Physics of semiconductor microcavity lasers

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Physics of semiconductor microcavity lasers
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My IOPscience
Semicond.
Sci.
Technol.
10
(1995) 739-751. Printed in
the
UK
TOPICAL
REVIEW
1
Physics
of
semiconductor
microcavity
iasers
i
c
,w
"--LI
e
,-L-b-L
---I 1M
\&I
CI-
^...
I
J
YY nuciiil,
r
uaiiiine(
ariu
VY
VI
wi uwt
t
Department of Phvsics and Materials Sciences Centre. Philipus Universitv.
..
..
5-35032
Marburg, Germany
i
Semiconductor Phvsics Department. Sandia National Laboratow, Albuaueraue.
.
..
NM
87185-0350,
USA
Received
29
December
1994,
accepted for publication
11
January
1995
Abstract. This review summarizes recent developments
and
successes
in
t he
theoretical modelling of
t he
characteristics of semiconductor microcavity lasers.
After a discussion of
the
basic laser properties,
results
of
a
quasi-equilibrium
microcavity laser operation
not
too
far
above the laser threshold. Non-equilibrium
phenomena, such
as
spectral and kinetic hole burning
as well
as
plasma heating
effects, are analysed using a quantum kinetic approach. Comparisons with
experimental observations
are
discussed, before open problems and
future
challenges are outlined.
many-body
Bw
pwseniea*
which
BR
v ~ ~ y
.useilji
undeEyanding ai
1.
What
are
microcavity lasers?
Semiconductor microcavity lasers are tiny semiconductor
lasers with
an
overall volume in the pm3 regime.
up
to
now, such miniature lasers have been realized
in
two
fundamentally different configurations. These are the
vertical cavity surface emitting lasers
(VCSEL)
[ I 4 1
and the
so-caiied microdisc iasers
i i j.
Both
structures are shown
schematically in figure 1.
A
microdisc laser consists of a disc of semiconductor
material with a typical diameter
of
2-5
pm, which is thinner
than the optical wavelength. Under laser conditions high-Q
whispering-gaiiery modes propagate inside the disc ciose
to
the circumference. These modes are confined inside the
disc due to the large effective reflection coefficient.
The
VCSEL
has a Fabry-P&ot resonator with very high
mirror reflectivity. This high reflectivity is achieved with
distributed Bragg retiectors (DBRsj, consisting
of
epitaxiaiy
grown layers
of
different refractive indices
[SI.
The
simplest
DBR
has alternating layers that are each
1/4
thick
so
as
to
provide constructive interference of the reflected
waves from each interface. For a typical arrangement
the resuiting frequency-dependent reflectivity
is
shown
in
figure
2.
The more refined DBRs have complicated layer
structures, which are designed to give graded potential
energy differences to improve carrier transport without
degrading the optical performance. Present
DBRs
consisting
of
as many
as
30
to
40
layers can achieve over
99.9%
reflectivity. For a review
of
the design details see,
for
example,
161.
0268-1242/95/060739+13$19.50
8
1995 IOP
Publishing
Ltd
The entire
VCSE
structure is that
of
an
effectively one
dimensional resonator with a length of a few micrometres.
The active region between the mirrors consists either
of
bulk semiconductor material or of one
or
a few
quantum wells embedded between cladding layers.
As
a
consequence of the short cavity length
it
is possible
to
design a
mode
structure in
Vcsm
where only one high-Q
region
of
the semiconductor medium. Such devices exhibit
nearly ideal single-mode operation.
iiangiiudinaij
cavity.
made
exists
within
he
specirai
gain
2.
Spontaneous emission properties
Besides a number
of
promising practical advantages, which
we will discuss in later sections
of
this review, the high-Q
cavities
in
semiconductor microlasers offer interesting
possibilities for the study of fundamental aspects
of
the
interaction between light and the semiconductor medium.
Many
of
these studies are extensions of investigations
dealing with the modified radiative properties
of
atoms
between highly reflecting
mirrors.
For example,
it
could be
demonstrated (see, for example,
[9])
that the,spontaneous
decay rate
of
excited atoms in high-Q cavities is enhanced
if the atomic transitions are in resonance with a cavity
mode. On the other hand the spontaneous decay
is
inhibited
if the transitions
of
excited atoms occur at frequencies
which are strongly suppressed because
of
the negative
feedback inside the cavity.
Besides their general importance, such studies of
spontaneous emission in optical microcavities are also of
practical relevance for the performance
of
laser devices.
739
S
W Koch et
a/
I
.2 m
0
2m
4 Oi N
LENGTH
(nm)
0.0
800
860
900
950
1wo
0.0
800
860
900
950
1wo
WAVE
LENGTH
(nm)
Figure
2.
Typical
VCSEL
design example with
top
mirror
(24
1/4
layers), active region (bulk material), bonom mirror
(45
A!4
layers) and substrate. The upper part shows the
calculated longitudinal intensity distribution and the lower
pari
is
t he
reflectivity of an
external
light beam.
For example, the intrinsic origin of
a
clear laser threshold
is the ioss caused by ihe sponianeous emission inio ihe
non-lasing modes. This part of the spontaneous carrier
recombination presents a net loss of inversion for the laser
process. The pump process has to overcome these losses
before laser action becomes possible. On the other hand,
spontaneous emission into the laser mode is necessary in
order to s mt the lasing. Hence, a characteristic number
for lasers
is
the spontaneous emission coupling,
,9,
which
defines the ratio of the spontaneous emission into the
laser mode to the total spontaneous emission.
It
can be
show
on
the bzsis
of
B
equa!inn
ana!ysis
[IOi
II!
that the amount of intensity increase at the laser threshold
is determined by the spontaneous emission coupling. If
740
all of the spontaneous carrier recombinations were to
lead to emission into the laser mode
(,9
=
1)
and if
non-radiative recombination processes were negligible, the
emitted intensity would linearly increase from zero with
increasing pump rate. This ideal configuration has been
called the thresholdless laser
[
121.
In conventional semiconductor lasers with low-Q
resonators in the length range
of
a few hundred p m one
has a large number of modes spectrally inside the gain
region. Correspondingly the spontaneous emission of the
active material is distributed over the large number of non-
lasing modes and the spontaneous emission coupling into
the
laser
mode
is
small. The factor
,3
is
typically
of
the
order
of
io-'
or even smaiier. Such iow
6
vaiues iead io
a pronounced laser threshold.
According
to
Fermi's golden rule the spontaneous
photon emission probability of the active medium is given
by
the electronic transition probability from the excited to
L l l r
&L"Y"Y
DLYLL. L a - 0 L..l
y""L"..
Y".*'L, "& nLyL".l
\"""/.
Therefore, in optical resonators the spontaneous emission
properties can be changed via changes of the photon
DOS.
A strongly enhanced photon
DOS
of the resonator mode
and reduced photon
DOS
of other modes increases the
spontaneous emission coupling efficiency.
In the case
of
an ideal
VCSEL,
where all but one
longitudinal mode in the gain region is suppressed,
this single laser mode has a very high cavity
Q
due to the high (larger than
99.9%)
effective mirror
reflectivities. Consequently, the spontaneous emission
coupling efficiency is mostly determined by the number
of transverse modes. Generally, in
VCSELs
the spontaneous
emission coupling is considerably improved in comparison
to conventional semiconductor lasers and drastic reductions
of
the threshold pump current have been reported
[ 1 4 ].
However, until now, only for whispering-gallery modes
in
microdisc lasers could an effective three-dimensional mode
confinement be achieved. With a cavity volume of the order
of the cubic wavelength only
a
few cavity modes remain
and spontaneous emission coupling larger than 10% has
been reported
[13].
Even though semiconductor microdisc lasers have
been used for investigations of fundamental importance,
most application interests are currently focused on
VCSEL
structures. These devices offer
a
number of interesting
posslbi!i!ies
re!a!ed
to
the
ease
of
fabrication and
the
potential for improved performance in comparison to that
of conventional semiconductor lasers. Therefore. we will
the
.,*-*--A
i t n t ~
ti
PC
thn
nhntrrn
rlpnritrr
nf
ot~tee
fnnr\
Physics
of
semiconductor microcavity lasers
independent of the gain layer cross section, there is more
flexibility in the shape and size of the transverse optical
mode. It is possible to make the beam circular (note that
this makes it no longer necessary to distinguish between
transverse
and
lateral dimensions).
With
an
approximately
6
p m diameter circular output aperture, lowest-order
(Gaussian-like) transverse mode operation with less than
10"
beam
divergence is possible
[151.
Furthermore, as
mentioned earlier, the very short cavity length makes
VCSEL
inherently single mode.
Other advantages
of
a
VCSEL
are even more practical
in nature. Unlike
an
edge emitter, the
VCSU
mirrors
are fabricated during the epitaxial growth
of
the entire
wafer, so that the mirrors of hundreds of
lasers
are
made
simultaneously. With edge emitters, we can only make
one-dimensional laser arrays, whereas
with
surface emitters
truly two-dimensional arrays are routinely possible.
To
test
an
edge emitter one needs
to
expose the edges, which
means cleaving the wafer and making the necessary facet
preparations. These
are
time-consuming and therefore
costly steps that have
to
be repeated for each laser. Surface
emitters can be fabricated, tested and operated at the wafer
level.
concentrate our discussion in the following sections mostly
on
VCSELS,
even
though many of the general considerations
are also valid for microdisc lasers.
3.
Problems of conventional semiconductor
lasers
To
show the
reasons
for the growing interest in
VCSELs
let
us
first mention a
few
problems associated with
conventional semiconductor lasers. The commonly used
laser structure is called an 'edge emitter', because the
laser emission occurs from an edge of the semiconductor
chip. Gain is achieved by current injection pumping of a
heterostructure, consisting of a semiconductor active layer
sandwiched between materials with wider bandgaps. A
common heterostructure
has
a
GaAs
active iayer between
AlGaAs barriers. One of the functions of a heterostructure
is the confinement of the charge caniers which are pumped
into the active region and which are
needed
to obtain
gain. This carrier confinement is achieved by designing
the iaser smcture
to
be very thin in the direction of curreni
flow, which is
also
the epitaxial growth direction. The
conventional laser configuration works very well, as is
evident from
its
wideranging applications in many electro-
optical systems.
On
tine oiher hand, ihe combinaiion of edge emission
and active layer shape gives rise to several disadvantages.
Edge emission makes the laser transverse and lateral
modes depend on the cross section of the gain region,
which is very thin in the transverse dimension for carrier
~U,,I,,,C,,,C,,,
a,,u
WLVS
111
ULS l d l Gl d l
"II,IG,:II>I"II
,U,
UUqJUL
power. The resulting near field of the laser emission
is then also highly elongated, which does not match
well, for example, to the circular cross section of an
optical fibre into which one might wish to couple the
!aser
eF2r::Gn. r i x e
(he
!%?stverse benm dimension
is very small, typically of
the
order
of
1 pm
or
less,
the transverse beam divergence is rather high (typically
x
50"
full angle) because of diffraction. Therefore, the
far field of the laser emission
is
also highly elongated.
The
ka!!
ztigmatism
!oge!her
with.
th.e
high divergence
makes design and fabrication of coupling optics rather
challenging. Due to the long
(m
1
mm) optical cavity, an
edge emitter typically operates multimode. Approaches to
single-mode operation involve embedded gratings
or
buried
heterostructures, which increase unit cost and limit output
power 1141.
-2-e
^_.I
...
:.I..
:..
.L^
,...,.-"a
a:...---:--
CA-
^..I.....
..
4.
How
VCSELS
can
help
The discussion in the previous section makes
it
piausibie
that a possible solution
to
the problems associated
with
the
beam profile of typical edge emitters lies
in
a decoupling
of the transverse and lateral optical modes from the cross
section of the gain region. Exactly this is achieved in
a
VCSEL.
Since the opticai cavity axis is in the verticai
(epitaxial growth)duection (see figure l), the laser emission
occurs from the surface
of
the wafer. This surface-emitting
configuration has several desirable features. By being
5.
Problems with
VCSELS
The advantages of
VCSELs
do not come without a price.
Most
apparent is the drastic reduction in the gain length,
from hundreds of micrometres in an edge emitter to a few
tens of nanometres in a
VCSEL.
This
gain length reduction
has
to
be compensated by the high cavity
Q,
putting great
demands on the quality of the
DER
mirrors.
However, there still remain complications of a more
fundamental nature. The short high-Q
VCSEL
optical cavity
has sharp, widely spaced resonances.
In
many cases, there
is only one resonance under the gain curve. As
a
result,
the laser threshold current
is
very sensitive
to
the position
of this resonance in relation to the peak of the material
gain spectrum.
To
achieve minimum threshold operation
the optical resonance should be well aligned with the gain
peak.
A
related complication arises since the cavity-gain
alignment can be maintained typically only within a small
temperature range, because the gain spectrum and the
cavity resonances have different temperature dependences.
The gain spectrum shifts in frequency because of
the
temperature dependences of the bandgap energy and the
carrier distributions. The cavity resonances
shifi
with
temperature because of thermally induced changes in the
refractive indices of the mirror material and the material
within the optical cavity. The result is a temperature
ucpcnuc,l~;c
U,
ULC
Lavlry-gan
ar,g,,o'cn,,
WI I KI I
pays
ar,
important role
in
the sensitivity
of
a
VCSEL
output
to
temperature variations.
1
.-.-
-c
<L.
-.:-
.I:
__I__.
L:.L
_ I
.-
6.
The
role
of
theory
Whereas the uniqueness
of
the microcavity structure offers
many desirable properties, it clearly requires precise
optimization
for
reliable operation and fabrication. Unless
74?
S
W
Koch
et
a/
the microcavity is well matched to the gain medium
(and we have yet to discover exactly what that really
entails), the
VCSEL
will not have the minimum achievable
threshold current and/or the output will be very sensitive to
temperature.
To
successfully design the microcavity, then,
requires detailed knowledge of
the
properties of the gain
medium. One may acquire that knowledge
by
performing
many experiments.
On
the other hand, an accurate theory
can do a more thorough job for a wider range of laser
configurations and in a shorter time.
In
addition, a detailed
comparison
of
theory and experiment can isolate the effects
due to the physical interactions in the device from those due
to growth processes.
In
this and the following sections we will review
microscopic approaches to study
the
laser gain medium and
the dynamical and non-equilibrium aspects of
VCSELs.
In
the first part of our studies
we
concentrate mostly
on
the
medium aspects of the laser, assuming quasi-steady-state
operation not too far above the laser threshold. Later. we
extend
the
theory to include dynamic and non-equilibrium
effects, which allows
us
to investigate the development of
lasing out of spontaneous emission, the laser dynamics, as
well as non-equilibrium phenomena such as kinetic and
spectral hole burning and carrier heating.
One goal of
our
quasi-equilibrium analysis is to treat
and compare different gain structures.
For
this purpose the
theory has to take into account band structure effects which
arise
as
a consequence of quantum confinement or strain, in
addition to the carrier and lighf interaction phenomena.
To
accomplish
this,
one has to work at the level of electron and
hole states.
In
contrast, rate equation approaches consider
only the total carrier density
[161.
Besides the band structure effects
the
theory has to
account for the band filling, i.e. the detailed population
of the electron and hole states, 'and for the Coulomb
interactions among the carriers. When only band structure
and band filling
are
taken into account, one has a so-called
freecarrier theory.
As
a justification
for
such an approach
one usually argues that for high temperature and high carrier
densities plasma screening effectively reduces the relevance
of interactions among carriers. The free-carrier theory is
widely used in modelling semiconductor lasers. However,
we will present examples that clearly show that,
even
with
plasma screening, the residual Coulomb interactions play
important roles in
VCSEL
behaviour. Since the Coulomb
interactions involve many carriers, we refer to their effects
as
many-body effects.
A
systematic description of these
effects requires a microscopic theory
[17,18].
We will now briefly discuss the calculation of gain
and refractive index spectra using microscopic
quasi-
equilibrium theory. Our intention is to show where the
many-body effects enter, and that the resulting expressions
are actually quite straightforward to apply. Under
typical laser conditions, i.e. high carrier density and
room temperature, the strong screening of the Coulomb
interaction justifies a perturbational treatment of the
Coulomb interaction between carriers.
As
shown in detail
in [17] the microscopic quasi-equilibrium theory then gives
the following expression for the intensity gain,
G,
and
742
carrier-induced refractive index,
6n
(MKS
units):
G
+
2iK0m
6n
where KO is the magnitude of the laser field wavevector,
the summations e and
h
are
over the various electron
and hole subbands,
n
is the average refractive index of
the medium,
U
is the laser frequency,
y
is the transition
linewidth,
c
and
CO
are the speed of light
and
permittivity
in vacuum, V is the volume of
the
active region,
~ (.k,~,h
is
the dipole matrix element and
fem,(k)
is the electron (hole)
population. A many-body effect is the carrier-density-
dependent correction of the transition energy,
h%e.
h
=
Eg
f
Ee,
k
-k
Eh.
k
(2)
where
Eg
=
E80
f
c(vs,q -
v q )
-
Vs.l k-k'l (f(k')
f
fh(k'))
9#0
k'#k
(3)
is
the renormalized bandgap energy,
k
is the electron
(hole) kinetic energy,
VS.,
and V,
are
the screened
and unscreened Coulomb potential energies respectively,
and
E@
is
the unexcited semiconductor bandgap energy.
We
treat
plasma screening in the static plasmon-pole
approximation for the longitudinal dielectric function
[18].
The corrections to the unexcited semiconductor
bandgap energy may
be
written in terms of a Coulomb-
hole contribution (second term in equation
(3))
and a
screened exchange contribution (third term
in
equation
(3)).
The second many-body effect originates in the Coulomb
attraction between an electron and a hole, and leads to
an excitonic
or
Coulomb enhancement of the interband
transition probability. In equation
(I),
the Coulomb
enhancement appears as the factor
(4)
Figure
3
shows the typical dependence
of
threshold current
on
VCSEL
wavelength. The points are experimental data
taken with a
group
of lasers from
the
same wafer
[19].
The wafer has not been rotated during growth,
so
that there
is sufficient variation in layer thicknesses
or
composition
to provide
VCSELs
with a range
of
lasing wavelength. One
clearly sees that there is an optimal wavelength where the
threshold current is minimum and that the threshold current
increases rapidly with deviation from this wavelength. Note
also that these
VCSELS
operate at visible wavelengths. They
are made with phosphidebased compounds and represent
one of the noteworthy advances made recently in the field
of microcavity lasers [20].
The ability to operate with visible wavelengths will
increase the number of applications for microcavity lasers,
Physics of semiconductor microcavity
lasers
6.
4 -
660
680
700
Lasing Wavelength
(nml
Figure
3.
VCSEL
threshold current
veffius
wavelength
according to experiment (dots)
and
theory (full curve). The
theoretical curve
was
computed for carrier transport
efficiency
q
=
0.7.
The
laser
structure
consists
of four
8
nm
ln0.56Ga~,44P quantum
wells
between
Ino,so(Alo.50Gao.so)asoP
barriers. The outer barriers are
40
nm
wide
and are
sandwiched between distributed Bragg reflector
mirrors
of
alternating
ALAS
and
Alo.5Gao.5As layers. The material
threshold gain is estimated to be
1000
cm-'
for
t he
experimental devices. The experimental measurements
were
performed by
M
Hageron Crawford,
Sandia
National
Labs.
for example in optical computing, printing and displays.
This is especially true if we can extend beyond the red
portion of the visible spectrnm.
The gain structure for the
VCSELS
in figure 3 consists
of three
8
nm
Ga0.44ho.56P quantum wells separated
by
(.A~&~.&LPI,~F'
bz-ers.
???e
e.!!
rxrre
i:
figure
3 his
been computed using equation
(1).
The band structure and
dipole transition elements necessary
for
the calculation are
obtained with Luttinger theory
[17,21],
which uses
as
input
the bulk material Luttinger parameters, unrenormalized
bandgaps? energy offsets, deformation potentials and lattice
constants.
To
convert
from
canier density to threshold
current, we solve, at steady state, the rate equation for
the total carrier density. This equation requires for
input the Auger coefficient. surface recombination velocity
(because we have air-post
VCSELS)
and a canier decay rate
representing all the non-radiative processes. The transport
of
injected carriers is not treated by the microscopic theory.
We approximate the net result of carrier transport with a
phenomenological parameter
q
that gives the probability of
an injected carrier ending up in a quantum well
or
barrier
state. Finally, the theoretical curve is computed for a
material threshold gain,
Gm,
which
is
determined largely
by the optical cavity design and growth-induced losses.
Dominating the latter are the absorption losses at the
DBRS
because a major portion
of
the laser field resides there.
Besides the transport efficiency,
q,
all input parameters
in the microscopic theory may be obtained directly or
indirectly through experiments. We use
q
as a fitting
parameter to match the experimental and theoretical values
for
the
minimum threshold current. Once that is done, the
microscopic theory predicts well the dependence of the
threshold current
on
lasing wavelength. The prediction
of the optimal laser wavelength depends
on
the value
of
the bandgap energy used
for
unexcited bulk Ga,,.441n,,.56P.
0
Gdf l
I
I
"7-
4
6
8
10
12
Quantum Well Width
(nml
Figure
4.
Optimal laser wavelength versus quantum
well
width
for
In,Ga,_,P-lno.5(AlyGa,-,)o.~As
at
300
K.
For
the
full
cuwe
x =
0.55
and
y
=
0.5,
for the
chain
curve,
x =
0.57 and y
=
0.5,
and
for
the
broken cuwe,
x =
0.56
and
y
=
0.7.
The experimental data
are
for
three
8
nm
quantum
well
devices
with
x
=
0.55 and y
=
0.5 (open
symbols), one
6
nm
quantum well device
with
x
=
0.57 and
y =
0.5
(full
triangle),
and
two
10
nm
quantum
well devices
with
x
=
0.56
and
y
=
0.7
(full square and circle).
We use Stringfellow's formula
I221
and obtain very
good
agreement with experiment. The free-carrier theory would
have predicted cumes that are red shifted by roughly
20
nm.
The example shows that the microscopic theory
correctly accounts for the changes in the gain spectrum
with changing carrier density.
It
accomplishes
that
with
a consistent treatment
of
band structure, band filling
and many-body Coulomb effects. One way to use
the microscopic theory is to first anchor
it
to
a
laser
configuration where experimental data exist. This is
necessary because we do not know the value of
7,
and
in
some cases
we
are uncertain
of
the exact values of threshold
gain, unexcited material bandgap energies,
or
band offsets
in heterostructures.
Next, we apply the theory
to
look
for
trends
in behaviour when we deviate
from
the experimental
laser configuration. Figure
4
shows an example of
such an exercise
1231.
We anchor the theory to the
three
8
nm
Ga,,.~~Ino.saP-(Alo.~Gao.3)o.sIno.sP
quantum
well devices, because the most experimental information
exists there. Then we allow the quantum well
widths and compositions to vary.
in
particular, we
considered the structures
G~.~~In,,,~~P-(Alo.~Ga,,.~)o.sIno.~P
and
G~~.~~I~o.~~P-(A~~.~G~.~)o.~I~,,.~P.
The agreement
of
the theoretical predictions with measurements
on
the other
laser structures is very encouraging. The discrepancy with
the
cw
measurement is probabiy due, to the reduction
in
bandgap energy with increasing temperature.
Our
calculations are performed for
T
=
300
K,
while the laser
is most likely at an elevated plasma temperature during
Cw
operation. The difference
in
behaviour between the two
supposedly similar
10
nm
devices, and the theory's inability
to explain the difference, point to the fact that much more
work still remains for the experiment and the theoretical
analysis.
On
the other hand, considering the wide range
743
S
W
Koch et
a/
L2
1
0.6
I I
60
’-
Free-Carrier Theory
0 ‘
I
270
350 430
T
(IO
Figure
5.
VCSEL
threshold current
versus
temperature
according
to
experiment (symbols),
free-carrier
theory
(broken curve) and many-body theory
(full
curve).
in the experimental laser structures, the agreement between
theory and experiment is very satisfactory, especially since
all theoretical curves are computed using the
same
bulk
material parameters.
No
attempt is made to optimize the
input parameters
to
provide a better
fit.
The temperature dependence of the threshold current
provides another
test
of the microscopic many-body theory.
The open squares in figure
5
show the experimental data
from an
InGaAs-GaAs
quantum well
VCSEL
[24]. The
broken curve is
the
best
fit
possible using the free-canier
theory. We see that the free-carrier theory significantly
overestimates the temperature sensitivity of the threshold
current at low temperatures. While
an
underestimation
of the threshold current by the theory often means that
some practical aspect of the experiment, for example
current leakage, has been ignored, an overestimation
suggests
some fundamental inadequacy in the theory. The
significantly better agreement between many-body theory
and
experiment shows this to be the case. The full curve
is the many-body result, which describes well the more
symmetrical dependence of
the
threshold current around
T
=
350
K
[25].
The agreement between many-body theory and
experiment helps to identify the cavity-gain misalignment
as the primary cause for the temperature sensitivity of a
V ~ E L
threshold current. This provides
us
with some clues
on
how to improve temperature stability. After using the
theory to closely examine the behaviour of the cavity-gain
alignment as a function
of
temperature, we discover that
the alignment can be made less sensitive to temperature
changes by operating away from the wavelength that
gives
the
minimum threshold. Figure 6(u) shows the
improvement that is possible.
In
addition,
we
find even
greater improvement when we make use of the higher-
lying quantum well subband transitions. Figure 6(b)
indicates a drastic reduction in the temperature dependence
of the threshold current density when the
n
=
2 subband
transitions significantly contribute to the gain [26].
744
0
U
0.6
!
0.4
0.2
0
‘\
\/
i
-20 0 20
h=; 1
0
1 2 3 4
Detuning
lnm)
Threshold
Gain
I103cm’
I
Figure
6.
(a)
Change
in
VCSEL
threshold current
versus
detuning for threshold gain,
Gh
=
1000
cm-’
(full
curve),
2000
cm-’
(long-dashed cuwe)
and
3000
cm-’
(sholt-dashed
curve).
(b)
Change
in
threshold current
versus
threshold gain. The temperature range
for
both
(a)
and
(b)
is
200
K
5
T
5
400
K,
and
the
currents
are
normalized
to
that
at
T
=
300
K.
The
results
in
(b)
are
computed using
the
values
of
detuning giving
the
smallest
change
in
threshold
current
over
the
temperature
range.
The gain media
are
(a)
10
nm
and
(b)
14
nm
G~As- AI ~.~G~~.~As
quantum
wells.
0.
Above-threshold behaviour
So
far
we have dealt with only the threshold properties
of a
VCSEL.
The behaviour above threshold is even
more interesting and different from that of edge emitters.
Whereas the
L-Z
curves of
an
edge emitter consist of
straight lines, those for
a
VCSEL
show appreciable saturation
behaviour (see figure 7).
For an edge emitter, both efficiency and temperature
stability improve at higher excitation. This is not the
case with a
VCSEL
because
of
the rollover in the
L-I
curves [27,28]. Since this rollover behaviour limits power
scalability and operating temperature range,
it
is
important
that
we
understand
its
origin so that we
can
attempt to delay
its onset.
Several groups
are
involved in modelling the
VCSEL
to study the interplay
of
the many physical mechanisms,
with the hope of finding some answers (see, for example,
[29-31]). The development of a
VCSEL
model turns
out
to
be very challenging. One quickly realizes that
a three-dimensional model
is
needed, and that carrier
transport, physical optics and gain physics, together with
Edge-Emitting
Laser VCSEL
Injection
Current
Injection Current
Figure
7.
Typical
L-l
curves
for
an
edge-emitting laser and
a
VCSEL.
Physics
of
semiconductor microcavity
lasers
spectra as well as the resonator properties. Optical
gain,
spontaneous emission and mode structure
are
dynamically coupled within a self-consistent treatment.
A first-principles derivation of such a theory can be
given within the framework of a non-equilibrium Green
function description [32,33], which
has
been applied to
semiconductor lasers in 134-371.
In
the following we start to discuss how the various
interaction processes influence the carrier system. During
the pump process, carriers are excited in the conduction
and valence bands due
to
either optical excitation or
carrier injection. Optical excitation creates an initial
non-equilibrium distribution of carriers, which is centred
around
the
pump energy. The fast carrier-canier Coulomb
scattering redistributes these carriers inside the bands
towards a quasi-equilibrium Fermi-Dirac distribution. The
corresponding scattering times are in the sub-picosecond
region. The whole process of establishing a quasi-
equilibrium situation may take, however, a few picoseconds
(see examples below).
Taking into account only carrier-carrier Coulomb
scattering, the resulting Fermi-Dirac functibn would have
a temperature that depends only on
the
excess energy
of pumped carriers, and usually exceeds the lattice
temperature.
In
addition the carriers can emit
or
absorb
phonons. In polar materials
the
coupling
to
longitudinal
optical phonons is especially strong with scattering times
of about 1 ps
or
even faster. The carrier-phonon scattering
provides, in addition to
the
relaxation of the initial non-
equilibrium distribution,
an
exchange of kinetic energy of
the carriers with the crystal lattice. The corresponding
reduction
of
the effective carrier temperature towards the
lattice temperature is particular important for laser operation
since increasing carrier temperature strongly reduces the
optical gain.
At
first sight the situation seems to be simpler for
carrier injection. Here we assume that
the
carriers have
thermalized with the crystal lattice on their way from some
electrical contacts
to
the active region. However, the
population of the bands is restricted by Pauli's exclusion
principle. This leads also for carrier injection
to
a carrier
momentum selective pump process so that the redistribution
of carriers by carrier-camier and carrier-phonon scattering
again plays a crucial role.
For a laser we also have to consider spontaneous and
stimulated recombination of carriers. The latter is restricted
to
the carriers around the laser resonance and tries to burn
a hole into the carrier distribution for high laser intensities.
To
consider the complex interplay of these processes
under transient as well
as
stationary conditions we use a
generalized Boltzmann equation
heat generation and transport, play important roles. Due
to the large number of interfaces,
the
carrier transport
problem has
so
far been solved only with the assumption
of separability of spatial coordinates. Diffraction in the
optical field
is
important because of
the
DBB,
and a well
behaved method for solving the physical optics equations
in the presence of the
DBRs
is needed. Furthermore, as
we will show in
the
following sections, the assumption of
quasi-equilibrium carrier distributions with a temperature
similar to the lattice temperature is usually invalid above
the lasing threshold. Depending
on
the conditions, we
find that
the
plasma temperature may be as much
as
a
few
hundred
degrees higher than that of the lattice. The
following sections describe a laser theory that can treat the
non-equilibrium problem. However, the computations are
sufficiently complicated that some simplifications must be
found before it can be incorporated into an easily usable
three-dimensional (3D)
V ~ E L
model. Both the injection
current and the optical field generate heat at different
locations in the laser. The heat generation and transport
create temperature variations within the microcavity, which
in
turn affect the optical field and carrier transport, via
thermally induced changes in refractive index and electrical
resistance.
In spite of the difficulties, progress is being made
in
the development of
VCSEL
models. Several of these
models are already producing useful results, ranging from
detailed descriptions of the optical field's longitudinal
spatial dependence, to the discovery of the importance of
thermal lensing [30]. Preliminary results from one 3D
VCSEL
code show good agreement between theory and
experiment on the onset of output power rollover in InGaAs
VCSELS
[30].
At present
the
rollover problem is more severe
in the phosphide-based
VCsELs,
but the model has
yet
to
identify the cause.
9.
Non-equilibrium
theory
The quasi-equilibrium theory outlined and applied in the
previous sections assumes Fermi-Dirac distributions for
the carriers with a kinetic energy corresponding to the
lattice temperature. Consequently, non-equilibrium effects,
such as carrier heating
or
kinetic hole burning. cannot
be modelled within this approach.
To
extend the quasi-
equilibrium analysis we outline in this and the following
sections a more sophisticated theory
for
the carrier-photon
system in a microcavity laser.
In
the framework of
this
approach, we study the possible deviations of the
system from quasi-equilibrium, the influence of altered (and
especially strong) spontaneous emission
in
microresonators
as well
as
the transient behaviour
of
the laser system.
To consider these points we use a fully quantum
mechanical treatment of the coupled carrier9hoton system
in the semiconductor microcavity, that incorporates both
many-body and mode confinement effects. The carrier
system
is
treated
on
the basis of a quantum Boltzmann
equation including carrier-photon, wrier-camer and
carrier-phonon scattering. The photon problem is described
by a kinetic equation for the spectral laser intensity,
which contains the semiconductor gain and luminescence
( 5)
The terms on the
RHS
describe the changes of the occupation
function
fe,h(k)
due to the various scattering processes.
The scattering contributions can be cast into the form of
a filling factor and a scattering time. For example the
74s
S
W
Koch
et
a/
pump process is blocked by a factor 1
-
f&)
-
f h( k)
for optical excitation (which has to be replaced with
1
-
f,( k )
for carrier injection). The intraband relaxation
is described through a scattering rate out of a state
c(
f,,(k)
and a scattering rate into that state
cx
1
-
f o( k).
The corresponding scattering times
rin,au,
are solutions of
Boltzmann integrals.
Spontaneous and stimulated emission act as a bridge
between the carrier and the photon system.
As
a sink of
carriers they
are
a source
for
the lasing and non-lasing
photon modes. These processes, together with the mode
selective properties of the resonator, are included in a
kinetic equation of photons, which schematically has the
form
The temporal changes of the spectral intensity are
determined by stimulated emission, cavity
loss
and
spontaneous emission. The characteristic functions of these
processes, which need
to
be computed from a solution
of the wri er problem,
are
the
optical gain,
g(o),
the
luminescence spectrum, W( o), and the renormalized index
of refraction. The latter enters
in
the cavity loss,
K,
the photon density of states,
S(w),
and the mode overlap
functions,
3g,s.
Equation
(6)
is only the special case of a general kinetic
equation which follows after averaging the spectral intensity
over the active region and using a resonant approximation
(for optical frequencies closed to the cavity resonances).
Neither assumption is necessary for practical calculations;
they are used here to considerably simplify the microscopic
complexity of the kinetic equation. Furthermore, the
formulation of a kinetic equation like equation
(6)
requires
an adiabatic treatment of the mode structure, which assumes
that the temporal changes of the spectral intensity are slow
compared with the time-scale on which the mode structure
is established. This time-scale
is
somewhat related to the
round-trip time of the light in the resonator and the effective
mirror reflectivities. Careful attention has to be paid also to
the transverse modes, which can be treated in the simplest
case within
an
expansion in terms of transverse eigenmodes.
For a detailed discussion of the kinetic equations of carriers
and photons see, for example,
[38].
10.
Carrier heating and hole burning
Numerical solutions
of
the Boltzmann equation allow
us
to
study
the
carrier dynamics in the running laser and to
check approximations often used in laser theories. The
following calculations are done
for
microcavity lasers with
a cavity length of 1
p n
and
99.9% mirror
reflectivity. For
the active region
we
consider bulk material
(GaAs)
and
assume a parabolic band structure with effective masses.
Figure
8
shows the stationary electron occupation
probability as a function of the carrier momentum for
stationary pumping through carrier injection. The three
different pump rates correspond
to
the arrows in figure
9.
For a low pump rate approximately
at
the laser threshold
(figure
8(a))
f&)
can be well fitted to a Fermi-Dirac
746
4
1.0
G3
U
w
0.0
3
8
0.5
l5
WAVE NUMBER
(units ofai l )
',\.
I
Y\ I
-
fb
I
(b)
t,
y.
0.0
3
0
2
4
6 8
10
WAVE NUMBER
(units
of
a,.')
1.0
l5
l5
2
!a
2
0.5
0
U
8
0.0
!a
Fr
(SO0
IQ
--.
Fk(S70Io
0
U
0
2
4
6 8 10
0
2
4
WAVE NUMBER
(units
of
a,.')
Figure
8.
Electron population
fk
in
the
conduction band
at
(a),
above
(b)
and
well
above
(c)
the
laser
threshold, and
Fermi-Dirac functions
Fk
with
the
same
carrier
density and
various temperatures. The
pump
rates
correspond
to
t he
arrows
in
figure
9.
function
F&)
whose temperature lies somewhat above the
lattice temperature of
300
K
(which
is
assumed
to
remain
constant, i.e. perfect heat sinking). With increasing pump
rate the effective carrier temperature increases strongly
(figure
8(b)),
and well above the laser threshold at the
k
value corresponding to the resonance
of
the laser emission
in
the
bands (marked by the arrows) a kinetic hole
appears. This hole is strongly smeared out due to the fast
intraband Coulomb scattering, which almost succeeds in
establishing Fermi-Dirac functions. Furthermore the sharp
laser emission in frequency space, which has been self-
consistently computed together with the carrier dynamics,
is connected
to
a region of carrier momenta whose width
is determined by the carrier dephasing rate
[35].
A
comparison of the carrier distribution function
f&)
with a lattice temperature Fermi-Dirac function with the
same carrier density reveals strong deviations. The carrier
temperature changes can
be
traced back to the carrier
momentum selective character of both the pump process
(blocked
for
the filled states according
to
the Pauli exclusion
principle) and the stimulated recombination (restricted to
transitions around the gain maximum). Under stationary
conditions, the removal of 'cold' carriers through the lasing
Physics
of
semiconductor microcavity
lasers
functions of electrons and holes.
In
particular the gain sat-
urates due to changes of
the
carrier occupation functions
in the spectral region around the lasing mode. However,
the increasing carrier temperature above the laser thresh-
old reduces the optical gain.
In
order to maintain gain-
clamping
(or
more strictly speaking a balance of stimulated
emission, cavity
loss
and spontaneous emission) above
the
laser threshold the carrier density has to increase too. This
clearly shows that under the present excitation conditions
the carrier density alone is
not
a good measure
for
the opti-
cal gain. The carrier density changes can be experimentally
verified via changes of the frequency-integrated lumines-
cence and the changed index of refraction
[39].
11.
Threshold reduction and emission anomalies
In
this
section we discuss the results of the mode con-
finement and altered spontaneous emission
in
semicon-
ductor microcavity lasers. Instead of performing a com-
plete three-dimensional (parameter-free) computation of the
mode structure we wish
to
discuss the emission properties
and the corresponding carrier dynamics in terms
of
confined
modes, which are characterized by the mirror reflectivi-
ties for the confinement direction, the spontaneous emission
coupling and a corresponding spontaneous carrier lifetime.
From these parameters and the self-consistently renormal-
ized index of refraction the spontaneous emission follows
as
a
dynamical quantity with strongly variable spectral and
temporal properties.
Figure
9(a)
shows
the
computed input-output charac-
teristics of a
VCSEL
with
99.9% mirror
reflectivity, a cavity
length of
1
Wm
and
various
spontaneous emission coupling
efficiencies. Generally the laser threshold is reduced if the
ratio between the spontaneous emission into the laser mode
and the total spontaneous emission increases up
to
unity.
On the other hand, possible spontaneous emission into non-
lasing modes causes a non-lasing de-excitation of the carrier
system which in turn reduces the output intensity below the
threshold.
From the solution of the photon kinetic equation we
also have information about the spectral laser intensity
together with the luminescence and gain spectra. A
comparison of the emission linewidth versus emission
8
600
500
4
400
U
300
*...a
-
:pi
2
.*..-
........
,..**'
/I
700
d
PUMP
RATE
(cm"
pi')
Figure
9.
Laser
intensity
(a)
and
carrier
density
(b)
for
a
wide range of pump rates and various spontaneous
emission couplings. The corresponding electron and hole
temperatures
(c)
are shown for
SC
=
0.4
.
process and the addition of 'hot' carriers through the
pumping leads to an increase
of
the kinetic plasma energy,
which
is
partially compensated through the coupling of the
carriers to the crystal lattice by
Lo
phonon emission. Note
that the very efficient carrier-carrier Coulomb scattering
leads to carrier redistribution in the form of quasi-
equilibrium Fermi-Dirac distributions, but
it
does not
dissipate kinetic energy from the carrier system. The
resulting state is therefore a non-equilibrium state where the
carrier temperature differs from that of the crystal lattice.
It
should be noted that the discussed carrier heating process
is due to the microscopic nature of the carrier interaction
processes and cannot be avoided by external cooling of the
laser devices.
An
interesting consequence
of
the carrier heating
is the behaviour of the total carrier density
N
=
(l/V)C,f,(k)
=
( l/V) x k
f h ( k )
shown in figure
9(b).
In
a simple rate equation theory one usually assumes
that the optical gain
is
directly proportional to the carrier
density. The fact that under stationary conditions the optical
gain cannot exceed the cavity loss leads to a canier density
clamping above the laser threshold. For many conditions
such a clamping is not obtained in
our
calculations.
In
our microscopic theory the optical gain (together
with the luminescence and the refractive index) is cal-
culated from the carrier-momentum-dependent occupation
..........................................
).
...................................
/
I
1o"o.o
0.25
0.5
0.75
1.0
1.25
LASER
INTENSITY (E:)
Figure
10.
Laser linewidth for various spontaneous
emission couplings corresponding to figure
9(a).
747
S
W
Koch
et
a/
p:
w
m
10-l
10'~
10.~
10.6
I
0
100 200
300
TIME
(ps)
1.Y
-
I
140
145
TIME
(ps)
PHOTON ENERGY (hW-EG)/EB
Figure
11.
Development
of
the
total
intensity
and spectral
intensity
after
the
onset
of
stationary pumping for various
spontaneous emission couplings
SC.
intensity in
figure
10 reveals the well-known Schawlow-
Townes line narrowing [40] only up
to
a certain level of
spontaneous emission coupling
Sc.
For large spontaneous
emission into the laser mode the gain-loss compensation
is reduced. This is due
to
the fact that under stationary
conditions gain, loss and spontaneous contributions have
to
compensate each other.
On
the other hand it is well
known from various laser theories that reduced gain-
loss compensation leads to an increasing laser linewidth.
For large spontaneous emission coupling, which can be
achieved in microdisc lasers, spontaneous emission is
no
longer a small perturbation
(as
often assumed in laser
theories).
Our
quantum-mechanical treatment of the laser
field, which is valid for arbitrarily strong spontaneous
emission, reveals the absence of the laser line narrowing for
large spontaneous emission coupling. This behaviour has
recently been observed in semiconductor microdisc lasers
[39].
It may restrict applications of low-threshold devices
in
situations where
a
narrow linewidth is required.
12.
Laser dynamics under non-equilibrium
conditions
The dynamical behaviour of the coupled carrier-photon
system from a starting
of
the pump process over the onset of
laser action
to
the stationary situation
is
characterized
by
an
748
interplay of processes acting
on
very different time-scales.
The Coulomb scattering and (for large laser intensities)
stimulated recombination are the fastest processes acting
on
a sub-picosecond time-scale. Cooling and equilibration
of the carrier system takes place within several picoseconds
and spontaneous emission has characteristic times of some
hundreds of picoseconds.
Aspects of the laser behaviour
on
the various time-
scales
are
shown in figure
11
after
the onset of stationary
pumping. For low spontaneous emission coupling the
total
laser intensity increases rapidly
to
its stationary value after
the system has established optical gain. For larger
Sc
the
intensity increases more gradually due
to
the accumulation
of spontaneous emission in the resonator. The smooth
intensity increase between 145 and
200
ps is connected with
the carriers cooling to their stationary temperature. Because
of
the
necessary balancing between stimulated emission,
resonator losses
and
spontaneous emission the spectral laser
emission narrows
on
the longest time-scale, which is that
of
the spontaneous emission.
For
low spontaneous emission
coupling this process is extended into the nanosecond
regime.
Another example for the dynamical interplay
of
the
various processes is the transient behaviour after a pulsed
excitation. Figure 12 shows the evolution
of
the canier
distribution functions after
an
optical excitation with a
100
I s
7.00
fs
--.
50Ofs
""""
1
P'
'.'"
IPS
---
4ps
-.
l ops
-
1
.o
E
0
1
.........
..
.-
.......
c
....
.....
.....
e
5
'a
0.5
.
0
....
v)
E
5
0
0
-
0.0
0
5
10
15
wave
vector
(kx%)
c
0
I
.*
c
.,--
---
4pr
m
c
0
5
10
15
wave
vector
(kx%)
Figure
12.
Electron and hole population
oi
a microcavity
!aser
ltter
a
shnrt-pu!~e
exci!a!.'inn
?vi!!!
an
excess e!er.;y
of
420
meV
above the unexcited bandgap.
I
0
10
20 30
40
time
(ps)
Figure
13.
Carrier density and laser intensity after a
sholt-pulse
excitation
of
a
VCSEL
at
zero
time.
100
fs
pulse high into the bands. The initial non-
equilibrium distribution thermalizes
by
canier-carrier and
carrier-phonon scattering. With increasing population
of
UlG
,"WCl
n
>LdlGjs
"IC
nvautpu,,
UI
"1s
SGLI"LUII""LLUL
decreases, and after the establishment of optical gain a
kinetic hole
is
burned into the distribution function. Then
the carrier density (figure 13) starts to decease, which in
turn reduces the laser intensity.
pulse carrier generation is is already partially blocked.
This blocking increases for increasing pump intensity or
for shorter pulse widths, since the carriers cannot be
scattered fast enough
out
of
the spectral region of the pump.
Figure
14
shows a comparison
of
the saturated versus non-
saturated pump rate for various pump pulse intensities and
100
fs pulses. The insets demonstrate the corresponding
transient Pauli blocking.
.L^
1..
.....
,.
......
.L^
-I."-.....:--
^C
*L"
"-...L.."J..".--
It
is dso
i.?t.rest;ng
!O
BO!e
!!!a!
dL!~!?g
&e
pllmp
Physics
of
semiconductor microcavity
lasers
time
(ps)
time
@s)
time
(ps)
Figure
14.
Temporal behaviour
of
the saturated pump rate
(full
curve)
and
non-saturaled pump
rate
(chain curve)
during a
100
fs
pulse
for
various pump pulse intensities
Q2,.
The insets
show
the
corresponding electron populations at
50
is (chain curve) and
i o0
is
(iuii curve).
13.
Future directions
The previous few sections showed that semiconductor
microlasers can serve as 'miniature laboratories'
for
studying high-density and non-equilibrium effects
in
a two-
component carrier plasma. After the discussion
of
several
of the interesting many-body effects we want
to
use this
final section to speculate about some
of
the possible future
developments
of
VCSEL
structures. As already briefly
mentioned earlier, the development of VCSELs operating
at visible
or
shorter wavelengths is important because
of
the many potential applications. At the red wavelengths,
one finds applications in flat panel and projection displays,
and
650
nm) data communication with optical fibres.
In
the blue-green region, there are applications involving
high-density optical storage, undersea communications and
imaging.
If
ultraviolet laser emission is achievable,
monitoring. For the ,red wavelength region, GaInP
VCSELS
have demonstrated lasing between
660
nm and 680 nm.
Room-temperature
cw
operation was first reported in mid-
1993
with
an
air-post device
[41].
Shortly afterwards,
an
order of magnitude improvement in optical power and
efficiency was achieved with an implanted smcture
[20].
In
terms of
optical confinement, implanted devices are
gain guided, while air-post devices are index guided. As
749
I n me -
nA-+:n- nlA
/*A- +ha
o--..:Gn
......
-I--..+h-
-F
CVC
_-
,'lac,
p,,,'Lu,~
aIIY
,I"'
U*=
"pcLLuc
"'l"C1CL,~U'"
"I
,I, ,1111
...............
wmirnnrlrrrtnr
laserc mi v
..._,
he
.............
Cnitihle
for
~ I E P
-_
i n
...
atmmnheric
......
S
W
Koch et a/
>
+
.-
Y)
C$--
Leakage
I
j
lnv,,si;;
J
,.I
I
0
0.2
0.4
.c
‘y
Leakage-’
+
O.....’.
AI
Concentration AI Concentration
Figure
15.
(a) Gain peak wavelength and
(b)
come-
sponding threshold current density
versus AI
concentration
in
6
nm AlGalnP quantum wells.
In
(b)
the long-dashed
curve
is
the contribution to the inversion and the
short-dashed
curve is
the
leakage current.
in edge emitters. gain guiding gives better output power
and efficiency because it allows an optical mode
to
take
whatever form is necessary to maximize modal gain. In
early 1994, epitaxial growth techniques for
the
phosphide
compounds improved sufficiently
to
provide the control
necessary for making short (l h) microcavities. This
results
in further improvement by eliminating from the microcavity
extraneous materials that would otherwise contribute to
the
optical losses. It is with a l k microcavity device that
the
present records of
3
mW output power and
10%
wall-plug
efficiency were made [42].
A present goal
is
to extend
VCSEL
emission to shorter
wavelengths. Edge emitters with tensile strained InGaP
quantum wells have operated with
TM
polarization to below
610 nm 1431. However, a microcavity can only support the
TE
mode,
which
has lower gain in tensile strained quantum
wells. Also, heating is a rather severe problem
in
a
vCSEL,
giving rise
to
higher current leakage and temperature-
induced bandgap energy shrinkage.
In
the worst case, all
these factors may combine to prevent a repeat of the edge
emitter’s success.
Another possibility for shorter wavelengths is to use
AlGaInP in the quantum wells. Regardless of the
approach, the ability
to
achieve shorter lasing wavelengths
is eventually limited by the available bandgap energy. For
example, the r-X crossing in (AI,Gal-x)o.51~.~P, which
occurs at
x
=
0.58,
places an upper limit of 2.31 eV
on
the
usable bandgap energy. Choosing (Alo.sGao.5)o.sIno.sP
for the barrier material, figure
15(n)
shows the possible
reduction in the gain peak wavelength with increasing AI
concentration in
a
6
nm (Al,Gal-x)O.~InO.~f,P quantum
well. However, the threshold current also increases
because
of
current leakage (see figure
15(b)).
The leakage
current rises because
of
the smaller electron and hole
confinement potentials resulting
from
the higher quantum
well bandgap energy. Note
that
the current density that
actually contributes
to
the creation of the inversion remains
relatively constant with AI concentration,
even
in the
presence of a discontinuity at
x
=
0.38 due to the transition
from two conduction subbands to a single conduction
subhand. Figure 15 is obtained using the microscopic quasi-
equilibrium theory discussed earlier
in
this review, and it
neglects the detrimental effects of oxygen which may be
introduced into
the
quantum well along with the Al.
750
600 650 700
Wavelength
(nm)
Figure
16.
Inversion
and leakage contributions to
the
threshold current density
versus
laser
wavelength for
different
In
concentration
in
6
nm
(AlxGal~x)l~ylnyP
quantum wells.
The
AI
concentration
is
varied to change
the
wavelength.
It
is
informative to plot the inversion and leakage
contributions to the threshold current density versus the
gain peak wavelength. This is done in figure 16, where
the different wavelengths are obtained by changing the
quantum well AI concentration, and each set
of
inversion
and leakage current density curves corresponds to a given
In
concentration
in
the quantum well. For an
In
concentration
of
0.50
the quantum well is unstrained, and larger (smaller)
values result
in
compressive (tensile)
strain.
While optimal
gain configurations exist, for example compressive strain
appears to give the lower threshold current density, every
set
of
curves shows a lower-wavelength limit
of
FZ
620 nm,
after which cument leakage accounts for over half of the
total current density. The same basic conclusion is also
reached if we had
varied
the quantum well width instead
of the strain.
The approach to shorter wavelength in microcavity
lasers may lie with the U-VI compounds
or
the nitride
based
m-V
compounds.
Acknowledgments
We thank R
E
Slusher,
H
M Gibbs and J McInerney for
many stimulating discussion on semiconductor microcavity
lasers.
Parts
of
this work were supported through the
Deutsche Forschungsgemeinschaft, through the Optical
Circuitry Cooperative, through the National Science
Foundation, as well as the
US
Department of Energy
through contract DEACO4-94DP85000. Furthermore, we
thank the
HRLZ
Jiilich for grants for
CPU
time.
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