Impurity size and heavy doping effects on the energy gap of silicon at 0 K

Solid State Communications, Vol. 58, No. 8, pp. 5 5 1 - 5 5 4 , 1986. Printed in Great Britain.

0 0 3 8 - 1 0 9 8 / 8 6 $3.00 + .00 Pergamon Journals Ltd.

IMPURITY SIZE AND HEAVY DOPING EFFECTS ON THE ENERGY GAP OF SILICON AT 0 K H. Van Cong and S. Brunet Laboratoire de Physique Appliqu6e, D6partement de Physique, Universit6 de Perpignan, Av. de Villeneuve, F-66025 Perpignan, France

(Received 12 October 1985: hi revised form 7 January 1986 by F. Bassani) The change in energy gap of degenerate silicon is expressed in terms of the impurity radius ri, the total impurity concentration N I and the type of crystal (n or p). It is applied to As, P, Sb, B, A1 or Ga impurity-silicon systems and also written as: AEg(rl, Nx) = AEu(rz) + AEg ~.(ND, where the first term represents the impurity size elfect and the second represents the rrdependence of the heavy doping effect. It is shown that this dependence only occurs in B-Si system. The present results are also compared with some other theories and experiments. 1. INTRODUCTION WHEN THE ENERGY GAP of degenerate silicon (DS) is defined as: Eg(r~, Nx) = E ° --AE~(rt, NI), its change yields:

--AEg(rI, N,) = --AEu, N~(rl)--AEg, r~(Nx).

(1)

Here, E~ = 1.166eV is the energy gap o f silicon at OK, NI the total impurity concentration (identical to N a and N a respectively for n- and p-type DS), rl the impurity radius in tetrahedral covalent bonds, AEu, NI(rl) the change in uperturbed band edge (Ec° or E ° ) due to an impurity-size effect depending on Nx, and finally &Eg, rt(N1) the change in energy gap produced by a heavy doping effect depending on rx. One further notes that, when rl = rsi and NI is sufficiently high, AEu, NI(rI) = 0 [1], and therefore &E~(rl = rsi, A~) = z~Eg, rx=rsi(Ni), which is replaced by &Eg(Nt), for simplicity [2, 3]. In the present paper, basing on our previous results [ 1 - 3 ] obtained in two limiting cases such as: AEg(rl, Nz ~ 0) and AEg(rl = rsi, N1), we will calculate/XEu. Nl(r~) and /XEg,rl(Nx) for n- and p-type DS, and also show that, when Nt is ranged i n [ 1 0 1 9 c m - 3 , 3 x 102° c m "-3 ] , AEu, NI(rl) is reduced to AEu(rl). 2. CALCULATION OF AEu, NI(rz) Since r z is usually either larger (or smaller) than rsi = 1.17A [4], a local mechanical strain is induced. The compression (or dilatation), which corresponds to a repulsive (or attractive) force acting on each nearest surrounding Si-atom, increases (or reduces) the energy gap, respectively. A change in unperturbed band edge (developed in [1 ] ) can also be applied to non-degenerate silicon. That gives: -- AEu(ri) = Bo Vo6z-lgu 1 In (1 + 6). 551

Here, Bo = 0.988 x 1012 d y n ' c m -2 is the isothermal modulus of Si-atom [4], z = 4 the coordination number, gu (equal to gc = 6 and gv = 1 respectively for n- and p-type silicon) the number of equivalent bands, Vo = 4rrr~i/3 the Si-atom volume, and finally 6 = ( V - Vo)/Vo = [(r~/rsi) 3 - 1] > - - 1 . Hence,

--2XEu(rx) in meV = 1034.04g?J [(rffl .17 A) 3 -- 1] x In ( r / 1 . 1 7 A ) 3 .

(2)

Then, in DS, we must have

-- AEu, Nt(rz) = -- AEu(rt) [1 -- (Nz/Nsi) (B/Bo)] , where N s i = 5 × 1 0 22 c m - 3 and B ~ Bo is the isothermal modulus of impurity atom [4]. That means AEu, NI(rz) is reduced to zero when Nt ~ N s i , in accordance with a discussion by Friedel for alloys [5]. However, because N t ~< 3 x 10 z° c m - 3 , which is the solid solubility concentration found in impurity-Si systems, one has: (NI/Nsi)'~ 1, and therefore: AEu,NI(rl) ~ AEu(rl) , determined by (2). Going back to equation (1), one can define the energy gap of lightly doped silicon by: Eg(rt,N t ~ O)= E~ -- AEu(rl). This change in energy gap may accompany the change in effective mass m i determined by: m!~)/mi = Eg(rD/E ° [6]. Here, the subscript i denotes respectively n or p for electrons or holes; m n -----0.33 rno and m p = 0.52 mo (heavy hole). Some numerical results are applied to P, As, Sb, B, A1 or Ga impurity-silicon systems and tabulated in Table I. It is concluded that Ee(rx) increases with increasing values of r I and that, in most impurity-Si systems, m! O-~mi, and in particular B-Si system, because o f r B ¢ rsi , m} O = 0.57 m i o r m (1) = 0.2 mo and " m i(t) can be due to mp(1) = 0.30 m0. Further, a change m

552

DOPING EFFECTS ON ENERGY GAP OF SILICON

a non-parabolic energy band effect [7] by: m}2)/m} 1) = 1 + [JiEi. H e r e , E i = E ° (rnai/ m al 0)~) = E° (mi/m}l)), with mai = migZu/a the density-of-states effective mass and E ° = 17.64(mo/mdi) (N1/1019) 2/3 the penetration of Fermi level into the appropriate unperturbed parabolic band, and Hi ~ E g l ( r B ) [1--2(m!X)/mo)], because the spin-orbit interaction equal to 44 meV is much smaller compared with Eg(rB). For example, even at Nl = 1 0 : ° c m -3, one has: E , = 1 3 2 m e V , Ep=275meV, ~n = 9 x 10 -4 m e V -1 and /3p = 6 x 10 -4 meV -1 , and therefore: m ~ ) = 0.22 mo ~ - m u) and m ( 2 ) = 0 . 3 5 mo--~m(p~). As a result, the change in effectice mass due to the non-parabolic energy band effect in B-Si system can thus be neglected. Finally, equation ( I ) can be rewritten by: -- AEg(ri, N I ) = -- AEu(rl) -- AE,,@ND,

(1)'

where AEu(ri) is independent of N1 as discussed above. In [2, 3], we calculated the band-gap narrowing (BGN) by heavy doping effect in n-type DS, in which ri = rsi, e0 = 11.8 (dielectric constant) and m} O = rni (or man = 1.09mo and map = 0.52mo (heavy hole)); it thus corresponds t o : ZXlz~g(r! = r s i , N d ) = z2~Eg,ri=rsi(Nd). An analogous result can also be applied in p-type DS to give: AEg(rl = r s i , N , ) = AEg, ri=rsi(N,). From Table 1 and equation (1)', one thus notes that, in silicon heavily doped with P, As, Sb, A1 or Ga-impurities, because of m} 1) --~ mi, AEg, r/(NI) is found to be almost independent upon r~, and in silicon heavily doped with B-impurities, because m } O = 0 . 5 7 m i depending on r BV~cln r ~ o ) ~ 0 . 6 2 m o and m dP O) ' ' 0.30mo), the result of AEu, ~ ( N , ) strongly depends on r B .

Vol. 58, No. 8

In [ 2 , 3 ] , based on two local screened donor potential models and taking into account the electronelectron interaction effect in the electron-donor interaction, AE~(Na) given in P, As or Sb impuritysilicon systems is found to be: AEg(Na) = AEg -e -- ((Vn))c -- (( Vp))c 4-

2 2 o on,en/4E no + ap/4Ep.

(3)

Here, AEff -e = 41re2eolNa d~ = 52.28 (Na/IOa9) 1/3 , with dl = 0.4 N~al/a, represents the electron-electron interaction effect [8], ((Vi)) c =--4ne2eoXNaq{i2Z and a~ =2rreaeo~Naq[i 1 are respectively the first and second-order cumulant. Further, Z is the screening charge (equal to 1 and - 1 respectively for electrons and holes) and q~a is the screening length of carriers determined by a second-order T h o m a s - F e r m i model: qli = qi [~/1 + (biZ/2) 2 + (biZ/2)], with q~ = 4he: eo' (3Nd/2E ° ) and bi = (e2 q~/4eo) (Z/2E°). Finally, one notes that qli is a physical solution of the following equation [2] : (( Vi))c -- Z(o~ /4E ° ) = -- 47re2 eol Naq72 Z = (4)

-Z(2E°I3),

where E ° is determined in Section 2. With the aid of equation (4), equation (3) becomes:

3. CALCULATION OF AE., q(Ni) We first express AEg, rI(Na) as a function of Na from a result of BGN in n-type DS, where r I = rsi , given in [2, 3]. As noted above, in P, As or Sb impurity-silicon systems, AE~,~I(Na) ~ AEg,~x=~si(Na) = z~Eg(Na), for simplicity.

AEg(Na) = aeg-o+ (2/3) (E° - F ° ) 4-

2 o (on/4En) IX/1 + (dx/qiln) 2 -- l]

+(@/2E°),

Table 1. Acceptors

Donors [4]

P

As 1.10

AEu, Ni(ri) (meV)

-

-

Eg(ri) (meV) (m!O/mi)

Sb 1.18

--5.4

0.1

44.4

1160.6

1166.1

1210.4

1

1

A1

B

1.36

0.88 --507 659.0 0.57

Ga 1.26

1.26

57

57

1223.0

1223.0

21

21

Vol. 58,No. 8

DOPING EFFECTS ON ENERGY GAP OF SILICON

553

Table 2 NI (1019 cm -3 )

q2 (ilk)

@1 (/~)

ql-ln (~t)

ql-~ (ilk)

d! (/~)

AEg(Nd) (meg)

AEg(Na) (meg)

1 2 10

8.06 7.18 5.50

11.67 10.40 7.95

4.64 4.81 4.58

9.64 9.08 7.48

18.57 14.74 8.62

77 84 98

86 111 203

Table 3 Nd in

1

1019 c m - 3

7

10

AEg(rv, Na) in meV

82 (our result) 27 [121 75 [13]

95 (our result)

103 (our result) 87 [12] 125 [13]

AEg(rAs , Nd) in meV

77 (our result) 148 [I0]

90 (our result) 205 [10] 115[13]

98 (our result) 217 [101

and replacing 02 and E ° defined above, on finally gets:

~E~(Nd) =

q~X]

2 qn t AEg -e + (2/3) (E ° _ ~ o ) + (e \12~o

\q{n]

1 6eo ]

X

(mdn/moe3o)1/2(Na/lO19) 1/6

{ (qlnl qn) + (1/2) (map/man) 1/2 qlp - 1 qp

~@x f Finally, as discussed in Section 2, the result of

(Nd/lO19) 2/3 "4- 9.6 x 102 × (mdp/moe3o) In

/ (Na/1019) v6 ((qlpqv) + (1/2) (man/mdp)l/2q~lnq n

1

l j.l

(5) Following the same treatment as that given above,

AEg,,~(N,) in B-Si system can also be determined from equation (6), with replacing man -- 1.09 mo and map = 0.52mo by man (1)~0.62mo and m ~ - - 0 . 3 0 m 0 respectively. Some numerical results of ~Eg(Na) and AEg(Na) in P, As, Sb, A1 or Ga impurity-silicon systems, useful to a comparision with other theories [9-11 ] and experiments [12, 13], are given in Table 2. Table 2 thus shows that, for a given Nx, AEg(Na) is found to be larger than AEgbVd) , because of map ~ man. 4. COMPARISON WITH OTHER THEORIES AND EXPERIMENTS In recent theories [9, 10], assuming that r z ---rsi

AEgiNa) obtained in AI or Ga impurity-silicon systems, and map = mdn = 1.089mo for n-type DS, one comin which the density-of-states effective mass mai = monly' obtained: AEu(Na) = e2 qn/eo = 148(Na/ mag~/3 is not changed by impurity size effect (see 1019) 1/6 in the first-order Thomas-Fermi approxiTable 1) and a model used to calculate the effective screening length of majority-hole carriers is determined by: qlp,-1et~ = ~ , yields:

AEg(Na) = 52.28(Na/1019) 1/3 + 11.75 ( m-~ap

m-~an)(Na/1019)2/3 + 9 . 6 x 102

mation. These assumptions reduce our result of secondorder band-edge shifts given in equation (5) to: 6.18 ( N d / 1 0 1 9 ) 1/6 -b 27.92 = 34.1 meV for Na = 1019 c m - 3 • This result is considerably smaller than the corresponding one given by above theories (148meV), in accordance with a conclusion by Abram et al. [9] when they used a Lindhard dielectric function method. Furthermore, Van Cong [11 ] recently showed that AEK(Na) = e2qn/eo

554

DOPING EFFECTS ON ENERGY GAP OF SILICON

can be obtained only in the non-degenerate case (see also [3]). Table 3 gives a comparison of our results of AEg(rp, Na) and AEg(rAs, Na) (determined from equation (1)' and Tables 1 and 2 with some corresponding theoretical [10] and experimental [12, 13] results. Table 3 shows that our results are in accordance with photoluminescence experiments in As or P impuritysilicon systems at 10K, obtained recently by Wagner [13]. Finally, as given in [1], the acceptor level or holetrap level is defined by:

EA(rt) = E~(r, = rs0 + AEv(rl), where E~ = 50.79meV is the unperturbed acceptor level in silicon. For example, in B--Si system (see Table 1), one obtains: EA(rB) = 557.79 meV. This level is well situated between two deep hole-trap levels in boronimplanted phosphorus-doped silicon, 471 and 671 meV, recently observed by Jackson and Sah [14] from the measurements of the thermal emission rates of holes and concentration profiles. 1.

REFERENCES H. Van Cong, S. Brunet & J.C. Martin, Solid State Commun. 49,697 (1984).

2. 3. 4.

5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

Vol. 58, No. 8

H. Van Cong, S. Charar & S. Brunet, Solid State Commun. 44, 1313 (1982). H. Van Cong & S. Brunet, Solid State Electron. 28,587 (1985). C. Kittel, Introduction to Solid State Physics, p. 33, 85 and 100 Wiley and Sone, Inc., New York (1976). J. Friedel,Adv. Phys. 3,446 (1954). J.D. Wiley, P.S. Peercy & R.N. Dexter,Phys. Rev. 181, 1181 (1969). E.O. Kane, J. Phys. Chem. Solids 1,249 (1957); H. Van Cong, S. Brunet & S. Charar, Phys. Status Solidi (b) 109, KI (1982). H. Van Cong, Phys. Status Solidi (a) 56, 395 (1979). R.A. Abram, G.J. Rees & B.L.H. Wilson, Adv. Phys. 27,799 (1978). J.C. Inkson, J. Phys. C9, 1177 (1976); P.T. Landsberg, A. Neugroschel, F.A. Lindholm & C.T. Sah, Phys. Status Solidi (b) 130,255 (1985). H. Van Cong, Phys. Status Solidi (b) 117, 575 (1983). M. Balkanski, A. Aziza & E. Amzallag, Phys. Status Solidi 31,323 (1969). J. Wagner, Phys. Rev. B29, 2002 (1984). D.B. Jackson & C.T. Sah, J. Appl. Phys. 58, 1270 (1985).