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IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 41, NO. 1, JANUARY 2005
The Impact of Energy Band Diagram and
Inhomogeneous Broadening on the Optical
Differential Gain in Nanostructure Lasers
Hanan Dery and Gadi Eisenstein, Fellow, IEEE
Abstract—We present a general theoretical model for the optical differential gain in semiconductor lasers. The model describes
self assembly quantum dots (QDs), self assembly quantum wires
(QWRs) and single quantum-well lasers. We have introduced the
inhomogeneous broadening due to size fluctuations in the assembly
cases. At each dimensionality, we have considered the carrier populations in the excited states and in the reservoirs, where conduction and valence bands are treated separately. We show that for
room temperature operation the differential gain reduction due to
increased size inhomogeneity is more pronounced in QDs than in
QWRs. We show this reduction to be smaller than the one-order
reduction attributed to state filling in conventional dot and wire assemblies operating at room temperature. The integration prefactor
coefficient of the differential gain in zero-dimensional cases exceed
one- and two-dimensional coefficients only for low temperatures
where the homogenous broadening is considerably smaller than
the thermal energy. The differential gain of QDs, QWRs, and compressively strained single quantum-well lasers operating at room
temperature and close to equilibrium is nearly the same.
Index Terms—Modulation, nonhomogeneous media, quantum
dots (QDs), quantum wires (QWRs), semiconductor lasers.
I. INTRODUCTION
T
HE differential gain of a semiconductor laser relates the
gain to the carrier density by the simple phenomenolog[1] where
is the transparency
ical equation
carrier density. In lasers based on simple bulk gain media, fast
is transintraband transitions ensure that any perturbation
is used
lated to a gain change at the laser transition energy.
then to describe the rate of change in gain at the lasing wavelength to changes in the carrier density occupying states at the
. Naturally,
plays a
corresponding energy,
key role in the formulation of diode laser modulation response
.
where the resonance frequency is known [1] to vary as
The situation is more complicated in advanced lasers based
on nanostructures, such as multiple or single quantum wells
(SQWs) [2], [3], self-assembled quantum dots (SAQDs) [4], [5]
Manuscript received March 25, 2004; revised September 1, 2004. This work
was supported in part by the BigBand project of the EC and by a grant from the
Ministry of Science, Israel. The work of H. Dery was supported in part by the
Vatat graduate student fellowships.
H. Dery was with the Department of Electrical Engineering, Technion-Israel
Institute of Technology, Haifa 32000, Israel. He is now with the Physics Department, University of California at San Diego, La Jolla, CA 92093 USA.
G. Eisenstein is with the Department of Electrical Engineering, TechnionIsrael Institute of Technology, Haifa 32000, Israel (e-mail: [email protected].
ac.il).
Digital Object Identifier 10.1109/JQE.2004.837953
or quantum dashes [6]–[8]. The latter have been shown to behave as a self-assembly quantum wires (SAQWRs) [9]. All such
nanostructure gain media comprise a reservoir through which
carriers are fed to the gain region via complex carrier relaxation
mechanisms [10]. There is a finite steady-state carrier population of reservoir states so that induced carrier perturbations are
distributed over a range of energies and contribute only partially
to the gain at the lasing energy. This amounts to an effective reduction in the differential gain.
In lasers based on SAQD and SAQWR, there are three additional problems. The energy separation between reservoir and
lasing state is often large complicating the coupling process between the two regions. Moreover, the spectra of such lasers is inhomogeneously broadened [11]–[16] and these structures may
also have a measurable carrier population of excited states. Any
carrier density perturbation is fed now to several states other
than the lasing state (which is most often the ground state) resulting effectively in a reduced differential gain.
Documented experimental results suggest that SAQD exhibit
rather moderate modulation bandwidths [17]–[22]. Examination of the details of these papers highlights a very large range
cm in [19] to
of extracted differential gain values, from 10
2 10
cm in [21]. These deviations originate from different
evaluation techniques as well as from very large discrepancies
in the total confinement factor value (heterostructure confinement factor times the dots filling factors), from 1.2 10 in [23]
to 3 10 in [24] per dot layer.
Standing out among reported experimental results is the work
of Bhattacharya et. al. [25] who used a tunnelling injection
scheme to demonstrate a bandwidth of 22 GHz. The principle
of tunnelling injection was previously used [26] to widen the
bandwidth of QW lasers. Similarly, n-type -doped regions
enhanced the bandwidth of QW lasers [27]. A recent paper
[28] employing a thin InGaP matrix buffer beneath the QDs
also achieved a respectable bandwidth of 12 GHz. All those
special laser structures circumvent limitation imposed by the
differential gain and the nonlinear gain coefficient.
In this paper, we present a general theoretical model for the
differential gain in lasers based on nanostructure gain media.
The formalism has a common framework for the three nanostructures we consider. The SAQD, SAQWR, and SQW gain
media differ only in their respective density of states (DOS).
In the two-dimensional (2-D) case, we derive the well known
analytical expression where we show results for a compressively strained separate confinement QW heterostructure. A
model was used to calculate the energy placement of the excited
0018-9197/$20.00 © 2005 IEEE
DERY AND EISENSTEIN: THE IMPACT OF ENERGY BAND DIAGRAM AND INHOMOGENEOUS BROADENING
subbands and the reservoirs. The expression we derive for the
zero-dimensional (0-D) case is the same as the one derived previously by Shchekin and Deppe [29] except that the confinement
factor is formulated differently here so as to enable a clear comparison with the other dimensionalities. Finally, we present here
the first calculation for the differential gain of a SAQWR structure. The calculation employs an analytical approximation for
the Fermi energy of an ideal one-dimensional (1-D) Fermi gas
under the condition of carrier densities which are relevant for
lasing conditions.
We point out three factors which affect the differential gain,
all of which are manifestations of the different DOS functional
forms. First, the state filling factor resulting from carrier energy
distribution. We address issues related to the reservoir and to excited states. We use realistic energetic structures for the different
geometries and highlight the reasons for hindered differential
gain in SAQDs. Second, we introduce the inhomogeneously
broadened nature of the gain in SAQD and SAQW structures.
Finally, we address the effects of the homogeneous broadening,
the temperature and the optical dipole matrix element.
We consider operation at room temperature and use
Fermi–Dirac statistics which implies equilibrium conditions. The equilibrium condition is unquestionably valid below
and near threshold. Far above threshold, where the largest
modulation bandwidths are achieved, there may be a measurable deviation from equilibrium due to intraband saturation
processes. The differential gain reduces however in the high
power regime. Deviations from equilibrium in SAQDs result in
multiline lasing [11]–[16] from different dot size populations
or by simultaneous lasing from the ground and excited states.
Recalling the physical interpretation of the optical differential
perturbation in the reservoir is spread now over
gain, the
different spectral regions which are separated by more than the
homogeneous line width hence reducing the differential gain.
The differential gain near threshold is, therefore, an upper limit.
II. THEORETICAL MODEL
The linear optical gain is derived from the linear susceptibility
using the density matrix formalism for a two-level system as in
[30]
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on the scale of the Lorentzian shaped homogeneously broadened gain function. The summation in (1) is replaced by a conbecomes
tinuous integration and
dot
wire
well
(2)
(3)
where
denotes the well width and
is the th wire crossand
are the th element energy gap and
section area.
the reduced mass. We have also considered the homogeneous
broadening to be independent of the element index (this assumption is not essential and is used for simplicity).
We define the differential gain as
(4)
and denote,
where is an index counting confined states,
respectively, the th element particle number of subband in the
band and the th element volume.
and
describe the
corresponding values in the reservoir.
In the following we consider SAQD, SAQWR, and SQW,
all operating under equilibrium conditions. The equilibrium assumption is valid below and slightly above threshold where carrier escape [3], [10], [32], [33] is fast compared with the spontaneous and stimulated emission rates. These conditions are easily
obtainable in SAQD and SAQWR operating near room temperature [34].
We use the chain rule to rewrite (4)
(5)
(1)
are the velocity of light, the refractive index, and
where
the angular photon frequency , respectively. is the quantum
number enumerating states (including spin) where for one and
two dimensional structures it corresponds to the wave vector. is
an index counting the gain element number (well, wire or dot),
being the th element volume,
the optical dipole
with
the inverse dephasing time.
and
matrix element and
are, respectively, electron and hole distribution functions.
We aim to formulate a general expression for the ground-state
which can serve to analyze all nanostructures. We
gain
assume parabolic bands and a wavevector independent matrix
. Doubly degenerate spin is inelement [31]
cluded and the distribution functions are assumed to vary slow
relates to the chemical potential and will be described
where
is the ground-state carrier density in the th element.
later.
For the simple case of infinite confinement and parabolic dispersion in the free dimensions, the state energies are proportional
, and the distributo the inverse value of the mass
tion function is
(6)
where
is a reference energy
which, for the SQW, denotes the gap energy and, hence, the
is calculated from the edge energy of
chemical potential
denotes the
each band. For the lower dimensional structures,
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IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 41, NO. 1, JANUARY 2005
average ground-state transition energy of the various elements.
Using (6) and (2), the first term of (5) becomes
(7)
being the coefficient of the phase filling factor
according to (2) where denotes the corresponding
dimensionality.
The second term in (5) involves the ground-state carrier density which is given by
with
A. Self Assembly Quantum Dots
The reservoir in the SAQD structure is assumed to be a two
dimensional wetting layer characterized by a single level which
above the th dot ground state of
is energetically placed
. We consider two disband
tributions for each band with one having an intifisimal increase
at the energy of the chemical potential. A simple algebric manipulation leads to
dot
wire
(8)
(12)
well.
The solution is analytical for the 0- and 2-D cases.
The quantum-wire case is more complicated. We solve it by
modifying the Joyce and Dixon [35] or Aguilera-Navaro et al.
[36] procedures. For symmetrical bands, one can make use of
the similarity between the wire energy integral and the bulk density integral and use the well known expression [36]. However,
in cases where the masses and total densities within each wire
are not equal (only the total assembly density is), the entire analysis must be repeated. We define
where
is the width of the reservoir well,
are, respectively, the occupation probabilities of the th state and edge
denotes the degeneracy
energy of the band reservoir.
ratio between the th state and the ground state. Considering
the SAQD to have a Gaussian-shaped inhomogeneously broadened function and inserting the dot terms into the differential
gain equation leads to
(9)
Using series inversion techniques [37] leads to
(10)
where the logarithmic term describes the classical result and where
. Retention of terms through
yields a good
approximation with an accuracy of better than one percent up
. The standard Padé approximation technique for the
to
Fermi–Dirac integral function yields poor results for the low
order polynomial ratios.
The second term of (5) can now be written as
dot
wire
well.
(11)
The third part of (5) which is referred to as the state filling reduction term is calculated separately for each dimension.
(13)
The square brackets term in (13) describes the state filling reduction. It is obvious that the differential gain degrades when
the ground-state population increases and for small energetic
spacing between the various transition. This is simply explained
by state filling and off-resonance state filling (
as
).
For an ideal assembly with identical dots, the Gaussian function is replaced by a delta function yielding an analytical expression. Similarly, for a narrow homogeneous broadening
the Lorentzian function reduces to a delta function.
Fig. 1 shows the state filling reduction of the conduction
term at room temperatures. The solid lines represent the ground
state and two excited states with equal spacing, which describe
the case of a parabolic potential. This choice agrees very well
with many published photo and electro luminescence measurements of GaAs–InAs QDs grown by molecular beam epitaxy
[24], [38]–[42] where the growth conditions determine the
wavelength and the state spacing. The separation between the
DERY AND EISENSTEIN: THE IMPACT OF ENERGY BAND DIAGRAM AND INHOMOGENEOUS BROADENING
Fig. 1. Differential gain reduction in SAQD due to state filling versus the
electronic ground-state occupation probability. Each of the solid lines denote
a different energetic spacing between the states characterized in units of room
temperature thermal energy. These are considering two excited states and a
wetting layer. The dashed line denotes the case of a single excited state and
a wetting layer (see text).
wetting layer and the second excited state was chosen to be the
same which fits reasonably well to luminescence measurements
in cases where the wetting layer emission is clearly detected.
A 6-Å wetting layer and excited-state degeneracy ratios of
were chosen according to the parabolic
potential approximation where the uniaxial nature is considered due to the known low height to diameter aspect ratio of
conventional QDs. The electron effective mass in the reservoir
and the dot volume is
cm
is
(which describes a pyramidal dot with 100-Å base length and
60-Å height). Fig. 1 shows five such energy separations. For
twice the room temperature thermal energy ( 50 meV) and
ground-state occupation of 0.85, the differential gain drops by
one order of magnitude.
In InGaAs–InAs QDs, whose confinement is shallower than
in GaAs–InAs QDs, photoluminescence experiments [33],
[43]–[46] reveal emission from a single excited state above
which the wetting layer emits. The energy separation between
the ground state and wetting layer transitions varies in those
structures in the range of 110–170 meV. Here, we describe the
state filling reduction by approximating the two conduction
band separations to be 50 meV each. This case, presented
by the dashed line in Fig. 1, is nearly identical to the case of
two excited states and a wetting layer with twice the room
temperature energy spacing.
The valance band state spectrum in SAQDs grown by the
Stranski–Krastanow (SK) technique, is very dense (with separations of the order of 10 meV). The conduction band states,
on the other hand, are separated by more than 50 meV. This
was demonstrated theoretically for pyramidal dots using an
empirical pseudopotential analysis [47] and by an eight-band
model [48]. The same conclusions are drawn from simple
parabolic confinement potential models that fit photoluminescence experiments [38], [49] and which qualitatively reflect
29
the shallow cross sectional shape of the QDs. The various
calculations reveal the dense valence band spectrum with state
separations of approximately 10 meV. Considering the first five
hole states along with their degeneracies as given by parabolic
potential for shallow dots, the state filling reduction for holes is
approximately 0.09, 0.06, 0.04 for ground-state populations of
0.25, 0.5, and 0.75, respectively. Considering the degeneracies
of the hole states to be s-, p-, d-like, etc. yields basically the
same results and trends. The degeneracies of the holes is most
likely more complex due to strong interband mixing. However,
the large number of discrete states keeps the state filling reduction in the valence band considerably lower than one.
Large differential gain in room temperature conventional
SAQD lasers may be predicted based on the conduction band
filling reduction whose peak values may exceed considerably
the valence band peaks as seen in the large spacing cases of
Fig. 1. However, the increase in differential gain is hindered
by neutrality conditions which are only satisfied for high
ground-state occupation of electrons. The electron-state filling
so that in order to
reduction term tends to zero as
maximize the electron-state filling term, the laser should operate close to transparency. This can be achieved by increasing
the waveguide confinement factor and/or the QD coverage.
Alternatively, reduced inversion levels may be achieved by
long cavities but this increases the photon lifetime which is
detrimental for fast modulation. Furthermore, high-energy
spacings in the conduction band slow the carrier capture [10],
[50]–[53] also affecting the modulation capabilities.
Equation (13) can be modified to partially incorporate excitonic populations in QDs at elevated temperatures. Considering
the binding energy of a pyramidal InAs QDs to be of the order
of 20 meV [54], room temperature exciton population becomes
non negligible. Dikshit and Pikal [55] have found the exciton
population to be one fifth of the total population at room temperature by modifying the Saha equation following [56]. For the
exciton population, we replace the free carrier phase filling term
. Considering the
with the corresponding excitonic term
higher energetic spacing in the conduction band, the exciton distribution can be approximated by the electrons distribution and
hence increasing the state filling reduction term by a factor of
two relative to the free carrier case. Two additional modifications are needed. One is factorization of (13) by the fraction of
the exciton population within the total population. The second
modification is due to the increased dipole matrix element [57],
[58] caused by the strong confinement in the excitonic regime.
Introducing p-doping modulation in the barriers was also suggested as a means to overcome the imbalance between the valence and conduction bands. Vastly improved static characteristics such as threshold current and characteristic temperature
were indeed demonstrated by Shchekin and Deppe [59]. The
barriers in these structures are partially ionized and hence the
electrons are rejected by Coulomb forces and tend to localize in
the QDs. This rejection reduces thermal escape from the QDs
and diminishes electron diffusion in the barriers leading to less
trapping in nonradiative centers.
The predicted enhancement of dynamical properties [60] has,
on the other hand, not been proven experimentally. A recent report [61] of such a p-doped laser shows but a mild enhancement
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IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 41, NO. 1, JANUARY 2005
compared with regular structures. We postulate that the discrepancy is related to spatial effects which are governed by the neutrality conditions. Coulumb forces between the ionized acceptors and the holes modify the energy band structure and keep
the acceptors only partially ionized. QDs are, therefore, populated by a small number of holes. In order to properly model
this effect, the Poisson equation has to be solved self consistently with (at least) a simplified 1-D Schrödinger equation. The
overall system neutrality can be derived then from the carrier
probability of the convergent wave
densities by using the
function. Such a calculation was reported for elevated injection
levels in intrinsic multiple QW structures [3] and doped single
QW structures [27]. Finally, even in extreme cases where imposing dozens of holes within each QD is a constraint, the differential gain of SAQDs does not exceeds that of tunneling barrier
QW structures as can be viewed from the resonance peak positions of the modulation response ([60] compared with [62]).
B. SAQWR
In the case of SAQWR structures, we consider once more a
2–D reservoir. Repeating the analysis with the corresponding
adjustments to the QWR configuration results in
(14)
where
(15)
Gathering all of the wire terms leads to the SAQWR differential gain
(16)
where
and
are taken from (3), (9), and (15).
We have simulated the state filling reduction, described by the
square bracket term in (16), by following the energy diagram of
a self assembled InP–InAs quantum dash structure as calculated
by Miska et al. [8]. These calculations fit well to photoluminescence experiments and present a rich and dense spectrum in both
bands with a similar energy spacing. The similarity stems from
the flat cross section of quantum dashes having an aspect ratio
Fig. 2. Differential gain reduction in SAQWR due to state filling versus the
conduction ground subband population at the average transition energy of the
various elements. The lower graph refers to InP–InAs quantum dashes. We have
considered four excited states as in [8]. The upper graph denotes a quantum dash
with a wetting layer placed 100 meV above a single subband (see text).
of about 0.1. The quantum dash resembles a highly biaxially
strained thin QW with a distinct light-hole–heavy-hole decoupling where the heavy hole and electron have almost identical
masses in the well plane. Due to the similarities of the masses
and band diagrams, the hole contribution to the state filling reduction almost equals the electron contribution, where the subband degeneracy is lifted by the highly asymmetrical geometry.
The large number of subbands has the same effect as the degeneracy in the SAQD cases. The dense conduction band spectrum
is similar in nature to that of V-groove GaAs–AlGaAs QWRs
[63], [64].
A second class of quantum dashes are the InAlGaAs–InAs
[6], [7] where the quaternary material is lattice matched to InP.
In this category, the dashes have considerably lower barriers
( 1.06 eV compared with 1.35 eV in the InP dashes, where the
wire luminescence in both is centered around 0.8 eV). Since no
rich luminescence spectra was observed [9], [65], we assume
a single subband with a wetting layer for the conduction band.
No sound analysis of the energy diagram in these structures was
carried out. We, therefore, do not refer to the holes contribution
but rather postulate that their contribution is limited compared
to the InP quantum dash case. This is supported by some indirect (calculated) evidence for additional hole subbands [9].
As no clear wetting layer emission was detected [9] for a wide
range of driving currents, we assume that the energy spacing
between the ground state and the reservoir is large. In addition,
considering the approximately 60/40 band offset for the conduction–valence band in this material compositions [66], and the
fact that the total ground state to barrier offset is 250 meV, we
use in our calculation a 100-meV spacing between the conduction band ground state and the wetting layer for this wire class.
The state filling reduction of the conduction band for both
quantum dash families is presented in Fig. 2. The abscissa denotes the conduction ground subband population at the average
ground-state transition energy of the various elements. These
DERY AND EISENSTEIN: THE IMPACT OF ENERGY BAND DIAGRAM AND INHOMOGENEOUS BROADENING
31
are also the energies for which the differential gain reduction
is calculated. The lower graph denotes the InP–InAs structure
where we have used the parameters calculated in [8]. The elec,
tron and reduced masses in the wire are
respectively, and the excited states are 15, 30, 55, and 80 meV
above the ground state. Considering more states has but a
minor effect on the results and due to the strong confinement,
the reservoir contribution to the reduction is also negligible.
The upper graph denotes the InAlGaAs–InAs structure where
we have used a 6-Å wetting layer with an electron mass of
, the wire reduced and electron masses are
and
, respectively. For this wire class, we have calculated
model following [67]
the masses by using an eight band
for a biaxially strained quantum-wire configuration [9].
C. SQWs
The reservoir in the SQW case is considered to have a 3-D
configuration. Repeating the analysis with some mathematical
manipulations yields the differential gain
(17)
with and denoting the distributions of the band edge and
reservoir energies (the chemical potential is measured from the
is the well width. We have assumed the
ground-state edge).
reservoir population to be low so we invoke the Maxwell–Boltzmann distribution. Using compressive strain QWs increases the
differential gain as the ratio between the reduced and heavy hole
mass increases. This enhances the weight of hole contribution
to the differential gain. We simulate a typical separate confinement heterostructure QW laser with the following parameters.
The total width of the heterostructure is 985 Å comprising a
single In Ga As 85-Å well with 100 ÅGaAs barriers on each
side. The rest of the structure comprises Al Ga As layers on
both sides. The temperature was fixed at 300 K and strain effect were included. The
diagrams and the confined wave
functions were computed following [67]. The particle masses
are
and the energies at the point
eV
eV
eV
eV.
The reservoir we consider is the GaAs separate confinement
region and the masses are
. The
filling reduction is presented in Fig. 3.
D. Inhomogeneous Broadening
The inhomogeneous broadening due to size fluctuations in
SAQDs and SAQWRs also introduces a reduction of the optical
Fig. 3. Differential gain reduction in a compressively strained AlGaAs–
GaAs–InGaAs SQW due to state filling versus the ground subband population
at the edge energy. The lower graph refers to holes and the upper to electrons.
Two hole subbands, one electron subband and bulk reservoirs are used.
gain and the differential gain. For SAQDs we make use of the
fact that the state filling reduction term, denoted by the square
bracket in (13) varies slowly within the Lorentzian-shaped homogeneous broadening function. Moreover, for the ground state
, taking the state filling term
central transition frequency
out of the integral in (13) amounts to a second-order correction
with a nearly identical reduction of the optical gain and the differential gain. The differential gain becomes
(18)
Fig. 4 shows the differential gain as a function of the inhomogeneous broadening in SAQDs. The energy spacing in the
conduction band is twice the room temperature thermal energy.
Other parameters we use are: the dipole matrix element
C M [68], [69]. The central transition energy
eV, the refractive index
and the dephasing energy
meV [70], [71]. The other parameters are the same
as in Fig. 1. Considering the state filling reduction as a constant
prefactor has a negligible effect on the results. Changing the
from
assembly full width half maximum
5 to 100 meV leads to a reduction by a factor of four in both
the gain and the differential gain. This specific reduction stems
from the fact that for a wide inhomogeneous broadening, the homogeneous width can be viewed as a delta function and hence
. This behavior dominates the
the differential gain scales as
32
Fig. 4. Differential gain versus inhomogeneous broadening in SAQDs with
conduction band energy spacing of twice the room temperature thermal energy.
Upper, middle, and lower sets denote different populations for the central
transition dots. Solid and dashed lines denote, respectively, results with the
state filling term kept inside and taken out of the integral.
right side of Fig. 4. For the opposite extreme case, where hypothetical identical dots are considered, the gain inhomogeneity
is basically a delta function and the differential gain scales as
. This behavior dominates the left-hand side of Fig. 4
and explains the fact that the two edges differ by a factor of four.
Applying other energy spacings between the states were found
to yield the same results (with the only differences being vertical
shifts of the lines) as the state reduction term is slowly varying
within the Lorentzian shape.
Fig. 5 shows the results for SAQWR based on an InP–InAs
structure whose state filling reduction is shown in the lower curve
eV and all
of Fig. 2. The central transition energy was
other parameters were unchanged. Placing the state filling term
outside the integral is somewhat less accurate in this case compared with the SAQD case. This is related to the highly nonsymmetrical integrand expression below and above the photon frequency which describes a broadened wire DOS function due to
the Lorentzian shape [9]. Changing the assembly FWHM from 5
to 100 meV reduces the gain and the differential gain by a factor
of two. Reaping the analysis as for the SAQDs reveals that the
for ideal uniform wire
differential gain scales as
assemblies (the left side of Fig. 5). For the opposite case where
the homogeneous broadening is considerably narrower than the
inhomogeneous broadening, one can not replace the functional
form which involves the expression in (16) with a delta func, the complex expression can be
tion. In this regime
replaced (in the limiting case where
) by
for most
of the integration interval where the Gaussian shape is nonnegligible. The integration interval is restricted to positive values
(as can be seen from the highly nonsymmetrical integrand).
Using these approximations (with the assumption of a slowly
varying filling term) the outcome is a numerical constant times
which describes the behavior on the right side of Fig. 5.
Therefore, the QD case reduction in both gain and differential
IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 41, NO. 1, JANUARY 2005
Fig. 5. Differential gain versus inhomogeneous broadening in InP–InAs
SAQWR. Upper, middle, and lower sets denote different populations for the
central transition wires. Solid and dashed lines denote, respectively, results
with the state filling term kept inside and taken out of the integral.
where an ideal assembly is regain is proportional to
placed by a wide assembly (with being a measure of the size
fluctuations). The wire proportionality scales as the square root
of this value. In this context SAQWRs are less sensitive for size
fluctuations. This behavior is essentially not influence by the excited state and reservoir positions as the filling term corrections
to the integral are of lower order (first– and second-order for 1and 0-D). The results for wires with different energetic diagrams
introduce minor modifications.
The physical interpretation for the above discussion is simple
since different dot sized populations do not simultaneously contribute to the optical gain due to their discrete DOS shape. On the
other hand, different sized wires share optical transitions on the
high energy side and hence the effect of inhomogeneous broadening is smaller.
Self-assembly laser structures with wide inhomogeneous
and
lines require higher threshold inversion levels
hence their overall reduction (compared with similar structures
having narrow lines) is the product of the state filling ratio and
the inhomogeneous linewidth ratio.
E. Integration Prefactor
The last difference between the three geometries relates to
the differential gain integration prefactor in (13), (16), and (17).
The SAQD, SAQWR, and SQW prefactors are, respectively,
and
. If the
proportional to
[72], the differential gain
dephasing energy varies as
, and
. Considering the
scales, respectively, as
dephasing to be 10 meV at room temperature and at elevated carriers densities, we note that the SAQD coefficient is larger than
the other coefficients, only at low temperatures where the homogeneous broadening is determined by the excitons lifetime
which yields a dephasing energy that is considerably smaller
than the thermal energy. At low temperatures, one should also
consider the random dot population effect [73] as the Fermi
model is invalid. The advantage of low temperatures may be
DERY AND EISENSTEIN: THE IMPACT OF ENERGY BAND DIAGRAM AND INHOMOGENEOUS BROADENING
somewhat diminished as the revised state filling reduction term
may not be ideal due to the random capture process as in higher
dimensional equilibrium systems where there is no reduction
.
when
Using the fact that the optical dipole matrix elements is approximately the same for all nanostructures near room temperC M [68], [69]), we find that the optimal
ature (
cm for lasing in the
achievable differential gain is 2 10
1–1.5 m regime (in the ideal case of no state filling reduction
and no size inhomogeneity). Even this idealized case yields a
differential gain value which is one order of magnitude lower
cm of Klotzkin and Bhatthan the reported value of 2 10
tacharya [21]. We believe this discrepancy to result from the
extracting technique as indicated by Riedl et al. [19]. Our argument is also supported by the narrow bandwidth and low resonance frequencies reported in the experiment.
The optimal value for the differential gain of SAQDs would
not increase if nonequilibrium distributions would have been
considered. This is based on the fact that the change in the phase
filling term in (2) with a change in the density is by definition bounded by unity (transforming the total perturbation into
a mere change of the ground-state populations).
III. CONCLUSION
This paper described a theoretical model which compares the
optical differential gain for practical 2-, 1-, and 0-D semiconductor lasers. The model assumes room temperature operation
and equilibrium conditions. We present a new expression for the
1-D case and use a unified formalism for the differential gain in
the three structures.
The differential gain has three main factors. First, the state
filling term which considers the excited states and subbands as
well as the reservoirs populations. Second, the inhomogeneous
broadening which affects 1- and 0-D assemblies. Third, the integration pre factor which includes the homogenous broadening
and the temperature. Our calculations estimate the differential
gain of the different geometries by using energy schemes which
rely on published luminescence experiments and their corresponding theoretical modeling. For the SAQDs, our calculations
yield conservative estimates for the differential gain. We postulate that exaggerated published values stem from the extracting
technique. We find that the differential gain of SAQD, SAQWR,
and compressively strained QW structures are of the same order
cm to 10
cm ). We believe that our analysis clari( 10
fies the disagreements in reported differential gains and previous
predictions.
In devices with low modal gain, where a high inversion population is imperative to reach threshold, the differential gain
can be reduced by more than one order of magnitude from its
optimal value. This effect is highly important in conventional
SAQD and SAQWR structures. In SAQDs, the imbalance between the bands energy spectra imposes neutrality constraints
which lead to strong deviations in the electron and hole ground
state populations, an effect that lowers the differential gain appreciably and also reduces the modal gain.
The inhomogeneous broadening was shown to have a larger
effect on conventional SAQDs as compared with SAQWRs.
33
The difference stems from the respective shapes of the DOS.
The overlap of high energy transitions between different populations in a SAQWR assembly reduces the impact of the size
inhomogeneity, whereas in SAQDs the atom like energy spectra
inhibit light amplification from different dot size populations.
Considering the FWHM of present assemblies to be 20 meV
8 meV) and a room-temperature homogeneous broadening
(
of 10 meV, leads to the conclusion that even if uniform dots
could be realized, the differential gain improvement would
be small. With respect to that, we emphasize that imbalanced
spectra are the critical issue for lowering the differential gain
in SAQD-based devices.
Another important parameter, the linewidth enhancement
factor ( parameter) is predicted to be small in SAQD lasers.
Indeed, small parameter values have been observed in several
experiments. The
parameter is defined as the ratio of the
refractive index derivative with respect to the carrier density to
the differential gain. The low parameter in SAQD lasers is
solely due to the small influence the carrier density has on the
refractive index in delta-like energy spectra (which in principle
can be zero). Reported parameter values are finite (although
very low) due to contributions from excited state populations.
In conventional SAQD lasers a differential gain which is larger
than that of lasers based on higher dimensional gain media is
not attainable at room temperature. The reason is the integration
pre factor which is of the same order for all three geometries.
Large differential gain values are only possible in conventional
SAQD lasers when they operate at low temperatures where
the dephasing energy is considerably smaller than the thermal
energy.
REFERENCES
[1] A. Yariv, Optical Electronics in Modern Communications. New York:
Oxford, 1997.
[2] R. Nagarajan, M. Ishikawa, T. Fukushima, R. S. Geels, and J. E. Bowers,
“High speed quantum-well lasers and carriers transport effects,” IEEE J.
Quantum Electron., vol. 28, no. 10, pp. 1990–2008, Oct. 1992.
[3] N. Tessler and G. Eistenstein, “On carrier injection and gain dynamics
in quantum well lasers,” IEEE J. Quantum Electron., vol. 29, no. 6, pp.
1586–1595, Jun. 1993.
[4] D. Bimberg, M. Grundmann, and N. N. Ledentsov, Quantum Dot Heterostructures. New York: Wiley, 1998.
[5] M. Sugawara, Self-Assembled InGaAs/GaAS Quantum Dots. New
York: Academic, 1999.
[6] R. H. Wang, A. Stintz, P. M. Varangis, T. C. Newell, H. Li, K. J. Malloy,
and L. F. Lester, “Room-temperature operation of InAs quantum-dash
lasers on InP (001),” IEEE Photon. Technol. Lett., vol. 13, no. 8, pp.
767–769, Aug. 2001.
[7] R. Schwertberger, D. Gold, J. P. Reithmaier, and A. Forchel,
“Long-wavelength InP-based quantum-dash lasers,” IEEE Photon.
Technol. Lett., vol. 14, no. 6, pp. 735–737, Jun. 2002.
[8] P. Miska, J. Even, C. Platz, B. Salem, T. Benyattou, C. Bru-Chevalier, G.
Guillot, B. Bermond, Kh. Moumanis, F. H. Julien, O. Marty, C. Monat,
and M. Gendry, “Experimental and theoretical investigation of carrier
confinement in InAs quantum dashes grown on InP(001),” J. Appl. Phys.,
vol. 95, pp. 1074–1080, 2004.
[9] H. Dery, E. Benisty, A. Epstein, R. Alizon, V. Mikhelashvili, G. Eisenstein, R. Schwertberger, D. Golg, J. P. Reithmaier, and A. Forchel,
“On the nature of quantum dash structures,” J. Appl. Phys., vol. 95, pp.
6103–6111, 2004.
[10] H. Dery and G. Eisenstein, “Self consistent rate equations of self assembly quantum wire lasers,” IEEE J. Quantum Electron., no. 10, pp.
1398–1409, Oct. 2004.
[11] L. V. Asryan and R. A. Suris, “Inhomogeneous line broadening and the
threshold current density of a semiconductor quantum dot laser,” Semicond. Sci. Technol., vol. 11, pp. 554–567, 1996.
34
[12] Y. Qiu, D. Uhl, R. Chacon, and R. Q. Yang, “Lasing characteristics
of InAs lasers on (001) InP substrate,” Appl. Phys. Lett., vol. 83, pp.
1704–1706, 2003.
[13] L. Harris, D. J. Mowbray, M. S. Skolnick, M. Hopkinson, and G. Hill,
“Emission spectra and mode structure of InAs/GaAs self-organized
quantum dot lasers,” Appl. Phys. Lett., vol. 73, pp. 969–971, 1998.
[14] M. Sugawara, K. Mukai, and Y. Nakata, “Light emission spectra of
columnar-shaped self-assembled InGaAs/GaAs quantum-dot lasers: Effect of homogeneous broadening of the optical gain on lasing characteristics,” Appl. Phys. Lett., vol. 74, pp. 1561–1563, 1999.
[15] H. Saito, K. Nishi, and S. Sugou, “Groung-state lasing at room temperature in long-wavelength InAs quantum-dot lasers on InP(311) substrates,” Appl. Phys. Lett., vol. 78, pp. 267–269, 2001.
[16] A. V. Platonov, C. Lingk, J. Feldmann, M. Arzberger, G. Böhm, M.
C. Amann, and G. Abstreiter, “Ultrafast switch-off of an electrically
pumped quantum-dot laser,” Appl. Phys. Lett., vol. 81, pp. 1177–1179,
2002.
[17] N. Hatori, M. Sugawara, K. Mukai, Y. Nakata, and H. Ishikawa,
“Room-temperature gain and differential gain characteristics of self-assembled InGaAs/GaAs quantum dots for 1.1–1.3 m semiconductor
lasers,” Appl. Phys. Lett., vol. 77, pp. 773–775, 2000.
[18] R. Krebs, F. Klopf, S. Rennon, J. P. Reithmaier, and A. Forchel, “High
frequency characteristics of InAs/GaInAs quantum dot distributed feedback lasers emitting at 1.3 m,” Electron. Lett., vol. 37, pp. 1223–1225,
2001.
[19] T. Riedl, A. Hangleiter, J. Porsche, and F. Scholz, “Small-signal modulation response of InP/GaInP quantum-dot lasers,” Appl. Phys. Lett., vol.
80, pp. 4015–4017, 2002.
[20] H. Saito, K. Nishi, A. Kamei, and S. Sugou, “Low chirp observed in
directly modulated quantum dot lasers,” IEEE Photon. Technol. Lett.,
vol. 12, no. 10, pp. 1298–1300, Oct. 2000.
[21] D. Klotzkin and P. Bhattacharya, “Temperature dependence of dynamic
and DC characteristics of quantum-well and quantum-dot lasers,” J.
Lightwave Technol. Lett., vol. 17, pp. 1634–1642, Sep. 1999.
, “Quantum capture times ar room temperature in high-speed
[22]
In Ga As-GaAs self-organized quantum-dot lasers,” IEEE Photon.
Technol. Lett., vol. 9, no. 10, pp. 1301–1303, May 1997.
[23] N. Kirstaedter, O. G. Schmidt, N. N. Ledentsov, D. Bimberg, V. M.
Ustinov, A. Yu. Egorov, A. E. Zhukov, M. V. Maximov, P. S. Kop’ev,
and Zh. I. Alferov, “Gain and differential gain of single layer InAs/GaAs
quantum dot injection lasers,” Appl. Phys. Lett., vol. 69, pp. 1226–1228,
1996.
[24] P. G. Eliseev, H. Li, G. T. Liu, A. Stintz, T. C. Newell, L. F. Lester, and K.
J. Malloy, “Ground-state emission and gain in ultralow-threshold InAsInGaAs quantum-dot lasers,” IEEE J. Sel. Topics Quantum Electron.,
vol. 7, no. 2, pp. 135–142, Mar.–Apr. 2001.
[25] S. Ghosh, S. Praghan, and P. Bhattacharya, “Dynamic characteristics
of high-speed In Ga As/GaAs self-organized quantum dot lasers at
room temperature,” Appl. Phys. Lett., vol. 81, pp. 3055–3057, 2002.
[26] P. Bhattacharya, J. Singh, H. Yoon, X. Zhang, A. Gutierrez-Aitken,
and Y. Lam, “Tunneling injection lasers: A new class of lasers with
reduced hot carrier effects,” IEEE J. Quantum Electron., vol. 32, no. 9,
pp. 1620–1629, Sep. 1996.
[27] O. Buchinsky, M. Blumin, M. Orenstein, G. Eisenstein, and D. Fekete,
“Strained InGaAs-GaAs single-quantum-well lasers coupled to n-type
-doping-improved static and dynamic performance,” IEEE J. Quantum
Electron., vol. 34, no. 9, pp. 1690–1697, Sep. 1998.
[28] S. M. Kim, Y. Wang, M. Keever, and J. S. Harris, “High-frequency modulation characteristics of 1.3-m InGaAs quantum dot lasers,” IEEE
Photon. Technol. Lett., vol. 16, no. 2, pp. 377–379, Feb. 2004.
[29] O. B. Shchekin and D. G. Deppe, “The role of p-type doping and the
density of states on the modulation response of quantum dot lasers,”
Appl. Phys. Lett., vol. 80, pp. 2758–2760, 2002.
[30] H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic
Properties of Semiconductor, Singapore: World Scientific, 1990.
[31] W. W. Chow, S. W. Koch, and M. Sargent III, Semiconductor-Laser
Physics. New York: Springer-Verlag, 1994.
[32] S. Sanguinetti, M. Henini, M. G. Alessi, M. Capizzi, P. Frigeri, and
S. Franchi, “Carrier thermal escape and retrapping in self-assembled
quantum dots,” Phys. Rev. B, vol. 60, pp. 8276–8283, 1999.
[33] M. Sugawara, K. Mukai, Y. Nakata, H. Ishikawa, and A. Sakamoto, “Effect of homogeneous broadening of optical gain on lasing spectra in
As/GaAs quantum dot lasers,” Phys. Rev. B,
self-assembled In Ga
vol. 61, pp. 7595–7603, 2000.
[34] H. D. Summers, J. D. Thomson, P. M. Smowton, P. Blood, and M. Hopkinson, “Thermodunamic balance in quantum dot lasers,” Semicond. Sci.
Technol., vol. 16, pp. 140–143, 2001.
IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 41, NO. 1, JANUARY 2005
[35] W. B. Joyce and R. W. Dixon, “Analytic approximations for the Fermi
energy of an ideal Fermi gas,” Appl. Phys. Lett., vol. 31, pp. 354–356,
1977.
[36] V. C. Aguilera-Navarro, G. A. Estévez, and A. Kostecki, “A note
on the Fermi–Dirac integral function,” Appl. Phys. Lett., vol. 63, pp.
2848–2850, 1988.
[37] Handbook of Mathematical Functions, M. Abramowitz and I. A.
Stegun, Eds., Dover, New York, 1970. sections: 3.6.24, 3.6.25.
[38] G. Park, O. B. Shchekin, and D. G. Deppe, “Temperature dependence
of gain saturation in multilevel quantum dot lasers,” IEEE J. Quantum
Electron., vol. 36, no. 9, pp. 1065–1071, Sep. 2000.
[39] H. Shoji, Y. Nakata, K. Mukai, Y. Sugiyama, M. Sugawara, N.
Yokoyama, and H. Ishikawa, “Lasing characteristics of self-formed
quantum-dot lasers with multistacked dot layer,” IEEE J. Sel. Topics
Quantum Electron., vol. 3, no. 2, pp. 188–195, Mar.–Apr. 1997.
[40] D. Bimberg, N. Kirstaedter, N. N. Ledentsov, Zh. I. Alferov, P. S. Kop’ev,
and V. M. Ustinov, “InGaAs-GaAs quantum-dot lasers,” IEEE J. Sel.
Topics Quantum Electron., vol. 3, no. 2, pp. 196–205, Mar.–Apr. 1997.
[41] A. Fiore, U. Oesterle, R. P. Stanley, R. Houdré, F. Lelarge, M. Ilegems,
P. Borri, W. Langbein, D. Birkedal, J. M. Hvam, M. Cantoni, and F.
Bobard, “Structural and electrooptical characteristics of quantum dots
emitting at 1.3 m on gallium arsenide,” IEEE J. Quantum Electron.,
vol. 37, no. 8, pp. 1050–1058, Aug. 2001.
[42] S. Kaiser, T. Mensing, L. Worschech, F. Klopf, J. P. Reithmaier, and A.
Forchel, “Optical spectroscopy of single InAsInGaAs quantum dots in
a quantum well,” Appl. Phys. Lett., vol. 81, pp. 4898–4900, 2002.
[43] K. Kamath, N. Chervela, K. K. Linder, T. Sosnowski, H.-T. Jiang, T.
Norris, J. Singh, and P. Bhattacharya, “Photoluminescence and timeAs/GaAs selfresolved photoluminescence characteristics of In Ga
organized single- and multiple-layer quantum dot laser structures,” Appl.
Phys. Lett., vol. 71, pp. 927–929, 1997.
[44] T. S. Sosnowski, T. B. Norris, H. Jiang, J. Singh, K. Kamath, and P.
Bhattacharya, “Rapid carrier relaxation in In Ga As/GaAs quantum
dots characterized by differential transmission spectroscopy,” Phys. Rev.
B, vol. 57, pp. R9423–R9426, 1998.
[45] P. Borri, W. Langbein, J. M. Hvam, F. Heinrichsdorff, M.-H. Mao, and
D. Bimberg, “Spectral hole-burning and carrier-heating dynamics in InGaAs quantum-dot amplifiers,” IEEE J. Sel. Topics Quantum Electron.,
vol. 6, no. 3, pp. 544–551, May–Jun. 2000.
[46] D. R. Matthews, H. D. Summers, P. M. Smowton, and M. Hopkinson,
“Experimental investigation of the effect of wetting-layer states on the
gain-current characteristic of quantum-dot lasers,” Appl. Phys. Lett., vol.
81, pp. 4904–4906, 2002.
[47] L. W. Wang, A. J. Williamson, A. Zunger, H. Jiang, and J. Singh, “Comparison of the k 1 p and direct diagnolization approches to the electronic
structure of InAs/GaAs quantum dots,” Appl. Phys. Lett., vol. 76, pp.
339–341, 2000.
[48] H. Jiang and J. Singh, “Strain distribution and electronic spectra of
InAs/GaAs self-assembled dots: An eight-band study,” Phys. Rev. B,
vol. 56, pp. 4696–4701, 1997.
[49] K. Mukai, N. Ohtsuka, M. Sugawara, and S. Yamazaki, “Self-formed
In Ga As quantum dots on GaAs substrates emitting at 1.3 m,”
Jpn. J. Appl. Phys., vol. 33, pp. 1710–1712, 1994.
[50] U. Bockelmann and G. Bastard, “Phonon scattering and energy relaxation in two-, one, and zero-dimensional electron gases,” Phys. Rev. B,
vol. 42, pp. 8947–8951, 1990.
[51] H. Benisty, C. M. Sotomsayor-Torrès, and C. Weisbuch, “Intrinsic mechanism for the poor luminescence properties of quantum-box systems,”
Phys. Rev. B, vol. 44, pp. 10 945–10 948, 1991.
[52] J. Urayama, T. B. Norris, J. Singh, and P. Bhattacharya, “Observation
of phonon bottleneck in quantum dot electronic relaxation,” Phys. Rev.
Lett., vol. 86, pp. 4930–4933, 2001.
[53] D. Fekete, H. Dery, A. Rudra, and E. Kapon, “Considerable enhancement of carrier relaxation channels by incorporating n-type -doping
region in quantum dot assemblies,” unpublished.
[54] M. Grundmann, O. Stier, and D. Bimberg, “InAs/GaAs pyramidal
quantum dots: Strain distribution, optical phonons, and electronic
structure,” Phys. Rev. B, vol. 52, pp. 11 969–11 981, 1995.
[55] A. A. Dikshit and J. M. Pikal, “Carrier distribution, gain, and lasing in
1.3-m InAs-InGaAs quantum-dot lasers,” IEEE J. Quantum Electron.,
vol. 40, no. 1, pp. 105–112, Jan. 2004.
[56] D. W. Snoke and J. D. Crawford, “Hysteresis in the Mott transition between plasma and insulating gas,” Phys. Rev. E, vol. 52, pp. 5796–5799,
1995.
[57] L. Claudio, A. Panzarini, G. Panzarini, and J. M. Gérard, “Strong-coupling regime for quantum boxes in pillar microcavities: Theory,” Phys.
Rev. B, vol. 60, pp. 13 276–13279, 1999.
DERY AND EISENSTEIN: THE IMPACT OF ENERGY BAND DIAGRAM AND INHOMOGENEOUS BROADENING
[58] T. H. Stievater, X. Li, D. G. Steel, D. Gammon, D. S. Katzer, D. Park,
C. Piermarocchi, and L. J. Sham, “Rabi oscillations of excitons in single
quantum dots,” Phys. Rev. Lett., vol. 87, pp. 133 603 1–133 603 4, 2001.
[59] O. B. Shchekin and D. G. Deppe, “Low-threshold high-to 1.3-m InAs
quantum-dot lasers due to P-type modulation doping of the active region,” IEEE Photon. Technol. Lett., vol. 14, no. 9, pp. 1231–1233, Sep.,
2002.
[60] D. G. Deppe, H. Huang, and O. B. Shchekin, “Modulation characteristics
of quantum-dot lasers: The influence of P-type doping and the electronic
density of states on obtaining high speed,” IEEE J. Quantum Electron.,
vol. 38, no. 12, pp. 1587–1593, Dec. 2002.
[61] N. Hatori, K. Otsubo, M. Ishida, T. Akiyama, Y. Nakata, H. Ebe, S. Okumura, T. Yamamoto, M. Sugawara, and Y. Arakawa, “20 C–70 C temperature independent 10 Gb/s operation of a directly modulated laser
diode using p-doped quantum dots,” in Proc. Eur. Conf. Opt. Comminicatiom, 2004, Paper No. Th4.3.4.
[62] P. Bhattacharya, “Quantum well and quantum dot lasers: From strainedlayer and self-organized epitaxy to high-performance devices,” Optical
and Quantum Electron., vol. 32, pp. 211–225, 2000.
[63] M. A. Dupertuis, F. Vouilloz, D. Y. Oberli, H. Weman, and E. Kapon,
“Band-mixing and coupling in single and double quantum wire structures,” Physica E, vol. 2, pp. 940–943, 1998.
[64] A. Sa’ar, S. Calderon, A. Givant, O. Ben-Shalom, E. Kapon, and C.
Caneau, “Energy subbands, envelope states, and intersubband optical
transitions in one-dimensional quantum wires: The local-envelope-states
approach,” Phys. Rev. B, vol. 54, pp. 2675–2684, 1996.
[65] R. Schwertberger, D. Gold, J. P. Reithmaier, and A. Forchel, “Epitaxial
growth of 1.55 m emitting InAs quantum dashes on InP-based heterostructure by GS-MBE for long-wavelength laser applications,” J.
Crystal Growth, vol. 251, pp. 248–252, 2003.
[66] T. Ishikawa and J. E. Bowers, “Band lineup and in-plane effective mass
of InGaAsP or InGaAlAs on InP strained-layer quantum well,” IEEE J.
Quantum Electron., vol. 30, no. 2, pp. 562–570, Feb. 1994.
[67] D. Gershoni, C. H. Henry, and G. A. Baraff, “Calculating the optical
properties of multidimensional heterostructures: Application to the modeling of quaternary quantum well lasers,” IEEE J. Quantum Electron.,
vol. 29, no. 9, pp. 2433–2450, Sep. 1993.
[68] P. G. Eliseev, H. Li, A. Stintz, G. T. Liu, T. C. Newell, K. J. Malloy, and
L. F. Lester, “Transition dipole moment of InAs/InGaAs quantum dots
from experiments of ultralow-threshold laser diodes,” Appl. Phys. Lett.,
vol. 77, pp. 262–264, 2000.
[69] K. L. Silverman, R. P. Mirin, S. T. Cundiff, and A. G. Norman, “Direct
measurement of polarization resolved transition dipole moment in InGaAs/GaAs quantum dots,” Appl. Phys. Lett., vol. 82, pp. 4552–4554,
2003.
[70] A. V. Uskov, A.-P. Jauho, B. Tromborg, J. Mrk, and R. Lang, “Dephasing
times in quantum dots due to elastic LO phononcarrier collisions,” Phys.
Rev. Lett., vol. 85, pp. 1516–1519, 2000.
35
[71] P. Borri, W. Langbein, J. M. Hvam, F. Heinrichsdorff, M.-H. Mao,
and D. Bimberg, “Time-resolved four-wave mixing in InAsInGaAs
quantum-dot amplifiers under electrical injection,” Appl. Phys. Lett.,
vol. 76, pp. 1380–1382, 2000.
[72] P. Y. Yu and M. Cardona, Fundamentals of Semiconductors. Berlin,
Germany: Springer-Verlag, 2001.
[73] M. Grundmann and D. Bimberg, “Theory of random population for
quantum dots,” Phys. Rev. B, vol. 55, pp. 9740–9745, 1997.
Hanan Dery received the B.Sc. and Ph.D. degrees in
electrical engineering from the Technion-Israel Institute of Technology, Haifa, Israel, in 1999 and 2004,
respectively. His Ph.D. research was in the field of
optoelectronic nanostructure devices specializing in
nonlinear gain processes and carrier dynamical properties.
He is currently with the Physics Department, University of California at San Diego, La Jolla. His current interests are in spintronics.
Gadi Eisenstein (S’80–M’80–SM’90–F’99) received the B.Sc. degree from the University of Santa
Clara, Santa Clara, CA, in 1975 and the M.Sc. and
Ph.D. degrees from the University of Minnesota,
Minneapolis, in 1978 and 1980, respectively.
In 1980, he joined AT&T Bell Laboratories,
Holmdel, NJ, where he was a member of the
Technical Staff in the Photonic Circuits Research
Department. His research at AT&T Bell Laboratories
was in the fields of diode laser dynamics, high-speed
optoelectronic devices, optical amplification, optical
communication systems, and thin film technology. In 1989, he joined the
faculty of the Technion Israel Institute of Technology, Haifa, Israel, where he
holds the Dianne and Mark Seiden Chair of Electro-Optics in Electrical Engineering and serves as the head of the Barbara and Norman Seiden Advanced
Optoelectronics Center. His current activities are in the fields of quantum-dot
lasers and amplifiers, nonlinear optical amplifiers, compact short-pulse generators, bipolar heterojunction photo transistors, wideband fiber amplifiers,
and broadband fiber optics systems. He has published over 250 journal and
conference papers, lectures regulary in all major fiber optics and diode laser
conferences, and serves on numerous technical program committees.
Prof. Eisenstein is an Associate Editor of the IEEE JOURNAL OF QUANTUM
ELECTRONICS.