- Email: [email protected]
Dynamic magnetic susceptibility of copper oxide superconductors in the metallic state
PHYSICAI ELSEVIER
Physica B 199&200 (1994) 344-346
Dynamic magnetic susceptibility of copper oxide superconductors in the metallic state F. Onufrieva*, J. Rossat-Mignod* Laboratoire L~on Brillouin, CEA-CNRS, CE Saclav, 91191 Gif-sur-Yvette cedex, France
Abstract The main features of the spin dynamics within CuO2-1ayers observed in the YBazCu306+~ system as a function of doping and temperature in the metallic state with magnetic short-range order are well accounted for by a theory based on the t - r - J model and a slave-fermion approach. Im z(q, co) looks like a two-spinon density of states and exhibits a gap at low energy and a cut offat high energy, the uniform susceptibility vanishes at ~"= 0. The shape of Im 7,(q, co) as a function of co is T-dependent.
During these last five years a large amount of experimental and theoretical works has been reported on highT~ copper oxide superconductors, but there is not yet a well accepted understanding of their unusual physical properties in the normal state and of the pairing mechanism. The investigation of the spin dynamics, i.e. the dynamic susceptibility x(q, co), is of crucial importance to reveal the specific properties of strongly correlated electronic CuO2-1ayers in the hole-doped high-7"~ superconductors. In this paper we summarize the general features of lm x(q, to) extracted from inelastic neutron scattering (INS) experiments for the YBa2Cu306+~ system as a function of hole doping and temperature, and we present a theory of the spin dynamics in the metallic state with magnetic short-range order (SRO) based on the t - t ' - J model. We show that the unusual shape of Im ;~(q,co) and its evolution as a function of doping and temperature in the metallic state with magnetic SRO * Corresponding author. t Decca ;ed 20 August 1993.
naturally arise within an approach using a slave-fermion representation (SFR) of Hubbard operators. It should be noted that for a state with long-range order (LRO) (doped AF state) the spin dynamics is well accounted for within the same t - t ' - J model using a generalized Maleyev-Dyson representation of Hubbard op:ators: critical hole doping n~- 2.7% and strong damping of long wavelength spin waws [1]. The main results of INS experiments on the YBa2Cu306+~ system [2, 3] are the following. For any doping, q-scans show that Im X is peaked around the AF wave vector qAF and has a q-width which is energy and temperature independent and, surprisingly, proportional to hole donin~
Fnero'v-~t'an~
fc~r ,a _-- q A F , allr~ll, IIC tra a a t
Im ;¢(q, ~,~)as a function of energy, as shown in Fig. I for two typical dopings corresponding to the weakly doped (x = 0.51, T~ = 50 K) and the heavily doped (x = 0.92, Tc = 91 K) regimes. At low temperatures, |m Z exhibits a spin gap at low energy which is sniall (E~"-kT¢) for the weakly doped regime and increases with doping. Im Z sharply decreases at high energies, revealing some encrgy cut off. Moreover,
0921-4526/94/$07.00 ~ 1994 Elsevier Science B.V. All rights reserved SSDI 0921-4526[93] E0327-D
F. Omt.frieva. J. Rossat-Mignod/ Physica B 199&200 (1994) 344-346
YBa~Cu~0 s,st
~ ,oo
A
~~.,~oo~_
f YBazCu, O~.s' "
6"=(v, ~, S.Zl
= -
.,=,oo,/
I
0
In Eq. (I), J(q) is the Fourier transform of the isotropic spin- spin interaction and Z(q, ¢o~is the irreducible part of diagrams which cannot be cut across the interaction line (Larkin's formalisml. In the lowest approximation Xlq. ~,) is given by Eq. (2), where E k represents the spinon dispersion, ua and vk are the coefficients of the Bogolubov transformation which diagonalizes the spinon Hamiltonian and n B is the Bose distribution function. The behavior of spinon and holon degrees of freedom was investigated in many papers within the MFA. The explicit expressions for Ek, u, and t,~ oepend on the actual model { t - J [ 4 ] , t ' - J [5] and t - t ' - J [6]) and on the nature of the SRO (AF, spiral, canted). The most general expressions which are valid for the t - t ' - J model and for different spiral states are given by
l-Er -- ~,ImeV
zoo Q=tvz,,/z,s,zl
10
20
30
~0
50
]' -I
60
Energy (rneV) Fig. I. Im g(Q,w) measured at Q = (1/2, i/2, 5.2) for YBazCu~O~,+~ in the weakly doped (x = 0.51, 7", = 50 K( and the heavily doped (x = 0.92, T, = 91 K) regimes.
E, = (flzk - BZ) I/z,
2 t,2k = (fl*/Ek + 1}/2, u*"
utv A = -- Bk/2E k,
lm Z exhibits a peak at an energy E, which depends on doping (E, = 7.5 and 41 me V for x = 0.51 and 0.92, respectively}. This peak is associated with a narrowing of the q-width. By increasing the temperature, the low energy part of Im Z is strongly depressed contrary to the high energy part which remains almost unchanged. At high temperatures, surprisingly, lm Z has a maximum for an energy 1~ 30 meV) which is independent on doping. So, the shape and the temperature dependence of lm ~ are quite unusual, not at all of Lorentzian type. Up to now no theory exists to account for the features described above. The theory that we present below allows us to understand some of these features. The quantity which is measured in INS and NMR experiments is the imaginary part of the dynamic magnetic susceptibility lm ~ ( q , e~) which is connected with the spin Green function G'=(q, t o ) = <>,~ 1~ = x, y, z) by the relation lm ;(~(q, ~) = lim,-o lm G :~ (q, ~o + iq)/2n. In the SFR, spin operators are compound operators consisting of two spinon operators and the spin Green function is represented in terms of spinon Green [unctions. Straightforward calculations give the following expression for ~(q, ~o): (I)
~(q.ml = Z(q.t,fl[I + d(q)L(q,,,~)]-
L'(q, m) = Zm~(q, co) = ~(ukv~k
,1 + ne(Ek) + nB(Ek + q) + q -- t,ku~ + q)'" Ek + E, + q - c,~ - iq nn(Ek) - nn(E~ + ¢)
•
Ek+q
-
-
k
-
-
345
(2)
(3)
f t t = 2 + Ht'bk -- sgn(i,6k)(2tH + dF/2)lPkl, Bk = 2J3'kA.
In Eq. (3), A, F and H are the amplitudes of spinon correlation functions, Air = (ai+ b ; - b? a ; ) and F u = ( a ~ a j + bi+bj), characterizing the AF and Ferro SRO, respectively, and of the holon correlation function H o = (f~+fj). t~ and PA are given in Ref. [6]. Pk depends on the symmetry of the SRO, for a l l, 1) spiral state Fk = i(sin k.~ + sin kr), ). is the Lagrange multiplier. The states which are realized at different dopings and temperatures have to be found self-consistently by solving the set of equations for A,j, F~j, H o and 2. The answer concerning the type of SRO in the spinon description is quite sensitive to the parameter values, an increase of t' suppresses the incommensurability arising from the spinon dispersion law [6]. The answer is also quite sensitive to the approximation (n~ = 0.6 in MFA and 0.275 taking into account fluctuations [5]) and we cannot rely op MFA results for a quantitative description. However, the main qualitative features of the spinon subsystem which arise from general symmetry considerations are reproduced quite well. These features are: Ill existence of a gap in the spinon spectrum with minimum value Eg at q = qo (q, = qAv for AF state and q o - q~.v characterizes the incommensurability for the spiral state), (it) increase of Eg wi',h T and n, lift) decrease of A with T and n. and fivt slight increase of F and H with n. Taking into account these general spinon features and analyzing the behavior of g(q, ~,~ given by Eqs. (1) and (2) we obtain the following results. Im Zlq. oJ) and hence Im/Jq,~ot look like a twoparticle density of states and so is not at all of Lorentzian type. At T = 0 only the first term of Eq. (2) contributes
346
F. Onufrieva. J. Rossat.Mignod / Physica B 199&200 (1994) 344-346
and therefore l m x exists only between too(q)= min(E~ + Ek + q) and toM(q) = max(Ek + Ek + q). Thus, for any q value Im X exhibits a gap at low energy and a cut-off at high energy. This gap has a minimum value for q = qo and increases with doping. The to width of lm X decrcasos with doping. All these features allow, for the first time, to understand the experimental shape of Imx, Equation (1) shows that J(q) strongly enhances Im X around qAF, however, Z(q, to) is enhanced around qo. So Im ;( can exhibit a maximum either at an incommensurate wave vector or at qAr. This may explain the difference between (La-Sr)2CuO4 and YBa2Cu306 +~ if we assume that t' is larger in YBa2Cu306+~ and hence q^F -- qo is smaller. The q width of Irn x results from a delicate balance between J(q) and Z (q, to). As no incommensurability is observed in YBa2Cu306+~ the q width around qAF must be related to qAv -- qo and so proportional to doping in agreement with the surprising experimental result. For q = 0 , ~ , = 0 a t T = 0 , and then x a t t o = 0 i n creases with 7 because the first term in Eq. (2) is always equal to zero whereas the second term is equal to zero at T = 0. This result accounts quite well for the anomalous T-dependence of the static uniform susceptibility observed in magnetization and NMR experiments. For q in the vicinity of qAF, the prefactor of the first term in Eq. (2) is maximum whereas the second term gives always a small contribution. The T-dependence of lmx is determined mostly by that of X which arises from a competition between the prefactor of the first term which is proportional to A and strongly decreases with 7",
and the second factor which slightly increases with T. Then the strong decrease of Imx as a function of T, observed at low energies, finds a natural explanation in the weakening of the nearest neighbor AF correlation. The second term in Eq. (2) increases with T and contributes at small energies to fill the gap. At high T the spinon dispersion is reduced and the characteristic energy is related to A which seems to be only weakly dependent on doping and 7". In conclusion we would like to emphasize that the main features of the spin excitation spectrum are well accounted for by this theory despite the fact of the rather rough approximation. In a next paper we will show that an additional Fermi liquid like contribution to X(q, ta) arises from the hole motion if we deal directly with Hubbard operators. The good agreement of the present theory with experiments seems to indicate that the spinon contribution of Im ;( is the dominant one in the INS energy scale.
References
[I ] F. Onufrieva, V. Kushnir and B. Teperverg, to be published. [2] J. Rossat-Mignod et al., Physica C |85-189 0991) 86; Physica Scripta T 45 (1992) 75; Physica B 186-188 (1993) 1. [3] J. Rossat-Mignod et al., in: Selected Topics in Superconductivity, Fronti~res in Solids Sciences, Vol. 1, eds. L.C. Gupta and M.S. Multani (1993) p.265. [4] D. Yoshioka, J. Phys. Soc. Japan 58 {1989) 1516. ES] K. Kuboki ct al,, J. Phys. Soc. Japan 61 0992) 621. [6] C.L. Kane ct al., Phys. Rcv. B 41 (1990) 2653.
Physica B 199&200 (1994) 344-346
Dynamic magnetic susceptibility of copper oxide superconductors in the metallic state F. Onufrieva*, J. Rossat-Mignod* Laboratoire L~on Brillouin, CEA-CNRS, CE Saclav, 91191 Gif-sur-Yvette cedex, France
Abstract The main features of the spin dynamics within CuO2-1ayers observed in the YBazCu306+~ system as a function of doping and temperature in the metallic state with magnetic short-range order are well accounted for by a theory based on the t - r - J model and a slave-fermion approach. Im z(q, co) looks like a two-spinon density of states and exhibits a gap at low energy and a cut offat high energy, the uniform susceptibility vanishes at ~"= 0. The shape of Im 7,(q, co) as a function of co is T-dependent.
During these last five years a large amount of experimental and theoretical works has been reported on highT~ copper oxide superconductors, but there is not yet a well accepted understanding of their unusual physical properties in the normal state and of the pairing mechanism. The investigation of the spin dynamics, i.e. the dynamic susceptibility x(q, co), is of crucial importance to reveal the specific properties of strongly correlated electronic CuO2-1ayers in the hole-doped high-7"~ superconductors. In this paper we summarize the general features of lm x(q, to) extracted from inelastic neutron scattering (INS) experiments for the YBa2Cu306+~ system as a function of hole doping and temperature, and we present a theory of the spin dynamics in the metallic state with magnetic short-range order (SRO) based on the t - t ' - J model. We show that the unusual shape of Im ;~(q,co) and its evolution as a function of doping and temperature in the metallic state with magnetic SRO * Corresponding author. t Decca ;ed 20 August 1993.
naturally arise within an approach using a slave-fermion representation (SFR) of Hubbard operators. It should be noted that for a state with long-range order (LRO) (doped AF state) the spin dynamics is well accounted for within the same t - t ' - J model using a generalized Maleyev-Dyson representation of Hubbard op:ators: critical hole doping n~- 2.7% and strong damping of long wavelength spin waws [1]. The main results of INS experiments on the YBa2Cu306+~ system [2, 3] are the following. For any doping, q-scans show that Im X is peaked around the AF wave vector qAF and has a q-width which is energy and temperature independent and, surprisingly, proportional to hole donin~
Fnero'v-~t'an~
fc~r ,a _-- q A F , allr~ll, IIC tra a a t
Im ;¢(q, ~,~)as a function of energy, as shown in Fig. I for two typical dopings corresponding to the weakly doped (x = 0.51, T~ = 50 K) and the heavily doped (x = 0.92, Tc = 91 K) regimes. At low temperatures, |m Z exhibits a spin gap at low energy which is sniall (E~"-kT¢) for the weakly doped regime and increases with doping. Im Z sharply decreases at high energies, revealing some encrgy cut off. Moreover,
0921-4526/94/$07.00 ~ 1994 Elsevier Science B.V. All rights reserved SSDI 0921-4526[93] E0327-D
F. Omt.frieva. J. Rossat-Mignod/ Physica B 199&200 (1994) 344-346
YBa~Cu~0 s,st
~ ,oo
A
~~.,~oo~_
f YBazCu, O~.s' "
6"=(v, ~, S.Zl
= -
.,=,oo,/
I
0
In Eq. (I), J(q) is the Fourier transform of the isotropic spin- spin interaction and Z(q, ¢o~is the irreducible part of diagrams which cannot be cut across the interaction line (Larkin's formalisml. In the lowest approximation Xlq. ~,) is given by Eq. (2), where E k represents the spinon dispersion, ua and vk are the coefficients of the Bogolubov transformation which diagonalizes the spinon Hamiltonian and n B is the Bose distribution function. The behavior of spinon and holon degrees of freedom was investigated in many papers within the MFA. The explicit expressions for Ek, u, and t,~ oepend on the actual model { t - J [ 4 ] , t ' - J [5] and t - t ' - J [6]) and on the nature of the SRO (AF, spiral, canted). The most general expressions which are valid for the t - t ' - J model and for different spiral states are given by
l-Er -- ~,ImeV
zoo Q=tvz,,/z,s,zl
10
20
30
~0
50
]' -I
60
Energy (rneV) Fig. I. Im g(Q,w) measured at Q = (1/2, i/2, 5.2) for YBazCu~O~,+~ in the weakly doped (x = 0.51, 7", = 50 K( and the heavily doped (x = 0.92, T, = 91 K) regimes.
E, = (flzk - BZ) I/z,
2 t,2k = (fl*/Ek + 1}/2, u*"
utv A = -- Bk/2E k,
lm Z exhibits a peak at an energy E, which depends on doping (E, = 7.5 and 41 me V for x = 0.51 and 0.92, respectively}. This peak is associated with a narrowing of the q-width. By increasing the temperature, the low energy part of Im Z is strongly depressed contrary to the high energy part which remains almost unchanged. At high temperatures, surprisingly, lm Z has a maximum for an energy 1~ 30 meV) which is independent on doping. So, the shape and the temperature dependence of lm ~ are quite unusual, not at all of Lorentzian type. Up to now no theory exists to account for the features described above. The theory that we present below allows us to understand some of these features. The quantity which is measured in INS and NMR experiments is the imaginary part of the dynamic magnetic susceptibility lm ~ ( q , e~) which is connected with the spin Green function G'=(q, t o ) = <
~(q.ml = Z(q.t,fl[I + d(q)L(q,,,~)]-
L'(q, m) = Zm~(q, co) = ~(ukv~k
,1 + ne(Ek) + nB(Ek + q) + q -- t,ku~ + q)'" Ek + E, + q - c,~ - iq nn(Ek) - nn(E~ + ¢)
•
Ek+q
-
-
k
-
-
345
(2)
(3)
f t t = 2 + Ht'bk -- sgn(i,6k)(2tH + dF/2)lPkl, Bk = 2J3'kA.
In Eq. (3), A, F and H are the amplitudes of spinon correlation functions, Air = (ai+ b ; - b? a ; ) and F u = ( a ~ a j + bi+bj), characterizing the AF and Ferro SRO, respectively, and of the holon correlation function H o = (f~+fj). t~ and PA are given in Ref. [6]. Pk depends on the symmetry of the SRO, for a l l, 1) spiral state Fk = i(sin k.~ + sin kr), ). is the Lagrange multiplier. The states which are realized at different dopings and temperatures have to be found self-consistently by solving the set of equations for A,j, F~j, H o and 2. The answer concerning the type of SRO in the spinon description is quite sensitive to the parameter values, an increase of t' suppresses the incommensurability arising from the spinon dispersion law [6]. The answer is also quite sensitive to the approximation (n~ = 0.6 in MFA and 0.275 taking into account fluctuations [5]) and we cannot rely op MFA results for a quantitative description. However, the main qualitative features of the spinon subsystem which arise from general symmetry considerations are reproduced quite well. These features are: Ill existence of a gap in the spinon spectrum with minimum value Eg at q = qo (q, = qAv for AF state and q o - q~.v characterizes the incommensurability for the spiral state), (it) increase of Eg wi',h T and n, lift) decrease of A with T and n. and fivt slight increase of F and H with n. Taking into account these general spinon features and analyzing the behavior of g(q, ~,~ given by Eqs. (1) and (2) we obtain the following results. Im Zlq. oJ) and hence Im/Jq,~ot look like a twoparticle density of states and so is not at all of Lorentzian type. At T = 0 only the first term of Eq. (2) contributes
346
F. Onufrieva. J. Rossat.Mignod / Physica B 199&200 (1994) 344-346
and therefore l m x exists only between too(q)= min(E~ + Ek + q) and toM(q) = max(Ek + Ek + q). Thus, for any q value Im X exhibits a gap at low energy and a cut-off at high energy. This gap has a minimum value for q = qo and increases with doping. The to width of lm X decrcasos with doping. All these features allow, for the first time, to understand the experimental shape of Imx, Equation (1) shows that J(q) strongly enhances Im X around qAF, however, Z(q, to) is enhanced around qo. So Im ;( can exhibit a maximum either at an incommensurate wave vector or at qAr. This may explain the difference between (La-Sr)2CuO4 and YBa2Cu306 +~ if we assume that t' is larger in YBa2Cu306+~ and hence q^F -- qo is smaller. The q width of Irn x results from a delicate balance between J(q) and Z (q, to). As no incommensurability is observed in YBa2Cu306+~ the q width around qAF must be related to qAv -- qo and so proportional to doping in agreement with the surprising experimental result. For q = 0 , ~ , = 0 a t T = 0 , and then x a t t o = 0 i n creases with 7 because the first term in Eq. (2) is always equal to zero whereas the second term is equal to zero at T = 0. This result accounts quite well for the anomalous T-dependence of the static uniform susceptibility observed in magnetization and NMR experiments. For q in the vicinity of qAF, the prefactor of the first term in Eq. (2) is maximum whereas the second term gives always a small contribution. The T-dependence of lmx is determined mostly by that of X which arises from a competition between the prefactor of the first term which is proportional to A and strongly decreases with 7",
and the second factor which slightly increases with T. Then the strong decrease of Imx as a function of T, observed at low energies, finds a natural explanation in the weakening of the nearest neighbor AF correlation. The second term in Eq. (2) increases with T and contributes at small energies to fill the gap. At high T the spinon dispersion is reduced and the characteristic energy is related to A which seems to be only weakly dependent on doping and 7". In conclusion we would like to emphasize that the main features of the spin excitation spectrum are well accounted for by this theory despite the fact of the rather rough approximation. In a next paper we will show that an additional Fermi liquid like contribution to X(q, ta) arises from the hole motion if we deal directly with Hubbard operators. The good agreement of the present theory with experiments seems to indicate that the spinon contribution of Im ;( is the dominant one in the INS energy scale.
References
[I ] F. Onufrieva, V. Kushnir and B. Teperverg, to be published. [2] J. Rossat-Mignod et al., Physica C |85-189 0991) 86; Physica Scripta T 45 (1992) 75; Physica B 186-188 (1993) 1. [3] J. Rossat-Mignod et al., in: Selected Topics in Superconductivity, Fronti~res in Solids Sciences, Vol. 1, eds. L.C. Gupta and M.S. Multani (1993) p.265. [4] D. Yoshioka, J. Phys. Soc. Japan 58 {1989) 1516. ES] K. Kuboki ct al,, J. Phys. Soc. Japan 61 0992) 621. [6] C.L. Kane ct al., Phys. Rcv. B 41 (1990) 2653.