- Email: [email protected]
Spin fluctuation spectra and high temperature superconductivity
Journal of Magnetism and Magnetic Materials 140-144 (1995) 2005-2006
~
Journalof magnetism and magnetic m~erlals
,~
,~
ELSEVIER
Spin fluctuation spectra and high temperature superconductivity Kazuo Ueda a,*, T6ru Moriya b a Institutefor Solid State Physics, University of Tokyo Roppongi, Tokyo, 106 Japan b Department of Physics, Faculty of Science and Technology, Science University of Tokyo, Noda, 278 Japan Abstract The spin fluctuation mechanism for the high Tc cuprates is investigated with the strong coupling theory for the spin fluctuations. The doping concentration dependence of the T~ in the best studied cuprates is explained in terms of the parameter values estimated from the normal state experiments.
A central issue for the high temperature superconductivity in cuprates is whether their anomalous normal state properties are directly related with the high transition temperatures. One of the promising scenarios which explains both in a unified way is the antiferromagnetic spin fluctuation theory of two dimensional metals. In this scenario the spin fluctuation mechanism leads to the unconventional superconductivity of d-wave character. Recently, many experimental investigations have been reported which indicate the d-wave symmetry of the Copper pairs for the cuprates. This means that the possibility is very high for the spin fluctuation mechanism to be responsible for high temperature superconductivity. Thus it seems to be worth while to extend our previous theory [1,2] to discuss systematic relations between the transition temperature of the d-wave superconductivity and the spin fluctuation spectra. In nearly and weakly antiferromagnetic metals the dynamical susceptibility (in units of 4/.~2) above TN for a wave vector near the antiferromagnetic ordering vector Q may generally be written as x(Q+q,
o2)
x ( Q + q) 1-io2/FQ+q'
(1)
with [ x(Q + q)]-i = [ x(Q)]-i
spatial correlation of the spin fluctuations. It is convenient to use TA = A q g / 2 ,
(3a)
To = F q 2 / 2 ~ ,
(3b)
y = [2TA Xfl(T)] -1,
(3c)
for the parametrization of the spin fluctuation spectra where qB is the cut-off wave vector. We first show in the middle column of Table 1 the SCR results for the anomalous temperature dependences of various physical quantities in two dimensional systems. These behaviors are realized above the crossover temperature below which the normal Fermi liquid-like behaviors as shown in the right hand side column in Table 1 are predicted. The crossover temperature tends to zero as the critical phase boundary for the magnetic long range order is approached. We find that these relsults for the two dimensional system just explain the reported anomalous behaviors of high Tc cuprates above To. In two dimensions the important parameters of the theory, TO and TA, may be determined from the slope of the electrical resistivity, (4)
d R ( T ) / d T = a/To, +Aq2 '
(2a)
FO+ q = F ( K2 + q2),
(2b)
K 2= 1 l A x ( Q ) ,
(2c)
where K-1 is the correlation length, and A and F are constants which specify the frequency spread and the
Table 1 Anomalous temperature dependences of physical quantities around the critical phase boundary for itinerant antiferromagnetism predicted by the SCR theory; XQ: magnetic susceptibility, R: electrical resistivity, /'1: nuclear spin-lattice relaxation time, CW (Curie-Weiss): T + 0 3-dim.
* Corresponding author. Fax: +81-3-3402-7326; [email protected].
email:
XQ1 R
T 3/2 ~ C W
r; 1
rx~/~
0304-8853/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0304-8853(94)00938-4
T 3/2
2-dim. T+ 0 T rx~
Normal FL a+ bT 2 T2
T
K. Ueda, T. Moriya /Journal of Magnetism and Magnetic Materials 140-144 (1995) 2005-2006
2006 Table 2
r o [K]
TA [K]
YBa2Cu306+ x
Tc = 90 K 60 K 30 K
1400-2300 1000 600
8000-9000 4500-5000
La2_ xSrxCuO4
x=
0.15 0.1
1000 ~ 500
3000-4000
and the saturation value of the NMR relaxation rate at high temperatures,
1 / T 1 = b / T A.
(5)
The constants in the above expressions are a = 1 × 10 -3 cm and b = 1 × 106 for the cuprates [3]. In Table 2 we tabulate the results of estimations for the best studied cuprates. Success of the spin fluctuation theory to describe various anomalous properties in the normal state of the cuprates has led us to the study of the mechanism of superconductivity due to the same spin fluctuations. First we used a weak coupling theory [1] and then proceeded to the strong coupling theory [2]. Since the spectrum of the spin fluctuations for each of the cuprates is determined from normal state properties, necessary inputs to perform the strong coupling calculations are the single particle energies, ~k = ek - / z , and the coupling constant I. Existing electronic band structure calculations by several groups for La2_xSrxCuO 4 [4] and YBa2Cu306+ x [5] are consistent among one another. A fitting to the relevant band by a tight binding form is obtained by the nearest neighbor hopping t 1 and the next nearest neighbor hopping t 2 in the square lattice. The total band width is W = 8t 1. We assume t 2 = 0 for La2_xSrxCuO 4 and t 2 = - t l / 2 for YBa2Cu306+ x. First we will discuss T~ of 90 K class YBa2Cu306+ x. We use TO = 2300 K, TA = 8000 K, y = 0.01, W = 10000 K and I = 7000 K as a representative set of parameters and obtain T¢ = 101 K for n h = 0.25. The total band width reported by LDA band calculations is about 2 eV. The reason we use W = 10000 K ~ 1 eV rather than 2 eV is the effect of band narrowing, which is not included in the LDA calculations. Now we turn to the 60 K class compounds of YBa2Cu306+ x. We keep the same values for y, W and I
as the 90 K class compounds and take the following values, TO= 1000 K and TA = 5000 K and n h = 0.12. For this set of parameters, T¢ = 58 K is obtained. It is remarkable that in the series of YBa2Cu306+ x the doping concentration dependence of the spin fluctuation spectra is correctly reflected on that of Tc. This fact is a strong support for the spin fluctuation mechanism as the origin of high Tc in cuprates. Next we will discuss La2_xSrxCuO 4. The LDA band calculations give a 50% larger value for W compared with YBa2Cu306+ x. A set of parameters, TO = 1000 K, TA = 4000 K, y = 0.01, W = 15000 K and I = 7000 K, gives T¢ = 63 K for La2_~SrxCuO 4 at the optimal concentration n h = 0.15. This number is about 50% higher than the experimentally observed value, an apparently poorer result than for YBa2Cu306+ x. However, the result seems satisfactory in view of the crudeness of estimated parameter values. We have seen that the difference of T~ between 90 and 60 K classes in YBa2Cu306+ x can be attributed to the difference in the spectra of the spin fluctuations. Concerning the doping dependence of Tc of La2_~Sr~CuO 4, first we observe that the slope of resistivity increases with decreasing doping concentration in the underdoped regions. We may conclude that the suppression of Tc in underdoped region originates from the decrease of T0. On the other hand, in the overdoped region the electron-electron interaction becomes effectively weaker, which naturally leads to smaller T¢. In conclusion the doping dependence of T¢ in both YBa2Cu306+ x and La2_~SrxCuO 4 can be understood in the context of antiferromagnetic spin fluctuation mechanism.
References [1] T. Moriya, Y. Takahashi and K. Ueda, J. Phys. Soc. Jpn. 59 (1990) 2905; Physica C 185-189 (1991) 114. [2] K. Ueda, T. Moriya and Y. Takahashi, in: Electronic Properties and Mechanisms of High-Tc Superconductors (Tsukuba Symposium, 1991), eds. T. Oguchi et al. (North-Holland, 1992) p. 145; J. Phys. Chem. Sol. 53 (1992) 1515. [3] T. Moriya and K. Ueda, J. Phys. Soc. Jpn. 63 (1994) 1871. [4] L.F. Mattheiss, Phys. Rev. Lett. 58 (1987) 1028; A.J. Freeman, J. Yu and C.L. Fu, Phys. Rev. B 36 (1987) 7111. [5] S. Massidda, J. Yu, A.J. Freeman and D.D. Koelling, Phys. Lett. 122 (1987) 198; T. Oguchi, K.T. Park, K. Terakura and A. Yanase, Physica B 148 (1987) 253.
~
Journalof magnetism and magnetic m~erlals
,~
,~
ELSEVIER
Spin fluctuation spectra and high temperature superconductivity Kazuo Ueda a,*, T6ru Moriya b a Institutefor Solid State Physics, University of Tokyo Roppongi, Tokyo, 106 Japan b Department of Physics, Faculty of Science and Technology, Science University of Tokyo, Noda, 278 Japan Abstract The spin fluctuation mechanism for the high Tc cuprates is investigated with the strong coupling theory for the spin fluctuations. The doping concentration dependence of the T~ in the best studied cuprates is explained in terms of the parameter values estimated from the normal state experiments.
A central issue for the high temperature superconductivity in cuprates is whether their anomalous normal state properties are directly related with the high transition temperatures. One of the promising scenarios which explains both in a unified way is the antiferromagnetic spin fluctuation theory of two dimensional metals. In this scenario the spin fluctuation mechanism leads to the unconventional superconductivity of d-wave character. Recently, many experimental investigations have been reported which indicate the d-wave symmetry of the Copper pairs for the cuprates. This means that the possibility is very high for the spin fluctuation mechanism to be responsible for high temperature superconductivity. Thus it seems to be worth while to extend our previous theory [1,2] to discuss systematic relations between the transition temperature of the d-wave superconductivity and the spin fluctuation spectra. In nearly and weakly antiferromagnetic metals the dynamical susceptibility (in units of 4/.~2) above TN for a wave vector near the antiferromagnetic ordering vector Q may generally be written as x(Q+q,
o2)
x ( Q + q) 1-io2/FQ+q'
(1)
with [ x(Q + q)]-i = [ x(Q)]-i
spatial correlation of the spin fluctuations. It is convenient to use TA = A q g / 2 ,
(3a)
To = F q 2 / 2 ~ ,
(3b)
y = [2TA Xfl(T)] -1,
(3c)
for the parametrization of the spin fluctuation spectra where qB is the cut-off wave vector. We first show in the middle column of Table 1 the SCR results for the anomalous temperature dependences of various physical quantities in two dimensional systems. These behaviors are realized above the crossover temperature below which the normal Fermi liquid-like behaviors as shown in the right hand side column in Table 1 are predicted. The crossover temperature tends to zero as the critical phase boundary for the magnetic long range order is approached. We find that these relsults for the two dimensional system just explain the reported anomalous behaviors of high Tc cuprates above To. In two dimensions the important parameters of the theory, TO and TA, may be determined from the slope of the electrical resistivity, (4)
d R ( T ) / d T = a/To, +Aq2 '
(2a)
FO+ q = F ( K2 + q2),
(2b)
K 2= 1 l A x ( Q ) ,
(2c)
where K-1 is the correlation length, and A and F are constants which specify the frequency spread and the
Table 1 Anomalous temperature dependences of physical quantities around the critical phase boundary for itinerant antiferromagnetism predicted by the SCR theory; XQ: magnetic susceptibility, R: electrical resistivity, /'1: nuclear spin-lattice relaxation time, CW (Curie-Weiss): T + 0 3-dim.
* Corresponding author. Fax: +81-3-3402-7326; [email protected].
email:
XQ1 R
T 3/2 ~ C W
r; 1
rx~/~
0304-8853/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0304-8853(94)00938-4
T 3/2
2-dim. T+ 0 T rx~
Normal FL a+ bT 2 T2
T
K. Ueda, T. Moriya /Journal of Magnetism and Magnetic Materials 140-144 (1995) 2005-2006
2006 Table 2
r o [K]
TA [K]
YBa2Cu306+ x
Tc = 90 K 60 K 30 K
1400-2300 1000 600
8000-9000 4500-5000
La2_ xSrxCuO4
x=
0.15 0.1
1000 ~ 500
3000-4000
and the saturation value of the NMR relaxation rate at high temperatures,
1 / T 1 = b / T A.
(5)
The constants in the above expressions are a = 1 × 10 -3 cm and b = 1 × 106 for the cuprates [3]. In Table 2 we tabulate the results of estimations for the best studied cuprates. Success of the spin fluctuation theory to describe various anomalous properties in the normal state of the cuprates has led us to the study of the mechanism of superconductivity due to the same spin fluctuations. First we used a weak coupling theory [1] and then proceeded to the strong coupling theory [2]. Since the spectrum of the spin fluctuations for each of the cuprates is determined from normal state properties, necessary inputs to perform the strong coupling calculations are the single particle energies, ~k = ek - / z , and the coupling constant I. Existing electronic band structure calculations by several groups for La2_xSrxCuO 4 [4] and YBa2Cu306+ x [5] are consistent among one another. A fitting to the relevant band by a tight binding form is obtained by the nearest neighbor hopping t 1 and the next nearest neighbor hopping t 2 in the square lattice. The total band width is W = 8t 1. We assume t 2 = 0 for La2_xSrxCuO 4 and t 2 = - t l / 2 for YBa2Cu306+ x. First we will discuss T~ of 90 K class YBa2Cu306+ x. We use TO = 2300 K, TA = 8000 K, y = 0.01, W = 10000 K and I = 7000 K as a representative set of parameters and obtain T¢ = 101 K for n h = 0.25. The total band width reported by LDA band calculations is about 2 eV. The reason we use W = 10000 K ~ 1 eV rather than 2 eV is the effect of band narrowing, which is not included in the LDA calculations. Now we turn to the 60 K class compounds of YBa2Cu306+ x. We keep the same values for y, W and I
as the 90 K class compounds and take the following values, TO= 1000 K and TA = 5000 K and n h = 0.12. For this set of parameters, T¢ = 58 K is obtained. It is remarkable that in the series of YBa2Cu306+ x the doping concentration dependence of the spin fluctuation spectra is correctly reflected on that of Tc. This fact is a strong support for the spin fluctuation mechanism as the origin of high Tc in cuprates. Next we will discuss La2_xSrxCuO 4. The LDA band calculations give a 50% larger value for W compared with YBa2Cu306+ x. A set of parameters, TO = 1000 K, TA = 4000 K, y = 0.01, W = 15000 K and I = 7000 K, gives T¢ = 63 K for La2_~SrxCuO 4 at the optimal concentration n h = 0.15. This number is about 50% higher than the experimentally observed value, an apparently poorer result than for YBa2Cu306+ x. However, the result seems satisfactory in view of the crudeness of estimated parameter values. We have seen that the difference of T~ between 90 and 60 K classes in YBa2Cu306+ x can be attributed to the difference in the spectra of the spin fluctuations. Concerning the doping dependence of Tc of La2_~Sr~CuO 4, first we observe that the slope of resistivity increases with decreasing doping concentration in the underdoped regions. We may conclude that the suppression of Tc in underdoped region originates from the decrease of T0. On the other hand, in the overdoped region the electron-electron interaction becomes effectively weaker, which naturally leads to smaller T¢. In conclusion the doping dependence of T¢ in both YBa2Cu306+ x and La2_~SrxCuO 4 can be understood in the context of antiferromagnetic spin fluctuation mechanism.
References [1] T. Moriya, Y. Takahashi and K. Ueda, J. Phys. Soc. Jpn. 59 (1990) 2905; Physica C 185-189 (1991) 114. [2] K. Ueda, T. Moriya and Y. Takahashi, in: Electronic Properties and Mechanisms of High-Tc Superconductors (Tsukuba Symposium, 1991), eds. T. Oguchi et al. (North-Holland, 1992) p. 145; J. Phys. Chem. Sol. 53 (1992) 1515. [3] T. Moriya and K. Ueda, J. Phys. Soc. Jpn. 63 (1994) 1871. [4] L.F. Mattheiss, Phys. Rev. Lett. 58 (1987) 1028; A.J. Freeman, J. Yu and C.L. Fu, Phys. Rev. B 36 (1987) 7111. [5] S. Massidda, J. Yu, A.J. Freeman and D.D. Koelling, Phys. Lett. 122 (1987) 198; T. Oguchi, K.T. Park, K. Terakura and A. Yanase, Physica B 148 (1987) 253.