TEMPERATURE COEFFICIENT OF THE REFRACTIVE INDEXOF DIAMOND- AND ZINCBLENDE-TYPE SEMICONDUCTORS

Peter Y. Yu and Manuel Cardona*Brown U niversity , Providence, Rhode Island , USA

and

Deutsches Elektronen-Synchrotron DESY, Hamburg, Germany

+ J.S . Guggenheim Memorial Foundation Fellow

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A bstract

We have calculated the temperature coefficient of the long-wavelength

refractive index of several group IV and III-V semiconductors using

the Penn model for the electronic contribution to the dielectric

constant. The isotropic band gap of this model is identified with

the band gap at the X point of the Brillouin zone3 which can be

simply expressed in terms of pseudopotential coefficients. The

explicit temperature dependence of this gap is calculated by

applying to these pseudopotential coefficients the appropriate

Debye-Waller factors. The thermal expansion effect is obtained in

the manner suggested recently by Van Vechten. Good agreement be­

tween the calculated and the observed temperature dependence of

the long-wavelength refractive index is found.

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1. In troductionRecently P h i l l ip s ’ theory of ionicity-^has been applied successfu lly to ca lcu la te various p ro p ertie s , namely ion iza tion p o te n tia l2, cohesive energy3 , non-linear o p tica l s u s c e p t ib i l i t ie s 4 »5 »5 e t c . , of a large number of covalent and ionic compounds. At the heart of th is theory is a very simple iso tro p ic one-gap model of the band s tru c tu re due to Penn7. Heine and Jones8 pointed out th a t one reason why such a simple model works for the diamond-type semiconductors is th a t the gap. on the surface of the Jones-zone is nearly constant over a large area. By id en tify ing the "Penn gap” with the band gap a t the center of the (lio) Jones-zone face, they calculated the hydrosta tic pressure co e ffic ien t of the d ie le c tr ic constant of Si and Ge. In th is paper we generalise Heine and Jones’ ca lcu la tio n to the III-V semi­conductors and obtain the "Penn gap” as a function of the pseudo- p o ten tia l co e ffic ie n ts . We then ca lcu la te with th is model the tempe­ra tu re co e ffic ien t of the re fra c tiv e index of several group IV and group III-V semiconductors.

2, Penn ModelThe e lec tron ic cen trib u tion to the long-wavelength d ie le c tr ic of a so lid in the Penn model is given by : 7 »8

oo 2e (o) 'v 1 + D —

constant

( n

where oo is the plasma frequency of the valence e lec tro n s , the. "Penn gap", and D is a parameter introduced by Van Vechton" to take in to account the e ffec t of d -lilte core e lec tron s. I t is di 1 i: ic u lt to speculate on how D w ill vary with temperature and volume. Since D is

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of the order of unity and i t s v aria tio n s are not expected to be large we sh a ll neglect th is facto r in our c a lc u la tio n ..

By d if fe re n tia tio n of Eq. (1) w ith respect to temperature (throughout th is paper pressure is assumed to be constant when the d-sign of d if fe re n tia tio n is used) we obtain:

J_ d£ _ £-1 £ dT ~ £

£— 1 E

do)dT “ 2 C

dm

m dT g )1

da)-3a - 2 ( — )w dT ' g( 2 )

where a = co e ffic ien t of lin ea r expansion. In Eq. (2) we have made useof the known p roportiona lity of to the valence e lectron concentration.We decompose the temperature co e ffic ien t of to which appears in Eq. (2)ginto i t s "e x p lic it" temperature dependence a t constant volume and i t s volume expansion e ffe c t:

1 dmdT- = “ (

3d)g3 T -fc 3a V 3o)( — & 3 V

(3)

3, E x p lic it Temperature Dependence of the Penn Gap

Following Heine and Jones8 we id en tify the "Penn gap" to with the gapgat the X point of the B rillou in zone. As is well known, i t s energy is close to th a t of the strongest peak in the o s c il la to r streng th and the re f le c t iv i ty of the group IV and group III-V semiconductors10. An expression for th is gap as a function of the pseudopotential form fac to rs v ( l l l ) , and v(220) has been ca lcu la ted by Heine and Jones8 for germanium- type m aterials using pertu rbation theory. This expression can be eas ily generalized to zincblende-type compounds. We fin d , in Rydbergs11

2 (v ( 1 1 1) ± v ( 1 1 1 ) } 2hi ~ = 2v (220) ± 2v (200) + g s a

(4)

5

where a is the la t t i c e constant in Bohr ra d i i . The subscrip t s and a of the pseudopotential co e ffic ien ts v (220) , v (200) and v ( l l l ) denotes th e ir symmetric and antisymmetric components respec tive ly . The a n t i ­symmetric co e ffic ien ts v (20 0) and v ( 1 1 1) are responsible for thea as p l i t t in g of the conduction bands (± sign in Eq. (4)) but does notchange, to f i r s t order, the average value of oo . Since the s p l i t t in g

ê

is known to be small for III-V compounds, i t can be neglected for the purpose of ca lcu la ting d ie le c tr ic p ro p ertie s .We thus obtain

(v ( 1 1 1 ) ) 2 + (v ( 1 1 1 ) ) 2 ,to — 2 ( v ( 2 2 0 ) +g S (2f ) 2 (5)

The values of the pseudopotential co e ffic ien ts vg(220), vg(111) and v (111) for several group IV and group III-V semiconductors are l is te d in Table I together w ith other parameters relevant for our ca lcu la tio n s .The values of oo obtained from these parameters with Eq. (5) are l is te d

O

in Table I I , together w ith the values obtained from the d ie le c tr ic constant (see Eq. (1)} by Van Vechten9 and the energy E^ of the corresponding r e f le c t iv i ty peak. The good agreement between these three se ts of values confirms the v a lid ity of Eq. (5).

Equation (5) can be used to ca lcu la te the temperature dependence of wê

by m ultiplying the Fourier component of the pseudopotential v ( g ) by the appropriate Debye-Waller fa c to r12»13:

e-B(T) | s | 2 =■ e <u2> (for cubic c ry s ta ls ) (6)

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For a c ry s ta l with r atoms per u n it c e ll the average square displacementi

of atom K i s 11*:

<e(SC,j,q) |2coth ■ j 3o). (q) _____£___J ~~

w. (q)J(7)

where 3 = kT. is the mass of the K-th atom, N the number of u n it c e lls per u n it volume, ok (q) is the frequency of the normal mode with wavevector q in the j th-branch and e (K, j , q) the p o la riza tio n vector of the corresponding displacement of atom K.

In group IV semiconductors the two atoms of the u n it c e ll are id en tic a l and Eq. (7) s im p lifie s to :

<u2> = j coth j 6 (q)2NM q,j 2 o).(q)

J —' (8 )

9B(T)we ca lcu la te w ith the Debye model for the phonon sppctrum andobtain V7ith Eq. (8):

9B(T) _ 3 _J__8T “ 2 MkT2

©F ( - s 2- )(?)

with

F(x) y dyJ - i

( 10)

O

We sh a ll use in our ca lcu la tio n values of the Debye temperature @ D 3t room tem perature, since most experinental data are obtained around room temperature. The function F(x) has been evaluated by Benson and G i l l15.

For the III-V compounds i t is no longer possib le to r e la te <u2> d ire c tly to the phonon spectrum with Eq. (7) since the masses of the two co n stitu en t

7

atoms are not equal. However for those of the III-V compounds we are considering we find th a t the ra t io of the heavier mass to the l ig h te r is smaller than two. In the case of the a lk a li halides i t was found th a t at room temperature the average square displacements for the anion and cation d if fe r by less than 10 % even for masses which d if fe r by a fac to r of 3 . le Hence we assume for the sake of sim plic ity th a t <u2^j^> = <u2y>. The v a lid ity of th is assumption cannot be checked u n ti l the Debye-Waller fac to rs for the III-V compounds have been determined. However the re su lts of our ca lcu la tio n seem to support th is assumption. Thus for the III-V compounds we use the formula:

9BI I I9T

3BV

9T . i l _ = 33T kT2 (Mm + My)

combining Eqs.(5) and (6) we obtain:

(1 0

9co

<5*“ > , 1 2vs (220)(- | g_(220) [2 ||) +

( 12)

(v (1 1 1 ) ) 2 + (v ( 1 1 1 ) ) 2

& ) 2 a(- 2 |g(lll)|2 ||)

In order to sim plify the evaluation of Eq. (12) we note tha t for the semiconductors we are considering v ( 22 0) is small and |g (220) j 2 v 2 | g ( l l l ) | 2 , Equation (12) can thus be approximated by the expression:

. 9a)_ L ( — § L )03 k 9T ; Vg -

2 | g ( ! ! 1 ) | 29B9T (13)

with, an error sm aller than 10 %.

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4. Volume Dependence of the "Penn gap’1

The volume co e ffic ie n t of co can be calcu la ted with Eq. (5) and the knowngderivatives of v(c^) with respect to |c} | . 8 Since th is volume dependence is only responsible for less than 20 % of the to ta l ca lcu la ted temperature dependence of w , we choose for i t s evaluation the sim pler, although possibly less accurate procedure given by Van Vechten9.

We s ta r t with the equation:

u) = (w ? + C ^ ) ^ (14)

where is the homopolar energy gap of the m ate ria l, and C the hete ro -polar energy gap. In order to ca lcu la te the volume dependence of oo i tSis necessary to know the corresponding dependence of a a ^ } 9

and C. As suggested by the value of ( ~ ) for the a lk a li h a lid e s9, 3 Cwe take (—),£ ~ 0 • Our re su lt does not depend c r i t i c a l ly on th is

assumption since C only accounts for a small fra c tio n of a) , whoseSvolume dependence, in tu rn , only accounts fo r a small fra c tio n of theto ta l temperature dependence of co . We thus find:

3oo

3V >T = - ° - 8 x r 0 5 )

5. Temperature Dependence of the long-wavelength R efractive Index• * - 1 3 nWe have l is te d m Table I the parameters used for our ca lcu la tio n of —(— )n o T ; P

based on Eqs. (2 ), (3 )j (13), and Cl5). We have used for the Debye temperature the values obtained from sp ec ific heat measurements. Batterman and Chipman17 found th a t the Debye temperatures obtained for Ge and Si

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from x-ray in te n s ity measurements were about 20 % lower than thosefrom sp ec ific h ea t. This fac t introduces an uncertain ty of about 40 %

. 3 toin our ca lcu la tio n of —— (-—§-) . We believe th is uncertain ty is ourto 3T V

8main source of e r ro r .

• • d cogIn Table I I we.show the temperature co e ffic ien t ca lcu la ted withEqs.(13) and (15) and those determined experim entally for E^. The agreement is s a tis fa c to ry .

1 d nIn Table I I I we show the calcu la ted value of — — a t 300° K comparedn dTwith experimental re s u lts . T heoretical I are the values obtained by

dtodTusing in Eq. (2) the calcu la ted co e ffic ien ts

m u ±d E LI I are the values obtained by using the experimental ~

& while Theoreticaland

assuming 1 dojg _ J dE2dT . We note th a t the trend in the experimentalco dT E0

g . 2values in going from one m aterial to another is b e tte r represented byTheoretical I than by T heoretical I I . A possib le explanation is th a t the

dE2values of dT determined by the wavelength modulation technique aresen sitiv e to the detailed c r i t i c a l point s tru c tu res while m representsthe somewhat d if fe re n t average gap. The agreement between the calcualted

1 dn ,values of — and the experimental values is always b e tte r than 40 %, which is the e r ro r of our ca lcu la tion due to u n ce rta in tie s in the Debyetemperature.

AcknowledgmentsWe wish to express our g ra titu d e to the members of group F 41 of DESY for th e ir h o sp ita li ty and to R .R .I. Zucca and Y.R. Shen for making availab le to us th e ir re s u lts p rio r to publication . One of us (P.Y.Y.) is indebted to General Telephone and E lectronics for a fellow ship.

References1. J.C. P h il l ip s , Phys.Rev.Letters 20, 550 (1968); Covalent Bonding in

Molecules and Solids (University of Chicago P ress, Chicago, to be published)

2. J.C. P h il l ip s , Phys.Rev. L e tte rs 22_, 705 ( 1969)3. J.C. P h il l ip s , Phys.Rev. L e tte rs 22 , 645 (1969)4. J.C. P h illip s and J.A.Van Vechten, Phys.Rev. 183, 709 (1969)5. B.F. Levine, Phys.Rev. L etters 2_2, 787 (1969)6 . J.A. Van Vechten, M. Cardona, D.E. Aspnes, and R.M. Martin,

to be published7. D. Penn, Phys.Rev. _1_28, 2093 (1962)8 . V. Heine and R.0. Jones, J.Phys.C (Solid St. Phys.) Ser. 2, 2_, 719

(1969)9. J.A. Van Vechten, Phys.Rev. 182, 891 (1969)10. M. Cardona, J.Appl. Phys. 3 6, 2181 (1965)11. In Ref. 8 Heine and Jones ca lcu la ted the band gap a t X both by pertu rba tion

theory and also more exactly by d iagonalisa tion of a 6x6 m atrix. They found th a t pertu rbation theory gave b e tte r agreement with experimentthan the more exact method.We found th a t th is is also true for the III-V compounds. As a re su lt we have used the expressions obtained by pertu rbation theory.

12. C. K effer, T.M. Hayes,and A. Bienenstock, Phys.Rev. L e tte rs 2J_, 1676 (1968)13. J .P . W alter, R.R.L. Zucca, M.L. Cohen, and Y.R. Shen (to be published)14. A.A. Maradudin, E.W. M ontroll, and G.H. Weiss, Solid State Physics

Suppl. 3, p. 237, edited by F. S eitz and D. Turnbull, Academic P ress,N.Y. (1963)

15. G.C. Benson and E.K. G ill , Canad. Journal of Phys. 44_, 614 (1966)16. W.J.L. Buyers and T. Smith, J . Phys. Chem. S o lid s, 29_, 105 (1968)17. B.W. Batterman and D.R. Chipman, Phys.Rev. 127, 690 (1962)

Table I

The various parameters used in th is a r t ic le for ca lcu la ting the temperaturedependence of the long wavelength semiconductors.

a Pseudopotential , v Coeff. (Ryd) £ °

v (220) v ( 1 1 1 ) v ( 1 1 1 ) (b)o o d

re frac tiv e index of group IV and I I I - V

Coeff. of lin ea r Debye Temp. L a ttice ConstantExpansion a t 0 a t (in Bohr ra d ii)room temperature ^3000 k (b)c g

C - 0 . 0 2 2 - 0 . 5 1 4 - 5 . 7 1 . l x l O ’"6 /° K 1 8 8 0 ° K 6 . 7 4

+ 0 . 3 3 7 - 0 . 8 1 1r c h

S i + 0 . 0 4 - 0 . 2 1 - 1 2 . 0 2 . 5 x l 0 " 6 6 4 7 1 0 . 2 6d h

G e + 0 . 0 1 - 0 . 2 3 — 1 6 . 0 5 . 7 x l 0 " 6 3 5 4 1 0 . 6 9

d dG a A s + 0 . 0 1 - 0 . 2 3 + 0 . 0 7 1 0 . 9 5 . 7 x l O " 6 3 6 0 10. 68

G a S b 0 . 0 - 0 . 2 2 + 0 . 0 6 1 4 . 4 d 6 . 3 x l 0 ~ 6 d 2 3 0 1 1 . 5 6

* dI n A s 0 . 0 - 0 . 2 2 + 0 . 0 8 1 2 . 3 e 5 . 4 x l 0 " 6 2 7 0 1 1 . 4 1

d dI n S b 0 . 0 - 0 . 2 0 + 0 . 0 6 1 5 . 7 5 . 1 x 1 0 “ 6 1 5 0 1 2 . 2 4

e dI n P + 0 . 0 1 - 0 . 2 3 + 0 . 0 7 9 . 6 4 . 9 x l O " 6 4 2 0 1 1 . 0 9

G a P + 0 . 0 3 - 0 . 2 2 + 0 . 1 2 9 . 1 f 5 . 8 1 x l 0 " 6 1 4 7 0 1 1 . 0 3

e5.3X1CT6

a M.L. Cohen and T.K. B erg stresser, Phys.Rev. 141, 789 (1966)L.R. Saravia and D. Brust, Phys.Rev. 170, 683 (1968)

b See J.A. Van Vecliten, Phys.Rev. _182_, 891 (1969) and references there inndc American In s t i tu te of Physics Handbook, 2 Ed. p. 4-66

Me Graw H ill, N.Y. (1963)d Semiconductors and Semimetals Vol. I I edited by R.K. W illardson

and A.C. Beer, Academic P ress, N.Y. (1966)e H. Welker and H. Weiss in Solid S tate Physics Vol. I l l edited by

F. Seitz and D. Turnbull, Academic Press, N.Y. (1956)

f E.D. P ierron , D.L. Parker and J.B . Me Neely, Journal of Appl. Phys. 4669 (1967)

g J.E . Desnoyers and J.A. Morrison, Phil.Mag. 3_, 42 (1958)h P. Flubacher, A .J. Leadbetter and J.A. Morrison, Phil.Mag. 4_,

273 (1959)i The Debye temperature of GaP a t room temperature is not a v a il­

ab le . The value given in Table I is estim ated from i t s 0° K Debye temperature (446° K) calcu la ted from the e la s t ic constants (E.F. Steigm eier, Appl.Phys. L e tte rs 3_, 6 (1963)V

Table II

Calculated and experimental values of the "Penn gap" to , energy E ofS 2

highest peak in the re f le c tio n spectrum, and temperature co e ffic ien tto (th eo re tic a l) and E? (experimental) .

theof

tOg ( th eo re tica l) from Eq.. , (5)

in eV

(0g (experimental) from E q .,(1)

in eV

Highest Peakin R eflection Spectrum(E2) in eV

dto r^ r - < c a k . ) .

in eV/°K¿iT (exr'- )in eV/° K

c 7.829.6

13.6

b

12. 6 -1. 17xlO“4 - 4 .6x 10 ~ 4

bSi 4.2 4.8

b

4.44 -1 .9xl0“ 4 - 2 . 2xlO" 4

d +0 . 5xl0~ 4

Ge 4.4 4.3

b

4.49 - 3 .3xl0~ 4 b - 2 . 4xl0“ 4

d - 2x 10 “ 4

GaAs 4.9 5.2

b

5.11 - 3 .5xl0“ 4 b - 3 .6xlO- 4

f -3 .3x l0 " 4

GaSb 4.8 4. 1

b

4.35 -4.7xlO~ 4 b - 4 . 1x 1 0 " 4

6 - 6 .2, \ ft' " 4

InAs 4.9 4.58b

4.74 -3.8x10" 4 b -5.6x10” 4

InSb 4.5 3.74

c

4.23 -6.9x10” '' b - 3 . 6x 10“ “' e . c: j. f -*■

InP 5.2 5.2c

5.0 -2.3x10“ 4

GaP 5.85 5.75 5.3 - 1 . 9x 1 0 " 4 f -4.5x10"“

3. W.C. Walker and J . Osantowski, Phys.Rev. 1_34_, A 153 (1964)b R. R. L. Zucca and Y.R.' Shen, (to be published). All these values

were measured at 5° K.

ci M. Cardona in Semiconductors and Semimetals, Vol. Ill, p. 140

edited by R.K. Willardson and A.C. Beer, Academic Press,

N.Y. 1967. Both are room temperature values.

d M. Cardona and H.S. Sommers, Jr., Phys.Rev. 122, 1382 (1961)

e F. Lukes and E. Schmidt, Proc. Intern. Conf. Phys. Semicond.,

Exeter, 1962, p. 389, Inst, of Phys. and Phys. Soc., London, 1962

f A.G. Thompson, J.C. Woolley and M. Rubenstein, Canad. Journ. of

Phys. 44, 2927 (1966)

Table III

Temperature coefficient of long wavelength refractive index of group IV

and III-V semiconductors.

— ~ (Theoretical I) — (Theoretical n ai n dl II) n cFF (Experimental)

c 1.lxlO“5 /° Ka 0.5xl0"5 1° K

Si 3.7x10“5 4.2x10“5 /° Kb

3.9x10“5 /° K

Ge 6. lxlO“5 4.2xlO”5b

6.9xlO"5

GaAs 5.4x10“5 5.7xlO”5 5.lxlO“5

c4.5x10“5

GaSb 9.3x10”5 8.0x10“5TZ 4 X \o~5

c8.2x10“5

InAs 6.7xlO~5 10.3x10-5 —

InSb ] 5.Ox 10~ 5 7.4x10“5c

11.9xlO"5

InP 3.6x10“5 —c

2.7xlü“5

GaP 2.2x10“5 6.8xl0“5d

3,7>: 10” 5

a P.T. Narasimhan, Proc.Phys. Soc. B 6_8, 315 (1955)

b M. Cardona, W. Paul, and H.. Brooks, J.Phys .Chem.Solids 8, 204 (1959)

c M. Cardona, Proc. Internat. Conf. Phys. Semiconductors, Prague, 1960

P* 388, Czech.Acad .Sci., Prague and Academie Press , N.Y . 1961

d A.N. Pikhtin and D .A. Yaskov, Fiz. Tverd. Tela 9_, 145 (1967)

(English Transi.: Soviet Physics - Solid State 9 , 107 (1967)}