Int. J. Che. Sci.: 1(), 1, 1157-1166 ISSN 97-768X www.sadgurupulications.co AN EVALUATION OF OPTICAL PARAMETERS OF HIGH- TEMPERATURE SUPERCONDUCTORS SANJU KUMARI a, SOBHA RANI, S. KUMAR c and LALIT K. MISHRA Departent of Physics, Magadh University, BODH GAYA 8434 (Bihar) INDIA a Departent of Physics, K. S. T. College, Shalpur, SOHSARAI 83118, Biharsarif, Nalanda (Bihar) INDIA Departent of Physics, G. D. Mahila College Harinaut, Biharsarif, NALANDA (Bihar) INDIA c Departnent of Physics, Nalanda College, Biharsarif, NALANDA (Bihar) INDIA ABSTRACT Frequency dependent scattering rate and effective ass of electron were evaluated for two high Tc-superconductors using extended Drude odel at different teperatures. Our theoretically evaluated result for scattering rate is large and effective ass is sall in coparison with experiental data for the given teperatures. However, the trend for oth the scattering rate and effective ass are in agreeent with the experiental data and also with other theoretical workers. Key words: Frequency dependent scattering rate, Effective ass, High Tc-superconductor, Optical paraeters, Dirty liit, K-K relation. INTRODUCTION The optical reflectivity easureent is a powerful proe for the study of the electronic structure of solids. It can provide uch inforation on the conduction ands as well as valence ands of the crystals through inter and transition. It is a interesting prole for Bi-ased cuprates. How the excitations within Bi -O layer affect those within Cu-O layer and how the Optical transitions differ fro those of Y-Ba-Cu-O and La-Sr-Cu-O. The optical reflectivity spectra of single crystal Bi-ased cuprates Bi Sr CaCu O 8+x and Bi Sr CuO 6+X were easured 1 in a wide energy range fro.5 to 4 ev and analyzed through the Kraers-Kronig (KK) relation. The otained spectra are different fro other superconducting cuprates such as (LaSr) CuO 4 and YBa CuO 4 ecause of existence of characteristic BiO layer. The optical excitation starts fro ev or higher energy so it is Author for correspondence; E-ail:
1158 S. Kuari et al.: An Evaluation of Optical Paraeters of. unlikely that the electrons in the BiO layer contriute to low energy excitations or dc conduction in this faily of high Tc superconductors. Recently, there are two works reported on the optical properties of high Tc superconductors HgBa CuO 4 (Hg-11) (Tc = 97 K) and optially doped Bi Sr CaCu O 8 (Tc = 88 K) 3. In superconductor Hg-11 in and out of plane optical spectra was presented. In plane noral incidence reflectivity easureent were perfored on a Fourier transfor spectroeter in the frequency range etween 1-7 c -1 (1-87 ev). Ellipsoetric easureent were also ade in the frequency range etween 6 and 3, c -1 (.75-3.7 ev). This easureent directly gives the real and iaginary parts of dielectric function ε(ω). Reflectivity was calculated fro the pseudo dielectric function. In addition, the c-axis reflectivity R c (ω) was easured on ac plane of different saples fro 3 to, c -1. The c-axis optical conductivity was otained fro Kraers-Kronig variational analysis 4. In this paper, we report the evaluation of optical paraeters naely the frequency dependent scattering rate 1 and effective ass ( ω) of Hg-11 and Bi 1. Using τ(ω) the extended Drude odel 5 and taking the contriution fro inter and transition in the infra red region ε,ir, we have evaluated the aove optical paraeters. Matheatical forulae used in the evaluation We have used extended Drude Lorentz odel 5 for the evaluation of frequency dependent optical paraeters 1 τω ( ) and ( ω ). We have also used a ter ε,ir which is a contriution to the dielectric function in the infrared region arising fro inter-and transitions. This ter has not een taken into account earlier in the single coponent approach. Drude theory was the theory of non-interacting electrons which assues a frequency independent scattering rate [ 1 = constant] is given y τ 1 ω p σω ( ) = (1) 4π 1 iω τ ω p is the are plasa frequency of the free charge carriers. This assuption does not hold in the syste where the charge carriers interact with osonic spectru or where strong correlations are iportant. In order to evaluate physical properties of high Tc superconductors Allen and Mikkelsen 5 extended the Drude odel y including a frequency dependent scattering rate -
Int. J. Che. Sci.: 1(), 1 1159 σ( ω, T ) 1 ωp = 4π 1 ( ω, T ) iω τ( ω, T ) () is the effective ass and is the and ass. 1 ( ω, T and ) oey τ ( ω, T ) Kraers Kroing (KK) relations 4 1 ( ω, T). and are siply related to the real and τ ( ω, T ) iaginary part of 1. One can also express these ters in ters of dielectric function σ ( ω) ε ( ω) = ε1( ω) + iε( ω) where ε 1 (ω) and ε (ω) are the real and iaginary part of the dielectric function, we have 1 ω p ε ( ω) = τ ( ω) ω [ ε ε ( ω)] ε ( ω), IR 1 + (3) ( ω) ω p [ ε ε ( ω)] = ω [ ε ε ( ω)] + ε ( ω) (4), IR 1, IR 1 Where ε,ir is the contriution to the dielectric function in the infrared region arising fro inter-and transition. The choice of ε,ir is not so iportant at low energies where ε1 ε,ir ut ecoes iportant at high energies. One also writes Where ε = ε +Σ S (5a),IR j j S j ω ω p, j, j = (5) These are the oscillator strength of the inter-and transition otained fro Drude Lorentz fit. The contriution to the dielectric function fro the polarizaility of oxygen is calculated using the Clausis-Mossotti relation 7,8. ε,ir α 4π N V α =1+ = 1 + 4π α 1 N 1 γα 3 V (6)
116 S. Kuari et al.: An Evaluation of Optical Paraeters of. Where N is the nuer of oxygen per unit cell. V is the unit volue and αis the 4π Nα polarizaility of the oxygen atos. α =. For high Tc-superconductor HgBa CuO 4+δ V (Tc = 97 K) putting α = 3.88 1-8 c 3 and unit cell paraeter for Hg -11 of a c = 3.85 3.85 9.5 Ǻ and four oxygen ato per unit cell, we find ε,ir = 3.56 and for superconductor Bi Sr CaCu O 8 (Bi 1, Tc = 88K) ε = 4.5. The other iportant quantity is the in-plane reflectivity at low teperature. For frequency ω p, one uses Hagen-Ruen approxiation 9 to descrie the reflectance. 1 τ ω R( ω) 1 ( ),IR 1/ = (7) πσ where σ is the dc conductivity. ε 1 (ω) and ε (ω) are calculated fro K-K relation 1 and xε ( ω) ε1( ω) 1 = P ( ) dx π (8) x ω ε 4πσ ε1( x) ( ω) = p dx π + (9) ( x ω ) ω Equations (8) and (9) cannot e applied directly since oth ε 1 and ε depend on the unknown phase θ of the coplex reflectivity. 1 ε r = = Rexp( iθ ) 1+ ε (1) R(ω) is the noral-incidence reflectivity - ln r( ω) = ln R( ω) + iθ( ω) (11) ω ln Rx ( ) θ ( ω) = P dx+ θ() π (1) ( x ω ) These equations are used to copute θ(ω) fro R(ω). Now, one can restore dielectric function y inverting equation (1) -
Int. J. Che. Sci.: 1(), 1 1161 (1 r) ε = (1 + r) (13) Now, one can also calculate the real part of the dielectric function in the superconducting state 11 y using the forulae ω p 1s 1( ) ε ω = (14) ω Where ω p,s is in-plane super fluid plasa frequency 1. In case of superconductor HgBa CuO 4+δ (Tc = 97 K). ω p,s = 96 ± 4 c -1 (1. ±.5 ev). In case of superconductor Bi Sr CaCu O 8 (Tc = 88 K) ω p,s = 95 c- 1. RESULTS AND DISCUSSION We have evaluated frequency dependent scattering rate τ( 1 ω) using equation (3) at different teperature for two high Tc-superconductor HgBa CuO 4+δ (Tc = 97 K) and Bi Sr CaCu O 8 (Tc = 88K). The results are shown in Tale 1 and along with the experiental data,3,13. Our theoretical results for HgBa CuO 4+δ indicate that the values of τ( 1 ω) is large in coparison to experiental data for all teperatures 5 K, K, 1 K and 5 K etween ω = 1 to 4 c -1. On the other hand the values for Bi Sr CaCu O 8 is lower against the experiental data. In the experiental analysis of the τ( 1 ω) data 13, it was entioned that scattering rate is strongly suppressed for teperatures elow Tc which is indicative of the opening of the gap. ARPES 14 easureents on Hg-11 show a axiu gap value, there is soe uncertainty in this value ecause no quasi-particle peak is oserved around the anti nodal direction 1. Fro the optical easureents, it is difficult to extract the gap, s-wave BCS superconductor in the dirty liit show an onset in the asorption associated with the superconducting gap at Δ. It was oserved that the onset seen in cuprates is shifted due to interaction of the electrons with the agnetic resonance 15. Due to this, one feels that the evaluated results of τ( 1 ω) do not atch with the agnitude of the experiental data. However the trend that τ( 1 ω) increases with ω for a given teperature has een noticed in our calculation for oth superconductors Hg-11 and Bi-1. We have also evaluated frequency dependent effective ass ( ω) at different teperatures for oth superconductors using equation (4). The results are shown in Tale 4 and 5. Our theoretically evaluated results show that up to ω = 6 c -1, ( ω) increases with ω and after that it
116 S. Kuari et al.: An Evaluation of Optical Paraeters of. decreases with ω = 4 c -1. In case of superconductor Hg-11 the agnitude of ( ω) is lower than the experiental data and for superconductor Bi-1 the evaluated results are larger with experiental data 13. However the trend is in agreeent with the experiental data,13. The arguent of this isatch is the sae as we have entioned aove. It has een pointed out that at onset of superconducting gap, K-K relations 4 gives axiu value ( ω). There is a quite uncertainty in the experiental data. In another work J. J. McGuire 16 studied the infra-red dissipation in the a-plane scattering rate. They oserved two separate effects. At high teperature there is a road dispersion of scattering rate elow 1 c -1 and at low teperature a sharp structure is seen which is associated with the scattering fro a ode at 3 c -1 in under doped high Tc-superconductor Bi Sr CaCu O 8. The various paraeters used in the evaluation are shown in Tale 1. Tale 1 ε =.53, ε, = 3.56 (HgBa CuO 4+δ, Tc = 97K) IR ε, IR = 4.5(Bi Sr CaCu O 8, Tc = 88K) ω p,s (in plane super fluid plasa frequency) = 96 ± 4 c -1 (1. ±.65 ev) HgBa CuO 4+δ, Tc = 97K) ω p,s = 95 c -1 (Bi 1) (Tc = 88K). Tale : An evaluated result of frequency dependent scattering rate [τ(ω)] 1 at different teperature for superconductor HgBa CuO 4+δ nuer [τ(ω)] 1 x 1 3 c -1 T = 5 K K 1 K 5 K c -1 Theory Expt. Theory Expt. Theory Expt. Theory Expt. 1.54.439.55.45.453.398.4.377.656.563.64.56.55.456.486.43 4.738.678.78.68.654.63.584.59 6.849.775.8.713.7.698.653.64 8.967.859.943.886.896.754.73.695 Cont
Int. J. Che. Sci.: 1(), 1 1163 nuer [τ(ω)] 1 x 1 3 c -1 T = 5 K K 1 K 5 K c -1 Theory Expt. Theory Expt. Theory Expt. Theory Expt. 1 1.18 1.98 1.16.986.954.863.853.78 15.74 1.863 1.963 1.55 1.133.955.936.845.365.59.14 1.354 1.586 1.14 1.139.957 3.967.353.759 1.798.54 1.863 1.386 1.116 4 3.547.958 3.18.154.846.58 1.754 1.458 Tale 3: An evaluated result of frequency dependent scattering rate [τ(ω)] 1 at different teperature for superconductor Bi Sr CaCu O 8 (1) Tc = 88 K nuer [τ(ω)] 1 x 1 3 c -1 T = 3 K K 15 K 5 K c -1 Theory Expt. Theory Expt. Theory Expt. Theory Expt. 1.183.4.154.16.98.17.55.67.196.3.167.174.115.11.74.88 4.6.36.188.193.18.137.83.97 5.17.47.195.5.139.144.94.15 6..55.4.14.146.15.16.11 8.38.63.15..155.167.118.16 1.46.7.4.37.167.178.1.134 15.87.9.49.56.199.7.144.155 17.35.316.56.67.15.6.158.166.346.35.64.77.4.37.167.174
1164 S. Kuari et al.: An Evaluation of Optical Paraeters of. (ω ) Tale 4: An evaluated result of frequency dependent effective ass at different teperature for superconductor HgBa CuO 4+δ (Tc = 97 K) ) (ω nuer c -1 T = 5 K K 1 K Theory Expt. Theory Expt. Theory Expt. 1 3. 3.5.53.66.46.54 3.86 3.97.95 3.3.58.68 4 4.5 4.7 3.84 3.64.67.77 6 5.5 5.8 4.5 4.7.86.94 8 4.3 4.54 4.8 4.34.48.53 1 3.75 3.86 3.86 3.9.34.39 15 3. 3.57 3.67 3.84.1.6 3.15 3.3 3.48 3.56.16. 5.86.97 3. 3.37.1.15 3.73.86.89.9.8.1 35.67.74.75.84.5.9 4.58.64.65.7..4 (ω ) Tale 5: An evaluated result of frequency dependent effective ass at different teperature for superconductor Bi Sr CaCu O 8 (1) Tc = 88 K (ω ) nuer c -1 T = 3 K K 15 K Theory Expt. Theory Expt. Theory Expt. 1.1.8 3.46 3.55 3.68 3.34.34.14 3.37 3.48 3.6 3.7 4.67.18 3.3 3.4 3.5 3.15
Int. J. Che. Sci.: 1(), 1 1165 ) (ω nuer T = 3 K K 15 K c -1 Theory Expt. Theory Expt. Theory Expt. 5.86.3 3. 3.36 3.47 3.9 6.9.9 3.15 3. 3.33 3. 8.94.34.9 3.5 3.17.97 1.99.38.75.95 3.9.84 1 3.4.4.64.86.95.73 15 3.7.56.53.74.87.65 17 3.9.67.47.58.7.54 3.1.75.38.49.63.44 35 3.15.79.45.86.4.1 4 3.3.98.54.36.44.9 REFERENCE Cont 1. I. Terasaki, S. Tajia, H. Eisaki, H. Takagi, K. Uchinokura and S. Uchida, Phys. Rev., B41, 865 (199).. E. Van Heuen, R. Lorentz, A. B. Kuzenko, F. Carone, D. Vander Marel, X. Zhao, G. Yu, Y. Cho, N. Barisie, M. Greven, C. C. Hoes and S. V. Dordevic Phys. Rev., B75, 545 (5). 3. A. B. Kuzenko, H. J. A. Molegraaf, F. Carone and D. Vander Marel, Phys. Rev., B75,14453 (5). 4. A. B. Kuzenko, Rev. Sci. Instru., 76, 8318 (5). 5. J. W. Allen and J. C. Mikkelsen, Phys. Rev. B15, 95 91977). 6. D. L. Cox and N. Grewe, Z. Phys., B71, 31 (1988). 7. A. El Azarak, R. Nahou, N. Bonteps, M. Guilloux-Viry and H. Raffy, Phys. Rev., B49, 9846 (1994).
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