Optical Gain Spectra in InGaN/GaN Quantum Wells with the Compositional Fluctuations
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Formalism Let us describe the localized states caused by the compositional fluctuations. We assume that the potential fluctuations in the well are written by the quantum-disk like potentials and that they are randomly distributed without any correlation among them, as shown in Fig. 1. The Hamiltonians Hi for the electronic states for the electron (i=e), the heavy hole (i=hh) and the light hole (i=lh) is given by h2 ∂ 2 Hi = h i (z ) − + U (r, z), 2mi ∂r 2 where h i (z ) = −
h2 ∂ 2 i + VQW (z),(i = e , h h , l h ) , 2 2mi ∂z
0 if | z |> 2L U(r, z) = v(r − r ,R ,u ) otherwise ∑ j j j j v(r - rj, Rj, uj) are the potentials by the compositional fluctuations at rj in the two dimensional system, where Rj and uj are the radius and the potential depth of the quantum-disk, respectively, as shown in Fig.2. Then, the eingenfunctions of the localized state can be approximated as Ψij, n, m (r, z) = φ ji ,n (r)ζ mi (z ), where h2 ∂ 2 (− + v(r − rj ,R j ,u j ))φ ji ,n (r) = E ji ,n φ ji ,n (r), 2 2mi ∂r i h i (z )ζ mi (z ) = E z,m ζ mi (z).
φij,n(r) and ζim(z) are the eigenfunctions in the lateral directions and z direction, respectively. Using the above eingenfunctions, the optical gain spectrum is given by 2
ζ ne ζ mh Γs 1 e h g(ω ) = ∑(1− f (En, k ) − f (Em ,k )) π (ω − En,e k − Emh ,k )2 + Γs2 i, n, m,k 2
+
∑ (1−
i,n ' ,m' ,n,m ,k
f (E ) − f (E e j,n'
h j, m'
2
e h e h 1 ζ n ζ m φn' φ m' Γl )) e π (ω − E j,n' − E hj, m' )2 + Γl2
where Γs and Γl are the energy widths of the broaden spectra for the continuum state transition and the localized state one, respectively. Γs is mainly caused by the carriercarrier scattering and Γl is caused by the inhomogeneities of Rj and uj of the potential.
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Results and Discussions Figure 3 (a) and (b) show the potential profiles and the bound state energies by the compositional fluctuations. The energy depths are 80 meV for the electron and 50 meV for the holes. The radii of the disks are assumed to be 20_ and 50_, respectively. The effective masses are 0.2 m0 for electron, 1.1m0 for heavy hole and 0.17 m0 for light hole, respectively. The quantum well length is 40_. In the case of R=20_, the number of the bound states are a few and it is almost like quantum-dots. However, the bound states become condensed, especially for the holes when the radius is more than R=50_. Then, these states become like band tailing by introducing the homogeneous or inhomogeneous broadening. Figure 4 (a) and (b) show the carrier density dependence of the optical gain spectra of quantum wells with the random potentials in the ca
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