OPTICAL-CONSTANTS MODEL FOR SEMICONDUCTORS AND INSULATORS

Optical-constants

model

for

semiconductors

and

insulators

Y.

F.

Chen and

C. M.

Kwei

Department

of

Eiectronics Engineering, National Chiao Tung University, Hsinchu, Taiwan, Republic

of

China

C.

J.

Tung*

Institute

_of

Nuclear Science, Nationa! Tsing Hua University, Hsinchu, Taiwan, Republic

of

China (Received 5April 1993)

The refractive index and the extinction coefficient, i.e., the real and imaginary parts ofthe complex in-dex ofrefraction, for semiconductors and insulators are derived as afunction ofphoton energy. These derivations apply f-sum rules and symmetry relations tocritical-point transitions from the valence band to the conduction band. Comparison with measured optical data reveals that present formulations are valid over awide range ofphoton energies immediately above the band gap tothe first ionization thresh-old for inner shells. The present work shows an improvement and extension over the theory ofForouhi and Bloomer, which applies for anarrow range ofphoton energies above the absorption edge.

I.

INTRODUCTION

Optical properties

of

solids can be characterized by quantities such as the _complex index

of

refraction n( )co= _i(i)co+i (tr)coand the complex dielectric function

s(co) a=i(co)+isa(co) T.hese quantities are photon energy

codependent and provide information on the propagation

of

radiation and on the electronic structure

of

solids.

'

Optical quantities are useful in the design

of

mirrors, prisms, multilayers,

etc.

The refractive index il(co) and the extinction coefficient tc(co) are commonly determined by

rejective

and absorp-tive spectroscopies. Utilizing modern synchrotron

ra-diation sources, these quantities can be measured over a wide range

of

photon energies, from the far infrared to the hard x-ray region. Theoretical formulations

of

g and

~ can be obtained by the energy-dependent dielectric

function through the relations

il=

_[

[(E,

+

sz)'~

+E, ]/2j'~

and

tc=

_j[(E,

+sz)'~

—

E,

]/2j'~

. The real and imaginary parts

of

the dielectric function, c.& and c.2,

can be derived by the quantum theory for the interaction

between radiation and matter. A widely accepted model

for the derivation

of

dielectric functions is the extended

Drude model ' _{for the description}

_of

_collective

oscilla-tions and interband transitions. Parameters in this model

are determined by a fit

of

the imaginary part

of

the

dielectric function tocorresponding experimental data.

However, it is the refractive index and the extinction

coefficient rather than the dielectric function which are

experimentally accessible. Therefore, analytical expres-sions for the former two quantities are useful in the deter-mination

of

fitting parameters and in the analysis

of

ex-perirnental data. Previously, such expressions for

semi-conductors and insulators were established' ' only for anarrow range

of

photon energies immediately above the band gap. Later, Forouhi and Bloomer

(FB)

(Refs. 14 and 15) derived a formula for the extinction coefficient to

extend the validity to photon energies in the vicinity

of

interband transitions. The

FB

formula sufFers at least two deficiencies,

i.

e.

, the extinction coefficient does not

II.

THEORY

A. f-sum rules

f-sum rules are important constraints for the analysis

of

energy-dependent optical quantities. These rules may be expressed in different forms involving the imaginary

part

of

the dielectric function, E2(co), the extinction

coefficient a.₍co), and the energy-loss function

Im[

—

1/E(co)]. They are explicitly expressed as'

coE (co )dco

=

co

oi K(co )dio

=

co (2)

where co =Pi(4mNZe

/m)'

is the free-electron plasma energy, N is the atomic (or molecular) density,

Z

is the satisfy the symmetry relation and it does not comply with

f-sum rules. As a consequence, their formula cannot be applied

to

photon energies around and above the plasmon resonant energies.

In the present work, we construct an analytical expres-sion for the extinction coefficient which satisfies the sym-metry relation and sum rules. We also derive an expres-sion for the refractive index by applying Krarners-Kronig

dispersion relations. These expressions are then applied

to the critical-point transitions for electrons from the

valence band to the conduction band

of

insulators and

semiconductors. Comparing the results between this work and experimental data on il(co), tc(co),

s,

(co),E2(co), and the energy-loss function Im[

—

1/s(co)],

it is

demon-strated that present formulas for the optical quantities

are valid for photon energies over a wide range from the

band gap to the edge

of

inner shells.

4374

Y.

F.

CHEN, C.M.KWEI,AND C.

J.

TUNG 48 total number

of

electrons per atom (or molecule), and e

and m are, respectively, electron charge and mass. Note that these sum rules apply to optical quantities contribut-ed by all absorption processes including valence-band

ex-citations and inner-shell ionizations over the entire ener-gyinterval:

co=0

—

+

~.

A typical spectrum

of

the optical quantity usually re-veals well-separated peaks, with little overlapping, due to

different absorption processes. In such a case, one can set the upper limit

of

integration in Eqs. (1)

—

(3)equal to an energy cutoff co;,large compared to the ith peak energy but small compared to other peak energies. Thus the above sum rules may be separated into individual

contri-bution from each isolated absorption process by the ap-proximate relations

f

,

(

')d

0 2

4me

I

W(co)=

_i(k;„,

ixik;„,

)

i 5(co

—

cop),

3Ac

where W(cp) is the energy absorbed by the solid in the range co to co+dco per unit time,

I

isthe incident photon intensity, cisthe speed

of

light, and the Dirac 5function isowing to the conservation

of

energy.

For

the moment, we consider a single critical-point transition from the valence band to the conduction band, with transition

ma-trix element

(

k;„,

~

x

~

k,

'„,

)

and the transition energy

ct)0=E

E .

The extinction coefficient isgiven by A'cp(co)W(co)

26)I

where p(cp) is the number

of

possible transitions in the range co to

co+de

per unit volume. Defining the dipole

oscillator strength

of

the transition as

and

2

f

coK(co )dcp

=

0 4 r]b

fo=

and substituting it into Eqs. (10)and

(11),

we obtain (12) 2 i

—

1 m ~P,I

co'Im,

dco'

=—

o E(co') (6) neff(co)

=

_{z ~ z}

f

cp'E2(co')dco'

2~'e'A'N

(7) and

n,

ff(co)

=

f

c'o(_Kco)d c'o,

~e

AN

where

co,

is the plasmon energy associated with the ith absorption process and qb is the background refractive index. Note that the above partial f-sum rules for K(co) and Im[

—

1/s(co)_{] yield} eff'ective oscillator strengths that are reduced by qb and gb due to the shielding

of

other absorption processes. A detailed discussion about the background refractive index will begiven below.

It

is useful to define an effective number

of

electrons per atom (or molecule) corresponding to a given optical

quantity according to

p

K(CO)

—

5(co cop)

4cop (13)

Equation (13)does not satisfy the symmetry relation, '

i.

e.

,K(co)

=

—

K(

—

co).

To

rectify this situation, we let

~fo

K(co)

=

[5(co

—

cop)

—

5(co+cop)]

.

4ct)p (14)

(15) The extinction coefficient

of Eq.

(14)also satisfies the sum rule

of

Eq. (2). In this case, however, the lower limit

of

integration must be extended from

0 to

—

~

with the right-hand side

of

this equation multiplying by

2.

Derivations so far assume zero energy breadth or infinite lifetime for the excited state in the critical-point

transi-tion. This assumption leads to the 5-function dependence

of

the extinction coefficient on photon energy. In reality, the spontaneous emission produces the damping

of

excit-ed states.

To

accommodate the damping effect, one may replace the 5 functions in

Eq.

(14) by the Lorentzian line-shape function according to

m CO

n,

ff(co)

=

co'Im,

dco' .

2me AN E Co

For

each absorption process, we expect that

(9) _where

_I

0 is the damping parameter relating to the full

width at half maximum

of

the K(co)spectrum. Therefore,

the extinction coefficient becomes

n,

ff(co)~,

:n,

ff(co)~,

:n,

ff(co)

~,

=

1:g&

'.

rib

f

pl

pco

K(CP)

=

_{2 2}

[(cp

—

cop)

+I

p][(co+coo)

+I

p]

(16) in the limit

co~co;.

Equations (7)—(9) provide guidance

for the determination

of

the number

of

valence electrons participating in critical-point transitions.

B.

Index ofrefraction

Consider the forward propagation

of

photons in a

solid. ' The first-order time-dependent perturbation theory under the electric dipole approximation gives'

g(co)

=qb+

—

1 P

f,

K(co')

dco',

7T —~CO CO

(17)

where P denotes the Cauchy's principal value.

Substitut-The refractive index is connected to the extinction coefficient by Kramers-Kronig

(KK)

dispersion relations.

These relations are direct consequences

of

the causality principle. Based on the Hilbert transform in

KK

rela-tions, itgives

ing

Eq.

(16)into

Eq.

(17),we obtain

f

o(m'

—

mo

—

r(')

2

_[(~

—

_~,

)'+

r,

'][(~+~,}'+

r2]

(18)

3

1,2

The above formulas for the refractive index and the

ex-tinction coefficient are approximate since they are de-rived by first neglecting the damping effect and then resuming this effect using a Lorentzian line-shape

func-tion. Corresponding approximate formulas for the real and imaginary parts

of

the dielectric function can also be derived using Eqs. (16) and (18) and the relations

e&(co)=q) (co)

—

Ic(co) and e2(co)=2qI(co)a(co). Alterna-tively, the quantum theory yields a Drude-type dielectric

function as ' 0.8 0.4

1.

4

1.

0—

0.2 I I I I I I I I

Fc

(co coo ) ,(

)=

b-(c& coo) +co l'0

Fpgpcs E,

(~)

=

( 2

~2)2+~2y2

(20) -0.2

0.

8 0.6

0.

4

0.

2 0.0 I I

«

i I 8&e

COI-f

=

3 '

l(k:.

I-Ik:"& I'P(-

) (21)

be the oscillator strength for the ith critical-point

transi-tion. One can generalize Eqs. (16)and (18)into

where c._& is the background dielectric function

of

the solid, I'p is related tothe osci11ator strength, and yp isthe damping parameter. Comparing the approximate

dielec-tric function with the Drude dielectric function, we find that they are equal under conditions E&=q)z,

Fc=fogb,

ye=21

c,

I

o«coo,

and

fo

«coo.

To

illustrate this com-parison, in

Fig.

1 we plot results in E,(co), E2(co), and

Im[

—

I/e(co)]

calculated using the approximate (solid curves) and the Drude (dashed curves) formulas under conditions

e&=qlb=l, In=coo/10,

and

fo=coo/10. It

is seen that fairly good agreement isfound for all functions

plotted. The _imposed conditions

I

0«coo

and

_fc«coo

make Eqs. (16) and (18) valid only for insulators and semiconductors where cop are large enough to satisfy

these conditions.

The above formulations are for a single critical-point

transition

of

electrons from the valence tothe conduction

bands. The generalization

of

these formulations to a gen-eral situation for several critical-point transitions can be made in a straightforward way.

It

isnot difficult to show

that each critical-point transition makes an independent contribution to the extinction coefficient. Thus, one can

divide the oscillator strength into fractions

_f;

with transi-tion energy co;and damping coefficient

I;.

Let

I I I I I I I I I I I I I I I I i i I

0Q

0.0 0.5

1.

0

1.

5 2.0

Flax. 1. Aplot ofe&(co},Ez(co},and Im[

—

I/E(c0}]calculated using Eqs. (16) and (18) and the relations c&

=

g

—

a and

c2=2qa (solid curves) and using Eqs. (19) and (20) (dashed curves) bytaking Eb=r)~

=1,

I

0=coo/10,and fo=c002/10.

C

f,

_{(co col}

I

.

)—

—

1.

=(

[(co

—

co,)

+I

1.

][(co+co.

)

+I

)

(24) where the first and second summation terms indicate,

re-spectively, contributions from the valence band (having q critical-point transitions) and from inner shells (with c

ex-citation groups). Since we are interested in the energy

re-gion co

«

cuj (

j

=

1to

c),

Eq.

(24) reduces to

Eq.

(23) with

where q is the total number

of

critical-point transitions from the valence to the conduction bands.

To

show that gb is indeed the background refractive index due to the contribution from core electrons in inner shells, one can rewrite

Eq.

(23) as

q

f

(~2 ~2~ F2~)

';

=)

[(

co

—

co; )

+

I";

]

[(co+

co, )

+

I,

]

f;

I,

co

;=i

[(co

—

co;)

+I;][(co+co;)

+I;]

(22)

C

rl„=

1+

—,'

g

co

+I

(25) and q

f

(~2 ~2 P2}

;

=

i

[(

co

—

co;)

+

I;

]

[(co+

co; )

+

I,

]

It

is seen that core electrons contribute to the

back-ground refractive index by an amount directly

propor-tional to their oscillator strengths but inversely

propor-tional to the square binding energies. Equation (25}will be applied toafew examples discussed below.

4376

Y.

F.

CHEN, C.M.KWEI,AND C.

J.

TUNG

III.

RESULTSAND DISCUSSION

We now fit

Eq.

(22) to optical data' for several in-sulators and semiconductors. Tables

I

—

IVlist the fitting parameters for MgO, SiOz, Si, and GaAs. In these fittings, we check the accuracy

of

optical quantities in-cluding ri, a, and Im(

—

1/E),

critical-point energies tu;, and sum rules

of

Eqs. (4)

—

(6) by comparing fitted values

to experimental data.

It

is confirmed that the total

oscil-lator strength

of

critical-point transitions satisfies the sum rule,

i.

e.,

gq=,

f;

=co~,

/rib, with co~, being the plasmon energy

of

valence electrons. Critical-point ener-gies measured by the optical reAectometry ' _and

calcu-lated by the pseudopotential band theory for Si and

GaAs are also listed in Tables

III

and IV for comparison.

For

MgO and

Si02,

our fits extend to

~=300

eV, which exceeds the L-shell threshold energies

of

Mg and Si. In

these cases, the first eleven terms

(i

=1

—

11)

in Tables

I

and

II

correspond

to

critical-point transitions

of

the valence band, whereas the last three terms (i

=

12,13, 14) are associated with L-shell excitations

of

Mg and Si. Neglecting the K-shell contribution, we find that the re-fractive index at the region well below the L-shell

excita-tion energy may be given by

Eq.

(24) with 14 terms orby

Eq.

(23)with 11 terms. In the latter equation we find that

qb=1.

035 and

1.

004 for MgO and Si02, respectively. Thus the background refractive index is indeed deter-mined by

Eq.

(25).

Figure 2 shows a comparison

of

q(cu_{), a}(co), and

Im[

—

1/E(co)] for semiconducting Si calculated presently (solid curves) using Eqs. (22) and (23) and the relations

c&

=

_g

—

~ and

e2=2g~,

measured experimentally

(dashed curves) and determined by

FB

(chain curves).

It

is seen that close agreement is found between present re-sults and experimental data for all quantities plotted.

It

is also seen that prominent interband transitions, represented by the strong resonant peaks in _g and ~

spec-tra, occur in the region

of

critical-point energies below

—

10 eV. Above this energy, the refractive index gradual-ly approaches the asymptotic value gb, and the extinction coefficient decreases with co

.

The corresponding

FB

re-1 2 3 4 5 6 7 8 9 10 11 12 13 14 1 2 3 4 5 6 7 8 9 10

TABLE

II.

Parameters in Eqs.(22)and (23) forSiO&.

gb=1

(1.004 if i

=1

to 11 are used)

f;

(eV')

I;

(eV) 6 15 10 38 51 110 45 41 33 70 63 16 58 170 0.16 0.60 0.69 1.31 1.70 3.96 4.97 5.00 4.98 8.00 12.94 5.95 14.90 59.82 10.2 11.6 12.4 14.0 16.9 20.0 25.0 30.0 40.0 51.0 72.0 108.0 134.6 189.5 0.2 0.4 11.0 16.0 10.0 21.0 23.0 70.0 70.0 50.0 0.01 0.01 0.36 0.25 0.30 0.50 0.70 1.60 2.30 2.80 3.43 3.48 3.72 4.35 4.75 5.45 6.40 7.80 10.60 14.00 3.40 3.45 3.66 4.30 4.57 5.48 3.42 3.48 4.47 4.60 5.56 TABLE

III.

Parameters in Eqs.(22)and (23) for Si.

Experimental Theoretical

gb

=

1.015 reAectivity critical-point

f;

(eV )

1;

(eV) co; (eV) structure (eV) analysis (eV)

TABLE

I.

Parameters in Eqs.(22) and (23) forMgO.

gb=1

(1.035 if i

=1

to 11 are used)

f,

(eV')

I;

(eV) co; (eV) TABLEIV. Parameters inEqs.(22)and (23) forGaAs.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 2 24 35 35 24 84 38 28 29 131 143 18 95 500 0.10 1.30 0.90 0.59 0.91 1.40 1.00 0.79 1.58 7.47 11.92 1.96 8.95 29.64 7.8 9.1 11.0 13.3 14.8 17.5 19.4 21.2 24.1 34.0 49.0 58.0 72.0 99.0 1 2 3 4 5 6 7 8 9 10 11 0.8 3.4 5.8 13.0 20.0 13.0 20.0 44.0 59.0 49.0 13.0 0.10 0.20 0.40 0.50 0.40 0.50 0.79 1.40 2.20 2.20 1.70 3.03 3.25 3.70 4.60 5.05 5.76 6.70 8.00 10.50 13.50 21.50 3.02 3.25 4.64 5.11 5.64 6.60 3.03 3.25 4.54 5.07 5.76 6.67 9.87 12.55 Experimental Theoretical gb

=1.

01 reAectivity critical-point

8 I I I I I I I II I 1 I II I 'I I I I I I I i I I I I I I I 5 6 I

Si

3

₂

3

₀

3

3

0

3

2 0

i2

18 24

30

—— 0

36

0 10 15 20 250

FIG.

2. A cornomparison ofg(co), a(co), and Im

—

1

Si calculated using

E

(22) & an

n c,&= _g~(solid curves), measured e

(.

.

.

h.

.

.

.

„.

.

)(=.

f

2)

.

.

".

..

'

(Ref.15).

e

.

and determined by FB(chain curves) es d 2 determined by

FB

(chain

suits, on the other hand, approach constant both optical quantities. Th

tinction coefficient

t

1

~ ~

iies. econstant value

of

the

FB

ex-the integr 1 th

n a arge co is clearl in

a in e sum

ruleofE

.

2

y 'ncorrect because q. ( ) will then diverge. ring wit the experimental data it '

th t th 1'd't

of

FB

1

b lo

-7

V

o results is restricted to h energy-loss function, where

B

resul

e

.

is is true es eciall

f

a energy. Figure 3 shows a lot

of

defined in Eqs. (7)

—

(9),for

or

SiSi calculated presently (solid

curves), measured experimentall determined uusing'

FB

optical uanti

n a y (dashed curves), and

It

is seen that th

quantities (chain curves). a t e e6'ective number

of

valenc

ormulas lead toto unphysical results for n z(co

3 I I 1 I I I I I I I I ' ' ' ' I ' ' ' ' 4 I I I I I II I I I I I MgO Z

=4

V v~b v~b 0 20 40

(e

60 80 0 15 I s s i s I

30

45 60 750

FKx.3. The results ofn,n,s(~)la.

co,

_{,n,a(col~„,}and n,a.

(co)~,

for

i calculated presentl ( 1

d,

'

ex

y soi curves), usin ex

(dashed curves) (Re

f

.21)and using

FB

o tica

'

g experimental data curves) (Ref. 15). The

.

.

edotted line atZ

=4

re r

'

g ptical quantities (chain ration value for the number

represents the satu-enum erofvalence electrons per atom.

~ p n ofg(co), a(co), and Im

—

1/c

FIG.

5. A corn arison

gO calculated using

E

s. (22)

an c2

=

g~(solid curves) and measured

4378

Y.

F.

CHEN, C.M.KWEI,AND C.

J.

TUNG 48

I I

20 s s s &

( &» &

( '' & & &

( s I I I I I ( I I I ( I I I ( I I

2.

0 Z +Z

=16

15

1.

5 10

3

3

0

3

—

_1.

₀ 50 ( & I I I I I I I I I I I I I I 100 150 200 250

300

(eV) —

_0.

₅

FIG.

6. The results ofn,ff(co)~, ,n,z(co)~,and n, (acyl~, for M

0

l lated resently (solid curves) and using experimental data (dashed curves) (Ref. 19). The dotted line at Z&

represents the saturation value for the number ofelectrons con-tributing toanisolated absorpting group.

I ( I I I I i I I

20 40 60 80

100

0.

0

FIG.

7. A comparison ofg(co), ir(co), and Im[

—

I/E(col] for Si02glass calculated using Eqs. (22) and (23)and the relations

El g K and E2=2gK (solid curves) and measured

experimen-tally (dashed curves) (Ref. 20). and

_,

n(sue)~, at large co. The

FB

results on

n,

fr(co)~

2

also deviate greatly compared to experimental data at co&15eV.

Similarly, in Fig. 4 we plot the results

of

q(co), ir(co,

and Im[

—

I/E(ru)] for semiconducting GaAs calculated presently (solid curves), measured experimentally (dashed curves), and determined by

FB

(chain curves).

It

isagain seen that close agreement is found between present re-sults and experimental data for all quantities plotted. The small deviations occurring at cu&22 eV are due to

the contribution from d-band excitations. The onset

of

these excitations is

-25

eV, above which optical data are

unavailable.

It

is also seen that the

FB

results are in agreement with experimental data only forco

(6

eV. The

failure

of

these results around and above plasmon ener-gies is clearly demonstrated.

As an example

of

insulators, in

Fig.

5 we plot

g(co,

ir(ro), and Im[

—

I/E(co)] for MgO. Here the agreement between present results (solid curves) and experimental

data (dashed curves) is quite good for all quantities

plot-ted. Figure 6 shows a plot

of n,

ff(co) calculated presently

( 1'd curves) and measured experimentally (dashed

f

curves) for MgO. Note that the saturation value o

( )~ is near 16instead

of

8, the number

of

valence

net 6) ~ Is nea

electrons. This is due to the onset

of

Mg L-shell

excita-tions near 55 eV; there only about

75% of

the oscillator

strength

of

valence electrons is exhausted. Thus a strong overlapping

of

oscillator strengths between the valence band and the Mg

L

shell exists above 55 eV. Also, the next filled

0

X

shell lies

-500

eV above the valence b

an.

d Thereforee the isolated absorption group should

in-2

.

2 ₂ 4

elude eight valence electrons

(3s of

Mg; 2s and _p o

0)

and eight Mg L-shell electrons (2s and 2p

of

Mg). The merge

of

saturation values

of

all three

n,

( a)scupectra

f

indicates that gb

=1.

It

means that absorption peaks o

Mg and

0

K

shells are far away from critical-point tran-sition peaks

of

the valence band.

Finally, in

Fig.

7we plot optical quantities for another insulator, Si02 glass. Good agreement is found between present results (solid curves) and experimental data

(dashed curves). In this case, the isolated absorption grou~roup involves 16 valence electrons

(3s

an 3_p o

of

Sii; 2s and 2p_p

of

o

0)

and 8Si L-shell electrons (2s2and 2_p6

=

300

of

Si). The saturation

_{of n,}

fr(co) to 24 occurs at

co=

eV.

IV. CONCLUSIONS

We have constructed analytical expressions for the ex-tinction coefficient and the refractive index

of

semicon-ductors and insulators. Our approach involved the

appli-cation

of

f-sum rules and symmetry relations. These

ex-' ~

pressions contain parameters charactenzing the oscillator

strength, the damping effect, and the transition energy

as-sociated with each critical-point transition from the valence to the conduction bands. Contributions from inner shells to the refractive index in the vicinity o critical-point energies were included as a background re-fractive index. Applications

of

present formulations were made for MgO, Si02, Si,and GaAs over a wide range

of

photon energies. Results were in very good agreement with experimental data for all optical quantities studied.

ACKNO%'LED GMENT

This work was supported by the National Science

Council

of

the Republic

of

China under Contract No. NSC82-0208-M-009-013.

*Present address: Departmerit of Nuclear Engineering, Texas A8cM University, College Station, TX77843.

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