- Email: [email protected]
Dual itinerant-localized nature of magnetism in high-Tc cuprates
Journal of Magnetism and Magnetic Materials 140-144 (1995) 1271-1272
,.,.
journal of magnetism and magnetic materials
~ ,i~ ELSEVIER
Dual itinerant-localized nature of magnetism in high-Tc cuprates F.
Onufrieva *, J. Rossat-Mignod
*
Lab. L~onBrillouin (CEA-CNRS), Saclay, 91191 Gif-sur-Yt~ette,France Abstract Spin dynamics in YBCO is analysed on a framework of a new theory (based on the t-t'-J model and the diagrammatic technique for Hubbard operators) which treats correctly strong electron correlations within CuO 2 layer.
Inelastic neutron scattering (INS) and NMR experiments have discovered many exotic features of the spin dynamics in high-Tc cuprates in the superconducting (SC) state and the metallic state above Tc, especially for YBa2Cu306+ ~. There is anomalous behaviour of the static uniform susceptibility for the weakly-doped regime ( x < 0.85) which evolves continuously to the normal one at x = 1 [1]. There is an unusual shape of lm x(k, to) versus to (revealed by INS [2]) that is neither of a Lorentzian type typical for localized magnetism, nor of a smooth extended type typical for itinerant magnetism. The spectral band Im x(k, to) is very narrow in to with an energy gap, E C, at low energy and a sharp cut-off at high energy. The energy scale is surprisingly small. A resonance peak, E R, is observed inside the spectral band. It disappears above Tc together with the spin gap, while the values of E G and E R vary differently with x (and so with Tc). Another striking feature is a puzzling evolution with doping of the shape of Im x(k, to)[2]. Based on the t-t'-J model and the diagrammatic technique for Hubbard operators (HO) we have developed a theory [3] which treats correctly the interrelation between localized spins on Cu and itinerant holes in a new narrow band that corresponds to a coherent motion of 'Zhang-Rice singlet'. This interrelation exists in the t-J model and is given by the multiplication rules for HO, but is lost when any representation of HO in terms of B o s e / F e r m i operators (the slave-boson (SB) or slave-fermion ones) is used. In the present work, we apply this theory to calculate the dynamic magnetic susceptibility, x(k, to), for YBCO. The obtained expression for x(k, to) has an RPA structure with two contributions to the 'zero susceptibility', instead of a single one in the weak-coupling theories [4] and in the strong-coupling theories based on the SB repre-
sentation [5]. The first contribution, I(k, to), arises from the subsystem of itinerant holes with a reduced bandwidth W (Wct t(1 + 6 ) / 2 , where 6 is the doped hole concentration). It has a BCS structure in the SC state and a Lindhard function structure in the normal state. The second contribution, L(k, to), arises from the subsystem of localized spins with SR AF correlations. It has a two-spinon structure [3]. The symmetry of the hole anomalous Green function reflects the symmetry of SC pairing (assumed to be of a d-wave type), the symmetry of the spinon anomalous Green function reflects the symmetry of SRO in the subsystem of localized spins. The intensity of the localized-spin contribution decreases with doping proportionally to the amplitude of SR AF order parameter. The intensity of the itinerant-hole contribution increases with doping. As a result of their competition, the different dynamics occurs in different doping regimes. Deeply in the overdoped regime, the former contribution becomes negligible, and the behaviour tends to a normal metal one. In the underdoped regime, both contributions are important. In the weakly-doped regime the 'localised' contribution is dominant. This general picture accounts for crossover effects that are observed experimentally for many magnetic properties in cuprates.
r
o YBCO o.e - 6 = 0 . 2 5 .06
r
i
--QAF=(W.~)
.......... (0.8~,0.8~) . . . . kn~=(061~,051~)
-
N
I 02
o.o
i:
..... ,"~"i
0
0~
i
018
'
1'2
t0/J * Corresponding author. Fax: +33-1-69 08 82 61; email: [email protected].
Fig. 1. Itinerant contribution for different wavevectors ( t / J = 1.8; t/t' = -0.45; Asc = 0.1J).
0304-8853/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0304-8853(94)01649-6
F. Onufriet:a, J. Rossat-M ignod / Journal of Magnetism and Magnetic Materials140-144 (1995) 1271 1272
1272
As to dynamic properties, details are different for different to ranges due to the different beaviour of Im I and Im L as a function of to. The plots of lm l(k, to) versus to for different wavevectors are shown in Fig. 1. (When calculating, we use the value of t ' / t equal to - 0 . 4 5 in order 1o model the shape of the experimentally observed Fermi surface.) At lowest to only a nodal contribution is nonzero. For higher w, the AF contribution is important due to existence of a sharp peak at low enough energy. lm L(k, to) represents an ideal two-spinon band (with a gap M ( k ) ) narrow in energy which becomes broader when spinon damping is taken into account. Spinon damping exists even at T - 0 in the presence of fermionic quasiparticles, however, due to the existence of a gap in the hole dispersion law in the SC state, it appears only above a threshold energy, to~, which value is proportional to ,4sc. Then, Im L versus w has a shape shown in Fig. 3. For any to, Im L and Re L are peaked at k = QAF. The resultant Im x ( k , to) has a k-structure peaked around QAI~"for all w belonging to the principal part of the spectral band, Im X (compare energies in Fig. 2 and Fig. 3 for 6 = 0.25). For lowest to, a very weak incommensurate (nodal) structure is observed. The shape of Im X versus to is determined by the competition between l ( k , to) and L(k, to) (Fig. 3). For the heavily-doped regime (HDR), 6 ~ 0.15, the ideal twospinon spectral band is situated on the left from the peak in lm I(QAF, to). For the weakly-doped regime (WDR), (5 7: 0.15, the situation is reversed. Then, Im X exhibits a gap which is equal to A L in the HDR and to A ~ in the WDR, when neglecting damping. When spinon and hole damping are taken into account, A L manifests itself as a pseudogap, a true gap in lm X is equal to to~ in the HDR and to w~ in the WDR and is proportional to Asc: (to~ is a threshold energy in the hole damping). The resonance peak in | m X for ~ = 0.20-0.25 corresponds to a sharp drop in Re I
6
F\
YBC " 6 : 0.25 -o~ --=,~°o, / \ .3 ....... (0/..1=0.23 .~" ---=/J =0.27 _//...._--~., \ . . . .
0.=o37 / L
'
"\
.I
/,¢
- 02
- -
J ImXIQAF.W )
----
JIm I (QAF,I,O)
YBCO
........... JIr. LIQ~,,~J
,aL/x 6=0.095 l
06
\..,
o 2 ~,
0
O
Z~tI 6 : 0 2 5 "
102
",
!
0.2
04
0
col
~-
tO/J
Fig. 3. Im X(QAF, t,,O) together with Im L(QAF, ~o) and lm
I(QA ~ , w) for YBCO at different dopings (T = 0). versus to which accompanies the sharp peak in Im 1. The latter is a consequence of d-wave superconductivity and the resonance peak in Im X occurs at to -=- A/(QA F) OC A SC. Thus, three important characteristics, the gap in the WDR, the gap in the HDR and the resonance peak in the HDR are related to the SC gap. However, the explicit forms of these relations are different and so, they vary differently with Tc (and with doping). At lowest 6 near the critical concentration, the spin dynamics is characterized by welldefined magnon-like excitations (Fig. 3, 6 = 0.08). These excitations appear as a pole of x ( k , to). As 6 decreases, the gap in magnon-like excitation tends to zero and at = 0.075, the pole occurs at to - 0 which corresponds to the instability of the considered phase ( X(QAF, O) = 3c). All these results account well for many unusual features of spin dynamics in YBCO [2]: for the gap and pseudogap effects, for the shape of k-scan, for the shape of Im X versus to and its unusual evolution with doping and others. In addition, the theory predicts a weak incommensurate 'in-gap' scattering which should exist in the case of d-wave superconductivity. This point has to be verified by subsequent INS experiments. References
zi ... 0
,
(~/2.x/2)
'-.\ ~
,
(~/x) Wawevector
A"
(3~/2.3~/2)
Fig. 2. Im X versus k in YBCO in increasing o) (T = 0).
[1] [2] [3] [4] [5]
H. Alloul et al., Phys. Rcv. Lctt. 63 (1989) 1700. J. Rossat-Mignod ct al., Physica B 192 (1993) 109. F. Onufrieva et al., Phys. Rev. B (1995), in press. N. Bulut et al., Phys. Rev. B 47 (1993) 3419. T. Tanamoto et al., J. Phys. Soc. Jpn. 61 (1992) 1886.
,.,.
journal of magnetism and magnetic materials
~ ,i~ ELSEVIER
Dual itinerant-localized nature of magnetism in high-Tc cuprates F.
Onufrieva *, J. Rossat-Mignod
*
Lab. L~onBrillouin (CEA-CNRS), Saclay, 91191 Gif-sur-Yt~ette,France Abstract Spin dynamics in YBCO is analysed on a framework of a new theory (based on the t-t'-J model and the diagrammatic technique for Hubbard operators) which treats correctly strong electron correlations within CuO 2 layer.
Inelastic neutron scattering (INS) and NMR experiments have discovered many exotic features of the spin dynamics in high-Tc cuprates in the superconducting (SC) state and the metallic state above Tc, especially for YBa2Cu306+ ~. There is anomalous behaviour of the static uniform susceptibility for the weakly-doped regime ( x < 0.85) which evolves continuously to the normal one at x = 1 [1]. There is an unusual shape of lm x(k, to) versus to (revealed by INS [2]) that is neither of a Lorentzian type typical for localized magnetism, nor of a smooth extended type typical for itinerant magnetism. The spectral band Im x(k, to) is very narrow in to with an energy gap, E C, at low energy and a sharp cut-off at high energy. The energy scale is surprisingly small. A resonance peak, E R, is observed inside the spectral band. It disappears above Tc together with the spin gap, while the values of E G and E R vary differently with x (and so with Tc). Another striking feature is a puzzling evolution with doping of the shape of Im x(k, to)[2]. Based on the t-t'-J model and the diagrammatic technique for Hubbard operators (HO) we have developed a theory [3] which treats correctly the interrelation between localized spins on Cu and itinerant holes in a new narrow band that corresponds to a coherent motion of 'Zhang-Rice singlet'. This interrelation exists in the t-J model and is given by the multiplication rules for HO, but is lost when any representation of HO in terms of B o s e / F e r m i operators (the slave-boson (SB) or slave-fermion ones) is used. In the present work, we apply this theory to calculate the dynamic magnetic susceptibility, x(k, to), for YBCO. The obtained expression for x(k, to) has an RPA structure with two contributions to the 'zero susceptibility', instead of a single one in the weak-coupling theories [4] and in the strong-coupling theories based on the SB repre-
sentation [5]. The first contribution, I(k, to), arises from the subsystem of itinerant holes with a reduced bandwidth W (Wct t(1 + 6 ) / 2 , where 6 is the doped hole concentration). It has a BCS structure in the SC state and a Lindhard function structure in the normal state. The second contribution, L(k, to), arises from the subsystem of localized spins with SR AF correlations. It has a two-spinon structure [3]. The symmetry of the hole anomalous Green function reflects the symmetry of SC pairing (assumed to be of a d-wave type), the symmetry of the spinon anomalous Green function reflects the symmetry of SRO in the subsystem of localized spins. The intensity of the localized-spin contribution decreases with doping proportionally to the amplitude of SR AF order parameter. The intensity of the itinerant-hole contribution increases with doping. As a result of their competition, the different dynamics occurs in different doping regimes. Deeply in the overdoped regime, the former contribution becomes negligible, and the behaviour tends to a normal metal one. In the underdoped regime, both contributions are important. In the weakly-doped regime the 'localised' contribution is dominant. This general picture accounts for crossover effects that are observed experimentally for many magnetic properties in cuprates.
r
o YBCO o.e - 6 = 0 . 2 5 .06
r
i
--QAF=(W.~)
.......... (0.8~,0.8~) . . . . kn~=(061~,051~)
-
N
I 02
o.o
i:
..... ,"~"i
0
0~
i
018
'
1'2
t0/J * Corresponding author. Fax: +33-1-69 08 82 61; email: [email protected].
Fig. 1. Itinerant contribution for different wavevectors ( t / J = 1.8; t/t' = -0.45; Asc = 0.1J).
0304-8853/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0304-8853(94)01649-6
F. Onufriet:a, J. Rossat-M ignod / Journal of Magnetism and Magnetic Materials140-144 (1995) 1271 1272
1272
As to dynamic properties, details are different for different to ranges due to the different beaviour of Im I and Im L as a function of to. The plots of lm l(k, to) versus to for different wavevectors are shown in Fig. 1. (When calculating, we use the value of t ' / t equal to - 0 . 4 5 in order 1o model the shape of the experimentally observed Fermi surface.) At lowest to only a nodal contribution is nonzero. For higher w, the AF contribution is important due to existence of a sharp peak at low enough energy. lm L(k, to) represents an ideal two-spinon band (with a gap M ( k ) ) narrow in energy which becomes broader when spinon damping is taken into account. Spinon damping exists even at T - 0 in the presence of fermionic quasiparticles, however, due to the existence of a gap in the hole dispersion law in the SC state, it appears only above a threshold energy, to~, which value is proportional to ,4sc. Then, Im L versus w has a shape shown in Fig. 3. For any to, Im L and Re L are peaked at k = QAF. The resultant Im x ( k , to) has a k-structure peaked around QAI~"for all w belonging to the principal part of the spectral band, Im X (compare energies in Fig. 2 and Fig. 3 for 6 = 0.25). For lowest to, a very weak incommensurate (nodal) structure is observed. The shape of Im X versus to is determined by the competition between l ( k , to) and L(k, to) (Fig. 3). For the heavily-doped regime (HDR), 6 ~ 0.15, the ideal twospinon spectral band is situated on the left from the peak in lm I(QAF, to). For the weakly-doped regime (WDR), (5 7: 0.15, the situation is reversed. Then, Im X exhibits a gap which is equal to A L in the HDR and to A ~ in the WDR, when neglecting damping. When spinon and hole damping are taken into account, A L manifests itself as a pseudogap, a true gap in lm X is equal to to~ in the HDR and to w~ in the WDR and is proportional to Asc: (to~ is a threshold energy in the hole damping). The resonance peak in | m X for ~ = 0.20-0.25 corresponds to a sharp drop in Re I
6
F\
YBC " 6 : 0.25 -o~ --=,~°o, / \ .3 ....... (0/..1=0.23 .~" ---=/J =0.27 _//...._--~., \ . . . .
0.=o37 / L
'
"\
.I
/,¢
- 02
- -
J ImXIQAF.W )
----
JIm I (QAF,I,O)
YBCO
........... JIr. LIQ~,,~J
,aL/x 6=0.095 l
06
\..,
o 2 ~,
0
O
Z~tI 6 : 0 2 5 "
102
",
!
0.2
04
0
col
~-
tO/J
Fig. 3. Im X(QAF, t,,O) together with Im L(QAF, ~o) and lm
I(QA ~ , w) for YBCO at different dopings (T = 0). versus to which accompanies the sharp peak in Im 1. The latter is a consequence of d-wave superconductivity and the resonance peak in Im X occurs at to -=- A/(QA F) OC A SC. Thus, three important characteristics, the gap in the WDR, the gap in the HDR and the resonance peak in the HDR are related to the SC gap. However, the explicit forms of these relations are different and so, they vary differently with Tc (and with doping). At lowest 6 near the critical concentration, the spin dynamics is characterized by welldefined magnon-like excitations (Fig. 3, 6 = 0.08). These excitations appear as a pole of x ( k , to). As 6 decreases, the gap in magnon-like excitation tends to zero and at = 0.075, the pole occurs at to - 0 which corresponds to the instability of the considered phase ( X(QAF, O) = 3c). All these results account well for many unusual features of spin dynamics in YBCO [2]: for the gap and pseudogap effects, for the shape of k-scan, for the shape of Im X versus to and its unusual evolution with doping and others. In addition, the theory predicts a weak incommensurate 'in-gap' scattering which should exist in the case of d-wave superconductivity. This point has to be verified by subsequent INS experiments. References
zi ... 0
,
(~/2.x/2)
'-.\ ~
,
(~/x) Wawevector
A"
(3~/2.3~/2)
Fig. 2. Im X versus k in YBCO in increasing o) (T = 0).
[1] [2] [3] [4] [5]
H. Alloul et al., Phys. Rcv. Lctt. 63 (1989) 1700. J. Rossat-Mignod ct al., Physica B 192 (1993) 109. F. Onufrieva et al., Phys. Rev. B (1995), in press. N. Bulut et al., Phys. Rev. B 47 (1993) 3419. T. Tanamoto et al., J. Phys. Soc. Jpn. 61 (1992) 1886.