Electron–phonon coupling constant of cuprate based high temperature superconductors

Solid State Communications 142 (2007) 587–590 www.elsevier.com/locate/ssc

Electron–phonon coupling constant of cuprate based high temperature superconductors R. Abd-Shukor ∗ School of Applied Physics, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia Received 29 September 2006; received in revised form 18 January 2007; accepted 7 April 2007 by D.D. Sarma Available online 19 April 2007

Abstract The electron–phonon coupling constant in two-dimensional cuprate high temperature superconductors has been determined by the ultrasonic method. The electron–phonon coupling constant in the Van Hove scenario λVH was found to increase with transition temperature Tc . λVH is in the range of 0.025–0.060 which is 10–100 times smaller than the conventional three-dimensional Bardeen–Cooper–Schrieffer coupling constant. The characteristic Debye temperature θ D does not correlate with Tc . These findings show that the interplay between the Debye frequency and electron–phonon coupling in the two-dimensional system and their variations have a combined effect in governing the transition temperature. c 2007 Elsevier Ltd. All rights reserved.

PACS: 63.20.Kr; 74.20.Fg; 74.25.Ld; 74.72.-h Keywords: A. Cuprate superconductors; D. Electron–phonon coupling; D. Debye temperature; D. Van Hove scenario

1. Introduction The origin of high temperature superconductivity in the cuprates continues to be a major topic in materials research since the discovery of this class of superconductors many years ago. The possible important role of phonons in the pairing mechanism in the materials has regained attention in the past few years following a number of experimental and theoretical evidences [1–3]. The two-dimensional aspect of the cuprates together with the Van Hove scenario and dwave pairing are found to be viable for high temperature superconductivity [4]. Interestingly, the possibility of high temperature superconductivity in two-dimensional systems with Van Hove singularity in the density of states had been proposed immediately after the discovery of high temperature superconductivity in the cuprates [5]. The Van Hove singularity in the density of states has been observed experimentally in the p-type HTSC superconductors such as through angle resolved photoemission experiments in YBa2 Cu3 O7−δ and YBa2 Cu4 O8 [6,7], as well as from scanning ∗ Tel.: +60 3 89215904; fax: +60 3 89213777.

E-mail address: [email protected]. c 2007 Elsevier Ltd. All rights reserved. 0038-1098/$ - see front matter doi:10.1016/j.ssc.2007.04.012

tunneling microscopy in Bi2 Sr2 CaCu2 O8−δ [8,9]. Various calculations in HgBa2 CaCu2 Od and HgBa2 Ca2 Cu3 Od [10] and also in the n-type Pr2−x Cex CuO4 superconductor [11] also showed such features. Although there is also evidence to the contrary, for example in the La2−x Srx CuO4−δ superconductor [12], this concept has been used successfully in the search of new superconducting materials, for example in the antimonide-sulfide, quite recently [13]. Some reports on the isotope effect have shown a null result. However, the null isotope effect in these materials has also been shown to be a natural consequence of the singularity in the density of states near the Fermi energy [14]. The various normal state properties of the superconducting cuprates such as the pseudogap have also been explained using the singularity in the density of states [15]. In this paper, we report on the electron–phonon coupling constant of various cuprate high temperature superconductors determined indirectly from sound velocity measurements. Acoustic phonon anomalies have been observed near Tc in the cuprate based high temperature superconductors (for example [16–19]). Hence, is it interesting to investigate further the role of acoustic phonons in the mechanism of these materials.

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By combining the two-dimensional nature of the structure together with the electron–phonon coupling mechanism, we found that the electron–phonon coupling is 10–100 times smaller than the conventional superconductors. This is observed in all the samples studied covering a broad range of materials and transition temperatures. According to the BCS theory, in the weak coupling limit, the transition temperature Tc and Debye temperature (θ D ) can be related using the equation Tc = 1.13θ D e−1/λ where λ ≡ λ(ω) = N (0)V, ω is the phonon frequency, N (0) is the density of states at Fermi level and V is the scattering matrix. In a twodimensional system where the Van Hove scenario is applicable, in the weak coupling limit (λ  1), Tc can be written as √ λ) (−1/ where TF = Ek BF (E F is the Fermi Tc = 2.72TF e energy) [20]. The characteristic ratio of TF /θ D is given as 10 for high Tc materials and 1000 for low Tc materials [20]. In the high Tc materials, for example the La–Sr–Ca–Cu–O system, TF /θ D ranges from 7.4 to 13.9 and in the Y–Ba–Cu–O system TF /θ D can be close to 11.7 [21]. Although some variation in TF /θ D ratio between the high Tc materials is expected, the ratio of TF /θ D = 10 is in the correct order of magnitude. Near the Van Hove singularity, the density of states diverges and even arbitrarily weak interactions can produce a large effect [22]. Hence in principle, a very small electron–phonon coupling is sufficient for the formation of the Cooper pairs. In this paper, however we show that a very small coupling constant (10–100 times smaller than the BCS type) emerges naturally from the Van Hove scenario and is sufficient for the formation of the Cooper pairs. 2. Experimental details Samples were prepared via solid-state reaction using highpurity (>99.9%) metal oxides and carbonates. Samples of about 12.5 mm diameter and about 2 mm thickness were sintered in air at various temperatures followed by annealing in oxygen and various gases. The sound velocity was measured using a Matec 7700 system which utilizes the pulse-echo-overlap technique. The samples were bonded to a transducer using Nonaq stopcock grease. The ultrasonic waves were propagated along the direction of pressing using X -cut (longitudinal) or Y -cut (shear) quartz transducers at 5–10 MHz. The velocity and measurement was performed in an Oxford Instruments liquid nitrogen cryostat model DN 1711. The absolute longitudinal and shear velocities were evaluated at 80 K. The temperature-dependent electrical resistance measurements were carried out using the four-point probe technique with silver-paint contacts. Powder XRD analyses with CuKα radiation using a Siemens D5000 diffractometer were used to confirm that the samples are single phased. The sound velocity in an ideal void-free material can be estimated q from the measured velocity using the relation, v I = v M ρρth , where v I is the void-free velocity, ρth is the theoretical density and v M is the measured velocity. The Debye temperature, θ D can be estimated by using the standard

formula, θ D =

h k




3N 1/3 vm , where ( 4π V)

3 3 vm

=

1 vl3

+

2 ,h vs3

is the

Planck constant, k is the Boltzmann constant, N is the number of mass point, V is the atomic volume and vm is the mean velocity. 3. Results and discussion The Debye temperature and electron–phonon coupling constant estimated in the weak coupling limit of the BCS theory as well as the Van Hove scenario are shown in Table 1. The Tc and θ D in Table 1 are from our previous reports on ultrasonic studies on high temperature superconductors (for example, [23– 25]). The uncertainty in θ D is about 2.5% of the absolute value. Tc and θ D do not show a linear relationship when comparing between samples of different compositions. For example the Tl2 Ba2 Ca2 Cu3 O10 with the highest Tc showed the lowest θ D . However, for samples of the same composition except for the oxygen content, for example ErBa2 Cu3 O6.9 and ErBa2 Cu3 O6.3 , the superconducting sample (O6.9 ) showed a higher θ D . This is also true for the EuBa2 Cu3 O6.98 and EuBa2 Cu3 O6.9 samples. In such cases higher oxygen content (higher Tc ) shows a higher θ D . In the Zn doped GdBaSrCu3 O7−δ samples, the disruption of the Cu spins by non-magnetic Zn resulted in a decrease in Tc although an increase in θ D was observed. Zn doped samples also showed a lowering of the electron–phonon coupling constant both in the BCS and the Van Hove scenario. The electron–phonon coupling constant calculated using the Table 1 Transition temperature Tc (K), Debye temperature θ D (K), BCS electron– phonon coupling λBCS and Van Hove electron–phonon coupling λVH constant of various copper oxide based high temperature superconductors Samples

Tc (K)

θ D (K)

λBCS

λVH

EuBa2 Cu3 O6.98 ErBa2 Cu3 O6.9 (Er0.9 Pr0.1 )Ba2 Cu3 O6.9 (Er0.8 Pr0.2 )Ba2 Cu3 O6.9 ErBa2 Cu2.99 Zn0.01 O6.9 ErBa2 Cu2.95 Zn0.05 O6.9 ErBa2 Cu3 O6.3 GdBaSrCu3 O7−δ GdBaSr(Cu2.99 Zn0.01 )O7−δ GdBaSr(Cu2.97 Zn0.03 )O7−δ GdBaSr(Cu2.94 Zn0.06 )O7−δ GdBaSr(Cu2.9 Zn0.1 )O7−δ DyBaSrCu3 O7−δ (Dy0.9 Pr0.1 )BaSrCu3 O7−δ (Dy0.8 Pr0.2 )BaSrCu3 O7−δ (Dy0.6 Pr0.4 )BaSrCu3 O7−δ (Dy0.3 Pr0.7 )BaSrCu3 O7−δ TlSr2 (Ca0.7 Y0.3 )Cu2 O7−δ TlSr2 (Ca0.5 Y0.5 )Cu2 O7−δ TlSr2 (Sr0.7 Y0.3 )Cu2 O7−δ TlSr2 (Sr0.5 Y0.5 )Cu2 O7−δ TlSr2 (Ca0.5 Pr0.5 )Cu2 O7−δ Tl2 Ba2 Ca2 Cu3 O10 RuSr2 GdCu2 O8 Pr2 CuO4 Pr2−x Cex CuO4

90 94 89 77 86 78 NS 87 84 82 73 NS 82 75 59 28 NS 71 73 81 87 90 123 40 NS 21

457 375 387 428 398 388 351 385 420 449 440 452 464 400 374 402 434 400 396 433 454 342 268 528 408 406

0.57 0.66 0.63 0.54 0.60 0.58 – 0.62 0.58 0.55 0.52 – 0.54 0.56 0.51 0.36 – 0.54 0.55 0.56 0.56 0.69 1.11 0.28 – 0.32

0.041 0.046 0.044 0.040 0.043 0.042 – 0.044 0.041 0.040 0.038 – 0.039 0.040 0.038 0.028 – 0.039 0.040 0.040 0.041 0.046 0.060 0.029 – 0.025

NS: non-superconducting.

R. Abd-Shukor / Solid State Communications 142 (2007) 587–590

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Fig. 1. BCS electron–phonon coupling constant versus transition temperature. Solid line is to guide the eye.

Fig. 2. Van Hove electron–phonon coupling constant versus transition temperature. Solid line is to guide the eye.

weak coupling limit, λBCS , is between 0.28 and 1.11 as shown in Table 1 and plotted versus Tc in Fig. 1. The position of the Van Hove singularity with respect to the Fermi level depends on the doping level. In addition a linear resistance versus temperature relation is evidence that the Fermi surface is near the Van Hove singularity [26]. All the samples state behavior listed in Table 1 showed the metallic normal √ where the equation Tc = 2.72TF e(−1/ λ) was employed to calculate the Van Hove electron–phonon coupling constant. The Van Hove scenario electron–phonon coupling constant, λVH , is between 0.025 and 0.060 which are proportional to the transition temperature of the samples studied (Fig. 2) and is much closer to the value of various electron–phonon coupling constants found in the Bi based and RBa2 Cu3 O7−δ superconductors (λ is centered at around 0.04) from Raman scattering data [27–29]. The λVH seems to agree well with other experimental data and should be relevant to superconductivity in the cuprates. If electron–phonon coupling is playing a role in the Cooper pair formation in the cuprate superconductors, then this small momentum transfer is important for the pair formation. Fig. 2 showed a slightly better correlation between Tc and electron–phonon coupling constant in the Van Hove scenario compared to the standard BCS theory as shown in Fig. 1. According to the conventional theory there are two ways that govern the electron–phonon coupling. Usually the coupling constant is determined by the density of states given as λ = N (0)V , with V reasonably constant. In the doped HTSC, this can be interpreted as the variation in the density of states near the Fermi level. The Van Hove singularity moves closer or further away from the Fermi level depending on doping. The coupling constant in the BCS–McMillan scheme can also be related to the characteristic phonon frequency ω, C with λ ∼ Mhωi , where C is a constant and M the ionic mass. In this case an elastically softer material will have a stronger electron–phonon coupling. The two expressions for

λ, however, are related because materials with high density of states are elastically softer [30]. However, in the cuprate HTSC, a decrease in the electron–phonon coupling constant does not necessarily result in lower Tc . The corresponding changes in the characteristic phonon frequency may instigate suppression of the transition temperature and vice versa. The variation in the characteristic phonon (Debye) frequency and the electron–phonon coupling constant has a combined effect on Tc as in the case of other types of two-dimensional superconductors [31]. Detailed studies on the phonon and oxygen vibration modes in the CuO2 planes have been widely reported [32,33]. It is interesting to note that the electron–phonon coupling constant for the breathing mode is λ = 0.02 [32], and is of the same order of magnitude with the λVH values determined in this work. Although the electron–phonon coupling constant in the Van Hove scenario is small, i.e. λVH  1, it may be very important in the formation of the Cooper pairs. The density of states diverges near the Fermi level and even very weak electron–phonon interactions can result in an unexpectedly large effect. A more direct method is necessary to probe the role of electron–phonon coupling in these materials. However, from this study, we suggest that the Van Hove scenario could be playing an important role in the mechanism of superconductivity in the cuprates. Acknowledgement This research has been supported by the Academy of Sciences, Malaysia under SAGA grant P07. References [1] A. Lanzara, P.V. Bogdanov, X.J. Zhou, S.A. Kellar, D.L. Feng, E.D. Lu, T. Yoshida, H. Eisaki, A. Fujimori, K. Kishio, J.-I. Shimoyama, T. Noda, S. Uchida, Z. Hussain, Z.-X. Shen, Nature 412 (2001) 510.

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