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On the applicability of the critical state model to the description of electromagnetic properties of high-Tc superconductors
Physica C 206 (1993) North-Holland
PHMCA B
195-201
On the applicability of the critical state model to the description of electromagnetic properties of high- TCsuperconductors L.M. Fisher,
N.V. Il’in and I.F. Voloshin
All-Russian Electrical Engineering Institute, Krasnokazarmennaya
N.M. Makarov
Str. 12, Moscow, I1 1250, Russian Federation
and V.A. Yampol’skii
Institute for Radiophysics and Electronics, Ukr. Acad. Sci., Acad. Proskura Str. 12, Karkov, 310085, Ukraine
F. Perez Rodriguez Intituto de Fisica, Universidad Autonoma de Puebla J-48, Puebla, Rue 725 70, Mexico
R.L. Snyder New York State College of Ceramics at Aljred University, Alfred, NY, 14802, USA Received
19 November
1992
The frequency dependence of the surface impedance of superconductors have been studied experimentally and theoretically in the radio frequency range. Its essential deviation was found from the linear law predicted by the usual critical state model. The character of this deviation depends qualitatively on the amplitude of the radio wave. We have established the frequency limits of applicability of the traditional critical state model. Results obtained print out an explanation in the frame of the modified model where we take into account the contribution of a dissipative term to the screening current. The value of this is connected with the V-I plot of the superconductor, so it is possible to obtain information about the V-i characteristics by the contactless method.
1. Introduction The critical state model [ 1 ] is commonly used to explain the magnetic properties of hard superconductors. According to ref. [ 11, the distribution of the magnetic induction B inside a superconductor is described by the equation curlB=(4x/c)J,(B)E/E,
(1)
where J, is the critical current density, c is the speed of light and E is the electric field. This model was proposed by Bean to describe static magnetization curves of homogeneous superconductors. The background of its application to granular systems (see, for example refs. [2] and (3)) concerns static conditions too. However, it is well known that many observations of electromagnetic properties of hard superconductors are interpreted in the frame of the model ( 1). For example, the magnetic field depen0921-4534/93/$06.00
0 1993 Elsevier Science Publishers
dence of the critical current density JC( B) calculated in the frame of the critical state model using results of measurements of the surface impedance y= 9- iy is in a good agreement with direct measurements of the critical current. Despite the popularity of the critical state model the question was raised as to whether this model can be used to describe the electromagnetic properties of superconductors. This problem has not been solved yet. It has been known that electric fields exist in the volume of a sample owing to the magnetic flux varying with time. In other words, a superconductor in the dynamic regime is always in the resistive state. This means that the superconducting critical current is accompanied by an existing dissipative component of current in the volume of the sample. This dissipative component arises owing to the electric field E. At such conditions the model (1) must be modified to take into account this dissipative cur-
B.V. All rights reserved.
196
L.M. Fisher et al. /Applicability ofthe criticalstate model
rent (see, for example, ref. [ 41). A corresponding equation describing the distribution of the magnetic induction may be written as follows curlB=
(4n/c)
[J,(B)+a(E)E]E/E,
(2)
where (7 is the dissipative conductivity of the medium which rapidly increases with E. It was shown [4] that the dissipative item in eq. (2) affects noticeably the harmonic composition of the electromagnetic field inside a superconductor. This item is discounted in the usual critical state model ( 1). This is possible in the case of low frequencies o and amplitudes X of the excited magnetic field. If the frequency o or amplitude SV arise, the electric field E increases and the role of the second item increases too. It is clear that the modification of the usual model ( 1) is inevitable. Nevertheless, many authors interpret their results obtained even in the microwave region using the simplest model. So, the special analysis of the response of a superconductor to the external electromagnetic signal in the frame of modified material equation (2) is needed to establish the frequency boundaries of the critical state model ( 1) application. This paper is devoted to solving this problem. We have carried out the theoretical and experimental study of the dependence of the surface impedance of granular highT, superconductors on the frequency of an external radio wave. The necessity of considering this study is conditioned by a great number of papers where the information about parameters of the superconducting state is extracted from the results of electromagnetic measurements. Besides, the frequency dependence of the surface impedance contains significant information about the V-Z plot of a superconductor in the resistive state. We have shown that the inclusion of the dissipative item in the current density in eq. (2) leads to deviation of the dependence P(o) from the linear law following from eq. ( 1). The character of this deviation depends on the amplitude X of the radio wave. At low .# the surface impedance increases with frequency lower than w. If the amplitude .% is so high that electromagnetic fields exist at any point of the sample volume, the surface resistance A will increase faster and the surface reactance 3” slower than o. We have found that it is possible to determine VI plot parameters for a sample having measured the
deviation of the function 9(w) from a linear law. The frequency boundaries were established where the traditional critical state model agrees satisfactorily with experimental data.
2. Theory Let us consider a plate with thickness d of a granular high-T, superconductor placed in an external AC magnetic field having the form H(t)=.#coswt.
(3)
Let this field H be directed along the surface and parallel to the z-axes. Let the x-axis be perpendicular to the plate. Assume that the origin of the coordinates is placed in the center of a plate. The system of Maxwell equations describing the AC field distribution inside a plate has the form II’=-
%
(J,+AIEI”)
signE,
E’ = -b/c.
(5)
Here a prime denotes the derivative with respect to X, the point signifies the time derivative. The parameter p is the effective magnetic permeability which takes into account intragranular currents [ 21. To simplify our consideration we shall neglect the dependence of J, on the magnetic induction B. The dissipative component of the current density in eq. (4) is assumed to be a power function of the electric field E with the exponent equal to y and the parameter A is the dimensional multiplier. This dependence of the dissipative current on E reflects the empirical form of the V-Z plot obtained in ref. [ 5 1. It was found in ref. [ 5 ] that y is equal to 0.15 and we shall use this value in our numerical calculation. The boundary conditions for eqs. (4) and (5 ) have the form B(d/2)
=/M
cos ot ,
E(0) =0 .
(6)
The surface impedance is defined as the relation of the first harmonics of electrical and magnetic fields on the plate surface: n/w
s
yy=8wdtE( C.H
0
-d/2,
t) exp(iwt)
.
(7)
L.M. Fisher et al. /Applicability of the critical state model
The following dimensionless low for convenience: p= ot,
C= 2x/d,
h=sf”lH,,
variables
HP = 2nJ,dlc
,
h=BlpH,,
e=2Ec/,umH,d, o. = ( 2c/pHpd)
are used be-
The substitution of eq. ( 12 ) in ( 11) gives the following equation for the surface impedance: z=(h/3)(1-3ni/4)+PF(h),
(8)
F(h) =ah’+‘Y,
(13)
/3= (o/o,,)“, (J,/A)“Y.
It should be pointed out that the fields are normalized by the value of HP which has a simple physical sense. HP is equal to the value of an external static magnetic field at the beginning of which a sample has no region where H=O. Using new variables we can rewrite eqs. (3)-(7) in the form h’=-(l+pleIY)
signE,
h(l)=kcosp,
E(O)=O,
zc4pud --z(~), C2
EL-h,
(9) (10)
(11)
z(w)=-(Llh)jdp.(l,p)enp(ip). 0
Recall that the real part of the surface impedance is proportional to losses dissipated in a sample per unit time. Accordingly, Re z(o) is proportional to losses per period 27~10 of the radio wave. Equations describing the distribution of an AC field in cylindrical samples are analogous to (9) and ( 10) and we shall not present them here. It should be pointed out only that the parameter din eq. (8) must be changed by the diameter 2R of a cylinder. Equations (9) and ( 10) are nonlinear and they cannot be solved analytically. However, it proves to be possible to analyze this system by the method of the perturbation theory at a low value of the dissipative term in comparison with unity in eq. (9) i.e. at low /I. We do not present this standard procedure here and give only an expression for the electric field on the surface of a plate in the case h < 1 taking into account the first order correction: e(l,q)=-
197
h2 Tsina,
(12)
where (Y is a complex number. The first term in eq. ( 13 ) does not depend on frequency and comprises the well known expression for the surface impedance in the frame of the traditional critical state model. The second item, proportional to oy, is connected with the dissipative term in eq. (2 ). An analogous equation may be obtained in the case h> 1. The same is true for cylindrical samples and the main term in the expansion of z is independent on frequency, and the correction is always proportional to WY. The expansion ( 13 ) is true at low 8, i.e. at O-=Xuo. To obtain the frequency dependence of the impedance in a wide range of o we solved eqs. (9 ) and ( 10) and the analogous system for cylindrical samples by numerical methods. Some results of calculations of the dependence r= Re z( w) for a cylindrical sample and two different amplitudes of the excited AC field are presented in fig. 1. The lower curve corresponds to X = H,/2. The upper curve is calculated for I = 3HJ2. Figure 1 shows that the function Re z(w) decreases monotonically with o at X= H,/2. However, the character of its behavior changes at X= 3HJ2. The value of r increases at low w, has a maximum, and decreases at high o. Changing the character of the behavior of r at low frequency takes place at some amplitude % which will be named the inversion amplitude. An analysis shows that in the case of a cylinder X zzHp. The imaginary part - Im z(w) decreases monotonically in any case independently of the amplitude X at a wide range of the frequency. The impedance behavior, as a function of w, is qualitatively the same in the case of plates. The inversion amplitude for plates coincides with 4HJ3. The results of our calculation are in contradiction with one of the main consequences of the traditional critical state model (1). As noted above, the value of z is independent on frequency at any field amplitude in the frame of this model.
198
L.M. Fisher et al. /Applicability of the criticalstate model
3. Experiment and discussion
plates with a thickness in the range 0.5-5 mm and cylindrical samples 3-5 mm in radius. Dependences of the surface impedance r= Re z on the external AC magnetic field amplitude R for the superconducting plate with thickness d=0.8 mm are presented in fig. 2. Curve I was obtained at the frequency w/2rc= 1 kHz and curve 2 at the frequency 100 kHz. We can see that every curve has a maximum of which the physical nature is well known. The surface impedance increases with X at low amplitudes owing to increasing penetration depth S of the AC field. However, at X > HP the induced electric fields penetrating from opposite sides of the plate are compensated. As a result, increasing 3% leads to an increase of 6 and to decreasing of the surface resistance. The surface resistance as a function of &! must therefore have a maximum. The difference between curves in fig. 2 shows the noticeable frequency dependence of the surface resistance r which cannot be explained in the simple critical state model ( 1). First, let us pay attention to the shift of the maximum position of Y(Z) to higher amplitudes at increasing frequency. This phenomenon obtains a natural explanation in the modified
The real and imaginary parts of the surface impedance of samples have been measured in the frequency range 10 Hz-100 kHz using standard inductive method. The pick-up coil was wound closely on the sample by a fine wire 30 pm in diameter. The AC magnetic field was applied using a coil wound by a copper wire. Pick-up signals proportional to &’ and X were extracted by a phase sensitive amplifier PAR124A. To increase the accuracy of our measurements we compensated the part of the signal connected with changing the magnetic flux in the gap between pickup coil and sample using an additional device. The measurement process was computer controlled, allowing rapid data acquisition and calculations to be made. The measurements were carried out at liquid nitrogen temperature. For this purpose the sample together with pick-up and AC coils was immersed into a liquid nitrogen vessel. To obtain the values Ret and Imz amplifier output signals were normalized by the amplitude and the frequency of the external AC field. We have measured granular Y-Ba-Cu-0 system samples prepared by a solid state reaction. They were
Fig. 2. Results of the measurements of the surface resistance r as a function of the radio wave amplitude X for a superconducting plate 0.8 mm in thickness at T= 77 K. Curve 1 corresponds to a frequency w/2x= 1 kHz, curve 2 to a frequency of 100 kHz.
Fig. 1. The calculated dependence of the surface resistance r(o) / r(oO) on the dimensionless frequency o/o,, for a superconducting cylinder at two amplitudes of X. These amplitudes are included in the figure. The value of y is taken to be equal to 0.15.
L.M. Fisher et al. /Applicability of the criticalstate model
model (2 ). As a matter of fact, the contribution of the dissipative term in eq. (2) to the screening current density increases with frequency and the penetration depth 6 of the AC field decreases. As a result, the maximum of r(X) reaches a higher amplitude of X. The inversion of the frequency dependence, observed in our experiment near the field L%$ (see figs. 1 and 2)) may be explained by the same reason. Decreasing 6 with o must lead to decreasing the resistance r at low amplitudes X < HP and to its increasing at ix> Hp. Such a character of the frequency dependence of the surface resistance is illustrated by the curves in fig. 3. These experimental data are in qualitative agreement with calculated results presented in fig. 1. As the frequency or field amplitude increase, a superconducting sample is a close match to a normal metal in electromagnetic properties owing to the increasing relative contribution of the dissipative term in the current density in eq. (2 ). In particular, the conductivity a(E) tends asymptotically to the conductivity of a normal metal. It is well known that the ratio of the imaginary part of the surface impedance of a metal plate to the real part tends to the conditions of the normal skin effect and &d/2 is equal
199
to unity. Therefore it is natural to expect a decreasing ratio Imz/Rez with w in the case of a superconducting plate at the condition X < Hp. This supposition agrees with the experiment as well as with our calculation in the frame of the modified model (see figs. 4 and 5, respectively). We emphasize that there is a good qualitative agreement between experimental observations and our calculation of the frequency dependence of the surface impedance. However, some discrepancies of our results are noteworthy. The most distinct between them is a maximum in the calculated dependence r( co) at 2 > HP which was not observed in our experiment. It is possible that our experiments have been done within a narrow frequency limit. The quantitative discrepancies observed may be connected with different circumstances. The critical current density J, was taken at constant value in our calculation whereas it is well known that J, decreases with the magnetic induction. The contribution of this dependence must lead to an essential influence of B on the frequency o. in eq. (8) with a consequent deformation of the scale along the x-axis for all calculated curves. Furthermore, we use in our calculation V-Z plot parameters taken from ref. [ 5 ] while
20
Fig. 3. Frequency dependences of the surface resistance r( o/2x) / r(w/2x= 10 Hz) for a cylindrical sample 10 mm in diameter measured at T= 77 K and at two amplitudes of the AC field which values are included in figure. The penetration field HP= 8 Oe.
Fig. 4. The measured relation 5’/ 9 as a function of frequency o/w0 for a superconducting plate 0.8 mm in thickness. .%=H,/ 3=0.6Oe, T=77 K.
200
L.M. Fisher et al. /Applicability of the critical state model
T~rrm-m~TTlmT IO -=
10 -’
10 -5
10
FREQUENCY, Fig. 5. The calculated dependence y/9 on the dimensionless frequency w/w, for a superconducting plate at %=H,/2 and WO.15.
these parameters are specific for every sample. Their role is very essential because they define the dissipative term in eq. (8) which controls the frequency dependence of the impedance z. It has been known that the surface impedance is one of the main characteristics of a conducting medium and contains information about its electrodynamic and kinetic properties. Hence, in principle, it is possible to obtain the parameters of a V-Z plot of a superconductor by measuring the frequency dependence of its surface impedance connected with these parameters. According to eq. ( 13), the frequency correction to the impedance is proportional to WY.This means that at moderate frequencies the value p= Ir(w)/r(O)-1) must change with frequency by a power law. This deduction is confirmed by the results of our calculations. Figure 6 shows the dependence p ( o/wO) presented in double logarithmic scale. Both straight lines calculated for different values of h and low w/w0 are parallel to each other and have the slope according to choosing a value of y= 0.15. The results of analogous data handling of the measured dependence Rez(w) for the plate at two different amplitudes 3y are presented in fig. 7. It is clear that for both amplitudes the experimental dependence r(w) may be described by a power func-
-iml- I ‘T’Ti
3
10
7
10
I
I
a/%
Fig. 6. The calculated dependence of the frequency addition p to the surface resistance of a superconducting cylinder on o/wO. The amplitudes of radio wave field are included in the figure. The value of y is 0.15.
I
I 0,
01
I
,A
//
A’
-A’
.
.
1 /’
,/ /
A’ .
/
,’
.
/’
/
*“’
/’
10 2
III/,
10 3
FREQUENCY,
10 ’
w/‘&r
10 5
, Hz
Fig. 7. The measured dependence of the frequency addition p to the surface resistance for a high-T, plate 0.8 mm in thickness, T= 77 K and a radio wave amplitude X= 0.6 Oe (curve 1) and X= 12 Oe (curve 2). The parameter p is defined in this figure by the relationp= lr(o/2x)/r(w/Zx=32 Hz)- 11.
L.M. Fisher et al. /Applicability of the critical state model
tion with the exponent equal to 0.37, correlated with data of direct measurements of the V-Z plot for this sample. In such a manner the contactless method is developed to determine one of the main parameters of the V-1 plot for a superconductor in the resistive state. To extract the value of y it is convenient to measure the impedance at Xc> Hp. Changing the impedance is very noticeable at this region of %. Moreover, the instrument error decreases at high amplitudes. We consider that it is possible to obtain more detailed information about the V-Z plot using impedance measurements. However, to restore the V-Z characteristics by this contactless method we need additional theoretical and experimental investigations. Generally speaking, our investigation of the frequency dependence of the surface impedance shows that the usual critical state model ( 1) has relevance for the description of the electrodynamic properties of superconductors. However, this model is of a limited applicability within the narrow limits of frequency and amplitude of excited fields. Figure 3 shows that the change of the surface resistance r does not exceed 10% at X
201
from electrodynamic measurements at low frequency and explained in the framework of the usual critical state model hold credibility. In particular, this is true for results of critical current density measurements by the contactless method [ 6 1.
Acknowledgements We would like to thank C.P. Bean for useful discussions. This work has been done in the framework of National Programs in high-T, superconductivity, project 90067 “Impedance” (Russia), “Collapse” (Ukraine) and grant 1050 E 91112 of the Mexican Technologies Science and Committee for ( CONACyT ) .
References [ 11 C.P. Bean, Phys. Rev. Lett. 8 (1962) 250. [ 21 H. Dersch and G. Blatter, Phys. Rev. B 38 ( 1988) 11391. [ 31 V.V. Bryksin, A.V. Goltsev and S.N. Dorogovtsev, Physica C 172 (1990) 352.
[ 41 C.P. Bean, Technical Report Grant 88FO34-NYSIS ( 199 1) [S] L.M. Fisher, N.V. Win, I.F. Voloshin, I.V. Baltaga, N.M. Makarov and V.A. Yampol’skii, Physica C 197 ( 1992) 161. [6] L.M. Fisher et al., Phys. Rev., to be published.
PHMCA B
195-201
On the applicability of the critical state model to the description of electromagnetic properties of high- TCsuperconductors L.M. Fisher,
N.V. Il’in and I.F. Voloshin
All-Russian Electrical Engineering Institute, Krasnokazarmennaya
N.M. Makarov
Str. 12, Moscow, I1 1250, Russian Federation
and V.A. Yampol’skii
Institute for Radiophysics and Electronics, Ukr. Acad. Sci., Acad. Proskura Str. 12, Karkov, 310085, Ukraine
F. Perez Rodriguez Intituto de Fisica, Universidad Autonoma de Puebla J-48, Puebla, Rue 725 70, Mexico
R.L. Snyder New York State College of Ceramics at Aljred University, Alfred, NY, 14802, USA Received
19 November
1992
The frequency dependence of the surface impedance of superconductors have been studied experimentally and theoretically in the radio frequency range. Its essential deviation was found from the linear law predicted by the usual critical state model. The character of this deviation depends qualitatively on the amplitude of the radio wave. We have established the frequency limits of applicability of the traditional critical state model. Results obtained print out an explanation in the frame of the modified model where we take into account the contribution of a dissipative term to the screening current. The value of this is connected with the V-I plot of the superconductor, so it is possible to obtain information about the V-i characteristics by the contactless method.
1. Introduction The critical state model [ 1 ] is commonly used to explain the magnetic properties of hard superconductors. According to ref. [ 11, the distribution of the magnetic induction B inside a superconductor is described by the equation curlB=(4x/c)J,(B)E/E,
(1)
where J, is the critical current density, c is the speed of light and E is the electric field. This model was proposed by Bean to describe static magnetization curves of homogeneous superconductors. The background of its application to granular systems (see, for example refs. [2] and (3)) concerns static conditions too. However, it is well known that many observations of electromagnetic properties of hard superconductors are interpreted in the frame of the model ( 1). For example, the magnetic field depen0921-4534/93/$06.00
0 1993 Elsevier Science Publishers
dence of the critical current density JC( B) calculated in the frame of the critical state model using results of measurements of the surface impedance y= 9- iy is in a good agreement with direct measurements of the critical current. Despite the popularity of the critical state model the question was raised as to whether this model can be used to describe the electromagnetic properties of superconductors. This problem has not been solved yet. It has been known that electric fields exist in the volume of a sample owing to the magnetic flux varying with time. In other words, a superconductor in the dynamic regime is always in the resistive state. This means that the superconducting critical current is accompanied by an existing dissipative component of current in the volume of the sample. This dissipative component arises owing to the electric field E. At such conditions the model (1) must be modified to take into account this dissipative cur-
B.V. All rights reserved.
196
L.M. Fisher et al. /Applicability ofthe criticalstate model
rent (see, for example, ref. [ 41). A corresponding equation describing the distribution of the magnetic induction may be written as follows curlB=
(4n/c)
[J,(B)+a(E)E]E/E,
(2)
where (7 is the dissipative conductivity of the medium which rapidly increases with E. It was shown [4] that the dissipative item in eq. (2) affects noticeably the harmonic composition of the electromagnetic field inside a superconductor. This item is discounted in the usual critical state model ( 1). This is possible in the case of low frequencies o and amplitudes X of the excited magnetic field. If the frequency o or amplitude SV arise, the electric field E increases and the role of the second item increases too. It is clear that the modification of the usual model ( 1) is inevitable. Nevertheless, many authors interpret their results obtained even in the microwave region using the simplest model. So, the special analysis of the response of a superconductor to the external electromagnetic signal in the frame of modified material equation (2) is needed to establish the frequency boundaries of the critical state model ( 1) application. This paper is devoted to solving this problem. We have carried out the theoretical and experimental study of the dependence of the surface impedance of granular highT, superconductors on the frequency of an external radio wave. The necessity of considering this study is conditioned by a great number of papers where the information about parameters of the superconducting state is extracted from the results of electromagnetic measurements. Besides, the frequency dependence of the surface impedance contains significant information about the V-Z plot of a superconductor in the resistive state. We have shown that the inclusion of the dissipative item in the current density in eq. (2) leads to deviation of the dependence P(o) from the linear law following from eq. ( 1). The character of this deviation depends on the amplitude X of the radio wave. At low .# the surface impedance increases with frequency lower than w. If the amplitude .% is so high that electromagnetic fields exist at any point of the sample volume, the surface resistance A will increase faster and the surface reactance 3” slower than o. We have found that it is possible to determine VI plot parameters for a sample having measured the
deviation of the function 9(w) from a linear law. The frequency boundaries were established where the traditional critical state model agrees satisfactorily with experimental data.
2. Theory Let us consider a plate with thickness d of a granular high-T, superconductor placed in an external AC magnetic field having the form H(t)=.#coswt.
(3)
Let this field H be directed along the surface and parallel to the z-axes. Let the x-axis be perpendicular to the plate. Assume that the origin of the coordinates is placed in the center of a plate. The system of Maxwell equations describing the AC field distribution inside a plate has the form II’=-
%
(J,+AIEI”)
signE,
E’ = -b/c.
(5)
Here a prime denotes the derivative with respect to X, the point signifies the time derivative. The parameter p is the effective magnetic permeability which takes into account intragranular currents [ 21. To simplify our consideration we shall neglect the dependence of J, on the magnetic induction B. The dissipative component of the current density in eq. (4) is assumed to be a power function of the electric field E with the exponent equal to y and the parameter A is the dimensional multiplier. This dependence of the dissipative current on E reflects the empirical form of the V-Z plot obtained in ref. [ 5 1. It was found in ref. [ 5 ] that y is equal to 0.15 and we shall use this value in our numerical calculation. The boundary conditions for eqs. (4) and (5 ) have the form B(d/2)
=/M
cos ot ,
E(0) =0 .
(6)
The surface impedance is defined as the relation of the first harmonics of electrical and magnetic fields on the plate surface: n/w
s
yy=8wdtE( C.H
0
-d/2,
t) exp(iwt)
.
(7)
L.M. Fisher et al. /Applicability of the critical state model
The following dimensionless low for convenience: p= ot,
C= 2x/d,
h=sf”lH,,
variables
HP = 2nJ,dlc
,
h=BlpH,,
e=2Ec/,umH,d, o. = ( 2c/pHpd)
are used be-
The substitution of eq. ( 12 ) in ( 11) gives the following equation for the surface impedance: z=(h/3)(1-3ni/4)+PF(h),
(8)
F(h) =ah’+‘Y,
(13)
/3= (o/o,,)“, (J,/A)“Y.
It should be pointed out that the fields are normalized by the value of HP which has a simple physical sense. HP is equal to the value of an external static magnetic field at the beginning of which a sample has no region where H=O. Using new variables we can rewrite eqs. (3)-(7) in the form h’=-(l+pleIY)
signE,
h(l)=kcosp,
E(O)=O,
zc4pud --z(~), C2
EL-h,
(9) (10)
(11)
z(w)=-(Llh)jdp.(l,p)enp(ip). 0
Recall that the real part of the surface impedance is proportional to losses dissipated in a sample per unit time. Accordingly, Re z(o) is proportional to losses per period 27~10 of the radio wave. Equations describing the distribution of an AC field in cylindrical samples are analogous to (9) and ( 10) and we shall not present them here. It should be pointed out only that the parameter din eq. (8) must be changed by the diameter 2R of a cylinder. Equations (9) and ( 10) are nonlinear and they cannot be solved analytically. However, it proves to be possible to analyze this system by the method of the perturbation theory at a low value of the dissipative term in comparison with unity in eq. (9) i.e. at low /I. We do not present this standard procedure here and give only an expression for the electric field on the surface of a plate in the case h < 1 taking into account the first order correction: e(l,q)=-
197
h2 Tsina,
(12)
where (Y is a complex number. The first term in eq. ( 13 ) does not depend on frequency and comprises the well known expression for the surface impedance in the frame of the traditional critical state model. The second item, proportional to oy, is connected with the dissipative term in eq. (2 ). An analogous equation may be obtained in the case h> 1. The same is true for cylindrical samples and the main term in the expansion of z is independent on frequency, and the correction is always proportional to WY. The expansion ( 13 ) is true at low 8, i.e. at O-=Xuo. To obtain the frequency dependence of the impedance in a wide range of o we solved eqs. (9 ) and ( 10) and the analogous system for cylindrical samples by numerical methods. Some results of calculations of the dependence r= Re z( w) for a cylindrical sample and two different amplitudes of the excited AC field are presented in fig. 1. The lower curve corresponds to X = H,/2. The upper curve is calculated for I = 3HJ2. Figure 1 shows that the function Re z(w) decreases monotonically with o at X= H,/2. However, the character of its behavior changes at X= 3HJ2. The value of r increases at low w, has a maximum, and decreases at high o. Changing the character of the behavior of r at low frequency takes place at some amplitude % which will be named the inversion amplitude. An analysis shows that in the case of a cylinder X zzHp. The imaginary part - Im z(w) decreases monotonically in any case independently of the amplitude X at a wide range of the frequency. The impedance behavior, as a function of w, is qualitatively the same in the case of plates. The inversion amplitude for plates coincides with 4HJ3. The results of our calculation are in contradiction with one of the main consequences of the traditional critical state model (1). As noted above, the value of z is independent on frequency at any field amplitude in the frame of this model.
198
L.M. Fisher et al. /Applicability of the criticalstate model
3. Experiment and discussion
plates with a thickness in the range 0.5-5 mm and cylindrical samples 3-5 mm in radius. Dependences of the surface impedance r= Re z on the external AC magnetic field amplitude R for the superconducting plate with thickness d=0.8 mm are presented in fig. 2. Curve I was obtained at the frequency w/2rc= 1 kHz and curve 2 at the frequency 100 kHz. We can see that every curve has a maximum of which the physical nature is well known. The surface impedance increases with X at low amplitudes owing to increasing penetration depth S of the AC field. However, at X > HP the induced electric fields penetrating from opposite sides of the plate are compensated. As a result, increasing 3% leads to an increase of 6 and to decreasing of the surface resistance. The surface resistance as a function of &! must therefore have a maximum. The difference between curves in fig. 2 shows the noticeable frequency dependence of the surface resistance r which cannot be explained in the simple critical state model ( 1). First, let us pay attention to the shift of the maximum position of Y(Z) to higher amplitudes at increasing frequency. This phenomenon obtains a natural explanation in the modified
The real and imaginary parts of the surface impedance of samples have been measured in the frequency range 10 Hz-100 kHz using standard inductive method. The pick-up coil was wound closely on the sample by a fine wire 30 pm in diameter. The AC magnetic field was applied using a coil wound by a copper wire. Pick-up signals proportional to &’ and X were extracted by a phase sensitive amplifier PAR124A. To increase the accuracy of our measurements we compensated the part of the signal connected with changing the magnetic flux in the gap between pickup coil and sample using an additional device. The measurement process was computer controlled, allowing rapid data acquisition and calculations to be made. The measurements were carried out at liquid nitrogen temperature. For this purpose the sample together with pick-up and AC coils was immersed into a liquid nitrogen vessel. To obtain the values Ret and Imz amplifier output signals were normalized by the amplitude and the frequency of the external AC field. We have measured granular Y-Ba-Cu-0 system samples prepared by a solid state reaction. They were
Fig. 2. Results of the measurements of the surface resistance r as a function of the radio wave amplitude X for a superconducting plate 0.8 mm in thickness at T= 77 K. Curve 1 corresponds to a frequency w/2x= 1 kHz, curve 2 to a frequency of 100 kHz.
Fig. 1. The calculated dependence of the surface resistance r(o) / r(oO) on the dimensionless frequency o/o,, for a superconducting cylinder at two amplitudes of X. These amplitudes are included in the figure. The value of y is taken to be equal to 0.15.
L.M. Fisher et al. /Applicability of the criticalstate model
model (2 ). As a matter of fact, the contribution of the dissipative term in eq. (2) to the screening current density increases with frequency and the penetration depth 6 of the AC field decreases. As a result, the maximum of r(X) reaches a higher amplitude of X. The inversion of the frequency dependence, observed in our experiment near the field L%$ (see figs. 1 and 2)) may be explained by the same reason. Decreasing 6 with o must lead to decreasing the resistance r at low amplitudes X < HP and to its increasing at ix> Hp. Such a character of the frequency dependence of the surface resistance is illustrated by the curves in fig. 3. These experimental data are in qualitative agreement with calculated results presented in fig. 1. As the frequency or field amplitude increase, a superconducting sample is a close match to a normal metal in electromagnetic properties owing to the increasing relative contribution of the dissipative term in the current density in eq. (2 ). In particular, the conductivity a(E) tends asymptotically to the conductivity of a normal metal. It is well known that the ratio of the imaginary part of the surface impedance of a metal plate to the real part tends to the conditions of the normal skin effect and &d/2 is equal
199
to unity. Therefore it is natural to expect a decreasing ratio Imz/Rez with w in the case of a superconducting plate at the condition X < Hp. This supposition agrees with the experiment as well as with our calculation in the frame of the modified model (see figs. 4 and 5, respectively). We emphasize that there is a good qualitative agreement between experimental observations and our calculation of the frequency dependence of the surface impedance. However, some discrepancies of our results are noteworthy. The most distinct between them is a maximum in the calculated dependence r( co) at 2 > HP which was not observed in our experiment. It is possible that our experiments have been done within a narrow frequency limit. The quantitative discrepancies observed may be connected with different circumstances. The critical current density J, was taken at constant value in our calculation whereas it is well known that J, decreases with the magnetic induction. The contribution of this dependence must lead to an essential influence of B on the frequency o. in eq. (8) with a consequent deformation of the scale along the x-axis for all calculated curves. Furthermore, we use in our calculation V-Z plot parameters taken from ref. [ 5 ] while
20
Fig. 3. Frequency dependences of the surface resistance r( o/2x) / r(w/2x= 10 Hz) for a cylindrical sample 10 mm in diameter measured at T= 77 K and at two amplitudes of the AC field which values are included in figure. The penetration field HP= 8 Oe.
Fig. 4. The measured relation 5’/ 9 as a function of frequency o/w0 for a superconducting plate 0.8 mm in thickness. .%=H,/ 3=0.6Oe, T=77 K.
200
L.M. Fisher et al. /Applicability of the critical state model
T~rrm-m~TTlmT IO -=
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FREQUENCY, Fig. 5. The calculated dependence y/9 on the dimensionless frequency w/w, for a superconducting plate at %=H,/2 and WO.15.
these parameters are specific for every sample. Their role is very essential because they define the dissipative term in eq. (8) which controls the frequency dependence of the impedance z. It has been known that the surface impedance is one of the main characteristics of a conducting medium and contains information about its electrodynamic and kinetic properties. Hence, in principle, it is possible to obtain the parameters of a V-Z plot of a superconductor by measuring the frequency dependence of its surface impedance connected with these parameters. According to eq. ( 13), the frequency correction to the impedance is proportional to WY.This means that at moderate frequencies the value p= Ir(w)/r(O)-1) must change with frequency by a power law. This deduction is confirmed by the results of our calculations. Figure 6 shows the dependence p ( o/wO) presented in double logarithmic scale. Both straight lines calculated for different values of h and low w/w0 are parallel to each other and have the slope according to choosing a value of y= 0.15. The results of analogous data handling of the measured dependence Rez(w) for the plate at two different amplitudes 3y are presented in fig. 7. It is clear that for both amplitudes the experimental dependence r(w) may be described by a power func-
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Fig. 6. The calculated dependence of the frequency addition p to the surface resistance of a superconducting cylinder on o/wO. The amplitudes of radio wave field are included in the figure. The value of y is 0.15.
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Fig. 7. The measured dependence of the frequency addition p to the surface resistance for a high-T, plate 0.8 mm in thickness, T= 77 K and a radio wave amplitude X= 0.6 Oe (curve 1) and X= 12 Oe (curve 2). The parameter p is defined in this figure by the relationp= lr(o/2x)/r(w/Zx=32 Hz)- 11.
L.M. Fisher et al. /Applicability of the critical state model
tion with the exponent equal to 0.37, correlated with data of direct measurements of the V-Z plot for this sample. In such a manner the contactless method is developed to determine one of the main parameters of the V-1 plot for a superconductor in the resistive state. To extract the value of y it is convenient to measure the impedance at Xc> Hp. Changing the impedance is very noticeable at this region of %. Moreover, the instrument error decreases at high amplitudes. We consider that it is possible to obtain more detailed information about the V-Z plot using impedance measurements. However, to restore the V-Z characteristics by this contactless method we need additional theoretical and experimental investigations. Generally speaking, our investigation of the frequency dependence of the surface impedance shows that the usual critical state model ( 1) has relevance for the description of the electrodynamic properties of superconductors. However, this model is of a limited applicability within the narrow limits of frequency and amplitude of excited fields. Figure 3 shows that the change of the surface resistance r does not exceed 10% at X
201
from electrodynamic measurements at low frequency and explained in the framework of the usual critical state model hold credibility. In particular, this is true for results of critical current density measurements by the contactless method [ 6 1.
Acknowledgements We would like to thank C.P. Bean for useful discussions. This work has been done in the framework of National Programs in high-T, superconductivity, project 90067 “Impedance” (Russia), “Collapse” (Ukraine) and grant 1050 E 91112 of the Mexican Technologies Science and Committee for ( CONACyT ) .
References [ 11 C.P. Bean, Phys. Rev. Lett. 8 (1962) 250. [ 21 H. Dersch and G. Blatter, Phys. Rev. B 38 ( 1988) 11391. [ 31 V.V. Bryksin, A.V. Goltsev and S.N. Dorogovtsev, Physica C 172 (1990) 352.
[ 41 C.P. Bean, Technical Report Grant 88FO34-NYSIS ( 199 1) [S] L.M. Fisher, N.V. Win, I.F. Voloshin, I.V. Baltaga, N.M. Makarov and V.A. Yampol’skii, Physica C 197 ( 1992) 161. [6] L.M. Fisher et al., Phys. Rev., to be published.