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Electric field effect on the critical current of SNS contact
PHYSICA ELSEVIER
Physica C 267 (1996) 233-242
Electric field effect on the critical current of SNS contact A.L. R a k h m a n o v
a,
A . V . R o z h k o v * ,a,b
a Scientific Center]or Application Problems in Electrodynamics, Russian Academy of Sciences, 174122, lzhorskaya Str. 13 / 19, Moscow, Russia b Department of Physics, University of California at San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0354, USA
Received 4 March 1996; revised manuscript received 14 June 1996
Abstract The effect of electric field on the critical current of SNS contact is studied in the framework of Ginzburg-Landau theory. We show that electric field can significantly enlarge the critical current of the contact. This effect is especially strong when the temperature of the sample is close to the superconducting transition temperature of normal layer Tcn. The electric field effect depends on film thickness: the thicker the film the weaker the effect. Nevertheless, it can be significant for a thick film when the temperature is close to T~n. The cooperative effect of electric and magnetic fields is also studied. PACS: 74.50 + r; 74.25.Nf
1. Introduction The electric field effect is commonly observed in high-T~ superconductors [1-9]. Different mechanisms are suggested to explain this observation [ 1 3,8,9]. One of the most appropriate explanations is that the electric field changes the concentration of the free charge carriers near the sample surface [1,2]. Recent experiments shows that this mechanism is appropriate at any rate for BSCO and YBCO superconductors [9]. It suggests that the electric field E applied to the surface of superconductor induces the change 8n in the free charge carriers concentration n o in the layer of the thickness of l D, where 1D is the Debye screening length. In the Thomas-Fermi approximation l D = (gd//47rUF e 2)I/2 [ 1,2], where e d is a dielectric constant, u F is the density of states at
* Corresponding author.
the Fermi level. A transverse electric field changes longitudinal current-voltage characteristic of superconductor if I o > ~ [1,2] where sc is the coherence length. The last inequality may be fulfilled for high-T~ superconductors [ 1,2] because of the small coherence length and relatively high 8 a. In papers [5-7] the observed changes in superconducting properties with the electric field are ascribed to the field effect on weak links that usually exists in high temperature superconductor materials. Indeed, in the experiments [5-7] it has been detected a strong dependence of current-voltage characteristics of the isolated weak links in YBCO films on the transverse electric field. The present paper is aimed to investigate the electric field effect on the critical current of SNS (superconductor-normal metal-superconductor) contact. We suppose that normal layer is characterized by the non zero temperature of superconducting transition To, and can be described in the framework of
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A.L. Rakhmanov, A.V. Rozhkov / Physica C 267 (1996)233-242
234
results in the growth of the critical current of the junction I c. The change in the critical current A I~ is equal to:
Y
AI c
L
lD ATcn
1--~-'" ~o------~ S S
v
X
z
I E,H
Fig. 1. SNS contact in electric and magnetic field: geometry of the problem.
Ginzburg-Landau (GL) theory. The sample temperature T is assumed to be between Ten and Te: Ten < T < T~ (where Te is the critical temperature of superconductor). It will be shown that it is a temperature range where the effect is the most pronounced. Let us consider SNS contact of the thickness L parallel to the yz-plane (Fig. 1). The transport current flows along x-axis and electric field E is directed along y-axis. We study here the usual experimental situation and assume that the electric field is equal to E at y = 0, decays at y = S, and It) a~ S,L.
The critical current density jc of SNS junction is a function of the ratio L / ~ n, where ~, is the coherence length in normal metal [10]. For L / ~ , : ~ 1 the critical current can be evaluated as Jc ~ e x p ( - L / ~ n ) , where in the framework of GL theory:
fn = f O / G ,
(r-- Te.)/Te.,
and ~0 is the coherence length in N layer at T = 0. If the applied electric field increases free charge concentration n 0, then, Ten becomes larger: Ten --¢ T c n + Arch , where AT~n> 0. The increase of T~n
Ten
(l)
As it follows from (1), this ratio can be of significance at small values of I D / S if L / ~ o : ~ 1. If we change the sign of the applied field E, the free charge density and, consequently, the critical current will decrease. In this case the ratio I A I c I/I~ cannot exceed the value of order of l D / S << 1. To clarify this let us consider a circuit of N = S / l D superconductors connected in parallel. If we increase the critical current of a single element, the enhancement of the total critical current of the system will be limited by our ability to increase the critical current of one particular superconductor. That is, the critical current of the system is an unbounded function of the critical current of one particular element. However, if we decrease the critical current of the single superconductor to zero, the total critical current will decrease for not more than of 1 / N part of its initial value, So, in the present approach the critical current is much more sensitive to the increasing of local critical current density than to its decreasing and we shall discuss mainly the case of critical current growth. We shall use the relations between the local values of the critical temperature Ten(Y) = Ten + ATen(Y) and the electric field E ( y ) obtained in [1,11]: ATe n Ten
E
4zr elDn o
gE * "
6d
a log Ten ) - 1 g=
0 log n o
(2)
The estimated value of gE* for YBCO-type superconductors is of ordero of 107 V / c m [11]. Taking for o o evaluations l D = 20 A, S = L = 100 A, ~0 = 10 A, E = 105 V / c m , T - Ten = 10 K, Te, = 50 K, we get A l c / l c - 10%. In the present paper we investigate also the case when both electric and magnetic fields are applied to SNS contact.
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2. M a i n e q u a t i o n s
The order parameter An in N layer is assumed to be small and obeys the linearized GL equation [10]: -- i)2An "b ~n 2An =" 0.
the order parameter magnitude on the current. This allows us to put A t = A. In the " d i r t y " limit the current density in N layer is given by the well-known formula [12]: [
(3)
. 0,a.
with the usual boundary conditions [10]: aA n
An= °tAs'
ax
OAn/by = O,
"B'eI~FD n cn = - - ,
aA s
Y ax
x = +L/2,
y = O, S,
(4) (5)
where As is the order parameter in S layer, a and y are phenomenological constants which characterize NS interface. The order parameter can be chosen to be real at zero current across the contact, and the solution of Eq. (3) is A n = A cosh(x/~n) / cosh(L/2~n) in the absence of the electric field. The constant A is defined from the boundary conditions (4). Approximating As near the NS interface as [12]:
(8)
2kBT
where D n denotes the diffusion coefficient of the free charges in N layer, k B is the Boltzman constant. The supercurrent through the contact reaches its highest value I = I c at ~ q~= 7r/2. So, we can write for Ic with the help of (7) and (8):
Cn A2SLz ~:n sinh(L/~n)
2CnSL,( ,ao )2
As(x) = a s ( L / 2 ) + ( x - L / 2 ) ×exp(-L/~,),
x [ I a ° I as(tl2)] -
where I A°I is the equilibrium value of the order parameter far from the NS interface, sos is the coherence length in S layer, one can rewrite the boundary conditions (4) in the form: ~A n - -
'~ A n +
I A01 =
0x
y
~
,
x =
+L/2.
(6)
"
T~,( y ) = T~n + ( E / g E * ) e x p ( - y / l o ) .
Then, we find at zero current across the contact: A=
vla°l
The coherence length in N layer becomes a function of y due to coordinate dependence of T~n [1]
(~s/~n) t a n h ( t / 2 ~ o ) + ~,/,~ "
Under the same approximation the solution to Eq. (3) with non zero current can be presented as:
- i
cosh( L / 2 ~n) sinh(x/~n) sin(Sq~/2) sinh( L / 2 ~n) ),
~ = 1 - 6 exp(--Y/lD)'
e
= ge*r"
Now (3) can be rewritten as:
c o s h ( x / ~ n ) COS(8~o/2) An = A,
where L a is the contact length in z direction. The Thomas-Fermi approximation for E(y) gives: E(y) = E exp(--Y/lD). This electric field changes the free charge density near the surface: g n / n o = E e x p ( - y / I D ) / E * . Following Refs. [1,11], we can introduce the local transition temperature in N layer Ten(y) under the condition l D > ~:0 and in the notations of Eqs. (2) one gets:
-a2An + (7)
where 8~o is a difference between phase of the order parameter on two sides of the junction. We imply that L >> s~n and the critical current density is relatively low. Then, we can neglect the dependence of
1 - 6 exp(-y/lo)
¢~
A, = 0,
(9)
where Sen= scn(oo). The applied electric field changes the order parameter in both S and N layers. We shall see that the critical current depends drastically on the variation of the order parameter in N layer. As for S layers, it can be readily shown that the change of the order
A.L. Rakhmanov, A.V. Rozhkov / Physica C 267 (1996) 233-242
236
parameter in S layers is not very important. Therefore, we can neglect the electric field effect in S layers and use the boundary conditions (5). (6) for Eq. (9). Let us seek the solution to the problem (9), (5), (6) as a sum
To calculate the critical current we use the same procedure as in the case of E = 0. The critical current across the contact is low and the A~ are independent of the current. This gives a result analogous to (7):
An = E f v ( X ) g . ( y ) ,
f.=a.
(10)
cosh( x~/~ + v/~n) cos(a ~p/2) c-~sh(L~ + 7 / 2 ~.)
11
where v and g. are the eigenvalues and eigenfunctions of the equation: g"+
e exp(--Y/lD) + v f2 g.=0,
(11)
with the boundary conditions g'~ = 0,
y = 0, S,
(12)
sinh(x~/1 + V/~n) s i n ( ~ p / 2 ) 1
-i
s-~X+7/2en)
]"
The supercurrent is maximum at 8~o = w / 2 and the maximum or critical value of the supercurrent is given by the sum: I¢ = )-~.I~,
(14)
V
and the functions fv(x) satisfy the equation:
K--[(l+v)/fn2]fv=O.
t~= (~d~n)¢l + " t ~ ( L ¢ l ; v / 2 f . ) + 3 ' / ~
The general solution of (11) is:
2
S
g . = J. [2 lDV/-88exp( -- y/2ID)/~n]
+ B.J_.[21DV~exp(--y/21D)/fn],
(13)
where J . are the Bessel functions, B~, are constants, and iz2 = - 4 1 ~ v / ~ . The constants B~, and eigenvalues v are defined from the boundary conditions (12). It is easy to show that the set of functions gv is complete and orthogonal. Then, substituting the sum (10) in (6), multiplying it by g~, and integrating the equation from 0 to S, we find the boundary conditions for f~ in the form: 3'
f" +--~f~=
~3"1fsA° ILs
g~ d y /
LS 2
g~ dy,
1 y= +sL.
So, we have at zero current cosh(x~/1 + v / f . )
LzC.~I1 + v × f. sinh(L~/1 + v / f . ) "
(15)
We assume that the coefficients A~ are independent of the current. This assumption is valid if I c is not too high. Direct calculation gives the applicability condition of this approximation in the form:
L2 I1 + vl >> f~.
(16)
The last inequality is true up to high electric field since it is supposed that L2 :~ f2. The condition (16) is violated in the electric fields at which the transition of the N layer to the superconducting state occurs. We shall restrict here our study by the case E.~C gE* or 6 << ( ¢ . / I D ) 2
(17)
since the characteristic field gE* is rather high.
f. = A~ cosh( Lx/T + v / 2 f n ) '
3"la°l a . = ( f j f . ) ¢ T + ,, tanh(L¢l + ,,/2f.) + 3'/o,
x fSg~ d y / fSg 2 d y.
3. Electric field dependence of critical current (thin films)
In the case of thin films if S 2 << L~n,
(18)
A.L. Rakhmanov, A.V. Rozhkov / Physica C 267 (1996) 233-242
237
the Taylor expansion of Bessel functions in (13) can be used
J.[2z exp(--y/21D) ] z" exp(--tzy/21D) ( =
r(1 +~)
t2 4.0
z2 exp(--y/ID) ) 1-
(1 + ~ )
"
.#. z 2 = -77e<< 1,
b
(19) 2.0
where F(x) is the gamma function. We get the equations for eigenvalues v and constants B~, on substituting (13) and (19) in the boundary conditions (12). It is easy to verify that in the case under study the set of eigenvalues v consists of one negative value Vm~n and an infinite number of positive values v(n) -~ (qr~n//S)2 n 2, where n is an integer. The terms in (14) corresponding to the positive eigenvalues are exponentially small, and only the term corresponding to Vm~n should be taken into account. The equations for l~min < 0 and for g(Vmi n) can be written as:
._IDStLS21D sc--~ = tanh (~S).~o ,
/-'min=
~:"2/zz412n,
g(umi,) = const X cosh[(/z/2/D) ( y-- S)].
(20)
We can find Umi" as a function of e with the help of the first of these equations. The ratio I¢(E)/I~(O)can be calculated numerically by means of Eqs. (14), (15) and (20). The results are shown in Fig. 2 as a function of E for different z. The influence of the electric field on the critical current increases with the approach of T to T~,, as was predicted in the introduction. If the electric field is as low as
<< ¢.~/StD,
3,1 a°l ( SCs/~n)V/1 + v tanh(L¢l + v / 2 ~ n ) +
y/a'
we get from (14) and (15) the explicit formula for
Ic(E) in the form I¢(E) ( L ID E ) - exp
I 0.00
I O.01
I
'
I 0.~
E/gE* r a t i o I¢(E)/Ic(O) v e r s u s E / g E * f o r thin ID/~O = 2, L/Co = 100, S / l D = 3; ( a ) r = 0 . 0 5 , ( b ) ~" = F i g . 2. T h e
f i l m at 0.1, ( c )
r = 0.2. It follows from (18), (21) and (22) that a relatively weak electric field can change the critical current of the SNS contact significantly. At higher electric fields
@2/Slr~<< ,~ << ~2/12,
(23)
we get from (20) that IAS//lD >> 1, then, Vmi n =
--
l~,
21/~:g,
(24)
and one derives from (15), in analogy with (22): Ic(E) It(O)
2~2gE* ¢1 - (E/Ec) 2 loSE sinh(L¢l-(E/E¢)2/£n)
(21)
it is obvious that tzS/l D<< 1 and Vmin = --ID6/S. Neglecting the slow electric field dependence of the factor
I¢(0)
00-
2~. s r gE*
(22) "
× s i n h ( ~ , ),
(25)
gE'.
(26)
E¢ = ( ~¢0Vr-~zlID)
Note here, that, together with e and ~n, the Debye screening length 1D is a temperature-dependent quantity. Indeed, the dielectric constant e D and, therefore, l D rapidly vary when the temperature comes close to T~n. We do not specify the function l D =/D(T). However, as it can be seen from (25), (26), this function is important for the temperature dependence of It(E).
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A.L. Rakhmanov, A.V. Rozhkov / Physica C 267 (1996) 233-242
expression (22) for the critical current remains true for E < 0 . For higher fields,
It follows from (25) that the critical current of SNS junction increases drastically when the applied electric field approaches the critical value Ec. This sharp growth of I¢ is due to the superconducting transition in N layer induced by the electric field [1] and E c is the field at which this transition occurs. Let us consider now the case of the opposite polarity of the applied electric field (E < 0). This corresponds to the depletion of the local carrier density and, therefore, to the decrement of the critical current. The spectrum of (11) in the case of negative E consists of the set of positive v. As was argued at the beginning of this section, only the smallest eigenvalue Vm~. should be taken into account. For negative e the equation for ltmin > 0 and g(Vmin) Can be written as:
I ,ls 21D
(l ,ls
/
tan - ~ D
tos
~lSto
Ic( E )
8
(28)
I:(0)
~r2
~/1 + ( E/E¢) 2 sinh(L~/1 + ( E / E : ) 2/~,)
x sinh
.
In contrast to the case of E > 0, nothing special occurs at E = -E¢. In accordance with our qualitative discussion in the introduction, the effect of the electric field on the SNS junction is not symmetric with respect to altering the field polarity. Fig. 3 shows the dependence of log Ic(E)/Ic(O) versus E/gE * for both positive and negative polarities. We can see this asymmetry explicitly: It(E) varies more rapidly in the region of positive E than in the region of negative E.
b ' m i n = ' ~ O l /J,]
gv = const × cos[([ p. [/2tD) ( y -- S)]. If the electric field is low enough. [ 6" [ ~g: ~ g / / S l o ,
Cll~,,
formula (25) must be replaced by
2
] = ~ 6so, .,
"~ l e I <<
(27)
,~,¢.J '0. ¢ ~.q¢O
o
- .
- .
-
- .
Z #
o.oos
o701' ' ~ 6.61~
"-0.2
-0.4
-0.6
Fig. 3. log lc(E)/lc(O) versus E/gE" for thin film at lv/C o = 2, L/Co = 100, S/Io = 3, r = 0.05 for both positive and negative values of E.
A.L. Rakhmanov, A.V. Rozhkov / Physica C 267 (1996)233-242
4. E l e c t r i c field d e p e n d e n c e ( t h i c k films)
o f critical c u r r e n t
If the film is relatively thick and
(29) the perturbation theory can be used to find the
S 2 ~ L~n,
solution of Eq. (11) in the range of weak electric fields under condition (21). The first order approximation with respect to e gives
LI D
It(E) - -
- -
=
.
g~ = const × cos " ~ o y - S) .
(31)
Then, in the approximations used, one can find ix(n) = ( l o ~ r l S ) ( 2 n + 1), n = O , 1, 2 . . . . . n ",~ S / 2 ~ ' l o . (32) With the help of (31), (32) and (15) we get for v(n) > 0 8SLz ..rr2(2n + I) 2
( X
~'l A°I )2 (~:,/~n) lanh(L41 + v(n)/2~.)+7/a C.~/1 + ~,(n)
X sinh( L¢I + ~,(n)/~:o) "
1) 2 la(0).
Let us define: nmax = min{S/zr~:., S/2zr/n} >> 1. Then, Plmax
~
E I,(,)= E /,(,) -- /c(0) E
The dependence (30) coincides qualitatively with the evaluation (1) and Taylor expansion of (22). We should use Eqs. (14), (15) if the inequality (21) is violated. It can be shown that upon condition (17) the eigenvalue spectrum of the problem (11), (12) has the same structure as in the case of thin films. It consists of a single negative eigenvalue Vm~n and an infinite set of positive eigenvalues u(n) > 0. However, the difference between two neighboring eigenvalues is not large and we cannot neglect the terms corresponding to v(n) > 0 in Eq. (14). The values lgrnin and I~,,(E)/I~(O) are given by (24) and (25) at sufficiently high electric fields [see condition (23)]. It follows from (13), (19) and (15) that the contribution to the sum (14) of the terms with large positive v is negligible. Indeed, I~ decays exponentially as ~, tends to infinity. For small [ IX[ we obtain in analogy with (20):
Iv(n)
8 I~,(,,) = 7r2( 2n +
(30)
Ic(0 ) = 1 + 2~0S7,~- gE*
tan /
We assume in our study that L :~ ~n" Then, the terms I~(,) are exponentially small if v ( n ) > 1 and can be neglected. At n << S/~SCn the eigenvalues ~'(n) << 1, and (33) reduces to:
E --
239
(33)
n=0
n=0
= It(0).
n=O
8
,n.2(2n+
1)2 (34)
Finally, we have: Zc(e) 2 gge* - (elec) 2 - - = l + - lc(O) IDSE s i n h ( L ¢ l - ( E / E c ) 2 / ~ n )
Eq. (35) is valid if E < E¢, at E > E¢ a part of N layer near the surface y = 0 transits to the superconducting state. The only difference between expressions for dimensionless critical current (35) (thick film) and (25) (thin film) is the first term in the right hand side of (35). This term is the contribution of the eigenfunctions with positive z,. Fig. 4 shows the electric field dependence of the critical current in the case of thick films. The curves are calculated using Eqs. (30) (low fields) and (35) high fields. The dashed line represents the interpolation between two field regions. The behavior of the function I c ( E ) / l c is qualitatively the same as in the case of thin films, however, the field effect is naturally less pronounced in the case of thick films. Now consider the case of the opposite polarity of the applied electric field. For the low field [see Eq. (27)] the formula (30) is valid. If the field satisfies (28) we should consider the spectrum of (11). For 6 < 0 the set of eigenvalues consists of the infinite sequence of positive numbers v(n) which satisfy (31). In contrast to the case of positive 8, there is no negative eigenvalue in the spectrum. The summation over v(n) has been done in (34). Therefore: /c(e)//o(o)
--- 1.
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A.L. Rakhmanov, A.V. Rozhkov / Physica C 267 (1996) 233-242
where I m = ¢@0/2rrH. This equation has two linearly independent solutions
(x) =exp
3.00-
:, Tm,
(l+(l+v)12m/~g 1 x2) ×M 4 ' 2' 12m '
2.00-
b
(x ) XexpC :)
1.00
f2
1-~' 1: = l m
"~m
(3+(l+v)12m/¢~ 3 x2) ×M 4 ' 2' 12m '
0.00
I 0.00
'
0.~
I
0.~
'
I 0.~
E/gE* 1¢(E)/I¢(0) versus E/gE* lo/~o = 2, L / ~ o = 100, S/lID = 100; (a) ~" =
Fig. 4. T h e ratio
for thick film at
0.05, (b) r = 0.1, (c) ~" = 0.2. D a s h e d line is the extrapolation o f the results obtayned at low electric fields to h i g h e r values o f E / g E *.
where M(a, b, z) is the confluent hypergeometric function, M(a, b, O)= 1. Following the same procedure as in the case H = 0, we find that the critical current is given by the sum (14), where the terms Iv can be presented in the form
[ Thus, just as was the case for the thin film, the effect of the electric field is more pronounced for positive polarity than for negative.
5. Influence of magnetic field Let the external magnetic field H be directed along the y-axis (see Fig. 1). To simplify the problem, we study here the case H :~ HcJI, where H i is the critical field of SNS junction. Under the latter condition the magnetic field is approximately uniform in N layer. To take the magnetic field into account we make a usual substitution in Eq. (9) [10]: O ~ O + i(2~r/@0) A, where A = (0, 0, Hx) is a vector potential. The boundary conditions to the problem remain unchanged. The solution of modified Eq. (9) can be presented in the form (10), where the functions gv are the solutions of Eq. (8). For fv we find:
:: -
k
.6)
[(1 +
TIAOI¢I_H/Hc2
]2
Iv= ~sf~(L/21m, v)/lmfl(L/2l m, u) + y/a 2
S
L~C~ X 21mf](L/21m, v)f2(L/21rn, v)"
(37)
On deriving (37), it is assumed that magnetic field dependence of the bulk value of the order parameter in S layer may be approximated as [12]: l As I = ] A° l t/1 -
H/Hcz,
where /-/ca is the upper critical field of S layers. The eigenvalue spectrum v is defined by the same equations as in the case of H = 0 and we can use the results obtained in Sections 3 and 4, Eq. (37) is valid if I1 +
vll2m/~ >> 1,
and the first argument of the function M(a, b, z) in expressions for fl and f2 is not small. Analogously to (16), this inequality means that the electric field does not induce the superconducting phase transition in N layer.
A.L. Rakhmanov, A.V. Rozhkov / Physica C 267 (1996) 233-242
241
totic properties of the confluent hypergeometric function M(a, c, x) at a ~ 0, we can find Ec(H)
4.0
= Ec(0)~/1
2
"at" ~ n / / l m
2
--- E¢(0)(1 + H~Z4/Trqbo),
2.0
0.0--
' --
I
'
E
'
I
'
0.01
0,00
I 0.02
E/gE* Fig. 5. T h e r a t i o lc(E)/lc(O) v e r s u s E / g E *
f o r thin film a t
1o/~o = 2, L/Go = 100, S / I o = 3, ~- = 0.05; (a) H / H o = 0.1, (b) H / H o = 10. T h e g r a p h f o r H / H o = 0 a l m o s t c o i n c i d e s w i t h
where E~(0) is defined by Eq. (26). To describe the magnetic field dependence of the field effect, we can introduce two characteristic magnetic fields: H 0 = qbo/2zrL2 and H n = (h0/2~r~:n2 (in the case ~n > ~s). The properties of the SNS contact are independent of magnetic field at H ~"~:H 0 [12] and the electric field effect is unaffected by H. The critical current of the SNS contact decreases at H >> H 0 [12] and in this field range the field effect is reduced. At high fields H > H n the electric field effect decreases significantly. The decrease of the field effect with growth of H is due to the suppression of the superconducting transition in N layer by the magnetic field.
(a). 6. Conclusions We calculate the ratio Ic(E, H)/Ic(O, H) in the case of thin films (18) assuming that the limitation (17) on the electric field value is fulfilled. In this case we get lc(E, H ) -- I~.,, and the eigenvalue ~'min is defined by Eq. (20). Then, disregarding in (37) the factor slowly dependent on E in the squared brackets, we obtain
lc( E, H) lc(0, H)
1 ft( L/2lm, O)f2( L/21 m, O) S f]( L/21 m, v)f2( L/21 m, ~') ×
(Sosg~ d y )/So sg~ dy,
(38)
The value I~(0, H ) has been calculated in Ref. [12]. The graphs on Fig. 5 show the results of the numerical calculation of It(E, H)/lc(O, H) by means of Eq. (38) at different values of the ratio H/H o where H 0 = q~0/27rL2. The value of the field effect decreases with the increase of magnetic field at any rate if H~l .~H..~Hc.~. In particular, the magnetic field shifts to higher values the critical electric field Ec under which the superconducting transition in N layer occurs. Using (38) and asymp-
We investigated the electric field effect on the critical current of SNS contact in the framework of GL theory. It was assumed that the transverse electric field influences the critical current across the contact due to the change in the concentration of the free carriers near the sample surface. If this mechanism is valid, the following conclusions can be made. The main contribution to the increase of the critical current of SNS junction results from the increasing of the critical temperature of superconducting transition of N layer. The field effect increases with the decrease of film thickness and the temperature approaching the critical temperature of N layer. The magnetic field suppresses the superconducting transition of N layer and, then, reduces the field effect in the SNS junction in the range of sufficiently high magnetic fields. If the electric field is applied to one of two surfaces of the junction, then, the field effect depends on the field polarity. Changing the sign of E one can a observe significant growth of the critical current and a relatively small decrease of it.
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Acknowledgments This work is supported by the Russian State Program on High-T~ superconductivity (Grant No 93027).
References [1] B.Ya. Shapiro and I.B. Khalfm, Phys. Lett. A 181 (1993) 179. [2] L. Budachkov, I.B. Khalfm and B.Ya. Shapiro, Phys. Rev. B 48 (1993) 1156. [3] J. Manhart, D.G. Schlom and J.G. Bednorz, Z. Phys. B 83 (1991) 307.
[4] A. Walkcnhorst, C. Doughty, S.N. Mao, X.X. Xi, Q. Li and T. Venkatesan, IEEE Trans. Appl. Supercond. 3 (1993) 2929. [5] J. Manhart, J Strobel, J.G. Bednorz and Ch. Gerber, Appl. Phys. Lett. 62 (1993) 630. [6] B.I. Smimov, T.S. Orlova and H.-J. Kaufman, Fizika Tverdogo Tela (Russian: Solid State Physics) 36 (1994) 461. [7] K. Nakajima, K. Yokota, H. Myoren, J. Chert and T. Yamashita, Appl. Phys. Lett. 63 (1993) 684. [8] N. Chandrasekhar, O.T. Vails and A.M. Goldman, Phys. Rev. Lett. 71 (1993) 1079. [9] T. Frey, J. Manhart, J.G. Bednorz and E.J. Williams, Phys. Rev. B 51 (1995) 3257. [10] P.G. de Germes, Superconductivity of Metals and Alloys (Benjamin, New York, 1966). [11] B.Ya. Shapiro and I.B. Khalfin, Physica C 219 (1994) 465. [12] L. Dobroslavlivijevic-Grujic, Z. Radovic, Physica C 185-189 (1991) 2313.
Physica C 267 (1996) 233-242
Electric field effect on the critical current of SNS contact A.L. R a k h m a n o v
a,
A . V . R o z h k o v * ,a,b
a Scientific Center]or Application Problems in Electrodynamics, Russian Academy of Sciences, 174122, lzhorskaya Str. 13 / 19, Moscow, Russia b Department of Physics, University of California at San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0354, USA
Received 4 March 1996; revised manuscript received 14 June 1996
Abstract The effect of electric field on the critical current of SNS contact is studied in the framework of Ginzburg-Landau theory. We show that electric field can significantly enlarge the critical current of the contact. This effect is especially strong when the temperature of the sample is close to the superconducting transition temperature of normal layer Tcn. The electric field effect depends on film thickness: the thicker the film the weaker the effect. Nevertheless, it can be significant for a thick film when the temperature is close to T~n. The cooperative effect of electric and magnetic fields is also studied. PACS: 74.50 + r; 74.25.Nf
1. Introduction The electric field effect is commonly observed in high-T~ superconductors [1-9]. Different mechanisms are suggested to explain this observation [ 1 3,8,9]. One of the most appropriate explanations is that the electric field changes the concentration of the free charge carriers near the sample surface [1,2]. Recent experiments shows that this mechanism is appropriate at any rate for BSCO and YBCO superconductors [9]. It suggests that the electric field E applied to the surface of superconductor induces the change 8n in the free charge carriers concentration n o in the layer of the thickness of l D, where 1D is the Debye screening length. In the Thomas-Fermi approximation l D = (gd//47rUF e 2)I/2 [ 1,2], where e d is a dielectric constant, u F is the density of states at
* Corresponding author.
the Fermi level. A transverse electric field changes longitudinal current-voltage characteristic of superconductor if I o > ~ [1,2] where sc is the coherence length. The last inequality may be fulfilled for high-T~ superconductors [ 1,2] because of the small coherence length and relatively high 8 a. In papers [5-7] the observed changes in superconducting properties with the electric field are ascribed to the field effect on weak links that usually exists in high temperature superconductor materials. Indeed, in the experiments [5-7] it has been detected a strong dependence of current-voltage characteristics of the isolated weak links in YBCO films on the transverse electric field. The present paper is aimed to investigate the electric field effect on the critical current of SNS (superconductor-normal metal-superconductor) contact. We suppose that normal layer is characterized by the non zero temperature of superconducting transition To, and can be described in the framework of
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A.L. Rakhmanov, A.V. Rozhkov / Physica C 267 (1996)233-242
234
results in the growth of the critical current of the junction I c. The change in the critical current A I~ is equal to:
Y
AI c
L
lD ATcn
1--~-'" ~o------~ S S
v
X
z
I E,H
Fig. 1. SNS contact in electric and magnetic field: geometry of the problem.
Ginzburg-Landau (GL) theory. The sample temperature T is assumed to be between Ten and Te: Ten < T < T~ (where Te is the critical temperature of superconductor). It will be shown that it is a temperature range where the effect is the most pronounced. Let us consider SNS contact of the thickness L parallel to the yz-plane (Fig. 1). The transport current flows along x-axis and electric field E is directed along y-axis. We study here the usual experimental situation and assume that the electric field is equal to E at y = 0, decays at y = S, and It) a~ S,L.
The critical current density jc of SNS junction is a function of the ratio L / ~ n, where ~, is the coherence length in normal metal [10]. For L / ~ , : ~ 1 the critical current can be evaluated as Jc ~ e x p ( - L / ~ n ) , where in the framework of GL theory:
fn = f O / G ,
(r-- Te.)/Te.,
and ~0 is the coherence length in N layer at T = 0. If the applied electric field increases free charge concentration n 0, then, Ten becomes larger: Ten --¢ T c n + Arch , where AT~n> 0. The increase of T~n
Ten
(l)
As it follows from (1), this ratio can be of significance at small values of I D / S if L / ~ o : ~ 1. If we change the sign of the applied field E, the free charge density and, consequently, the critical current will decrease. In this case the ratio I A I c I/I~ cannot exceed the value of order of l D / S << 1. To clarify this let us consider a circuit of N = S / l D superconductors connected in parallel. If we increase the critical current of a single element, the enhancement of the total critical current of the system will be limited by our ability to increase the critical current of one particular superconductor. That is, the critical current of the system is an unbounded function of the critical current of one particular element. However, if we decrease the critical current of the single superconductor to zero, the total critical current will decrease for not more than of 1 / N part of its initial value, So, in the present approach the critical current is much more sensitive to the increasing of local critical current density than to its decreasing and we shall discuss mainly the case of critical current growth. We shall use the relations between the local values of the critical temperature Ten(Y) = Ten + ATen(Y) and the electric field E ( y ) obtained in [1,11]: ATe n Ten
E
4zr elDn o
gE * "
6d
a log Ten ) - 1 g=
0 log n o
(2)
The estimated value of gE* for YBCO-type superconductors is of ordero of 107 V / c m [11]. Taking for o o evaluations l D = 20 A, S = L = 100 A, ~0 = 10 A, E = 105 V / c m , T - Ten = 10 K, Te, = 50 K, we get A l c / l c - 10%. In the present paper we investigate also the case when both electric and magnetic fields are applied to SNS contact.
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A.L. Rakhmanov, A.V. Rozhkov / Physica C 267 (1996) 233-242
2. M a i n e q u a t i o n s
The order parameter An in N layer is assumed to be small and obeys the linearized GL equation [10]: -- i)2An "b ~n 2An =" 0.
the order parameter magnitude on the current. This allows us to put A t = A. In the " d i r t y " limit the current density in N layer is given by the well-known formula [12]: [
(3)
. 0,a.
with the usual boundary conditions [10]: aA n
An= °tAs'
ax
OAn/by = O,
"B'eI~FD n cn = - - ,
aA s
Y ax
x = +L/2,
y = O, S,
(4) (5)
where As is the order parameter in S layer, a and y are phenomenological constants which characterize NS interface. The order parameter can be chosen to be real at zero current across the contact, and the solution of Eq. (3) is A n = A cosh(x/~n) / cosh(L/2~n) in the absence of the electric field. The constant A is defined from the boundary conditions (4). Approximating As near the NS interface as [12]:
(8)
2kBT
where D n denotes the diffusion coefficient of the free charges in N layer, k B is the Boltzman constant. The supercurrent through the contact reaches its highest value I = I c at ~ q~= 7r/2. So, we can write for Ic with the help of (7) and (8):
Cn A2SLz ~:n sinh(L/~n)
2CnSL,( ,ao )2
As(x) = a s ( L / 2 ) + ( x - L / 2 ) ×exp(-L/~,),
x [ I a ° I as(tl2)] -
where I A°I is the equilibrium value of the order parameter far from the NS interface, sos is the coherence length in S layer, one can rewrite the boundary conditions (4) in the form: ~A n - -
'~ A n +
I A01 =
0x
y
~
,
x =
+L/2.
(6)
"
T~,( y ) = T~n + ( E / g E * ) e x p ( - y / l o ) .
Then, we find at zero current across the contact: A=
vla°l
The coherence length in N layer becomes a function of y due to coordinate dependence of T~n [1]
(~s/~n) t a n h ( t / 2 ~ o ) + ~,/,~ "
Under the same approximation the solution to Eq. (3) with non zero current can be presented as:
- i
cosh( L / 2 ~n) sinh(x/~n) sin(Sq~/2) sinh( L / 2 ~n) ),
~ = 1 - 6 exp(--Y/lD)'
e
= ge*r"
Now (3) can be rewritten as:
c o s h ( x / ~ n ) COS(8~o/2) An = A,
where L a is the contact length in z direction. The Thomas-Fermi approximation for E(y) gives: E(y) = E exp(--Y/lD). This electric field changes the free charge density near the surface: g n / n o = E e x p ( - y / I D ) / E * . Following Refs. [1,11], we can introduce the local transition temperature in N layer Ten(y) under the condition l D > ~:0 and in the notations of Eqs. (2) one gets:
-a2An + (7)
where 8~o is a difference between phase of the order parameter on two sides of the junction. We imply that L >> s~n and the critical current density is relatively low. Then, we can neglect the dependence of
1 - 6 exp(-y/lo)
¢~
A, = 0,
(9)
where Sen= scn(oo). The applied electric field changes the order parameter in both S and N layers. We shall see that the critical current depends drastically on the variation of the order parameter in N layer. As for S layers, it can be readily shown that the change of the order
A.L. Rakhmanov, A.V. Rozhkov / Physica C 267 (1996) 233-242
236
parameter in S layers is not very important. Therefore, we can neglect the electric field effect in S layers and use the boundary conditions (5). (6) for Eq. (9). Let us seek the solution to the problem (9), (5), (6) as a sum
To calculate the critical current we use the same procedure as in the case of E = 0. The critical current across the contact is low and the A~ are independent of the current. This gives a result analogous to (7):
An = E f v ( X ) g . ( y ) ,
f.=a.
(10)
cosh( x~/~ + v/~n) cos(a ~p/2) c-~sh(L~ + 7 / 2 ~.)
11
where v and g. are the eigenvalues and eigenfunctions of the equation: g"+
e exp(--Y/lD) + v f2 g.=0,
(11)
with the boundary conditions g'~ = 0,
y = 0, S,
(12)
sinh(x~/1 + V/~n) s i n ( ~ p / 2 ) 1
-i
s-~X+7/2en)
]"
The supercurrent is maximum at 8~o = w / 2 and the maximum or critical value of the supercurrent is given by the sum: I¢ = )-~.I~,
(14)
V
and the functions fv(x) satisfy the equation:
K--[(l+v)/fn2]fv=O.
t~= (~d~n)¢l + " t ~ ( L ¢ l ; v / 2 f . ) + 3 ' / ~
The general solution of (11) is:
2
S
g . = J. [2 lDV/-88exp( -- y/2ID)/~n]
+ B.J_.[21DV~exp(--y/21D)/fn],
(13)
where J . are the Bessel functions, B~, are constants, and iz2 = - 4 1 ~ v / ~ . The constants B~, and eigenvalues v are defined from the boundary conditions (12). It is easy to show that the set of functions gv is complete and orthogonal. Then, substituting the sum (10) in (6), multiplying it by g~, and integrating the equation from 0 to S, we find the boundary conditions for f~ in the form: 3'
f" +--~f~=
~3"1fsA° ILs
g~ d y /
LS 2
g~ dy,
1 y= +sL.
So, we have at zero current cosh(x~/1 + v / f . )
LzC.~I1 + v × f. sinh(L~/1 + v / f . ) "
(15)
We assume that the coefficients A~ are independent of the current. This assumption is valid if I c is not too high. Direct calculation gives the applicability condition of this approximation in the form:
L2 I1 + vl >> f~.
(16)
The last inequality is true up to high electric field since it is supposed that L2 :~ f2. The condition (16) is violated in the electric fields at which the transition of the N layer to the superconducting state occurs. We shall restrict here our study by the case E.~C gE* or 6 << ( ¢ . / I D ) 2
(17)
since the characteristic field gE* is rather high.
f. = A~ cosh( Lx/T + v / 2 f n ) '
3"la°l a . = ( f j f . ) ¢ T + ,, tanh(L¢l + ,,/2f.) + 3'/o,
x fSg~ d y / fSg 2 d y.
3. Electric field dependence of critical current (thin films)
In the case of thin films if S 2 << L~n,
(18)
A.L. Rakhmanov, A.V. Rozhkov / Physica C 267 (1996) 233-242
237
the Taylor expansion of Bessel functions in (13) can be used
J.[2z exp(--y/21D) ] z" exp(--tzy/21D) ( =
r(1 +~)
t2 4.0
z2 exp(--y/ID) ) 1-
(1 + ~ )
"
.#. z 2 = -77e<< 1,
b
(19) 2.0
where F(x) is the gamma function. We get the equations for eigenvalues v and constants B~, on substituting (13) and (19) in the boundary conditions (12). It is easy to verify that in the case under study the set of eigenvalues v consists of one negative value Vm~n and an infinite number of positive values v(n) -~ (qr~n//S)2 n 2, where n is an integer. The terms in (14) corresponding to the positive eigenvalues are exponentially small, and only the term corresponding to Vm~n should be taken into account. The equations for l~min < 0 and for g(Vmi n) can be written as:
._IDStLS21D sc--~ = tanh (~S).~o ,
/-'min=
~:"2/zz412n,
g(umi,) = const X cosh[(/z/2/D) ( y-- S)].
(20)
We can find Umi" as a function of e with the help of the first of these equations. The ratio I¢(E)/I~(O)can be calculated numerically by means of Eqs. (14), (15) and (20). The results are shown in Fig. 2 as a function of E for different z. The influence of the electric field on the critical current increases with the approach of T to T~,, as was predicted in the introduction. If the electric field is as low as
<< ¢.~/StD,
3,1 a°l ( SCs/~n)V/1 + v tanh(L¢l + v / 2 ~ n ) +
y/a'
we get from (14) and (15) the explicit formula for
Ic(E) in the form I¢(E) ( L ID E ) - exp
I 0.00
I O.01
I
'
I 0.~
E/gE* r a t i o I¢(E)/Ic(O) v e r s u s E / g E * f o r thin ID/~O = 2, L/Co = 100, S / l D = 3; ( a ) r = 0 . 0 5 , ( b ) ~" = F i g . 2. T h e
f i l m at 0.1, ( c )
r = 0.2. It follows from (18), (21) and (22) that a relatively weak electric field can change the critical current of the SNS contact significantly. At higher electric fields
@2/Slr~<< ,~ << ~2/12,
(23)
we get from (20) that IAS//lD >> 1, then, Vmi n =
--
l~,
21/~:g,
(24)
and one derives from (15), in analogy with (22): Ic(E) It(O)
2~2gE* ¢1 - (E/Ec) 2 loSE sinh(L¢l-(E/E¢)2/£n)
(21)
it is obvious that tzS/l D<< 1 and Vmin = --ID6/S. Neglecting the slow electric field dependence of the factor
I¢(0)
00-
2~. s r gE*
(22) "
× s i n h ( ~ , ),
(25)
gE'.
(26)
E¢ = ( ~¢0Vr-~zlID)
Note here, that, together with e and ~n, the Debye screening length 1D is a temperature-dependent quantity. Indeed, the dielectric constant e D and, therefore, l D rapidly vary when the temperature comes close to T~n. We do not specify the function l D =/D(T). However, as it can be seen from (25), (26), this function is important for the temperature dependence of It(E).
238
A.L. Rakhmanov, A.V. Rozhkov / Physica C 267 (1996) 233-242
expression (22) for the critical current remains true for E < 0 . For higher fields,
It follows from (25) that the critical current of SNS junction increases drastically when the applied electric field approaches the critical value Ec. This sharp growth of I¢ is due to the superconducting transition in N layer induced by the electric field [1] and E c is the field at which this transition occurs. Let us consider now the case of the opposite polarity of the applied electric field (E < 0). This corresponds to the depletion of the local carrier density and, therefore, to the decrement of the critical current. The spectrum of (11) in the case of negative E consists of the set of positive v. As was argued at the beginning of this section, only the smallest eigenvalue Vm~. should be taken into account. For negative e the equation for ltmin > 0 and g(Vmin) Can be written as:
I ,ls 21D
(l ,ls
/
tan - ~ D
tos
~lSto
Ic( E )
8
(28)
I:(0)
~r2
~/1 + ( E/E¢) 2 sinh(L~/1 + ( E / E : ) 2/~,)
x sinh
.
In contrast to the case of E > 0, nothing special occurs at E = -E¢. In accordance with our qualitative discussion in the introduction, the effect of the electric field on the SNS junction is not symmetric with respect to altering the field polarity. Fig. 3 shows the dependence of log Ic(E)/Ic(O) versus E/gE * for both positive and negative polarities. We can see this asymmetry explicitly: It(E) varies more rapidly in the region of positive E than in the region of negative E.
b ' m i n = ' ~ O l /J,]
gv = const × cos[([ p. [/2tD) ( y -- S)]. If the electric field is low enough. [ 6" [ ~g: ~ g / / S l o ,
Cll~,,
formula (25) must be replaced by
2
] = ~ 6so, .,
"~ l e I <<
(27)
,~,¢.J '0. ¢ ~.q¢O
o
- .
- .
-
- .
Z #
o.oos
o701' ' ~ 6.61~
"-0.2
-0.4
-0.6
Fig. 3. log lc(E)/lc(O) versus E/gE" for thin film at lv/C o = 2, L/Co = 100, S/Io = 3, r = 0.05 for both positive and negative values of E.
A.L. Rakhmanov, A.V. Rozhkov / Physica C 267 (1996)233-242
4. E l e c t r i c field d e p e n d e n c e ( t h i c k films)
o f critical c u r r e n t
If the film is relatively thick and
(29) the perturbation theory can be used to find the
S 2 ~ L~n,
solution of Eq. (11) in the range of weak electric fields under condition (21). The first order approximation with respect to e gives
LI D
It(E) - -
- -
=
.
g~ = const × cos " ~ o y - S) .
(31)
Then, in the approximations used, one can find ix(n) = ( l o ~ r l S ) ( 2 n + 1), n = O , 1, 2 . . . . . n ",~ S / 2 ~ ' l o . (32) With the help of (31), (32) and (15) we get for v(n) > 0 8SLz ..rr2(2n + I) 2
( X
~'l A°I )2 (~:,/~n) lanh(L41 + v(n)/2~.)+7/a C.~/1 + ~,(n)
X sinh( L¢I + ~,(n)/~:o) "
1) 2 la(0).
Let us define: nmax = min{S/zr~:., S/2zr/n} >> 1. Then, Plmax
~
E I,(,)= E /,(,) -- /c(0) E
The dependence (30) coincides qualitatively with the evaluation (1) and Taylor expansion of (22). We should use Eqs. (14), (15) if the inequality (21) is violated. It can be shown that upon condition (17) the eigenvalue spectrum of the problem (11), (12) has the same structure as in the case of thin films. It consists of a single negative eigenvalue Vm~n and an infinite set of positive eigenvalues u(n) > 0. However, the difference between two neighboring eigenvalues is not large and we cannot neglect the terms corresponding to v(n) > 0 in Eq. (14). The values lgrnin and I~,,(E)/I~(O) are given by (24) and (25) at sufficiently high electric fields [see condition (23)]. It follows from (13), (19) and (15) that the contribution to the sum (14) of the terms with large positive v is negligible. Indeed, I~ decays exponentially as ~, tends to infinity. For small [ IX[ we obtain in analogy with (20):
Iv(n)
8 I~,(,,) = 7r2( 2n +
(30)
Ic(0 ) = 1 + 2~0S7,~- gE*
tan /
We assume in our study that L :~ ~n" Then, the terms I~(,) are exponentially small if v ( n ) > 1 and can be neglected. At n << S/~SCn the eigenvalues ~'(n) << 1, and (33) reduces to:
E --
239
(33)
n=0
n=0
= It(0).
n=O
8
,n.2(2n+
1)2 (34)
Finally, we have: Zc(e) 2 gge* - (elec) 2 - - = l + - lc(O) IDSE s i n h ( L ¢ l - ( E / E c ) 2 / ~ n )
Eq. (35) is valid if E < E¢, at E > E¢ a part of N layer near the surface y = 0 transits to the superconducting state. The only difference between expressions for dimensionless critical current (35) (thick film) and (25) (thin film) is the first term in the right hand side of (35). This term is the contribution of the eigenfunctions with positive z,. Fig. 4 shows the electric field dependence of the critical current in the case of thick films. The curves are calculated using Eqs. (30) (low fields) and (35) high fields. The dashed line represents the interpolation between two field regions. The behavior of the function I c ( E ) / l c is qualitatively the same as in the case of thin films, however, the field effect is naturally less pronounced in the case of thick films. Now consider the case of the opposite polarity of the applied electric field. For the low field [see Eq. (27)] the formula (30) is valid. If the field satisfies (28) we should consider the spectrum of (11). For 6 < 0 the set of eigenvalues consists of the infinite sequence of positive numbers v(n) which satisfy (31). In contrast to the case of positive 8, there is no negative eigenvalue in the spectrum. The summation over v(n) has been done in (34). Therefore: /c(e)//o(o)
--- 1.
240
A.L. Rakhmanov, A.V. Rozhkov / Physica C 267 (1996) 233-242
where I m = ¢@0/2rrH. This equation has two linearly independent solutions
(x) =exp
3.00-
:, Tm,
(l+(l+v)12m/~g 1 x2) ×M 4 ' 2' 12m '
2.00-
b
(x ) XexpC :)
1.00
f2
1-~' 1: = l m
"~m
(3+(l+v)12m/¢~ 3 x2) ×M 4 ' 2' 12m '
0.00
I 0.00
'
0.~
I
0.~
'
I 0.~
E/gE* 1¢(E)/I¢(0) versus E/gE* lo/~o = 2, L / ~ o = 100, S/lID = 100; (a) ~" =
Fig. 4. T h e ratio
for thick film at
0.05, (b) r = 0.1, (c) ~" = 0.2. D a s h e d line is the extrapolation o f the results obtayned at low electric fields to h i g h e r values o f E / g E *.
where M(a, b, z) is the confluent hypergeometric function, M(a, b, O)= 1. Following the same procedure as in the case H = 0, we find that the critical current is given by the sum (14), where the terms Iv can be presented in the form
[ Thus, just as was the case for the thin film, the effect of the electric field is more pronounced for positive polarity than for negative.
5. Influence of magnetic field Let the external magnetic field H be directed along the y-axis (see Fig. 1). To simplify the problem, we study here the case H :~ HcJI, where H i is the critical field of SNS junction. Under the latter condition the magnetic field is approximately uniform in N layer. To take the magnetic field into account we make a usual substitution in Eq. (9) [10]: O ~ O + i(2~r/@0) A, where A = (0, 0, Hx) is a vector potential. The boundary conditions to the problem remain unchanged. The solution of modified Eq. (9) can be presented in the form (10), where the functions gv are the solutions of Eq. (8). For fv we find:
:: -
k
.6)
[(1 +
TIAOI¢I_H/Hc2
]2
Iv= ~sf~(L/21m, v)/lmfl(L/2l m, u) + y/a 2
S
L~C~ X 21mf](L/21m, v)f2(L/21rn, v)"
(37)
On deriving (37), it is assumed that magnetic field dependence of the bulk value of the order parameter in S layer may be approximated as [12]: l As I = ] A° l t/1 -
H/Hcz,
where /-/ca is the upper critical field of S layers. The eigenvalue spectrum v is defined by the same equations as in the case of H = 0 and we can use the results obtained in Sections 3 and 4, Eq. (37) is valid if I1 +
vll2m/~ >> 1,
and the first argument of the function M(a, b, z) in expressions for fl and f2 is not small. Analogously to (16), this inequality means that the electric field does not induce the superconducting phase transition in N layer.
A.L. Rakhmanov, A.V. Rozhkov / Physica C 267 (1996) 233-242
241
totic properties of the confluent hypergeometric function M(a, c, x) at a ~ 0, we can find Ec(H)
4.0
= Ec(0)~/1
2
"at" ~ n / / l m
2
--- E¢(0)(1 + H~Z4/Trqbo),
2.0
0.0--
' --
I
'
E
'
I
'
0.01
0,00
I 0.02
E/gE* Fig. 5. T h e r a t i o lc(E)/lc(O) v e r s u s E / g E *
f o r thin film a t
1o/~o = 2, L/Go = 100, S / I o = 3, ~- = 0.05; (a) H / H o = 0.1, (b) H / H o = 10. T h e g r a p h f o r H / H o = 0 a l m o s t c o i n c i d e s w i t h
where E~(0) is defined by Eq. (26). To describe the magnetic field dependence of the field effect, we can introduce two characteristic magnetic fields: H 0 = qbo/2zrL2 and H n = (h0/2~r~:n2 (in the case ~n > ~s). The properties of the SNS contact are independent of magnetic field at H ~"~:H 0 [12] and the electric field effect is unaffected by H. The critical current of the SNS contact decreases at H >> H 0 [12] and in this field range the field effect is reduced. At high fields H > H n the electric field effect decreases significantly. The decrease of the field effect with growth of H is due to the suppression of the superconducting transition in N layer by the magnetic field.
(a). 6. Conclusions We calculate the ratio Ic(E, H)/Ic(O, H) in the case of thin films (18) assuming that the limitation (17) on the electric field value is fulfilled. In this case we get lc(E, H ) -- I~.,, and the eigenvalue ~'min is defined by Eq. (20). Then, disregarding in (37) the factor slowly dependent on E in the squared brackets, we obtain
lc( E, H) lc(0, H)
1 ft( L/2lm, O)f2( L/21 m, O) S f]( L/21 m, v)f2( L/21 m, ~') ×
(Sosg~ d y )/So sg~ dy,
(38)
The value I~(0, H ) has been calculated in Ref. [12]. The graphs on Fig. 5 show the results of the numerical calculation of It(E, H)/lc(O, H) by means of Eq. (38) at different values of the ratio H/H o where H 0 = q~0/27rL2. The value of the field effect decreases with the increase of magnetic field at any rate if H~l .~H..~Hc.~. In particular, the magnetic field shifts to higher values the critical electric field Ec under which the superconducting transition in N layer occurs. Using (38) and asymp-
We investigated the electric field effect on the critical current of SNS contact in the framework of GL theory. It was assumed that the transverse electric field influences the critical current across the contact due to the change in the concentration of the free carriers near the sample surface. If this mechanism is valid, the following conclusions can be made. The main contribution to the increase of the critical current of SNS junction results from the increasing of the critical temperature of superconducting transition of N layer. The field effect increases with the decrease of film thickness and the temperature approaching the critical temperature of N layer. The magnetic field suppresses the superconducting transition of N layer and, then, reduces the field effect in the SNS junction in the range of sufficiently high magnetic fields. If the electric field is applied to one of two surfaces of the junction, then, the field effect depends on the field polarity. Changing the sign of E one can a observe significant growth of the critical current and a relatively small decrease of it.
242
A.L. Rakhmanov, A.V. Rozhkov / Physica C 267 (1996) 233-242
Acknowledgments This work is supported by the Russian State Program on High-T~ superconductivity (Grant No 93027).
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