Induction of superconductivity by strong electric field

Solid State Communications, Printed in Great Britain.

0038-1098185 $3.00 + .OO Pergamon Press Ltd.

Vol. 56, No. 1, pp. 149-15 1, 1985.



B.Ya. Shapiro The Institute

of Inorganic Chemistry,

Siberian Division of the Academy of Sciences, Novosibirsk,

630090, USSR

(Received 28 February 1985 by E.A. Kaner)

It is shown that strong electric field perpendicular to the surface of a solid with free charges creates the surface inhomogeneous superconducting state with critical parameters depending on the surface electrical potential only. The critical temperature and magnetic field of the surface inhomogeneous state can be high in the strong electric field.

THE INFLUENCE of the electrical field upon the superconductivity has been the object of intensive investigation for a long time. In 1960 Clover III and Sherill [l] observed a small increase of the critical temperature AT= lo-’ K of a tin film placed in the strong electric field E =Z10’ V/sm. In 1965 Stadler [2] used very strong electric field (the inner field of the ferroelectric) and observed the critical temperature increasing AT% 10e3 K in the tin film with film thickness of about 150 A. The reason for the critical temperature increasing is the electron density increasing near the surface. This was pointed out in 1965 by Sandomirskii [3] . In 1975 the superconductivity was observed in the semiconductor InAs placed in the electric field [4]. (In the absence of the electric field, this semiconductor is not a superconductor). The critical temperature in this case was rather high T, = 1.9 K. The theories devoted to induction of superconductivity by the external electric field are based on the supposition about the potential well for electrons (holes) near the surface. In this potential well two types of the energy level exist: the localized electron states in the potential well, and free electron states over the potential well. The detailed theory of the surface, two-dimensional superconductivity based on the supposition about the existence of a single surface electron zone is discussed in ]51- [71 . The theory of the surface superconducting state arizing due to the increasing of the density of electron states over the potential well has been constructed in [ 8]- [ 1 I] . This theory is devoted to the case of a weak electric field (e& < po, c#+ is the electric potential on the surface, ~0 is the Fermi energy in the bulk, e is the electron charge). However, in the strong electric field the electric potential has to satisfy to the condition (e& >>po). In this case the main role in the superconductivity is played

by the quasi-classical electrons in the surface potential well. To obtain the formal solution of this problem one has to use the Hamiltonian of the interacting electrons in the potential well






ti+oE”V’) Jlod3r




Z(P) = P2 /2m* - [email protected](x) kimp = ; Iti: Here A is the action, is the potential. As usual, linear equation



constant of the electron-electron interFermi operator, V(r - ro) is the impurity we obtain from the Hamiltonian for the order parameter

x(C;~~)17(r)+(V(x)--e)9(r) x(z) = w/2

(1) the

= 0

+ z/2) - NW

E = In (1,14wo/T); &


= vl/6aT;

V(x) = (N(x)A)-’

(2’ = (-iv--?A)I,

with the boundary



= 0.


on the surface (4)

Here J/ is the digamma-function, v is the Fermi velocity, I is the mean free path of the electron, wo is the Debye frequency, T is the temperature, A is the vector potential, c is the speed of light. The model with a constant value of the electronelectron interaction A, used here, takes into account only the increase of the electron state density at the 149




Fermi level N(x) near the surface due to the electric field. Using the Thomas-Fermi method under condition e& % p. we have for e(x) the equation 4.e (2Ae)3’2

[email protected] -= &2


[email protected]

e(o) = tik ;

ww, =+. = 0.

The solution of the equation

(5) is

G(x) = &z (XIX0 + 1)-4 * 314 l/4 112 $k e x0 == 0, 4h3”/(2me) N(x)



([email protected])1’2/7r2 (x/x0 + 1)’ .


Fig. 1. The critical temperature of the surface superconducting state vs electrostatic potential on the surface.


can be done in equation (3). In this case, the equation obtain the critical magnetic field Hk is

To obtain the critical temperature of the surface superconducting state Hk one has to calculate the lowest energy level of equation (3) under condition A = 0. Due to the condition x0 3 d % ET (d is the characteristic length of the surface superconducting state) one has to expand function X and “potential” V(x). The equation obtained has the Airy equation form

x Afh(x); C

(an,/ax)l = 1,14wD [email protected]

exp (- G-’ (&));


= N(O)A


x’ = dX/du.


equation in this case

= 0. x=+0


As usual, we obtain from (14)-( 1.5) (9)

= 0.


AH = Hk -&(h);

u = %$Hc2

The boundary


Vol. 56. No. 1


AH g c/e (&X’(U)Xo)2’3 G(&) The critical field H,, is determined

a r$;a .


from the equation

From equations, (8)-( 10) we obtain X(u) + In T

Tk = Tok(l -c&/L2);

cl - 1



> .&., .


= (Xo &0kG(&))‘i3

= 0,


[email protected]‘k

and has the form

2 c TGb- T It should be noted that in the main approximation -; T’T,++k (&r/d is the small parameter) the critical temperature [email protected] Hc2 (gk) = ‘2 “‘@k (18) depends only on the electric potential on the surface. 2 c The proximity effects decrease the critical temperature --’ T+O. T, in the case of the “weak electric field”. In our case, n2 eGok ’ the proximity effect is absent and, hence, critical The function Hc2 (&) is shown in Fig. 2. temperature Tk may be quite high. The characteristic length of the surface superIf the substance in the absence of the electric field is a superconductor, then the critical temperature Tk conducting state in this case is is a function of the critical temperature in the bulk dHI - (%-X0 X(U))“” Z+ TV. (19) Tk z wr, (u,D/TJ([email protected]~)“7 . (13) If the external magnetic field is parallel to the surface, the superconducting state is not localized by the (The dependence Tk (&) is shown in Fig. 1). magnetic field. In this case, to obtain the critical If the external magnetic field is applied to the magnetic field one has to use the perturbation theory. surface together with the electric field then the superconducting state arises localized along the magnetic Here different situations are possible depending on the different relations between the characteristic parameters: field. Under condition A, = Hz an approximate the Armour radius aH = (&!eH)” the characteristic separation of the variables (&& is the small parameter)


Vol. 56, No. 1





= Hc3(&)-6H

6H =

Fig. 2. The critical magnetic field of the surface superconducting state vs electrostatic potential on the surface at various temperature values. (Curve 1 relates to the case T = 0, curves 2 and 3 relate to values Tz and T3, respectively, (T3 > Tz)). size of the surface superconducting state d, and the temperature. Under condition A, =H, (the magnetic field is parallel to the surface) equation (3) has the form length of the (d HI exe) (&,, is the characteristic sur t’ace superconducting state in the parallel magnetic field)

151 (23)






(24) ’

The structure of the surface superconducting state depends on the direction of the external magnetic field. If the magnetic field is perpendicular to the surface then the inhomogeneous superconducting state has the form of the lattice localized along the magnetic field. The characteristic size along the magnetic field is dHI. The period of the superconducting structure is -aH. If the magnetic field is parallel to the surface then inhomogeneous superconducting state has the plate form with characteristic size of about min (d, an}. For SrTiOa with characteristic parameters T, = 0,3K, e&

wD = lo2 K,

/.I0 q

lo3 K,

= 104K.

(The electric field on the surface = 10’ V/sm) one obtains from (13). For experimental observation of the effects pointed out here the wide class of substances can be used in particular sandwiches consisting of ferroelectric and superconducting semiconductor. However, in such experiments two condictions are to be satisfied. The surface has to be perfectly smooth, because inhomogeneities may change the electron distribution in such a way that the superconducting state does not arise. The dielectric layer between the metal and superconductor (usually of MOS or MOMi structure) has to be without the acceptors because the electrons from the superconductors can pass to the dielectric layer [ 121.

xjf:(-a: +~)Hvx-Yo++(--$&-)+ +lnL

n = 0;


= (ic/2eH)a,.


% 1. If aH 9 d 9 .&r then one has to take into account the magnetic field using the perturbation theory. In this case the critical magnetic field Hk has the form

(21) 2. If d % an % .& then in the main approximation the function V(x) in equation (3) can be neglected. As a result we obtain a well-known equation to calculate a surface critical magnetic field Hc3 for the superconductor with the constant value of the electronelectron interaction G = G(&). Representing the magnetic field H in the form H = Hc3 - 6H 6H = H,, H in equation (3) and multiplying it by the function (-rx2), r = 0.7eH/c one obtains after $0 - exp integrating for SH

(22) 3. The condition ET % d takes place in the low temperature region. In this case $Q” is not a small parameter. Substituting magnetic field in the form H = Hc3 - 6H into function x and expanding x by SH we obtain

Acknowledgement - I would like to thank Prof. Yu.V. Kopaev for helpful discussion.

REFERENCES 1. 2. 3. 4. 2. 7: 8. 9. 10. 11. 12.

R.E. Clover & M.D. Sherill, Phys. Rev. Lett. 5, 248 (1960). H.L. Stadler, Phys. Rev. Lett. 14,979 (1965) V.B. Sandomirskii, Pis’ma v Zh. Eksp. Teor. Fiz. 32,543 (1965). S. Kawaji, S. Miki & T. Kinoshita, J. Phys. Sot. Japan. 39,163l (1975). Y. Takada, J. Phys. Sot. Japan 45,786 (1978). Y. Takada, J. Phys. Sot. Japan 49,1713 (1980). M.J. Kelly & W. Hanke, Phys. Rev. B23, 112 (1981). B.Ya. Shapiro, Phys. Lett. A105,7 (1984). B.Ya. Shapiro, Solid State Commun 53, 673 (1985). B.Ya. Shapiro, Phys. Status Solidi. 129, 1 (1985). B.Ya. Shapiro, Zh. Exp. i Theor. Fiz. 88,5 (1985). W. Ruhl, Z. Phys. 186,190 (1965).