Buried topological edge state associated with interface between topological band insulator and Mott insulator

The electronic structure at the interface between a topological band insulator and a Mott insulator is studied within layer dynamical mean field theory. To represent the bulk phases of these systems, we use the generalized Bernevig-Hughes-Zhang model that incorporates the Hubbard-like on-site Coulomb energy $U$ in addition to the spin-orbit coupling term that causes band inversion. The topological and Mott insulating phases are realized by appropriately choosing smaller and larger values of $U$, respectively. As expected, the interface is found to be metallic because of the localized edge state. When the Coulomb energy in the Mott insulator is close to the critical value, however, this edge state exhibits its largest amplitude deep within the Mott insulator rather than at the interface. This finding corresponds to a new type of proximity effect induced by the neighboring topological band insulator and demonstrates that, as a result of spin-orbit coupling within the Mott insulator, several layers near the interface convert from the Mott insulating phase to a topological insulating phase. Moreover, we argue that the ordinary proximity effect, whereby a Kondo peak is induced in a Mott insulator by neighboring metallic states, is accompanied by an additional reverse proximity effect, by which the Kondo peak gives rise to an enhancement of the density of states in the neighboring metallic layer.

Figure 1

One-dimensional interface between two-dimensional topological band and Mott insulators. The Coulomb energies on the left and right side of the interface are defined as $U_L$ and $U_R$, respectively. The electronic properties in the embedding region are calculated self-consistently within the layer DMFT. The properties of the asymptotic bulk regions are taken into account via embedding potentials.

Figure 2

Schematic bulk phase diagram of correlated topological band insulator, derived within generalized BHZ two-orbital model and single-site DMFT. In the region limited by the lines $U_{c1}\left(T\right)$ and $U_{c2}\left(T\right)$, band and Mott insulating states may coexist. The true phase boundary defining the relative stability of these phases is indicated by the line $U_b\left(T\right)$.

Figure 3

Variation of edge state at bare surface of correlated topological band insulator as a function layer index $y$ for various Coulomb energies $U$, as calculated within inhomogeneous DMFT. Plotted is the function $-\beta g_{11}(y,\beta /2)$, defined in Eq. (20), which represents the partially integrated density of states within a few $T$ of the chemical potential ($T=0.01$). The embedding region at the surface comprises $N=12$ atomic layers beyond which bulk behavior is assumed.

Figure 4

Intensity plot of interacting DOS summed over two orbital components with spin $\sigma =1$ in surface layer of semi-infinite topological insulator as a function of parallel momentum $k_x$ for several values of $U$ ($N=12$ and $T=0.01$).

Figure 5

Interacting DOS, $N_{\alpha }\left(\omega \right)$, in first three surface layers of topological insulator at $U=2$ ($N=12$ and $T=0.01$). Solid (red) curves: orbital $\alpha =1$; dashed (blue) curves: $\alpha =2$. These spectra are derived by extrapolating the lattice Green's function from Matsubara frequencies to $\omega +i\gamma$ with a small imaginary energy $\gamma =0.05$ via the routine ratint. The spectral weight near $\mu =0$ in panel (a) is due to the metallic edge state, while the DOS in panel (c) approaches the one characteristic of the bulk band gap.

Figure 6

(a) Partially integrated spectral weight, $-\beta g_{11}(y,\beta /2)$, and (b) occupancy $n_1$ of orbital $\alpha =1$ as functions of layer index near the interface between two correlated topological band insulators at $T=0.01$. The systems on the left and right sides have local Coulomb energies $U_L=2$ and $U_R=6...12$, respectively. The location of the interface is indicated by the dashed line. The embedding region consists of 10 layers on the left and 20 layers on the right side of the interface ($N=30$). The asymptotic behavior is derived from DMFT calculations for the respective bulk materials.

Figure 7

(a) Edge state at interface between topological band insulator (left) and Mott insulator (right). Shown is the partially integrated density of states $-\beta g_{11}(y,\beta /2)$ of orbital $\alpha =1$ as a function of layer index at $T=0.01$. The embedding region consists of $N=21$ layers: 6 layers at $U_L=2$ and 15 layers in the range $U_R=13...15$. Outside the embedding range, bulk behavior is assumed. The location of the edge state is seen to be a sensitive function of the Coulomb energy in the Mott insulator. The maximum value $-\beta g_{11}(\beta /2)\approx 0.33$ is associated with the Kondo peak. (b) Amplitude of the edge state as a function of layer index at $U_R=12.75$ for increasing numbers of iterations in the self-consistency procedure. The maximum due to the Kondo peak is seen to shift toward the right-hand side of the embedding region. The solid vertical line locates the phase boundary between topological and Mott insulating phases, with both having the same Coulomb repulsion $U=12.75$, at the 160th DMFT iteration.

Figure 8

(a) Interacting DOS, $N_{\alpha }\left(\omega \right)$, at interface between topological band insulator ($U_L=2$) and Mott insulator ($U_R=15$) with $N=21$ at $T=0.01$ as in Fig. 7. Solid (red) curves: orbital $\alpha =1$; dashed (blue) curves: orbital $\alpha =2$. Panels (a) and (b) correspond to two surface layers on the right of the interface ($y=14,15$ in Fig. 7) and panels (c) to (f) to four surface layers on the left of the interface ($y=16...19$). These spectra are obtained by extrapolating the lattice Green's function to $\omega +i\gamma$ with a small imaginary energy $\gamma =0.05$ via the routine ratint. The central spectral feature in panel (b) corresponds to the Kondo peak induced in the Mott insulator via the usual proximity effect due to the metallic edge state, while in panel (c) the low-energy feature is induced in the surface layer of the topological insulator via a reverse proximity effect caused by the Kondo peak. Inset of panel (b) shows $N_{\alpha }\left(\omega \right)$ in a smaller $\omega$ region corresponding to the Kondo peak obtained with smaller $\gamma =0.01$.

Figure 9

Edge state at interface between topological band insulator and Mott insulator in the coexistence region with $U_L=U_R=13$ and $N=20$ at $T=0.01$. The Mott gap at this Coulomb energy is much larger than the topological band gap. Thus the asymptotic value of $-\beta g_{11}(y,\beta /2)$ on the right-hand side is much lower than on the left-hand side, and the decay of the edge state within the Mott insulator is more rapid. The vertical bar denotes the Kondo peak in the surface layer of the Mott insulator at the effective boundary with the topological band insulator.

Figure 10

(a) Temperature dependence of partially integrated density of states $-\beta g_{11}(y,\beta /2)$ as a function of layer index for an interface between topological insulator with $U_L=2$ ($y\ge 16$) and a Mott insulator with $U_R=15$ ($y\le 15$). (b) $-\mathrm{Im}{g}_{11}(y,i{\omega }_n)/\pi$ of the same interface system as a function of Matsubara frequency for four lattice layers near the interface ($y=14...17$).