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Superconductivity stimulation in anisotropic superconductors
PHYSICA ELSEVIER
Physica C 259 (1996) 222-226
Superconductivity stimulation in anisotropic superconductors A.V. Galaktionov LE. Tamm Department of Theoretical Physics, P.N. Lebedev Physics Institute, Leninsky Prospect 53, Moscow 117924, Russia
Received 11 October 1995; revised manuscriptreceived 30 January 1996
Abstract
The superconductivity stimulation under high-frequency electromagnetic pumping in a thin film of anisotropic superconductor is examined. The frequency dependence of the nonequilibrium term in the Ginzburg-Landau equation differs from the isotropic case. This term may be represented as the integral transformation of the density of states in a superconductor. Keywords: d-Wavesuperconductor;Thin films; Microwaveabsorption
The AC properties of high-Te materials have been a subject of intense study since their discovery. In conventional low-To superconductors such studies provide useful information on the superconducting gap as well as on the phonon mediated pairing interaction a2 Fph (oJ), plasma frequency, impurity scattering rate etc. For instance, the frequency dependence of conductivity o-(~o) (absorptive part of the conductivity is implied here and below) can be used for the determination of a2Fph (oJ). For a review of experimental studies in HTSC see [1], where as one of the interpretations of the data, the two-component response (consisting of a free-carrier part and contribution of "bound" mid-infrared carriers) was discussed. More recent microwave experiments on Y-Ba-Cu-O [2-5] were compared to the predictions of the theory of the electromagnetic response of a d-wave superconductor [6] and a partial correspondense was observed. The study of AC properties of HTSC at frequencies lower than those of the "bound" mid-infrared carriers seems to be advantageous, since the response in this case is governed by the interplay of the superconducting gap and frequency, while other factors may be omitted. This case may be realized for temperatures close
tO the temperature of the superconducting transition. Namely this temperature range will be discussed below while analyzing the superconductivity stimulation by an alternating electromagnetic field. The theory of the superconductivity stimulation under the influence of the alternating electromagnetic field for the case of the isotropic superconductors was worked out by Eliashberg [7,8] (for a review see also [9,10] ). The explanation of the enhancement of the superconducting properties is quite transparent, when the frequency oJ of the electromagnetic field is less than the doubled superconducting gap. Indeed, new excitations cannot be born, while the existing ones are redistributed to the higher energies. So in the essential region of energies for the gap self-consistency equation, to D
1 [ = g J
(1)
A
the distribution function nE is diminished, that corresponds to the effective reduction of the temperature and increase of the gap. The frequency dependence of this effect is governed by the ratio t o / A . I f the pairing
0921-4534/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved PII S0921-4534(96)001 12-8
1 - 2n, de z / ~ ,
A.V. Galaktionov/Physica C 259 (1996) 222-226
is anisotropic the latter ratio will be variable over the Fermi-surface, and one can expect that the frequency dependence of the stimulation effect will be different from the conventional case. In HTSC materials d-wave symmetry type of pairing is currently widely discussed (the anisotropic pairing also seems to realize in some of the heavy-fermion compounds). The experimental support in favour of this symmetry is provided in particular by measurements of the magnetic flux penetrating the three junctions circuit [ 11-13], paramagnetic behaviour of the granular HTSC compounds [ 14] etc. Nevertheless the experimental situation remains ambiguous and s-wave pairing cannot be excluded, so in the given article another test is proposed, which is based on the measurement of the frequency dependence of the nonequilibrium term under electromagnetic pumping. Below it is shown, that for frequencies exceeding the one for which the effect has its maximum, the stimulation is diminished essentially steeper than for the conventional case, if the gap averaged over the Fermi-surface is zero (due to the change of sign as for e.g. d-wave pairing). Besides, in some cases this frequency dependence can be reduced to the integral transformation of the density of states, hence the energy dependence of the density of states can also be characterized. Below the Fermi-surface will be assumed to be either spherical or cylindrical for simplicity. The nature of the interaction responsible for the superconducting pairing appears to be unimportant under certain conditions, so we shall say that it is carried by "bosons". Let us consider a thin film on a substrate with a good thermal conductivity, being in a good acoustic contact with the film. So the substrate can be regarded as a thermostat. The thickness of the film d is taken to be much less both the skin depth and the London penetration depth at the given temperature. Hence the electromagnetic field is homogeneous across the sample. Nevertheless d should be much larger than the temperature dependent coherence length sO(T) (this is compelled by the spatial variation of the superconducting order parameter of the anisotropic superconductor near the boundary with characteristic scale ~:(T) [ 15,16] ). Thus the latter condition enables one to neglect the surface effects and assume the gap to have its bulk value. The temperature is supposed to be close to the temperature of the superconducting transition Te, so the
223
gap is much less than T. The frequency to of the incident wave is also taken to be much less than T, at the same time it should be much larger than the energy relaxation rate 7'- The latter condition guarantees that the absorption is mainly one-photon [ 8,10]. Since the elastic scattering on impurities impairs the anisotropic pairing due to the averaging over the Fermi-surface, the elastic relaxation rate 1/T is taken to be 1/T << T, but it is assumed to be much larger than % The contributions to 7, originate from electron-electron interactions ,,~ T2/eF or electron-boson ones, also characterized by higher powers of temperature (T3/to2D for phonons), hence for low enough temperature, corresponding to small enough effective superconducting coupling constant g the hierarchy 7+ << 1/T << T is possible. The value of y in low-Te materials is usually 107-109 s -1. Assuming the relevence of phononcoupling mechanism in Y-Ba-Cu-O, the value of may be estimated [17] as 1013 s -1 in the optimally doped case. This is due to the proportionality of y to T for T > too/27r and to T3/to2 for the opposite inequality. Since too "~ 600 K in this compound, the estimate 7, N T holds in the optimally doped case [ 17]. So, if phonon coupling prevails, the hierarchy T << T is realizable only when the doping deviates from optimal. The plasma frequency in HTSC usually has the order of 1 eV [ 1 ], hence the contribution ,-., T2/eF to y is much less. The problem of the finding of the change of the distribution function under electromagnetic pumping is equivalent to the problem of conductivity, which was considered for the case of anisotropic superconductors in [ 18]. The results of [ 18] can be understood within qualitative conventional approach leading to MattisBardeen formula [ 19]. Indeed, the Hamiltonian of electromagnetic interaction has the structure:
H' c<~ A(p + p')a~ape -i°'' + c.c.
(2)
p
Here A is the vector-potential. The direction of the electric field E in the case of the cylindrical Fermisurface is supposed to coincide with the radius of the cylinder. Now one can as usually rewrite Eq. (2) in terms of Bogolubov quasiparticles (if A(T) >> 1/T they are well defined), using factors up, vp, and come to the conventional coherence factors for the electromagnetic absorption (singlet pairing) [ 19 ]:
224
A.V, Galaktionov/Physica C 259 (1996) 2 2 2 - 2 2 6
l(p, f ) = (p + pt)k(UpU*p, + VpVp,), f (p, p') = (p + p') k( veu*e, - u~ve, ).
I= (3)
The subscript k stands for the direction of electric field, the first of these factors applies when the number of the quasiparticles is not changed; the second, when it is changed by two. The free electrons cannot absorb radiation since conservation laws of energy and momentum cannot be satisfied simultaneously, The absorption processes are accompanied by the impurity scattering, hence the coherence factors should be averaged over Fermi-surface. The following procedure is analogous to that in [20], one should average the factors ]112, ]f[2 over quasiparticle energy levels e, e t, taking into account the momentum dependent density of quasiparticle states. In the case of isotropic pairing these factors are reduced to (when account of the states above and below Fermi-surface is taken): 1112 = (p~ + p~2) (1 + a2/eet),
Ifl 2= (p~ + p~2) (1 - a2/ee').
(4)
But if the pairing is anisotropic and under a certain symmetry transformation the gap changes its sign, these factors can be substituted by: ifl2
=
ill2 = (p2
+
Pk,2) •
(5)
Note that the term, containing PkP~ vanishes due to (pkd(p)) = 0, where the average is taken over the Fermi-surface, under the same condition (i.e. for singlet pairing) the electromagnetic vertex renormalization due to isotropic impurity scattering is unimportant [ 18]. In considering the transitions in the linear approximation under pumping from initial i to the final state f one must also take into account the factors due to the occupancy of states n. The decrease of the distribution function due to the absorption will be proportional to - h i ( 1 - ny), and increase due to the radiation to nf ( 1 - n i ) , SO the total effect will be proportional to n f - - n i ( t h e spontaneous radiation is neglected). Hence in the case of the isotropic pairing the pumping source I (in the linear approximation over the field intensity) for the quasiparticle density at the energy level ~, normalized by the density of states per spin in normal metal N(0) will be:
°'(w)E2
2N(0) to2 t,~
[0(e -
r o ) u , _ , o ( n ° o, - n °)
A2 x (1 + e ( ~ - - w ) ) -ne+~o)
+0(~o
-
~),,~_,(1
l+e(e+w), -
~g_,
-
no,)
A2 (6) in agreement with [9]. Here o'(w) is the normal metal conductivity, ue = e 0 ( e - 4 ) / v / ~ - A2 is the quasiparticle density of states normalized by N(0) and n o = 1/( 1 + e x p ( e / T ) ). The coefficient of proportionality is determined from the absorption in a normal metal. In order to write out the expression for I in the case of anisotropic pairing it is covenient to introduce the quantities:
/~(e,d) = [(,,(p, E)p~)(,,(p', d))
+
(7)
Here (g) = f g d2S/ f d2S, where integration is taken over the Fermi-surface, u ( p , e ) = e 0 ( e A( p ) ) / Vie2 - A( p ) 2 and we shall denote (u (p, e ) ) = v,. The pumping source acquires the form: I = °'(a0E2 [0(e - w ) f l ( e -- w,~)(n°_~, -- nO) 2N(0)w 2
_ ~ ( e + ~ , , , ) ( n o _ ~o+~) +O(o~-e)~(o~-e,e)(1-n°_,-n°)].
(8)
D e f i n i n g / 3 ( + e , + e ' ) -- 13(e,e'), one can similarly obtain the expression for conductivity in anisotropic superconductor: O0
o's(w) = -1- f tr(o~)
w
d ~ p ( e - w , e ) [n~_o, o - n ~o] ,
--00
(9) in agreement with [ 18]. One can see that I is characterized by the energy scales w, A. When the change of the distribution function is localized in the energy interval much less than T near Fermi-surface the applicability of the
A.V. Galakfionov/Physica C 259 (1996) 222-226
r-approximation to the collision integral was proved by Eliashberg [8,9]. The same arguments apply also to the case under consideration. Indeed, the change of the distribution function enters the collision integral either in a multiplicative way or is integrated over energies. Since the energy integration of the unperturbed distribution function extends over the region ,-~ T, the terms with integrated perturbation are characterized by additional smallness to/T, A/T as compared to the terms which the perturbation enters in a multiplicative way. As the elastic impurity scattering rate is much larger than y, the latter should be momentum independent. Besides 9' does not involve coherence factors, because the particles with energy ,~ T are employed in the energy transfer, for which d2/ee I ~ O. Hence, the change of the distribution function 6n,~ will be:
y,Sn, v,~ = L
225
around the axis of the cylinder (e.g. when the gap transforms according to the one-dimensional non-unit irreducible representation of D4h). In this case:
fl(e, e I) = vet',,.
(13)
Hence taking account of the possibility to expand the Fermi-distribution function, since characteristic values of the energy are much less than T, one may come to the following equation:
Tc - T Tc
bA~(T) + KF(Ao(T)/to) = 0,
(14)
where A0 (T) is the amplitude of the gap, so A (p, T) = Ao(T)qb(p) (with Icb(p)l < 1). The function F is given by: oo
(I0)
0
-v,+~, + 0(to - e)v~,_,] de,
Substituting 6n,~ into self-consistency equation:
(15)
and K is defined as:
A(v) = -½
V(t',v')A(v ') p'
x ( 1 - 2ne, - 26n~,)e~ 1,
(11)
one will obtain:
Zc - Z (la(P)12 )
rS
7~'(3) ([A(p) 14)
P
-2/[ev~
- 4i~] cSn~ = 0.
(12)
J
0
In Eq. ( 11 ) % = (~:2 + iA(p ) 12)1/2 (here ~: = vvlP PF]) and in Eq. (12) ~ , = for,,, de'. In deriving Eq. (12) the identity (lal2/s~) = av, - ~ , was used. The averaging in the latter expression is implied at the given energy level e. F-xl. (12) has the form of the usual Ginzburg-Landau equation with addition of the nonequilibrium term. The dynamic term A2IAI 2 can be neglected if tot << Tc/T (in deriving this condition the relation (or >> 1 was used). The latter condition does not impose new restrictions, because it is the consequence of the previous ones to << Tc and r << The function/~(e, e') simplifies in the ease of cylindrical Fermi-surface, when the modulus of the gap remains unchanged under rotation over the angle ¢r/2
o-(to)E 2 K = 49,toTN(0) (l~b(p)12)"
(16)
Thus the frequency dependence of the nonequilibrium term is governed by the integral transformation of the density of states. From the expression for the nonequilibrium term one can derive that the inequality toT/Y' << T should hold so that the superconductivity suppression under heating does not exceed the stimu, lation effect. Here 9,~ is the energy transfer rate to the thermostat which may be less than 9'. The asymptotics of F(x) for large values of x is: oo
F(x) = x1 f ( t ' y
1) 2 dy.
(17)
0
In the latter equation the density of states is represented as the the function of dimensionless variable y = e/A0, besides the identity J o ( V y - 1) dy = 0 was used. The integral in Eq. (17) converges in the case of anisotropic pairing, while for the isotropic one it diverges, hence the appropriate asymptotics in the conventional case is different: F ( x ) oc lnx/x. For small values of x the asymptotics for anisotropic pairing with the logarithmic accuracy is given by:
F(x) = ½(l~b2 (p)I)2x 2 I n ( I / x ) ,
(18)
226
A.V. Galaktionov/Physica C 259 (1996) 222-226 the film below Te (the measurements above Te are unfavourable since the thickness o f the film should be larger than ~:(T), hence the gap may vary across the film due to metastability).
0.60
0.50
0.40
Acknowledgements
0.30
0.20
0.10
/
0.00 Q.O0
0.50
Ao/ m 1.00
1.50
2.00
2.50
3.00
The valuable discussions with Yu.S. Barash and E.G. Maksimov are gratefully acknowledged. The research described in this publication was made possible in part by the Russian Foundation for Fundamental Research under Grant N o 94-02-05306. The author also thanks the ISI Foundation for hospitality.
Fig. 1. The nonequilibrium term F in the case of the cylindrical Fermi-surface and a(~o) = Ao cos2~o. while in the conventional case F ( x ) c< x. The reason o f this difference is (AA') = 0 in calculating coherence factors Ill 2, lf[ 2 in the anisotropic case. One can show that the asymptotics cx x ( i f x >> 1) and c< x 2 I n ( 1 / x ) ( i f x << 1) will remain in the general anisotropic case i f (A) = 0. Hence for small x the nonequilibrium term is much smaller than in the conventional case. The F ( x ) dependence for cylindrical Fermi-surface and A ( ~ ) = Ao c o s 2 ~ is given in the Fig. 1. The steep decrease o f F ( x ) for x < Xmax as compared to the isotropic case is observable in this figure. A l s o note, that while in isotropic case the effect is m a x i m u m at oJ = 2A0, in the case under consideration the m a x i m u m takes place at ~o = 2(1~21)a0. A t the same time in tunnel junctions between two superconductors the conductance d I / d V in the anisotropic case has its m a x i m u m at e V = 2z10 both in the case o f diffusive [21 ] and specular reflection [ 16] (for some o f the orientations) at the interface. So t h e superconductivity stimulation in the anisotropic case is characterized by a much smaller value o f the effect (as compared to the conventional case) for frequencies exceeding the one at which the stimulation is maximum. The frequency o f the maxi m u m is governed not by the amplitude but by the average value o f the gap. The frequency dependence o f the nonequilibrium term can be used for determining the energy dependence o f the density o f states. This effect can be studied experimentally, as usual, by determining the increase o f the critical current o f
References [ 1] T. Timusk and D.B. Tanner, in Physical Properties of High Temperature Superconductors, ed. D.M. Ginsberg (World Scientific, Singapore), Vol. I (1989), p. 339; Vol. III (1992), p. 363. [2] D.A. Bonn et al., Phys. Rev. Lett. 68 (1992) 2390. [3] D.A. Bonn et al., Phys. Rev. B 47 (1993) 11314. [4l D.A. Bonn et al., J. Supercond. 6 (1993) 219. [5] D.A. Bonn et al., Appi. Phys. Lett. 62 (1993) 3019. [6] EJ. Hirshfeld, W.O. Putikka, D.J. Scalapino, Phys. Rev. B 50 (1994) 10250. [7] G.M. Eliashberg, Pis'ma Zh. Eksp. Teor. Fiz. 11 (1970) 186. [8] G.M. Eliashberg, Zh. Eksp. Teor. Fiz. 61 (1971) 1254. [9] G.M. Eliashberg and B.I. Ivlev, Theory of superconductivity stimulation, in: Nonequilibrium Superconductivity,eds. D.N. Langenberg and A.I. Larkin (North-Holland, Amsterdam, 1986). [ 10] A.M. Gulyan and G.E Zharkov, Superconductors in External Fields (Nonequilibrium Properties) (Nauka, Moscow, 1990). [ I l l D.A. Wollman et al., Phys. Rev. Lett. 71 (1993) 2134. [12] D.A. Brawner and H.R. Ott, Phys. Rev. B 50 (1994) 6530. [13] C.C. Tsuei et al., Phys. Rev. Lett. 73 (1994) 593. [ 14] W. Braunisch et al., Phys. Rev. Lett. 68 (1992) 1908 ; Phys. Rev. B 48 (1993) 4030. [15] E.A. Shapoval, Zh. Eksp. Teor. Fiz. 88 (1985) 1073. [16] Yu.S. Barash, A.V. Galaktionov and A.D. Zaikin, Phys. Rev. B 52 (1995) 665. [17] S.V. Shulga, O.V. Dolgov and E.G. Maksimov, Physica C 178 (1991) 266. [18] P.J. Hirshfeld et al., Phys. Rev. B 40 (1989) 6695. [ 19] J.R. Schrieffer,Theory of Superconductivity(Benjamin, New York, 1964). [20] C.J. Pethick and D. Pines, Phys. Rev. Lett. 57 (1986) 118. [21] H. Won and K. Maki, Phys. Rev. B 49 (1994) 1397.
Physica C 259 (1996) 222-226
Superconductivity stimulation in anisotropic superconductors A.V. Galaktionov LE. Tamm Department of Theoretical Physics, P.N. Lebedev Physics Institute, Leninsky Prospect 53, Moscow 117924, Russia
Received 11 October 1995; revised manuscriptreceived 30 January 1996
Abstract
The superconductivity stimulation under high-frequency electromagnetic pumping in a thin film of anisotropic superconductor is examined. The frequency dependence of the nonequilibrium term in the Ginzburg-Landau equation differs from the isotropic case. This term may be represented as the integral transformation of the density of states in a superconductor. Keywords: d-Wavesuperconductor;Thin films; Microwaveabsorption
The AC properties of high-Te materials have been a subject of intense study since their discovery. In conventional low-To superconductors such studies provide useful information on the superconducting gap as well as on the phonon mediated pairing interaction a2 Fph (oJ), plasma frequency, impurity scattering rate etc. For instance, the frequency dependence of conductivity o-(~o) (absorptive part of the conductivity is implied here and below) can be used for the determination of a2Fph (oJ). For a review of experimental studies in HTSC see [1], where as one of the interpretations of the data, the two-component response (consisting of a free-carrier part and contribution of "bound" mid-infrared carriers) was discussed. More recent microwave experiments on Y-Ba-Cu-O [2-5] were compared to the predictions of the theory of the electromagnetic response of a d-wave superconductor [6] and a partial correspondense was observed. The study of AC properties of HTSC at frequencies lower than those of the "bound" mid-infrared carriers seems to be advantageous, since the response in this case is governed by the interplay of the superconducting gap and frequency, while other factors may be omitted. This case may be realized for temperatures close
tO the temperature of the superconducting transition. Namely this temperature range will be discussed below while analyzing the superconductivity stimulation by an alternating electromagnetic field. The theory of the superconductivity stimulation under the influence of the alternating electromagnetic field for the case of the isotropic superconductors was worked out by Eliashberg [7,8] (for a review see also [9,10] ). The explanation of the enhancement of the superconducting properties is quite transparent, when the frequency oJ of the electromagnetic field is less than the doubled superconducting gap. Indeed, new excitations cannot be born, while the existing ones are redistributed to the higher energies. So in the essential region of energies for the gap self-consistency equation, to D
1 [ = g J
(1)
A
the distribution function nE is diminished, that corresponds to the effective reduction of the temperature and increase of the gap. The frequency dependence of this effect is governed by the ratio t o / A . I f the pairing
0921-4534/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved PII S0921-4534(96)001 12-8
1 - 2n, de z / ~ ,
A.V. Galaktionov/Physica C 259 (1996) 222-226
is anisotropic the latter ratio will be variable over the Fermi-surface, and one can expect that the frequency dependence of the stimulation effect will be different from the conventional case. In HTSC materials d-wave symmetry type of pairing is currently widely discussed (the anisotropic pairing also seems to realize in some of the heavy-fermion compounds). The experimental support in favour of this symmetry is provided in particular by measurements of the magnetic flux penetrating the three junctions circuit [ 11-13], paramagnetic behaviour of the granular HTSC compounds [ 14] etc. Nevertheless the experimental situation remains ambiguous and s-wave pairing cannot be excluded, so in the given article another test is proposed, which is based on the measurement of the frequency dependence of the nonequilibrium term under electromagnetic pumping. Below it is shown, that for frequencies exceeding the one for which the effect has its maximum, the stimulation is diminished essentially steeper than for the conventional case, if the gap averaged over the Fermi-surface is zero (due to the change of sign as for e.g. d-wave pairing). Besides, in some cases this frequency dependence can be reduced to the integral transformation of the density of states, hence the energy dependence of the density of states can also be characterized. Below the Fermi-surface will be assumed to be either spherical or cylindrical for simplicity. The nature of the interaction responsible for the superconducting pairing appears to be unimportant under certain conditions, so we shall say that it is carried by "bosons". Let us consider a thin film on a substrate with a good thermal conductivity, being in a good acoustic contact with the film. So the substrate can be regarded as a thermostat. The thickness of the film d is taken to be much less both the skin depth and the London penetration depth at the given temperature. Hence the electromagnetic field is homogeneous across the sample. Nevertheless d should be much larger than the temperature dependent coherence length sO(T) (this is compelled by the spatial variation of the superconducting order parameter of the anisotropic superconductor near the boundary with characteristic scale ~:(T) [ 15,16] ). Thus the latter condition enables one to neglect the surface effects and assume the gap to have its bulk value. The temperature is supposed to be close to the temperature of the superconducting transition Te, so the
223
gap is much less than T. The frequency to of the incident wave is also taken to be much less than T, at the same time it should be much larger than the energy relaxation rate 7'- The latter condition guarantees that the absorption is mainly one-photon [ 8,10]. Since the elastic scattering on impurities impairs the anisotropic pairing due to the averaging over the Fermi-surface, the elastic relaxation rate 1/T is taken to be 1/T << T, but it is assumed to be much larger than % The contributions to 7, originate from electron-electron interactions ,,~ T2/eF or electron-boson ones, also characterized by higher powers of temperature (T3/to2D for phonons), hence for low enough temperature, corresponding to small enough effective superconducting coupling constant g the hierarchy 7+ << 1/T << T is possible. The value of y in low-Te materials is usually 107-109 s -1. Assuming the relevence of phononcoupling mechanism in Y-Ba-Cu-O, the value of may be estimated [17] as 1013 s -1 in the optimally doped case. This is due to the proportionality of y to T for T > too/27r and to T3/to2 for the opposite inequality. Since too "~ 600 K in this compound, the estimate 7, N T holds in the optimally doped case [ 17]. So, if phonon coupling prevails, the hierarchy T << T is realizable only when the doping deviates from optimal. The plasma frequency in HTSC usually has the order of 1 eV [ 1 ], hence the contribution ,-., T2/eF to y is much less. The problem of the finding of the change of the distribution function under electromagnetic pumping is equivalent to the problem of conductivity, which was considered for the case of anisotropic superconductors in [ 18]. The results of [ 18] can be understood within qualitative conventional approach leading to MattisBardeen formula [ 19]. Indeed, the Hamiltonian of electromagnetic interaction has the structure:
H' c<~ A(p + p')a~ape -i°'' + c.c.
(2)
p
Here A is the vector-potential. The direction of the electric field E in the case of the cylindrical Fermisurface is supposed to coincide with the radius of the cylinder. Now one can as usually rewrite Eq. (2) in terms of Bogolubov quasiparticles (if A(T) >> 1/T they are well defined), using factors up, vp, and come to the conventional coherence factors for the electromagnetic absorption (singlet pairing) [ 19 ]:
224
A.V, Galaktionov/Physica C 259 (1996) 2 2 2 - 2 2 6
l(p, f ) = (p + pt)k(UpU*p, + VpVp,), f (p, p') = (p + p') k( veu*e, - u~ve, ).
I= (3)
The subscript k stands for the direction of electric field, the first of these factors applies when the number of the quasiparticles is not changed; the second, when it is changed by two. The free electrons cannot absorb radiation since conservation laws of energy and momentum cannot be satisfied simultaneously, The absorption processes are accompanied by the impurity scattering, hence the coherence factors should be averaged over Fermi-surface. The following procedure is analogous to that in [20], one should average the factors ]112, ]f[2 over quasiparticle energy levels e, e t, taking into account the momentum dependent density of quasiparticle states. In the case of isotropic pairing these factors are reduced to (when account of the states above and below Fermi-surface is taken): 1112 = (p~ + p~2) (1 + a2/eet),
Ifl 2= (p~ + p~2) (1 - a2/ee').
(4)
But if the pairing is anisotropic and under a certain symmetry transformation the gap changes its sign, these factors can be substituted by: ifl2
=
ill2 = (p2
+
Pk,2) •
(5)
Note that the term, containing PkP~ vanishes due to (pkd(p)) = 0, where the average is taken over the Fermi-surface, under the same condition (i.e. for singlet pairing) the electromagnetic vertex renormalization due to isotropic impurity scattering is unimportant [ 18]. In considering the transitions in the linear approximation under pumping from initial i to the final state f one must also take into account the factors due to the occupancy of states n. The decrease of the distribution function due to the absorption will be proportional to - h i ( 1 - ny), and increase due to the radiation to nf ( 1 - n i ) , SO the total effect will be proportional to n f - - n i ( t h e spontaneous radiation is neglected). Hence in the case of the isotropic pairing the pumping source I (in the linear approximation over the field intensity) for the quasiparticle density at the energy level ~, normalized by the density of states per spin in normal metal N(0) will be:
°'(w)E2
2N(0) to2 t,~
[0(e -
r o ) u , _ , o ( n ° o, - n °)
A2 x (1 + e ( ~ - - w ) ) -ne+~o)
+0(~o
-
~),,~_,(1
l+e(e+w), -
~g_,
-
no,)
A2 (6) in agreement with [9]. Here o'(w) is the normal metal conductivity, ue = e 0 ( e - 4 ) / v / ~ - A2 is the quasiparticle density of states normalized by N(0) and n o = 1/( 1 + e x p ( e / T ) ). The coefficient of proportionality is determined from the absorption in a normal metal. In order to write out the expression for I in the case of anisotropic pairing it is covenient to introduce the quantities:
/~(e,d) = [(,,(p, E)p~)(,,(p', d))
+
(7)
Here (g) = f g d2S/ f d2S, where integration is taken over the Fermi-surface, u ( p , e ) = e 0 ( e A( p ) ) / Vie2 - A( p ) 2 and we shall denote (u (p, e ) ) = v,. The pumping source acquires the form: I = °'(a0E2 [0(e - w ) f l ( e -- w,~)(n°_~, -- nO) 2N(0)w 2
_ ~ ( e + ~ , , , ) ( n o _ ~o+~) +O(o~-e)~(o~-e,e)(1-n°_,-n°)].
(8)
D e f i n i n g / 3 ( + e , + e ' ) -- 13(e,e'), one can similarly obtain the expression for conductivity in anisotropic superconductor: O0
o's(w) = -1- f tr(o~)
w
d ~ p ( e - w , e ) [n~_o, o - n ~o] ,
--00
(9) in agreement with [ 18]. One can see that I is characterized by the energy scales w, A. When the change of the distribution function is localized in the energy interval much less than T near Fermi-surface the applicability of the
A.V. Galakfionov/Physica C 259 (1996) 222-226
r-approximation to the collision integral was proved by Eliashberg [8,9]. The same arguments apply also to the case under consideration. Indeed, the change of the distribution function enters the collision integral either in a multiplicative way or is integrated over energies. Since the energy integration of the unperturbed distribution function extends over the region ,-~ T, the terms with integrated perturbation are characterized by additional smallness to/T, A/T as compared to the terms which the perturbation enters in a multiplicative way. As the elastic impurity scattering rate is much larger than y, the latter should be momentum independent. Besides 9' does not involve coherence factors, because the particles with energy ,~ T are employed in the energy transfer, for which d2/ee I ~ O. Hence, the change of the distribution function 6n,~ will be:
y,Sn, v,~ = L
225
around the axis of the cylinder (e.g. when the gap transforms according to the one-dimensional non-unit irreducible representation of D4h). In this case:
fl(e, e I) = vet',,.
(13)
Hence taking account of the possibility to expand the Fermi-distribution function, since characteristic values of the energy are much less than T, one may come to the following equation:
Tc - T Tc
bA~(T) + KF(Ao(T)/to) = 0,
(14)
where A0 (T) is the amplitude of the gap, so A (p, T) = Ao(T)qb(p) (with Icb(p)l < 1). The function F is given by: oo
(I0)
0
-v,+~, + 0(to - e)v~,_,] de,
Substituting 6n,~ into self-consistency equation:
(15)
and K is defined as:
A(v) = -½
V(t',v')A(v ') p'
x ( 1 - 2ne, - 26n~,)e~ 1,
(11)
one will obtain:
Zc - Z (la(P)12 )
rS
7~'(3) ([A(p) 14)
P
-2/[ev~
- 4i~] cSn~ = 0.
(12)
J
0
In Eq. ( 11 ) % = (~:2 + iA(p ) 12)1/2 (here ~: = vvlP PF]) and in Eq. (12) ~ , = for,,, de'. In deriving Eq. (12) the identity (lal2/s~) = av, - ~ , was used. The averaging in the latter expression is implied at the given energy level e. F-xl. (12) has the form of the usual Ginzburg-Landau equation with addition of the nonequilibrium term. The dynamic term A2IAI 2 can be neglected if tot << Tc/T (in deriving this condition the relation (or >> 1 was used). The latter condition does not impose new restrictions, because it is the consequence of the previous ones to << Tc and r << The function/~(e, e') simplifies in the ease of cylindrical Fermi-surface, when the modulus of the gap remains unchanged under rotation over the angle ¢r/2
o-(to)E 2 K = 49,toTN(0) (l~b(p)12)"
(16)
Thus the frequency dependence of the nonequilibrium term is governed by the integral transformation of the density of states. From the expression for the nonequilibrium term one can derive that the inequality toT/Y' << T should hold so that the superconductivity suppression under heating does not exceed the stimu, lation effect. Here 9,~ is the energy transfer rate to the thermostat which may be less than 9'. The asymptotics of F(x) for large values of x is: oo
F(x) = x1 f ( t ' y
1) 2 dy.
(17)
0
In the latter equation the density of states is represented as the the function of dimensionless variable y = e/A0, besides the identity J o ( V y - 1) dy = 0 was used. The integral in Eq. (17) converges in the case of anisotropic pairing, while for the isotropic one it diverges, hence the appropriate asymptotics in the conventional case is different: F ( x ) oc lnx/x. For small values of x the asymptotics for anisotropic pairing with the logarithmic accuracy is given by:
F(x) = ½(l~b2 (p)I)2x 2 I n ( I / x ) ,
(18)
226
A.V. Galaktionov/Physica C 259 (1996) 222-226 the film below Te (the measurements above Te are unfavourable since the thickness o f the film should be larger than ~:(T), hence the gap may vary across the film due to metastability).
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Acknowledgements
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0.20
0.10
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0.00 Q.O0
0.50
Ao/ m 1.00
1.50
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The valuable discussions with Yu.S. Barash and E.G. Maksimov are gratefully acknowledged. The research described in this publication was made possible in part by the Russian Foundation for Fundamental Research under Grant N o 94-02-05306. The author also thanks the ISI Foundation for hospitality.
Fig. 1. The nonequilibrium term F in the case of the cylindrical Fermi-surface and a(~o) = Ao cos2~o. while in the conventional case F ( x ) c< x. The reason o f this difference is (AA') = 0 in calculating coherence factors Ill 2, lf[ 2 in the anisotropic case. One can show that the asymptotics cx x ( i f x >> 1) and c< x 2 I n ( 1 / x ) ( i f x << 1) will remain in the general anisotropic case i f (A) = 0. Hence for small x the nonequilibrium term is much smaller than in the conventional case. The F ( x ) dependence for cylindrical Fermi-surface and A ( ~ ) = Ao c o s 2 ~ is given in the Fig. 1. The steep decrease o f F ( x ) for x < Xmax as compared to the isotropic case is observable in this figure. A l s o note, that while in isotropic case the effect is m a x i m u m at oJ = 2A0, in the case under consideration the m a x i m u m takes place at ~o = 2(1~21)a0. A t the same time in tunnel junctions between two superconductors the conductance d I / d V in the anisotropic case has its m a x i m u m at e V = 2z10 both in the case o f diffusive [21 ] and specular reflection [ 16] (for some o f the orientations) at the interface. So t h e superconductivity stimulation in the anisotropic case is characterized by a much smaller value o f the effect (as compared to the conventional case) for frequencies exceeding the one at which the stimulation is maximum. The frequency o f the maxi m u m is governed not by the amplitude but by the average value o f the gap. The frequency dependence o f the nonequilibrium term can be used for determining the energy dependence o f the density o f states. This effect can be studied experimentally, as usual, by determining the increase o f the critical current o f
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