The higher harmonics of magnetization in the critical state of ceramic high-Tc superconductors

Volume 141, number 1,2

PHYSICS LETTERS A

23 October 1989

THE HIGHER HARMONICS OF MAGNETIZATION IN THE CRITICAL STATE OF CERAMIC HIGH-T~,SUPERCONDUCTORS I.D. LUZYANIN, S.L. GINZBURG, V.P. KHAVRONIN and G.Yu. LOGVINOVA Leningrad Nuclear Physics Institute, Gatchina, Leningrad 188350. USSR Received 31 July 1989; accepted for publication 8 August 1989 Communicated by V.M. Agranovich

The study of field dependences ofthe dynamic susceptibility first and higher harmonics in ultra-low fields is performed. The results show a satisfactory fit with the conclusions of the critical state theory of the low field electrodynamics of high-T. superconductors.

At present a wide range of irreversible and nonlinear phenomena is discovered and being intensively studied in most of the ceramic high-Ta superconductors (HTCS) in low fields essentially lower than the first critical intragranular field (He ig. ~ 100 Oe). These phenomena are the subject of the lowfield electrodynamics of high-Ta superconductors [1—4]. According to the current ideas a ceramic high-ternperature superconductor is a multiply connected systern. The penetration of the low field into this system starts at the true H~ 1,which is rather low for weakly coupled grains. For example, according to ref. [5] H~1 el/a, where a is the grain size, I is the characteristic energy of the random weak couplings. The small values of I result in quite low values thesense true 2 (n~ 10 Oe) hasofthe H~1 1—10critical mOe. Hg~~ of upper field ~0/a in granular systems (here cII~J is the flux quantum). One can expect that such a superconductor behaves as a classical type-I! superconductor with ic—’ (H~ 2/H~1 ) 1/2 100>> 1. Due to the small H~1the vortices’ size will be large, their dimensions depend on the true penetration depth into the multiply connected 1 (22 ~/ 2) where /tefffn medium, +f~g.is the effective H~1 / ~ff>> a magnetic permeability of the ceramic taking into account that the low fields do not penetrate into the grains. Here J is the part of the volume filled by the superconductor, .f,~ = 1 —J and ~tg. is the mean magnetic permeability of the grains, dependent on the

grain size, their configurationand the London depth 2L• IfAL <> 1 and h ~ H~ 1,Beq( h)

0375-9601/89/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

/.tefIh.

Therefore the problem is re85

Volume 141, number 1,2

PHYSICS LETTERSA

23 October 1989

duced to determination of inhomogeneous magnetic field h (r) for which the critical state equation has

This results in a strong nonlinear susceptibility fielddependence. Let the field h ( I) = H+ h0 cos wt be ap-

the form [6—8]

plied to the sample. In this case B(l) can be expanded in a Fourier series: B(t)=~ao+>(a~cosnwt+b~sinnwt). (3)

jdh/dxI=4iij~(h),

(1)

where j~(h)=a(h)/h and the mean induction d/2

2 =

~‘

l~eff j

.

h

(x) dx.

We would like to call attention to the difference of the equation in refs. [6—8] from eq. (1) since the last contains the effectivemagnetic permeability ~ From here we will consider the case of the simplest geometry namely for the slab of thickness d and the magnetic field parallel to its plane. !neq. (1) a(h) isthepinningforceandj~(h)has the meaning of a critical current and isisaa phenomenological function of h. This function very important characteristic ofthe system investigated since it may be employed for a calibration certificate of ceramic superconductors. Note that for this function different models are proposed. In particular, I~ (h) in the form 1~(h) =j 0H0/ (I hp + H0) (2)

Having performed some calculations, similar to ref. [6], we obtain in adiabatic approximation at h0 < 2 2— 32itd dHj~(H)’ 2k— h3 d / 1 “ k b 2—1)(k22) 2k= i6~2ddII(,JC (H)) (k 4 4 a2k+1 =

0, _

—

-

-

86

(4)

One can see from eq. (4) that all even harmonics are proportional to h odd ones to h~ and their coefficients differ only in the harmonic’s number. This means strong nonlinearity. Note that the expressions for odd harmonics in eq. (4) conserve their form at H= 0 while even harmonics naturally turn to zero. Note that in Bean’s model there should be no even harmonics at H~0. This means the existence of random symmetry here, resulting in even harmonics vanishing. Let us proceed to the experiments. We studied the dependence of imaginary and real parts of the first harmonic of induction (a1, b1) as well as the moduli of higher harmonic amplitudes c~= (a~+ b~)1/2 on the intensity of the ac and dc fields. Note that according to theory (see eq. (4)) at n ~ 3 all a~= 0, though in our experiments this statement was not checked. The experiments were performed with a 3.ceramic In orY—Ba—Cu—O sample with density 4.8the g/cm der to avoid demagnetization effects sample has toroidal shape. A single-layer toroidal coil was wound uniformly around the sample. The measurements, by using a technique very much like that previously ~,

is used in ref. [17]. (Here H0 is some characteristic field.) The case H0—~ results in Bean’s model [61 in whichj~( h) is independent of the magnetic field. The case of H0=0, JoI~Io= const, corresponds to the Kim— Anderson model [8], in which the pinning force a ( h) does not depend on h. Generally speaking there does not exist a choice ofthe JC(h) function based on theory. That is why the necessity of the experimental determination of this function form is quite evident, In the present paper we show that the behavior of the nonlinear dynamic susceptibility observed within the low field range (lO_2_l Oe) is well described by this simple critical state theory. We succeeded in determining an experimentally evident aspect of./~(H) which coincides with eq. (2). This experiment has yielded that H0 is about 3 Oe. In order compare with experiment we make some simpletotheoretical predictions. As was mentioned above, at h> ~ the granular superconductor is essentially a nonequilibrium system, which in particular is manifested by the existence of hysteresis.

.

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PHYSICS LETTERS A

described in ref. [4], were made at 78 K within the frequency range l02_ l0~Hz and l0—2~h0,H~1 Oe. The background field did not exceed 10 mOe. Note that it is not a1 or b1 that is usually determined in the experiment, but the real and imaginary parts of the susceptibility x’ and x”. It is clear that in the limit h0—~0,x’ = l/47t. Therefore x’ can be represented asx’(ho)=—l/4ir+x’a(h0). Then —

=

4ithox” (h0)

(5)

‘

Let us turn now directly to the experimental results. Fig. 1 shows the dependences of a1, b~,C3~C5 on h0 obtained for H= 0. One can see that the values studied show the h~ dependence, the amplitude h0 being changed by two orders of magnitude. As follows from the theory the quadratic dependence is characteristic for the critical state. Eqs. (4) for a~ and b~contain two independent values ~ and Jo which cannot be determined in our experiments in an independent way. Therefore to compare the theoretical and experimental results it seems reasonable to consider the ratios a1/b1, b1/c3, etc., which according to eqs. (4) should not depend on l~effand I~ (H). Here one should have in mind that eqs. (4) a1 U—U

b1

23 October 1989

Table 1 Ratio

_______________________________ a1

c3

H+H0 C2 h0 c3 __________________________________________________ theory 2.3 5 7 0.56 experiment 1.8±0.2 3.1±0.3 9.8±1.2 0.6±0.3

_____________________________________

were obtained in adiabatic approximation. However as was shown in ref. [41, in Y—Ba—Cu—O ceramic samples the real part of the susceptibility depends only slightly, in a logarithmic way, on the ac field frequency whereas the imaginary part does not depend on w. Nevertheless (see table 1) a rather good fit of the experimental data with the theoretical expectations is observed. This is quite natural as the part of x’~connected with logarithmic dependence on w and being apparently due to low field flux creep, is only a small correction. The studies in dc fields have shown that h ~ and h ~ dependences for even and odd harmonics respectively at h0 <
en

10_2 (Ce)

a1

a.U.

H~1.25

10_i

1

~

10

C3 __________________

~

C2

io_2

10_2

10_i

li0 (Oe)

10°

Fig. 1. Dependences ofa~,b1, c3 and c5 on h0 at H= 0; a1 and b1 were measured at f 100 kHz; c3 and c5 were measured at a frequency of the ac field of 20 kHz.

10_i

100

h 0 (Ce) Fig. 2. Dependences of the amplitudes of the second (c3) and third (c5) harmonics on the ac field amplitude h0 (f= 20 kHz) at H= 1.25 Oe. The quantities c2 and c3 are normalized to their values at h0= 1 Oe.

87

Volume 141, number 1,2

C

2

PHYSICS LETTERS A

(a.u.)

23 October 1989

H=1.25 Oe

(c~—c~)/c~

h0=o.i Oe I

._-1_______________ 1.0

—101-

1.0

Ce 0.5

0

—20~ 0.5

0.5

1.0

H (Oe)

2 3 4 5 6 7 8 9 10

Fig. 3. Dependences of c2 and (ci’ —c~)/c~ on H at h0=0. 1 Oe. The quantity c2 is normalized to its value at H= 1.25 Oe; c~’and ci are the third harmonic amplitudes at H~0 and H= 0, respectively;f= 20kHz.

proportional to H+H0. The dependence (c~ C03)/c03 (= IHI /H0) on H (at h0=const) is given in 0 fig. 3 as an example, where c~’and C 3 are the third harmonic amplitudes at H 0 and H= 0, respectively. Here the second harmonic amplitude is independent of H (fig. 3). According to eqs. (4) even harmonics should be proportional to d [1/j~(H)J / dH. Comparison of these facts regarding to the behavior of c2 and c3 in dc field results in the self-consistent determination of the evident aspect of j~(H) IoHo/ (I HI + H0), which is in accord with the model [7].The estimation for our case gives H0 3 Oe. Note that with this H0 value the experimental ratio of the amplitudes of second and third harmonics appeared to be close to the one expected theoretically (see table 1). The strongly pronounced nonlinear properties of the granular superconductor in low fields show the spectrum of harmonics having slightly decreasing amplitudes (see fig. 4). This is also the confirmation of the critical state picture, considered above. The results of the study performed enable us to assert that the critical state really occurs in ceramic high-Ta superconductors in the low field region (10 2 1 Oe), therefore the ideas of critical state the-

fl

Fig. 4. The spectrum of higher harmonics: f= 10 kHz; H= 1.25 Oe, h0 = 0.1 Oe.

ory may be used for the low field electrodynamics of HTCS.

—

88

This work is supported by the Scientific Council on the Problems of HTCS and is carried out within the framework of Project No 2 “Shkala” of the State Program “High-temperature superconductivity”. It is the authors’ pleasant duty to thank S.V. Maleev, B.P. Toperverg, A.I. Sibilev and S.M. Bezrukov for constant interest, for helping to perform the experiments and discussions of results.

References [1] H. Maletta, A.P. Malozemoff, D.C. Cronemeyer, C.C. Tsuei, R.L. Greene, J.G. Bednorz and K.A. Muller, Solid State Commun. 62 (1987) 323. [2] S. Senoussi, M. Oussena and S. Hadjoudj, J. AppI. Phys. 63 (1988) 4176. [31C. Jeifries, Q.M. Lam, Y. Kim, L.C. Bourne and A. Zettl, Phys. Rev. B 37 (1988) 9840. [4] V.P. Khavronin, I.D. Luzyanin and S.L. Ginzburg, Phys. Lett. A 129 (1988) 399. [5] J.R. Clem, Physica C 153—155 (1988) 50. [6] C.P. Bean, Rev. Mod. Phys. 36 (1964) 31. [7] Y.B. Kim, C.F. Hempstead and AR. Strnad, Phys. Rev. 131 (1963) 2486. [81 P.W. Anderson and Y.B. Kim, Rev. Mod. Phys. 36 (1964)