Photoreflectance Study of the Multilevel Energy Transitions of Quantum Wires in the AlGaAs/GaAs/AlGaAs/GaAs(631) Heterostructure
E. Cruz-Hernández1, J.S. Rojas-Ramírez1, J. Hernández-Rosas1,3, M. Ramírez-López1, S. Gallardo-Hernández2, I. Martínez-Velis1 y M. López-López1
Physics Department, Centro de Investigación y de Estudios Avanzados del IPN, Apartado Postal 14-740, México, D. F. 07000. 2 AEES, Electrical Engineering Department, Centro de Investigación y de Estudios Avanzados del IPN, Apartado Postal 14-740, México, D. F. 07000. 3 Physics Department, Universidad Autónoma Metropolitana-Iztapalapa, Apartado Postal 55-534, México, D. F. 09340.
1
Introduction
We present a photoreflectance (PR) study of the energy transitions between the quantum levels of a system of self-assembling GaAs Quantum Wires (QWRs) embedded in AlxGa1-xAs barriers. The energy transitions are calculated from two different ways, (1) Experimentally, from the PR spectra employing the model of Aspnes [1] and, (2) Theoretically, with the calculation of the permitted transitions in the subband structure of QWR of cylindrical geometry [2]. From this comparison we have deduced the most-probable origin of each PR-transition.
Here, Jnj (xnjmj r/r0) is the Bessel function of order nj, xnjmj denotes its zeros, Jnj (xnjmj) = 0 and J'nj (xnjmj) is its derivative. On the other hand, the energy levels are determined by 2
2 k zj + Ej = 2µ j
2
xn j m j r 0
Eq. (2)
Results and discussion
Below is presented the experimental PR spectrum (black points) together with the fitted (red) line resulting from the seven-signals version of Eq. (1). The found Ej value positions are also shown in the spectrum. In the table is shown the experimental Ej values obtained from Eq. (1) and the transitions between electron (e) levels and hard and light hole (hh and lh, respectively) levels as calculated from Eq. (2). We found only permitted transitions for n= 0,1, 2 and m=1, 2 for cylindrical wires of r0 = 9.5nm (the average radius of the wires in the sample). In the last column we match both experimental and theoretical energy values and therefore we define the type of transitions.
The Sample
The sample was grown by Molecular Beam Epitaxy on GaAs(631) substrates. In the growth process we have induced the self-assembling of Wire-like structures [3] and then we have embedded it between barriers of AlxGa1-xAs with x=0.23. The grooved interface AlxGa1-xAs/GaAs to induce the one-directional confinement of charge carriers and the formation of quantum levels in the band energy of the heterostructure.
Photoreflectance setup
PR measurements were carried out at room temperature employing a tungsten lamp and a He-Ne laser as the modulation source.
PR lineshape model
In order to obtain energy values of features observed in the PR spectrum we have employed the third-derivative functional for developed by Aspnes [1]:
∆R −7 2 iθ = Re Ce ( ω − Ei + iΓ ) R
[
]
Eq. (1)
GaAs Quantum wires
AlGaAs Second barrier
PR setup
AlGaAs First barrier GaAs(631) substrate
Experimental Energy (eV) E1 = 1.424 GaAs band gap E2 = 1.444 E3 = 1.468 E4 = 1.526
n
n=0
Calculated Energy Energy (eV) Type of Transition 1.4542 m1e-m1hh 1.47217 m1e-m2hh 1.56228 m2e-m1hh 1.58021 m2e-m2hh 1.46966 1.54659 1.5777 1.65463 1.50037 1.64014 1.53858 1.67835 1.56008 1.62438
m2e
Assigned E2 Energy
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E5
Model of Cylindrical QWR
In this model QWR geometry is cylindrical with circular cross section of radius r0 and length L. We consider a single conduction (valence) band split into a subband system due to electron confinement within the structure. The solution of the Schrodinger equation, in the envelope function approximation, leads to
E5 = 1.592 E6 = 1.669 E7 = 1.749 Al0.23Ga0.77As band gap n=1
m1e-m1lh m1e-m2lh m2e-m1lh m2e-m2lh m1e-m1hh m2e-m1hh m1e-m1lh m2e-m1lh m1e-m1hh m1e-m1lh
E3
-------
--E6 --E4
---
n=3
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Final schematic subband energy distribution in the QWR structures. Each type of transition detected from the PR spectrum is shown. and j = 1 (2) is used to denote the electron (hole). Jnj and Knj are the Bessel and modified Bessel functions, µ1 (µ2) are the effective masses in the wire (surrounding medium) and V0 represents the band offset. The states are described by the quantum numbers: nj = 0,1,…; mj = 1,2,…; and kzj. From the condition of continuity of Ψ and (1/µj)δΨ/δr at the surface, the confinement energies are determined by the secular equation:
m1e
E2
E3
E4 E5
E6
n=0 n=0 n=1 n=0 n=0 m1hh m1lh m2hh m2lh
Al0.23Ga0.77As band gap GaAs band gap
Conclusions
From (a) the PR experimental technique, (b) the model of PR lineshape of Aspnes and, (c) a theoretical model of cylindrical quantum wires, we have defined the subband energy distribution of a AlGaAs/QWR-GaAs/AlGaAs/GaAs(631) heterostructure. We have also determined the kind of each energy transition in the QWR system. We have seen that the experimental and theoretical results match is very well.
′ ′ µ 1k yj K n j (k yj ) J n j (k xj ) = µ 2 k xj J n j (k xj ) K n j (k yj )
The total energies are
2 E j (k xj , k yj , k zj ) = En j m j (k xj , k yj ) + E zj (k zj ) E zj (k zj ) = k zj 2µ
2
References
[1] F.H. Pollak, Surf. Interface Anal. 31 (2001) 938 [2] J M Bergues, J. Phys.: Condens. Matter 12 (2000) 7983-7998 [3] E. Cruz-Hernández et al. Journal of Crystal Growth 301-302 (2007) 884-888
Finally, uj is the Bloch function taken at k0 = 0, where (by assumption) the band extreme are located. For an infinitely high potential barrier the wavefunction −1 r becomes ′ Ψ j = π L r0 J n j ( xn j m j ) J n j xn j m j exp(− i (n jφ + k zj z ))uc r0
[
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Acknowledgments
This work was partially supported by CONACyT-Mexico.