Calculation of electronic and optical properties of zinc blende GaP1 − xNx

Superlattices and Microstructures, Vol. 23, No. 2, 1998

Calculation of electronic and optical properties of zinc blende GaP1−x Nx F. Benkabou, J. P. Becker†, M. Certier‡, H. Aourag Computational Materials Science Laboratory, University of Sidi BelAbbes 22000, Algeria

(Received 4 July 1997) In order to clarify the electronic properties of the ternary compound semiconductor GaPN, in a zinc-blende structure, a simple pseudopotential scheme (EPM), within an effective potential (VCA), is proposed. The effects of disorder and spin–orbit coupling are neglected. Various quantities, such as energy levels, charge densities, ionicity character, transverse effective charge, and refractive index are obtained for this alloy. Moreover, the crossover of the direct and indirect band gaps is predicted. c 1998 Academic Press Limited

Key words: GaPN, EPM, VCA, ionicity character, transverse effective charge, refractive index.

1. Introduction The III–V nitrides have long been viewed as a promising system for semiconductor device applications in the blue and ultraviolet (UV) wave lengths. These semiconductors have received considerable attention both experimentally [1–8] and theoretically [9–13]. GaN is one of the candidates for making blue-light-emitting devices, due to its wide band gap, E g = 3.4 eV [14] at room temperature (RT). A number of reviews on GaN were given by Strite and Morkoc [15], David et al. [16] and Pankove [17]. It is well known that an interpolation of the band gap between GaAs and GaP has been achieved by forming the alloy GaAs1−x Px . However, can a similar alloy be realized for the GaPN system? There are two conspicuous problems with this subject. First large lattice mismatch between the two components, GaN and GaP, and secondly the fact that they crystallize in different lattice structures: GaP in a zinc-blende (ZB) but GaN in a wurtzite (W) structure. The reason is that most of the research on III–V nitrides has been grown using sapphire substrates which generally transfer their hexagonal symmetry to the nitride film. Therefore, the problem of creating an alloy of the nitride film (GaN, GaP) is the very low solubility of N in GaP (1019 cm−3 at 1700 ◦ C) [18]. It seems impossible to make solid GaP1−x Nx solutions with x > 0.1 in equilibrium. Baillargeon et al. [19] reported that they have formed GaP1−x Nx alloy with x < 0.03 in GaPN samples grown by molecular beam epitaxy (MBE). On the other hand, Igarashi [20] produced aGaP1−x Nx alloy with x near 1 by vapour-phase epitaxy. Very recently, by using the ion implantation technique, Xiashan et al. .

.

.

.

.

.

.

.

.

.

.

† Present address: Structure e´ lectroniques et modelisations, Ecole Centrale des Arts et Manufactures, 92295, Chatenay Malabry Cedex, France. ‡ Present address: L.S.O.M., Institut de physique e´ lectronique chimie, Universit´e de Metz, Technopˆole 200, France.

0749–6036/98/020453 + 13 $25.00/0

sm970503

c 1998 Academic Press Limited

454

Superlattices and Microstructures, Vol. 23, No. 2, 1998 Table 1: Pseudopotential form factors and lattice constants used in the calculation. V s (3) (Ryd)

a ˚ (A)

5.451 −0.2460 4.4953 −0.2218

GaP GaN

V s (8) (Ryd)

V s (11) (Ryd)

V a (3) (Ryd)

V a (4) (Ryd)

V a (11) (Ryd)

0.0300 0.0180

0.0767 0.0989

0.0884 0.3360

0.0700 0.2430

0.0200 0.0400

.

Table 2: Comparison of the calculated levels spacings (eV) with experiment and other calculations. GaP

v 015 − 01c v 015 − X 1c v γ15 − L c1

GaN

Cal

Exp

Cal

Exp, The

2.8 2.30 2.789

2.8840 2.3543 2.74543

3.200 4.700 6.200

3.242 , 3.141

.

[21], calculated a phase diagram for the alloy system formed by the GaP-like phase (in the ZB structure) and the GaN-like phase (in the W structure). For an equilibrium system in the high-temperature region, the ZB phase exists only within a low composition region with x near 0 (x = 0.02 to x = 0.07), while the phase exists only within a high-composition region with x near 1(x ≈ 0.99). As the temperature decreases, a case occurs in which the two phases can co-exist. Ion implanted samples are usually in a nonequilibrium state. Nevertheless, interest in ZB nitrides has been growing recently. The ZB GaN has a higher saturated electron-drift velocity [15] and a somewhat lower energy band than W GaN. Mizuta et al. [22] first reported bulk ZB GaN grown on (001) GaAs. There have been several recent studies of ZB GaN [6, 7, 23, 24] and GaN in a ZB structure with a direct gap of 3.5 eV has been reported by Bloom [9]. One aim of this paper is to review the established future prospect of the wide-band gap device concepts and applications based on ZB III–V nitride semiconductors, particularly GaP1−x Nx alloy. The problem of GaP1−x Nx in ZB phases has been treated theoretically by Van Vechten using the dielectric method [25], where calculations made in the virtual-crystal approximation are assumed to have linear dependence on the alloy concentration of the lattice constant and parameters. Using the dielectric method for the critical composition xc , for the alloy system (GaN)1−x (GaP)x the predictions where the lowest and indirect gaps will be equal in energy are xc = 0.76 and E c = 3.23 eV. In this paper, we have calculated the electron band structure of ZB of the gap, by using the empirical pseudopotential method (EPM) [26]. We report a pseudopotential calculation of the band structure of GaP1−x Nx in an extended virtual-crystal approximation (VCA) [27, 28], which treats an alloy as a perfectly periodic crystal with an average potential at the anionic sublattice sites and does not include the effects of the aperiodic fluctuations in the crystal potentials. The use of the EPM has generally proven to produce reasonably good band structures, in comparison with the self-consistent pseudopotential method in the local-density approximation (LDA) which underestimates the energy band [29], with the quasiparticle method [29] which is reliable but very time consuming. .

.

.

.

.

.

.

.

.

.

.

.

.

.

Superlattices and Microstructures, Vol. 23, No. 2, 1998

455

Table 3: Energy values (eV) for the valence bands for GaP, GaN and GaP1−x Nx alloy at x = 0.27. All values are in reference to the v valence-band maximum (015 ). GaP

GaP0.73 N0.27

01v −12.858 −14.738 v 015 0.00 0.00 2.800 0.00003 01c c 015 5.248 2.965 X 1v −10.728 −13.118 X 3v −5.982 −5.371 X 5v −2.310 −2.218 2.300 2.960 X 1c X 3c 3.066 4.156 L v1 −11.348 −13.541 L v1 −6.001 −5.716 L v3 −0.929 −0.828 L v1 2.789 3.784 5.622 6.992 L c3

GaN −21.64 0.00 3.20 12.14 −20.45 −5.47 −2.29 4.70 7.46 −20.74 −6.19 −0.780 6.20 11.19

.

The main goal of this study is to clarify the electronic and optical properties of the ZB alloys, GaPN, not yet synthesized.

2. Calculations Let us define our empirical pseudopotential parameters (EPP) of the semiconductor as a superposition of the pseudo-atomic potential of the form V = VL + VNL , where VL and VNL are local and nonlocal parts, respectively. In this calculation we have omitted the nonlocal part. We regard the Fourier components of VL (r) as the local EPP. We determine the EPP by the nonlinear least squares method, in which all the parameters are simultaneously optimized under a defined criterion of minimizing the root mean square (rms) deviation. The experimental electronic band structure is used. Our nonlinear least squares method requires that the rms deviation of the calculated level spacings (LSs) from the experimental ones defined by δ=

X m



1E

 (i, j) 2

1/2 /(m − N )

(2.1)

(i, j)

should be minimum. (i, j)

(i, j) − E cal , 1E (i, j) = E exp (i, j)

(i, j)

where E exp and E cal are the observed and calculated LSs between the ith state at the wave vector k − ki and the jth state at k − k j , respectively, in the m chosen pairs (i, j) and N is the number of the EPP. The calculated energies given by solving the EPP secular depend nonlinearly on the EPP. The starting values of the parameters are improved step by step by iterations until δ is minimized. Let us denote the parameters by Pu (u = 1, 2, . . . , N ) and write as Pu (n + 1) = Pu (n) + 1Pu , where Pu (n) is the value at the nth iteration.

456

Superlattices and Microstructures, Vol. 23, No. 2, 1998 20 A GaP 15

10 L3

Energy (eV)

5

X3

L1

0

15

01

X1

0

015

L3 –5

X5

–10

–15

–20 0

X

W

L

0

K

X

Wavevector Fig. 1. (A) The band structure of GaP. .

These corrections 1Pu are determined simultaneously by solving a system of linear equations.  N X m m h i X X j (i, j) j (i, j) (Q iu − Q uj )(Q iu 0 − Q u 0 ) 1Pu = − E cal (n) (Q iu 0 − Q u 0 ), E exp u−1

(i, j)

(2.2)

(i, j) (i, j)

where u 0 = (1, 2, . . . , N ), E cal (n) is the value at the nth iteration, Q u is given by   X ∗ ∂ H (ki ) Cqi 0 (ki ), Q uj = Cqi (ki ) ∂ P 0 u 0 q,q q,q

(2.3)

H (ki ) is the pseudo-Hamiltonian matrix at k = ki in the plane-wave representation, and the ith pseudowave function at k = ki is expanded as X Cqi (ki )ei(ki +Gq )·r , (2.4) 9ki i (r) = q

Gq being the reciprocal lattice vector. Equation (2.2) shows that all of the parameters are determined automatically in an interdependent way.

Superlattices and Microstructures, Vol. 23, No. 2, 1998

457

30 B

GaN

20

Energy (eV)

10

L3

015

L1

01

L3

015

L

0

X3 X1

0 X5 –10

–20

0

X

W

K

X

Wavevector Fig. 1. (B) The band structure of GaN. .

The adjustable parameters are then the symmetric V s (G) and antisymmetric V a (G) form factors of GaP, and GaN. The lattice constant of the alloys A1−x Bx (A: GaP, B: GaN) are determined by using the Vegard’s rule as a(x) = (1 − x)aA + xaB ,

(2.5)

where aA , aB are the lattice constants of the pure semiconductors A and B respectively. The alloy potential is calculated within the virtual crystal approach (VCA), where we add a compositional disorder as an effective periodic potential: Valloy = VVCA + Vdis A B VA + x VB VVCA = (1 − x) alloy alloy

(2.6a) (2.6b)

where A , B and alloy are the volumes of the pure semiconductors A, B and their alloy respectively. By adding this effective disorder potential to the virtual crystal potential, we have the final expression for the potential [30]:   p A AC VA (r) − VB (r) (2.6c) Vdis (r) = − p x(1 − x) alloy alloy .

where the disorder parameter p which simulates the disorder effect, is treated in our calculations as an adjustable parameter. This parameter cannot arbitrarily be varied when a particular choice of experimental band-bowing parameter is used. For example, p can be varied only in a very narrow region to get a reasonable fit to experimental data.

458

Superlattices and Microstructures, Vol. 23, No. 2, 1998 6.5 L

GaP1–xNx

6.0 5.5

Energy (eV)

5.0 X 4.5 4.0 3.5

0

3.0 2.5 2.0 0.0

0.2

0.4

0.6

0.8

1.0

Concentration of N Fig. 2. Compositional dependence of the lowest direct (0 − 0) and indirect (0 X, 0L) gaps. .

3. Results 3.1. Electronic properties The pseudopotential form factors for the pure components GaP and GaN are given in Table 1 which gives reasonable agreement with experiment for the principal energy gaps (see Table 2). The eigenvalues from the EPM calculation for GaP and GaN in the ZB structure are listed in Table 3 for high symmetry points of the v ). In Fig. 1(A) and (B), the Brillouin zone. All energies are in reference to the top of the valence band (015 EPM band structures for GaP and GaN in the ZB phase are given. The results show that GaP is an indirect gap semiconductor with the minimum of the conduction band at point X , while GaN has a direct gap. The calculated energy gaps of GaP and GaN are 2.30 eV and 3.20 eV, respectively, which are in good agreement with the experimental results and other calculations as listed in Table 2. The VCA potential for intermediate compositions is obtained from that of the pure components. Furthermore, a linear variation of the lattice constant with composition has been assumed. The results of our calculation are presented in Fig. 2(A). A least square fit of the curve exhibits a sublinearity, yielding for the linear and quadratic energy coefficient: E g0 = 2.775 + 0.916x − 0.479x 2 E gX = 2.253 + 2.854x − 0.387x 2 E gL = 2.745 + 4.147x − 0674x 2 The quadratic term stands for the bowing parameter predicted for GaPN.

Superlattices and Microstructures, Vol. 23, No. 2, 1998

459

20 GaP0.73N0.27 15

10 L3 X3

Energy (eV)

5

L1

015 01

X1

0

L3

015

X5

–5

–10

–15

–20 0

W

X

L

0

K

X

Wavevector Fig. 3. Energy band structure of GaP1−x Nx calculated at x = 0.27 within the virtual crystal approximation. .

The crossing point of band gaps E 00 and E 0 X occurs near xc ≈ 0.27 and E c = 2.963 eV, which predicts a range of concentration where the gaps are indirect. The complete VCA electronic band structure of GaPN is shown in Fig. 3 for x ≈ 0.27., i.e. very close to the theoretically calculated crossover composition [25]. Our VCA calculation indicates that complete electronic band structure of GaP1−x Nx can be considered as a linear function of composition (see Fig. 2). This linearity can be easily understood if we consider that the alloy potential form factors calculated within the scheme VCA turn out to be linear functions of composition. In order to better understand the physical mechanism behind the calculated band gap trends, we give the results of calculation of the conduction band charge densities at 01C point in Fig. 4(A). The calculated charge densities show a strong antibonding contribution. This antibonding distribution is indicated by the minimum of the charge density approximately halfway along the bond. However, the electronic charge density surrounding the anion is larger than that surrounding the cation, this charge distribution is predominantly s-like, most of the charge is localized in the anion site. We notice that this amount increases as we increase the volume. The overall trend is also observed for the total valence charge densities. As shown in Fig. 4(B) the charge distribution shows that covalent charge build up is even greater and that the charge distribution favors the hP1−x Nx i ion. It is well known that the ionicity character is highly dependent of the total valence-charge densities. It is .

460

Superlattices and Microstructures, Vol. 23, No. 2, 1998 25 A GaP1–xNx

Charge density (arbitary units)

20

15

10

x = 0.0 x = 0.25 x = 0.50 x = 0.75 x = 1.0

5

Ga –0.50

–0.25

P1–xNx 0.00

0.25

0.50

Atomic position (a.u.) Fig. 4. (A) The variation of the conduction band charge densities at 0 point for GaPN with different volumes. .

interesting to see how the ionicity character moves for the alloy GaPN. We have calculated the variation of the ionicity character using the model which uses the pseudopotential parameters [31]. We find the dependence of the ionic character with N concentration as: .

f i = 0.356 + 0.447x − 0.225x 2 . Figure 5 displays this nonlinear variation versus the N fraction x. We notice that the ionicity value increases with increasing volume. This result confirms the overall trends observed for the charge densities. Another feature linked to the ionicity character is the gap between the first and second valence band at X . This gap is related to the different pseudopotentials for the cation and anion potentials. This ‘antisymmetric’ gap has been proposed as a measure of crystal ionicity [32]. The calculation of this gap of the alloy is shown in Fig. 6. We notice that the trends follow that of the ionicity character obtained above. .

3.2. Optical properties It is also of interest to estimate the transverse effective charge eT∗ because it gives some insight into the infrared activity of phonons in materials. The transverse dynamical effective charges eT∗ are fundamental quantities in the lattice dynamics of semiconductors, determining the long-range part of the force constants in

Superlattices and Microstructures, Vol. 23, No. 2, 1998

461

25 B GaP1–xNx

Charge density (arbitary units)

20

15

x = 0.0 x = 0.25 x = 0.50 x = 0.75 x = 1.0

10

5 –0.50

Ga –0.25

P1–xNx 0.00

0.25

0.50

Atomic position (a.u.) Fig. 4. (B) The variation of the total valence charge densities for GaPN with different volumes. .

the long-wavelength limit, the Froehlich electron–optical-phonon coupling, parts of the piezoelectric coefficients, etc. [33]. A number of the empirical and semi-empirical models have been developed for eT∗ [34, 37]. In this paper, the procedure of determining the transverse effective charge, is based on the model of Vogl [38]. .

.

.

.

eT∗ = 1Z −

8α 1 − α2

where 1Z = −Z A + (1 − x)Z B + x Z C and V a (3) . V s (3) The composition dependence of effective charge determined from these procedures are shown in Fig. 7. One feature of this curve is the strong dependence of eT∗ with volume, since the effective charge increased continuously with increasing volume. We find the dependence of effective charge with N concentration as: α=−

eT∗ = 3.951 + 0.676x + 0.152x 2 . This increase suggests the strong influence of anharmonicity on the effective charge.

462

Superlattices and Microstructures, Vol. 23, No. 2, 1998 0.65 GaP1–xNx 0.60

0.55

ƒi

0.50

0.45

0.40

0.35 0.0

0.2

0.4

0.6

0.8

1.0

Concentration of N Fig. 5. The variation of the ionicity character versus the N fraction x for GaPN. .

0.9 GaP1–xNx

Heteropolar gap (eV)

0.8

0.7

0.6

0.5

0.4

0.3 0.0

0.2

0.4

0.6

0.8

1.0

Concentration of N Fig. 6. The variation of the ‘antisymmetric’ gap versus the N fraction x for GaPN. .

Superlattices and Microstructures, Vol. 23, No. 2, 1998

463

4.8 GaP1–xNx 4.6

e* T

4.4

4.2

4.0

3.8 0.0

0.2

0.4

0.6

0.8

1.0

Concentration of N Fig. 7. The variation of the effective charge with alloy composition for GaPN. .

2.42 GaP1–xNx 2.40

Refractive index

2.38 2.36

2.34 2.32 2.30 2.28 0.0

0.2

0.4

0.6

0.8

Concentration of N Fig. 8. The variation of the refractive index with alloy composition for GaPN. .

1.0

464

Superlattices and Microstructures, Vol. 23, No. 2, 1998

In order to study the optical properties of the ternary alloy GaPN, we have used the Herv´e and Vandamme model [39] for the determination of the refractive index with concentration x. The refractive index is related to the energy band gap of the semiconductor, as follows: s  2 A n = 1+ Eg + B .

where E g is the energy band gap, A = 13.6 eV and B = 3.4 eV. This expression is valuable for ω  ω0 , the ultraviolet resonance frequency. Using these models we have calculated the variation of refractive index with alloy concentration. The results are displayed in Fig. 8 where composition dependence of refractive index was determined by polynomial fitting. Our best fit yields: n = 2.417 − 0.279x + 0.149x 2 . Again, we observe the strong nonlinear dependence of alloy properties with N concentration.

4. Conclusion The aim of this work was the tentative determination of a semiconductor with a large direct energy gap for light-emitting-diode and laser applications. Since all III–V compounds have an indirect edge with the conduction-band minimum located near the X point (GaP) of the Brillouin zone, alloys with a smallestdirect- gap material will give a mixed crystal with a direct gap (GaN) of intermediate magnitude to obtain large-direct-gap devices with emitted radiation in the region of maximum eye sensitivity. The indirect to direct energy gap transition is predicted at x ≈ 0.27 for GaP1−x Nx .

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]

}W. C. Johnson, J. B. Parsons, and M. C. Crew, J. Phys. Chem. 36, 2561 (1932). }E. Tiede, M. Thimann, and K. Sensse, Chem. Berichte 61, 1568 (1928). }H. P. Maruska and J. J. Tietjen, Appl. Phys. Lett. 15, 327 (1969). }C. G. van de Walle (ed) Proceedings of the 7th Trieste ICTP-IUAP Semiconductor Symposium, Physica B 185, R9 (1993). }T. Lei, T. D. Moustakas, R. J. Graham, Y. He, and S. J. Berhowitz, J. Appl. Phys. 71, 4933 (1992). }T. Lei, M. Fanciulli, R. J. Molnar, T. D. Moustakas, R. J. Graham, and J. Scanlon, Appl. Phys. Lett. 95, 944 (1991). }M. J. Paisley, Z. Sitar, J. B. Posthill, and R. F. Davis, J. Vac. Sci. Technol. A7, 701 (1987). }Z. Sitar, M. J. Paisley, J. Rnan, J. W. Choyke, and R. F. Davis, J. Mater. Sci. Lett. 11, 261 (1992). }S. Bloom, G. Harbeke, E. Meier, and I. B. Ortenburger, Phys. Status, Solidi B66, 161 (1974). }P. E. Van Camp, V. E. Van Doren, and J. T. Deverse, Phys. Rev. B44, 9056 (1991). }A. Munoz and K. Kunc, Phys. Rev. B44, 10372 (1991). }A. Rubio, J. L. Corkill, M. L. Corkill, M. L. Cohen, E. L. Shirley, and S. G. Louie, Phys. Rev. B48, 11810 (1993). }A. F. Wright and J. S. Nelson, Phys. Rev. B50, 2159 (1994). }H. P. Maruska and J. J. Tietjen, Appl. Phys. Lett. 15, 327 (1969). }S. Strite and H. Morkoc, J. Vac. Sci. Technol. B10, 1237 (1992). }R. F. Davis et al. Mater. Sci. Eng. B1, 77 (1988). }J. I. Pankove, Mater. Res. Soc. Symp. Proc. 97, 409 (1987); 162, 515 (1990). }J. M. Stringfellow, F. Electrochem. Soc. 119, 1780 (1972). }J. N. Baillargeon, K. Y. Cheng, G. E. Hofler, P. J. Pearah, and K. C. Hsieh, Appl. Phys. Lett. 60, 2540 (1992). .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

Superlattices and Microstructures, Vol. 23, No. 2, 1998 [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43]

465

}O. Igarashi and Y. Okada, Japan. J. Appl. Phys. 24, L792 (1985). }X. Wu, B. Ma, X. Yang, and Z. Lin unpublished. }M. Mizuta, S. Fujieda, Y. Matsumoto, and T. Kawamara, Japan. J. Appl. Phys. 25, L945 (1986). }Z. Sitar, M. J. Paisley, B. Yan, and R. F. Davis, Mater. Res. Soc. Symp. Proc. 162, 537 (1990). }T. P. Humphreys, C. A. Sukow, R. J. Nemanich, J. B. Posthill, R. A. Radder, S. V. Hattangadady, and R. J. Markunas, Mater. Res. Soc. Symp. Proc. 162, 53 (1990). }J. A. Van Vechten, T. K. Bergstresser, Phys. Rev. B6, 3351 (1970). }M. L. Cohen and T. K. Bergstresser, Phys. Rev. 141, 789 (1996). }F. Bassani and D. Brust, Phys. Rev. 11, 1524 (1963). }M. Z Huang and W. Y. Ching, Superlat. Microstruct. 1, 137 (1984). }M. S. Hybertsen and S. G. Louie, Phys. Rev. Lett. 55, 1418 (1985); Rev. B32, 7005 (1985); B34, 5390 (1986). }S. J. Lee, T. S. Kwon, K. Nahm, and C. K. Kim, J. Phys. Condens. Matter. 2, 3253 (1990). }A. Zaoui, M. Ferhat, B. Khelifa, J. P. Dufour, and H. Aourag, Phys. Stat. Sol. (b) 185, 163 (1994). }J. R. Chelikowski, J. J. Wagener, J. H. Weaver, and A. Jin, Phys. Rev. B40, 9644 (1989). }A. A. Maradudin Dynamical Properties of Solids, Vol. 1 edited by G. K. Horton and A. A. Maradudin, North-Holland, Amsterdam (1974), pp. 1–82. }S. H. Wemple, Phys. Rev. B7 4007 (1973). }P. Lawaetz, Phys. Rev. Lett. 26 697 (1971). }K. Hubner, Phys. Stat. Solidi. (b) 68, 223 (1975). }W. A. Harrison and S. Ciraci, Phys. Rev. B10, 1516 (1974). }P. Vogl, J. Phys. C11, 252 (1978). }P. J. L. Herv´e and L. K. J. Vandamme J. Appl. Phys. 77, 5476 (1995). }D. M. Roessler and D. E. Swets, J. Appl. Phys. 49, 804 (1978). }A. Rubio, J. L. Corkhill, M. L. Cohen, E. L. Shirley, and S. G. Louie, Phys. Rev. B48, 11810 (1993). }T. Lei, M. Franciulli, R. J. Molnan, and T. D. Moustakas, Appl. Phys. Lett. 95, 944 (1991). }A. C. Carter, P. J. Dean, M. S. Skolnick, and R. A. Stradling, J. Phys C10, 5111 (1977). .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.