- Email: [email protected]
Dynamics of order parameter fluctuations in gapless superconductors below Tc
Solid State Communications,Vol. 15, pp. 757—760, 1974.
Pergamon Press.
Printed in Great Britain
DYNAMICS OF ORDER PARAMETER FLUCTUATIONS IN GAPLESS SUPERCONDUCTORS BELOW T~ G. Brieskorn,M. Dinter and H. Schmidt Institut für Theoretische Physik der Technischen Universität Munchen, München, Germany (Received 30 April 1974 by B. Muhlschlegel)
The dynamical pairfield susceptibility of gapless superconductors has been calculated using the theory of Gorkov and Eliashberg and improvements thereof. The result differs drastically from the usual Ginzburg—Landau theory and agrees qualitatively with recent experiments on the I—V characteristic of tunnel junctions with Al films.
FERRELL’ and Scalapino2 have suggested a method to determine the frequency and wave number dependent pairfield susceptibility x(q, w) of a superconductor above its transition temperature T~by measuring the I—V-characteristic of a tunnel junction with a second superconductor well below its transition temperature.
(3) At sufficiently high frequencies there appears another peak, which depends sensitively on temperature near T~but is not affected by the magnetic field in contrast to the low frequency part of the I—V characteristic. The diffusive G—L equation in fact is ill founded as has been pointed out by Gorkov and Eliashberg (G—E),’2 who have presented a set of non-linear equations to describe the dynamics of superconductors in the gapless regime. We calculate the pairfield susceptibiity of a superconductor described by the G—E equations and improvements thereof.13
Above T~Anderson et al.3’4 found excellent agreement of x(O~w) with the predictions of the diffusive Ginzburg—Landau5 (G—L) equations, which also provided a satisfactory description6’7 of the dynamical conductivity of Pb-films.8 Below T~ Carlson and Goldman’s9 I—V curves drastically deviate from the predictions based on G—L equations, which have been tested before with the dynamical conduc10” The deviations can be summarized as tivity. follows (see Fig. 1 and compare Fig. 1 and Fig. 2 of
After linearization with respect to the deviations of the order parameter ~ = z~+ + iL~ 1from its equilibrium value ~ and with respect to the anomalous term”2 U = U~ 1+ ill1, which describes the coupling between order parameter and density fluctuations, equations (9)—(12) of reference 12 can be Fourier-transformed with respect to time and space. Thus for neutral superconductors one obtains two decoupled sets of equations for the ‘longitudinal’ modes 2 4m~) —4irTi~ f—ic~,+D(q 0/iJi’(p+ i~ 2 1/2)\/ /~U~ ~_iw~og(p)/(2~T)2 —iw + Dq 1I)=°
reference 9): (1) The main peak of I(q, w) = const. x”(q, ~ as a function of w = 2 eV with fixed q = 2eH(d/2 + X’) is much narrower than in G—L theory. (d is the thickness of the film, X’ the penetration depth of the superconducting electrode, V andH are the d.c. bias and magnetic field, respectively, and we use h = 1, ka l.).~ (2) On the low frequency (voltage) side of the main peak there develops a shoulder or even a small additional peak as the temperature decreases below T~.
—
)t
(1) and for the ‘transverse’ modes 757
758
DYNAMICS OF ORDER PARAMETER FLUCTUATIONS
I—lw +
2
4~T~
Dq
0/~’(p
1/2)
—
+ 1/2)/A)
~‘(pA
+
1/2)~( ~ = 0
4~T~0/\
a and f3 are the first expansion coefficients of the free energy in powers of &; yielding ~ = —a/il: =
At T~ ~ vanishes and the expression for x merges with the expression x> above T~where the usual time dependent G—L equations are applicable: 4irT
x>(q,w)
=
N(0)~’(p+ 1/2) [—iw
+ D(q2
+ 2ma)1~
-
(10)
The tunnelling current due to order parameter fluctuations is proportional to the imaginary part
4(T—T
(5)
0)f(p)/(irmD);
(3=—
where 2’. is the strength of the effective electron— electron contact interaction.
U1,
(2) respectively. We have used the notation of reference 12 with 2)))~2 (3) A = (1 + ~/(irpT(—iw + Dq and g(p) = (~‘(p+ 1/2) + p ~“(p + l/2))/(2p). (4)
a
Vol. 15, No.4
l/2)/3
~j”(p+ 1/2)+p~h”’(p+ __________________________________________________ ,
(6)
p~~‘(p + 1/2))/~(p+ 1/2)
(7)
8irmTTh,V(p + 1/2)
55 (0)
where f(~)
(~2/~)(~ —
=
4 governs the decrease of the order parameter relaxation rate above T0 due to pair breaking parameter p.’ D is the diffusion constant. Neglecting the off-diagonal elements of the above matrices yields the G—L equations. Equations 1 and 2 together with the static pair field susceptibility’0 yield:
x(q, w)
=
2 iTT N(0)~’(p+ 1/2) x
50
________________
5
10
+
4ma)
—
~iw
iw/~’(p+1/2)
[D~i_A/~t(A
+ l/2)+B
j
.
—~
64~-X
~
0
~‘
dl
.~
o .500 10
--
p-0~ P -02
(8)
1/2)/(irT))”
~
~ 9009C
0°)
vollog,
IXIN(0)&0
~)1V,
1. (a) Imaginary part x”(q, w) of the pair field susceptibility forH= l000e and (T~ T)/T~= 15.7 X l0-~according to experiment9 (o o o o), diffusive G-—L equations —),‘° G—~Eequations’2 (equation 8 with B = 0) ) and Ladder approximation’3 (equations 8 and 9) ( (b)I~’(e)forH= l500e according to G—L equations and Ladder approximation ‘I (o o o o). (c) x (q, w)/w for H = 100 Oe, p = 0.074 and values of e =l5.7X (T~ l0~ T)/T~. x”(q, w) for H 3l000e,e= and(d) 3valuesofp, both FIG.
—
(—
=
-&
~j
242
1/2~I
We compare the result for x also with that of calculating the fourpoint functions below T~in ladder approximation13 yieldmg an expression of the same structure as equation (8) with B
-2~~’
-1
N(0) is the density of electronic states at the Fermisurface and B = 0 in the G—E theory. The first term in the curly brackets represents the contribution of the longitudinal modes, essentially due to the relaxation of the order parameter, and the second term is that of the propagating transverse modes,’5 that become fourth sound in the hydrodynamic regime with the velocity c = ~ 2. 0(D1I’(p +
25
20
ci —
-
+ D(q2 —
15
voltoge 1)101
~
iwLX~g(p) + + Dq2)xT~’(p
1)
—
(.
(—
—- —
—)
—
(9)
(c) and (d) according to equations (8) and (9).
Vol. 15, No.4
DYNAMICS OF ORDER PARAMETER FLUCTUATIONS
2 one finds the x” ~16For fixed values of p, T and Dq maximum ~ at Wm =
Dq2 + 8(T— T~)f(p)/ir.
(11)
The data of reference 9 for the peak voltage ~, confirm equation (11) and allow us to determine p independent of the uncertainty in T~.We find p = 0.074. It may be caused by a contamination of paramagnetic impurities wIth a concentration of the order of 10~.It is large enough to make the superconductor gapless in the temperature regime (T~ T)/T~~ 2.10-2. The value ofDq2 for the various fields used9 can be determined from the minimum of t~,(see Fig. 2 of reference 9). We find Dq2 = cH2 with c = 1.52 X 106 Hz/0e2. We use9 T~=1.955 K for H = 100 Oe and take IXN(O) = 0.18 as the value for bulk Al. —
With these parameters we have calculated xt according to equation (8). The results are presented in Fig. 1(a) and compared with the G—L result that we obtain from equation (8) by formally setting = 0 and A = 1. Equation (8) in contrast to the usual G—L theory yields a nonmonotonous vanation of the peak-currentI~(q)with the temperature —
—
below T~at fixed magnetic field [Fig. 1(b)] in qualitative agreement with experiments (cf. Fig. 3 of reference 9). The increase ofI’ at lower temperatures occurs outside the region of gaplessness, where this theory is not valid. The low frequency behaviour is most clearly exhibited by plotting x11(q, w)/w [Figs. 1(c) and (d)]. Its characteristic feature is the maximum at finite frequencies. It does not exist in the G—L theory and represents the propagating mode observed in reference 9. It involves fluctuations of the superfluid density (4th sound) at such a low frequency that the ions and the normal electron fluid screen completely the associated electric field, thereby making the theory of uncharged superconductors applicable.The spectral weight can be shifted by addition of paramagnetic impurities, [Fig. 1(c)] allowing a check of the present theory.
The second (high frequency) peak observed in reference 9 is found to be independent of the magnetic field and to scale with ~ suggesting that it is caused by quasiparticle processes,’7 rather than coherent pair transfer.
REFERENCES 1. 2.
FERRELL R.A.,J. Low Temp. Phys. 1,423 (1969). SCALAPINO D.J.,Phys. Rev. Lett. 24, 1052 (1970).
3.
ANDERSON J.T. and GOLDMAN A.M.,Phys. Rev. Lett. 25, 743 (1970).
4.
ANDERSON J.T.,CARLSON R.V. and GOLDMAN A.M.,J. Low Temp. Phys. 8,29(1972).
5.
SCHMID A.,Phys. kondens. Materie 5, 302 (1966).
6. 7.
SCHMIDT H.,Z. Physik 216, 336(1968). ASLAMAZOV L.G. and LARKIN A.I.,Phys. Lett. 26A, 238 (1968).
8. 9.
759
LEHOCZKY S.G.(A.) and BRISCOE C.V.,Phys. Rev. Lert. 23, 695 (1969). CARLSON R.V. and GOLDMAN A.M.,Phys. Rev. Lett. 31, 880 (1973).
10.
SCHMIDT H.,Z. Physik 232, 443 (1970).
11.
LEHOCZKY S.L.(A.) and BRISCOE C.V.,Phys. Rev. Lett. 24,880(1970).
12. 13.
GOR’KOV L.P. and ELIASHBERG G.M.,J Low Temp. Phys. 2, (1970) and references given therein. DINTER M., Thesis TU MUnchen, and to be published.
14.
SCHMIDT H.,Phys. Lett. 27A, 658 (1968); FULDE P. and MAKI K.,Phys. kond. Materie 8,371(1969).
760
DYNAMICS OF ORDER PARAMETER FLUCTUATIONS
Vol. 15, No.4
15.
The form x used in reference 9 for comparison with the experiments below T0 is the same as that above T~ and therefore does not exhibit the difference between longitudinal and transverse modes, which is already present in the G—L theory.
16.
The calculation of references 1 and 2 are easily extended to the case T< T~.
17.
SIMANEK
E,Phys. Lett. 47A, 109 (1974).
Die dynamische Paarfeld-Suszeptibilität von Supraleitern ohne EnergielQcke ist im Ralimen der Gorkov- und Eliashberg-Theorie und leichten Verallgemeinerungen davon berechnet worden. Das Ergebnis ist wesentlich verschieden von der üblichen Ginzburg Landau Theorie und stimmt mindestens qualitativ mit experimentellen Strom-Spannungs-Charakteristkien von Tunnelkontakten mit Al-Filmen überein.
Pergamon Press.
Printed in Great Britain
DYNAMICS OF ORDER PARAMETER FLUCTUATIONS IN GAPLESS SUPERCONDUCTORS BELOW T~ G. Brieskorn,M. Dinter and H. Schmidt Institut für Theoretische Physik der Technischen Universität Munchen, München, Germany (Received 30 April 1974 by B. Muhlschlegel)
The dynamical pairfield susceptibility of gapless superconductors has been calculated using the theory of Gorkov and Eliashberg and improvements thereof. The result differs drastically from the usual Ginzburg—Landau theory and agrees qualitatively with recent experiments on the I—V characteristic of tunnel junctions with Al films.
FERRELL’ and Scalapino2 have suggested a method to determine the frequency and wave number dependent pairfield susceptibility x(q, w) of a superconductor above its transition temperature T~by measuring the I—V-characteristic of a tunnel junction with a second superconductor well below its transition temperature.
(3) At sufficiently high frequencies there appears another peak, which depends sensitively on temperature near T~but is not affected by the magnetic field in contrast to the low frequency part of the I—V characteristic. The diffusive G—L equation in fact is ill founded as has been pointed out by Gorkov and Eliashberg (G—E),’2 who have presented a set of non-linear equations to describe the dynamics of superconductors in the gapless regime. We calculate the pairfield susceptibiity of a superconductor described by the G—E equations and improvements thereof.13
Above T~Anderson et al.3’4 found excellent agreement of x(O~w) with the predictions of the diffusive Ginzburg—Landau5 (G—L) equations, which also provided a satisfactory description6’7 of the dynamical conductivity of Pb-films.8 Below T~ Carlson and Goldman’s9 I—V curves drastically deviate from the predictions based on G—L equations, which have been tested before with the dynamical conduc10” The deviations can be summarized as tivity. follows (see Fig. 1 and compare Fig. 1 and Fig. 2 of
After linearization with respect to the deviations of the order parameter ~ = z~+ + iL~ 1from its equilibrium value ~ and with respect to the anomalous term”2 U = U~ 1+ ill1, which describes the coupling between order parameter and density fluctuations, equations (9)—(12) of reference 12 can be Fourier-transformed with respect to time and space. Thus for neutral superconductors one obtains two decoupled sets of equations for the ‘longitudinal’ modes 2 4m~) —4irTi~ f—ic~,+D(q 0/iJi’(p+ i~ 2 1/2)\/ /~U~ ~_iw~og(p)/(2~T)2 —iw + Dq 1I)=°
reference 9): (1) The main peak of I(q, w) = const. x”(q, ~ as a function of w = 2 eV with fixed q = 2eH(d/2 + X’) is much narrower than in G—L theory. (d is the thickness of the film, X’ the penetration depth of the superconducting electrode, V andH are the d.c. bias and magnetic field, respectively, and we use h = 1, ka l.).~ (2) On the low frequency (voltage) side of the main peak there develops a shoulder or even a small additional peak as the temperature decreases below T~.
—
)t
(1) and for the ‘transverse’ modes 757
758
DYNAMICS OF ORDER PARAMETER FLUCTUATIONS
I—lw +
2
4~T~
Dq
0/~’(p
1/2)
—
+ 1/2)/A)
~‘(pA
+
1/2)~( ~ = 0
4~T~0/\
a and f3 are the first expansion coefficients of the free energy in powers of &; yielding ~ = —a/il: =
At T~ ~ vanishes and the expression for x merges with the expression x> above T~where the usual time dependent G—L equations are applicable: 4irT
x>(q,w)
=
N(0)~’(p+ 1/2) [—iw
+ D(q2
+ 2ma)1~
-
(10)
The tunnelling current due to order parameter fluctuations is proportional to the imaginary part
4(T—T
(5)
0)f(p)/(irmD);
(3=—
where 2’. is the strength of the effective electron— electron contact interaction.
U1,
(2) respectively. We have used the notation of reference 12 with 2)))~2 (3) A = (1 + ~/(irpT(—iw + Dq and g(p) = (~‘(p+ 1/2) + p ~“(p + l/2))/(2p). (4)
a
Vol. 15, No.4
l/2)/3
~j”(p+ 1/2)+p~h”’(p+ __________________________________________________ ,
(6)
p~~‘(p + 1/2))/~(p+ 1/2)
(7)
8irmTTh,V(p + 1/2)
55 (0)
where f(~)
(~2/~)(~ —
=
4 governs the decrease of the order parameter relaxation rate above T0 due to pair breaking parameter p.’ D is the diffusion constant. Neglecting the off-diagonal elements of the above matrices yields the G—L equations. Equations 1 and 2 together with the static pair field susceptibility’0 yield:
x(q, w)
=
2 iTT N(0)~’(p+ 1/2) x
50
________________
5
10
+
4ma)
—
~iw
iw/~’(p+1/2)
[D~i_A/~t(A
+ l/2)+B
j
.
—~
64~-X
~
0
~‘
dl
.~
o .500 10
--
p-0~ P -02
(8)
1/2)/(irT))”
~
~ 9009C
0°)
vollog,
IXIN(0)&0
~)1V,
1. (a) Imaginary part x”(q, w) of the pair field susceptibility forH= l000e and (T~ T)/T~= 15.7 X l0-~according to experiment9 (o o o o), diffusive G-—L equations —),‘° G—~Eequations’2 (equation 8 with B = 0) ) and Ladder approximation’3 (equations 8 and 9) ( (b)I~’(e)forH= l500e according to G—L equations and Ladder approximation ‘I (o o o o). (c) x (q, w)/w for H = 100 Oe, p = 0.074 and values of e =l5.7X (T~ l0~ T)/T~. x”(q, w) for H 3l000e,e= and(d) 3valuesofp, both FIG.
—
(—
=
-&
~j
242
1/2~I
We compare the result for x also with that of calculating the fourpoint functions below T~in ladder approximation13 yieldmg an expression of the same structure as equation (8) with B
-2~~’
-1
N(0) is the density of electronic states at the Fermisurface and B = 0 in the G—E theory. The first term in the curly brackets represents the contribution of the longitudinal modes, essentially due to the relaxation of the order parameter, and the second term is that of the propagating transverse modes,’5 that become fourth sound in the hydrodynamic regime with the velocity c = ~ 2. 0(D1I’(p +
25
20
ci —
-
+ D(q2 —
15
voltoge 1)101
~
iwLX~g(p) + + Dq2)xT~’(p
1)
—
(.
(—
—- —
—)
—
(9)
(c) and (d) according to equations (8) and (9).
Vol. 15, No.4
DYNAMICS OF ORDER PARAMETER FLUCTUATIONS
2 one finds the x” ~16For fixed values of p, T and Dq maximum ~ at Wm =
Dq2 + 8(T— T~)f(p)/ir.
(11)
The data of reference 9 for the peak voltage ~, confirm equation (11) and allow us to determine p independent of the uncertainty in T~.We find p = 0.074. It may be caused by a contamination of paramagnetic impurities wIth a concentration of the order of 10~.It is large enough to make the superconductor gapless in the temperature regime (T~ T)/T~~ 2.10-2. The value ofDq2 for the various fields used9 can be determined from the minimum of t~,(see Fig. 2 of reference 9). We find Dq2 = cH2 with c = 1.52 X 106 Hz/0e2. We use9 T~=1.955 K for H = 100 Oe and take IXN(O) = 0.18 as the value for bulk Al. —
With these parameters we have calculated xt according to equation (8). The results are presented in Fig. 1(a) and compared with the G—L result that we obtain from equation (8) by formally setting = 0 and A = 1. Equation (8) in contrast to the usual G—L theory yields a nonmonotonous vanation of the peak-currentI~(q)with the temperature —
—
below T~at fixed magnetic field [Fig. 1(b)] in qualitative agreement with experiments (cf. Fig. 3 of reference 9). The increase ofI’ at lower temperatures occurs outside the region of gaplessness, where this theory is not valid. The low frequency behaviour is most clearly exhibited by plotting x11(q, w)/w [Figs. 1(c) and (d)]. Its characteristic feature is the maximum at finite frequencies. It does not exist in the G—L theory and represents the propagating mode observed in reference 9. It involves fluctuations of the superfluid density (4th sound) at such a low frequency that the ions and the normal electron fluid screen completely the associated electric field, thereby making the theory of uncharged superconductors applicable.The spectral weight can be shifted by addition of paramagnetic impurities, [Fig. 1(c)] allowing a check of the present theory.
The second (high frequency) peak observed in reference 9 is found to be independent of the magnetic field and to scale with ~ suggesting that it is caused by quasiparticle processes,’7 rather than coherent pair transfer.
REFERENCES 1. 2.
FERRELL R.A.,J. Low Temp. Phys. 1,423 (1969). SCALAPINO D.J.,Phys. Rev. Lett. 24, 1052 (1970).
3.
ANDERSON J.T. and GOLDMAN A.M.,Phys. Rev. Lett. 25, 743 (1970).
4.
ANDERSON J.T.,CARLSON R.V. and GOLDMAN A.M.,J. Low Temp. Phys. 8,29(1972).
5.
SCHMID A.,Phys. kondens. Materie 5, 302 (1966).
6. 7.
SCHMIDT H.,Z. Physik 216, 336(1968). ASLAMAZOV L.G. and LARKIN A.I.,Phys. Lett. 26A, 238 (1968).
8. 9.
759
LEHOCZKY S.G.(A.) and BRISCOE C.V.,Phys. Rev. Lert. 23, 695 (1969). CARLSON R.V. and GOLDMAN A.M.,Phys. Rev. Lett. 31, 880 (1973).
10.
SCHMIDT H.,Z. Physik 232, 443 (1970).
11.
LEHOCZKY S.L.(A.) and BRISCOE C.V.,Phys. Rev. Lett. 24,880(1970).
12. 13.
GOR’KOV L.P. and ELIASHBERG G.M.,J Low Temp. Phys. 2, (1970) and references given therein. DINTER M., Thesis TU MUnchen, and to be published.
14.
SCHMIDT H.,Phys. Lett. 27A, 658 (1968); FULDE P. and MAKI K.,Phys. kond. Materie 8,371(1969).
760
DYNAMICS OF ORDER PARAMETER FLUCTUATIONS
Vol. 15, No.4
15.
The form x used in reference 9 for comparison with the experiments below T0 is the same as that above T~ and therefore does not exhibit the difference between longitudinal and transverse modes, which is already present in the G—L theory.
16.
The calculation of references 1 and 2 are easily extended to the case T< T~.
17.
SIMANEK
E,Phys. Lett. 47A, 109 (1974).
Die dynamische Paarfeld-Suszeptibilität von Supraleitern ohne EnergielQcke ist im Ralimen der Gorkov- und Eliashberg-Theorie und leichten Verallgemeinerungen davon berechnet worden. Das Ergebnis ist wesentlich verschieden von der üblichen Ginzburg Landau Theorie und stimmt mindestens qualitativ mit experimentellen Strom-Spannungs-Charakteristkien von Tunnelkontakten mit Al-Filmen überein.