Large shift current, $\pi$ Zak phase and unconventional nature in Se and Te

Recently, unconventional materials (or obstructed atomic insulators) have attracted much attention owing to the unconventional feature of mismatch between Wannier centers and atomic positions. In this paper, we demonstrate that the trigonal selenium and tellurium host an unconventional nature in both electronic and phonon spectra. In electronic band structures, the band representation (BR) decomposition for occupied bands has to contain the essential BR of $A@3b$, and the real-space invariant is $\delta_1@3b=-1$. The $\pi$ Zak phase suggests that the one-dimensional Se/Te chain is a chiral Su-Schrieffer-Heeger chain. The effective magnetism can be induced by $p$ states at ends. More importantly, a large shift current is obtained in Se quantum well. In addtion, in phonon spectra, three sets of phonon bands are well separated and assigned to $B@3b$, $B@3a$, and $A@3b$ BRs, respectively. Thus, the obstructed phonon states are predicted on the (0001) surface. As the prototypes of unconventional materials in both electronic and phonon spectra, our findings could create much interest in the study of obstructed surface electronic and phonon states in these novel materials.

Recently, unconventional materials (or obstructed atomic insulators) have attracted much attention owing to the unconventional feature of mismatch between Wannier centers and atomic positions. In this paper, we demonstrate that the trigonal selenium and tellurium host an unconventional nature in both electronic and phonon spectra. In electronic band structures, the band representation (BR) decomposition for occupied bands has to contain the essential BR of A@3b, and the real-space invariant is δ1@3b = −1. The π Zak phase suggests that the one-dimensional Se/Te chain is a chiral Su-Schrieffer-Heeger chain. The effective magnetism can be induced by p states at ends. More importantly, a large shift current is obtained in Se quantum well. In addtion, in phonon spectra, three sets of phonon bands are well separated and assigned to B@3b, B@3a, and A@3b BRs, respectively. Thus, the obstructed phonon states are predicted on the (0001) surface. As the prototypes of unconventional materials in both electronic and phonon spectra, our findings could create much interest in the study of obstructed surface electronic and phonon states in these novel materials.
Introduction. Although topological quantum chemistry (TQC) [1][2][3] and symmetry-indicator classification theory [4][5][6][7][8] have achieved great success in the searching for topological band structures in both nonmagnetic and magnetic crystal materials [9][10][11][12][13][14][15][16][17][18], a significant proportion of materials are topologically trivial, but host the type I unconventional feature of mismatch between the electronic Wannier centers and atomic positions, which are named as unconventional insulators or obstructed atomic insulators (OAIs) [19][20][21][22]. Among them, various properties have been well studied already, such as electrides [19], thermoelectronics [21], catalysis [23], electrochemistry [24], solid-state hydrogen storage [21], ferroelectronics and superconductivity [25][26][27]. Given an OAI with cleavage terminations cutting through an obstructed Wannier charge center, metallic surface states could possibly exist within the bulk gap, which is referred to as the filling anomaly [28][29][30][31]. Such metallic states on the surface/interface are key to many interesting physical and chemical properties of these unconventional materials [21,22]. On the other hand, the concept of unconventionality can be extended to phonon spectra, when phonon modes are well separated in frequency (by a frequency gap) [32]. For an unconventional phonon band structure, the surface phonon modes are expected in the frequency gap. So far, no material has been predicted to be unconventional yet in both electronic and phonon band structures.
In this work, we demonstrate that the trigonal selenium and tellurium are unconventional in both electronic * These authors contributed equally to this work. † zpguo@iphy.ac.cn ‡ wzj@iphy.ac.cn and phonon spectra. The essential band representation is A@3b (3b is an empty site). The real-space invariant (RSI) is computed δ 1 @3b = −1. The π Zak phase suggests that the 1D Se/Te chain is a chiral Su-Schrieffer-Heeger (SSH) chain. The electronic gap can be modified by quantum-well structures, with electrons and holes being at two different interfaces, which yields a large shift current. On the other hand, in their phonon bands, the band representations are assigned to three sets of separated phonon bands by the IR2PH [33] and UnconvMat packages [21,34,35]. Accordingly, the obstructed surface phonon modes are obtained on the (0001) surface. Se and Te are the prototypes of unconventional materials in both electronic and phonon band structures. Crystal structures and aBR analysis. The Se and Te crystals crystallize in the structure of SG 152 (righthanded). The atoms are located the 3a Wyckoff site (WKS ; Table I   for any isolated set of electronic bands, which tells the Wannier center (q) and local orbital character (ρ; labeled by q site symmetry) [19]. The atomic valence-electron BRs (aBRs) are tabulated in Table I. Based on the computed irreducible representations, the aBR decomposition for the occupied bands is solved online [21,34] to be B@3a(p y ) + A@3b, with the essential BR A@3b responsible for charge mismatch. The RSI for the empty 3b site is computed to be π Zak phase and surface states. With the understanding of the essential BR at 3b, we will expect the metallic state on the top surface while no such state on the side surface. Thus, we have performed the (0001)-and (0110)slab calculations. In Fig. 1(f), we clearly see metallic surface states on the (0001) surface. Moreover, we notice that the unconventional nature is closely related to the Zak phase [36]. The Zak phase is defined by the integral of the Berry connection along a reciprocal lattice vector (G). In trigonal Se crystals, the Zak phase is defined by where |u n (k) is the periodic part of the bulk Bloch wave function in the nth band, with the gauge choice |u n (k) = e iG·r |u n (k + G) and k ≡ (k x , k y ). Due to the presence of C 2x symmetry on Γ -K/K -M and TC 2x symmetry on Γ -M, the Zak phase is quantized to be zero or π(mod 2π). The Zak phase along ΓMKΓ is computed, being π. Away from the high-symmetry lines, it would slightly vibrate around π. The π/0 values of the Zak phase correspond to the presence/absence of in-gap boundary states, which are similar to those in the SSH model [37] . The 1D Se chains are chiral SSH chains. As shown for the (0001) slab, polarization charges are equal to e/2 (mod e) at the ends of the 1D insulating chain. As these chiral SSH chains are vdW interacted, they result in metallic surface states, being consistent with ab-initio calculations.
Quantum wells, interfaces and shift current. Due to existence of the π Zak phase, we can generate in-gap states on different (0001) interfaces, to modify the band gap and properties though the quantum-well structures. Here, we twisted two different chiral structures by 60 degrees to form two interfaces 1 and 2 and 180 degrees to form interfaces 3. We have performed the calculations of two quantum wells (QWs). QW-I contains interfaces 1 and 2, while QW-II has identical interface 3. The top and side views of the structures are given in Fig. 9 in Appendix C. The band structures of the quantum wells are shown in Figs. 2(a) and 2(b). In Fig. 2(b), the flat in-gap interface states are also obtained on interface 3. In Fig.  2(a), the interface states are clearly seen in the bulk gap, and their surface weights have been denoted by the size of the dots for interfaces 1 and 2, respectively. We find that the indirect band gap becomes smaller in the QW-I structure. More surprisingly, electron and hole carriers accumulate at different interfaces, which can prohibit the recombination of electrons and holes.
Thus, we have calculated the shift current of this quantum well. In second-order optical response, a shift current J c (ω) can be generated by an injected linear polarized light via where E a (ω) is electrical field alongâ and the strength of the response can be described by the magnitude of second-order photoconductivity σ c ab (ω) [38][39][40]. The results show a large shift current at E( ω) ∼ 1. making the Se quantum well a promising photodetection device. We further analyzed the origin of the strong response. From the electronic band structure, one can easily find that the bands from two different interfaces (1 and 2) with direct band gaps ∼ 1.2 eV in a large area [see Fig. 2(a)]. The optical transition between the interfaces 1 and 2 contributes to the peak value of photoconductivity and can generate a strong in-plane shift current via the tenser elements σ x zz (ω) and σ y zz (ω). The details of the second-order optical conductivity are given in Appendix A.
Magnetism at the ends of the chiral SSH chain. To investigate the magnetic property at the ends of the chiral SSH chain, we induce a vacancy defect on the surface of the (0110) slab and passivate an end by the F atom. The relaxed crystal structure is obtained in Fig.  3(a). The non-spin-polarized and spin-polarized band structures are calculated. In the non-spin-polarized band structure of Fig. 3(b), an isolated state is localized at the end of the chain, being half filling and metallic. The partial charge distribution of this end state is plotted in Fig.  3(a). After considering spin polarization, it becomes insulating with 1µ B at the end with a lower total energy. The magnetism is induced by the unpaired p electron at the end of a SSH chain. The polarized end state is confirmed by the calculations with two (or more) vacancies on a chain to reduce the interaction between the two ends of the chain in the slab structures.
Then we simulate the magnetic ordering on the ideal (0001) termination, which is an arrangement of the ends of the chiral SSH chains. We have performed the calculations for different magnetic configurations in a 2 × 2 supercell. The results are shown in Fig. 3(d). We use the energy mapping method to estimate the nearest-neighbor exchange interaction. Here, we consider an Ising-type model, Because there are three types of nearest-neighbor exchange interactions, the NN J <i,j> can be divided into J a alongâ, J b alongb, and J ab alongâ+b. We consider one ferromagnetic (FM) and three antiferromagnetic (AFM) configurations, as indicated by the arrows in Fig. 3(d).
We find that the FM and AFM-I states converge to paramagnetic states during the self-consistent calculations and the converged magnetic moments in AFM-II and AFM-III states are presented in Fig. 3(d). The converged energies are given in Table II. According to the Ising-type model, the energies of the Se magnetic configurations can be written as, We can solve J b = 15.7 meV. Then, assuming |J a | = |J ab |, we get J a = −J ab = 1. is much less then J b and these J values are consistent with the most stable configuration AFM-III. Unconventional phonon spectra. The phonon modes are originated from the vibration of the atoms, which would induce the atomic vibration band representations as the p-orbitals in the phonon spectrum. Similarly, the phonon spectra with a set of separated phonon bands being not a sum of aBRs are defined as the unconventional phonon spectra [32]. As a result, the obstructed phonon modes emerge on terminations of finite-size samples. The phonon spectra are computed and shown in Fig. 4 for Se and Te respectively. We use IR2PH to deal with the phonon modes from the DFPT calculations and obtain the irreps [33]. Through tqc.data, the BR decomposition for each set of phonon bands is solved online [34], as shown in Figs. 4(a) and 4(c). As expected, the obstructed phonon states are obtained in Figs. 4(b) and 4(d). Based on the full list of symmetry-protected Weyl phonons in Ref. [50], the depicted A and H points are double and monopole Weyl points, respectively. The arc states of Weyl points are shown in Fig. 10 in Appendix D.
Discussion. In summary, we present that the trigonal Se and Te are unconventional elements in both electronic and phonon spectra. The essential BR of the occupied electronic bands is B@3b and it is characterized by a nonzero RSI δ@3b = −1. Since 3b site is the center of bonds in the 1D chiral Se/Te chain in the z direction (π Zak phase), the metallic in-gap states are obtained on the (0001) surfaces and interfaces. Noting that the topological property of the Se/Te chain was investigated very recently [51]. As semiconductors, the band gap can be modified by the quantum-well structure with electrons and holes being at two different interfaces, which benefits to the application of the solar cells. A super-high shift current is obtained in the quantum well. Additionally, based on the BR decomposition of the phonon bands, we conclude that the phonon bands are unconventional and the obstructed surface phonon modes are obtained on the (0001) surface/interface. As the prototypes of a novel kind of materials with both electronic and phonon unconventional band structures, our study can create future interest in the study of surface charge and phonons in these kind of materials. The same band topology, we mainly focus on the discussion of Se in the main text hereafter.
It is physically different from that of the p-n junction mechanism. In bulk optical response, the linear response can be understood as J a (ω) = σ ab (ω)E b (ω), where σ ab (ω) is the first-order optical conductivity. A current J a (ω) along the a direction can be generated by the applied optical-electrical filed E b (ω) along the b direction. One can see that J a (ω) and E b (ω) are with the same frequency; therefore, the bulk photovoltaic effect is forbidden in the first-order response and dominated by the second-order response. The bulk photovoltaic effect can be considered as a joint effect of the frequencydependent density of states and the wave function.
We calculate the second-order optical conductivity by  where n and m are band indices, f nm = f n − f m is the Fermi-Dirac distribution difference, r a mn = iA a mn (1−δ nm ) is the interband Berry connection, r a;b nm = ∂r a nm ∂k b the term relates to the mediate virtual states, and E nm = E n −E m is the energy difference. We deal with the delta function δ( ω − E mn ) by a Gauss distribution with a smearing factor of 0.01eV.

B. ORBITAL-RESOLVED BAND STRUCTURES AND TB MODEL
The orientations of the defined local orbitals at each Se atoms according to 3b site symmetry are presented in Fig. 6(d). The orbital-resolved band with respect to this orbital setup are shown in Figs. 6(a)-(c), from which we can find that the defined p y orbitals are fully occupied forming the aBR B@3a.
Though the Slater-Koster method [59], the band structures can be well reproduced by a p−orbital tight-binding (TB) model with two kinds of bonds: the nearestneighbor (NN) bonds and the next-nearest-neighbor (NNN) bonds in Fig. 7. Their ppσ and ppπ parameters are given in Table III. The band structures of the TB model are given in in Fig. 7.  The crystal structures of the quantum wells of trigonal Se. QW-I has two interfaces (1 and 2) and QW-II has two identical interfaces (3).

D. WEYL POINTS AND SURFACE STATES IN PHONON SPECTRUM
The Weyl points between 8th and 9th phonon bands in the phonon spectrum of Se are listed in Table IV. The corresponding surface/arc states are obtained in Fig. 10 for (0001) surface.