Ginzburg-Landau theory of superconducting surfaces under the influence of electric fields

Abstract

A boundary condition for the Ginzburg-Landau wave function at surfaces biased by a strong electric field is derived within the de Gennes approach. This condition provides a simple theory of the field effect on the critical temperature of superconducting layers.The critical temperature of a thin superconducting layer is increased or lowered by an electric field applied perpendicular to the layer. [1][2][3][4][5] Similarly to the conductivity of inverse layers in semiconductors, superconductivity of thin metallic layers can thus be controlled by a gate voltage, which makes these structures attractive for applications.In this paper we show that the phase transition in a thin metallic layer is conveniently described by the Ginzburg-Landau (GL) theory, where the electric field E enters the GL boundary condition asBriefly, the logarithmic derivative of the GL function ψ or the gap function ∆ at the surface is a sum of the zerofield part 1/b 0 and the field induced correction E/U s . The zero-field part has been derived by de Gennes 6 from the BCS theory. A typical value b 0 ∼ 1 cm is large on the scale of the GL coherence length, therefore this contribution is usually neglected. This approximation, 1/b 0 ≈ 0, corresponds to the original GL condition ∇ψ = 0.Here we employ the de Gennes approach to derive the field induced correction E/U s . The correction becomes important for the above mentioned experiments, where fields of the order of 10 7 V/cm are realized. Small electric fields appearing e.g. in Josephson junctions do not require such corrections.We start from the conditionderived by de Gennes (Eq. (7-62) in Ref. 6). Here N 0 is the density of states of a bulk material, V is the BCS interaction, and N (x) is the local density of states at position x. The actual gap function ∆(x) has a nontrivial profile close to the surface at x = 0, but it has only slow variation at distances exceeding the BCS coherence length ξ 0 = 0.18hv F /k B T c . For x ∼ ξ 0 it is crudely linear ∆(x) ≈ ∆ 0 (1 + x/b), so that ∆ 0 is not the true surface value but the extrapolation of the gap function towards the surface. In Eq. (2) we have used the GL coherence length at zero temperature ξ(0) = 0.74 ξ 0 for pure metals. In measurements of the field effect on the transition temperature, the zero-field term b 0 is included in the reference zero-bias transition temperature. Accordingly, we can assume a model of the crystal for which 1/b 0 = 0. The simplest model of this kind is a semi-infinite jellium, where for zero field the density of states is steplike, N (x) = N 0 for x > 0 and N (x) = 0 elsewhere. Using that the gap function is restricted to the crystal, ∆(x) = 0 for x < 0, one can check that from (2) follows 1/b 0 = 0. Now we include the electric field. According to the Anderson theorem 7 , the electric field does not change the thermodynamical properties directly but only via the density of states. The change of the density of states is also indirect. The penetrating electric field induces a deviation δn of the electron density. The den...

Two-component Ginzburg–Landau theory of superconducting surfaces in transverse electric field

“…A different situation is met if the direction of the electric field is reversed. If L = 500 Å for the parameters of [14] we obtain T ⁎ − T c = 0.9 K in reasonable agreement with the reported shift by 1 K and the results of [9].…”

“…In the one-band model [9], it can be shown that for |EL/ U s | = 6.9 the shift T ⁎ − T c = 0.1 K in the sample Sm 0.7 Ca 0.3 Cu 3 O y and is overdoped with T c ≈ 50 K. Matijasevic et al [14] reported no suppression of the superconductivity. The estimate in [9] shows that the suppression of the superconductivity T ⁎ − T c = 0.1 K is below the sensitivity of the measurement method [14]. A different situation is met if the direction of the electric field is reversed.…”

“…The critical temperature of a thin superconducting layer is lowered or increased by a static electric field applied perpendicular to the layer [1][2][3][4][5][6][7][8][9][10][11]. Thus, the superconductivity of thin metallic layers can be controlled by gate voltage which makes these structures useful for applications.…”

“…Since the beginning of the 1960s there has been an ongoing effort to use the transverse static electric field to change the superconducting properties of films [1][2][3][4][5][6][7][8][9][10][11]. The critical temperature of a thin superconducting layer is lowered or increased by a static electric field applied perpendicular to the layer [1][2][3][4][5][6][7][8][9][10][11].…”

Guiding structures with multiply connected cross sections: Evolution of propagation in external fields at complex Robin parameters

“…Apart from mentioned above acoustical and electromagnetic systems, complex extrapolation lengths can be realized for the superconducting materials too [1]. Namely, it was argued recently [80][81][82] that the electric field E applied perpendicularly to the surface should be accounted for in the total de Gennes distance Λ tot by the addition to the inverse zero-field extrapolation length 1/Λ of the extra term E/U s…”