Taek Jung Kim,Min Yong Jeong,Myung Joon Han
First principles investigation of screened Coulomb interaction and electronic structure of low-temperature phase TaS2
Condensed matter physics,Magnetism,Nanomaterials
Summary By means of ab initio computation schemes, we examine the electronic screening, Coulomb interaction strength, and the electronic structure of a quantum spin liquid candidate monolayer TaS2 in its low-temperature commensurate charge-density-wave phase. Not only local () but non-local () correlations are estimated within random phase approximation based on two different screening models. Using GW + EDMFT (GW plus extended dynamical mean-field theory) method, we investigate the detailed electronic structure by increasing the level of non-local approximation from DMFT () to EDMFT and GW + EDMFT.
First principles investigation of screened Coulomb interaction and electronic structure of low-temperature phase TaS2

By means of ab initio computation schemes, we examine the electronic screening, Coulomb interaction strength, and the electronic structure of a quantum spin liquid candidate monolayer TaS2 in its low-temperature commensurate charge-density-wave phase. Not only local () but non-local () correlations are estimated within random phase approximation based on two different screening models. Using GW + EDMFT (GW plus extended dynamical mean-field theory) method, we investigate the detailed electronic structure by increasing the level of non-local approximation from DMFT () to EDMFT and GW + EDMFT.

Understanding the effect of Coulombic interaction between electrons in solid has long been a central theme of condensed matter physics. Notably, the relevant material systems are being extended to include various two-dimensional (2D) van der Waals materials,,,,,,, and more recently their twisted combinations.,,,,,,,, The local on-site interaction (conventionally represented by in Hubbard model) can induce electron localization, magnetic moment and thereby leading to metal-to-insulator and magnetic phase transition.,,, Many of most intriguing phenomena have been explored within this picture such as unconventional superconductivity and quantum spin liquid.,,,,, Non-local Coulomb interaction (denoted by in extended Hubbard model) also plays an important role. For example, it can enhance the itinerancy of electrons by screening Coulomb interactions or broadening the effective bandwidth., Out of its competition and cooperation with electron hoppings and local correlations, various instabilities can be induced toward the charge-ordered or charge density wave (CDW) phase and many others including high angular momentum superconductivity.,,,,,,,,,, Recently, the effect of non-local Coulomb interaction receives increasing attention in 2D material.,,,, Among many correlated van der Waals materials, TaS2 provides an intriguing case by displaying multiple ordered phases and their transitions.,,, At high temperature, TaS2 is known to be metallic and has 1T structure shown in Figure 1A. As temperature is lowered, it exhibits so-called ‘nearly commensurate CDW (NCCDW)’, and then finally becomes insulating., Superconductivity is also observed below K by applying pressure, doping or electric field.,, Importantly, the stabilization of the low temperature insulating phase is accompanied by commensurate CDW (CCDW) transition for which the long-range ‘star of David (SOD)’ pattern of atomic rearrangement is well identified in both experiments and simulations (see Figures 2A and 2B).,,,, This phase hosts a flat band in the close vicinity of chemical potential because of the reduced hopping integral in between SOD molecular orbitals. Therefore, unveiling the electronic property of CCDW SOD phase is important to understand the metal-insulator transition and the other related phases observed or suggested in this material at low temperature.,,,,,, Given that Ta atom has the 4+ formal valance, the SOD unit cell should have odd number of electrons (i.e., ), and CCDW-TaS2 is expected to be metallic according to the standard band picture. It is however in a sharp contrast to experiments. Several recent studies focusing on this issue suggested that either Mott mechanism or interlayer dimerization is responsible for metal-to-insulator transition.,,,,,,, The important challenge here is to estimate the strengths of Coulomb interactions. Another intriguing possibility in this material is quantum spin liquid (QSL). By noting that the low-temperature CCDW phase forms a triangular lattice and carries nominally S = ½ local moment, the low temperature TaS2 was suggested as a QSL candidate. This idea is supported by the absence of long-range magnetic order down to low temperature and the gapless spin excitations.,, Notably, a recent study highlights the effect of non-local Coulomb interaction () on the magnetic nature related to QSL. In this paper, we examine the detailed electronic behaviors of monolayer TaS2. Based on two different models, the electronic screenings are taken into account within random phase approximation (RPA). Correlation strengths, in terms of both on-site and inter-site , are successfully quantified for the low-temperature CCDW phase. By adopting GW + EDMFT,,,,,,, we calculate the electronic structure with three different levels of approximation for inter-site correlation. Although the Mott gap is mainly governed by the local interaction, non-local Coulomb interaction is also sizable. Its effect on the electronic screening, band dispersion and another possibility toward charge-ordered phase are discussed.

Figures 1A and 1B shows the crystal structure of 1T-TaS2 and the local TaS6 unit, respectively. The corresponding non-interacting band structure within generalized gradient approximation (GGA; i.e., eV) is presented in Figure 1C where different colors represent the different orbital characters as projected onto the local coordinates as defined in Figure 1B. Ta- states dominate the near Fermi energy () region and the configuration is well identified. The higher energy bands above eV and below eV are mainly composed of Ta- and S- character, respectively, whereas the greenish bands reflect the mixture of Ta- and S-. The overall electronic feature is in good agreement with previous studies.,,, By lowering temperature, 1T-TaS2 is known to undergo CDW structural transitions. So-called nearly commensurate charge density wave (NCCDW) phase is stabilized below K and followed by CCDW at K.,, Figure 2A depicts the CCDW structure, so-called SOD pattern, in which twelve Ta atoms move toward the central one forming a 13-Ta-atom unit cluster.,,, In this phase, the band structure also changes accordingly.,,, As shown in Figure 2C, a well-separated single electron band is solely responsible for the low energy region near whereas six bonding orbital states are fully occupied by twelve Ta-d electrons well below . It is noted that the near band is fairly flat and nondispersive. It is therefore prone to metal-to-insulator transition and other exotic possibilities such as quantum spin liquid phase when interactions come in to play.,,, The green-colored line in Figure 2C shows the band produced by maximally localized Wannier function (MLWF), based on which we estimated a reliable set of parameters for the low energy effective (non-interacting) Hamiltonian of CCDW-TaS2. Its real space visualization is given in Figure 2B. The calculated TB parameters are as follow: the first and the second neighbor hopping is and meV, respectively, well compared with the previous reports., It is interesting to note that the third neighbor hopping meV is larger than and comparable with .

Now we try to calculate the interaction parameters from first-principles. Quantitative information of correlation strength is often crucial for exploring the possible exotic quantum phases and understanding their properties. In the debate over the band gap origin of TaS2,,,,,,,, for example, the size of on-site is of key importance. Non-local is of particular interest in the systems close to the CDW instability as it can induce charge-ordering instability in the extended Hubbard systems.,, In general, the constructed phase diagrams can be qualitatively different depending on the non-local term. As such, there have been several attempts to directly estimate the interaction strengths of various transition metal dichalcogenides (TMDs).,,,,,, However, ab initio calculation of these parameters corresponding to CDW phase is challenging especially for non-local . We calculate both and for CCDW-TaS2 based RPA/cRPA method. Ideally, it is desirable to compute these parameters from the direct cRPA calculation of CCDW unitcell. However, a difficulty arises from the demanding computation cost in dealing with the large cell size. Here we extract the interaction parameters from the undistorted 1T-structure,, and consider two different approaches. First, the CCDW value can be estimated as the average of atomic Coulomb interactions of 1T structure whose screening process can be considered within RPA. This first way of computation gives rise to:where is the Fourier transformed leading eigenvalue of RPA-screened Coulomb matrix. Namely, where and is the eigenvalue of RPA-screened Coulomb matrix represented in the eigenbasis of bare Coulomb matrix. The screened Coulomb matrix is calculated for manifold; namely, the red-colored bands in Figure 1C. and are the lattice cell vectors corresponding to 1T structure, and is the lattice momentum. is the grid number of lattice momentum. The white-star symbol represents the indices of Ta atoms in the same SOD (which is composed of 13 Ta atoms; see Figure 2A). In this first approach, the inter-site correlation is also computed within the same spirit:where the black-star symbol refers to the indices of Ta atoms belonging to the nearest-neighboring SOD (indicated by the unit cell vector and in Figure 2A). The calculated and following this first approach are presented in Table 1; referred to as ‘3 bands (RPA)’. The second approach considers the weights of Ta atoms by projecting them onto the CCDW molecular orbital. This idea itself is not necessarily combined with (c)RPA but can go with any other method of computing atomic Coulomb interaction . The molecular orbital then can be estimated from the atomic Ta-: By calculating the projected weights of Ta- orbitals () with respect to CCDW molecular orbitals (),where is the atomic Ta index. Ta atoms are classified into three different types considering their symmetry (see the inset of Figure 2A). Our calculations give rise to and 0.035 for Ta-I, Ta-II, and Ta-III, respectively. and are obtained as follow: Here we calculate based on cRPA. In Table 1, this set of results are denoted by ‘5 bands (cRPA)’ as 5-band model of 1T-TaS2 (both and ) is adopted. The calculation results are summarized in Table 1. The values are different in two different approaches: eV from ‘3-band RPA’ and 0.653 eV from ‘5-band cRPA’. It is not surprising but rather expected because different ‘models’ shall take care of different screenings. For example, the weaker correlation in the former model is naturally attributed to the more screening channels taken into account. Both results therefore carry useful information for further study whereas the larger supercell of CCDW phase is not directly computable within cRPA. It is also noted that our results are considerably larger than that of a previous study based on linear response calculation; eV. From the point of view of the extracting method of CCDW- value from 1T structure, the previous study by Darancet et al. is the same with our second approach (i.e., ‘five-band model’). Thus the ∼0.5 eV difference certainly reflects the difference between the linear response adopted by Darancet et al. and our RPA method whereas only the latter takes the electronic screenings into account directly. Interestingly, however, the calculated atomic itself (that is, corresponding to Ta- on-site interaction) is larger in the linear response calculation, eV, than our cRPA eV. It is therefore indicative of that the difference is not simply attributed to the difference between cRPA and linear response. We presume that the projection also plays an important role. As shown in Equation 4, our projection method not just takes all of Equation 3 portions into account but it also includes some inter-site contributions between different Ta atoms in an SOD molecular orbital. It is also useful to compare the correlation strength of TaS2 to that of NbSe2. A previous study of Kamil et al. reports eV for NbSe2 and their estimation was based on RPA and three-band model, corresponding to our first approach. It is noted that for CCDW-TaS2 is significantly larger than CCDW-NbSe2 by about 0.07 eV (see Table 1), which is likely attributed to the different screenings as discussed in the previous studies on the comparison of TaSe2 with TaS2., Here we also note that a recent ARPES study suggests the weaker correlation for NbSe2 than TaSe2. Table 1 also presents the inter-site correlation strength which can be important for understanding the possible exotic phases relevant to this system, but has not been quantitatively estimated before. First of all, sizable value is noticed; ∼50% of on-site correlation in both models. It can therefore be important to take this effect into account within the extended Hubbard model. It is obvious that Hubbard-type correlation is indispensable for overcoming the DFT-GGA limitation which gives rise to metallic band structure. Previous theoretical studies showed that the inclusion of on-site correlation within DFT+ and single-site DMFT indeed well reproduces the insulating state even though the former incorrectly predicts the magnetically ordered ground state., According to the DMFT calculation, Mott transition starts to occur at ∼0.7 eV in bulk TaS2. Note that both sets of our results in Table 1 provide the large enough interaction with respect to the bandwidth eV. Importantly, however, the effect of non-local has remained unexplored.

To see the effect of both local and non-local on the electronic structure, one needs to go beyond the single-site DMFT. Here we adopt GW + EDMFT for that purpose. Computation details are given in STAR Methods. In Figures 3A–3C, we present the calculated spectral functions as obtained by analytic continuation of Matsubara Green function using maximum entropy method.,, In sharp contrast to the metallic DFT-GGA band structure, GW + EDMFT result clearly shows the insulating gap being consistent with experiment, DFT+ and single-site DMFT. In the case of DFT+ , the long-range spin order naturally comes in as the ground state configuration. With our first set of interaction parameters ( eV, eV; shown in Figures 3A and 3B), the upper and the lower Hubbard peak is observed at about 0.2 eV, respectively, which is well compared with a recent STM experiment of monolayer TaS2., Another set of 5-band model produces a slightly larger gap as presented in Figure 3C. Although this stronger correlation set of parameters also reproduces the insulting phase as expected, the notable difference is the position of Hubbard bands which are in this case found inside the uncorrelated states (thin black line). The detailed spectroscopic characterization of the valence and conduction band can therefore be useful to determine the correlation strength whereas the band gap size itself is probably too small to be precisely compared. To unveil the other aspects of electronic correlations, we present the imaginary part of local Green function in Figure 3D. With both parameter sets, it is clearly observed that Im goes to zero as , indicative of the zero spectral weight at , namely, the insulating phase. From the fact that minimum frequency point is lower in the 3-band-model case (red line), it also confirms the observed gap size difference in Figures 3A–3C. The calculated self-energies highlight the same feature: In Figure 3E, the slope of at shows the vanishing quasiparticle weight , indicating Mott phase.

To gain further insights on the electronic correlations, we investigate the screened Coulomb interaction and the charge susceptibility. Figure 4A shows the real part of screened Coulomb interaction . The effect of non-local Coulomb interaction can be systematically examined by changing the level of approximations. Here we fix the value of on-site interaction eV, and compare the results of two different inter-site and 0.208 eV. GW self-energy can also be taken into account on top of that. In high-frequency limit, the effective Coulomb interaction approaches to its ‘bare’ value eV in all three cases. In low frequency limit, gradually decreases as the more non-local interactions get involved. It is a general feature of extended Hubbard model, although the screening effect by is not significant as the charge fluctuation is largely suppressed in Mott phase. Figure 4B also shows that gradually decreases as increases whereas it does not vary much when is small. In large limit, on the other hand, its effect can become pronounced. As shown in Figure 4B, the screened Coulomb interaction is significantly reduced, indicating that charge fluctuations caused by further screen the Coulomb interactions., In this regime of 0.5 eV, gets rapidly reduced, and simultaneously, the local charge susceptibility diverges (red line):,where and . To perform the detailed investigation of this limit, we calculate the momentum-dependent static charge susceptibility at eV which is deemed a point close enough to the charge-ordering-type instability (corresponding to the black star point in Figure 4B). In Figure 4C, a diverging feature is clearly identified at ()=(0.333,0.333) indicative of a charge-ordering instability ( is fractional coordinate of reciprocal lattice vector ). Although it implies the further charge instability over the CCDW phase, the critical value is far from being realistic in the current system.

We investigated the electronic screening and the spectral property of monolayer TaS2 by the combination of DFT, RPA, MLWF and GW + EDMFT. Both on-site and inter-site correlation strengths are estimated based on two different ab initio RPA approaches. Although its Mott insulating gap is mainly determined by on-site interaction, the inter-site is also found to be sizable. The comparison of DMFT, EDMFT, and GW + EDMFT exhibits a systematic change of the electronic behavior because the levels of non-local self-energy computations are varied. Our work provides useful information and insight for understanding TaS2 and the other related systems.

Our computations of screened Coulomb interaction are based on random phase approximation. The electronic structure calculations take account of non-local as well as dynamical self-energy within GW + EDMFT. The singlet formation caused by interlayer dimerization,,, could not be considered in our calculation setups.



Further information and requests for resources should be directed to and will be fulfilled by the lead contact, Myung Joon Han (mj.han@kaist.ac.kr).

This study did not generate new unique reagents.

We performed DFT calculations of monolayer 1T- and CCDW-TaS2 with Vienna Ab initio Simulation Package (VASP)., The cell parameters and the internal coordinates were optimized with the force criterion of 0.1 meV/. The GGA-PBE functional was used and the 25 vacuum taken into account to simulate the monolayer. () k-grid and 500 eV (400 eV) cutoff energy were adopted for 1T-TaS2 (CCDW-TaS2). To describe the effect of electronic correlations of low-energy state in CCDW-TaS2, we consider the single-band extended Hubbard model:where and is the creation and annihilation operator of electron, respectively, with spin at site . is the electron number operator at site , and denotes the nearest-neighbor summation. The hopping amplitude between two site and was calculated with the MLWF method., and is the local and non-local Coulomb interaction, respectively. The realistic estimation of these two interaction parameters is one of the main issues of current study. For this purpose, we used (c)RPA method as implemented in RESPACK code.

The Hamiltonian is solved within GW + EDMFT whose standard self-consistent loop can be briefly summarized as below., (i) Calculate the lattice green function and the screened Coulomb interaction where and are the crystal momentum vectors. and refers to the fermionic and bosonic Matsubara frequency, respectively. The momentum space corresponds to the Fourier transformed . and represents the electron self-energy and the polarization function, respectively. The bare Coulomb interaction is given by . In practice, we start the first loop from 0. (ii) Compute the fermionic and bosonic Wiess fields and : where for any , and is the number of -points. (iii) Solve the following impurity model to obtain the impurity self-energy and polarization : where is the Fourier transormation of , and and are the Grassmann field for electrons. Here we took the continuous-time hybridization expansion quantum Monte Carlo impurity solver as implemented in ComDMFT package. (iv) Construct the new self-energy and polarization function from the calculated and : where , and (v) Go back to the step (i) and repeat the calculations until and are converged.