VNU JOURNAL OF SCIENCE, Mathematics - Physics, T xx, N01 - 2004 STUDY OF SELF-DIFFUSION IN SEMICONDUCTORS BY STATISTICAL MOMENT METHOD Vu Van H ung, N g u y en Q uan g Hoc and P h a n Thi T h an h H ong Hanoi Un iversity o f Education Abstracts Using the statistical moment method, self-diffusion in semiconductors is studied including the anharmonic effects of lattice vibrations The interaction energies between atoms in semiconductors are estimated by applying many-body potential The activation energy Q and pre-exponential factor D{) of the self-diffusion coefficient are given in closed forms The values of Q and D0 are calculated for Si and GaAs at high temperature region near the melting temperatures and they shown to be in good agreement with the experimental data I n t r o d u c tio n The physical properties of crystalline solids, like electrical conductivity, atomic diffusivity and mechanical stre n g th are generally influenced quite significantly by the presence of lattice defects [1] The point defects like the vacancies and in te rstitials, play an im portant role in determ ining the atomic diffusions in crystals [2] It is known th a t the self-diffusion in close-packed crystals is alm ost completely conducted by the therm al lattice vacancies On the other hand, the mechanical properties of the m aterials, e.g., creep, aging, recrystallization, precipitation h a rd e n in g and irradiation effects (void swelling), are also extensively controlled by atomic diffusions [ 1J Therefore, it is of great significance to establish a theoretical scheme for tre a tin g atomic diffusion in crystalline solids The theory of atomic diffusion in solids has a long history In 1905, Einstein used incidental chaotic model for investigating the diffusion [3] Bardeen Hering impoved this model so as to include the correlation effect [4] Using the transition state theory [5], Glestom et al have derived the diffusion coefficient and showed th a t the self-diffusion obeys the A rrhen ius’s law Kikuchi discussed the atomic diffusion in m etals and alloys by applying the path probability method [6 ] In general the atomic diffusion have been studied w ithin the fram ework of the phenomenological theories and based on the simple theory of the th erm al lattice vibrations In the p resen t study, we establish a theoretical scheme to tr e a t the self­ diffusion in sem iconductors tak in g into account the anharm onicity of lattice vibrations We use the m om ent method in statistical dynam ics in order to calculate the pre-exponential factor D and the activation energy Q for self-diffusion in semicoductor with diam ond cubic and zincblende ZnS stru c tu re s We compare the calculated resu lts of self-diffusion in semiconductors with the experim ental data 24 Vu Van Hung , Nguyen Quang Hoc , Phan Thi Thanh Hong T h e o r y o f s e lf-d iffu s io n in s e m e c o n d u c t o r s In the case of the self-diffusion conducted by a vacancy mechanism, it has been generally assum ed th a t the diffusion coefficient D is simply given as D = av exp [- Q/ RT)], Q = gvf + gvm, (1) where a and V are the jum p distance and a tte m p t frequency of the atom, respectively The activation energy Q of the self-diffusion is the sum of the changes in the free energy for the form ation gvf and m igration gvmof the vacancy In this paper, we investigate the self-diffusion in sem iconductors by using the moment method in sta tistica l dynamics We consider the self-diffusion via vacancy mechanism and not take into account the contribution from di-vacancies and direct atomic exchange m echanism s We take into account the global lattice expansion originated from the anharm onicity of th erm al lasttice vibrations, but not co n sid er th e d e ta ile d local la ttic e re la x a tio n aro u n d th e vacancies In o rd e r to study the atomic diffusion in semiconductors, one m ust firstly determ ine the equilibrium lattice spacing and the free energy of the perfect crystal because the atomic diffusion occur at finite tem p eratu res The calculational procedure for obtaining therm odynam ic q u an tities of the perfect crystals has been given in our previous studies [7,8] We then derive the therm odynam ic q u an tities of the crystal containing th erm al lattice vacancies, which play a central role in the self-diffusion of semiconductors Let us consider a monoatomic crystal consisting of N atom s and n lattice vacancies By assum ing N » n the Gibbs free energy of the crystal is given as G(T, p) = Go(T,p) + n gvf(T,p) - T SC, (2 ) where T and p denote the absolute tem p e tu re and hydrostatic pressure, respectively G 0(T,p) is the Gibbs free energy of pefect crystal of N atoms, gvf(T,p) is the change in the Gibbs free energy due to the form ation of a single vacancy and s c is the entropy of mixing c _ 1 (N + n)! Sc = k Bln ~T,' N!n! where k B denotes th e B oltzm ann constant It is noted here th a t gvf(T,p) contains contribution from vibrational entropy of the system The equilibrium cocentration of a vacancy n v in semiconductors can be calculate from the Gibbs free energy of the system To obtain the equilibrium concentration nv, we use the m inim ization condition of the free energy with respect to nv und er the condition of constant p, T and N as (ỠG/