Diamagnetic susceptibility of cubic SiC

Volume 115, numbea" 4

PHYSICS LETTERS A

7 April 1986

DIAMAGNETIC SUSCEPTIBILITY OF CUBIC SiC Trinath SAHU Department of Physics, Berhampur University, Berhampur 760007, Orissa, India Received 10 January 1986; accepted for publication 4 February 1986

The diamagnetic susceptibility (X) of cubic SiC is calculated by a model recently developed by us, which consists of using a linear combination of hybrids. Our expression for X, which is derived using a general expression for the X of intrinsic semiconductors, is independent of the choice of origin and free from any scaling parameter, unlike earlier theories.

In recent years, there have been several attempts [1-4] to develop chemical-bond theories of magnetic susceptibility (X) of tetrahedrally bonded semiconductors. Hudgens, Kastner and Fritz.sche [1 ] (HKF) first showed that chemical bonding and diamagnetism of tetrahedral semiconductors are related in a simple way. Sukhatme and Wolff [2] (SW) and Chadi, White and Harrison [4] (CWH) have independently developed chemical-bond models for the × of tetrahedral semiconductors. However, in the SW theory there is large disagreement between the theoretical and experimental results. In the CWH theory, arbitrary scale factors have been used to fit with the experimental results. Further, the complete problem of the magnetic susceptibility of solids [5-7] has not been considered in these theories. In fact, recent experiments [8] on the temperature dependence of X of tetrahedral semiconductors indicate that there are substantial deficiences in these theories [4]. Recently we have developed a model [9,1()] for tetrahedral semiconductors which consists of a linear combination of hybrids. We have constructed a basis set for the valence bands which is a linear combination of sp 3 hybrids forming a bond in which their relative phase factor, heretofore neglected, has been properly included, We have also constructed a basis set for the conduction bands which are orthogonal to the valence band functions. We have constructed Wannier functions [11 ] from our Bloch functions and have shown that the bond orbitals used in earlier theories [3,4] are not a proper choice for the Wannier functions of the valence band. We have derived [9,10] a general expression for X of intrinsic semiconductors using a finite temperature Green function formalism and have shown that terms of the same order have been missed in the earlier theories [3,4] of X. We have used our basis states in our general expression for X and have calculated X of group IV elemental [9] and HI-V compound [12] semiconductors. Our results have agreed fairly well with the experimental results [1]. We note that our expression for × is origin independent and is free from any scaling parameter, unlike earlier theories [4]. Furthermore our expression for the van Vleck type terms (Xp) is proportional to the overlap integral (S) and tends to zero in the no bonding limit. It may be noted that since S and hence Xp decreases with increase of disorder, it is possible to explain diamagnetic enhancement in amorphous semiconductors by suitably extending our formulation to the disordered case. In this paper we calculate the magnetic susceptibility of IV-IV compound semiconductor SiC using our chemical bond formalism. We note that both C and Si are group IV elements and the bonding force in these elemental semiconductors is purely covalent. However, due to asymetric mixture of orbitals, the bonding force in SiC is partially covalent and partially ionic.We consider a zinc blende structure where each Si atom is surrounded tetrahedrally by four C atoms and vice versa. The primitive cell contains two basic atoms. At each site i, we construct four sp 3 hybrids hC(r -Ri) pointing from the atom C to the nearest-neighbour atom Si along the direction/(/= l, ..., 4) and four ~ther sp 3 hybrids h~i(r - R i -d/) pointing from these nearest neighbours to the atom C. R i is a lattice 0.375-9601/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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PHYSICS LETTERS A

7 April 1986

vector for site i and d/is a nearest-neighbour vector. We construct Bloch-like tight binding sums for the valence band basis functions by taking linear combination of hybrids forming a bond

~;(r,k) = ~ N N / ~ C~(k)exp(ik "Ri) ~,/(r, k),

(1)

where C/V(k) = [1 + X2 + 2~S cos(k • d])] -1/2,

~bVj(r,k) = hC(r - R i ) + hh~(r - R i -d/) exp(ik • d]).

(2)

S is the overlap integral and X accounts for the partial ionic character of SiC via Coulson's ionicity parameter fc = (1 -$2) 1/2 (1 -X2)/(1 + X2 + 2~g). The basis functions for the conduction bands are obtained by constructing functions orthogonal to ~7(r, k): ~C(r'k) =~N-N ~1. CT(k)exp(ik " R i ) ~ , ( r , k ) ,

(3)

where

(

X+Sexp(-ik.d/) 11/2 C~/(k) = (1 - $ 2 ) ( 1 + X2 +2LS cosk.dl)(X+Sexp(ik 'd]))]

(4)

and ¢~?,](r,k) = [X + S exp(ik "d/)] hC(r - R i ) - [XS + exp(ik .d/)] hSi(r - R i -d]).

(5)

We construct the Bloch eigen functions for the valence (n) and conduction (m) bands as

XPn(r,k)= ~. ctV]n(k)~l(r,k), 1

~m(r,k)= ~. C~n(k)~l(r,k ). 1

(6,7)

~'s are elements of 4 × 4 matrices which satisfy the unitary properties ~n c~ (k)o~+(k) = 5M, and T,m Ct~n(k)Ct~m/,(k) = 8//,. We use the Bloch functions (eqs. (6), (7)) to evaluate the matrix elements in the expression for × of intrinsic semiconductors recently derived by us [10], which can be written as

x = (2hc~ah-t~/B2) -

[-(~i2/rn)Qn~mQmn86#+Qn~mZmn ' nm Zmn +P~nmQmn, Q~,Qm,n

O+a 107 +~ Q~. p~._, Q~,. - Qnm +a p#mm , p'l, ~nm mn' 0+~ ~n m' Qm'n6 + 2Qnm ............... m m ,,Q6m "n/Era'n

+ 2Pnn'Q~mP'rmm' Qm'n 6 [E m'n' + pann,o+~ p~mm ,Q~m'n/E:mn_] = XI+X2 "'" +X8. -.'nm

(8)

Here

Qmn = -

fUmV~Un dr,

h o=eag,./h'r ,

h=eB[2hc,

Emn = E m k - E n k ,

and Pnm ffi are the matrix elements of the velocity operator between o/n and ~m" Un and Um are periodic part of the Bloch function, (n, n', ...) and (m, m', ...) denote the valence and conduction bands and repeated indices imply summation. We have used the completeness relation to express the matrix elements in terms of valence band functions and the HaU-Weaire approximation [ 13,14] to calculate the matrix elements between different hybrids. Furthermore, we make an average-energy-gap ansatz and obtain ×1 + ×2 + ×5 = ×1 + Xv 2' where 174

(9)

Volume 115, number 4

PHYSICS LETTERS A

7 April 1986

Xv 1 = _ (4e2/mc 2) [al/((X2)c + ~20C2)Si) + 2 ~a2/Oc(x- d7))12 ] ,

(10)

X~v = - ( 4 e 2 / m c 2) {X[-Xa3/+ a4/((x2-xy) C + ?~2(x2 - xY)Si) + 2Xa5/Oc(x - d; - y + dY))12 ] (11)

+ [a6/(M C - X2Msi) + a7/(P C - X2Psi)] }. Further, _1+2 )(6 + X7 + )(8 - Xp Xp,

X3 + )(4 = O,

(12)

where

Xlp = [2e2~i2S2/m2c2Eg(1 - $2)] [(1 + X2)2 A1/f - 4X2A2M,] ,

(13)

X~ = [2e2-h2S/m2c2Eg(1 - $2)] {.4 3/]' + [A4#'(20¢)12 - S d / ~ ) (14)

+ A S#S(1 - X2 ) - A 6#,(Mc - MSi ) + A 7#,(MC - )~2Msi) + A 8#,(Pc - X2Psi)] ) .

Here/is chosen in the (111) direction,/' ~ ] , (0)~ = (hT(r)lOlhT(r)) (a = C, Si), (0)12 = (hC(r)lOIh~(r - d / ) ) , M a = / ! ,e~ = ~ Id/dxl Pxa> and al/' a21 , ... and A 1#,s A 2# t .... are functions of the above parameters involving summation over k. sa and Pxa are atomic orbitals and Eg is the average-energy-gap. We note that X1 and X1 are the results which can be obtained by using our basis functions in the Langevin and van Vleck t e r ~ of W ~ e [3]. X~vand X2 are additional terms obtained by us. We further note that our expressions for xlv, X2, Xp 1 and X~pare independent of the choice of origin and free from any scaling parameters. We have calculated X of cubic SiC using Hartree-Fock orbitals from Clementi's table [15]. We have used a spheroidal transformation technique [16] to calculate the two-centered integrals. The coefficients al/, a2/ ... and A 1#', A 2#' "" are evaluated by carrying out the k integration over a sphere of volume equal to that of the Brillouin zone. Unfortunately, to our knowledge, experimental results are not available for X of cubic SiC to compare with our theoretical results. However, as noted by Bailly and Manca [17], we have compared our results with the geometrical mean of the values of X of Si and C. We note that there is excellent agreement between our results and their predictions (X = - 8 . 6 9 X 10 -6 cm3/mole). In order to know the accuracy of our present results we have further compared them with those of C and Si obtained in previous calculations [9]. In table 1, we present the value of d, ?~,Eg and S together with the results of various components of X of SiC, C and Si. We note that ionicity plays an important role in the results of various components of X of SiC. Due to the specific dependence of ~ on difference terms, the value of Xlv for SiC is nearly equal to that of C. However, in X2, the first group of terms are proportional to ?~. In the homopolar limit, the first term of the first group predominates over the other two terms and the second group of terms vanishes, Therefore, X2 is weakly paramagnetic for C and Si whereas in case of SiC it is diamagnetic. Further X~ and X~ are proportional to S/Eg. Since the values o r s for C, Si and SiC are nearly equal Table 1 Diamagnetic susceptibility of cubic SiC (× in 10 -6 cruZ/mole). 2 SiC C

Si

1.88

0.404

1.55 2.35

1.0 1.0

a) ref. [18].

b)

ref. [8].

7.94

12.2 4.4

0.627

0.646 0.67

-2.5 -0.3

-22.9 -19.2

-4.6

-42.9

-5.1

1.7 3.5

c)

12.7

9.1

-8.7

4.4 14.0

3.5 22.5

-9.9 -7.5

-11.8 - 6.4

C) ref. [1]. 175

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and the second group o f terms in Xp 2 vanish in the homopolar limit, the results o f Xlp and X~ essentially depend upon the values o f Eg and )~. Thus our comparison reveals that the present results follow a similar trend for tetrahedrally b o n d e d semiconductors [10]. It is interesting to compare our results for the X o f cubic SiC with experimental results, when available.

References [1 ] S. Hudgens, M. Kastner and H. Fritsehe, Phys. Rev. Lett. 33 (1974) 1552. [2] V.P. Sukhatme and P.A. Wolff, Phys. Rev. Lett. 35 (1975) 1369. [3] RAI. White, Phys. Rev. B 10 (1974) 3426. [4] D.J. Chadi, R.M. White and W.A. Harrison, Phys. Rev. Lett. 35 (1975) 1372. [5] E.I. Blount, Phys. Rev. 126 (1962) 1636. [6] L.M. Roth, J. Phys. Chem. Solids 23 (1962) 433. [7] P.K. Misra and L. Kleinman, Phys. Rev. B 5 (1972) 4581. [8] R.M. Candea, SJ. Hudgens and M. Kastner, Phys. Rev. B 18 (1978) 2733. [9] T. Sahu and P.K. Mista, Phys. Left. A 85 (1981) 165. [10] T. Sahu and P.K. Mista, Phys. Rev. B 26 (1982) 6795. [11 ] R.N. Nucho, J.G. Ramos and P.A. Wolff, Phys. Rev. B 17 (1978) 1843. [12] T. Sahu and P.K. Misra, Phys. Lett. A 87 (1982) 475. [13] G.G. Hall, Philos. Mag. 43 (1952) 3381; 3 (1958) 429. [14] D. Weaire and M.F. Thorpe, Phys. Rev. B 4 (1971) 2508. [ 15 ] E. Clementi, Table of atomic functions (IBM, San Jos6, 1965). [16] J.A. Pople and D.L. Beveridge, Approximate molecular orbital theory (McGraw-Hill, New York, 1970). [17 ] F. Bailly and P. Manca, in: Chemical bond in solids, Vol. 1, ed. N.N. Sirota (Plenum, New York, 1972) p. 30. [18] W.A. Harrison and S. Clraei, Phys. Rev. B 10 (1974) 1516.

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