Photon Green’s functions theory for Coulomb correlated quantum well lasers
A photon Green’s function theory is used to incorporate Bethe-Salpeter-like many body corrections in the computations of output spectra of semiconductor quantum well lasers. Coulomb, quantum-confinement, multiple valence band coupling and cavity resonator effects are consistently included in the the...
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| Формат: | Стаття |
| Мова: | English |
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2004
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| Назва видання: | Semiconductor Physics Quantum Electronics & Optoelectronics |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Photon Green’s functions theory for Coulomb correlated quantum well lasers / M.F. Pereira Jr. // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2004. — Т. 7, № 2. — С. 138-140. — Бібліогр.: 6 назв. — англ. |
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irk-123456789-1181582017-05-29T03:05:04Z Photon Green’s functions theory for Coulomb correlated quantum well lasers Pereira Jr., M.F. A photon Green’s function theory is used to incorporate Bethe-Salpeter-like many body corrections in the computations of output spectra of semiconductor quantum well lasers. Coulomb, quantum-confinement, multiple valence band coupling and cavity resonator effects are consistently included in the theory. Numerical results are presented for multiple quantum wells, under different design, and excitation conditions. 2004 Article Photon Green’s functions theory for Coulomb correlated quantum well lasers / M.F. Pereira Jr. // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2004. — Т. 7, № 2. — С. 138-140. — Бібліогр.: 6 назв. — англ. 1560-8034 PACS: 85.35.Ве http://dspace.nbuv.gov.ua/handle/123456789/118158 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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A photon Green’s function theory is used to incorporate Bethe-Salpeter-like many body corrections in the computations of output spectra of semiconductor quantum well lasers. Coulomb, quantum-confinement, multiple valence band coupling and cavity resonator effects are consistently included in the theory. Numerical results are presented for multiple quantum wells, under different design, and excitation conditions. |
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Article |
| author |
Pereira Jr., M.F. |
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Pereira Jr., M.F. Photon Green’s functions theory for Coulomb correlated quantum well lasers Semiconductor Physics Quantum Electronics & Optoelectronics |
| author_facet |
Pereira Jr., M.F. |
| author_sort |
Pereira Jr., M.F. |
| title |
Photon Green’s functions theory for Coulomb correlated quantum well lasers |
| title_short |
Photon Green’s functions theory for Coulomb correlated quantum well lasers |
| title_full |
Photon Green’s functions theory for Coulomb correlated quantum well lasers |
| title_fullStr |
Photon Green’s functions theory for Coulomb correlated quantum well lasers |
| title_full_unstemmed |
Photon Green’s functions theory for Coulomb correlated quantum well lasers |
| title_sort |
photon green’s functions theory for coulomb correlated quantum well lasers |
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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2004 |
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http://dspace.nbuv.gov.ua/handle/123456789/118158 |
| citation_txt |
Photon Green’s functions theory for Coulomb correlated quantum well lasers / M.F. Pereira Jr. // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2004. — Т. 7, № 2. — С. 138-140. — Бібліогр.: 6 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
| work_keys_str_mv |
AT pereirajrmf photongreensfunctionstheoryforcoulombcorrelatedquantumwelllasers |
| first_indexed |
2025-07-08T13:28:16Z |
| last_indexed |
2025-07-08T13:28:16Z |
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1837085570118975488 |
| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics. 2004. V. 7, N 2. P. 138-140.
© 2004, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine138
PACS: 85.35.Be
Photon Green�s functions theory
for Coulomb correlated quantum well lasers
M.F. Pereira Jr.*
Instituto de Fisica, Universidade Federal da Bahia, 40210-340, Salvador BA Brazil
Abstract. A photon Green�s function theory is used to incorporate Bethe-Salpeter-like many body
corrections in the computations of output spectra of semiconductor quantum well lasers.
Coulomb, quantum-confinement, multiple valence band coupling and cavity resonator ef-
fects are consistently included in the theory. Numerical results are presented for multiple
quantum wells, under different design, and excitation conditions.
Keywords: semiconductor lasers; quantum wells, many body effects, Green�s functions.
Paper received 28.01.04; accepted for publication 17.06.04.
1. Introduction
The simulation of semiconductor lasers is a fascinating
challenge to modern physics [1]. They operate in a highly
excited regime, where many-particle effects play a domi-
nant role [2].
A previous approach for quantum well lasers has suc-
cessfully described the combination of resonant cavity,
band structure and many body effects in a temperature re-
gime and for materials where a vertex-type of approximation
still successfully describes the excited media [3]. There are
however experimental conditions, especially at low tempera-
tures where higher order Coulomb corrections must be dealt
with [4,5].
In this paper, we present solutions of a Bethe-Salpeter
type of equation for quantum wells within a photon Green�s
functions approach capable of handling the resonant cav-
ity microscopically, as well as quantum confinement and
band structure effects. The microscopic approach consist-
ently describes Pauli-blocking, the screening of the Cou-
lomb interaction, Coulomb enhancement of the polarization
function, and band gap shrinkage.
The many-body approach used here is based on
Keldysh Green�s functions for carriers (G), photons (D),
and plasmons (W) to describe the coupled light-excited semi-
conductor system [3,6]. Here, we just outline the method
with words and a few representative equations.
2. Main equations
The Keldysh Green�s function�s time evolution is descri-
bed by Dyson equations, characterized by free propaga-
tors G0
�1, D0
�1, W0
�1, and selfenergies, S, P, and p, being
the carrier selfenergy, the transverse, and the longitudinal
polarization functions, respectively.
Detailed band-structure and quantum-confinement ef-
fects, obtained by solving the Luttinger Hamiltonian, are
included in the free-carrier propagator G0
�1, and serve as
the input for the solution of our many-body problem. Each
of the selfenergies changes the bare into dressed propaga-
tors in a specific way.
The transverse polarization, P describes the optical
response of the system, and can be written as a sum of an
RPA term and a Coulomb-correlation contribution, ex-
pressed by the solutions of the Bethe-Salpeter Eq. (4),
)2,2,4,3()1,4()3,1()1,2()2,1()2,2,1,1( PGGGGP += . (1)
In this paper, we consider Coulomb correlations beyond
RPA in the spectral density of photons but restrict the car-
rier selfenergies to RPA, i.e., )2,1()2,1()2,1( WG=Σ .
In order to describe light emission, we refer to the quan-
tum mechanical Poynting vector,
2121
)12()1()2()2()1(
8
)1( ==
≈×−×= DEBBE
c
S
rrrrr
π
, (2)
M.F. Pereira Jr.: Photon Green�s functions theory for Coulomb correlated quantum well lasers
139SQO, 7(2), 2004
where D is the photon Green�s function introduced above.
For linearly polarized light, the expression reduces to
∫
∞
=
0
1 ,)()( ωω dIRSx
r
(3)
),()(tan
4
)( 2
4
32
ωωθ
π
ωω LLFiP
c
I <= h
(4)
where è is a small aperture angle characterizing paraxial
propagation, and the mode structure is characterized by
the cavity function,
)(
)(1
1
)(
1
1
1
)(
)(
2222
2
ω
ωω
ω
ωω
L
Ln
c
L Je
nnr
F
−
+−
= , (5)
with the slowly varying form factor and complex propaga-
tion wavenumber given respectively by
,
)sin(
)(
)sinh(
)1()(
1
1*
2
22
Lq
Lq
rr
Lq
Lq
rJL +++=ω (6)
and
2
22
2
c
n
q
ω= , )(
)(2 2
2
2 ω
εω
rP
c
n ℑ
∞
−≈ , with )(1 ωn
obtained by a Kramers-Kronig transformation.
The polarization function satisfies the Kubo-Martin-
Schwinger (KMS) sum rule [2,4].
So, once )(ωrPℑ is computed, the carrier recombi-
nation spectra is immediately obtained, (µ is the total,
electron + hole, chemical potential),
))(exp(1
)(2
)(
µωβ
ωω
−−
ℑ−=<
h
rPi
P . (7)
3. Numerical results and discussion
Figure 1 depicts the TE absorption/gain of 3 nm ZnCdSe/
ZnSSe quantum well at 50K. From top to bottom, the carrier
densities are N = 0, 0.555, 1.11, 2.22 ⋅ 1018 carriers/cm3. Fi-
gure 2, is similar to Fig. 1, but for 15 nm ZnSe / (Zn,Mg)
(S,Se) / (Zn,Mg)(S,Se) quantum well at 200K. From top to
bottom, the carrier densities are N = 0, 0.01, 0.05, 0.1, 0.5, 1.0,
1.5, 2.0, 2.5, 3.0, 4.0, 5.0, and 6.0 ⋅ 1018 carriers/cm3.
Figure 3 shows an application for the TE emission spec-
tra of the quantum well of Figure 2, with 21 nm barriers and
total resonator lengths from left to right, L = 60, 80, 90,
123 µm. The carrier density is fixed at N = 3⋅1018 carriers/cm3.
The spectra are normalized to 1, but as the resonator
length used in the computation increases, the gain equal to
loss condition is reached and the output increases and
evolves from multi-mode to essentially mono-mode.
The idea is to illustrate the change from multi-to
single-mode operation, which in practice would happen
for a fixed resonator length in a given structure as the
density of carriers in the active region increases, and
the gain equal loss condition is approached. The scal-
ing ratios with respect to the output at L = 123 µm are
Imax,L = 123/Imax,L = 60 = 534, Imax,L = 123/Imax,L = 80 = 231,
Imax,L = 123/Imax,L = 90 = 95.
Fig. 1. TE absorption/gain of a 3 nm ZnCdSe/ZnSSe quantum well
at 50K. From top to bottom, the carrier densities are N = 0, 0.555,
1.11, 2.22⋅1018 carriers/cm3.
Fig. 2. TE absorption/gain of 15 nm ZnSe / (Zn,Mg) (S,Se) /
/ (Zn,Mg)(S,Se) quantum well at 200 K. From top to bottom, the
carrier densities are N = 0, 0.01, 0.05, 0.1, 0.5, 1.0, 1.5, 2.0, 2.5,
3.0, 4.0, 5.0, and 6.0⋅1018 carriers/cm3.
0
0.2
0.4
2.8 2.84 2.922.88
E n erg y, eV
A
bs
or
pt
io
n,
m
µ
–1
0
1
2
2.55 2.65 2.75
E n erg y, eV
A
bs
or
pt
io
n,
m
µ
–1
140
SQO, 7(2), 2004
M.F. Pereira Jr.: Photon Green�s functions theory for Coulomb correlated quantum well lasers
strong Coulomb correlations, which cannot be described
by vertex corrections only.
In both cases, T-matrix correlations beyond RPA are
included in the spectral density of photons, but the car-
rier self-energies are restricted to RPA, with the same
level of approximations discussed in Ref. [4].
The Coulomb correlations are so strong that optical
gain co-exists with features, which are well described by
a strongly interacting electron-hole plasma. This interpre-
tation is consistent with recent experimental results [5].
In summary, the approach presented here allows for
the description of optical properties of semiconductor la-
sers using a Green�s functions technique. It has the ad-
vantage of describing cavity and many body effects in a
consistent way.
Although all the results presented are for steady state,
the general theory outlined is valid for general
nonequilibrium conditions. The approach has the fur-
ther advantage of allowing the inclusion of high order
correlation effects consistently. Laser output spectra are
presented for different cavity, and excitation conditions
in the presence of strong Coulomb correlations.
Although low-dimensional II�VI semiconductors have
been used, the method is useful in the description of other
dielectric materials, relevant for basic and applied physics
alike.
Acknowledgments
Research supported by Conselho Nacional de Pesquisas,
CNPq of Brazil.
References
1. P. Zory, Quantum Well Lasers, Academic Press, San Diego, 1993.
2. R. Zimmermann, Many-Particle Theory of Highly Excited Semi-
conductors, �Teubner Texte zur Physik�, Leipzig (1987).
3. M.F. Pereira Jr., and K. Henneberger // Phys. Rev. B, 53, p. 16485
(1996).
4. M.F. Pereira Jr., and K. Henneberger // Phys. Rev. B, 58, p. 2064
(1998).
5. P. Michler, M.Vehse, J. Gutowski, M. Behringer, D. Hommel,
M.F. Pereira Jr., and K. Henneberger // Phys. Rev. B, 58, p. 2055
(1998).
6. L.V. Keldysh // Zh. Eksp. Teor. Fiz., 20(4) (1965).
Furthermore, in order to obtain a clearer plot, the
curves are displaced in the units of the plot, from left to
right, respectively by (0,0), (0.1,0.10), (0.2,0.2),
(0.3,0.3). In this case, the corresponding version of the
vertex corrections of Ref. [3] cannot be used, since as
depicted in Figs 1 and 2, the pronounced exciton-like
bumps in the absorption spectra are a clear evidence of
Fig. 3. TE emission spectra of the quantum well of Figure 2, with
21 nm barriers and total resonator lengths from left to right, L = 60,
80, 90, 123 µm. The carrier density is fixed at N = 3⋅1018 carriers/cm3.
The scaling ratios with respect to the output at L = 123 ìm are
Imax,L = 123/Imax,L = 60 = 534, Imax,L = 123/Imax,L = 80 = 231, Imax,L = 123/
/Imax,L = 90 = 95. The curves are displaced in the units of the plot,
from left to right, respectively by (0,0), (0.1,0.10), (0.2,0.2),
(0.3,0.3).
0.4
0.8
1.2
2.8 2.9 3.13
E n erg y, eV
O
ut
pu
t i
nt
en
si
ty
, a
rb
. u
ni
ts
–1
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