Exotic odd-even parity effects in transmission phase, (Andreev) conductance, and shot noise of a dimer atomic chain by topology

TL;DR

This paper explores how topologically nontrivial dimer chains exhibit unique odd-even parity effects in transmission, conductance, noise, and phase, driven by zero-energy edge states, with implications for hybrid normal-superconductor junctions.

Contribution

It reveals the topological origin of odd-even parity effects in transport properties and the role of zero-energy edge states in hybrid junctions, a novel insight into topological quantum transport.

Findings

Opposite odd-even parity in conductance and noise for topological chains

Zero-energy edge states influence Andreev bound states and conductance anomalies

Distinct $2 ext{ extpi}$ phase variation at zero-energy resonance

Abstract

We investigate the transport properties through a finite dimer chain connected to two normal leads or one normal and one superconductor (SC) leads. The dimer chain is described by the Su-Schrieffer-Hegger model and can be tuned into a topologically nontrivial phase with a pair of zero-energy edge states (ZEESs). We find that if the dimer chain is of nontrivial topology, (1) it will show apparent but opposite odd-even parity of the number of sites, in comparison with the topologically trivial and plain chains, in the (Andreev) transmission probability at the Fermi energy (i.e. the conductance and the Andreev conductance), the noise Fano factor in the zero bias limit, and even the transmission phase due to the coupled ZEESs; (2) the ZEES can determine appearance of the Andreev bound states at the site connected to the SC lead, and thereby induces a nonzero-bias-anomaly in the Andreev differential conductance of the hybrid junction; (3) the transmission phase of the normal junction has a unique $2\pi$ continuous phase variation at the zero-energy resonant peak that is also different from the usual phase shift in resonant point in usual systems.

Equations

$$H_{1D}=\sum _{j\sigma }\left(t_1{d}_{a\sigma ,j}^{†}{d}_{b\sigma ,j}^{}+t_2{d}_{a\sigma ,j+1}^{†}{d}_{b\sigma ,j}^{}+H.c.\right),$$

$$H_T=\sum _{\mathbf{k}\sigma }({\gamma }_L{c}_{L\mathbf{k}\sigma }^{†}{d}_{a\sigma ,1}+{\gamma }_R{c}_{R\mathbf{k}\sigma }^{†}{d}_{a(b)\sigma ,N}+H.c.).$$

$$I$$

$$S$$

$$T(0)=\{\begin{array}{ll}\frac{(\Gamma /t_2{)}^2(t_2/t_1{)}^N}{[(\Gamma /2t_2{)}^2(t_2/t_1{)}^N+1{]}^2},& \text{N is even};\\ \frac{4(t_2/t_1{)}^{(N-1)}}{[(t_2/t_1{)}^{(N-1)}+1{]}^2},& \text{N is odd},\end{array}$$

$${\cal K}=\left(\begin{array}[]{cc}K_{0}&K_{S}\\ K_{S}&-K_{0}\end{array}\right),$$

$$K_{0}=\left(\begin{array}[]{cccccc}0&t_{1}&0&\cdots&\cdots&0\\ t_{1}&0&t_{2}&0&\cdots&0\\ 0&t_{2}&0&\ddots&\ddots&\vdots\\ \vdots&\ddots&\ddots&\ddots&\ddots&\vdots\\ \vdots&\ddots&\ddots&\ddots&\ddots&t_{1(2)}\\ 0&\cdots&\cdots&\cdots&t_{1(2)}&0\end{array}\right),$$

$${\cal G}^{r}(\omega)=\left[\omega{\bm{I}}-{\cal K}-\left(\begin{array}[]{cc}\Sigma^{r}&0\\ 0&\Sigma^{r}\end{array}\right)\right]^{-1},$$

$$I_A=-\int \frac{d\omega }{\pi }{T}_A(\omega )\left(1-f_L-f_{-L}\right),$$

$$S_A=2{\int }_{-V}^V\frac{d\omega }{\pi }{T}_A(\omega )[1-T_A(\omega )].$$

$$T_A(0)=\{\begin{array}{ll}\frac{({\Gamma }_L{\Gamma }_S/2t_2^2{)}^2(t_2/t_1{)}^{4N}}{[({\Gamma }_L{\Gamma }_S/4t_2^2{)}^2(t_2/t_1{)}^{4N}+1{]}^2},& \text{N is even};\\ \frac{{\Gamma }_L^2{\Gamma }_S^2(t_2/t_1{)}^{4N}}{4[({\Gamma }_L/2{)}^2(t_2/t_1{)}^{4N}+({\Gamma }_S/2{)}^2{]}^2},& \text{N is odd}.\end{array}$$

Full text

I Introduction

Much intention has been paid, in recent years, to the physics of topological state in solid-state physics.Hasan The experimental realization of the topology in one-dimensional (1D) systems,Atala ; Xiao by measuring the quantized Zak phase,Zak has stimulated over again extensive investigation on the Su-Schrieffer-Hegger (SSH) model,Su since this simple system has two topologically different phases and thus is a $Z_{2}$ topological insulator.Ryu ; Delplace ; Lang ; Kraus ; Li ; Ganeshan ; Gomez ; Asboth The two topological phases can be distinguished via the presence or absence, controlled by tuning two different dimerization strengths, of twofold degenerate zero-energy edge states (ZEESs) under the open boundary condition (OBC).Ryu ; Delplace It is therefore an intriguing issue to provide experimental confirmation for the existence of the ZEESs in the topological dimer chain. Recently, transport properties have been studied to sign the edge states of a finite dimer chain, when it is connected to two normal leads.Benito ; Niklas ; Ruocco These works have discussed incoherent tunneling of the chain subject to a large bias voltageBenito and/or a high frequency ac electric field.Niklas ; Ruocco In this paper, we examine the linear and nonlinear (Andreev) tunneling in the coherent regime for both a normal junction and a hybrid junction involving superconductor (SC).

II Model of dimer chain

We consider a 1D dimer lattice of $N$ sites, which contains two sublattices $a$ and $b$ in each unit cell with alternatingly modulated nearest-neighbor hopping amplitudes between them. It is just the celebrated SSH Hamiltonian,

[TABLE]

where $d_{a\sigma,j}^{\dagger}$ $(d_{a\sigma,j})$ and $d_{b\sigma,j}^{\dagger}$ $(d_{b\sigma,j})$ are the fermion creation (annihilation) operators of electrons on the sublattices $a$ and $b$ of the $j$th unit cell with spin-$\sigma$, respectively. $t_{1(2)}=t_{0}\mp\delta t$ denote the hopping amplitudes in the unit cell and between two adjacent unit cells, $\delta t$ being the dimerization strength. Notice that the dimer chain showing topologically either trivial or nontrivial property can be easily controlled by simply tuning the relative strength of the intracell-to-intercell couplings, $\delta t$.Ryu ; Delplace ; Gomez ; Asboth Throughout we will measure all energies in units of $t_{0}$ and use units $e=\hbar=k_{B}=1$.

III Normal junction

In this paper, we propose two kinds of transport measurements for detecting the zero-mode by connecting the left and right ends of the dimer chain respectively to two electronic reservoirs. We consider the first setup that two leads are both normal metals (NCN junction), $H_{\eta}=\sum_{{\bf k}\sigma}(\varepsilon_{\eta{\bf k}\sigma}-\mu_{\eta})c_{\eta{\bf k}\sigma}^{\dagger}c_{\eta{\bf k}\sigma}^{\phantom{\dagger}}$, where $c_{\eta{\bf k}\sigma}^{\dagger}$ ($c_{\eta{\bf k}\sigma}$) is the creation (annihilation) operator of an electron with momentum ${\bf k}$ and spin-$\sigma$, energy $\varepsilon_{\eta{\bf k}}$, and chemical potential $\mu_{\eta}$ in the lead $\eta$ ($\eta=\{L,R\}$). The tunneling Hamiltonian for the coupling between the chain and the leads are

[TABLE]

The right end site of the chain could be sublattice either $a$ or $b$ depending on that the number $N$ of the sites is even or odd. Here $\gamma_{\eta}$ describes the tunnel-coupling matrix element between the QD and lead $\eta$ and the corresponding coupling strength is defined as $\Gamma_{\eta}=2\pi\sum_{k}|\gamma_{\eta}|^{2}\delta(\omega-\varepsilon_{\eta{\bf k}})$. Without loss of generality, we use the wide band limit and assumed that $\Gamma_{L}=\Gamma_{R}=\Gamma$ is independent of energy to avoid undesirable effects from the conduction band edge. In addition, in order not to disturb the zero-energy mode of the dimer chain as far as possible, we set the tunnel-coupling $\Gamma=0.1$ in our following calculations to guarantee that it is much weaker than hopping amplitude, $\Gamma\ll t_{0}$.

Applying the nonequilibrium Green function (NGF) method, we can evaluate the current from the left lead to the chain and its shot noise asHaug

[TABLE]

with the Fermi distribution $f_{\eta}=[e^{(\omega-\mu_{\eta})/T}-1]^{-1}$ at the temperature $T$ and the transmission probability $T(\omega)={\Gamma_{L}\Gamma_{R}}|G_{1N}^{r}(\omega)|^{2}$. Here, $G_{1N}^{r}(\omega)$ is the retarded GF between the first and last sites of the chain.

It is seen that the transport properties are completely determined by the transmission coefficient. Therefore we first examine how the transmission probability evolves when the atomic chain changes from linear to topological. In Fig. 1(a), we plot the transmission spectrum $T(\omega)$ as functions of the number of sites for the dimer chains with $\delta t=0.2$. It is observed that the transmission probability at $\omega=0$ shows (1) a nearly perfect transmission even for the even number of sites, $N=16$, but (2) a rapid decrease with increasing length of the chain, and (3) on the contrary, a nearly perfect reflection for the case of odd-site dimer chain. These behaviors are clearly different from those of the plain chain ($\delta t=0$). In fact, we can obtain the exact analytical expressions for $T(0)$.Zeng ; Kim For the plain chain, the transmission probability has the well-known odd-even parity dependence: $T(0)=16\Gamma_{L}\Gamma_{R}t_{0}^{2}/(\Gamma_{L}\Gamma_{R}+4t_{0}^{2})^{2}\simeq 0$ for an even-site chain and $T(0)=4\Gamma_{L}\Gamma_{R}/(\Gamma_{L}+\Gamma_{R})^{2}=1$ for an odd-site chain. In contrast, for the case of dimer chain, we have

[TABLE]

showing an opposite odd-even parity. For instance, the dimer chain with $\delta t=0.2$ has a nearly perfect transmission at $\omega=0$ only if $N=16$, whereas it has a perfect reflection if $N$ is odd.

These peculiar behaviors can be ascribed to the appearance of the ZEESs of the dimer chain. As shown in Fig. 1(b) for $N=18$, the dimer chain has a topologically nontrivial phase in the regime of $\delta t>0$ characterized by the presence of zero-energy states under the OBC, whereas no edge states exist in the regime of $\delta t<0$. From the distribution probability of the zero-energy states along the chain in the case of topological regime, $\delta t=0.2$ [Figs. 1(d,e)], we find that these states are indeed most probably occupied at the two endpoints of the chain. In addition, we find that the overlap integral of the wave function between the two coupled ZEESs can not be negligible for the short chain, $N=18$ [Fig. 1(d)], which induces a nonzero coupling between the two edge states. As a result, electron can be transferred coherently from one end to the other via the two ZEESs when the energy of the incident electron is zero, rather than via the scattering states as the odd-site plain chain does. Nevertheless, the overlap integral is quite sensitive to the chain length. It decays rapidly and becomes infinitesimal for the long chain, for instance, $N=28$ [Fig. 1(e)]. In this case, the two ZEESs become localized at the two endpoints respectively, and the transport channel closes. Moreover, for the odd-site chain, one ZEES exists at the whole regime of $\delta t$ [Fig. 1(c)]. But this state is always localized at one end of the chain [Fig. 1(f)] (the left end for $\delta t>0$ whereas the right end for $\delta t<0$), except for the plain chain ($\delta t=0$). Of course, the single localized edge state can not support electron tunneling, indicating strong localization in the odd-site dimer atomic wire.

The transmission probability at the Fermi energy can be detected in experiments by measuring the linear conductance, $G=(2e^{2}/h)T(0)$, at zero temperature. We then calculate $G$ and plot them, in Fig. 2, as functions of chain length for various dimerization strengths. It is evident that the dimer chain displays the opposite odd-even parity to the plain chain. Moreover, the conductance of the dimer chain shows an exponential decay with increasing chain length, scaled properly as $\exp(-2N\delta t/t_{0})$, while the plain chain does not. This fact confirms that the long dimer chain is an ideal Anderson insulator. In addition, the shot noise Eq. (5) can be approximated as $S=(4e^{2}/h)T(0)[1-T(0)]eV$ in the limit of small bias voltage, $V=\mu_{L}-\mu_{R}$, at zero temperature. Therefore, it is expected that the Fano factor $F=S/2I$ of the dimer chain will also display the different odd-even behavior from the plain chain, as shown in Fig. 3.

We now analyze the transmission phase since it provides a complementary information to depict transmission coefficient, $t(\omega)=\sqrt{\Gamma_{L}\Gamma_{R}}G_{1N}^{r}(\omega)$.Lee ; Taniguchi For the plain chain, irrespective of the number $N$ of sites, the transmission resonant peaks are all out of phase, i.e. the phase increase continuously but rapidly by $\pi$ from one resonant peak to the next, as seen by the dashed thin lines in Fig. 4(a).Lee ; Taniguchi ; Zhai The odd-site dimer chain has the similar phase variation behavior for both the bulk states and the ZEES, even though the transmission amplitude is nearly zero at $\omega=0$. This behavior is illustrated in Fig. 4(c), showing that the path of $t(\omega)$ as a function of energy encircles the origin of the complex plane but always stay away from it. As the energy of incident electron is close to the bulk states, the circle radii are nearly equal to $1$ [red and blue lines in Fig. 4(c)], which are the typical trajectory of transmission amplitude for a single-channel symmetric double-barrier structure.Taniguchi ; Zhai Nevertheless, the radius of the trajectory becomes very small near the edge state [black line in Fig. 4(c)]. While for the case of even-site dimer chain, transmission amplitude starts from the positive real axis with $|t|=1$ and then evolves along the black line as seen in Fig. 4(b) to approach the origin. This indicates a continuous phase variation from [math] to $2\pi$ when the energy of incident electron sweeps through the zero-energy resonance, and manifests that the two sides of the zero-energy resonance are in phase, which is just opposite to the regular resonances in the plain chain and even the resonances via the bulk states in the dimer chain. It can thereby be regarded as a unique signal for the appearance of the ZEESs.

IV Hybrid junction

We turn to the second transport setup at which the right lead is replaced with a SC (NCS junction), $H_{R}=\sum_{{\bf k}\sigma}\varepsilon_{R{\bf k}}c_{R{\bf k}\sigma}^{\dagger}c_{R{\bf k}\sigma}^{\phantom{\dagger}}+\sum_{{\bf k}}(\Delta c_{R{\bf k}\downarrow}^{\dagger}c_{R-{\bf k}\uparrow}^{\dagger}+{\rm H.c.})$, where $\Delta$ is the SC gap. Since we are interested only in the Andreev tunneling in the subgap region, $\Delta$ can be set as the biggest energy scale and thereby the role of the SC lead is solely to induce $s$-wave pairing for the site connected to the SC lead, i.e. the right end site in the NCS junction. To account for the proximity effect, we make use of the site$\otimes$Nambu space, $\psi=(d_{a\uparrow,1},\cdots,d_{a(b)\uparrow,N},d_{a\downarrow,1}^{\dagger},\cdots,d_{a(b)\downarrow,N}^{\dagger})^{T}$, to rewrite the effective Hamiltonian for the SSH model in matrix form as $\widetilde{H}_{1D}=\frac{1}{2}\psi^{\dagger}{\cal K}\psi$, where the $2N\times 2N$ matrix ${\cal K}$ is given by

[TABLE]

in terms of the two $N\times N$ matrices,

[TABLE]

and $K_{S}$ having only one nonzero element for the right end site of the chain, $K_{S,NN}=\Gamma_{S}/2$ (Here we use $\Gamma_{S}$ instead of $\Gamma_{R}$ to signify the SC hybrid effect). Without loss of generality, we set $\Gamma_{S}=3\Gamma$ in the calculations such that a pair of the Andreev bound states (ABSs) is established unambiguously. The retarded GF ${\cal G}^{r}(\omega)$ of the chain, defined as ${\cal G}^{r}(t,t^{\prime})=-i\theta(t-t^{\prime})\langle[\psi(t),\psi^{\dagger}(t^{\prime})]\rangle$, can be formally given by the Dyson equation

[TABLE]

where ${\bm{I}}$ is a $2N\times 2N$ unit matrix and the $N\times N$ retarded self-energy $\Sigma^{r}$ is due to the coupling to the left norm lead. It has only one nonzero diagonal element, $\Sigma_{11}^{r}=-i\Gamma_{L}/2$.

Employing NGF method, we can derive the Andreev current for the subgap tunneling asKim ; Dong

[TABLE]

with the hole distribution function $f_{-L}=[e^{(-\omega-\mu_{L})/T}-1]^{-1}$, and the Andreev reflection probability $T_{A}(\omega)=\Gamma_{L}^{2}|{\cal G}_{1,N+1}^{r}(\omega)|^{2}$. In this kind of hybrid transport device, the chemical potential of the SC lead is usually set as a reference, and the external bias voltage is applied to the norm lead, $\mu_{L}=V$. The noise of the Andreev current can therefore be calculated at zero temperature asDong ; Muzykantskii

[TABLE]

We then analyze the topological effect on the Andreev reflection spectrum $T_{A}(\omega)$ of the dimer chain. We have the exact expressions for $T_{A}(0)$.Kim The odd-even parity is also evident in the Andreev conductance of the plain chain as in the NCN junction: $T_{A}(0)=(\Gamma_{L}\Gamma_{S}t_{0}^{2}/2)^{2}/[(\Gamma_{L}\Gamma_{S})^{2}+t_{0}^{4}]^{2}\simeq 0$ if $N$ is even, while $T(0)=4\Gamma_{L}^{2}\Gamma_{S}^{2}/(\Gamma_{L}^{2}+\Gamma_{S}^{2})^{2}=0.36$ if $N$ is odd. For the dimer chain, we obtainKim

[TABLE]

An opposite odd-even parity is also found, for instance, $T_{A}(0)=0.29$ for the short even-site chain ($N=16$), while $T_{A}(0)\simeq 0$ for the $N=17$ chain with $\delta t=0.2$. More interestingly, we find from Fig. 5(a) that $T_{A}(\omega)$ exhibits triple peaks, say one zero-energy peak and two sharp peaks located equally at the left and right sides of the central peak, for the even-site dimer chain in the topologically nontrivial phase. In addition, the height of the two side peaks is nearly unity for short chains and decreases with increase of the chain length also but relatively much slowly compared with the central peak. The side peaks can be ascribed to the emergence of a pair of ABSs adhered to the ZEES due to SC proximity effect, as illustrated in Fig. 5(c). While in the topologically trivial case ($\delta t<0$), no ZEES means no ABS, and thus no side peaks. Another intriguing result is that for the odd-site dimer chain, $T_{A}(\omega)$ shows either a single central peak if $\delta t>0$ or two side peaks if otherwise [Fig. 5(b)]. This is because the isolated ZEES is located at the left end of the odd-site chain in the case of $\delta t>0$, thereby no electronic state exists at the right end to support the ABS; on the contrary, the degenerate single ZEES at the right end is splitting owing to the SC proximity effect and becomes a pair of ABSs in the case of $\delta t<0$[Fig. 5(d)].

The ZEES supported ABS can be detected through two quantities in the nonlinear regime. The first quantity, for example, is the nonzero-bias-anomaly in the differential Andreev conductance. Indeed, from the inset figure in Fig. 6(a), it is observed a upward jump in the Andreev current $I_{A}$-$V$ curves stemming from the sharp peak of the ABS for the short even-site dimer chains in the topologically nontrivial phase. Correspondingly, the ABS induces a sharp downward jump in the current noise, i.e. the Fano factor $F=S_{A}/2I_{A}$. Likewise, a rapid downward jump is also found for the noise of the odd-site chain, $N=17$, in the case of $\delta t<0$. While in the case of $\delta t>0$, the noise remains Poissonian, i.e. $F=1$, until the bulk state begins to contribute [Fig. 6(b)].

V Summary

Using the SSH tight-binding model, we have analyzed the topological effect on the transport properties of a 1D dimer chain when the chain is connected to two normal leads or to one normal and one SC leads by means of NGF method. It has been found that the topologically nontrivial chain possesses an opposite odd-even parity dependences of the (Andreev) conductance and the noise Fano factor in the zero-bias limit with respect to the number of sites, compared with results of the plain chain. Moreover, we have predicted a nonzero-bias-anomaly in the Andreev differential conductance of the NCS junction, and ascribed it to the emergence of ABSs due to the joint effects of the ZEES and the SC proximity effect. Besides, we have also analyzed the transmission phase of the NCN junction and found a unique $2\pi$ continuous phase variation at the zero-energy resonant peak only when the dimer chain is in the topologically nontrivial phase. Finally we would like to mention that we propose, in this paper, two simple transport experiments to detect the ZEES, which could provide useful information to identify the topological quantum phase transition in the 1D atomic wire.

Acknowledgements.

This work was supported by Projects of the National Basic Research Program of China (973 Program) under Grant No. 2011CB925603, and the National Science Foundation of China under Grant No. 11674223.