Energy gaps and optical properties for the quaternary AlxGayIn1−x−yN matched to GaN substrate

Materials Chemistry and Physics 115 (2009) 122–125

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Energy gaps and optical properties for the quaternary Alx Gay In1−x−y N matched to GaN substrate A. Hafaiedh, N. Bouarissa ∗ Department of Physics, Faculty of Science, King Khalid University, P.O. Box 9004, Abha, Saudi Arabia

a r t i c l e

i n f o

Article history: Received 19 January 2008 Received in revised form 6 November 2008 Accepted 17 November 2008 PACS: 71.15.−m 71.20.−b 78.20.Ci

a b s t r a c t Information on the energy band gaps, the lattice parameters and the lattice matching to available substrates is a prerequisite for many practical applications. A pseudopotential formalism within the virtual crystal approximation is used to the zinc-blende Alx Gay In1−x−y N quaternary alloys lattice matched to GaN substrate to predict their energy band gaps and optical properties. The range of compositions for which the alloy is lattice-matched to GaN is determined. Very good agreement is obtained between the calculated values and the available experimental data. The compositional dependence of direct and indirect band gaps and dielectric constants has been investigated. The information derived from this study may be useful for ultraviolet optoelectronic applications. © 2008 Elsevier B.V. All rights reserved.

Keywords: Electronic structure Optical properties Quaternary nitrides Lattice matched and mismatched alloys Pseudopotential calculations

1. Introduction Alloying in the group-III nitrides has diversified the properties of semiconductor materials, enabling their production with wide band gaps. These semiconductors are used to produce commercially important high-performance electronic and optoelectronic devices and systems, such as light emitting devices covering many regions of the visible spectrum. The availability of the quaternary alloy permits an extra degree of freedom by allowing independent control of the band gap, Eg , and the lattice constant, a0 . Tailoring of these compounds could lead to new semiconductor materials with desired band gaps over a continuous broad spectrum of energies [1–9]. The potential of the III-nitride system can be significantly enhanced with the ability to grow the quaternary alloys Alx Gay In1−x−y N. These alloys have emerged as interesting materials for device applications because of their energy band structure and lattice parameters. They are a better choice over AlGaN in many UV optoelectronic applications because of their high emission intensity which can be much higher than that of AlGaN [10]. Recently, the use of quaternary AlInGaN alloys has been proposed as a replacement to the GaN in multiple quantum wells involving semiconductor-based

∗ Corresponding author. E-mail address: n [email protected] (N. Bouarissa). 0254-0584/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.matchemphys.2008.11.036

nitrides [11]. Furthermore, the use of quaternary films enhances the experimental capability for investigating the effects of strain and piezoelectric fields in quantum wells [12]. From experimental point of view, progress has been already achieved in the study of nitride quaternaries [4,12,13]. However, only few attempts have been reported on theoretical predictions of their physical properties [4,10,14]. So far, most of the work has been conducted on hexagonal (wurtzite) structures. Fernández-Garrido et al. [15] have measured the photoluminescence and optical absorption spectra in wurtzite Inx Aly Ga1−x−y N molecular beam epitaxy (MBE)-grown layers. They have reported the enhancement of the photoluminescence in quaternary three-nitrides by increasing the Al content. Their results suggested that the efficiency of visible and ultraviolet light emitting diodes and light diodes may be improved by taking advantage of the Al-enhanced exciton localization in quaternary Inx Aly Ga1−x−y N layers. Recently, research efforts towards a more complete understanding of the cubic zinc-blende nitride derived heterostructures have increased [14]. Androulidaki et al. [16] have used the transmission electron microscope to study quaternary In0.12 Al0.29 Ga0.59 N and In0.10 Al0.02 Ga0.8 N layers, 200 nm thick, grown on (0 0 0 1) GaN/sapphire composites by metalorganic chemical vapor deposition. Their results showed that the quaternary layers contain high density stacking faults, which have zinc-blende structures bounded by Shockley partials. In order to help understand and control the materials and device properties, we have carried out a

A. Hafaiedh, N. Bouarissa / Materials Chemistry and Physics 115 (2009) 122–125

123

Table 1 Band gaps and lattice constants of the zinc-blende structure, for AlN, GaN, and InN alloys.

E (eV) Ex (eV) EL (eV)

´˚ Lattice constant a (A) a b c

AlN

GaN

InN

6.0a 4.9a 9.3a

3.3b 4.57b 6.04b

0.78c 2.51c 5.82c

4.38a

4.52b

4.986c

Ref. [19]. Ref. [20]. Ref. [21].

theoretical study of the energy band gaps and optical properties of the quaternary Alx Gay In1−x−y N, lattice matched to GaN substrate. The electronic structure of the quaternary alloy system is calculated using the empirical pseudopotential method (EPM) within the virtual crystal approximation (VCA). Calculations are performed over the entire composition range of x and y. The zinc-blende polytype is considered as a model system. This paper is organized as follows. A brief introduction was given in Section 1. The computational method we have adopted for the calculations is described in Section 2. We present our results and compare them to the available experimental data and other previously published theoretical results in Section 3. Then, our work is summarized in Section 4. 2. Computational method The electronic structure of the material under investigation is calculated using the EPM within the VCA. In the EPM, the actual atomic potential is replaced by a pseudopotential and a set of atomic form factors Va (G), which are treated as adjustable parameters that can be fitted to available experimental data. Generally for zincblende structures, just three form factors are sufficient to determine each atomic potential. In the present work, the empirical pseudopotential parameters are optimized using the nonlinear least-squares method of Kobayasi and Nara [17]. More details about the method can be found also in Ref. [18]. The experimental and theoretical (when the experimental data is not available) band gap energies for parent compounds of interest used in the fitting procedure are shown in Table 1. The final adjusted symmetric (Vs ) and antisymmetric (VA ) pseudopotential form factors along with the used lattice constants for these compounds are given in Table 2. For a quaternary system of the form Ax By C1−x−y D, the form factors are obtained as Valloy (G) = x(VAD (G) − VCD (G)) + y(VBD (G) − VCD (G)) + VCD (G)

(1)

Hence, in case of Alx Gay In1−x−y N, the form factors may be described as Valloy = x(VAlN (G) − VInN (G)) + y(VGaN (G) − VInN (G)) + VInN (G)

(2)

where G is the reciprocal lattice vector, and VAlN , VGaN , and VInN are the form factors for AlN, GaN and InN, respectively. Recently, Marques et al. [10] have reported a linear behavior of the lattice parameter of Alx Gay In1−x−y N as a function of the com-

Fig. 1. Dependence of Ga content (y) as a function of Al content (x) for which the alloy is lattice matched to the GaN substrate.

position x, y. Thus, the Vegard’s rule has been assumed for the calculation of the lattice constant of quaternary alloys under study. a(x, y) = xaAlN + yaGaN + (1 − x − y)aInN

(3)

The lattice matching conditions for Alx Gay In1−x−y N quaternary systems on the GaN substrate is: y = 1−1.270833x. In other words, the quaternary Alx Gay In1−x−y N alloy with an Al/In mole fraction ratio of 3.7 will be lattice-matched to GaN. The composition range for which the lattice constants of Alx Gay In1−x−y N alloys are latticematched to GaN substrate is represented by the solid line in Fig. 1. The refractive index, n, was determined using the empirical expression proposed by Herve and Vandamme [22].



 1+

n=

A Eg + B

2 (4)

where A = 13.6 eV and B = 3.4 eV. This model has been preferred to other models [23,24] because it is found to give better agreement with known data obtained for n in the case of III–V semiconductors [25,26]. The optical high frequency dielectric constant (ε∞ ) is estimated according to the relation, ε∞ = n2

(5)

The static dielectric constant (ε0 ) is related to ε∞ through the Harrison model [27]: ε0 − 1 =1+ ε∞ − 1

(6)

where  is given by [27]: =

˛2p (1 + 2˛2c )

(7)

2˛4c

Table 2 Symmetric and anti-symmetric form factors V(G2 ) in Ryd and lattice constants (a) used in the calculations for AlN, GaN and InN. Material

a (a.u.)

VS (3)

VS (8)

VS (11)

VA (3)

VA (4)

VA (11)

AlNa GaNb InNc

8.26087 8.50662 9.41399

−0.309603 −0.347240 −0.330117

0.112783 −0.016 0.00393

0.067538 0.21217 −0.031435

0.28 0.159988 0.089383

0.33 0.2 0.32

0.015 0.135 −0.006892

a b c

Ref. [19]. Ref. [20]. Ref. [21].

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A. Hafaiedh, N. Bouarissa / Materials Chemistry and Physics 115 (2009) 122–125

where ˛p is the polarity defined by Vogl [28] as ˛p = −

VA (3) VS (3)

(8)

VS (3) and VA (3) are the symmetric and antisymmetric pseudopotential form factors at G(1 1 1), respectively. ˛c is the covalency of the material of interest defined as ˛2c = 1 − ˛2p

(9)

3. Results The band structure of Alx Gay In1−x−y N lattice matched to GaN was calculated through the high-symmetry points  , X, and L in the Brillouin zone. The Ga composition energy variations of the conduction band edges at  , X, and L with respect to the top of the valence band are obtained. Results are plotted in Fig. 2, which shows the dependence of direct and indirect band-gap energies, in the lattice matched Alx Gay In(1−x−y) N/GaN structure, on the Ga content. Note that both energy band-gaps namely,  −  and  − L decrease monotonically with increasing the composition y. This is not the case for the indirect band-gap  − X which varies non-monotonically. This behavior shows that up to y = 0.2, the absorption at the optical gaps in the alloy system of interest is indirect ( − X). A direct to indirect transition occurs at a Ga concentration of about 0.2 (y = 0.2), which corresponds to a band gap energy in the order of 4.45 eV. Beyond this concentration (i.e. y = 0.2), the material under study becomes a  −  direct gap semiconductor. One should also note that in this composition range the fundamental band-gap becomes narrower as far as y increases towards the value of 1. This behavior allows important technological properties of the alloy of interest in the ultraviolet region. In order to provide analytical expressions for the energy band gaps of the Alx Gay In1−x−y N quaternary system which could be of practical use and of easy access, we obtained the following analytical expressions by fitting our energy gaps data using a least squares procedure:  (eV) = 4.9 − 2.5y + 0.89y2 E

(10)

X E (eV) = 4.5 − 0.85y + 0.88y2

(11)

L E (eV) = 8.7 − 0.32y + 0.55y2

(12)

Fig. 3. Direct-to-indirect gap transition projected in the x − y plane for zinc-blende Alx Gay In(1−x−y) N for lattice mismatching condition.

The quadratic terms are referred to as band-gap bowing parameters. Note that the band bowing behavior occurs similarly for the  , X, and L energy gaps, since all energy gaps show a downward bowing. Androulidaki et al. [16] investigated the band gap energy in many ternary and quaternary nitride alloy MBE grown thin films, namely Alx Ga1−x N, Inx Ga1−x N, Inx Al1−x N, and Inx Aly Ga1−x−y N. They noticed that the bowing parameter depends strongly on the In concentration, suggesting a more general trend in all the In-containing nitride alloys. The direct-to-indirect band gap transition is represented by the solid line in Fig. 3, and is predicted to occur at x = −0.33445y + 0.69763. Using the first principle total energy method and an extension of the generalized quasi-chemical approach to treat disorder and compositional fluctuations, Marques et al. [10] have studied the energy band gap as a function of the alloy composition in quaternary Alx Gay In1−x−y N nitride alloys. Their results showed that the direct-to-indirect band gap transitions is predicted to occur at y = 1.4 − 1.7x, which corresponds to x = −0.59y + 0.82. This relation is qualitatively similar to that found in our calculations. From the quantitative point of view, the coefficients reported by Marques et al. [10] are somewhat larger than those shown in the present work. It is worth mentioning that this direct-to-indirect gap transition is not expected to occur for the wurtzite phase of the alloy, since wurtzite AlGaN is known to be a direct semiconductor alloy [10]. In Table 3, we compare our results for the energy band gap Eg with experimental data as obtained from photoluminescence measurements [13]. A very good agreement is seen between the theoretical and experimental values.

Table 3 Comparison of the calculated to the theoretical direct energy band gaps for the Alx Gay In(1−x−y) N alloy.

Fig. 2. Direct and indirect band-gap Alx Gay In(1−x−y) N/GaN structure.

energies

in

the

lattice

matched

(x, y)

Eg (eV) (experimental)

Eg (eV) (calculated)

(0.0, 0.90) (0.03, 0.87) (0.16, 0.74) (0.18, 0.72) (0.22, 0.68) (0.25, 0.65)

2.85a 2.98a 3.23b 3.37b 3.42b 3.35a

2.87 2.94 3.24 3.39 3.45 3.47

a b

Ref. [13]. PL for thin w-films (∼60 nm). Ref. [13]. PL for thick w-films (∼430 nm).

A. Hafaiedh, N. Bouarissa / Materials Chemistry and Physics 115 (2009) 122–125 Table 4 Values of the refractive index, the static dielectric constant and the high frequency dielectric constant for different values of the Ga concentration for the lattice matched Alx Gay In(1−x−y) N/GaN structure. Ga(y) concentration

Refractive index

Static dielectric constant

High frequency dielectric constant

0 0.1 0.3 0.5 0.7 0.9 1

1.98 2 2.05 2.13 2.19 2.24 2.26

9.94 8.02 5.81 4.61 3.75 3.09 2.81

3.93 3.99 4.19 4.52 4.81 5.03 5.11

125

by our calculations was found to fit best by a cubic least squares procedure giving the following relations: ε0 = −10y3 + 22y2 − 19y + 9.8 3

2

ε∞ = −1.8y + 2.7y + 0.22y + 3.9

(13) (14)

These expressions may be useful for obtaining ε∞ and ε0 for any y concentration in Alx Gay In1−x−y N alloy lattice matched to GaN. 4. Conclusion In conclusion, we studied the composition dependence of energy band gaps and dielectric constants of Alx Gay In1−x−y N quaternary nitride alloys, lattice matched to GaN substrate, by means of the empirical pseudopotential approach under the virtual crystal approximation. Based on the calculated lattice constants, we predicted the range of compositions for which AlGaInN is lattice matched to GaN. Very good agreement was obtained between the band gap predictions and available experimental data. The absorption at the optical gaps suggested that a transition from an indirect to direct band gap may occur in Alx Gay In1−x−y N/GaN at y equal 0.2. Our findings indicate that the high-frequency dielectric constant increases monotonically with increasing y from 0 to 1, whereas the static dielectric constant decreases monotonically. The expressions derived for direct and indirect energy band gaps and static and high frequency dielectric constants as a function of Ga concentrations are useful for tailoring electric and opto-electric devices based on cubic Alx Gay In1−x−y N quaternary alloys. Due to the lack of experimental and theoretical data regarding the refractive index and high frequency and static dielectric constants, our results are predictions and may serve as reference for future experimental work. References

Fig. 4. Static and high frequency dielectric constants as a function of Ga concentration (y) for the lattice matched Alx Gay In(1−x−y) N/GaN structure.

A good knowledge of the full electronic structure is an essential feature in order to get the best understanding of the optical properties of semiconductors. Thus, using the expression (4), the refractive index which is essential in the design of heterostructure lasers, in optoelectronic devices as well as in solar cell applications has been calculated. Our results for various concentrations y are listed in Table 4. The knowledge of the refractive index makes it possible to proceed with the calculations of the dielectric constants. Using the relation (5), the high frequency dielectric constant has been calculated and the obtained data are depicted in Table 4. The calculations in the present work were extended to include the static dielectric constant. Based on the relations (6–9), we have calculated the values of ε0 at various Ga concentrations y, for the lattice matched Alx Gay In1−x−y N/GaN structure. Our results are presented in Table 4. For the fact that both experimental and theoretical data regarding n, ε∞ , and ε0 for Alx Gay In1−x−y N lattice matched to GaN were not available, our results are predictions and may serve for reference. The variations of both static and high frequency dielectric constants as a function of alloy composition are plotted in Fig. 4. As described by Fig. 4, the high frequency dielectric constant increases slowly for Ga concentrations varying from 0 to 2% then shows a rapid increase. This may be due to the fact that in the first region the gap is indirect whereas in the second region the gap is direct. The static dielectric constant decreases rapidly as a function of Ga concentration. At a concentration of Ga equal to 0.515, the two dielectric constants have equal values. In order to provide analytical expressions for both ε∞ and ε0 of the material of interest, the data obtained

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