Transport through Quantum Anomalous Hall Bilayers with Lattice Mismatch

In recent decades, topology protected states in two dimensions attract extensive researches due to their robust quantum transports and entanglements. Among them, the quantum anomalous Hall (QAH) effect is a two-dimensional (2D) Chern insulator without an external magnetic fieldHaldane , which possesses quantized transport in its bulk gap, leading to potential applications in dissipationless and quantum computing devices. After proposals based on different materialsQAH_CXLiu ; QAH_HJZhang ; QAH_QZWang ; QAH_RYu ; QAH_SCWu ; QAHQSH1 , the QAH effect has been experimentally observed QAH_Exp1 ; QAH_Exp2 ; QAH_Exp3 ; QAH_Exp4 ; QAH_Exp5 .

As an effect in 2D, the minimal model of QAH effect can be realized on a monolayer lattice with two orbitals at each siteHaldane ; Bernevig2006 . On the other hand, bilayer systems have been shown to be a platform for richer phenomena. For example, a graphene bilayer, even regularly stacked as a periodic lattice, has possessed electronic structures and transport properties significantly different from those of a monolayer oneGrapheneRMP . By twisting bilayers into generic non-commensurate or quasi-periodic lattices, more nontrivial physics emerge, including strong correlation, superconductivity and nontrivial topologyBistritzer2011 ; YuanCao ; XiDai2019 ; Twistl ; Twist2 .

Quasi-periodicity is a delicate pattern between periodicity and disorder, which may give rise to interesting behaviors of electronic wavefunctions, even in the single particle pictureFibonacciRMP ; JFanReview . For example recently, plenty of interesting localization properties have been found in one-dimensional (1D) lattices with a quasi-periodic potential, such as well-defined mobility edges MobilityEdge1D1 ; MobilityEdge1D2 , and a critical region consisting of critical statesYCZhang2022 . Insightful predictions on other rich physics in 1D quasi-periodic lattices are proposed, for example, topology, non-Hermeticity, superconductivity and superfluidityUpdate01 ; Update02 ; Update03 ; Update04 . Some interesting experimental realizations of quasicrystalline physics have also been discussed recentlyHofstadterModel ; Quasicrystal .

In this manuscript, for the QAH effect, we consider another method of creating quasi-periodicity, by coupling two QAH layers with arbitrary ratios of lattice constants. Experimentally, this can be realized by using state-of-the-art fabrication technologies of topological hetero-structuresFabrication1 ; Fabrication2 ; Fabrication3 , ultracold atomicFabricationOL1 ; FabricationOL2 ; FabricationOL3 or photonic systemsFabricationPhot . Firstly, by comparing the numerical and analytical results in the simple case of commensurate bilayer, we confirm that the fundamental physics underlying this system is governed by two coupled 1D Dirac equations, if no bulk states have been considered. Then, the quantum transports are investigated for different lattice constant ratios, by using a tight-binding model that naturally includes both bulk and edge states. The central plateau of the quantized conductance is not affected since the coupling has not been able to close and re-open the bulk gap. On the other hand, the phase boundary of this quantized conductance plateau consists of a fractal transitional region, with small and fractured islands of quantized and unquantized conductances penetrating into each other. This picture is distinct from that of the commensurate system, where the boundaries of the topological region are clear. Moreover, we find that these unquantized states in the transitional region are extending across the sample. This is also distinct from that of the disordered topological systems (topological Anderson insulators) where they are localized. As a result, the details of this transitional region is sensitive to disorder, shape of the sample, and even the configuration of leads connected to it. The effects from nonzero disorder and varying coupling strength are also discussed.

The rest of the paper is organized as follows. In Sections II, we introduce the model and the method we use to describe the electronic transport properties of bilayer QAH system. Section III is dedicated to compare the results of commensurate lattices obtained by two different approaches: a tight-binding model using the Landauer-Büttiker formalism and a mode-matching calculation in the continuum Dirac-like Hamiltonian approximation. Section IV is dedicated to present numerical results of the generic case with a full tight-binding simulation: mismatching lattice constants in two layers. Finally, we conclude in Sec. V summarizing our main results.

The general form of a spinless bilayer QAH system can be expressed as

where $H_{\ell}$ ($\ell=1,2$) is the Hamiltonian for the $\ell$-th layer and $H_{c}$ is the coupling between them. Here, we choose each layer to be the spin up component of the Bernevig-Hughes-Zhang (BHZ) model Bernevig2006 defined on a square lattice, with one $s$ orbital and one $p$ orbital on each site. In this manuscript, we fix the lattice constant of the first layer $d_{1}=1$ as the length unit, while vary that of the second layer $d_{2}$ continuously. The layer Hamiltonian $H_{\ell}$ in the momentum space is

where $c^{\dagger}_{\ell;\bm{k}\alpha}$ ($c_{\ell;\bm{k}\beta}$) creates (annihilates) an electron with wave number $\bm{k}$ and orbital $\alpha\in{\{s,p\}}$ in the $\ell$-th layer. For the BHZ model, $h_{\ell;\alpha\beta}$ is a $2\times 2$ matrix defined as Bernevig2006

with $\sigma_{x,y,z}$ the Pauli matrices acting on the orbital space $\{s,p\}$. We assume that these model parameters of each layer can be tuned independently. In the absence of the inter-layer coupling $H_{c}$, the $\ell$-th layer is in the QAH state if $B_{\ell}\cdot M_{\ell}>0$QAH2010 . The real space version of the layer Hamiltonian $H_{\ell}=\sum_{ij,\alpha\beta}h_{\ell;\alpha\beta}(i,j)c^{\dagger}_{\ell;i\alpha},c_{\ell;j\beta}$ can be obtained from Eq.(2) and (3) by performing a straightforward inverse Fourier transformation $c_{\ell;\bm{k}\beta}=\frac{1}{\sqrt{V}}\sum_{j}c_{\ell;j\beta}e^{-i\bm{k}\cdot\chi_{j}}$, where $j$ is the site index.

With the real space layer Hamiltonian at hand, we can define the inter-layer Hamiltonian $H_{c}$. In most of this manuscript (i.e., except Section III), we adopt a simple form as InterlayerCoupling1 ; InterlayerCoupling2

where $t$ is the strength of the inter-layer coupling and $d_{ij}$ is the distance between sites $i$ and $j$ in the first and second layer respectively. To simplify numerical calculations, we only count in inter-layer couplings $\frac{t}{d_{ij}}\geqslant 0.01t$. By varying the lattice constant of the second layer, the lattice mismatch can be felt by the electron through $H_{c}$.

We concentrate on the transport properties of the QAH bilayer connected to two semi-infinite leads, as shown in figure 1. At zero temperature, the two-terminal conductance can be calculated within the Landauer-Büttiker formalism $G=\frac{2e^{2}}{h}T$Landauer1957 ; Buettiker ; TransmissionRMP ; Datta , where $e$ is the elementary charge, $h$ is the Planck constant and 2 is the spin degeneracy. This transmission $T$ at Fermi energy $E$ can be expressed in terms of Green’s functions asDHLee ; QuantumTransport ,

where the dressed retarded (advanced) Green’s function $G^{r(a)}_{D}(E)=\big{[}E-H-\Sigma^{r(a)}_{L}-\Sigma^{r(a)}_{R}\big{]}^{-1}$, $\Gamma_{p}=i\big{[}\Sigma^{r}_{p}-\Sigma^{a}_{p}\big{]}$ , and $\Sigma^{r(a)}_{p}$ is the retarded (advanced) self-energy from lead $p(=L,R)$, i.e., the left and right leads.

The local current from site $i$ to $j$ along the bond isJiang2009

where $H_{ij}$ the matrix element of the bare Hamiltonian $H$, and $G^{n}(E)=G^{r}_{D}(E)\Gamma_{L}(E)G^{a}_{D}(E)$ is the correlation function. Since it is defined in the linear response regime, we simply take the voltage difference $V_{S}-V_{D}$ between the source and drain leads to be unity.

To understand more physics underlying transport properties, we rely on the density of states (DOS) YYZhang2013 and the normalized participation ratio (NPR) NPR2021 ; NPR2020 ; NPR2017 . With the real space representation of the Green’s function $G^{r}(E;i,j)$, the single particle local density of states (LDOS) $\rho(i)$ at each site $i$, and the total density of states (DOS) can be calculated respectively as

where $N$ is the total number of sites, and the summation is over all sites of the sample. Here the Green’s function $G^{r}$ can be the dressed one $G^{r(a)}_{D}(E)$ defined above, or the bare one $G^{r(a)}_{B}(E)=\big{[}(E\pm i\eta)I-H\big{]}^{-1}$, depending on whether one wants to include the effects from leads. With the LDOS normalized over the sample $\tilde{\rho}(i)$, the NPR can be defined as followsNPR2021 ; NPR2020 ; NPR2017

The NPR is a significant diagnostic tool to characterize the localization transition. The localized (extended) phases are characterized by $\mathrm{NPR}=0$ ($\mathrm{NPR}\neq 0$) in the large $N$ limit NPR2021 ; NPR2020 ; NPR2017 .

To visualize and distinguish edge states from bulk ones, the edge-locality marker is introduced asHofstadterInsulators

where the summation runs over four edges of the bilayer QAH sample. In this work, we take the width of the edge as 10% of that of the bilayer sample.

In the topological phase of the bilayer, although the backscattering is forbidden, the inter-layer (forward) scattering is not. The nontrivial topology only guarantees the robustness of the total transmission through the bilayer, but the details of inter-layer scattering are not clear yet. As a starting point, let us gain some physical insights from the simplest bilayer with identical lattice constants but different model parameters. To further simplify the model so that an analytical continuum model can be easily obtained, we adopt a simpler form of inter-layer coupling in this section as

where the inter-layer coupling $t$ [green line in figure 1(a)] only exists between nearest sites from different layers.

In figure 2, we present the band structures of the QAH bilayer in the ribbon geometry, without (a) and with (b) the inter-layer coupling respectively. With remarkably different model parameters in two layers, the velocities of two groups of edge states are different too. In figure 2(b), the inter-layer coupling lifts the trivial degeneracy of edge states at the $\Gamma$ point.

For this commensurate system with the inter-layer coupling (12), the low-energy properties of the QAH edge states can be captured by a Dirac-like continuum model from $k\cdot p$ approximation Dirac-likeHamiltonian1 ; Dirac-likeHamiltonian2 . This effective Hamiltonian has the formBilayerHamiltonian1 ; BilayerHamiltonian2 :

where $k_{x}$ is the $x$ component of the momentum relative to the Dirac point, and $v_{i}>0$ is the velocity of edge states in the $i$-th layer. For each layer, only the forward channel should be considered here, due to the forbidden of backscattering. The inter-layer coupling coefficient $F$ depends on the concrete realization of the bilayer. It can be verified that, when the model parameters of the bilayer QAH system satisfy $\frac{A_{2}}{A_{1}}=\frac{B_{2}}{B_{1}}=\frac{D_{2}}{D_{1}}=\frac{M_{2}}{M_{1}}$, $F$ just equals the coupling strength $t$ in Eq. (12). In the rest of this section, we adopt the model parameters satisfying this condition. In the low energy limit around $k_{x}=0$, we have $v_{\ell}=\mathrm{sgn}(B_{\ell})A_{\ell}\sqrt{1-\frac{D_{\ell}^{2}}{B_{\ell}^{2}}},\ell=1,2$ Dirac-likeHamiltonian1 ; Dirac-likeHamiltonian2 . The expressions of eigenvalues and eigenfunctions of this coupled Dirac Hamiltonian are listed in the Appendix.

We use the transport configuration illustrated in figure 1(a), with one layer as the source lead while a decoupled bilayer as the drain lead, so that the intra- and inter-layer transports can be conveniently distinguished. Since the source lead is a monolayer carrying Chern number 1, the total transmission through the central sample will be quantized as 1, If it is in the topological state. Then due to the inter-layer coupling of the sample, an electron from the source lead of monolayer can be transmitted into either monolayer of the drain lead, corresponding to the intra-layer and inter-layer transmissions respectively. It is straightforward to solve this tunnelling problem. Due to one monolayer (with one pair of edge states) as the source, only one active channel (positive-velocity branch of this pair) is propagating in the central QAH bilayer, with the length $L$ and width $W$. The wave functions in different spatial regions can be written as

where $k_{\ell}=\frac{E}{v_{\ell}},\ell=1,2$ at the incoming energy $E$. In the central bilayer sample the wave functions are linear combinations of the eigenfunctions given in Eq. (A2) in Appendix, with $M$ and $N$ the combination coefficients. Due to the perfect transmission of Dirac-like fermions, there is no reflection herePerfectTransport1 ; PerfectTransport2 ; PerfectTransport3 . We only need to match the wave functions at boundaries because the Dirac-like Hamiltonian is first-order differentialKleinParadox ; GrapheneProperties . From this continuity of the wave functions at the $x=0$ and $x=L$, i.e., boundaries of the central bilayer, we obtain coefficients $M$, $N$, $t_{1}$ and $t_{2}$. According to current conservation of the Dirac-like Hamiltonian, we have $T_{1\rightarrow 1}=t_{1}t_{1}^{\ast},T_{1\rightarrow 2}=\frac{v_{1}t_{2}t_{2}^{\ast}}{v_{2}}$Current1 ; Current2 ; Current3 ; Current4 , where $T_{1\rightarrow 1}$ ($T_{1\rightarrow 2}$) represents the intra-layer (inter-layer) transmission.

In figure 3, red curves are results calculated from this effective continuum model, for the intra-layer (left column) and inter-layer (right column) transmissions, with the central sample length $L=50$ (upper row) and $L=100$ (lower row) respectively. For comparison, corresponding results from the tight-binding simulations are also plotted as the black curves. The perfect quantization of the total transmission shown in figure 3 (c) and (f) confirm that the central sample is in the topological phase.

Now let us scrutinize details of the layer resolved transmissions $T_{1\rightarrow 1}$ and $T_{1\rightarrow 2}$. Curves in both colors agree very well, especially when $E\rightarrow 0$. So in the low energy limit, the Dirac continuum approximation contains most of the physics. The intra-layer and inter-layer transmissions show a complementarity: the transmission minima in one configuration coincide with the maxima of the other one, since the sum of them is robustly quantized as 1.

Using the above mentioned analytical formalism, we can also obtain the dependence on the sample length $L$ as

where $q_{1}=\frac{F}{\sqrt{v_{1}v_{2}}}$ by Eq. (A2). In figure 4, we can clearly see that for a fixed energy $E$, the transmission varies periodically with $L$ from . At zero energy. the period $\frac{\pi\sqrt{v_{1}v_{2}}}{F}$ can be obtained from Eq. (15), which has an excellent agreement with the tight-binding result shown in figure 4(a).

In Fig. 5, we plot the spatial distribution of local currents [obtained from equation (6)] of the sample at a typical parameter setting (see the caption), which clearly show that the currents are carried by edge states.

In brief, in the topological phase, if the influence from bulk states are ignored, the transmission through the bilayer is quantized. Now the only interesting effects from the inter-layer coupling are just the resonant traversing between forward propagating waves in two layers.

In the rest of this manuscript, we will come back to the generic case of mismatching lattice constants between two layers, with the distance dependent inter-layer coupling Eq. (4). As mentioned in Section II, the lattice constant of the first layer $d_{1}$ is fixed to be the length unit, and that of the second layer $d_{2}$ will be varied. Now even in the topological phase, a low energy approximation based analytical treatment will be difficult, because, for example, the inter-layer coefficient $F$ in the effective Hamiltonian (13) will have a very complicated dependence on model parameters and lattice configurations. Moreover, the effective Dirac Hamiltonian (13) only involves edge states and therefore it cannot describe any effects from bulk states, or the transition into topologically trivial phase, which we are focusing in the following. As a result, we will rely on numerical calculations based on the tight-binding model, as introduced in Section II.

The transport configuration will be taken to be a more “natural” setup as illustrated in figure 1(b). Each lead (source or drain) consists of two decoupled monolayers, otherwise the coupling between incommensurate bilayers will break the translation symmetry of the lead and make the self energies incomputable. Since we are not interested in what happens in the leads, this setup will not influence the main physics of the results. The model parameters of each monolayer lead are identical to those of the central sample layer it is connected to, to decrease unwanted scattering on the contact boundaries. The total transmission is a sum of all four possible transmissions between these monolayer leads, i.e., $T_{\rm{total}}=T_{1\rightarrow 1}+T_{1\rightarrow 2}+T_{2\rightarrow 1}+T_{2\rightarrow 2}$.

In this work, we studied the quantum transports of a bilayer QAH system. In the case of lattice match, we compared the transmissions from the tight-binding approach and the continuum Dirac model. Results from both methods have an excellent agreement especially in low-energy limit. We found that the intra-layer and inter-layer conductances show a complementarity since the sum of them is robustly quantized as 1 and the intra-layer transmission varies periodically with the central sample length. In this simple case, in the topological phase, the inter-layer coupling just leads to the resonant traversing between forward propagating waves in two layers.

In the case of lattice mismatch, the quantum transports are investigated by a full tight binding simulation. In the phase diagram on the $E-d_{2}$ plane, the central plateau of quantized conductance corresponds to the bulk gap which has not been influenced by the lattice mismatch. On the other hand, outside the central plateau of quantization, there is a transitional region with small and fractured islands of quantized and unquantized conductances penetrating each other. We identify this region as in-gap extended states penetrating with topological edge states. Details of this transitional region is sensitive to disorder, shape of the sample, and even the configuration of leads connected with it, due to the bulk and topologically trivial nature of these in-gap states. In phase diagram on the $E-t$ plane, the quantized conductance persists at weak inter-layer coupling and there is also a similar transitional region. Our results offer a comprehensive view of transport through the QAH bilayer with lattice mismatch.

This work was supported by National Natural Science Foundation of China under Grant Nos. 11774336, 12104108, 61427901, and 12147112, and the Starting Research Fund from Guangzhou University under Grant Nos. RQ2020082 and 62104360.

The eigenvalues of the coupled Dirac Hamiltonian equation(13) are

Correspondingly, the normalized wave function takes the form

where $q_{1}=\frac{\sqrt{E^{2}(v_{1}-v_{2})^{2}+4F^{2}v_{1}v_{2}}+E(v_{1}+v_{2})}{2v_{1}v_{2}}$ and $q_{2}=\frac{-\sqrt{E^{2}(v_{1}-v_{2})^{2}+4F^{2}v_{1}v_{2}}+E(v_{1}+v_{2})}{2v_{1}v_{2}}$.