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PHYSICAL REVIEW B 67, 144504 2003

Surface impedance anisotropy of YBa2 Cu3 O6.95 single crystals: Electrodynamic basis of the measurements
Yu. A. Nefyodov, M. R. Trunin, A. A. Zhohov, I. G. Naumenko, and G. A. Emel'chenko
Institute of Solid State Physics RAS, 142432 Chernogolovka, Moscow district, Russia

D. Yu. Vodolazov and I. L. Maksimov
Nizhny Novgorod University, 23 Gagarin Avenue, Nizhny Novgorod 603600, Russia Received 5 June 2002; published 4 April 2003 An electrodynamic technique is developed for determining the components of surface impedance and complex conductivity tensors of HTSC single crystals on the basis of measured quantities of a quality factor and a resonator frequency shift. A simple formula is obtained for a geometrical factor of a crystal in the form of a plate with dimensions b a c in a microwave magnetic field H ab . To obtain the c-axis complex conductivity from measurements at H ab , we propose a procedure that takes into account of sample size effects. With the aid of the technique involved temperature dependences of all impedance and conductivity tensors components of YBa2 Cu3 O6.95 single crystal, grown in BaZrO3 crucible, are determined at a frequency of f 9.4 GHz in its normal and superconducting states. All of them proved to be linear at T T c /2, and their extrapolation to zero temperature gives the values of residual surface resistance R ab (0) 40 and R c (0) 0.8 m and magnetic-field penetration depth ab (0) 150 nm and c (0) 1.55 m. DOI: 10.1103/PhysRevB.67.144504 PACS number s : 74.25.Nf, 74.72.Bk

I. INTRODUCTION

Microwave measurements of the temperature dependence of the complex conductivity tensor ^ ( T ) ^ ( T ) i ^ ( T ) of high-T c superconductors HTSC have advanced considerably our understanding of the mechanisms of quasiparticles transport along crystallographic axes of these anisotropic compounds. The real part ^ ( T ) is susceptible to the scattering rate of quasiparticles, as well as their density of states. The imaginary part ^ ( T ) is related to the magnetic-field penetration depth ( T ) . In the local electrodynamics, which can be applied to HTSC,

^T

i

0

^ /Z2 T ,

1

^ where Z ( T ) is the surface impedance tensor of the sample, ^ 2 f and 0 4 10 7 H/m. In HTSC the tensors Z and ^ are characterized by two components: Z ab R ab iX ab or ab ab i ab ) in weakly anisotropic ab planes CuO2 and Z c R c iX c ( c c i c ) perpendicular to these planes. In the temperature range T T c , R ab ( T ) X ab ( T ) in the ab plane of the optimum-doped YBa2 Cu3 O6.95 Refs. 1 ­ 5 Refs. 5 ­ 7 single crystals, and this and Bi2 Sr2 CaCu2 O8 relation is equivalent to the condition of the normal skin effect. The common features of these crystals are the linear temperature dependence of the surface resistance R ab ( T ) X ab ( T ) T and of the surface reactance ab ( T ) T at temperatures T T c see Refs. 8 ­ 11 and references therein . The difference is that the linear resistivity region and terminates extends to near T c /2 for Bi2 Sr2 CaCu2 O8 near or below T T c /3 for YBa2 Cu3 O6.95 single crystals. At higher temperatures, R s ( T ) of YBa2 Cu3 O6.95 has a broad
0163-1829/2003/67 14 /144504 9 /$20.00

peak. In addition, the ab ( T ) curves of some YBa2 Cu3 O6.95 single crystals have unusual features in the intermediate temperature range.4,12 In comparison with the microwave response of the cuprate layers of HTSC, the data concerning their microwave properties in the direction perpendicular to these layers are scarce. Moreover, the available experimental data are controversial. In this connection, the major electrodynamic problem is the accuracy of the techniques used in determination of Z c ( T ) and c ( T ) in HTSC. The most convenient technique for measurements of the surface impedance of small HTSC samples in the X -W microwave frequency bands is the so-called ``hot-finger '' method.9,13 The underlying idea of the method is that a crystal is set on a sapphire rod at the center of a superconducting cylindrical cavity resonating at the frequency f in the H 011 mode, i.e., at the antinode of a quasihomogeneous microwave magnetic field. In the experiment the real R and imaginary X parts of the surface impedance are derived from the following relations:

R

1/Q ,

X

2

f/f.

2

Here is the sample geometrical factor; (1/Q ) is the difference between the values 1/Q of the cavity with the sample inside and the empty one; f is the frequency shift relative to that which would be measured for a sample with perfect screening and no penetration of microwave fields. In the experiment we measure the difference f ( T ) between resonant frequency shifts versus temperature of the loaded f ( T ) f 0, and empty cavity, which is equal to f ( T ) where f 0 is a constant. In HTSC single crystals the constant f 0 can be determined from measurements of the surface im©2003 The American Physical Society

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FIG. spect to entation, indicate

1. the H the

Two experimental orientations of the sample with remicrowave magnetic-field H , a longitudinal L oriab , and b transverse T orientation, H ab . Arrows direction of the high-frequency currents.

pedance in the normal state.14 Another quantity essential for determining the values of R ( T ) and X ( T ) from Eq. 2 is the sample geometric factor 2W , W
0

2

H 2 dV ,
V s

H 2 ds , t

3

where the subscripts of Z denote the directions of the screening currents. While deriving formula 4 we neglect not only weak anisotropy in the ab plane but also a contribution of the crystal ac faces which is apparently minute in comparison with the summands in the numerator in Eq. 4 due to the areas difference ac bc ab . The cleaving of the crystal along the b edge into several needles multiplies up the contribution from the c axis, so that measurements of Z ab c before and after the cleaving allowed to extract Z c in the superconducting state of YBa2 Cu3 O6.95 single crystal.15 However, this procedure has the following disadvantages: i it assumes abZ ab term in Eq. 4 to be nonalterable, which results in an uncontrolled inaccuracy due to nonideal sample cleaving into rectangular needles, ii as it will be shown below, the size effect takes place at temperatures T 0.9T c in the L orientation; this restricts applicability range of Eq. 4 within low temperatures and does not allow to extract c (0) value, and iii in many cases one needs to save the initial sample for further study, e.g., for investigation of evolution of its anisotropic properties with doping level. Therefore, consecutive measurements of the crystal at first in transverse T orientation H c Fig. 1 b to obtain Z ab and then in longitudinal L one appear to be a more natural way to obtain Z c value. A difficulty in determining the geometrical factor or the integral in Eq. 3 in the T orientation of the crystal arises while using this technique. As 2 H 2 a 2 ln(a/c) 1 proves to be a mentioned in Ref. 9, 0 reasonable estimation for a square sample with a b c . It is also known that the approximation of a rectangular plate to an ellipsoid inscribed in it results in an overestimated value of . The purpose of this paper is i to calculate for a typical HTSC crystal in the T orientation, ii to generalize formula 4 for the L orientation of the crystal to the range of higher temperatures T 0.9T c , and iii to report on the measurement results for all surface impedance components of highquality YBa2 Cu3 O6.95 single crystals, grown in BaZrO3 crucibles, in the normal and superconducting states.
II. GEOMETRICAL FACTOR IN THE T ORIENTATION

where W is the energy stored in the cavity, V is the volume of the cavity, H is the microwave magnetic field generated in the cavity, s is the total sample surface area, and Ht is the tangential component of H on the sample surface. The energy W is easily obtained for the resonator mode under use, therefore the task of deriving the impedance value is reduced to defining the integral in Eq. 3 . The task simplifies if a typical HTSC crystal in the form of a rectangular plate with dimensions b a c and volume v 0.1 mm3 is radiated by the microwave magnetic field H b L orientation, Fig. 1 a . If this is the case, in the superconducting state at T 0.9T c , when the magnetic-field penetration depth is smaller than the characteristic sample dimensions, the quantity H t H 0 can be taken out of the integral , where H 0 is 2 H 2 ( ab bc ), so an amplitude of H . Thus we obtain 0 the surface impedance Z ab c for the sample in the L orientation will be equal to Z abZ ab bcZ c , ab bc 4

Let us consider a rectangular ideal conductor with dimensions L y L x , L z placed in a constant magnetic field H z Fig. 2 . The problem of obtaining the field distribution around such a conductor becomes two dimensional and to solve it one can apply the method suggested in Ref. 16. The magnetic field outside the conductor satisfies Maxwell equations " H 0 and "·B 0. The former equation allows to introduce a scalar potential , and the latter allows to intro" A/ 0 . Let A be duce a vector potential A: H "· directed along the y axis: A (0,A ,0) . Then the magneticfield components will be as follows:17 H
x

iH

z

d , dw

5

ab c

iA / 0 is an analytical where the complex potential function of w x iz variable, which determines the conformal mapping of ( x , z ) plane into ( , A / 0 ) plane. Schwarz transformation

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FIG. 2. Complex planes ( x , z ) and ( , A / mapping.

0

) used for conformal

dw d

i H

3 2

3 2

6

specifies the relationship between the unshaded areas in into plain ( , A / 0 ) point of Fig. 2. Placing the point 0 and A 0 along the path origin, we get 123 3 2 1 . Owing to the task symmetry 2 2 and 3 3 . Integrating Eq. 6 , we obtain the following relationship between x , z coordinates and potential on the conductor surface:
3

FIG. 3. The values of M /vH 1 dashed line and / sH 2 1 solid line calculated from Eqs. 10 and 11 versus the ratio L z / L x . The upper inset represents the magnetic-field distribution on the surfaces L z L y upper three curves and L x L y lower curves for the values of L z / L x 0.25 dotted lines , 0.1 dashed lines , and 0.03 solid lines . The lower inset, / sH 2 as a function 0 of c / a . Solid and dashed lines correspond to the calculations of Eqs. 11 and 14 , respectively. Dotted line is the upper limit of / sH 2 estimated from Eq. 15 for b 4 a . 0

4L

L x /2 y 0

H 2 dx x

L z /2 0

2 H z dz .

9

In order to obtain this we change from integration over coordinates in Eqs. 8 and 9 to integration over potential, taking formulas 5 and 6 into account, M L yH 4 H
3 2

x

H

1 k2 E k

2

1

2 2

,

1 1k
2

xd
2 z

Lx 2 Ek
2 2 3

3

d

0

( z L z /2, 0 x L x /2) , z
3

1 k2 L 4 7 L yH
2

1 k2 K k
0 2 2 2 3

2

,

10

H

E
2

,

1 k

x L x /2,0 z L z /2 ,

where k 3 / 2 , and E ( u , v ) is an incomplete elliptic integral of the second kind. Equations 5 and 7 define an implicit dependence of the magnetic field against coordinates on the conductor surface. The upper inset in Fig. 3 displays the distribution of H z ( z ) and H x ( x ) magnetic-field components in L z L y and L x L y planes, respectively, for three different L z / L x ratios. From the formulas 5 ­ 7 one can easily calculate the magnetic moment of the conductor M 1 2 j r dv 4L
L x /2 y 0

4 H 2 Lx f
x

3 2

2 2 2

2 2

d
3

d 11

Lz f z ,

where f K
x

1k 1k
2

2

E k 2K

1k

2 2

E f

1k ,

,

Ek
z

x H x dx

Lx 2

L z /2 0

H z dz 8

Ek

1 k2 K k

12

and the integral

in Eq. 3 ,

with K ( v ) and E ( v ) being complete elliptic integrals of the first and second kinds. Formula 10 coincides with the result obtained previously in Ref. 18. Furthermore, the relationship

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between the ratio L z / L x and k is required to calculate the values of the moment M / v ( v L x L y L z is a conductor volume and the factor / s s 2 L y ( L z L x ) is a conductor surface area , which we obtain substituting the values x L x /2 and z L z /2 into the left side of Eq. 7 , L L
z x

Ek E 1k
2

1 k2 K k k 2K 1k
2

.

13

M and dependences on L z / L x ratio, computed from formulas 10 and 11 , are presented in Fig. 3. If L z L x , the magnitude of the magnetic field on the sample surface tends to that of the applied one along with the values of M and tending to ( v H ) and sH 2 , respectively. Presently we apply the formulas obtained to determe the HTSC crystal geometrical factor when transversely placed with respect to H field Fig. 1 b . The field distribution on the ideal conductor surface will coincide with alternating magnetic-field distribution on the surface of a superconducting sample of the same dimensions ( L x a , L y b , L z c ) placed in microwave field H H, provided that the penetration depth is smaller than the sample dimensions. It is easy to obtain a simple estimation of in case of a very thin crystal ( c a b ) . Upon keeping the first term from the expansion into k 1 series of the right side of Eq. 13 , we get the value 4 L z / L x . Therefore, small values of k correspond to k 4 c / a into c / a 1 . The subsequent substitution of k asymptotic forms f x ( k ) K ( 1 k 2 ) 1, f z ( k ) 2/k 2 with k 1 in Eq. 12 gives the following from Eq. 11 : 2 H 2 ab 0 1 14a . ln 2 c 14

mentally obtained values of M reached up to 20% in case of short ( b 3 a ) samples and decreased to less than 5% for substantially long samples ( b 6 a ) . In contrast to magnetic moment, the values of proved to be in better agreement with the theory. In fact, divergence have never exceeded 5%, probably, due to weaker logarithmic dependence of on a / c ratio. Both M and integrals in Eqs. 8 and 9 are convergent. At the same time, as follows from Eqs. 5 and 6 , the magnetic field diverges 1/r 1/3 at small distances r from the edges point 3 in the Fig. 2 . In this connection the question arises about possible nonlinearities on the edges of the superconducting slab. In the resonant circuits we applied small amplitude ( H 0 0.1 Oe) of the ac magnetic field. The oneorder increase of H 0 did not give rise to any nonlinear effects under the measurements of YBa2 Cu3 O6.95 crystals.

III. ELECTRODYNAMICS OF ANISOTROPIC CRYSTAL

2

The lower inset in Fig. 3 represents the comparison of values obtained from the general formula 11 and the asymptotic one 14 . Let us now consider the influence of a finite crystal length ( b dimension on the quantity. Taking an arising additional contribution from ac faces of the crystal Fig. 1 b into account and assuming a tangential field component on these faces to be the same as that on the other lateral bc faces, from Eq. 11 we obtain 2 H 2 ab f 0
x

The electrodynamics of a layered anisotropic HTSC is characterized by components ab and c of the conductivity tensor. In the normal state ac field penetrates along the c axis through a skin depth ab 2/ 0 ab and in the CuO2 plane through c 2/ 0 c . In the superconducting state all parameters ab , c , ab c ab i ab , and c i c are complex. At T T c , if the field penetra1/ 0 ab , c tion depths are given by formulas ab 1/ 0 c . In the close neighborhood of T c , decay of the magnetic-field in a superconductor is characterized by functions Re( ab) and Re( c ) , which turn to ab and c at T T c , respectively. In the T orientation the surface impedance Z ab is directly connected with the in-plane penetration depth ab ( T ) at T T c and the skin depth ab ( T ) at T T c . Both lengths are smaller than the typical crystal thickness. Hence, when the crystal is in the T orientation and at an arbitrary temperature the surface impedance Z ab is defined as a coefficient in Leontovich boundary condition,17 and is correlated with the conductivity ab through the local relation i
ab ab 1/2

bc f

z

ac f z .

15

Z

ab

R

ab

iX

0

.

16

However, estimation 15 gives an overestimated value of in the case of a real three-dimensional sample. Indeed, the limitation of the crystal length b will result in a decrease of the magnetic-field tangential component Ht a on the sample surface. The appropriate decrease in will not be compensated by the appearance of Ht b component, which is absent when b . Thus, formula 15 is an upper limit of , and its c / a dependence is shown in the lower inset of Fig. 3. In order to check the accuracy of the above calculations we have measured both magnetic moment M and geometrical factor of superconducting slabs with different a / c and b / a ratios using ac susceptibility and cavity perturbation techniques. The discrepancy between theoretical and experi-

In case the HTSC microwave conductivity is real in the normal state the real and imaginary parts of the surface impedance are equal. Hence, in the T orientation the constant f 0 , essential to determine X ab ( T ) in Eq. 1 , may be found as a result of R ab ( T ) and X ab ( T ) coincidence at T T c . It should be pointed out that thermal expansion of the crystal may essentially affect the shape of X ab ( T ) curve in the T orientation. Since the resonance frequency depends on the volume occupied by the field, the crystal expansion is equivalent to a reduction in the magnetic-field penetration depth and results in an additional frequency shift f l ( T ) of the cavity:9

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fl T

f0 8W f
0

s

l i T H 2 ds t
2 0

2 n

c

2

2 ab

i 2

2

4a

2 c 2

n

2

,

20

vH

4W

cfx

a

b

fz ,

17

where the sum is performed over odd integers n 0. If in the superconducting state, we get 8
2 n

where i is a relative change l i / l i of the sample dimension l i ( a , b , c ) resulting from the thermal expansion, and the functions f x and f z are defined according to Eq. 12 . In Ref. 9 the contribution 17 to the overall frequency shift is shown to be negligible at low temperatures, however, it becomes noticeable at T 0.9T c in the T orientation. In the L orientation at T 0.9T c the penetration depth in an HTSC crystal is still smaller than characteristic sample dimensions. It allows to treat the experimental data in terms of impedance Z ab c averaged over the sample surface in accordance with Eq. 4 . In particular, taking account of the X ab ( T )/ measurements of ab ( T ) 0 in the T orienta(T) X ab c ( T )/ tion and the measured value ab c 0 in the L orientation, we obtain
c

1 tanh ~ n2 ~n a
2 2 c

n

tanh ~ ~n
2 ab 2

n

,

~

2 n

1 4 1 4

2

4c
2

n

2

,

~

2 n

c

2

2 ab

4a

2 c 2

n

2

.

21

In particular, ab c and c a at T 0.9T c , so we derive from Eq. 21 a simple expression for the real part of : 2
c

/a 2

ab

/c.

22

ac

ab c

a

ab

/c.

18

This technique of c ( T ) determination was used in microwave experiments3,6,15,19 ­ 23 at low temperatures T T c . Nevertheless, this approach to investigation of the surface impedance anisotropy in HTSC crystals at T T c does not allow to determine the value of c ( T ) from the measurements of quality factor and resonance frequency shift, nor may it be extended to the range of higher temperatures. The point is that the size effect provides with an essential influence in the L orientation at T 0.9T c , when the penetration depths c and c turn out to be comparable with the crystal width. As a result, the R ab c ( T ) temperature dependence measured in the normal state does not coincide with X ab c ( T ) , which makes the previous method of determining f 0 nonapplicable. In this case in order to analyze our measurements in both the superconducting and normal states we shall use the formulas for field distribution in an anisotropic long strip ( b a , c ) in the L-orientation.24 These formulas neglect the effect of the bc faces of the crystal Fig. 1 a , but allow for the size effect correctly. At an arbitrary temperature the meaf ( T ) f 0 are exsured quantities (1/Q ) and f ( T ) (T) pressed in terms of a complex function ( T ) i ( T ): 7,25 1 Q 2i f f i
0

One can easily check up that in the range of low temperatures the change in c ( T ) prescribed by Eq. 22 is identical to that in Eq. 18 . . If the sample In the normal state the conductivity dimensions were much more than the penetration depths, we would obtain the following from Eqs. 19 and 20 : 1 Q 2i f f 1i
0

vH

2 0

c

ab

2W

a

c

,

23

i.e., the temperature dependences of (1/Q ) and ( 2 f / f ) would be identical at T T c , and the values R ab c ( T ) and X ab c ( T ) , derived from Eqs. 2 ­ 4 , would be equal. In practice, the value c 0.1 mm ( f 10 GHz) proves to be comparable with the crystal width a even for YBa2 Cu3 O6.95 crystals, which can be referred to as weakly anisotropic in comparison with other layered HTSC's. So to determine the surface impedance components from the measurements in the L orientation it is necessary to use general formulas 19 and 20 , as shown below.
IV. EXPERIMENTAL RESULTS

vH

2 0

2W

,

19

which is controlled by the components ab ( T ) and c ( T ) of the conductivity tensor through the complex penetration depths ab and c : 8
2 n

1 tan n2 a
2 2 c

n n 2

tan
n 2 ab 2

n

,

2 n

i 2

4c

n

2

,

YBa2 Cu3 O6.95 single crystals were grown using the method of slow cooling from a solution-melt with the use of a BaZrO3 crucible. The initial mixture was prepared from a mixture of oxides with mass portions Y2 O3 : BaO2 : CuO 1:25:24 and subsequent pressing of the compound into a tablet of 40 mm in diameter under the pressure of 200 MPa. The initial components purities were 99.95% for both yttrium and copper oxides and 99.90% for barium peroxide. Crucible material porosity 2% was taken into account when choosing a heating regime and homogenization time. Preliminary experiments have demonstrated that the melt under use saturates the crucible walls through the whole width 3 mm during the period of 5 ­ 7 h at the working temperature. In 10 h crystals growth terminates due to a complete vanishing of the melt from the crucible. To reduce the melt homog-

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FIG. 4. R ab ( T ) and X ab ( T ) of YBa2 Cu3 O6.95 single crystal ( T orientation . The upper inset shows the measured temperature dependences 2 f / f triangles and (1/Q ) squares . Taking the constant f 0 and thermal expansion into account we get 2 f / f circles . The lower inset displays R ab ( T ) and ab ( T ) dependences at low T.

enization time which amounts to 10 ­ 20 h at 1030 °C, according to Ref. 26, the method of accelerated-decelerated rotation of the crucible27 was used, which made intensive mixing of the melt possible. The homogenization time of the melt at 1010 °C did not exceed 1 h. Crystals growth time amounted to 2 h at a cooling rate of 3 ­ 4 °C/h, after which the remaining melt was decanted at 950 °C and cooled down to room temperature at a rate of 15 ­ 20 °C/h. The crystals obtained were saturated with oxygen at 500 °C in an oxygen flow, after which their critical temperature was equal to 92 K. The measurements of the dynamic susceptibility showed that the width of the superconducting transition in the samples did not exceed 0.1 K at 100 KHz. The surface impedance was measured using the hot-finger technique9 at a frequency of f 9.42 GHz in the T and L orientations. Figure 4 displays the typical temperature dependences of R ab ( T ) and X ab ( T ) for YBa2 Cu3 O6.95 single crystal in the normal and superconducting states measured in the T orientation. The sample represented a prolate parallelepiped with dimensions a b c 0.4 1.6 0.1 mm3 . The sample geometrical factor 90 k was calculated from Eqs. 11 and 3 . The upper inset in Fig. 4 displays the measured tempera2 f / f triture dependences of Q 1 squares and angles in the normal state of the crystal. The curves 2 f ( T )/ f 2 f (T) f l( T ) f 0 / f Q 1 ( T ) and circles coincide after taking into account the additional frequency shift f l ( T ) from Eq. 17 , which arises due to the sample thermal expansion,9,28, along with the constant f 0 , which is independent of temperature. The coincidence of 2 f ( T )/ f curves and equality R ab ( T ) Q 1 ( T ) and X ab ( T ) at T T c according to Eq. 2 demonstrate the

FIG. 5. a Q 1 squares and 2 f / f circles measured in the L orientation of YBa2 Cu3 O6.95 single crystal at T 100 K. Solid line shows the temperature dependence of 2 f / f derived from Eqs. 19 and 20 . The constant 2 f 0 / f is indicated by arrow. b Q 1 and 2 f / f calculated from Eqs. 19 and 20 solid and dashed lines, respectively and from Eq. 25 dash-dotted and dotted lines for a sample width a 0.2 mm. c The same as b but for a 0.1 mm.

fulfilling of the normal skin effect condition in the ab planes of YBa2 Cu3 O6.95 crystal in the T orientation. A linear temperature dependence of resistivity ab ( T ) 1/ ab ( T ) cm in the range 100 T 200 K together with 0.63T the skin depth ab ( 150 K) 5 m are derived from Eq. 16 . The temperature dependence R ab ( T ) has a broad peak in the range T T c /2, characteristic of YBa2 Cu3 O6.95 crystals in the superconducting state. The dependences R ab ( T ) and ab ( T ) X ab ( T )/ 0 are linear at T T c /3 lower inset in Fig. 4 . Their extrapolation to T 0 K results in the values of and penetration residual surface resistance R ab (0) 40 depth ab (0) 150 nm. Equations 19 and 20 are used to analyze the experimental data in the L orientation of the crystal. In the normal state the real part of Eq. 19 defines the relationship between Q 1 ( T ) and the skin depths ab ( T ) and c ( T ) . Upon measuring the dependence Q 1 ( T ) squares in Fig. 5 a at T Tc in the L orientation and taking the dependence ab ( T )

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2 ab ( T )/ 0 measured in the T orientation into account, from Eq. 20 we obtain the function c( T ) 2 c ( T )/ 0, c ( 150 K) 0.06 mm, and the temperacm in the range ture dependence c ( T ) 104 24T 100 T 200 K. Using already determined dependences (T) c ( T ) and ab ( T ) and having computed the real part in Eq. 20 , from Eq. 19 we calculate ( 2 f / f ) versus temperature at T T c , which is shown by solid line in Fig. 5 a . This line is approximately parallel to the experimental curve 2 f ( T )/ f in the L orientation triangles in Fig. 5 a f )/ f yields the adat T 110 K. The difference 2( f ditive constant f 0 . Given f 0 and f ( T ) we obtain f ( T ) in the entire temperature range in the L orientation. It should be emphasized that the discrepancy of the curves Q 1 ( T ) and 2 f ( T )/ f at T T c in the L orientation does not originate from the thermal expansion of the crystal essential in the T orientation, but arises due to the size effect. This discrepancy becomes more noticeable with the decrease in the crystal width a, when it becomes equal to the skin depth c . The computational result for Q 1 ( T ) solid line and 2 f ( T )/ f dashed line from Eqs. 19 and 20 for the above-mentioned dependences ab ( T ) and c ( T ) is shown in Fig. 5 b and 5 c for a 0.2 mm and a 0.1 mm. It should be noticed that the surface impedance components R ab c and X ab c in Eq. 4 cannot be found from the values of (1/Q ) and f / f measured at T 0.9T c in the L orientation with the use of Eq. 2 due to the size effect in anisotropic HTSC crystals. Moreover, it is also incorrect to substitute the values R and X in these formulas by their effective values R ef f ( d ) and X ef f ( d ) for a thin metal plate of , placed in microwave field H parallel to its width d infinite surfaces R
ef f

FIG. 6. The conductivities ab / ab (0) and reduced temperature T / T c . The insets display dependences.

c ab

/ c ( 0 ) versus ( T ) and c ( T )

d

R

sinh cosh sinh cosh

sin cos sin cos

,

X

ef f

d

R

,

24

where 0 d /2R , R 0 /2. The point is that though allowing the usage of formulas 24 in Eqs. 2 and 4 , the solution of Maxwell equations results in an incorrect onedimensional distribution of high-frequency currents in the crystal. Indeed, for b H we obtain from Eqs. 2 ­ 4 , 1 Q 2i f f H
2 0

W

abZ

ef f ab

c

bcZ

ef f c

a,

25

ef ef ef where the effective values Z abf ( c ) R abf ( c ) iX abf ( c ) and ef f ef f ef f Z c ( a ) R c ( a ) iX c ( a ) are defined by Eq. 24 . Figures 5 b and 5 c display the computational result for Q 1 ( T ) and 2 f ( T )/ f from Eqs. 24 and 25 , and above determined dependences ab ( T ) and c ( T ) in the case of two plates of dimensions a b c 0.2 1.6 0.1 mm3 and half the width a 0.1 mm. Presently, having compared these results with these obtained from Eqs. 19 and 20 , we can see that at a 3 c Fig. 5 b approximation 25 proves to be practically insensitive to the size effect, giving rise to weakly 2 f ( T )/ f at T differing dependences Q 1 ( T ) and

100 K. Only in the case of the crystal width a 0.1 mm a c , Fig. 5 c does approximation 25 give rise to the result resembling that obtained with the aid of formulas 19 and 20 . Upon finding the dependences Q 1 ( T ), f ( T ) and, hence, the function ( T ) in Eq. 19 in the normal and superconducting states of YBa2 Cu3 O6.95 crystal in the L orientation and using the conductivities ab ( T ) and ab ( T ) found from Eq. 16 , we get the c-axis conductivity components c ( T ) and c ( T ) from Eq. 20 . All the conductivity tensor components obtained are shown in Fig. 6. In case of local relationship between the electric field and the current along the c axis, the surface impedance Z c ( T ) R c ( T ) iX c ( T ) is related to the conductivity c ( T ) c ( T ) i c ( T ) through Eq. 1 . Figure 7 displays the components of Z c ( T ) obtained in this manner. R c ( T ) and c ( T ) X c ( T )/ 0 dependences insets in Fig. 7 demonstrate linear behavior at T T c /2. The value of the penetration depth along cuprate planes of YBa2 Cu3 O6.95 single crystal is equal to c (0) 1.55 m when extrapolated at zero temperature. From Eq. 4 one can easily estimate that in the L orientation of our crystal the contribution of the c-axis currents into measurable quantities is about two times greater than that of the ab plane ones: bcZ c 2 abZ ab . Taking the accu5% and racy of determination of R ab ( T ) ab ( T ) a few angstroms values into account, we conclude that linear behavior of R c ( T ) and c ( T ) at low temperatures is the property of optimum-doped YBa2 Cu3 O6.95 , and is not due to the inaccuracy of the method used. The linear temperature dependence of c at T T c /2 is also confirmed by the previous microwave20,23 and low-frequency29 measurements of YBa2 Cu3 O6.95 . In contrast to the result of Ref. 15 we did not observe an upturn in R c ( T ) at low temperatures. Our surface

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FIG. 7. R c ( T ) and X c ( T ) of YBa2 Cu3 O6.95 single crystal. The insets demonstrate R c ( T ) and c ( T ) at low T.

resistance R c ( T ) measurements are in qualitative agreement with other microwave data.3,20
V. CONCLUSION

the quantities measured in the L orientation ( H ab ) at T 0.9T c . We have shown that both these problems may be solved in the case of a crystal in the form of a plate with dimensions b a c . Furthermore, in this case we have calculated the magnetic-field distribution on the crystal surface in the T orientation and found simple expression 11 for its geometric factor. In the L orientation of the crystal, we have shown that only the use of general formulas 19 and 20 allows to take into account of the size effect correctly and to determine the complex conductivity and the surface impedance in the c direction of the crystal in the normal and superconducting states. The experimental technique of measuring all the components of the conductivity tensor is described in detail, and we hope that this technique will be useful in comprehensive studies of anisotropic characteristics of HTSC crystals. The reported electrodynamic approach has been successfully applied to the analysis of the microwave ( f 9.4 GHz) response measurements in both the T and L orientations of YBa2 Cu3 O6.95 single crystal, grown in BaZrO3 crucible. The temperature dependences of all conductivity and surface impedance tensors components proved to be linear at T T c /2. Their extrapolation to zero temperature gives and the values of residual surface resistance R ab (0) 40 and magnetic-field penetration depth R c (0) 0.8 m m. ab (0) 150 nm and c (0) 1.55
ACKNOWLEDGMENTS

In conclusion, we have developed the electrodynamic approach for HTSC surface impedance anisotropy measurements. The major problems in analyzing these measurements in HTSC single crystals are i determining the crystal geometrical factor in the T orientation ( H ab ) , which is dependent on the microwave field distribution on the sample surface; ii allowing for the size effect, which influences

We thank P. Monod for helpful discussions. This research has been supported by RFBR Grants Nos. 00-02-17053, 0202-06578, 02-02-08004 and Government Program on Superconductivity Contract No. 540-02 . M.R.T. thanks Russian Science Support Foundation.

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