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JETP Letters, Vol. 77, No. 10, 2003, pp. 592­597. Translated from Pis'ma v Zhurnal èksperimental'nooe i Teoreticheskooe Fiziki, Vol. 77, No. 10, 2003, pp. 696­701. Original Russian Text Copyright © 2003 by Trunin, Nefedov.

Anisotropy of Microwave Conductivity in the Superconducting and Normal States of YBa2Cu3O7 ­ x: 3D­2D Crossover
M. R. Trunin* and Yu. A. Nefedov
Institute of Solid State Physics, Russian Academy of Sciences, Chernogolovka, Moscow region, 142432 Russia *e-mail: trunin@issp.ac.ru
Received April 23, 2003

The imaginary parts of microwave conductivity '' (T < Tc) and resistivity (T) = 1/(T > Tc) along ( '' and ab ab) and across ( '' and c) the cuprate ab planes of a YBa2Cu3O7 ­ x crystal with the oxygen doping level x c varying from 0.07 to 0.47 were measured in the temperature range 5 T 200 K. In the superconducting state, the '' (T)/ '' (0) and '' (T)/ '' (0) curves coincide for an optimally doped (x = 0.07) crystal, but, with an ab ab c c increase in x, the slopes of the '' (T)/ '' (0) curves decrease noticeably at T < Tc /3, on the background of small c c changes happening to the '' (T)/ '' (0) curves. The two-dimensional (2D) transport along the ab planes in ab ab the normal state of YBa2Cu3O7 ­ x is always metallic, but there is a crossover (at x = 0.07) from the Drude to hopping (at x > 0.07) conductivity along the c axis. This is confirmed both by the estimates of the lowest metallic and the highest tunneling conductivities along the c axis and by quantitative comparison of the measured c(T) curves with the curves calculated in the polaron model of quasiparticle transport along the c axis. © 2003 MAIK "Nauka / Interperiodica". PACS numbers: 74.25.Fy; 74.72.Bk

In recent years, growing interest has been shown in the evolution of transport properties of high-temperature superconductors (HTSCs) upon changing the level of doping with oxygen and other substitutional impurities or, in other words, upon changing the hole concentrations p per one copper atom in the CuO2 plane. The p value and the superconducting transition temperature Tc in HTSC are related by the empirical formula [1] Tc = Tc, max[1 ­ 82.6(p ­ 0.16)2]. A narrow region in the phase diagram of an optimally doped HTSC (p 0.16) with maximal critical temperatures Tc = Tc, max has received most attention. In the normal state of an optimally doped HTSC, the resistivity ab(T) in the cuprate ab planes increases linearly with temperature, ab(T) T. The quantity ab(T) is much smaller than the resistivity c(T) in the perpendicular direction, which also has a metallic character (the derivatives of ab(T) and c(T) with respect to temperature are positive). The exception is provided by the most anisotropic HTSC compound Bi-2212 (the corresponding ratio is c/ab 105 at p 0.16), for which the resistivity c(T) increases as T approaches Tc (dc(T)/dT < 0). This property of Bi-2212 agrees with the estimate of the lowest possible metallic conductivity in the c direction for anisotropic three-dimensional (3D) Fermi-liquid model [2]:
3D c, min

where n 1021 cm­3 is the carrier concentration, d is the lattice constant along the c axis, and h is Planck's con3D stant. In Bi-2212, the conductivity c = 1/c c, min at T = Tc, but, in other optimally doped HTSCs, c(T) > c, min (Tc). The conductivity c, min in Eq. (1) is lower than the two-dimensional Ioffe­Regel limiting value IR = e2kF/h: c, min ab / c IRd/a IR (a 2/kF is the lattice constant in the CuO2 plane), whereas ab, min IR [2]. The ratio of the superconducting liquid densities in the cuprate planes and in the perpendicular direction serves as a measure of HTSC anisotropy in the superconducting state. This ratio equals '' (0)/ '' (0) = ab c
3D 2 2 c (0) / ab(0) , where '' and '' are the imaginary ab c parts of the corresponding conductivities and ab and c are the microwave-field penetration depths for the currents flowing, respectively, in the ab planes and perpendicularly to them. It is well known that, in high-quality optimally doped HTSC single crystals, ab(T) T at T < Tc/3, and this experimental fact suggests a d x2 ­ y2 symmetry of the order parameter in them [3]. There is no agreement in the literature about the low-temperature behavior of c(T). Both the linear dependence c(T) T at T < Tc/3 [4­6] and the quadratic dependence [7] have been observed for the most studied YBa2Cu3O6.95 (Tc 93 K) single crystals.

=

ab / c ne d / h ,
22

(1)

0021-3640/03/7710-0592$24.00 © 2003 MAIK "Nauka / Interperiodica"


ANISOTROPY OF MICROWAVE CONDUCTIVITY IN THE SUPERCONDUCTING

593
­x

Annealing temperatures, doping parameters, and characteristics of the superconducting and normal states of YBa2Cu3O7 Annealing Critical temperature temperature T, °C Tc, °C 500 520 550 600 720 92 80 70 57 41 Doping parameters p 0.15 0.12 0.105 0.092 0.078 x 0.07 0.26 0.33 0.40 0.47 values at T = 0 ab, nm 152 170 178 190 198 c, µm 1.55 3.0 5.2 6.9 16.3 c(T) T 1.0 1.1 1.2 1.3 1.8 c /ab at T = 0 10 18 29 36 83

c / ab at T = 200 K 11 18 16 16 35

A broad region of pseudogap states arising in the HTSC phase diagram at concentrations p < 0.16 has been studied to a much lesser extent. It follows from the measurements of dynamic susceptibility of oriented HTSC powders at T < Tc [8] that, at T 0, the slopes of '' (T)/ '' (0) for '' (T)/ '' (0). The nonmetallic c c ab ab behavior of resistivity c(T) as T approaches Tc, the deviations from the linear dependence ab(T) T, and a dramatic increase in the ratio c/ab with decreasing concentration p are common properties of underdoped HTSCs in their normal state. Although many theoretical models have been proposed for the explanation of these properties, none of them describes in full measure '' '' the evolution of the ab (T), c (T), ab(T), and c(T) curves over a wide range of concentrations and temperatures. The transport mechanism along the c axis has also not been established, and, in particular, it still remains unclear whether it can be metallic (of the Drude type) or whether the conductivity for any p is caused by the quasiparticle tunneling between the cuprate layers with scattering both within the layers and between them. In this work, the anisotropy and evolution of temperature dependences of the conductivity components of YBa2Cu3O7 ­ x with oxygen doping in the range 0.07 x 0.47 were measured and the measurement results were analyzed. The crystal was grown in a BaZrO3 crucible and had a rectilinear shape with sizes 1.6 â 0.4 â 0.1 mm. Measurements were performed at a frequency /2 = 9.4 GHz and temperatures 5 T 200 K. The oxygen content in the sample changed through the controlled annealing in air at different temperatures T 500°C (listed in the table). Measurements of the conductivity anisotropy were carried out for each of the five crystal states, in which the superconducting transition width, according to the susceptibility measurements at a frequency of 100 kHz, was 0.1 K in the optimally doped (x = 0.07) state and increased with x to reach 4 K at x = 0.47. The superconducting transition temperatures were Tc = 92, 80, 70, 57, and 41 K. The full cycle of microwave studies included (i) measurements of the temperature dependences of the Q value and the frequency shift for a superconducting niobium cavity with crystal samples in two, transverse and lonJETP LETTERS Vol. 77 No. 10 2003

gitudinal, orientations about the microwave magnetic field; (ii) determination of the surface resistance Rab(T), reactance Xab(T), and conductivity ab(T) of the cuprate planes in the normal and superconducting states from the measurements in the first orientation; and (iii) determination of c(T), Xc(T), and Rc(T) using the data obtained for the longitudinal orientation. The entire measurement procedure for the optimally doped YBa2Cu3O6.95 crystal is described in detail in [6]. The temperature dependences of the components of surface impedance of YBa2Cu3O7 ­ x at different x were reported in our short communication [9]. The '' (T)/ '' (0) ab ab (light symbols) and '' (T)/ '' (0) (dark symbols) curves at T Tc are prec c sented in Fig. 1 for the YBa2Cu3O7 ­ x crystal in the states with Tc = 92, 70, and 41 K. The field penetration depths ab(0) and c(0) at T = 0 are also given in the table. The overall temperature behavior of the '' (T)/ '' (0) curves changes only slightly upon varyab ab ing p. A distinctive feature of the optimally doped YBa2Cu3O6.93 state is that the temperature dependences '' (T)/ '' (0) and '' (T)/ '' (0) coincide to a good ab ab c c accuracy. This fact can be rigorously explained only in the theory of linear response of an anisotropic 3D superconductor [8]. As p decreases, the '' (T)/ '' (0) c c dependence at T < Tc/3 becomes noticeably weaker than '' (T)/ '' (0). ab ab The model proposed in [10] is most suitable for a comparison with the experimental data of our work. In this model, the following contributions to the quasiparticle transport along the c axis in the superconducting and normal HTSC states are considered: (a) direct hopping between the cuprate planes and (b) hopping with inelastic scattering from impurities located between the planes. The conductivity within the cuprate planes is assumed to be of the Drude type: n 2D e e 2D D ab = --------------------- = --------------- , md d
2 2

ab

(2)


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YBa2Cu3O7 ­ x with an oxygen deficiency x > 0.07 clearly demonstrates that the slopes of the '' (T)/ '' (0) c c curves strongly decrease with increasing x, while the '' (T)/ '' (0) curves change only slightly. The fact ab ab that the experimental curves at T > Tc/2 are steeper than the theoretical ones may be caused by the strong electron­phonon interaction [3], which was not taken into account in [10]. The dashed line in Fig. 1 coincides also '' '' with the c (T)/ c (0) curve calculated in [10] for the case where there is no diffuse tunneling (b) and the remaining mirror-tunneling regime (a) along the c axis becomes identical with the transport along c in an anisotropic 3D superconductor. This exceptional situation corresponds to the optimally doped YBa2Cu3O6.93. The real and imaginary parts of the surface impedance measured for the YBa2Cu3O7 ­ x crystal at T > Tc coincided with each other; i.e., Rab(T) = Xab(T) and Rc(T) = Xc(T) for each x from the table [9]. Because of this, the resistivities ab(T) and c(T) were derived from Rab(T) and Rc(T) using the standard formulas for the normal skin effect: ab(T) = 2 R ab(T ) /µ0 and c(T) =
2

Fig.

1.

The



'' (T)/ '' (0) (light symbols) and ab ab

'' (T)/ '' (0) (dark symbols) measured curves for three c c states of a YBa2Cu3O7 ­ x crystal with Tc = 92, 70, and 41 K. The solid and dashed lines correspond, respectively, to the '' (T)/ '' (0) and '' (T)/ '' (0) dependences calculated c c ab ab in [10] for oxygen-deficient YBa2Cu3O7 ­ x.

where
2

2D

= m/

2

is the two-dimensional density of
2

states per unit area and Dab = v F /2, vF, , and n2D = k F /2 are the diffusion coefficient, Fermi velocity, relaxation time, and two-dimensional quasiparticle density in the ab plane, respectively. The total Hamiltonian of the electron system in model [10] is the sum H m of the Hamiltonians of individual (m) CuO2 m layers and the interplane Hamiltonian H, which is H m . As a result, assumed to be small compared to m the second-order perturbative quasiparticle transport between the neighboring weakly bonded layers proves to be analogous to the tunneling through the SIS junction at T < Tc and through the NIN junction at T > Tc. In this case, the ab component of electron momentum is conserved in process (a) (mirror tunneling) and is not conserved in process (b) (diffuse tunneling) [11].

2 R c (T ) /µ0. The evolution of the ab(T) and c(T) curves with changing x is shown in Fig. 2 for the temperature range Tc < T 200 K, and the (c/ab)1/2 values at T = 200 K are given in the last column of the table. The ab(T) and c(T) dependences have a metallic character only in optimally doped YBa2Cu3O6.93, and the c/ab ratio approximately corresponds to the anisotropy of charge-carrier effective masses mc/mab =
2





c (0) / ab(0) in a pure 3D London superconductor, to which YBa2Cu3O6.93 belongs. In all other YBa2Cu3O7 - x states with a lower hole concentration, the resistivity c(T) increases with temperature decreasing, demonstrating the nonmetallic behavior. In Fig. 3, the experimental c(T) dependences are compared with the
2 2

c, min values calculated by the YBa2Cu3O7 ­ x crystal: Tc 70 K (dotted line), and Tc = Over the entire temperature conductivity along c is the
3D

Eq. (1) for three states of = 92 K (dashed line), Tc = 41 K (dot-and-dash line). interval, the YBa2Cu3O6.93 only one that exceeds the .

minimal metallic value of

3D c, min

The calculations of the anisotropy of the superconducting HTSC state were carried out in [10] using the BCS model with a d-symmetry order parameter in the '' '' CuO2 layers. The c (T)/ c (0) curve numerically calculated with allowance for both processes (a) and (b) is shown in Fig. 1 by the solid line and the same for '' (T)/ '' (0) is shown by the dashed line. A compariab ab son with the experimental data obtained at T < Tc/2 for

Thus, it is natural to assume that, as in the case of the superconducting state of YBa2Cu3O7 ­ x, a small decrease in the carrier concentration from its optimal level in the normal state leads to a crossover from the 3D metallic conduction to the 2D Drude conduction in the CuO2 layers and tunneling conduction between the layers (3D­2D crossover). To analyze this assumption, it is convenient to again use model [10]. If t is the hopJETP LETTERS Vol. 77 No. 10 2003


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595

Fig. 3. Symbols correspond to the experimental dependences for three YBa2Cu3O7 ­ x states with Tc 70, and 41 K. The dashed, dotted, and dot-and-dash
3D

c(T) = 92, lines

are for the corresponding c, min (T) values obtained from Eq. (1) using the measured ab(T) and c(T) presented in Fig. 2. The solid line corresponds to c(T) calculated for YBa2Cu3O6.67 by the formulas given in [10].

mately equal to the minimal metallic conductivity 3D c, max given by Eq. (1). In the case of diffuse quasiparticle tunneling (processes (b)) in model [10], the conductivity along the c axis equals [11, 14]
Fig. 2. Evolution of the sured for YBa2Cu3O7 ­ ab(T) and c(T) dependences meax with differing oxygen content.
diff c

e 2D d e 2D D c = ------------------- = ---------------- , c d
2 2

(4)

where Dc = d2c is the diffusion coefficient and 1/c is the scattering probability between the cuprate planes. diff As in the preceding case, we find that c, max = IR ab / c c, min for c , and, using Eqs. (2) and (4), we arrive at the following alternative form of the criterion for a 3D­2D crossover:
3D

ping matrix element, the quasiparticle conductivity along c in process (a) will be [10­13]
dir c

t 2 td 2 2 = 2 e 2D --- = 4 ab --------- , v F )2

(3)
c, max

where 2(t/ is the direct-tunneling rate between the neighboring CuO2 planes and ab is the conductivity along these planes (Eq. (2)). In this case, the characteristic hopping time /t appreciably exceeds the in-plane relaxation time [11]: /t . In the reverse limit /t , the conductivity is of the Drude type in all directions, as in the case of an anisotropic 3D metal. The crossover occurs when /t . At this point, the tunneling conductivity along c (Eq. (3)) reaches its dir maximum c, max = 2IR ab / c , which is approxiJETP LETTERS Vol. 77 No. 10 2003

n 2D e 2 2 ab ------- ---- .

(5)

From Eq. (5) it follows that, at n2D = n/d 1014 cm­2, the 3D­2D crossover occurs upon reaching the value cab 10­6 ( cm)2. Returning to the data in Fig. 2, we make sure that the product cab 10­6 ( cm)2 only at x = 0.07, thereby substantiating the applicability of the anisotropic 3D Fermi-liquid model for explaining the properties of optimally doped YBa2Cu3O6.93. Equations (3) and (4) account for the basically different temperature dependences of the conductivity


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TRUNIN, NEFEDOV

phonons to form polaron [17] that only weakly affects the transverse ab transport. For the Einstein spectrum of c-polarized phonons, one has exp [ g tanh ( 0 /4 T ) ] c(T ) ab(T ) -------------------------------------------------- , sinh ( 0 /2 T )
2

(6)

Fig. 4. Comparison of the (symbols) experimental and (solid lines) calculated (by formula (6)) c(T) dependences for YBa2Cu3O7 ­ x.

along the c axis at T Tc; for the direct tunneling, c (T) ab(T) increases with increasing (T) as T
dir

approaches Tc, whereas c (T) decreases with increasing c(T). According to model [10], the total conductivity c along the c axis is the sum of conductivities caused by each of the above-mentioned processes ((a) diff and (b)). Near the Tc temperatures, c is mainly due to the quasiparticle scattering from the impurities located between the cuprate planes and, hence, is indediff pendent of T, because the phonon contribution to c is frozen out. Quite the reverse, the phonon contribution becomes dominant at T Tc. As a result, the temperature dependence of the conductivity c(T) takes an approximate form, A/T + C + BT (A, B, and C are independent of T), that does not describe the experimental data; an example of c(T) calculated by the formulas given in [10] is shown by the solid line in Fig. 3 for the YBa2Cu3O6.67 sample.
diff

where g (g > 1) is the parameter characterizing the electron­phonon coupling strength. The comparison of the experimental data (symbols) with the c(T) dependences calculated by Eq. (6) (solid lines) is demonstrated in Fig. 4. In the calculations, the data given for ab(T) in Fig. 2 were used. The parameter g was almost identical (g 3) for all curves in Fig. 4, and 0 increased from 110 K (75 cm­1) to 310 K (215 cm­1) upon decreasing the oxygen content (7 ­ x) in YBa2Cu3O7 ­ x from 6.93 to 6.53. It seems not surprising that the anomalies of the optical c conductivity were observed for a YBa2Cu3O7 ­ x crystal with oxygen deficiency just in the indicated range of frequencies 0 [18]. In summary, the anisotropy of microwave conductivity was measured for a YBa2Cu3O7 ­ x crystal in which the hole concentration p was varied in the range 0.08 p 0.15. An analysis of the temperature dependences of the imaginary parts of the conductivity tensor ^ ''(T ) in the superconducting state and the resistivity ^ (T ) in the normal state indicates that optimally doped YBa2Cu3O6.93 is a three-dimensional anisotropic metal. A decrease in the carrier concentration leads to a crossover from the Drude-type to hopping conduction along the c axis. In order to quantitatively describe the evolution of the ''(T ) and c(T) dependences with changing c p, the effects of strong electron­phonon interaction must be taken into account. We are grateful to V.F. Gantmakher and A.F. Shevchun for helpful discussions. This work was supported by the Russian Foundation for Basic Research, project nos. 03-02-16812, 03-02-06386, and 02-02-08004. REFERENCES
1. J. L. Tallon, C. Bernhard, H. Shaked, et al., Phys. Rev. B 51, 12 911 (1995). 2. Y. B. Xie, Phys. Rev. B 45, 11 375 (1992). 3. M. R. Trunin and A. A. Golubov, in Spectroscopy of High-Tc Superconductors. A Theoretical View (Taylor and Francis, London, 2003), p. 159. 4. J. Mao, D. H. Wu, J. L. Peng, et al., Phys. Rev. B 51, 3316 (1995). 5. H. Srikanth, Z. Zhai, S. Sridhar, et al., J. Phys. Chem. Solids 59, 2105 (1998). 6. Yu. A. Nefyodov, M. R. Trunin, A. A. Zhohov, et al., Phys. Rev. B 67, 144 504 (2003). 7. A. Hosseini, S. Kamal, D. A. Bonn, et al., Phys. Rev. Lett. 81, 1298 (1998).
JETP LETTERS Vol. 77 No. 10 2003

However, all c(T) dependences shown in Fig. 2 can be described by the c-transport model that was recently proposed in [15]. Contrary to [10], where the electron­ phonon effects appeared in the second-order of the perturbation theory, the model Hamiltonian [15] includes them through the canonical transformation [16], after which the interplane quasiparticle tunneling can be considered as a perturbation of the originally strongly coupled electron­phonon system. This approach applies if F 0 t, where F is the Fermi energy and 0 is the characteristic phonon energy. Both inequalities are fulfilled for the layered anisotropic HTSCs, in which, according to [15], an electron moving in the c direction is enveloped by a large number of


ANISOTROPY OF MICROWAVE CONDUCTIVITY IN THE SUPERCONDUCTING 8. T. Xiang, C. Panagapoulos, and J. R. Cooper, Int. J. Mod. Phys. B 12, 1007 (1998). 9. Yu. A. Nefyodov and M. R. Trunin, Physica C (Amsterdam) (2003) (in press). 10. R. J. Radtke, V. N. Kostur, and K. Levin, Phys. Rev. B 53, R522 (1996); R. J. Radtke and K. Levin, Physica C (Amsterdam) 250, 282 (1995); R. J. Rojo and K. Levin, Phys. Rev. B 48, 16 861 (1993). 11. M. Turlakov and A. J. Legget, Phys. Rev. B 63, 064518 (2001). 12. N. Kumar and A. M. Jayannavar, Phys. Rev. B 45, 5001 (1992).

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13. L. B. Ioffe, A. I. Larkin, A. A. Varlamov, et al., Phys. Rev. B 47, 8936 (1993). 14. M. J. Graf, D. Rainer, and J. A. Sauls, Phys. Rev. B 47, 12 089 (1993). 15. A. F. Ho and A. J. Schofield, cond-mat/0211675. 16. I. G. Lang and Yu. A. Firsov, Zh. èksp. Teor. Fiz. 43, 1843 (1962) [Sov. Phys. JETP 16, 1301 (1962)]; Zh. èksp. Teor. Fiz. 45, 378 (1963) [Sov. Phys. JETP 18, 262 (1963)]. 17. T. Holstein, Ann. Phys. (N.Y.) 8, 343 (1959). 18. T. Timusk and B. Statt, Rep. Prog. Phys. 62, 61 (1999).

Translated by V. Sakun

JETP LETTERS

Vol. 77

No. 10

2003