İzotropik yapılarda sıfır dış alan için enerjiye bağımlı süperiletken yasak bandı denklemlerinin kritik bölgede çözümü

Abstract

Süper iletken-Normal iletken geçiş sıcaklığı T nin altındaki kritik sıcaklık bölgesi içinde, enerji bağımlı yasak band için integral denklemler varyasyonel bir yaklaşımla çözüldü. Daha i5nc& yapılan birçok çalışmadaki gibi lineer cebrik bir denklem aramak yerine, T için c grafik yöntemlerle çözülmek Üzere transandantal bir bağıntı bulundu. Bulunan kritik sıcaklık bağıntısı kullanılarak izotop etkisi için de bir bağıntı türetildi. Bir grup malzeme için deneysel veriler ve bunlara ait literatürde bulunan parametreler kullanılarak, çalışmamızda elde edilen bağıntılar sayısal olarak test edildi. Ayrıca, süper iletken parametrelerinin birbirleri ile ilişkileri çeşitli grafikler çizilerek incelendi.

The purpose of this study is to obtain solutions for the superconductor - normal conductor transition temperature (critical temperature) T e and the isotope effect in terms of the mean phonon frequency u. the electron-phonon coupling parameter X' and, the fj which represents the average of the repulsive Coulomb interactions between electrons over the Fermi surface. For this, we made use of the results of the previous studies based on the Eliashberg-NaiTibu formalism. Therefore, Te and the isotope effect calculations have been carried out in terms of v o % X' and p, which can be obtained from the data on various experimental methods such as the electron tunneling and the neutron diffraction experiments and also from the knowledge of the specific heat at the normal phase, the Debye temperature and, the Fermi energy. Although there exist a number of studies similar to the present one, this study has two interesting points. The first point is that we have been able to evaluate the strong electron-phonon coupling for which, McMillan's still useful Te formula fails. The second one is that this study simplifizes the method of solution in comparison with Allen-Dynes work, which is rather complicated, but it takes into account the materials where the electron-phonon coupling is strong. In the present study, we have not seeked for linear algebraic equations for Te and the isotope effect, rather we have found it the most convenient to obtain the parameters by graphical methods and of course by using computer. After the discovery of iiigdal that, the usual Feynman Dyson perturbation series method is applicable to the normal metals to an accuracy of the order of the square root of the electron ion mass ratio, and that the same can also be done for superconductors, Eliashberg and Nambu have developed the formalism for the superconducting phase. Electron field operator in the Nambu scheme is given by *'p - -P4, The phonon field operator is: The Hamiltonian of the system is then given as follows t "qX qX - X H = I = W T V' P P 3 P 2>. q,X qX qX qX /. wpp'X *X *>- a *> pp-X - 7T- ) (y t y D Cyt r\c D pip2 p3p4 where e is the Bloch enegy measured from Fermi energy P VI O % g. and V are the bare phonon energies, electron phonon matrix elements and the bare Coulomb interaction respectively. The self energy function Z is defined as 1 X = E G(p,i<* ) G (p,4ö ) where G refers to the Green's function and the indice o stands for the system of noninteracting particles. As can be seen easily, the self energy function represents the contribution of the interactions to noninteracting system at any instant, and thus to the thermodynamic properties of the real system. Am beg o a kar and Tewordt have obtained explicit integral equations for superconducting energy gap which is of interest to us. The phases are chosen so that the t component of the self energy becomes zero. Making use of the expression £(pto>)= Il+Ztpsco) Jo>I + $(pt")T X(p?û>)T two integral equations are obtained which describe the gap and the renormalisation functions. -> -+ -v The gap is defined by A(p, <*>)=$ ( p,d>)/Z( p,o>) Restricting ourselves to the cases where the gap is isotropic, in which the crystalline anisotropy effects are washed out by impurity scattering, the momentum dependence reduces to an average over the Fermi surface and hence to the p dependence. The excitation energy spectrum is important mainly for excitations of energy of the order of the Debye energy. vai By taking into account these restrictions, we give the integral equations for the gap and the renormalisation functions: at '-J A(os) = 1 dw'Re Mos' ) [w'2-AZ(w' )J J X II dl\WFx(V N(vx)+/( -W',jf + | J ^W ' +VX +0S W +1». -U>) N(vx)+/( -.'IN - H N(0)V Z(6> -IV p dw'Re A(w' ) fw'2-A2(w* )1 i/2 tanh fw*/2Tİ' VI 11 Z(ü>)=İ --H CO dw' Re w w ' *-~A"" ( W ) -I J- J x x"x(VFx(V [N(*> ) +f ( -w ' ) w +i>x+o> w +vx-W| N(i>x)+f (w' -w X -w +u ~u> In this study we have carried out the variational method in close similarity with the McMillan's approach. However, our results are more meaningful 1 and the limitations imposed in this study seen to be more realistic. The most important simplicity of our variational procedure is that, we choose for ourself that the boundary for a two component trial function that McMillan has determined in his solution. Our choice is the one given below: A(to) = A 0 < u> < 2 v 1| O D 00 iii > 2 V In section 2 the mathematical tools introduced in solving the gap equations are briefly considered. In section 3 the physical approaches are discussed. In section 4 the gap equation is solved in zero external magnetic field at temperatures close to Tc and below Tc. In the final section our results are presented.