Effects of hydrostatic pressure on the Coulomb-bound states in GaAs–Ga1−xAlxAs semiconductor superlattices

Superlattices and Microstructures 44 (2008) 809–813

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Effects of hydrostatic pressure on the Coulomb-bound states in GaAs–Ga1−x Alx As semiconductor superlattices J. Vargas a , N. Raigoza a , A.L. Morales a , C.A. Duque a,b , E. Reyes-Gómez a,∗ a

Instituto de Física, Universidad de Antioquia, AA 1226, Medellín, Colombia

b

Instituto de Física, Unicamp, CP 6165, Campinas, São Paulo, 13083-970, Brazil

article

info

Article history: Received 18 July 2008 Received in revised form 17 September 2008 Accepted 19 September 2008 Available online 26 October 2008 Keywords: Hydrostatic pressure Superlattice Impurity Exciton

a b s t r a c t The effects of hydrostatic pressure on the Coulomb-bound states in GaAs–Ga1−x Alx As and GaAs–AlAs semiconductor superlattices are theoretically studied. Calculations of the impurity binding energies for different configurations of the system and for various values of the hydrostatic pressure are performed in the framework of the parabolic-band and effective-mass schemes, and within the variational procedure. The hydrostatic-pressure dependence on the exciton energy is also obtained, and theoretical results are compared and found in good agreement with available experimental measurements. © 2008 Elsevier Ltd. All rights reserved.

1. Introduction The effects of hydrostatic pressure on the Coulomb states in semiconductors and their heterostructures have been widely studied in the past few decades. Hydrostatic pressure has been proven to be a powerful tool in order to investigate the electrical and optical properties of nanostructures, and has been successfully used to modulate the electronic, impurity and exciton states in such systems. For example, Venkateswaran et al. [1,2] reported the pressure dependence of the photoluminescence (PL) spectra of GaAs–Ga1−x Alx As quantum wells (QWs) and superlattices (SLs) at various values of the temperature, and the pressure coefficients associated with the observed transitions were also obtained. Burnett and co-workers [3] investigated the energies of the electronic states in GaAs–Ga1−x Alx As strongly coupled double QWs and uncoupled multiple QWs heterostructures as a function of the hydrostatic pressure, and their experimental results were found to be in fair agreement with an envelope-function-approximation model. Elabsy [4] studied the

∗

Corresponding author. Tel.: +57 4 2195630; fax: +57 4 2195666. E-mail address: [email protected] (E. Reyes-Gómez).

0749-6036/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.spmi.2008.09.009

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enhancement on the donor binding energies in single QWs as the hydrostatic pressure is increased, and found that the donor binding energies decrease when the hydrostatic pressure approaches to the crossover pressure between the well 0 states and the barrier X states. Spain et al. [5] and Guha et al. [6] carried out PL spectroscopy experiments in GaAs–AlAs SLs, and analyzed the recombination lifetimes as functions of the hydrostatic pressure. Itskevich et al. [7] performed PL measurements of self-assembled InAs quantum dots in strong magnetic fields and under hydrostatic pressure, and presented results for bulk GaAs. More recently, we reported the effects of hydrostatic pressure on the exciton states in GaAs–Ga1−x Alx As QWs under an electric field applied along the growth axis [8], and calculated the transition energies associated to the correlated electron–hole (e–h) pair for different QW-widths [9]. Theoretical results were found to be in good agreement with the available experimental data. From the theoretical point of view, the study of the Coulomb-bound states in periodic SLs under hydrostatic pressure have received much less attention than QWs. The purpose of the present work is to theoretically investigate the effects of the hydrostatic pressure on shallow-impurity states in GaAs–Ga1−x Alx As semiconductor SLs. We also calculate the PL-peak energy associated to the correlated e–h pair in GaAs–AlAs and GaAs–Ga1−x Alx As semiconductor SLs, and compare with the experimental data reported by Venkateswaran et al. [1] and by Spain et al. [5]. 2. Theoretical framework Within the effective-mass and non-degenerate parabolic-band approximations, the Schrödinger equation for a shallow impurity in a GaAs–Ga1−x Alx As semiconductor SL with growth axis along the z direction is given by

ˆ Ψ (Er) = E Ψ (Er), H

(1)

with

ˆ = pEˆ H

1 2m∗ (P , T , z )

Eˆ + V (P , T , z ) − p

e2

(P , T )|Er − Eri |

,

(2)

where m∗ (P , T , z ) and V (P , T , z ) are the pressure-, temperature-, and position-dependent effective mass [9] and SL confining potential [3,4], respectively, of the impurity electron (for donors) or hole (for acceptors), Eri is the impurity position [we assume Eri = (0, 0, zi )], and (P , T ) is the static dielectric constant, which is also a function of the pressure and temperature [9] and is taken as constant, for simplicity, throughout the heterostructure. Note that the effects of the hydrostatic pressure lead to changes in the structural parameters of the SL, i. e., the hydrostatic-pressure dependencies of the well and barrier widths (a and b, respectively) and of the impurity position are given by a(P ) = a0 [1 − (S11 + 2S12 ) P] ,

(3)

b(P ) = b0 [1 − (S11 + 2S12 ) P] ,

(4)

zi (P ) = z0 [1 − (S11 + 2S12 ) P] ,

(5)

and respectively, where a0 , b0 , and z0 are the well width, the barrier width, and the impurity position, respectively, in the absence of hydrostatic pressure [10], and S11 = 1.16 × 10−3 kbar−1 and S12 = −3.7 × 10−4 kbar−1 are the compliance constants of bulk GaAs [10,11]. Here, for simplicity, we have considered the compliance constants to be the same as in bulk GaAs throughout the heterostructure. The Eq. (1) may be solved via a variational procedure by proposing the ground-state trial wave function as

Ψ (Er) = Nf0 (z )e−λr ,

(6)

where N is a normalization constant, f0 is the ground-state solutions of the Schrödinger equation for the electrons or holes in their corresponding SL confining potentials in the absence of the Coulomb interaction, and λ is a variational parameter obtained from the minimization of the functional ˆ |Ψ (Er)i, which allows one to obtain the impurity-ground state energy for donors or E (λ) = hΨ (Er)|H for acceptors.

J. Vargas et al. / Superlattices and Microstructures 44 (2008) 809–813

811

By using a similar procedure, one may calculate the energy associated to the correlated e–h pair [8,9] in semiconductor heterostructures. For simplicity, we have not taken into account the complex structure of the valence band, and we have used a simple parabolic and isotropic band scheme in order to describe the heavy hole states in the semiconductor SL. The heavy-hole exciton Hamiltonian is given by

ˆ ex = pEˆ e H

1 2m∗ (P , T , z e

e

)

Eˆ e + pEˆ h p

1 2mh (P , T , zh ) ∗

+ Ve (P , T , ze ) + Vh (P , T , zh ) −

Eˆ h p e2

(P , T )|Ere − Erh |

+ Eg (P , T ),

(7)

Eˆ i , m∗i , and Vi are the vector position, momentum operator, effective mass, where Eri = (xi , yi , zi ), p and confining potential, respectively, associated with the 06c conduction electron and 08v heavy hole (i = e and i = h, respectively) in the SL, and Eg is the GaAs fundamental gap which is a function of the temperature and hydrostatic pressure. The ground-state exciton wavefunction may then be chosen as [8,9] Φ (Er) = Nf0e (ze )f0h (zh ) e−λr ,

(8) f0e

f0h

where N is a normalization constant, and are the ground-state wavefunctions, in the absence of the Coulomb potential, associated with the conduction electron and heavy hole, respectively, and r = |Ere − Erh | is the electron–hole distance. As for shallow impurities, the variational parameter λ and the ground-state exciton energy Eex may be obtained from the minimization of the functional hΦ |Hˆ ex |Φ i. 3. Results and discussion In Fig. 1 we display the donor binding energy as a function of the well width a0 , the barrier width b0 , the impurity position z0 , and the hydrostatic pressure P in GaAs–Ga0.75 Al0.25 As semiconductor SLs. Calculations in Fig. 1(a), (b), and (c) were performed for three different values of the hydrostatic pressure, whereas in Fig. 1(d) we considered three different positions of the shallow donor at the SL axis. The behavior of the binding energy may be understood in terms of the wave function localization, which is determined by the hydrostatic-pressure effects on the electronic states and by the anisotropy caused by the SL potential. The impurity binding energy is larger for donors at the well center than for donors at the barrier region, where the donor-electron wave function becomes partially trapped in the well region, far from the impurity nucleus, leading to a smaller confinement and binding energy. In addition, as the pressure is increased, both the structural parameters of the SL and the static dielectric constant diminish, the impurity becomes more confined, and as a consequence, the impurity-binding energy increases. Similar results may be obtained for excitons in semiconductor SLs. In the case of the excitons, the observation of exciton features in the PL spectra is, in part, determined by the radiative recombination probability from the conduction to the valence band. At low temperatures and in the absence of the hydrostatic pressure, only direct transitions at the 0 point are experimentally observed, and the exciton states behave as direct-exciton states. However, as the pressure is increased, a crossing between the 0 and X conduction subbands takes place, and also radiative-indirect transitions from the X point of the conduction band to the 0 point of the valence band may occur, besides the direct transitions already observed, as was discussed by Guha et al. [6] in short-period semiconductor superlattices. In Fig. 2 we display the PL-peak energies, as functions of the hydrostatic pressure, associated to the correlated e–h transitions (solid symbols) in semiconductor SLs. Solid triangles are the experimental data, measured at T = 1.5 K, reported by Spain et al. [5] in a GaAs–AlAs SL with a0 = 35 Å and b0 = 23 Å, whereas solid circles correspond to the experimental results from Venkateswaran et al. [1] in a GaAs–Ga0.75 Al0.25 As SL with a0 = 150 Å and b0 = 100 Å at T = 80 K. In addition, we have shown as open circles the electron-acceptor (e–A0 ) recombination energies in the SL studied by Venkateswaran et al. [1]. Solid lines are our theoretical calculations obtained within the variational procedure above

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Fig. 1. (Color online) Donor binding energy in GaAs–Ga0.75 Al0.25 As semiconductor SLs (a) as a function of the well width for b0 = 20 Å and z0 = 0, (b) as a function of the barrier width for a0 = 300 Å and z0 = 0, (c) as a function of the impurity position for a0 = 150 Å and b0 = 100 Å, and (d) as a function of the hydrostatic pressure for a0 = 150 Å and b0 = 100 Å. Solid, dashed and dot lines in (a), (b) and (c) correspond to the hydrostatic-pressure values of 0, 10 kbar and 20 kbar, respectively, whereas in (d) solid, dashed, and dot lines correspond to the impurity position at the well center, at the well/barrier interface, and at the barrier center, respectively.

described. For direct excitons and shallow acceptors, calculated results indicate that the pressure dependencies of the PL-peak energies are approximately linear functions of P. Deviations from the linear behavior are due to the slight quadratic dependence of the GaAs energy gap as a function of the hydrostatic pressure [12], and to the pressure-dependent confining potential [3,4]. From Fig. 2 one may note that our theoretical results are in quite good quantitative agreement with the experimental measurements [1,5]. 4. Conclusions In summary, we have theoretically studied the effects of the hydrostatic pressure on the impurity and exciton states in GaAs–Ga1−x Alx As and GaAs–AlAs semiconductor SLs. In the framework of the effective-mass and non-degenerate parabolic-band approximations, and within the variational

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Fig. 2. (Color online) PL-peak energies associated to various transitions, as functions of the hydrostatic pressure, in two different semiconductor superlattices. Solid triangles correspond to the experimental measurements, reported by Spain et al. [5], associated to direct-exciton transitions in a GaAs–AlAs SL at T = 1.5 K. Solid and open circles correspond to the experimental results reported by Venkateswaran et al. [1], associated to direct-exciton transitions and to e–A0 transitions in a GaAs–Ga0.75 Al0.25 As SL at T = 80 K, respectively. Present theoretical calculations are displayed as solid lines.

procedure, we obtained the binding energy associated with shallow donors as functions of the structural parameters of the SL as well as of the hydrostatic pressure. We also calculated the hydrostatic-pressure dependence of the recombination energy for excitons in a GaAs–AlAs SL at T = 1.5 K, and the exciton and e–A0 transition energies in a GaAs–Ga0.75 Al0.25 As SL at T = 80 K. Our theoretical calculations were found to be in good agreement with the experimental measurement reported by Venkateswaran et al. [1] and Spain et al. [5]. Acknowledgments This research was partially supported by Colombian COLCIENCIAS, CODI-Universidad de Antioquia Agencies, and the Excellence Center for Novel Materials/COLCIENCIAS under contract N0 043-2005. References [1] U. Venkateswaran, M. Chandrasekhar, H.R. Chandrasekhar, T. Wolfram, R. Fischer, W.T. Masselink, H. Morkoç, Phys. Rev. B 31 (1985) 4106. [2] U. Venkateswaran, M. Chandrasekhar, H.R. Chandrasekhar, B.A. Bojak, F.A. Chambers, J.M. Meese, Phys. Rev. B 33 (1986) 8416. [3] J.H. Burnett, H.M. Cheong, W. Paul, E.S. Koteles, B. Elman, Phys. Rev. B 47 (1993) 1991. [4] A.M. Elabsy, J. Phys.: Condens. Matter 6 (1994) 10025. [5] I.L. Spain, M.S. Skolnick, G.W. Smith, M.K. Saker, C.R. Whitehouse, Phys. Rev. B 43 (1991) 14091. [6] S. Guha, Q. Cai, M. Chandrasekhar, H.R. Chandrasekhar, H. Kim, A.D. Alvarenga, R. Vogelgesang, A.K. Rambas, M.R. Melloch, Phys. Rev. B 58 (1998) 7222. [7] I.E. Itskevich, M. Henini, H.A. Carmona, L. Eaves, P.C. Main, D.K. Maude, J.C. Portal, Appl. Phys. Lett. 70 (1997) 505. [8] N. Raigoza, C.A. Duque, E. Reyes-Gómez, L.E. Oliveira, Physica B 367 (2005) 267. [9] N. Raigoza, C.A. Duque, E. Reyes-Gómez, L.E. Oliveira, Phys. Status Solidi (b) 243 (2006) 635. [10] P.Y. Yu, M. Cardona, Fundamentals of Semiconductors, Springer, Berlin, 1998. [11] S. Adachi, J. Appl. Phys. 58 (1985) R1. [12] E. Herbert Li, Physica E 5 (2000) 215.