- Email: [email protected]
Nested spin-fluctuation theory for spin-gap behaviors of thermodynamic properties in the normal state of high-Tc cuprates
Journal of Magnetism and Magnetic Materials 177-181 (1998) 507 508
~ l ~ Journalof
magnetism and magnetic J ~ i materials
ELSEVIER
Nested spin-fluctuation theory for spin-gap behaviors of thermodynamic properties in the normal state of high- cuprates O.
Narikiyo*
Department ~?fMaterial Physics, Faculty of Engineering Science, Osaka UniversiO,, Machikanevama-cho 1-3, Toyonaka 560, Japan
Abstract On the basis of the itinerant-localized duality model the collective contribution of nested spin fluctuations to the uniform spin susceptibility and the specific heat is calculated. Such a calculation well accounts for the so-called 'spin-gap" behaviors of these quantities in the normal state of high-To cuprates. © 1998 Elsevier Science B.V. All rights reserved. Keywords: Spin fluctuations; Superconductors- high-To; Susceptibility; Specific heat
In the normal state of high-To cuprates the uniform spin susceptibility and the specific heat show the socalled 'spin-gap' behaviors [1 4]. For the susceptibility we have two different scenarios: a crossover in local spin system for Lae-~SrxCuO4 [1, 3] and a gap in quasiparticle density of states for YBa2Cu306 +x [4]. The specific heat has a smaller characteristic temperature than the susceptibility [4]. In this paper we try to understand the above-mentioned experiments in terms of nested spin fluctuations on the basis of the itinerant-localized duality model [5], the third scenario, by reexamining the temperature dependence of the susceptibility and the specific heat, since the estimation of these quantities is not unambiguous in our previous study [5]. The dynamical susceptibility for the coherent part of spin fluctuations in our model is given by [5]
NF ~¢oh(Q + q,o-O = A(q2 + K2) _ i c e ) '
(1)
where N v is the renormalized density of states at the Fermi energy and ~c is the temperature-dependent measure of the antiferromagnetic correlation around the
*Corresponding author. Fax: + 81 6 845 4632; e-mail: [email protected].
antiferromagnetic wave vector Q. The incoherent part is neglected in the following discussions of temperature dependences, since it gives only temperature-independent background. Here A consists of the contributions from itinerant fermions Ar and localized spins As: A = Af + As. Af depends strongly on temperature, if the Fermi surface of the renormalized quasiparticle is technically nested: Afq 2 = af/r 2 where af is a dimensionless constant determined by the dispersion of itinerant fermions, qc is a cut-off wave number of the order of the inverse of the lattice constant and z2 = t 2 + h 2 with t = T/~v and h = H/2~ev. The temperature T and the degree of the deviation from the perfect nesting H are normalized by the renormalized Fermi energy ev. As is essentially temperature-independent and proportional to the superexchange interaction J. A is a monotonically decreasing function of temperature and has a crossover temperature T* given by the condition Af =As: A strongly decreases as temperature increases for T < T* where the itinerant contribution dominates and A depends weakly on temperature for T > T* where the localized contribution dominates. As has been discussed in our previous study [5] the existence of T* gives a natural explanation to the experimentally observed crossover by various probes [6]. C also depends strongly on temperature: Ce.F = cdT where cf is a dimensionless constant determined by the dispersion of itinerant fermions.
0304-8853/98/$19.00 i ' 1998 Elsevier Science B.V. All rights rescrved PII S 0 3 0 4 - 8 8 5 3 ( 9 7 ) 0 0 9 1 6-5
508
O. Narikiyo / Journal o f Magnetism and Magnetic Materials 177-181 (1998) 507 508 --
T=Tt T=T~
~1
f /
T
/
0.04
....
0.02 O
.
. - .-
0
.
.
.
.
.
.
q
Fig. 1. The static component of the susceptibility.
0.1
.
.
.
.
i
.
.
.
.
0
'
'
015.
'
h=0.3
- -
h=0.1
' t'
'
The coefficient 7cob = C v / T of the specific heat Cv due to the coherent part of spin fluctuations is given by 7~oh = -- O2Fcoh/OTZ with the free energy [5, 7] ......... h = 0 ~
1
'
- -
h=0.1
' t
'
Fcoh = --
zf
q
de)
cothtan
A(q 2 + K2)'
(3)
2
Fig. 2. Temperature dependence of the uniform spin susceptibility (in arbitrary unit) for La2 xSr~CuO4 system.
The uniform spin susceptibility Z¢oh due to the coherent part of spin fluctuations is given by NF
A(Q2 + K2),
'
h=0.5
.......
Fig. 3. Temperature dependence of the uniform spin susceptibility (in arbitrary unit) for YBa2Cu306 +x system.
i
0.05
Zcoh
'
---
(2)
where x is determined self-consistently I-5, 7] and approximately obeys the Curie-Weiss law: Oc/q~) 2 = a + bt with dimensionless constants a and b. Schematically, the static component of the susceptibility behaves as shown in Fig. 1. In the case of Fig. 2 K is small and depends strongly on temperature as La2 xSrxCuO4 system and we take (Q/q~)2 = 1, a = 0.03, b = 0.3 and Af/A~ = r 2/ 10 in accordance with previous studies [5,7]. In the case of Fig. 3 ~: is of the order of Q and its temperature dependence is weak as YBa2Cu306+x system and we take ( Q 2 - t - f i e ) / q 2 = 2 for clarity and A f / A s = r z/10. Here we have assumed that cv ~ 1000 K and that the doping dependence can be simulated by varying the degree of the nesting h, since h is a monotonically increasing function of the doping concentration. The qualitative feature of temperature and doping dependences in Figs. 2 and 3 is consistent with experiments I-1-4]: at a temperature around T* the datum for La2-:,SrxCuO4 system in Fig. 2 has a peak and the one for YBazCu306+~ in Fig, 3 changes its curvature. These results could not be obtained in our previous study [5] where the susceptibility was determined by the magnetic field dependence of the nesting parameter h: the susceptibility depended on both A and C and had a peak at a temperature below T* as shown in Fig. 14 of Ref. 1,5].
where we take Aq2 = 1 and C(5~ = 1. We have evaluated the free energy numerically in two dimensions and obtained the specific heat by the numerical derivative. The results are essentially the same as those reported in our previous study obtained by an approximate analytic formula: 7~oh has a peak at a temperature below T* as shown in Fig. 12 of Ref. [5] in accordance with experiments I-4]. Thus, the characteristic temperatures for the spin susceptibility and the specific heat are different. In our theory, the difference arises from the fact that the former only depends on A as seen in Eq. (2) and the latter A and C as in Eq. (3) apart from the dependence on ~:. In conclusion, we have presented a unified point of view to understand the apparent difference of the temperature dependence of the uniform spin susceptibility between La2-xSrxCuO4 and YBa2Cu306+x systems and the different characteristic temperatures of the susceptibility and the specific heat. The gross feature of the 'spin-gap' behaviors is c o m m o n to two-dimensional high-T~ cuprates and three-dimensional V2 yO3 [8].
References
[1] D.C. Johnston, Phys. Rev. Lett. 62 (1989) 957. [2] H. Alloul, T. Ohno, P. Mendels, Phys. Rev. Lett. 63 (1989) 1700. [3] T. Nakano, M. Oda, C. Manabe, N. Momono, Y. Miura, M. Ido, Phys. Rev. B 49 (1994) 16000. [4] J.W. Loram, K.A. Mirza, J.M. Wade, J.R. Cooper, W.Y. Liang, Physica C 235-240 (1994) 134. I-5] K. Miyake, O. Narikiyo, J. Phys. Soc. Japan. 63 (1994) 3821. [6] B. Batlogg, H.Y. Hwang, H. Takagi, R.J. Cava, H.L. Kao. J. Kwo, Physica C 235 240 (1994) 130 and references therein. [7] T. Moriya, Y. Takahashi, K. Ueda, J. Phys. Soc. Japan. 59 (1990) 2905. [8] O. Narikiyo, K. Miyake, J. Phys. Soc. Japan. 64 (1995) 2730.
~ l ~ Journalof
magnetism and magnetic J ~ i materials
ELSEVIER
Nested spin-fluctuation theory for spin-gap behaviors of thermodynamic properties in the normal state of high- cuprates O.
Narikiyo*
Department ~?fMaterial Physics, Faculty of Engineering Science, Osaka UniversiO,, Machikanevama-cho 1-3, Toyonaka 560, Japan
Abstract On the basis of the itinerant-localized duality model the collective contribution of nested spin fluctuations to the uniform spin susceptibility and the specific heat is calculated. Such a calculation well accounts for the so-called 'spin-gap" behaviors of these quantities in the normal state of high-To cuprates. © 1998 Elsevier Science B.V. All rights reserved. Keywords: Spin fluctuations; Superconductors- high-To; Susceptibility; Specific heat
In the normal state of high-To cuprates the uniform spin susceptibility and the specific heat show the socalled 'spin-gap' behaviors [1 4]. For the susceptibility we have two different scenarios: a crossover in local spin system for Lae-~SrxCuO4 [1, 3] and a gap in quasiparticle density of states for YBa2Cu306 +x [4]. The specific heat has a smaller characteristic temperature than the susceptibility [4]. In this paper we try to understand the above-mentioned experiments in terms of nested spin fluctuations on the basis of the itinerant-localized duality model [5], the third scenario, by reexamining the temperature dependence of the susceptibility and the specific heat, since the estimation of these quantities is not unambiguous in our previous study [5]. The dynamical susceptibility for the coherent part of spin fluctuations in our model is given by [5]
NF ~¢oh(Q + q,o-O = A(q2 + K2) _ i c e ) '
(1)
where N v is the renormalized density of states at the Fermi energy and ~c is the temperature-dependent measure of the antiferromagnetic correlation around the
*Corresponding author. Fax: + 81 6 845 4632; e-mail: [email protected].
antiferromagnetic wave vector Q. The incoherent part is neglected in the following discussions of temperature dependences, since it gives only temperature-independent background. Here A consists of the contributions from itinerant fermions Ar and localized spins As: A = Af + As. Af depends strongly on temperature, if the Fermi surface of the renormalized quasiparticle is technically nested: Afq 2 = af/r 2 where af is a dimensionless constant determined by the dispersion of itinerant fermions, qc is a cut-off wave number of the order of the inverse of the lattice constant and z2 = t 2 + h 2 with t = T/~v and h = H/2~ev. The temperature T and the degree of the deviation from the perfect nesting H are normalized by the renormalized Fermi energy ev. As is essentially temperature-independent and proportional to the superexchange interaction J. A is a monotonically decreasing function of temperature and has a crossover temperature T* given by the condition Af =As: A strongly decreases as temperature increases for T < T* where the itinerant contribution dominates and A depends weakly on temperature for T > T* where the localized contribution dominates. As has been discussed in our previous study [5] the existence of T* gives a natural explanation to the experimentally observed crossover by various probes [6]. C also depends strongly on temperature: Ce.F = cdT where cf is a dimensionless constant determined by the dispersion of itinerant fermions.
0304-8853/98/$19.00 i ' 1998 Elsevier Science B.V. All rights rescrved PII S 0 3 0 4 - 8 8 5 3 ( 9 7 ) 0 0 9 1 6-5
508
O. Narikiyo / Journal o f Magnetism and Magnetic Materials 177-181 (1998) 507 508 --
T=Tt T=T~
~1
f /
T
/
0.04
....
0.02 O
.
. - .-
0
.
.
.
.
.
.
q
Fig. 1. The static component of the susceptibility.
0.1
.
.
.
.
i
.
.
.
.
0
'
'
015.
'
h=0.3
- -
h=0.1
' t'
'
The coefficient 7cob = C v / T of the specific heat Cv due to the coherent part of spin fluctuations is given by 7~oh = -- O2Fcoh/OTZ with the free energy [5, 7] ......... h = 0 ~
1
'
- -
h=0.1
' t
'
Fcoh = --
zf
q
de)
cothtan
A(q 2 + K2)'
(3)
2
Fig. 2. Temperature dependence of the uniform spin susceptibility (in arbitrary unit) for La2 xSr~CuO4 system.
The uniform spin susceptibility Z¢oh due to the coherent part of spin fluctuations is given by NF
A(Q2 + K2),
'
h=0.5
.......
Fig. 3. Temperature dependence of the uniform spin susceptibility (in arbitrary unit) for YBa2Cu306 +x system.
i
0.05
Zcoh
'
---
(2)
where x is determined self-consistently I-5, 7] and approximately obeys the Curie-Weiss law: Oc/q~) 2 = a + bt with dimensionless constants a and b. Schematically, the static component of the susceptibility behaves as shown in Fig. 1. In the case of Fig. 2 K is small and depends strongly on temperature as La2 xSrxCuO4 system and we take (Q/q~)2 = 1, a = 0.03, b = 0.3 and Af/A~ = r 2/ 10 in accordance with previous studies [5,7]. In the case of Fig. 3 ~: is of the order of Q and its temperature dependence is weak as YBa2Cu306+x system and we take ( Q 2 - t - f i e ) / q 2 = 2 for clarity and A f / A s = r z/10. Here we have assumed that cv ~ 1000 K and that the doping dependence can be simulated by varying the degree of the nesting h, since h is a monotonically increasing function of the doping concentration. The qualitative feature of temperature and doping dependences in Figs. 2 and 3 is consistent with experiments I-1-4]: at a temperature around T* the datum for La2-:,SrxCuO4 system in Fig. 2 has a peak and the one for YBazCu306+~ in Fig, 3 changes its curvature. These results could not be obtained in our previous study [5] where the susceptibility was determined by the magnetic field dependence of the nesting parameter h: the susceptibility depended on both A and C and had a peak at a temperature below T* as shown in Fig. 14 of Ref. 1,5].
where we take Aq2 = 1 and C(5~ = 1. We have evaluated the free energy numerically in two dimensions and obtained the specific heat by the numerical derivative. The results are essentially the same as those reported in our previous study obtained by an approximate analytic formula: 7~oh has a peak at a temperature below T* as shown in Fig. 12 of Ref. [5] in accordance with experiments I-4]. Thus, the characteristic temperatures for the spin susceptibility and the specific heat are different. In our theory, the difference arises from the fact that the former only depends on A as seen in Eq. (2) and the latter A and C as in Eq. (3) apart from the dependence on ~:. In conclusion, we have presented a unified point of view to understand the apparent difference of the temperature dependence of the uniform spin susceptibility between La2-xSrxCuO4 and YBa2Cu306+x systems and the different characteristic temperatures of the susceptibility and the specific heat. The gross feature of the 'spin-gap' behaviors is c o m m o n to two-dimensional high-T~ cuprates and three-dimensional V2 yO3 [8].
References
[1] D.C. Johnston, Phys. Rev. Lett. 62 (1989) 957. [2] H. Alloul, T. Ohno, P. Mendels, Phys. Rev. Lett. 63 (1989) 1700. [3] T. Nakano, M. Oda, C. Manabe, N. Momono, Y. Miura, M. Ido, Phys. Rev. B 49 (1994) 16000. [4] J.W. Loram, K.A. Mirza, J.M. Wade, J.R. Cooper, W.Y. Liang, Physica C 235-240 (1994) 134. I-5] K. Miyake, O. Narikiyo, J. Phys. Soc. Japan. 63 (1994) 3821. [6] B. Batlogg, H.Y. Hwang, H. Takagi, R.J. Cava, H.L. Kao. J. Kwo, Physica C 235 240 (1994) 130 and references therein. [7] T. Moriya, Y. Takahashi, K. Ueda, J. Phys. Soc. Japan. 59 (1990) 2905. [8] O. Narikiyo, K. Miyake, J. Phys. Soc. Japan. 64 (1995) 2730.