Modeling of the out-of-plane resistivity of cuprate superconductors

Physica C 471 (2011) 1598–1601

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Modeling of the out-of-plane resistivity of cuprate superconductors S.H. Naqib a,⇑, M. Borhan Uddin a,b, J.R. Cole c a

Department of Physics, University of Rajshahi, Rajshahi 6205, Bangladesh Department of CSE, International Islamic University, Chittagong, Bangladesh c Cambridge Flow Solutions Ltd., Histon, Cambridge CB24 9AD, UK b

a r t i c l e

i n f o

Article history: Received 21 July 2011 Received in revised form 27 August 2011 Accepted 1 September 2011 Available online 10 September 2011 Keywords: c-Axis charge transport Pseudogap Quantum critical point

a b s t r a c t The out-of-plane (c-axis) resistivity, qc(T), of high-Tc cuprates have been modeled in this study. The nonFermi liquid like temperature dependence of qc(T) has been described by considering (i) the full impact of the pseudogap (PG) in the electronic density of states (EDOS) and (ii) the presence of a quantum critical point (QCP) beneath the superconducting dome at slightly overdoped region. This simple phenomenological model describes the experimental qc(T) data over a wide range of hole content (from the underdoped to slightly overdoped regions) remarkably well. The PG energy scale, eg (dominated by the anti-nodal parts of the Brillouin zone) extracted from the analysis of qc(T) data was found to decrease almost linearly with increasing hole concentration, p, in the CuO2 planes. We have also discussed about the possible origin of more conventional behavior of qc(T) observed in the deeply overdoped side of the T–p phase diagram in this paper. Ó 2011 Elsevier B.V. All rights reserved.

1. Introduction Elucidation of unconventional superconductivity in strongly correlated electronic systems such as cuprates has been one of the central issues both in experimental and theoretical research fields of condensed matter physics in recent decades. In high-Tc cuprates anomalous charge transport behaviors in the normal state have provided with many challenging issues and stimulated much interest [1–3]. It is widely believed that understanding of these strange normal state properties will lead eventually to an understanding of the mechanism giving rise to high-Tc superconductivity itself. One of the key questions to be answered is that why the (out-of-plane) c-axis resistivity differs so much from the (in-plane) ab-plane resistivity in the normal state. In the underdoped (UD) region, the temperature dependence of the c-axis resistivity, qc, is ‘‘semiconductor’’ like (dqc/dT < 0), in contrast to the in-plane resistivity, qab, whose temperature dependence is metallic (dqab/ dT > 0). This dramatic difference between qc and qab is not what one might expect within the conventional Fermi-liquid theory [4]. Such different T-dependences and very large (105 in Bi2212) value of the resistivity anisotropy (qc/qab) has stimulated vigorous theoretical and experimental investigations on the interlayer charge dynamics of high-Tc cuprates. However, a complete description of the c-axis charge transport in hole doped cuprates is still lacking. ⇑ Corresponding author. Tel.: +880 721 750288; fax: +880 721 750064. E-mail address: [email protected] (S.H. Naqib). 0921-4534/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2011.09.001

A number of previous studies have attempted to model the outof-plane charge transport in cuprates by considering different physical scenarios [5–9]. For example, Rojo and Levin [5] proposed a microscopic model incorporating interplanar disorder with incoherent c-axis conduction. Ratan Lal et al. [6] proposed a phenomenological model based on the analysis of the role of the Cu 3d3z2 r2 orbitals on c-axis charge transport. In a model somewhat similar to that proposed in Ref. [5], Turlakov and Leggett [7] proposed a picture where c-axis conductivity is described by incoherent tunneling between the CuO2 planes. Su et al. [8], on the other hand, modeled the c-axis resistivity by taking into consideration of the k-dependent out-of-plane hopping integral and the pseudogap (PG) in the electronic density of the states (EDOS). Very recently Levchenko et al. [9] have modeled the c-axis conductivity by considering a phenomenological Greens function and PG in the quasi-particle (QP) energy spectrum. All these theoretical models fit the available experimental data qualitatively or sometimes quantitatively with varying degree of success [10]. In this study, we have proposed a simple phenomenological model capable of describing the temperature and doping dependences of the c-axis resistivity for hole doped cuprates. The model is based on two experimentally observed phenomena that are generic to all the hole doped cuprate superconductors, namely (i) T-linear in-plane and out-of-plane resistivities in compounds with planar hole content p  0.19 and (ii) the presence of a p-dependent PG in the EDOS when p < 0.19 [11–15]. We have developed this model in the subsequent sections and have used it to fit the high-quality qc(T) data obtained for pure Y123 and 6% Ca

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substituted Y123 single crystals. We have also extracted the pdependent PG energy scale from the analysis of the resistivity data.

has been used successfully to extract eg-values from the analysis of electronic specific heat and Pauli magnetic susceptibility data in previous studies [23,24].

2. Modeling of qc(T) 3. Experimental samples and analysis of the qc(T) data

qc1 ¼ aT

ð1Þ

where a is a doping dependent parameter that determines the highT slope of the qc(T). The second term, qc2, is related to the interplane tunneling of the charge carriers. Following the t–J model calculations by Prelovsek et al. [21], Cooper et al. [22] modeled this incoherent inter-plane conductivity in an earlier study. By taking into consideration of these earlier works, we have modified this EDOS dependent term as

qc2 ¼ h

1



2kB T

eg



b n  oi2 e ln cosh 2kgB T

ð2Þ

where b ¼ 1=At2c N 20 . Here tc is the c-axis tunneling matrix element and N0 is the flat EDOS at high energies outside the PG region. These two parameters are expected to be p-dependent, whereas A is a constant (almost) independent of doping [22]. Therefore, we have the following expression for the out-of plane normal state resistivity for cuprates

qc ðTÞ ¼ aT þ h

1



2kB T

eg



b n  oi2 e ln cosh 2kgB T

ð3Þ

The term in the denominator represents the thermal average (within ±2kBT around the chemical potential) of the square of the EDOS [23]. In this formalism eg represents the PG energy scale with the EDOS profile depleted by a triangular gap. Such a model EDOS

High-quality Y123 single crystals used in this study were grown using the self-flux method in ultra-pure BaZrO3 crucibles. Whereas the 6% Ca doped Y0.94Ca0.06Ba2Cu3O7d single crystals were grown using commercial YSZ crucibles with ultra-pure chemicals. To control the in-plane hole content, these crystals were annealed at different oxygen partial pressures and temperatures. The hole content for these compounds were obtained from room-temperature thermopower [17,25] and Tc values [17,26]. The annealing conditions were as follows: pure Y123, p = 0.123 (annealed in 20%O2 in N2 at 500 °C for 120 h); p = 0.148 (in O2 at 500 °C for 120 h); p = 0.164 (in O2 420 °C for 120 h), 6% Ca–Y123, p = 0.122 (in 2%O2 in N2 at 500 °C for 240 h); p = 0.131 (in 1%O2 in N2 at 450 °C for 240 h); p = 0.149 (in O2 at 500 °C for 240 h); p = 0.187 (in O2 at 420 °C for 240 h). The p-values reported here are accurate within ±0.004. qc(T) was measured by employing the method due to Montgomery [27] and Logan et al. [28]. Details of sample preparation, characterization and anisotropic resistivity measurements can be found in Ref. [17]. For analysis of the resistivity data, the low-T region near Tc was omitted. This is because in this region the resistivity is strongly affected by the pairing fluctuations and Eq. (3) does not include any term to describe paraconductivity. We have used a temperature range of Tc + 20–300 K for the fitting of qc(T). Tc was taken as the zero resistivity temperature. The lower temperature limit was set following some previous studies where it was shown that strong superconducting fluctuations persist till around 20 K above Tc [12,29,30]. Results of fitting of the experimental qc(T) data are shown in Figs. 1 and 2 for pure Y123 and 6% Ca doped Y123 (nominally Y0.94Ca0.06Ba2Cu3O7d), respectively. 6% Ca substitution gave access to the slightly overdoped region (up to p = 0.187). The agreement between the modeled and the experimental qc(T) is excellent. From this analysis we have extracted the values of the coefficients a, b, and the characteristic PG energy (expressed in temperature) as a function of hole content. These values are tabulated in Table 1. We have shown the eg(p)/kB behavior with respective Tc(p) in Fig. 3.

20 p = 0.123 p = 0.148 p = 0.164

15 ρc (μOhm-cm)

PG affects the in-plane and out-of-plane charge dynamics in different ways. In the presence of the PG, qab(T) shows a gradual downward deviation from the high-T T-linear behavior at temperatures much above Tc [11,16] in the underdoped region. The onset temperature of this deviation gives a measure of the average value of the PG energy scale in the underlying EDOS. Whereas PG leads to a much stronger upturn in qc(T) [15,17], starting at certain temperatures significantly above Tc in the underdoped compounds. At higher-T, like qab(T), qc(T) also shows linear behavior [15,17]. As hole content increases, the characteristic PG temperature (obtained both from qab(T) and qc(T)) decreases and extrapolates to zero at p  0.19 [13,17]. At p  0.19, both qab(T) and qc(T) show T-linear behavior over the entire experimental temperature range (except in the vicinity of Tc, where strong superconducting fluctuations induces an accelerating downturn in the resistivity curve). This strange metal T-linear resistivity has been explained as a consequence of a quantum critical (QCP) point in T–p phase diagram of cuprates at p  0.19 [13,14,18]. QCP, although defined by a ground state (T = 0) order parameter, affects various finite temperature properties over a wide region of the T–p electronic phase diagram [19,20] centered on the critical point. In this scenario, above the QCP lies an extended region in the T–p phase diagram where various magnetic and transport properties are dominated by quantum critical fluctuations. In this region, except temperature, all other energy scales are excluded from the charge scattering phenomenon. This can naturally produce a T-linear resistivity above certain characteristic temperature. At lower T, competition between neighboring electronic ground states (e.g., quantum ordered and disordered [20] residing at the two sides of the QCP) produces deviation from the T-linear resistive behavior. Therefore, our phenomenological model for qc(T) consists of two terms with different T-dependences. The first term, qc1, is linear in T and can be expressed as

10

5

0

0

100

200

300

T (K) Fig. 1. Experimental qc(T) for pure Y123 and the respective fits (full black lines). To show the fitted curves clearly only one in ten experimental data points are plotted in this figure. The planar hole contents are given in the figure.

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35

4. Discussion and conclusions p = 0.122

30

p = 0.131 p = 0.149

ρc(μOhm-cm)

25

p = 0.187

20 15 10 5 0

0

100

200

300

T (K) Fig. 2. Experimental qc(T) for 6% Ca substituted Y123 and the respective fits (full black lines). To show the fitted curves clearly only one in ten experimental data points are plotted in this figure. The planar hole contents are given in the figure.

Table 1 Extracted values of a(p), b(p), and eg(p)/kB from the fits to qc(T) data. Hole content (p)

a (lX cm/

b (lX cm)

eg/kB

K)

YBa2Cu3O7d

0.123 0.148 0.164

0.0260 0.0163 0.0150

2.913 0.871 0.0752

266 172 114

Y0.94Ca0.06Ba2Cu3O7d

0.122 0.131 0.149 0.187

0.0261 0.0183 0.0158 0.0082

4.540 4.181 3.800 2.770

282 230 171 12

Sample

(K)

300

250 εg/kB

Tc; εg/kB (K)

200

150

100

T

c

50

0 0.1

0.12

0.14

p

0.16

0.18

0.2

Fig. 3. Tc(p) (j for pure Y123 and d for 6% Ca substituted Y123) and eg(p)/kB (h for pure Y123 and s for 6% Ca substituted Y123). The dashed and the full lines show the trends for Tc(p) and eg(p)/kB, respectively.

In the previous section, we have seen that this simple phenomenological model accounts for the p and T dependent normal state out-of-plane resistivity remarkably well. The nearly linear decrease in the extracted PG energy scale also agrees quite well with previous studies [11–13,23,24]. It should be noted that the eg(p)/kB values obtained here for underdoped compounds are somewhat (20%) larger than those obtained from the analysis of the in-plane resistivity data [11,12]. This is expected as Su et al. [8] suggested that c-axis tunneling should be dominated by the QPs near the anti-nodal regions in the Brillouin zone. The PG energy is maximum at these anti-nodal points. Therefore, the average PG energy obtained from the analysis of qc(T) data is dominated by these large PG regions in the k-space. With increasing hole content, eg(p)/kB decreases and the strange-metal linear region in qc(T) extends to lower temperatures. As PG vanishes around p = 0.19, qc(T) becomes completely linear up to the onset of strong superconducting fluctuations. These experimental features are captured well by the present formalism. It should be also noted that the high-T slope of qc(T), a, decreases with p, this is a direct consequence of increasing hole content in the CuO2 plane and also of the decrease in the PG energy scale. It is seen from Table 1 that the value of a is insensitive to out-of-plane disorder (Ca) for a given value of p. This is not surprising because Ca does not change the PG energy scale at a fixed p [11,12]. On the other hand, the parameter b varies strongly for samples with and without Ca. For both Ca-free and Ca-doped Y123, b decreases with increasing p, but their magnitudes are very different, highly dependent on Ca content. The decrement of b with increasing p can be well-explained by noting that the value of this parameter is inversely proportional to the square of the c-axis tunneling matrix element that increases with increasing hole content (since N0 only varies weakly with hole content [23]). Unlike a, the value of b depends significantly on extrinsic effects. Out-of-plane Ca2+ acts as a strong source of disorder scattering for charge transport along the c-direction and therefore reduces the value of tc irrespective of the value of p. In this study we have analyzed the available qc(T, p) data for up to p = 0.187. Considering the generic T–p phase diagram proposed in Ref. [20], we can predict the qc(T, p) behavior for more overdoped compounds, within the present formalism as follows – we assume that the underlying QCP still affects the normal state charge dynamics for p > 0.19 and the temperature crossover line separates a high-T region (e.g., marginal Fermi-liquid) with linear qc(T) and a low-T quantum disordered (Fermi-liquid like) state where the carrier scattering rate varies as T2. Such a picture can describe the essential features of the in-plane and out-of plane resistivity data reasonably well for the overdoped compounds [11,14,31]. In this situation the resistivity ratio, qc(T, p)/qab(T, p) should become T-independent as was found in experimental studies [31,32]. In summary, we have proposed a simple phenomenological framework based on the presence of a QCP at p  0.19, separating a quantum ordered (a state with broken symmetry) and a Fermiliquid like regions. In this model, the PG present for p < 0.19 induces the strong upturn in qc(T, p). The high-T linear behavior is seen in the strange metal region due to the underlying quantum critical fluctuations. From the analysis of the qc(T, p) data we have extracted the values of eg(p)/kB which shows the same qualitative and quantitative behavior as found in previous studies [23,24]. Acknowledgements One of the authors (J.R. Cole) acknowledges the funding from the Engineering and Physical Sciences Research Council (EPSRC), UK, for his Ph.D. research at the University of Cambridge, UK. S.H. Naqib thanks the AS-ICTP, Trieste, Italy, for the hospitality.

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