~ Solid State Communications, Vol. 74, No. 12, pp. 1327-1332, 1990. Printed in Great Britain.

0038-i098/8903.00+.00 Pergamon Press plc

LOW CRITICAL FIELD AND LONDON PENETRATION DEPTHS IN,MULTI-SHEET SUPERCONDUCTING STATES

Zhen Ye, H. Umezawa and R. Teshima The Theoretical Physics Institute

University of Alberta Edmonton, Canada T6G 2J1

(Received 5 February 1990 by M.F. Collins)

Temperature dependence of the low critical field ~tcl and London penetration depths AL for a mnlti-sheet superconducting system is studied through calculation of the Meissner current. The results are compared with those for the usual bulk super- conductors and also with experiments. Both H,1 and 1/AL decrease with increasing temperature faster than the result of the conventional BCS -model. This is an effect of enhancement of the thermal fluctuation due to low dimensionality. The behavior of He1 shows that the Meissner effect is dramatically diminished in a small domain of temperature immediately below T,. This indicates that the electron tunnelling along the c - axis is drastically curtailed by the thermal fluctuation.

1 I n t r o d u c t i o n

The London penetration depth, )~L, a measure of the at- tenuation of a magnetic field near the surface of a su- perconductor, is one of the most important properties of a superconductor. Since the discovery of the high Tc superconductors several experimental results have been reported about the temperature dependence of this quan- fit ly for Y-Ba-Cu-0 [1,2,3]. Some of these results are qualitatively consistent with the conventional BCS the- ory outside the low temperature domain. This quantity has also at tracted the attention of the theorists. Chi and Nagi [4] for instance calculated the dependence for a model of electron pairing induced by holes. Within the Ellashberg theory, authors in [5] studied this quan- t i ty for several coupling values and showed that AL(T) depends mainly on the energy of the boson responsible for the electron interaction. All of these theories give re- sults qualitatively similar to the usual BCS theory. In this report we try to study how the multi-sheet structure influences the London penetration depth with an antic- ipation that the enhancement of the fluctuation by the low dimensional structure may tend to accelerate the in- crease of AL with increasing temperature. Recent exper- iments of He1 of twinned and untwinned YBa2CusOz-, in the Argonne National Lab. [fi] and of a single crys- tal Bi2.2Sr2Cao.sCu2Os+, by [7] shows a deviation from the conventional BCS theory at low temperature, de- creasing faster with increasing temperature in the low temperature domain. We wish to see if such a devia- tion can be at least partly explained by the effect of the multi-two dimensionality in the high T, materials. In this paper we study AL and H,, versus temperature. It is well kvown that there cannot exist any superconducting

state in an isolated two dimensional system because the low dimensionality enhances the infrared effect of quan- tum fluctuations of long range correlation which quench the superconducting order. This motivates us to study a multi-sheet model to see whether or not the electron tunnelling among the sheets is able to compensate the infrared catastrophy, developing a long range correlation which stabilizes a superconducting state. Our model is a system of multi-sheets with equal spacing. The realistic multi-sheets in high T, materials are of a more compli- cated structure. However, in order to s tudy general and qualitative features of multi - sheet e - ~ c ~ i we use the simple model and plan to refine the model in our future study. Even in our simple model there are two possi- ble kinds of Cooper pair. One is called the bridge pair in which one electron resides on a layer while its part- ner is on one of the two nearest layers. Another kind of Cooper pair is called the in-sheet pair in which both electrons reside on the same layer. There have been some reports[8,9] suggesting that there could be two gaps co- existing in some high Tc materials. A general analysis of a system consisting of both pairings will be left to the future. Since our interests lie in the nature of the long range correlation extending over the multi - sheets, here we study the superconductivity due to the bridge pairs only.

Our method heavily depends on the quantum field theoretical method for the treatment of ordered states which emphasizes the role of the bosons associated with the long range order. This method is particularly suitable for the calculation of electric currents (see for example [10]). Since the superconducting state is a phase ordered state, the Goldstone theorem states that there appears the energy - gapless boson which maintains the phase in-

1327

1328

variance and which is called the phase boson. Thus the set of the quasi-particles consists of the quasi-electrons and the phase bosom As it is well known the quantum excitations of the phase boson act as the longitudinal part of the electromagnetic gauge potential and becomes unobserved. However, when the phase boson condenses in the ground state and the form of the condensation carries a certain topological structure, this condensation becomes observable. These are the vortices in the mixed states of superconductors. We can mathematically relate different condensation states with a family of invariant transformations which is called the boson transformation. Therefore a central part of our task in the boson method [11,10] is the study of the Green's function for the phase bosom Since we make full use of quantum field theoret- ical thechniques, the temperature effects are considered by the thermo field dynamics (TFD) [10]. Since there are many publications on TFD, we omit any detailed sum- mary on this subject.

We consider infinite sheets in parallel to the ab plane with spacing a. The sheets can move up and down along c-axis in order to absorb the recoil momenta of the elec- trons tunnelling out. We call this kind of structure quasi- two dimension. We use the coordinate system such that the ab-plane is the xy- plane and the z - axes is along the c - axes. The model Hamiltonian we use is

H = Ho + Hi,,.

MULTI-SHEET SUPERCONDUCTING STATES Vol. 74, NO. 12

together with the gap equation

4A o o r ,:,,2 cos2x. ] 1= [Jo (8)

The dk - integration is performed by the de - integra- tion by means of the relation de = ( k / m ) d k and the de - integration is cut at e = w¢. In this paper w~ is an unknown parameter because we do not specify what me- diates the interactions among electrons. We used also the notation x = kza. The k, belongs to the Brillonin zone - v / a < kz < v / a . Note that there are gapless points in this domain, because the energy gap is A cos k~a which vanishes at k~ = +r / (2a) .

The gap at T = 0 is found to be

× e ~ ( - 4 ~ m ) . (9)

At the critical temperature we have

A oo 1 1 - 2(2¢r)2/'-00 d2k2-~ek(1 - 2f(ek)), (10)

from which we have the critical temperature

kBT, = A~, exp(-4~rAm), A --- 2"r/r ~ 1.13, ( n )

where 7 is Euler's constant. The ratio of T, and energy gap is

Ao (1) c = ~ ~ 2.15, 02)

which is larger than the BCS value. This value of c coin. c_ides with the one suggested by an experimantal study of the low temperature penetration depth in [12]. For any T below or equal to T¢ we have

- - d x c o s 2 x

(') + <., + ( , , , / . <o.,

where we used the reduced temperature t = T/Tc, and the reduced energy gap A' and reduced energy y are de- fined as

= (14)

We plot the gap with the reduced temperature in Figure a. We find that the shape is similar to the usual one of the BCS model but we note a slight reduction due to the fluctuation.

with the free and interaction parts

Ho = ~ ¢~,i(x, i)e(-iV)¢~,, i( x, i), (2) a,i

and

1 t • t • H~., = (-;9 ~ ~ ][¢l(x, ~)~T(x, ~ + I)¢*(x, / ÷ i)¢i(x, i)

+%bI(x , i)¢I(x, i - 1)el(X, i - 1)%bl(x, i)],

where

= 1 2 e(- - iV)e i~ {~m (kx + kv 2) - #Y} e e;~. (4)

Here ~b~(x, i) is the electron Heisenberg field with spin a on the i-th layer. The coupling constant, denoted by A, is positive.

2 Q u a s i E l e c t r o n

The bridge pairing order parameter is

A = A < O( f l ) l¢ l ( x , i )¢T(x , i+ 1)1003) > (5)

= A < O(/3) l¢l(x , i )¢r(x , i - 1)10(/3) >, (6)

where the z-reflection symmetry is considered. Here t0(fl) > is the thermal vacuum at temperature T = 1//3 in TFD. By applying the usual self-consistent calculation, we find the energy spectrum of the quasi-electron

E~ = ( 4 + ~ cos~ k,a (7)

3 P h a s e B o s o n

We can avoid a considerable amount of tedious computa- tion by making use of several general theorems associated with the spontaneous breakdown of the phase symmetry. Here we summarize these theorems which will be used in the following consideration. Introduce the Heisenberg operator

• (x , i ) = [¢ lCx, i )¢ l (x , i ) + ¢~(x,i)¢l(x,i)l, (15)

I/O1. 74, NO. 12

where

1 ¢,(x , i) = ~[¢,(x, i + 1) + ¢~(x, i - 1)]. (16)

With this operator we can rewrite (5) and (6) as

< 0(~) l¢(x , i )10(~) > = 2A, (17)

Now perform the electric phase transformation ¢~ --+ exp[(i/2)0]¢~ with infinitesimal & Denote the change of ¢ by this transformation by 0 X. Then we have

X(X, i) = i[¢t(x, i )¢,(x, i) - ¢~(x, i)¢~(x, i)1. (18)

Now consider the Green's function for X:

Dij(x, y) = < O(B) lT[x~(x)Xi(Y)]lO(B) > . (19)

The Fourier transform of this will be written as D(k). Then the Goldstone theorem states that D(k) has a pole singularity at a certain value of k0, say k0 = wB(k) and that we is gapless, meaning we(0) = 0. The following computation explicitly confirms this as we are going to show. The WB is the phase boson energy which is the Goldstone boson. In the following the phase boson free field will be denoted by B(x). We normalize B in such a manner t ha t / 3 is the canonical conjugate of B, as we usually normalize the free boson field. Indicating the residue of D(k) at k0 = wB(k) by Ze(k), we have a gen- eral theorem (see, for example, p.285 in [10]) states that the boson transformation

B(x) ~ B(x) + 4AZ-V2(-iV)O (20)

induces the electric phase transformation ¢~ ---} exp[i(1/2)8]¢~. This indicates that the phase boson bet- ter be normalized as

Bp(x) = £ Z I / 2 ( - i V ) B ( x ) , (21)

because the transformation Bp ~ Bp + 6 induces the elec- tric phase transformation. Then the canonical conjugate of Bp is

~rp(x) = (4A) 2 Z-l(-iV)13p(x). (22)

Both B and B~ satisfy the free boson equation

+ w~(V) B,(x) = 0. (23)

Recall now that the charge (or the electron number) is the canonical conjugate of the phase. We thus roughly put p(x) = erp(x) for the charge density operator p. Then the charge conservation law together with the phase bo- son equation (23) determines the structure of the current operator as

1 ji(x) = 4~rA2L,ieCi(-iV)ViBp(x) (24)

with i =-1, 2, 3. We put the suffix i by considering the spatial anisotropy. We will use the notation All = Aa with the third axis in the c-axes and A j_ = A2 = A1. In general [[ means the c-direction while _k is in parallel to the ab -

MULTI-SHEET SUPERCONDUCTING STATES 1329

pla.~e. Note that the suffix of A L corresponds, not to the direction of the magnetic field, but to the direction of the current. The c - function is normalized as c(0) = 1. This normalization determines AL. The AL and the c-function are given by

(A-~.i) 2 =41re2(4A)2Z-l(O)v~,,(O) (25)

and

z- l (k)4 , , (k) c,(k) = Z_a(0)v~,i(0), (26)

where the phase boson velocity vs,i is defined by

w~(k) = vk , (k )k l + vk~(k )k l . (27)

To create vortices, we perform the boson transformation Bn(x ) ---+ Bp(x) + f(x) with a c-number function satisfy- ing the phase boson equation (23). In a static situation this equation becomes the Laplace equation

vV(x) = 0. (28)

The boson transformation creates the macroscopic cur- rent

1 Y,(x) = ~ ( - i V ) V , f ( x ) . (29)

When we take into account the electromagnetic field, the gauge invariance dictates that the macroscopic current becomes

1 J,(x) = 4reA2L, Ci(-iV)[Ai(x) - Vif(x)], (30)

because f(x) is the phase. This indicates that f(x) be- comes the unobserved longitudinal component of the vec- tor potential when f(x) has its Fourier transform. The Coulomb effect modifies the phase boson equation (23) in such a manner that the phase boson becomes a super- position of the plasmon field and a boson satisfying the Laplace equation. Being interested only in low energy excitations, we ignore the plasmon. Then equation (28) holds true in any situation. For example, when a system of vortices moves with a velocity ~7, f(x) is a function of ( I - ~Tt). In the following we consider only a static situation. When f(x) is Fourier transformable, (28) has only the trivial solution f(x) = constant. However, this equation has non - trivial solutions which have a cer- tain topological singularity. To illustrate this we recall the cylindricM angle O which satisfies the Laplace equa- tion, though V x V0 # 0. This creates a vortex current. With the knowledge of the form of the current in (30) we can write the Maxwell equation which is the basis for the analysis of magnetic properties. This considera- tion is completed with the calculation of Z(k) and we(k) by solving the Bethe - Salpeter equation for the Green's function D(k). The linear A - term in (30) is the Meiss- net term and, therefore, AL is the London penetration depth.

Our next task is to obtain D(k). To write its Bethe -

1330

Salpeter equation we need the unper turbed Green's func- t ion

D°t(x, y) = < O(~)lT[x°(x)Xo(Y)]lO(fl ) > (31)

with

X°(X) = i{¢~(x, i)¢?(x, i) - ¢~t(x, i )¢~t(x, i)}. (32)

Here Co is the quasi electron free field which is the un- per turbed free electron field and ¢o is obta ined from ¢~ in (16) with ¢ being replaced by ¢o.

To calculate D(k) we need two more D-functions:

D,,~j(x - y) = < O(/~)IT[¢~(z, i)¢T(x, i)Xj(Y)]IO0) >,(33)

D2.,~(z - y) = < 0(~)lT[¢~(z, i)¢~(z, i)xj(Y)l[O(fl) >,(34)

because we have

Dij(x - U) = i[Da,ij(x - U) - D2,ii( x - U)] (35)

The random phase approximat ion together with the Bo- goliubov approximat ion which considers the simplest loop (consisting of spin up and down pair) leads to the follow- ing Bethe - Salpeter equation in momentum space:

D~(k) = D°(k) + Q~(k)Dl(k) - Q2(k)D2(k), D2(k) = D°(k) + Qs(k)D2(k) - Q2(k)D~(k), D(k) = i[Dl(k) - D2(k)], (36)

where the Q - functions are given by

Q,, , . , (~ - z) = ~ , , , . , ( = - z)gO.,(~ - ~),

Q2,i,,(x - z) = iAf~i,,.(x - z)j~i,,,(x - z), Q~,,.,(~ - ~) = iAG°,~,~(z - ~ ) g ° m ( z - x ) . (37)

Here G °, fo and gO are the unper turbed electron Green's functions defined by

G~¢(x-y) = < O(~)[T[¢°~(x,i)¢~'(y,j)]lO(~) >, (38)

g ° ( x - y) = < O(~)[T[¢?(x,i)¢~l(x,j)]lO(~) >, (39)

f~i i (x-y) = < O(~)[T[¢°~(x,i)¢?(x,j)]lO(~) > . (40)

A calculat ion leads to

1 Dl(k) - 1 - Q(k)[DO(k){1 - Qa(k)} - Q2(k)DO(k)],(41)

1 D2(k) - 1 - Q(k)[D°(k){1 - Ql(k)} - Q2(k)D°(k)],(42)

i D(k) = 1 - Q(k) {[1 + Q2(k)]D°(k) - DO(k)Qa(k)

+ D°Ck)Q~Ck)}, (43)

where

1 - Q(k) -- 1 - Qi(k) - Q3(k) + Ql(k)Q3(k) - Q~(k)(44)

The phase boson energy spectrum is determined by

1 - Q(k) = 0 (45)

MULTI-SHEET SUPERCONDUCTING STATES Vol. 74, No. 12

For the sake of convenience we also introduce

Qo(k) = l [ Q l ( k ) + Qa(k)]. (46)

We expand all the quantit ies in powers of k and keep up to k 2 - order. In this approximation we have

Q(k) = Qo(k) + Q2(k). (47)

Wi th a help of the gap equation (8) we can derive

1 - Qo(k) - Qz(k) = [w~(k) - k02]Ro,a (48)

with

w~(k) = 1.2z.2 l_a2k2all.~ (49) 2 ~F~'I" + 4 II Ro,,~ '

where

R,,~ = (~m) Jo T : - . 2-7

cos x (E 2 + e 2) - e~ + ½5 2 cos 2 2Ea x [ 1 - 2fF(E)]

A 2 c o s 2 x ( c o s 2 x - 3 + E3 ~)flfF(E)[1 - - rE(E)]) (50)

and

Ro,~ = " o de [ . dx (cos2x[1-- 2f f (E)] f

Jo 7- J - . ~ k SES

cos 2 x fF( E)]'~) - - ~ .]3IF(E)[1 -- ( 5 1 )

Here E = Vie 2 + A2 cos 2 x. Use of integrat ion by par ts leads to

RII,Z = 2A2Ro,~. (52)

Under the small k approximation, we obtain from the Bethe - Salpeter equation the result

1 D(k) = 1 - Q(k) D°(k) (53)

with

D 0 ( k ) = 2i -~ - [Q0(k) + Q2(k)]. (54)

The residue of D(k) at k0 = ~B(k) is

2 Z(k) = y ~ , [ 1 + O(k2)], (55)

where O(k 2) stands for a term of order of k 2. Wi th the knowledge of Z(k) and wB(k), the form of the current, the London penetra t ion depth and the c - function are obtained from the general consideration presented above.

Denoting the London penetra t ion depth at tempera- ture T by AL(T), we find

( ~'~' ' (°)~ - " ~ ' ~ ' (561 AL,.(T)] A2Ro.oo '

Vol. 74, NO. 12

1.00

0.75

O

0.50

0.25

0 0

MULTI-SHEET SUPERCONDUCTING STATES

04

I I I 0.25 0.50 0.75 1.110

T/T c

Fig:l~aThe reduced order parameter ACT)/A(O) v s . re- duced temperature. The upper curve for the BCS - model and the lower one for the multi - sheet model.

~,~,,(T) ) ~, ~ T ) ) where we have made use of equation (53). We can arrange the first equation as

~ ) ~ ) (58)

We plot the temperature dependence of inverse of square of the London penetration depths,

2 2 • • ()~L,u(O)/)~L,H(T)) a n d ()~L,J.(O)/)~L,.L(T)) i n Figure b, in which the results are compared with the one of the BCS model. We see that they are dit~erent from the result of the conventional BCS model. As T goes to zero, both curves retain a nonvauishing slop. The de- crease of these curves with increasing temperature tends to be much faster than the result of the BCS model. We feel that this is due to the enhancement of thermal fluctuation due to low dimensionality. To illustrate the anisotropic behavior of the multi - sheet model under s tud~ we present the ratio of the London penetration depths in both parallel and perpendicular to the c - axes at T -- 0. We have

)~L,J. ( 0 ) a A o = (59) )~L,II(O) ~VF

_ 2 . 1 5 a k v T ¢ (60) ~VF '

where (12) is considered. Choosing T¢ = 100K, a =10/~ and VF = l O ~ c m / s e c , we find XL,±(0)/XL,II(0) around 0.3 which is not bad when it is compared with experiments[l].

Since the behavior of the London penetration depths of this system is quite different from the one for the bulk superconducting system, the magnetic properties too are expected to be different. Knowing the current, we

London Penetration Depths

1331

1.0 - - - - - - ~....,,... 7 - - a b plane

X j ' ~ ~ ~ - - C direction

0.6 "~x//A\.

0 4 " k \ \ "~,~\\

0.2 "~ ~X~\ 0 t I i I t I I I ~'-~

0 0.2 0.4 0.6 0.8 1.0

T /T c

i

1.2

Fig~l~The ( A L ( O ) / A L ( T ) ) 2 vs. reduced temperature for r the BCS - model, and for the ab - plane and c - di-

rection in the multi - sheet model. Note that AL for the ab - plane (i.e. ~L~ ) does not mean the pen- etration depth for the magnetic field in parallel to the ab - plane. Rather, it means that the strength of the current in the ab - plane is determined by the factor (1/XL±) 2 as in equation (30). A similar note is applied to )% for the c - direction, (i.e. XLtl).

can calculate the internal magnetic field by solving the Maxwell equation. We then obtain the strength of the magnetic field at the vortex center which will be denoted by h. Then minimizing the free energy leads to the well known formula

Hol = 1-h + Eo, (61) 2

where Ec is the core energy. Since the Ginzburg - Landau parameter is supposed to be large, we may safely ignore the core energy. Then the calculation of h immediately leads us to H¢1. Here we present our result for H¢1 in parallel to the ab - plane which at temperature T will be called Hel,ab( T ):

Hcl,ob(T) )~L,H (0))~L,.L (0) (62) H=l,,b(O) = AL,I I (T)AL, . ( T ) "

This is plotted in Figure c. As is expected, this decreases with increasing temperature faster than the result of the usual BCS - model. A noticeable feature of our result is the sudden'decrease of He1 in vicinity of To, indicat- ing a practically vanishing Meissner effect in a small do- main below To. Intuitively speaking, the current in the z-direction is caused by the electron tunnelling among the sheets. Our result indicates that as the temperature goes up close to T¢, the tunnelling seems to be drastically diminished by the thermal fluctuation. A similar behav- ior was observed in an experiment which was reported in a recent preprint [13]. In our calculation of h we approx- imately replaced the Meissner form factor c(k ) by unity.

1332

¢'~ .O

1.2

1.0

0.8

0.6

0.4

0.2

MULTI-SHEET SUPERCONDUCTING STATES VOI. 74, No. 12

Hcl Parallel to ab Plane There are two reasons for the difference in the behav- ior of the multi - sheet model from the conventional bulk model. One is the periodicity which confines the momen- tum in the c - direction to the Brillouin zone. Another is the enhancement of the fluctuation effects due to low di- mensionality. As a result the ratio Ao/(ksTc) is enlarged. The infrared catastrophe due to the fluctuation, which quenches the superconducting states in an isolated two - dimensional sheet, is avoided by the tunnelling electrons which maintain the superconducting states in the multi - sheet system. Still the fluctuation effects are larger than those in the bulk model. This can be seen, for example, from the fact that the energy gap is not constant in the

I 0 momentum space and there even appear gapless points. 0 0.2 0.4 0.6 0.8 1.0 1.2 We also see the increase of the thermal effect which tends

T/Tc to diminish the tunnelling current. This is the reason

Fig. l 'cThe Hd(T)/Hd(O) vs. reduced temperature in the direction parallel to the ab - plane.

The effect of c(k) will further reduce Hcl,ab, but this ef- fect is expected to be very small because the coherence length is small. Calculation of/'/ca along the c - axes is more tedious and will be reported elsewhere.

why Hcl,ab(T) and 1/AL(T) drop faster with increasing temperature than those in the bulk model. This effect becomes so strong for the temperature just below Tc that H¢x,~b rapidly disappears.

Acknowledgements - - We like to thank Dr. G. Crabtree and Mr. A. Umezawa for valuable comments and ad- vice. We would also like to thank Dr. J. Whitehead for stimulating discussions.

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