Electronic Structure Calculation and Superconductivity in λ -(BETS) 2 GaCl 4
Hirohito Aizawa, Takashi Koretsune, Kazuhiko Kuroki, Hitoshi Seo
aa r X i v : . [ c ond - m a t . s up r- c on ] J u l Journal of the Physical Society of Japan
LETTERS
Electronic Structure Calculation and Superconductivity in λ -(BETS) GaCl Hirohito Aizawa ∗ , Takashi Koretsune , , Kazuhiko Kuroki , and Hitoshi Seo , Institute of Physics, Kanagawa University, Yokohama, Kanagawa 221-8686, Japan Department of Physics, Tohoku University, Sendai 980-8578, Japan JST PRESTO, Kawaguchi, Saitama 332-0012, Japan Department of Physics, Osaka University, Toyonaka, Osaka 560-8531, Japan Condensed Matter Theory Laboratory, RIKEN, Wako, Saitama 351-0198, Japan Center for Emergent Matter Science (CEMS), RIKEN, Wako, Saitama 351-0198, Japan
Quasi-two-dimensional molecular conductor λ -(BETS) GaCl shows superconductivity (SC) below 5.5 K,neighboring the dimer-type Mott insulating phase. To elucidate the origin of SC and its gap function, wecarry out first-principles band calculation and derive a four-band tight-binding model from the maximallylocalized Wannier orbitals. Considering the spin-fluctuation-mediated mechanism by adding the Hubbard U -term to the model, we analyze the SC gap function by applying the random phase approximation. Weshow that the SC gap changes its sign four times along the Fermi surface (FS) in the unfolded Brillouinzone, suggestive of a d -wave-like SC gap, which only has two-fold symmetry because of the low symmetry ofthe crystal structure. Decomposing the SC gap into the pairing functions along the crystal axes, we comparethe result to similar analysis of the well-studied κ -type molecular conductors and to the experiments. The quasi-two-dimensional (Q2D) molecu-lar conductor λ -(BETS) GaCl , where BETS isbis(ethylenedithio)tetraselenafulvalene, exhibits super-conductivity (SC) below 5.5 K. It has attractedinterest as a candidate for realizing the FFLO stateunder magnetic field owing to its highly two-dimensionalelectronic structure.
Another interest is that itsisostructural compound λ -(BETS) FeCl shows afield-induced SC phase under strong magnetic field,
7, 8) considered to be connected to that of the Ga salt (atzero field) as indicated by the measurements of alloyedsamples. Despite extensive experimental works, the-oretical investigation of SC in this compound from amicroscopic viewpoint has been lacking, which is thepurpose of this study.In the λ -type structure, the BETS molecules stackalong the a direction, forming a triclinic unit cell withspace group P¯1.
1, 2)
There are dimers of BETS moleculeswith large intradimer transfer integrals (termed t A ),which show further dimerization, i.e., a tetramer ofBETS forms the unit cell. The GaCl − closed-shell an-ion sheets lead to the highest-occupied molecular orbital(HOMO) of BETS forming a Q2D quarter-filled sys-tem in terms of holes. From its dimerized structure,whose limit of large dimerization will be a half-filled sys-tem, the electronic structure has an analogy with thewell-studied Mott transition system κ -(ET) X [ET =bis(ethylenedithio)tetrathiafulvalene]. In fact, by chem-ical substitution in the anions Ga X z Y − z ( X, Y =F, Cl,Br)
10, 11) or by choosing different donor molecules, theSC phase is suggested to locate next to the Mott insu-lating phase as in κ -(ET) X . Although the nature of the insulating state just nearthe SC phase remains to be clarified, i.e., whether it is ∗ [email protected] non-magnetic
10, 14) or antiferromagnetic,
12, 15) a recentNMR measurement in λ -(BETS) GaCl reports the de-velopment of spin fluctuations above the SC transitiontemperature. As for the SC gap function, early mea-surements show a two-fold symmetry within the conduc-tive plane, by means of the anisotropy of the upper criti-cal field H c217) and of the flux-flow resistivity. Ref. 18observes a dip structure in the angle dependence of theresistivity under magnetic field, when the magnetic fieldis applied parallel to the c axis. More recently, a heatcapacity measurement indicated the line-nodal gap of d -wave pairing, whereas a µ SR measurement reports apossible mixture of the extended s - and d -wave gaps. The electronic structure of λ -(BETS) GaCl has beendiscussed within the tight-binding model based onthe HOMO of the BETS molecule, where the trans-fer integrals are calculated using the extended H¨uckelmethod.
11, 21, 22)
The band structure near the Fermi en-ergy shows four bands since the unit cell contains fourBETS molecules as mentioned above. The calculatedFermi surface (FS) is similar to that of the κ -(ET) X , which consists of a pair of open and closed FS, despitethe difference in their molecular packings. One issue isthat, since the extended H¨uckel method contains semi-empirical parameters, there are estimates with apprecia-ble discrepancies.In this study, we present the band structure obtainedfrom first-principles calculations, and estimate the trans-fer integrals of the four-band model from the maximallylocalized Wannier orbitals (MLWO). Then, consideringthe pairing mechanism mediated by the spin fluctuations,we apply the random phase approximation (RPA) to thefour-band Hubbard model of λ -(BETS) GaCl . The re-sults show a d -wave-like SC gap and we will discuss itsorigin related to the spin susceptibility.
1. Phys. Soc. Jpn.
LETTERS
The first-principles band calculations were performedwithin density functional theory (DFT) with gen-eralized gradient approximation using WIEN2k, and a tight-binding model was derived by applyingMLWO
26, 27) scheme using wannier90 package.
Fig-ure 1(a) shows the two-dimensional band dispersionsnear the Fermi level for the experimental structuredata.
Dispersion along the interlayer direction is small,of the order of 0.1 meV. There are four bands originatedfrom HOMO of BETS, corresponding to the extendedH¨uckel bands. One point we note is that, since nearly flatband dispersions are present near Z point, the densityof states (DOS) exhibits a van-Hove singularity (vHS)slightly below the Fermi level, as shown in Fig. 1(b). InFig. 1(c), we show the FS obtained from the DFT calcu-lation, consisting of open and closed portions, which wecall FS0 and FS1 in the following. The former comes fromthe top band and the latter comes from the second-to-top band. The shape of the FS is similar to the extendedH¨uckel results.
11, 21, 22)
We regard these four bands asthe target bands and derive a tight-binding model byconstructing a MLWO on each molecule. As shown inFig. 1(a), the band structure of the four-band model,which includes the distant transfer integrals, accuratelyreproduces the DFT band dispersion.
Table I.
Transfer integrals and site-energy difference in meVfor λ -(BETS) GaCl , where the site-energy difference between theBETS-1(4) and BETS-2(3) is defined as ∆ E ≡ E − E . Thesuperscript, eH, stands for the extended H¨uckel results
11, 21, 22) andthe subscript, Fe, represents the results of λ -(BETS) FeCl , havingthe same crystal structure.Label t t eH 11) t eHFe 21) t eHFe 22) A 233 238 747 336B − − − − − − − − − − − − − − − − E −
29 – – –
We summarize the obtained transfer integrals and theenergy difference between the nonequivalent BETS, ∆ E ,in Table I, together with the extended H¨uckel results inthe literature. The notation of inter-molecular bonds isas shown in Fig. 1(d). The other transfer integrals notlisted here have absolute values less than 13 meV. In theRPA analysis below, we use all the obtained transfer inte-grals in the two-dimensional plane. As a common fea-ture among our results and the previous extended H¨uckelcalculations, t A is the largest, which is the intradimertransfer integral. This gives the splitting between the up-per two and lower two bands, approximately correspond-ing to the antibonding and bonding combinations of theHOMO. The transfer integrals along the stacking direc-tion t B and t C have close values in contrast with previousdata, which indicates that the degree of tetramerizationis smaller than previously discussed. Fig. 1. (a) Band structure obtained from the DFT calculation(red dotted curves) and the four-band model (blue solid curves),where the Fermi level is taken as zero energy. X, Z, U and U’represent (1/2, 0, 0), (0, 0, 1/2), (1/2, 0, 1/2) and ( −
21, 22)
The BETS molecules numbered in the bold(normal) characters are related through inversion symmetry.
The effective transfer integrals between the anti-bonding combination of HOMO of BETS dimers alongthe a direction can be approximated from the largedimerization limit as ˜ t B ≡ t B / t C ≡ t C /
2; thatin the transverse direction is t ⊥ ≡ ( t p + t q + t r ) /
2. Ourresults show a relation | ˜ t B | ≃ | ˜ t C | ≃ t ⊥ . Then, the BETSdimers possess a square-lattice-like network along the a and c directions, with weaker diagonal transfer integrals˜ t s ≡ t s / ≃ . t ⊥ or ˜ t t ≡ t t / ≃ . t ⊥ along the a + c direction. We can interpret the large DOS to be origi-nated from this relation since the ideal square lattice hasa singularity of the DOS at half-filling. Another recentDFT calculation based on the pseudopotential shows thesame result. Next, by introducing the on-site Coulomb interaction U to the four-band model, we study the spin susceptibil-ity χ sp and the SC gap function within the frameworkof the spin-fluctuation-mediated pairing mechanism. TheHamiltonian is described as H = X h iα : jβ i ,σ n t iα : jβ c † iασ c jβσ + H . c . o + ∆ E X i,α =2 , n iα + X i,α U n iα ↑ n iα ↓ , (1)
2. Phys. Soc. Jpn.
LETTERS where i and j are unit-cell indices, α and β specify thesites 1–4 in a unit cell [see Fig. 1 (d)], c † iασ ( c iασ ) isthe creation (annihilation) operator for spin σ at site α in unit cell i . t iα : jβ is the transfer integral between site( i, α ) and site ( j, β ), estimated as above, and h iα : jβ i represents the site pairs. n iασ is the number operatorfor electrons with spin σ on site α in unit cell i and n iα = n iα ↑ + n iα ↓ .To deal with the effect of the Coulomb interaction U ,we apply the multisite RPA, e.g., described in Ref. 30;here we focus on SC state in a situation where otherinstabilities are weaker. The Green’s function, as well asthe susceptibilities, pairing interaction, and SC gap func-tion are all 4 × ϕ ( k , iε n ) andits eigenvalue λ are obtained by solving the linearizedEliashberg equation. The critical temperature T c corre-sponds to the temperature where λ reaches unity. Be-cause we consider only the on-site interaction U , the spinsusceptibility is much larger than the charge susceptibil-ity. Therefore, we will show the spin susceptibility χ sp obtained from the largest eigenvalue for the lowest Mat-subara frequency. The SC gap function is presented in theband representation at the lowest Matsubara frequency.In the present calculation, we take 96 × k -point meshesand 16384 Matsubara frequencies. The on-site interac-tion is chosen as U = 0 . T = 0 .
006 eV ( ≃
70 K), the spin susceptibility χ sp has the maximum valuearound Q = ( Q a , Q c ) ≃ ( − π/ , π/
8) and a broadsubstructure around Q = ( Q a , Q c ) ≃ ( − π/ , π/ χ sp in the extended zone along the Γ-Z direction is shownand we define ˜ Q = − Q + (0 , π ). In Fig. 2(b), weshow the FS on the left and the SC gap function for thetop (second-to-top) band, namely, for FS0 (FS1) on thecenter (right) for λ ≃ . . The FS can approximatelybe regarded as an ellipse in the unfolded Brillouin zone,whereas FS0 and FS1 are slightly disconnected aroundthe k points where they approach to each other. In thefollowing, we will call this point as the crossing pointand the elliptic FS in the unfolded zone as the ‘extendedFS’. As we can see in the figure, wave-number vectors ˜ Q and Q correspond to the FS nesting. As for the SCgap function, first we note that for FS0 (FS1) it has apositive (negative) sign along almost the whole Brillouinzone. This gives rise to the large s -wave components inthe analysis below.To clarify the relation between the electronic structureand the SC gap, in Fig. 2(c) we plot the SC gap func-tions for k vectors within ± .
01 eV from the Fermi level,which corresponds to be about 1% of the band width (offour bands), from the Fermi level in the extended zone.Then, we can see that the SC gap changes its sign fourtimes along the extended FS reminiscent of a d -wave-like gap, which possesses two kinds of nodes, from whichgentle/steep increase of the SC gap is seen. We call themas gentle/steep node structures. In the figure, the twonesting vectors ˜ Q and Q connect the portions of FSwhere the SC gap has a large amplitude and shows signchanges. The positions of the FS where the SC gap am-plitude is large almost coincide with the positions giving rise to the vHSs, resulting in a high stability of the gapstructure. In fact, a similar analysis based on the effec-tive two-band model (the upper two bands) also givesrise to similar angular dependence. Fig. 2. (a) Spin susceptibility , whose unit is 1/eV, at T =0.006 eV, where the black solid (dashed) arrow represents thenesting vector Q ( Q ) and the arrow ˜ Q = − Q + (0 , π ). (b)The left panel is the FS and its nesting vectors, where “c.p.” standsfor the crossing point (see text) and the solid (dashed) curves rep-resent the FS0 (FS1). The center (right) panel shows the SC gapfor FS0 (FS1), where the red (blue) contours represent the positive(negative) SC gap sign, where the plotted SC gap represents theratio to its maximum value. In the center and right panels, thethick black curves represent the FS for which the gap function isplotted. (c) The SC gap function within ± .
01 eV from the Fermilevel, where “g.n. (s.n.)” stands for “gentle (steep) node” structure(see text).
Next, we attempt to decompose the SC gap into dif-ferent components, as has been done for κ -(ET) X . In the λ -type structure, the point group symmetry is lowas C i , therefore naturally different components mix. Therefore, we decompose the d -wave-like gap into pairingcomponents along the crystal axes. Here we take crystal c and a directions as x and y axes, respectively, to makethe correspondence between other systems clearer. In thischoice of axes, the SC gap structure in Fig. 2 (c) appar-ently looks close to a d x − y -wave gap since the nodesare nearly along the diagonal directions. We introducethe fitting function given as ϕ f ( k ) = C f + C fc cos( k x ) + C fa cos( k y )+ C fc + a cos( k x + k y ) + C fc − a cos( k x − k y )+ · · · (up to 20th nearest neighbors) , (2)where f is for the choice of the two bands representedby FS0 or FS1, the subscript represents the pairing di-rection, and the fitting variables C f are the weights ofthe basis function on the FS “ f ”. Longer range pairingstates as represented in Eq. (2) are also considered in theactual calculation, but have small contributions.We summarize the ratio of the fitting variables for thebasis function in Table II. In the case of FS0 [center
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LETTERS of Fig. 2 (b)], although the ratio in the c − a directionis the largest, the pairing ratio along the a direction issubdominant and comparable with that of the intra-unitcell. It is suggestive that the SC gap of the FS0 is af-fected by the pairing along the a direction, in which theBETS molecules stack. By contrast, for FS1, the threecomponents, namely the intra-unit cell as well as c and a directions, are comparable. As expected, the SC gap ofFS1 exhibits a two-dimensional pairing. Table II.
Ratio of the fitting variables of the basis function onthe FS “ f ”, from the d -wave-like gap function. We take the intra-unit-cell component C f as unity; to stress the different sign be-tween the two bands, we put different signs.Fitting variable FS0 FS1 C f − C f c − − C f a − C f c + a − C f c − a − − To compare with the previous studies of κ -(ET) X , we rewrite the ratio of the well-known SCgap, as d x − y -, d xy -, extended s -wave, which is thepairing with the same sign between the first (second)nearest neighbors. Note that we decompose the d -wave-like gap into the well-known SC gap and confirm that thesame components are obtained. We list the componentsof the SC gap in Table III. Several SC-gap componentsof the FS0 are comparable. By contrast, for the FS1,the components of the isotropic s - and extended s -wavepossess large negative value. We should note that, eventhough the component of the isotropic s -wave, which issame as the intra-unit-cell pairing, is large, this doesnot mean that the pairing, in real space picture, occurson the same BETS molecule, since the SC componentsare obtained in the “folded” Brillouin zone. Namely, ananisotropic pairing, e.g., the nearest neighbor pairing be-tween BETS-2 and BETS-1 or BETS-3 within the sameunit cell, is converted to an isotropic s -wave componentin the folded Brillouin zone because the pairing occurswithin the unit cell. Table III.
Ratio of the component of the well-known SC gap onthe both FSs based on Table II.SC gap component FS0 FS1isotropic s -wave 1.00 − s -wave 0.53 − d x − y -wave − s -wave − − d xy -wave 0.77 0.16 The results here that multiple components have com-parable values are noticeably different from the case of κ -(ET) X . In that case, the effective half-filled dimer Hubbard model shows the instability toward d x − y -wave SC in the extended zone
32, 37–42) while for the 3/4-filled model realistic parameters provide d xy -type -wavegap but with considerable extended s -wavecomponent. We can attribute such difference to thedifferent crystal structure geometries: κ -type has a D h point group symmetry, so that pure d x − y -wave can bestabilized but not pure d xy -wave in the extended zone. κ -(ET) X has parameters close to the triangular latticegiving rise to geometrical frustration effect, while ouranalysis here provides a square-lattice like network, as inthe high T c cuprates producing the stability of d x − y -wave, but with large mixing with other components ofthe well-known SC gap due to the low symmetry of thecrystal structure.Finally let us discuss the experimental works from theviewpoint of our results giving the d -wave-like gap. Theresults in the transport measurements indicating the two-fold symmetry of the angular dependence of the SC gapin this compound are compatible with our results sincethe d -wave-like gap only possesses the two-fold symme-try.
17, 18)
The existence of the nodal SC gap is suggestedby a recent measurement of the heat capacity, whichis in accordance with our results showing nodes along thediagonal directions. As for the nodal position, the flux-flow resistivity measurement suggests a dip structure ofthe resistivity when the magnetic field is applied parallelto the c axis. This is consistent with the d -wave-likegap we obtained, namely, the large SC gap around vHSand the steep node structure are present. A recent µ SRmeasurement suggests that the SC of this compound isa mixture of the extended s -wave and d -wave SC. Adirect comparison between our results might be difficultsince the method of decomposing the SC gap is differ-ent from ours here, magebut nevertheless, the mixture ofdifferent components is indeed consistent.In conclusion, we have obtained the DFT band struc-ture and the four-band model of the Q2D molecular con-ductor λ -(BETS) GaCl . Within the spin-fluctuation-mediated pairing mechanism, we study the SC gap func-tion and its properties by applying the RPA. The net-work of the BETS dimers shows a square-lattice-likestructure, giving rise to large DOS near the Fermi level.We propose that the FS nesting within this characteris-tic electronic structure results in the d -wave-like SC gap,which changes its sign four times along the extended FSand possesses the two-fold symmetry.To elucidate the pairing components of the d -wave-likegap, we have decomposed this gap function into the pair-ing components along the crystal axes, and estimate thepairing ratio for each FS. We have shown that the SC gapof FS0 is affected by the pairing in the stacking directionof the BETS and that the gap of FS1 exhibits a two-dimensional pairing. To compare the previous studies of κ -(ET) X , we transform the component of the pairing inthe crystal axes to that of the well-known SC gap func-tions, and show that the several SC gap components canbe comparable in both FSs.The effect of strong electronic correlation beyond RPA,which is expected to play a role since the system is con-
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LETTERS sidered to be located near the Mott transition, is an in-teresting issue left for future studies.
Acknowledgments
The authors acknowledge D. P. Sari and I. Watanabefor valuable discussions. HA is grateful to S. Yasuzukaand S. Imajo for useful discussions. This work is sup-ported by the Japan Society for the Promotion of Sci-ence KAKENHI Grants No. 16K17754, 18K03442 and26400377, Grants-in-Aid from the Yokohama AcademicFoundation, and the RIKEN iTHES Project.
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