Based on a single band Hubbard model and the fluctuation exchange
approximation, the effective mass and the energy band renormalization in
Na$_{0.33}$CoO$_2$ is elaborated. The renormalization is observed to exhibit
certain kind of anisotropy, which agrees qualitatively with the angle-resolved
photoemission spectroscopy (ARPES) measurements. Moreover, the spectral
function and density of states (DOS) in the normal state are calculated, with a
weak pseudogap behavior being seen, which is explained as a result of the
strong Coulomb correlations. Our results suggest that the large Fermi surface
(FS) associated with the $a_{1g}$ band plays likely a central role in the
charge dynamics.Comment: 5 pages, 5 figure

J. Kunes

Jian-Xin Li

Z. D. Wang

Zi-Jian Yao

English

arXiv

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Band structure renormalization and weak pseudogap behavior in
                    
                      
                        
                          Na

Based on a single-band Hubbard model and the fluctuation exchange approximation, the effective mass and the energy band renormalization in Na0.33 Co O2 is elaborated. The renormalization is observed to exhibit certain kind of anisotropy, which agrees qualitatively with the angle-resolved photoemission spectroscopy measurements. Moreover, the spectral function and density of states in the normal state are calculated, with a weak pseudogap behavior being seen, which is explained as a result of the strong Coulomb correlations. Our results suggest that the large Fermi surface associated with the a1g band plays likely a central role in the charge dynamics. © 2007 The American Physical Society.published_or_final_versio

Yao, ZJ

Wang, ZD

Li, JX

HKU Scholars Hub

TitleBand structure renormalization and weak pseudogap behavior inNa0.33CoO2: Fluctuation exchange study based on a single-band modelAuthor(s) Yao, ZJ; Li, JX; Wang, ZDCitation Physical Review B - Condensed Matter And Materials Physics,2007, v. 76 n. 21, article no. 212506Issued Date 2007URL http://hdl.handle.net/10722/57332Rights Creative Commons: Attribution 3.0 Hong Kong LicenseBand structure renormalization and weak pseudogap behavior in Na0.33CoO2: Fluctuationexchange study based on a single-band modelZi-Jian Yao,1,2 Jian-Xin Li,1 and Z. D. Wang2,11National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China2Department of Physics and Center of Theoretical and Computational Physics, The University of Hong Kong, Pokfulam Road,Hong Kong, ChinaReceived 27 August 2007; revised manuscript received 5 November 2007; published 19 December 2007Based on a single-band Hubbard model and the fluctuation exchange approximation, the effective mass andthe energy band renormalization in Na0.33CoO2 is elaborated. The renormalization is observed to exhibitcertain kind of anisotropy, which agrees qualitatively with the angle-resolved photoemission spectroscopymeasurements. Moreover, the spectral function and density of states in the normal state are calculated, with aweak pseudogap behavior being seen, which is explained as a result of the strong Coulomb correlations. Ourresults suggest that the large Fermi surface associated with the a1g band plays likely a central role in the chargedynamics.DOI: 10.1103/PhysRevB.76.212506 PACS numbers: 74.70.b, 71.18.y, 71.27.a, 74.25.JbThe layered oxide NaxCoO2 has attracted much attentiondue to its possible connection to the high-Tc superconductiv-ity since the discovery of superconductivity in it.1,2 This ma-terial consists of two-dimensional CoO2 layers, where Coform a triangular lattice, making the NaxCoO2 a possiblerealization of Anderson’s resonating valence bond state.3 Byhydration, it becomes a superconductor with Tc5 K inNa0.35CoO2·1.3H2O.1 The unhydrated compound NaxCoO2exhibits a rich phase diagram: a paramagnetic metal x0.5, a charge-ordered insulator x=0.5, a “Curie-Weissmetal” x0.7, and a magnetic order state x0.75.4Experimentally, it is indicated that NaxCoO2 seems to bestrongly electronic correlated.2,5,6 Angle-resolved photoemis-sion spectroscopy ARPES measurements show a strongmass renormalization the effective mass is about five to tentimes larger than the bare mass for x=0.6.7 Decreasing theNa concentration to x=0.3, the cobaltates appear to be lesselectron correlated with a weaker but still apparent energyband renormalization factor 2.8 For the hydrated com-pounds, the effective mass is estimated to be 2–3.9 In ad-dition, recent experiments report that Na0.33CoO2·1.3H2Odisplay certain pseudogap behaviors such as the decreasingof the Knight shift10 and the density of states DOS at theFermi level below 300 K.11 Optical spectroscopy measure-ments for Na0.2CoO2 and Na0.5CoO2 also suggest the incipi-ent formation of pseudogap.12 However, unlike thepseudogap effect observed in the high-Tc cuprates, thepseudogap behavior here is rather weak. The renormalizationof the energy band and the pseudogap formation are directlyrelated to the electronic structure, such as the quasiparticlespectral function, the quasiparticle dispersion, and the Fermisurface FS topology. Therefore, it is of importance and sig-nificance to study the normal state quasiparticle dynamics.It is believed that the topology of the FS plays an impor-tant role in the unconventional superconductivity. The localdensity approximation LDA calculations13,14 predict a largeFS associated with the a1g band and six pockets associatedwith the eg band. However, intriguingly, the small pocketshave not been observed in the ARPES measurements.8,9,15The unexpected inconsistency of the topology of the FS be-tween the LDA results and the ARPES measurements hasaroused much controversy. One possibility might be due tothe surface effect in the ARPES measurements. From anotherviewpoint, Zhou et al. suggested that the strong Coulombinteractions may induce significant renormalization of theband structure which pulls the eg band down from the Fermienergy.16 Furthermore, the eg band is suggested to be rel-evant to the spin-triplet superconductivity in some theoreticaland numerical studies.17,18 On the other hand, the possibilityof the spin-singlet superconductivity is also suggested ac-cording to recent Knight-shift measurements.19,20 Taking theabove considerations into account, a single-band model fo-cusing on the a1g band may be a starting point as a minimalmodel of NaxCoO2.In this Brief Report, we study the normal state electronicstructure in Na0.33CoO2 on the basis of the single-band Hub-bard model and the fluctuation exchange FLEX approxima-tion. Several experimental features, such as the band renor-malization and the weak pseudogap behavior, are wellreproduced, which suggests an important role of the holelikeFermi surface centered around the  point in the quasiparti-cle dynamics. We start with the two-dimensional single-bandHubbard model given byH = ij,tijci† cj + H.c. + Uini↑ni↓ − ini, 1where tij denotes the hopping term in the following, we willuse t1, t2, t3, and t4 to denote the hoppings along the nearestto the fourth nearest neighbors, U the on-site Coulomb re-pulsion, and  the chemical potential. To reproduce the largeFS around the  point, we set the parameters t1 , t2 , t3 , t4 ofthe bare dispersion k to be −1,0 ,0 ,0.2, where t1 is set tobe the unit hereafter. According to the numerical result forthe bandwidth,13,21 one can get t1=130 meV. We note thatthe main feature of the bare dispersion is similar to that ofthe band with an a1g character in the LDA calculation by Leeet al.14The FLEX approximation is employed in our study,which has been applied to the two-dimensional HubbardPHYSICAL REVIEW B 76, 212506 20071098-0121/2007/7621/2125064 ©2007 The American Physical Society212506-1model in the literatures.22–25 As a self-consistent approxima-tion, the FLEX approximation solves the Dyson’s equationwith a primarily random-phase approximation+ladder-typeeffective interaction self-consistently. Based on the scenarioof the FLEX approximation, the self-energy is given by	k =TNq Vef fk − qGq , 2whereVef fq =32U2sq +12U2cq . 3The spin susceptibility issq =¯q1 − U¯q, 4and the charge susceptibility iscq =¯q1 + U¯q, 5with the irreducible susceptibility,¯q = −TNk Gk + qGk . 6The electron Green’s function is give byGk = in − k − 	k−1. 7In the above equations, kk , in and qq , in are used,and T is the temperature. These equations are solved self-consistently, where N=6464 k-point meshes, and up to2048 Matsubara frequencies n= 2n+1T are taken. Theelectron density is determined by the chemical potential from the equation 	n=1− 2TN kGk.As an important physical quantity related closely to thestrength of the electronic correlations, the effective mass m*is given bymm*= Z1 + mkF k Re 	k,0k=kF , 8whereZ = 1 − Re 	kF,=0−1 9is the renormalization constant. The self-energy with real fre-quencies 	k , is obtained with the Padé approximants,26which analytically continue 	k , i from the Matsubara fre-quencies to the real-frequency axis. The ratio of the effectivemass to the bare mass m* /m with different on-site Coulombrepulsions U near the FS point A in Fig. 1 is plotted in Fig.2, where m* /m increases monotonously with the increasingof U. It is noticeable that when U is less than 4, the enhance-ment of the effective mass is inapparent 1.2. With astrong Coulomb repulsion U60.8 eV, the effectivemass is enhanced significantly. Comparing the calculated ef-fective mass with the ARPES measurements 2 for x0.3, we set U6 in the present simple model to reflectthe strong correlation effect in Na0.33CoO2.27 The strongelectronic correlations are evident from Fig. 3a, in whichthe spectral function of quasiparticles defined as Ak ,=−1 Im Gk , at the FS point A in Fig. 1 is plotted. Onecan see that the sharp quasiparticle peak is suppressed withstrong Coulomb repulsion. For U=2 and T=0.1, the spectralfunction shows a sharp peak near the Fermi energy. Increas-ing the Coulomb repulsion to U=6, the spectral function isbroadened and the spectral weight at the Fermi level is re-duced greatly.For U=6, we now investigate the renormalization of thequasiparticle energy band. The renormalized band dispersionis determined by the position of the peak in Ak ,. Thecalculated result at T=0.1 is shown in Fig. 3b, where thesolid and the dashed lines denote the renormalized and thebare energy bands, respectively. One can see that the band-width is compressed obviously due to the strong Coulombinteractions. From Fig. 3b, one can determine that thebandwidth of the bare energy band is 7.7 and the renormal-ized one is 5.0, which give a band renormalization factorof 1.54. This is consistent with the ARPES measurements forx=0.3.8 Moreover, we find that the band renormalizationshows an anisotropy along the −K and −M directions,namely, the renormalization along the −K direction isFIG. 1. The Fermi surface of Na0.33CoO2 for t1 , t2 , t3 , t4= −1,0 ,0 ,0.2. The gray area denotes the unoccupied electronstates. Point A near the Fermi surface indicates the k point wherem* is calculated.FIG. 2. The ratio of the effective mass to the bare mass m* /m atpoint A as indicated in Fig. 1 versus the on-site Coulomb repulsionU at T=0.1.BRIEF REPORTS PHYSICAL REVIEW B 76, 212506 2007212506-2stronger than that along the −M direction. A similar aniso-tropy has been found in the ARPES experiments for x=0.6,7though the experimental result seems stronger than what weget here. We note that the compound NaxCoO2 with x=0.6 is,in fact, in the range of the Curie-Weiss metal, in which thecorrelation is stronger than that in the paramagnetic metalwith x=0.3. So a weaker anisotropy in the renormalizationreported here is expected. This anisotropy originates fromthe nesting of the Fermi surface along the -K direction, asshown in Fig. 1.We now turn to address the weak pseudogap behavior.This is manifested in the suppression of DOS at the Fermilevel. We present the  dependence of DOS at different tem-peratures in Fig. 4. With the decrease of temperature fromT=0.7 dash-dotted through T=0.5 dotted and T=0.3dashed to T=0.1 solid, a weak suppression of DOS nearthe Fermi energy is evident, suggesting the opening of aweak pseudogap. The weak pseudogap behavior is alsomanifested in the quasiparticle spectral function Ak ,which measures the probability to find quasiparticles at mo-mentum k and frequency . As shown in Fig. 5a, with thedecrease of temperature from the dashed line to the solidline, the spectral weight Ak , at the k-point B shown inFig. 1 is transferred away from the region around =0,producing a weak secondary maximum at EF. Thus, aweak pseudogap is formed.This weak pseudogap behavior is a consequence of strongCoulomb repulsion. To show this, we also present the resultswith a smaller U=2 at T=0.1,0.3,0.5,0.7 in the inset of Fig.4. Different from the case of U=6, the density of states is notsuppressed with the decreasing of temperature at all in fact,there is no appreciable difference between them, so only oneline can be seen from the figure. For a more detailed discus-sion, we also present the spectral function at the k-point Bshown in Fig. 1 with a fixed temperature T=0.1 for U=2dotted, U=4 dash-dotted, and U=6 solid in Fig. 5a. Itis seen that the spectral weight near the Fermi energy issuppressed, and consequently, a secondary maximum showsup gradually with the increase of Coulomb correlations. Notethat since the depression and the secondary maximum disap-pears for U=2, no pseudogap is expected for this case andthat for U2. The weak pseudogap we refer here is based onFIG. 3. a The effect of the on-site Coulomb repulsion on thespectral function at point A of Fig. 1. Solid line: U=6, T=0.1.Dashed line: U=2, T=0.1. b Solid line: the renormalized banddispersion for U=6 and T=0.1. Dashed line: the bare band disper-sion. Dotted line: the Fermi energy.FIG. 4. The total density of states DOS versus energy for U=6. Solid line: T=0.1; dashed line: T=0.3; dotted line: T=0.5; anddash-dotted line: T=0.7. Inset: DOS plots for U=2 at T=0.1,0.3,0.5,0.7.FIG. 5. a The energy dependence of the spectral function at themomentum indicated as point B in Fig. 1. Below the Fermi energy,there is a small peak in the spectral function for U=6 and T=0.1solid line. Dashed line: U=6 and T=0.7; dash-dotted line: U=4,T=0.1; and dotted line: U=2, T=0.1. The arrow points to the loca-tion of the secondary maximum. Inset: zoom-in view of the solidrectangle. b and c The real and the imaginary parts of theself-energy at T=0.1 for U=6 solid and U=2 dotted. The arrowindicates the position of the extrema in Im 	k ,.BRIEF REPORTS PHYSICAL REVIEW B 76, 212506 2007212506-3the observation that the spectral function consists of a peakand a weak secondary maximum not a peak. In fact, thereal part of the denominator of the Green’s function has onlyone pole, which can be seen from the real part of the self-energy shown in Fig. 5b. It is the extremum of the imagi-nary part of the self-energy around =−4 that produces theweak maximum of the spectral function, as shown in Fig.5c. In the case of a smaller Coulomb repulsion U, such asU=2, a usual Fermi-liquid form of the self-energy is pre-served the dotted lines in Figs. 5b and 5c, so it gives awell-defined quasiparticle peak in the spectral function thedotted line in Fig. 5a. This secondary maximum mightsuggest a “shadow band”,28 which occurs when a short-rangespin correlation is developed with the increase of Coulombrepulsion. Therefore, the weak pseudogap behavior reportedhere may be due to the spin fluctuation in the strong corre-lated regime.29A similar weak pseudogap behavior was reported by Yadaand Kontani30 based on a multiorbital model with the ab-sence of small hole pockets. Our results based on the single-band model provide further support for the sinking down ofthe small hole pockets. In our opinion, the agreement be-tween the two models suggests that the multiorbital effectmay play a minor role in the mechanism of the pseudogapformation.In summary, we have studied the quasiparticle band renor-malization and the pesudogap behavior in Na0.33CoO2 basedon the single-band Hubbard model on a two-dimensional tri-angle lattice. The renormalization of the band structure andits anisotropy of the -K and -M directions have beenelaborated. The estimated effective mass is consistent withthe ARPES measurements. In the meantime, a weakpseudogap behavior is reproduced, which is explained as theresult of the strong spin fluctuations. Our results are qualita-tively consistent with experiments as well as the theoreticalcalculations based on a multiorbital model.We thank Q.-H. Wang for helpful discussions. The workwas supported by the NSFC 10525415, 10474032, and10429401, the RGC grants of Hong Kong HKU7045/04Pand HKU-3/05C, the State Key Program for Basic Researchof China 2006CB921800 and 2006CB601002, and MOENCET-04-0453.1 K. Takada, H. Sakurai, E. Takayama-Muromachi, F. Izumi, R. A.Dilanian, and T. Sasaki, Nature London 422, 53 2003.2 Y. Wang, N. S. Rogado, R. J. Cava, and N. P. Ong, Nature Lon-don 423, 425 2003.3 G. Baskaran, Phys. Rev. Lett. 91, 097003 2003.4 M. L. Foo, Y. Wang, S. Watauchi, H. W. Zandbergen, T. He, R. J.Cava, and N. P. Ong, Phys. Rev. Lett. 92, 247001 2004.5 R. Jin, B. C. Sales, P. Khalifah, and D. Mandrus, Phys. Rev. Lett.91, 217001 2003.6 F. C. Chou, J. H. Cho, P. A. Lee, E. T. Abel, K. Matan, and Y. S.Lee, Phys. Rev. Lett. 92, 157004 2004.7 H.-B. Yang, S.-C. Wang, A. K. P. Sekharan, H. Matsui, S. Souma,T. Sato, T. Takahashi, T. Takeuchi, J. C. Campuzano, R. Jin etal., Phys. Rev. Lett. 92, 246403 2004.8 H.-B. Yang, Z.-H. Pan, A. K. P. Sekharan, T. Sato, S. 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Lett. 98,136401 2007 and Ref. 14.28 J. J. Deisz, D. W. Hess, and J. W. Serene, Phys. Rev. Lett. 76,1312 1996.29 We note that with stronger Coulomb repulsion, e.g., U=10, noqualitative difference is found. However, as expected, thesecondary-maximum structure in Ak , becomes more obviousas well as the pseudogap effect.30 K. Yada and H. Kontani, J. Phys. Soc. Jpn. 74, 2161 2005.BRIEF REPORTS PHYSICAL REVIEW B 76, 212506 2007212506-4

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Band structure renormalization and weak pseudogap behavior in Na0.33CoO2: Fluctuation exchange study based on a single-band model

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Band structure renormalization and weak pseudogap behavior in Na_{0.33}CoO_2: Fluctuation exchange study based on a single band model

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