99

Chapter 5

Free carrier absorption in III – nitride

semiconductors

5.1 Introduction

The absorption of electromagnetic radiation, due to its interaction with electrons in

semiconductors, is essentially determined by two distinct processes: interband and intraband

transitions. The electronic transitions may be direct or indirect according as the band extrema

occur at the same k or different k points in the Brillouin zone. The minimum photon energy

for interband absorption defines the fundamental absorption edge. In the second process,

namely, free-carrier absorption (FCA), electrons in the conduction band and holes in the

valence band are excited to higher energy states in the same band.

FCA, is essentially an intraband indirect transition phenomenon. It accounts for the

absorption of electromagnetic radiation of frequencies lower than those which give rise to

interband transitions and necessitates the mediation of phonons or imperfections to conserve

Part of the work presented in this chapter has appeared in

Journal of the Physical Society of Japan, 82, 043706 (2013)

AIP Conf. Proc., 1391, 72 (2011)

Proc. of 15th, International Workshop on Physics of Semiconductor Devices, XV, 579 (2009)

100

overall momentum. In the presence of phonons or imperfections a free carrier may absorb a

photon via a second order process in which the carrier changes its momentum by scattering.

FCA, intrinsically connected to scattering, is therefore a powerful means of determining the

possible scattering mechanisms operative and understanding transport properties in a system

[5.1].

In the last few years III–nitrides and their alloys have attracted attention as potential

material systems for use in high speed opto–electronic devices such as blue light-emitting

diodes (LEDs), blue lasers and solar blind ultraviolet photodetectors [5.2, 5.3]. The commonly

studied nitride–based compound semiconductors are GaN, InN and AlN whose room

temperature minimum band gaps range from 0.7 eV for InN through 3.4 eV for GaN to 6.2 eV

for AlN [5.3]. Among these, GaN, because of the large direct band gap and its chemical and

physical stabilities, has emerged as an attractive candidate for devices operating at high–

temperatures, high–voltages and high–power at microwave frequencies.

However, nitride semiconductor material systems, including GaN, InN and AlN are

known to be characterized by large built-in piezoelectric fields as well as the presence of

unintentional imperfections such as threading dislocations introduced during growth process,

due to lattice mismatch between substrate and active material. Threading dislocations can be

of the screw, edge and mixed types. Edge dislocations formed at the substrate-buffer interface

with mainly vertical orientation thread to the epilayer surface whereas the number of screw

and mixed dislocations are known to decrease with distance from the interface. The order of

edge dislocations is found to range from 107 cm

-2 to 10

10cm

-2 [5.2, 5.3]. However, in nitrides,

the dislocations parallel to the c – axis do not couple any piezoelectric potential [5.4]. These

edge dislocations are found to limit carrier transport properties considerably in nitride

101

quantum well systems [5.4] and are expected to influence performance of optical devices as

well.

The influence of dislocations on the optical absorption spectra of nitrides is found to

manifest in significant loss coefficients, and in enhancement of optical absorption near

fundamental absorption edge and a red shift of the absorption edge [5.5 – 5.10]. In GaN and

InGaN systems, Kioupakis and coworkers [5.11] employing the first principles approach, and

considering scattering by phonons, charged-defect and alloy scattering, have found FCA an

important loss mechanism. The peculiarities of FCA in InN epitaxial layers with wide range of

electron concentrations have been investigated by Nargelas et al [5.12] and Wu et al [5.13].

With a view to estimate the influence of dislocations on the FCA in nitride systems, we

have, in this chapter, made a study of FCA first in 2D QW systems and then in bulk nitride

systems. In section 5.3, we present our theory, developed for the first time, for FCA assisted

by dislocation scattering via Coulomb and strain fields in quantum wells. The absorption of

radiation by a free carrier is treated by a second-order perturbation in which the interaction of

the 2DEG with dislocations and with the radiation field is considered simultaneously.

Investigations of FCA in bulk nitrides are lacking. In section 5.4, we present a systematic

analysis of FCA coefficient in bulk GaN data considering electrons to be scattered by acoustic

phonons via deformation and piezoelectric interactions, impurities, optical phonons and

dislocations.

5.2 Theory

The FCA coefficient, K, is given by [5.1]

, (5.1)

102

where κ, is the dielectric constant of the medium, no the number of photons in the radiation

field and fi the carrier distribution function. The sum is over all initial states ‘i’ of the system.

Wi+ and Wi

- represent the transition probabilities for the absorption and emission of photons,

respectively and can be calculated using the standard second order Born golden rule

approximation. These are defined by

. (5.2)

In (5.2) the transition matrix elements are given by

, (5.3)

where Hrad is the electron-photon interaction Hamiltonian and VD the electron-dislocation

scattering potential. ħΩ is photon energy and Ei , Ej and Ef denote the initial, intermediate and

final state energies of electrons, respectively. The sum is over all the intermediate states j of

the system.

The expression for K can be evaluated using the wavefunction, eigenvalues and the

appropriate distribution function. The total FCA coefficient, K, due to various scattering

mechanism (s) is given by [5.1]: .

5.3 Free carrier absorption in nitride quantum wells

The edge dislocations are known to grow normal to the plane of the quantum well. The

electrons in quantum well are scattered by the dislocations via strain and coulomb interactions.

The scattering from the long range strain field surrounding the dislocations may be via

deformation and piezoelectric potentials.

In the case of 2D systems, FCA has been theoretically studied extensively in

GaAs/AlGaAs structures [5.14 - 5.16]. There are no systematic studies of FCA due to

103

dislocation scattering even in nitride quantum structures except for the investigation of Wu et

al in InN layers [5.13].

We consider a 2DEG in a square QW system with the electrons assumed to be

confined to move in the x-y plane. For simplicity, we consider the QW to be of infinite depth.

The electron wave functions and energy eigen values are given by [5.14]

, n = 1, 2… (5.4)

and

, (5.5)

where, is position vector, is electron wave vector, d is width of

the QW, n is the subband index and .

For a non-degenerate quasi 2D electron gas, the distribution function can be expressed

as [5.14]

, (5.6)

with,

, and ns is the sheet carrier concentration.

Assuming electromagnetic radiation to be polarized along the plane of the QW the

matrix elements of the electron–photon interaction Hamiltonian can be expressed as [5.14]

, (5.7)

where, e is unit vector in the direction of polarization of the radiation field.

5.3.1 Expression for dislocation-assisted FCA coefficient

FCA, which is intrinsically associated with carrier scattering through second order

process, requires a quantitative description of the dislocation scattering processes in a 2D

104

system. The scattering of the carriers by the edge dislocations in these systems is due to the

coulomb potential produced by the charges on the dislocation lines and the long-range strain

field surrounding the dislocation lines [5.4]. Dislocation scattering of 2DEG in QW systems

has been studied by many workers, considering the interaction of electrons with the strain

fields surrounding edge dislocations via deformation potential coupling [5.17, 5.18] and with

charged dislocations via coulomb potential [5.18]. Since the electric fields generated by

dislocations do not extend over large distances and are rather localized around the core of

dislocations, the screening of the interaction is weak. In the present work we consider the

interactions to be unscreened.

5.3.1.1 Dislocation scattering via strain field

The effect of strain field around dislocations is to shift the conduction and valence

band edges. The perturbing potential for electron scattering can be expressed as [5.17]

, (5.8)

where θ is the polar angle with respect to Burgers vector, b along QW plane, q is in-plane

wave vector, ac, the conduction band offset and γ the Poisson ratio.

Using eqns. (5.4) and (5.8) the expression for the matrix elements of electron-

dislocation scattering via strain interaction is expressed as

, (5.9)

where

, and is the angle between b and q. The

screening function with

. In the absence of screening

.

105

Using (5.1), (5.4), (5.6) and (5.9), we obtain the following expression for FCA

coefficient due to scattering by dislocations (of density Nd), via unscreened strain field

, (5.10)

where

,

,

and is exponential integral function and Nd is

dislocation density.

5.3.1.2 Dislocation scattering via Coulomb interaction

Modelling a threading dislocation, growing perpendicular to the QW plane, as a line of

charge with charge density ρL, we obtain the expression for interaction potential of electrons

with charged line dislocations as

,

(5.11)

where κw and κb are the dielectric constants of the material in well and barrier, respectively.

Assuming. κw= κb and using (5.11), we obtain the expression for the matrix element of

electron-dislocation scattering via coulomb field as

(5.12)

Here , , , and

.

106

Using (5.1), (5.4), (5.6) and (5.12), we obtain expression for FCA coefficient due to

scattering by dislocations (of density Nd) via unscreened coulomb interaction as

. (5.13)

Here

and

5.3.2 Results and Discussion

We have performed numerical calculations of FCA coefficient, K using equations

(5.10) and (5.13) considering scattering of electrons by edge dislocations. Here, we present the

results for three nitride quantum well systems of GaN/AlGaN, InN/AlN and AlN/AlGaN. The

material parameters, characteristic of GaN, AlN and InN, used in our calculations are given in

table 5.1.

Table 5.1:Material parameters of GaN, InN and AlN [5.3]

Parameter GaN InN AlN

Effective mass, m* (mo) 0.22 0.115 0.48

Conduction band offset, ac(eV) - 8.0 - 3.0 - 4.31

Burgers vector, b (Ǻ) 3.189 3.540 3.110

Poisson ratio, γ 0.3 0.42 0.287

Dielectric constant, ε 9.5 15.3 8.5

Lattice constant, c (Ǻ) 5.185 5.705 4.98

107

We choose to illustrate the behavior of FCA for quantum wells of width d = 100 Ǻ

and carrier concentration ns = 1 x 1011

cm-2

. The dislocation line charge density, , is taken as

where f is the fraction of filled acceptor states and co , the lattice spacing in the (0001)

direction. We take Nd = 108 cm

-2, and assume f =1.

Figures 5.1, 5.2 and 5.3 depict respectively, the frequency dependence of the

dislocation–mediated FCA coefficient in GaN/AlGaN, InN/AlN and AlN/AlGaN quantum

well systems at T = 300K. In each of the figures 5.1, 5.2 and 5.3, curves a and b represent the

contributions to FCA from the strain field and the Coulomb field, respectively. Curves 1

denote the overall contributions. The following common features are noticed. For the

parameters and range of frequencies considered, the FCA coefficient decreases with increase

in photon frequency, . A kink is observed whenever the photon frequency equals

corresponding to transition to the second subband. It may be noted that, this kink occurring

along with the FCA process is peculiar to QW THz systems. This is incontrast to the bulk case

where, FCA and interband transitions occur as separate processes. The dominant contribution

to overall dislocation-mediated FCA is due to scattering via the strain field of the dislocations.

In the case of GaN QW, the large contribution to the total FCA (curve 1) from the strain field

(curve a) may be due to large conduction band deformation potential as compared to that of

InN and AlN. It is also noticed that, the position of the kink shifts towards higher (lower)

energy region for InN (AlN), as compared to that in GaN. This shift may be due to the varying

values of the effective masses of electron in the three systems (see table I). The value of the

FCA coefficient is found to be enhanced at the subband transition energy with the increase in

108

4 6 8 10 12 14 16 18 20

0.1

1

10

100GaN

b

1a

K (

10

3 c

m-1)

(1013

s-1)

Figure 5.1: Variation of dislocation mediated FCA coefficient, K, as function of photon frequency, ,

for the GaN quantum well of width d = 100 Ǻ, carrier concentration ns = 1 x 1011

cm-2

and dislocation

density Nd = 1 x 108 cm

-2. Curves a and b depict K due to dislocation scattering via strain and coulomb

interactions, respectively. Curve 1 represents the total K.

4 6 8 10 12 14 16 18 20

2

4

6

8

10

12

1

InN

b

a

K (

10

3cm

-1)

( x 1013

s-1)

Figure 5.2: Variation of dislocation mediated FCA coefficient, K, as function of photon frequency, ,

for the InN quantum well of width d = 100 Ǻ, ns = 1 x 1011

cm-2

and Nd = 1 x 108 cm

-2. Curves a and b

represent K due to dislocation scattering via strain and coulomb interactions, respectively. Curve 1

depicts the total K.

109

4 5 6 7 8 9 10

0.5

1.0

1.5

2.0

2.5

3.0

3.5

1

AlN

b

a

K (

10

3cm

-1)

( x 1013

s-1)

Figure 5. 3: Variation of dislocation mediated FCA coefficient, K, as function of photon frequency, ,

for the AlN quantum well of width d = 100 Ǻ , ns = 1 x 1011

cm-2

and Nd = 1 x 108 cm

-2. Curves a

and b depict K due to dislocation scattering via strain and coulomb interactions, respectively. Curve 1

represents the total K.

absorption coefficient near the kink found to be larger for InN compared to that of AlN and

GaN. A comparison of the behavior of FCA in QWs due to scattering of 2DEG by dislocation

strain with that in bulk systems [5.13] shows the magnitudes of K to be larger in QW system.

This may be because the increased localization of charge within the 2DEG, could enhance

scattering of the 2DEG [5.17].

We have also investigated the influence of QW width and dislocation density on FCA.

In figure 5.4, curves 1, 2 and 3 show variation of FCA coefficient in GaN QWs with

Nd =108

cm-2

for well widths, d = 125 Ǻ, d = 100 Ǻ and

d = 75 Ǻ, respectively. It may be noted

that, the position of kink at redshifts with increase in well width. This is because the

110

Figure 5.4: Variation of dislocation mediated FCA coefficient, K as function of photon frequency, ,

for the GaN quantum well of ns = 1 x 1011

cm-2

. Curves 1, 2 and 3 depict total K for Nd = 1 x 108 cm

-2

well widths, d = 125 Ǻ, d=100 Ǻ and d = 125 Ǻ, respectively. Curves 2, 4 and 5 depict the variation

of total K with dislocation density, Nd = 1 x 108 cm

-2, Nd = 1 x 10

9 cm

-2 and Nd = 1 x 10

10 cm

-2,

respectively for QW of width d = 100 Ǻ.

confined states of the well are closer in energy for larger well widths. Our results are

consistent with experimental studies of Bayram in AlGaN/GaN superlattices [5.19]. Curves 2,

4 and 5 depict the variation of FCA coefficient for the GaN QW of width d = 100 Ǻ for

dislocation densities, Nd = 1 x 108 cm

-2, Nd = 1 x 10

9 cm

-2 and Nd = 1 x 10

10 cm

-2,

respectively. An increase in number of dislocations results in an increase in FCA. This follows

from eqns. (5.10) and (5.13), indicating that larger loss coefficient can result from higher

dislocation densities [5.6]. It may be noted that a similar dependence is exhibited by 2DEG

scattering rate [5.4, 5.18], a first order scattering process.

In conclusion, we have developed a theory of edge dislocation assisted FCA in QWs

assuming the scattering via coulomb and strain fields. Calculations of frequency dependence

111

of FCA coefficients in QWs of three nitride systems – GaN ,InN and AlN – are presented.

The dominant contribution to the FCA is found to be due to the strain field of the dislocations.

The frequency dependence of FCA coefficients exhibits kinks whenever the frequency

corresponds to transition to the second subband. The position of kinks shows redshift with

increase in quantum well width. The theory shows larger loss (absorption) coefficients due to

increase in dislocation density [5.7]. It may be mentioned here that, the inclusion of the

screening of the interactions is expected to reduce the FCA coefficients. An estimate of the

effect of screening for large angle scattering (q = 2kF; , being Fermi

wavevector) in the QWs of GaN, InN and AlN indicates a reduction in the FCA coefficients

by approximately 20%, 13% and 42% respectively. It may be mentioned that, with proper

control of the parameters characterizing dislocations, one may obtain information about the

band structure of the QW system especially in those regions of the subbands might be

explored which cannot be reached by the electrons in an experiment on transport phenomena.

5.4 Free carrier absorption in bulk nitrides

Light absorption in nitrides arises because of scattering of free carriers from inevitable

acoustic phonons, via deformation potential and piezoelectric interaction, optical phonons and

unintentional impurities, and dislocations introduced during growth process. In particular,

FCA depends on the individual contributions due to various scattering mechanisms operative

in the system. Here, we are interested in investigating the influence of dislocation scattering on

FCA in bulk nitrides.

In literature, there exist, investigations of FCA in bulk semiconductors with regard to

dislocation scattering [5.6, 5.7]. However, the role and importance of the contribution from

112

dislocation scattering to FCA in GaN seems to be unclear. Cunninghaum et al [5.20] have

measured room temperature IR spectra for 3 GaN samples in the region of 1 < λ < 3.5μm.

Their measurements showed a characteristic FCA described by a wavelength dependence of

absorption coefficient s , with s ranging from 2.2 to 3.9, characteristic of optical mode(s

~ 2.5) and impurity (s ~ 3.5) scatterings. Vignaud and Farvacque [5.6] proposed that the low-

energy component of optical absorption in GaAs induced by strong electric fields resulting

from charged dislocations, may be observed only at low temperature. Ambacher et al [5.21]

used photothermal deflection spectroscopy to study the sub-band gap absorption of GaN thin

films in the range 0.6 – 3.6eV. They correlate the FCA below 1.5eV with electron

concentration and find s ~ 2. Hasegawa et al [5.22], who have recently measured optical

absorption spectra for plastically deformed n-GaN, have observed, in the long wavelength

region (λ > 1μm), a decrease in FCA by deformation.

The electron wavefunction and eigenvalues for a bulk semiconductor are (see (1.2) and

(1.3))

(5.14)

and

. (5.15)

The distribution function for a non-degenerate electron gas is

(5.16)

The FCA coefficient K, for bulk semiconductor can be obtained using expressions

(5.1), (5.2), (5.3) and (5.16) as [5.23]

113

(5.17)

where,

,

,

and being photon and phonon energies, respectively, and

.

In (5.17) the sum is over all scattering mechanisms, s, and , represents the matrix elements

for electron–phonon, electron–impurity, or electron–dislocation scattering interaction

Hamiltonian. The expressions for the electron-radiation and electron-imperfection scattering

interaction Hamiltonians are documented in literature [5.23, 5.24].

5.4.1 Analysis of FCA data in bulk GaN

We have performed numerical calculations of FCA coefficient, K using (5.17) for

parameters characteristic of bulk GaN (see table 5.1) [5.1] at T=300K in range of wavelength,

λ, 1 – 50μm. The other parameters used are typical of the GaN sample of Hasegawa [5.22]:

impurity concentration Ni = 5 x 1024

m-3

, dislocation density Nd = 1014

m-2

, and optical

phonon energy, ħωq = 91 meV.

114

100

101

102

103

104

105

106 d

1e

c

b

a

K (

m-1)

(m)

Figure 5.5: Variation of FCA coefficient, K as a function of wavelength of incident radiation for bulk

GaN. Curves a, b, c, d and e respectively show individual contribution to FCA due to polar optical

phonons, dislocations, acoustic phonons via deformation, piezoelectric couplings, and impurities.

Curve 1 represents total contribution.

Figure 5.5 depicts variation of FCA coefficient as a function of wavelength, λ of the

incident radiation. Curves a, b, c, d and e represent the variation of FCA coefficient due to

scattering electrons by polar optical phonons, dislocations, acoustic phonons via deformation

and piezoelectric couplings, and impurities, respectively. Curve 1 represents the total

contribution to FCA. We find FCA increases with increase in the wavelength of incident

radiation. Dependence of absorption coefficient K, on λ is found to be s ~ 2.3. For the

parameters chosen, absorption due to acoustic phonon scattering via both deformation and

piezoelectric couplings is dominant in the range of wavelength, 1–5μm. Absorption due to

impurity scattering becomes important at wavelengths, λ > 5μm. Effect of dislocations on

FCA is minimal. For the wavelength, λ > 30μm, absorption due to polar optical phonons

becomes important.

115

100

101

103

104

105

106

107

108

109

b

c

a

K (

m-1)

(m)

Figure 5.6: Variation of FCA coefficient in bulk GaN at 300K for three impurity concentration: 1019

m-3

(curve a), 1023

m-3

(curve b) and 1025

m-3

(curve c), m-3

.

In figure 5.6, curves a, b, and c show variation of FCA coefficient for impurity

concentration 1019

, 1023

and 1025

, m-3

at T = 300K respectively. For the parameters

considered, we find that, an increase of impurity concentration from 1023

to 1025

m-3

increases,

the absorption by one order of magnitude at λ = 1μm. The effect of impurity concentration is

large for higher wavelengths.

Figure 5.7 shows a comparison of our numerical results of FCA coefficient with the

measured data of Hasegawa [5.22] in GaN in the range of wavelengths 1 – 2 μm. For the

parameters considered a good fit is obtained. For the GaN sample considered we take impurity

concentration, Ni = 5 x 1024

m-3

and dislocation density Ndis = 1014

m-2

[5.2]. Curves a – e

denote individual contributions to FCA coefficient due to scattering of electrons from acoustic

116

1.0 1.2 1.4 1.6 1.8 2.0

0

1

2

3

4

5

c

b

a

d e

1

K (

mm

-1)

(m)

Figure 5.7: Variation of FCA coefficient in bulk GaN. Curves a, b, c, d and e represent the individual

contribution due to acoustic deformation, piezoelectric, impurity, dislocation and polar optical phonons

scatterings, respectively. Curve 1 denotes total contribution. Circles denote experimental data of [5.22].

phonons via deformation, and piezoelectric, impurities, dislocations and polar optical phonons,

respectively. Curve 1 represents total contribution. Circles denote measured data. K increases

with increase in wavelength of incident radiation. Dependence of K on wavelength is 2.3 and

it agrees with experiment. K, is found to be of the order of few mm-1

.

We find that, for the range of wavelengths the considered, contribution from acoustic

phonon scattering via deformation potential coupling is large compared to that from

piezoelectric phonon and impurity scatterings. The wavelength dependence for acoustic

phonons via deformation potential and piezoelectric couplings is same. FCA due to dislocation

and polar optical phonon scattering is small. It is also found that, the effect of impurity

scattering is more at higher wavelengths.

117

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