Re-entrant superconductivity due to Landau level quantization?

Physica C 170 (1990) 195-210 North-Holland

Re-entrant superconductivity due to Landau level quantization? C.T. Rieck and K. Scharnberg Abteilung f~r Theoretische Festkrrperphysik, Fachbereich Physik der Universitllt Hamburg, Jungiusstrasse 11, D-2000 Hamburg 36, FRG

R.A. K l e m m Solid State Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA Received 27 February 1990 Revised manuscript received 13 July 1990

Conditions are examined under which superconductivity in type-II superconductors can persist in magnetic fields far above the upper critical field calculated from the semiclassical approximation. While orbital diamagnetism will not destroy superconductivity when only a few Landau levels are occupied, magnetic moments of the charge carriers together with scattering by impurities will strongly suppress this phase transition, rendering its observation in real singlet superconductors highly unlikely.

1. Introduction

Through a recent letter [ 1 ] the interest in the effects of Landau level quantization on the properties of typeII superconductors [2 ] has been resuscitated. The authors of this letter state that a) the transition temperature To2 is a nonmonotonic function of the external magnetic field H, b) To2(H) remains finite in arbitralily strong field H, c) for fields so high that only one Landau level is occupied (quantum limit), To2 is an increasing function of the field, and, most importantly, d) statements a), b) and c) remain valid in the presence of Pauli paramagnetism and disorder, provided the field is high enough so that only very few Landau levels are occupied. It is the purpose of the present communication to elucidate the extent to which these statements are correct and to which extent they are new. The latter question is of interest because Te~anovi6 et al. [ 1 ] base their calculation on the same model of a weak coupling, s-wave superconductor employed by all authors working in this field. In the course of this discussion we shall present a number of detailed analytical and numerical results which have not been published before. We find that orbital diamagnetism in sufficiently high fields indeed does not suppress superconductivity and we thus agree with statements a), b) and c). Statement d), however, is essentially incorrect. In the quantum limit finite magnetic moments of the charge carriers together with a finite lifetime will suppress the superconducting transition. The sharp peaks in Tc2(H) predicted for fields below the quantum limit are suppressed either by deviations of the g-factor from even integers or by the presence of scattering processes. The deleterious effects of Pauli paramagnetism would not be present if the pairing were of the triplet type with equal spin pairing, and it has been suggested before that, if such a superconducting state were to exist in thin films or layered compounds, arbitrarily large parallel magnetic fields would not destroy it [ 3 ]. In this case, orbital diamagnetism is absent because the parallel geometry strongly impedes the flow of screening currents. Similarly, Burlachkov et al. [4] have argued that the highly anisotropic dispersion ~(k) in organic supercon0921-4534/90/$03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)

C.T. Rieck et al. / Re-entrant superconductivitydue to Landau levelquantization?

196

ductors will lead to vanishing orbital diamagnetism for certain field directions and thus allow the formation of a stable superconducting state in arbitrarily high fields. It is interesting to note that Hc2 (T) obtained in this model is qualitatively similar to fig. 1 of Te~anovi6 et al. [ 1 ]. Both groups [3,4 ] would interpret observation of such high field superconductivity as evidence for triplet pairing. 2. Weak coupling theory of the upper critical field Assuming weak coupling and isotropic pairing, Tc2(H) is obtained in the clean limit from the well-known linearized selfconsistency equation [5,6,2,1 ] A ( r ) = V T ~ ~ d3r ' Gff(r, r'; mn)GH_,(r, r'; --mn)3(r')

(1)

ton

for the order parameter 3(r). The BCS pairing interaction Vis assumed to be field- and spin-independent. The Green's function Gff (r, r'; ~o~) in the presence of the uniform magnetic field H depends on both position variables separately. However, using the phase factor i.e. f;, A.ds, familiar from the semiclassical approximation, one can write [2,7] r

Gff ( r, r' ; ran) = e x p ( i e ] A.ds )Gff ( I r - r ' l; ran) •

(2)

r'

Without any approximation the kernel in eq. ( 1 ) can, therefore, be written in terms of the semiclassical phase factor and a function which depends only on [r - r' I. It has been pointed out by Gruenberg and Gunther [ 2 ] that the exact form in which the kernel depends on Ir - r' I is not relevant for the determination of the eigenfunctions A(r). This can easily be seen from the work of Helfand and Werthamer [ 5,6 ]. We have, therefore, the important result that the ground-state wave function for a particle with charge 2e moving in a constant magnetic field remains a solution of eq. ( 1 ) even in the presence of Landau level quantization. This wave function is not uniquely determined, which is the reason why various Abrikosov vortex lattices can be constructed. In fact, the set of functions representing the ground state can be written in terms of a continuous momentum variable py (eq. (23) of ref. [ 8 ] ). Multiplying with an arbitrary bounded function of py and integrating with respect to py we obtain (in the Landau gauge) e x p ( - e H x 2 ) f ( x - i y ) . According to this construction, f i s an analytic function. This property of f also has to be assumed when one wants to show directly that eq. ( 1 ) can be solved with this general ansatz for the order parameter. Furthermore, it ensures that the order parameter does not change when the coordinate system is rotated around the z-axis by 2~. The set of order parameters suggested by Te~anovi6 et al. [ 1 ] as solution of eq. ( 1 ), therefore, does not differ in any way from that used by everybody else. It is not surprising that the eigenvalue employed by Te~anovi6 et al. is exactly the one first obtained by Rajagopal et al. [ 9 ]. If each Cooper pair carries a moment q in the z-direction, i.e. if the order parameter is of the form zJ(r) =eiqzA(x, y ) ,

(3)

then the eigenvalue is given by

E(H, T) = V rnc°c T ~ 4n

~

~

(v+v'),

o,, ~=o ~,=o 2"+~'v!v'!

fdpz

1

2n ira, + (k~+ - ( p z + q / 2 ) z ) / 2 m

1 - i r a , + (k 2,_ - ( p z - q / 2 ) 2 ) / 2 m

'

(4)

C.T. Rieck et al. / Re-entrantsuperconductivitydue to Landau levelquantization?

197

with 2----m-=Ev(coc, y) - ( v + ½+ Y)coc = ~ + Ycoc,

(5)

coc= eH/m is the cyclotron frequency and yco¢is the Zeeman energy. Ordinarily, 7~ 1/2. k~o is the wavevector of an electron at the Fermi energy EF (coo) when it occupies the Landau level (va). The field-dependent Fermi energy at zero temperature is determined from 3 co~

~ /EF(CO~,y)

1= ~ - v Re ~ ~=ot~/

-E~

CO~

/Ev(co~, y)

(£+½+y)-- + ~F ~ /

-~F

(~+½-y)

y~} ~

'

(6)

with ~F=k~/2m= (3n:n)2/3/2m the Fermi energy in zero field. Below, parameters are chosen such that ~v/ Too= 1000 where Too is the superconducting transition temperature in the absence of the magnetic field. Since the transition temperatures obtained in the presence of the magnetic field in most cases do not exceed T~o, it suffices to consider the low temperature limit of the Fermi energy. We have chosen such a large ratio ~F/T~o in order to ensure the applicability of the weak coupling approximation, which requires Tc << COD<< ~F" COt)is the usual cut-off energy for the pairing interaction. The requirement COo<< ~F can actually be relaxed in the present study. The regime of magnetic fields of greatest interest here is the one for which, at zero temperature, the lowest landau level is occupied with electrons of both spin orientations, i.e. (½ +7)co¢
+4

1 4 ( ~ ) ~F+ ~ 6 Ev(COc, y)=~COc+~

7 2(e-~)

{~F

(7)

for 8

1

9 1 - 7 + ~

<(CO~+3< 8

key/

97"

The expression for the eigenvalue can be reduced further by either performing the sum over Matsubara frequencies COn [ 2 ] or by integrating with respect to Pz- Te~anovi6 et al. seem to have followed neither of these routes and instead replaced the p=-integral by an energy integral taking the density of states as slowly varying function, so that it can be replaced by its value at the Fermi surface. While this is generally a good approximation it leads to incorrect conclusions in the present case since the density of states is a rapidly oscillating function which diverges at the Landau level. Gruenberg and Gunther [2 ] decided to do the sum over Matsubara frequencies first. Note that, if the limits of integration in the remaining pz-integral are taken to be symmetric around p~= 0, as the authors seem to imply [ 10 ], then their expression for the eigenvalue is too large by a factor of two. To ensure convergence it is necessary to introduce a cut-off. If the weak coupling approximation is applicable, then the cut-off can be introduced either in the frequency sum or in the m o m e n t u m integral without affecting the results. In the present case, this is true only in the absence of Pauli paramagnetism. If the Landau levels are spin-split, the integrand of the p~-integral in eq. ( 2.6 ) of ref. [ 2 ] becomes negligible at low temperatures in the interval (EF ( CO¢,y) -- 7CO~, EF(coc, 7 ) + Yco~). Hence, there is no dominant contribution from the vicinity of the Fermi energy when only very few Landau levels are occupied. In this case there is some ambiguity as to which range of momenta to include in the integral. These problems do not arise if the m o m e n t u m integral is done exactly and a cut-off COD is introduced in the frequency sum. Following this path which, in view of the strong coupling theory of superconductivity, also seems to be the more physical one, we can write the condition E(H; T) = 1, from which T~2 is determined in the form

c.T. Rieck et al. / Re-entrant superconductivitydue to Landau levelquantization?

198 I

n

E/-E~-vCO~ o%

~

~o [i'~ "+"' ( v + v ' ) !

VN(O) - ~ k / Tco ev ,o~>ov~=ov~__o~)

v ! v , ~ F~.,(CO~,~,CO.),

(8a)

where, beside the normal state density of states N ( 0 ) = mkv/2n z, we have introduced the abbreviation F~, (COc, Y, co.)

(8b)

T iT3d 2 Re 2 T~o x / i c o . + k . + / 2 m x / - i c o . + k ~ , _ / Z m

-

x/ico. +kZ~+/2m-x/-iCO. +k~._/2m (x/ico.+k2+/Zm-x/-iCO.+kZ,_/Zm)Z-q2/Zm

"

Unlike the corresponding equation obtained in the semiclassical approximation, eq. (8) bears no resemblance to the T : e q u a t i o n in the absence o f the magnetic field. It is, therefore, unfortunately not possible to eliminate both parameters F N ( 0 ) and CODin favour o f the zero-field transition temperature Tco. Using the BCS result

1/ VN ( O) = ~ l + 2~-~coco) - ~( ½) ~ ln l.14COD/ Tco , which follows from eq. (8) for H--,0 and COD>> T~o, we keep Tco and CODas parameters. Note that, as none of the usual approximations are employed in the evaluation o f the integral over quasiparticle energies, there is no need to require COD<< eF. AS we shall see below, Tcz(H ) resulting from eq. (8a) varies rapidly over several orders of magnitude. If T~2(H) is very small, the sum over Matsubara frequencies contains so m a n y terms that we have to resort to an approximate evaluation of the sum. Fortunately, application o f the Euler-MacLaurin formula leads to integrals that can be done analytically: '°D

1

~

1

~, F,,,,,(coc, 7, CO,,)~- ~ F,,,,,(COc, Y, COD)+ 8nff""'(CO~'Y'nT) '

ton>0

with P ~ , (co¢, 7, x) =

T~o

[2'w'wr-]l/2+Cr[(qZ/4m)wr-]'/2

_~+ {arctanh

N[ ~,'," =-

[x,/~ ' + aq/x/8m ]2 + x / ( ~ - ~ ' C O c ) 2 + x 2 [2~.,,,,'wr'+]l/2+cr[(q2/4m)w~'+] 1/2 ~ + a r c t a n h [~fe,,. + aq /x/ 8m ] 2 + x/ ( e~. + 7CO~)2+ x 2)

for

~v.,=½(~v+Ev,)--q2/8m>O

P~.,(COc,~,,x)=

~

Tf~-e°

4 ~

and

{ [--2~vv'wr-]l/2+a[(q2/4m)wi-]l/2 Y~ k n + a r c t a n ,,= +_ E,,,,, - q 2 / 8 m + x/ ( ~,-ycoc)2 + x 2 r ]l/2+a[(q2/4m) +arctan [-2~'w~'+

~.,--qZ/8m+x/(e~,

W,,+] i 1/2}

"3L70)c) 2"~X 2

for ~,~, < 0. We have introduced the abbreviations w~+ r = x/( e~ _+7coc)2 + x 2 +~-+~coc, i = w~_+

% / ( ~ v "+"~COC) 2 "~-x 2

- (~, + ~,co~).

Multiples o f n have to be added to ensure that the integral ff~, (co¢, 7, x) is a continuous function o f x . When the argument ot+_.o(x) of one o f the arctan diverges for any real X=Xo~ (nT, COD), then n has to be subtracted (added) if arctan ot ±,o ( x + fi) > 0 and arctan ot_+,~( x - d) < 0 (arctan ot +_,o( x + d) < 0 and arctan a_+ ,~( x - d ) > 0 ).

C.T. Rieck et aL / Re-entrant superconductivity due to Landau level quantization?

199

Here, ~ is a small positive number. For E,=~,, = 0 the contribution from a = - vanishes.

3. Orbital diamagnetism Peculiar features of the eigenvalue E(H, T) (eq. ( 4 ) ) , which arise when the sum is dominated by a single term, i.e. when only very few Landau levels are occupied, have already been noted by Gruenberg and Gunther [2]. It is obvious from their eqs. (2.7) and (2.8) that To2 should be large whenever a Landau level crosses the Fermi energy, so that Tc2 is nonmonotonic. The authors state "that at sufficiently low temperatures there is a stable superconducting state, no matter how strong the field". If only one Landau level is occupied then it follows immediately from eq. (2.7) of ref. [2] that Tc2 increases monotonically, but the authors did not specifically state this result. They do, however, indicate a physically appealing explanation for why at high fields orbital diamagnetism does not cost energy and thus will not destroy superconductivity: The lattice constant of the Abrikosov vortex lattice is approximately equal to the orbit radius of the lowest Landau level. If only one Landau level is occupied, the currents required to shield the flux lines can be made to coincide with the orbital motion of the electrons in this Landau level if the degeneracy of this eigenstate with respect to the choice of origin is suitably exploited. Note that this argument heavily relies on the presence of a conventional vortex lattice so that contrary to Te~anovi6 et al. we see no reason to invoke a new superconducting state. The supercurrent can be calculated from the usual expression [ 11 ] but in very high fields this does not reduce to the well-known Ginzburg-Landau equation [ 11 ]. Instead, one obtains a nonlocal relationship between the screening currents and the order parameter because the range of the Green's functions is comparable to the distance between vortices, this will introduce some quantitative changes but does not represent a qualitatively new behaviour. In particular, this calculations shows immediately that there are no supercurrents parallel to the large externally applied magnetic field as proposed by Te~anovi6 et al.. The electromagnetic response of this vortex state has not been investigated, because this requires first of all an explicit form of the order parameter. It probably differs from the response of the vortex lattice in low fields for the same reasons that the normal state conductivity in high magnetic fields differs from that in zero field. Gruenberg and Gunther did not pursue the anomalous behaviour of To2 in the high-field regime any further because they had qualms about the applicability of the BCS pairing mechanism at such high fields. They proceed to study the regime of experimentally measured upper critical fields, where the number of occupied Landau levels is very large, in order to answer the important question of whether Landau level quantization can affect the theoretical interpretation of H¢2 (T) data. Their answer to this question is essentially negative. With the advent of new superconducting materials with low electron densities and high transition temperatures there is some justification in re-examining the high-field case in order to provide more detailed insight than has been obtained previously. In the absence of an analytical solution of eq. (8) we have to rely on numerical calculations. To better understand the numerical results, it is, however, very useful to approximate eq. (8) to such an extent that analytical calculations become feasible. We first ignore the magnetic moment of the electron and concentrate on the diagonal terms in the double sum over Landau levels. With y = 0 and q = 0 , these reduce to T 3/2

l

F ~ ( m ¢ ' Y=0' °9")-N/-/--~r v / ~ 2 + , ~ 2 I m ~

"

(9)

In general, these expressions cannot be summed analytically with respect to Matsubara frequencies. If, however, the ~oth Landau level crosses the Fermi energy, then ~o =EF(09¢)--(£o+ ½)tOe=0 and the frequency sum actually converges so that no cut-off is required:

200

C.T. Rieck et al. / Re-entrant superconductivity due to Landau level quantization?

#

~,>o

~,>o

x/2T

(1-2-3/2)((3/2)

.

(10)

Frequency sums of this kind cannot be evaluated accurately using the Euler-MacLaurin formula because of the divergence for T~O. In this particular case, the approximate result would contain a factor of 1 instead of ( 1 - 2 -3/2) ((3.2) = 1.69. If the Euler-MacLaurin formula is generalized to include the next term in the asymptotic expansion, in particular, the integrand at the lower limit of integration, then the factor turns out to be 1.50. In the regime of overlap, numerical results obtained with the use of this generalized Euler-MacLaurin formula cannot be distinguished from the results obtained by direct evaluation of the sums in the graphical display of figs. 1-5. The term ( l 0) certainly constitutes the largest contribution to the sum over Landau levels and if it actually dominates the sum, we can obtain an analytical expression for the transition temperature. Because of the condition E~o= 0 the magnetic field is fixed at eF

~ = o ~o~//£~-~-]

(11)

which one might expect to be that field at which T¢2 peaks. The transition temperature at this field is then found to be

(£o)=0.06607'F[22~,i (2£o)!

Tc2

2 to-1 ] [~o ~ J

1.14C0D]-2 L l n ~ j

-]-4,3[-

£°>--i;

(12)

which is a rapidly decreasing function of £o. Note that T~2ocCF SO that only low transition temperatures can be expected for low carrier density systems. In fig. 1, where the numerical results for 7 = 0 are displayed, these approximate values o f T~2 are indicated by open circles. As one can see, the maxima in Tca ( H ) obtained from numerical calculations are higher than those given by eq. (12) and they occur at fields slightly below those defined through eq. (11 ). This shows that taking the contribution from all Landau levels into account can change the results by several percent. At least as far as the numerical calculations have been extended, T ~ X / T~o is found to vary as (o&/eF)3. T~2(£o) (eq. ( 12 ) ) follows this power law over many more orders o f magnitude.

uE ~F

N N D O 3 O

10-1' 10-12

10-10

10-8

10.6 TE2/TE0

10-#

10-2

1

Fig. 1. Double logarithmic plot of the upper critical field, expressed as cyclotron frequency oJ¢= 2/IBHand normalized to the zero-field Fermi energyeF, VS.reduced temperature in the absence of spin paramagnetism. Parameters are chosen as ev/T~= 1000 and o~/T¢o= 100. Open circles indicate the results of the approximate analytic expression ( 12 ). Open triangles are obtained from eq. ( 14a, c).

C.T. Rieck et al. / Re-entrantsuperconductivitydue to Landau levelquantization?

201

If none of the Landau levels is close to crossing the Fermi energy and if ~F>> O)O one can expand the square roots in F.v(09c, 7=0, to.) using E~>> 09. for all occupied Landau levels U<£o. One thus finds oJo 1 T~c° ~ ~°D/2nT 1 1 ~ o F " ~ ( t ° c ' 7 = O ' ° ~ " ) = ~ n N I ~ .~=o n + l / 2 - 2 n ' ~

T ~ ° l n l'14t°D e,

(13)

Approximating the .eigenvalue E(H, T) by the sum over diagonal terms, one obtains the following expression for the transition temperature

/1.14 D/

~(1

-- 1 / G ( t o c ) )

Tc0 = L ~ - ~ - ~ o J

'

(14a)

with 4 N/ ev ~=0 x/EF((Oc)

-

(£+ 1/2)oA 22~(£! )2"

(14b)

The divergences of eqs. (10) and ( 13 ) of F ~ for small temperatures have been noted before by Gruenberg and Gunther [ 2], which led them to the conclusion that a finite solution T~2 must exist no matter how large the magnetic field H, but they did not give explicit expressions for T~2. The result (14) has been published by Te~anovi6 et al. (eq. (5) of ref. [ 1 ] ) without any restriction on the value of e, = EF ( O~) -- ( U+ 1/ 2 )tOc. For e~o~ 0, eq. ( 14 ) yields Tc2(£o) = 1.14tnD. This unphysical result is simply due to the fact that the 1D density of states for the £th Landau level m m Nl~(0)-- 2nk~ - 2 n ~ diverges when this Landau level crosses the Fermi energy. Equation ( 14a, b) can be usefully employed to estimate the relative minima of T~2(H) which occur when the £oth Landau level has just been depopulated, i.e. for fields given b), eq. ( 11 ) but with the divergent £oth term excluded from the sum in eq. (14b). T~2~"(£o) is thus still given by'eq. (14a), but with G(to~) replaced by

1~°~ 1 1 (2£)! [3~°~ 1 (7(£o)= ~ ~=o ,,//£-o-£ 22~(£!)2 L2 ~=o ~

1/3

1-

(14c)

The results for T ~ 'n (£o) are indicated in fig. 1 as open triangles. In the quantum limit, where only one Landau level is (doubly) occupied, we have from eqs. (14) and (6) T3L

[1.14~OD] {'- (8/3)~/~°2}

T~o - L--T~-~o J

"

(15)

However, long before T ~ L comes close to its limiting value 1.14rOD the weak coupling approximation breaks down. This breakdown shows up in the numerical calculations as discontinuous jumps in o~c( T ~ L ) which occur when the small number n~=toD/2nT~ L of terms in the frequency sum changes by one (see fig. 5).

4. Pauli spin paramagnetism Even if orbital diamagnetism does not destroy superconductivity in very high fields, one would expect that Pauli paramagnetism will. However, Gruenberg and Gunther [ 2 ] noted that the divergences of eqs. (10) and (13) persist if 7= 1/2. In this case, the dominant contribution to the eigenvalue (4) is

202

C.T. Rieck et al. / Re-entrant superconductivity due to Landau level quantization?

F~_,,~(rn¢, y = ½, o9,) =

rf~7

T3/2

1

/ "--~° N/ T x/oo 2 + (EF(Ogc) -- V09c)2 2 Imx/ito . +EF(tO¢) -- VOOc

(16)

For fields such that Er(co¢)=£otO~ for some integer £o, i.e. for fields given by O9c( £ o ) [ 4

]2/3

l

The eigenvalue again diverges as T - 1 / 2 so that a solution to eq. (8) will exist. It differs only slightly from eq. (12):

Tc2(£o, 7= ½) =0.06607EF

t (2£o)!/222O(£o!)2 ] 2 tr*z2°-I r / 2=0 £ 2~V o - £ - 1 + x / £ o - £ } ] 4 / 3 [ l n 1.14too/Too] 2'

%>_1

"

(18)

Again, this is an approximation to the m a x i m u m Tc2(H), indicates as open circles in fig. 2. An estimate for the m i n i m u m T~2(H), attained at fields marginally above those determined from EF(O&) =£o09~, can be obtained as before by expanding the square roots in eq. (16). The result is of the form of eq. ( 14a) with

1 .~1

1

(2£)! [3 2~;71~ _

3

]-1/3

This gives T ~ " (£o, Y= 1/2) several orders of magnitude lower than T~2 mi" (£o, 7 = 0 ) in agreement with the numerical results presented in fig. 2. In the q u a n t u m limit no result corresponding to eq. ( 15 ) is found because with 7= 1/2 the spins of all electrons populating the Landau level £o = 0 are aligned, so that no spin singlet Cooper pairs can be formed. If the pairing interaction is highly anisotropic so that triplet pairs would have a sufficiently large condensation energy, it is conceivable that in this q u a n t u m limit triplet superconductivity in an equal spin pairing state can persist. The orbital structure of such a triplet state would have to be of the polar type [ 3 ] because of the reduced availability of m o m e n t u m states. If 7 deviates from those values n / 2 , n integer, at which Landau levels for spin t and spin ~ electrons having

to tF

o o o

o o° o° o

lo12

1 1o

#4

lo2

1

Tc2/tco Fig. 2. Same as fig. 1 for charge carriers with magnetic moments g#a, if g= 4y= 2. Open circles indicate the results of the approximate analytic expression ( 18 ).

C.T. Rieck et al. / Re-entrant superconductivity due to Landau level quantization?

203

the same energy can be found, the transition temperature is rapidly suppressed to zero. This is shown in fig. 3, where TeE(H, 7), normalized to TeE(H, 7 = 0 ) is plotted versus 7 for three different fields at which To2 is large when 7= 0 or 7= 1/2. The critical value for 7 can be estimated from the condition (4)

E(H, T = 0 ; 7) = 1 = E ( H , Tc2; 7 = 0 ) . Assuming 709~<< Ogn,which is justified a posteriori, the same approximations that let to the analytic expressions for the transition temperature ( 12, 15) can be used to show that 7criticalOc ~

Tc2 (H; 7=0) ,

(19)

which is in good agreement with the numerical results presented in fig. 3. We would conclude, therefore, that Pauli paramagnetism does destroy superconductivity in the quantum limit, even if the magnetic moments of the electrons are very small. Only if the Zeeman energy were zero or equal to the energy difference between Landau levels with a precision not likely to occur in nature could the Abrikosov vortex lattice persist in fields well above the semiclassical upper critical field.

5. The Fulde and Ferrell state

Fulde and Ferrell [ 12 ] were the first to note that the destructive influence of Pauli paramagnetism on superconductivity can be mitigated by displacing the Fermi sphere of spin t and spin ~ electrons relative to each other by a wave vector whose length is roughly given by kvT--kF~ [ 13 ]. In this way, the pairing condition, which requires that opposite spin electrons with equal and opposite momenta and equal energy should be paired, can be fulfilled with much improved accuracy over part of the Fermi surface. On other parts of the Fermi surface it may then not be possible to pair electrons at all, but the resulting pair state describing pairs with momentum q can nonetheless be more stable than the isotropic solution [ 12-14 ], if the Zeeman energy is large enough. This kind of pair state with spatial variation along the direction of the magnetic field, which we have already

1

TE2(H,'~) T£2(H,0)

""..... ii .i

10-1

%.

"-..

a

'I 10-2

10-3

-~4

i0=3

"6

I0"~

I0-~

Fig. 3. Dependence of transition temperature on quasiparticle magnetic m o m e n t ggB=47#B. Full curve: to¢/¢v=2 ( q u a n t u m limit). Dashed curve: toJ¢v= (2/3)2/3, corresponding to ~o= 1 in eq. ( 11 ). Dash-dotted curve: ~o¢/¢F= (4/3)2/3, corresponding to ~o = 1 in eq. ( 17 ). In this case, values o f ? plotted on the abscissa represent deviations of the g-factor from the free electron value 1/2. Dotted curves: Fulde-Ferrell states with o p t i m u m pair m o m e n t u m for the fields to~/¢F= 2, (2/3)2/3.

C.T. Rieck et al. / Re-entrant superconductivity due to Landau level quantization?

204

introduced above (eq. ( 3 ) ) , has been invoked by Fulde and Ferrell [ 12] to describe the ground state of a superconductor containing paramagnetic impurities which order ferromagnetically. The homogeneous exchange field resulting from this ordering is included in the system's Hamiltonian through the Zeeman energy while its effect on the orbital motion is neglected [ 12-14 ]. Including orbital diamagnetism within the framework of the semiclassical theory, Gruenberg and Gunther [ 15 ] have shown that in the clean limit the FuldeFerrell state is the most stable state at sufficiently high fields such that the Maki parameter ot=x//2Hc2o/Hp satisfies the condition o~> a c = 1.8. Hp is the Clogston limit field and Hc2o is the upper critical field at T = 0 in the absence of Pauli spin paramagnetism (eq. (26)). In the presence of Landau level quantization, we can see from the eigenvalue (4) how a finite pair momentum q can compensate for the Zeeman energy. For q given by

1

--pzq=(v'-v-2y)m~,

(20)

m

we have

12m[k~+_(pz+q/2)z]=~m[

1

k2

"'-

_

(Pz-q/2)2]

1

(21a)

1

=Ev(m¢, y)-- ~ (V+ u ' + l ) 0 9 c - - ~mm [pz2+ (q/2)2] '

(21b)

so that spin ~ and spin ~ electrons do indeed have the same energy. Since q is a constant, this can be achieved only for one particular pair of Landau levels (u, v' ) and one particular value #z of the momentum. #z should be chosen such that eq. (21 ) vanishes for pz=pz, because the neighbourhood of this m o m e n t u m gives the largest contribution to the integral in eq. (4). If [pz2 + (q/2)2] is approximated by p Zz+_Pzq + (q/2)2 and if the constant energy shift q2/8m is omitted, then the Zeeman energy appears to have been removed completely from one of the terms (v, nu' ) in the eigenvalue (4) [ 1 ]. However, since one has to integrate over positive and negative Pz, the pairing condition (21 a) is more closely followed by only half the electrons. Contrary to statement d) in the introduction it will be impossible, therefore, to retrieve the large values of T~2(H) obtained in sections 3 and 4 by introducing Fulde-Ferrell type states (eq.

(3)). While in the absence of Landau level quantization it is usually a good approximation to set (p+q/ 2)z =p2 +Pv'q+ q2/4, the accuracy of the corresponding approximation introduced above, where only the zcomponent of the momentum is involved, remains uncertain. We therefore use eq. (8) to study the effect of a finite pair momentum. In order to obtain an approximate analytic result we assume that, as before, the maximum T~2 is attained when one Landau level coincides with the Fermi energy. This happens at a field ~Oc~ such that k, + = 0 and consequently k 2_ / 2 m = ( v - v' + 2y)o9~. For small values of the g-factor g = 4y, only diagonal terms in eq. (8a) need to be considered. Choosing qR/2m=2yoJ~, eq. (8b) then reduces to F..(~o~, 7, 00.) = ~

T

I1 l 1 1 1 Re iT3~ 2 ] . x/_io)" +2?a)~ - i x / ~ . -io). +2yo)~ "

A simple result is obtained only if co. can be neglected against 2y~o~, so that the final formula TV2

/'co

r l . 1 4 O J D T l-(4~(v')=/(2v)')8(2~/c°g)'/2}

=L~J

will not permit taking the limit 7--+0. We see that for v> 1, i.e. in the field regime where the sharp maxima in Tc2 occur (fig. 1 ), Tg2 is reduced strongly by Pauli paramagnetism if y exceeds a certain value. For v = 1, for example, we have T c~=~ = T~o for 7= (1/512) ~og=~/EF~ 10-3. For larger values of ~, T~2=~ decreases rapidly.

C. 72.Rieck et al. / Re-entrant superconductivity due to Landau level quantization?

205

In the quantum limit (u = 0 ), a finite pair m o m e n t u m is considerably more effective in reducing the influence of Pauli paramagnetism to /'ca. From eq. (7) we find that ko+ vanishes for

( 8 ~ 1/3 ~°~=°

(22)

so that the maximum transition temperature is estimated to be

Tc~L

T~o

FI. 14(.0D1 {'-8(3e2)1/3} =

k~J

(23)

Hence, for 7 up to 0.025 ( g = 0.1 ) the maximum TcQ2L will be of the same order of magnitude as the zero-field transition temperature T~o. For larger g-factors, T g g will become vanishingly small. For fields much less than the limiting field (eq. (22)) for which total spin polarization sets in, but still in the quantum limit, both energies

2

3 2

ka°-+2m- 94e( e FF] L [l-T\ o ~)( ~ )c J ] are comparable, if Ion [ << k2+_/2m for all n, we can approximate eq. (8b) according to r iT3~ 2 Foo(O)~, Y, con) = 2T~o Re ( k ~ + / 2 m ) ( k 2 _ / 2 m )

1 x/iog, + k 2 + / 2 m - x / - i o ) n

+kZo_/2m+q/x~

"

Expanding the square roots and choosing q x/~-

3709~ 2 ,2F x/~F

(24)

again gives a logarithmic singularity for small temperatures. This leads to

TcQ2L

r"... -1{1--(16/3) ~:2/t°2} / l . lt40)D/

~o - L ~ c o _1

(25)

which should be compared with eq. ( 15 ). We see that even when a Fulde-Ferrell state is invoked a considerable reduction in T ~ L results from the presence of the electron magnetic moment. Note that eq. (25) gives a larger value for T ~ L than eq. (23) when the limiting field (eq. ( 2 2 ) ) is inserted. Hence, we can expect that Tc2 is a nonmonotonic function in the quantum limit. Since the errors introduced through the approximations leading to eqs. (23) and (25) are hard to estimate, we present in figs. 3-5 numerical results based on the exact evaluation of eq. (8). In fig. 4 Tea is shown as a function of pair m o m e n t u m q for a given field O)c=2EF and various values of Y. There exists an optimum value for q, which varies linearly with y in agreement with eq. (24), at which T¢2 (H, 7, q) has a maximum. This maximum value has been plotted versus 7 in fig. 3. We see from this figure that for values of Y in excess of the critical value (eq. ( 1 9 ) ) at which superconductivity in a state with zero pair m o m e n t u m would be destroyed, the Fulde-Ferrell state persists with transition temperatures gradually decreasing with increasing 7 until complete spin polarization sets in at a value of 7 given by eq. (22). Close inspection of fig. 3 actually shows that for very small 7 the most stable state has q=0. Thus, qmax in the inset of fig. 4 vanishes for finite 7~ 1.6× 10 -3. For O2c~ (2/3)2/3EF the effect of introducing a finite q can be seen to be very small as anticipated. In fig. 5 we show results for variable fields and fixed g-factor. Without Pauli limiting, Tcz/T~o increases rapidly with field so that with the choice O2D/T¢o= 100 the weak coupling condition Tc2 >> CODceases to hold. As the number of terms in the sum over Matsubara frequencies decreases one by one, the eigenvalue E ( H , T) (eq. ( 4 ) ) from which T¢2 is determined, is actually represented by different functions, each giving a different

206

C.T. Rieck et al. / Re-entrant superconductivity due to Landau level quantization?

0.5

00/

Tc2(H,'G,q) TC2(H,O,O)

\

qm°xT/

01/

0"

0.05

1'

0.25!

-0.2

-0.1

0

q- qmax

0.1

0.2

Fig. 4. Dependence of the transition temperature on the dimensionless pair momentum q'= q / 2 x / ~ of the Fulde-Ferrell state for various values of 7. The innermost curve belongs to 7= 0.1. The inset shows that the pair momentum q'~x which, for a given y gives the maximum To2,and increases linearly with y. relationship between To2 a n d toc. The solid section o f each o f the curves represents the relevant solution o f E (H, T) = 1 because there toD (2n + 3 ) n < To2 < t o o / ( 2 n + 1 ) n. The last section shown for g = 1 corresponds to n = 1. The p r o b l e m also occurs for finite but small electron magnetic m o m e n t s 7 = 0.0125. F o r the next higher value 7 = 0 . 0 2 5 the m a x i m u m T¢2 is small enough to yield a curve which is continuous on the scale o f the drawing. As a result o f a strong coupling calculation we would expect a curve smoothly interpolating between these segments. F o r 7 = 0 such a curve can also be p r o d u c e d by s u m m i n g over all M a t s u b a r a frequencies and introducing a suitable cut-off in the pz-integral [ 2 ]. Increasing toD/Too to 500, SO that eF/toD = 2, has very little effect on the m a x i m u m Tc2/T¢o o b t a i n e d for finite 7 a n d thus gives a s m o o t h curve for 7 = 0.0125 represented by the d a s h e d line. The s u d d e n destruction o f the superconducting state at fields toe given by is due the large ratio ~F/To2 which prevents t h e r m a l p o p u l a t i o n o f an otherwise e m p t y L a n d a u level. Finally, we note that a F u l d e - F e r r e l state o f the form o f eq. ( 3 ) carries a supercurrent. This conflict with Bloch's theorem, according to which the g r o u n d state carries no current, has been resolved by Fulde a n d Ferrel by invoking a n o r m a l current due to u n p a i r e d electrons, which exactly balances the supercurrent. To m a i n t a i n this balance at varying i m p u r i t y concentrations the n o r m a l current must r e m a i n unaffected by i m p u r i t y scattering, which causes some p r o b l e m s with intuition [ 13 ]. Since, according to eq. ( 8 b ) , the eigenvalue does not d e p e n d on the sign o f q, it is clear that we can a v o i d these difficulties by choosing a real o r d e r p a r a m e t e r A ( r ) = A ( x , y ) cos qz .

A variety o f such real i n h o m o g e n e o u s o r d e r p a r a m e t e r s have been considered by Larkin et al. [ 14 ] in the context o f ferromagnetic superconductors as described above.

6. Impurities Since the large transition t e m p e r a t u r e s o b t a i n e d in section 3 originate in part from the high values which the density o f states at the F e r m i level attains periodically as 1 / H is reduced, we would expect the supercon-

C.T. Rieck et al. / Re-entrant superconductivity due to Landau level quantization?

207

~. - - - - - - I

z~.O,

"'.,

"".,.I "-

I ""

I

wE EF / // y.-

3.0" .....

...... I

2.0"

1.0~

......

I

"

...... ......

5

10

Tc2/Tco

Fig. 5. Dependence of the transition temperature on the magnetic field for three values of the g-factor, g = 0 , g = 0 . 0 5 (dotted curve) and g = 0 . 1 . Parameters are the same as in fig. 1, except for the dashed curve for which a ~ / T ¢ o = 5 0 0 has been chosen. For details see section 5.

ducting state at these high fields to be very sensitive to impurity scattering. In a diagrammatic approach, the smearing of the singularities in the density of states, which intuition demands, is obtained only if the self energy is calculated in a self-consistent Born or T-matrix approximation [ 16,17 ]. While for nearly free electrons with spherical Fermi surface both self-consistent and non-self-consistent Born and t-matrix approximations give a constant lifetime z for electrons at the Fermi level, in the presence of Landau level quantization z diverges in a non-self-consistent t-matrix approximation [ 16 ] and vanishes in a non-self-consistent Born approximation when a Landau level coincides with the Fermi energy. A fully self-consistent calculation leads to a fairly complicated system of equations for the energy-dependent real and imaginary parts of the self energy. The result of such a calculation is a density of states and a lifetime z which are well behaved at all fields [ 16,17 ]. If, as a crude approximation, z would be taken as constant, i.e. if ogn would be replaced by ogn+sgn o9,/2z as has been done by Gruenberg and Gunther [ 2 ], then the impurities would act as pair breaker in very much the same way as the Zeeman energy (eq. (19) ) and superconductivity would be destroyed at 1/'C~TtO c ~ Tc2(H , 7=0, ~'=oo) .

This approach leads to a T¢-reduction for any field H and hence is in conflict with Anderson's theorem. It is, therefore, necessary to take vertex corrections into account and Te~anovi6 et al. [ 1 ] have shown, apparently using a non-self-consistent Born approximation, that in the quantum limit the self energy effects are indeed cancelled by vertex corrections. The calculation is essentially the same as for the field-free case. The quantum limit seems to be, therefore, the most likely field regime in which to observe this type of reentrant superconductivity provided, of course, that the effective moments of the electrons vanish. If this is not

208

C.T. Rieck et aL / Re-entrant superconductivity due to Landau level quantization?

the case, the superconducting state would have to be of the Fulde-Ferrel type which in turn is rather sensitive to impurity scattering [ 18,19 ].

7. Connection with the semiclassical approximation Most calculations of the upper critical field employ the semiclassical approximation which consists in the replacement of G~ ( I r - r ' l ; ogn) in eq. (2) by G ~ = ° ( l r - r ' l ; o~n). In the clean limit, a reduced field

2 (VF] hc2=eHc2 ~ \L~Ico}

1 O)c~F hc2 , 2 hc2

2re 2 Tco

(26)

h'c2

is introduced [5,6 ] whose slope at Too is hc2 = 6 / 7 ( ( 3 ). Without Pauli paramagnetism one finds at T--0 that hcE/h'~z =0.7272 so that the maximum field in the semiclassical approximation is

2 (/)~emi)( 'TF= 0 -- 10"24 (~F0) " Since the field regime we are interested in here is ¢oc~ eF, we have oggemi ( T = 0) several orders of magnitude smaller than the fields required to see re-entrant superconductivity. For high-To superconductors with their large T¢o'S and comparatively small eF'S even a favourable estimate would put T~o/eF only somewhere in the region of 10 -2, SO that at ojgemi(T=0) at least 1000 Landau levels are occupied. Due to the extremely poor convergence of the double sum (8a), it is out of the question to calculate T¢2( H ) for such low fields directly from eq. (8) and thus retrieve the semiclassical result, modified possibly by small oscillations [2 ]. Analytical approximations to the eigenvalue (4) are required to make the problem tractable. The steps leading directly from eq. (4) to the corresponding semiclassical result E ( H , T) = V N ( O ) 2 ~ T

~

f~n>0

Re ./dO sin 0

0

ds exp{~o~eF sin 2 0 s 2 - 2 [m, +i7o)¢]s}

0

have been outlined by Gruenberg and Gunther [2]. The maximum transition temperatures T ~ ax ( H ) just above ~og~m~( T = 0 ) can be estimated from fig. 1, according to which we have T~"X/Tcooc (o)~/~F) 3. When this is extrapolated to o)~m~( T = 0 ) , it gives T'g"x/ T~o= 10 -8, starting from the Nifty large value T~o/EF= 10 -2. Since the peaks in Tc2(H) are very sensitive to disorder and Pauli paramagnetism, we actually expect Tc2(H) = 0 for ~og~mi( T = 0 ) < O)c< eF/3. This is a range of magnetic fields spanning several orders of magnitude. There seems to be no possibility, therefore, to observe the kind of smooth transition from the semiclassical to the re-entrant regime as sketched in fig. 1 of the letter by Teganovid et al. [ 1 ]. Even though it is only of academic interest because of an unphysical range of parameters, we tried to study Tc2(H) for the case when the semiclassical and the re-entrant regimes are not separated by a wide gap. The question we wanted to answer was whether in such a case Landau level quantization severely suppresses H~2 in the semiclassical regime. Taking T~o/eF= 10-1 we had to choose e)o ~ eF in order to avoid conflict with the weak coupling approximation. Because of this large cut-off frequency pairing is not confined to the vicinity of the Fermi energy so that a large number of unoccupied Landau levels can contribute to the sum in eq. (8). For fields below the ones calculated semiclassically we took as many as 10 6 terms in the double sum (8a) into account but were still not able to estimate the limiting value of this sum. The only possible way to investigate this regime in parameter space seems to be the one used by Gruenberg and Gunther [ 2].

C. T. Rieck et al. / Re-entrant superconductivity due to Landau level quantization?

209

8. Conclusions While it is true that orbital diamagnetism does not destroy superconductivity in fields high enough that only very few Landau levels are occupied, Pauli paramagnetism and disorder will strongly suppress this particular phase transition. Only for clean materials in which the charge carriers have magnetic moments less than 0.1/ZB can one expect to see re-entrant superconductivity in high fields provided, of course, the theoretical framework within which these conclusions have been derived remains applicable in high fields. It follows from the condition o9c~ eF that the fields required are in excess of 10 4 T for most metals. Only for low carrier density semimetals and semiconductors could one hope to reduce these fields to a level which would be accessible experimentally [ 2,1 ]. The superconductor with the lowest carrier density known to us is SrTiO3 with n ~ 10 ~9 [ 20]. In this case, re-entrant superconductivity would occur for fields as low as 150 T. However, reduced or Nb-doped SrTiO3 is a dirty superconductor (~/~o << 1 [20] ) and its g-factor is close to two, so that it will certainly not show superconductivity in the quantum limit. When looking for new materials it should be remembered that, according to the model treated here, re-entrant superconductivity can be expected only if there is a superconducting transition in zero field. Unless the g-factor is exactly zero, the maximum transition temperature in high fields is predicted to be of the same order of magnitude as the zero-field transition temperature. Hence, before subjecting a truly low-carrier density semiconductor like Ge or Si to a high-field study, the presence of a superconducting phase transition in zero field should be established. Finding superconductors with carrier densities much below n = 1019 does not seem to be very likely because in all the known superconducting semiconductors the transition temperature Tc eventually decreases with decreasing carrier density [20]. This may be due to a decrease in the density of states but could also reflect the fact that a sharp Fermi surface is a prerequisite of the Cooper instability. At carrier densities low enough for the re-entrant regime to be easily accessible experimentally, the model of a degenerate nearly free electron gas used here is no longer adequate. The temperature dependence of the chemical potential would be important, but this could easily be taken into account. It would be much more difficult to assess the importance of effects due to the band structure. If such effects are included simply through replacing the electron mass m by an effective mass m*, then the theory presented here remains essentially unchanged. For m * < m, fields required to reach the quantum limit are reduced and hence the Zeeman energy yoJc is reduced relative to the distance between Landau levels. Thus, the limit on the g-factor mentioned above would be raised by a factor m / m * . If the g-factor is large but not exactly equal to 2, then the Zeeman energy could be matched to the difference between Landau levels by a suitable effective mass, so that the result (18) (cf. fig. 2) remains valid. However, a single isotropic effective mass is certainly inadequate. If the system under consideration does have a nonspherical Fermi surface, then the order parameter considered here ceases to be a solution of the linearized selfconsistency equation ( 1 ) [ 21 ]. Since a quantitative theory of Bloch electrons in a magnetic field providing exact and explicit expressions for the Green functions (2) is not available, one can only speculate about the extent to which the order parameter A ( r ) is affected by the periodic lattice potential. The pairing interaction is also likely to be affected by the application of large magnetic fields. In the present paper, as well as in all the work quoted here, a BCS-type pairing interaction of strength V has been assumed for all fields considered. The electron-phonon interaction, generally acknowledged to be responsible for the superconducting phase transition in most compounds, is affected by magnetic fields in several ways: First, the dielectric properties of the electron gas are changed which can alter the phonon dispersion relations and thus the phonon density of states. For semiconductors this is probably not important because the phonon dispersion relations are determined primarily by covalent bonding. Second, the electron-phonon interaction will be changed dramatically when the separation of Landau levels becomes comparable to the maximum phonon frequency because the electronic system is then essentially one-dimensional and thus can absorb and emit only phonons with small wave-vectors parallel to the magnetic field. Most of the large-q phonons responsible for the attractive

210

C T. Rieck et al. / Re-entrant superconductivity due to Landau level quantization?

pairing interaction in c o n v e n t i o n a l superconductors can no longer be exchanged between electrons if these occupy a single L a n d a u level. It is not only the restriction of the available phase space that will reduce the effective coupling constant F. Matrix elements are also likely to be reduced as can be seen, by way of an example, from a consideration of the deformation potential coupling, relevant for longitudinal acoustic phonons. It follows from eq. (7) that the interaction energy in the q u a n t u m limit is 8 / 9 x 2 ( 1 - (97xa/8)a)~F with X=oJcleF, which should be compared with 2eF/3. The phase space a r g u m e n t also applies to the electron-electron interaction so that one could expect the Coulomb pseudo-potential/t* to be reduced also. In view of the complexity of the problem it is impossible to say whether the net result of L a n d a u level q u a n t i z a t i o n is an increase or a decrease of the BCS parameter V. It is most unlikely, though, that a magnetic field will increase V (by a reduction of/~*?) to such an extent that a well-known n o n s u p e r c o n d u c t o r (pure Ge, To< 50 m K ) will become superconducting in the q u a n t u m limit.

Acknowledgements Extensive discussions with S.H. Liu a n d M. N o r m a n are gratefully acknowledged. This work was supported in part by the US D e p a r t m e n t of Energy, Division of Basic Energy Sciences, u n d e r contract no. DE-AC00584OR21400 with M a r t i n Marietta Energy Systems.

References [ 1l Z. Te[anovi6, M. Rasolt and L. Xing, Phys. Rev. Lett. 63 (1989) 2425. [2 ] L.W. Gruenberg and L. Gunther, Phys. Rev. 176 ( 1968) 606. [3] R.A. Klemm and K. Scharnberg, Phys. Rev. B24 ( 1981 ) 6361. [4] L.I. Burlachkov, L.P. Gor'kov and A.G. Lebed', Europhys. Lett. 4 (1987) 941. [5] E. Helfand and N.R. Werthamer, Phys. Rev. 147 (1966) 288. [6] N.R. Werthamer, E. Helfand and P.C. Hohenberg, Phys. Rev. 147 (1966) 295; N.R. Werthamer, in: Superconductivity,ed. R.D. Parks (Marcel Dekker, New York, 1969) p. 321. [7] G. Eilenberger,Phys. Rev. 153 (1967) 584. [8 ] K. Scharnbergand R.A. Klemm, Phys. Rev. B22 (1980) 5233. [9] A.K. Rajagopal and R. Vasudevan, Phys. Lett. 20 (1966) 585; ibid., 23 (1966) 539. [ 10] L. Gunther, in: Quantum Fluids, eds. N. Wiser and D.J. Amit (Gordon and Breach, New York, 1970) p. 579. [ 11 ] A.L. Fetter and J.D. Walecka, Quantum Theory of Many-Particle Systems (McGraw-Hill, New York, 1971 ) p. 473. [ 12] P. Fulde and R.A. Ferrell, Phys. Rev. 135 (1964) 550. [ 13] S. Takada and T. Izuyama, Prog. Theor. Phys. 41 (1969) 635. [ 14] A.I. Larkin and Yu.N. Ovchinnikov, Zh. Eksp. Teor. Fiz. 47 ( 1964 ) 1136 [Sov. Phys. JETP 20 ( 1965) 762]. [ 15] L.W. Gruenberg and L. Gunther, Phys. Rev. Lett. 16 (1966) 996. [ 16] S. Tanaka and T. Morita, Prog. Theor. Phys. 47 (1972) 378. [ 17] R. Gerhardts and J. Hajdu, Z. Phys. 245 ( 1971 ) 126. [ 18 ] L.G. Aslamazov:Zh. Eksp. Teor. Fiz. 55 (1968) 1477 [Sov. Phys. JETP 28 (1969) 773]. [ 19] S. Tadaka, Prog. Theor. Phys. 43 (1970) 27. [20 ] J.K. Hulm, M. Ashkin, D.W. Deis and C.K. Jones, in: Progress in Low Temperature Physics,Vol. 6, ed. C.J. Gorter (North-Holland, Amsterdam, 1970) p. 205. [21 ] C.T. Rieck and K. Scharnberg, Physica B 163 (1990) 670.