Analysis of Optical Gain of Strained Wurtzite In x Ga 1-x N/GaN Quantum Well Lasers
- PDF / 432,050 Bytes
- 6 Pages / 414.72 x 648 pts Page_size
- 7 Downloads / 185 Views
F -K
-K' G
-H* H
0 0
H= -H
-H'
X
0
A
0 0 0
0 0 A
0 A 0
F -K' H'
-K G -H
0 - Jul) = -[(X+ il)/T1A 1U2) = (X- iY) T)/ V2 0 1U 3 ) = 1Z T) H Iu4) = I(X- iY),)/N/2 -H" IUs= -J(X+ iY) ,)/'/2 X U6)= 1Z 1)
where F=A,+A 2 +?'+0,
G=Al-A
2
+,+0
,
A=Vf2A 3 ,
1119 Mat. Res. Soc. Symp. Proc. Vol. 482 0 1998 Materials Research Society
(1)
D,= + D2+(s +ey) X = (h2 / 2mo)[A k,' + A2A(k2 + k 2)]+ =(h2/2m)[A3k2 + A4 (k' +k2)]+D 3e, +DA(,Ex + s), K=(h2 /2mo)As(k, +ikY)
2
+De, , H =(h 2 /2mo)A 6 (kx +iky)kz +D 6 ,÷..
(2)
In (2), A1 is the crystal-field split energy, A2 and A3 account for the spin-orbit interaction, ki is Ais the wave vector, e0. is an element of the strain tensor, e+= P, + 2i-,y - , z+=z +i are the effective-mass parameters, and Dis are the deformation potentials. For A, > A2 > 0, the three bands from top to bottom are labeled as HH, LH, and CH respectively. The strain tensor e%= = (a 0 - a)/a, cz, = -2(C13 /C33)s•, and 6xy = z= in a biaxial-strained (0001) QW contains e, = 0. a0 and a are the lattice constants of the barrier and well respectively. C13 and C33 are the well stiffness constants. The valence subbands are evaluated by diagonalizing
Z(Hi + 8iE,•E(z))*J)(z,k) = E"(k)4()(z,k)
(3)
j=l
for i = 1 to 6, where m indexes the valence subbands and Eo(z) is the periodic MQW potential. AE,:AE, = 70:30 is assumed [6]. 4,j)(z,k) is the six-dimensional envelope function. When the barrier in the MQW is wide enough, the energy dispersion in the k, direction is negligible, giving a SQW energy dispersion, E'(kx,ky). Details of the calculation is found in Ref. 8.
Parameters used are summarized in Table I. The cubic approximation [5],[7] has been used. In Fig. 1, the valence subbands of the Ino 2Gao0 SN/GaN QW are shown for (a) Lw = 25 A and (b) Lw = 50 A with (solid) and without (dashed) strain accounted. The energy dispersion is plotted against the in-plane wavevector, k, The inset shows the corresponding densities-ofstates. Inclusion of the biaxial strain does not change the subband structures significantly and the valence band-edge density-of-states remains relatively unaltered. This is because the C6 , symmetry of the WZ crystal is not reduced by the biaxial strain. The energies of [X) and 1Y)are not differentiated and the HHi and LHi subbands remain closely spaced. Only the CHi subbands are distinguished and they move towards lower electron energy with increasing biaxial compressive strain. Comparing Fig. l(a) and (b), we see that a larger quantum size effect in a narrower well does not effectively separate HHi and LHi. This is because the quantum size effect is isotropic in the in-plane directions and does not separate the energies of MA) and 1Y) constituting LHi and HHi. In Fig. 2, we plot the valence subband dispersion for two well In mole fractions (x = 0.1 and 0.2). An increase in x introduces a larger strain and a larger QW potential. Features of the density-of-states shift deeper into the valence band with an increase in x. In general, it is observed that the decrease of Lw, increa
Data Loading...
