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Specific Heat Jump in Anisotropic Yba2cu307−Δ Superconductor

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Specific Heat Jump in Anisotropic Yba2cu307−Δ Superconductor Indian Journal of Pure & Applied Physics Vol. 45, December 2007, pp. 991-993 Specific heat jump in anisotropic YBa 2Cu 307−δ superconductor K M Khanna a, M S Karap Kirui b, T W Sakwa c, P K Torongey a, K Y Ayodo c & S Rotich a aDepartment of Physics, Moi University, P O Box 3900, Eldoret, Kenya bDepartment of Physics, Egerton University, P O Box 001,Egerton, Kenya cDepartment of Physical Sciences, Masinde Muliro University, P O Box 190, Kakamega, Kenya Received 13 November 2006; revised 22 August 2007; accepted 12 September 2007 Using the exotic pairing model, the jump in the specific heat ∆C/Tc in YBa 2Cu 307−δ, for both breathing mode and buckling mode has been calculated. Its value was compared with the experimental values and it has been found that the specific heat jump in superconductor and the density of states were one fifth of the earlier reported values. This is indicative of the small fraction of carriers close to the Fermi level, which are paired and exotic pairing due to anharmonic perturbation of the apical oxygen ions leads to a lowering of ∆C/Tc. Keywords : Specific heat, Anisotropic, YBa 2Cu 307−δ superconductor, Density of states IPC Code : C04B10/00 1 Introduction Exotic pairing is assumed 3 to be contributing to the YBa 2Cu 3O7−δ, 0 ≤ δ ≤0.1, is a superconductor that high value of the transition temperatures in has perovskite-like orthorhombic structure. It is made YBa 2Cu 3O7−δ. Specifically it was found that up of two corrugated CuO 2 planes separated by a anharmonic perturbation of phonons with quadratic layer of Yttrium atoms and intercalated with two BaO temperature dependence significantly increases the layers and one CuO layer. It is assumed that this transition temperature. Furthermore 4, the shape of the compound is moderately anisotropic with anisotropy specific heat graph was found to be typical of the parameter 1 of 5 . Superconductivity of this material superconducting state and gave support to the exotic and hence the transition temperature is strongly pairing, due to anharmonic perturbation, as dependent on the content and ordering of oxygen contributing to the electron-phonon coupling. The deficiencies.Although there is no clear theory for the graph was found to consist of two sections. First, an microscopic understanding of high-Tc superconduc- exponential term for 0 ≤ T ≤ 0 .7T c, which is indicative tivity, it is believed that the existence of several of an existence of the energy gap in the electronic equivalent positions of oxygen atoms causes a strong energy levels. Secondly, a linear term for 0.7 Tc<T<T c. anharmonic perturbation, which can increase the Therefore, the properties of the specific heat electron-phonon coupling 2,3 leading to increase in discontinuities and specific heat coefficient and their transition temperature Tc. dependence on Tc may be essential to the Important properties of a superconducting material understanding of the nature of superconductivity in high Tc ceramic oxides. In normal metals, the low include the specific heat discontinuity ∆C at Tc due to temperature behaviour of the specific heat is CV = aT the second-order transition of the normal state to 3 3 superconducting state and the electronic specific heat + bT , where bT term is associated with the Debye coefficient γ. This specific heat coefficient is vibrations of lattice. At very low temperatures, the aT proportional to the density of electronic states of the linear term dominates, and this term arises from Fermi surface and is one of the three parameters the kinetic energy of the heat motions of the electron gas. In the superconducting state, the phonon specific (others being the effective attraction coupling 3 −∆/T constant and the energy shell around the Fermi heat Cph ∝ T and the electron specific heat Ces ∝ e surface) which specify the interactions of the and this vanishes exponentially in the limit of very low temperatures. Thus, the specific heat Cs in the electrons and thus, determine Tc. Understanding of 3 2 these electronic properties may lead to a clue to high superconducting state is Cs= [3 γ T/ T c ] . The Tc mechanism. electronic contribution to specific heat varies 992 INDIAN J PURE & APPL PHYS, VOL 45, DECEMBER 2007 exponentially and approaches zero at very low 3 Results temperatures. As a rule at very low temperature, The value of the specific heat coefficient has been phonon contribution will be negligible in the calculated with the aid of the specific heat data 3 using superconducting state, and the specific heat will be Eq. (3). due to electronic motion. We should get the known value of Ces /Cen as C es /Cen =2.43 at T=T c where Ces is (a) Breathing mode electronic specific heat in the superconducting state Kirui 3 worked out the value of the specific heat for and Cen is the electron contribution in the normal this mode as: state. In this paper, the specific heat coefficient and the step in the specific heat capacity are calculated C= e−1160/ T (1.06372 ×0−21 + 1.834 ×10 -24 T together with related parameters. −8.3868 ×10 −27 T2 −2.892 ×10 −29 T3) … (5) 2 Theory The specific heat at the critical temperature ( Cmax ) In the normal state the specific heat is composed of was calculated and found to be 5.38 ×10 -24 JK −1. the lattice contribution and the electronic contribution Hence, the transition temperature ( Tc) has been easily and is given by: worked out to be 258.5K. A graph of specific heat ( C) 3 3 against temperature cubed ( T ) for 220< T<233 is Cn = c 1T + c 2T ...(1) obviously a straight line whose gradient is 4×10 −31 JK −4. Using Eq. (3), γ = 8.91 ×10 −27 JK −2 = 5.36 At very low temperatures lattice contribution and JK −2mol −1. Since the electronic specific heat is related the electronic contribution are negligible and the to the free carriers in the CuO plane and there are expression is reduced to: 2 three Cu atoms in YBa Cu30 , then: 2 7−δ C = γT …(2) n æmJ ö γ ç ÷ … (6) = 1.79 ç 2 ÷ where γ is the specific heat coefficient (Sommerfeld èçK. g- at ø÷ gamma). Hence, the density of states 6 is given by : In the superconducting state æ ö æstates ÷ ö ç mJ ÷ 3 v(0)ç ÷= 0.414 γç ÷ 3γT èç ø÷ ç 2 ÷ C = , 0.7 Tc<T<Tc eV.atomè K . g- at ø s T 2 c æstates ö The specific heat can be assumed to have the form = 0.74 ç ÷ …(7) èçeV .Cu atom ø÷ - ∆ C µ exp because of the existence of an energy s kT At the transition temperature ( Tc), the specific heat gap in the temperature range 0 ≤T≤0.7 Tc. in the normal state ( Cn) is: - ∆ −24 −1 C µ exp , 0< T<0.7 Tc …(3) Cn = γTc = 2.30 ×10 JK …(8) s kT Hence, at the phase transition temperature Tc, the where ∆ represents the gap in the allowed energy specific heat jump is given by: states and k is the Boltzmann constant. The jump in 5 the specific heat capacity at the transition ∆C C- C temperature is given by = S N =1.19 ×10 −26 JK −2 = 7.164mJK −2/mol T T C C ¶S ¶ 2 G …(9) ∆CCCT=- = ∆ =- T ∆ …(4) and s n ¶T ¶ T 2 ∆c where G is the Gibbs free energy density and S is the = 1.33 …(10) entropy. γTc KHANNA et al .: ANISOTROPIC YBa 2Cu 307−δ SUPERCONDUCTOR 993 (b) Buckling mode concluded that this model describes an ideal Again, from Ref. (3), the specific heat under this superconductor that has no impurities. mode has been derived as: In the breathing mode, the density of states is 0.74 −580/ T −21 −23 (states/eV.Cu atom) and the specific heat jump is C = e (4.2514 ×10 + 1.466 ×10 T 2 ∆C/T = 7.16 mJ/mol K while the buckling mode, the −2.697 ×10 −25 T2−1.86 ×10 −27 T3) …(11) c density of state is 1.56 (states/eV Cu atom) and the 2 Again, the transition temperature(Tc) has been specific heat jump is ∆C/Tc = 11.38 mJ/molK . These computed to be 96.8K while the specific heat at values are about one fifth of the reported values 7,8 . critical temperature ( Cmax ) is found to be The small value of the specific heat discontinuity 3.64 ×10 −24 JK −1. Thus, the graph of C versus T3 is indicates that in the type of pairing model assumed, again linear and similarly,we have calculated the the density of the electronic states of the Fermi level value of γ to be 1.87 ×10 −26 JK −2 = 11.28mJK −2/mol. for this compound is low which means that only a very small fraction of carriers close to the Fermi level For Cu atom are paired. It is argued, therefore, that the formation of exotic pairs, due to the anharmonic perturbation is æmJ ö state due to a small number of free carriers in the CuO γ =3.76ç ÷ Þv (0) = 1.56 2 èçK2. g- at ø÷ eV .Cu atom plane. 9 …(12) It has been pointed out that the quasi particle excitation spectrum has two branches depending on At the critical temperature, the specific heat in the the spin orientation. At the transition, it is only the normal state is: spin-up Fermi surface that open up the superconducting gap and this reduces the specific heat −24 −1 Cn = γTc = 1.81 ×10 JK …(13) jump from the BCS value of 1.43.
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