Quantized states in homogenous polarized GaInN/ GaN quantum wells C. Wetzel1, S. Kamiyama1, H. Amano2, and I. Akasaki2
1 High Tech Research Center, Meijo University, 1-501 Shiogamaguchi, Tempaku-ku, Nagoya 468-8502, Japan, e-mail:
[email protected]
2 High Tech Research Center and Department of Materials Science and Technology, Meijo University, 1-501 Shiogamaguchi,
Tempaku-ku, Nagoya 468-8502, Japan
Abstract The electronic bandstructure and interband tran-
sition energies in hetero polarization Ga1,x InxN/GaN quantum wells is calculated in a single particle model. The acting electric eld in the wells and the band gap energy are directly taken from absorption-type experiments. Including strain and non-degeneracy the quantized states are predicted and respective interband transitions are compared with experiment. This provides a quantitative basis for the description of models of Stark splitting and carrier localization.
Ga1-xInxN/GaN cb
_
cb+
x = 0.12 Lw = 30 Å T = 300 K
e1
1 Introduction
Lingering puzzlement surrounds the nature of the electronic bandstructure in the GaInN/GaN active region of ecient all-color light emitting diodes and sought-for laser diodes. [1] While signi cant e ort has been centered around concepts of spatial inhomogeneities within the two-dimensional well layers our approach is based on induction from results obtained at the binary GaN barriers and thin lm ternary GaInN. We pay especial attention to the hetero polarization nature of the GaInN/GaN layer sequence manifested by the quantum con ned Stark e ect[2] and quanti ed by Franz-Keldysh oscillations.[3] In GaInN/GaN quantum wells of variable optimization levels pronounced and characteristic luminescence bands, absorption edges, and ionization thresholds have been derived in photoluminescence, photovoltage, and photore ectance experiments.[4{7]
hh1 lh1 _ vb
hh2 vb+
Fig. 1 Calculated band structure in a polarized 3 nm Ga1,x Inx N/GaN QW for electrons and the three sets of holes. The electric eld strength is assumed at the value directly derived form Franz Keldysh oscillations in absorptiontype spectroscopy of such wells. Crystal- eld split-of states are found to lie in resonance with the valence band. The charge distribution is shown superimposed on the respective energy levels.
fect and polarization induced asymmetry of the barriers. We solve the Schrodinger equation for electrons ( ), 2 Band structure model heavy holes ( ), light holes ( ) and crystal- eld ( ) Direct readings of the electric eld strength in GaInN/ GaN split-o holes within a single QW structure subjected to quantum wells from Franz-Keldysh oscillations[3] at the the measured electric eld strengths across the well. For GaN barrier band gap, the associated polarization dipoles simplicity and the sake of generality we defer consideraas well as the DOS band gap energy derived in thin lm tion of lateral uctuations in the model. material[9] are used as important input parameters for a self-consistent perturbation calculation of the quantized 3 Results states. We include the full valence band non-degeneracy Fig. 1 depicts the calculated bandstructure in a deviceand consider all excited states in a single-particle model. typical QW of well width 3 nm for = 0 12. We also Parameters of the model are based on established GaN include higher excited states of the quantized levels. The data and GaInN data where available. The band disper- wave functions are superimposed on the respective ension is assumed xed at the theory values of GaN.[10] Bi- ergy levels. All single particle levels bound to the well axial strain is considered on the basis of linear interpola- by more than 30 meV are depicted. For this small bandtion of the elastic constants of the binaries.[11] Linear in- o set at the valence band the crystal- eld split-o hole terpolations are also used for the theoretical deformation is expected to be in resonance with the valence band potentials.[12] The relative bando set c g = 0 803 forming a type-II staggered interface. is assumed at the value of rst principles theory for reDue to the presence of the piezoelectric polarization laxed material.[12] The model automatically accounts and its associated electric eld symmetry with respect for the higher orders of the quantum con ned Stark ef- to inversion of the growth direction is lifted. Respece
hh
lh
ch
x
dE =dE
:
z
:
C. Wetzel et al. Interband Transition Energy (eV)
2 Transition Energy (meV)
3.6
Ci 3.2
2.8
3.4
3.2
3.0
∂ ∂
2.8 0.00
0.05
0.10
0.15
0.20
x in Ga1-xInxN/GaN
0.05 0.10
0.15
0.20
x in Ga1-xInxN/GaN
0.05
0.10
0.15
0.20
x in Ga1-xInxN/GaN
Fig. 3 Dependence of interband transition scheme on the
relative conduction band o set as found in the literature. a) Ref. [14], b) Ref. [12] (also shown in b) and c) for reference), c) Ref. [15]. Only the higher transitions are sensitive to the di erences.
2.4 0
0.05
0.1
0.15
x in Ga1-xInxN well
0.2
Fig. 2 Interband transition scheme in dependence of the InN-fraction of the well. The piezoelectric dipole splits cb+ vb, from cb, vb, and induces low ionization thresholds. Transitions appear in close lying sets near the ionization thresholds. Only e1 hh1 and e1 lh1 are strongly bound with both states.
tive electric dipole interband transitions therefore may include any combination of even and odd states in the standard labeling. Fig. 2 depicts the full composition dependence in the range 0 0 2 together with the experimental results of extrema in photore ection and photoluminescence spectroscopy.[13] To test the sensitivity of the calculation results on the value of the band o set we compare them (Fig. 3, dashed lines) with those obtained using an earlier value derived in rst principles theory for relaxed material[14] (Fig. 3a, solid lines) and an experimental value derived from the variation of deep defect luminescence in pseudomorphic material (Fig. 3c, solid lines).[15] x
:
4 Discussion
The calculations result in a sequence of interband transition energies that re ect the large polarization dipole induced across both interfaces of the well. This is apparent by the large eld controlled splitting of , , , conduction band ( ) to valence band ( ) on the left of the well with respect to + , between conduction band on the right and valence band on the left of the polarization dipole. This splitting replicates in low energy ionization thresholds involving those band edges and either of the quantized states. The result is that the pair of 1 1 , 1 1 are the only transitions where both states are strongly bound to the well. The transition next to that is 1 2 , which lies close to the ionization threshold to the valence band. The comparison with experimental results[13] (Fig. 2) shows that only the upper range of extrema are covered in this model. 1 1 therefore has to be assigned close to a pronounced photore ection (PR) maximum that coincides with the level of stimulated emission. This nding cb
cb
vb
cb
e hh
e lh
e hh
e hh
vb
vb
is independent of the choice of published bando sets in the model. This shows that the lowest level in PR and PL requires a higher order model to explain its splitting from 1 1 which appears to replicate that of , , and + , . e hh
cb
cb
vb
vb
5 Conclusion
Our model induced from the experimental eld conditions and band gap energy therefore provides a solid basis for the quantitative description of the electronic bandstructure in GaInN/GaN QWs. This work was supported in part by the Japan Society for the Promotion of Science "Research for the Future Program in the Area of Atomic Scale Surface and Interface Dynamics" under the project of "Dynamic Process and Control of the Bu er Layer at the Interface in a Highly-Mismatched System (JSPS96P00204)" and the Ministry of Education, Science, Sports and Culture of Japan (contract numbers 11450131, 12450017 and 12875006). C. W. thanks the A. C. Bindereif foundation for a time grant.
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