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Correlated electron–hole transition energies in quantum-well wires: Effects of hydrostatic pressure
ARTICLE IN PRESS
Physica B 371 (2006) 153–157 www.elsevier.com/locate/physb
Correlated electron–hole transition energies in quantum-well wires: Effects of hydrostatic pressure N. Raigozaa, C.A. Duquea,, N. Porras-Montenegrob, L.E. Oliveirac a
Instituto de Fı´sica, Universidad de Antioquia, AA 1226, Medellı´n, Colombia Departamento de Fı´sica, Universidad del Valle, AA 25360, Cali, Colombia c Instituto de Fı´sica, Unicamp, CP 6165, Campinas-SP, 13083-970, Brazil
b
Received 18 August 2005; received in revised form 21 September 2005; accepted 18 October 2005
Abstract We perform a theoretical study, using a variational approach, of the effects of hydrostatic pressure on the exciton binding energies and photoluminescence peak positions in GaAs2Ga1x Alx As cylindrical quantum-well wires within the effective-mass approximation. Calculated results suggest that future device applications may be realized by properly varying the wire dimensions and applying hydrostatic pressure. r 2005 Elsevier B.V. All rights reserved. PACS: 71.55.Eq; 73.20.Dx; 73.20.Hb Keywords: Quantum-well wire; Hydrostatic pressure; Excitons
1. Introduction Theoretical studies predicted that low-dimensional semiconductor heterostructures, such as quantum-well wires (QWWs) and quantum dots (QDs), would offer superior optical and electrical characteristics that may be used for potential applications to high-performance devices [1]. These heterostructure systems may offer the advantage of lower switching energy and enhanced oscillator strength over the confined region. Real quantum wires may be obtained by techniques such as electron beam lithography and etching, growth on the substrate with V grove, and selective growth on SiO2 patterned GaAs (1 1 0) substrate, etc. In situ fabrication techniques embedding QWWs in barriers are promising for reducing damage or impurities and the resulting non-radiative recombination at the surfaces or the interfaces. QWWs obtained by metal–organic chemical vapor deposition techniques (MOCVD) exhibited clear cathodoluminescence (CL) or photolumi-
Corresponding author. Tel.: +57 4 2105630; fax: +57 4 2330120.
E-mail address: cduque@fisica.udea.edu.co (C.A. Duque). 0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2005.10.096
nescence (PL) spectra [2], and an anisotropic polarization dependence of the PL excitation (PLE) spectra. The enhanced binding energy of 0D and 1D excitons has been systematically studied in GaAs QWWs and QDs [3]. It has been shown that by increasing the confinement the binding energy is enhanced by up to 47 times the bulk value. Ham and Spector [4] have calculated the exciton linewidth in semiconducting cylindrical QWWs and found that when the exciton is scattered by polar optical phonons the confinement will increase the linewidth of the exciton peaks. By a perturbative approach, the polaron self-energy and correction to the electron effective mass in a freestanding GaAs QWW have been investigated by Xie [5] who showed that, for small radius, the effects of the confined longitudinal optical phonon on the polaron selfenergy are depreciable and that the larger effects are due to the surface optical phonon modes. Here we note that, in a previous work, Buonocore et al. [6] calculated the polaron effect in a cylindrical QWW, and concluded that the polaron effect of the confined longitudinal optical phonon modes decreases as the radius of the wire increases. A recent study used a Gaussian-type orbital variational function in a rectangular-shaped GaAs QWWs and
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evaluated the dependence with hydrostatic pressure of the interband optical absorption and the exciton binding energy [7]. In the present study we have used a variational procedure (and a hydrogenic-type trial wave function) within the effective-mass approximation, and calculated the effects of hydrostatic pressure on the binding energy and PL peak energy for the confined heavy-hole exciton in cylindrical-shaped GaAs2Ga1x Alx As QWWs. The work is organized as follows. In Section 2 a brief theoretical framework of the problem is given. Section 3 presents calculated results and discussion, and finally Section 4 presents our conclusions. 2. Theoretical framework The Hamiltonian for a heavy-hole exciton in a cylindrical-shaped GaAs2ðGa; AlÞAs QWW under the effect of hydrostatic pressure ðPÞ is given by [8] _2 1 2 1 2 e2 H¼ r þ r e h mh 2 me r þ V e ðre ; PÞ þ V h ðrh ; PÞ, ð1Þ !! r j2 þ z2 Þ1=2 is the electron–hole distance, where r ¼ ðjr me
mh
e
h
and are the electron and hole effective masses, respectively [9], is the static dielectric constant [10], and V e ðre ; PÞ, V h ðrh ; PÞ are the potential barriers that confine the carriers in the QWW heterostructure [11]. In our calculations, we follow Brown and Spector [8] and use a variational wave function for the heavy-hole exciton in a cylindrical QWW of radius R as Cðr; PÞ ¼ N
8 J 0 ðg re Þ J 0 ðh rh Þ gðrÞ > > > > > < a K 0 ðj re Þ J 0 ðh rh Þ gðrÞ
if re ; rh pR; if re XR; rh pR;
> b J 0 ðg re Þ K 0 ðm rh Þ gðrÞ if re pR; rh XR; > > > > : d K ðj r Þ K ðm r Þ gðrÞ if r ; r XR: 0 0 e h e h ð2Þ
where gðrÞ ¼ expðlrÞ is a 1s-like hydrogenic-type function and J 0 ðzÞ and K 0 ðzÞ are the usual Bessel functions. The parameters a, b, d, g, h, j, and m are obtained from the continuity of the non-correlated electron and hole wave functions and its derivatives at the wire frontier. The exciton binding energy is obtained from the usual definition, i.e., hCjHjCimin Eb ¼ E0 , hCjCi
One should note that the inclusion of the electron and hole mass mismatches as well as the dielectric constant mismatch in the Ga1x Alx As QWW barriers may be important, especially in the regime of small radii: for example, Gerlach et al. [12] have studied exciton properties in GaAs2Ga1x Alx As QW heterostructures, and found ˚ the dielectric constant that for a well width of 50 A, mismatch gives origin to a larger binding energy by 1 meV, whereas inclusion of the mass mismatches diminishes the binding energy by 1 meV, and the combined effect of mass and dielectric constant mismatches essentially leaves the exciton binding energy unaltered [12]. Here, therefore, for simplicity, we have considered the dielectric constants and the electron and hole effective masses as constants throughout the QWW heterostructure, and equal to the GaAs-bulk values. In what follows, the conduction (valence) band-offset of the GaAs2Ga1x Alx As heterostructure was taken as 60% (40%) of the total band-gap difference. The E Gg temperature- and pressure-dependent band-gap at the G point is [13] E Gg ðP; TÞ ¼ E 0g þ aP þ bP2 þ bT 2 =ðT þ cÞ,
where E 0g is the T ¼ 0, P ¼ 0 energy gap, and we take E 0g ðGaAsÞ ¼ 1:519 eV, and E 0g ðGa1x Alx AsÞ2E 0g ðGaAsÞ ¼ ð1:155x þ 0:37x2 Þ eV. The pressure coefficients [9,11] for the well and barrier regions are a ¼ 10:7 meV=kbar and a ¼ ð11:5 1:3xÞ meV=kbar, with b ¼ 0:0377 meV=kbar2 , and b ¼ 0, respectively. The temperature coefficients for both the well and barrier regions are taken as b ¼ 0:5405 meV=K and c ¼ 204 K [11]. The pressure-dependent conduction and valence effective masses are given by [11,14] me ðPÞ m0 " ¼ 1þ
E GP
2 1 þ E Gg ðP; TÞ E Gg ðP; TÞ þ D0
where E 0 is the eigenvalue of the Hamiltonian in Eq. (1) without the Coulomb potential term and, in the second term in the RHS of Eq. (3), the expected value of the Hamiltonian is minimized with respect to the variational parameter l.
!#1 ð5Þ
and mh ðPÞ ¼ a1 þ a2 P þ a3 P2 , m0
(6)
where m0 is the free electron-mass, E GP ¼ 7:51 eV, D0 ¼ 0:341 eV, a1 ¼ 0:30242, a2 ¼ 0:1 103 kbar1 , and a3 ¼ 5:56 106 kbar2 . For the low-temperature and pressure-dependent static dielectric constant we take [10] ðP; TÞ ¼ 0 ed1 ðTT 0 Þ ed2 P ,
(3)
(4)
(7)
where 0 ¼ 12:74, d1 ¼ 9:4 105 K1 , d2 ¼ 1:67 103 kbar1 , and T 0 ¼ 75:6 K. The pressure-dependent radius of the GaAs QWW may be obtained from the fractional change in volume associated with the hydrostatic pressure [15], i.e., DV =V 0 ¼ 3PðS 11 þ 2S12 Þ,
(8)
ARTICLE IN PRESS N. Raigoza et al. / Physica B 371 (2006) 153–157
with V ðPÞ ¼ pR2 ðPÞL, V 0 ¼ pR20 L, where L is the (infinite) length of the wire, and therefore RðPÞ ¼ R0 ½1 3PðS 11 þ 2S 12 Þ1=2 ;
ð9Þ
where R0 is the radius of the wire at atmospheric pressure and S 11 and S 12 are the compliance constants [15] given by S 11 ¼ ðC 11 þ C 12 Þ=½ðC 11 C 12 ÞðC 11 þ 2C 12 Þ,
(10)
S 12 ¼ C 12 =½ðC 11 C 12 ÞðC 11 þ 2C 12 Þ,
(11)
and C 11 and C 12 are the elastic constants [11]. Results in next section were obtained at a low T ¼ 4 K temperature. 3. Results and discussion In Fig. 1 we present our results for the heavyhole exciton binding energy in a cylindrical-shaped GaAs2Ga0:7 Al0:3 As QWW. We note that as the radius decreases the binding energy grows up to a maximum that
120 kbar
40
Eb (meV)
90 kbar
60 kbar
20
155
depends on the value of the pressure and then decreases to the bulk limit that depends on the pressure. This wellknown behavior of the increase in the exciton binding energy is associated with the reduction of the electron–hole distance with decreasing radii of the wire. In the case the QWW radius is t20 A˚ the exciton wave function spreads into the barrier region, and the binding energy reduces to the GaAs bulk limit (as the electron and heavy-hole effective masses were taken equal to the GaAs values throughout the heterostructure). From the results in Fig. 1(b), it is clear that the exciton binding energy increases with pressure. The pressure-related changes are mainly due to the pressure dependence of both the static dielectric constant and the wire radius. The behavior is as follows. As the pressure increases: (1) the dielectric constant decreases leading to an increase of the energy of the correlated e–h pair, and (2) the radius of the wire decreases and confining effects reduce the e–h distance leading to an increase of the exciton binding energy. Here we also note that the present exciton binding energies in cylindrical QWWs are larger (see Fig. 2) than the ones calculated by Kasapoglu et al. [7] in the case of square (of side L) cross-section pffiffiffiQWWs with the same cross-section area, i.e., R0 ¼ L0 = p. It is important to mention that our theoretical results are expected to be more reliable as the hydrogenic-type trial wave function used in the present calculations is more appropriate [16] in describing an exciton envelope wave function than their one-variable Gaussian-type orbital trial function. 25
30 kbar P=0 0
50
100
150
200
R0 (Å)
(a) 35
20 Å
25 Eb (meV)
Eb (meV)
120 kbar 15
40 kbar
50 Å 40 kbar
P=0 15
150 Å P=0
300 Å 5
5 80 0
(b)
120 kbar
40
80
120
100
120
140
160
R0 (Å)
Pressure (kbar)
Fig. 1. Heavy-hole exciton binding energies in cylindrical GaAs2 Ga0:7 Al0:3 As QWWs. In (a) results are as a function of the radius of the wire and for different values of the hydrostatic pressure whereas in (b) results are as a function of the hydrostatic pressure for different values of the radius of the wire.
Fig. 2. Heavy-hole exciton binding energies in GaAs2Ga0:7 Al0:3 As QWWs as a function of the radius of the wire and for different values of the hydrostatic pressure: comparison between present calculations (full curves) and theoretical results by Kasapoglu et al. [7] (dotted lines) in square (of side L) cross-section QWWs with the same cross-section area, pffiffiffi i.e., R0 ¼ L0 = p.
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2.8
9.9 120 kbar 90 kbar
2.4
2.0
9.8
30 kbar
PL peak energy (eV)
1.6
P=0 1.2
50
100
150
200
R0 (Å)
(a) 2.8
2.4
Pressure coefficient (meV / kbar)
60 kbar
9.7
0
50
100
150
200
R0 (Å)
(a) 9.9
20 Å 2.0
150 Å
1.6
1.2 (b)
9.8
50 Å
0
40
80
9.7 1500
120
Pressure (kbar)
(b)
1600 PL peak energy (meV)
1700
Fig. 3. Heavy-hole exciton peak position in cylindrical-shaped GaAs2Ga0:7 Al0:3 As QWWs as functions of (a) the radius of the wire, and (b) the hydrostatic pressure.
Fig. 4. Pressure coefficient for the heavy-hole exciton PL peak in cylindrical GaAs2Ga0:7 Al0:3 As QWWs. Calculated results are as function of: (a) the radius of the wire and (b) the PL exciton peak energy at P ¼ 0.
Fig. 3 displays the present results for the correlated e–h pair PL peak energy as a function of both the GaAs2ðGa; AlÞAs QWW radius and hydrostatic pressure. For a given value of the applied hydrostatic pressure, one observes that the exciton peak energy is essentially constant as the QWW radius diminishes. Also, for a given pressure ˚ we note an increase and diminishing QWW radius t50 A, in the energy of the exciton PL peak due essentially to the larger confining effects of the electron and hole potential barriers leading to larger energies of the non-correlated confined electron and hole states. The different large QWW radius limiting values correspond to the exciton PL peak for the bulk GaAs exciton under hydrostatic pressure. In Fig. 3(b) we observe that the energy of the exciton PL peak shows a linear growth for Pt30 kbar essentially following the pressure coefficient of the bulk GaAs energy gap and, for larger values of the hydrostatic pressure, the quadratic term in the pressure [cf. Eq. (4)] begins to be of importance, leading to a slower increase of the PL peak energy. In Fig. 4 we present our results for the pressure coefficient of the GaAs2ðGa; AlÞAs QWW exciton-related PL peak as a function of both the radius of the wire and of
the PL energy at the atmospheric limit (calculated at low pressures, i.e., Pt30 kbar). From Fig. 4(a) it is clear that the confining effects are mainly in the low range of the QWW dimensions, as we have previously discussed. With increasing QWW radii, the pressure coefficient increases to the a ¼ 10:7 meV=kbar GaAs bulk limit in which the barrier effects disappear. This behavior is also shown in Fig. 4(b). The changes in the pressure coefficient with the PL exciton peak energy at atmospheric pressure, or with the radius of the wire, may be explained in the following way: very low values of the P ¼ 0 exciton PL peak energy are associated to QWWs with large radii (confining effects are quite small) and one essentially obtains a bulk GaAs limiting value for the pressure coefficient, whereas increasing values of the PL exciton peak energy correspond to decreasing values of the QWW radii and larger confinement effects of the non-correlated e2h pair transition energy, with considerable effects both in the exciton binding energy as well as in the pressure coefficient. Results displayed in Fig. 4 are in qualitative agreement with the previously observed behavior in GaAs QW heterostructures (see Fig. 6 by Venkateswaran et al. [18]).
ARTICLE IN PRESS N. Raigoza et al. / Physica B 371 (2006) 153–157
We do hope that the present calculations would be of interest in future experimental work in the future. Here we should note that, in the present calculations, we have altogether neglected the effects of crossing between G- and X -related states [19–21]: the relative energies of the bulk G and X minima change with hydrostatic pressure and, for bulk GaAs, the G X crossover occurs at P 35 kbar. As we are interested in G-electron + heavyhole excitons which can be observed even above the G–X crossover [20], it is expected [21] that crossing effects will not modify much of the present results, unless the pressure is very close to P . 4. Conclusions We have investigated the effects of hydrostatic pressure on the binding energy and PL peak energy for a heavy-hole exciton in cylindrical-shaped GaAs2Ga1x Alx As QWWs. Calculations were performed via a variational procedure within the effective-mass approximation. As it is well known, size effects show that as the QWW radius decreases the exciton binding energy increases up to a limit and then decreases to the GaAs bulk limit. Pressure effects show that the exciton binding energy increases with applied hydrostatic pressure mainly due to the pressure variation of the dielectric constant and radius of the wire. Variations of the pressure coefficient for the PL exciton peak energy have been calculated and are in qualitative agreement with experiment in QWs [18], as one should expect. Moreover, although we are not aware of any experimental data with respect to GaAs2Ga1x Alx As QWW heterostructures, we do hope that the present calculations would be of interest in future experimental work in the future. Finally, we should mention that the present theoretical results may be of importance for future device applications such as infrared electro-absorption modulators, detectors, and tunable wavelength lasers, in the sense that convenient optoelectronic behavior may be achieved by properly varying either the wire dimensions or applying hydrostatic pressure. Acknowledgements This research was partially supported by Colombian COLCIENCIAS (Grants 1106-05-13828 and 1115-0511502), CODI-Universidad de Antioquia Agencies, and by the Excellence Center for Novel Materials/COLCIENCIAS (contract No. 043-2005). We are also grateful to Brazilian Agencies CNPq, FAPESP, Rede Nacional de Materiais Nanoestruturados/CNPq, and Millenium Insti-
157
tute for Quantum Computing/MCT, for partial financial support. References [1] Y. Arakawa, H. Sakaki, Appl. Phys. Lett. 40 (1982) 939; H. Temkin, G.J. Dolan, M.B. Panish, S.N.G. Chu, Appl. Phys. Lett. 50 (1987) 413; Y. Arakawa, Solid-State Electron. 37 (1994) 523. [2] E. Kapon, K. Kash, E.M. Clausen Jr., D.M. Hwang, E. Colas, Appl. Phys. Lett. 60 (1992) 477; S. Tsukamoto, Y. Nagamune, M. Nishioka, Y. Arakawa, J. Appl. Phys. 71 (1992) 533. [3] T. Someya, H. Akiyama, H. Sakaki, Phys. Rev. Lett. 76 (1996) 2965; M. Bayer, S.N. Walck, T.L. Reinecke, A. Forchel, Phys. Rev. B 57 (1998) 6584. [4] H. Ham, H.N. Spector, Phys. Rev. B 62 (2000) 13599. [5] H.-J. Xie, Physica E 22 (2004) 906. [6] F. Buonocore, G. Iadonisi, D. Ninno, F. Ventriglia, Phys. Rev. B 65 (2002) 205415. [7] E. Kasapoglu, H. Sari, M. Gunes, I. So¨kmen, Physica B 353 (2004) 345. [8] J.W. Brown, H.N. Spector, Phys. Rev. B 35 (1987) 3009. [9] S. Adachi, J. Appl. Phys. 58 (1985) R1. [10] G.A. Samara, Phys. Rev. B 27 (1983) 3494; R.F. Kopf, M.H. Herman, M. Lamont Schnoes, A.P. Perley, G. Livescu, M. Ohring, J. Appl. Phys. 71 (1992) 5004. [11] A.M. Elabsy, Physica Scripta 48 (1983) 376; A.M. Elabsy, J. Phys.: Condens. Matter 6 (1994) 10025. [12] B. Gerlach, J. Wu¨sthoff, M.O. Dzero, M.A. Smondyrev, Phys. Rev. B 58 (1998) 10568. [13] Z.M. Fang, K.Y. Ma, D.H. Jaw, R.M. Cohen, G.B. Stringfellow, J. Appl. Phys. 67 (1990) 7034. [14] M. Boucenna, N. Bouarissa, Mater. Chem. Phys. 84 (2004) 375. [15] P.Y. Yu, M. Cardona, Fundamentals of Semiconductors, Springer, Berlin, 1998. [16] We note that our hydrogenic-type trial envelope wave function for the exciton ground state leads to the appropriate bulk limit for both small and large QWW radii, in contrast with the choice by Kasapoglu et al. [7]. Moreover, variational calculations with hydrogenic-type trial wave functions have been widely used [17] in the case of Coulomb-bound states in semiconductor heterostructures with excellent agreement with experiment. [17] L.E. Oliveira, J. Lo´pez-Gondar, Phys. Rev. B 41 (1990) 3719; A. Latge´, N. Porras-Montenegro, L.E. Oliveira, Phys. Rev. B 47 (1983) 13864; A. Latge´, N. Porras-Montenegro, M. de Dios-Leyva, L.E. Oliveira, Phys. Rev. B 53 (1996) 10160; E. Reyes-Go´mez, A. Matos-Abiague, C.A. Perdomo-Leiva, M. de Dios-Leyva, L.E. Oliveira, Phys. Rev. B 61 (2000) 13104; C.A. Duque, C.L. Beltran, A. Montes, N. Porras-Montenegro, L.E. Oliveira, Phys. Rev. B 61 (2000) 9936. [18] U. Venkateswaran, M. Chandrasekhar, H.R. Chandrasekhar, B.A. Vojak, F.A. Chambers, J.M. Meese, Phys. Rev. B 33 (1986) 8416. [19] P. Lefebvre, B. Gil, H. Mathieu, Phys. Rev. B 35 (1987) 5630; J.H. Burnett, H.M. Cheong, W. Paul, E.S. Koteles, B. Elman, Phys. Rev. B 47 (1993) 1991. [20] D.J. Wolford, T.F. Keuch, J.A. Bradley, M.A. Gell, D. Ninno, M. Jaros, J. Vac. Sci. Technol. B 4 (1986) 1037. [21] D.Z.-Y. Ting, Y.-C. Chang, Phys. Rev. B 36 (1987) 4359.
Physica B 371 (2006) 153–157 www.elsevier.com/locate/physb
Correlated electron–hole transition energies in quantum-well wires: Effects of hydrostatic pressure N. Raigozaa, C.A. Duquea,, N. Porras-Montenegrob, L.E. Oliveirac a
Instituto de Fı´sica, Universidad de Antioquia, AA 1226, Medellı´n, Colombia Departamento de Fı´sica, Universidad del Valle, AA 25360, Cali, Colombia c Instituto de Fı´sica, Unicamp, CP 6165, Campinas-SP, 13083-970, Brazil
b
Received 18 August 2005; received in revised form 21 September 2005; accepted 18 October 2005
Abstract We perform a theoretical study, using a variational approach, of the effects of hydrostatic pressure on the exciton binding energies and photoluminescence peak positions in GaAs2Ga1x Alx As cylindrical quantum-well wires within the effective-mass approximation. Calculated results suggest that future device applications may be realized by properly varying the wire dimensions and applying hydrostatic pressure. r 2005 Elsevier B.V. All rights reserved. PACS: 71.55.Eq; 73.20.Dx; 73.20.Hb Keywords: Quantum-well wire; Hydrostatic pressure; Excitons
1. Introduction Theoretical studies predicted that low-dimensional semiconductor heterostructures, such as quantum-well wires (QWWs) and quantum dots (QDs), would offer superior optical and electrical characteristics that may be used for potential applications to high-performance devices [1]. These heterostructure systems may offer the advantage of lower switching energy and enhanced oscillator strength over the confined region. Real quantum wires may be obtained by techniques such as electron beam lithography and etching, growth on the substrate with V grove, and selective growth on SiO2 patterned GaAs (1 1 0) substrate, etc. In situ fabrication techniques embedding QWWs in barriers are promising for reducing damage or impurities and the resulting non-radiative recombination at the surfaces or the interfaces. QWWs obtained by metal–organic chemical vapor deposition techniques (MOCVD) exhibited clear cathodoluminescence (CL) or photolumi-
Corresponding author. Tel.: +57 4 2105630; fax: +57 4 2330120.
E-mail address: cduque@fisica.udea.edu.co (C.A. Duque). 0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2005.10.096
nescence (PL) spectra [2], and an anisotropic polarization dependence of the PL excitation (PLE) spectra. The enhanced binding energy of 0D and 1D excitons has been systematically studied in GaAs QWWs and QDs [3]. It has been shown that by increasing the confinement the binding energy is enhanced by up to 47 times the bulk value. Ham and Spector [4] have calculated the exciton linewidth in semiconducting cylindrical QWWs and found that when the exciton is scattered by polar optical phonons the confinement will increase the linewidth of the exciton peaks. By a perturbative approach, the polaron self-energy and correction to the electron effective mass in a freestanding GaAs QWW have been investigated by Xie [5] who showed that, for small radius, the effects of the confined longitudinal optical phonon on the polaron selfenergy are depreciable and that the larger effects are due to the surface optical phonon modes. Here we note that, in a previous work, Buonocore et al. [6] calculated the polaron effect in a cylindrical QWW, and concluded that the polaron effect of the confined longitudinal optical phonon modes decreases as the radius of the wire increases. A recent study used a Gaussian-type orbital variational function in a rectangular-shaped GaAs QWWs and
ARTICLE IN PRESS N. Raigoza et al. / Physica B 371 (2006) 153–157
154
evaluated the dependence with hydrostatic pressure of the interband optical absorption and the exciton binding energy [7]. In the present study we have used a variational procedure (and a hydrogenic-type trial wave function) within the effective-mass approximation, and calculated the effects of hydrostatic pressure on the binding energy and PL peak energy for the confined heavy-hole exciton in cylindrical-shaped GaAs2Ga1x Alx As QWWs. The work is organized as follows. In Section 2 a brief theoretical framework of the problem is given. Section 3 presents calculated results and discussion, and finally Section 4 presents our conclusions. 2. Theoretical framework The Hamiltonian for a heavy-hole exciton in a cylindrical-shaped GaAs2ðGa; AlÞAs QWW under the effect of hydrostatic pressure ðPÞ is given by [8] _2 1 2 1 2 e2 H¼ r þ r e h mh 2 me r þ V e ðre ; PÞ þ V h ðrh ; PÞ, ð1Þ !! r j2 þ z2 Þ1=2 is the electron–hole distance, where r ¼ ðjr me
mh
e
h
and are the electron and hole effective masses, respectively [9], is the static dielectric constant [10], and V e ðre ; PÞ, V h ðrh ; PÞ are the potential barriers that confine the carriers in the QWW heterostructure [11]. In our calculations, we follow Brown and Spector [8] and use a variational wave function for the heavy-hole exciton in a cylindrical QWW of radius R as Cðr; PÞ ¼ N
8 J 0 ðg re Þ J 0 ðh rh Þ gðrÞ > > > > > < a K 0 ðj re Þ J 0 ðh rh Þ gðrÞ
if re ; rh pR; if re XR; rh pR;
> b J 0 ðg re Þ K 0 ðm rh Þ gðrÞ if re pR; rh XR; > > > > : d K ðj r Þ K ðm r Þ gðrÞ if r ; r XR: 0 0 e h e h ð2Þ
where gðrÞ ¼ expðlrÞ is a 1s-like hydrogenic-type function and J 0 ðzÞ and K 0 ðzÞ are the usual Bessel functions. The parameters a, b, d, g, h, j, and m are obtained from the continuity of the non-correlated electron and hole wave functions and its derivatives at the wire frontier. The exciton binding energy is obtained from the usual definition, i.e., hCjHjCimin Eb ¼ E0 , hCjCi
One should note that the inclusion of the electron and hole mass mismatches as well as the dielectric constant mismatch in the Ga1x Alx As QWW barriers may be important, especially in the regime of small radii: for example, Gerlach et al. [12] have studied exciton properties in GaAs2Ga1x Alx As QW heterostructures, and found ˚ the dielectric constant that for a well width of 50 A, mismatch gives origin to a larger binding energy by 1 meV, whereas inclusion of the mass mismatches diminishes the binding energy by 1 meV, and the combined effect of mass and dielectric constant mismatches essentially leaves the exciton binding energy unaltered [12]. Here, therefore, for simplicity, we have considered the dielectric constants and the electron and hole effective masses as constants throughout the QWW heterostructure, and equal to the GaAs-bulk values. In what follows, the conduction (valence) band-offset of the GaAs2Ga1x Alx As heterostructure was taken as 60% (40%) of the total band-gap difference. The E Gg temperature- and pressure-dependent band-gap at the G point is [13] E Gg ðP; TÞ ¼ E 0g þ aP þ bP2 þ bT 2 =ðT þ cÞ,
where E 0g is the T ¼ 0, P ¼ 0 energy gap, and we take E 0g ðGaAsÞ ¼ 1:519 eV, and E 0g ðGa1x Alx AsÞ2E 0g ðGaAsÞ ¼ ð1:155x þ 0:37x2 Þ eV. The pressure coefficients [9,11] for the well and barrier regions are a ¼ 10:7 meV=kbar and a ¼ ð11:5 1:3xÞ meV=kbar, with b ¼ 0:0377 meV=kbar2 , and b ¼ 0, respectively. The temperature coefficients for both the well and barrier regions are taken as b ¼ 0:5405 meV=K and c ¼ 204 K [11]. The pressure-dependent conduction and valence effective masses are given by [11,14] me ðPÞ m0 " ¼ 1þ
E GP
2 1 þ E Gg ðP; TÞ E Gg ðP; TÞ þ D0
where E 0 is the eigenvalue of the Hamiltonian in Eq. (1) without the Coulomb potential term and, in the second term in the RHS of Eq. (3), the expected value of the Hamiltonian is minimized with respect to the variational parameter l.
!#1 ð5Þ
and mh ðPÞ ¼ a1 þ a2 P þ a3 P2 , m0
(6)
where m0 is the free electron-mass, E GP ¼ 7:51 eV, D0 ¼ 0:341 eV, a1 ¼ 0:30242, a2 ¼ 0:1 103 kbar1 , and a3 ¼ 5:56 106 kbar2 . For the low-temperature and pressure-dependent static dielectric constant we take [10] ðP; TÞ ¼ 0 ed1 ðTT 0 Þ ed2 P ,
(3)
(4)
(7)
where 0 ¼ 12:74, d1 ¼ 9:4 105 K1 , d2 ¼ 1:67 103 kbar1 , and T 0 ¼ 75:6 K. The pressure-dependent radius of the GaAs QWW may be obtained from the fractional change in volume associated with the hydrostatic pressure [15], i.e., DV =V 0 ¼ 3PðS 11 þ 2S12 Þ,
(8)
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with V ðPÞ ¼ pR2 ðPÞL, V 0 ¼ pR20 L, where L is the (infinite) length of the wire, and therefore RðPÞ ¼ R0 ½1 3PðS 11 þ 2S 12 Þ1=2 ;
ð9Þ
where R0 is the radius of the wire at atmospheric pressure and S 11 and S 12 are the compliance constants [15] given by S 11 ¼ ðC 11 þ C 12 Þ=½ðC 11 C 12 ÞðC 11 þ 2C 12 Þ,
(10)
S 12 ¼ C 12 =½ðC 11 C 12 ÞðC 11 þ 2C 12 Þ,
(11)
and C 11 and C 12 are the elastic constants [11]. Results in next section were obtained at a low T ¼ 4 K temperature. 3. Results and discussion In Fig. 1 we present our results for the heavyhole exciton binding energy in a cylindrical-shaped GaAs2Ga0:7 Al0:3 As QWW. We note that as the radius decreases the binding energy grows up to a maximum that
120 kbar
40
Eb (meV)
90 kbar
60 kbar
20
155
depends on the value of the pressure and then decreases to the bulk limit that depends on the pressure. This wellknown behavior of the increase in the exciton binding energy is associated with the reduction of the electron–hole distance with decreasing radii of the wire. In the case the QWW radius is t20 A˚ the exciton wave function spreads into the barrier region, and the binding energy reduces to the GaAs bulk limit (as the electron and heavy-hole effective masses were taken equal to the GaAs values throughout the heterostructure). From the results in Fig. 1(b), it is clear that the exciton binding energy increases with pressure. The pressure-related changes are mainly due to the pressure dependence of both the static dielectric constant and the wire radius. The behavior is as follows. As the pressure increases: (1) the dielectric constant decreases leading to an increase of the energy of the correlated e–h pair, and (2) the radius of the wire decreases and confining effects reduce the e–h distance leading to an increase of the exciton binding energy. Here we also note that the present exciton binding energies in cylindrical QWWs are larger (see Fig. 2) than the ones calculated by Kasapoglu et al. [7] in the case of square (of side L) cross-section pffiffiffiQWWs with the same cross-section area, i.e., R0 ¼ L0 = p. It is important to mention that our theoretical results are expected to be more reliable as the hydrogenic-type trial wave function used in the present calculations is more appropriate [16] in describing an exciton envelope wave function than their one-variable Gaussian-type orbital trial function. 25
30 kbar P=0 0
50
100
150
200
R0 (Å)
(a) 35
20 Å
25 Eb (meV)
Eb (meV)
120 kbar 15
40 kbar
50 Å 40 kbar
P=0 15
150 Å P=0
300 Å 5
5 80 0
(b)
120 kbar
40
80
120
100
120
140
160
R0 (Å)
Pressure (kbar)
Fig. 1. Heavy-hole exciton binding energies in cylindrical GaAs2 Ga0:7 Al0:3 As QWWs. In (a) results are as a function of the radius of the wire and for different values of the hydrostatic pressure whereas in (b) results are as a function of the hydrostatic pressure for different values of the radius of the wire.
Fig. 2. Heavy-hole exciton binding energies in GaAs2Ga0:7 Al0:3 As QWWs as a function of the radius of the wire and for different values of the hydrostatic pressure: comparison between present calculations (full curves) and theoretical results by Kasapoglu et al. [7] (dotted lines) in square (of side L) cross-section QWWs with the same cross-section area, pffiffiffi i.e., R0 ¼ L0 = p.
ARTICLE IN PRESS N. Raigoza et al. / Physica B 371 (2006) 153–157
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2.8
9.9 120 kbar 90 kbar
2.4
2.0
9.8
30 kbar
PL peak energy (eV)
1.6
P=0 1.2
50
100
150
200
R0 (Å)
(a) 2.8
2.4
Pressure coefficient (meV / kbar)
60 kbar
9.7
0
50
100
150
200
R0 (Å)
(a) 9.9
20 Å 2.0
150 Å
1.6
1.2 (b)
9.8
50 Å
0
40
80
9.7 1500
120
Pressure (kbar)
(b)
1600 PL peak energy (meV)
1700
Fig. 3. Heavy-hole exciton peak position in cylindrical-shaped GaAs2Ga0:7 Al0:3 As QWWs as functions of (a) the radius of the wire, and (b) the hydrostatic pressure.
Fig. 4. Pressure coefficient for the heavy-hole exciton PL peak in cylindrical GaAs2Ga0:7 Al0:3 As QWWs. Calculated results are as function of: (a) the radius of the wire and (b) the PL exciton peak energy at P ¼ 0.
Fig. 3 displays the present results for the correlated e–h pair PL peak energy as a function of both the GaAs2ðGa; AlÞAs QWW radius and hydrostatic pressure. For a given value of the applied hydrostatic pressure, one observes that the exciton peak energy is essentially constant as the QWW radius diminishes. Also, for a given pressure ˚ we note an increase and diminishing QWW radius t50 A, in the energy of the exciton PL peak due essentially to the larger confining effects of the electron and hole potential barriers leading to larger energies of the non-correlated confined electron and hole states. The different large QWW radius limiting values correspond to the exciton PL peak for the bulk GaAs exciton under hydrostatic pressure. In Fig. 3(b) we observe that the energy of the exciton PL peak shows a linear growth for Pt30 kbar essentially following the pressure coefficient of the bulk GaAs energy gap and, for larger values of the hydrostatic pressure, the quadratic term in the pressure [cf. Eq. (4)] begins to be of importance, leading to a slower increase of the PL peak energy. In Fig. 4 we present our results for the pressure coefficient of the GaAs2ðGa; AlÞAs QWW exciton-related PL peak as a function of both the radius of the wire and of
the PL energy at the atmospheric limit (calculated at low pressures, i.e., Pt30 kbar). From Fig. 4(a) it is clear that the confining effects are mainly in the low range of the QWW dimensions, as we have previously discussed. With increasing QWW radii, the pressure coefficient increases to the a ¼ 10:7 meV=kbar GaAs bulk limit in which the barrier effects disappear. This behavior is also shown in Fig. 4(b). The changes in the pressure coefficient with the PL exciton peak energy at atmospheric pressure, or with the radius of the wire, may be explained in the following way: very low values of the P ¼ 0 exciton PL peak energy are associated to QWWs with large radii (confining effects are quite small) and one essentially obtains a bulk GaAs limiting value for the pressure coefficient, whereas increasing values of the PL exciton peak energy correspond to decreasing values of the QWW radii and larger confinement effects of the non-correlated e2h pair transition energy, with considerable effects both in the exciton binding energy as well as in the pressure coefficient. Results displayed in Fig. 4 are in qualitative agreement with the previously observed behavior in GaAs QW heterostructures (see Fig. 6 by Venkateswaran et al. [18]).
ARTICLE IN PRESS N. Raigoza et al. / Physica B 371 (2006) 153–157
We do hope that the present calculations would be of interest in future experimental work in the future. Here we should note that, in the present calculations, we have altogether neglected the effects of crossing between G- and X -related states [19–21]: the relative energies of the bulk G and X minima change with hydrostatic pressure and, for bulk GaAs, the G X crossover occurs at P 35 kbar. As we are interested in G-electron + heavyhole excitons which can be observed even above the G–X crossover [20], it is expected [21] that crossing effects will not modify much of the present results, unless the pressure is very close to P . 4. Conclusions We have investigated the effects of hydrostatic pressure on the binding energy and PL peak energy for a heavy-hole exciton in cylindrical-shaped GaAs2Ga1x Alx As QWWs. Calculations were performed via a variational procedure within the effective-mass approximation. As it is well known, size effects show that as the QWW radius decreases the exciton binding energy increases up to a limit and then decreases to the GaAs bulk limit. Pressure effects show that the exciton binding energy increases with applied hydrostatic pressure mainly due to the pressure variation of the dielectric constant and radius of the wire. Variations of the pressure coefficient for the PL exciton peak energy have been calculated and are in qualitative agreement with experiment in QWs [18], as one should expect. Moreover, although we are not aware of any experimental data with respect to GaAs2Ga1x Alx As QWW heterostructures, we do hope that the present calculations would be of interest in future experimental work in the future. Finally, we should mention that the present theoretical results may be of importance for future device applications such as infrared electro-absorption modulators, detectors, and tunable wavelength lasers, in the sense that convenient optoelectronic behavior may be achieved by properly varying either the wire dimensions or applying hydrostatic pressure. Acknowledgements This research was partially supported by Colombian COLCIENCIAS (Grants 1106-05-13828 and 1115-0511502), CODI-Universidad de Antioquia Agencies, and by the Excellence Center for Novel Materials/COLCIENCIAS (contract No. 043-2005). We are also grateful to Brazilian Agencies CNPq, FAPESP, Rede Nacional de Materiais Nanoestruturados/CNPq, and Millenium Insti-
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