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Collective modes in layered superconductors
PHYSICA
N
ELSEVIER
Physica C 253 (1995) 373-382
Collective modes in layered superconductors S.N. Artemenko *, A.G. Kobel'kov Institute Jor Radioengineering and Electronics of Russian Academy of Sciences, Mokhovaya, 11, 103907, Moscow, Russian Federation
Received 6 July 1995
Abstract Weakly damped collective oscillations in layered superconductors are examined theoretically. The calculation is based on kinetic equations for Green functions generalized to the case of layered superconductors with weak interlayer coupling. We show that weakly damped plasma oscillations of superconducting electrons can exist in such superconductors and calculate the spectrum of the mode. The oscillations are reminiscent of Josephson plasma oscillations in a superconducting tunnel junction. As the temperature approaches the critical one, these oscillations transform into Carlson-Goldman modes. Possibilities for detecting the plasma mode experimentally are discussed. We found distinctive features in the reflection coefficient for an electromagnetic wave incident at an arbitrary angle on a plane parallel to the layers. The presence of the mode changes also the dispersion relation for the waves propagating along a thin film of a layered superconductor.
1. Introduction
Weakly damped oscillations in superconductors were studied since the late fifties [1,2]. It was shown in the pioneering papers by Bogolubov and Anderson that free oscillations of the phase of the superconducting order parameter would have an acoustic spectrum, provided that the electron charge is ignored. However, the perturbations of the electron density create Coulomb forces and, therefore, transform such oscillations into a plasma mode whose energy is normally much larger than the superconducting energy gap A. Thus, the resulting oscillations are nearly identical to usual plasma oscillations in a normal metal. In 1975 weakly damped collective oscillations with frequency below the energy gap were discovered experimentally by Carlson and Goldman [3].
* Corresponding author.
The theoretical explanation of the Carlson-Goldman mode was presented for both dirty [4] and clean [5] superconductors. This mode has no analogy in normal metals. In distinction to usual plasma oscillations the perturbations of the charge density in the Carlson-Goldman mode may by neglected, and the mode was shown to be associated with oscillations of a charge imbalance between electron-like and hole-like excitations (for a review see Ref. [6]). This mode exists only in a limited frequency range near the transition temperature T~ and has a linear spectrum. In this mode perturbations of the supercurrent (associated with oscillations of the phase) are compensated by a normal current (associated with oscillations of the longitudinal electric field). In addition to the bulk modes there are weakly damped plasma oscillations with frequency oJ < A in superconducting thin films, predicted in Ref. [7] and observed in Ref. [8]. Such a mode is possible, because the electric field created by charge oscillations is concentrated mainly outside the film.
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374
S.N. Artemenko, A.G. Kobel'kov / Physica C 253 (1995)373-382
Weakly damped modes may appear also in higly anisotropic layered superconductors (LS's) like highT~ ones or intercalated transition metal dichalcogenides. These materials can be considered as a stack of quasi-2D superconducting layers with Josephson interlayer coupling [9,10], which was confirmed by experimental observation of the Josephson effect [ 11] in layered superconductors. Therefore, plasma oscillations analogous to those in Josephson tunnel junction arrays may be present in LS. As interlayer coupling may be extremely small, one can expect that the plasma frequency of LS for the direction perpendicular to the layers is smaller than the gap A. The experimental evidence of plasma oscillations was obtained by Tamasaku et al. [12], who detected the plasma edge of the reflection coefficient at frequencies below the gap. Plasma oscillations were studied in several theoretical papers [13-18]. They were predicted by Mishonov [13] based on a phenomenological approach. Modifications of 2D plasma oscillations due to interlayer coupling in a stack of 2D superconductors were studied in Refs. [15] and [14] by means of a diagrammatic approach and numerical calculations, and the possibility of plasma oscillations below the superconducting gap was demonstrated. Josephson plasma oscillations in LS were considered by Tachiki et al. [17] using a phenomenological model, and an interpretation of the experimental data of Ref. [12] was given. Bulaevskii et al. [18] obtained the plasma oscillations using the Lawrence-Doniach model [9]. But still some questions remain to be clarified. The plasma oscillations are related to an electric field inside a superconductor, and the electric field creates a nonequilibrium electron distribution. Since the frequency of the oscillations may be small, the scattering processes must be considered in order to describe the response of the electrons to the electric field. Therefore, to take these effects into account, the spectrum of the collective modes must be calculated using the kinetic approach as it was done in the case of conventional superconductors [4,5]. We shall use kinetic equations for the Green functions in the Keldysh [19] representation. Such an approach allows one to take strictly into account the influence by the normal electrons on the spectrum of the mode and on its damping. In addition, this approach permits us to elucidate the relation between
the plasma mode and the Carlson-Goldman mode, which appears near the transition temperature. Some preliminary results of our study of collective modes in LS using kinetic theory were published in our letter [16]. In Section 2 we derive expressions for current and charge densities using kinetic equations for the Green functions in the Keldysh representation. In Section 3 we derive the dispersion relation for free oscillations and analyze the spectra of weakly damped modes, concentrating on the limits of low temperatures (T << A) and temperatures near T~ (T >> A). In Section 4 we consider some effects related to the plasma modes in thin films and possibilities of their experimental observation. We calculate the reflection coefficient for an electromagnetic wave incident at an arbitrary angle on a superconducting film in the frequency range where the modes are weakly damped, and then we consider the propagation of waves along a film. The cumbersome calculations with the Green functions are collected in the Appendix.
2. Basic equations To describe the behavior of the phase of the order parameter and of the electric field inside a superconductor we use the equations for nonequilibrium Green functions which have proven to be a powerful tool in studying the transport phenomena in conventional superconductors. These equations were derived in Ref. [20] and generalized to the case of LS in Ref. [21]. A Wannier site representation is used for the z direction perpendicular to the layers, while for directions parallel to the layers the continuous coordinate representation is used. The equations simplify strongly in the tight-binding limit, when only coupling between the neighboring layers is taken into account. The one-electron spectrum in the normal state near the Fermi energy ~v is modelled as -- eV + v ( I p l --PF) + 61 COS(pzd), where ~1 << sF is related to the tunneling rate, v and PF are the velocity and the momentum at the 2D Fermi surface, p and Pz are the electron's momenta in the plane of the layers and perpendicular to the layers, respectively, d is the lattice period in the z
S.N. Artemenko,A.G. Kobel'kov/Physica C 253 (1995)373-382 direction. In the following we use units such that h = k B = e = 1. Then the equation obtained in Ref. [21] has the form
ipVGnm-t-~1 E
i=_1
(Ann+iGn+im-Gnm+iam+im) 0 ^
q-io-z-~tGnm q-iVGnmo-z-PnGnm q-GnmPm 2r
........
;
(1)
where P,, = -iO'yA, + hn, h n = IX, + CrzVpn, tz, = l/2OXn/Ot+ ~,, IX, is the gauge-invariant scalar potential, qb is the electric potential, Xn is the order parameter phase, p, is the superconducting m o m e n tum parallel to the layers, v is the 2D Fermi velocity vector, and
Ar'm=C°S( Xn-Xm'
2
]] + i ° ' z
sin(Xn--Xm) "2
(Indices m, n refer to the layers). All these functions depend on the coordinates along the layers. The Green functions G,m are matrices
=
0
gA
'
(2)
where the retarded function g,m, R the advanced function g,m A and the function g~m K introduced by Keldysh [19] which is related to the distribution function of the electrons, are in turn matrices in layer and in spin indices. The Pauli matrices act on the spin indices of the Green functions. The r.h.s, of Eq. (1) is an elastic-collision integral. We neglect inelastic collisions considering frequencies larger than the inverse energy relaxation time. The bar over the Green's function denotes averaging over the directions of the momentum parallel to the layers. The current and charge densities are calculated from the solution of Eq. (1) for the off-diagonal component of the Green function (2) using the following expressions:
k2f2~
Jll = 8vrv Jo
dO Tr
O'zi,'gnn, K
375
with arguments t = t' in the Green functions. Here 0 is the polar angle and ko 1 is the Thomas-Fermi screening radius; subscripts II and 3_ denote the directions relative to the layers. To calculate the spectrum of the normal modes, we find the Green functions in linear approximation in Ps, /x and ~b, = X , , - 1 - X,. Then, using Eqs. (3-5), we derive the expressions for the current and charge densities and, combining them with Maxwell's equations, obtain the dispersion relations. The details of the solution of Eq. (1) can be found in the Appendix. Corrections of different symmetry to the equilibrium Green's function gnn'( K ~, S') are found in the Fourier representation, in which they are functions of the wave vector q along the layers and of the wave number k (J k[ < w / d ) related to the discrete Fourier transformation with respect to the layer index n. We use the limit of q < < A / v ~ ~(T) -1, ~(T) being the coherence length. Substituting Eqs. (A.7-A.9) in Eqs. (3-5), and performing the integration over the energy (the integration is equivalent to calculation of g at t = t' in the time representation) we may obtain the expressions for current and charge density valid for arbitrary temperature and frequency. However, in simple analytic form the integrals can be calculated in the limit of small frequencies o9 << A and in the cases of low (T<< A) or high (T>> A) temperatures. We shall concentrate on these limits. First we consider temperatures near the critical value To. The integrals are calculated using the smallness of the factor 6 = ~ A / 4 T << 1. In the limit o9 >> DIIq2 q_ D . k 2, in which low-damping modes exist we find
¢2 jll = 4--~1 p - io-iI[q/x + ogp(1 + J ) ] ,
j.-
(3) P--
¢2 4.rrA-----fpz-iO'.[klx+ogpz(l+J)],
(5)
(6)
k2 6Ix + i D t l q 2 + D ± k 2 IX 4"rr co
K j ± = f : ~ d0 Tr o-z(An+,ngK,+,-- A ,n+,gn+,n),
(4) P'=--~
1f2
/~'+-8Jo
)
d0 T r g . K ,
+ i( DiIqp + D . kpz) ] .
(7)
Here p is the superconducting momentum along the layers in the Fourier representation, ~'11 and A± are
S.N. Artemenko, A.G. Kobel'kov / Physica C 253 (1995) 373-382
376
magnetic field penetration depths for the magnetic fields perpendicular and parallel to the layers, respectively; Pz = qS/d corresponds to the component of the superconducting momentum across the layers. For All we have found the usual expression, and A± is related to the Josephson critical current Jc by
A2 = c2/(87rdjc). Conductivities o-Ii and o-± near T~ within the corrections of the order of 82 coincide with the conductivities in the normal state parallel and perpendicular to the layers, respectively. DII = v2~'/2 and D± = 2e2d2~- are diffusion coefficients along and across the layers, and (---Aln---~ ] 2T co'
A~->>I
~ln-~,
At<< 1.
J= [ A
Zl
(8)
The factor J is small, but it gives an important contribution to the damping of the collective modes. Eqs. (5) and (6) are similar to those obtained for isotropic superconductors (see Ref. [6]). These expressions for the current densities can be considered as a combination of superconducting currents (terms with a~,±), normal currents (the first two terms inside the square brackets) and interference currents (the last terms with factors J). This interpretation becomes more transparent when the normal current densities are written in the form of Ohm's law j = o-E, where the electric field is related to a superconducting moment p and scalar potential /x as 0p E = - - - V/x. 0t
(9)
The terms describing superconducting currents in Eqs. (5) and (6) originate from the regular (retarded and advanced) Green functions, while normal and interference currents originate from the anomalous function, which is related to the perturbation of the quasiparticle distribution function by an electric field. If quasiparticles were absent, the charge density would be equal to
which corresponds to the shift of the chemical potential of condensed electrons by/z. Near T~ the density of quasiparticles is high and modifies strongly the charge density, resulting in expression (7). At low temperatures T << A the density of quasiparticles is proportional to n ~ exp( - A/T), and, therefore, their contribution to the charge and current densities is exponentially small. In the main approximation the charge density is given by Eq. (10), and the current density is described by the first terms in Eqs. (5) and (6). The quasiparticle contribution, though small, cannot be neglected in the damping of the collective modes. The quasiparticle contribution to the current cannot be reduced to the Ohmic current, because the coefficients of the terms with V/, and Op/Ot are different. In the frame of the BCS model the larger contribution is given by the terms proportional to Op/Ot in the expressions for the current, the other contributions being small by factors T / A or oo/A. Thus at low temperatures the expressions (5) and (6) are substituted by = ( ill \
c2 4,rrA~I
J±=
-
4'rr
(lO)
)
iwo'± Pz"
47rA~
(11)
For o'11 and or± we obtained the following expressions: °'ll'±=~r~'±--(1-e-'°/ CO
e-a/rg
7'
r/ , (12)
where o-~ '± are the DC conductivities which would be in the normal state at the same temperature,
= [ ~ exp - --~x
1
l+,(,/l+x+
1+ o =
iw°'ll)P'
1,ff-;-;x - ,ff)2
and rl = 2wA~"2.
dx,
S.N. Artemenko, A.G. Kobel'kov/ Physica C 253 (1995) 373-382 3. S p e c t r u m o f t h e n o r m a l m o d e s
by A ± / ~ / I
Effects specific for a layered structure are expected, provided there is a nonzero component of electric field perpendicular to the layers. So we consider a propagating wave, the electric field of which has components both perpendicular and parallel to the layers, and the magnetic field is parallel to the layers. To calculate the spectrum of the collective oscillations we substitute the expressions for the current and charge densities obtained in the previous section into Maxwell's equations. We express the electric field in terms of/x,
-
(j/jc)
377
2 , and as a result we get for the
plasma frequency 0)p(j)= 0)p~/1 - ( j / j c ) 2 . Near T~, where A << T, quasiparticles play a major role and must be taken into account carefully. Using now the expressions (5-7) for the current and charge densities and the Maxwell equations, we obtain another system of linear algebraic equations. Zeros of its determinant yield the dispersion relation for the eigenmodes
(k2~l q- QII)(q 2~2 q-Qm)
( 5)(2 io t
= k 2 q 2 A~ - i
S
AL
0)6
]'
where
Ell = - - i ( q / x + 0)p),
0)2
E L = -i(k/x + 0)qb/d).
(lS)
Then we exclude /x, p=, and ~b, obtaining a system of three homogeneous linear equations for Ell and E L , which has nontrivial solutions provided the determinant of the system is equal to zero. Having expanded the determinant, we find the dispersion relation for the modes. In the limit of low temperatures, we obtain a single low-damping mode, which describes the plasma oscillations of the superconducting electrons:
092=
2
(l-{- =Lk2/kl)(1-{-k2/~]
0)P
+qZa2)
(1 _1_k2Al)
- 2iT0),
(14)
where 0)p = C / ( / ~ Z ~ ) is the plasma frequency, and the damping constant is given b y
QI = 1 - i w % - -U
2
="All +
i DII Sq 2,
0)2 Q . = 1 - i0)r, - -c-T ~ ± A2 + i
S k 2,
S = 1 - i0)%J,
with r I which is, according to Eq. (15), determined by the conductivities and penetration lengths near To. Eq. (16) has solutions corresponding to weakly damped modes. The frequency of dielectric relaxation 0)r is proportional to the quasiparticle density, while o)2 is proportional to the density of condensed electrons. At lower temperatures when 0),. << 0)p, the weakly damped oscillations correspond to the plasma mode
22
2 1 +qA± +
0)2
0)P
D'k2 ) (Dr ~
-i
0)0)r D±k 2
•
1 + - -
0)1"~
"Y= "2"0)27"1 l -]- k2a~ -{- 1 q- k2al -]- q2a2
"
(17)
Here e L is a background dielectric constant for the electric field perpendicular to the layers, and
,7-1 __
4 q-rAilo-ii
C2
0),./0)p2,
('Or
4-fro- l
6" L
(15)
The plasma frequency can be diminished if a direct current j
Being proportional to Aj_1, 0)p decreases rapidly with temperature increasing, while 0),. remains about the same value. At higher temperatures when the condition 0)p << 0)r starts to hold, the weakly damped solution of Eq. (16) transforms into the CarlsonGoldman mode 0)2 =
D . k2 -I- Dlq2 6"rl - 2iT0),
(18)
S.N. Artemenko, A.G. Kobel'kov /Physica C 253 (1995) 373-382
378
where the damping coefficient is given by
+ O,,q2) +
with q = f ~ a ( 0 ) / c ) s i n 0. Here l determines the decaying length
1l-1
As in the usual case of isotropic superconductors, the region of weak damping is bounded from below and from above and exists as long as J ~ A / T In A / w is small.
Al0)p2/
0)2(6a sin 0 2 - 1)
0)2--_COp2 •
Using Eqs. (11) we obtain the relation for the fields at the film surfaces 0)
ex = ;/4,
;=
(20) C
4. Plasma modes in thin films Now we consider some consequences of the presence of a low-frequency plasma mode of superconducting electrons. The presence of the plasma mode may be detected by the frequency dependence of the reflection coefficient at 0)>/0)p. A sharp plasma edge in the reflection coefficient has been observed in the experiment by Tamasaku et al. [12], where polarized light was incident on the sample surface, which was perpendicular to the layers. Using the phenomenological two-fluid model, Tachiki et al. [17] interpreted these data as the result of the excitation of collective modes with low damping. We shall calculate the reflection coefficient for the case when the surface of the LS film is parallel to the layers, while a polarized electromagnetic wave is incident at an angle 0. Interference of the lowdamping waves inside the film leads to an oscillating dependence of the reflection coefficient on frequency and on angle 0. To calculate the reflection coefficient for light incident at an angle 0 we must match the magnetic and electrical fields inside and outside of the superconductor at the surfaces of the film. We denote the dielectric constant of the surrounding medium by 6a. There are five different waves to be taken into account: incident, reflected, and transmitted waves outside the film, and two waves inside the film. All waves have the same wave vector component parallel to the layers. Two waves inside the superconductor differ only by the sign of the component k of the wave vector perpendicular to the layers. The value of k is determined from Eq. (14): k2A~ =
0)2(1 + q2A~) -- 0)2 0)2 2 -- 2ikA~l/l -- 0)p
(19)
Using expression (20) and boundary conditions, we obtain for the reflection coefficient 2~cos Rref =
1 + a 2 cot 2 kw '
0
a = e a cos 2 0 - ~ a (21)
One can see that due to the term cot 2 k0) in Eq. (21), in which k depends on 0) and 0 according to Eq. (19), the reflection coefficient oscillates strongly with frequency and angle at 0)> 0)0" If Im kw = w / l << 1 may be neglected, the minimum value of Rr~f= 0 is achieved at kw = wn (n is an integer), while its maximum value R r e f = 1 is achieved at kw = ~ ( n - ½). When the thickness w of the film becomes comparable to the damping length 1 the oscillating behavior disappears, but R still has a minimum near 0)p, provided 0 v~ 0. Next we consider the dispersion relation 0)(q) for waves propagating along a thin superconducting film with surfaces parallel to the layers, surrounded by isolating media with dielectric constants 61 and 62. To solve this problem we match two waves inside and two waves outside the film. For the two waves inside the film the relation between the components of the wave vector is determined by Eq. (14), and the component q along the layers is the same for all four waves. The electromagnetic field must vanish at large distances from the surface. Using Maxwell's equations outside the film, we obtain the relations for the z components of the wave vector of the waves outside the film defined as k = iKl and k = - i K 2, so that K1 > 0 and K2 > 0; we have 0) 2
K? = q2 __ 6 1 - 7 '
(.0 2
K2 = q2 __ 6 2 7 .
(22)
379
S.N. Artemenko, A.G. Kobel'kov / Physica C 253 (1995)373-382
Matching the electric and magnetic fields inside and outside the film and using relation (19) we obtain an uniform linear system of equations• Its determinant is zero, provided A1 + A 2
tan k w -
1 -AIA 2 '
(23)
From Eq. (19) one can see that the wave number k is imaginary at 03 < 03w Thus, the tangent in Eq. (23) transforms into the hyperbolic one, which results in a single low-damping mode at 0) < 03p. We consider first the case I k l w >> 1 satisfied provided either w>>Aii o r 03>>W03p 8~--~/(AIIV~ ). Then, from A = 1 we obtain
where
032 q2 =
KI,2C
A1,2
el ,2 03~ "
__
2 0)2 0)p--
8 C2 top2 _ 0)2(1 + I )
,
(27)
where
We consider the case ~1 = e2 = 8 ( A 1 = A 1 = A ) . The case c 1 v~ ~2 should be qualitatively the same. Then Eq. (23) can be rewritten in the form A = tan k w / 2 and A = - c o t k w / 2 .
0)2A 032
I= - I o° ± C2
03p2
<<1•
(24)
The opposite case I k l w << 1 requires both w <<
Some solutions can be found analytically if we assume A >> 1. This condition is easy to meet if the frequency is not too large due to the fact that c2//(0)pAH) 2 is large, e.g. about 10 5 for T1 based superconductors. Then we have
All a n d 0) << W0)p8V~-±//(Aii!/r~'g)• Then the expression
03: 03p /l + q2A /(1 + ( s,,)2) ,
(25)
where s = ~Aii/w and n is any integer• Note that the solutions with n = 0 is exact• All these solutions correspond to low-damping modes propagating along the film only if the electric field decays outside the film, i.e. when K given by Eq. (22) is real, that is if the frequencies of the modes satisfy the condition
03 < q c / v/~.
5. Conclusion
There is another low-damping mode. At kw << 1 (but k v~ 0) we obtain
092(
0)2A4 )
q2 = 8--~- 1 + 4~c--~wII2 .
(26)
The approximation kw << 1 is valid provided Aii/w >> ~ and 03<< c ~--~l /(All6). At large frequencies, expression (26) resembles the well-known [7] square-root law for 2D plasma oscillations
~ wq 0) =
for this mode coincides with Eq. (26). Thus we found that the single mode below 0)p splits into a number of modes above 03p. This fact arises from the interference between the waves reflected from both surfaces of the film• Note that at frequencies above 0)p the signs of the film's effective dielectric constants along and across the layers are opposite. That is, at 0) > 03p the LS film has the properties of a superconductor and of an isolator in the direction along and across the layers, respectively.
c
2 e All
When the frequency exceeds c ~ / A i l ~ the mode deviates from the law (26) and approaches the solution (25) with n = 1.
We applied a nonequilibrium Green function technique to find the dispersion relation for low-damping collective oscillations in a LS with weak interlayer coupling• This approach allows one to consider the contribution of quasiparticles strictly, to calculate the damping of the collective modes more accurately and to show the relation between the collective modes at low temperatures and at temperatures near Tc. The calculations were performed within the BCS theory, and all the results of this article are directly applicable to LS with BCS pairing mechanism• But the equations obtained in Section 2 have a clear physical meaning and may be considered as phenomenological irrespective to the pairing mechanism. So we believe that the main results (excluding,
380
S.N. Artemenko, A.G. Kobel'kov /Physica C 253 (1995) 373-382
maybe, the magnitude of the damping) are qualitatively correct for non-BCS superconductors too. An extremely large anisotropy of LS leads to a low plasma frequency COp for an electric field perpendicular to the layers. In some materials this plasma frequency drops far below the superconducting gap A and the low-damping plasma mode of the superconducting electrons exists. For Bi and T1 based high-Tc superconductors the plasma frequency is estimated as COp~ 10 THz. We have shown that the low-damping CarlsonGoldman mode, which exists in LS near T~, continuously transforms into plasma oscillations at lower temperatures. We considered some effects arising from the interference of low-damping plasma oscillations inside the film. The reflection coefficient from the film has an oscillating dependence on the angle and frequency just above plasma frequency COp.In the frequency range below Wp the 2D Plasma mode has the usual square root behavior at low frequencies and saturates when frequency approaches to cop. Many modes propagating along a thin film may appear above the plasma frequency.
Acknowledgements We are grateful to F. Gleisberg for reading the manuscript and useful comments. This work is supported in part by INTAS grant No. A92-056.
A) __~ ( 8 0 . z _}_ i A o . v ) / ~ R ( A ) ,
~R(A)(8 ) =
+
s i g n ( 8 ) O ( 8 - A) + i f - A T - ~ - ~ O ( A ] 8 I) "---- ___((8 -t- i 0 ) 2 -- A 2 ) 1/2, where __.i0 reminds us of the direction of encircling the singular points 8 = + A while moving along the real axis. We consider p, /x and ~b as perturbations and calculate the deviations of the Green functions from their equilibrium values. Eq. (1) for 4 × 4 matrices corresponds to three equations for 2 × 2 matrices, namely for gn,n' R gn,, A and g,m K from Eq. (A.1). Here we seek g~m K in the form 8r
8
g~, = g,m R t a n h - - - grim a t a n h - - + g nam '
2T
2T
(A.2)
where the anomalous Green function g a is introduced. We present the perturbative part of both anomalous and regular Green functions in the form (A.3)
i __ --i -t- ~gn,n i cos 0, gnm -- gnm
where i = a, R, or A, the bar over the Green function denotes averaging over the angle 0 between the directions of v and q. To simplify we shall omit the index a of the anomalous Green function. Then we substitute Eq. (A.2) into the equation for g,,,, K from g,,,, multiplied by Eq. (1), subtract the equation for R tanh 8 ' / 2 T , add the equation for gA multiplied by tanh 8 / 2 T , linearize the equation keeping terms up to first order in the perturbation and perform the 2D Fourier transformation with respect to the in-plane coordinates, introducing the wave vector q. So we obtain an equation for gnm: 81(gn-lm
+ gn+lm-
gnm+l --grim-l)
--(8¢r z -I- i Acry ) gnm + gnrn( g~O'z"t- i Ao'y) i +qv c o s Ogn,.- ~-~r(ggg,,m- go"--gram 6n,n
Appendix We present here the detailed solution of Eq. (1). Our goal is to find the expressions for the current and for the charge density in linear approximation. The unperturbed equilibrium solution of Eq. (1) (with h, = 0, th, = 0) has the form
-A -I- g nn go ~nm -S--CO = tanh 2T
g,mgoA) tanh 8 / ( 8 1 ( A n n
2T } ~ +A,~n+lgA6n+X,,) + hngA~nm) + tanh
2T
lgA~n-l,n
tanh
8(8-- /8')~nm × (el(g0RA,~+i,~ 6.m+1 (A.1)
+goRAm- l,, t~,,,,,- 1) + gRhm6nm ).
(A.4)
S.N. Artemenko, A.G. Kobel'kov /Physica C 253 (1995) 373-382
Besides, we use the orthogonality relations [20]
Q 1 __gin ) _ l(gn+ln+l
2 +
--
'~k ,Snk 6km = R K K sumk(gnkgkm+gnkgAm)=O,
381
~ g oR-g n . = oLStx , 1
R
Qgnn + e l ( 3 g n + l n + l -- 6 g ~ . ) -- - - g o 6 g . .
Y
which in the linearized form yields
= aSp, (A.6)
(co- z + i A % , ) g .... = ~ R g R g . m , gnm( e'o-z + i A%,) = ~A gn,ng A • NOW we separate the equations for the functions which are symmetrical and antisymmetrical with respect to cos 0. Integrating Eq. (A.4) first over cos 0, and then integrating it after multiplication by cos 0, we find Q --~6g,,n + e l ( g . - 1 . -- g . . - I + g n + l n -- g n n + l )
g~n =
Og, nn + Sl( S g n - l n -- 8gnn-1 + 6gn+ln -- 8gnn+l) 1
= aSp,
Q
1
= C ~ S A -+ , 1
egnn+ 1 + el(tSg.+l.+, - 6g..)
-- ~ g R 6 g . . + l
-gn-ln,
6g,l,n = t3gnn-1 -- ~ g n - l n '
(A.5)
the first of which determines the current density across the layers. The functions (A.5) obey the equations Q ~ 1 --f6g.. - _~gRgl
g ( k ) exp(ikdn)
+ 2 e l ( ~ n - ln-I -- gnn) = OtSA, 1
Qgl n + 2 e , ( 6 g n_ ~n-, - t~gnn) -- --gRtSgln = O, 3'
2~r
.
It results in a linear system of equations, which can be easily solved. Substituting its solution and solutions of the analogous equations for the retarded and advanced Green functions into Eq. (A.1) we get expressions for the deviation of the Keldysh function gK from its equilibrium value. We denote these functions by ~K, 6gK and glK. Being integrated over energy these functions determine the linear response of layered superconductors. In order to obtain simple equations which may be integrated analytically we assume the low-frequency and lowwavelength limit ~o =
=0,
where Q = q v , 1 / y - - i / z + ~ R ( e ) + ~A(e'), S ~ = / x . [ g R ( e ) -- g A ( e ' ) ] , Sp = p . v [ g R ( e ) ~ r z -O-zgR(~')], SA-+ = ~ l ( g R ( ~ ) a n n + l - - a n n _ + l g 0 A ( 8 ' ) ) and a = tanh e / 2 T - t a n h e'/2T. Then we define the functions glv = g , . . - ,
f\ -
- - ( ~ : R ( 6 ) + ~ : A ( 8 ' ) ) g R ~ . . = aS~,
-- - - g g 6 g . . "y
where SA = S2 - $2. Analogous equations were obtained for the retarded (advanced) Green functions. We can easily get them substituting a in Eq. (A.6) for 1 and supplying all Green functions with the superscripts R (A). To solve the equations we perform a Fourier transformation with respect to the discrete variable n
8 -
~' << A ,
qv
<<
A,
1
e l ( e i k d - 1) << --. '7"
It is convenient to replace e = e+, e ' = e_ with e_+ = 8 + oJ/2. Then we can write the solution of Eq. (A.6) in the form ~K = O~i~'(2~btrzk'd + v 2 q p / 2 ) ( gR+ ~ -- ~ g A _ ) _ /z(l -- g+~ g_A) ~:+R+ ~:_A+ iT(qZ 02/2 + k2e~d z)
(A.7) [ ~ _ gR+ 6g K = - pv
A
E_
-~-°'zg- tanh [ -~+~R+~-A 2T
~r~- g~+~ g R_ i _ + ~+R+ ~A
T [ trz - gR+ trzgR_ I gR _ gA-- apv | ~ - J --iaqv'rtz ~_A " \ -; + ~+ + ~ ~+~+
(A.8)
S.N. Artemenko, A.G. Kobel'kov / Physica C 253 (1995) 373-382
382
R
A
R
A]
gli< = l'~b[ "= i - ---+--~R+-g+ --~_Atanh "~-2T °'z -- g+ ~+I~+ ~_A
--
+i o~4~
.
i' _ + ~+R + ~:_A
T
g + - - g -
-2i~lkd%~ ~ ~+ +
~-
'
(A.9)
where gR(A): g R ( A ) ( g + )
a n d ~R(A)= ~R(A)(~_+).
References
[1] N.N. Bogolyubov, V.V. Tolmachev and D.N. Shirkov, New Methods in Superconductivity Theory (Izd. Akad. Nauk, SSSR, Moscow, 1968). [2] P.W. Ande~on, Phys. Rev. 112 (1958) 1900. [3] R.V. Carlson and A.M. Goldman, Phys. Rev. Lett. 34 (1975) 11. [4] A. Schmid and G. Sch~Sn, Phys. Rev. Lett. 34 (1975) 941. [5] S.N. Artemenko and A.F. Volkov, Zh. Eksp. Teor. Fiz. 69 (1975) 1764 [Sov. Phys. JETP 42 (1975) 896].
[6] S.N. Artemenko and A.F. Volkov, Usp. Fiz. Nauk 128 (1979) 3. [7] J.E. Mooij and G. Sch~Sn, Phys. Rev. Lett. 55 (1985) 114. [8] O. Buisson, P. Xavier and J. Richard, Phys. Rev. Lett. 73 (1994) 3153. [9] W.E. Lawrence and S. Doniah, in: Proc. Conf. on Low Temperature Physics (LT-12), Kyoto (1970) p. 361. [10] L.N. Bulaevskii, Usp. Fiz. Nauk 116 (1975)449. [1l] R. Kleiner and P. Miiller, Phys. Rev. B 49 (1994) 1327. [12] K. Tamasaku, Y. Nakomura and S. Ushida, Phys. Rev. Let. 69 (1992) 1455. [13] T. Mishonov, Phys. Rev. Lett. B 44 (1991) 12033. [14] H.A. Fertig and S. Das Sarma, Phys. Rev. Lett. 65 (1990) 1482; H.A. Fertig and S. Das Sarma, Phys. Rev. B 44 (1991) 4480. [15] R. C6t6 and A. Griffin, Phys. Rev. B 48 (1993) 10404. [16] S.N. Artemenko and A.G. Kobel'kov, Pis'ma Zh. Eksp. Teor. Fiz. 58 (1993) 435. [17] M. Tachiki, T. Koyama and S. Takahashy, Phys. Rev. 50 (1994) 7065. [18] L.N. Bulaevskii, M. Zamora, D. Baeriswyl, H. Beck and J.R. Clem, Phys. Rev. B 50 (1994) 12831. [19] L.V. Keldysh, Zh. Eksp. Teor. Fiz. 47 (1964) 1515 [Sov. Phys. JETP 20 (1965) 1018]. [20] A.I. Larkin and Yu.N. Ovchinnikov, JETP 61 (1977) 299. [21] S.N. Artemenko, Zh. Eksp. Teor. Fiz. 79 (1980) 162.
N
ELSEVIER
Physica C 253 (1995) 373-382
Collective modes in layered superconductors S.N. Artemenko *, A.G. Kobel'kov Institute Jor Radioengineering and Electronics of Russian Academy of Sciences, Mokhovaya, 11, 103907, Moscow, Russian Federation
Received 6 July 1995
Abstract Weakly damped collective oscillations in layered superconductors are examined theoretically. The calculation is based on kinetic equations for Green functions generalized to the case of layered superconductors with weak interlayer coupling. We show that weakly damped plasma oscillations of superconducting electrons can exist in such superconductors and calculate the spectrum of the mode. The oscillations are reminiscent of Josephson plasma oscillations in a superconducting tunnel junction. As the temperature approaches the critical one, these oscillations transform into Carlson-Goldman modes. Possibilities for detecting the plasma mode experimentally are discussed. We found distinctive features in the reflection coefficient for an electromagnetic wave incident at an arbitrary angle on a plane parallel to the layers. The presence of the mode changes also the dispersion relation for the waves propagating along a thin film of a layered superconductor.
1. Introduction
Weakly damped oscillations in superconductors were studied since the late fifties [1,2]. It was shown in the pioneering papers by Bogolubov and Anderson that free oscillations of the phase of the superconducting order parameter would have an acoustic spectrum, provided that the electron charge is ignored. However, the perturbations of the electron density create Coulomb forces and, therefore, transform such oscillations into a plasma mode whose energy is normally much larger than the superconducting energy gap A. Thus, the resulting oscillations are nearly identical to usual plasma oscillations in a normal metal. In 1975 weakly damped collective oscillations with frequency below the energy gap were discovered experimentally by Carlson and Goldman [3].
* Corresponding author.
The theoretical explanation of the Carlson-Goldman mode was presented for both dirty [4] and clean [5] superconductors. This mode has no analogy in normal metals. In distinction to usual plasma oscillations the perturbations of the charge density in the Carlson-Goldman mode may by neglected, and the mode was shown to be associated with oscillations of a charge imbalance between electron-like and hole-like excitations (for a review see Ref. [6]). This mode exists only in a limited frequency range near the transition temperature T~ and has a linear spectrum. In this mode perturbations of the supercurrent (associated with oscillations of the phase) are compensated by a normal current (associated with oscillations of the longitudinal electric field). In addition to the bulk modes there are weakly damped plasma oscillations with frequency oJ < A in superconducting thin films, predicted in Ref. [7] and observed in Ref. [8]. Such a mode is possible, because the electric field created by charge oscillations is concentrated mainly outside the film.
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374
S.N. Artemenko, A.G. Kobel'kov / Physica C 253 (1995)373-382
Weakly damped modes may appear also in higly anisotropic layered superconductors (LS's) like highT~ ones or intercalated transition metal dichalcogenides. These materials can be considered as a stack of quasi-2D superconducting layers with Josephson interlayer coupling [9,10], which was confirmed by experimental observation of the Josephson effect [ 11] in layered superconductors. Therefore, plasma oscillations analogous to those in Josephson tunnel junction arrays may be present in LS. As interlayer coupling may be extremely small, one can expect that the plasma frequency of LS for the direction perpendicular to the layers is smaller than the gap A. The experimental evidence of plasma oscillations was obtained by Tamasaku et al. [12], who detected the plasma edge of the reflection coefficient at frequencies below the gap. Plasma oscillations were studied in several theoretical papers [13-18]. They were predicted by Mishonov [13] based on a phenomenological approach. Modifications of 2D plasma oscillations due to interlayer coupling in a stack of 2D superconductors were studied in Refs. [15] and [14] by means of a diagrammatic approach and numerical calculations, and the possibility of plasma oscillations below the superconducting gap was demonstrated. Josephson plasma oscillations in LS were considered by Tachiki et al. [17] using a phenomenological model, and an interpretation of the experimental data of Ref. [12] was given. Bulaevskii et al. [18] obtained the plasma oscillations using the Lawrence-Doniach model [9]. But still some questions remain to be clarified. The plasma oscillations are related to an electric field inside a superconductor, and the electric field creates a nonequilibrium electron distribution. Since the frequency of the oscillations may be small, the scattering processes must be considered in order to describe the response of the electrons to the electric field. Therefore, to take these effects into account, the spectrum of the collective modes must be calculated using the kinetic approach as it was done in the case of conventional superconductors [4,5]. We shall use kinetic equations for the Green functions in the Keldysh [19] representation. Such an approach allows one to take strictly into account the influence by the normal electrons on the spectrum of the mode and on its damping. In addition, this approach permits us to elucidate the relation between
the plasma mode and the Carlson-Goldman mode, which appears near the transition temperature. Some preliminary results of our study of collective modes in LS using kinetic theory were published in our letter [16]. In Section 2 we derive expressions for current and charge densities using kinetic equations for the Green functions in the Keldysh representation. In Section 3 we derive the dispersion relation for free oscillations and analyze the spectra of weakly damped modes, concentrating on the limits of low temperatures (T << A) and temperatures near T~ (T >> A). In Section 4 we consider some effects related to the plasma modes in thin films and possibilities of their experimental observation. We calculate the reflection coefficient for an electromagnetic wave incident at an arbitrary angle on a superconducting film in the frequency range where the modes are weakly damped, and then we consider the propagation of waves along a film. The cumbersome calculations with the Green functions are collected in the Appendix.
2. Basic equations To describe the behavior of the phase of the order parameter and of the electric field inside a superconductor we use the equations for nonequilibrium Green functions which have proven to be a powerful tool in studying the transport phenomena in conventional superconductors. These equations were derived in Ref. [20] and generalized to the case of LS in Ref. [21]. A Wannier site representation is used for the z direction perpendicular to the layers, while for directions parallel to the layers the continuous coordinate representation is used. The equations simplify strongly in the tight-binding limit, when only coupling between the neighboring layers is taken into account. The one-electron spectrum in the normal state near the Fermi energy ~v is modelled as -- eV + v ( I p l --PF) + 61 COS(pzd), where ~1 << sF is related to the tunneling rate, v and PF are the velocity and the momentum at the 2D Fermi surface, p and Pz are the electron's momenta in the plane of the layers and perpendicular to the layers, respectively, d is the lattice period in the z
S.N. Artemenko,A.G. Kobel'kov/Physica C 253 (1995)373-382 direction. In the following we use units such that h = k B = e = 1. Then the equation obtained in Ref. [21] has the form
ipVGnm-t-~1 E
i=_1
(Ann+iGn+im-Gnm+iam+im) 0 ^
q-io-z-~tGnm q-iVGnmo-z-PnGnm q-GnmPm 2r
........
;
(1)
where P,, = -iO'yA, + hn, h n = IX, + CrzVpn, tz, = l/2OXn/Ot+ ~,, IX, is the gauge-invariant scalar potential, qb is the electric potential, Xn is the order parameter phase, p, is the superconducting m o m e n tum parallel to the layers, v is the 2D Fermi velocity vector, and
Ar'm=C°S( Xn-Xm'
2
]] + i ° ' z
sin(Xn--Xm) "2
(Indices m, n refer to the layers). All these functions depend on the coordinates along the layers. The Green functions G,m are matrices
=
0
gA
'
(2)
where the retarded function g,m, R the advanced function g,m A and the function g~m K introduced by Keldysh [19] which is related to the distribution function of the electrons, are in turn matrices in layer and in spin indices. The Pauli matrices act on the spin indices of the Green functions. The r.h.s, of Eq. (1) is an elastic-collision integral. We neglect inelastic collisions considering frequencies larger than the inverse energy relaxation time. The bar over the Green's function denotes averaging over the directions of the momentum parallel to the layers. The current and charge densities are calculated from the solution of Eq. (1) for the off-diagonal component of the Green function (2) using the following expressions:
k2f2~
Jll = 8vrv Jo
dO Tr
O'zi,'gnn, K
375
with arguments t = t' in the Green functions. Here 0 is the polar angle and ko 1 is the Thomas-Fermi screening radius; subscripts II and 3_ denote the directions relative to the layers. To calculate the spectrum of the normal modes, we find the Green functions in linear approximation in Ps, /x and ~b, = X , , - 1 - X,. Then, using Eqs. (3-5), we derive the expressions for the current and charge densities and, combining them with Maxwell's equations, obtain the dispersion relations. The details of the solution of Eq. (1) can be found in the Appendix. Corrections of different symmetry to the equilibrium Green's function gnn'( K ~, S') are found in the Fourier representation, in which they are functions of the wave vector q along the layers and of the wave number k (J k[ < w / d ) related to the discrete Fourier transformation with respect to the layer index n. We use the limit of q < < A / v ~ ~(T) -1, ~(T) being the coherence length. Substituting Eqs. (A.7-A.9) in Eqs. (3-5), and performing the integration over the energy (the integration is equivalent to calculation of g at t = t' in the time representation) we may obtain the expressions for current and charge density valid for arbitrary temperature and frequency. However, in simple analytic form the integrals can be calculated in the limit of small frequencies o9 << A and in the cases of low (T<< A) or high (T>> A) temperatures. We shall concentrate on these limits. First we consider temperatures near the critical value To. The integrals are calculated using the smallness of the factor 6 = ~ A / 4 T << 1. In the limit o9 >> DIIq2 q_ D . k 2, in which low-damping modes exist we find
¢2 jll = 4--~1 p - io-iI[q/x + ogp(1 + J ) ] ,
j.-
(3) P--
¢2 4.rrA-----fpz-iO'.[klx+ogpz(l+J)],
(5)
(6)
k2 6Ix + i D t l q 2 + D ± k 2 IX 4"rr co
K j ± = f : ~ d0 Tr o-z(An+,ngK,+,-- A ,n+,gn+,n),
(4) P'=--~
1f2
/~'+-8Jo
)
d0 T r g . K ,
+ i( DiIqp + D . kpz) ] .
(7)
Here p is the superconducting momentum along the layers in the Fourier representation, ~'11 and A± are
S.N. Artemenko, A.G. Kobel'kov / Physica C 253 (1995) 373-382
376
magnetic field penetration depths for the magnetic fields perpendicular and parallel to the layers, respectively; Pz = qS/d corresponds to the component of the superconducting momentum across the layers. For All we have found the usual expression, and A± is related to the Josephson critical current Jc by
A2 = c2/(87rdjc). Conductivities o-Ii and o-± near T~ within the corrections of the order of 82 coincide with the conductivities in the normal state parallel and perpendicular to the layers, respectively. DII = v2~'/2 and D± = 2e2d2~- are diffusion coefficients along and across the layers, and (---Aln---~ ] 2T co'
A~->>I
~ln-~,
At<< 1.
J= [ A
Zl
(8)
The factor J is small, but it gives an important contribution to the damping of the collective modes. Eqs. (5) and (6) are similar to those obtained for isotropic superconductors (see Ref. [6]). These expressions for the current densities can be considered as a combination of superconducting currents (terms with a~,±), normal currents (the first two terms inside the square brackets) and interference currents (the last terms with factors J). This interpretation becomes more transparent when the normal current densities are written in the form of Ohm's law j = o-E, where the electric field is related to a superconducting moment p and scalar potential /x as 0p E = - - - V/x. 0t
(9)
The terms describing superconducting currents in Eqs. (5) and (6) originate from the regular (retarded and advanced) Green functions, while normal and interference currents originate from the anomalous function, which is related to the perturbation of the quasiparticle distribution function by an electric field. If quasiparticles were absent, the charge density would be equal to
which corresponds to the shift of the chemical potential of condensed electrons by/z. Near T~ the density of quasiparticles is high and modifies strongly the charge density, resulting in expression (7). At low temperatures T << A the density of quasiparticles is proportional to n ~ exp( - A/T), and, therefore, their contribution to the charge and current densities is exponentially small. In the main approximation the charge density is given by Eq. (10), and the current density is described by the first terms in Eqs. (5) and (6). The quasiparticle contribution, though small, cannot be neglected in the damping of the collective modes. The quasiparticle contribution to the current cannot be reduced to the Ohmic current, because the coefficients of the terms with V/, and Op/Ot are different. In the frame of the BCS model the larger contribution is given by the terms proportional to Op/Ot in the expressions for the current, the other contributions being small by factors T / A or oo/A. Thus at low temperatures the expressions (5) and (6) are substituted by = ( ill \
c2 4,rrA~I
J±=
-
4'rr
(lO)
)
iwo'± Pz"
47rA~
(11)
For o'11 and or± we obtained the following expressions: °'ll'±=~r~'±--(1-e-'°/ CO
e-a/rg
7'
r/ , (12)
where o-~ '± are the DC conductivities which would be in the normal state at the same temperature,
= [ ~ exp - --~x
1
l+,(,/l+x+
1+ o =
iw°'ll)P'
1,ff-;-;x - ,ff)2
and rl = 2wA~"2.
dx,
S.N. Artemenko, A.G. Kobel'kov/ Physica C 253 (1995) 373-382 3. S p e c t r u m o f t h e n o r m a l m o d e s
by A ± / ~ / I
Effects specific for a layered structure are expected, provided there is a nonzero component of electric field perpendicular to the layers. So we consider a propagating wave, the electric field of which has components both perpendicular and parallel to the layers, and the magnetic field is parallel to the layers. To calculate the spectrum of the collective oscillations we substitute the expressions for the current and charge densities obtained in the previous section into Maxwell's equations. We express the electric field in terms of/x,
-
(j/jc)
377
2 , and as a result we get for the
plasma frequency 0)p(j)= 0)p~/1 - ( j / j c ) 2 . Near T~, where A << T, quasiparticles play a major role and must be taken into account carefully. Using now the expressions (5-7) for the current and charge densities and the Maxwell equations, we obtain another system of linear algebraic equations. Zeros of its determinant yield the dispersion relation for the eigenmodes
(k2~l q- QII)(q 2~2 q-Qm)
( 5)(2 io t
= k 2 q 2 A~ - i
S
AL
0)6
]'
where
Ell = - - i ( q / x + 0)p),
0)2
E L = -i(k/x + 0)qb/d).
(lS)
Then we exclude /x, p=, and ~b, obtaining a system of three homogeneous linear equations for Ell and E L , which has nontrivial solutions provided the determinant of the system is equal to zero. Having expanded the determinant, we find the dispersion relation for the modes. In the limit of low temperatures, we obtain a single low-damping mode, which describes the plasma oscillations of the superconducting electrons:
092=
2
(l-{- =Lk2/kl)(1-{-k2/~]
0)P
+qZa2)
(1 _1_k2Al)
- 2iT0),
(14)
where 0)p = C / ( / ~ Z ~ ) is the plasma frequency, and the damping constant is given b y
QI = 1 - i w % - -U
2
="All +
i DII Sq 2,
0)2 Q . = 1 - i0)r, - -c-T ~ ± A2 + i
S k 2,
S = 1 - i0)%J,
with r I which is, according to Eq. (15), determined by the conductivities and penetration lengths near To. Eq. (16) has solutions corresponding to weakly damped modes. The frequency of dielectric relaxation 0)r is proportional to the quasiparticle density, while o)2 is proportional to the density of condensed electrons. At lower temperatures when 0),. << 0)p, the weakly damped oscillations correspond to the plasma mode
22
2 1 +qA± +
0)2
0)P
D'k2 ) (Dr ~
-i
0)0)r D±k 2
•
1 + - -
0)1"~
"Y= "2"0)27"1 l -]- k2a~ -{- 1 q- k2al -]- q2a2
"
(17)
Here e L is a background dielectric constant for the electric field perpendicular to the layers, and
,7-1 __
4 q-rAilo-ii
C2
0),./0)p2,
('Or
4-fro- l
6" L
(15)
The plasma frequency can be diminished if a direct current j
Being proportional to Aj_1, 0)p decreases rapidly with temperature increasing, while 0),. remains about the same value. At higher temperatures when the condition 0)p << 0)r starts to hold, the weakly damped solution of Eq. (16) transforms into the CarlsonGoldman mode 0)2 =
D . k2 -I- Dlq2 6"rl - 2iT0),
(18)
S.N. Artemenko, A.G. Kobel'kov /Physica C 253 (1995) 373-382
378
where the damping coefficient is given by
+ O,,q2) +
with q = f ~ a ( 0 ) / c ) s i n 0. Here l determines the decaying length
1l-1
As in the usual case of isotropic superconductors, the region of weak damping is bounded from below and from above and exists as long as J ~ A / T In A / w is small.
Al0)p2/
0)2(6a sin 0 2 - 1)
0)2--_COp2 •
Using Eqs. (11) we obtain the relation for the fields at the film surfaces 0)
ex = ;/4,
;=
(20) C
4. Plasma modes in thin films Now we consider some consequences of the presence of a low-frequency plasma mode of superconducting electrons. The presence of the plasma mode may be detected by the frequency dependence of the reflection coefficient at 0)>/0)p. A sharp plasma edge in the reflection coefficient has been observed in the experiment by Tamasaku et al. [12], where polarized light was incident on the sample surface, which was perpendicular to the layers. Using the phenomenological two-fluid model, Tachiki et al. [17] interpreted these data as the result of the excitation of collective modes with low damping. We shall calculate the reflection coefficient for the case when the surface of the LS film is parallel to the layers, while a polarized electromagnetic wave is incident at an angle 0. Interference of the lowdamping waves inside the film leads to an oscillating dependence of the reflection coefficient on frequency and on angle 0. To calculate the reflection coefficient for light incident at an angle 0 we must match the magnetic and electrical fields inside and outside of the superconductor at the surfaces of the film. We denote the dielectric constant of the surrounding medium by 6a. There are five different waves to be taken into account: incident, reflected, and transmitted waves outside the film, and two waves inside the film. All waves have the same wave vector component parallel to the layers. Two waves inside the superconductor differ only by the sign of the component k of the wave vector perpendicular to the layers. The value of k is determined from Eq. (14): k2A~ =
0)2(1 + q2A~) -- 0)2 0)2 2 -- 2ikA~l/l -- 0)p
(19)
Using expression (20) and boundary conditions, we obtain for the reflection coefficient 2~cos Rref =
1 + a 2 cot 2 kw '
0
a = e a cos 2 0 - ~ a (21)
One can see that due to the term cot 2 k0) in Eq. (21), in which k depends on 0) and 0 according to Eq. (19), the reflection coefficient oscillates strongly with frequency and angle at 0)> 0)0" If Im kw = w / l << 1 may be neglected, the minimum value of Rr~f= 0 is achieved at kw = wn (n is an integer), while its maximum value R r e f = 1 is achieved at kw = ~ ( n - ½). When the thickness w of the film becomes comparable to the damping length 1 the oscillating behavior disappears, but R still has a minimum near 0)p, provided 0 v~ 0. Next we consider the dispersion relation 0)(q) for waves propagating along a thin superconducting film with surfaces parallel to the layers, surrounded by isolating media with dielectric constants 61 and 62. To solve this problem we match two waves inside and two waves outside the film. For the two waves inside the film the relation between the components of the wave vector is determined by Eq. (14), and the component q along the layers is the same for all four waves. The electromagnetic field must vanish at large distances from the surface. Using Maxwell's equations outside the film, we obtain the relations for the z components of the wave vector of the waves outside the film defined as k = iKl and k = - i K 2, so that K1 > 0 and K2 > 0; we have 0) 2
K? = q2 __ 6 1 - 7 '
(.0 2
K2 = q2 __ 6 2 7 .
(22)
379
S.N. Artemenko, A.G. Kobel'kov / Physica C 253 (1995)373-382
Matching the electric and magnetic fields inside and outside the film and using relation (19) we obtain an uniform linear system of equations• Its determinant is zero, provided A1 + A 2
tan k w -
1 -AIA 2 '
(23)
From Eq. (19) one can see that the wave number k is imaginary at 03 < 03w Thus, the tangent in Eq. (23) transforms into the hyperbolic one, which results in a single low-damping mode at 0) < 03p. We consider first the case I k l w >> 1 satisfied provided either w>>Aii o r 03>>W03p 8~--~/(AIIV~ ). Then, from A = 1 we obtain
where
032 q2 =
KI,2C
A1,2
el ,2 03~ "
__
2 0)2 0)p--
8 C2 top2 _ 0)2(1 + I )
,
(27)
where
We consider the case ~1 = e2 = 8 ( A 1 = A 1 = A ) . The case c 1 v~ ~2 should be qualitatively the same. Then Eq. (23) can be rewritten in the form A = tan k w / 2 and A = - c o t k w / 2 .
0)2A 032
I= - I o° ± C2
03p2
<<1•
(24)
The opposite case I k l w << 1 requires both w <<
Some solutions can be found analytically if we assume A >> 1. This condition is easy to meet if the frequency is not too large due to the fact that c2//(0)pAH) 2 is large, e.g. about 10 5 for T1 based superconductors. Then we have
All a n d 0) << W0)p8V~-±//(Aii!/r~'g)• Then the expression
03: 03p /l + q2A /(1 + ( s,,)2) ,
(25)
where s = ~Aii/w and n is any integer• Note that the solutions with n = 0 is exact• All these solutions correspond to low-damping modes propagating along the film only if the electric field decays outside the film, i.e. when K given by Eq. (22) is real, that is if the frequencies of the modes satisfy the condition
03 < q c / v/~.
5. Conclusion
There is another low-damping mode. At kw << 1 (but k v~ 0) we obtain
092(
0)2A4 )
q2 = 8--~- 1 + 4~c--~wII2 .
(26)
The approximation kw << 1 is valid provided Aii/w >> ~ and 03<< c ~--~l /(All6). At large frequencies, expression (26) resembles the well-known [7] square-root law for 2D plasma oscillations
~ wq 0) =
for this mode coincides with Eq. (26). Thus we found that the single mode below 0)p splits into a number of modes above 03p. This fact arises from the interference between the waves reflected from both surfaces of the film• Note that at frequencies above 0)p the signs of the film's effective dielectric constants along and across the layers are opposite. That is, at 0) > 03p the LS film has the properties of a superconductor and of an isolator in the direction along and across the layers, respectively.
c
2 e All
When the frequency exceeds c ~ / A i l ~ the mode deviates from the law (26) and approaches the solution (25) with n = 1.
We applied a nonequilibrium Green function technique to find the dispersion relation for low-damping collective oscillations in a LS with weak interlayer coupling• This approach allows one to consider the contribution of quasiparticles strictly, to calculate the damping of the collective modes more accurately and to show the relation between the collective modes at low temperatures and at temperatures near Tc. The calculations were performed within the BCS theory, and all the results of this article are directly applicable to LS with BCS pairing mechanism• But the equations obtained in Section 2 have a clear physical meaning and may be considered as phenomenological irrespective to the pairing mechanism. So we believe that the main results (excluding,
380
S.N. Artemenko, A.G. Kobel'kov /Physica C 253 (1995) 373-382
maybe, the magnitude of the damping) are qualitatively correct for non-BCS superconductors too. An extremely large anisotropy of LS leads to a low plasma frequency COp for an electric field perpendicular to the layers. In some materials this plasma frequency drops far below the superconducting gap A and the low-damping plasma mode of the superconducting electrons exists. For Bi and T1 based high-Tc superconductors the plasma frequency is estimated as COp~ 10 THz. We have shown that the low-damping CarlsonGoldman mode, which exists in LS near T~, continuously transforms into plasma oscillations at lower temperatures. We considered some effects arising from the interference of low-damping plasma oscillations inside the film. The reflection coefficient from the film has an oscillating dependence on the angle and frequency just above plasma frequency COp.In the frequency range below Wp the 2D Plasma mode has the usual square root behavior at low frequencies and saturates when frequency approaches to cop. Many modes propagating along a thin film may appear above the plasma frequency.
Acknowledgements We are grateful to F. Gleisberg for reading the manuscript and useful comments. This work is supported in part by INTAS grant No. A92-056.
A) __~ ( 8 0 . z _}_ i A o . v ) / ~ R ( A ) ,
~R(A)(8 ) =
+
s i g n ( 8 ) O ( 8 - A) + i f - A T - ~ - ~ O ( A ] 8 I) "---- ___((8 -t- i 0 ) 2 -- A 2 ) 1/2, where __.i0 reminds us of the direction of encircling the singular points 8 = + A while moving along the real axis. We consider p, /x and ~b as perturbations and calculate the deviations of the Green functions from their equilibrium values. Eq. (1) for 4 × 4 matrices corresponds to three equations for 2 × 2 matrices, namely for gn,n' R gn,, A and g,m K from Eq. (A.1). Here we seek g~m K in the form 8r
8
g~, = g,m R t a n h - - - grim a t a n h - - + g nam '
2T
2T
(A.2)
where the anomalous Green function g a is introduced. We present the perturbative part of both anomalous and regular Green functions in the form (A.3)
i __ --i -t- ~gn,n i cos 0, gnm -- gnm
where i = a, R, or A, the bar over the Green function denotes averaging over the angle 0 between the directions of v and q. To simplify we shall omit the index a of the anomalous Green function. Then we substitute Eq. (A.2) into the equation for g,,,, K from g,,,, multiplied by Eq. (1), subtract the equation for R tanh 8 ' / 2 T , add the equation for gA multiplied by tanh 8 / 2 T , linearize the equation keeping terms up to first order in the perturbation and perform the 2D Fourier transformation with respect to the in-plane coordinates, introducing the wave vector q. So we obtain an equation for gnm: 81(gn-lm
+ gn+lm-
gnm+l --grim-l)
--(8¢r z -I- i Acry ) gnm + gnrn( g~O'z"t- i Ao'y) i +qv c o s Ogn,.- ~-~r(ggg,,m- go"--gram 6n,n
Appendix We present here the detailed solution of Eq. (1). Our goal is to find the expressions for the current and for the charge density in linear approximation. The unperturbed equilibrium solution of Eq. (1) (with h, = 0, th, = 0) has the form
-A -I- g nn go ~nm -S--CO = tanh 2T
g,mgoA) tanh 8 / ( 8 1 ( A n n
2T } ~ +A,~n+lgA6n+X,,) + hngA~nm) + tanh
2T
lgA~n-l,n
tanh
8(8-- /8')~nm × (el(g0RA,~+i,~ 6.m+1 (A.1)
+goRAm- l,, t~,,,,,- 1) + gRhm6nm ).
(A.4)
S.N. Artemenko, A.G. Kobel'kov /Physica C 253 (1995) 373-382
Besides, we use the orthogonality relations [20]
Q 1 __gin ) _ l(gn+ln+l
2 +
--
'~k ,Snk 6km = R K K sumk(gnkgkm+gnkgAm)=O,
381
~ g oR-g n . = oLStx , 1
R
Qgnn + e l ( 3 g n + l n + l -- 6 g ~ . ) -- - - g o 6 g . .
Y
which in the linearized form yields
= aSp, (A.6)
(co- z + i A % , ) g .... = ~ R g R g . m , gnm( e'o-z + i A%,) = ~A gn,ng A • NOW we separate the equations for the functions which are symmetrical and antisymmetrical with respect to cos 0. Integrating Eq. (A.4) first over cos 0, and then integrating it after multiplication by cos 0, we find Q --~6g,,n + e l ( g . - 1 . -- g . . - I + g n + l n -- g n n + l )
g~n =
Og, nn + Sl( S g n - l n -- 8gnn-1 + 6gn+ln -- 8gnn+l) 1
= aSp,
Q
1
= C ~ S A -+ , 1
egnn+ 1 + el(tSg.+l.+, - 6g..)
-- ~ g R 6 g . . + l
-gn-ln,
6g,l,n = t3gnn-1 -- ~ g n - l n '
(A.5)
the first of which determines the current density across the layers. The functions (A.5) obey the equations Q ~ 1 --f6g.. - _~gRgl
g ( k ) exp(ikdn)
+ 2 e l ( ~ n - ln-I -- gnn) = OtSA, 1
Qgl n + 2 e , ( 6 g n_ ~n-, - t~gnn) -- --gRtSgln = O, 3'
2~r
.
It results in a linear system of equations, which can be easily solved. Substituting its solution and solutions of the analogous equations for the retarded and advanced Green functions into Eq. (A.1) we get expressions for the deviation of the Keldysh function gK from its equilibrium value. We denote these functions by ~K, 6gK and glK. Being integrated over energy these functions determine the linear response of layered superconductors. In order to obtain simple equations which may be integrated analytically we assume the low-frequency and lowwavelength limit ~o =
=0,
where Q = q v , 1 / y - - i / z + ~ R ( e ) + ~A(e'), S ~ = / x . [ g R ( e ) -- g A ( e ' ) ] , Sp = p . v [ g R ( e ) ~ r z -O-zgR(~')], SA-+ = ~ l ( g R ( ~ ) a n n + l - - a n n _ + l g 0 A ( 8 ' ) ) and a = tanh e / 2 T - t a n h e'/2T. Then we define the functions glv = g , . . - ,
f\ -
- - ( ~ : R ( 6 ) + ~ : A ( 8 ' ) ) g R ~ . . = aS~,
-- - - g g 6 g . . "y
where SA = S2 - $2. Analogous equations were obtained for the retarded (advanced) Green functions. We can easily get them substituting a in Eq. (A.6) for 1 and supplying all Green functions with the superscripts R (A). To solve the equations we perform a Fourier transformation with respect to the discrete variable n
8 -
~' << A ,
qv
<<
A,
1
e l ( e i k d - 1) << --. '7"
It is convenient to replace e = e+, e ' = e_ with e_+ = 8 + oJ/2. Then we can write the solution of Eq. (A.6) in the form ~K = O~i~'(2~btrzk'd + v 2 q p / 2 ) ( gR+ ~ -- ~ g A _ ) _ /z(l -- g+~ g_A) ~:+R+ ~:_A+ iT(qZ 02/2 + k2e~d z)
(A.7) [ ~ _ gR+ 6g K = - pv
A
E_
-~-°'zg- tanh [ -~+~R+~-A 2T
~r~- g~+~ g R_ i _ + ~+R+ ~A
T [ trz - gR+ trzgR_ I gR _ gA-- apv | ~ - J --iaqv'rtz ~_A " \ -; + ~+ + ~ ~+~+
(A.8)
S.N. Artemenko, A.G. Kobel'kov / Physica C 253 (1995) 373-382
382
R
A
R
A]
gli< = l'~b[ "= i - ---+--~R+-g+ --~_Atanh "~-2T °'z -- g+ ~+I~+ ~_A
--
+i o~4~
.
i' _ + ~+R + ~:_A
T
g + - - g -
-2i~lkd%~ ~ ~+ +
~-
'
(A.9)
where gR(A): g R ( A ) ( g + )
a n d ~R(A)= ~R(A)(~_+).
References
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