transport properties of SSAR when STM tip contact with the point, which is 1 nm away from the vortex core (the results of the vortex core are shown in the Supplementary Material, which are consistent with the previous studies[14]). As shown in Fig. 2, when $\theta =0^{\circ}$, the conductance line has one broad peak with a maximum value of two at zero energy. Moreover, there are four sharp peaks near the energy of the first and second excited states. Then, with the decrease of $\theta$, the width of the broad peak is much smaller. Therefore, the influence of the excited states decreases as $\theta$ increases. To determine the influence of the first excited states, we calculate the conductance with different $\theta$ for two cases. In the first case, we ignore the first excited states, and in the second case, we consider the first excited states.
cpl-36-10-107102-fig2.png
Fig. 2. The angular dependence of the conductance in the SSAR effect. The experimental data of the tunneling conductance consists of two parts, the first term is the contribution from the normal tunneling and the second term from the Andreev reflection.[17] Our results focus on the Andreev reflection part. The maximum value of the conductance at zero energy is 2. This value is the same as two times the ratio of the electron spin projected in the direction of MZM spin polarization. In addition, when $\theta$ is small, there are four more peaks in the conductance line. These two peaks may be due to the contribution of the excited states.
cpl-36-10-107102-fig3.png
Fig. 3. The conductance when considering the first and second excited states (solid line) or not (dotted line). The excited states can induce four more peaks near the energy of the first and the second excited states. These two conductances become similar when $\theta >{90}^{\circ}$.
From Fig. 3, we can see that the excited states can induce four more peaks on conductance of the SSAR when $\theta =0^{\circ}$. However, under the noncollinear condition, the excited states have several novel features with different $\theta$. When $\theta$ is small, the first and second excited states can induce additional peaks at the energy of them, while the excited states cannot induce them in the case of large $\theta$. The different properties between the two conditions can be explained using Green's function of the system (see the Supplementary Material). The above analysis is performed in the case of zero temperature. Here we study the transport properties under finite temperature conditions in a specific case with $\theta =0^{\circ}$. As shown in Fig. 4, the conductance decreases greatly with increasing temperature, while the width of the conductance increases with the temperature. Due to the thermal broadening under finite temperature, the width of the conductance increases while the maximum value of the conductance decreases.
cpl-36-10-107102-fig4.png
Fig. 4. The temperature dependence of the conductance in the SSAR effect when $\theta =0^{\circ}$. With increasing temperature, the conductance at zero energy is greatly decreased and the width of the conductance line increases.
cpl-36-10-107102-fig5.png
Fig. 5. The angular dependence of the shot noise power, When $\theta =0^{\circ}$, ${45}^{\circ}$, ${90}^{\circ}$ and ${135}^{\circ}$, there are shifts of, 0.5, 1 and 1.5. The shot noise power forms a valley near the zero energy with some peaks near some energies of excited states. Moreover, the shot noise power at zero energy is zero. In addition, the width of the valley is smaller with increasing $\theta$.
At zero energy, as shown in Fig. 5, we find that the shot noise is zero due to the full reflection of the electrons. Let us focus on the formula of the shot noise power. As mentioned above, all of the spin-up electrons can be reflected as holes with the same spin while the spin-down electron can be reflected as themselves at zero temperature. The term that contains $(r_{\rm eh(he)}^{+}a_{\rm h(e)}^{+}(E)r_{\rm hh(ee)}a_{\rm h(e)}(E))$ should be zero at zero temperature. Therefore, the shot noise power can be simplified to $P_{\rm s}=\frac{2e^{3}V}{h}[(r_{\rm he}^{+}a_{\rm e}^{+}(E)r_{\rm he}a_{\rm e}(E))(r_{\rm hh}^{+}a_{\rm h}^{+}(E)r_{\rm hh}a_{\rm h}(E))+(r_{\rm ee}^{+}a_{\rm e}^{+}(E)r_{\rm ee}a_{\rm e}(E))(r_{\rm eh}^{+}a_{\rm h}^{+}(E)r_{\rm eh}a_{\rm h}(E))]$. In other words, the shot noise power is proportional to the probability of an electron being reflected as a hole times the probability of an electron being reflected as itself at zero energy. Now, let us look at the shot noise power non-zero energy. We find that, in the collinear case with increasing the absolute energy, the shot noise power is firstly increases and then decreases to form a major valley. Furthermore, the shot noise powers have some peaks near the energy of excited states. In addition, the width of the valley is decreased with the increase of $\theta$.
cpl-36-10-107102-fig6.png
Fig. 6. The angular dependence of the Fano factor in the SSAR effect. As opposed to normal Andreev reflection, the Fano factor at zero energy is zero.
As shown in Fig. 6, the Fano factor is similar to two minus the value of the conductance. The Fano factor almost always increases with the absolute energy which forms a wide valley near the zero energy with some peaks near the energy of excited states. Moreover, the Fano factor is always zero at zero energy, which is very different from normal Andreev reflection,[23–25] where the Fano factor equals two.[19] This can be explained by the probability of an electron being reflected as a hole. The Fano factor is proportional to the probability that an electron is reflected as itself. This probability always increases from 1 to 0 with increasing the absolute energy. Moreover, as mentioned above, the first excited states will influence the probability that an electron is reflected as a hole. This means that the first excited states also have an influence on the shot noise power and the Fano factor. In more detail, the excited states can cause more peaks due to the new peak of the conductance induced by the excited states. In summary, we have built a model to study the charge transport properties of the SSAR effect. Considering both the MZM and the excited states, we have studied the conductance and the shot noise of the SSAR effect using Green's function combined with scattering theory. Firstly, we numerically calculate the angular dependence of the conductance in the SSAR effect. It is found that the excited states can increase the conductance at zero energy. Then, we calculate the influence of the temperature under finite temperature conditions. With increasing temperature, the maximum of conductance decreases, while the width of conductance increases. At low temperatures, the influence of the excited states is also obvious. Finally, we study the shot noise power and the Fano factor of the SSAR effect. We find that, at zero energy, the shot noise power is zero, meanwhile, the Fano factor is also zero. This is very different from normal Andreev reflection where the Fano factor is two.[19] In addition, the shot noise power is also influenced by the excited states of the vortex state. These transport properties can provide more information concerning the detection of MZM via the SSAR effect. References Nonabelions in the fractional quantum hall effectPaired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effectUnpaired Majorana fermions in quantum wiresFault-tolerant quantum computation by anyonsNon-Abelian anyons and topological quantum computationHelical Liquids and Majorana Bound States in Quantum WiresColloquium : Majorana fermions in nuclear, particle, and solid-state physicsMajorana fermions and topological quantum computationProposal to stabilize and detect half-quantum vortices in strontium ruthenate thin films: Non-Abelian braiding statistics of vortices in a p x + i p y superconductorSuperconducting Proximity Effect and Majorana Fermions at the Surface of a Topological InsulatorMajorana Fermions and a Topological Phase Transition in Semiconductor-Superconductor HeterostructuresMajorana fermion realization and relevant transport processes in a triple-quantum dot systemSelective Equal-Spin Andreev Reflections Induced by Majorana FermionsArtificial Topological Superconductor by the Proximity EffectEvolution of Density of States and a Spin-Resolved Checkerboard-Type Pattern Associated with the Majorana Bound StateMajorana Zero Mode Detected with Spin Selective Andreev Reflection in the Vortex of a Topological SuperconductorTheory of spin-selective Andreev reflection in the vortex core of a topological superconductorShot noise in mesoscopic conductorsSpin-polarized transport through a quantum dot: Anderson model with on-site Coulomb repulsionRelation between conductivity and transmission matrixExplanation of subharmonic energy gap structure in superconducting contactsTunnelling effects on surface bound states in unconventional superconductors
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