Quantum confinement effect induced topological phase transitions in anisotropic Weyl semimetal

Journal Pre-proof Quantum confinement effect induced topological phase transitions in anisotropic Weyl semimetal Mei-Mei Wu, Yanmei Sun, Hui Zhao, Hui Pan

PII: DOI: Reference:

S0749-6036(19)31876-2 https://doi.org/10.1016/j.spmi.2019.106386 YSPMI 106386

To appear in:

Superlattices and Microstructures

Received date : 4 November 2019 Accepted date : 23 December 2019 Please cite this article as: M.-M. Wu, Y. Sun, H. Zhao et al., Quantum confinement effect induced topological phase transitions in anisotropic Weyl semimetal, Superlattices and Microstructures (2019), doi: https://doi.org/10.1016/j.spmi.2019.106386. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.

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Quantum confinement effect induced topological phase transitions in anisotropic Weyl semimetal Mei-Mei Wua , Yanmei Suna , Hui Zhaoa , Hui Pana,∗ a School

of Physics, Beihang University, Beijing 100191, People’s Republic of China

Abstract

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The quantum anomalous Hall effect can be realized in the Weyl semimetal based on a simple cubic lattice. On the basis of the system with broken time-reversal symmetry, we find the energy spectrum is isotropic in kx and ky direction but anisotropic in kz direction. And the quantum confinement effect is strongly anisotropic in different direction, exhilaratingly, the periodic modulation of en-

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ergy gap with the film thickness along z direction is found. We also find out that the topological phase transition is happened between a topological trivial insulator and a quantum anomalous Hall insulator due to confinement effect. These results can not only promote a better understanding of Weyl Semimetal, but also provide a good platform to design topological quantum devices based

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on the Weyl semimetal in the future. Keywords: Weyl semimetal, the quantum confinement effect, the quantum

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anomalous Hall effect

1. Introduction

The research of topological insulators (TIs) have been widely studied in past several years [1–3], which attracted much interest in the peculiar electronic properties different from the ordinary insulators [2–4]. Graphene and silicene,

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the novel two dimensional (2D) electronic materials, which are densely packed I This

document is a collaborative effort. author Email address: [email protected] (Hui Pan)

∗ Corresponding

Preprint submitted to Superlattices and Microstructures

October 31, 2019

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into a hexagonal honeycomb crystal structure, have attracted much attention in theory and experiment as the representatives of 2D TIs [5–8]. With the

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further investigation of TIs, the three dimensional (3D) topological materials have been discovered and researched both theoretically and experimentally, such as Dirac semimetal (DSM) [9–20], Weyl semimetal (WSM) [21–26], quantum anomalous Hall insulator (QAHI) [2, 3, 27–32], etc. As a topological nontrivial semimetal without bulk band gap, the DSM understood on the characteristies of electronic band structure transport, has two Dirac symmetric points which

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are located along kz direction [11, 33]. Due to the existence of time reversal (T ) and inversion symmetry (P), the Dirac points are double degenerate at high symmetry place in the Brillouin zone [34]. By breaking T or P, each Dirac point can separate into a pair of Weyl points with opposite chirality, this implements a transition from DSM to WSM [11, 24–27, 34, 35]. Although the

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concept about Weyl fermions was firstly introduced in high-energy physics, it has received extensive attention and research in condensed-matter physics. The WSM, which is a novel topological state of 3D quantum matters with much interesting and fascinating physical properties [24–27, 36–41], has been discovered in experiment [22]. As an especial topological material, the physical

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properties of WSM are much different from other 3D TIs [2, 3, 42–44], such as surface states [19, 45–51], electromagnetic response [34, 52–56], charge-spin response [34, 57–59] and other transport properties [60–65]. These singular

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characters derive from the unique electronic properties. For WSM, the two Weyl points which are located along z direction and central symmetric can be annihilated in pair of opposite chirality by short range scatters or adding definite perturbation [60]. And the confinement-induced energy gap as function of the thin-film thickness has been studied in the low energy effective model of DSM [33], and further research shows that quantum confinement effect has been

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found in a simple two-band tight-binding model in WSM [28]. Compared with the previous studies on WSM, the strongly anisotropic of quantum confinement effect in different direction and the transition of topological phase by the film thickness will attract more attention. 2

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Our work is discussed by following sections. In Sec. II, the universal lowenergy effective model and analytic analysis are represented. In Sec. III, we show

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the results of numerical calculation. We find that the topological phase transformation has a periodic modulation with the increase of film thickness along the z direction. The alternation of topological trivial phase and QAHI phase in the topological phase transformation is also found in paper. And we clearly observe the change of the number of edge states by the surface local density of state (LDOS). Finally, we give some discussions and potential applications for

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WSM by analyzing our conclusions in Sec. IV.

2. Physical Model and Analytic Analysis

Our analysis is based on the low-energy effective model depicting a 3D TI of Bi2 Se3 family [3, 66]. This model as a simple cubic lattice has been proposed

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in previous work, and the effective Hamiltonian in the momentum space can be written as [34]

H0 (k) = 2λσz (sx sin ky − sy sin kx ) + 2λz σy sin kz + σx Mk ,

(1)

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where σi and si (i = x, y, z) are the Pauli matrices in orbital and spin space, P and Mk =  − 2t α cos kα (α = x, y, z). This model can be described as a strong topological insulator with the Z2 index, when the constants λ and λz are

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all larger than 0, and 2t <  < 6t [34]. In the following calculation and analysis, we mainly focus on the around of topological phase transition that happens at  = 6t.

The Weyl semimetal can be easily presented when we add the following

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perturbation to H0 [34]

H1 (k) = b0 σy sz + b · (−σx sx , σx sy , sz ),

(2)

Here, nonzero b can break T , nonzero b0 can break P. Each Dirac point can divided into a pair of Weyl points with opposite chirality in momentum or energy when T or P is broken. The two symmetries can be generally expressed as, P: 3

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sx H(k)sx = H(−k) and T : sy H ∗ (k)sy = H(−k). The Hamiltonian of the system is H(k) = H0 (k) + H1 (k). In the following calculation, we set b0 = 0,

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b 6= 0(bx = by = 0, bz 6= 0) to break T but protect P. Thus, a pair of Weyl points is located at (0, 0, ±(bz /2λz )). And this model can be viewed as a layered 2D QAHI with interaction between layers. In the concrete calculation, we use Taylor expansion Eq. 3 (only one order is preserved) for trigonometric functions in H(k)

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1 sin(ki ai ) → ki , cos(ki ai ) → 1 ai

(3)

where i = x, y, z for lattice discretization. The low-energy effective Hamiltonian H(k) for a 3D WSM around the Γ point in the Brillouin zone can be given by

where

X

X

c†i H0 ci +

X

c†i Hx ci+x +

c†i Hz ci+z + H.c.,

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H=

X

c†i Hy ci+y +

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H0 = σx + b0 σy sz + bz sz ,

(4)

(5)

Hx = iλσz sy − tσx ,

(6)

Hy = −iλσz sx − tσx ,

(7)

Hz = −iλz σy − tσx .

(8)

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For bulk Weyl semimetal, the energy spectrum of Hamiltonian H can be solved q E(k) = ± (bz ± 2λz kz )2 + 4kx2 λ2 + 4ky2 λ2 .

(9)

The two Weyl points are on the kz -axis, and they are centrosymmetrical with opposite chiralities[9]. As shown in Fig. 1, the location of the two Weyl points 4

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and the band structure of Hk are displayed by Eq. 9. In the process of calculation, we take λ = 1, λz = 1, t = 0.5, and bz = 0.3.

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In order to study the properties of the Weyl semimetal, the surface local density of state (LDOS) ρ(ki ) = −T r[ImG00 (ki )]/π is also calculated, where G00 is the retard Green’s function for surface of lattice. G00 can be showed by using transfer matrix through the standard iterative method[67], G00 = (E − H00 − H01 T )−1 ,

(10)

the surface LDOS is gained.

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where H00 and H01 are Hamiltonian matrixes, and T is inverse matrix. Thus,

3. Numerical results and discussions

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Generally speaking, the confinement effect of system is different along different directions when there are anisotropic effects in system. To study the quantum confinement effect of WSM, we discretize H(k) on a 3D simple cubic lattice. We set lattice constants as ax = ay = 1 and az = 1.002 in the numerical calculation. Firstly, let’s make a analysis for Weyl points and the energy spectrum. The two Weyl points are located at kz -axis and symmetrical about

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Γ, shown in Fig. 1(a). In Fig. 1(b), Fig. 1(c) and Fig. 1(d), the energy spectrum of bulk along kx , ky and kz is plotted respectively. We find that the energy

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spectrum is isotropic in kx and ky direction but is anisotropic in kz direction, and the dispersion is linear around the two Weyl points along kz -axis direction. Then, we investigate the quantum confinement effect of the WSM in different directions. A quasi-2D structure of WSM can be arisen with confinement. We consider the case that the system is confined along x and z direction. First of all, the bulk confinement-induce gap as a function of film thickness Wy a-

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long y-direction is calculated, as shown in the Fig. 2(a). The gap decreases monotonously with the increase of Wy . According to the quantum well ap-

proximation and location of the Weyl points, the scale of induced gap should be as Eg ∼ Wy−1 . This characteristic is consistent with numerical calculation 5

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of Fig. 2(a). We can find that the confinement effect along the y direction is relatively trivial in the model of WSM. Next, we study the system with con-

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finement only along z direction. Similarly, it is easy to find out that the period of the oscillation should be π/kw · az ≈ 21 by using the quantum well approximation. And the numerical calculation is shown about the confinement-induced gap with the film thickness Wz along z-direction, as shown in Fig. 2(b). We see that there is contrary case in Fig. 2(a), now the energy gap has a periodic modulation with a period of 21 layers. The confinement effect along the z direction

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is rather interesting in the model of WSM. This phenomenon is similar with the confinement in Dirac semimetal, which has been studied in previous works [20, 33]. One observes that the critical film thickness is about 9 layers and 30 layers corresponding to the first and second energy gap closed and reopened, respectively.

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Before and after the first gap closed is analyzed particularly, that is marked by point A, B and C in Fig. 2(b). In Fig. 3(a), we show the energy spectrum along kx (ky = 0) with a gap when thickness of film is about 7 layers (corresponding to point A in Fig. 2(b)). With increasing of film thickness, the energy gap is closed at a critical film thickness 9 layers, as shown in Fig. 3(b) (corre-

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sponding to point B in Fig. 2(b)). Then, the energy gap will be reopened when the film thickness is more than 9 layers. The energy spectrum is calculated for film thickness 11 layers, as shown in Fig. 3(c) (corresponding to point C in

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Fig. 2(b)). We know that the energy gap closes at Γ point when the system reaches the critical situation, it can be marked as boundary of topological phase which differs between topological trivial and topological nontrivial phases. To further confirm transformation of topological phase and to visualize the edge state, we calculate the surface LDOS for the side surface. Since kx and ky of model H(k) have isotropic properties, without loss of generality, only the surface

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LDOS along kx is discussed in our work. The surface LDOS ρ(kx ) to show the variation of edge states can be calculated with ρ(kx ) = −T r[ImG00 (kx )]/π. The

surface LDOS of before and after the topological phase transition (corresponding to state A and C in Fig. 2(b)) are plotted in Fig. 4. One observes that there 6

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are different situations in the two cases. In Fig. 4 (a), when the film thickness of system is 7 layers (before topological phase transition), there exists two pairs

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of overlapping bright lines crossing gap, which corresponding to the degenerate edge states for the C nontrivial QAHI. This can be characterized by nonzero Chern number C = 2. However, there are no edge states in the gap after topological phase transition (Wz > 9) in Fig. 4(b), so this is a topological trivial insulator. This can be characterized by Chern number C = 0. The topological phase can transform topological nontrivial to trivial by changing film thickness.

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Then, the situation of before and after the second energy gap closed is discussed. In Fig. 5(a), Fig. 5(b) and Fig. 5(c), we plot the energy spectra corresponding to the point D, point E and point F of Fig. 2(b). We can see that the energy gap is closed and reopened in the process, and Wz = 30 (corresponding to point E in Fig. 2(b)) is critical value about closing and opening the energy

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gap. Similarly, we also show the surface LDOS of before and after the second energy gap closed (corresponding to point D and F) in Fig. 6. As shown in Fig. 6(a), We find that there are no edge states between the energy gap, which can be characterized by Chern number C = 0. This is a topological trivial phase. However, there exists two pairs of overlapping bright lines and crossing

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gap in Fig. 6(b), which corresponding to the degenerate edge states for the C nontrivial QAHI. This can be characterized by nonzero Chern number C = 2. The topological phase changes from topological trivial insulator to topological

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nontrivial QAHI when the energy gap is from closing (corresponding to point E in Fig. 2(b)) to reopening (corresponding to point F in Fig. 2(b)). The above calculations are very sufficient to prove that the oscillation behavior of the energy gap with film thickness is interesting in the WSM. And the topological phase has a transformation between topological trivial and QAHI

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with variation of film thickness along z direction.

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4. Conclusion This work expounds the transition of topological phase owing to variation

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of film thickness along z direcion in a WSM. We all know that the WSM is a 3D topological insulator with the opposite chirality Weyl points, which can be obtained by the Dirac point splitting in the case of destroying T or P. We find that the WSM has similar characteristics to the DSM in the aspect of the topological phase transition caused by thickness. In this work, the QAHI can be realized in the WSM based on a simple cubic lattice by changing film thickness.

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On the basis of system with broken T , we find that the quantum confinement effect is strongly anisotropic in different directions of system. And we show the oscillation behavior of the energy gap with film thickness along z direction in the WSM, which is periodic of 21 layers. Due to confinement effect, the topological phase will be transformed from topological trivial insulator into QAHI. In order

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to better exhibit the interesting physical phenomenon, we calculate the energy spectrum and their corresponding the surface LDOS before and after the topological phase transition. And the change of the number of edge states can be used to characterize the transformation of topological phase. The confinement effect owing to the z-direction film thickness can provide a good platform to

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design topological quantum devices based on WSM in the near future.

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Acknowledgement

The authors thank P.-T. Xiao for helpful discussions. This work was financially supported by the NSFC under Grants No. 11574019, and the Fundamental Research Funds for the Central Universities.

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Figure 1: (color online) (a) Schematic diagram shows the locations of two

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Weyl points in Brillouin zone. The two points are located along kz -axis (at k= (0, 0, ±bz /(2λz ))) and symmetric about Γ point. (b), (c) and (d) The energy spectrum plotted as a function of kx , ky and kz . The energy spectrum as a function of kz around the two Weyl points is enlarged in upper right in (d). The first and the second sub-bands in the bulk are labeled with red, green colors.

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The parameters are described in the main text.

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Figure 2: (color online) The case of confinement along y− and z−direction. (a) Confinement induced gap versus the confinement thickness Wy . (b) Confinement induced gap versus the confinement thickness Wz , where A, B, C, D, E, and F represents different film thickness along z direction about before and after the

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first and second energy gap closing.

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Figure 3: (color online) (a), (b), and (c) Energy spectra corresponding to point A(Wz = 7), B(Wz = 9), and C(Wz = 11) as marked in Fig. 2(b) are calculated

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along kx (ky = 0). The first, the second and the third sub-bands are labeled with red, green, and blue colors respectively.

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Figure 4: (color online) (a) and (b) The LDOS for the side surface corresponding to point A(QAHI) and C(trivial insulator) as marked in Fig. 2 are calculated.

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The calculation is for a slab which is semi-infinite along y direction.

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Figure 5: (color online) (a), (b), and (c) Energy spectra corresponding to point D(Wz = 28), E(Wz = 30), and F(Wz = 32) as marked in Fig. 2 are calculated along kx (ky = 0). The energy gap closing and reopening is shown. The first,

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the second and the third sub-bands are labeled with red, green, and blue colors respectively.

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Figure 6: (color online) (a) and (b) The LDOS for the side surface corresponding to point D(trivial insulator) and F(QAHI) as marked in Fig. 2 are calculated.

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The calculation is for a slab which is semi-infinite along y direction.

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1: On the basis of breaking the system’s time inversion symmetry, the energy spectrum of Weyl semimetal is isotropic in kx and ky direction but anisotropic in kz direction. 2: The quantum confinement effect is strongly anisotropic in different direction, and the energy gap with film thickness along z direction is regularly periodic oscillation.

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3: The topological phase transition is happened between a topological trivial insulator and a quantum anomalous Hall insulator due to confinement effect.