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Journal of Superconductivity and Novel Magnetism, Vol. 19, Nos. 7­8, November 2006 ( C 2007 ) DOI: 10.1007/s10948-006-0212-5

Microwave Surface Impedance and Complex Conductivity of High-Tc Single Crystals: Current State and Unsolved Problems
Yu. A. Nefyodov,1 A. F. Shevchun,1 A. M. Shuvaev,1 and M. R. Trunin
Published online: 9 February 2007

1

Common and distinctive features of the temperature dependences of microwave surface impedance Z(T) = R(T) + iX(T) and conductivity (T) in the ab-plane and along the c-axis of high-Tc single crystals (HTSC) are discussed. The main attention is focused on an evolution of these dependences in YBaCuO crystal with the oxygen deficiency. Comparison of YBaCuO with other HTSC crystals reveals a number of peculiarities, namely, the linear dependence Rab(T) T up to Tc /2 in HTSC single crystals with tetragonal lattice and up to Tc /3 in YBaCuO where Rab(T) shows also a broad peak at T Tc /2; breakdown of normal skin effect in some HTSC crystals; dramatic effect of pseudogap on the superfluid density in the heavily underdoped YBCO; several orders of magnitude higher residual surface resistance in HTSC when compared to conventional superconductors; etc. Possible explanations are discussed in the context of the specific features of HTSC structure and in the framework of recent models of quasiparticles' c-transport and pseudogap state in HTSC.
KEY WORDS: high-temperature superconductors; surface impedance; microwave conductivity; anisotropy; pseudogap; single crystals.

1. INTRODUCTION In past years, a lot of interest has been focused on investigations of transport properties evolution in high-Tc single crystals (HTSC) with different level of doping by oxygen and other substitutional impurities; in other words, the dependence upon the number p of holes per Cu atom in the CuO2 plane. The p value and the critical temperature Tc of a superconducting transition in HTSC satisfy the following empirical relationship 1 : Tc = Tc
, max

[1 - 82.6(p - 0.16)2 ].

The narrow band of HTSC phase diagram corresponding to the optimal doping (p 0.16) and maximum values of the critical temperature Tc = Tc, max
1

Institute of Solid State Physics, Russian Academy of Sciences, 142432 Chernogolovka, Moscow, Russia; e-mail: nefyodov@ issp.ac.ru.

is the most studied area. In the normal state of optimally doped HTSC, the resistivity ab(T) of cuprate ab-planes increases proportionally to temperature viz., ab(T) T. The resistivity c (T) in perpendicular direction substantially exceeds the value of ab(T) and also has metallic behavior (both ab(T) and c (T) increase with T). However, BiSrCaCuO, which is the most anisotropic HTSC (having c /ab 105 at p 0.16) represents an exception: its resistivity c (T) increases as T approaches Tc (dc (T)/dT < 0). A measure of HTSC anisotropy in the superconducting state is the ratio ab(0)/c (0) = 2 (0)/2 (0), c ab where ab and c are imaginary parts of conductivity, ab and c are penetration depths of high-frequency field for currents running in the ab-planes and perpendicular to them, respectively. It is a common knowledge that ab(T) T at T < Tc /3 in highquality HTSC crystals and this experimental fact provides strong evidence for dx2 -y2 symmetry of the order parameter in these materials. However, there is no consensus in literature about c (T) behavior at 579
1557-1939/06/1100-0579/0
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2007 Springer Science+Business Media, LLC


580 low temperatures. Even YBa2 Cu3 O6.95 (Tc 93 K), the most thoroughly studied single crystals have shown both linear ( c (T) T) [2­4], and quadratic dependences 5 in the range T < Tc /3. Pseudogap states appearing at p < 0.16 occupy a wide band of HTSC phase diagram, which is far less investigated. Measurements of ac-susceptibility of oriented HTSC powders at T < Tc show 6 that their c (T)/c (0) dependences have smaller slope at T 0 than ab(T)/ab(0) dependences. The normal state of underdoped HTSC is characterized by nonmetallic behavior of the resistivity c (T) at T approaching Tc , by deviation of the ab-plane resistivity from its linear dependence ab(T) T and by c /ab ratio rising dramatically with the decrease of p . A lot of theoretical models have been proposed to explain these properties, but there are none fully describing the evolution of the dependences ab(T), c (T) and ab(T), c (T) in the wide range of concentration p and temperature T. Among the experimental methods of studying these problems, there are the measurements of the temperature dependences of the surface impedance Z(T) = R(T) + iX(T). In the local electrodynamics, which applies to HTSC, Z(T) defines the complex conductivity tensor (T) = (T) - ^ ^ i (T) = i µ0 /Z2 (T) characterized by the compo^ nents ab and c . At microwave frequencies, the imaginary part of the surface impedance--the surface reactance X--reflects mainly the response of the superconducting carries. The real part--the surface resistance R--is proportional to the microwave losses and is due to the normal carriers. In the present paper, we will analyze the common and distinctive features of Z(T) and (T) curves ^ in HTSC single crystals. Our main attention in this analysis will be focused on the evolution of these curves in YBa2 Cu3 O7-x crystal with the oxygen deficiency varied in the range 0.07 x 0.47.

Nefyodov, Shevchun, Shuvaev, and Trunin 2. EXPERIMENT The experiments were performed by the "hotfinger" technique 7 at the frequency of /2 = 9.4 GHz and in the temperature range 5 T 200 K. The initial YBa2 Cu3 O6.93 crystal was grown in BaZrO3 crucible and had a rectangular shape with dimensions being 1.6mm â 0.4mm â 0.1 mm. To change the carrier density, we successively annealed the crystal in the air at various temperatures T 500 C and quenched it with liquid nitrogen. Finally, five crystal states with critical temperatures Tc = 92, 80, 70, 57, 41 K and, correspondingly, the concentrations p = 0.16, 0.12, 0.106, 0.092, 0.078 were investigated (Table I). Anisotropy was measured for each of the five states. The whole cycle of the microwave anisotropy measurements includes the following 4 : (i) we measure the temperature dependences of the quality factor and of the frequency shift of the superconducting niobium resonator with the sample inside in the two crystal orientations with respect to the microwave magnetic field, namely, transversal (T) and longitudinal (L); (ii) measurements in the T-orientation gave the surface resistance Rab(T), reactance Xab(T) and conductivity ab(T) = i µ0 /Z2 (T) of the crystal cuprate planes, both in ab the normal and superconducting states; and (iii) measurements in the L-orientation gave c (T), Xc (T), Rc (T).

3. NORMAL STATE The temperature dependences of surface impedance components in the ab-plane, Rab(T) and Xab(T), and along c-axis, Rc (T) and Xc (T) are shown in Fig. 1 for each of the five crystal states indicated in Table I. The Xab(T) dependences in Fig. 1(a) are constructed with allowance made for both (i) the

Table I. Annealing and Critical Temperatures, Doping Parameters, Characteristics of the Microwave ab-plane Response [Transition Width Tc , Residual Surface Resistance Rres , Relaxation Time ab(Tc ), Parameter ab(Tc )/ab(0)] and Penetration Depths in YBa2 Cu3 O7-x Crystal Annealing temperature T ( C) 500 520 550 600 720 Critical temperature Tc (K) 92 80 70 57 41 Doping parameters p x 0.16 0.12 0.106 0.092 0.078 0.07 0.26 0.33 0.40 0.47 ab-Plane characteristics Rres (µ ) ab(Tc ) â1013 (s) 5 5 5 3 3 5 4 0 5 0 0 0 1 1 1 .7 .9 .0 .4 .5 values at T = 0 ab (nm) c (µm) 152 170 178 190 198 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

Tc (K) 0.5 1 1.5 1.5 4.0

0.05 0.06 0.10 0.13 0.20


Microwave Surface Impedance and Complex Conductivity

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Fig. 1. (a) real Rab(T) (solid symbols) and imaginary Xab(T) (open symbols) parts of the ab-plane surface impedance of the five states of YBa2 Cu3 O7-x single crystal; (b) the components of the c-axis surface impedance.

contribution Xth (T) of thermal expansion of the ab crystal, which essentially affects the measured reactance shift Xab(T)at T > 0.9 Tc , and (ii) the additive constant X0 , which is equal to the difference between the values of [ Xab(T) + Xth (T)] and Rab(T) at ab T > Tc : Xab(T) = Xab(T) + Xth (T) + X0 . The ab detailed procedure of the Zc (T) determination from the quantities measured is described in 4 . In the normal state, we have Rab(T) = Xab(T) and Rc (T) = Xc (T), which implies the validity of the normal skin-effect condition. Therefore, the resistivities ab(T) and c (T) can be found from Rab(T) and Rc (T) curves at T > Tc , as shown in Fig. 1, applying the standard formulas of the normal skin effect: ab(T) = 2R2 (T)/µ0 , c (T) = 2R2 (T)/µ0 . Figure 2 shows c ab the evolution of the dependences ab(T) and c (T) in the temperature range Tc < T 200 K for YBa2 Cu3 O7-x crystal with the change of x. Only the optimally doped YBa2 Cu3 O6.93 shows that both dependences ab(T) and c (T) have a metallic behavior, and the ratio c /ab approaches the anisotropy of the effective mass of charge carriers mc /mab = 2 (0)/2 (0) in the pure c ab three-dimensional (3D) London superconductor of the type YBa2 Cu3 O6.93 . The other states of YBa2 Cu3 O7-x with lower concentration of holes have the resistivity c (T) increasing with the decrease of temperature, which shows its nonmetallic

behavior. A crossover from Drude conductivity (at x = 0.07) along the c-axis to the tunneling one (at x > 0.07) takes place. This is proved in 8 by both estimating minimum metallic c-conductivity and maximum tunneling c-conductivity and quantitatively comparing the measured dependences c (T) with the ones calculated in the polaron model of quasiparticles' c-transport 9 . According to this model, the interplanar tunneling of quasiparticles is considered as a perturbation of a strongly coupled electron­phonon system. In the c-direction, an electron is surrounded by a large number of phonons, forming a polaron 10 , which influences the transversal ab-transport weakly. The following analytic expression was obtained in 9 for Einsteinian spectrum of c-polarized phonons in the temperature range T 0 : c (T) ab(T) exp[g2 tanh(0 /4T)] sinh(0 /2T) , (1)

where 0 is a typical phonon energy, g is a parameter characterizing strength of electron­phonon interaction and g > 1. Figure 3 represents our result of comparing experimental dependences c (T) (symbols) and those calculated from Eq. (1) (solid lines). To calculate them, we used ab(T) data in Fig. 2; g was almost the same for all the dependences in Fig. 3: g 3; 0 increased from 110 K (75 cm-1 ) to 310 K




582

Nefyodov, Shevchun, Shuvaev, and Trunin

Fig. 3. Comparison of the experimental dependences c (T) in YBa2 Cu3 O7-x (symbols) and those calculated from Eq. (1) (solid lines).



µ

µ

Assuming that scattering processes in this liquid are similar to those in metals, in 11 we proposed the ¨ Bloch­Gruneisen formula (electron­phonon interaction) for the function ab(T) in the normal and superconducting states of HTSC and retained the temperature-independent impurity relaxation time ab(0), which is present in the Gorter and Casimir two-fluid model: 1 t5 J5 (/t)/J5 () 1 = 1+ , ab(t) ab(0) J5 (/t) =
0 /t







z5 ezdz , (ez - 1)2

(2)

Fig. 2. The evolution of the measured ab(T) and c (T) dependences in YBa2 Cu3 O7-x with different oxygen content.

(215 cm-1 ) when the oxygen content (7 - x) was decreased in YBa2 Cu3 O7-x from 6.93 to 6.53. At the same time, the ab-plane transport in the normal state of YBa2 Cu3 O7-x crystal always remains metallic. The value of ab(T) 50 µ ·cm is indicative of high quality of the crystal. As in the normal metals, in the ab-planes of optimally doped HTSC, the resistivity increases proportionally to temperature, ab(T) 1/ab(T) T, where ab is a quasiparticle relaxation time in the cuprate planes.

where t T/Tc , = /Tc ( is the Debye temperature) and is a numerical parameter, which equals (as it follows from Eq. (2)) = ab1 (Tc )/[ab(0) - ab(Tc )]. Following the formal analogy with metals, one can say that the parameter is the characteris1 if tic of the "HTSC purity": ab(Tc )/ab(0) ab(Tc ). The parameter in HTSC can be ab(0) estimated at several hundred degrees. At T < /10 ( > 10t), the second summand in the square brackets in Eq. (2) is proportional to T5 ; in the region T > /5( < 5t), it is proportional to T. Thus, the reciprocal relaxation time is almost constant and equal to 1/ab(0) over the interval 0 < T < Tc /3, and at higher temperatures it increases gradually, starting as the power function T5 in the region T < Tc /2 and changing later to function T around Tc ; at T > Tc we have linear dependence ab(T) 1/ab(T) T.


Microwave Surface Impedance and Complex Conductivity Whereas the conditions Rab(T) = Xab(T) at < T Tc and Xab(T) < Rab(T) at T Tc were experimentally proved for YBaCuO [12­17], BiSrCaCuO [12,15,17­19], TlBaCaCuO 17 ,LaSrCuO 20 , and BaKBiO 11 in the crystals of TlBaCuO 21,22 , HgBaCuO, and HgBaCaCuO 22 the reactance variation Xab(T) is considerably larger than that of the resistance Rab(T) in the entire temperature range such that Rab(T) = Xab(T) at T Tc . In 22 , this breakdown of the normal skin-effect condition was treated in terms of a collective phason mode. In model 23 , the appearance of the strong inequality Rab(T) = Xab(T) was due to the presence of superconducting islands embedded in a normal metallic matrix. Both concepts are related to the pseudogap phenomenon in HTSC. As an alternative explanation, we propose to allow for the shielding effect of the microwave field by roughnesses (cleavage plane traces), which may crop out at TlBaCuO crystal surface. If the penetration depth is much less than roughness sizes, both components of the effective surface impedence, that are measured will increase in comparison with their values for a flat surface by the same factor equal to the ratio of the real and flat surface areas. If the roughness sizes are comparable to the penetration depth the situation may occur when the microwave magnetic field H is slightly distorted by the roughness, whereas the high-frequency current j caused by the field decays noticably 24 . In 2 this case, the effective reactance ( µ0 H dV) will exceed the sample surface resistance ( j E dV). It is likely that this is the case for TlBaCuO crystal whose roughness' dimensions proved to be comparable with its skin depth. Indeed, scanning electron micrographs demonstrated 25 that the traces of the cleavage planes form juts (grooves) in the form of parallel channels at the surface of TlBaCuO crystal and the sizes (height, width, spacing) of these roughnesses are about of 1­2 µm.

583

Fig. 4. Low-temperature dependences of ab(T) (open symbols) measured for five states of YBa2 Cu3 O7-x crystal with Tc = 92 K, Tc = 80 K, Tc = 70 K, Tc = 57 K, and Tc = 41 K. Dashed lines are linear extrapolations at T < Tc /3.

In YBa2 Cu3 O6.93 (p = 0.16), the temperature dependence Rab(T) has a broad peak in the range T Tc /2, which is the characteristic for high-quality optimally doped YBaCuO crystals. The peak gradually vanishes with decreasing value of p . Note that such a peak has not been observed in any crystals with tetragonal (BiSrCaCuO, TlBaCaCuO, TlBaCuO, LaSrCuO) or cubic (BaKBiO) lattices. The conductivity components (T) and (T) are not measured directly, but can be derived from measurements of R(T) and X(T): = 2µ0 RX , (R2 + X2 )2 = µ0 (X2 - R2 ) . (R2 + X2 )2 (3)

4. SUPERCONDUCTING STATE Figure 4 shows the low-temperature sections of the measured ab(T) = Xab(T)/µ0 curves for five states of YBa2 Cu3 O7-x crystal. The error in the absolute value of ab(T) is mainly caused by the accuracy of the additive constant X0 determination. In our experiments the root-mean-square difference between Rab(T) and Xab(T) in the normal state corresponded to about 5 nm accuracy in ab(T) value.

At temperatures not very close to Tc , R(T) X(T) in HTSC (Fig. 1) and, hence, (T) curve is determined by the function X(T) = µ0 (T) alone and reflects the properties of the magnetic field penetration depth. The shape of (T) curve depends on the value of the residual surface resistance Rres = R(T 0), which is obtained by the extrapolation of R(T) curve to zero temperature in the region T < Tc /3. It follows from Eq. (3) that (T) curve has a peak at T < Tc if the value of Rres is sufficiently small 26 : Rres < X(0) dR(T) 3 dX(T) .
T0



µ

(4)

This peak shifts to lower temperatures as Rres increases, and when the inequality Eq. (4) is violated, it


584 completely disappears. Figure 5 shows ab(T) curves for five states of YBa2 Cu3 O7-x with different oxygen content (Table I). Usually (T) in HTSC single crystals has also a narrow peak near Tc (Fig. 5). Its width equals the width of the superconducting transition on the curve of R(T) at microwave frequencies.

Nefyodov, Shevchun, Shuvaev, and Trunin Even though the form chosen for the function ab(T) in Eq. (2) is oversimplified in the case of HTSC materials with complex electronic spectra, it is turned out that all common and even specific features of Rab(T) and ab(T) curves for both T < Tc and T > Tc are adequately described by the modified two-fluid model (MTFM) with the only free-fitting parameter in Eq. (2). 1, which is typIndeed, in the case of (ab)2 ical for HTSC crystals at all temperatures in the frequency band around 10 GHz or below, the components of the complex conductivity = - i can be written in a very simple form: nn e2 ab ns e2 , = , (5) m m where nn,s (T) are the densities of the normal and superconducting carriers (both having the same charge e and effective mass m). At temperatures T Tc , the total carrier concentration n = ns + nn is equal to the concentration of quasiparticles in the normal state. From the values of Rab(Tc ) and Xab(0), and the slopes Tc , one can easily dedRab/dT and dXab/dT at T rive the parameters 26 = ab(Tc ) = X2 (0) ab , 2R2 (Tc ) ab ab(0) = dRab dXab (6)
T0

Fig. 5. The conductivities ab(T) (symbols) of the five states of YBa2 Cu3 O7-x crystal extracted from the surface impedance measurements of Fig. 1(a) and the calculations (solid lines) based on the modified two-fluid model, which takes into account the inhomogeneous broadening Tc of the superconducting transition and residual surface resistance Rres (Table I).

and determine the parameters in Eq. (2), as indicated in Table I. Now, if we use the dependence ns (T)/n = (T)/ (0) = 2 (0)/2 (T) determined in the same experiment and, thus, derive the function nn (T)/n = 1 - ns (T)/n, we can fit the measurements of Rab(T) using Eqs. (2), (5) and free parameter by the general formula R =Re[ i µ0 /( - i )]. Then, we derive the real part of the conductivity ab(T) using Eq. (3). Figure 5 shows the comparison of the experimental data with the calculations by the procedure given above. In addition, we took account of the inhomogeneous broadening Tc of the superconducting transition (Table I) following the approach 26 that describes the peak in the conductivity (T) at a temperature Tm = Tc - Tc , which is close to the critical temperature. The narrower the superconducting transition (the smaller Tc ), the smaller the peak amplitude. Furthermore, when comparing the calculations with the measurements of the surface resistance, we added to the functions R(T), determined by the given above general formula, the values Rres from Table I. For this reason, the curves of ab(T) in Fig. 5, which were calculated by Eq. (3), do not tend to zero as T 0, even though the carrier density nn = 0 at T = 0, according to MTFM, and as it


Microwave Surface Impedance and Complex Conductivity follows from Eq. (5), the conductivity should tend to (0) = 0. The origin of the residual losses in HTSC materials has remained unclear. In some works [27], these losses were attributed to the presence of a fraction n0 of carriers that remain unpaired at T = 0. The magnitude of Rres was estimated by formula R = 2 µ2 3 /2, following from the general one at T < Tc , 0 with the nonzero conductivity (0) = n0 e2 ab(0)/m from Eq. (5). One can easily prove, however, that this approach requires inequality Eq. (4) to be satisfied. If this condition is not met, which may occur in HTSC crystals 17,26 , the number n0 needed to describe Rres value will be larger than the entire concentration n of the carriers. In developing the traditional approach to the problem, which attributed the residual surface resistance to various imperfections of the surface, the researchers took account of the losses due to weak links [28­30], twinning planes 30,31 , clusters with normal conductivity 32 , etc. Numerical estimates, however, indicate that the contribution of these mechanisms to the residual surface resistance is much smaller than Rres measured in HTSC materials. Moreover, a very remarkable fact is that the residual surface resistance measured at a frequency of 10 GHz was approximately the same, Rres 100 µ , in the ab-plane of all high-quality copper-oxide HTSC crystals of different chemical compositions and grown by different methods, irrespective of whether the samples contained twinning planes or not, and whether measurements were performed on freshly cleaved or as-grown surfaces. This fact, apparently, indicates that the presence of residual losses is the inherent feature of all HTSC which originates from their structure, namely, from their conspicuously layered nature. In other words, a fraction of current flowing in the surface layer of an HTSC crystal may run through regions of the layer that are in the normal state and have a finite resistivity n . In the phenomenological model under consideration, this contribution to the impedance can be included as a circuit element n connected in parallel to the two-fluid circuit characterized by Eq. (5), i.e., a resistor = 1/ shunted by a kinetic inductor l = 1/ (the parallel connection of and l is in conformity with the formula relating the current to the field in the twofluid model). Obviously, the complex impedance of the circuit consists of the imaginary part iX = i µ0 at T < Tc and the sum of two real components: the ordinary R = 2 µ2 3 /2 and R0 = 2 µ2 3 /2n .At T = 0 0 0, when R(0) = 0, only R0 value defines the resid-

585

ual resistance Rres . At a frequency of 10 GHz, and using Rres 100 µ and (0) 0.2 µm as an estimate, which are typical parameters for the ab-plane of HTSC crystals, we obtain n (0) 25 µ · cm, that is quite usual value for normal metals. To sum up this section, one can describe characteristic features of Rab(T) and ab(T) curves in HTSC crystals by generalizing the well-known Gorter and Casimir two-fluid model. For this, we introduce a temperature dependence of quasiparticles relaxation ¨ time according to the Bloch­Gruneisen law. We conclude that ab(T) curves can be well described by MTFM with the use of the only one free-fitting parameter, namely, = /Tc , while the other parame1 can be estimated directly ter = ab(Tc )/ab(0) from the experimental data with the aid of Eq. (6). < The ab(t) curve passes through a maximum at t0.5 if the inequality Eq. (4) is valid. This peak is due to the competition of two effects, namely, the decrease of normal carriers density as the temperature decreases and the increase of the relaxation time, which saturates at t 1/5 , where the impurity scattering starts to dominate. From MTFM, it also follows that the absence of the broad peak of Rab(t) at t 0.5 in YBa2 Cu3 O7-x with oxygen deficiency and in high-quality tetragonal HTSC single crystals is due to a less rapid increase of ab(T) with decreasing temperature. In other words, the value of is the smallest for the case of optimally doped YBaCuO crystals.

5. EFFECT OF PSEUDOGAP Currently, the origin of the pseudogap remains unclear. Proposed theoretical scenarios can be divided into two categories. One is based on the idea that the pseudogap is due to precursor superconductivity, in which pairing takes place at the pseudogap transition temperature T > Tc but achieves coherence only at Tc . The other assumes that the pseudogap state is not related to superconductivity per se, but rather competes with it. This magnetic precursor scenario of the pseudogap assumes dynamical fluctuations of some kind, such as spin, charge, structural, or so-called staggered flux phase. These two scenarios treat anomalies of electronic properties in underdoped HTSC observed at temperatures both above Tc and in its vicinity [33­36]. In the heavily underdoped HTSC, at T Tc , a competition of pseudogap and superconducting order parameters develops most effectively and results


586 in the peculiarities of the superfluid density ns (T, p ) as a function of temperature T and concentration p . In a clean Bardeen, Cooper, and Schrieffer (BCS) d-wave superconductor (DSC) the dependence ns (T) ns (T) - n0 is linear on temperature T Tc : ns (T) (-T/ 0 ), where n0 = ns (0) and (0) are the superfluid density and the su0= perconducting gap amplitude at T = 0. This dependence is confirmed by the measurements of the abplane penetration depth ab(T) = m /µ0 e2 ns (T) in the optimally doped (p = 0.16) HTSC single crystals: ab(T) T at T < Tc /3. The derivative | dns (T)/dT| at T 0 determines n0 / 0 ratio. If thermally excited fermionic quasiparticles are the only important excitations even at p < 0.16, then the slope of ns (T) curves at T Tc is proportional to n0 (p )/ 0 (p ) ratio: | dns (T)/dT|T0 n0 (p )/ 0 (p ). In Fig. 4, the linear extrapolation (dashed lines) of the ab(T) dependences at T < Tc /3 gives the following ab(0) values 37 : 152, 170, 178, 190, 198 nm for p =0.16, 0.12, 0.106, 0.092, 0.078, respectively (Table I). As follows from Fig. 6, halving of the concentration (namely, from p = 0.16 to p = 0.078) results in approximately two times decrease of -2 (0) value. ab Similar behavior n0 (p ) p within the range 0.08 < p 0.16 was observed by other groups 38,39 . It is easily seen that this dependence contradicts Uemura's relation n0 (p ) Tc (p ) 40 . To the best of our knowledge, there are no data on the superfluid density in HTSC at p < 0.08. As for theoretical predictions, n0 linearity on p extends down to p = 0 in the generalized Fermi-liquid models [41­43], while in the magnetic precursor d-density wave (DDW) scenario of pseudogap [44­46] it exists in the underdoped range of the phase diagram where the DSC order parameter grows from zero to its maximal value, moreover, n0 (p ) is nonzero as 0 (p ) vanishes (Fig. 1 from 46 ). The latter agrees with our data. In Fig. 6 we also show the slopes | d-2 (T)/dT|T0 | dns (T)/dT|T0 of -2 (T) curves ab ab obtained from ab(T) dependences at T < Tc /3 in Fig. 4. The value of | d-2 (T)/dT| changes slightly ab at 0.1 < p 0.16 in accordance with model 41 , and measurements of YBa2 Cu3 O7-x single crystals 47 and oriented powders 48 with the holes concentration p 0.1. However, it grows drastically at p 0.1, namely, -2 (T) slope increases 2.5 times < ab with the decrease of p from 0.12 to 0.08. The | d-2 (T)/dT| p -2 dependence 42 is shown by solid ab line in Fig. 6 and roughly fits the data at p 0.12. The dotted line drawn through | d-2 (T)/dT| experab

Nefyodov, Shevchun, Shuvaev, and Trunin

µ



Fig. 6. The values of -2 (0) = ns (0)µ0 e2 /m (right scale) and ab slopes | d-2 (T)/dT|T0 = µ0 e2 /m | dns (T)/dT|T0 (left scale) ab as a function of doping p = 0.16 - (1 - Tc /Tc, max )/82.6 with Tc, max = 92 K in YBa2 Cu3 O7-x . Error bars correspond to experimental accuracy. The dashed and dotted lines guide the eye. The solid line is | dns (T)/dT| p -2 dependence.

imental points in Fig. 6 qualitatively agrees with the behavior of this quantity in the DDW model 45,46 . The temperature dependence of the superfluid density ns (T) at low T in the heavily underdoped YBa2 Cu3 O7-x proves to be one more check-up of the DDW scenario of pseudogap. Also, 2 (0)/2 (T) = ab ab ns (T)/n0 dependences obtained from the data in Fig. 4 are shown in Fig. 7(a) for different p values. The solid line represents the DSC result. The evident peculiarities in Fig. 7(a) are the concavity of ns (T)/n0 curves corresponding to the heavily underdoped states (p = 0.078 and p = 0.092) and their deviation from DSC and the curves for p = 0.16, 0.12, 0.106. It should be noted that these peculiarities do not strongly depend on ab(0) values. This is demonstrated in the inset of Fig. 7(a), where ns (T)/n0 experimental curve for p = 0.092 is compared to the ones obtained using ab(0) increased (open stars) and decreased (solid stars) by 40 nm. Actually, the latter value is much higher than the experimental uncertainty. The behavior of the superfluid density ns (T)/n0 , as shown in Fig. 7(a), contradicts the conclusions of the precursor pairing model 49,50 based on the formation of pair electron excitations with finite momentum at T > Tc , but agrees with the DDW scenario 45 . According to 45 , at temperatures much smaller than the relevant energy scales, a DDW (W0 ) and DSC





µ


Microwave Surface Impedance and Complex Conductivity

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Fig. 7. (a) 2 (0)/2 (T) = ns (T)/ns (0) at T < Tc /2 in YBa2 Cu3 O7-x with different doping. The solid line is the 2 (0)/2 (T) dependence ab ab ab ab in BCS d-wave superconductor (DSC). The inset shows ns (T)/n0 experimental curve for p = 0.092 and the ones obtained using ab(0) increased (open stars) and decreased (solid stars) by 40 nm; (b) comparison of experimental -2 (T) ns (T) curves (symbols) with linear ab -2 (T) (-T) (dashed lines) and root -2 (T) (- T) (solid lines) dependences for moderately doped (p = 0.106, x = 0.33) and ab ab heavily underdoped (p = 0.092, x = 0.40; p = 0.078, x = 0.47) YBa2 Cu3 O7-x .

( 0 ) order parameters at T = 0, only the nodal regions close to the points (/2,/2) and symmetryrelated points on the Fermi surface will contribute to the suppression of the superfluid density. In a wide range of temperatures, ns (T) dependence will be linear for the optimally and moderately doped samples, in which 0 is larger than or comparable to W0 (Fig. 8) and plays a leading role in the temperature dependence of the superfluid density. However, for the heavily underdoped samples, the situation is quite different. As the DDW gap is much larger than the superconducting gap in these heavily underdoped samples, W0 becomes dominant around the nodes. Though in the asymptotically lowtemperature regime the suppression of the superfluid density is linear on temperature, there is an intermediate temperature range over which the suppression actually behaves as T. It is worth emphasizing that the authors of 45 state that these features are independent of the precise W0 (p ) and 0 (p ) functional forms. The only input that is needed is the existence of DDW order, which diminishes with increase in p and complementary development of the DSC order. The DDW order eats away part of the superfluid density from an otherwise pure

DSC system. Actually, in the intermediate tempera< ture range 0.1 Tc < T0.5 Tc , the experimental ns (T) curves in YBa2 Cu3 O6.60 and YBa2 Cu3 O6.53 with p < 0.1 are not linear but similar to T-dependences. This is confirmed by Fig. 7(b), where the measured curves -2 (T) ns (T) are compared to the linab ear (T) dependences in YBa2 Cu3 O (p = 0.106) 6.67 and T-dependences -2 (T) = -3 T (ab and T ab are expressed in µm and K) YBa2 Cu3 O6.60 (p = in 0.092) and -2 (T) = -3.5 T in YBa2 Cu3 O6.53 ab (p = 0.078). Dashed lines in Fig. 7(b) correspond to the linear dependences of ab(T) at T < Tc /3, as shown in Fig. 4, and extended to higher temperatures. It is also interesting to note that these deviations of -2 (T) in YBa2 Cu3 O6.60 and YBa2 Cu3 O6.53 ab are accompanied by inflection of the resistivity ab(T) curves in the normal state of these samples. These inflections are seen at two lower ab(T) curves in Fig. 2 around T 100 K. Furthermore, the evolution of the temperature dependences of c (T) with doping in Fig. 2 correlates with those of the c-axis penetration depth c (T). Solid symbols of Fig. 9(a) show the dependences 2 (0)/2 (T)at T Tc /2 for YBa2 Cu3 O7-x c c states with Tc = 92, Tc = 70, and Tc = 41 K. Table I


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Nefyodov, Shevchun, Shuvaev, and Trunin good coincidence of 2 (0)/2 (T) (open circles in ab ab Fig. 9(a) and 2 (0)/2 (T) temperature dependences. c c With the decrease of p , the temperature dependence of 2 (0)/2 (T) becomes substantially weaker than c c that of 2 (0)/2 (T). Model 52 associates the reducab ab tion in the low-T slope of 2 (0)/2 (T) curves and c c the appearance of semiconducting-like temperature dependence of c (T) with a decrease of the interlayer coupling in the crystal. Dashed line in Fig. 9(a) represents numerical result 52 for this case. On the other hand, in the optimally doped YBa2 Cu3 O6.93 , the interlayer coupling is strong and quasiparticle transport along the c-axis becomes identical to one in the anisotropic 3D superconductor 6 . Solid line in Fig. 9(a) is 2 (0)/2 (T) dependence, calculated in 52 c c for this particular case. So, the low-T dependences of c (T) are well-described without taking pseudogap effects into consideration. Let us consider now their possible manifestations in the doping dependence of the c-axis penetration depth. From the inset of Fig. 9(a), it follows that reciprocal value of the zero-temperature penetration depth 1/c (0, p ) is roughly linear on p . Note that it vanishes at p 0.07, which is near the value where Tc does too 46,51 . There are several theoretical models 52,53 and experimental confirmations 54 of the

Fig. 8. The temperature versus doping p schematic phase diagram based on calculations of 51 . AF is the 3D antiferromagnetic phase. The system is an isolator in the AF state, a metal in the DDW and DDW + AF states, and a superconductor in the DSC and DDW + DSC states 51 .

contains the values of the penetration depth c (0) at T = 0 and the exponents in the measured c (T) - c (0) = c (T) T dependences at T Tc /3. The peculiarity of the optimally doped YBa2 Cu3 O6.93 is

Fig. 9. (a) dependence 2 (0)/2 (T) (open symbols) in YBa2 Cu3 O6.93 and 2 (0)/2 (T) (full symbols) measured for three states of the c c ab ab YBa2 Cu3 O7-x crystal with Tc = 92 K, Tc = 70 K and Tc = 41 K. Solid and dashed lines stand for the dependences 2 (0)/2 (T) calculated c c 52 for YBa Cu O in 2 3 7-x with different oxygen deficiency. The inset shows 1/c at T = 0 as a function of doping p ; (b) doping dependences of 2 (p )/2 (0.16) at T = 0and c (p )/c (0.16) at T = Tc .Their ratio U0 (p )/U0 (0.16) is shown in the upper inset. The lower inset is c (0) versus c c c (Tc ) plot.


Microwave Surface Impedance and Complex Conductivity direct proportionality of -2 (0) to the c-axis conducc tivity c (Tc ) in HTSC. In the simplest theory, this correlation is caused by -2 J c relation, where J c c is the c-axis critical current in the d-wave superconductor with anisotropic interlayer scattering and weak interlayer coupling. The value of J c (0) is determined by both the superconducting gap 0 and conductivity c (Tc ). The symbols in the lower inset of Fig. 9(b) show our data fitted by the dashed line log c (0)[µm] = -0.5log c (Tc )[ -1 m-1 ] + 2.1. The latter constant defines a proportionality factor U0 (p ) in -2 (0, p ) = U0 (p ) c (Tc , p ) relation. In the framec work of DDW model, the value of U0 (p ) is determined by the doping dependences of 0 (p ), W0 (p ) and chemical potential µ(p ). As shown in 55 , the opening of DDW gap can lead to increase as well as to decrease of U0 (p ). This depends on the position of the Fermi surface with respect to DDW gap, but in any case U0 (p ) changes less than twice in the whole range of doping. The values of 2 (0.16)/2 (p ) c c at T = 0 and c (p )/c (0.16) at T = Tc are shown in Fig. 9(b). Their ratio U0 (p )/U0 (0.16) is demonstrated in the upper inset of Fig. 9(b). This weak doping dependence indicates that strong decrease of the interlayer coupling integral t (p ) c (Tc , p ) with lowering p 56 in YBa2 Cu3 O7-x dominates over effects of the DDW order on c (0, p ) 55 . 6. CONCLUSION This paper attempts to review the main results and unsolved problems of microwave investigations of HTSC single crystals. Measurements of the surface impedance and complex conductivity of the ab-plane of the crystals with different chemical compositions are well-described in terms of the phenomenological modified two-fluid model. It seems quite natural that a consistent microscopic theory of HTSC should include the aspects of the model discussed. Namely, the theory can be based on the Fermi-liquid approach accounting for three essential factors: (i) strong electron­phonon coupling in the cuprate planes providing high-Tc values (Tc 100 K), the linear temperature dependence of the resistivity ab(T) 1/ab(T) T at T > Tc , and a broad peak in the real part of conductivity ab(T) at T < Tc ; (ii) predominant d-wave component of the superconducting order parameter leading to the linear low-temperature dependence of the penetration depth ab(T) T; (iii) layered structure and pronounced anisotropy of HTSC.

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As for the underdoped states of HTSC, this wide band of HTSC phase diagram is far less investigated. Four main experimental observations of this paper viz., (i) linear dependence of n0 (p ) in the range 0.078 p 0.16, (ii) drastic increase of lowtemperature ns (T) slope at p < 0.1, (iii) the deviation of ns (T) dependence from universal BCS behavior ns (T) (-T) at T < Tc /2 toward ns (T) (- T) with decreasing p < 0.1, and (iv) very weak influence of pseudogap on the low-T and doping dependences of the c-axis penetration depth evidence the DDW scenario of electronic processes in underdoped HTSC.

ACKNOWLEDGMENTS This research was supported by RFBR Grants numbers 03-02-16812, 02-02-08004 and 04-0217358. Yu. A. N. thanks Russian Science Support Foundation.

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