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In the present paper, the linear Eliashberg equation is converted into a new integral equation and from it, the following Tc formula is derived analytically in the weak coupling limit:, where ln γ=C=0.5772 is the Euler constant.

The energy levels of the ideal divacancy states in band gap for more than 10 cubic semiconductors are invesgated by using the basic equations given in ref. [1] and the tight binding Hamiltonian given by vogl et al. A simple physical model is used to describe the basic physics of the divacancy problem.

By using Koster-Slater Green's function method and on-site defect potential approximation, a systematic method of A1 symmetric deep level wavefunction calculation of substitutional defect pairs is worked out. The method is applied to the spectator-oxygen pairs in GaP successfully. The Zn-O and Cd-O pair wavefunctions are obtained for the first time. The wavefunction is mainly concentrated on four bonds which locate at four nearest neighbour sites of oxygen center and point toward the oxygen. Nevertheless, the four bonds are not in the equal weight. The component of the bond which links spectator with oxygen is reduced in comparison with other three and it is more reduced for more electropositive spectator. The rest part of the wavefunction extends diversely over a wide range of space.

A complete set of equations for determining the energy levels and the wavefunc-tions with different symmetry of electronic states of the ideal divacancy in cubic semiconductors is obtained using the Koster-Slater Green's function method. Using an empirical tight binding Hamiltonian for the band structure of the host and the on-site pertubation approximation for the defect, only the on-site and the off-site Green's functions of the host and their derivatives with respect to energy E are involved in these equations.

The present letter shows that the Tc-λ curve is composed of three parts with entirely different properties, namely, "McMillan region", "λ1.55 region" and "series solution region".

A systematic investigation and numerical results of calculation of T2 symmetric deep level wave function induced by short range defect potential in Si are described, based on a recently developement on site defect potential Green's function method. [5,6] Such a complete information of T2 symmetric wave functions in Si is presented for the first time. The occupation probability P1 of the wave function located around four nearest neighbour sites of the defect center has a peak exceeding 50%. This part of wave function could be described by T2 symmetric combination of four hybrid orbital quasi dangling bonds located at four nearest neighbour sites and pointing toward the defect center. The total occupation probability of wave function located on the 0,1,2 shells arround the defect center is about 70%. The rest part of the wave function extends diversely over a wide range of space. The characteristics of the wave function are insensitive to the defect energy over most part of the energy range in the gap. Only when the defect energy level approaches very closely to the band edge of either conduction (Ec) or valence (Ev) band, the aforementioned peak P1 disappears and the wave function extends smoothly all over the space. A T2 symmetric deep level due to ideal vacancy is found at 0.51 eV up to the Ev.

In this paper, an example is given, through which we have arrived at the following conclusion: the convergence radius of the first order approximate T c series solutionfor μ * =0 is determined neither by nor by z ph =-∫ 0 (ω ph ) dωg(ω)dω(ω 2 )/(ω ph 2 -ω 2 ) nor by z ph =∫ 0 (ω ph ) dωg(ω)dω(ω 2 )/(ω ph 2 +ω 2 ).

By using numerical method, the superconducting critical temperature Tc is cal-culated from the Eliashberg equation and the dependence of Tc on λ and the shape of effective phonon spectrum is investigated. In this study, α2F(ω) is taken as the double delta-function spectrum and the spectrum parameters are permited to change over a wide range. It is found with surprising that in the regime of λc series solution, Tc depends, not only on λ and ratios of moments but also on Λ, the reciprocal of the convergence radius of the Tc series solution, and the relation between them shows a certain regularity. Under the light of these results, an approximate Tc formula for the case of μ*= 0 is constructed in the regime of λ 0, or Λ is greater than λ0 but λ is smaller than λ0. Here, λ0 is a parameter depending on the shape of spectrum and its definition is given in this paper. For other type A.superconductors (λ0c is determined is not λ, but δ=1/∧0.5(ω1/2/ωlog)5.5λ1.55.

A simple method in the framework of Koster-Slater Green's function calculation and central cell defect potential (CGDP) approximation is used to calculate the distribution of deep level wavefunctions in Bloch space for Si, GaAs and GaP for the first time. Convergence properties are discussed. It is shown that for definite number of energy bands in the host crystal model, the convergence of deep level wavefunction calculations is faster than that of deep energy calculations.

The wavefunctions of the Eu, state and the Eg state are evaluated for the ideal divacancy in Si, using the basic equations given in [1] and the tight binding Hamiltonian given by P. Vogl et al. Comparison between the culculated results and the ESR and ENDOR spectra of V2+ and V2- in Si is discussed.

Based on the Eliashberg equation, a formula for calculating the convergence radius Λ-1 of the Tc series solution is proved. Practically, it is identical with that which was conjectured in a previous paper by one of the authors and his co-workers. Therefore, the method for calculating Λ suggested by them is justified.

The wavefunctions of the A1 and E states of the ideal divacancy in GaAs, GaP are calculated using the basic equations given in ref. [1] and the tight-binding Hamiltonian given by P. Vogl et al. It is shown that the largest part of the wavefunctions of divacancy states is on the three nearest Ga atoms of the As or P vacancy, the rest part of the wavefunction decreases slowly and non-monotonically as the distance from the defect increases.

Based on the OS model, the Harris singularity of the Josephson current of a junction which is driven into non-equilibrium, is investigated.