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Electronic structure of two-dimensional hexagonal diselenides: charge density waves and pseudogap behavior

We present theoretical study of electronic structure (spectral functions and Fermi surfaces) for incommensurate pseudogap and charge density wave (CDW) and commensurate CDW phases of quasi two dimensional diselenides 2H-TaSe2 and 2H-NbSe2. Incommensurate pseudogap regime is described within the scenario based on short-range order CDW fluctuations, considered within the static Gaussian random field model. In contrast e.g. to high-Tc cuprates layered dichalcogenides have several different CDW scattering vectors and electronic spectrum with two bands at the Fermi level. To this end we present theoretical background for the description of multiple scattering processes within multiple bands electronic spectrum. Thus obtained theoretical spectral functions and Fermi surfaces are compared with recent ARPES experimental data, demonstrating rather good qualitative agreement.


INTRODUCTION
Quasi-two-dimensional dichalcogenides TX 2 (T=Nb,Ta,Mo,Hf; X=S,Se) and their different polymorhic modifications long time ago attracted the attention of scientific community [1]. This was connected with: (i) early suggestions to look for high-T c superconductivity in layered compounds; (ii) the discovery of phase transitions with formation of charge density waves (CDW) [1]. In particular, in 2H-TaSe 2 (2H means -hexagonal structure with two Ta layers in the unitcell) the second order transition into incommensurate CDW phase is observed at temperature =122.3K. At 90K there is another transition to commensurate CDW phase [1,2]. In 2H-NbSe 2 transition to incommensurate CDW phase happens at much lower temperature of 33.5K [2] and no commensurate CDW phase is observed.
Above the temperature of incommensurate CDW transition in these systems there might be the range of temperatures where short-range order CDW fluctuations with finite, but pretty large, correlation length ξ may exist due to low-dimensional nature of these systems (and in analogy with antiferromagnetic fluctuations in cuprates). This indeed is experimentally observed in angular resolved X-ray photoemission (ARPES) experiments [3][4][5].
In this paper we present band structure calculation results for 2H-TaSe 2 and 2H-NbSe 2 with analysis of possible topologies of the Fermi surfaces upon doping, showing possibility of formation of "bone"-like Fermi sheets. Further we describe the details of theoretical description of multiband electronic multiple scattering on CDW in a multiple band systems, as applied to pseudogap, incommensurate and commensurate CDW phases for both 2H-TaSe 2 and 2H-NbSe 2 . As an outcome, we obtain spectral functions and Fermi surface maps, which are compared with a number of recent ARPES results [3,4].

BAND STRUCTURE
The 2H-TX 2 layered compounds have hexagonal crystal structure with the space group of symmetry P6 3 /mmc with lattice parameters for Ta system a=3.436Å and c=12.7Å.
Obtained band structure and Fermi surfaces are in good agreement with similar LDA cal-culations by other authors [7]. We do not present any LDA results on 2H-NbSe 2 since its crystal structure [2] and corresponding band structure are very close to those of 2H-TaSe 2 .
In accord with the previous works [7] in our LDA calculations the Fermi level in 2H-TaSe 2 is crossed by two Ta-5d bands with 3z 2 −r 2 symmetry (see Fig. 1a), which are well separated from other bands.
Fermi surface (FS) of 2H-TaSe 2 has three (in some works two [7]) hole-like cylinders near the Γ-point and two hole cylinders around K-point. Our results are presented in Fig. 1b.
Here we observe three hole-like cylinders around the Γ-point.
During the recent years several ARPES studies detected the experimental FS of 2H-TaSe 2 .
In particular, in Ref. [8] the electronic structure of the valence band was studied in 1T-TaS 2 and 2H-TaSe 2 . For 2H-TaSe 2 it was shown that along Γ-K direction there are four crossings with the FS. Similar picture is also seen in LDA results (Fig. 1a,b). In later ARPES works [3,4,9] it was observed that FS of 2H-TaSe 2 has more complex topology. Namely, along Γ-K direction there appear the "bone"-like FS sheets. Within the LDA picture one can obtain such "bones" by the shift of the Fermi level down by about 0.1 eV (Fig. 1a,c).
To improve over simple LDA, in Fig. 2 we show the "experimental" bands with dispersions: with hopping integrals t i obtained from the fit to experimental Fermi surfaces [10]. Corresponding values of t i (in eV) for Ta system are: for the band forming barrels around Γ and K points t 0 =-0.027, t 1 =0.199, t 2 =0.221, t 3 =0.028, t 4 =0.013, for the band forming Thus an electron with momentum k is scattered by any of thirteen different momenta (see Table I): 0 -preserving its initial momentum k; Q (Q l andQ l ); X (X l andX l ).
Thus for one band case to find the diagonal Green's function of an electron G(k, k) and twelve off-diagonal (G(k ± Q l , k) and (G(k ± X l , k)) single-electron Green's functions we have to solve the system of thirteen linear equations (17) (see Appendix). Such an approach can be generalized for a multiple band case with simplifying assumption [11] that intraand interband CDW scattering amplitudes are just the same (see Appendix) Solving these equations we can finally find the diagonal Green function G ij (k, k) (i, j = 1, 2 -band indices) and corresponding spectral function: determining the effective electron dispersion.

Incommensurate CDW phase
As was pointed above at temperature T = 90 K 2H-TaSe 2 (and 2H-NbSe 2 at 33.5K) undergoes phase transition into incommensurate CDW phase with scattering vector Q ∼ 0.58 − 0.6ΓM. Similar to the commensurate case discussed above, this vector corresponds to six independent scattering vectors Q l ,Q l l = 1, 2, 3. Let us consider single scattering of an electron with momenta k near the FS by vector Q(Q l ,Q l ). For general values of k, such scattering act moves an electron quite far away from the FS, the only exception is an electron in the vicinity of the "hot-spots" where ǫ(k+Q) = ǫ(k)). Among multiple scattering processes most probable will be successive scattering processes by vectors Q l andQ l since in this case the scattered electron is back again to initial point with momenta k close to the Fermi surface. To this end further we will work in the so called two-wave approximation, when scattering act consists of two successive scattering processes by vectors Q l andQ l .
Assuming that scattering amplitude is the same for intra-and interband transitions, for diagonal Green function's we obtain (corresponding diagrammatic representation see in Fig.   4): where Σ = ∆ 2 jl (g j (k + Q l ) + g j (k − Q l )) and g j (k) = 1 E−ǫ j (k)+iδ is bare retarded Green's function for the n-th band. Summing Eq. (3) over i, one can get: Then using Eq. (3) again we obtain: which can provide us with the spectral function (2) for the case of incommensurate CDW scattering.

CDW pseudogap fluctuations
Above the temperature of incommensurate CDW transition there is no long-range charge ordering, but due to low-dimensionality of the system there are rather well developed shortrange order CDW fluctuations, with finite correlation length ξ and characteristic wavevector Q which becomes rather quick commensurate with Q = 2 3 ΓM [4] as temperature lowers. In analogy to incommensurate electronic CDW scattering we will employ two-wave approximation with pair of vectors (Q l ,Q l ). Diagrammatically such scattering processes are show in Fig. 5, where three types of interaction lines correspond to three characteristic transfer momenta l = 1, 2, 3.
Let us assume fluctuations Gaussian. Then averaging over such fluctuations corresponds to all possible interconnections of incoming and outgoing interaction lines of the same type [12][13][14], producing appropriate effective interactions, assumed to be of the form discussed in these works. For the case of high enough temperatures one can neglect dynamics of fluctuations and average over static random field of Gaussian pseudogap fluctuations [12][13][14].
Let us mention that the number of different diagrams is defined by product of number of ways to interconnect vertices of type 1, 2 and 3. Since only outgoing and incoming lines of each type can be connected, combinatorics corresponds to incommensurate case [12]. Following Refs. [12][13][14] we use the basic property of the diagrams of this model: any diagram with crossing interaction lines is equal to some noncrossing diagram of the same order. Thus only noncrossing diagrams can be considered, while contributions of all diagrams can be accounted by combinatorial prefactors. And for each type of interaction lines (1,2,3) we will have its own incommensurate combinatorial prefactors, same as in Refs. [12,13].

Recurrent procedure for Green's function: single band case
Within some straightforward generalization of the approach of Refs. [12,13], for single band case one-electron Green's function can be obtained via recurrent procedure, which is shown diagrammatically in Fig. 6. Here n l is the number of interaction lines of type l surrounding the "bare" electron line. Analytically this procedure can be written as: where n 1 , n 2 , n 3 -are even and Σ n 1 +1,n 2 ,n 3 = ∆ 2 s(n 1 + 1)[G n 1 +1,n 2 ,n 3 (k + Q 1 ) + G n 1 +1,n 2 ,n 3 (k − Q 1 )].
The other self-energies Σ in (6) can be found similarly to (7), but n 2 or n 3 should be increased by one and vectors Q 2 or Q 3 should be added (subtracted) to (from) k, while and Σ n 1 +2,n 2 ,n 3 = ∆ 2 s(n 1 + 2)G n 1 +2,n 2 ,n 3 (k). Here and κ = 1/ξ is the inverse correlation length of pseudogap fluctuations, n = n 1 + n 2 + n 3 , This recurrent procedure is applied in analogy with refs. [12,13]. As a first step, one takes large enough n = n 1 + n 2 + n 3 , for example even, and assume that all Green functions G n 1 ,n 2 ,n 3 with even n 1 , n 2 , n 3 , such that n = n 1 + n 2 + n 3 are equal to zero. Then from (9) one can find that all Σ n 1 ,n 2 ,n 3 for the same indices are equal to zero too. Then using recurrent procedure, one can get all new values for G n 1 ,n 2 ,n 3 with even n 1 , n 2 , n 3 , such that n 1 + n 2 + n 3 = n − 2, and repeat the recurrent procedure until one obtains physical Green function: Multiple bands pseudogap model for quasi two dimensional hexagonal structures.
In hexagonal diselenides TaSe 2 and NbSe 2 , as we have seen, the Fermi level is crossed by two bands. Thus, our recurrent procedure should be generalized for multiple bands. We will follow Ref. [11], devoted to description of possible pseudogap behavior in iron based superconductors, and assume that intra and interband pseudogap scattering amplitudes are identical. This simplifies further analysis and the recurrent procedure for diagonal elements of the general matrix (over band indices) Green's function G ij can be drawn diagrammatically as in Fig. 7. For our two band model each of the band indices run over two possible values, and there is summation over all possible values of indices p, m, l in the vertices (Fig.   7). Thus, the self-energy in these diagrams has no dependence on band indices at all and we can obtain the recurrent procedure for G n 1 ,n 2 ,n 3 = i,j G ij n 1 ,n 2 ,n 3 , which is identical to Eqs. (6)-(9) as in single band case, with only replacement: and at the end of the procedure we define the physical matrix Green function as: where Σ = Σ 1,0,0 +Σ 0,1,0 +Σ 0,0,1 . This Green's function allows us to find the spectral function (2) in the presence of CDW pseudogap fluctuations.

RESULTS AND DISCUSSION
In our calculations we used rather typical estimate of CDW potential ∆ =0.05 eV, and for correlations length of pseudogap fluctuations we assumed the value of ξ=10a (where a is the lattice spacing). To mimic for the experimental ARPES resolution we broadened our spectral functions with Lorentzian of width γ = 0.03eV , practically it means that we made substitution E → E + iγ during all calculations.
In Fig. 8 Fig.8) those gaps become even stronger and we can see much more pronounced shadow bands. In Fig. 12 we present comparison of experimental and theoretical Fermi surfaces for 2H-TaSe 2 . In the middle part of Fig. 12 we show experimental ARPES data from Ref. [4].
Data at 180K corresponds to the pseudogap phase, while those at 30K are for commensurate CDW phase.
Upper panel of Fig. 12 shows our theoretical Fermi surface in the incommensurate pseudogap regime for 2H-TaSe 2 . In general it more or less reminds LDA Fermi surface from Fig.   1c. However there are obvious signatures of partial destruction of the Fermi surface sheets.
Namely, cylinder around K-point and "bones" along K-M direction are partially smeared out. It is seen that this picture agrees well with the experimental ARPES data of Ref. [3,4].
For the commensurate CDW phase (lower panel of Fig. 12) Fermi surface stays close to that obtained in LDA and shown in of Fig. 1c. In contrast to incommensurate pseudogap phase Fermi surface sheets here are more sharp both in experiment and in theory. The cylinder around K-point is now continuous. In the middle of the "bones" we observe the start of formation of small triangles as shown in the center of middle panel. Thus, here in commensurate CDW phase of 2H-TaSe 2 we again obtain an overall agreement between theory and experiment.
In Fig. 13 we show comparison between experimental (middle panel) and theoretical Fermi surfaces (lower and upper panel) for 2H-NbSe 2 . Experimental data on the Fermi surface are available only for commensurate CDW phase [5]. Thus we can compare these with theoretical picture shown on lower panel of Fig. 13. In general, both Fermi surfaces remind those from Fig. 1b. Account of electron scattering on commensurate CDW leads to a small regions of Fermi surface destruction, namely, along Γ-K and K-M directions. If there exists incommensurate pseudogap phase for 2H-NbSe 2 at high enough temperatures, its Fermi surface will not be changed much by pseudogap fluctuations, as seen in the upper panel of Fig. 13).

CONCLUSION
To conclude, here we presented theoretical results on electronic structure of twodimensional diselenides 2H-TaSe 2 and 2H-NbSe 2 within different CDW phases.
First of all we formulated a theoretical approach to account for multiple scattering of electrons on different types of CDW, also for the multiple bands case. Further we investigated spectral functions and Fermi surfaces for the pseudogap, incommensurate CDW and commensurate CDW phases. Calculated theoretical spectral functions within the pseudogap phase demonstrate "hot regions", where spectral function is additionally broadened. In incommensurate CDW and commensurate CDW phases in a place of these "hot regions" we obtained opening of the number of gaps at the intersections with rather pronounced "shadow bands". Comparing experimental and theoretical Fermi surfaces in the pseudogap phase we observe rather clear signs of partial Fermi surface destruction with formation of a number of typical "Fermi arcs", separated by pseudogap regions. In commensurate CDW phase Fermi surfaces are rather similar to initial LDA picture, with pretty small features due to CDW.
The overall agreement between theory and ARPES experiments is rather satisfactory.
Appendix: Scattering on commensurate CDW

One band scattering
First let us consider single band case with electronic "bare" spectrum ǫ(k). These "bare" electrons are scattered on CDW potential, written as: "Bare" retarded Green's function is: Let us introduce short notations: g(k) = g; g(k + Q l ) = f l ; g(k − Q l ) = fl; g(k + X l ) = φ l ; g(k − X l ) = φl. Then for diagonal Green's function G = G(k, k) and twelve off-diagonal k); Φl = G(k−X l , k)) one can get the following system of thirteen linear equations (see Table I of scattering vectors summation): where F = 3 l=1 (F l + Fl). Solving Eqs. (17), one can the get diagonal Green's function G = G(k, k): where α = ∆(f 2 +f1 +f3), β = ∆(f 1 +f 3 +f2), a = ∆ 2 [f1(φ1 +φ2)+f 2 (φ 2 +φ 3 )+f3(φ3 +φ 1 )], Following the approach of Ref. [11] we assume that CDW scattering amplitude ∆ is identical for intra and interband transitions. One can define short notations: , where i -band index. Diagonal and offdiagonal Green's functions will have additional band indices The rest of notations are the same as for single band case. For diagonal Green's function, in analogy with first equation of the system (17), one can obtain: l(l) and summing Eq. (19) over i we get: The rest of other twelve equations for F j l(l) and Φ j l(l) are completely equivalent to corresponding equations of one band case (17). Thus, we immediately obtain: where K is defined in Eq. (18). However, now the quantities g, f l(l) , φ l(l) are summed up over all band indices. From Eq. (19) and (20), using Eq. (21) we finally obtain: which allows us to calculate the spectral function (2) with the account of scattering on commensurate CDW.