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Energy gaprefractive index relations in semiconductors An overview

N.M. Ravindra *, Preethi Ganapathy, Jinsoo Choi

Department of Physics, New Jersey Institute of Technology, Newark, NJ 07102, USA

Received 23 September 2005Available online 12 June 2006

Abstract

An overview of the understanding of correlations between energy gap and refractive index of semiconductors is presented here. Theatomic approach of Moss, the nearly free electron model of Penn, the oscillator concept of Wemple and the optical polarizabilityapproach of Finkenrath are considered in this study. The Ravindra relation is discussed in the context of alternate approaches that havebeen presented in the literature. Case studies of applications of these relations to infrared materials and wide band gap semiconductorsare presented. 2006 Elsevier B.V. All rights reserved.

1. Introduction

The refractive index and energy gap of semiconductors

represent two fundamental physical aspects that character-ize their optical and electronic properties. The applicationsof semiconductors as electronic, optical and optoelectronicdevices are very much determined by the nature and mag-nitude of these two elementary material properties. Theseproperties also aid in the performance assessment of bandgap engineered structures for continuous and optimalabsorption of broad band spectral sources. In addition,devices such as photonic crystals, wave guides, solar cellsand detectors require a pre-knowledge of the refractiveindex and energy gap. Application specific coating technol-ogies (ASPECTTM) [1] including antireflection coatings and

optical filters [2] rely on the spectral properties of materials.The energy gap determines the threshold for absorptionof photons in semiconductors. The refractive index in thesemiconductor is a measure of its transparency to incidentspectral radiation. A correlation between these two funda-mental properties has significant bearing on the band struc-ture of semiconductors. In 1950, Moss [3] proposed a basic

relationship between these two properties using the generaltheory of photoconductivity which was based on the photoeffect studies of Mott and Gurney [4], Smekal [5], Zwicky

[6], Gudden and Pohl [7] and Pearson and Bardeen [8].According to this theory, the absorption of an opticalquantum will raise an electron in alkali halides to anexcited state rather than freeing it from the center. Thermalenergy then moves this electron to the conduction bandfrom the lattice. Such a photo effect takes place in imper-fections at certain lattice points, and thus, the electronbehaves similar to an electron in an isolated atom with adielectric constant of the bulk material. As a result of thiseffective dielectric constant, eEff, the energy levels of theelectron are scaled down by a factor of 1=e2Eff whichapproximately corresponds to the square of the refractive

index, n. This factor, thus, should be proportional to theenergy required to raise an electron in the lattice to anexcited state as given by the Bohr formula for the ioniza-tion energy, E, of the hydrogen atom, E= 2p2m*e4/e2h2,where, m* is the electron effective mass, e is the electroniccharge, e is the relative permittivity and h is the Plank con-stant. This minimum energy determines the thresholdwavelength, ke, which then varies as the fourth power ofthe refractive index. Experimental data on different photo-conductive compounds show that the values of n4/ke wereclose to 77 throughout a range of refractive indices. The

1350-4495/$ - see front matter 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.infrared.2006.04.001

* Corresponding author.E-mail address: [email protected] (N.M. Ravindra).

www.elsevier.com/locate/infrared

Infrared Physics & Technology 50 (2007) 2129
mailto:[email protected]:[email protected]

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similarities in the quotient show that the photoelectronsstem from the same type of lattice imperfections, or alter-natively, the binding energies to the different types ofhydrogen-like centers are similar. Thus the Moss relationwas formulated as [3]:

n4

ke 77=lm 1

where n is the refractive index and ke is the wavelength cor-responding to the absorption edge. In terms of energy gap,this is [9]:

n4Eg 95 eV 2

According to this relation, the refractive index of a semi-conductor can be determined with a known energy gap,Eg. This relation, again, was based on the general assump-tion that all energy levels in a solid are scaled down by afactor of 1=e2eff.

2. Penn model

In 1962, Penn [10] proposed a simple model for an iso-tropic semiconductor with electrons in a sphere of momen-tum space and are characterized by an isotropic energygap. In his investigation of a nearly free electron modelfor a semiconductor, Penn showed that two commonassumptions were flawed in Callaways approximation ofthe static dielectric constant [11]. Previous semiconductormodels did not allow for the formation of standing wavesin the Brillouin zone. The Umklapp process was neglected.The energy, E, and wave function, w, with respect to the

state k for this model are given by,Ek

12fE0k E

0k0 E

0k E

0k0

2 E2g1=2g

wk eikr ak e

ik0r=1 ak 21

=2

where

ak 12Eg=E

k E

0k0

E0k h2=2mk2; k0 k 2KFk

3

Here, E is the electron energy, k is the wave vector, KF isthe Fermi wave vector and a represents an averaged Jonesboundary [12]. The electron energy Eas a function of wavevector k for an isotropic three-dimensional free electronmodel is plotted in Fig. 1. This model of placing the energygap above the Fermi surface was first suggested by Call-away [11] in his investigation of the correlation energy ofelectrons.

Fig. 2 shows the density of states vs. the energy anddescribes the treatment of the Penn model. The dashed linerepresents Callaways model and the solid line representsthe Penn model. According to the Penn model, the statesfrom the energy gap are removed and piled up at the topof the valence band and the bottom of the conduction band[10]. The resulting graph is thus asymptotic.

Based on this model, Penn describes the dielectric func-

tion as:

q 1 8pe2

q2

Xk

Nk1 Nkqjhkjeiqrjk qij2

Ekq Ek

"

X

k

Nk1 Nk0 jhkjeiqrjk0ij2

Ek0

Ek

#4

In this equation, k 0 = k + q (2KF)(k + q)/jk + qj whereK is the reciprocal lattice vector and q is a wave number.N is the occupation number for the states k and k + q. eis the electronic charge. If the Brillouin zone is divided into

sections, then the dielectric function can be written as:

q 1 8pe2

q2

Xki

dikjMij

2

Ei5

where dik 1 when kis in the iregion and 0 otherwise. Mi isthe matrix element. For the specific case when q = 0,

e0 % 1 hxp=E02 1 Eg=EF=4 Eg=EF

2 48

6

where xp is the plasma frequency and EF is the Fermi en-ergy. The expression in the brackets is nearly equal to 1 formaterials with band gaps in the commonly occurring range

where (Eg/EF) = 0.3 [12]. Since the most significant varia-

Fig. 1. Electron energy as a function of k for isotropic three-dimensionalnearly free electron model [10].

Fig. 2. Density of states vs. energy. The solid curve represents the three-dimensional nearly free electron model while the dashed curve represents

Callaways model [10].

22 N.M. Ravindra et al. / Infrared Physics & Technology 50 (2007) 2129

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tion occurs in the expression before the square brackets, (5)can be written as,

e0 % 1 hxp=E02 S0 7

where S0 represents the quantity in brackets in Eq. (6).Penn approximates S0 to be nearly 1, thereby neglecting

the smaller quantities of Eg/EF. The dielectric function isthus:

e0 % 1 hxp=E02 8

Grimes and Cowley [12] found a more accurate value ofS0.In their investigation, Grimes and Cowley found that thevalue of S0 is only weakly dependent on the band gapand that a value of 0.62 is a fairly good representation ofS0. Thus, with this slightly more accurate value for S0,the energy gap can be determined by using appropriate val-ues of the dielectric constant.

In the Penn model, the dielectric constant was found to

have reasonable values for small values of wave number(q). This model is quite simplistic in design, though itaccounts for the formation of standing waves near theBrillouin zone. It does not consider the degeneracy thatmay occur in the Brillouin zone. In spite of the simplicityof the Penn model, the mere isotropy of the system allowsthe model to be applied to a liquid or amorphoussemiconductor.

3. Ravindra relation

The Ravindra relation, which was initially proposedempirically [13] was shown to be an approximation [14]of the Penn model [10]. One feature common to all thesemiconductor band structures is that the valence and con-duction bands are more or less parallel to each other atleast along the symmetry directions. This formed the basisfor Gupta and Ravindra [14] to define:

Ep Eg K 9

Using Eq. (9) in the Penn model in Eq. (8) and consideringthe situation for which Eg/K< 1, Gupta and Ravindra ar-rived at an expression for the refractive index:

n K1 K2Eg K3E2g K4E

3g 10

where the values of K are calculated by Gupta and Ravin-dra [14] and the higher values are neglected since they aretoo small to be considered. Eq. (10) then reduces to a form:

n 4:16 1:12Eg 0:31E2g 0:08E

3g 11

The oscillator model from Wemple [15] defines the dielec-tric constant as: e = 1 + Ed/E0 where e is the real part ofthe complex dielectric constant, Ed is the oscillator strengthand E0 is the average energy gap which is approximatelyequal to the Penn energy gap, Ep.

Performing a similar treatment with the dielectric func-tion in the optical region, they [14] show a similar equation

for the refractive index with the constants evaluated,

n 4:16 0:85Eg 12

Other forms of the Ravindra relation include a variation ofthe relation with respect to temperature and pressure [16]:

dn

dT 0:62

dEgdT

13

dndp

0:62 dEgdp

14

3.1. Moss relation and its implications

It is perhaps noteworthy to mention that other opticalproperties of semiconductors are merely integral relationsto the energy gap and refractive index. Moss [9] pointedout that the absorption edge can be calculated using therefractive index. Moss [9] showed that the refractive indexand absorption edge are related by:

n 1 12p2

Z

K dk1 k

2

k2e

0@ 1A 15which for long wavelengths at zero frequency becomes,

n 1 1

2p2

ZKdk 16

Here, K is the absorption coefficient and ke is the wave-length corresponding to the absorption edge. This functionis integrated over the wavelength rather than energy. Thisexpression shows that the long wavelength refractive indexis determined simply by the total area under the curve of

absorption coefficient vs. wavelength it is independentof the absorption spectrum [18]. As a result, if a certain le-vel of absorption persists over a given energy interval as itoccurs where absorption is due to transitions between twoallowed energy bands then, the smaller the width of theforbidden zone, the greater will be the spread of absorptionin wavelength, resulting in a greater value of the integraland larger n [9].

Moss [9] provided an illustration of this relation in hispaper. This analysis shows that much of the refractiveindex originates from narrow bands of intense absorptions.The absorption occurs at wavelengths below 1 lm whichcorresponds to the very short wavelength compared tothe absorption edge.

With these observations, Moss concluded that the nar-row absorption band of semiconductors in the UV regionis the factor which determines the refractive index. Twoconclusions followed from this assumption: (1) the relationbetween the energy gap and UV oscillator frequency arefactors in determining the relationship between the refrac-tive index and the long wavelength absorption edge; (2) anymodifications of the absorption edge that do not vary atthe same time are not expected to produce a change inthe refractive index.

Moss compared his relation with the Ravindra relation

[9]. The Ravindra relation predicts no results of indices

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beyond 4.1. However, very reliable infrared materials, suchas PbSe and PbSnTe, do exist at refractive indices of 4.7and 7.0 respectively. The Moss formula, however, closelyestimates these values with the formula: n4Eg = 108 eVwhere the constant has been revised. This formula yieldsrefractive indices of 4.6 for PbSe and 6.03 for PbSnTe.

For low refractive indices, the Ravindra relation predictsimpractical results. It predicts that the refractive index willbecome zero when the energy gap is 6.6 eV. The Mossformula on the other hand requires that the energy gapbe infinity for the same result. For the index to be unity,the Ravindra relation predicts the energy gap to be 5 eV.However, there exist materials which exceed this value.The Moss relation requires that the energy gap shouldbe greater than 100 eV to obtain a refractive index of 1.This is much larger than the refractive index of anymaterial.

In a theoretical derivation of optical polarizability, Fin-kenrath [17] described a treatment in the transitions from

the valence band to the conduction band. These transitionsare represented by differential oscillator densities that occurat distributed resonant frequencies. The combined polariz-abilities are summed and replaced by an integral over therelevant frequencies. These differential oscillators are iden-tified through the wave number and occupation probabilitybased on spherical energy surfaces. The result of this studyshowed that (eeff 1)

2Eg is a complex expression that isbasically constant. From this expression, one can see thatneglecting unity compared to the dielectric constant givesthe Moss relation. According to Finkenraths estimate,the average value of the constant is nearly 95 eV. These

observations are made for direct interband transitions. Itmust be understood that, if the transitions are not direct,then deviations could arise from ignoring the transitionsfrom other bands and the varying densities of states andthe matrix elements.

Moss [9] thus showed the interrelations between therefractive index and the absorption. He showed that theimportant factor that determines the index is the frequencyof the UV absorption peak. Reasonable values for therefractive index can be found using the model of a singleclassical oscillator at this frequency. The relation betweenthe refractive index and the absorption edge band energyarises from the comparative nature of the band edge andthe resonance energy. Assuming a constant energy differ-ence, a relation similar to the linear Ravindra relationcan be obtained. However, the Ravindra relation is shownto have a few shortcomings at high and low index values.The Moss relation out performs the Ravindra relation inthese areas and is further supported by formal theory fromFinkenrath. From the different semiconductors known, onecan understand that a relationship between refractive indexand energy gap, or for that matter, other optical constantsthat are integrally related, can only be general and approx-imate at best. From Moss [9], it is seen that where the Rav-indra relation fails, the Moss relation holds approximately

true.

4. Alternate approaches and interpretations

Further relations were developed as a modification oraddition to the Moss and Ravindra relations. While theMoss formula is limited by the structure of the material,the Ravindra relation is restricted by the refractive index.

From the Ravindra relation, the refractive index cannotbe greater than a value of 4.1, which corresponds to anenergy gap of 6.587 eV. In an effort to broaden the applica-tion of these two concepts, several authors have presentedvariations of the Moss and Ravindra relations.

In 1992, a relation similar to the Moss relation was pro-posed by Reddy and Anjaneyulu [19]. According to theirformula, the relation is:

Egen 36:3 17

This relation holds true for energy gaps greater than 0 eV.Reddy and Ahammed [20] proposed an empirical rela-

tion that was a modification of the Moss formula [18]

n4Eg 0:365 154 18

This relation was proposed to overcome some of the draw-backs of the Moss relation. However, it is not valid whenthe energy gap is less than 0.36 eV nor does it hold forinfrared materials such as lead salts and InSb.

Further derivations based on the Moss and Ravindrarelations include approaches by Herve and Vandamme[21]. They separated the semiconductors into a covalentlybonded group and ionically bonded group. For a cova-lently bonded crystal, the dielectric function is:

ex 1 4pNq2

=mx2

0 x2

19

where Nis the density of valence electrons, m the rest massand x0 the UV resonance frequency. Eq. (19) describes thedielectric constant for elements like Si and Ge. For the caseof binary compounds such as NaCl or GaAs, the dielectricfunction needs to account for the ionic bonding. Thus thedielectric function in an ionic diatomic lattice is,

e0x e0xop X2p=x

2T x

2 20

where Xp is similar to the plasma frequency for ions and xTis the infrared resonance frequency. To account for materi-

als in the optical range, xT < xop < x0, Herve and Vand-amme proposed,

n e0xop1=2 n

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

A

Eg B

2s21

where A is the hydrogen ionization energy %13.6 eV and Bis 3.47 eV. Herve and Vandamme based this equation onoscillatory theory, assuming the UV resonance energy hasa constant difference with band gap energy,

hx0 Eg B 22

Herve and Vandamme claim that their model provides the

lowest deviation for IIIV, IVII and chalcopyrites. This


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model is accurate for most materials used in optoelectronicdevice structures and high band gap materials. Yet it doesnot explain the behavior of the IVVI group.

In the Moss, Ravindra and Herve and Vandamme rela-tions, the variation of the energy gap with temperatureleads to three relations. By differentiating their relation

with respect to temperature, we arrive at:dn

dT

13:62

nEg B3

dEgdT

B1

23

where B1 dBdT

Allowing, Eg B ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

13:62

n2 1

q,

1

n

dn

dT

n2 13=2

13:6n2Eg B3

dEgdT

B1

24

Similarly, the Moss and Ravindra relations were differenti-ated to arrive at temperature coefficient for the energy gap.

A few values of (dEg/dT) and (dn/dT) calculated usingthis relation for some semiconductors are given in Table 1.

In conducting this study, Herve and Vandamme [22]found two particular temperature dependences. Generally,the energy gap and temperature are inversely proportionalfor most semiconductors, that is, as T increases, the energygap decreases, and, as a result, the refractive indexincreases. This occurs in diamond and similar structures.On the other hand, infrared detector materials includingthe lead sulphide, lead selenide and lead telluride exhibita positive temperature coefficient of energy gap and nega-tive refractive index temperature dependence.

Herve and Vandamme [22] found that the Moss relationshowed the strongest deviation at lower energy gaps(

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independent of temperature in the measured range. Theslope of the curvature of the absorption coefficient changesdramatically with the increase of Sr in the compound. Sucha change in curvature of the absorption coefficient repre-sents a change in band gap.

The absorption coefficients for direct and indirect tran-sitions are calculated as follows:

adirect A

ht1=2

ht Eg1=2

aindirect B

htht Eg Ep

2

25

Here, A and B are constants, ht is the photon energy andEp is the phonon energy. The phonon energy is much lowerthan the energy band gap and so is neglected. These equa-tions only hold true in the region where the photon energyis greater than the energy band gap. Taking the secondderivative of the absorption coefficient (a) (25) with respectto the photon energy yields,

d2adirect

dht2

AEg

4f4ht3 3ht2Eggfht

4

ht3Egg

3=2; 26

which is negative for ht > Eg and,

d2aindirect

dht2 2

BE2g

ht327

which is positive for ht > Eg. As the absorption coefficientchanges from negative to positive, the band gap changesfrom direct to indirect due to the increase in strontiumcomposition [23]. Fig. 5 shows the direct and indirect bandgap plotted against the strontium composition at 77 K forPb1xSrxSe.

The following equations were found to describe the bestfit for the curve of the calculated direct and indirect band

gap energies:

Egdirect 0:278 1:356x 1:040x2 1:144x3 eV

Egindirect 0:29 1:527x eV

28

As can be seen, the direct band gap shows a dependence ona polynomial function of the strontium composition (x)while the indirect band gap is linear in nature with respectto the strontium composition. In this experiment, Majum-dar et al. found the indirect band gap of SrSe at 77 K to be1.82 eV and the direct band gap to be 3.81 eV. For PbSe,the direct band gap occurred at 0.278 eV at 77 K. Majum-dar et al. assumed that the lowest indirect band gap energyfor all strontium compositions (0.431) occurs at similarbands which separate linearly as the amount of strontiumincreases. This occurs as the indirect band gap of

Pb1xSrxSe changes linearly with the strontium composi-tion. When the indirect band gap energy curve of lowerSr composition is extrapolated, the direct energy bandgap curve crosses at x $ 0.2. This suggests that atx $ 0.2 1, the lowest energy band gap of Pb1xSrxSe isdirect.

Majumdar et al. have thus, determined the refractiveindices and direct and indirect band gaps of Pb1xSrxSefor strontium compositions ranging from 0 to 1. Usingthe data obtained in this experiment, it would be possibleto aid further development of optoelectronic devices basedon Pb1xSrxSe.

5.2. HgZnTe (MZT)

Ali Omar and El-Akkad [24] performed a study on theoptical parameters of mercury zinc telluride (MZT) in solidsolution. MZT is a direct band gap material with an energygap between 0.15 and 2.20 eV depending on the compo-sition ratio. An equation describing the energy gap is,

Eg a0 a1x1=2 a2x a3x

2 a4x3 eV 29

The best fit was found to be,

Eg 0:1016 1:978x 0:3144x2 30

Fig. 4. Refractive index of Pb1xSrxSe for different strontium composi-tions at room temperature and liquid nitrogen temperature (77 K) at awavelength of 4 lm. Inset shows refractive index of SrSe at 77 K [23].

Fig. 5. Direct and indirect band gap energies for Pb1xSrxSe at differentcompositions of strontium at 77 K [23].


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where x represents the percent composition. According toNag [25], an empirical formula illustrating the dependency

of energy gap on percent composition is given by,

Eg a bZ2 31

where a and b depend on the semiconductor group. Whenthis formula is applied to the ternary solution, Eg(x)becomes:

Egx 0:15 0:8637x 1:245x2 eV 32

Fig. 6 describes the results of Eqs. (29)(32).Relating the energy gap to the refractive index, Ali

Omar and El-Akkad refer to the Ravindra relation. With

reference to the best fit, this equation is:

n 4:036 1:127x 0:262x2 33

In the range of 0 6 x 6 1 the quadratic approximation is

nCM 3:84 1:56x 0:428x2 34

From graphical illustrations of the refractive index vs. per-cent composition, as in Fig. 7, the refractive index is seen todecrease with x.

Studying the relation between these parameters, compo-sition and Eg for MZT allows for an understanding and useof the material for IR detectors and solar cells.

5.3. GaN

Bourissa [26] performed a study on the pressuredependence of optoelectronic properties of GaN in thezinc-blende structure. The refractive index was calculatedat different pressures with the Moss, Ravindra andHerve Vandamme relations. On a graph of the band lineupvs. pressure, the conduction band minimum is seen toincrease with the increase in pressure at C yet it is seen todecrease at X. The valence band maximum decreases at C(Fig. 8).

The lowest direct energy gap increases with increas-ing pressure while the indirect gap is seen to have asmaller dependence on pressure. As a result, the directband gap of zinc-blende GaN cannot be expected to cross-

over to indirect behavior in the pressure range studied(Fig. 9).

The dependency of band gap on pressure can be repre-sented by,

E a bDa

a0

c

Da

a0

235

where Da = ap + a0 and ap and a0 are lattice constants atpressure p and zero pressure. Applying this to the equationdescribing least square fit, namely,

Ep E0 dEp

dpp 36

Fig. 6. Eg vs. percent composition. TE refers to the energy gap equation(29) and QF refers to the best fit equation (30). N refers to the ternarysolution equation (32) [24].

Fig. 7. Refractive index vs. composition ratio. R refers to refractive index

with respect to best fit, CM refers to the quadratic approximation [24].

Fig. 8. Band lineup in eV vs. pressure in (kbar). The valence bandmaximum C is indicated by () while the conduction band minimum C is

indicated ( ). (-%-) indicates the conduction band X bottom [26].


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is then,

E0 3:2 23:83Da

a0

195:62

Da

a0

2

EXX 6:91 3:24Da

a0

240:65

Da

a0

2

ELL 7:23 23:60Da

a0

106:10

Da

a0

2

E

X

g 4:70 0:16

Da

a0

1:22

Da

a0 2

ELg 6:20 25:48Da

a0

166:76

Da

a0

2

37

These dependencies show a non-linear behavior. FromFig. 9, the valence band width is shown to increase withpressure. This can be attributed perhaps to decreasing ion-icity under hydrostatic compression which is typical ofsemiconductors.

The refractive index obtained by the three models is alsoplotted as a function of pressure in Fig. 10.

In Fig. 10, the refractive index is seen to decrease line-arly with respect to increasing pressure, as opposed to theenergy gap which shows the opposite behavior. It can be

concluded that for GaN and other IIIV compounds, thesmaller the energy band gap, the larger the refractive index.

In Table 2 are the refractive indices at zero pressure andthe first pressure derivative. The refractive index obtainedby the Moss relation shows a large difference when com-pared with the Ravindra and Herve Vandamme relations.Since no known experimental data has been published onthe refractive index of GaN, Bourissa calculated the indexusing e

1= n2. The results of the refractive index for the

three relations were 5.81, 4.41 and 5.24 respectively.Compared to the experimental value of 5.15 for the

dielectric constant obtained by Yu et al. [27], the value

5.24 as obtained by the Herve Vandamme relation is a bet-ter result. However, the value of the dielectric function of5.8, theoretically calculated by Zheng et al. [28] is almostin agreement with the Moss relation.

The pressure derivative of the refractive index is anothersignificant parameter which explains pressure induced dis-tortion. The first pressure derivative calculated throughthe Ravindra relation indicates a greater effect of appliedhydrostatic pressure to the refractive index. Results of cal-culations show that the pressure derivative is negative andthis appears to be the trend among the IIIV group. How-ever, a study by Johanssen et al. [29] shows a direct linearrelation between refractive index and increase in pressure.Thus, it is believed that ionicity of the material plays a sig-nificant role in determining the magnitude and sign of thepressure coefficient of refractive index.

6. Conclusions

The various energy gaprefractive index relations andtheir applications to semiconductors have been summa-rized in the above study. The analysis examines the contri-butions of Moss, Penn, Finkenrath, Wemple andRavindra. Applications of these relations to IR detectormaterials such as Pb1xSrxSe and Hg1xZnxTe and UV

detector material such as GaN have been discussed.

Fig. 9. Direct and indirect band gaps of GaN as a function of pressure in(kbar) [26].

Fig. 10. Refractive index as a function of pressure for GaN. () indicatesMoss relation, ( ) represents Ravindra relation and (-%-) is the Herve

Vandamme relation [26].

Table 2Refractive indices at zero pressure and the first pressure derivative forGaN [26]

Model n dndp 103 kbar1

Moss relation 2.41 0.67Ravindra et al. relation 2.10 2.40Herve and Vandamme relation 2.29 1.13


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Acknowledgements

N.M. Ravindra is indebted to Dr. T.S. Moss for hisenormous appreciation of the fundamental physics of semi-conductors and beyond. He appreciates very much hisinteractions with Dr. Moss for over 15 years which contin-

ued without any impediment from physical or geographicalboundaries. Dr. Moss will be dearly missed but his contri-butions to fundamental semiconductor physics will go along way. The work reported in this paper is dedicated toDr. T.S. Moss.

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