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Electronic and positronic properties of Al1−xInxN with zincblende structure
Physica B 324 (2002) 72–81
Electronic and positronic properties of Al1xInxN with zincblende structure S. Bounaba,b, Z. Charifia, N. Bouarissaa,* b
a Physics Department, University of M’sila, 28000 M’sila, Algeria Faculty of Science, Physics Department, University of Constantine, 25000 Constantine, Algeria
Received 4 March 2002; received in revised form 19 April 2002
Abstract The electronic and positronic properties of the zincblende ternary alloy semiconductor Al1xInxN are calculated and analyzed by using the pseudopotential formalism within the virtual crystal approximation. To make allowance for the disorder effect, the compositional disorder is added to the virtual crystal approximation as an effective potential. The positron wave functions are obtained employing the point-core approximation for the ionic potential. The entire composition variations of electronic and positronic quantities such as band-gap energies, and effective masses have been calculated and found to be generally in good agreement with the available experimental data and ab initio calculations. r 2002 Elsevier Science B.V. All rights reserved. Keywords: Electronic properties; Positronic properties; Zincblende structure; Al1xInxN
1. Introduction The III–V nitrides are promising for optoelectronic device application [1–3]. Indium nitride (InN) is an actively studied semiconductor that has promise for the construction of blue light emitting diodes and other high-temperature optoelectronic devices [4]. Aluminum nitride (AlN) is an insulator and is a promising thin-film piezoelectric material for its high-ultrasonic velocity and fairly large electromechanical coupling coefficient [5]. There is much current interest in alloys of the group III nitrides, because of the possibilities they offer for optoelectronic devices operating over a wide spectrum of wavelengths from the visible to *Corresponding author. Fax: +213-35-686213. E-mail address: n [email protected] (N. Bouarissa).
the deep ultraviolet [6]. The ternary semiconductor alloy Al1xInxN with parent compounds AlN and InN is very attractive and useful and has a great promise for applications in light-emitting diodes and laser diodes operating in the ultraviolet to orange spectrum [7]. The vast majority of research on III–V nitrides has been focused on the wurtzite crystal phase. The reason is that most of III–V nitrides have been grown on sapphire substrates which generally transfer their hexagonal symmetry to the nitride film [8]. Because of the recent progress in crystal growth, one can now also produce thin films of zincblende crystals, which opens up the further technological applications involving the III nitrides [9]. The ternary alloys, Ga1xInxN and Al1xGaxN have been extensively studied [3,10–15] and found
0921-4526/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 2 ) 0 1 2 7 6 - 0
S. Bounab et al. / Physica B 324 (2002) 72–81
to be useful as well layers and cladding layers in quantum well (QW) laser diode (LD) structures, respectively [16]. In contrast, the ternary alloy Al1xInxN, is less investigated because of difficulty in its growth from the thermodynamic view point [16]. This has motivated us to study such an alloy with zincblende structure. The experimental activity has stimulated theoretical investigations of the electronic properties of the materials. On the other hand, the electron– positron annihilation method can provide information on the electron structure of a crystal in a region where positrons are annihilated, on the structure of defects, and so on. However, interpretation of the annihilation spectra is a difficult task that requires a theoretical study of the states of positrons in a given material [17–21]. The latter quantities can be directly compared with the experimental data. The present theoretical study deals with electronic and positronic properties of zincblende Al1xInxN. Our aim is to carry out a semiempirical calculation of the electron structure and positron states of the material of interest in order to provide a basis for understanding future wideenergy gap device concepts and application based on zincblende III–V nitride semiconductor alloys, in particular AlInN alloy lattice matched to GaN that is expected to serve as an insulating barrier in GaN-based electronic devices [22,23].
2. Calculation method
t ¼ 18ð1; 1; 1Þ in units of the lattice constant. V S and V A are the symmetric and antisymmetric form factors, respectively. Since the potential is spherically symmetric, we take V ðjGjÞ: The pseudowave function for the valence electrons is thus taken as 1=2 X 1 Cnk ðrÞ ¼ Cnk ðGÞeiðKþGÞr ; ð3Þ O G where O is the unit cell volume. The coefficients Cnk ðGÞ are found out by solving the electron secular equation X ½ðjK þ G 0 j2 Enk ÞdG;G0 þ V ðjG G 0 jÞ G0
Cnk ðG 0 Þ ¼ 0:
a
where Sa ðGÞ represents the structure factor. G are the reciprocal lattice vectors and Va ðGÞ are the atomic form factors. In the case of zincblende compounds AN B8N, the relation (1) becomes V ðGÞ ¼ V S ðGÞcosðGtÞ þ iV A ðGÞsinðGtÞ;
ð2Þ
ð4Þ
They are complex because the potential is complex. Hence we can express Cnk ðGÞ in terms of real and imaginary parts R I Cnk ðGÞ ¼ Cnk ðGÞ þ iCnk ðGÞ:
ð5Þ
The electron wave functions are calculated at each iteration using all the plane waves with G satisfying _2 ðjK þ Gj2 jKj2 ÞpEmax : ð6Þ 2m A choice of Emax ¼ 14222 in units of ð_2 =2mÞð2p=aÞ2 turns out to be practical. In the present work, Emax ¼ 14 is adopted. The pseudopotential form factors for the ternary alloy of Al1xInxN being studied here are evaluated according to the virtual crystal approximation (VCA) as follows: S;A S;A V S;A ðGÞ ¼ ð1 xÞVAlN ðGÞ þ xVInN ðGÞ:
The electron wave function is evaluated in the pseudopotential formalism. The pseudopotential in momentum space is given by X V ðGÞ ¼ Sa ðGÞVa ðGÞ; ð1Þ
73
ð7Þ
However, it has been reported that the VCA predictions are not in quantitative agreement with the experimental data [15,24,25]. Recent experimental and theoretical studies on several semiconductor alloys indicate that the VCA breaks down whenever the mismatch between the electronic properties of the constituent atoms exceeds a certain critical value [24]. This is because the VCA does not take into account the compositional disorder. It is probably fair to assume that in general deviations from the VCA predictions remain insignificant in systems where the differences between the constituent atoms and/or lattice
S. Bounab et al. / Physica B 324 (2002) 72–81
74
constants of the binary compounds are small (e.g. GaAlAs) [24], a fact which let us believe that the compositional disorder effects become more important when the lattice mismatch becomes larger. In our case, we have followed the approach of Lee et al. [26] and added to the VCA a non-periodic potential due to the compositional disorder. Thus, the expression for the pseudopotential form factors becomes
In the point-core approximation, the ionic potential is expressed as
S;A S;A V S;A ðGÞ ¼ xVInN ðGÞ þ ð1 xÞVAlN ðGÞ
The electron–positron potential Vep ðrÞ is a slow function of the electron density. It is generally flat in the interstitial region and swamped by Vi ðrÞ and VC ðrÞ in the ion core region. Hence it is not considered in the present work. The positron wave function is expressed in terms of plane waves, 1=2 X 1 Cþ ðrÞ ¼ AðGÞexpðiGrÞ: ð13Þ O G
P½xð1
S;A ðGÞ xÞ 1=2 ½VInN
S;A VAlN ðGÞ ;
ð8Þ where P is treated as an adjustable parameter. According to Lee et al. [26], this scheme does not include the lattice relaxation effect and hence it is believed to be most suitable for the materials with small lattice mismatches. Recent studies of Charifi and Bouarissa on Si1x Gex alloys [27] showed that the inclusion of the lattice relaxation effect affects significantly the value of P which was found to become very weak as compared to that found when the lattice relaxation effect was not considered but both values can lead to the reach of the experimental band-gap optical bowing parameter. We do believe then that as P is an adjustable parameter which simulates the disorder effect, its value varies on going from one alloy to another one following generally, the same trend as that of the lattice mismatch. This makes possible the use of the above approximation even when the lattice mismatch is large. The lattice constant of the ternary alloy under investigation is obtained by using Vegard’s law [28], aAl1x Inx N ¼ ð1 xÞaAlN þ xaInN :
ð9Þ
Since at each moment a crystal can contain just one positron (because of the low intensity of the positron sources used at present), the exchange part is not considered. Thus, the total positron potential can be expressed as Vþ ðrÞ ¼ Vi ðrÞ þ VC ðrÞ þ Vep ðrÞ;
ð10Þ
Za e2 ; r where Za is the valence charge of atom a: The Coulomb potential is given by Z rðr0 Þ 3 0 d r: VC ðrÞ ¼ 2 jr r0 j
Via ðrÞ ¼
ð12Þ
The coefficients AðGÞ are found by solving the secular equation for the positron X ðG 02 EÞdG;G0 þ Vþ ðjG G 0 jÞ AðG0 Þ ¼ 0 ð14Þ G0
3. Results The available experimental and theoretical energy band gaps at the high-symmetry points G, X and L in the Brillouin zone used in the fitting procedure are given in Table 1. Table 2 shows the final local adjusted pseudopotential form factors for zincblende InN and AlN. The used lattice
Table 1 Band-gap energies of zincblende InN and AlN Compound
Band-gap energy (eV) EgX
EgG InN AlN a
1.90a,b 6.0a,e
Ref. [29]. Wurtzite structure. c Present work. d Ref. [10] e Ref. [30]. b
where Vi ðrÞ; VC ðrÞ and Vep ðrÞ are the ionic, Coulomb and electron–positron correlation potentials, respectively.
ð11Þ
1.94c 6.0c
2.8d 4.9a,e
EgL 2.51c 4.9c
5.12a 9.15a
5.82c 9.21c
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Table 2 Pseudopotential form factors for zincblende InN and AlN Compound
InN AlN
Form factors (Ry) VS (3)
VS (8)
VS (11)
VA (3)
VA (4)
VA (11)
0.172195 0.309603
0.01555 0.112783
0.044698 0.067538
0.238742 0.28
0.2321 0.33
0.03149 0.015
Fig. 1. Direct and indirect band-gap energies in Al1xInxN as a function of the indium concentration x:
constants for the latter materials are 4.98 [10] and ( [31], respectively. 4.37 A The variation of the direct and indirect bandgap energies (EgG ; EgX and EgL ) as a function of In concentration x for Al1xInxN with zincblende structure is plotted in Fig. 1. As one can see, all studied band-gap energies decrease non-linearly when going from x ¼ 0 to 1 showing a downward band-gap bowing parameters. The curves in Fig. 1 correspond to the best fit of our results to the following expression: EðxÞ ¼ a þ bx þ cx2 :
ð15Þ
EgX The resulting values of a; b and c for the L and Eg band-gap energies are given in Table 3. In view of Table 3, we note that our obtained values EgG ;
of a are roughly in good agreement with those reported in Refs. [8,22,32]. The same good agreement could be noticed for our estimated value of b and that reported in Ref. [22] for the EgG band-gap energy. Of particular importance are the c values which represent the optical bowing parameters for the band gaps at G, X and L high-symmetry points. These parameters are induced by composition disorder, in which the mixing ratio of the cations and anions is the main factor controlling the extent of disorder. Van Schilfgaarde et al. [10] reported a large value of c in zincblende Al1xInxN. The experimental results of Kubota et al. [5] have shown also a large value of c in the alloy of interest. However, theoretical calculations of the band-gap energy of Al1xInxN
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Table 3 Values of the parameters a; b and c obtained by fitting the dependence of the direct and indirect band-gap energies of zincblende Al1xInxN on x to E ¼ a þ bx þ cx2 Band-gap energy (eV)
a (eV)
b (eV)
c (eV)
EgG ðG GÞ
5.93a 4.3b 6.02b 5.94c 6.28d
6.67a 6.92d
2.53a 3.5e 2.53d
EgX ðG XÞ
4.86a 3.1b 4.92b 5.10c
3.95a
1.45a
EgL ðG LÞ
9.24a 9.42c
5.69a
2.15a
a
Present calculations. Ref. [32]. c Ref. [8]. d Ref. [22]. e Ref. [10]. b
using the ab initio pseudopotential method performed by Wright and Nelson [22] predicted that the band gap of Al1xInxN has a bowing parameter of 2.53 eV. The latter is much smaller than that reported in Refs. [5,10]. Recently, the experimental results of Kim and co-workers [23] showed that the band-gap variation is observed to follow the predicted bowing curve of Wright and Nelson [22] fairly well and that the larger discrepancy with the results of Kubota et al. [5] are believed to be due to their sample quality which might be so poor that the band gap and/or composition determination was disturbed. We have adopted the experimental value of c reported in Ref. [23] and thus our obtained value of c for EgG is found to be 2.53 eV for p ¼ 0:281: This value is greater, in magnitude, than that reported for the semiconductor ternary alloy Gax In1x As [25] where the lattice mismatch is smaller than that in the alloy of interest suggesting therefore that the value of P is expected to be higher in magnitude as the lattice mismatch becomes larger. It should be noted also that the c values for EgG ; EgX and EgL are larger in magnitude as compared with other III–V ternary alloys [33–35]. This is consistent with the larger
energy gaps found in nitride compounds. However, the c value for EgG seems to be the largest one as compared to those for EgX and EgL indicating therefore that the effect of the alloy disorder is more important on the direct band-gap EgG : Likewise, our calculated band-gap bowing parameters are higher than those in zincblende Ga1xInxN ternary alloys [15] suggesting thus that the Al1xInxN semiconductor alloy is more altered by the disorder effect than the Ga1xInxN one. This is believed to be due to the lattice mismatch that is larger in Al1xInxN. On the other hand, the indirect to direct band-gap transition has been examined. It is found that the X to G energy gap transition occurs at xD0:5: Hence, the zincblende Al1xInxN alloys should have an indirect gap for small values of x and a direct gap for large values. Using first-principles calculations, Wright and Nelson [22] predicted that the indirect to direct transition occur at x ¼ 0:18 which disagree with our results. The origin of such a discrepancy is not clear. In Table 4, we show the resulting energy band gaps in zincblende Al1xInxN for some selected indium concentrations. Available experimental data and other theoretical results are also presented for comparison. For the band-gap energy of Al0.92In0.08N, our result seems to be in good agreement with the experimental one of 5.26 eV measured by optical absorption spectroscopy [23]. In addition, our calculated value for the band-gap
Table 4 Calculated band-gap energies of zincblende Al1xInxN for some selected composition x compared with available data in the literature Ternary alloy
Band-gap energy (eV) EgG
EgX
EgL
Al0.92In0.08N
5.29a 5.26b
4.45a
8.70a
Al0.83In0.17N
4.87a 4.7b
4.23a
8.34a
Al0.50In0.50N
3.30a
3.30a
6.99a
a b
Present calculations. Ref. [23].
S. Bounab et al. / Physica B 324 (2002) 72–81
energy of Al0.83In0.17N is in good agreement with that of 4.7 eV roughly estimated by extrapolating the data using the same bowing parameter to x ¼ 0:17 [23]. According to Ref. [23], the band-gap energy results of Al0.83In0.17N imply a possible application of the ternary as an efficient barrier material for GaN. Moreover, Yamaguchi et al. [16] have reported that Al0.83In0.17N can be lattice matched to GaN and consequently, Al1xInxN can be used as a cladding layer with no strain on the laser diode structure, leading to reduction of defects. Thus, we were interested to the electronic and positronic band structure of the ternary semiconductor alloy Al1xInxN at composition x ¼ 0:17: In Fig. 2, we plot the computed electronic energy band structure of zincblende Al0.83In0.17N at the high-symmetry points and along the highsymmetry direction in the Brillouin zone. The valence band maximum is the zero reference. The conduction band minimum occurs at point X. Thus, the material Al0.83In0.17N is an indirect gap semiconductor. For the lowest conduction band at point G, the wave functions exhibit the largest degree of localization and are s-like. At point X or
77
L, however, the states are more free-electron like. Similar shape has been observed for the electronic structure of zincblende Ga0.5In0.5N with less dispersive bands and band crossings [15]. Considering the free-electron-like behavior of the conduction bands, this similarity is not surprising. The small difference in the bands dispersion is believed to be due to the difference of the energy band gaps, that is related to the difference in the ionicity. The effective electron mass is a fundamental quantity of semiconductors, used in numerous analyses of experiments and theoretical modellings. It can be obtained from the band structure of the material under investigation. The simplest approximation corresponds to the parabolic EðkÞ dependence, that is the case in the AIIIBV compounds where the conduction-band EðkÞ relation is approximately parabolic close to its k ¼ 0 extremum (see for example, Fig. 2). Thus, we may use one scalar electron effective mass m
defined as 1 4p2 q2 EðkÞ ¼ : m
h2 qk2
Fig. 2. Electron band structure for Al0.83In0.17N.
ð16Þ
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Our results regarding the electron effective masses in the composition range x (0pxp1) for the material of interest along with available data in the literature are given in Table 5. In view of Table 5, it is clearly seen that our results are in good
Table 5 Conduction band-edge electron effective mass (in units of the free electron mass) in zincblende AlN, InN and some of their ternary alloys mne ðGÞ=m0
Material
a
AlN
0.21 0.21b 0.19c 0.21d 0.20a 0.19a 0.16a 0.12a 0.10c 0.14d
Al0.92In0.08N Al0.83In0.17N Al0.50In0.50N InN
a
Present calculations. Ref. [8]. c Ref. [32]. d Ref. [36]. b
agreement with those reported in the literature suggesting therefore that while the method used is simple it gives results that are in good agreement with those obtained by more complicated methods. The indium concentration dependence of the electron effective mass in the range 0–1 for zincblende Al1xInxN is shown in Fig. 3. As one can see, increasing the composition x from 0 up to 1 leads to the decrease of the electron effective mass according to the following expression: m e ðGÞ ¼ 0:21 0:13x þ 0:04x2 m0
ð17Þ
showing nearly a non-linear behavior with a downward bowing parameter of 0.04. We now turn our attention to the positronic properties of zincblenbe Al1xInxN and display the computed positron band structure in Fig. 4. Accordingly, the positron energy spectra are similar to those of the electron ones, with the exception that the positron energy spectrum does not exhibit a band gap at the G point between the fourth and the fifth bands (Fig. 2). This is consistent with the fact that the positron energy
Fig. 3. Effective mass of the electron in Al1xInxN as a function of the indium concentration x:
S. Bounab et al. / Physica B 324 (2002) 72–81
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Fig. 4. Positron band structure for Al0.83In0.17N.
Table 6 Calculated positron band effective mass (in units of the free electron mass) in zincblende AlN, InN and some of their ternary alloys Material
mnb =m0
AlN Al0.92In0.08N Al0.83In0.17N Al0.50In0.50N InN
1.27 1.27 1.28 1.30 1.32
band resembles the s states of the valence electrons, in agreement with the theoretical ideas on the nature of positron states [17]. The distribution of positrons corresponds to a positron position in the region of interstices (for the ground state) since the positron is repelled by the positively charged ion cores, whose wave functions are being concentrated in the outer parts of the unit cell. The positron in a semiconductor, in analogy with the hole, may be considered as a carrier of
positive charge. However, due to its higher effective mass, it is considerably less mobile than its hole counterpart [37]. This mass is an important parameter for studying positron diffusion in semiconductors. Similarly to the electron part, such parameter can be obtained from the band structure of material using the relation (16). However, the obtained mass is termed as the positron band mass. The latter is found to have the largest contribution to the positron effective mass [19,21,37] and has been obtained in our case by fitting parabolas to the positron band energy (Fig. 4) calculated in the vicinity of the G1 point (the lowest positron level). Our results for zincblende AlN, InN and some of their ternary alloys are listed in Table 6. It has been reported in the literature that the positron effective mass m ¼ 1:5 m0 ; where m0 is the positron free-particle mass, has been considered as a compromise over the various theoretical and experimental determinations [38]. Our calculated positron band effective masses are smaller than m : The difference between the positron effective mass m and the positron band effective masse m b is believed
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Fig. 5. Band effective mass of the positron in Al1xInxN as a function of the indium concentration x:
to arise from the neglect of the positron–phonon and positron–plasmon interactions. The variation of the positron band effective mass as a function of the indium concentration x in the range 0–1 is displayed in Fig. 5. It is to be noted that the positron band effective mass increases with increasing the composition x from 0 to 1. A least-quares fit was made to these data giving the following expression: m b ¼ 1:27 þ 0:05x þ 0:006x2 m0
introduced as an effective disorder potential. The positron wave functions have been calculated employing the point-core approximation for the ionic potential. Our results are roughly in good agreement with the available experimental data and ab initio calculations. Some predicted electronic and positronic properties for the ternary alloy Al0.83In0.17N which might be used as a future lattice matched barrier for GaN, have also been reported.
ð18Þ
showing a very weak band mass bowing parameter of 0.006 suggesting thus a nearly linear behavior of mnb :
4. Conclusion Theoretical calculations of electronic and positronic properties of zincblende Al1xInxN have been performed in the virtual crystal approximation using the empirical pseudopotential scheme. In this scheme, the compositional disorder is
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Electronic and positronic properties of Al1xInxN with zincblende structure S. Bounaba,b, Z. Charifia, N. Bouarissaa,* b
a Physics Department, University of M’sila, 28000 M’sila, Algeria Faculty of Science, Physics Department, University of Constantine, 25000 Constantine, Algeria
Received 4 March 2002; received in revised form 19 April 2002
Abstract The electronic and positronic properties of the zincblende ternary alloy semiconductor Al1xInxN are calculated and analyzed by using the pseudopotential formalism within the virtual crystal approximation. To make allowance for the disorder effect, the compositional disorder is added to the virtual crystal approximation as an effective potential. The positron wave functions are obtained employing the point-core approximation for the ionic potential. The entire composition variations of electronic and positronic quantities such as band-gap energies, and effective masses have been calculated and found to be generally in good agreement with the available experimental data and ab initio calculations. r 2002 Elsevier Science B.V. All rights reserved. Keywords: Electronic properties; Positronic properties; Zincblende structure; Al1xInxN
1. Introduction The III–V nitrides are promising for optoelectronic device application [1–3]. Indium nitride (InN) is an actively studied semiconductor that has promise for the construction of blue light emitting diodes and other high-temperature optoelectronic devices [4]. Aluminum nitride (AlN) is an insulator and is a promising thin-film piezoelectric material for its high-ultrasonic velocity and fairly large electromechanical coupling coefficient [5]. There is much current interest in alloys of the group III nitrides, because of the possibilities they offer for optoelectronic devices operating over a wide spectrum of wavelengths from the visible to *Corresponding author. Fax: +213-35-686213. E-mail address: n [email protected] (N. Bouarissa).
the deep ultraviolet [6]. The ternary semiconductor alloy Al1xInxN with parent compounds AlN and InN is very attractive and useful and has a great promise for applications in light-emitting diodes and laser diodes operating in the ultraviolet to orange spectrum [7]. The vast majority of research on III–V nitrides has been focused on the wurtzite crystal phase. The reason is that most of III–V nitrides have been grown on sapphire substrates which generally transfer their hexagonal symmetry to the nitride film [8]. Because of the recent progress in crystal growth, one can now also produce thin films of zincblende crystals, which opens up the further technological applications involving the III nitrides [9]. The ternary alloys, Ga1xInxN and Al1xGaxN have been extensively studied [3,10–15] and found
0921-4526/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 2 ) 0 1 2 7 6 - 0
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to be useful as well layers and cladding layers in quantum well (QW) laser diode (LD) structures, respectively [16]. In contrast, the ternary alloy Al1xInxN, is less investigated because of difficulty in its growth from the thermodynamic view point [16]. This has motivated us to study such an alloy with zincblende structure. The experimental activity has stimulated theoretical investigations of the electronic properties of the materials. On the other hand, the electron– positron annihilation method can provide information on the electron structure of a crystal in a region where positrons are annihilated, on the structure of defects, and so on. However, interpretation of the annihilation spectra is a difficult task that requires a theoretical study of the states of positrons in a given material [17–21]. The latter quantities can be directly compared with the experimental data. The present theoretical study deals with electronic and positronic properties of zincblende Al1xInxN. Our aim is to carry out a semiempirical calculation of the electron structure and positron states of the material of interest in order to provide a basis for understanding future wideenergy gap device concepts and application based on zincblende III–V nitride semiconductor alloys, in particular AlInN alloy lattice matched to GaN that is expected to serve as an insulating barrier in GaN-based electronic devices [22,23].
2. Calculation method
t ¼ 18ð1; 1; 1Þ in units of the lattice constant. V S and V A are the symmetric and antisymmetric form factors, respectively. Since the potential is spherically symmetric, we take V ðjGjÞ: The pseudowave function for the valence electrons is thus taken as 1=2 X 1 Cnk ðrÞ ¼ Cnk ðGÞeiðKþGÞr ; ð3Þ O G where O is the unit cell volume. The coefficients Cnk ðGÞ are found out by solving the electron secular equation X ½ðjK þ G 0 j2 Enk ÞdG;G0 þ V ðjG G 0 jÞ G0
Cnk ðG 0 Þ ¼ 0:
a
where Sa ðGÞ represents the structure factor. G are the reciprocal lattice vectors and Va ðGÞ are the atomic form factors. In the case of zincblende compounds AN B8N, the relation (1) becomes V ðGÞ ¼ V S ðGÞcosðGtÞ þ iV A ðGÞsinðGtÞ;
ð2Þ
ð4Þ
They are complex because the potential is complex. Hence we can express Cnk ðGÞ in terms of real and imaginary parts R I Cnk ðGÞ ¼ Cnk ðGÞ þ iCnk ðGÞ:
ð5Þ
The electron wave functions are calculated at each iteration using all the plane waves with G satisfying _2 ðjK þ Gj2 jKj2 ÞpEmax : ð6Þ 2m A choice of Emax ¼ 14222 in units of ð_2 =2mÞð2p=aÞ2 turns out to be practical. In the present work, Emax ¼ 14 is adopted. The pseudopotential form factors for the ternary alloy of Al1xInxN being studied here are evaluated according to the virtual crystal approximation (VCA) as follows: S;A S;A V S;A ðGÞ ¼ ð1 xÞVAlN ðGÞ þ xVInN ðGÞ:
The electron wave function is evaluated in the pseudopotential formalism. The pseudopotential in momentum space is given by X V ðGÞ ¼ Sa ðGÞVa ðGÞ; ð1Þ
73
ð7Þ
However, it has been reported that the VCA predictions are not in quantitative agreement with the experimental data [15,24,25]. Recent experimental and theoretical studies on several semiconductor alloys indicate that the VCA breaks down whenever the mismatch between the electronic properties of the constituent atoms exceeds a certain critical value [24]. This is because the VCA does not take into account the compositional disorder. It is probably fair to assume that in general deviations from the VCA predictions remain insignificant in systems where the differences between the constituent atoms and/or lattice
S. Bounab et al. / Physica B 324 (2002) 72–81
74
constants of the binary compounds are small (e.g. GaAlAs) [24], a fact which let us believe that the compositional disorder effects become more important when the lattice mismatch becomes larger. In our case, we have followed the approach of Lee et al. [26] and added to the VCA a non-periodic potential due to the compositional disorder. Thus, the expression for the pseudopotential form factors becomes
In the point-core approximation, the ionic potential is expressed as
S;A S;A V S;A ðGÞ ¼ xVInN ðGÞ þ ð1 xÞVAlN ðGÞ
The electron–positron potential Vep ðrÞ is a slow function of the electron density. It is generally flat in the interstitial region and swamped by Vi ðrÞ and VC ðrÞ in the ion core region. Hence it is not considered in the present work. The positron wave function is expressed in terms of plane waves, 1=2 X 1 Cþ ðrÞ ¼ AðGÞexpðiGrÞ: ð13Þ O G
P½xð1
S;A ðGÞ xÞ 1=2 ½VInN
S;A VAlN ðGÞ ;
ð8Þ where P is treated as an adjustable parameter. According to Lee et al. [26], this scheme does not include the lattice relaxation effect and hence it is believed to be most suitable for the materials with small lattice mismatches. Recent studies of Charifi and Bouarissa on Si1x Gex alloys [27] showed that the inclusion of the lattice relaxation effect affects significantly the value of P which was found to become very weak as compared to that found when the lattice relaxation effect was not considered but both values can lead to the reach of the experimental band-gap optical bowing parameter. We do believe then that as P is an adjustable parameter which simulates the disorder effect, its value varies on going from one alloy to another one following generally, the same trend as that of the lattice mismatch. This makes possible the use of the above approximation even when the lattice mismatch is large. The lattice constant of the ternary alloy under investigation is obtained by using Vegard’s law [28], aAl1x Inx N ¼ ð1 xÞaAlN þ xaInN :
ð9Þ
Since at each moment a crystal can contain just one positron (because of the low intensity of the positron sources used at present), the exchange part is not considered. Thus, the total positron potential can be expressed as Vþ ðrÞ ¼ Vi ðrÞ þ VC ðrÞ þ Vep ðrÞ;
ð10Þ
Za e2 ; r where Za is the valence charge of atom a: The Coulomb potential is given by Z rðr0 Þ 3 0 d r: VC ðrÞ ¼ 2 jr r0 j
Via ðrÞ ¼
ð12Þ
The coefficients AðGÞ are found by solving the secular equation for the positron X ðG 02 EÞdG;G0 þ Vþ ðjG G 0 jÞ AðG0 Þ ¼ 0 ð14Þ G0
3. Results The available experimental and theoretical energy band gaps at the high-symmetry points G, X and L in the Brillouin zone used in the fitting procedure are given in Table 1. Table 2 shows the final local adjusted pseudopotential form factors for zincblende InN and AlN. The used lattice
Table 1 Band-gap energies of zincblende InN and AlN Compound
Band-gap energy (eV) EgX
EgG InN AlN a
1.90a,b 6.0a,e
Ref. [29]. Wurtzite structure. c Present work. d Ref. [10] e Ref. [30]. b
where Vi ðrÞ; VC ðrÞ and Vep ðrÞ are the ionic, Coulomb and electron–positron correlation potentials, respectively.
ð11Þ
1.94c 6.0c
2.8d 4.9a,e
EgL 2.51c 4.9c
5.12a 9.15a
5.82c 9.21c
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Table 2 Pseudopotential form factors for zincblende InN and AlN Compound
InN AlN
Form factors (Ry) VS (3)
VS (8)
VS (11)
VA (3)
VA (4)
VA (11)
0.172195 0.309603
0.01555 0.112783
0.044698 0.067538
0.238742 0.28
0.2321 0.33
0.03149 0.015
Fig. 1. Direct and indirect band-gap energies in Al1xInxN as a function of the indium concentration x:
constants for the latter materials are 4.98 [10] and ( [31], respectively. 4.37 A The variation of the direct and indirect bandgap energies (EgG ; EgX and EgL ) as a function of In concentration x for Al1xInxN with zincblende structure is plotted in Fig. 1. As one can see, all studied band-gap energies decrease non-linearly when going from x ¼ 0 to 1 showing a downward band-gap bowing parameters. The curves in Fig. 1 correspond to the best fit of our results to the following expression: EðxÞ ¼ a þ bx þ cx2 :
ð15Þ
EgX The resulting values of a; b and c for the L and Eg band-gap energies are given in Table 3. In view of Table 3, we note that our obtained values EgG ;
of a are roughly in good agreement with those reported in Refs. [8,22,32]. The same good agreement could be noticed for our estimated value of b and that reported in Ref. [22] for the EgG band-gap energy. Of particular importance are the c values which represent the optical bowing parameters for the band gaps at G, X and L high-symmetry points. These parameters are induced by composition disorder, in which the mixing ratio of the cations and anions is the main factor controlling the extent of disorder. Van Schilfgaarde et al. [10] reported a large value of c in zincblende Al1xInxN. The experimental results of Kubota et al. [5] have shown also a large value of c in the alloy of interest. However, theoretical calculations of the band-gap energy of Al1xInxN
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Table 3 Values of the parameters a; b and c obtained by fitting the dependence of the direct and indirect band-gap energies of zincblende Al1xInxN on x to E ¼ a þ bx þ cx2 Band-gap energy (eV)
a (eV)
b (eV)
c (eV)
EgG ðG GÞ
5.93a 4.3b 6.02b 5.94c 6.28d
6.67a 6.92d
2.53a 3.5e 2.53d
EgX ðG XÞ
4.86a 3.1b 4.92b 5.10c
3.95a
1.45a
EgL ðG LÞ
9.24a 9.42c
5.69a
2.15a
a
Present calculations. Ref. [32]. c Ref. [8]. d Ref. [22]. e Ref. [10]. b
using the ab initio pseudopotential method performed by Wright and Nelson [22] predicted that the band gap of Al1xInxN has a bowing parameter of 2.53 eV. The latter is much smaller than that reported in Refs. [5,10]. Recently, the experimental results of Kim and co-workers [23] showed that the band-gap variation is observed to follow the predicted bowing curve of Wright and Nelson [22] fairly well and that the larger discrepancy with the results of Kubota et al. [5] are believed to be due to their sample quality which might be so poor that the band gap and/or composition determination was disturbed. We have adopted the experimental value of c reported in Ref. [23] and thus our obtained value of c for EgG is found to be 2.53 eV for p ¼ 0:281: This value is greater, in magnitude, than that reported for the semiconductor ternary alloy Gax In1x As [25] where the lattice mismatch is smaller than that in the alloy of interest suggesting therefore that the value of P is expected to be higher in magnitude as the lattice mismatch becomes larger. It should be noted also that the c values for EgG ; EgX and EgL are larger in magnitude as compared with other III–V ternary alloys [33–35]. This is consistent with the larger
energy gaps found in nitride compounds. However, the c value for EgG seems to be the largest one as compared to those for EgX and EgL indicating therefore that the effect of the alloy disorder is more important on the direct band-gap EgG : Likewise, our calculated band-gap bowing parameters are higher than those in zincblende Ga1xInxN ternary alloys [15] suggesting thus that the Al1xInxN semiconductor alloy is more altered by the disorder effect than the Ga1xInxN one. This is believed to be due to the lattice mismatch that is larger in Al1xInxN. On the other hand, the indirect to direct band-gap transition has been examined. It is found that the X to G energy gap transition occurs at xD0:5: Hence, the zincblende Al1xInxN alloys should have an indirect gap for small values of x and a direct gap for large values. Using first-principles calculations, Wright and Nelson [22] predicted that the indirect to direct transition occur at x ¼ 0:18 which disagree with our results. The origin of such a discrepancy is not clear. In Table 4, we show the resulting energy band gaps in zincblende Al1xInxN for some selected indium concentrations. Available experimental data and other theoretical results are also presented for comparison. For the band-gap energy of Al0.92In0.08N, our result seems to be in good agreement with the experimental one of 5.26 eV measured by optical absorption spectroscopy [23]. In addition, our calculated value for the band-gap
Table 4 Calculated band-gap energies of zincblende Al1xInxN for some selected composition x compared with available data in the literature Ternary alloy
Band-gap energy (eV) EgG
EgX
EgL
Al0.92In0.08N
5.29a 5.26b
4.45a
8.70a
Al0.83In0.17N
4.87a 4.7b
4.23a
8.34a
Al0.50In0.50N
3.30a
3.30a
6.99a
a b
Present calculations. Ref. [23].
S. Bounab et al. / Physica B 324 (2002) 72–81
energy of Al0.83In0.17N is in good agreement with that of 4.7 eV roughly estimated by extrapolating the data using the same bowing parameter to x ¼ 0:17 [23]. According to Ref. [23], the band-gap energy results of Al0.83In0.17N imply a possible application of the ternary as an efficient barrier material for GaN. Moreover, Yamaguchi et al. [16] have reported that Al0.83In0.17N can be lattice matched to GaN and consequently, Al1xInxN can be used as a cladding layer with no strain on the laser diode structure, leading to reduction of defects. Thus, we were interested to the electronic and positronic band structure of the ternary semiconductor alloy Al1xInxN at composition x ¼ 0:17: In Fig. 2, we plot the computed electronic energy band structure of zincblende Al0.83In0.17N at the high-symmetry points and along the highsymmetry direction in the Brillouin zone. The valence band maximum is the zero reference. The conduction band minimum occurs at point X. Thus, the material Al0.83In0.17N is an indirect gap semiconductor. For the lowest conduction band at point G, the wave functions exhibit the largest degree of localization and are s-like. At point X or
77
L, however, the states are more free-electron like. Similar shape has been observed for the electronic structure of zincblende Ga0.5In0.5N with less dispersive bands and band crossings [15]. Considering the free-electron-like behavior of the conduction bands, this similarity is not surprising. The small difference in the bands dispersion is believed to be due to the difference of the energy band gaps, that is related to the difference in the ionicity. The effective electron mass is a fundamental quantity of semiconductors, used in numerous analyses of experiments and theoretical modellings. It can be obtained from the band structure of the material under investigation. The simplest approximation corresponds to the parabolic EðkÞ dependence, that is the case in the AIIIBV compounds where the conduction-band EðkÞ relation is approximately parabolic close to its k ¼ 0 extremum (see for example, Fig. 2). Thus, we may use one scalar electron effective mass m
defined as 1 4p2 q2 EðkÞ ¼ : m
h2 qk2
Fig. 2. Electron band structure for Al0.83In0.17N.
ð16Þ
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78
Our results regarding the electron effective masses in the composition range x (0pxp1) for the material of interest along with available data in the literature are given in Table 5. In view of Table 5, it is clearly seen that our results are in good
Table 5 Conduction band-edge electron effective mass (in units of the free electron mass) in zincblende AlN, InN and some of their ternary alloys mne ðGÞ=m0
Material
a
AlN
0.21 0.21b 0.19c 0.21d 0.20a 0.19a 0.16a 0.12a 0.10c 0.14d
Al0.92In0.08N Al0.83In0.17N Al0.50In0.50N InN
a
Present calculations. Ref. [8]. c Ref. [32]. d Ref. [36]. b
agreement with those reported in the literature suggesting therefore that while the method used is simple it gives results that are in good agreement with those obtained by more complicated methods. The indium concentration dependence of the electron effective mass in the range 0–1 for zincblende Al1xInxN is shown in Fig. 3. As one can see, increasing the composition x from 0 up to 1 leads to the decrease of the electron effective mass according to the following expression: m e ðGÞ ¼ 0:21 0:13x þ 0:04x2 m0
ð17Þ
showing nearly a non-linear behavior with a downward bowing parameter of 0.04. We now turn our attention to the positronic properties of zincblenbe Al1xInxN and display the computed positron band structure in Fig. 4. Accordingly, the positron energy spectra are similar to those of the electron ones, with the exception that the positron energy spectrum does not exhibit a band gap at the G point between the fourth and the fifth bands (Fig. 2). This is consistent with the fact that the positron energy
Fig. 3. Effective mass of the electron in Al1xInxN as a function of the indium concentration x:
S. Bounab et al. / Physica B 324 (2002) 72–81
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Fig. 4. Positron band structure for Al0.83In0.17N.
Table 6 Calculated positron band effective mass (in units of the free electron mass) in zincblende AlN, InN and some of their ternary alloys Material
mnb =m0
AlN Al0.92In0.08N Al0.83In0.17N Al0.50In0.50N InN
1.27 1.27 1.28 1.30 1.32
band resembles the s states of the valence electrons, in agreement with the theoretical ideas on the nature of positron states [17]. The distribution of positrons corresponds to a positron position in the region of interstices (for the ground state) since the positron is repelled by the positively charged ion cores, whose wave functions are being concentrated in the outer parts of the unit cell. The positron in a semiconductor, in analogy with the hole, may be considered as a carrier of
positive charge. However, due to its higher effective mass, it is considerably less mobile than its hole counterpart [37]. This mass is an important parameter for studying positron diffusion in semiconductors. Similarly to the electron part, such parameter can be obtained from the band structure of material using the relation (16). However, the obtained mass is termed as the positron band mass. The latter is found to have the largest contribution to the positron effective mass [19,21,37] and has been obtained in our case by fitting parabolas to the positron band energy (Fig. 4) calculated in the vicinity of the G1 point (the lowest positron level). Our results for zincblende AlN, InN and some of their ternary alloys are listed in Table 6. It has been reported in the literature that the positron effective mass m ¼ 1:5 m0 ; where m0 is the positron free-particle mass, has been considered as a compromise over the various theoretical and experimental determinations [38]. Our calculated positron band effective masses are smaller than m : The difference between the positron effective mass m and the positron band effective masse m b is believed
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Fig. 5. Band effective mass of the positron in Al1xInxN as a function of the indium concentration x:
to arise from the neglect of the positron–phonon and positron–plasmon interactions. The variation of the positron band effective mass as a function of the indium concentration x in the range 0–1 is displayed in Fig. 5. It is to be noted that the positron band effective mass increases with increasing the composition x from 0 to 1. A least-quares fit was made to these data giving the following expression: m b ¼ 1:27 þ 0:05x þ 0:006x2 m0
introduced as an effective disorder potential. The positron wave functions have been calculated employing the point-core approximation for the ionic potential. Our results are roughly in good agreement with the available experimental data and ab initio calculations. Some predicted electronic and positronic properties for the ternary alloy Al0.83In0.17N which might be used as a future lattice matched barrier for GaN, have also been reported.
ð18Þ
showing a very weak band mass bowing parameter of 0.006 suggesting thus a nearly linear behavior of mnb :
4. Conclusion Theoretical calculations of electronic and positronic properties of zincblende Al1xInxN have been performed in the virtual crystal approximation using the empirical pseudopotential scheme. In this scheme, the compositional disorder is
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