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AlGaN quantum wells
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Physica E 17 (2003) 238 – 239 www.elsevier.com/locate/physe
Electronic structure of cubic GaN/AlGaN quantum wells J. Arriagaa;∗ , H. Hern*andez-Cocoletzia , D.A. Contreras-Solorioa; b a Instituto
de F sica, Luis Rivera Terrazas, Universidad Aut onoma de Puebla, Apartado Postal J-48, 72570 Puebla, Mexico b Escuela de F sica, UAZ, Av. Preparatoria 301, 98060 Zacatecas, Mexico
Abstract For cubic Alx Ga1−x N=GaN=Alx Ga1−x N quantum wells we calculated the 2rst energy transition 1h–1e as a function of x and the well width. The nearest neighbour sp3 s∗ empirical tight binding approximation, including spin-orbit interaction, together with the Surface Green Function Matching method is used. ? 2002 Elsevier Science B.V. All rights reserved. PACS: 73.21.Fg; 31.15.−P; 71.20.Nr Keywords: Quantum wells; Nitrides; Electronic structure
1. Introduction GaN, AlN, and their alloys have received considerable attention due to their potential applications in electroluminescent devices as well as for high-power and high-temperature devices. They crystallize in hexagonal wurtzite structure as the stable phase. However, it is also possible to grow them in cubic zincblende structure. In fact, LEDs have been prepared using cubic GaN [1]. Although there is considerable reported work for hexagonal quantum wells (QWs) made of III-nitrides in both experiments and theory, there is little work conducted on cubic-phase QWs [2]. In this work we calculated the energy of the 1h–1e transition, depending on the x concentration of Al and the well width for (0 0 1) zincblende Alx Ga1−x N=GaN=Alx Ga1−x N strained QWs. We used the empirical tight binding approximation (ETBA)
∗
Corresponding author. Fax: +52-22-448947. E-mail address: [email protected] (J. Arriaga).
with the sp3 s∗ basis, nearest neighbours and the spin–orbit (S–O) interaction, together with the surface green function matching (SGFM) method.
2. Theoretical aspects We have calculated 2rst the band structure of bulk cubic GaN and AlN. For the calculation of the electronic structure of the Alx Ga1−x N alloy we use the virtual crystal approximation. While theory predicts that cubic GaN is a direct gap semiconductor like hexagonal AlN and GaN, we found like other theoretical works that cubic AlN is indirect. The QWs energy levels are calculated using the SGFM theory [3]. The diFerence between lattice constants of the two materials produces strain. We assumed that the GaN lattice constant accomodates to that of the alloy, which is relaxed. Due to the bigger lattice constant of GaN, the well is subjected to a biaxial compressive strain. This eFect has been incorporated through scaling of the bulk ETBA parameters.
1386-9477/03/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S1386-9477(02)00781-6
J. Arriaga et al. / Physica E 17 (2003) 238 – 239
3. Results
239
3.8
Table 1 ETBA and S–O parameters (in eV)
3.7
energy (eV)
Lattice and elastic constants data for GaN and AlN were taken from Ref. [4]. The ETBA parameters obtained are given in Table 1. For the alloy gap, we show in Fig. 1 its dependence on x at the ; X , and L points. In the calculation of the QWs we assumed a valence band oFset KEv of 60%. We have calculated the energy of the 1h–1e transition for x = 0:12 and 0.3, and we show in Fig. 2 their dependence on the well width. At these concentrations the alloy is still a direct gap semiconductor. The compressive strain,
0.3
3.6
3.5
0.12 3.4
3.3
0
1
2
3
4
5
well width (nm) Fig. 2. Transition energies 1h–1e as a function of the well width for x = 0:12 and 0.3.
ETBA
GaN
AlN
E(s; a) E(p; a) E(s; c) E(p; c) E(s∗ ; a) E(s∗ ; c) V (s; s) V (x; x) V (x; y) V (sa; pc) V (sc; pa) V (s∗ a; pc) V (s∗ c; pa) a c
−12:9156
−11:5047
3.2697 −1:5844 9.1303 14.00 14.00 −8:8996 5.4638 8.7208 6.7152 7.3524 7.8440 2.3827 0.003 0.015
3.1815 0.5047 9.0184 17.20 17.20 −9:8077 5.3565 7.6630 8.0734 7.3218 9.9763 1.052 0.003 0.008
6
relative to the alloy’s lattice constant, is 0.004 and 0.01 at x = 0:12 and 0.3, respectively. The energy transition reduces when the well is wider because the hole and electron energy levels go closer to the bottom of the well potential. Note the crossing of the transition at large widths. We ascribed this eFect to the strain present in the wells. 4. Conclusions We have calculated the 1h–1e transition energy for the cubic GaN=Alx Ga1−x N QWs. At large widths and for two diFerent values of x we observed a crossing in the transition energies. We attributed this eFect to the strain in the wells. Acknowledgements
L
energy (eV)
This work was partially supported by CONACYT (M*exico), under Grants 27736E and 33808E. One of the authors (DACS) wishes to acknowledge IFUAP for a sabbatical year.
X
5
Γ
4
References
Al x Ga 1–x N
3
0
0.2
0.4
0.6
0.8
1
concentration x Fig. 1. Evolution of the gap with concentration x at ; X , and L points for the alloy.
[1] J. Wu, et al., Appl. Phys. Lett. 71 (1997) 2067. [2] D.J. As, et al., Mater. Res. Soc. Symp. Proc. 639 (2001) G5.9.1. [3] F. Garc*Qa-Moliner, V.R. Velasco, Theory of Single and Multiple Interfaces, World Scienti2c, Singapore, 1992. [4] K. Shimada, T. Sota, K. Suzuki, J. Appl. Phys. 84 (1998) 4951.
Physica E 17 (2003) 238 – 239 www.elsevier.com/locate/physe
Electronic structure of cubic GaN/AlGaN quantum wells J. Arriagaa;∗ , H. Hern*andez-Cocoletzia , D.A. Contreras-Solorioa; b a Instituto
de F sica, Luis Rivera Terrazas, Universidad Aut onoma de Puebla, Apartado Postal J-48, 72570 Puebla, Mexico b Escuela de F sica, UAZ, Av. Preparatoria 301, 98060 Zacatecas, Mexico
Abstract For cubic Alx Ga1−x N=GaN=Alx Ga1−x N quantum wells we calculated the 2rst energy transition 1h–1e as a function of x and the well width. The nearest neighbour sp3 s∗ empirical tight binding approximation, including spin-orbit interaction, together with the Surface Green Function Matching method is used. ? 2002 Elsevier Science B.V. All rights reserved. PACS: 73.21.Fg; 31.15.−P; 71.20.Nr Keywords: Quantum wells; Nitrides; Electronic structure
1. Introduction GaN, AlN, and their alloys have received considerable attention due to their potential applications in electroluminescent devices as well as for high-power and high-temperature devices. They crystallize in hexagonal wurtzite structure as the stable phase. However, it is also possible to grow them in cubic zincblende structure. In fact, LEDs have been prepared using cubic GaN [1]. Although there is considerable reported work for hexagonal quantum wells (QWs) made of III-nitrides in both experiments and theory, there is little work conducted on cubic-phase QWs [2]. In this work we calculated the energy of the 1h–1e transition, depending on the x concentration of Al and the well width for (0 0 1) zincblende Alx Ga1−x N=GaN=Alx Ga1−x N strained QWs. We used the empirical tight binding approximation (ETBA)
∗
Corresponding author. Fax: +52-22-448947. E-mail address: [email protected] (J. Arriaga).
with the sp3 s∗ basis, nearest neighbours and the spin–orbit (S–O) interaction, together with the surface green function matching (SGFM) method.
2. Theoretical aspects We have calculated 2rst the band structure of bulk cubic GaN and AlN. For the calculation of the electronic structure of the Alx Ga1−x N alloy we use the virtual crystal approximation. While theory predicts that cubic GaN is a direct gap semiconductor like hexagonal AlN and GaN, we found like other theoretical works that cubic AlN is indirect. The QWs energy levels are calculated using the SGFM theory [3]. The diFerence between lattice constants of the two materials produces strain. We assumed that the GaN lattice constant accomodates to that of the alloy, which is relaxed. Due to the bigger lattice constant of GaN, the well is subjected to a biaxial compressive strain. This eFect has been incorporated through scaling of the bulk ETBA parameters.
1386-9477/03/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S1386-9477(02)00781-6
J. Arriaga et al. / Physica E 17 (2003) 238 – 239
3. Results
239
3.8
Table 1 ETBA and S–O parameters (in eV)
3.7
energy (eV)
Lattice and elastic constants data for GaN and AlN were taken from Ref. [4]. The ETBA parameters obtained are given in Table 1. For the alloy gap, we show in Fig. 1 its dependence on x at the ; X , and L points. In the calculation of the QWs we assumed a valence band oFset KEv of 60%. We have calculated the energy of the 1h–1e transition for x = 0:12 and 0.3, and we show in Fig. 2 their dependence on the well width. At these concentrations the alloy is still a direct gap semiconductor. The compressive strain,
0.3
3.6
3.5
0.12 3.4
3.3
0
1
2
3
4
5
well width (nm) Fig. 2. Transition energies 1h–1e as a function of the well width for x = 0:12 and 0.3.
ETBA
GaN
AlN
E(s; a) E(p; a) E(s; c) E(p; c) E(s∗ ; a) E(s∗ ; c) V (s; s) V (x; x) V (x; y) V (sa; pc) V (sc; pa) V (s∗ a; pc) V (s∗ c; pa) a c
−12:9156
−11:5047
3.2697 −1:5844 9.1303 14.00 14.00 −8:8996 5.4638 8.7208 6.7152 7.3524 7.8440 2.3827 0.003 0.015
3.1815 0.5047 9.0184 17.20 17.20 −9:8077 5.3565 7.6630 8.0734 7.3218 9.9763 1.052 0.003 0.008
6
relative to the alloy’s lattice constant, is 0.004 and 0.01 at x = 0:12 and 0.3, respectively. The energy transition reduces when the well is wider because the hole and electron energy levels go closer to the bottom of the well potential. Note the crossing of the transition at large widths. We ascribed this eFect to the strain present in the wells. 4. Conclusions We have calculated the 1h–1e transition energy for the cubic GaN=Alx Ga1−x N QWs. At large widths and for two diFerent values of x we observed a crossing in the transition energies. We attributed this eFect to the strain in the wells. Acknowledgements
L
energy (eV)
This work was partially supported by CONACYT (M*exico), under Grants 27736E and 33808E. One of the authors (DACS) wishes to acknowledge IFUAP for a sabbatical year.
X
5
Γ
4
References
Al x Ga 1–x N
3
0
0.2
0.4
0.6
0.8
1
concentration x Fig. 1. Evolution of the gap with concentration x at ; X , and L points for the alloy.
[1] J. Wu, et al., Appl. Phys. Lett. 71 (1997) 2067. [2] D.J. As, et al., Mater. Res. Soc. Symp. Proc. 639 (2001) G5.9.1. [3] F. Garc*Qa-Moliner, V.R. Velasco, Theory of Single and Multiple Interfaces, World Scienti2c, Singapore, 1992. [4] K. Shimada, T. Sota, K. Suzuki, J. Appl. Phys. 84 (1998) 4951.