Inhomogeneous Electronic Distribution in High-Tc Cuprates
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Inhomogeneous Electronic Distribution in High-T c Cuprates
Shigeru Koikegami ∗ , Masaru Kato , and Takashi Yanagisawa Second Lab, LLC, Tsukuba, Ibaraki 305-0045, Japan Department of Mathematical Sciences, Osaka Prefecture University, Sakai 599-8531, Japan Electronics and Photonics Research Institute, AIST Tsukuba Central 2, Tsukuba, Ibaraki 305-8568,Japan
We theoretically investigate the doping evolution of the electronic state of high-T c cuprateon both sides of the half-filling on the basis of the three-dimensional three-band Hubbardmodel with a layered structure using the Hartree-Fock approximation. Once a small amountof holes or electrons are doped into the half-filled state, our model exhibits the charge-transferinsulator-to-metal transition along with a chemical potential jump. At the same time, thedoped holes or electrons are inhomogeneously distributed, and they tend to form clusters inthe vicinity of the half-filling. This suggests the possibility of microscopic phase separationwith the separation between the metallic and the insulating regions.
1. Introduction
As shown by the photoemission spectroscopy, the undoped high-T c superconductingcuprate (HTSC) is a charge-transfer insulator, where the Cu 3d electrons are almost lo-calized by the strong electron correlation. When the rare-earth element is substituted outof the two-dimensional (2D) CuO layers and the number of holes or electrons doped intothe CuO layers is increased, Cu 3d electrons hybridized with O 2p electrons achieve itin-erancy and display superconductivity. Moreover, in slightly hole-doped La − x Sr x CuO with0 < x < .
12, the chemical potential shift suppression is observed by photoemission spec-troscopy (PES).
2, 3
The electronic phase separation between the antiferromagnetic insulatingphase and the superconducting phase is considered to be one of the reason for the suppres-sion of the shifting of the chemical potential. The phase separation assumes that the dopedcarriers are inhomogeneously distributed due to the strong electron correlation.Many experimental findings have suggested that the electrons under such circumstancesfavor some types of spontaneous ordering in certain doped regions. For instance, in or-der to elucidate the anomalous suppression of the superconducting transition tempera-ture in La . Ba . CuO , the spin and charge correlations in La . Ba . CuO or(La,Nd) − x Sr x CuO with x ≈ .
125 have been intensively studied, and the stripe order hasbeen observed by neutron scattering.
The stripe order has also been observed by x-rayscattering.
Furthermore, an electron paramagnetic resonance study showed that micro- ∗ E-mail: [email protected] 1/14 . Phys. Soc. Jpn.
Full Paper scopic electronic phase separation occurs in La − x Sr x Cu . Mn . O with 0 . ≤ x ≤ . and a Cu nuclear magnetic resonance (NMR) study suggested that a large charge droplet(’blob’) is formed in the electron-doped Nd . Ce . CuO − δ . Much theoretical works has also been performed to study the behavior of the doped carri-ers in HTSC. Pioneering works adopting the Hartree-Fock approximation (HFA) have studiedthe stripe order in La . Ba . CuO on the basis of the 2D one-band Hubbard model
20, 21 or the 2D two-band Hubbard model. Furthermore, dynamical mean field theory (DMFT)has been exploited to study the stripe phase on the basis of the 2D Hubbard model with L non-equivalent sites, where L = 8 , . . . , The DMFT approach has also been adopted toanalyze the three-band Hubbard model.
In some of these works,
25, 27 the DMFT approachwas combined with the local density approximation (LDA). Another study considered thepossibility of two-sublattice antiferromagnetism. All of these works have successfully repro-duced the Zhang-Rice singlet band. This shows that the three-band Hubbard model is anappropriate model for HTSC near the half-filling and that the DMFT is a powerful tool foranalyzing its electronic state. However, when we investigate the inhomogeneous electronic dis-tribution near the half-filling, we need to adopt the model for a large number of non-equivalentsites. In general, the DMFT costs much more than the HFA to analyze the model with a largenumber of non-equivalent sites.In this paper, we analyze the normal ground state of the 3D three-band Hubbard modelwith a single-layered perovskite structure in order to study the evolution of the electronic statewhen holes or electrons are doped into the undoped HTSC. We consider 256 non-equivalentcopper sites for each rectangular parallelepiped super cell, and adopt the HFA for these con-ditions. We performed the calculation without any assumptions about the electronic distribu-tion, and we obtained fully self-consistent solutions except near the 1/8-filling. These solutionsshowed the chemical potential jump at half-filling, which means that the electron suddenlybecomes itinerant when a small number of holes or electrons are doped. Moreover, the dopedholes or electrons tend to form clusters in the vicinity of the half-filling. These clusters are con-sidered to form a metallic region, and are surrounded by the insulating region. This suggeststhe possibility of microscopic electronic phase separation in HTSC near half-fillig.
2. Formulation
Our 3D three-band Hubbard model Hamiltonian, ˆ H , is composed of d -electrons at eachCu site and p -electrons at each O site. To consider the spatial inhomogeneity, we introducethe rectangular parallelepiped super cell containing N c Cu and 2 N c O sites as a unit cell.Thus, ˆ H is defined as follows:ˆ H = N c X i =1 N c X j =1 X k σ ˆ C † i k σ ˆ H ij k ˆ C j k σ + UN N c X i =1 X kk ′ q d † i k + q ↑ d † i k ′ − q ↓ d i k ′ ↓ d i k ↑ − µ N c X i =1 X k σ ˆ C † i k σ ˆ C i k σ . (1) . Phys. Soc. Jpn. Full Paper
Here we use the abbreviations ˆ C † i k σ ≡ ( d † i k σ p x † i k σ p y † i k σ ) and ˆ C i k σ ≡ t ( d i k σ p xi k σ p yi k σ ), where d i k σ ( d † i k σ ) and p x ( y ) i k σ ( p x ( y ) † i k σ ) are the annihilation (creation) operators for the d -orbital and p x ( y ) -orbital electron on the i -th site, as specified by the momentum k and spin σ = {↑ , ↓} , respectively. U , N , and µ are the on-site Coulomb repulsion between the d -orbitals, thenumber of k -space lattice points in the first Brillouin zone (FBZ), and the chemical potential,respectively. The FBZ is defined in the reciprocal space to the lattice whose unit cell contains N c Cu and 2 N c O sites. The unit cell is schematically shown in Fig. 1. The two non-equivalentCuO layers, indicated with L L
2, are alternatively stacked along z -axis. On each CuO layer, the size of the unit cell along x -axis and y -axis is N x and N y Cu sites, respectively.Thus, when we set N x = 8 and N y = 16, N c = 2 × ×
16 = 256. We take the primitivetranslation vectors for the unit cell to be ± a (ˆ x + ˆ y ), ± a (ˆ x − ˆ y ), and ± c ˆ z , where ˆ x , ˆ y , andˆ z are the unit vectors for the x -axis, y -axis, and z -axis, respectively. ˆ H ij k in Eq. (1) is definedas follows: ˆ H ij k = ∆ dp δ R id R jd + ζ zij k ζ xij k ζ yij k ζ x ∗ ji k ζ pij k ζ y ∗ ji k ζ p ∗ ji k , (2)where ∆ dp is the hybridization gap energy between the d - and p x ( y ) -orbitals. R iϕ , where ϕ ∈ ( d, p x , p y ), is the coordinate of the ϕ -orbital electron on the i -th site. δ R iϕ R jϕ ′ is Kronecker’sdelta, i.e., it is 1 for R iϕ = R jϕ ′ and 0 for R iϕ = R jϕ ′ . In the following, we take both a and c to be a unit of length and set a = c = 1. Then, we can represent ζ pij k = t pp h e i ( k x / − k y / δ R ipx R jpy +ˆ x / − ˆ y / + e − i ( k x / − k y / δ R ipx R jpy − ˆ x / y / − e i ( k x / k y / δ R ipx R jpy +ˆ x / y / − e − i ( k x / k y / δ R ipx R jpy − ˆ x / − ˆ y / i ,ζ xij k = t dp h e ik x / δ R id R jpx +ˆ x / − e − ik x / δ R id R jpx − ˆ x / i ,ζ yij k = t dp h e ik y / δ R id R jpy +ˆ y / − e − ik y / δ R id R jpy − ˆ y / i , and ζ zij k = t ⊥ h e i ( k x / − k y / k z ) δ R id R jd +ˆ x / − ˆ y / z + e − i ( k x / − k y / − k z ) δ R id R jd − ˆ x / y / z + e i ( k x / k y / k z ) δ R id R jd +ˆ x / y / z + e − i ( k x / k y / − k z ) δ R id R jd − ˆ x / − ˆ y / z + e i ( k x / − k y / − k z ) δ R id R jd +ˆ x / − ˆ y / − ˆ z + e − i ( k x / − k y / k z ) δ R id R jd − ˆ x / y / − ˆ z + e i ( k x / k y / − k z ) δ R id R jd +ˆ x / y / − ˆ z + e − i ( k x / k y / k z ) δ R id R jd − ˆ x / − ˆ y / − ˆ z i , where t pp is the transfer energy between a p x -orbital and a p y -orbital, t dp is that between a d -orbital and a p x ( y ) -orbital, and t ⊥ is that between d -orbitals, respectively. In this study, t dp is the unit of energy. . Phys. Soc. Jpn. Full Paper
L1 L2 L1L2 xz c N x aN x a N y a N y a y Fig. 1. The schematic figure of the unit cell. L L layersalternatively stacked along z -axis. On each CuO layer, the size of the unit cell along x -axis and y -axis is N x and N y Cu sites, respectively. In total, the unit cell contains N c Cu and 2 N c O sites,where N c = 2 × N x × N y . We adopt the HFA with respect to every N c Cu site and two spin states, and we onlyconsider collinear spin states. Thus, we define1 N D d † i k σ d i k ′ σ ′ E ≡ n idσ δ kk ′ δ σσ ′ , (3)and we approximate Eq. (1) as follows:ˆ H ≈ N c X i =1 N c X j =1 X k σ ˆ C † i k σ ˆ H U ij k σ ˆ C j k σ − µ N c X i =1 X k σ ˆ C † i k σ ˆ C i k σ , (4)ˆ H U ij k σ = [∆ dp + U n id − σ ] δ R id R jd + ζ zij k ζ xij k ζ yij k ζ x ∗ ji k ζ pij k ζ y ∗ ji k ζ p ∗ ji k . (5)Then, we conduct a self-consistent calculation of Eqs. (3), (4), and (5) and obtain self- . Phys. Soc. Jpn. Full Paper consistent fields n idσ , n ip x σ , and n ip y σ , where we define1 N D p x † i k σ p xi k ′ σ ′ E ≡ n ip x σ δ kk ′ δ σσ ′ , (6)1 N D p y † i k σ p yi k ′ σ ′ E ≡ n ip y σ δ kk ′ δ σσ ′ . (7)These satisfy N c X i =1 " − X σ ( n idσ + n ip x σ + n ip y σ ) = N c δ h (8)for a given total number of doped holes δ h .
3. Results and Discussion
In the numerical calculations, we divide the FBZ into N = 16 × × t dp = 1 . t pp = − . t ⊥ = 0 .
005 eV, ∆ dp = 0 . U = 6 . n idσ , n ip x σ , and n ip y σ and carry out iterative calculation untilall of these fields have sufficient accuracy. Here, we chose the initial values for the fields fromthe uniform-distributed random numbers. In this manner, we obtained the solutions for thedoping region near half-filling; 0 . ≤ δ h ≤ . . ≤ δ e ≤ .
1, where δ e ≡ − δ h . For thesesolutions, all of the obtained fields n idσ , n ip x σ , and n ip y σ have three digits of accuracy. Oncethey get three digits of accuracy, all of these fields rapidly converge. Thus, we can considerthe solutions with these fields as fully self-consistent. We also tried to obtain the solutionsfor other doping region e.g. δ h ∼ .
125 but failed. For these doping regions, the ground stateaccompanied by certain types of long-period superlattice structure would be stable. The fieldscorresponding to such ground state could be hardly obtained by our iterative calculation. Wehenceforth concentrate our discussion on the doping region near half-filling.The doping dependence of the chemical potential µ for our fully self-consistent solutions isshown in Fig. 2. The chemical potential changes rapidly at half-filling, which is consistent withexperimental results obtained by comprehensive PES studies on HTSC. The magnitude ofthis change is about 3 . U/ − ∆ dp , which is equal to U/ δ e = 0 .
094 in Fig. 3 is the mean DOS over theseven solutions for which we found δ e = 0 . ± .
000 and µ = 6 . ± .
015 as a mean withstandard error of the mean of doping and chemical potential, respectively. In the slightlyelectron-doped case, δ e = 0 . , . , . d -band, while in the slightly hole-doped case, δ h = 0 . , . , . . Phys. Soc. Jpn. Full Paper µ ( e V ) δ e δ h Fig. 2. (Color online) The doping dependence of the chemical potential µ . δ e and δ h are the numbersof doped electrons and holes per CuO unit, respectively. -300 0 300 -8 -4 0 4 D O S ( a r b i t r a r y un i t ) Energy (eV) δ e =0.094 δ h =0.032 δ e =0.067 δ h =0.066 δ e =0.031 δ h =0.095 δ h =0.000 Fig. 3. (Color) The doping dependence of the mean DOS. The Fermi level at zero energy is indicatedby the vertical dashed line. The lines of upper half and lower half indicate the DOS for spin upand spin down, respectively. In the ascending orders of energy, we attribute each three blocks ofDOS to p -band plus lower Hubbard d -band, p -band, and upper Hubbard d -band, respectively. the p -band. The upper d -level is about U n idσ / p -level when ∆ dp = 0 . p -band and the upper Hubbard d -band, the density of states is almost zero. In the undoped case, δ h = 0 . U/ Thus,the chemical potential jump at half-filling is the characteristic behavior of the multi-bandHubbard model composed of both Cu 3d electrons and O 2p electrons independent of thedimensionality of the model. . Phys. Soc. Jpn.
Full Paper N δ (e , h) ± SEM µ ± SEM δ e = 0 .
094 7 0 . ± .
000 6 . ± . δ e = 0 .
067 8 0 . ± .
000 6 . ± . δ e = 0 .
031 8 0 . ± .
000 6 . ± . δ h = 0 .
000 5 0 . ± .
000 4 . ± . δ h = 0 .
032 8 0 . ± .
000 3 . ± . δ h = 0 .
066 8 0 . ± .
000 2 . ± . δ h = 0 .
095 8 0 . ± .
000 2 . ± . Table I. The statistical values of the solutions with similar doping δ (e , h) and chemical potential µ ,which are selected from the ones in Fig. 2. N and SEM mean number of solutions and standarderror of the mean, respectively. These labels are commonly used in Figs. 3, 4, 5, 6, and 7. The inhomogeneous distribution of every obtained field n iϕσ can be observed in our so-lutions. The doping dependence of the distribution of the obtained fields can be shown byhistogram, as in Fig. 4. In this figure, we show the accumulated histograms over the solutionswith similar doping and chemical potential as the histogram for each doping state, and weindicate each doping state with the mean doping over these solutions as well as in Fig. 3. Athalf-filling, where the ground state is insulating, almost all n idσ have the same value near 1and almost all n ip x σ and n ip y σ have the same value near 2. In the electron-doped case, with theincrease of the doped electrons, only the peak for n idσ broadens and its center shifts higher.In contrast, in the hole-doped case, with increasing hole density, not only the peak for n idσ but also the peaks for n ip x σ and n ip y σ broaden and their centers shift lower. Except for theirvariances, the doping dependence of the average of n idσ is similar to that of n d obtained bythe LDA+DMFT calculation, and the doping dependence of the average of n ip x σ or n ip y σ issimilar to that of n p . The spacial distribution of the magnetic moment m iϕ ≡ n iϕ ↑ − n iϕ ↓ also becomes inhomo-geneous by doping. In the same manner as in Fig. 4, the doping dependence of the distributionof the magnetic momenta magnitude can be shown by histogram in Fig. 5. At half-filling, onehalf of m id is exactly at 1 and the other half is exactly at −
1. This indicates that the d -electronspins are fully polarized. Both in the electron-doped case and in the hole-doped case, with anincrease of the doped carrier, the two peaks for m id become broader and their centers shifttoward zero. This means that the number of the fully-polarized d -electron spins decreases andthat the number of the doubly-occupied Cu sites increases. We should note that the tails of m id in the electron-doped case extend more than those in the hole-doped case.The difference between the electron-doped case and the hole-doped case is caused by thedoped carriers being differently distributed in the unit cell. In order to show this distribution, . Phys. Soc. Jpn. Full Paper F r equen cy Number of electrons per site δ e =0.094 δ h =0.032 δ e =0.067 δ h =0.066 δ e =0.031 δ h =0.095 δ h =0.000 Fig. 4. (Color) The doping dependence of the histogram for the number of electrons per site. We canattribute the peaks whose number of electrons per site are at or near 1 to the ones for n idσ andthose whose number of electrons per site are at or near 2 to the ones for n ip x σ or n ip y σ . F r equen cy Magnetic moment per site δ e =0.094 δ h =0.032 δ e =0.067 δ h =0.066 δ e =0.031 δ h =0.095 δ h =0.000 Fig. 5. (Color) The doping dependence of the histogram for the magnetic moment per site. We canattribute the peaks whose magnetic moment per site are at or near ± m id andthose whose magnetic moment per site are at or near 0 to the ones for m ip x or m ip y . we define the distribution function of the doped carriers as follows: δ h ( r ) ≡ X i (1 − n id ↑ − n id ↓ ) δ rR id + X ϕ ∈ ( p x , p y ) (2 − n iϕ ↑ − n iϕ ↓ ) δ rR iϕ . (9)The doping dependences of δ h ( r ) for the electron-doped case and for the hole-doped cases areshown in Figs. 6 and 7, respectively. In these figures, the red spots indicate the location ofdoped holes and the blue ones indicate the location of doped electrons . In the electron-dopedcase, as shown in Fig. 6, the doped electrons form blobs even in the slightly electron-dopedcase, with δ e = 0 . . Ce . CuO − δ , . Phys. Soc. Jpn. Full Paper suggested by the Cu NMR study. On the other hand, as shown in Fig. 7, the doped holesstay within a single CuO cluster in the slightly hole-doped case, with δ h = 0 . Inour lattice model, the size of such a cluster depends on the number of adjacent orbitals wherethe doped carriers are allowed to occupy. In the electron-doped case, the doped electrons haveroom to sit only on the Cu sites, because all O sites are fully filled by electrons. On the otherhand, in the hole-doped case, the doped holes have room to sit on both Cu sites and O sites.Therefore, when the number of doped electrons is as many as the number of doped holes, thecluster formed by the doped electrons is larger than the one formed by the doped holes.
4. Conclusion
In this paper, we conducted HFA calculations for the 3D three-band Hubbard model witha single-layered perovskite structure considering a large number of non-equivalent sites. Weobtained the fully self-consistent solutions both for the electron-doped and hole-doped cases ator near half-filling. Our solutions show the chemical potential jump at half-filling. The jumpcan be explained by the DOS dependence on doping, which is characteristic to the multi-band Hubbard model composed of both Cu 3d electrons and O 2p electrons independent ofthe dimensionality of the model. Inhomogeneous electronic distributions near half-filling areobserved in our solutions. There is a remarkable difference in the inhomogeneous electronicdistributions between the electron-doped and hole-doped cases. That is, the clusters formedby doped carriers extend more in the electron-doped cases than in the hole-doped cases. Thedifference between the electron-doped and hole-doped case is caused by the difference in thespecies of orbitals the electron and hole are allowed to occupy, which should be explained onlyon the basis of the multi-band Hubbard model. Thus, the theoretical approach on the basisof the multi-band Hubbard model can explain both the chemical potential jump at half-fillingand inhomogeneous electronic distributions near half-filling in a comprehensive way.
Acknowledgments
The authors are grateful to Dr. N. Nakai and Prof. Y. Aiura for their stimulating discus-sions. We are also grateful to an anonymous reviewer of the first manuscript for providing . Phys. Soc. Jpn.
Full Paper (a) δ e =0.094L1 L2 -0.2-0.1 0 0.1 0.2(b) δ e =0.067L1 L2(c) δ e =0.031L1 L2 Fig. 6. (Color online) The doping dependence of δ h ( r ) for the electron-doped case. Each figure is asingle snapshot of the solutions attributed to each doping state, indicated with the mean dopingover these solutions. . Phys. Soc. Jpn. Full Paper (a) δ h =0.032L1 L2 -0.2-0.1 0 0.1 0.2(b) δ h =0.066L1 L2(c) δ h =0.095L1 L2 Fig. 7. (Color online) The doping dependence of δ h ( r ) for the hole-doped case. Each figure is a singlesnapshot of the solutions attributed to each doping state, indicated with the mean doping overthese solutions. . Phys. Soc. Jpn. Full Paper N u m be r o f s i t e s δ e δ h Having more than 0.1 excess electronsHaving more than 0.1 excess holes
Fig. 8. (Color online) The doping dependence of the number of the sites having more than 0.1 excesscarriers. insightful comments and directions which have resulted in the revised manuscript. . Phys. Soc. Jpn.
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