Electron-phonon interaction in the three-band model

Transcription

1 Electron-phonon interaction in the three-ban moel O. Rösch an O. Gunnarsson Max-Planck-Institut für Festkörperforschung, Postfach , D Stuttgart, Germany We stuy the half-breathing phonon in the three-ban moel of a high temperature superconuctor, allowing for vibrations of atoms an resulting changes of hopping parameters. Two ifferent approaches are compare. From the three-ban moel a t-j moel with phonons can be erive, an phonon properties can be calculate. To make contact to ensity functional calculations, we also stuy the three-ban moel in the Hartree-Fock (HF) approximation. The paramagnetic HF solution, appropriate for the ope cuprates, has similarities to the local-ensity approximation (LDA). However, in contrast to the LDA, the existence of an antiferromagnetic insulating solution for the unope system makes it possible to stuy the softening of the half-breathing phonon uner oping. We fin that although the HF approximation an the t-j moel give similar softenings, these softenings happen in quite ifferent ways. We also fin that the HF approximation gives an incorrect oping an q epenence for the softening an too small a with for the (half-)breathing phonon. I. INTRODUCTION The electron-phonon coupling for high-t c cuprates has recently attracte much interest. Lanzara et al. iscovere strong coupling to a moe at 70 mev in many cuprates. The coupling was ascribe to a half-breathing phonon along the (,0,0) irection. This is an in-plane bon-stretching moe, where the vibrations are primarily ue to two of the four O atoms surrouning a Cu atom in the CuO 2 plane. Lanzara et al. euce a rather strong apparent electron-phonon coupling λ. The half-breathing phonon shows an anomalous softening when the cuprates are ope, in particular towars the zone bounary. 2 5 The softening of other phonons upon oping can be explaine as a screening of the ions in the ope system. 2 The softening of the half-breathing moe, however, cannot be escribe in a shell moel with conventional parameters. 2 This supports the iea that this phonon has substantial electron-phonon coupling, which woul lea to a reuction of the frequency in the ope but not in the unope system. This phonon has an appreciable broaening, 3 which can also be explaine in terms of a substantial electron-phonon coupling. From the broaening one coul estimate an electron phonon coupling for this moe of the orer of λ , using the formula of Allen 6 an the ensity of states of Mattheiss. 7 Anomalous behavior of bon-stretching moes has also been observe in other compouns. 8 0 The half-breathing phonon has been stuie 4 within the t-j moel, 5 an a substantial softening was foun.,2,4 Phonons an the electron-phonon interaction have also been stuie 6 8 extensively within the local-ensity approximation (LDA). 9 Bohnen et al. 8 foun phonon frequencies in goo agreement with experiment for YBa 2 Cu 3 O 7. In particular, the frequency of the half-breathing moe along the (,0,0) irection was foun to be anomalously soft, in agreement with experiment. Since LDA oes not escribe antiferromagnetism in the unope system, however, it cannot properly escribe phonons in the unope case. It is therefore not clear how much the phonon is softene uner oping in the LDA. Furthermore, LDA calculations show a weak electron-phonon coupling to the half-breathing phonon, with λ at the zone bounary being This is in isagreement with the large with of the halfbreathing phonon, which is believe to be ue to a rather strong electron-phonon coupling. The weak coupling raises questions about the reason for the low frequency of the half-breathing phonon in the LDA calculation for the ope system. To aress these issues, we here stuy the three-ban moel, 20 incluing a Cu x 2 y 2 an two O p-orbitals per unit cell. We allow isplacements of the atoms from their equilibrium positions an take the corresponing changes of hopping integrals into account. In this moel we can stuy phonons. We solve the moel using a Hartree- Fock (HF) mean-fiel approximation, which may be expecte to simulate features of the LDA for the ope system. For instance, we fin a similar with for the halfbreathing phonon as in the LDA. In contrast to LDA, however, this approximation gives an antiferromagnetic solution for the unope system. We can therefore obtain the softening upon oping within this framework. The softening of the half-breathing phonon is inee foun to be of the same orer of magnitue as the experimental result, supporting the iea that the LDA can escribe the softening. Alternatively, a t-j moel with phonons can be erive from the three-ban moel. This moel is solve using exact iagonalization. 4 Thereby, many-boy effects are inclue, an the interplay between electron-phonon an electron-electron interactions are consiere in a more explicit way. In this paper we compare the results in these two approaches. We fin that the HF solution of the three-ban moel an the t-j moel give a comparable softening of the half-breathing phonon. This happens, however, in a very ifferent way in the two approaches. Furthermore, we fin that the epenence on oping an on q is rather ifferent in the two approaches. In the t-j moel, the softening is δ for small δ, while the oping epenence

2 2 is weaker in the HF solution. The t-j moel gives a smaller softening for the q = (, )π/a breathing moe than for the q = (, 0)π/a half-breathing moe, while the opposite is foun in the HF approximation. In both cases t-j results are in better agreement with experiment. In Sec. II we introuce the moels an in Sec. III the methos. The results are presente in Sec. IV. The results of the HF approximation for the three-ban moel are compare with the exact iagonalization results of the t-j moel in Sec. V. II. MODELS We stuy the three-ban moel with N unit cells H = ε n iσ + ε p n jpσ + U n i n i () iσ jσ i + t p ij ψ iσ ψ jpσ + t pp jj ψ jpσ ψ j pσ + h.c., i,j σ j,j σ where n iσ an n jpσ are the occupation numbers for the N 3-orbitals an the 2N 2p orbitals, respectively, an ψ iσ an ψ jpσ are the corresponing annihilation operators. The site energies are ε an ε p. There is a Coulomb integral U, acting between two 3-electrons on the same Cu site. The moel inclues hopping between nearest neighbor Cu an O atoms an between the O atoms which are nearest neighbors of a particular Cu atom. 2 These hopping integrals are given by t p an t pp, respectively, with the signs etermine by the relative orientations of the orbitals involve. Displacing an atom leas to changes of both electrostatic potentials an hopping integrals. Electrostatic potentials are screene ifferently in the ope an unope systems. It was foun, however, that the change of screening oes not strongly influence the half-breathing phonon. 3 We therefore assume here that the changes of hopping integrals are the ominating effects. 22 We assume the hopping integral t p has a power epenence on the atomic separation, i.e., ( ) 3.5 a/2 t p () = t p0 (2) where a is the lattice parameter. The exponent 3.5 for t p was estimate from LDA ban structure calculations. 23 We neglect the phonon moulation of t pp. This moel can be solve irectly as it stans within some approximation, here the Hartree-Fock approximation. We can, however, also erive a t-j moel with phonons, an then solve this moel. The t-j moel has one site per Cu atom. In the unope system, there is one hole per site. Doping introuces aitional holes, which primarily sit on O sites. These O holes form Zhang- Rice singlets with neighboring Cu holes. 5 It is then assume that each site is occupie by either a -hole or a Zhang-Rice singlet. Expressing the atomic isplacements in terms of phonon operators an working to linear orer in the isplacements leas to the t-j moel with phonons 4 H t-j = J ( S i S j n ) in j t ] [ c 4 iσ c jσ + h.c. <i,j> <i,j>σ + ( ω ν (q) b qν b qν + ) 2 qν + c iσ c jσ gij t-j (q, ν)(b qν + b qν ). (3) ijσ qν Here c iσ creates a -hole on site i if this site previously ha no hole, i.e., if it ha a Zhang-Rice singlet. The operator b qν creates a phonon with wave vector q, inex ν an frequency ω ν (q). The formulas for gij t-j (q, ν) have been given elsewhere. 4 While we treat the three-ban moel in the electron picture, we treat the t-j moel in the hole picture. The erivation 4 of a t-j moel with electron-phonon interaction was performe for the case t pp = 0. The inclusion of a finite t pp woul lea to a more complicate moel an make the analysis less transparent. Within the HF approximation for the three-ban moel, we show below that t pp oes not substantially change the results for the half-breathing phonon. Since we are primarily intereste in this phonon, we mainly consier the simpler case t pp = 0 below. For the breathing phonon, however, results in the three-ban moel for t pp = 0 are not meaningful, since the system is close to nesting. We therefore use t pp =. ev when comparing the half-breathing an breathing moes in the three-ban moel. In eriving this moel, we assume that t p ε ε p, U. This is a rather poor approximation for realistic parameters. Here we have somewhat artificially reuce the value of t p0 to.2 ev, which gives a realistic 24 value of t = 0.47 ev if perturbation theory is use, an probably also more realistic values for gij t-j, by compensating for some of the effects of perturbation theory. 4 This treatment neglects quaratic terms, although these also can give a oping epenent contribution to the phonon softening. III. METHODS We use a frozen phonon metho to calculate the phonon frequency in the HF approximation for the threeban moel. This gives the effective Hamiltonian H eff = (ε + U n i σ )n iσ + ε p n jpσ (4) iσ jσ U n i n i i + t p ij ψ iσ ψ jpσ + i,j σ j,j σ jj ψ jpσ ψ j pσ + h.c., t pp

3 where n i σ is the expectation value of n i σ. We also introuce the effective level ε eff iσ = ε + U n i σ. (5) A static istortion is built into the lattice an the change of the total energy is calculate. From this the interaction with the electrons can be euce an the softening of the phonon obtaine. The isplacements of the atoms in the lattice are enote by {u i } with u i = u cos(qr i ) for a phonon with wavevector q. The energy of the electronic system is then E[{u i }] = Tr(H[{u i }]ρ[{u i }]), (6) where ρ[{u i }] is the ensity matrix, obtaine in some approximation, e.g. the HF approximation. Using the Hellmann-Feynman theorem, it follows that 2 E u 2 ( H = Tr u ) ( ρ 2 ) H +Tr u u 2 ρ ( 2 E u 2 ) () + (7) From the total energy calculation, we can euce both 2 E/ u 2 an ( 2 E/ u 2 ) (i), i=, 2. ( 2 E/ u 2 ) () escribes how a first orer change in the external Hamiltonian leas to a first orer change in the ensity matrix, which acts back at the Hamiltonian. This contribution can be calculate in linear response. Schematically, for a one-ban moel with u-epenent site energies ε i as in Sec. V, an H u = i ε i u n i g q cos(qr i )n i, ( ) ρ Tr n i = n i = u u j i χ ij ε eff j u, (8) where χ ij is the response function for noninteracting electrons an the erivative of the effective site energies is obtaine from the total energy calculation. Thus we have ( 2 ) () E u 2 ε i u χ ij ij ε eff j u We introuce a ielectric function = g qχ(q)g eff q. (9) ɛ(q) = g q /g eff q. (0) The result in Eq. (9) can also be obtaine from a iagrammatic technique. It is escribe by a bubble iagram, as shown in Fig.. This is the only iagram which enters in the HF approximation. We observe that the screening of the perturbation, escribe by g eff, enters at one of the vertices. Using g eff at both vertices woul lea to ouble counting. By using the iagrammatic approach, we can obtain not only a contribution to the energy but also the with of the phonon. g g eff FIG. : Diagram escribing the linear response. g an g eff escribe the bare an screene perturbations ue to a phonon. IV. RESULTS Here we present results for the three-ban moel in the Hartree-Fock approximation, using the parameters t p0 =.6 ev, t pp = 0 an U = 8 ev. The lattice parameter is a = 3.8 Å. We have ajuste ε p so that the separation between the effective 3-level an 2p-levels ( 2 ) E (2) εeff p ε + U n σ = 3 ev, () u 2. where n σ is the average occupation of the 3-levels per spin. This is a typical LDA estimate for ε eff εeff p in ε eff a three-ban moel. 2 We perform two calculations, one for the unistorte lattice an one for a lattice where a phonon has been built in. This gives the secon erivative, 2 E/ u 2. From this erivative we obtain the softening of the phonon ue to the interaction with the electrons in the moel, reucing the frequency ω ph0 to ω ph. Here ω ph0 is assume to be ue to forces not inclue in the moel in Eq. (), e.g., electrostatic forces an corecore overlap effects. The calculations were performe for a cluster of CuO 2 units an perioic bounary conitions. The oping was chosen in such a way that egenerate levels were either completely full or completely empty, i.e., all shells were either full or empty. We first consier the half-breathing phonon for q = (, 0)π/a, where the two O atoms at (a/2,0) an (-a/2,0) move towars (or away from) from the Cu atom at (0,0). We perform a calculation for the unope system, having five electrons per unit cell, an allowing for spin-polarization. We ajust ω ph0 so that the softene frequency ω ph is ev for the zone bounary halfbreathing phonon, as is foun experimentally. The result as a function of the hole oping δ (δ < 0 means electron oping) is shown in Fig. 2. The oping epenence is relatively weak. Since the unope system is antiferromagnetic but the ope system is (assume to be) paramagnetic, as foun experimentally, the softening is not necessarily small for small opings. The spinpolarize system has a large gap of about 4.6 ev. Due to this gap, the response of the system to a phonon is substantially weaker than for the paramagnetic state. For instance, ( 2 E/ u 2 ) () contributes a softening of only about 4 mev for the spin-polarize system but 6 mev for a non spin-polarize system with a similar number of electrons. Similarly, the contributions of ( 2 E/ u 2 ) (2) are about 8 an 3 mev, respectively. We fin that screening reuces the quantity ( 2 E/ u 2 ) () by about a factor of two, i.e., ɛ 2 3

4 4 - ω ph [mev] HF exp. t-j FIG. 2: Softening ω ph for the zone bounary half-breathing phonon. Results are given in the HF approximation (full line), the t-j moel (ashe line) an accoring to experiment (ash-otte line) as a function of the hole oping δ. The lines serve as a guie for the eye between the few points in the t-j moel an accoring to experiment. The HF approximation refers to results for the shift in a paramagnetic calculation for oping δ relative to an antiferromagnetic calculation for δ = 0. The ashe part of the HF line inicates schematically that the systems goes antiferromagnetic for small opings. in Eq. (0). By evaluating the iagram in Fig., we fin that the zone bounary half-breathing phonon is broaene by about 0.4 mev for δ = 0.6. This is similar to what Bohnen et al. 8 foun in an LDA calculation. It is interesting to compare the half-breathing phonon an the q = (, )π/a breathing phonon. In the latter case all four O atoms surrouning a Cu atom move towars this atom. In the moel consiere above (t pp = 0) an for δ = 0, the moel has nesting for q = (, )π/a. As a result of this, there woul be a very strong response for small δ, making the calculation rather meaningless. Instea we consier t pp =. ev. 2 This changes the Fermi surface an avois an unrealistically strong nesting. In this case we ajust ω ph0 so that ω ph = ev in the spin-polarize calculation for the unope system, as foun experimentally. For the oping δ = we fin that the q = (, )π/a breathing phonon is softene by mev. This is a larger softening than was foun in Fig. 2 for this oping. We have also performe a calculation for the q = (, 0)π/a half-breathing phonon in this moel an for this oping. The softening is 7 mev, which again is smaller than for the breathing phonon. This is contrary to experiment, where a larger softening is foun for the half-breathing phonon. Finally, we observe that the softening of the half-breathing phonon is only change from about 9 mev for t pp = 0 to 7 mev for t pp =. ev. This justifies the neglect of t pp for the qualitative analysis of the half-breathing phonon. δ V. COMPARISON OF THE THE t-j MODEL AND HF SOLUTION OF THE THREE-BAND MODEL To make contact with the t-j moel, we transform the three-ban moel to a one-ban moel. We assume that the O 2p-levels are far below the Cu 3-levels (in the electron picture). We can then project out the O 2plevels an obtain a moel with just effective Cu 3-levels. This moel can be compare with the t-j moel, since both moels have one ban. Here we focus on the linear response term ( 2 E/ u 2 ) () in Eq. (9), which are of the same orer of magnitue in the t-j moel an the HF approximation for a typical δ 0.. We introuce a projection operator P, which projects out the Cu 3-levels, an its complement Q = P. Consiering the resolvent operator of the HF Hamiltonian, we obtain P (z H) P (2) = [P (z H)P P HQ(z QHQ) QP H], where z is some typical energy. We then obtain the effective one-ban parameters iσ t One ii = ε + j (t p ij )2 z, = t p ij tp ji z, (3) where the sum over j for iσ runs over the nearest O neighbors of Cu an j for t One refers to the common ii nearest neighbor O atom of the Cu atoms i an i. We first consier the unperturbe (no phonon) system. We choose z to be in the mile of the ban an solve the self-consistent equations = ε + 4(tp ) 2 t One ii = (tp ) 2, (4), where the factor 4 for comes from the four O neighbors of a Cu atom. We then introuce a phonon in the system an ask for the linear response of the electronic system. A perturbation term is introuce in the one-ban Hamiltonian δε iσ = j 2 tp ij δtp ij, (5) where δt p ij are the changes in the Cu-O hopping integrals. To linear orer there is no change in t One. ii For a half-breathing phonon at the zone bounary, the on-site perturbation is u u = 4t t p p u u, (6)

5 TABLE I: Contributions to the phonon softening in the t-j moel an in the HF solution of the three-ban moel. Here = ε. Source t-j HF Ratio Coupling [(2λ 2 )/ + 2λ 2 /(U )] 2 (/ ) 2 3 Sum rule 2δπN πn 2δ Screening Denominator Prouct 2δ where u is the absolute value of the phonon amplitue. The quantity / u enters in the calculation of ( 2 E/ u 2 ) () in Eq. (9). In the t-j moel, the on-site perturbation is 4 ( t p 2λ 2 4t p u ε + 2λ 2 ) u, (7) U ε where λ = The first term comes from the hopping of a 3-hole into the O 2p-states an the secon term from the hopping of a O 2p-hole into the Cu 3-state. The secon term has no corresponence in Eq. (6). Equation (7) has an aitional factor 2 coming from a phase coherence factor in the Zhang-Rice singlet. This results from the singlet being explicitely written as a sum of two terms. Both these effects are genuine many-boy effects. The in the first term in Eq. (7) results from taking the ifference in the energy gain of a Zhang-Rice singlet an a single 3-hole. We fin that this on-site coupling ominates over the off-site coupling. 4 In the qualitative iscussion we therefore only consier the on-site coupling gii t-j (q) g(q). The phonon self-energy in the t-j moel is Π(q, ω) = g(q) 2 χ t-j (q, ω) + g(q) 2 χ t-j (q, ω)d 0 (q, ω), (8) where D 0 (q, ω) is the noninteracting phonon Green s function. We fin that the secon term in the enominator is small for the parameters consiere here. In the formal iscussions below we therefore neglect it. The phonon self-energy is then proportional to the response function. This result can then be irectly compare with the result in Eq. (9) for the one-ban moel. In both cases the response function is multiplie by the appropriate coupling constant square, given by Eq. (6) an Eq. (7), respectively. To compare the t-j moel with the one-ban moel, we put ε p = ε ε p = 3 ev an U = 8 ev. 25 We fin that the square of Eq. (7) is about a factor of three larger than the square of Eq. (6). This ifference ue to ifferences in coupling constants is shown in Table I. The linear response of the one-ban moel is given by χ HF (q, ω) = 2 k [ f(k + q)]f(k) (9) ( ) ω ω(k, q) + i0 + ω + ω(k, q) i0 + where f(k) is the Fermi function for a state with wave vector k an energy ε(k) an ω(k, q) = ε(k + q) ε(k). We have the sum-rule N q 0 Imχ HF (q, ω + i0 + ) ω = 4πn( n)n πn, (20) where N is the number of sites an n is the fractional filling of the ban. Typically, we are intereste in a system with n = ( δ)/2 0.5 electrons per site an spin, which leas to the right han sie of Eq. (20). For the t-j moel we ask for the carrier-carrier response function χ t-j (q, ω) = ν ( ν ρ(q) 0 2 (2) ω ω ν + i0 + ω + ω ν i0 + ), where ν is an excite many-electron state with the excitation energy ω ν an ρ(q) is carrier ensity operator. One fins 2 N q 0 Imχ t-j (q, ω + i0 + ) ω = 2πNδ( δ). (22) Eqs. (20) an (22) iffer by approximately a factor of 2δ for small δ (see Table I). The two approaches further iffer by the screening in the HF approach, iscusse in Eq. (0). This reuces the HF result by roughly a factor of two. Finally, we have to consier that a typical energy enominator in Eq. (20) an Eq. (2) are ifferent. Calculating the average /ω, we fin comparable results for the two moels. As can be seen from Table I, the ifference between the two approaches results in a ratio of about 2δ. For δ 0., the two results are then similar. This inicates why the t-j moel an the HF solution of the three-ban moel can give similar softening of the halfbreathing moe although the physics is quite ifferent. We next consier the imaginary part of the phonon self-energy Π, which gives the phonon with. As an orientation, we first consier a simple moel. Since Im χ(ω) ω for small ω, we assume Im Π(q, ω) = where A is some constant. relation, we can then erive { Aω, if ω W ; 0, otherwise, 5 (23) From the Kramers-Kronig γ ω ph = π ω ph W, (24) where γ = 2Im Π(q, ω) is the full with at half maximum of the phonon an ω ph is its shift. Figure 3 compares Im Π(k, ω) for the half-breathing phonon in the one- an three-ban moels an the

6 6 breathing phonon in the three-ban moel. The oneban moel was constructe to escribe what happens close to E F an therefore the one- an three-ban moels agree very well for small ω. For ω 0, the one-ban moel gives a larger Im Π(q, ω) than the three-ban moel, an Re Π(q, ω) is overestimate corresponingly in the one-ban moel. Appropriate numbers for the the half-breathing phonon an the one-ban moel are W = 2 ev, ω ph = 0.07 ev, ω ph = ev. Inserting this in Eq. (24) leas to γ = 0.8 mev. This is about twice the with actually calculate. The reason for this overestimate is that Im Π(q, ω) is actually smaller than assume in Eq. (22) for small ω. Figure 3 also shows the HF result for the q = (, )π/a breathing phonon. These results were obtaine for t pp =. ev. There is then no strong nesting of the Fermi surface. The HF approximation, nevertheless, gives a larger broaening for the q = (, )π/a breathing phonon than for the q = (, 0)π/a half-breathing phonon, whether the latter is calculate for t pp = 0 or t pp =. ev. As is well-known, 6 this is ue to the fact that the wave vector q = (, )π/a fits better to the traces of nesting left over for t pp =. ev. To stuy the phonon with in the t-j moel, we calculate the phonon self-energy. 26 We calculate the phonon spectral function B(q, ω) using exact iagonalization. A Hilbert transform can then be use to obtain the phonon Green s function D(q, ω). The phonon selfenergy Π(q, ω) is calculate by inverting D (q, ω) = D0 (q, ω) Π(q, ω), (25) where D 0 (q, ω) is the noninteracting phonon Green s function. This approach has important avantages for small systems. B(q, ω) has too few structure to etermine the phonon with. A broaene version of Π(q, ω), however, can give such information. 26 Results for Π(q, ω) are shown in Fig. 4. In the view of Eq. (24), one might have expecte the with of the half-breathing phonon to be similar in the HF approximation an the t-j moel, since the shifts are similar. This is not true, however, since the frequency epenence iffers strongly from the linear epenence assume in Eq. (23). This is illustrate in Fig. 4. The figure shows that for the t-j moel with J = 0, some spectral weight has been move to small frequencies ue to the hopping constraint, which creates low energy excitations. This is even more true for the finite J case (J/t = 0.3). The present clusters are too small to give reliable results, in particular for the low-lying excitations, an the results above shoul be consiere as qualitative. They illustrate the general tren, however, of transferring some spectral weight to small frequencies, which tens to lea to a substantially increase with of the halfbreathing phonon. Im Π(ω) (ev) Three-ban q=(,0) One-ban q=(,0) Three-ban q=(,) ε (ev) FIG. 3: Im Π(q, ω) for the q = (, 0)π/a half-breathing moe in the three-ban moel (full line) an the one-ban moel (ashe line) an for the q = (, )π/a breathing moe (otte line). The oping is δ = 0.. Im Π(ω) (ev) ε (ev) One-ban t-j, J=0.3t t-j, J=0 q=(,0) FIG. 4: Im Π(q, ω) for the q = (, 0)π/a half-breathing moe in the t-j moel for J/t = 0.3 (full line) an J/t = 0 (ashe line) an in the one-ban moel (otte line). The results for the t-j moel were obtaine for a 4 4 cluster. The oping is δ = VI. SUMMARY We have stuie the properties of the half-breathing phonon in the three-ban moel of a high-t c cuprate. The results in a t-j moel with phonons an the HF approximation of the three-ban moel were compare. Although the two approaches give similar softenings at typical opings δ, the unerlying physics is quite ifferent. The hopping constraint in the t-j moel, resulting from the strong Coulomb repulsion, leas to a strong reuction of the response to a phonon. This reuction is, however, partly compensate by several other effects. In particular, the coherent hopping in the formation of the Zhang-Rice singlet tens to increase the coupling in the t- J moel. We fin that the oping an q epenences for

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