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Edge states and integer quantum Hall effect in topological insulator thin films.

Zhang SB, Lu HZ, Shen SQ - Sci Rep (2015)

Bottom Line: We examine the formation of the quantum plateaux of the Hall conductance and find two different patterns, in one pattern the filling number covers all integers while only odd integers in the other.We focus on the quantum plateau closest to zero energy and demonstrate the breakdown of the quantum spin Hall effect resulting from structure inversion asymmetry.The phase diagrams of the quantum Hall states are presented as functions of magnetic field, gate voltage and chemical potential.

View Article: PubMed Central - PubMed

Affiliation: Department of Physics, The University of Hong Kong, Pokfulam Road, Hong Kong, China.

ABSTRACT
The integer quantum Hall effect is a topological state of quantum matter in two dimensions, and has recently been observed in three-dimensional topological insulator thin films. Here we study the Landau levels and edge states of surface Dirac fermions in topological insulators under strong magnetic field. We examine the formation of the quantum plateaux of the Hall conductance and find two different patterns, in one pattern the filling number covers all integers while only odd integers in the other. We focus on the quantum plateau closest to zero energy and demonstrate the breakdown of the quantum spin Hall effect resulting from structure inversion asymmetry. The phase diagrams of the quantum Hall states are presented as functions of magnetic field, gate voltage and chemical potential. This work establishes an intuitive picture of the edge states to understand the integer quantum Hall effect for Dirac electrons in topological insulator thin films.

No MeSH data available.


Related in: MedlinePlus

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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f2: 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).

Mentions: 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.


Edge states and integer quantum Hall effect in topological insulator thin films.

Zhang SB, Lu HZ, Shen SQ - Sci Rep (2015)

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).
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4548229&req=5

f2: 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).
Mentions: 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.

Bottom Line: We examine the formation of the quantum plateaux of the Hall conductance and find two different patterns, in one pattern the filling number covers all integers while only odd integers in the other.We focus on the quantum plateau closest to zero energy and demonstrate the breakdown of the quantum spin Hall effect resulting from structure inversion asymmetry.The phase diagrams of the quantum Hall states are presented as functions of magnetic field, gate voltage and chemical potential.

View Article: PubMed Central - PubMed

Affiliation: Department of Physics, The University of Hong Kong, Pokfulam Road, Hong Kong, China.

ABSTRACT
The integer quantum Hall effect is a topological state of quantum matter in two dimensions, and has recently been observed in three-dimensional topological insulator thin films. Here we study the Landau levels and edge states of surface Dirac fermions in topological insulators under strong magnetic field. We examine the formation of the quantum plateaux of the Hall conductance and find two different patterns, in one pattern the filling number covers all integers while only odd integers in the other. We focus on the quantum plateau closest to zero energy and demonstrate the breakdown of the quantum spin Hall effect resulting from structure inversion asymmetry. The phase diagrams of the quantum Hall states are presented as functions of magnetic field, gate voltage and chemical potential. This work establishes an intuitive picture of the edge states to understand the integer quantum Hall effect for Dirac electrons in topological insulator thin films.

No MeSH data available.


Related in: MedlinePlus