Introduction

The discovery of the integer quantum Hall effect in two-dimensional electron gas opens a window to explore topological phases in quantum matter1,2. In the quantum Hall effect the longitudinal conductance vanishes while the Hall conductance σxy is quantized at ve2/h, where e is the elementary charge and h is the Planck constant. It is known that the integer v is the topological invariant of a quantum phase, it counts the number of conducting chiral channels at the edges of the system and is insensitive to the geometry of the sample, impurity and interactions of electrons3,4,5,6. Three-dimensional topological insulator is a new class of topological materials and is characterized by the formation of the Dirac fermion gas covering its surfaces7,8,9,10. Soon after the discovery of three-dimensional topological insulators, the formation of the Landau level (LL) of the surface Dirac electrons in a strong magnetic field has been observed by the scanning tunneling microscope11,12,13 and Shubnikov-de Haas oscillations in the longitudinal conductance14,15,16. It has been known since 1980’s that the Hall conductance of massless Dirac fermions is quantized as half integers v = n + 1/2, where n = 0, ±1, ±2,….17,18,19,20,21 and 1/2 is attributed to the Berry phase of the massless Dirac fermions acquiring from a cyclotron motion around the Fermi surface19,22,23. Usually each LL carries one conducting channel near the edge due to the geometric distortion of the cyclotron motion of electrons in a magnetic field. The quantum Hall conductance is determined by the number of the edge states4,5. Thus the relation between the half-quantized Hall conductance and the number of the conducting edge channel is an open issue and has attracted a lot of studies20,21,24,25,26,27,28,29,30,31,32,33,34,35,36. Very recently the integer quantum Hall effect has been measured in three-dimensional topological insulator thin films by two independent groups37,38. One group measured a series of plateaux of v = −1, 0, 1, 2, 3 and v = 1, 3 in BiSbTeSe237 and the other measured the plateaux of v = 0, ±1 in (Bi1−xSbx)2Te338. These plateaux of the Hall conductance are attributed to the addition of the top and bottom surface electrons and always yield integers in units of e2/h.

In this work we present solutions of the LLs and edge states of the surface electrons and explore the formation of the quantum Hall effect in a topological insulator thin film. The Hall conductance is calculated by means of the Kubo formula at zero temperature, which is well quantized when the chemical potential lies between two LLs. Two distinct patterns of the quantum Hall conductance are found: v is an odd integer when the LLs of the top and bottom surface electrons are degenerate in a thick film and v is an integer when structure inversion symmetry is broken in the film or the top and bottom surface electrons are coupled to open an energy gap due to the finite-size effect39,40. The absence of the v = 0 plateau is caused by the degeneracy of two sets of LLs and the v = 0 plateau emerges when the degeneracy is lifted by the finite-size effect or structure inversion asymmetry (SIA, which can be produced by the energy difference between the two surfaces). The width of the v = 0 plateau is determined by the energy difference between the two LLs closest to zero energy. The state of v = 0 can be either in the quantum spin Hall phase or trivial band insulator phase and the quantum spin Hall effect can be easily broken down by SIA.

Results

Model

Consider a thin film of three-dimensional topological insulator. Starting from the bulk Hamiltonian for topological insulators10,41, the low-energy effective Hamiltonian for the surface electrons has been derived by solving the differential equation for the bulk bands exactly39,40

where k± = kx ± iky and with kx,y the wave vector in the surface plane. The mass term Δ/2 − Bk2 is generated by the hybridization between the wave functions of the top and bottom surface electrons. Both B and Δ depend on the film thickness and approach zero simultaneously in a thick film. with vF the effective velocity and the reduced Planck constant. 2V is the potential difference between the top and bottom surfaces due to SIA as shown in Fig. 1a, which can be induced in a realistic thin film by the potential difference between the substrate and vacuum surfaces and tunable by a gate voltage. The interplay of SIA and top-bottom hybridization gives the Rashba-like splitting in the band structure (see Fig. 1b). The physics of this model has been confirmed by the angle-resolved photoemission spectroscopy experiments on topological insulator thin films42,43.

Figure 1
figure 1

Two distinct quantum Hall conductance patterns.

(a) Schematics of LLs of the surface states in a thick topological insulator film under a perpendicular magnetic field. Each of the top and bottom surfaces hosts an independent set of LLs. 2V is the energy offset between the top and bottom surfaces, usually induced by SIA and tunable with a gate voltage. (b) The same as (a) except for a thin film, where the LLs reside in the whole film rather than at specific surfaces. Δ is the finite-size gap opened by the hybridization of the top and bottom surface states. (c) The pattern of integer quantum Hall conductance plateaux as a function of the chemical potential μ in a fixed magnetic field μ0H for a general case. (d) The pattern of odd integer quantum Hall plateaux in the case where the two sets of LLs of the top and bottom surface states are degenerate.

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Two patterns of quantum Hall plateaux

For a thin film of topological insulator in the presence of a perpendicular magnetic field, the Hall conductance is usually quantized as an integer v as shown in Fig. 1c, or can only be an odd integer v when the LLs of the top and bottom surface electrons coincide exactly as shown in Fig. 1d. Consider a topological insulator thin film. The surface states cover the top and bottom surfaces as illustrated in Fig. 1a,b. The lateral side is also covered by the surface electrons, but is ignored in the present work. For a film with a relatively large thickness, the top and bottom surface states are well separated. Thus the system consists of two decoupled massless Dirac electrons. The two Dirac points can be separated by the potential difference 2V between the top and bottom surfaces. The value of V then can determine the quantized pattern of the Hall conductance such as the width of quantum plateaux and the existence of v = 0 plateau. For V = 0, the LLs of the top and bottom surface electrons are degenerate and the Hall conductance is quantized as odd integers v = 2n + 1, which can be regarded as the addition of two sets of half-quantized Hall conductance, i.e., . This is very similar to that in graphene, in which the Hall conductance is , where the factor 4 is attributed to the spin and valley degrees of freedom44,45. For , the degeneracy of the LLs is removed and the Hall conductance becomes quantized as integers in Fig. 1c. Another factor leading to the lift of the degeneracy is the finite-size effect. When the thickness of a thin film is comparable with the spatial distribution of the wave functions of the surface states, the overlap of the wave functions will open an energy gap Δ at the Dirac points of the two surface states39, leading to the presence of the v = 0 plateau and a possible topological phase of the quantum spin Hall effect.

Landau levels and edge states

In Figs 2 and 3, we present the LLs in a magnetic field μ0H normal to the thin film, the LL energies or edge states near one edge of the system and the corresponding patterns of the quantum Hall conductance. In the absence of SIA, i.e., V = 0, four possible typical cases are shown in Fig. 2. Case (i) is for a thick film, i.e., Δ = B = 0. In the bulk, the LLs of zero energy are degenerate and are insensitive to the field, as shown in Fig. 2a. However they split into two branches when approaching one edge: one branch goes upward (called electron-like) and the other goes downward (called hole-like), as shown in Fig. 2e. The position is the guiding center of the wave packages of surface LLs and is proportional to the wave vector kx, where the magnetic length . The slope of the energy dispersion near the edge is proportional to the effective velocity of the edge states , which defines the current flow of the edge states. The Hall conductance is equal to v = 1 when the chemical potential is below the LLs of n = 0 and v = −1 when the chemical potential is above the LLs of n = 0. The plateau of v = 0 is absent and other plateaux are . For a thinner film, there exist three cases, (ii)Δ > 0, (iii) Δ = 0 and (iv) Δ < 0, all with the parameter B < 0 (without loss of generality we assume negative B). In case (ii) with ΔB < 0, the two LLs of n = 0 are separated in a finite field as shown in Fig. 2b. When approaching the edge the LLs with positive energies go upward while the LLs with negative energies go downward, indicating that the edge electrons with opposite energies move in opposite directions. One of the key features of the quantum Hall conductance is the emergence of the v = 0 plateau and the width of the plateau is determined by the energy difference between the two LLs of n = 0 (see Fig. 2j). In case (iii) with Δ = 0 and B < 0, the two LLs near the Dirac point are degenerate at zero field, as shown in Fig. 2c. The degeneracy is lifted by a finite field μ0H. The energies of edge states (see Fig. 2g) and the Hall conductance (see Fig. 2k) are very similar to those in case (ii). In case (iv) with ΔB > 0, two LLs near zero energy are separated in weak fields and cross at a finite field as shown in Fig. 2d, indicating a quantum phase transition. In a weak field, the pattern of the Hall conductance (see Fig. 2l) is also similar to that of case (ii), while the dispersions of the edge states (see Fig. 2h) are different. The two LLs of n = 0 cross near the edge, which is a key feature of the quantum spin Hall effect in a finite field46. At magnetic fields higher than the energy crossing in Fig. 2d, the LLs near zero energy never cross near the edge, similar to those in Fig. 2 f and g, indicating the breakdown of the quantum spin Hall effect in a magnetic field. However, the plateau of v = 0 still survives.

Figure 2
figure 2

Landau levels and edge states in the absence of SIA.

The four columns compare cases with different Δ and B, two parameters in the mass term of the model. From left to right, (i) Δ = 0 and B → 0; (ii) ΔB < 0; (iii) Δ = 0 and B ≠ 0; (iv) ΔB > 0. The top row is for the fan diagrams, i.e., the energies of LLs as functions of the magnetic field μ0H. The calculation of the Landau level spectra assumes no boundary. The two LLs of n = 0 are highlighted. The middle row is for the energy dispersions of LLs at μ0H = 5 T near an open edge at y = 0. y0 is the position of guiding center in units of magnetic length . The bottom row is for the Hall conductance σxy as a function of the chemical potential μ at μ0H = 5 T. For ΔB > 0, the two LLs of n = 0 cross at a critical magnetic field in (d) showing a field-induced quantum phase transition from the quantum spin Hall phase in weak field to trivial band insulator phase in strong field; correspondingly in (h) the higher hole-like LL and the lower electron-like LL of n = 0 cross when approaching the edge, contributing to two conducting channels with opposite velocities when the chemical potential crosses them. This characterizes the quantum spin Hall phase with no charge Hall conductance but a finite quantized spin Hall conductance. For both cases (ii) and (iii), all electron-like LLs are above all hole-like LLs, therefore there is no quantum spin Hall phase; For case (i) of Δ = 0 and B → 0, all LLs in (a) are two-fold degenerate in the bulk, but the degeneracy is lifted in (e) when approaching the edge. In all cases B = −500 meVnm2 and γ = 300 meVnm for comparison, while Δ = 0 in cases (i) and (iii), 20 meV in case (ii) and −20 meV in case (iv).

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Figure 3
figure 3

Integer quantized Hall plateaux due to SIA.

For case (i) with Δ = 0 and B → 0 but a finite V = 6 meV, (a) the fan diagram, (b) the energies of the two LLs of n = 0 near the edge and (c) the Hall conductance as a function of the chemical potential μ. SIA breaks the degeneracies of all LLs in the thick film where Δ = 0 and B → 0. As a result, even integer Hall conductance plateaux also appear.

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In the presence of SIA, , the relative positions of two sets of LLs from the top and bottom surfaces can be tuned by the value of V. A typical pattern of the quantum Hall conductance is presented in Fig. 3. The Hall conductance covers all integers. In the case that the chemical potential crosses some accidental crossing points of two LLs, the Hall conductance can change by .

Quantum spin Hall state and its breakdown

A question arises from the appearance of the quantum spin Hall effect in case (iv) with ΔB > 0. Usually the quantum spin Hall effect is protected by time reversal symmetry47,48. Applying a magnetic field breaks time reversal symmetry, but it is known from the calculation of the Hall conductance that the quantum spin Hall effect can be stabilized in a finite field when the system possesses an intrinsic and hidden sz symmetry which guarantees the decoupling of the two blocks with spin up and down in the Hamiltonian49. In the presence of SIA, , the situation changes. The energy crossing of the two n = 0 LLs in a finite field in Fig. 2d becomes an anti-crossing for a finite V in Fig. 4a, showing that there is no field-induced quantum phase transition from the quantum spin Hall insulator to the trivial band insulator in the presence of SIA and the system remains in the trivial phase at all magnetic fields. In other words, the quantum spin Hall effect breaks down as long as SIA is present. The breakdown of the quantum spin Hall phase can also be seen from the energies of the edge states in Fig. 4b, in which the energies of the n = 0 LLs cross for V = 0 while open a finite gap for a nonzero V. The gap increases with increasing V. For a small V, the corresponding energy gap of the edge states is also small. Although the edge states are no longer protected topologically, it can still produce some physical phenomena such as the spin accumulation, or the intrinsic spin Hall effect near the edge when an electric current is applied. However these effects will be weakened for a large V. Therefore, the quantum spin Hall effect does not survive in the presence of both a magnetic field and SIA.

Figure 4
figure 4

SIA-induced breakdown of the quantum spin Hall phase.

(a) The fan diagram in the presence of SIA, i.e., V ≠ 0. SIA turns the crossing between the two LLs of n = 0 in Fig. 2d into an anti-crossing. (b) The energies of the two LLs of n = 0 at 5 T near the edge for different V. In the presence of SIA, the two LLs do not cross near the edge and open an energy gap. Inset: the gap opened between the two LLs of n = 0 as a function of V. The parameters are γ = 300 meVnm, Δ = −20 meV and B = −500 meVnm2.

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Phase diagrams

In the absence of SIA, i.e., V = 0, we plot in Fig. 5a–d the corresponding phases diagrams of the Hall conductance as a function of the chemical potential μ and magnetic field μ0H for the four cases. Different quantum Hall phases are denoted by the quantized Hall conductances and are separated by the boundaries (marked by the white dotted lines). The Hall conductance is antisymmetric with respect to μ. In case (i), there are only odd integer quantum Hall phases (see Fig. 5a). The spacings of the phases grow with increasing magnetic field or . Thus the quantum Hall plateaux are more visible by varying μ in a stronger magnetic field, or by varying the magnetic field with a larger fixed . In contrast, not only odd but also even integer quantum Hall phases are possible in the presence of a finite mass, i.e., or (see Fig. 5b–d). When μ = 0, all negative LLs are filled and all positive LLs are empty no matter how large the magnetic field is and the corresponding Hall conductance is zero, regardless of whether there is energy crossing near the edge of the system. The quantum spin Hall effect only appears in the case of ΔB > 0 and V = 0 in a weak field and disappears in a stronger field as shown in Fig. 5d.

Figure 5
figure 5

Phase diagrams of the quantum Hall effect in topological insulator films.

(a–d) The phase diagrams as functions of the chemical potential μ and magnetic field μ0H in the absence of SIA, i.e., V = 0. Different phases are denoted by corresponding Hall conductance σxy in units of e2/h. The white dotted lines are the boundaries between different phases. The four columns compare cases with different finite size gap Δ and B. From left to right, (i) Δ = 0 and B → 0; (ii) ΔB < 0; (iii) Δ = 0 and B ≠ 0; (iv) ΔB > 0. The parameters for different cases are the same as those in Fig. 2. σxy is antisymmetric with respect to μ. In (a) there are only odd integer quantum Hall phases. In (b–d) there are both odd and even integer quantum Hall phases. In (d) the quantum spin Hall (QSH) phase is marked. (e–h) The phase diagrams as functions of V and μ0H while fixing the chemical potential μ = −V − 0+. Both odd and even integer quantum Hall phases can be induced by changing V in all the four cases.

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Moreover, we calculate the Hall conductance as a function of V and magnetic field μ0H in Fig. 5e–h, where we set μ = −V − 0+. An infinitesimal value 0+ is introduced to avoid the alignment of the chemical potential μ with one of the n = 0 LLs (E0 = −V) in case (i) in Fig. 5e. The asymmetry of the Hall conductance with respect to V reflects the fact that both of the n = 0 LLs (E0 = ±V) are empty for a positive V while only one level (E0 = +V) is filled for a negative V. Figure 5f–h are for the cases with a finite mass, the Hall conductance is antisymmetric with respect to V and show very similar phase diagrams. Both odd and even integer quantum Hall phases can be induced by changing V in all the four cases. If setting μ = 0, we find that the Hall conductance is always vanishing, which is expected since the chemical potential μ is fixed at the center between the electron-like LLs that would go upward and the hole-like LLs that go downward when approaching the edge.

Discussions

The formation of the edge states related to the two LLs near zero energy can be understood from the model that the top and bottom surfaces are separated by the metallic lateral surface. In the presence of a perpendicular magnetic field, the lateral surface electrons only experience an in-plane field and the two Dirac points will shift oppositely by a constant if the Zeeman field is taken into account30. The edge states for the two LLs near the zero energy are divided into three parts: the parts of the wave function at the top and bottom surfaces decay exponentially away from the edge while they are connected by the part of the lateral surface electron. This is very similar to the case of the quantum anomalous Hall effect in a topological insulator thin film with a perpendicular Zeeman field30. A similar calculation can be found in a recent paper50. As the conductance of lateral surfaces is nonzero for a thin film with a finite thickness, the longitudinal conductance no longer vanishes. In this case the Hall resistance can be quantized perfectly only when the residual conductance from the lateral effect can be suppressed completely.

Mathematically the coupling between the top and bottom surfaces has been reasonably taken into account in the model in Eq. (1). Although it was shown rigorously that the Hall conductance for an ideal massless Dirac fermion gas with is quantized in a magnetic field as half integers, it is found that the Hall conductance is modified into integers once a quadratic correction is introduced to HD, where σx,y,z are the Pauli matrices. This is case (iii) in Fig. 2c. The conclusion is valid even if the parameter B is in the infinitesimal limit and the solution of the edge states always exists. In other words, for an infinitesimal B, we obtain two well-defined edge states mathematically for two LLs of n = 0 as shown in Fig. 2e. The physical meaning of the term B is attributed to the coupling between the top and bottom surfaces according to the exact solution for a bulk model39,40.

Having in mind the picture of edge states of LLs for Dirac fermions we come to make some comments on the two experiments on the quantum Hall effect in topological insulator thin films. In the experiment by Xu et al.37, a series of the Hall conductance plateaux of v = −1, 0, 1, 2, 3 are measured as a function of the gate voltage Vbg. The absence of the v = −2, −3 plateaux can be understood as the particle-hole symmetry breaking in the band structure of BiSbTeSe2, which is also confirmed by the transport measurement. The role of Vbg is to control the relative positions of the two Dirac points at the top and bottom surfaces, or the SIA term V in the thin film model in Eq. (1). The nonzero conductance σxx indicates that the Hall resistance has not yet been quantized completely. One of the reasons is that the thicknesses of the samples are 80 nm and 160 nm, respectively. In this case the lateral conductance does not vanish completely even at low temperatures. Suppression of the lateral effect will be a key to realize high precision of quantum Hall conductance in this experiment. On the odd integer quantum Hall effect, the plateaux of v = 1 and 3 are observed. As one surface of the thin film is grown on the substrate while the other is exposed to the vacuum, the boundary conditions are quite different and the effective velocities of the surface electrons may not be identical. Thus it will be a hard task to make two sets of the LLs of the surface electrons degenerate completely by tuning the gate voltage only. However it is relatively easy to have only two specific LLs degenerate such that the plateau of v = 0 disappears.

In the experiment by Yoshimi et al.38, only two plateaux of v = −1, + 1 or v = 0, + 1 are measured in two different samples of (Bi1−xSbx)2Te3. The presence of v = 0 plateau is attributed to the potential difference between the two surfaces, i.e., the SIA term . As the thickness of two samples is 8 nm, the finite-size effect is also relatively weak, i.e., . This should correspond to the case in Fig. 3. Thus the width of the plateau is determined by the value of V and the pattern of edge states indicates that the state of v = 0 is simply a band insulator.

Methods

Landau levels

When a uniform field is applied perpendicular to the thin film, the wave vector is replaced by , where the vector potential under the Landau gauge is . kx remains a good quantum number. The LLs can be found by defining two ladder operators and , where the magnetic length and assuming the trial solution,

where and are Hermite polynomials. The energies of the LLs are found as

where , , , , s = ±1, and . For each given , there are four LLs, two of them are electron-like while the other two are hole-like. The expressions for the corresponding eigenstates can be found in the supplementary information.

The magnetic field can also induce a Zeeman energy described by where μB is the Bohr magneton and g is the g-factor. It is weak and thus we neglect it in this work. The Landau levels for the bulk states of a topological insulator was studied in a similar way51.

Hall conductance

The Hall conductance can be found from the Kubo formula

where is the eigenstates in the LL , is the Fermi function with kB the Boltzmann constant. and are two velocity operators.

Edge states near open boundary

To solve the wave function of edge states at an open boundary, we employ the properties of the two standard solutions and to the Weber equation52,

where the dimensionless quantity is defined by . With the trial wave functions in the y direction

we can find the corresponding λ for a given eigen energy E. There are four λ’s for each given eigenenergy E, so a general solution to the wave function is a linear combination of eight eigenstates. The allowed eigen energies, the superposition coefficients as well as the wave function can be found by applying the open boundary conditions at the edge.

For a system with a semi-infinite geometry , the wave function can only contain the components since is exponentially divergent while are vanishing as ξ approaches +∞ The boundary condition at y = 0 further provides an equation for E. In the weak coupling limit , the equation for E is reduced as

where and . If E is a solution, then −E is also a solution, reflecting the particle-hole symmetry, as expected. Thus we shows that the solutions of edge states exist even if .

Additional Information

How to cite this article: Zhang, S.-B. et al. Edge states and integer quantum Hall effect in topological insulator thin films. Sci. Rep. 5, 13277; doi: 10.1038/srep13277 (2015).