Peculiarities of the vortex dynamics in YBa2Cu3Ox single crystals as revealed by irreversible microwave absorption

Physica C 300 Ž1998. 239–249

Peculiarities of the vortex dynamics in YBa 2 Cu 3 O x single crystals as revealed by irreversible microwave absorption T. Shaposhnikova, Yu. Vashakidze, R. Khasanov, Yu. Talanov

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Institute for Technical Physics, Kazan 420029, Russian Federation Received 9 September 1997; revised 19 January 1998; accepted 8 March 1998

Abstract We report the results of a wide experimental study of the irreversible modulated microwave absorption as a function of temperature, applied magnetic field and modulation amplitude in the YBa 2 Cu 3 O x single crystals. To analyze the experimental data the model of the microwave power dissipation by the flux lines has been developed taking into account thermal fluctuations, the distributions of currents and vortices over a sample. We have obtained the information about regimes of vortex motion in the different areas of the H–T phase diagram and estimated the values of flux flow viscosity and critical current density. q 1998 Published by Elsevier Science B.V. All rights reserved. PACS: 74.25.Nf; 74.60.Ge; 74.60.Jg Keywords: YBa 2 Cu 3 O x single crystals; Microwave absorption; Vortex matter phase diagram

1. Introduction Barriers to the vortex motion, such as pinning centers, edge and surface barriers, result in the irreversible behavior of magnetic properties of the type II superconductors and control the value of critical current density. The most direct way to studying irreversibility is the registration of magnetic hysteresis loop upon varying temperature or field with the help of any magnetometer ŽSQUID, Hall sensors, torque etc... Using AC methods Žsuch as AC suscep-

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Corresponding author. Kazan Institute for Technical Physics, Sibirskii tract 10r7, 420029 Kazan, Russian Federation. Tel.: q7-8432-761154; fax: q7-8432-765075; e-mail: [email protected].

tibility, vibrating reed etc.. gives a useful knowledge which considerably complements data obtained by DC methods, but it requires a special consideration to extract valid information from the analysis. Modulated microwave absorption ŽMMWA. measurements occupy the particular place among other AC methods. It has great advantages such as high sensitivity and the possibility to create a large variety of external conditions during measurements. However it was repeatedly mentioned Žsee for example Refs. w1–3x. that a pinning is ineffective if measurements are performed at a frequency higher than that of so-called ‘intravalley oscillations’. It is in this range of frequencies that most of MWWA measurements are performed Ž10 10 Hz or higher.. Therefore it may seem that this technique is useless for studying the irreversibility due to pinning effects. At the

0921-4534r98r$19.00 q 1998 Published by Elsevier Science B.V. All rights reserved. PII S 0 9 2 1 - 4 5 3 4 Ž 9 8 . 0 0 1 4 0 - 3

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T. ShaposhnikoÕa et al.r Physica C 300 (1998) 239–249

same time there are many publications reporting the observations of irreversible behaviour of microwave absorption in high-Tc superconductors Žfor example see Refs. w4–10x.. The matter is that the statement about an ineffective pinning is valid only for the microwave absorption by single vortex. In reality, upon relatively slow changing field or temperature the irreversible variations of current and vortex distributions take place and the corresponding transformations of power dissipation occur. That is the reason why MMWA exhibits a hysteretic behavior and serves as useful instrument for the investigation of irreversibility. Concrete information obtained by MMWA measurements depends highly on the underlying theoretical model used for data analysis. There are several models describing the origin of hysteretic microwave absorption in the literature w4–7,9,11,12x. Some of them w6,12x, related to low field area, are based on the field dependence of critical current of the Josephson weak links. Other models w7,9,11x relate to the high field area and treat the MMWA hysteresis as due to the change of the absorption signal phase because of the transformation of the current distribution at the moment of the field sweep reversal. Here we present the results of detailed investigation of the MMWA hysteresis in the YBa 2 Cu 3 O x single crystals in a wide range of temperature and magnetic field. The dependence of the hysteresis loop on the modulation amplitude has also been studied. The analysis of the experimental data is based on the model proposed in Ref. w9x. In order to obtain the expressions for microwave absorption by vortices and its hysteresis the authors of Ref. w9x have solved the equation of vortex motion in the anharmonic pinning potential under the action of the Lorentz force due to the currents induced by sweeping, modulating and microwave fields. Analyzing the experimental MMWA hysteresis they have defined the dependence of pinning potential Up on the magnetic field strength. However such significant facts, as inhomogeneities of the field and current distributions, affecting the irreversible behavior of superconductors, as well as thermal fluctuations of the flux lines have been neglected. Here we show that the combined account of all these features enables us to obtain the consistent picture of the microwave dissipation in any conditions. This approach makes the

method more powerful and gives one the possibility to expand the technique of MMWA measurements to investigation of the phase diagram of the vortex lattice. Preliminary results of our study have been published in Ref. w13x. 2. Experimental The studied samples were the YBa 2 Cu 3 O x single crystals with typical sizes of 1 = 2 = 0.1 mm3, Tc , 92–93 K, and the developed twin structure. To measure microwave absorption we used the commercial electron spin resonance spectrometer ŽBruker BER-418s. operating at a frequency 9.4 GHz ŽXband.. During measurement a sample was placed into a spectrometer cavity inside the helium flow cryostat. Temperature was varied from 10 K to the superconducting transition temperature. A crystal was oriented with the applied DC magnetic field Ha parallel to the c-axis, and the microwave magnetic field h1 lying in the basic Ž ab . plane. The DC field was modulated with a frequency of 100 kHz and an amplitude h m from 0.1 to 10 Oe. The detection and the amplification of a MMWA signal are performed by the lock-in technique on the frequency of the first harmonic of modulation. No higher harmonic was registered. The ‘standard’ MMWA hysteresis loop was recorded from the initial field value Hi by sweeping magnetic field up and down. The sweeping range D H was equal to 150 or 200 Oe. Hi was varied from 0 to 9 kOe. By varying the external conditions the dependence of the loop amplitude on temperature, field Hi , and modulation amplitude was obtained. These measurements of the MMWA hysteresis loop probe the existence and magnitude of irreversibility in any point of the H–T phase diagram. Points of the irreversibility line were determined according to disappearance of the MMWA hysteresis. 3. Theoretical model Our approach to the calculation of the irreversible microwave absorption is similar to that of Kessler et al. w9x. The original aspects of our consideration are

T. ShaposhnikoÕa et al.r Physica C 300 (1998) 239–249

an inclusion of the field and current distributions and the account of the influence of thermal fluctuations on the vortex dynamics. On the one hand these innovations make the microwave dissipation picture more clear and on the other hand they give possibility to obtain a large amount of information on the vortex matter state. It is well known that the energy dissipation in a superconductor upon its interaction with the AC field is due to the vibrations of vortex lines having ‘normal’ Žnonsuperconducting. cores w1x. These vortices oscillate around the equilibrium position determined by the action of the pinning force and the Lorentz force. The pinning force depends on the magnitude and the shape of pinning potential of a certain pinning center and in general may change from one center to another. The Lorentz force is the result of the vortex interaction with the currents flowing through a superconductor. In the case discussed here, in a sample there are currents due to the critical state originated at the DC field Ž Ha . sweep and the low frequency alternation of modulation field h m . By contrast, the vortex oscillations under the Lorentz force, connected with the current induced by microwave field h1 , are insensitive to pinning and occur in the flux flow regime w1–3x. Let us consider the pictures of field and current distributions and their variations in more detail. In Fig. 1 the sketches of these distributions, generated upon sweeping the applied field up, are shown as solid lines. Modulating the applied field with the frequency vm , which is several orders of magnitude lower than the depinning frequency, gives rise to the periodical change of the field and current distributions in the sample. In Fig. 1 the changes of distributions at half period after the modulation is switched on are shown as dashed lines. The penetration depth of the modulation field is proportional to the ratio of the modulation amplitude to the complete penetration field H ) , which in turn is proportional to the value of the critical current density. Usually h m < H ) Žin YBa 2 Cu 3 O x of a typical size with Tc G 90 K at T F 65 K H ) ) 100 G w14x and h m F 10 Oe.. So the modulation field penetrates only into thin layer near the crystal edges, and the vortex distribution in the most part of a sample remains intact. In contrast, the current distribution is transformed by modulation in the whole volume of sample ŽFig. 1b.. It is due to

241

Fig. 1. Distributions of field Ža. and current Žb. across the sample, calculated in accordance to Ref. w15x for increasing applied field Ha with addition of modulation field h m at positive Žsolid curves. and negative Ždashed curves. half-periods.

the B x component of field, inducing a current along with Bz , changes over the entire sample surface w15,16x. Therefore the Lorentz forces acting on all vortices change with the modulation frequency. Thus the modulation serves not only for monitoring an absorption, but it gives rise to the significant changes in absorbing system as well. This considerably complicates the microwave absorption picture and its description. We solved the problem of determining the microwave absorption in the geometry shown in Fig. 2. The slowly sweeping DC field Ha with the modulation field h m s h m 0 e i v m t is applied along the crystal c-axis Žthe z-axis of the coordinate system.. Correspondingly the vortex lines in a sample are of the same direction. The total current J Ž Ha , h m . induced by fields Ha and h m has only x- and y-components. The calculation of the current and vortex distributions in this geometry is not a trivial problem because of the applied magnetic field distortion around a superconductor. Such calculations had been carried out in several works, see for example Refs. w15,16x. In our treatment we use their results. The microwave magnetic field h1e i v 1 t is applied along the y-axis. It results in a microwave current

T. ShaposhnikoÕa et al.r Physica C 300 (1998) 239–249

242

where u 0 Ž r, t . is the equilibrium position of vortex determined by the pinning potential and by the local value of current density j Ž Ha , h m .. u thŽ r, t . is the displacement from equilibrium as a result of thermal random fluctuations. The vortex oscillates with microwave frequency around the position u˜ s u 0 Ž r, t . q u t hŽ r, t .. u1Ž r .e i v 1 t is the vortex displacement under the action of microwave current. For the sake of simplicity we assume that the probability of thermal fluctuations is described by a normal distribution with the vortex mean-square displacement ² u 2 :: Fig. 2. The sketch of the geometry of experiment.

W Ž u th . s

density j1e i v 1 t flowing along the x-axis near the plane surfaces in the skin layer of the depth d . The Lorentz force due to this current produces oscillations of the parts of vortices, passing through a skin layer, with the frequency v 1. The equation of motion of a flux line in a coordinate system, localized on the pinning site, is as follows: mu˙ q h u˙ s Fp Ž u . q

Ž j Ž Ha ,h m . q j1e i v t . Ž 3.1 .

where u is the vortex displacement from the pinning site, m is the inertia of flux line per unit length, h is viscosity per fluxon unit length. Fp is the restoring force due to pinning, F 0 is the quantum of magnetic flux, k is the unit vector along the vortex, f th Ž u, t . is the thermal Langevin force with correlation function ² fa Ž u,t . fb Ž uX ,tX . : s

2T

h

(2p ² u : exp 2

ž

y

Ž u th .

2

2² u 2 :

/

.

Ž 3.4 .

The mean-square thermal displacement of the position of a vortex line ² u 2 : is proportional to the temperature, as was shown in Ref. w17x ² u 2 : s 4p

Tl2

F 03r2 B 1r2

.

Ž 3.5 .

For a small microwave displacement u1 Fp Ž u . s Fp Ž u˜ . q

1

=F 0 k q f th Ž u,t . ,

1

ž

E Fp

/

P u1 .

Eu

u˜

Ž 3.6 .

Then from Eqs. Ž3.1., Ž3.3. and Ž3.6. one can obtain the system of two equations: ym v 12 u1 q i v 1h u1 y

ž

E Fp Eu

/

P u1 s w j 1 = F 0 k x , u˜

Ž 3.7 . Fp Ž u˜ . q j Ž Ha ,h m , x . = F 0 k q f th Ž u,t ˜ . s 0.

Ž 3.8 . da b dr r X d t tX .

Ž 3.2 .

Eq. Ž3.1. is similar to the equation of motion from Ref. w9x, but there are two differences: Ža. the local value of current density jŽ Ha , h m . depends on the x-coordinate in our equation; Žb. the random thermal force is present. The modulation frequency vm is by several orders of magnitude less than the microwave frequency v 1 , therefore the solution of the equation can be represented in the form u Ž r ,t . s u 0 Ž r ,t . q u th Ž r ,t . q u1 Ž r . e i v 1 t ,

Ž 3.3 .

Solving Eq. Ž3.7. we can find the expression for the displacement u1. The absorption power per unit length of a vortex line is P s 12 Re Ž w j 1 = F 0 k x u1 . 1 s

hv 12 j12F 02

4 h 2v 2 q Ž E F rE u 1 p

2 u˜

.

Ž 3.9 .

.

To obtain a certain analytical form of the absorption P one has to know the value and the shape of the potential well of all pinning centers. It is cer-

T. ShaposhnikoÕa et al.r Physica C 300 (1998) 239–249

tainly impossible. However in many calculations the using of a simple sinusoidal shape of potential relief appeared to be highly efficient Žfor example see Refs. w1,9x.. We use such assumption too and express the local symmetry of the pinning potential as

ž

Up Ž u . s Up 0 1 y cos

2p u

ž // a

,

Ž 3.10 .

where a is the mean distance between the pinning sites. Then Fp ' < Fp < s y

E Up Eu

s ysin

2p u 2p Up 0

ž / a

a

s rn B Ž x,t .rBc2 is the flux flow resistivity. After integration of the expression Ž3.9. with Eq. Ž3.14. over the sample volume, one finds the total absorption Ptot . In order to use the solution for the experiments with lock-in technique we have to extract the Fourier component of the absorption with the modulation frequency v m . Thus, vm Ptot s

1

hv 12F 02 j12

4

2p Up 0 a

sin

ž

2p u 0

/

a

vm Ptot s

If j equals the critical current density jc , u 0 increases up to ar4 and Eq. Ž3.12. becomes 2p Up 0 a

E Fp Ž t .

s

2p jcF 0 a

u˜

s

2pF 0

cos

ž

2p

2 c

2p u th

Ž u 0 q u th .

a

ž (j y j

a

4

2 Bc2 v 13

rn m 0

F 02 j12

2

cos

ž

2p u th a

q`

Hy` HS (B

2

/ /

Ls

1 4

pup s

//

.

3

Ž x ,t .

d xd yd u th d t.

u˜ .

The amplitude of the signal hysteresis L is obtained as the difference between the absorption levels at field sweep up and sweep down:

To take account of the thermal fluctuations of the vortex displacement the result must be averaged over u with normal distributions W Ž u., and to obtain the absorption by unit area the result must be multiplied by the number of vortices per unit area n s B Ž x,t .rF 0 , where B Ž x,t . is the local magnetic field induction calculated in accordance with Ref. w15x. The microwave current density j1 flows in the skinlayer with thickness d s Ž2 r frm 0 v 1 .1r2 , where r f

a

Ž 3.15 .

Ž 3.16 .

where

ž

2p r v m

H0

W Ž u th . cos Ž v m t .

=

Ž 3.14 .

yj sin

dV d u th d t.

Ž 3.13 .

Then

Eu

)

1

h 2v 12 q Ž E FprE u s jc Ž Ha . F 0 .

2 u˜ .

Here we use the expression of viscosity for the flux flow regime h s F 0 Bc2rrn . Substituting the analytical expressions h and d in Eq. Ž3.15. and performing the integration over z one gets

q j Ž Ha ,h m ,t . F 0 s 0.

Ž 3.12 .

Hy` HVexp Ž y2 zrd .

h 2v 12 q Ž E FprE u

Ž 3.11 .

y

q`

2p r v m

H0

W Ž u th . cos Ž vm t .

=

.

To derive the expression for the derivative of Fp we solved the equation for the local equilibrium state of a vortex:

243

)

2 Bc2 v 13

rn m 0

HbbŽ hhqh, j Ž

m

s

c.

(H

m , jc .

H0 2p

Hp

q

Ž 3.17 .

q` 3 up

Ž x . d xH

W Ž u th . d u th

y`

cos Ž t . d t

p

=

F 02 j12 Ž pup y pdown . ,

2

2

2

h v q Ž fra . A21 cos Ž t . d t 2

h 2v 2 q Ž fra . A22

Ž 3.18 .

T. ShaposhnikoÕa et al.r Physica C 300 (1998) 239–249

244

and pdown s

Hyh q2 h HbbŽ DDHqh ,j . Ž

s

m

(Ž H

m ,2 j c .

max up

pdown s

3

c

=

=

Hy` W Ž u

. d u th = 2

(

=

3 Hdown

m ,2 j c .

H0

2

2

. d u th

2

2

2

h v q Ž fra . A25 cos Ž t . d t

2p

Hp

q

2

h v q Ž fra . A26

.

Ž 3.19 .

Here f s 2pF 0 jc and Ž fra. A k is the local value of the second derivative of the pinning potential with respect to the vortex displacement from the pinning centre. The definitions of other parameters involved in Eqs. Ž3.18. and Ž3.19. are presented in Appendix A. The magnitudes of pup and pdown are slightly different in modules, but they have opposite signs. The oscillations of vortex positions under the modulation field h m cosŽ vm t . are in phase with the modulation field for the increasing applied field Hup , but they differ in phase by p for the decreasing field Hdown . Eqs. Ž3.18. and Ž3.19. may be simplified if one takes into account that for microwave frequency hv 1 4 2pF 0 A kra: pup s

2p 2F 02 jc2 Ž H ,T . a 2h 4 Ž T . v 14

HbbŽ hhqh, j

=

Ž

m

s

p

=

c.

2 1 cos

H0 I

(H

m , jc .

ž

exp y 3 up

m

2 5 cos

H0 I

a2

(Ž H

m ,2 j c .

max up

8p 2 ² u 2 : a2

/ 3

Ž x. . d x

c

cos Ž t . d t q c.

8p 2 ² u 2 :

2p 2

Hp

I4 cos Ž t . d t

(H

Ž x. d x

3 down

m ,2 j c .

2p

Ž t . d t q H I62 cos Ž t . d t . p

Ž 3.21 .

cos Ž t . d t

p

=

th

2 3

ž

exp y

Ž x. d x

q`

Hy` W Ž u

Ž

p

2

=

H0 I

s

h 2v 2 q Ž fra . A24

s

s

,j HbbŽDhHyh q2 h

q

cos Ž t . d t

HbŽD Hyh q2 h

Ž

m

b Ž h m , jc .

q

Hyh q2 h HbbŽDDHqh ,j .

p

h 2v 2 q Ž fra . A23

2p

Hp

q

th

cos Ž t . d t

p

H0

a2h 4 Ž T . v 14

Ž x.. d x

q`

=

4p 2F 02 jc2 Ž H ,T .

As can be seen from above expressions, the variation of L with temperature is determined by T-dependences of h , jc and ² u 2 :. The field dependence of L is due to the jc Ž H . function. It is impossible to obtain simple analytical expressions for pup and pdown as the expressions under the integrals are too complex. That is why we had to do numerical calculations of LŽT, H . to compare the model functions with the experimental data. 4. Experimental results and discussion In Fig. 3 the experimentally observed loop of the MMWA hysteresis is shown. Its shape is very similar to that published before by other authors w5,7,19x.

/

Ž x. d x 2p

Ž t . d t q H I22 cos Ž t . d t , p

Ž 3.20 .

Fig. 3. The shape of the modulated microwave absorption hysteresis loop at temperature 70 K and Hi s1000 Oe. Dashed lines present the hysteresis loop calculated by the formulas Ž3.18. and Ž3.19..

T. ShaposhnikoÕa et al.r Physica C 300 (1998) 239–249

In this figure it also indicated the hysteresis loop calculated in accordance with the theoretical model described above ŽEqs. Ž3.18. and Ž3.19... Note that both experimental and calculated loops are asymmetric, that is the transient regions near the starting point and that of the field sweep reversal have different extent in field. This asymmetry is due to more considerable and prolonged transformation of the flux profile after the sweep reversal than after the field start from equilibrium state. The loop shape changed weakly when varying external conditions ŽT, Hi ., while its amplitude L experienced considerable transformations. Hereafter, the loop amplitude is measured as the difference between the up-field and down-field signal levels as indicated in Fig. 3. The temperature dependence of the hysteresis amplitude L, obtained at two magnitudes of Hi , is presented in Fig. 4. One can distinguish three temperature areas with different character of dependence. The first region ŽI. is from 10 to 50 K, where L is small and increases slightly with temperature. The second region ŽII. is from 50 to 70 K, L rises steeply and in the third region ŽIII. Ž70–88 K. L sharply decreases. In the frame of the above model such behavior of hysteresis loop may be understood as follows. The part of microwave absorption contributing to hysteresis depends strongly on critical

current and viscosity of vortex matter Žsee Eq. Ž3.17.., and the actions of current and viscosity on absorption is opposite. Absorption is proportional to jc2 , but the increase of viscosity results in the MMWA decrease Žapproximately as hy5 r2 .. The latter is due to reduction of the forced vortex oscillations with increasing viscosity. One can see that viscosity action is more considerable than that of critical current. Taking into account the temperature dependence of jc and h w18x one gets the rising function LŽT .. This is indeed observed in the regions I and II. It is worth to note that we cannot draw the unified theoretical curve in all temperature regions as the jc ŽT . functions are different due to different character of collective motion of vortices. The third area differs from others in the following: here the thermal fluctuations become pronounced and affect considerably the vortex pinning potential. As a result, the contributions to the absorption signal from different vortices become out of phase, followed by a consequent reduction of the MMWA loop. Upon raising a temperature above 70 K the effect of thermal fluctuations enhances, and the loop amplitude falls rapidly down to zero. The hysteresis vanishes at T lower than Tc by several degrees for all Hi ) 0. Note that the similar behavior of the MMWA hysteresis was observed earlier in works w5,20x. The authors w20x had also ascribed the hysteresis decrease with temperature to the variation of viscosity, although they discussed the case of low magnetic fields Žless than 200 Oe. and the Josephson vortex motion. The fitting of theoretical functions Ž3.17. to the experimental data are shown in Fig. 4 by solid and dashed lines. The parameters of fitting were the critical current density jc Ž H, T ., viscosity h ŽT ., the mean-square displacement by thermal fluctuations ² u 2 :, and the mean distance between pinning centers a, with ² u 2 : and a involved in equations as ratio. Moreover viscosity h ŽT . and the distance a are associated parameters as well. The functional dependence jc Ž H, T . is different for various regimes of pinning. The corresponding formulas are presented in Appendix B. As was established from the field dependence LŽ H . Žsee below., at T - 50 K the single vortex pinning took place, and the pinning by vortex bundles occurred in the re-

(

Fig. 4. The temperature dependence of the MMWA hysteresis loop amplitude L at Hi equal to 500 Oe and 1000 Oe. The dashed and solid lines are calculated by expression Ž3.17. with the different jc Ž H .-functions Žsee text..

245

T. ShaposhnikoÕa et al.r Physica C 300 (1998) 239–249

246

gions of higher temperatures ŽII and III. at Ha - 2000 Oe. We took the corresponding expressions of jc Ž H, T . from w18x. Viscosity depends on temperature as h s h 0 Ž1 y t 2 .rŽ1 q t 2 . with t s TrTc . To evaluate the values of h 0 and the mean-square displacement of vortices ² u 2 : one has to know the magnitude of the parameter a. We have made attempt to estimate a from the dependence Mtrap Ž Hcool . —the magnetization trapped by sample vs. the cooling field at fixed and low enough temperature Ž20 K.. This dependence has the form of increasing function coming onto saturation. We assume that the saturation occurs at matching the vortex density and that of pinning centers. The estimation of the distance between pinning centers from the obtained Mtrap Ž Hcool . depen˚ dence and the above assumption yields a , 1000 A. This value is a rough approximation and has to be taken as the upper limit of a. Accordingly, we can get the upper limit of the vortex displacement by thermal fluctuations and the lower limit of viscosity: ˚ and h0 G 10y8 N s my2 at T s 75 ² u 2 : F 150A K. This agrees well with values known from the literature w21,22x. The dependence of L on the applied magnetic field is a monotonously decreasing function Žsee Fig. 5.. At relatively low temperature ŽT - 50 K. this dependence is very weak, and at higher temperature ŽT ) 60 K. the loop amplitude L decreases drastically. Taking into account that the field dependence

(

(

Fig. 5. The field dependence of the MMWA hysteresis loop amplitude L at temperatures 50, 70 and 75 K. Solid curve is calculated by Eq. Ž3.17. with jc Ž H .-function for the small bundle pinning regime and dashed curve is that for large bundle pinning regime.

of loop amplitude is completely determined by the corresponding dependence of the critical current density, the LŽ H . plot can provide rather interesting information about the vortex dynamics. Independence of L on H implies that jc Ž H . s const, that is vortices pin individually and do not interact with each other in that area of the H–T phase plane ŽT F 50 K, 200 Oe - H - 9000 Oe.. It is due to the decrease of the vortex size at low temperature. As the temperature increases, the magnitudes of coherence length j and the London penetration depth l rise. Vortices become interacting and moving collectively as bundles. This results in strong dependence of jc Ž H . and corresponding LŽ Ha .. However our attempt to describe the experimental data using Eq. Ž3.17. with the jc Ž H . expression for small vortex bundles Žsee Appendix B or Ref. w18x. gives satisfactory result only in the narrow range of low fields, H F 1000 Oe Žsolid line in Fig. 5.. It is possibly due to crossover to the large bundle pinning regime as the field increases. The fitting with corresponding jc Ž H . Žfrom Ref. w18x. is shown by dashed line in Fig. 5 and does not repeat well the experimental dependence too. One further explanation of weakening the LŽ H . dependence in high fields is a possible availability of ‘peak effect’ in jc Ž H . or ‘fishtail’ in M Ž Ha . functions. Our measurements of the Ž B y Ha . value, which is proportional to M Ž Ha ., with the help of Hall probes have revealed out the presence of ‘fishtail’ in the fields Ha ) 6000 Oe and confirmed this assumption. The dependence of the hysteresis value L on the amplitude of modulation field is shown in Fig. 6. It has a nonmonotonic form. L increases at small magnitudes of h m and decreases at larger ones, after it passes over a maximum at some h m . The position of maximum depends on temperature and it shifts toward larger h m as the temperature lowered. In order to understand such behavior one has to return to Fig. 1. One can see that increasing h m results in two effects on the field and current distribution. Firstly, it magnifies the current variations over all volume of sample, resulting in increase of the MMWA signal. Secondly, upon increasing h m the depth of modulation field penetration grows too. This part of sample volume does not contribute to the hysteretic absorption because the vortex distribution changes many times during the course of the

T. ShaposhnikoÕa et al.r Physica C 300 (1998) 239–249

247

tudes of the complete penetration field and the critical current density, and, on the other hand, it serves as a clear demonstration of validity of the theoretical model used in present work. 5. Conclusions

Fig. 6. The hysteresis loop dependence on the modulation amplitude at temperatures 60, 65, 70 and 75 K. Solid curves are calculated according to the theoretical model with fitting parameter H ) equal to 100, 36, 16 and 4 Oe correspondingly.

applied field variation and reversal. Therefore, when the modulation field penetration is small in comparison with the sample width Žsmall h m ., the first effect is dominant and L is enhanced. But when the modulation field penetration is comparable to the sample size, the continuous transformations of the most part of vortex distribution result in averaging the absorption and reducing its difference upon applied field reversal. Both these effects were taken into account when calculating the LŽ h m . dependence curves Žsee Fig. 6.. The modulation field penetration depth is determined by the ratio of the modulation amplitude to the complete penetration field H ) . Ž H ) is the magnitude of applied field variation which provides that the flux profile change reaches the crystal center.. When h m matches with H ) , the MMWA hysteresis loop becomes negligible. This gives a further independent way for estimating the critical current density since H ) is defined just by jc . The change of the LŽ h m . with temperature is caused by the temperature dependence of jc . Here from fitting we have obtained H ) to be equal to 100, 36, 16 and 4 Oe at 60, 65, 70 and 75 K correspondingly. These values agree well with those obtained by independent method—via measurement of the ESR signal shift of the surface paramagnetic probe w14x: 110 " 30 Oe at 60 K and 19 " 7 Oe at 65 K. Thus the LŽ Hm . dependence makes it possible to estimate the magni-

We have presented a systematic study of the irreversible microwave absorption in the YBa 2 Cu 3 O x single crystals. The large set of experimental data on the behavior of hysteresis loop upon variation of temperature, applied magnetic field and modulation amplitude is obtained. The analysis of experimental results on the basis of the theoretical model, proposed in this work, made it possible to gain interesting physical results, in particular, the types of vortex dynamics in different areas of the H–T phase diagram. Namely, at low temperature ŽT - 50 K. the single vortex pinning regime is observed. At higher temperatures Ž50 K T - Tc . the pinning by vortex bundles occurs, with small bundles Žtransverse size is less than the London penetration depth l L . at Ha - 2000 Oe and large ones Žtransverse size is more than lL . at Ha ) 2000 Oe. In the temperature range from 70 K to 88 K Žbelow the irreversibility line. the thermal fluctuations have a dramatic effect on the vortex dynamics and irreversible properties. Moreover the theoretical description developed here gives the possibility to evaluate such parameters as the complete penetration field H ) , the critical current density jc ŽT, H ., the viscous drag coefficient h , the mean value of vortex displacement by thermal fluctuations. The estimates obtained are in good agreement with those known from the literature. This serves as verification of the adequacy of the developed model and gives one more evidence for usefulness of the MMWA measurements in studying the irreversible properties of superconductors. Acknowledgements The authors are grateful to Prof. G.B. Teitel’baum for very helpful discussions. We express our gratitude to Dr. L.V. Mosina for assistance during the preparation of the manuscript. This work was supported by the Russian Ministry of Science and by the

T. ShaposhnikoÕa et al.r Physica C 300 (1998) 239–249

248

Council on Superconductivity through project No. 940045.

I3 s 1 y J Ž x ,h max y h q 2 h m ,2 jc . q J Ž x ,h m , jc . y J Ž x ,h m Ž 1 y cos Ž t . . ,2 jc . ,

Ž A9.

I4 s 1 y J Ž x ,h max y h q 2 h m ,2 jc . Appendix A

q J Ž x ,h m Ž 1 q cos Ž t . . ,2 jc . ,

Here we show the values of variables involved in max Ž . expressions Ž3.18. – Ž3.21.. Hup Ž x ., Hup x and HdownŽ x . are local values of induction for increasing field, for the maximum value of increasing field Hi q D H and for the decreasing field, correspondingly. Hi is here the induction of cooling field. The definitions of parameters are the following: Hup Ž x . s Hi q H Ž x ,h s q h m , jc . ,

Ž A1.

Hupmax

Ž A2.

Ž x . s Hi q H Ž x ,D H q h m , jc . ,

max Hdown Ž x . s Hup Ž x . y H Ž x ,D H y h s q 2 h m ,2 jc . ,

Ž A3. where h s s Ha-Hi . The local field value H Ž x, h, jc . and the variables b, c and Hc were calculated using expressions obtained from Ref. w15x:

(x y b Ž h, j . 1q 2

H Ž x ,h, jc . s Hc arctan

b Ž h, jc . s and Hc s



(x y b Ž h, j . 2

2

c

c Ž h, jc . x

, c Ž h, jc . s tanh

2 cosh Ž hrHc .

q J Ž x ,h m Ž 1 q cos Ž t . . ,2 jc . ,

h

p J Ž x ,h, jc . s

.

Hc

/

k s 1,6,

ž

.

Ž A13.

Bsb Ž 0 .

ž

bsb Hc2

1q

3c 2

2

T Tdps

/ 3

T

aq

Tdps

//

.

Ž B1.

For the pinning of small bundles, Bsb ŽT . - B Blb ŽT . jcsb

Ž B,T . s j0

B

bsb Hc2

ž

1q

 ž

=exp y2 c1

/

ž

= 1q

Ž A6.

1r2q2 c1rc

T Tdp

/

B Bsb Ž 0 . 2 3r2

T Tdp Ž 0 .

//

0

,

Ž B2.

and for the pinning of large bundles, B ) B lb ŽT .

where I1 s 1 y J Ž x ,h m Ž 1 y cos Ž t . ,2 jc . ,

/

2

ž ž

Ž A5.

ž

ž(

b Ž h, jc . y x 2

=exp y

,

c Ž h, jc . x

When calculating the dependence of the hysteresis amplitude on temperature and field value we use the expressions of the critical current density jc Ž B, T . for various regimes of pinning, taken from Ref. w18x. For the single vortex pinning, at B - Bsb ŽT . it is as follows:

p Here x 0 and d are the width and the thickness of the sample. 2p u th 2p u th A k s 1 y Ik2 cos q Ik sin , a a

(

4

arctan

Appendix B

,

ž /

Ž A11. Ž A12.

I6 s 1 y J Ž x ,h m Ž 1 q cos Ž t . . ,2 jc . ,

jcsy Ž B,T . s j0

Ž A4.

x0 jc d

0

c Ž h, jc . x

1y

I5 s 1 y J Ž x ,h m , jc .

2

c

Ž A10.

Ž A7.

I2 s 1 y J Ž x ,h m , jc . q J Ž x ,h m Ž 1 q cos Ž t . . ,2 jc . , Ž A8.

jclb

Ž B,T . s j0

1 k2

ž

Bsb Ž 0 . B

3

/ž

Tdp Ž 0 . Tdp Ž 0 . q T

11r2

/

.

Ž B3.

T. ShaposhnikoÕa et al.r Physica C 300 (1998) 239–249

Here Tdp s Ž BrBsb Ž0..1r2 Tc is the depinning temperature. The higher critical field Hc2 Ž0. s 220 T, the depairing current density j0 , 10 9 Arm2 , the critical temperature Tc s 93 K and the Ginzburg– Landau parameter k s 100. The area of the single vortex pinning regime is bounded by the Bsb ŽT . curve and by the single vortex depinning temperatures Tdps . The Blb ŽT . curve is the boundary between areas of the small bundle pinning and the large bundle pinning. bsb , a and c are constants of order of unity. The magnitudes of parameters Bsb Ž0. s 6 T, Blb Ž0. s 10 T, Tdps s 60 K, bsb s 5, a s 1, c s 1.1 were taken from Ref. w18x. The values of c1 s 4 and j0 is found from the fitting.

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