UDC 537.312.62

PHYSICS

R. A. VOLKOV, V. I. SKOBELKIN

ON THE MOTION OF THE INTERFACE BETWEEN TWO PHASES IN SUPERCONDUCTING THIN FILMS

(Presented by Academician S. A. Vekshinskii, 11 III 1970)

For superconductors of the second kind, which include thin films (the film thickness (d) is much smaller than the penetration depth of the magnetic field into the superconductor (\delta)), the Maxwell–London equations (\left(^{1}\right)) are valid:

[
\operatorname{rot}\mathbf H=\frac{\varepsilon}{c}\frac{\partial \mathbf E}{\partial t}
+\frac{4\pi}{c}\mathbf j,\qquad
\operatorname{rot}\mathbf E=-\frac{\mu}{c}\frac{\partial \mathbf H}{\partial t},\qquad
\operatorname{div}\mathbf E=\frac{4\pi}{\varepsilon}\rho,\qquad
\operatorname{div}\mathbf H=0,
]

[
\rho_s+\rho_n=\rho,\qquad
\mathbf j_s+\mathbf j_n=\mathbf j,\qquad
\mathbf j_n=\sigma\mathbf E,
\tag{1}
]

[
\mathbf E=\frac{d}{dt}(\lambda \mathbf j_s),\qquad
\lambda=\frac{m}{n_s e^2}\geq \frac{4\pi\mu\delta^2}{c^2}=\lambda_0,
]

where (\mathbf E) is the electric-field strength; (\mathbf j) is the current-density vector; (\sigma) is the normal conductivity, finite at all temperatures; (\lambda) is the London constant, depending on the absolute temperature (T) and on the magnetic-field strength (\mathbf H); (n_s) is the concentration of superconducting electrons (Cooper pairs); (e) is the electron charge; (\rho) is the charge density; (m) is the electron mass; (\mu) is the magnetic permeability; (c) is the speed of light in vacuum; (\varepsilon) is the dielectric permittivity. The subscripts (s) and (n) refer, respectively, to the superconducting and normal states. In what follows it is assumed that (\rho=0); (\operatorname{div}\mathbf j_n=0); (\operatorname{div}\mathbf j_s=0).

Superconducting plane films permit the introduction of current functions (\left(^{2}\right)) (\psi_s(x,y,t)), (\psi_n(x,y,t)), (\psi=\psi_s+\psi_n), with respect to the coordinates (x,y) in the plane of the film (the (z)-axis is directed normal to the film):

[
j_{sx}=\partial\psi_s/\partial y,\qquad
j_{sy}=-\partial\psi_s/\partial x,
]

[
j_{nx}=\partial\psi_n/\partial y,\qquad
j_{ny}=-\partial\psi_n/\partial x,
\tag{2}
]

[
j_x=\partial\psi/\partial y,\qquad
j_y=-\partial\psi/\partial x.
]

Fig. 1. (xx_2) — normal phase; (x_1x_2) — intermediate state

The Maxwell–London equations for such films take the form

[
\Delta\psi_n-\frac{1}{\sigma}\frac{d\sigma}{dT}(\nabla T\cdot\nabla\psi_n)
=
\frac{4\pi\mu\sigma}{c^2}
\left(
\frac{\partial\psi}{\partial t}
+\frac{\varepsilon}{4\pi\sigma}\frac{\partial^2\psi_n}{\partial t^2}
\right),
]

[
\Delta\psi_s+
\frac{1}{\lambda}
\left(
\frac{\partial\lambda}{\partial T}\nabla T+
\frac{\partial\lambda}{\partial\psi}\nabla\psi
\right)\nabla\psi_s
=
-\frac{4\pi\mu}{c^2\lambda}
\left(
\psi+\frac{\varepsilon}{4\pi\sigma}\frac{\partial\psi_n}{\partial t}
\right),
\tag{3}
]

where (\Delta) is the two-dimensional Laplace operator; (\psi=\dfrac{c}{4\pi}H_z), (H_x=H_y=0). The boundary conditions on the contour of the film are (\partial\psi/\partial l=\partial\psi_s/\partial l=\partial\psi_n/\partial l=0) ((l) is the direction of the contour).

Let (x_1x_2) be the layer separating the superconducting region from the normal one (Fig. 1). Suppose that the fields are quasistationary, the problem is one-dimensional ((x,t)), and (\lambda=\lambda(\mathbf H)) (the supercritical region with respect to (T)). Under these assumptions equations (3) take the form

[
\Delta\psi_n=-\frac{4\pi\mu\sigma}{c^2}\frac{\partial\psi}{\partial t},
]

[
\lambda\Delta\psi_s+\frac{d\lambda}{d\psi}\frac{\partial\psi}{\partial x}\frac{\partial\psi_s}{\partial x}
=-\frac{4\pi\mu}{c^2}\psi .
\tag{4}
]

Using the relation (\psi=\psi_s+\psi_n), one can transform (4) to a form containing only (\psi_s), or only (\psi_n), and their derivatives.

Integrating (4) with respect to (x), we find

[
-\frac{\partial\psi_n}{\partial x}=j_n
=-\frac{4\pi\mu\sigma}{c^2}\frac{\partial}{\partial t}
\int_{-\infty}^{x}\psi\,dx .
\tag{5}
]

Multiplying the first equation (4) by (\lambda) and adding it to the second, we obtain

[
\lambda\Delta\psi+\frac{\partial\lambda}{\partial x}\frac{\partial\psi_s}{\partial x}
=-\frac{4\pi\mu\sigma}{c^2}
\left(\lambda\frac{\partial\psi}{\partial t}+\frac{1}{\sigma}\psi\right).
\tag{6}
]

Further,

[
\frac{\partial\psi_s}{\partial x}
=\frac{\partial\psi}{\partial x}-\frac{\partial\psi_n}{\partial x}
=\frac{\partial\psi}{\partial x}
-\frac{4\pi\mu\sigma}{c^2}\frac{\partial}{\partial t}
\int_{-\infty}^{x}\psi\,dx .
\tag{7}
]

Substituting (7) into (6), we find

[
\Delta\psi+\frac{\partial\ln\lambda}{\partial x}
\left(
\frac{\partial\psi}{\partial x}
-\frac{4\pi\mu\sigma}{c^2}\frac{\partial}{\partial t}
\int_{-\infty}^{x}\psi\,dx
\right)
=-\frac{4\pi\mu\sigma}{c^2}
\left(
\frac{\partial\psi}{\partial t}+\frac{1}{\sigma\lambda}\psi
\right).
\tag{8}
]

We restrict ourselves to stationary solutions of the form (\psi(x,t)=\psi(x-vt)). Figure 1 shows the dependence of (H) on (x). In the linear approximation,

[
\left(\frac{\partial H}{\partial x}\right)1
=\frac{H}-H_{k1}}{x_2-x_1
=\frac{H_{k2}-H_{k1}}{-v\tau},
\tag{9}
]

where (v) is the propagation velocity of the normal phase, and (\tau) is the time of the phase transition. The point (x_1) corresponds to the start of the reaction and to the magnetic field (H_{k1}) (the first critical field), while the point (x_2) corresponds to the end of the reaction and to the magnetic field (H_{k2}) (the second critical field).

Equation (8) in the region to the left of (x_1) takes the form

[
\frac{\partial^2\psi}{\partial x^2}
-\frac{4\pi\mu\sigma}{c^2}
\left(
\frac{\partial\psi}{\partial t}+\frac{1}{\sigma\lambda_0}\psi
\right)
=\ddot{\psi}+\alpha\dot{\psi}+\beta\psi=0,
\tag{10}
]

where (\dot{\psi}=d\psi/dz), (\ddot{\psi}=d^2\psi/dz^2), (\partial\psi/\partial t=-v\,d\psi/dz), (\alpha=4\pi\mu\sigma v/c^2), (\beta=-4\pi\mu/\lambda_0c^2), (z=x-vt).

The characteristic equation for (10) gives the roots

[
2p_1=-\alpha+\sqrt{\alpha^2-4\beta}>0,
\qquad
2p_2=-\alpha-\sqrt{\alpha^2-4\beta}<0.
]

The general solution of equation (10) is (\psi=C_1e^{p_1z}+C_2e^{p_2z}). Since for points to the left of (x_1) the coordinate (x<0), and at (x=-\infty), (\psi=0), it follows that (C_2=0) and (\psi=C_1e^{p_1z}=\psi_{k1}e^{p_1z}), where

[
\psi_{k1}=\frac{c}{4\pi}H_{k1},\qquad
\psi=\frac{c}{4\pi}H .
]

Differentiating the solution found with respect to (x), we obtain

[
2\frac{\partial H_{k1}}{\partial x}
=2H_{k1}p_1
=H_{k1}
\left(
-\frac{4\pi\mu\sigma}{c^2}v
+\sqrt{
\left(\frac{4\pi\mu\sigma}{c^2}\right)^2v^2
+\frac{16\pi\mu}{c^2\lambda_0}
}
\right).
\tag{11}
]

Squaring the last relation, using (9), and setting (H_{k2}=\eta H_{k1}), we find

[
v=\frac{(1-\eta)c}{2}\sqrt{\frac{\lambda_0}{\pi\mu\tau\bigl(\tau+\delta\lambda_0(\eta-1)\bigr)}} .
\tag{12}
]

The quantity (v) does not exceed the velocity of propagation of phonon perturbations and is of the order of the speed of sound. In the phase transition of a type-II superconductor from the superconducting to the normal state, in contrast to type-I superconductors, the negative sign of the surface energy makes equilibrium of the normal and superconducting phases impossible. The normal phase propagates through the superconducting phase with velocity (v).

To calculate (v) from formula (12), it is necessary to know the phase-transition time (\tau). To calculate (\tau), let us consider the motion of the normal phase as the motion of the surface (H=H_{k1}). The boundary conditions on such a surface will be

[
p_1^{+}=\left(\frac{\partial H}{\partial x}\right)_1^{+}
=-\frac{4\pi}{c}j_1^{+}
=-\frac{4\pi}{c}(j_n^{+}+j_s^{+})_1 ,
\tag{13}
]

where the sign (+) indicates the value of the corresponding physical quantities on the surface (H=H_{k1}) from the right (Fig. 1).

Relations (13) follow from the Maxwell–London equations (1). Differentiating (13) with respect to (t), we obtain

[
\frac{\partial}{\partial t}\left(\frac{\partial H}{\partial x}\right)1^{+}
=-v\left(\frac{\partial^2 H}{\partial x^2}\right)_1^{+}
=-\frac{4\pi}{c}\left(\sigma\frac{\partial E_1^{+}}{\partial t}
+\frac{\partial j\right)}^{+}}{\partial t
=-\frac{4\pi}{c}\left[-\sigma v\left(\frac{\partial E}{\partial x}\right)_1^{+}
+\frac{E_1^{+}}{\lambda_0}\right]
]

[
=-\frac{4\pi}{c}\left[\frac{\mu\sigma v^2}{c}
\left(\frac{\partial H}{\partial x}\right)_1^{+}
+\frac{E_1^{+}}{\lambda_0}\right]
=-\frac{4\pi}{c}\left(-\frac{4\pi\mu\sigma^2 v^2}{c^2}E_1^{+}
+\frac{E_1^{+}}{\lambda_0}\right).
\tag{14}
]

In (14), ((\partial E/\partial x)1^{+}) is expressed in terms of (\partial H_1^{+}/\partial t) from the second equation (1), and the operator (\partial/\partial t) is replaced by (-v\,\partial/\partial x) (because on the surface (H=H). Neglecting the penetration depth (\delta) of the magnetic field into the superconducting part of the film in comparison with the characteristic size (diameter) of the film, we obtain ((^3,^4))}) the value of the magnetic field is conserved in time). Considering the surface (H=H_{k1}) as a surface of strong discontinuity between the purely superconducting phase (to the left of (x_1) in Fig. 1) and the remaining part of the film, one can establish a relation between (E_1^{+}) and (H_{k1}^{+

[
E_1^{+}=\frac{v}{c}\,\mu H_{k1}.
\tag{15}
]

Substituting (E_1^{+}) from (15) into (14), noting that ((\partial^2 H/\partial x^2)1^{+}=(p_1^{+})^2H\ne0), we find}), and canceling by (v\ne0) and (H_{k1

[
(p_1^{+})^2=4\pi\mu/\lambda_0c^2-(4\pi\mu\sigma/c^2)^2v^2 .
\tag{16}
]

On the other hand, by continuity (p_1^{+}=p_1^{-}), where (p_1^{-}) is the value of (p) on the surface (H=H_{k1}) from the left (Fig. 1). From (11) we have

[
(p_1^{-})=\frac{1}{4}\left(-\frac{4\pi\mu\sigma v}{c^2}
+\sqrt{\left(\frac{4\pi\mu\sigma}{c^2}\right)^2v^2
+\frac{16\pi\mu}{\lambda_0c^2}}\right).
\tag{17}
]

Equating (p_1^{+}) from (16) and (p_1^{-}) from (17), we obtain

[
v=-\frac{c}{2\sigma\sqrt{2\pi\mu\lambda_0}}
=-\frac{ec}{2\sigma}\sqrt{\frac{n}{2\pi\mu m}},
\tag{18}
]

where (n) is the total concentration of electrons in the metal (in (1\ \mathrm{cm}^3)). Formula (18) contains no explicit magnetic fields (H_{k1}) and (H_{k2}). Substituting into (18) (\sigma=)

(=10^{20},\ c=3\cdot10^{10},\ e=4.8\cdot10^{-10},\ n\sim10^{22},\ \mu\approx1,\ m\cong10^{-27}), we obtain (v\sim10^5\ \text{cm/sec}).

Comparing (18) with (12), we find

[
\tau=\delta\lambda_0(\eta-1)=\frac{\delta m}{ne^2}\left(\frac{H_{k2}}{H_{k1}}-1\right).
\tag{19}
]

For type-II superconductors (\eta=H_{k2}/H_{k1}\approx10^3\gg1), and therefore one may write

[
\tau=\frac{\delta m}{ne^2}\frac{H_{k2}}{H_{k1}}.
\tag{20}
]

For the values of the physical quantities given above, (\tau\sim10^{-7}\ \text{sec}). Depending on (\sigma, n, \eta), for various superconductors (\tau), calculated from (20), varies within the range (10^{-6})—(10^{-8}\ \text{sec}). The width of the reaction zone (x_2-x_1=v\tau) fluctuates within the range (10^{-1})—(10^{-3}\ \text{cm}). In any case (x_2-x_1\gg\xi), where (\xi) is the parameter of the Cooper correlation. The condition (x_2-x_1\gg\xi) is in full agreement with the requirement for type-II superconductors and with the admissibility of the Maxwell–London equations.

The result obtained on the motion of a disturbance in the form of a normal phase in a type-II superconductor must be taken into account in the design of thin-film cryotrons. Although the film is stable with respect to a disturbance of the magnetic field, during the magnetic relaxation time (\tau_H=4\pi\mu\sigma:c^2\gamma_1), where (\mu) is the magnetic permeability, (c) is the speed of light in vacuum, and (\gamma_1) is the smallest eigenvalue of the equation (\Delta\psi+\gamma\psi=0) with boundary conditions (\psi=0) on the contour of the film (l) ((^2)), the normal phase will have time to propagate over considerable distances. For a film with conductivity (\sigma\cong10^{20}), thickness (0.3\ \mu), width (0.03\ \text{mm}), and length (2\ \text{mm}), (\tau\cong10^{-7}\ \text{sec}), and assuming (v=10^5\ \text{cm/sec}), we obtain the region of expansion of the normal phase (v\tau_H=0.1\ \text{mm}), which considerably exceeds the width of the film.

The phase-transition time (\tau), determined by (20), limits the speed of response of thin-film cryotrons. Technical realization of considerable speed is impossible if the switching time is less than the phase-transition time (\tau). Consequently, the problem of increasing the speed of response of a cryotron reduces to decreasing the ratio (H_{k2}/H_{k1}), i.e., to obtaining optimal magnetic properties of the film (provided that the reserves for decreasing (\tau) by means of (\sigma) and (n) have been used).

Moscow Institute of Electronic
Technology

Received
7 III 1970

REFERENCES CITED

1. P. de Gennes, Superconductivity of Metals and Alloys, Moscow, 1968.
2. V. I. Skobelkin, E. A. Zhilkov, V. I. Puchkov, FTT, 11, 10 (1969).
3. L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media, 1957, p. 227.
4. A. B. Pippard, Phil. Mag., 41, No. 314 (1950).