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Radio-frequency surface impedance of type-II superconductors: Dependence upon the magnitude and angle of an applied static magnetic field
Physica C 185-189 (1991) 1915-1916 North-Holland
RADIO-FREQUENCY SURFACE IMPEDANCE OF TYPE-H SUPERCONDUCTORS: DEPENDENCE UPON THE MAGNITUDE AND ANGLE OF AN APPLIED STATIC MAGNETIC FIELD John R. CLEM and Mark W. COFFEY Ames Laboratory-USDOE* and Department of Physics and Astronomy, Iowa State University, Ames, IA 50011, U.S.A. The application of a static magnetic field introduces vortices into a type-II superconductor. The dynamic response of these vortices strongly affects the surface impedance (surface resistance and surface reactance) of the superconductor. Using a self-consistent approach for including the influence of the vortices upon the complex-valued radio-frequency penetration depth, we calculate in this paper how the surface impedance depends upon the static magnetic field's magnitude and angle relative to the surface and to the rf magnetic field. The surface resistance is maximized when the induced rf currents are perpendicular to the static field. 1. INTRODUCTION In this paper we briefly describe a theoretical calculation of the radio-frequency (rf) surface impedance Zs = Rs + iXs of an isotropic type-II superconductor. In particular, we show how Zs depends upon the magnitude and direction of the internal magnetic induction Bo generated by a uniform array of vortices in the superconductor. The method is based on that of Refs. I and 2 but ignores flux creep effects. 2. METHOD AND RESULTS We consider a superconductor filling the h a l f space x > 0. The interior contains an array of vortices which generate a uniform magnetic flux density Bo. Let a denote the angle between the x axis and Bo, as sketched in Fig. 1. The Bx plane (the plane containing the vectors BO and 3) is inclined at the angle V relative to the z axis. It is ,',,,~,,,~,',~o,"~ ~" define the "'~'~ -'e '~+'" ~,.~a ~ ~ = ^
y cos
A
z sm V, and
=
x
=
sm
+
cos
V.
An applied rf magnetic field hrf(t) = ~ hrf(t) [hrf(t) = h0 exp(icot)] generates an rf current den-
f e oa
o e
•° 6
o° oe e •
e e'
o
Y
e
e
e
Q ee qJ ~o
e
V
FIGURE 1 Sketch of magnetic flux density vector BO, angles a and V, and unit vectors 3, ~, and ~. sity J(x,t), which produces a Lorentz force per unit length f(x,t) on the vortices. The resulting vortex motion, described by the displacement field u(x,t),
*Ames Laboratory is operated for the U.S. Department of Energy by Iowa State University under Contract No. W-7405-Eng-82. This research was supported in part by the Director for Energy Research, Office of Basic Energy Sciences, and in part by the Midwest Superconductivity Consortium through D.O.E. grant #DE-FG02-90ER45427. 0921-4534/91/$03.5e © 1991 - Elsevier Science Publishers B.V. All rights reserved.
1916
Y.l~ Clem, M.W. Coffey / Radio-frequency surface in~edance of type-II superconductors
causes the vortex density distribution to have a time-varying component, which in turn drives a n r f magnetic flux density distribution b(x,t). All these effects must be calculated self-consistently as follows. The resulting rf magnetic flux density b(x,t) in the region x > 0 can be resolved into independent components, b = ~ b~ +~ by, where bp(x,t) = - 90hrf(t) sin ¥ exp(-x/kp), (1) b~(x,t) = p0hrf(t) cos V exp(-x~), (2) and ~.p and ~.~ are complex-valued p e n e t r a t i o n depths. Similarly, the net r f current density can be expressed as J = ~ J~ +~ J¥,where Ampere's law yields J~ = b~/ttoL/ and J r = -b~/tto),~. This current density generates a Lorentz force per , n l t length f = J × (~0~0) [the unit vector in the vortex direction is ~0 = Bo/Bo]. The force balance between this force, the viscous drag force per unit length -~v = -iC0TlU, and a restoring pinning force per u n i t length -gpU yields the vortex displacement field u = f](K:p 4- i(o~),where u is resolved into independent components, u = ~ u f} + ~ us, where ~ = ~ x ]~0 = A • A x sm a - T cos a , u~ = [~oJ~l(gp + i(otl)] cos a, and u s = ~oJ~/(~p + i n ) .
The corresponding change in the vortex density, expressed as by in units of magnetic flux density, is obtained from the vortex continuity equation by = - V x (Box u), which can be resolved into components bv = ~ bvp +~ b~, where bv~ = - ( B o u ~ p ) cos a, and b ~ = B o u t . The r f supercurrent density Js js obtained from the London equation, supplemented by a vortex source term (by), V × Js = - (b - bv)/~to~2, where ~. is the London penetration depth. The r f electric field is obtained from Faraday+s law, V x E = - ~b/~t = - i~b. We obtain E = ~ E~ + ET, where E~ = i m b ~ and E T = - icob~)~. Using the two-fluid model, we obtain the normal current density J n = onfE, where ~nf is the conductivity of the normal fluid. The total r f current density is J = J s + Jn-
Combining the above equations results in consistency equations determining ~.~ and )~. For example, all the terms in the equation Jp = Jsp + Jn~ can be shown to be proportional to by a n d exp(-x~), and the resulting equation yields 1 + 2iX2/~}
"
(3)
Here 8v = (Sf"2 - i~.c "2/2)'1/2 is the complex-valued effective skin depth arising from vortex motion [Sf = (2Bo~o/~o~oo)1/2 is the flux-flow skin depth and = (B0~0/l~0~:p)1/2 is the Campbell penetration depth] and 8nf = (2/~t0O(Ynf)l/2 is the normal-fluid skin depth. Similarly, the equation Jr = Js~ + Jn~ yields the result
(.~2 kS =
-'2
\I12
- i(SJ2)cos2a|
!
m
The surface impedance Zs = Rs + LXs, obtained from Faraday's law and the definition Zs = Ey/hrf, is Zs = i(op0(~ cos2¥ + ~.~ sin2v).
(5)
3. CONCLUSIONS We have derived the surface impedance in a type-II superconductor for arbitrary orientation of the vortices relative to the surface and the rf magnetic field. The surface resistance and reactance are maximized when the induced rf currents are perpendicular to the vortices.
REFERENCES M. W. Coffey and J. R. Clem, IEEE Trans. Magn. 27 (1991) 2136. In Eqs. (12) and (13) of this paper (for the electric field), bo/k should replace Bouo. The factor 1 - k2~.2 is extraneous in Eq. (13), as is the factor 1 - ~,2f£,2 in Eqs. (15b) and (15c). 2. M. W. Coffey and J. R. Clem, Phys. Rev. Lett. 67 (1991) 386 and the references therein.
RADIO-FREQUENCY SURFACE IMPEDANCE OF TYPE-H SUPERCONDUCTORS: DEPENDENCE UPON THE MAGNITUDE AND ANGLE OF AN APPLIED STATIC MAGNETIC FIELD John R. CLEM and Mark W. COFFEY Ames Laboratory-USDOE* and Department of Physics and Astronomy, Iowa State University, Ames, IA 50011, U.S.A. The application of a static magnetic field introduces vortices into a type-II superconductor. The dynamic response of these vortices strongly affects the surface impedance (surface resistance and surface reactance) of the superconductor. Using a self-consistent approach for including the influence of the vortices upon the complex-valued radio-frequency penetration depth, we calculate in this paper how the surface impedance depends upon the static magnetic field's magnitude and angle relative to the surface and to the rf magnetic field. The surface resistance is maximized when the induced rf currents are perpendicular to the static field. 1. INTRODUCTION In this paper we briefly describe a theoretical calculation of the radio-frequency (rf) surface impedance Zs = Rs + iXs of an isotropic type-II superconductor. In particular, we show how Zs depends upon the magnitude and direction of the internal magnetic induction Bo generated by a uniform array of vortices in the superconductor. The method is based on that of Refs. I and 2 but ignores flux creep effects. 2. METHOD AND RESULTS We consider a superconductor filling the h a l f space x > 0. The interior contains an array of vortices which generate a uniform magnetic flux density Bo. Let a denote the angle between the x axis and Bo, as sketched in Fig. 1. The Bx plane (the plane containing the vectors BO and 3) is inclined at the angle V relative to the z axis. It is ,',,,~,,,~,',~o,"~ ~" define the "'~'~ -'e '~+'" ~,.~a ~ ~ = ^
y cos
A
z sm V, and
=
x
=
sm
+
cos
V.
An applied rf magnetic field hrf(t) = ~ hrf(t) [hrf(t) = h0 exp(icot)] generates an rf current den-
f e oa
o e
•° 6
o° oe e •
e e'
o
Y
e
e
e
Q ee qJ ~o
e
V
FIGURE 1 Sketch of magnetic flux density vector BO, angles a and V, and unit vectors 3, ~, and ~. sity J(x,t), which produces a Lorentz force per unit length f(x,t) on the vortices. The resulting vortex motion, described by the displacement field u(x,t),
*Ames Laboratory is operated for the U.S. Department of Energy by Iowa State University under Contract No. W-7405-Eng-82. This research was supported in part by the Director for Energy Research, Office of Basic Energy Sciences, and in part by the Midwest Superconductivity Consortium through D.O.E. grant #DE-FG02-90ER45427. 0921-4534/91/$03.5e © 1991 - Elsevier Science Publishers B.V. All rights reserved.
1916
Y.l~ Clem, M.W. Coffey / Radio-frequency surface in~edance of type-II superconductors
causes the vortex density distribution to have a time-varying component, which in turn drives a n r f magnetic flux density distribution b(x,t). All these effects must be calculated self-consistently as follows. The resulting rf magnetic flux density b(x,t) in the region x > 0 can be resolved into independent components, b = ~ b~ +~ by, where bp(x,t) = - 90hrf(t) sin ¥ exp(-x/kp), (1) b~(x,t) = p0hrf(t) cos V exp(-x~), (2) and ~.p and ~.~ are complex-valued p e n e t r a t i o n depths. Similarly, the net r f current density can be expressed as J = ~ J~ +~ J¥,where Ampere's law yields J~ = b~/ttoL/ and J r = -b~/tto),~. This current density generates a Lorentz force per , n l t length f = J × (~0~0) [the unit vector in the vortex direction is ~0 = Bo/Bo]. The force balance between this force, the viscous drag force per unit length -~v = -iC0TlU, and a restoring pinning force per u n i t length -gpU yields the vortex displacement field u = f](K:p 4- i(o~),where u is resolved into independent components, u = ~ u f} + ~ us, where ~ = ~ x ]~0 = A • A x sm a - T cos a , u~ = [~oJ~l(gp + i(otl)] cos a, and u s = ~oJ~/(~p + i n ) .
The corresponding change in the vortex density, expressed as by in units of magnetic flux density, is obtained from the vortex continuity equation by = - V x (Box u), which can be resolved into components bv = ~ bvp +~ b~, where bv~ = - ( B o u ~ p ) cos a, and b ~ = B o u t . The r f supercurrent density Js js obtained from the London equation, supplemented by a vortex source term (by), V × Js = - (b - bv)/~to~2, where ~. is the London penetration depth. The r f electric field is obtained from Faraday+s law, V x E = - ~b/~t = - i~b. We obtain E = ~ E~ + ET, where E~ = i m b ~ and E T = - icob~)~. Using the two-fluid model, we obtain the normal current density J n = onfE, where ~nf is the conductivity of the normal fluid. The total r f current density is J = J s + Jn-
Combining the above equations results in consistency equations determining ~.~ and )~. For example, all the terms in the equation Jp = Jsp + Jn~ can be shown to be proportional to by a n d exp(-x~), and the resulting equation yields 1 + 2iX2/~}
"
(3)
Here 8v = (Sf"2 - i~.c "2/2)'1/2 is the complex-valued effective skin depth arising from vortex motion [Sf = (2Bo~o/~o~oo)1/2 is the flux-flow skin depth and = (B0~0/l~0~:p)1/2 is the Campbell penetration depth] and 8nf = (2/~t0O(Ynf)l/2 is the normal-fluid skin depth. Similarly, the equation Jr = Js~ + Jn~ yields the result
(.~2 kS =
-'2
\I12
- i(SJ2)cos2a|
!
m
The surface impedance Zs = Rs + LXs, obtained from Faraday's law and the definition Zs = Ey/hrf, is Zs = i(op0(~ cos2¥ + ~.~ sin2v).
(5)
3. CONCLUSIONS We have derived the surface impedance in a type-II superconductor for arbitrary orientation of the vortices relative to the surface and the rf magnetic field. The surface resistance and reactance are maximized when the induced rf currents are perpendicular to the vortices.
REFERENCES M. W. Coffey and J. R. Clem, IEEE Trans. Magn. 27 (1991) 2136. In Eqs. (12) and (13) of this paper (for the electric field), bo/k should replace Bouo. The factor 1 - k2~.2 is extraneous in Eq. (13), as is the factor 1 - ~,2f£,2 in Eqs. (15b) and (15c). 2. M. W. Coffey and J. R. Clem, Phys. Rev. Lett. 67 (1991) 386 and the references therein.