Fermi Surface Deformation near Charge-Ordering Transition
aa r X i v : . [ c ond - m a t . m t r l - s c i ] N ov Typeset with jpsj3.cls < ver.1.1 > L etter Fermi Surface Deformation near Charge-Ordering Transition
Kazuyoshi Yoshimi , , Takeo Kato ∗ , , and Hideaki Maebashi Department of Physics, University of Tokyo, Tokyo 113-8656 Nanosystem Research Institute “RICS”, National Institute of Advanced Industrial Science and Technology, Ibaraki 305-8568 Institute for Solid State Physics, University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa 277-8581
We study the deformation of a Fermi surface (FS) near charge-ordering (CO) transition. By applying afluctuation-exchange approximation to the two-dimensional extended Hubbard model, we show that theFS is largely modified by strong charge fluctuations when the wave number of the CO pattern does notmatch the nesting vector of the FS in a noninteracting system. We also discuss the enhanced anisotropy inquasiparticle properties in the resultant metallic state.
KEYWORDS: organic conductors, charge ordering, Fermi surface, anisotropic electron transport, magne-toresistance
The shape of a Fermi surface (FS) is an important factor indetermining the electronic properties of a metal. For many or-ganic conductors, band structures and FSs have been obtainedby the extended H¨uckel method
1, 2) and first-principles calcu-lations.
3, 4)
The band structures thus obtained provide a basisfor constructing e ff ective tight-binding models, which explainthe various electronic phases in di ff erent types of organicsalts in a unified way. The shape of the FS is probed withhigh sensitivity by transport measurements such as angle-dependent magnetoresistance oscillations, the Shubnikov-deHaas e ff ect and magnetoresistance. However, the shape of the FS is modified by strong elec-tronic exchange-correlation e ff ects, which are neglected inband calculations. Deformation of the FS has already beendiscussed on the basis of a single-band Hubbard model tostudy high- T c superconductors. Theoretical studies using thesecond-order perturbation theory, the one-loop approxi-mation and the fluctuation-exchange (FLEX) approxima-tion
11, 12) indicate that the FS is gradually deformed as antifer-romagnetic (AF) spin fluctuations develop near the AF tran-sition. In these studies, the FS is deformed so that its nestingcondition improves at the wavenumber of the AF ordering.However, the degree of deformation obtained is small.
In organic conductors with a 3 / ff er-ent from that of AF spin fluctuations. The e ff ect of chargefluctuations has been clarified in recent theoretical studies of,for example, superconductivity mediated by charge fluctu-ations, non-Fermi-liquid behavior,
18, 19) anomalous en-hancement of Pauli paramagnetism and instability towardinhomogeneous electronic states.
Although the electronexchange-correlation e ff ect is also expected to modify the FSshape via charge fluctuations, such an e ff ect has not yet beenstudied.In this paper, we study how charge fluctuations a ff ect theFS shape. By employing a FLEX approximation, we demon-strate that in contrast to AF spin fluctuations, FS is largelymodified near the CO transition. We emphasize that this re-markable change in FS originates from the large discrepancy ∗ [email protected] Fig. 1. (a) Schematic of the extended Hubbard model with hopping integral t and intersite Coulomb interactions V and V ′ considered in this paper and(b) its relation to the crystal structure of θ -ET salts. between the CO wave vector and the nesting vector in a nonin-teracting system. As a result of FS deformation, quasiparticleproperties become more anisotropic near the CO transition.We also discuss the experimental relevance of our results.We consider the extended Hubbard model on a two-dimensional square lattice. The Hamiltonian is H = X h i , j i ,σ t ( c † i σ c j σ + H . c . ) + U X i n i ↑ n i ↓ + V X h i , j i n i n j + V ′ X hh i , j ii n i n j , (1)where c † i σ ( c i σ ) is the creation (annihilation) operator of anelectron on site i with spin σ = ↑ or ↓ , n i σ = c † i σ c i σ and n i = n i ↑ + n i ↓ . Here, t is the hopping integral between neighbour-ing sites, and U and V are the onsite and nearest-neighbourCoulomb interactions (denoted with h i , j i ), respectively. In thelast term of Eq. (1), we have added an intersite Coulomb re-pulsion V ′ between next-nearest-neighbour sites in one diag-onal direction, as shown in Fig. 1 (a).The present Hamiltonian (1) is an e ff ective model for COin the series θ -ET MM ′ (SCN) [ET = BEDT-TTF, M = Tl, Rb,M ′ = Co, Zn] (abbreviated as θ -ET salts),
22, 23) which exhibitsthe quasi-two-dimensional molecular arrangement shown inFig. 1 (b). Owing to the crystal structure of θ -ET salts, theoverlap between molecular orbits is approximately restrictedto the x and y directions, whereas strong intersite Coulombrepulsion remains along one diagonal direction. The Hamil-tonian (1) has been adopted as a minimum model in several etter Author Name Q Fig. 2. Static charge susceptibility χ c ( q ,
0) for U = k B T = .
1. TheCO transition occurs at V = . Q = (2 π/ , π/
3) is the wave vectorcharacterizing the CO pattern. theoretical studies of the CO phenomena in θ -ET salts. We introduce here a FLEX approximation to study theFS deformation caused by intersite Coulomb interactions. Theone-particle Green’s function G ( k ) is related to the self-energy Σ ( k ) through the Dyson equation G ( k ) = i ω n + ˜ µ − ε k − Σ ( k ) , (2)with a combined notation k of the wave number k and thefermionic Matsubara frequency i ω n . Here the noninteractingband dispersion is given by ε k = t (cos k x + cos k y ) and theHartree term is absorbed in the chemical potential as ˜ µ = µ − ( U / + V + V ′ ) n with the average number n of electrons persite. In the FLEX approximation, the self-energy is expressedas Σ ( k ) = − Z q G ( k − q ) (cid:2) V c ( q ) + V s ( q ) (cid:3) , (3) V c , s ( q ) = v c , s ( q )1 + v c , s ( q ) χ ( q ) , (4)where V c , s ( q ) represents the exchange-correlation interactionpotential, v c ( q ) = U + V (cos q x + cos q y ) + V ′ cos( q x + q y ), v s ( q ) = − U , χ ( q ) = − R k G ( k + q ) G ( k ) and R q = T P ω n R π − π R π − π d q / (2 π ) give the combination of the sum andintegral with respect to the bosonic Matsubara frequency andmomentum. We omitted the particle-hole and particle-particleladder diagrams, because SU(2) symmetry of the spin sectorof electrons is not relevant if the system is far from the AF in-stability.
15, 16)
This simple approximation is su ffi cient to studynontrivial exchange-correlation e ff ects on the FS.Throughout this paper, we fix the hopping integral, the on-site interaction, and the temperature as t = U =
5, and k B T = .
1, respectively, and choose V ′ = V for simplicity.We checked that no AF instability occurs in the entire param-eter region considered. By combining Eqs. (2)-(4), we self-consistently determine G ( k ) at 3 / n = /
2) with arelative precision of 10 − . We adopt the units e = ~ = χ c ( q , = χ ( q , + v c ( q ) χ ( q , . (5) -0.5 0 0.5 1 1.5 2 2.5 non-interacting k x π π k y π /2 π /2 π -π-π π k y k x k y (a)(b) Q k x π -π-π π Fig. 3. (a) V dependence of FS. One quarter of the FS is plotted in the mainpanel with the entire FS shown in the inset. (b) Contour plot of Re Σ R ( k , V = . As V ( = V ′ ) increases, the charge susceptibility increases ata specific wave vector Q = (2 π/ , π/ V = V c = . χ c ( Q ,
0) indicates a phasetransition into three-fold-type CO.
In the following discus-sion, we focus on the region V . V c , in which charge fluctua-tions become large enough to modify the shape of the FS.In Fig. 3 (a), we show a FS determined by ε k F + Re Σ R ( k F , − ˜ µ = , (6)where k F is the Fermi wave vector and Σ R ( k F ,
0) is obtainedfrom the analytic continuation of Σ ( k F , i ω k ) in the upper plane( ω k >
0) by using the Pad´e approximant. For weak inter-site Coulomb repulsion ( V = V ( = V ′ ) in-creases, the FS shape gradually changes. Near the CO tran-sition ( V = . Q ofthe CO pattern is also shown by the thick arrow in the insetof Fig. 3 (a). The FS shape is clearly modified to assist thenesting condition; a flat part of the FS, which is spanned withthe wave vector Q , grows as the system approaches the COtransition. This is the main result of this study.In the present study, the CO wave vector Q is a large mis-match for the nesting vector, which is determined by the shapeof the FS for a noninteracting system, because Q is deter- . Phys. Soc. Jpn. L etter Author Name 3 mined not only by χ (q) but also by v c ( q ) = U + V (cos q x + cos q y ) + V ′ cos( q x + q y ). However, because Q AF is not af-fected by the bare spin interaction v s ( q ) = − U and is deter-mined only by χ ( q ,
0) [Eq. (4)], the wave vector Q AF of the AFfluctuations already properly spans the FS of the noninteract-ing system. As a result, the modification of the FS induced byAF spin fluctuations is small compared with that induced bycharge fluctuations. Note that the FS is already modified within the Hartree-Fock (HF) approximation, in which the self-energy does notinclude any fluctuation and is taken as Σ ( k ) = ( − / R q G ( k − q )( v c ( q ) + v s ( q )). In the HF approximation, the FS is modi-fied simply by changing the e ff ective hopping integrals ˜ t i j = t i j − P σ V i j h c † i σ c j σ + h . c . i , where t i j and V i j are the hopping in-tegral and Coulomb interaction between sites i and j , respec-tively. Thus, in the HF approximation the FS retains its roundshape, whereas the FS in the FLEX approximation nests betterbecause of its flat part, which appears only when fluctuationsare fed back into Σ ( k ).To gain a deeper understanding of how the FS deforma-tion depends on charge fluctuations, we expand Eq. (6) with˜ µ = µ + δµ and k F = k F , + δ k F ˆ n F , , where µ and k F , are the chemical potential and the Fermi wave vector for thenoninteracting system, respectively, and ˆ n F , = k F , / | k F , | isa unit vector specifying a position on the FS. A shift in thechemical potential due to Coulomb repulsion is then given by δµ ≃ R d k δ ( ε k − µ )Re Σ R ( k , / R d k δ ( ε k − µ ), which cor-responds to an average of Re Σ R ( k ,
0) on the noninteractingFS. Similarly, a shift of FS is given by δ k F ≃ δµ − Re Σ R ( k F , , −| v F , | , (7)where v F , = ∂ε k /∂ k | k = k F , is the Fermi velocity. Here weused the fact that the direction of v F , is always directed op-posite to k F , for the present hole-type FS. From Eq. (7), thedeformation direction of FS (i.e., the sign of δ k F ) is deter-mined by the relative magnitude of Re Σ R ( k ,
0) on the FSof the noninteracting system ( k = k F , ). In Fig. 3 (b), weshow a contour plot of Re Σ R ( k ,
0) for V = .
43, where theFSs for the interacting and noninteracting systems are shownby the solid line and dashed line, respectively. The resultsshow that the deformation of the FS is mainly induced by therelative increase of the exchange-correlation energy to holes(i.e., the decrease in Re Σ R ( k ,
0) in terms of electrons) around k = ( π/ , π/ = Q /
2, which corresponds to a hot spot su ff er-ing strong electron scattering because of charge fluctuations.As a result of FS deformation, the quasiparticle-scatteringanisotropy strengthens because the anisotropic self-energy isenhanced near CO. In Fig. 4, we show the quasiparticle scat-tering rate Im Σ R ( k F ,
0) as a function of k F . The inset of Fig. 4shows three FS points: the − k x direction (A), + k y direction(B) and + k x direction (C). The results show that Im Σ R ( k F , hot region BC (the flat part of theFS) as the system approaches the CO transition, whereas itremains small in the cold region AB. Formation of hot andcold regions reflects anisotropy in the exchange-correlationinteraction potential, which becomes strong near the CO tran-sition.This anisotropic quasi-particle scattering can be observedvia the in-plane anisotropy of the electron transport such as Fig. 4. Plot of − Im Σ R ( k F ,
0) as a function of the position on the FS. Insetshows the shape of the FS for V = . k y ≥
0. Threerepresentative positions on the FS are denoted by A, B, and C. -8-4 0 4-8-4 0 4 (b)(a) -8-4 0 4 0 0.5 1 1.5 2-8-4 0 4 0 0.5 1 1.5 2
Fig. 5. Spectral weight A ( k , ω ) calculated for (a) V = V = . A ( k , ω ) along the line k x = k y ( k x = − k y ). Solid white lines are the band dispersions determined by ω = ε k + Re Σ R ( k , ω ). electronic and thermal conductivities. Another experimen-tal probe suitable for observing both the FS deformationand anisotropic quasiparticle scattering is magnetoresistance.For example, out-of-plane conductivity σ zz under an in-planemagnetic field B may be calculated from the Boltzmann equa-tion in the relaxation-time approximation: σ zz ∝ t z Z d k F | v F | τ k F + ( Ω k F τ k F ) , (8) J. Phys. Soc. Jpn. L etter
Author Name where t z ( ≪ t ) is the hopping integral between conductionplanes, 1 /τ k F = | Im Σ R ( k F , | is the quasiparticle lifetime, v F is the Fermi velocity and Ω k F = | v F × B | / | k F | is the cyclotronfrequency. Eq. (8) indicates that the dependence of σ zz on theorientation of the in-plane magnetic field reflects the shape ofthe FS (through v F and k F ) as well as the anisotropy in thequasiparticle lifetime.
36, 37)
The calculation presented in thispaper thus indicates that magnetoresistance in organic con-ductors is significantly a ff ected by charge fluctuations nearthe CO transition.Finally, we discuss the spectral weight A ( k , ω ) = − Im G R ( k , ω ) /π , which can be measured directly byangle-resolved photoemission spectroscopy (ARPES) experi-ments. We show the spectral weights for V = V = . k x = k y (in the hot region) is largely suppressed nearthe Fermi energy when the system approaches the CO transi-tion, while that on the line k x = − k y (in the cold region) nearthe Fermi energy remains unchanged. Note that, below theFermi energy, a kink structure appears on the line k x = − k y .We attribute this high-energy kink structure to high-energycharge fluctuations. We end this paper with a brief remark on CO phenom-ena in θ -ET salts. In actual experiments on θ -ET salts, thehorizontal-type CO, which is characterized by the wave vec-tor Q h = ( π/ , π/
2) in our notation, is stabilized below the COtransition temperature.
The horizontal-type CO cannot bereproduced by the RPA (or mean-field) calculation on the ba-sis of the simple extended Hubbard model,
24, 25) and its originhas been discussed theoretically by considering the electron-phonon interaction.
Although the horizontal-type CO can-not be obtained in our calculation, we expect that our con-clusion on the FS deformation holds true also for this type ofCO, because the rapid increase of V c ( q ) toward the CO tran-sition and the disagreement between the charge pattern andthe nesting vector are essential to our conclusion, regardlessof the detailed mechanism of the CO phenomena.In summary, we have studied the shape of the FS nearthe CO transition within the FLEX approximation. We haveshown that, as the system approaches the CO transition, theFS is remarkably modified from that obtained from band cal-culations. This phenomenon is induced by strong charge fluc-tuations with a wave vector which is a large mismatch withthe nesting vector of the FS for a noninteracting system. Onthe resultant FS, there appears a hot and cold region with dif-ferent quasiparticle lifetimes in each region. Our result maybe observed experimentally via ARPES as well as by electrontransport measurements such as those of in-plane magnetore-sistance anisotropy. Note that when the system is close to theCO transition, the FS may be sensitive to changes in temper-ature and / or pressure because strength of charge fluctuationschanges rapidly. Our calculation can also be extended to moregeneral CO models including those of inorganic systems.T. K. thanks H. Mori, T. Osada, S. Shin, S. Sugawara,and M. Tamura for providing useful information on exper-iments with organic conductors. This study was supportedby a Grant-in-Aid for Scientific Research in Priority Area of Molecular Conductors (No. 21110510, 20110003, 20110004)from the Ministry of Education, Culture, Sports, Science andTechnology, Japan.
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