Bismuth antiphase domain wall: A three-dimensional manifestation of the Su-Schrieffer-Heeger model

Polyacetylene, Heeger (2001) (CH)_x, is an infinite one-dimensional (1D) carbon chain whose trans configuration has two degenerate dimerized structures consisting of alternating double and single bonds which can be interchanged by symmetry. Interestingly, polyacetylene exhibits finite electric conductivity even though its intrinsic band structure is insulating. This can be understood in terms of the migration of electrically-charged antiphase domain walls (DWs) between two structures (domains) with opposite dimerization as illustrated in Fig. 1 (a-c). The Su-Schrieffer-Heeger (SSH) model, Su et al. (1979, 1980) introduced to describe polyacetylene, yields a transition from a trivial to topological non-trivial phase depending on the relative hopping amplitudes between the two distinct types of bondings, where the so-called “winding number” undergoes a discontinuous change from $0\rightarrow 1$. The “winding number” is closely related to the Zak phase Zak (1989) which is quantized to be 0 or $\pi$ for systems with space inversion symmetry. Moreover, the DW in the SSH model leads to the emergence of a boundary localized zero-energy mode in the middle of the energy gap with charge accumulation of $\pm e/2$, analogous to the fractionally charged excitations in quantum field theory. This midgap state is understood as a topologically protected boundary mode and the SSH model serves as a paradigmatic example of topological insulator protected by a chiral (i.e., sublattice) symmetry.

The SSH model is the simplest and one of the most important models in describing band topology in condensed matter physics, and has been the subject of intense investigations such as Majorana zero mode in a finite atomic chain Kitaev (2001); Yu et al. (2020) and an extension to two-dimensional (2D) systems, including grapheneDelplace et al. (2011) and four-basis-Liu and Wakabayashi (2017); Obana et al. (2019) and two-basis-Zhu et al. (2019) square-lattice models. The latter studyZhu et al. (2019) explicitly characterized several topological phases with distinct winding numbers upon uniaxial strain and sublattice dimerization where zero-energy flat bands were predicted to emerge on 1D antiphase DW if the winding numbers (equivalently the Zak phases) of the two facing domains are different. This is the 2D analogue of the SSH model.

In three-dimensional (3D) systems, non-trivial $\pi$ Zak phases have drawn less attention and only a few systems have been found to exhibit them. Hirayama et al. (2018); Aihara et al. (2020) Sc₂C, a designed inorganic electride, Zhang et al. (2017) was predicted to exhibit a $\pi$ Zak phase with consequent surface states inside its insulating band gap.Hirayama et al. (2018) Surface drum-head states of topological nodal-line semimetals are also known to originate from the $\pi$ Zak phase, Ryu and Hatsugai (2002); Burkov et al. (2011); Fang et al. (2016) where the drum-head states are bounded by surface-projected bulk nodal lines, in contrast to the $\pi$ Zak phase insulator. For example, Sc₂C (Y₂C) is a $\pi$ Zak phase insulator (topological nodal-line semimetal) where the surface states cover 100% (90.4%) of the surface BZ. Hirayama et al. (2018) The $\{111\}$ surfaces of silicon and diamond host surface bands from the $\pi$ Zak phase, Aihara et al. (2020) where each surface unit cell accumulates one half of an electron Vanderbilt and King-Smith (1993) leading to half-filled metallic surface bands. An insulating surface can be achieved only by even-number (such as $2\times 2$) surface reconstructions that allow an integer number of surface electrons and hence fully filled bands. Aihara et al. (2020); Smeu et al. (2012) The 3D $\pi$ Zak phase systems that have been reported so far involve no atomic displacement that can be described as a dimerization. In this work, we show that each (111) atomic layer of Bi corresponds to a single site of the 1D SSH model, and dimerization of the atomic layers in the ground state results in $0$ or $\pi$ Zak phases depending on the dimerization sign, $\delta=\pm 1$, as illustrated in Fig. 1(d-e).

The Zak phase Zak (1989) is a special form of the Berry phase Berry (1984) and is equivalent to the electronic part of the polarization, King-Smith and Vanderbilt (1993); Vanderbilt and King-Smith (1993); Resta (1994)

where $-e$ is the electron charge, $c$ is the lattice constant of the unit cell, and $p$ is the dipole moment of the bulk unit cell that can be in turn expressed in terms of the Wannier functions of the occupied bands,

Here, $r_{i}$ is the center of $i$-th Wannier function. In the presence of inversion symmetry, the dipole moment is quantized such that the Zak phase can only take on values $\phi_{Z}=0$ or $\pi$, corresponding to whether the net Wannier center $\bar{r}=\sum_{i}r_{i}$ is located at the center ($\bar{r}=0$) or the boundary ($\bar{r}=c/2$) of the bulk unit cell, respectively (see Fig. 1(b,c)). This definition depends on the choice of inversion center for the placement of the origin, which we assume to have been decided once and for all. The origin-dependent Zak phase has been discussed in detail by introducing the “intercellular Zak phase”.Rhim et al. (2017) The Zak phase can be easily computed from the product of the parity eigenvalues of the occupied bands at time-reversal invariant momenta (TRIM) Fu et al. (2007); Fu and Kane (2007) in the 1D momentum space, where $\phi_{Z}=0$ ($\pi$) corresponds to positive (negative) product as shown in Fig. 1 (b) (Fig. 1 (c)).

In 2D and 3D systems, the Zak phase can be defined on a closed 1D path such as a periodic $k_{z}$ string with a fixed in-plane momentum ($k_{x},k_{y}$) as shown in Fig. 1(d). Under inversion and time-reversal symmetry and in the absence of SOC, an insulating bulk has a constant and quantized Zak phase on such strings normal to a given surface regardless of the specific in-plane momentum ($k_{x},k_{y}$). This implies that a single pair of surface-projected TRIM is enough to determine the Zak phase of the entire surface, $\phi_{Z}(k_{x},k_{y})=\phi_{Z}(0,0)=\{0,\pi\}$. Turning on the SOC, however, allows modulation of the Zak phase which is no longer quantized at generic surface momenta except at the surface TRIM. Because of the strong SOC of Bi, we focus on the four surface TRIM where the Zak phase is quantized to be 0 or $\pi$ (see shaded lines in Fig. 1(d,e) connecting the four pairs of TRIM).

Figure 2 illustrates schematically the boundary states of various topological phases and the corresponding Zak phase configurations at the surface TRIM points. Hirayama et al. Hirayama et al. (2018) demonstrated that the surface states of the 3D $\pi$ Zak phase (Fig. 2(d)) is a full-BZ extension of the drum-head states of a nodal-line semimetal (Fig. 2(b)). This can be understood as a continuous shift of $k_{c}\rightarrow\pi$ that accompanies a band inversion at $k_{y}=\pi$ and switching of the Zak phase $\phi_{Z}(0,\pi)$ from $0$ to $\pi$. In the presence of SOC, the degeneracy of the bulk nodal lines is lifted and the system becomes a strong topological insulator (STI) as shown in Fig. 2(c). Note that the Zak phases $\phi_{Z}(0,0)=\pi$ and $\phi_{Z}(0,\pi)=0$ do not change. In contrast to the STI phase where a robust surface state is guaranteed by the “switch partners” band connectivity between TRIM, Fu and Kane (2007); Fu et al. (2007) the surface state induced by the $\pi$ Zak phase is rather isolated in energy from the valence and conduction bands (Fig. 2(e)). The surface band can be pushed into the valence or conduction bands via surface modifications unless it is protected by a chiral (or particle-hole) symmetry which in turn pins the non-trivial surface state at the Fermi level. Since the Bi $p$ bands are well separated from the lower energy $s$ bands and the inter-sublattice hopping matrix elements ($\sigma_{w,v}$ in Appendix A) are dominant there is an effective chiral symmetry which retains the non-trivial state within the bandgap of the Bi antiphase DW.

In this work we propose that the $\alpha$-phase of bulk Bi in the rhombohedral structure is a 3D analogue of the 1D SSH system. In Sec. II using DFT-parameterized tight-binding model calculations we investigate the topological properties of the two dimerized states of bulk Bi. We find that the dimerization reversal induces parity sign flip at four TRIM (without changing the bulk topology) which in turn induce a transition of the Zak phase from $\pi\rightarrow$ 0, consistent with the emergence of odd or even number of Wannier charge centers (WCCs) at the cell boundary. In Sec. III.1 we consider two types of (111) DWs sandwiched between two oppositely dimerized states and show the emergence of topologically-protected DW localized states, in contrast to the trivial DW states for the (11$\bar{2}$) DWs discussed in Sec. III.2. In Sec. III.3 we derive criteria for the emergence of $\pi$ and $0$ Zak phases for a DW of arbitrary orientation and identify those DW orientations that host non-trivial DW states. Sec. III.4 discusses a plausible experimental realization of the dimerization reversal in pristine Bi and the formation of DW using intense femtosecond laser excitations that can alter the interatomic forces and energy barriers between the two dimerized states. Fritz et al. (2007) Conclusions are summarized in Sec. IV and Sec. V describes the methodology used. Our findings suggest a novel band engineering concept for topologically protected states using antiphase DWs where the parity sign flip can occur without the assistance of strong spin-orbit coupling of heavy ions.

Atomic structure – The $\alpha$-phase of bulk Bi in the rhombohedral structure (space group $R\bar{3}m$, No. $166$) is shown in Fig. 3(a), where the conventional unit cell has a bilayer (BL) structure with an ABC stacking sequence along the $[111]$ direction consisting of three BLs. There is strong covalent bond within each BL (intra-BL bonding), with a Bi atom forming three $\sigma$ bonds with its nearest neighbors, and weak van der Waals bonding between two nearest-neighbor BLs (inter-BL bonding). The intra- and inter-BL sequence of bonds alternate along the $[111]$ stacking direction, which is exactly analogous to the alternating double and single bonds in polyacetylene shown schematically in Fig. 1(b-e).

Furthermore, as shown in Fig. 3(a), the intra- and inter-BL bonds can be interchanged, resulting in two degenerate dimerized ground states with opposite dimerization parameters $\delta\equiv\Delta/\Delta_{0}=\pm 1$. Here, $\Delta$ is the displacement of the two Bi atoms in the primitive cell along $[111]$ [Fig. 3(a)] in units of the lattice vector $c=|\vec{a_{1}}+\vec{a_{2}}+\vec{a_{3}}|$ ($\vec{a_{i}}$, $i=$1-3 are primitive lattice vectors), and $\Delta_{0}$ is the equilibrium displacement. The positively dimerized state can be obtained from the negatively dimerized state via a translation by a half lattice vector, or vice versa. In sharp contrast to 2D and 3D topological orders, the Zak phase is not invariant under such a translation.

Electronic Structure– Fig. 3(b) shows the tight-binding (see Appendix A) band structure with ($\delta=\pm 1$, red lines) and without ($\delta=0$, blue lines) dimerization. The direct band gaps at the TRIM points $L$ and $T$ close at $\delta=0$ where the parity eigenvalues of the states near the Fermi level reverse sign by the dimerization sign reversal, indicating band inversions at these TRIM points.

Parity– The number of negative-parity eigenstates at the TRIM points is listed in Table 1 for the two different dimerization states, $\delta=\pm 1$. The change of parity states upon dimerization reversal is also related to the multiple choices of inversion center. For instance, if one takes $(0,0,0)$ (the diagonal corner of the primitive cell) to be the inversion center instead of $(1/2,1/2,1/2)$ (the center of the primitive cell), the parity of the state changes as if the dimerization is reversed. This is because a structure with reversed dimerization is equivalent to one that is translated by half the cell diagonal.

Topological phases protected by time-reversal or crystalline symmetries should be independent of the choice of inversion center as well as the sign of dimerization. Even though there is a parity sign flip at the $L$ and $T$ points, we show below that the well-known topological phases of bulk Bi are indeed intact under dimerization reversal by calculating the various topological indices: (i) $\nu_{0}$ for STI under time-reversal symmetry, (ii) $\{\nu^{(\pi)}$, $\nu^{(\pm\pi/3)}\}$ for higher order topological insulator (HOTI) under the three-fold rotational symmetry $\hat{C}_{3}$, and (iii) $\{\nu^{(\pi/2)}$, $\nu^{(-\pi/2)}\}$ for crystalline topological insulator (CTI) under the two-fold rotational symmetry $\hat{C}_{2}$.

First, the STI $\mathbb{Z}_{2}$ phase, protected by time-reversal symmetry, is expressed in terms of the parity eigenvalues of the occupied states at the TRIM points as,

where $n^{-}_{\lambda}$ is the number of occupied states with negative parity at the TRIM point $\lambda$. For both dimerized phases, we find $\nu_{0}=0$ corresponding to the trivial phase that is consistent with previous reports. Schindler et al. (2018); Hsu et al. (2019)

Next, the HOTI phase of Bi is verified by grouping occupied states at TRIM points into $\hat{C}_{3}$ symmetry subspace according to the rotation eigenvalues of exp$(i\pi)$ and exp$(\pm i\pi/3)$. Schindler et al. (2018) The fact that each subspace is closed under time-reversal symmetry allows the $\mathbb{Z}_{2}$ classification for each subspace. Among the TRIM points, only $\Gamma$ and $T$ points are invariant under the $\hat{C}_{3}$ rotation. On the other hand, for the remaining ($F_{1,2,3}$ and $L_{1,2,3}$) TRIM which are not invariant under $\hat{C}_{3}$, one can construct linear combination of these three states (which transform into each other under three-fold rotation; $F_{1}\rightarrow F_{2}\rightarrow F_{3}\rightarrow F_{1}$ as well as $L_{i}$) to render them $\hat{C}_{3}$ eigenstates (see Ref.Schindler et al. (2018) for more details). The number of linearly combined states with negative parity for the two subspaces are $n^{-}_{\alpha,\pi}=n^{-}_{\alpha}$ and $n^{-}_{\alpha,\pm\pi/3}=2n^{-}_{\alpha}=0$ (mod 2), where $\alpha\in\{F,L\}$. Thus, the topological invariant for the two subspaces are given by

The dimerization sign reversal changes $n^{-}_{T,\pi}$ and $n^{-}_{L}$ by two, while $\nu^{(\pi)}$ and $\nu^{(\pm\pi/3)}$ do not change under modulo 2. Hence, we confirm that the HOTI phase, $\nu^{(\pi)}=\nu^{(\pm\pi/3)}=1$ is intact under the dimerization reversal.

Finally it was predicted that bismuth is also a first-order CTI protected by a two-fold rotational symmetry $\hat{C}_{2}$ around the [1$\bar{1}$0] axis or its symmetric copies [01$\bar{1}$] and [$\bar{1}$01].Hsu et al. (2019) Similarly, with the classification above, the parity states can be divided into the $\hat{C}_{2}$ subspace according to the symmetry eigenvalues of exp$(i\pi/2)$ and exp$(-i\pi/2)$. In contrast to the HOTI classification, the $\hat{C}_{2}$ subspaces are mapped to each other by time-reversal symmetry, indicating $\nu^{(\pi/2)}=\nu^{(-\pi/2)}$. The four TRIM points $\{\Gamma,T,F_{1},L_{1}\}$ are invariant under $\hat{C}_{2}$. The remaining states at the $F_{2,3}$ and $L_{2,3}$ points, which are not invariant under $\hat{C}_{2}$, can be linearly combined so that they become $\hat{C}_{2}$ eigenstates. The number of negative parity eigenvalues $n^{-}_{F,L}$ contributes equally to $\nu^{(\pi/2)}$ and $\nu^{(-\pi/2)}$ with a weighting factor of one. Thus, the topological indices are given by

Each subspace is found to have a strong topology since $\nu^{(\pi/2)}=\nu^{(-\pi/2)}=5$ (mod 2) and 7 (mod 2) for positive ($\delta=+1$) and negative ($\delta=-1$) dimerizations, respectively. Therefore, the rotational-symmetry-protected CTI phase is well reproduced and is confirmed to be intact under the dimerization reversal.

It is important to note that in contrast to the topological phases that are invariant under dimerization sign reversal, the Zak phase depends on the sign of dimerization (i.e., choice of the unit cell). For example, the $\Gamma$ and $T$ points are projected at the $\tilde{\Gamma}$ point of the (111) surface BZ [Fig. 3(c)] where the Zak phase at $\tilde{\Gamma}$ is determined by the parity eigenvalues,

Here, $\phi_{Z}(\tilde{\Gamma})=\pi$ and $0$ for positive ($\delta=+1$) and negative ($\delta=-1$) dimerization, respectively. Furthermore, the $F$ and $L$ points are projected on the other surface TRIM point $\widetilde{M}$, and the Zak phase at $\widetilde{M}$,

is calculated to be identical to $\phi_{Z}(\tilde{\Gamma})$ for each dimerized state. Note that systems with a strong topological order exhibit different Zak phases at the two surface TRIM points, exp$[i\phi_{Z}(\tilde{\Gamma})+i\phi_{Z}(\widetilde{M})]=-1$, or equivalently $\nu_{0}=1$ from Eq. (4). Fu et al. (2007); Fu and Kane (2007) The right panels in Fig. 3(a) show the (111) surface terminations of the two dimerized states where the surface with low cleavage energy corresponds to the positive dimerization with $\pi$ Zak phase. Surprisingly, the non-trivial phase emerges on the surface that cuts the weak bonds ($\delta=+1$) rather than the strong bonds ($\delta=-1$). This is counter-intuitive, especially when compared to the original 1D SSH model.

To corroborate the parity analysis, the hybrid Wannier charge centers (WCCs) are computed for the two dimerized states in the hexagonal structure having six Bi atoms (18 valence electrons) as shown in Fig. 4. Because of the inversion and time-reversal symmetries, the WCCs are mapped to symmetric copies as $r_{i}(\mathbf{k})\rightarrow-r_{i}(\mathbf{k})$ and $r_{i}(\mathbf{k})\rightarrow r_{i}(\mathbf{-k})$, respectively. For $\delta=+1$, there are two WCCs crossing the cell boundary $z/c=\pm 0.5$ at $\widetilde{\Gamma}$ ($\widetilde{M}$) which is equal to the negative parity difference, $|n^{-}_{\Gamma}-n^{-}_{T}|=2$ ($|n^{-}_{F}-n^{-}_{L}|=2$). Similarly, for $\delta=-1$, four (zero) WCCs cross the cell boundary at $\widetilde{\Gamma}$ ($\widetilde{M}$), which also agrees well with the difference of negative parity states. The factor 2 from the spin degeneracy can be decomposed by grouping the WCCs based on the mirror eigenvalues $\pm i$ on the $\widetilde{\Gamma}$-$\widetilde{M}$ plane (see Fig.4). The WCCs with mirror eigenvalues $-i$ and $+i$ are denoted by blue and red lines, respectively. It is clearly seen that the number of boundary-crossing WCCs in each subspace is reduced by half. For example, a single blue line passes the boundary in Fig. 4(a). It agrees well with the mirror-symmetry-classified parity states in Table 1 where the negative parity states are divided in half ($n^{-}_{\lambda,\pm i}=(1/2)n^{-}_{\lambda}$) as well as their difference. Note that the number of negative parity states only provides an upper limit on the number of WCCs crossing the boundary, with the exact number of crossings being determined by the symmetry protection. Alexandradinata et al. (2014) The net Wannier center $\bar{r}=\sum_{i}r_{i}$ is shown in Fig. 4(d), where the half polarization of the $\pi$ Zak phase $(\delta=+1)$ is clearly seen at the two TRIM, consistent with the parity results.

The corresponding boundary states of the $\pi$ Zak phase can be realized on the (111) surface by appropriate choice of the surface termination that is usually hard to control. Fortunately, Bi is found to exhibit the $\pi$ Zak phase on the low-cleavage-energy surface. In general, however, the surface with $\pi$ Zak phase is susceptible to reconstruction and contamination. Aihara et al. (2020) Thus, instead of the bare surface, we consider antiphase DWs across which the sign of dimerization is reversed, $\delta=\pm 1\rightarrow\mp 1$, as shown in Fig. 5 and Fig. 7. The Bi (111) DWs, hosting the $\pi$ Zak phase, is indeed the 3D analogue of the 1D SSH model. The DW is tolerant to chemical contamination and can easily be found in a system exhibiting charge density wave.

In order to study the DW state without the interference from the neighbor DW, we use the interface Green’s function method Sancho et al. (1985); Ozaki et al. (2010) where the central DW structure is sandwiched between two semi-infinite pristine Bi with opposite dimerizations, as is shown in Fig. 5(b,c) for the (111) DW and Fig. 7(c,d) for the (11$\bar{2}$) DW. The construction of the Hamiltonian matrices is described in Appendix B. Throughout the remaining manuscript, the tilde ($\sim$) and bar ($-$) symbols over the $k$-point labels denote TRIM points on the (111) and (11$\bar{2}$) DW BZ, respectively.

We propose that the $\alpha$-phase of bulk Bi is a 3D manifestation of the SSH model. We demonstrate that while the HOTI and CTI phases of bulk Bi remain invariant under dimerization sign reversal, the Zak phase undergoes a transition from $\pi$ to 0. The (111) antiphase DW is found to host metallic DW bands, which are topologically protected due to the difference in polarization between the two oppositely dimerized domains (i.e., $\pi$ Zak phase), which is the 3D analogue of the SSH model. Although the (11$\bar{2}$) DW has no such polarization difference, the DW localized states exhibit interesting behavior related to an effective symmetry that reveals itself in thicker DWs.

To the best of our knowledge, this result is the first demonstration of the non-trivial Zak phase in 3D antiphase DWs. Unlike the bare surface being vulnerable to doping, contamination, or reconstruction, antiphase DWs offer a relatively stable platform for the manifestation of a non-trivial Zak phase. Furthermore, the common presence of DWs in charge-density-wave states offers a novel venue for investigating the potential of the non-trivial Zak phase.

The tight-binding parameters are extracted from the Wannier Hamiltonian obtained by using VASP-Wannier90 interface.Kresse and Furthmüller (1996a, b); Mostofi et al. (2014) The pseudopotentials are of the projector-augmented-wave type as implemented in VASP,Blöchl (1994); Kresse and Joubert (1999) with valence configurations 6$s^{2}$6$p^{3}$ for Bi. The exchange-correlation functional is described by the Perdew-Burke-Ernzerhof generalized gradient approximation (PBE). Perdew et al. (1996) The plane-wave cut-off energy is set to $300$ eV and the Brillouin zone sampling grid is 12 $\times$ 12 $\times$12. The structure is relaxed with a constraint of being FCC for an insulating band gap. The twelve strongest hopping terms are then used in the calculation together with atomic spin-orbit coupling for the $p$ orbitals. The spectral density of DW-localized states is calculated using the interface Green’s function method Sancho et al. (1985); Ozaki et al. (2010) where two semi-infinite surface Green’s functions are first calculated for the two dimerized phases and then combined with the central DW Hamiltonian.