The complex fluctuation conductivity in high-tc superconductor at zero magnetic field

Nội dung text: The complex fluctuation conductivity in high-tc superconductor at zero magnetic field

1. 152 TRƯỜNG ĐẠI HỌC THỦ ĐÔ HÀ NỘI THE COMPLEX FLUCTUATION CONDUCTIVITY IN HIGH-TC SUPERCONDUCTOR AT ZERO MAGNETIC FIELD Bui Duc Tinh1, Pham Gia Hien, Le Minh Thu, Nguyen Quang Hoc Hanoi National University of Education Abstract: The time-dependent Ginzburg-Landau (TDGL) equation with thermal noise is used to calculate the complex fluctuation conductivity in high-Tc superconductor in three dimensional (3D) model at zero magnetic field. The nonlinear interaction term in the TDGL is treated within self-consistent Gaussian approximation we go beyond the often used lowest Landau level approximation. The expressions of the complex fluctuation conductivity including all Landau levels is presented in explicit form which is applicable essentially to both temperature above and below Tc. Our results are in good agreement with experimental data on high- Tc superconductor YBa2Cu3O7-δ . Keywords: Complex conductivity; High-Tc superconductors; Ginzburg-Landau equation 1. INTRODUCTION The investigation of thermal fluctuation on complex conductivity in strongly high-Tc superconductors (HTSC) have been a subject for active research for many years, mainly due to discovery of the HTSC in which the effects are enhanced by the short coherence length and the anisotropy and high-Tc. The fluctuation conductivity was calculated by Aslamasov and Larkin in the framework of the microscopic (BCS) theory [1], the calculation fast becomes too cumbersome in more complicated situations involving external magnetic field, layered structure etc and a more phenomenological Ginzburg- Landau approach is more effective. In the Gaussian fluctuations regime (ignoring the nonlinear interaction term in the TDGL for above the mean-field transition temperature), the formulas for the superconducting complex fluctuation conductivity in the normal phase at zero magnetic field have been very early obtained within the TDGL equation [2]. 1 Nhận bài ngày 20.12.2015, gửi phản biện và duyệt đăng ngày 25.01.2016. Liên hệ tác giả: Bùi Đức Tĩnh; Email: bdtinh@hnue.edu.vn.
2. TẠP CHÍ KHOA HỌC SỐ 2/2016 153 In this paper we will calculate the complex fluctuation conductivity including all Landau levels in a 3D superconductor at zero magnetic field by using TDGL approach with thermal fluctuations conveniently modeled by the Langevin white noise. The nonlinear interaction term in the TDGL is treated in self-consistent Gaussian approximation. A main contribution of our paper is an explicit form of the Green function incorporating all Landau levels. One of the main result of our work is that analytically explicit expression for the complex fluctuation conductivity at zero magnetic field is applicable essentially to both above and below the mean-field transition temperature (unlike Ref. [1]). The result is compared with experimental data [4] on YBa2Cu3O7-δ (YBCO). 2. THEORY 2.1. The Ginzburg - Landau Model in 3D We can start with the Ginzburg - Landau (GL) free energy in 3D at zero magnetic field: 22 hh2 2 2b 4 F d3 r    a   , (1) GL * z 2mm 2c 2 mf mf mf We assume linear dependence a = αTc (t −1), t ≡ T/Tc . The “mean field” critical mf temperature Tc depends on UV cutoff, τc, of the “mesoscopic” or “phenomenological” GL description, specified later. Effective Cooper pair mass in the ab plane is m* (disregarding for simplicity the anisotropy between the crystallographic a and b axes) while along the c 2 2 * axis it is much larger mc. The two scales, the coherence length ξ =  /(2 m αTc); and the 2 2 * *2 penetration depth,  c m b'/(4 e Tc ) define the GL ratio    / , which is very large for HTSC. The electric current, J=Jns J , includes both the Ohmic normal part: JEnn  , (2) and the supercurrent: * ie h JD-Ds *     . (3) 2m In order to study transport phenomena in superconductor, one uses the time dependent Ginzburg-Landau (TDGL) equation [5,6]: * 1  e  FGL 0 i   *  , (4) h
3. 154 TRƯỜNG ĐẠI HỌC THỦ ĐÔ HÀ NỘI  1 where 0 is the order parameter relaxation time [7],  E()y is the scalar electric potential describing the driving force in a purely dissipative dynamics. The electric field is coordinate independent but is a monochromatic periodic function of time E( ) Eexp i . Throughout most of the paper we use the coherence length ξ as a unit of length and 2 Hc2 0 / 2  as a unit of the magnetic field. In analogy to the coherence length and the penetration depth, one can define a characteristic time scale. In the superconducting phase * 1 2 2 a typical “relaxation” time is GL m 0 / h . It is convenient to use the following unit of the electric field and the dimensionless field: EGL Hc2 / c GL , ()()/  EE  GL . The TDGL Eq. (4) written in dimensionless units reads: 2 2 2 mf 1  1  1 t 1 2 2 2 2 iyε( )  . (5)  2 x 2  y 2  z 2 22mf The order parameter field and the thermal noise were rescaled:  2 Tbc / ' , mf 3/2 1/2 mf  (2 Tbc ) / . The “mean field” critical temperature Tc depends on UV cutoff. This temperature is higher than measured critical temperature Tc due to strong thermal fluctuations on the mesoscopic scale, and it will be renormalized later. The Langevin white-noise forces  are correlated through *(r,  )  (r',  ') 2 tmf  (r-r')  (   ') 1 2 2mf 2 2 2 with  2Gi , where the Ginzburg number is defined by Gi 8/ e  Tc c h 2 2 * with  m / mc being an anisotropy parameter. The dimensionless current density is Js JGL js where i js ( D   D  ), (6) 2 2 with JGL cH2 c / (2 ) being the unit of the current density. Consistently the 22 conductivity will be given in units of GL J GL/ E GL c  ' / (4  ) . This unit is close to the normal state conductivity n in dirty limit superconductors [8]. In general there is a factor k of order one relating the two: σn = kσGL. 2.2. The self-consistent Gaussian approximation
4. TẠP CHÍ KHOA HỌC SỐ 2/2016 155 The cubic term in the TDGL Eq. (5) will be treated in the self-consistent Gaussian 2 approximation [9] by replacing 2 with a linear one 2 1 2 1  2 1  2 2 2 2 ar  ,  2 x 2  y 2  z (7) leading the “renormalized” value of the coefficient of the linear term: mf t 1 2 a 2 . (8) r 2 The relaxational linearized TDGL equation with a Langevin noise, Eq. (7), is solved 0 0 using the retarded (G for τ<τ’) Green function (GF) G k z (r,;r',') : dk (,)r  z e ikz z d r ' d  ' G0 (,;',')(,) r  r   r  . (9) 2 kkzz The GF satisfies: 222 11   kz 0 22 aGrk(r , r ';  ')  ( r r ')  (   ') . (10)  2 xy 2  2 z The solution of Eq. (10) is: 2 22 0 1 kz XY Gakr(rr , '; '')  (  '')exp  ''  '' , (11) z 2  2 2 '' with X x x', Y y y ', ''   '.  (  '')is the Heaviside step function. The thermal average of the superfluid density in 3D (density of Cooper pairs) can be expressed via the Green functions. mf 2 2 t 1  (,)2r   tmf G0 (,';'')' r r  d  2 a . krz 2 3/2  '' c c (12) The critical temperature Tc is defined as 2' TTmf 1, (13) cc 3 c 1 2 2 2 2 2 where  ' 2Gi ' with Gi' 8 e  Tc / c h . 2 Then Eq. (8) can be solved for ar :
5. 156 TRƯỜNG ĐẠI HỌC THỦ ĐÔ HÀ NỘI 1  '2t 2 2  ' t  ' 2 t 2 a t 1 t 1 , (14) r 22 2 2 4 mf where t T/ Tc and note that tt ' . 3. THE COMPLEX CONDUCTIVITY AT ZERO MAGNETIC FIELD 3.1. Theoretical calculation We can express the supercurrent density, defined by Eq. (6), via the Green functions as following: dk  js ()(,,)(,,)., i  tz e ikz z dr'''' d  G r r''   G r r   c c (15) y 2 kzzy k where G (r,r', -') as the Green function of the linearized TDGL Eq. (5) in the presence kz of the scalar potential. One finds correction to the Green function to linear order in the electric field: 0 0 G (r,r', '') G (r,r', '') i dr d 'G (r,r , ' ) (τ1) yG1k (r 1 ,r' , ' 2 ), (16) kz k z 1 1 k z 1 1 z where (τ1) are the scalar electric potential and electric field in dimensionless units respectively, 1'  1 , and  2 ' 1  '. Substituting the full Green function (16) into expression (15), then doing the Fourier transform this current with respect to frequency one then obtains complex conductivity: j() ()()()    i   , (17) s () 12 Where: 3/4 't 2 2 22 3 1(  ) 2  2aarr 4  cos arctan , (18) 8  3 2 2a  r 3/4 't 222 3 2 (  ) 2  2aarr  4  sin arctan . (19) 8  3 2 2a  r The result (18) and (19) are consistent with the results of Larkin and Varlamov [1] if quartic term, |Ψ|4, in GL free energy is neglected for for above the mean-field transition temperature. Our results are applicable essentially to both above and below the mean-field transition temperature. 3.2. Comparison with experiment
6. TẠP CHÍ KHOA HỌC SỐ 2/2016 157 We compare our results with the experimental results of M. S. Grbic, et al. [4] at  / 2 = 15.15GHz on an underdoped YBa2Cu3O7-δ (YBCO) single crystal with Tc = 87K. The comparison is presented in Fig.1 in which the real part and the imaginary part of the complex conductivity curves were fitted simultaneously to Eq. (18) and Eq. (19) with the 6 -1 normal-state conductivity measured in Ref. [4] to be σn = 1.25 × 10 (Ωm) . The parameters we obtain from the fit are: H (0) T dH (T)/ dT | 186T (corresponding to c2 c c2 Tc o ξ = 13.3 A ), the GL parameter  = 48.7, and the factor k= σn /σGL=0.62, where we take γ= 7.8 for optimally doped YBCO in Ref. [11]. Using those parameters, we obtain G’i = 1.62×10-3 (corresponding to ’ = 0.177). Note that the real part and the imaginary part of the complex conductivity were fitted to Eq. (18) and Eq. (19) with using the same fitting parameters. Figure 1. Points are the real part and the imaginary part of the complex conductivity of an underdoped YBCO in Ref. [4] in zero magnetic field. The solid lines are the theoretical values calculated from Eq. (18) and Eq. (19) with fitting parameters (see text). 4. CONCLUSION Time dependent Ginzburg-Landau equations with thermal noise describing the thermal fluctuations is used to investigate the complex conductivity in HTSC in 3D at zero magnetic field in the presence of strong thermal fluctuations on the mesoscopic scale in linear response. We use the self-consistent Gaussian approximation to treat the nonlinear interaction term in TDGL. Therefore the analytically explicit expression for the complex conductivity including all Landau levels we obtain is valid for both above and below the mean-field transition temperature. The real part and imaginary part of the conductivity are in good qualitative and even quantitative agreement with experimental data on YBCO.
7. 158 TRƯỜNG ĐẠI HỌC THỦ ĐÔ HÀ NỘI REFERENCES 1. A.Larkin and A.Varlamov, Theory of uctuations in superconductors, (Clarendon Press, Oxford, 2005). 2. H.Schmidt, Z.Phys. 216 (1968) 336. 3. D.P.Li, B.Rosenstein and V.Vinokur, J. of Superconductivity and Novel Magnetism 19 (2006) 369. 4. M. S. Grbic, M.Pozek, D. Paar, V.Hinkov, M.Raichle, D.Haug, B.Keimer, N.Barisic, and A. Dulcic, Phys. Rev. B 83 144508 (2011). 5. J. B. Ketterson and S. N. Song, Superconductivity (Cambridge University Press, Cambridge, 1999). 6. B. Rosenstein and V. Zhuravlev, Phys. Rev. B 76 (2007) 014507. 7. R. J.Troy and A. T. Dorsey, Phys. Rev. B 47 (1993) 2715. 8. N. Kopnin, Vortices in Type-II Superconductors: Structure and Dynamics (Oxford University Press, Oxford, 2001). 9. B. D. Tinh, D. Li and B. Rosenstein, Phys. Rev. B 81 (2010) 224521. 10. D. Li and B. Rosenstein, Phys. Rev. B 65 (2002) 220504(R). 11. B. D. Tinh and B. Rosenstein, Phys. Rev. B 79 (2009) 024518. ĐỘ DẪN ĐIỆN XOAY CHIỀU CỦA VẬT LIỆU SIÊU DẪN NHIỆT ĐỘ CAO KHI KHÔNG CÓ TỪ TRƯỜNG NGOÀI Tóm tắt: Chúng tôi sử dụng phương trình Ginzburg-Landau phụ thuộc thời gian có kể đến thăng giáng nhiệt để tính toán độ dẫn điện xoay chiều của vật liệu siêu dẫn nhiệt độ cao trong mô hình ba chiều khi không có từ trường ngoài. Số hạng tương tác trong phương trình được tuyến tính hóa bằng phương pháp gần đúng Gaussian và chúng tôi tính đóng góp của các mức Landau cao hơn vào độ dẫn điện. Chúng tôi thu được biểu thức giải tích chính xác độ dẫn điện xoay chiều bao gồm tất cả các mức Landau, biểu thức này áp dụng cho nhiệt độ trên và dưới nhiệt độ tới hạn Tc. Kết quả của chúng tôi phù hợp tốt với số liệu thực nghiệm của vật liệu siêu dẫn nhiệt độ cao YBa2Cu3O7-5. Từ khóa: Độ dẫn điện xoay chiều; Siêu dẫn nhiệt độ cao; Phương trình Ginzburg- Landau.