Disordered Quantum Transport in Quantum Anomalous Hall Insulator-Superconductor Junctions

TL;DR

This paper numerically investigates how disorder affects quantum transport in quantum anomalous Hall insulator-superconductor junctions, revealing the critical role of multiple edge modes in Andreev conversion processes.

Contribution

It introduces a numerical study of disordered transport in QAH insulator-superconductor junctions focusing on multiple edge modes and their impact on Andreev conversion.

Findings

Multiple chiral edge modes enhance Andreev conversion.

Non-chiral metallic modes also promote Andreev conversion.

Single chiral edge mode suppresses Andreev conversion.

Abstract

In this communication, we numerically studied disordered quantum transport in a quantum anomalous Hall insulator-superconductor junction based on the effective edge model approach. In particular, we focus on the parameter regime with the free mean path due to elastic scattering much smaller than the sample size and discuss disordered transport behaviors in the presence of different numbers of chiral edge modes, as well as non-chiral metallic modes. Our numerical results demonstrate that the presence of multiple chiral edge modes or non-chiral metallic modes will lead to a strong Andreev conversion, giving rise to half-electron half-hole transmission through the junction structure, in sharp contrast to the suppression of Andreev conversion in the single chiral edge mode case. Our results suggest the importance of additional transport modes in the quantum anomalous Hall insulator-superconductor junction and will guide the future transport measurements.

Equations

$$H=\left(\begin{array}{cc}{H}_{\text{QAH}}& H_{\Delta }\\ H_{\Delta }^{†}& -H_{\text{QAH}}^{\ast }\end{array}\right)$$

$$H_{\text{eff}}=\left(\begin{array}{cc}{v}_{f}k-\mu & -v_{\Delta }k\\ -v_{\Delta }k& v_{f}k+\mu \end{array}\right)$$

$$T_{13}=1-\frac{2{ϵ}^2{v}_{\Delta }^2{\sin }^2\left(\frac{\sqrt{{ϵ}^2{v}_{\Delta }^2+(v_f^2-v_{\Delta }^2){\mu }^2}}{v_f^2-v_{\Delta }^2}L\right)}{{ϵ}^2{v}_{\Delta }^2+(v_f^2-v_{\Delta }^2){\mu }^2},$$

$$\begin{array}{rl}{H}_{\text{m}}& =\left(\begin{array}{cccc}{v}_{f1}k-{\mu }_1& 0& 0& \Delta \\ 0& v_{f2}k-{\mu }_2& -\Delta & 0\\ 0& -\Delta & v_{f1}k+{\mu }_1& 0\\ \Delta & 0& 0& v_{f2}k+{\mu }_2\end{array}\right)\end{array}$$

$$H_{\text{orig}}=\left(\begin{array}{cc}{H}_{\text{elec}}(k)& H_{\Delta }\\ H_{\Delta }^{†}& -H_{\text{elec}}^{\ast }(-k),\end{array}\right)$$

$$H_{\text{elec}}=\left(\begin{array}{ccc}{v}_{f}k-{\mu }_1& {\xi }_1& {\xi }_2\\ {\xi }_1& v_{1}k-{\mu }_2& 0\\ {\xi }_2& 0& -v_{1}k-{\mu }_2\end{array}\right)$$

$$H_{\Delta }=\left(\begin{array}{ccc}0& {\Delta }_1& {\Delta }_2\\ -{\Delta }_1& 0& 0\\ -{\Delta }_2& 0& 0\end{array}\right).$$

Full text

*Introduction: * The interplay between superconductivity proximity effect and chiral edge modes (CEMs) in a two-dimensional heterostructure with a quantum Hall (QH) or quantum anomalous Hall (QAH) system coupled to a superconductor (SC) has been a long-standing problem Fu and Kane (2008, 2009); Qi et al. (2010); Takagaki and Ploog (1998); Takagaki (1998); Hoppe et al. (2000); Chtchelkatchev and Burmistrov (2007); Khaymovich et al. (2010); Sun and Xie (2009); Akhmerov and Beenakker (2007). Recently, a strong resurgence of the research interest in this system results from the possible realization of topological superconductor (TSC) phaseHe et al. (2017). A TSC possesses a full bulk SC gap and gapless quasi-particle excitations, such as Majorana zero modes Fu and Kane (2008); Sun et al. (2016) or parafermions Alicea and Fendley (2016); Lindner et al. (2012); Vaezi (2014); Mong et al. (2014); Clarke et al. (2014), at the boundary. Non-Abelian statistics of these quasi-particle excitations enables the possibility of performing topological quantum computation based on TSC Nayak et al. (2008); Alicea et al. (2011).

Early theoretical studies on the QH/SC junction focus on the Andreev reflection process occurring at the QH/SC interface Takagaki and Ploog (1998); Takagaki (1998); Hoppe et al. (2000); Chtchelkatchev and Burmistrov (2007); Khaymovich et al. (2010); Sun and Xie (2009); Akhmerov and Beenakker (2007) and the supercurrent flowing through CEMsVan Ostaay et al. (2011); Zyuzin (1994). Several early experiments on SC/semiconductor heterostructure have revealed evidence of Andreev reflections for two-dimensional electron gas in the high-Landau-level states under the magnetic fields.Eroms et al. (2005); Moore and Williams (1999); Batov et al. (2007). Low-Landau-level states are challenging in these systems due to the limitation of the upper critical field of SCs and the high electron densityWan et al. (2015). Recent experiments have shown that the low-Landau-level states at a relatively low magnetic field can be achieved in the graphene system by tuning electron density through gate voltages, thus leading to significant progress in inducing SC correlation via the proximity effect into the graphene in the QH regime. Transport evidences, including supercurrents carried by CEMsAmet et al. (2016); Calado et al. (2015), enhanced QH plateau conductance due to Andreev process Rickhaus et al. (2012), cross Andreev conversion (AC)Cayssol (2008) and inter-Landau-level Andreev reflection Sahu et al. (2018), have been found in graphene QH systems in contact to SC electrodes under an external magnetic field. These encouraging experimental progresses lay the foundations for the theoretical proposals of realizing chiral Majorana modes (CMMs)Qi et al. (2010); Chung et al. (2011) and Majorana zero modesChen et al. (2018); Alicea (2012); Zeng et al. (2018) in the SC/QH (or QAH) junctions.

More recently, an observation of $e^{2}/2h$ conductance kink is claimed to be the transport evidence of CMMs in the TSC phase of a QAH/SC hetero-structureHe et al. (2017). However, this claim is still under debates Kayyalha et al. (2019); Zhang et al. (2019) because the early theoretical prediction was based on the calculation of the clean QAH/SC hetero-structure model with the Landauer-Buttiker formalism Chung et al. (2011) while the QAH samples in experiments, particularly the SC/QAH interface, are highly disordered Ji and Wen (2018); Huang et al. (2018); Lian et al. (2018). The mean free path $l_{\textrm{mfp}}$ around the SC/QH(QAH) interface region is greatly reduced due to interface roughness and normally much smaller than the typical length scale of the SC/QH(QAH) interface. For example, in Ref. Lee et al. (2017), $l_{\textrm{mfp}}$ is around 0.3nm, much smaller than the width of SC ribbon (around $50\sim 600$nm). Theoretically, the disorder effect in the SC/QH interface was investigated for the high-Landau-level systems based on either the semi-classical skipping orbit picture or the Landau level picture in the early literature Takagaki and Ploog (1998); Takagaki (1998); Chtchelkatchev and Burmistrov (2007); Khaymovich et al. (2010). More recently, several theoretical models are developed for the disorder-induced bulk topological phase transition in the QAH/SC hetero-structures Huang et al. (2018); Ji and Wen (2018); Lian et al. (2018); Kramer et al. (2005); Chalker et al. (2001). The disorder effect from elastic scattering is normally not important for CEM transport in the QH or QAH regime. However, the conductance oscillation can be induced by AC of CEMs propagating through the SC/QH (or QAH) interface in the ballistic regime Lian et al. (2016); Gamayun et al. (2017), which is sensitive to elastic scattering. Therefore, understanding disorder effect in the SC/QH(QAH) junctions Wang and Lian (2018); Lian and Wang (2019) is essential for any reliable theoretical interpretation.

In this work, we focus on the disordered transport through the QAH/SC junction. We consider a theoretical model for the setup with SC partially covering the QAH system, forming a planar junction between a pure QAH region (Region I) and a SC/QAH vertical junction region (Region II), as shown in Fig. 1a. We assume SC proximity effect is weak in the QAH system and thus the region II is topologically equivalent to the QAH phase. As a result, the CEM at the boundary of the region I is transformed into two CMMs along the boundary of the region II (Fig. 1a). Previous theoretical studies on the similar configurations have revealed conductance oscillationLian et al. (2016); Gamayun et al. (2017) in the ballistic transport regime. We focus on the elastic-scattering-dominated transport regime, in which the edge length $L$ of the QAH/SC junction is much larger than the mean free path $l_{\textrm{mfp}}$. In this transport regime, the disordered transport behavior sensitively depends on the number $N$ of CEMs and the existence of other transport channels. In the single CEM ($N=1$) case, we find that the transmission through the boundary of QAH/SC junction region approaches unity without AC even in the presence of disorders when the incident electron energy approaches 0 (or the zero-bias limit in experiments). This is a consequence of the $p$-wave nature of the allowed SC proximity effect in the single CEM case. In contrast, the transmission will quickly decay away from one and reach certain saturating values when considering the finite incident electron energy (or the finite bias), or multiple CEMs ($N>1$), or the coexistence of a CEM and a non-chiral metallic mode. These results bring new insights into the existing experiments of QAH/SC junction and reveal the important role of the interplay between elastic scattering and CMM transport.

*Model Hamiltonian and Transport of a single CEM: * Due to topological equivalence between the QH and QAH states, we consider a two-band model of QAH insulator and couple it to a SC. The Bogoliubov-de Gennes (BdG) Hamiltonian for a QAH/SC junction can be written as

[TABLE]

with $H_{\textrm{QAH}}=(M+B(\partial_{x}^{2}+\partial_{y}^{2}))\sigma_{z}+\alpha\sigma_{x}(-i\partial_{x})+\alpha\sigma_{y}(-i\partial_{y})-\mu$ and $\sigma_{x,y,z}$ are the Pauli matrices for spin Lian et al. (2016). We choose the superconducting gap $H_{\Delta}=i\sigma_{y}(\Delta_{0}+\Delta_{z}(-i\partial_{x}))$ with both spin-singlet component $\Delta_{0}$ and triplet component $\Delta_{z}$. The reason to include triplet components is because only triplet SC is allowed for the single CEM case, as discussed in the edge theory below. The parameters $\mu$, $\Delta_{0}$ and $\Delta_{z}$ are in general spatially dependent, to incorporate the spatial configuration of junctions and disorder-induced variations. We apply the Hamiltonian (1) to the configuration in Fig. 1a on a retanglar geometry and adopt the Recursive Green’s function method combined with Landauer-Büttiker formalism (see Appendix A and Appendix F for details). The leads 1, 2 and 3 are attached on the edge of the sample, while the lead 4 is connected to the SC bulk and grounded.

We first study the transport behavior of the Hamiltonian (1) in the clean limit and consider the transmission from the lead 3 to 1, labeled as $T_{13}$, which accounts for the transmission through two CMMs at the boundary of the region II (see Fig. 1a). Through this region, electrons can transmit as an electron or convert to a hole through the AC process. We denote the electron-electron transmission as $T_{ee}$ and the electron-hole transmission of AC as $T_{eh}$, thus $T_{13}=T_{ee}-T_{eh}$. In Fig. 1c and d, the calculated $T_{13}$ as a function of the length $L$ is shown as circles, triangles and diamonds for different incident electron energies $\epsilon$ and different chemical potentials $\mu$, respectively. The oscillation behavior of transmission comes from the AC at the SC/QAH boundary Lian et al. (2016); Gamayun et al. (2017). The amplitude of the oscillation increases with $\epsilon$ (or Fermi momentum of CEMs), but decreases with $\mu$. In the $\epsilon=0$ case, the transmission $T_{13}$ always keeps 1, suggesting the suppression of AC in the zero-bias limit, which is consistent with the literatureVan Ostaay et al. (2011).

Since we focus on the edge transport regime with the chemical potential within the bulk gap of $H_{\textrm{QAH}}$, the full Hamiltonian of the QAH/SC junction can be projected into the subspace spanned by CEMs, giving rise to

[TABLE]

on the CEM basis of $\left|\phi_{e}\right>$ and $\left|\phi_{h}\right>$, where $v_{f}$ is the Fermi velocity of CEM and $v_{\Delta}$ is the coefficient of triplet pairing component. Detailed derivation of the effective model can be found in Appendix B. We notice that only the triplet component can contribute to the pairing term for CEMs Van Ostaay et al. (2011), as guaranteed by the particle-hole symmetry. As described in Appendix C, the transmission $T_{13}$ can be directly computed through the scattering matrix approach and given by

[TABLE]

from which one finds $T_{13}$ oscillates with the amplitude $\frac{2\epsilon^{2}v_{\Delta}^{2}}{\epsilon^{2}v_{\Delta}^{2}+(v_{f}^{2}-v_{\Delta}^{2})\mu^{2}}$ and the period $\frac{v_{f}^{2}-v_{\Delta}^{2}}{2\sqrt{\epsilon^{2}v_{\Delta}^{2}+(v_{f}^{2}-v_{\Delta}^{2})\mu^{2}}}$. The oscillation amplitude increases when increasing $\epsilon$, but decreases when increasing $\mu$, while the oscillation period decreases with increasing either $\epsilon$ or $\mu$. All these features are consistent with the numerical simulations above. In addition, as the incident electron energy $\epsilon$ approaches zero, the oscillation disappears according to Eq. (3), and thus the transmission $T_{13}$ always stays at 1, independent of chemical potential $\mu$. Physically, this behavior originates from the $p$-wave nature of the pairing in the effective edge model. By choosing appropriate parameters in Eq. 3, the calculated transmission $T_{13}$ (solid lines in Fig. 1c,d) can fit well with that from the numerical simulations of the full model (circles, triangles and diamonds in Fig. 1c,d), thus justifying the validity of the effective edge Hamiltonian (2).

We next consider the influence of disorder scattering on the transmission $T_{13}$, particularly the on-site fluctuation of chemical potential $\mu$, based on the effective edge Hamiltonian (2). To do that, we divide the QAH/SC interface into $N$ segments with the chemical potential $\mu_{i}$ ($i=1,2...,N$) chosen to be a random variable from a uniform distribution on $[-\mu_{\textrm{imp}}/2,\mu_{\textrm{imp}}/2]$ (See Fig. 1b). For each disorder configuration $\{\mu_{i}\}_{i=1,...,N}$, we compute the transmission $T_{13}$ through the transfer matrix formalism. The physical transmission, denoted as $\bar{T}_{13}$, is obtained by averaging multiple independent disorder configurations.

Fig. 2a shows that the averaged transmission $\bar{T}_{13}$ decays exponentially with respect to the length $L$. The decay behavior of $\bar{T}_{13}$ can be understood as the decoherent interference between different trajectories of electron-hole oscillation as varying chemical potential. The decay length, denoted as $\lambda$, can be extracted from Fig. 2a and its dependence on $\epsilon$ is shown in Fig. 2b. As $\epsilon$ approaches zero, $\lambda$ increases rapidly, and thus $\bar{T}_{13}$ almost remains 1 when increasing $L$, indicating the suppression of AC in this limit, even when including disorder scattering. The dependence of $\bar{T}_{13}$ on $\epsilon$ and $\mu$ for a fixed $L$ is discussed in Appendix D.

*Disordered transport of multiple modes at the QAH/SC interface: * In real experiment devices, multiple transport channels may exist at the QAH/SC interface, including (1) high-Landau level states with multiple CEMs ($N>1$) for the QH states Amet et al. (2016); Cayssol (2008) and (2) the coexistence of a CEM and other non-chiral metallic modes Wang et al. (2013); Chang et al. (2015). Therefore, we next go beyond the single CEM case and study the influence of multiple modes on disordered transport.

We first consider the case with two CEMs ($N=2$) for simplicity. We denote the basis of the effective Hamiltonian as $\{\left|e_{1}\right>,\left|e_{2}\right>,\left|h_{1}\right>,\left|h_{2}\right>\}$, where the subscript labels two channels. In addition to the $p$-wave pairing between $\left|e_{1(2)}\right>$ and $\left|h_{1(2)}\right>$, the pairing between $\left|e_{1(2)}\right>$ and $\left|h_{2(1)}\right>$ also exists and can be of $s$-wave nature. Therefore, if we only keep the lowest order terms, the effective Hamiltonian can be written as

[TABLE]

with the pairing term $\Delta$. Transport simulations can be performed for the Hamiltonian (4) with the on-site chemical potential disorders. Fig. 2c shows the averaged transmission $\bar{T}_{13}$ as a function of $L$, which shows an exponential decay with the decay length $\lambda$ depending on $\epsilon$ in Fig. 2d. In sharp contrast to the single-CEM case, the decay length increases with $\epsilon$ and is generally quite small for the multiple-CEMs case. As a consequence, the transmission $\bar{T}_{13}$ is always close to zero for a range of $\epsilon$ and $\mu$ when the length $L$ becomes large. (See Fig. D.1c,d in Appendix D. Physically, the constant $\Delta$ term can induce a large oscillation in the clean limit and thus disorder can induce a strong dephasing of the oscillation pattern, giving rise to the decay behavior. This conclusion is consistent with the previous studies on the transport through the QH/SC interface for the high-Landau-level QH state Takagaki and Ploog (1998); Takagaki (1998); Chtchelkatchev and Burmistrov (2007); Khaymovich et al. (2010).

We next consider the coexistence of the CEM and the non-chiral metallic mode, and this scenario has been theoretically proposed Wang et al. (2013) and later experimentally demonstrated Chang et al. (2015) in magnetically doped TI films. We consider the BdG Hamiltonian

[TABLE]

where

[TABLE]

and

[TABLE]

Here $H_{\textrm{elec}}$ is written on the basis $\{\left|e_{c}\right>,\left|e_{nL}\right>,\left|e_{nR}\right>\}$, where $\left|e_{c}\right>$ labels the CEM while $\left|e_{nL}\right>$ and $\left|e_{nR}\right>$ together represent the non-chiral mode due to their opposite velocities. In magnetic TI films, non-chiral modes originate from the quasi-helical gapless modes at the side surfaces of TI films Wang et al. (2013). The coupling between the CEM and non-chiral mode is described by the parameters $\xi_{1,2}=\xi$. The $s$-wave pairing gap can exist between the CEM and non-chiral mode, chosen as $\Delta_{1,2}=\Delta$ in $H_{\Delta}$, making this system similar to the multiple CEMs case. On the other hand, the non-chiral nature also suggests the existence of backward propagating channel and thus will strongly affect the transport behaviors in the QAH system Wang et al. (2013). To understand its influence, we below discuss two different scenarios, both may occur in actual experiments. The details of the edge models can be found in Appendix E, with the transmissions/reflections of both models shown in Fig. E.3 and Fig. E.4.

In the first scenario, we assume disorders exist in the QAH/SC junction (region II) while the transport in the QAH side (region I) is ballistic (or quasi-ballistic), later referred as the “clean QAH” case. In this situation, the non-chiral modes have been experimentally shown to induce non-local transport signal in the QAH system Chang et al. (2015). In our setup, the ballistic transport of non-chiral modes in region I leads to a negligible backscattering for $\bar{R}_{33}$ and $\bar{R}_{11}$ at small $L$, both saturating to certain values when increasing $L$ in Fig. 3b. The transmissions $\bar{T}_{13}$ and $\bar{T}_{31}$ starts from 2 and 1 for a small $L$, respectively, reflecting the number of the left and right moving modes. For a large $L$, both transmissions decay to zero in Fig. 3a. The existence of non-zero $\bar{R}_{33}$ and $\bar{R}_{11}$ makes this situation quite different from other situations.

For the second scenario, disordered transport is assumed for the whole QAH insulator, spanning over both regions I and II in Fig. 1a, and later referred as the “disordered QAH” case. Due to the Anderson localization, the 1D non-chiral mode is completely localized, as manifested by the reflection $\bar{R}_{33}\approx\bar{R}_{11}\approx 1$ and $\bar{T}_{31}\approx 0$ in Fig. 3b, c and d. Even though getting localized, the non-chiral mode still mediates the AC at the QAH/SC interface (region II in Fig. 1a). Consequently, the transmission $\bar{T}_{13}$ exponentially decays to zero for a large $L$, even when $\epsilon=0$, making this situation more similar to the multiple CEMs case.

*Experimental Relevance: * Finally, we examine the behaviors of resistance in the experimental setup of Fig. 1a. We consider the current driven from the leads 2 to 4 and discuss the resistance $R_{24,14}\equiv V_{14}/I_{24}$ and $R_{24,34}\equiv-V_{34}/I_{24}$ for four different cases: (i) the single CEM case, (ii) the multiple CEMs case, (iii) the coexistence of the CEM and non-chiral metallic mode, with a clean QAH region and (iv) with a disordered QAH region. The case (i) shows a unique insulating behavior for $\epsilon\sim 0$ in Fig. 4a, while $R_{24,14}$ and $R_{24,34}$ are insensitive to the energy $\epsilon$ for all the other cases (see Fig. 4b and Fig. E.5a and b in Appendix E). For the cases (i) and (iv), $R_{24,34}$ and $R_{24,14}$ are related by $R_{24,14}+R_{24,34}=-h/e^{2}$, while for the case (ii), $R_{24,14}$ by $R_{24,14}+R_{24,34}=-h/2e^{2}$ (see Fig. 4c and d, and Fig. E.5 in Appendix E). For the case (iii), due to ballistic transport of non-chiral metallic mode, $R_{24,14}$ and $R_{24,34}$ are independent of each other in Fig. 4c and d. Therefore, measuring $R_{24,14}$ and $R_{24,34}$ simultaneously can distinguish these different cases.

*Acknowledgement: * We are thankful for the helpful discussion with Biao Lian and Jiabin Yu. We acknowledge the support of the Office of Naval Research (Grant No. N00014-18-1-2793), the U.S. Department of Energy (Grant No. DESC0019064) and Kaufman New Initiative research grant KA2018-98553 of the Pittsburgh Foundation.