Two-Band Electronic Reconstruction Induced via Correlation and CDW Order Effects

Abstract

The emergence of a charge density wave (CDW) in transition-metal dichalcogenides opens up a route to charge order, followed by superconductivity at low temperatures. A key question here concerns how many particle electron–electron interations govern the low-energy electronic structure in the normal and CDW states. Using dynamical mean-field theory, we explore the many-body properties of an extended, two-band Hubbard model applicable to $2H$-TaSe₂. We reveal the electronic structure reconstruction in the normal and CDW states driven by two-band dynamical correlations. Our results demonstrate a remarkable renormalization of the Ta-$5d$ bands crossing the Fermi level, showing a continuous reduction in the CDW gap up to an incomplete gapping, followed by a CDW to a CDW–Mott phase transition pertinent to strongly correlated transition-metal dichalcogenides.

1. Introduction

The relationship between the dimensionality of a lattice and the correlation between electrons is a long-standing issue in the fields of condensed matter and material physics, particularly in regard to the emergence of diverse quantum phases. A correlated electron system is defined by the presence of significant to pronounced many-particle Coulomb interactions between electrons, which give rise to a distinct set of physical properties that are not observed in a weakly interacting electron system. The most significant consequence of electron correlation is the Mott metal–insulator transition [1,2,3], which occurs when a half-filled paramagnetic metal is converted into a correlated insulator when the on-site Coulomb interaction, U, exceeds the bare bandwidth, W (i.e., the effective Coulomb interaction to bandwidth $U/W$ ratio). Consequently, following decades of fundamental and applied research, it has been established that the impact of electron–electron interactions in correlated electron systems is contingent upon the ratio of energy scales between U and W. In the limit of small $U/W$, the electronic properties are predominantly governed by electron hopping and electron–phonon interactions [4]. Conversely, when $U/W\approx 1$, electron–electron interactions become a dominant factor, and in the limit of large $U/W$, the double occupancy of electrons on a single site is prohibited [5]. This results in a Mott insulating state at half-filling [1,2,3], which could lead to the emergence of novel electronic states such as superconductivity (unconventional or not) upon external perturbations [6,7,8,9,10,11,12,13]. Interesting examples in this context are the transition-metal dichalcogenides ($1T,2H$)-$M{X}_2$ (M = Ta X = S, Se), which are considered to be systems with similar U and W energy scales [14,15,16,17]. As common to this material class, charge density wave (CDW)-phase instabilities [17,18,19,20,21] coexisting with superconductivity [21,22,23] emerge at low temperatures (T). Interestingly, while the low-T commensurate CDW phase of $1T$-TaS₂ is considered to be a Mott insulator [24,25] because of the half-filling insulating behavior [12,26,27,28,29], the monolayer $1T$-TaSe₂ shows a CDW–Mott phase transition around 530 K [17], which has also been reported for heterogeneous Ta-dichalcogenide bilayers [30]. Also noteworthy is that both the commensurate and incommensurate CDW phases lead to incomplete gapping or pseudogapped electronic excitations at low energies [21,22,31,32,33] similar to that observed in high-$T_C$ cuprates [34,35,36,37] due to their proximity to Mottness [38]. Motivated thereby, here, we show the emergence of a pseudogap regime as the precursor to the CDW–Mott phase [17] in a two-band ($\mathcal{TB}$) model relevant to $2H$-TaSe₂ [39].

In recent decades, transition-metal dichalcogenides have been the subject of extensive study due to the intrinsic electronic properties resulting from the interplay between the lattice structure, CDW ordering, disorder, and electron–phonon and electron–electron correlation effects. These effects have been the subject of numerous studies, as evidenced by the extensive literature on the topic [17,20,21,24,25,28,29,31,39,40,41,42,43,44,45,46,47,48]. These systems have a layered structure, and they are characterized by the formula $M{X}_2$; M represents a transition metal ion (M = Ti, Zr, Hf, V, Nb, Ta, Mo, W, and Re), while X denotes a chalcogen atom (X = S, Se, Te) [18,37]. Each layer is constituted by a hexagonal transition-metal sheet sandwiched between two analogous chalcogen layers, which are coupled to each other via weak van der Waals (vdW) forces. Within the layers, they form-bonded, two-dimensional $X-X$ layers, while M has either trigonal prismatic or octahedral coordination with X [22]. In some of these systems, the induction of superconductivity can be achieved through the intercalation of a variety of elements into the van der Waals gaps [8,9,10], the application of pressure [11,12], or the application of gate voltages [13]. Importantly, the electronic phase diagram, as a function of external perturbations like chemical doping or pressure, is analogous to those of high-$T_C$ cuprates, as well to some iron-based superconductors, suggesting the role of Mottness in the emergence of superconductivity at a low T in these vdW systems. However, the physical origin for stabilizing the pseudogapped state [21,22,31,32,33], which is the precursor to the CDW–Mott [17] phase, has not been well understood, and this is one of our focuses here.

Similar to cuprates and some Fe-based superconductors, the $2H$-polymorph TaSe₂ shows a strange metal, T-linear resistivity [22] in the normal state at a high T, which enters into an incommensurate CDW phase at a T close to 122 K, followed by a transition to a commensurate CDW phase near to 90 K [49,50]. Angle-resolved photoemission spectroscopy (ARPES) and optical spectroscopy data suggest the presence of a low-energy pseudogap for T above 122 K, similar to that observed in high-$T_C$ cuprates and Fe-based superconductors. Notably, from room-T down to the CDW-phase transition, the resistivity of Pd-intercalated $2H$-TaSe₂ [22] and $2H$-TaS$e_{1-x}$$S_x$ [23] systems decreases nearly linearly with T [22], a characteristic akin to the strange-metal phase in which the resistivity varies linearly over a broad T range [51]. Upon entering the CDW phase, the resistivity shows a Fermi liquid (FL)-like $T^2$ dependence at a low T [22], implying the importance of self-energy corrections [20] in the CDW phase. Therefore, clarifying the role of electron correlations [17,46,52] in the delicate balance between the CDW and non-CDW phases is necessary. Motivated by this and the fact that $2H$-TaSe₂ is considered to be a promising material for nano-electronic devices, as well as a material for flexible, two-dimensional ($2D$) optoelectronic applications [53], in this work, we derive the electronic properties of a $\mathcal{TB}$ [39] extended Hubbard model similar to that proposed for bilayer cuprate superconductors [54], showing the role of dynamical correlations in the non-CDW and CDW ordered states of $2H$-TaSe₂. Our results are relevant to understanding the electronic structure modification that could result in the creation of BCS-like s-wave [33] superconductivity at a low T, as well as the emergence of CDW–Mott localization [22] in the vicinity of a correlation-induced quantum phase transition [2].

2. Theory and Results

Based on Ref. [39], the two-band $\left(\mathcal{TB}\right)$ tight-binding model considered in this work reads as

$$H_{TB}\left(\mathbf{k}\right)=f(t_0,t_1,t_2;\mathbf{k}){\sigma }_0+f({\tilde{t}}_0,{\tilde{t}}_1,{\tilde{t}}_2;\mathbf{k}){\sigma }_x,$$

(1)

$$\begin{array}{ccc}\hfill f(\alpha ,\beta ,\gamma ;\mathbf{k})& =& \alpha +2\beta \left[cos{k}_{x}a+2cos{\displaystyle \frac{k_{x}a}{2}}cos{\displaystyle \frac{\sqrt{3}{k}_{x}a}{2}}\right]\hfill \\ & +& 2\gamma \left[cos\sqrt{3}{k}_{y}a+2cos{\displaystyle \frac{3k_{x}a}{2}}cos{\displaystyle \frac{\sqrt{3}{k}_{y}a}{2}}\right],\hfill \end{array}$$

where ${\sigma }_0$ is a $(2\times 2)$ identity matrix, ${\sigma }_x$ is the x Pauli matrix, $a=3.43 \AA$ is the lattice constant of $2H$-TaSe₂, $t_0=0.113$ eV is the Ta on-site energy, ${\tilde{t}}_0=0.184$ eV is the direct interlayer coupling, and $t_1=0.073$ eV (${\tilde{t}}_1=0.029$ eV) and $t_2=0.142$ eV (${\tilde{t}}_2=0.038$ eV) are the nearest- and next-nearest intralayer (interlayer) couplings. In Ref. [39], the one-particle parameters were fixed using a hybrid approach that fits the tight-binding $\left(TB\right)$ model to the ARPES data below the Fermi level $\left(E_F\right)$ and density functional theory (DFT) calculations above it. It is noteworthy that, while the $f\times {\sigma }_0$ contribution to $H_{TB}\left(\mathbf{k}\right)$ (in Equation (2)) stands for the diagonalized band dispersion, ${\epsilon }_a\left(\mathbf{k}\right)$, of the $a(\ne b)=1,2$ bands, the $f\times {\sigma }_x$ term in $H_{TB}\left(\mathbf{k}\right)$ corresponds to a non-local, $\mathbf{k}$-dependent hybridization [55,56,57] (here denoted as $\mathcal{V}\left(\mathbf{k}\right))$ between the two a-bands of $2H$-TaSe₂ crossing $E_F$, as reported in Ref. [39]. Finally, we shall mention here that, according to the ARPES data of Ref. [39], the dispersions ${\epsilon }_a\left(\mathbf{k}\right)$ are split by an energy $\Delta$ of the order of 0.15 eV; thus, in our study, we set ${\epsilon }_1\left(\mathbf{k}\right)=\Delta +{\epsilon }_2\left(\mathbf{k}\right)$. These are the relevant one-particle inputs to $\mathcal{TB}$+DMFT [54], which generates a strongly renormalized $\mathcal{TB}$ electronic state due to correlation and CDW effects. The local $(U,U^{\prime })$ and non-local (V) interactions of $2H$-TaSe₂ are contained in $H_{int}=U{\sum }_{i,a}{n}_{i,a,\uparrow }{n}_{i,a,\downarrow }+U^{\prime }{\sum }_{i,a\ne b}{n}_{i,a}{n}_{i,b}+V{\sum }_{i\ne j}{n}_i{n}_j$. Here, the indices i run over the lattice sites in the $a,b=1,2$ bands. U and $U^{\prime }$ are the on-site intra- and inter-band Coulomb interactions and V is the nearest-neighbor Coulomb repulsion, all responsible for the many-particle electronic structure reconstruction in the extended $(V\ne 0)$ $\mathcal{TB}$ Hubbard model considered here for $2H$-TaSe₂: without a loss of generality, here, we set $U^{\prime }=V=0.8U$. Finally, as in earlier works [54,58,59], we decouple the intersite Coulomb interaction term of $H_{int}$ in the Hartree approximation, which is exactly under the large D limit [5].

Inspired by earlier theory studies on bilayer cuprates [55,60,61], in this work, we consider an extended $\mathcal{TB}$ Hubbard model [54] for $2H$-TaSe₂ using hopping integrals introduced in Ref. [39] for the parent compound, as described above. We evaluate the retarded, one-particle Green’s functions

$$G_{a,\sigma }(\omega ,\mathbf{k})={\left[{\xi }_{a,\sigma }\left(\omega \right)-{\epsilon }_a\left(\mathbf{k}\right)-{\displaystyle \frac{{\mathcal{V}}^2\left(\mathbf{k}\right)}{{\xi }_{b,\sigma }\left(\omega \right)-{\epsilon }_b\left(\mathbf{k}\right)}}\right]}^{-1},$$

(2)

where ${\xi }_{a,\sigma }\left(\omega \right)\equiv \omega +i\eta -{\Sigma }_{a,\sigma }(\omega +i\eta )$ [62], of the hybrid $\mathcal{TB}$ system at zero T and real frequencies using the $\mathcal{TB}$ [62] iterated perturbation theory as an impurity solver for DMFT. The detailed formulation of this algebraic DMFT solver for correlated electron systems has been introduced in the context of real materials with different charge, orbital, and spin degrees of freedom (see Refs. [63,64]). It has also been used to study the evolution of the one-particle spectra of the periodic Anderson model with the incorporation of inter-band Coulomb correlations [62]. Therefore, we do not repeat the equations here. It should be noted, however, that this interpolative ansatz is based on second-order perturbation theory, which takes into account all dynamical scattering processes arising from intra- and inter-band Coulomb interactions in a self-consistent manner. Furthermore, for the sake of clarity, it should be noted that, in the $\mathcal{TB}$ Hubbard model, the one-particle Green’s functions of each band are coupled via the inter-band (inter-orbital in the multi-orbital case) Coulomb interaction $U^{\prime }$, as well as the k-dependent interband hybridization $\mathcal{V}\left(\mathbf{k}\right)$ [55,56,57]. In general, our formalism can be regarded as an extension of the periodic Anderson model [5], in which the two channels are dispersive and subject to on-site electron–electron interactions, which are proximitized [65] by k-dependent interband hopping integrals [39]. As demonstrated below, the integration of realistic $\mathcal{TB}$ [39] inputs with multi-particle, many-body effects [5] provides a comprehensive account of the electronic properties of $2H$-TaSe₂ in both the normal and CDW states.

Let us now discuss our $\mathcal{TB}$ and $\mathcal{TB}$+DMFT results. We begin by considering the many-body Hamiltonian $H=H_{TB}\left(\mathbf{k}\right)+H_{nt}$ within the normal, non-CDW ordered state. In this regime, the two non-equivalent CDW sites $A,B$ of the bipartite $(2\times 2)$ superlattice [39] considered here will have the same electronic structures and the total on-site band fillings $n=n_A=n_B$. In Figure 1, we show the TB-based electronic state at $U=0.0$ eV. Several features are noteworthy in this limit. Due to finite band splitting $\Delta$ [39], the unperturbed density of states (DOS) shows different lineshapes, with band two being more populated as compared to the higher-energy band, as shown in the lower and upper panels of Figure 1, respectively. From the band-resolved DOS up to the Fermi level $(E_F=\omega =0.0)$, we obtain $n_{2,\sigma }=0.56$ and $n_{1,\sigma }=0.44$, which would correspond to half-filled bands at $\Delta =0.0$, i.e., one electron per band. Also notably, in the bare $\mathcal{TB}$ DOS is the particle-hole asymmetry characteristic $TB$ models with sizable nearest and next-nearest neighbor hopping integrals [54], an aspect consistent with extant band structure calculations for $2H$-TaSe₂ [14]. Also interesting in the bare DOS of Figure 1 is the van Hove-like peak at 0.52 eV, binding energy in ${\rho }_{2,\sigma }\left(\omega \right)[=-{\displaystyle \frac{1}{\pi }}Im{\sum }_{\mathbf{k}}{G}_{2,\sigma }(\omega ,\mathbf{k})]$, and the valence band edge, which almost coincides with that reported in Ref. [14], providing support for the $TB$ model proposed in Ref. [39]. It is also noteworthy that the local spectral functions ${\rho }_{a,\sigma }\left(\omega \right)$ undergo changes with an increasing on-site Coulomb interaction, U, in the non-CDW limit, as illustrated in Figure 1. As can be observed in the main panel of Figure 1, the results presented correctly reproduce the expected behavior for the correlated spectral functions within the DMFT approximation. Of particular interest are the emergence of the lower (LHB) and upper (UHB) Hubbard bands at high energies and the Kondo–quasiparticle resonances [5] near $E_F$, which narrow with an increasing U. Furthermore, as a result of particle-hole asymmetry and the interplay between sizable U, $U^{\prime }$, and V, the pinning of the correlated spectral function to its bare value at $E_F$ [5] obtained for the one-band Hubbard model is no longer valid in the extended $\mathcal{TB}$ Hubbard model of $2H$-TaSe₂. This occurs despite the fact that the self-energy imaginary parts show ${\omega }^2$ dependence, which is characteristic of good FL metals [5], as can be seen in the right insets of Figure 1. However, in accordance with previous research [54,55,66], the non-CDW state of the extended $\mathcal{TB}$ Hubbard model exhibits a significant influence of many-body effects on the $U=0.0$ DOS, resulting in a notable transfer of spectral weight from low to high energies and a discernible $\omega$-dependence in the self-energy real and imaginary parts, as seen in Figure 1.

Let us now elucidate the evolution of the electronic structure of the correlated spectral function of $2H$-TaSe₂ by focusing on the total DOS $[{\rho }_{total}\left(\omega \right)={\sum }_{a,\sigma }{\rho }_{a,\sigma }\left(\omega \right)]$. To this end, we have obtained the total DOS for two U values, as shown in Figure 2. As anticipated within the framework of the dynamical mean-field theory (DMFT) [5], the narrowing of the coherent Kondo-quasiparticle resonance is attributed to dynamical correlations, which also result in the enhancement of the Hubbard satellites due to the increased dynamical transfer of spectral weight with an increasing U. Of particular interest is the peak deep hump observed in the DFT+DMFT total DOS below $E_F$, as illustrated in the main panel of Figure 2. This finding aligns well with the energy-distribution curve (EDC) reported in Ref. [17], where the EDC curve of the pristine $1T$-TaSe₂ at the $\Gamma$ point displays a comparable behavior, exhibiting a deep binding energy of approximately 1.0 eV, as illustrated in Figure 2. It is also noteworthy that the peak positions of the low-energy Kondo-quasiparticles blueshift [67,68] with an increasing U, a behavior associated with dynamical changes in the correlated spectral functions. Furthermore, the inset of Figure 2 presents a direct comparison between ${\rho }_{total}\left(\omega \right)$ and EDC curves [32] obtained at $T=290$ K, which corresponds to the non-CDW ordered state of $2H$-TaSe₂. As can be observed, our results for a U between 2.0 and 2.5 eV provide a qualitative account of the main lineshape seen in the EDC curves of Ref. [32], particularly in terms of the energy position of the shoulder features and the depth of the EDC below 0.1 eV binding energy. This provides support for our modeling and parameter choice.

To provide additional insights into the $\mathcal{TB}$ electronic structure reconstruction $2H$-TaSe₂, we have extended our DMFT Green’s function formalism above (see Equation (2)) to incorporate the CDW degree of freedom. For the sake of simplicity, we consider here the $(2\times 2)$ CDW ordered state, [39] where, similar to an antiferromagnetic spin-density wave (SDW) state [69], the four-component lattice Green’s functions relevant to $2H$-TaSe₂ can be written as

$$G_{a,\gamma ,\sigma }(\omega ,\mathbf{k})={\left[{\xi }_{a,\sigma }\left(\omega \right)-{\displaystyle \frac{{\epsilon }_a^2\left(\mathbf{k}\right)}{{\xi }_{\overline{\gamma },\sigma }\left(\omega \right)}}-{\displaystyle \frac{{\mathcal{V}}^2\left(\mathbf{k}\right)}{{\xi }_{b,\gamma ,\sigma }\left(\omega \right)-{\epsilon }_b\left(\mathbf{k}\right)}}\right]}^{-1},$$

(3)

where $\gamma (\ne \overline{\gamma })=A$ or B labels the sublattice [70] of the $(2\times 2)$ CDW-induced superlattice [39].

In the main panel of Figure 3, we show the correlated spectral functions that emerge when considering a $(2\times 2)$ charge ordering state [39] and $U=2.0$ eV as a representative Coulomb interaction parameter value. To derive the CDW ordered state, we consider polarized site-resolved band fillings as the starting point towards fully self-consistent DMFT calculations, where the site B site is set to be more populated with 1.4 electrons per spin, as compared to the A site, where $n_A=0.6$. As seen, the correlated spectral functions display site differentiation as compared to the non-CDW state where the spectral functions of sites A and B coincide. In spite of the small changes between the correlated A and B DOS in the CDW ordered state as a result of strong dynamical, particle-hole scattering processes induced via U and $U^{\prime }$, an energy gap opening associated with the occurrence of CDW [71,72,73,74] spans near $E_F$, resulting in the splitting of the Kondo–quasiparticle resonance into two narrow peaks for all spectral functions. Notably, a similar effect also takes place at high energies where the Hubbard bands also split into two branches. Also remarkable in Figure 3 is the stability of the self-energy real and imaginary parts against CDW, suggesting that the CDW state emerges as a result of one-particle nesting [17,71,72] due to the bipartite lattice structure of the $(2\times 2)$ CDW order.

In this study, we further examine the impact of additional electron–electron interaction effects induced via dynamical many-particle correlation effects on the pseudogapped [21,22,31,32,33] state of the $\mathcal{TB}$ system. In contrast to the approach taken in Refs. [17,47], where different U values were considered for transition-metal dichalcogenide systems, Figure 4 presents the results obtained for $U/W$ from 0.8 up to 1.07. Given that $U^{\prime }$ scatters electrons with opposite and equal spins, which significantly enhances electron–electron correlation effects, in Figure 4, we commence by presenting our results for the total DOS, ${\rho }_{total}\left(\omega \right)={\displaystyle \frac{1}{2}}{\sum }_{a,\gamma ,\sigma }{\rho }_{a,\gamma ,\sigma }\left(\omega \right)$, for $U=2.25$ eV. As can be observed, while the splitting of the LHB is robust against the combined effect of intra- and inter-band Coulomb repulsion, a reverse trend emerges at low energies with the concomitant suppression of the CDW bandgap splitting. Moreover, upon increasing U, the spectral weight transfer from quasiparticle excitations to incoherent Hubbard bands is evident in the correlated spectral function of the CDW $\mathcal{TB}$ model of $2H$-TaSe₂. It is noteworthy that the pseudogapped state obtained for $U=2.5$ eV diminishes, resulting in metallicity at low energies in proximity to $E_F$, as evidenced by our findings for $U=2.75$ eV. Finally, to illustrate the role played by local dynamical correlations, we present in the inset of Fig. reffig4 a comparison between the DFT+DMFT total spectral function displayed in the main panel of Figure 4 and EDC data taken from Refs. [20,32]. This comparison demonstrates good qualitative agreement between the two sets of data. In particular, the shoulder features near $E_F$ and the $\omega$-dependence of the correlated spectra at slightly higher binding energies are accurately reproduced using the DMFT approximation for the $\mathcal{TB}$ model of $2H$-TaSe₂.

Ultimately, to provide new fundamental insights into the correlated phenomenon that may emerge in strained $2H$-TaSe₂ crystals, where $U/W\gg 1$ [2], in Figure 5, we demonstrate the $\omega$ dependence of the correlated spectra as we approach the CDW–Mott localized state [17]. Figure 5 illustrates the occurrence of gradual alterations in the total spectral functions, which suggests the existence of a second-order metal-to-insulator transition from a CDW metal to a CDW–Mott localized state. These responses are characteristic of strongly correlated systems in which the changes to the spectral functions are linked to collective electronic (charge and spin) fluctuations in the reconstructed spectral function [75]. As observed for $U=3.5$ eV, which is in close proximity to the critical value for the continuous CDW–Mott transition, the narrow low-energy peak that was present at $U=3.25$ eV has been almost entirely suppressed, and only the Hubbard bands are clearly discernible at high energies above and below $E_F$. A comparable quantum-phase transition, with the coexistence of Mott localized and metallic electronic states, has been documented in the context of the three-orbital problem of ruthenate oxides [76] and in a theoretical investigation of the $T=0$ phase transition in two dimensions, from a Fermi liquid metal to a paramagnetic Mott insulator with a spinon Fermi surface [77]. Furthermore, the demonstration of a continuous Mott transition in MoTe₂/WSe₂ Moiré superlattices is also relevant in this context. This transition is induced by varying an out-of-plane electric field, which modifies the Moiré potential depth and, thus, the $U/W$ ratio [78], and it is observed to occur as a result of the metal–insulator transition.

3. Conclusions

In conclusion, this study has investigated the electronic structure reconstruction of an extended two-band Hubbard model, which is applicable to $2H$-TaSe₂. In accordance with earlier studies, this study found that intra- and inter-band many-body effects strongly renormalize electronic structure of $2H$-TaSe₂ within the normal and CDW ordered states. Due to sizable dynamical two-band correlations, we demonstrated the emergence of pseudogapped electronic state consistent with experimental observations [20,32]. Moreover, in the high correlation-to-bandwidth $(U/W)$ limit, we predict a gap-closing [79] scenario where a continuous metal–insulator transition from a CDW-gapped insulating state [73,74] followed by a CDW–Mott [17] insulating state is demonstrated. Taken together, the results in this work constitute a step forward in understanding the manifestation of band-selective pseudogaped and CDW–Mott phases that might, respectively, be the precursors to superconductivity in S-doped $2H$-TaSe₂ [23] and to the dimensionality-driven insulator–metal transition reported for $1T$-TaSe₂ [80].