Electrically Tunable Conductance and Edge Modes in Topological Crystalline Insulator Thin Films: Tight-Binding Model Analysis

We propose a minimal tight-binding model for thin films made of topological crystalline insulator (TCI) on the basis of the mirror and discrete rotational symmetries. The basic term consists of the spin-orbit interaction describing a Weyl semimetal, where gapless Dirac cones emerge at all the high symmetry points in the momentum space. We then introduce the mass term providing gaps to Dirac cones at our disposal. They simulate the thin films made of the [001], [111] and [110] TCI surfaces. TCI thin films are topological insulators protected by the mirror symmetry. We analyze the mirror-Chern number, the edge modes and the conductance by breaking the mirror symmetry with the use of electric field. We propose a multi-digit topological field-effect transistor by applying electric field independently to the right and left edges of a nanoribbon. Our results will open a new way to topological electronics.

Topological insulator is one of the most fascinating concept found in this decade 1,2 . Recent flourish of the study of topological insulator is based on the finding of the timereversal invariant topological insulator 3-5 . Very recently, a new class of topological insulator, topological crystalline insulator (TCI), attracts much attention [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] . The best example is given by Pb x Sn 1−x Te, which has been found to be a TCI experimentally [8][9][10] . There are three types of the surface states, the [001], [111] and [110] surfaces, which have discrete rotation symmetries C N with N = 4, 6 and 2, respectively. Gapless Dirac cones emerge inevitably on the surface of a topological insulator. Indeed, it has been experimentally observed that there are gapless Dirac cones at the X and Y points in the [001] surface [8][9][10] , and at the Γ and three M points in the [111] surface 11,12 . Theoretical studies have been performed with the use of first-principle calculation and low-energy effective Dirac theory.
Although there are yet no experimental measurements, theoretical studies 13,14 have been presented also on the thin film made of a TCI. It is characterized by the same discrete rotation symmetry C N , and additionally by the mirror symmetry about the 2D plane. When the film is thin enough, a gap opening occurs due to hybridization between the front and back surfaces, and it turns the system into a topological insulator. Now, electronics is based on circuits in two dimensions. It is important and urgent to make a further investigation of a TCI thin film, since it may well be a good candidate for designing nanodevices in topological electronics.
The TCI thin film is a topological insulator protected by the mirror-symmetry. Accordingly the topological number is the mirror-Chern number 21,22 . A prominent feature is that we can break the mirror symmetry simply by applying external electric field. This is highly contrasted to the case of the time-reversal invariant topological insulator, where the timereversal breaking should be caused by magnetic field or exchange field induced by ferromagnet. Magnetic field and exchange field are hard to control precisely. On the other hand, a precise control of the electric field is possible within current technique.
The aim of this work is to explore the physics of the TCI thin film by constructing a minimal tight-binding model based on the discrete rotation symmetry C N and the mirror symmetry about the 2D plane. The tight-binding model is essential to make a deeper understanding of the system, which is not attained by first-principle calculation and low-energy effective Dirac theory. For instance, according to the low-energy theory 13 , the [001] thin film made of Pb x Sn 1−x Te has two Dirac cones at the X and Y points, where the chirality is identical. However, the Nielsen-Ninomiya theorem 23 dictates that the total chirality is zero. Consequently, there must be two additional Dirac cones with the chirality opposite to that of the X and Y points. In our tight-binding model these two Dirac cones emerge at the Γ and M points, though they are removed from the low-energy spectrum. The tight-binding model is also useful to analyze the edge states, which transport the ordinary electric current reflecting the topological properties of the thin film.
We start with a system where the spin-orbit interaction (SOI) dominates the transfer term. We demonstrate that such a system is a Weyl semimetal consisting of multiple Dirac cones at all the high symmetric points. We next introduce mass terms to give gaps to Dirac cones at our disposal. The system may well describe the TCI thin film made of Pb x Sn 1−x Te with an appropriate choice of phenomenological mass parameters. Then we break the mirror symmetry by introducing a perpendicular electric field E z . The conductance is calculated in the presence of E z . The conductance is switched off by the electric field. Namely, it acts as a topological field-effect transistor 24 . By attaching two independent gates to the sample, we can separately control the right and left edge states. The conductance can be 0, 1 and 2, which forms a multidigit topological field-effect transistor, where the conductance is quantized and topologically protected. Our results open a new way to electric-field controllable topological electronics.

Main Results
Our main results consist of a phenomenological construction of minimal tight-binding Hamiltonians for the TCI thin film, and the analysis of electrically controllable conductance and edge modes of a nanoribbon in view of the bulk-edge correspondence.
FIG. 1: Nearest and next-nearest neighbor sites in real space. We give the value of σ · d ℓ ij for the nearest (ℓ = 1, magenta) and nextnearest (ℓ = 2, cyan) neighbor sites for the square lattice (a) and the triangular lattice (b) in the Hamiltonian (2). We also give the value of νij for the nearest neighbor site (ℓ = 1, magenta) for the square lattice (c) and the triangular lattice (d) in the Hamiltonian (7).
Tight-binding Hamiltonians. The SOI plays a key role in the physics of topological insulators. We consider a model where the SOI dominates the system. A simplest example would be the Rashba SOI, Alternatively we may think of or even take a sum of them. Here, σ = (σ x , σ y , σ z ) represents the Pauli matrix for the spin, and d ℓ ij = r i − r j connects a pair of the ℓ-th nearest neighbor sites i and j in the lattice with λ ℓ the coupling strength. As is easily shown, the results based on the Hamiltonian (1) are obtained from those on the Hamiltonian (2) simply with the replacement of a set of momenta (k x , k y ) by (k y , −k x ). Furthermore, the low-energy theory derived from a first-principle calculation 13 supports the choice of (2). Hence we concentrate on (2) hereafter.
Let N ℓ be the number of the ℓ-th neighbor sites. In the momentum representation the Hamiltonian is rewritten as withĤ where d ℓ ij is determined by requiring the invariance under the discrete rotational symmetry C N . (See the Methods with respect to C N .) For instance, d ℓ n for the nearest neighbor sites (ℓ = 1) is expressed as for the triangular (N = 3) and square (N = 4) lattices: See Fig.1(a) and (c). We shall soon see that this model has multiple Dirac cones at the high symmetry points known such as the X, Y , Γ and M points in the square lattice and the Γ, K, K ′ , M 1 , M 2 , M 3 points in the triangular lattice. Thus the Hamiltonian (1) describes a Weyl semimetal. The minimal tight-binding Hamiltonian of a TCI thin film would be a four-band model due to the spin and pseudospin (surface) degrees of freedom. Let τ = (τ x , τ y , τ z ) be the Pauli matrix to describe the pseudospin representing the front (τ z = 1) and back (τ z = −1) surfaces. When the film is thin enough, the symmetric state becomes the ground state, opening a gap to all Dirac cones due to hybridization. We employ the Hamiltonian (1) to describe the symmetric state. Furthermore we apply the electric field E z between the two surfaces.
These effects are realized by considering the four-band effective tight-binding Hamiltonian, together withĤ SO given by (4) andĤ m obtained from where ν ℓ ij is a number characteristic to the lattice structure and determined by the vector d ℓ ij so as to preserve the crystalline symmetry, and i, j runs over the ℓ-th nearest neighbor sites. We take ν 0 ij = δ ij and show ν 1 ij in Fig.1(c) and (d) for the square and triangular lattices. As we shall soon see, the gap at each Dirac point is adjusted by choosing the mass parameters m ℓ appropriately.
In the absence of the external electric field (E z = 0), the Hamiltonian (6) is invariant under the mirror symmetry about the 2D plane, where the mirror operator is given by The mirror symmetry is broken by the external electric field When the system is an insulator, the mirror-Chern charge is defined and calculable even for E z = 0 according to a general scheme 25 .
Square lattice with C 4 symmetry. We first consider the square lattice with the C 4 symmetry. Let us set E z = 0. First, taking the contributions from the nearest neighbor sites (ℓ = 1) and the next-nearest neighbor sites (ℓ = 2), we obtain from the Hamiltonian (2) aŝ with See the illustration in Fig.1(a) and (b). The energy spectrum is given by There are gapless Dirac cones at the X, Y , Γ and M points, as illustrated in Fig.2, where the band structure is shown. The Hamiltonian describes a Weyl semimetal. The effective low-energy Hamiltonian is given by (11) near each Dirac point with where v x and v y are the velocities andk x andk y are the renormalized momentã as follows from (12). A set of numbers (n x , n y ) is (−1, 1) for X, (1, −1) for Y , (1, 1) for Γ, (−1, −1) for M . The chirality of the Dirac cone is give by n x n y at each point. An anisotropy (v x = v y ) has been introduced into the system by introducing the nearest and next-nearest neighbor contributions (λ 1 = 0, λ 2 = 0). We illustrate the spin direction around each Dirac point in Fig.2. The spin direction yields one negative chirality at the X and Y points apiece, while it yields one positive chirality to the Γ and M points apiece. The total chirality is zero over the Brillouin zone, as required by the Nielsen-Ninomiya theorem 23 .
We proceed to consider the total Hamiltonian (6) with E z = 0, which readsĤ with (12) and The energy spectrum is now given by We see that m 1 opens a gap at the Γ and M points, while m 0 opens a gap at all Dirac points, as illustrated in Fig.2(b). Note that, if we set m 0 = 0, massless Dirac cones appear at the X and Y points [ Fig.2(b)]. The term m 0 τ x is understood to simulate the effect of a gap opening due to hybridization between the front and back surfaces in a thin film. The Dirac cones at the Γ and M points are removed from the low-energy theory when we take a large value of m 1 . The low-energy Dirac theory is extracted from (17) around the X and Y points as with the velocities v 1 and v 2 being given by It is worthwhile to notice that this low-energy Hamiltonian agrees with the one derived based on a first-principle calculation and the Dirac theory of the TCI surface 13 . Our Hamiltonian is capable to simulate various models by controlling m ℓ . For instance, a massless Dirac cone emerges only at the Γ point by setting m 0 = −2m 1 , as illustrated in Fig.3(a3) and (b3). Similarly a massless Dirac cone emerges only at the M point by setting m 0 = 2m 1 , as illustrated in Fig.3(a4) and (b4). The horizontal axes are −π < kx ≤ π, −π < ky ≤ π in (a), −π < kx ≤ π in (b), −π < k ≤ π in (c) and (d). (e) We present the topological phase diagram in the m0-m1 plane. A green line represents a phase boundary. The numbers 0 and ±2 are the mirror-Chern numbers. A circle with symbol such as c3 shows a point where the band structure is calculated in (c3).
The thin film is an insulator, since a gap is given to all Dirac points by the term m 0 τ x . It is a topological insulator indexed by the mirror-Chern number in the absence of the electric field (E z = 0). It is a symmetry protected topological number.
As we derive in the Methods, the mirror-Chern charge may be calculated 25 even for E z = 0, and is given by for each Dirac cone possessing the chirality n x n y with m given by (18). When E z = 0, it is reduced to The total mirror-Chern number is quantized and given by We show the topological phase diagram in the (m 0 , m 1 ) plane in Fig.3(e). Nanoribbons: With the tight-binding Hamiltonian at hand, we are able to demonstrate the band structure of nanoribbons, as shown in Fig.3 for various values for parameters m 0 and m 1 . We have still set E z = 0. We take the direction of nanoribbon as x-axis. The momentum component k x in the bulk band gives the momentum k of nanoribbon, while the momentum component k y is quantized. Accordingly, the X and M points are projected to the same momentum k = π, while the Y and Γ points are projected to k = 0. The projected view of the bulk band shown in Fig.3(b) is the same as the band structure of nanoribbon except for the edge states. Namely we can identify the edge states by comparing the projected band structure of the bulk band and the band structure of nanoribbon. The edge states are shown in magenta in Fig.3(c) and (d).
The bulk-edge correspondence works perfectly well. Indeed, gapless edge modes emerge when a nanoribbon has a nonzero mirror-Chern number. It is interesting that the gapless edge states emerge at k = 0 for C M > 0 and at k = π for C M < 0. There are no edge states when the system is trivial (C M = 0). There is exactly one to one correspondence between the mirror-Chern number and the appearance of edge states in the band structure of a nanoribbons.
The nonzero mirror-Chern number indicates "quantum mir-ror Hall effects". However it is a highly nontrivial problem to experimentally detect the "mirror-Hall conductivity" since "mirror-Hall currents" convey neither charge nor spin. On the other hand, there emerge |C M | gapless states in the edges of a nanoribbon made of a topological insulator with the mirror-Chern number C M . Without the electric field, these edge states transport merely the mirror charge M . Once we apply external electric field parallel to the nanoribbon direction, one edge state contributes one quantum unit to the electric conductance, as we show in Fig.4(d1). Hence we are able to determine the absolute value of the mirror-Chern number by measuring the conductance. Electric field: We now switch on the electric field E z between the front and back surfaces to control the edge modes and the conductance in nanoribbons [ Fig.4(a)]. The mirror symmetry is broken by the electric field as in (10).
We show the band structure of a nanoribbon under the electric field E z in Fig.4(b) and (c). The edge states become gapped due to the mixing of the right and left going edge states as a result of the mirror-symmetry breaking.
The gapless edge mode transports the electric current. We have calculated the conductance in the presence of E z , which we show in Fig.4(d): See the Methods for derivation. The conductance near the Fermi energy is 2 for E z = 0 [ Fig.4(d1 The horizontal axes are −π < kx ≤ π, −π < ky ≤ π in (d), −π < kx ≤ π in (e), −π < k ≤ π in (f) and (g). since the edge states are doubly degenerate. Once we turn on the electric field, the conductance falls to zero since the edge states disappear due to the anticrossing [ Fig.4(d2)]. Namely, it acts as a field-effect transistor 13 . It is possible to apply different electric fields E z1 and E z2 to the right and left edge states [ Fig.4(a)]. The conductance can be 0, 1 and 2, which forms a multi-digit field-effect topological transistor [ Fig.4(d4)]. The conductance is quantized and topologically protected.
Triangular lattice with C 6 symmetry. We proceed to study the triangular lattice with the C 6 symmetry. Note that the triangular lattice has the hexagonal symmetry. By substituting N = 3 into the Hamiltonian (2), and taking only the nearest neighbor sites (ℓ = 1), we obtain We show the band structure in Fig.5. There are six massless Dirac cones, in which one Dirac cone resides at Γ, three Dirac cones at M points and two Dirac cones at K and K ′ points.
In the vicinity of each Dirac cone, we obtain the low-energy Dirac theory with a set of velocities (v x , v y ) to be (−3λ 1 , −3λ 1 ) for Γ, (−λ 1 , 3λ 1 ) for M 1 , (3λ 1 /2, 3λ 1 /2) for K and K ′ . The chiralities of the Dirac cone at the Γ, K and K ′ points are identical, while three M points have opposite chirality. It is contrasted to the case of graphene, where the chiralities of K and K ′ points are opposite. We show the band structure of nanoribbons in Fig.5. It is interesting that there exists a flat band in the region −π ≤ k ≤ − 2π 3 and 2π 3 ≤ k ≤ π: See Fig.5(f1) . The one is connecting the K and M points, and the other is connecting K ′ and M points.
In the similar manner to the square lattice, we introduce the mass term (7) to the Hamiltonian. The leading and the next leading terms are By introducing the mass term, the Dirac cones at the K and K ′ points become gapped. The resultant spectrum has four Dirac cones, in which one Dirac cone resides at the Γ point and the other three Dirac cones at the M points.
The total mirror-Chern number is given by This should be compared with the mirror-Chern number (24) in the square lattice with the C 4 symmetry. It follows that the phase diagram is essentially given by the same one as Fig.3(e).
We show the band structure of nanoribbons in Fig.5 in the presence of the mass term (28). The flat bands turn into the dispersive edge modes. The position of the edge modes is between the K (K ′ ) and M 2 = M 3 points when C M < 0 (C M > 0). As in the case of the square lattice, there is a perfect agreement between the mirror-Chern number and the edge states of nanoribbons, as dictated by the bulk-edge correspondence.

Discussions
The minimal tight-binding Hamiltonian of a TCI thin film is a four-band model in order to take into account the spin and pseudospin (surface) degrees of freedom. We have constructed such a model based on the symmetry analysis. The prominent features are that gapless Dirac cones emerge at all the high symmetric points and that we can provide them with gaps phenomenologically at our disposal.
We have analyzed the square lattice with the C 4 symmetry and the triangular lattice with the C 6 symmetry in details. The results may well describe the thin films made of the [001] surface (C 4 symmetry) and the [111] surface (C 6 symmetry) made of Pb x Sn 1−x Te, by choosing the mass parameters appropriately. According to experimental observations and a first-principle calculation there are large gaps at the Γ and M points in the [001] surface [8][9][10]13 . This is realized by taking a large value of m 1 in our model. On the other hand there are small gaps at the X and Y points, which is taken care of by introducing a small value of m 0 .
We may similarly discuss the square lattice with the C 2 symmetry with the mass term being m = m 0 + m 1 cos k x . When m 0 = m 1 , there are Dirac cones only X and M points, as is consistent with theoretical results 13,18 on the [110] surface of Pb x Sn 1−x Te. The model with the C 3 symmetry is also constructed on the honeycomb lattice. We find Dirac cones at the K and K ′ points and two degenerated Dirac cones at the Γ point.
Our basic Hamiltonian consists of the SOI of the type σ·d ℓ ij as in (2). We have made this choice since it reproduces the low-energy Dirac theory 13 . The same spectrum is obtained even if we take the SOI of the Rashba type σ × d ℓ ij as in (1), although the low-energy Dirac theory now reads and different from (20). We predict that another TCI may be found in future, where the Rashba-type Hamiltonian (1) plays the basic role.
A TCI thin film may be used to design a nanodevice for topological electronics. Edge states can be gapped by applying electric field independently to the right and left edges. We have proposed a multi-digit field-effect topological quantum transistor with the use of gapless edge states of a TCI thin film nanoribbon. This could be a basic component of future topological quantum devices.

Methods
In this section we explain the discrete rotational symmetry C N . We also describe how to calculate the mirror-Chern number and the conductance.
Symmetry. We have constructed the tight-binding Hamiltonian so that it is invariant under the discrete rotation symmetry C N in addition to the mirror symmetry (9). The generator of C N is with the 2π N -rotation of the momentum We note that the C N rotation rotates the direction of spin with π/N . The rotation angle is restricted to be N = 2, 3, 4, 6 due to the crystal group of the lattice symmetry. They corresponds to the rectangular lattice for N = 2, the hexagonal lattice for N = 3, the square lattice for N = 4, and the triangular lattice for N = 6. Mirror-Chern number. According to a general scheme 25 , the mirror-Chern charge is defined even for E z = 0. With the use of the Matsubara Green function, with iω referring to the Matsubara frequency (ω: real), the mirror-Chern charge is calculated by 25 Here, Ω M = 1 6 ε µυρ Tr[GΓ µ GΓ ν GΓ ρ ] and with M the mirror-symmetry generator (9). The result is given by (22) for E z = 0.
Conductance. In terms of single-particle Green's functions, the low-bias conductance σ(E) at the Fermi energy E is given by 26 with the Hamiltonian H D for the device region. The selfenergy Σ L(R) (E) describes the effect of the electrode on the electronic structure of the device, whose the real part results in a shift of the device levels whereas the imaginary part provides a life time. It is to be calculated numerically 24,[27][28][29][30] .