Topological phase transition of single-crystal Bi based on empirical tight-binding calculations

The main framework used to calculate the bulk electronic structure was the same as that in [15] , and is briefly explained herein. Single-crystal Bi has an A17 rhombohedral lattice, but is also characterized by hexagonal lattice parameters, a and c, together with an additional parameter μ (see figure 2). The primitive (rhombohedral) unit cell contains two atoms and the relative position of the second atom is (0, 0, $2\mu c$), where μ = 0.2341. The hopping parameters of the sp³ orbitals between first-, second- and third-nearest-neighbor atoms were taken, and the SOI was included via the spin–orbit coupling parameter λ = 1.5 eV. The resulting (16 × 16) matrix is shown in the appendix of [15] .

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Next, the TB parameters were modified so that the calculated surface-state dispersion between $\bar{{\rm{\Gamma }}}$ and $\bar{M}$ could be reproduced without any artificial crossing such as was obtained in [16] . In addition, the new parameters were tuned to maintain the energies of the electron/hole pockets and the size of the bandgap at L at nearly the same value as the original parameter, which agrees with the experimental values. Table 1 represents two sets of the new TB parameters, labeled as TBP-1 and TBP-2 in the following, obtained by the methods above. Table 2 represents the parity invariants at each TRIM in the Brillouin zone of the bulk Bi crystal, calculated with the parameters given in table 1. As shown by the ${\nu }_{0}$ values in table 2, the major difference between TBP-1 and TBP-2 is the difference of topological order: trivial for TBP-1 but non-trivial for TBP-2. For such parameter tuning, we had to modify some parameters to a large extent: e.g. the sign of ${V}_{{sp}\sigma }^{\prime }$ is inverted from the original parameter. While it might appear strange, we do not focus in detail to the physical meaning of such parameters in this work, since they are merely an empirical set to reproduce the experimental results. Some 'not natural' values might be due to the imperfect modeling constructed only by the sp³ orbitals up to third-nearest-neighbor hoppings.

Table 1.  The new sets of tight-binding (TB) parameters for single-crystal Bi tuned so that the they could reproduce the surface states which agree with the experimental results. The definitions of each parameter are the same as in [15].

Parameter	Reference [15]	TBP-1	TBP-2
(eV)	(trivial)	(trivial)	(non-trivial)
Es	− 10.906	−10.906	−10.710
Ep	−0.486	−0.336	−0.366
${V}_{{ss}\sigma }$	−0.608	−2.860	−2.860
${V}_{{sp}\sigma }$	1.320	1.308	1.340
${V}_{{pp}\sigma }$	1.854	1.855	1.844
${V}_{{pp}\pi }$	−0.600	−0.600	−0.600
${V}_{{ss}\sigma }^{\prime }$	−0.384	−0.384	−0.384
${V}_{{sp}\sigma }^{\prime }$	0.433	−0.100	−0.050
${V}_{{pp}\sigma }^{\prime }$	1.396	1.396	1.382
${V}_{{pp}\pi }^{\prime }$	−0.344	−0.344	−0.344
${V}_{{ss}\sigma }^{\prime\prime }$	0	0	0
${V}_{{sp}\sigma }^{\prime\prime }$	0	0	0.300
${V}_{{pp}\sigma }^{\prime\prime }$	0.156	0.156	0.156
${V}_{{pp}\pi }^{\prime\prime }$	0	−0.050	−0.040

Table 2.  Parity invariants (δ) at each time-reversal-invariant momentum (TRIM; ${\rm{\Gamma }},L,X,T$ ) and the Z₂ topological invariants (${\nu }_{0};{\nu }_{1}{\nu }_{2}{\nu }_{3}$) calculated with the tight-binding parameters shown in table 1.

	δ(Γ)	δ(L)	δ(X)	δ(T)	(${\nu }_{0};{\nu }_{1}{\nu }_{2}{\nu }_{3}$)
Reference [15]	− 1	−1	−1	−1	(0;000)
TBP-1	−1	−1	−1	−1	(0;000)
TBP-2	−1	+1	−1	−1	(1;111)

The TM method is used to calculate the surface electronic states on a semi-infinite crystal from the bulk Hamiltonian and the TM $T({k}_{\parallel },E)$, as proposed in [22] . In this TM, k_∥ is the in-plane wavevector and E is the binding energy . The procedure reported in a previous paper [16] was followed, as described below, but the bulk TB parameters were changed.

Because the Bi crystal can be regarded as a stack of BLs, as depicted in figure 2(b), the bulk electronic states of a semi-infinite Bi crystal can be written as

$$\begin{eqnarray}&&\left(\begin{array}{c}{\phi }_{n+\mathrm{1,1}}\\ {\phi }_{n+\mathrm{1,2}}\end{array}\right)=T({k}_{\parallel },E)\left(\begin{array}{c}{\phi }_{n,1}\\ {\phi }_{n,2}\end{array}\right),\end{eqnarray} \tag{ 1 }$$

where ${\phi }_{n,a}$ is a basis of the states in the BL plane localized on the a = 1, 2 monolayer of the nth BL. In addition, each ${\phi }_{n,a}$ has eight components associated with the eight atomic orbitals. The TM, $T({k}_{\parallel },E)$, is given by equations (3.1)–(3.3) in [16] , together with the appendix in [15] . Any bulk states are the eigenstates of the 16 × 16 TM with unimodular eigenvalues. For each E in the projected bulk bandgap, $T({k}_{\parallel },E)$ has eight eigenvalues with moduli larger than 1, which correspond to the electronic states whose amplitude decays in the $-z$ direction. With the boundary condition ${\phi }_{\mathrm{0,1}}=0$, the surface states should also decay outside the crystal. These surface states are determined by forming an 8 × 8 matrix $M({k}_{\parallel },E)$ composed of the eight components of ${\phi }_{\mathrm{0,1}}$ for each of the eight decaying states. The detailed procedure to generate $M({k}_{\parallel },E)$ is shown in [22]. Finally, the surface-state band dispersion ($E({k}_{\parallel })$) is determined by solving $\det [M({k}_{\parallel },E)]$ = 0.

Slab geometry is widely used for DFT calculations of surface electronic structures, such as used in [20, 21, 23, 25]. This model can mimic the surface without breaking the three-dimensional periodicity and can therefore be easily applied toward various surface systems. However, sometimes this model generates artificial states owing to the interaction between the top and bottom surfaces. In this work, we followed the method reported in the recent paper [25], as is described below. Varying from the previous work, however, we used a new set of bulk TB parameters and assumed no surface hopping term.

The total Hamiltonian of the slab H is represented as

$$\begin{eqnarray}H=\left(\begin{array}{cccccc}{H}_{11} & {H}_{12}^{(1)} & 0 & 0 & \cdots & 0\\ {H}_{21}^{(1)} & {H}_{11} & {H}_{21}^{(2)} & 0 & \cdots & 0\\ 0 & {H}_{12}^{(2)} & {H}_{11} & {H}_{12}^{(1)} & \cdots & 0\\ \vdots & & \ddots & \ddots & \ddots & \\ 0 & \cdots & 0 & {H}_{12}^{(2)} & {H}_{11} & {H}_{12}^{(1)}\\ 0 & \cdots & 0 & 0 & {H}_{21}^{(1)} & {H}_{11}\end{array}\right),\end{eqnarray} \tag{ 2 }$$

where H₁₁ is the hopping term for inter-monatomic-layer hopping (i.e., in the same monatomic layer) and ${H}_{12}^{(1)}$ (${H}_{21}^{(2)}$) are the intra-BL (inter-BL) hopping terms. The ${H}_{12}^{(1)}$, ${H}_{21}^{(2)}$, and H₁₁ terms are given in the bulk TB Hamiltonian [15] as the first-, second- and third-nearest-neighbor hopping terms, respectively. The size of the matrix is therefore $16n\times$ 16n, where n is the number of the BLs in the slab.

In the following, the obtained states were plotted with Gaussian broadening (FWHM = 5 meV) to mimic the ARPES intensity plots produced with a typical instrumental energy resolution.

Figure 3 plots the surface states and the edge of the projected bulk bands generated using the TM method with three different sets of bulk TB parameters. To plot the surface-state dispersion, we plotted ln(1/det[$M({k}_{\parallel },E)$]), so that the locations $({k}_{\parallel },E)$ where surface states lie possess much smaller intensity values (i.e., darker) than the others .

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The surface bands calculated with the TB parameters given in [15] (figures 3(a) and (b)) agree with those given in the previous paper exhibiting an artificial crossing of the surface bands between $\bar{{\rm{\Gamma }}}$ and $\bar{M}$ owing to an incorrect mirror Chern number [16]. This artificial surface-band crossing disappears when the new TB parameters are used. Using the parameters that generate topologically trivial bulk bands (TBP-1 in table 1), the surface-band dispersion qualitatively agrees with the ARPES results except for the k region around $\bar{M}$, as shown in figures 3(c) and (d). The two branches of the surface bands degenerate with each other at $\bar{M}$, and this surface-state dispersion is therefore topologically trivial, as is expected from the topological order of bulk bands. This surface-state dispersion agrees with that reported in a previous paper [23]. Around $\bar{{\rm{\Gamma }}}$, the TB parameters generating non-trivial bulk bands (TBP-2 in table 1) does not significantly alter the surface-state dispersion from the trivial dispersion, as shown in figure 3(e). Around $\bar{M}$,however, the surface-state dispersion is different from those calculated with the other TB parameters. Specifically, the upper branch merges into the BCB while the lower merges into the BVB, showing a good agreement with the experimental results [17–20] (see figure 3(f)).

These good agreements of surface-state dispersions with the previous experimental and theoretical results except for the proximity of $\bar{M}$ implies that the TB calculation can no longer be the evidence of topological order of single-crystal Bi. In order to judge such balancing two possibilities, one requires the experimental data. Based on the ARPES results, it should be topologically non-trivial, while ARPES does not provide any direct information about the parity eigenvalues of bulk bands at L. The other, bulk-sensitive and accurate experimental method would be helpful to make a firm conclution on this controversial issue, topological order of single-crystal Bi.

In order to examine the topological phase transition driven by structural distortion based on [23, 24, 26], we surveyed the topological order and surface-state dispersion around $\bar{M}$ using the TM calculation and the new TB parameters.

Figure 4(a) shows the bulk band evolution at L with an in-plane lattice distortion inserted for calculations using the two TB parameter sets in table 1, TBP-1 and TBP-2. Irrespective of the topological order generated with zero lattice distortion, a topological phase transition occurs in both cases. The only difference observed is whether the topological phase transition occurs when one uses a lattice strain (TBP-1, trivial at zero lattice distortion) or a lattice expansion (TBP-2, non-trivial at zero lattice distortion). Figures 4(b)–(e) show the surface-state dispersions around $\bar{M}$ obtained using the TM calculation as figure 3. Note that the $\bar{M}$ position of each plot changes according to the in-plane lattice constant. The plots in figures 4(b) and (d) show that the surface-state dispersion is topologically non-trivial with lattice strain (−5%) for both TB parameter sets. In addition, with an in-plane lattice expansion (+5%), the surface states for both TB parameter sets exhibit a trivial dispersion, as seen in figures 4(c) and (e), that form a Kramers-degeneracy point at $\bar{M}$ in the projected bulk bandgap. It should be noted that the non-trivial surface-state dispersion observed by ARPES experiments with the presence of in-plane lattice strain [23, 26] exhibits no conflict with the calculated results from both of the new TB parameters, TBP-1 and TBP-2.

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Based on these results, we propose to trace the bulk bandgap of single-crystal Bi with in-plane lattice distortion in order to conclude the topological order of Bi. If the bandgap closes with the in-plane lattice expantion (strain), it would be the smoking-gun evidence of non-trivial (trivial) topological order.

Figure 5 shows the electronic structure calculated with the slab geometry, where the slab thickness is 20 BL and the dashed lines represent the edge of the projected bulk bands. The surface-state bands dispersing in the projected bulk bandgap are obtained together with the discrete quantum well (QW) states corresponding to the bulk bands in the projected bulk-band region. Around $\bar{{\rm{\Gamma }}}$, the surface-state dispersions are almost identical to those calculated by the TM method (see figures 3(c) and (e)). Note that no additional surface hopping terms are needed to obtain these surface bands, which is in contrast with a previous study [25]. The fact that no additional hopping terms are needed is possibly owing to the different TB parameter set used in this study; however, the surface-state dispersions are quite different around $\bar{M}$. Even when using the TB parameter that generates a trivial topology of bulk bands (TBP-1 in table 1), the surface-state dispersion obtained by the slab geometry suggests a non-trivial topological order wherein the upper branch merges into the BCB while the lower merges into the BVB. Such behavior is the same as that reported in previous studies [20, 25], wherein it has been explained as the influence of the interaction between the top and bottom surfaces owing to the finite slab thickness. Even when using the trivial TB parameter set (TBP-1 in table 1), the inter-surface interaction would re-invert the bulk bands at L, where the bulk bandgap is the smallest, and thus cause the topological phase transition. Similar topological phase transitions driven by the film thickness have been observed in the QWs of HgTe [27, 28].

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To explore the finite-thickness effect in more detail, we calculated the electronic states around $\bar{M}$ at different thicknesses. Figures 6(a) and (b) plot the electronic structure around $\bar{M}$ in a 200 BL slab calculated with the TB parameters TBP-1 and TBP-2, respectively, which are given in table 1. The number of the QW states in these plots are much larger than those of the 20-BL slab (see figure 5), reflecting the increased slab thickness. In the proximity of $\bar{M}$, however, one can find the difference between figures 6(a) and (b). The two surface-state branches dispersing out of the projected bulk bands in figure 6(b) enter the projected bulk bands and become QW states. However, in figure 6(a), these branches do not enter the projected bulk bands but remain in the bulk bandgap at $\bar{M}$, as is the case for the semi-infinite TM calculation (see figure 3(d)). Figure 6(c) plots the energy evolution of these two surface states at $\bar{M}$, which are connected to the surface states away from $\bar{M}$, together with the energy positions of the projected BVB and BCB. As shown in figure 6(c), the surface states calculated with TBP-1 (trivial bulk bands) disperse out of the projected bulk bands at a slab thickness greater than ∼150 BL. In contrast, with TBP-2 (topologically non-trivial bulk bands), the surface states never appear out of the projected bulk bands at $\bar{M}$. This result suggests that a much thicker slab is required to accurately measure the topological order of single-crystal Bi than has been used in previous studies [23, 26]. With a slab that is insufficiently thick, the surface-state dispersion always exhibits the topologically non-trivial behavior irrespective of the topological order of the bulk Bi.

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Recently, two groups have reported ARPES experimental results of the Bi(111) surface states using a thickness above 100 BL [29, 30]. In both cases, the surface-state bands disperse as that seen in figure 6(b), and hence these experimental results indicate the non-trivial topological order of Bi, which is in good agreement with the bulk single-crystal case [17].