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February 7, 2008 Hole Photoproduction in Insulating Copper Oxide O. P. Sushkova,b, G. Sawatzky, R. Eder, and H. Eskes Materials Science Center, University of Groningen Nijenborgh 4, 9747 AG Groningen 7 9 The Netherlands 9 1 n a J Abstract 1 3 To explain the experimental spectra for angle resolved photoemission we con- 1 v sider a modified t J model. The modified model includes next nearest (t′) 2 − 0 and next next nearest (t′′) hopping as well as Hubbard model corrections to 0 2 the spectral weights.A Dyson equation which relates the single hole Green’s 0 7 functions for a given pseudospin and given spin is derived and is compared to 9 t/ experimental results. a m PACS numbers: 75.50.Ee, 75.10.Jm, - d n o c : v i X r a Typeset using REVTEX 1 I. INTRODUCTION Recent angle resolved photoemission (ARPES) measurements by Wells et al1 and by Pothuizen et al2 for insulating Copper Oxide Sr CuO Cl provide a unique possibility to 2 2 2 experimentally determine the single hole properties. In the frameworks of a t t′ J model − − the problem has been analyzed by Nazarenko et al3 using a cluster method and Bala, Oles, and Zaanen4 using a self-consistent Born approximation (SCBA). It was demonstrated in these works that including t′ hopping improves agreement with experiment, but a complete description of the ARPES spectra especially the k dpendent intensity was not achieved. In the present work we consider a t t′ t′′ J model. A Dyson equation which relates − − − the one electron Green’s function measured in experiment to the hole Green’s function found in the self-consistent Born approximation is derived. We also introduce into this equation the corrections which originate from finite U as in the Hubbard model. With parameters taken from LDA band calculations by Andersen et al5 we got a reasonably good description of the experimental ARPES spectra . The results are sensitive to t′′. The parameter t′ is less important. The importance of t′′ was also pointed out by Belinicher and Chernyshev6. For the explanation of width of ARPES spectra we need to introduce additional broadening and possible origins of this are discussed . The Hamiltonian of t t′ t′′ J model is of the form − − − H = t c† c t′ c† c t′′ c† c +J S S . (1) − iσ jσ − iσ j1σ − iσ j2σ i j hXijiσ hiXj1iσ hiXj2iσ hXijiσ c† isthecreationoperatorofanelectronwithspinσ (σ = , )atsiteiofthetwo-dimensional iσ ↑ ↓ square lattice,the ij represents nearest neighbor sites, ij - next nearest neighbor (diago- 1 h i h i nal), and ij represents next next nearest sites. The spin operator is S = 1c† σ c . The h 2i i 2 iα αβ iβ size of the exchange measured in two magnon Raman scattering7,8 is J = 125meV. The most recent calculation of the hopping matrix elements has been done by Andersen et al5. They consider a two-plane situation and the effective matrix elements are slightly different for symmetric and antisymmetric combinations of orbitals between planes. After averaging over these combinations we get: t = 386meV, t′ = 105meV, t′′ = 86meV. Below we set − J = 1, in these units t = 3.1, t′ = 0.8, t′′ = 0.7 (2) − 2 II. HOLE GREEN’S FUNCTION WITH FIXED PSEUDOSPIN. SELF-CONSISTENT BORN APPROXIMATION (SCBA) It is well known that at half filling (one electron per site) the model under considera- tion is equivalent to a Heisenberg model. It represents a Mott insulator with long range antiferromagnetic order. We denote the corresponding ground state wave function by 0 . | i We are interested in the situation when one electron is removed from this state, so a single hole is produced. The dynamics of a single hole in the antiferromagnetic background can be described by SCBA9,10. This approximation works very well due to the absence of a single loop correction to the hole-spin-wave vertex11–13. Let us recall the idea of this approximation. The bare hole operator d is defined so that d† c on the sublattice and c on i i i↑ i↓ ∝ ↑ ∝ the sublattice. In the momentum representation ↓ 2 2 d† = c eikri, d† = c eikrj. (3) k↓ sN(1/2+m) i↑ k↑ sN(1/2+m) j↓ i∈↑ j∈↓ X X N is number of sites, m = 0 S 0 0.3 is the average sublattice magnetization . The iz |h | | i| ≈ quasi-momentum k is limited to be inside the magnetic Brillouin zone: γ = 1(cosk + k 2 x cosk ) 0. In this notations it looks like d has spin σ = 1/2, but actually rotation y kσ ≥ ± invariance is violated and σ is a pseudospin which denotes the sublattice. Nevertheless the pseudospin gives the correct value of the spin z-projection: S = σ = 1/2. The coefficients z ± in (3) provide the correct normalization: 2 1 1 0 d d† 0 = 0 c† c 0 = 0 +S 0 = 1. (4) h | k↓ k↓| i N(1/2+m) h | i↑ i↑| i 1/2+mh |2 iz| i i∈↑ X The hole Green’s function is defined as G (ǫ,k) = i 0 Td (τ)d† (0) 0 eiǫτdτ (5) d − h | kσ kσ | i Z The t′, t′′ terms in the Hamiltonian (1) correspond to the hole hopping inside one sublattice. This gives the bare hole dispersion ǫ = 4t′cosk cosk +2t′′(cos2k +cos2k ) β γ2 +β (γ−)2, (6) 0k x y x y → 01 k 02 k where γ− = 1(cosk cosk ), β = 4(2t′′+t′), andβ = 4(2t′′ t′). In equation (6) we took k 2 x− y 01 02 − into account that the sign of a hole dispersion is opposite to that for an electron (maximum of electron band correspond to minimum of hole band), and omitted some constant. The bare hole Green’s function is 3 1 G (ǫ,k) = . (7) 0d ǫ ǫ +i0 0k − For spin excitations the usual linear spin-wave theory is used (see, e.g. review paper14). It is convenient to have two types of spin-waves, α† with S = 1, and β† with S = +1, q z q z − and q restricted to be inside the magnetic Brillouin zone. 2 S+e−iqri u α +v β† , (8) sN i ≈ q q q −q i∈↑ X 2 S−eiqrj v α† +u β . sN j ≈ q q q −q j∈↓ X The spin-wave dispersion and parameters of Bogoliubov transformation diagonalizing the spin-wave Hamiltonian are: ω = 2 1 γ2, q q − q 1 1 u = + , (9) q sωq 2 1 1 v = sign(γ ) . q q − sωq − 2 Hopping to nearest a neighbor in the Hamiltonian (1) gives an interaction of the hole with spin-waves. H = g d† d α +d† d β +H.c. , (10) h,sw k,q k+q↓ k↑ q k+q↑ k↓ q Xk,q (cid:16) (cid:17) with vertex g given by k,q g 0 α d H d† 0 = k,q ≡ h | q k↑| t| k+q↓| i 2 = 0 α e−ikrjc† tc† c c ei(k+q)ri 0 = N(1/2+m)h | q j↓ − iσ jσ i↑ | i hi∈X↑,j∈↓i (cid:16) (cid:17) 2 = t e−ikrj+i(k+q)ri 0 α c† c c† c 0 + 0 α c† c c† c 0 N(1/2+m) h | q j↓ j↑ i↑ i↑| i h | q j↓ j↓ i↓ i↑| i ≈ hi∈X↑,j∈↓i (cid:16) (cid:17) 2 2 t e−ikrj+i(k+q)ri 0 α S− +S− 0 = 4t (γ u +γ v ). (11) ≈ N h | q j i | i sN k q k+q q hi∈X↑,j∈↓i (cid:16) (cid:17) In this calculation we have used the usual mean field ground state factorization approximation: 0 α c† c c† c 0 0 α c† c 0 0 c† c 0 = 0 α S− 0 (1/2+m). The vertex h | q j↓ j↑ i↑ i↑| i ≈ h | q j↓ j↑| ih | i↑ i↑| i h | q j | i g is independent of t′, t′′ because these parameters correspond to hopping inside one sub- k,q lattice. The form of the vertex g is well known. The actual purpose of calculation (11) k,q 4 is to demonstrate that g is valid in a more general situation than it is usually believed. k,q It is independent of the particular value of the sublattice magnetization m. Therefore, for example, g remains the same in the presence of strong additional frustrations. k,q One can easily prove that the spin structure of the interaction (10) forbids single loop corrections to the hole-spin-wave vertex and, as usually, the two loop correction is small numerically11–13. So, due to the spin structure we have an analog of the well known Migdal theorem for electron-phonon interactions. This justifies SCBA according to which the hole Green’s function satisfies a simple Dyson equation −1 G (ǫ,k) = ǫ ǫ g2 G (ǫ ω ,k q)+i0 . (12) d − 0k − q k−q,q d − q − ! X The anomalous Green’s function i 0 Td (t)d† (0) 0 vanishes because the z-projection of − h | k↑ k↓ | i spin is conserved. Due to the definition of the operators (3) the Green’s function (5) is invariant under translation with the inverse vector of the magnetic sublattice Q = ( π, π) ± ± G (ǫ,k+Q) = G (ǫ,k). (13) d d The numerical solution of equation (12) is straightforward. As usual, to avoid poles we replace i0 iΓ/2 = i 0.1. The energy scale consists of 300 points with variable density → (concentrated near sharp structures of G ). The number of points in the magnetic Brillouin d zone is 104 which is equivalent to the lattice 140 140. The plots of 1 Im G (ǫ,k) as a × −π d functions of ǫ for k = (π/2,π/2), k = (π/2,0), k = (π,0), and k = (0,0) are presented in Fig.1. We recall that we use the set of parameters (2) based on Ref.5. The position of the lowest peak gives the quasiparticle energy. Results of the calculation can be fitted by the formula ǫ = const+β γ2 +β (γ−)2 +β′(γ−)4 (14) k 1 k 2 k 2 k β 3.0, β 3.8, β′ 1.5. 1 ≈ 2 ≈ 2 ≈ − The dispersion has minima at (k = ( π/2, π/2). The hole pockets are slightly stretched 0 ± ± along the direction to the zone center, and it is very different from a pure t J model11,12. − The quasiparticle residue Zd can be found as the area under the peak. At the dispersion k minimum it equals Zd = 0.38. So it is larger than in a pure t J model11,12, but away k0 − from the dispersion minimum it drops down very rapidly. Plots of the residue Zd as a k function of k for k [(π/2,π/2) (0,0)] and k [(π/2,π/2) (0,π)] are given at Fig. ∈ − ∈ − 2. Throughout the Brillouin zone the residue is fitted by Zd = Zd 1 0.9γ2 1.52(γ−)2 +0.05γ4 +0.69(γ−)4 +0.5(γ γ−)2 . (15) k k0 − k − k k k k k (cid:16) (cid:17) 5 III. HOLE GREEN’S FUNCTION WITH FIXED SPIN. DYSON EQUATION RELATING TWO GREEN’S FUNCTIONS The operators d , d discussed in the previous section are defined at different sublat- k↑ k↓ tices. However, when a photon kicks out an electron from the system it does not separate the sublattices. Therefore for this process we have to define the operator as a simple Fourier transform 2 c = c eikri. (16) kσ iσ sN i X The normalization is chosen in such a way that 0 c† c 0 = 2 0 c† c 0 = 1. We can h | k↑ k↑| i Nh | i i↑ i↑| i consider c as an external perturbation, and the corresponding GrPeen’s function is kσ G (ǫ,k) = i 0 Tc† (τ)c (0) 0 eiǫτdτ. (17) c − h | kσ kσ | i Z This is the Green’s function measured in ARPES. Let us now find the relation between G (ǫ,k) and G (ǫ,k). c d The operator c acting on the vacuum (ground state of the Heisenberg model) can kσ produce a single hole state. We denote the corresponding amplitude by a and show it in k Fig. 3a as a cross.The thick line corresponds to the Green’s function G (17) and the thin c line corresponds to the G (5). The amplitude a equals d k 2 2 a = 0 d c 0 = 0 c† e−ikrj c eikri 0 = 1/2+m. k h | k↑ k↓| i h |sN(1/2+m) j↓ sN i↓ | i Xj∈↓ Xi q    (18) The operator c acting on the vacuum can also produce a hole + spin-wave state. This kσ amplitude is shown in Fig. 3b as a circled cross with the dashed line being a spin-wave. We denote this amplitude by b k,q 2 2 b = 0 β d c 0 = 0 β c† e−i(k−q)ri c eikrj 0 k,q h | q k−q↓ k↓| i h | qsN(1/2+m) i↑ sN j↓ | i ≈ i∈↑ j X X    2 2 0 β S+eiqri 0 = v . (19) ≈ Nh | q i | i sN q i∈↑ X   We stress that (19) is a bare vertex. It corresponds to the instantanious creation of a hole + spin wave, but not the creation of a hole with a subsequent decay into hole + spin-wave. To elucidate this point look at Fig.4.The upper part of this figure describes the wave function 6 of the initial Neel state: a - component without spin quantum fluctuations, b - component with spin quantum fluctuation. The lower part of Fig.4 arises imediately from the upper one after kicking out an electron with spin up. Part a does not contain a spin flip, and it corresponds to the amplitude a (18). Part b does contain a spin flip, and it corresponds to k the amplitude b (19). Note that b at q 0. The reason is that the operator (16) k,q k,q → ∞ → does not correspond to any quasiparticle of the system with long-range antiferromagnetic order, and therefore the usual Goldstone theorem is not applicable. Let us denote by a dot (Fig. 3c) the usual hole-spin-wave vertex g given by eq. (11). k,q In the leading in t approximation the amplitude of single hole creation by the external perturbation (16) is given by diagrams presented at Fig. 5, with the thin solid line in this case the bare hole Green’s function (7). If we set t′ = t′′ = 0 and ǫ = ǫ = 0 calculation of 0k this amplitude can be easily done analytically b g 8t v (γ u +γ v ) M (ǫ = 0,k) = a + k,q k−q,q = √0.8 q k−q q k q = 1 k ǫ ǫ ω − N ω q 0k 0k−q q q q X − − X 4t 1 1 = √0.8+ γ = √0.8(1+0.45 t γ ). (20) k k N q ωq − 2! · · X M2 is the quasiparticle residue of the Green’s function (17). The eq. (20) agrees with the 1 result obtained using a string representation15. We stress that even at t = 0 the residue is 0.8 due to the spin quantum fluctuation in the ground state of the Heisenberg model16,17 Now we can find the relation between Green’s functions G (17) and G (5). In the c d leading in t approximation it is given by diagrams presented at Fig. 6 with the thin solid linebeinginthiscasethebareholeGreen’sfunctionG (7). Nowletusdressthesediagrams 0d by higher orders in hopping t. As we already discussed there is no single loop correction to the “dot”. We neglect double loop correction to the “dot” as it has been done in SCBA. Therefore the only possibility is an introduction of a self energy corrections. An example of the correction to diagram Fig. 6c is shown at Fig. 7. To take into account all these corrections we need just to replace at Fig. 6 all bare hole Green’s functions (7) by dressed hole Green’s function given by eq. (12). So, the Fig. 6 actually represents a Dyson equation relating G (17) and G (5). In analytical form it is c d G (ǫ,k) = a2G (ǫ,k)+ b2 G (ǫ ω ,k q)+ c k d k,q d − q − q X + 2a G (ǫ,k) b g G (ǫ ω ,k q) + (21) k d k,q k−q,q d q " q − − # X 2 + G (ǫ,k) b g G (ǫ ω ,k q) . d k,q k−q,q d q " q − − # X 7 So as soon as we have found G using SCBA (12) we can calculate the Green’s function G d c defined by eq. (17). The imaginary part of G (ǫ,k) gives directly the spectra measured in c ARPES experiments. We now discuss sum rules. All singularities of Green’s functions are in the lower half plane of complex ǫ. Therefore if we integrate eq.(12) over ǫ from to + , this integral −∞ ∞ can be replaced by the integral over an infinite semi-circle in the upper ǫ half plane.For infinite ǫ, G = G , and we get the well known sum rule d 0d 1 ∞ Im G (ǫ,k)dǫ = 1, (22) d − π Z−∞ which agrees with with eq.(4). If we integrate now eq.(21) in the same limits, the terms which contain more than one Green’s function give zero contribution, because the integral can be transfered into the upper complex ǫ half plane. And we find 1 ∞ 1 2 Im G (ǫ,k)dǫ = Im G (ǫ,k)dǫ a2 + b2 = 0.8+ v2 = 1. (23) − π Z−∞ c (cid:18)−π Z d (cid:19) k q k,q! N q q X X Thus the equation (21) reproduces the correct normalization: 0 c† c 0 = 1. h | k↑ k↑| i The vertex b (19) is invariant under translation with the inverse vector of magnetic k,q sublattice Q = ( π, π): b = b . At the same time the vertex g (11) changes k+Q,q k,q k,q ± ± sign with this translation: g = g . Therefore the diagrams Fig. 6c,d change sign at k+Q,q k,q − k k+Q and → G (ǫ,k+Q) = G (ǫ,k). (24) c c 6 Dueto thesame properties of vertices b and g the diagrams Fig. 6c,d,e (square brackets k,q k,q in eq. (21)) vanish at the face of magnetic Brillouin zone (γ = 0). The diagram presented k at Fig. 6b (termwith b2 in eq. (21)) is small numerically. Therefore at theface of magnetic k,q Brillouin zone G (ǫ,k) G (ǫ,k). However away from the face they differ significantly. The c d ≈ plots of 1 Im G (ǫ,k) as a functions of ǫ for k = (π/2,π/2), k = (π/2,0), k = (π,0), and −π c k = (0,0) are presented at Fig.8. A plot of the quasiparticle residue Zc as a function of k k along (1,1) direction is given at Fig. 9. The quasiparticle residue outside the magnetic zone is smaller than that inside. For comparison we also present a plot of Zd. k IV. HUBBARD MODEL CORRECTION The picture considered above corresponded to a modified t J model. It means that − double electron occupancy was forbidden. Now we want to take into account the fact that 8 the t t′ t′′ J model originates from the Hubbard model. We assume that it is a simple − − − one band Hubbard model with on site repulsion U. First of all this gives some corrections to the “bare” hole dispersion (6), see, e. g. Ref.4. However we assume that renormalization is done and these corrections are already included in the values of effective hopping amplitudes t′, t′′ given in (2). There are also some corrections to the hole-spin-wave vertex4, but they are small at t′,t′′ U. The really important effect is the renormalization of the vertex a k ≪ (18). In t J model this vertex is given by the process shown at Fig. 10a: an electron is − removed from corresponding sublattice. In Hubbard model there is an additional possibility shown at Fig. 10b: first the electron hops to occupied nearest site, and then it is removed from this site. Simple calculation shows that this gives 4t J a a 1+ γ ) = 1/2+m 1+ γ , (25) k k k k → × U t (cid:18) (cid:19) q (cid:18) (cid:19) 4t 2 J b b 1+ γ = v 1+ γ . k,q k,q k q k → × U sN t (cid:18) (cid:19) (cid:18) (cid:19) We took into account that J = 4t2/U. The magnitude of the t/U correction in (25) is obvious, however one should be careful with the sign. To find it one needs to commute fermionic operators in order corresponding to Fig. 10b. The Dyson equation (21) remains valid. So we can easily find the Green’s function GH, where index H indicates that the c Hubbard model correction is taken into account. The plots of 1 Im GH(ǫ,k) as functions −π c of ǫ for k = (π/2,π/2), k = (π/2,0), k = (π,0), and k = (0,0) are presented in Fig.11. A plot of the quasiparticle residue ZcH as a function of k along (1,1) direction is given in Fig. k 9. We see that the “Hubbard correction” causes the decrease of the residue outside of the magnetic Brillouin zone to be steeper. The sum rule (23) is changed. Now we have 1 ∞ J 2 J Im GH(ǫ,k)dǫ 1+ γ 1+2 γ . (26) − π c ≈ t k ≈ t k Z−∞ (cid:18) (cid:19) Let us comment on the definition (16) of the operatorc . Its normalization is adjusted for a kσ system with strong antiferromagnetic correlations and it is close to that for d (see eq.(3)). kσ However as a result the definition (16) differs from that usually accepted for a normal Fermi liquid by a factor √2. This is the reason why the sum rule (26) can be larger than unity. Generally the normalization can be chosen arbitrally. It is a question of convenience only. However, let us prove that the sum rule for the total number of electrons in the system is fulfilled. According to definition (16) c† c = 2N , (27) kσ kσ kσ 9 where 1 1 N = c† e−ikri c eikrj (28) kσ sN iσ sN iσ  i j X X    is the operator for the number of electrons. Due to the definition (17) of Green’s function G one has the standard relation c 1 ∞ Im GH (ǫ,k)dǫ = 0 c† c 0 = 2 0 N 0 . (29) − π c,σ h | kσ kσ| i h | kσ| i Z−∞ Comparing with (26) we find 1 J 0 N 0 = (1+2 γ ). (30) kσ k h | | i 2 t The operator for the total number of electrons is equal to Nˆ = N . (31) kσ σ,k∈full X We put a “hat” to distinguish this operator from the number of sites N. Note that in all equations before we assumed summation over momenta inside the magnetic Brillouin zone. But in the eq. (31) we must sum over the full Brillouin zone. Finally from eqs. (30),(31) one finds that the sum rule for the total number of electrons 1 J 0 Nˆ 0 = (1+2 γ ) = N (32) k h | | i 2 t σ,k∈full X is fulfilled. In conclusion of this discussion we would like to note that the origin of all these complications with normalization is very simple:The natural zone for the operator d is the kσ magnetic Brillouin zone. On the other hand the natural zone for c is the full Brillouin kσ zone. This is the reason why one should be careful comparing these two operators. V. COMPARISON WITH EXPERIMENT Many of the experimental features observed are reproduced with the theory described here. The large dispersion between (0,0) and (π/2,π/2) and the assymetric quasi particle weight about(π/2,π/2)withtheverystrongdcrease inweight onmoving beyond(π/2,π/2). Also the lack of dispersion along (0,0) and (π,0) as well as the very low quasiparticle weight is well reproduced. A major discrepancy between theory and experiment concerns the width of the quasi particle peak.The theoretical spectra (Figs. 8,11) have narrow peaks. On the 10