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Chow W.W., Koch S.W. Semiconductor-Laser Fundamentals (Springer 1999)(K)(T)(253s)(600dpi)_EO_.djvu |
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Size 3.0Mb Date Jun 27, 2005 |
calculation of gain and refractive index
4.4 Formulation of Numerical Problem 119
two terms in D.33)...
We plotted the carrier distributions
fek (solid curve) and fhk (dashed curve) for a carrier density iV2d = 1.6 x
1012 cm, and a two-band system with effective masses, rae = O.O71mo and
mh = 0.167m0...
Moving to the diagonal Hartree-Fock contribution, we write C.23) in
terms of a continuous momentum distribution as
гоо />2тг 2
= - / dk" к" / Аф" (Wk>, + 1)
Jo Jo 8ir2ebhq(k, к", ф")
' dk"ti(k",k){wk»+l) , D.48)
124 4...
In the present formulation, C.17) becomes
\
,D.53)
where we use D.33) without the correlation contributions and introduce
a small phenomenological damping coefficient 7 = 10~13s to facilitate the
numerical solution...
D.75)
Jo
The fourth-order Runge-Kutta method provides a stable finite difference
method for solving differential equations such as D.71, 72)...
The calculation of Qk and Rk consumes a substantial amount of computer
time because it has to be performed for each time step...
The full calculation
predicts a blue shift in the gain peak with increasing carrier density, while
the relaxation rate results show negligible shift (see dotted lines)...
They are N2<i = 1.32 x 1012 cm and 7 = lO^s for the
012 cm and 7 = lO
Lorentzian lineshape function and N2d = 1.30 x 1012 cm and 7 = 10
sech lineshape function, respectively
142 4...
In describing the carrier probabilities nak themselves, the carrier-carrier
scattering rates 7ak{n} dominate the response on subpicosecond time scales...
There, it is fairly well-known
that the T\ given by
) ¦ <4Л04)
describes steady-state saturation correctly, although it fails to account for
the transient response in general...
Translational symmetry in the lattice dictates that the energy eigenfunctions
obey the Block theorem
where r+R describes translation by a lattice vector...
Furthermore, for bulk III-V semiconductors,
l<n|p*H|2 = \{n\py\m)\2 = \(n\pz\m)\2 , E.40)
because of the symmetry about the zone center...
We assume that the potential Vcon is sufficiently small
in comparison to Vo and varies sufficiently little within a unit cell, so that
6.1 Envelope Approximation Method 167
the new eigenstates may be approximated as the linear superposition of the
bulk-material eigenstates
\фГ) = Е\^)(Фп,к\ФГ) • F-2)
7l,fc
Here \фп,к) is the bulk eigenstate satisfying E.1) and (фп^Ф^) is the pro-
projection of the quantum-well eigenstate \ф^ ) on the bulk eigenstate \фп,к)-
In the coordinate representation,
<MO , F.3)
n,k
where we used the Bloch theorem and |n, k) is a lattice periodic eigenstate
of the bulk-material...
, we get
for the holes
HhhWxlk + RWX2k + yV^bWxlk' = exWxlk ,
, = exWX3k ,
k'
HhhWX4k + R*WX3k + YlVk°-k'WXAk' = exWX4k , F.26)
k'
where we used the states given by F.5) and i/ьь -ffih9 and R are defined in
E.77, 79, 91), respectively...
The diagonalization of F.30) then gives
the hole band structure (see, for example, the solid curves in Fig...
Hence the well material is under
strain, which is compressive (tensile) if the lattice constant of the barrier is
smaller (larger) than that of the well material...
Hole band structure for 4nm Ino.5Gao.5P/(Alo.5Gao.5)o.5lno.5P quantum-
well structure with (solid curves) and without (dashed curves) split-off states taken
into account
The effects of mixing between the j = 3/2 and j = 1/2 states are more sig-
significant in the optical dipole matrix elements...
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