STUDY OF SELF-DIFFUSION IN SEMICONDUCTORS BY STATISTICAL MOMENT METHOD

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 e n Q u a n g Hoc and P h a n Thi T h a n 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

1. I n t r o d u c t io n

The physical p roperties of crystalline solids, like electrical conductivity, atomic diffusivity and m echanical 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 rstitia ls , play an im p o rtan t role in d e te rm in in g the atomic diffusions in c ry stals [2]. It is known th a t th e self-diffusion in close-packed crystals is alm ost completely conducted by the th erm al lattice vacancies. On th e o th er hand, the m echanical p ro p erties of the m aterials, e.g., creep, aging, recrystallization, precipitation h a rd e n in g and irrad iatio n effects (void swelling), are also extensively controlled by atomic diffusions [ 1J. Therefore, it is of g rea t significance to establish a theoretical scheme for tr e a tin g atomic diffusion in cry stallin e solids.

The theory of atomic diffusion in solids h as a long history. In 1905, Einstein used incidental chaotic model for in v estig atin g the diffusion [3]. B ardeen và Hering impoved th is model so as to include the correlation effect [4]. Using the tran sitio n sta te theory [5], Glestom et al. have derived the diffusion coefficient and showed th a t the self-diffusion obeys the A rrh e n iu s’s law. Kikuchi discussed the atomic diffusion in m etals and alloys by applying the p a th probability method [6]. In general th e atomic diffusion have been studied w ith in the fram ew ork of the phenomenological th eo ries a n d based on the simple theory of the th e rm a l lattice vibrations. In th e p re s e n t study, we e stab lish a theoretical schem e to t r e a t the self­

diffusion in sem iconductors ta k in g into account th e an h arm o n ic ity of lattice vibrations. We use th e m om ent method in sta tistica l dynam ics in order to calculate the pre-exponential factor D0 and the activation energy Q for self-diffusion in semicoductor w ith diam ond cubic and zincblende ZnS stru c tu re s . We compare the calculated re s u lts of self-diffusion in sem iconductors w ith th e e x p erim en tal data.

24 Vu Van Hung

,

Nguyen Quang Hoc

,

Phan Thi Thanh Hong 2. T h e o r y o f s e lf - d if f u s io n in s e m e c o n d u c t o r s

In th e case of th e self-diffusion conducted by a vacancy m echanism , it h as been generally assu m e d t h a t the diffusion coefficient D is sim ply given as

D = av exp [- Q/ RT)], Q = gvf + gvm, (1) where a and ^V are th e jum p distance and a tte m p t frequency of the atom, respectively. The activation energy Q of the self-diffusion is th e sum of the changes in the free energy for th e form ation gvf and m igration gvm of th e vacancy.

In th is paper, we investigate the self-diffusion in sem iconductors by using the moment method in sta tis tic a l dynamics. We consider th e self-diffusion via vacancy m echanism and do not tak e into account the contribution from di-vacancies and direct atomic exchange m echanism s. We tak e into account the global lattice expansion originated from th e an h arm o n icity of th e rm a l lasttice vibrations, but do n o t co n sid e r th e d e ta ile d local la ttic e re la x a tio n a ro u n d th e v acan cies. In o rd e r to study the atomic diffusion in semiconductors, one m u st firstly determ in e the equilibrium lattice spacing and the free energy of the perfect crystal because the atomic diffusion occur a t finite te m p e ra tu re s . The calculational procedure for obtaining th erm o d y n am ic q u a n titie s of the perfect cry stals h as been given in our previous stu d ies [7,8]. We th en derive the therm odynam ic q u a n titie s of the crystal containing th e rm a l lattice vacancies, which play a c en tral role in the self-diffusion of semiconductors.

Let us consider a monoatomic crystal consisting of N atom s and n lattice vacancies. By assu m in g N » n th e Gibbs free energy of th e cry stal is given as

G(T, p) = Go(T,p) + n gvf(T,p) - T S C , (2) where T and p denote th e absolute te m p e ra tu re and hydrostatic pressure, respectively. G0(T,p) is th e Gibbs free energy of pefect crystal of N atoms, gvf(T,p) is the change in th e Gibbs free energy due to the form ation of a single vacancy and s c is th e entropy of mixing

c _ 1 1 (N + n)!

Sc = k Bln ~T,' N!n!

where k B denotes th e B oltzm ann constant. It is noted h ere t h a t gvf(T,p) contains contribution from v ib ratio n al entropy of the system.

The equilibrium cocentration of a vacancy n v in sem iconductors can be calculate from th e Gibbs free energy of the system. To obtain the equilibrium concentration n v, we use the m inim ization condition of the free energy with respect to n v u n d e r the condition of c o n stan t p, T and N as

(ỠG/ <9nv)p T N = 0. (4)

Study of self-diffusion in semiconductor by. ²⁵

This m inim ization condition leads to the concentration of th e vacancy as

g fv(T,p)!

n v = exp

e (5)

w ith 0 = k BT. Then, th e Gibbs free energy of the crystal c o n tain in g equilibrium th e rm a l vacancies can be given by

g j = - (n, + n 2) V|I0" + 11,1)/!* + n2i|/2* + Aụ0\ (6) where

V|V = 3{U0/ 6 + 0[x + In (1 - e ■2x)]}, (7a)

u 0 = I (Poi ( lr, I), (7b)

r, = r0 + A r ,. (7c)

H ere, X = ỈKÙỈ 20, CO d e n o te s th e ato m ic v ib ra tio n a l fre q u e n c y , n, a n d n 2 den o te n u m b ers of the first and second nearest-n eig h b o u rs, respectively. i|/0* = \ụ0l N denotes the H elmholtz free energy per single atom in th e perfect c ry stal[6],vị/j and vị/2* re p re s e n t the free energy of the atom s located a t the n e arest-n e ig h b o u r and next n e a re s t-n e ig h b o u r s ite s of th e v acan cy , re sp e c tiv e ly , cpoi is th e in te ra c tio n en erg y betw een zeroth and i-th atom s, r, indicates th e position of th e i-th atom located at the neighbouring sites of the c en tral 0-th atom or the n e a r e s t distance of the i-th atom a t te m p e r a tu re T, r0 d e te rm in e s the n e a re s t distan ce of the i-th atom at te m p e r a tu re OK, Ar, indicates the displacem ent of the i-th atom from the equilibrium position a t te m p e r a tu re T or the th e rm a l expansion depending on te m p e r a tu re of lattice and it is d eterm in ed as in [9]. It m u s t be noted t h a t we take into account the a n h arm o n ic ity of the th erm al lattice v ib ratio n a n d therefore the t e m p e r a tu re d e p en d e n t th e rm a l lattice expansion and v ib ratio n al force co n stan ts are considered.

To calculate the in teractio n energy U() of th e perfect crystal, we use the em pirical p a ir p o ten tials and ta k e into account the c o n trib u tio n s up to the second n earest-n eig h b o u rs. AVỊ/0‘ denotes the change in the H elm holtz free energy of the c en tral atom which c re ate s a vacancy by moving itself to th e c ertain sinks ( e.g., cry stal surface, or to th e core region of the dislocation a n d g rain boundary) in the cry stal

= v|/0*’ - Vị/q* = (B - 1) ụ 0* , (8)

w here v|/0*' denotes th e free energy of the c en tral atom a fte r moving to a c ertain sink sites in the crystal. In th is respect, it is noted th a t the vacancy form ation energies

2 6 Vu Van Hung, Nguyen Quang Hoc, Phan Thi Thanh Hong of the real cry stals are m easu red experim entally as an average value over all those values corresponding to th e possible sink sites.

In th e theoretical analysis, it has been often assu m ed t h a t th e central atom originally located a t th e "vacancy site" moves to the special atom ic sites, i.e., k in k sites on th e surface or in the core region of the edge dislocation in th e bulk c ry sta l which are th erm odynam ically eqivalent to bulk atom s [10]. T his assu m p tio n sim ply leads to B = 1 in the above eq. (8). In the p resen t study, we tak e th e average v alu e for B as

c 1 t - ( l + n, + n 2)vồ + n i\y] + n 2v|/2

~ 2 - 2v|/J

This is eq u iv alen t to the condition th a t the h a lf of th e broken bonds are recovered at the sink sites. We do not tak e into account the lattice relax atio n around a vacancy, because the change in the free energy due to the lattice relaxation is a m inor contribution compared to the form ation en ergy of a vacancy, especially for high te m p e r a tu re region n e ar the m elting te m p e ra tu re .

We now derive th e therm odynam ic q u a n titie s of th e sem iconductor lattice containing th e rm a l vacancies a n d discuss the self-diffusion via vacancy m echanism . From (2), the Gibbs free energy of the semiconductor lattice co n ta in in g th e rm a l vacancies can be w ritte n in the form

G = H - T S , (10)

where s = - (ỠG/ 5T)p is th e entropy and H re p re se n ts the e n th a lp y of the system . Thus, the change in the Gibbs free energy gvf due to th e creation of a vacancy can be w ritten as

g j (T,p) = G(T,P) - Go(T,p) = h vf(T,p) - TSvf(T,p), (11) where h vf and Svf are th e e n th a lp y and entropy of form ation of a vacancy.

The diffusion coefficient D of the sem iconductor lattice can be obtained by assum ing t h a t it is proportional to the vacancy co n cen tratio n n v and the jum p frequency r [2], W hen th e am p litu d e of the atomic vibration e x ceed s certain critical value in the n e a re s t neighbour sites of the vacancy, one can expect t h a t atomic exchange process w ith a vacancy occurs. The n u m b er of ju m p s r per u n it tim e is p ro p o rtio n a l to th e v ib ra tio n a l freq u en cy of th e ato m ^CO a n d th e s q u a re of th e diffusion length a ( or distance of jumping)

r* ~ r^co/ (27i) = (r0 + Ar,)2co/ (2n). (12) The general expression of diffusion coefficient D can th e n be w ritten in the form

Study of self-diffusion in semiconductor by.. 27

D = g r n v a : (13)

where g I S a coefficient which depends on th e cry stallin e stru c tu re and the mechanism of self-diffusion. It is given with the correlation factor f as

g = n , f . (14)

The a tte m p t frequency

r

of the atomic ju m p is proportional to

r*

^{and the}

tran sitio n probability of an atom

r _ I = ——expm

271

/ * \

A ị ị / Ị

0 (15)

The change in th e Gibbs free energy associated with th e exchange of the vacancy with the neighbouring atom s is equal to the inverse sign of A\\I* and

gvm = - A\\)y = - (B' - 1) 1|V. (16) where B' is simply reg ard ed as a num erical factor, which is analogous to the factor B defined for formation energy of the vacancy.

S u m m arizin g eqs. (12)-(16), one can derive th e diffusion c o n sta n t D of sem iconductors via th e vacancy m echanism as

T \ r G5 2

D = n ,i — a exp 2n

, f A * A / K - Av|/

exp TS[

0 (17)

The above formula of th e diffusion coefficient can be re w ritte n as D = D0 exp -

Q

V k„T^B . Q = h[ + h™, Dq = r ijf - ^ - a2 exp

Zn

s (1 8)

where the correlation factor of the self-diffusion f = 0,5 [14] for sem iconductors with diam ond cubic and zincblend s tru c tu re s.a n d the activation energy Q is given by

Q = - (n 1 + n 2) Vo + n iVi* + n2V‘2* + (B - 1) vị/0* - ( B' - 1) y !* + pAV. (19) It is noted here t h a t the contribution from th e entropy of m igration Svm is included in the - (B' - 1)Vị/!* term , and not se p era ted as Svf in eq.(18). On the other h a n d , th e e n tro p y S vf for th e fo rm a tio n of a v acan cy can be g iv en in th e n e x t-n e a re s t neighbour approxim ation as

Si. 1 , , (N + 1)!

---k Bln

n, + n2 N! n, + n 2ln(N +1) (20)

With th e use of eqs.(18)-(20) one can d eterm in e the activation e n e r g y 'Q and th e diffusion coefficients D0 at te m p e ra tu re T and p re ssu re p.

2 8 Vu Van Hung, Nguyen Quang Hoc, Phan Thi Thanh Hong

In the following section we shall use the above res u lts for finding the diffusion coefficient D0, th e activation energy Q for Si and GaAs sem iconductors and compare them with e x p erim en tal data.

3. R e s u lt s o f n u m e r i c a l c a l c u l a t i o n s a n d d i s c u s s i o n s

Recently th e th eo rists developed extensively th e in te rac tio n potentials between atom s in the form of simple model in order to calculate directly the stru c tu ra l and therm o d y n am ic properties for complex system s, especially for semiconductors [11,12], The p a ir potentials like the L en n a rd -Jo n es p o ten tial a n d và the Morse po ten tial have been applied to study the in e rt gas, m etal a n d ion crystal, b u t completely used to the stro n g valence system s like sem iconductors. To study the valence system s it is necessary to use the many-body in te rac tio n p otentials, e.g., the p otentials were p resen ted by Stillinger, W eber [11], Tersoff [12], One of the emperical many-body p o ten tials for Si h as the following form [13]

K J i < j<k

\ 12

- 2

v r ij J

ijk

( 2 1 )

Wljk = G 1 + 3 c o s 0 : COS0:COS9_{1 J}

(rijrjkrki/

This po ten tial firstly is p a ra m eteriz ed for Si. The p a ra m e te r s are fitted in with the cohesive len g h th of dim er and trim er, th e lattice p a r a m e te r and the cohesive energy of th e diam ond stru c tu re . Sam e p o ten tials are expanded for the system s of two com ponents and th re e components like GaAs, SiAs, SiGa. SiGaAs,...

Applying th e many-body p o ten tials (21), we calculate the n e arest-n eig h b o u r interaction and th e n ex t-n earest- neighbour in teractio n and tak e th e interaction energy in sem iconductors as

u,

12 12

v rw

-2A, v rw

0.07811^ 1.375n2G2 ri9 + (v2r1)9

(2 2 )

where A6, A12 are th e s tr u c tu r a l sum s for semiconductor.

The n e a r e s t neighbour distance r^Q) at te m p e r a tu re T = OK is obtained by m inim irizing th e to tal energy of sem iconductor or tak in g derivative

( ổUq/ ổr ) = 0 (23)

and is equal to

Study of self-diffusion in semiconductor by. 29

R,(0) = 2 e A 12r 0¹²

(a! + 4 A 6A 1!£V ' P - A

1/ 3

(24)

A = 0.234G1 + 0.0898G2. (25)

Using the ex p erim e n tal d a ta for Si and GaAs ( Table 1) and the formulae in th e previous section, we obtain the values of the activation energy Q, the p re ­ exponential factor D0 a t various te m o e ra tu re s for Si and GaAs. The num erical calculations are sum m irized in Tables 2 and 3. It is noted t h a t the theoretical resu lts can apply to GaAs tho u g h the analysis is more complex because of non-equal m asses of atoms in the s tr u c tu r e of ZnS type

T able 1: The p a ra m e te r s of the many-body p otential for Si and GaAs [13 ]

Q u a n titie s Si GaAs

eA( eV) 2.817

r0AA(A) 2.295

GAAA(eV.Â9) 3484

eAB(eV) 1.738

roAB (Ả) 2.448

GAAB(eV. Ả 9) 1900

GABB(eV. Ả 9) 4600

T ab le 2: The activation energy Q and the pre-exponential factor D() for Si

T(K) 2 00 400 600 800 1000 1200 1400

Q(kcal/mol) 85.95 87.56 89.21 90.88 92.565 94.25 95.96

D0(10 ’nr/s) 6.18 6.55 6.95 7.24 7.44 7.58 7.72

a(Ả ) 5.1745 5.1840 5.1922 5.2001 5.2078 5.2162 5.2262

T able 3: The activation energy Q and the pre-exponential factor D0 for GaAs

T(K) 2 00 400 600 800 1000 1200

Q(kcal/mol) 57.26 58.81 60.44 62.45 65.47 69.70

Do(l0'1m"/s) 3.83 4.09 4.64 ^{- - -}

a(Ả) 5.5861 5.6002 5.6261 5.6767 5.7748 5.9389

For Si, from e x p erim e n tal d a ta Q = 110 kcal/mol, D0 = 1.8. 10'4m 2/s [15], Q = 107.05 kcal/mol in th e in te rv al from 1128K to 1448K a n d Q = 109.82 kcal/mol in

30 Vu Van Hung

,

Nguyen Quang Hoc

,

Pha n Thi Thanh Hong the in terv al from 1473K to 1673K[16]. Therefore, th e c alcu latio n r e s u lts coincide relatively well with th e e x p erim en t data.

For GaAs, from e x p erim en tal d a ta Q = 59.86 kcal/m ol in th e in terv al from 1298K to 1373K and Q = 128.92 kcal/mol in the in te rv a l from 1398K to 1503KỊ16].

The num erical re s u lts also agree relatively well w ith e x p e rim e n ts . Both the activation energy and th e diffusion coefficient for Si a n d G aA s in c re a se s when the te m p e ra tu re in creases and th is coincides with ex p erim en ts.

This p ap er is finished by the financial sponsorship from th e N ational Basic R esearch P ro g ram m e in N a tu ra l Sciences..

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