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Influence of electron-phonon scattering on the properties of high Tc superconductors
~Solid State Communications, Vol. 72, No. i, pp. 81-83, 1989. Printed in Great B r i t a i n .
0038-i098/8953.00+. 00 Pergamon Press plc
INFLUENCE OF ELECTRON-PHONON SCATTERING ON THE PROPERTIES OF HIGH T c SUPERCONDUCTORS
O.V.Dolgov, A.A.Golubov*, A.E.Koshelev* P.N.Lebedev Physical Institute, Academy of Sciences of the USSR, Moscow, USSR *) Institute of Solid State Physics, Academy of Sciences of the USSR, Chernogolovka, USSR ( Received 29 June 1989 by V.M. Agranovich )
The system of electrons interacting with phonons and excitons is studied. In the standard phonon model modes with low and high frequencies are considered, the latter being interact strongly with electrons. The combined model takes into account low frequency modes and excitons interacting weakly with electrons.
rph =
1. INTRODUCTION The dominant role of the oxygen vibrations in the pairing mechanism of high T c oxides was discussed
(2Vph)--i = 2 7 r [ Sph(W) dw sh(w/T)
(2)
by Weber and Mattheiss. The tunneling 1, optical 2 and magnetoresistance data show large values of the electron-phonon coupling constant ~ ~ 1 - 3. In the present work the model is studied, in which the superconducting pairing is due mainly to high-frequency mechanism (possibly non-phonon), which satisfies the weak-coupling condition (T c < < Wo). We note that in the Eliashberg model the boundary energy between low and high frequency modes is the value 2rT c. The electron-phonon interaction may results both in the strong-coupling effects and the temperature dependent renormalization of the electron energy (strong electron scattering) 3. The strong-coupling effects were studied, in particular, for the upper critical field, Hc2(T) 4, and for the magnetic penetra-
It follows from (2) that at T _>WD/3 the relation rDh-1 = 2r~phT holds, which, in particular, results in the linear temperature dependence of the resistivity just above T c (see, for example,6). For Aph > 1 the electron
tion depth, ~(T) 5, and were shown to lead to large deviations from the BCS theory. Let us consider the clean superconductor (elastic scattering length 1e > > 4o) with the spectral density of the electron-phonon interaction in the form:
2. RESULTS AND DISCUSSION Let us discuss firstly the results obtained in the framework of the combined model which takes into account low frequency phonon modes and excitons interacting weakly with electrons. To calculate the temperature dependence of the upper critical field HC2 we use the general strong coupling relations. If we neglect Pauli limiting and Coulomb repulsion this relations takes the following form:
mean free path, lph(Tc) = vF/2rTcAph ~ 4o, i.e. the dirty limit condition holds. On cooling below T ~ WD/2r the value lph grows and the clean limit is approached. Moreover, the inelastic electron-phonon scattering leads to the smearing of the singularity in the electronic density of states which results in qualitative changes in the temperature dependence of the NMR relaxation rate just below T c.
S(w) ~ a2(w)F(w) = Sph(W) + So(W) = w 2
Ao%
= ~ph( ~D ) 0(~D - ~) + ---~ ~(~- %)'
(1) m
(2rtr)-l]-lA(wm),
where wD u Tc, w° > > T c. In this case the low-frequency phonons with w < wD do not contribute to A
~
~--'1 ~.J
and T c according to the cancellation theorem 3. However, at finite temperatures the thermal phonons determine the electron scattering time:
(3a) + U
111
m
(2rtr)-lsgnwn , 81
(3b)
111
82
INFLUENCE OF ELECTRON-PHONON SCATTERING
x(x) =
2 __ ]~dq e-q2tan-l(qal/2/x),
1
al/25 a = rHc2VF2/¢o , [
(3c)
a2(f~)F(~)
d•
A(%-~m) = 2
~2 wn = rT(2n-1).
, +
(w n
Wm)2 (3d)
_
E
05
Vol.
72, No.
BCS~
=
Here rtr is the transport electron scattering time. In the considered combined weak coupling model the equations (3) are essentially simplified: )t(Wn-Wm) = A, A(Wm) = A. Moreover for sufficiently high temperatures T ~ ~vD
0.5
we have rtr -~ rph (in general rtr < rph ). For ), > 1 we are in the dirty limit due to the strong electron-phonon scattering. Qualitatively for dirty superconductor from the relations ~ = (~olph)1/2
T/Tc 1
FIGURE
and HC2 ~ ¢ O / C ( T ) we get at T ~ Tc: Tc ) [---dHc2/dW]Tc ~ 0.7/Aph , Aph _>1, (4) HC 2 ( 0 i.e. the slope grows with Aph. For arbitrary temperatures and Aph values the exact solution of the problem for HC2 with rph (2) was obtained and the results of calculations are presented at Fig.1 (hc2 = HC2/[TcHc2'(Tc)]). It is seen that for Aph _>1 the function Hc2(T ) have a maximum at Tph - WD/27r. With the lowering T below Tph the thermal phonons are freezing out and HC2(0 ) does not depend on Aph. The temperature dependence of the magnetic penetration depth, ~(T), at T < T c can be calculated from the electrodynamical kernel Q(k) in the London limit (for k = 0): /~ 2 n N e22rTE + Wn2)3/2 q(k=0) = ~ n (/kn 2
. (5)
where the quantities An are the solutions of nonlinear version of the Eliashberg equations (3) in zero magnetic field. The penetration depth is determined by the following relation: = (c/4~q) 1/2
(6)
At T ~ T c and Aph > 1 the dirty limit relation is: = gL(~o/lph)l/2' (gL is the London penetration depth). Again the quantity ~T=0) (clean limit) does not depend on the Aph, and the slope of the lower critical field H c I ( T ) - g-2(T) at T z T c decreases with the increase of Aph. Fig.2 shows the results of the calculations in the framework of the microscopic theory for
% ~.
o.5 -
~h:2 ~ 0.5 I 0.5
T/T¢ FIGURE
2
arbitrary mean free path using the ABCS(T) and rph from (2). For large enough Aph values the positive curvature of the E-2(T) takes place at T _>Tph. Note that this picture would take place in the weak coupling limit ( % > > Tc). If this inequality is not fulfilled the strong coupling effects take place, which have opposite sign both for Hc2(T ) and fl(T) 4,5 The selfconsistent calculations including both strong coupling and renormalization effects were done for Hc2(T ) and g(T) in the model (1) at wo = 5wD and Aph = 1 by the numerical solution of the Eliashberg equations. The quantity Ao z 2 was found from the condition T c = wD. The results are shown in Figs.l,2 (curves SC - strong coupling). It is seen that for the given parameters both effects are nearly compensated and the curves SC in both cases are close to the BCS clean limit curves. In a number of experiments (see for
Vol. 72, No. I
INFLUENCE OF ELECTRON-PHONON SCATTERING
83
example ref. 7 and references therein) on high T c oxides
~"Ph=O ~ .
.
.
.
.
.
.
.
.
.
the b'-'2(T)/8"-2(0) dependence close to the BCS universal curve was observed. It is the argument in favour of the applicability of the phonon model to these oxides. Meanwhile the validity of these experimental results requires justification. We have also analyzed the NMR relaxation rate using the inelastic smearing of the density of states Fph according to eq.(2). The results of calculations are shown in Fig.3. It is seen that for '~ph -> 0.2 the inelastic smearing is large enough such that the hump in T1N/T1s just below T c disappears. Note, that the re-
.
z 0.5
Z, i
cent experiments indeed have not shown such hump 8.
0.5
Acknowledgment s. Useful discussions with L.N.Bulaevskii, G.M.Eliashberg and I.F.Schegolev axe gratefully acknowledged.
T/To FIGURE 3
REFERENCES 1. L.N.Bulaevskii, O.V.Dolgov, I.P.Kazakov, S.N.Maksimovskii, M.O.Ptitsyn, V.A.Stepanov, and S.I.Vedeneev, Superc. Sci. Technol. 1 (1988) 205. 2. R.T. Collins, Z. Schlesinger et al., Phys.Rev. 39B (1989) 6571. E.G. Maksimov in : High-Temperature Superconductivity, ed. by V.L. Ginzburg and D.A.
5. J. Rammer, Europhys.Lett. 5 (1988) 77 6. G.M.Eliashberg, PisVma ZhETF 48 (1988) 275. 7. L. Krusin-Elbaum, R.L. Greene, F. Holtzberg, A.P. Malozemoff, and Y. Yeshurun, Phys.Rev.Lett 62 (1989) 217.
3.
Kirzhnitz (Consultants Bureau, N.Y., 1982)chapter 4. 4. L.N.Bulaevskii, O.V.Dolgov, Phys.Rev. 38B (1988) 11290.
and
M.O.Ptitsyn,
8. Y. Kitaoka, S. Hiramatsu et al., Physica 153--155C (1988) 83
0038-i098/8953.00+. 00 Pergamon Press plc
INFLUENCE OF ELECTRON-PHONON SCATTERING ON THE PROPERTIES OF HIGH T c SUPERCONDUCTORS
O.V.Dolgov, A.A.Golubov*, A.E.Koshelev* P.N.Lebedev Physical Institute, Academy of Sciences of the USSR, Moscow, USSR *) Institute of Solid State Physics, Academy of Sciences of the USSR, Chernogolovka, USSR ( Received 29 June 1989 by V.M. Agranovich )
The system of electrons interacting with phonons and excitons is studied. In the standard phonon model modes with low and high frequencies are considered, the latter being interact strongly with electrons. The combined model takes into account low frequency modes and excitons interacting weakly with electrons.
rph =
1. INTRODUCTION The dominant role of the oxygen vibrations in the pairing mechanism of high T c oxides was discussed
(2Vph)--i = 2 7 r [ Sph(W) dw sh(w/T)
(2)
by Weber and Mattheiss. The tunneling 1, optical 2 and magnetoresistance data show large values of the electron-phonon coupling constant ~ ~ 1 - 3. In the present work the model is studied, in which the superconducting pairing is due mainly to high-frequency mechanism (possibly non-phonon), which satisfies the weak-coupling condition (T c < < Wo). We note that in the Eliashberg model the boundary energy between low and high frequency modes is the value 2rT c. The electron-phonon interaction may results both in the strong-coupling effects and the temperature dependent renormalization of the electron energy (strong electron scattering) 3. The strong-coupling effects were studied, in particular, for the upper critical field, Hc2(T) 4, and for the magnetic penetra-
It follows from (2) that at T _>WD/3 the relation rDh-1 = 2r~phT holds, which, in particular, results in the linear temperature dependence of the resistivity just above T c (see, for example,6). For Aph > 1 the electron
tion depth, ~(T) 5, and were shown to lead to large deviations from the BCS theory. Let us consider the clean superconductor (elastic scattering length 1e > > 4o) with the spectral density of the electron-phonon interaction in the form:
2. RESULTS AND DISCUSSION Let us discuss firstly the results obtained in the framework of the combined model which takes into account low frequency phonon modes and excitons interacting weakly with electrons. To calculate the temperature dependence of the upper critical field HC2 we use the general strong coupling relations. If we neglect Pauli limiting and Coulomb repulsion this relations takes the following form:
mean free path, lph(Tc) = vF/2rTcAph ~ 4o, i.e. the dirty limit condition holds. On cooling below T ~ WD/2r the value lph grows and the clean limit is approached. Moreover, the inelastic electron-phonon scattering leads to the smearing of the singularity in the electronic density of states which results in qualitative changes in the temperature dependence of the NMR relaxation rate just below T c.
S(w) ~ a2(w)F(w) = Sph(W) + So(W) = w 2
Ao%
= ~ph( ~D ) 0(~D - ~) + ---~ ~(~- %)'
(1) m
(2rtr)-l]-lA(wm),
where wD u Tc, w° > > T c. In this case the low-frequency phonons with w < wD do not contribute to A
~
~--'1 ~.J
and T c according to the cancellation theorem 3. However, at finite temperatures the thermal phonons determine the electron scattering time:
(3a) + U
111
m
(2rtr)-lsgnwn , 81
(3b)
111
82
INFLUENCE OF ELECTRON-PHONON SCATTERING
x(x) =
2 __ ]~dq e-q2tan-l(qal/2/x),
1
al/25 a = rHc2VF2/¢o , [
(3c)
a2(f~)F(~)
d•
A(%-~m) = 2
~2 wn = rT(2n-1).
, +
(w n
Wm)2 (3d)
_
E
05
Vol.
72, No.
BCS~
=
Here rtr is the transport electron scattering time. In the considered combined weak coupling model the equations (3) are essentially simplified: )t(Wn-Wm) = A, A(Wm) = A. Moreover for sufficiently high temperatures T ~ ~vD
0.5
we have rtr -~ rph (in general rtr < rph ). For ), > 1 we are in the dirty limit due to the strong electron-phonon scattering. Qualitatively for dirty superconductor from the relations ~ = (~olph)1/2
T/Tc 1
FIGURE
and HC2 ~ ¢ O / C ( T ) we get at T ~ Tc: Tc ) [---dHc2/dW]Tc ~ 0.7/Aph , Aph _>1, (4) HC 2 ( 0 i.e. the slope grows with Aph. For arbitrary temperatures and Aph values the exact solution of the problem for HC2 with rph (2) was obtained and the results of calculations are presented at Fig.1 (hc2 = HC2/[TcHc2'(Tc)]). It is seen that for Aph _>1 the function Hc2(T ) have a maximum at Tph - WD/27r. With the lowering T below Tph the thermal phonons are freezing out and HC2(0 ) does not depend on Aph. The temperature dependence of the magnetic penetration depth, ~(T), at T < T c can be calculated from the electrodynamical kernel Q(k) in the London limit (for k = 0): /~ 2 n N e22rTE + Wn2)3/2 q(k=0) = ~ n (/kn 2
. (5)
where the quantities An are the solutions of nonlinear version of the Eliashberg equations (3) in zero magnetic field. The penetration depth is determined by the following relation: = (c/4~q) 1/2
(6)
At T ~ T c and Aph > 1 the dirty limit relation is: = gL(~o/lph)l/2' (gL is the London penetration depth). Again the quantity ~T=0) (clean limit) does not depend on the Aph, and the slope of the lower critical field H c I ( T ) - g-2(T) at T z T c decreases with the increase of Aph. Fig.2 shows the results of the calculations in the framework of the microscopic theory for
% ~.
o.5 -
~h:2 ~ 0.5 I 0.5
T/T¢ FIGURE
2
arbitrary mean free path using the ABCS(T) and rph from (2). For large enough Aph values the positive curvature of the E-2(T) takes place at T _>Tph. Note that this picture would take place in the weak coupling limit ( % > > Tc). If this inequality is not fulfilled the strong coupling effects take place, which have opposite sign both for Hc2(T ) and fl(T) 4,5 The selfconsistent calculations including both strong coupling and renormalization effects were done for Hc2(T ) and g(T) in the model (1) at wo = 5wD and Aph = 1 by the numerical solution of the Eliashberg equations. The quantity Ao z 2 was found from the condition T c = wD. The results are shown in Figs.l,2 (curves SC - strong coupling). It is seen that for the given parameters both effects are nearly compensated and the curves SC in both cases are close to the BCS clean limit curves. In a number of experiments (see for
Vol. 72, No. I
INFLUENCE OF ELECTRON-PHONON SCATTERING
83
example ref. 7 and references therein) on high T c oxides
~"Ph=O ~ .
.
.
.
.
.
.
.
.
.
the b'-'2(T)/8"-2(0) dependence close to the BCS universal curve was observed. It is the argument in favour of the applicability of the phonon model to these oxides. Meanwhile the validity of these experimental results requires justification. We have also analyzed the NMR relaxation rate using the inelastic smearing of the density of states Fph according to eq.(2). The results of calculations are shown in Fig.3. It is seen that for '~ph -> 0.2 the inelastic smearing is large enough such that the hump in T1N/T1s just below T c disappears. Note, that the re-
.
z 0.5
Z, i
cent experiments indeed have not shown such hump 8.
0.5
Acknowledgment s. Useful discussions with L.N.Bulaevskii, G.M.Eliashberg and I.F.Schegolev axe gratefully acknowledged.
T/To FIGURE 3
REFERENCES 1. L.N.Bulaevskii, O.V.Dolgov, I.P.Kazakov, S.N.Maksimovskii, M.O.Ptitsyn, V.A.Stepanov, and S.I.Vedeneev, Superc. Sci. Technol. 1 (1988) 205. 2. R.T. Collins, Z. Schlesinger et al., Phys.Rev. 39B (1989) 6571. E.G. Maksimov in : High-Temperature Superconductivity, ed. by V.L. Ginzburg and D.A.
5. J. Rammer, Europhys.Lett. 5 (1988) 77 6. G.M.Eliashberg, PisVma ZhETF 48 (1988) 275. 7. L. Krusin-Elbaum, R.L. Greene, F. Holtzberg, A.P. Malozemoff, and Y. Yeshurun, Phys.Rev.Lett 62 (1989) 217.
3.
Kirzhnitz (Consultants Bureau, N.Y., 1982)chapter 4. 4. L.N.Bulaevskii, O.V.Dolgov, Phys.Rev. 38B (1988) 11290.
and
M.O.Ptitsyn,
8. Y. Kitaoka, S. Hiramatsu et al., Physica 153--155C (1988) 83