Spontaneous emission factor in oxide confined vertical-cavity lasers
D. V. Kuksenkov
a)
and H. Temkin
Department of Electrical Engineering, Texas Tech University, Lubbock, Texas 79409
K. L. Lear and H. Q. Hou
Sandia National Laboratories, Albuquerque, New Mexico 87185
~Received 3 September 1996; accepted for publication 5 November 1996!
We report on measurements of the spontaneous emission factor for oxide-confined InGaAs vertical
cavity surface emitting lasers. The spontaneous emission factor is determined as a function of the
active layer volume from the measurement of small-signal harmonic distortion at threshold. For a
333
m
m oxide aperture device we obtain spontaneous emission factor of 4.2• 10
2 2
at room
temperature. © 1997 American Institute of Physics. @S0003-6951~97!03501-8#
The possibility of controlling spontaneous emission in
semiconductor microcavities has attracted considerable at-
tention due to predictions of essentially thresholdless lasing
and enhanced modulation bandwidth.
1
The spontaneous
emission factor
b
has been measured for buried mesa VC-
SELs with dielectric mirrors,
2
gain-guided VCSELs,
3,4
opti-
cally pumped lasers with planar resonators,
5
and for air-post
mesa VCSELs at low temperatures.
6
The
b
factor of oxide-
confined VCSELs was estimated from the spectral and angu-
lar width of the spontaneous emission.
7
The results reported
thus far differ significantly from structure to structure and the
measurement conditions, with the highest values of
b
;10
2 2
observed at low temperatures.
5,6
The enhancement in the value of
b
is expected to sig-
nificantly affect the laser performance when the lateral di-
mensions of the cavity are reduced to less than 1
m
m.
8,9
Unfortunately, VCSEL size scaling is limited by increasing
optical loss and consequently the threshold carrier density. In
addition, the enhancement of spontaneous emission in a mi-
crocavity formed by distributed Bragg reflectors is limited by
the finite width of the reflection band and the optical field
penetration into the mirrors.
10
Oxide confined VCSELs are characterized by signifi-
cantly reduced internal loss and have already demonstrated
record performance levels in terms of the threshold current
and power conversion efficiency.
11
Lasers with the lateral
cavity dimensions below ;1
m
m appear feasible.
Devices used in our measurements are based on three
InGaAs quantum wells in the active region, and are designed
to emit at 980 nm. Two quarter-wavelength Ga
0.02
Al
0.98
As
layers, one above and one below the active region, are par-
tially oxidized to form the current aperture. Devices with the
aperture sizes in the range of 3–25
m
m exhibit threshold
currents from 0.27 to 3.3 mA, and threshold voltages of 2.6–
1.4 V.
The conventional method of estimating the
b
factor is
based on the rate equations fitting the measured light-current
~L-I! curve.
12
When applied to VCSELs, this technique is
likely to be inaccurate. In very small devices, the tempera-
ture of the active region, and therefore the threshold and
efficiency, are dependent on the drive current as a result of
self-heating, and these effects require careful compensation.
Larger index guided devices lase in multiple transverse
modes @see Fig. 1~b!#, and the modal distribution is also
dependent on the drive current, even though calculations
usually assume it to be constant.
An alternative technique is used here to determine
b
.It
is based on the measurement of small-signal harmonic dis-
tortion at threshold, first suggested by Goodwin and
Garside
13
and recently applied to VCSELs.
5
This method can
be very accurate since all the measurements are made at or
near threshold, and therefore all the current induced changes
are insignificant. To make the measurement even more pre-
cise we modify the previously derived expressions for
b
13
to
take into account the nonzero transparency current.
We restrict our analysis to the case of a single transverse
mode. As illustrated in Fig. 1, the number of transverse
modes in the VCSEL spectrum at threshold can be quite
large, but the zeroth-order mode always dominates. Single-
mode rate equations are written as:
dN
dt
5
h
i
I
eV
a
2
N
t
n
2A
~
N2N
0
!
S,
~1!
dS
dt
5GA
~
N2N
0
!
S2
S
t
p
1G
b
N
t
r
,
a!
Electronic mail: DKuksenkov@coe2.coe.ttu.edu
FIG. 1. Near-threshold spectra of 434 and 10310
m
m oxide aperture de-
vices.
13Appl. Phys. Lett. 70 (1), 6 January 1997 0003-6951/97/70(1)/13/3/$10.00 © 1997 American Institute of Physics
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where N(S) are the carrier ~photon! densities,
h
i
is the in-
jection efficiency, I is the drive current, e is the elementary
charge, V
a
is the active region volume, A is the differential
gain, G is the optical confinement factor, N
0
is the carrier
density at transparency,
t
p
is the photon lifetime in the cav-
ity,
t
n
is the carrier lifetime, and
t
r
is the radiative carrier
lifetime.
In general,
t
n
,
t
r
, and we use the following definition
of the radiative efficiency
h
r
:
h
r
5
t
n
t
r
5
radiative recombination rate
total recombination rate
. ~2!
After that, our analysis closely follows the guidelines of Ref.
13. We assume small-signal modulation in the form of x
5 x
0
1
g
sin(
v
t), where x is the normalized pump variable
and
g
is the modulation index
x5
I2 I
tr
I
th
2 I
tr
;
g
5
I
m
I
th
2 I
tr
, ~3!
where I
th
is the threshold current, I
tr
is the transparency cur-
rent, and I
m
is the modulation current amplitude. From har-
monic analysis of the steady-state solution of ~1!,wefind
that the second harmonic peaks very close to threshold at
x
peak
5 12 2
b
h
r
, and the ratio of the amplitudes of the sec-
ond and fundamental harmonics at x
peak
is given by
u
R
u
5
1
4
g
A
4
b
h
r
~
11 x
tr
!
, ~4!
where x
tr
5 I
tr
/(I
th
2 I
tr
)is the normalized transparency cur-
rent. Finally, we obtain
b
5
1
4
h
r
~
11 x
tr
!
S
g
4R
D
2
. ~5!
It is interesting to compare ~5! to the
b
8
5 (
g
8
/4R)
2
,
obtained in Ref. 13 @note that
g
8
5 I
m
/I
th
is different from
g
in our Eq. ~5!#. For
h
r
5 1 and the transparency current equal
to one third of the threshold current, we have
b
'1.5
b
8
.
When the approach of Ref. 13 is used without a correction
for the nonzero transparency current,
5
the resulting values of
the
b
factor are underestimated by at least 50%.
Small-signal sinusoidal modulation used in our experi-
ments is produced by a low-distortion synthesized rf genera-
tor. The modulation signal at 30 kHz is mixed with the dc
bias using a standard bias-T. To ensure the validity of the
small-signal approximation the modulation index is kept be-
low
g
52%. The laser is mounted on a heat-sink and tem-
perature stabilized at T520 °C. A cleaved end of standard
multimode ~50
m
m core! optical fiber, positioned several
millimeters away from the laser surface, is used to collect the
light and to spatially filter the spontaneous emission. The
light is then put through an optical spectrum analyzer acting
as a filter with the resolution bandwidth set at 0.2 nm, select-
ing only the zeroth-order transverse mode of the laser. The
spatially and spectrally filtered emission is detected by an
InGaAs p-i-n photodiode, and the amplitudes of the funda-
mental and second harmonics are measured by a lock-in am-
plifier.
The measured harmonic amplitudes are plotted in Fig. 2
as a function of drive current for two lasers, 535 and 10310
m
m in size. It is clear from Fig. 2 that the measured harmon-
ics behave as expected from theoretical analysis.
The knowledge of three additional parameters, I
th
, I
tr
,
and
h
r
, is needed in order to extract the
b
factor. The thresh-
old currents were determined from the kink position in the
measured differential current-voltage ~I-V! characteristics
IdV/dI(I). The transparency current I
tr
is determined from
the measurement of photoinduced current dependence on dc
bias, using the device under test as a photodetector and a
second VCSEL, of the same kind, as a source of light.
14
The
value of radiative efficiency
h
r
can be obtained from a direct
measurement of the differential carrier lifetime dependence
on the drive current below threshold,
15
or estimated from
comparison of the actual injection current density at thresh-
old with the one predicted for an ideal laser with
h
r
5 1, as
done in Ref. 6. We assume
h
r
5 1 for all devices meaning
that it is the product of
h
r
b
, rather than the
b
factor itself
being reported on.
The measured values of the spontaneous emission factor
are plotted in Fig. 3 as a function of the cavity width. It is
clear that
b
scales as the inverse of the active layer volume.
This type of scaling is predicted by the classical electromag-
netic theory and is attributed to a decrease in the number of
available cavity modes. A rigorous calculation of spontane-
ous emission coupling into the lasing mode of a distributed
Bragg reflector surface emitting microcavity laser with quan-
tum well ~QW! active region
16
shows that for cavity widths
a@ l/n
eff
, where l is the resonant wavelength and n
eff
is the
effective refractive index, one can approximate:
b
QW
'
j
m
b
bulk
. ~6!
The coefficient m accounts for the dipole radiation enhance-
FIG. 2. Harmonic amplitude dependence on the normalized drive current for
two VCSELs of different sizes. The amplitude of the fundamental harmonic
is normalized by
g
and the amplitude of the second harmonic is normalized
by
g
2
/4.
14 Appl. Phys. Lett., Vol. 70, No. 1, 6 January 1997 Kuksenkov
et al.
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ment in the direction perpendicular to the plane of the QW
and is equal to 1.5. The coefficient
j
accounts for the change
in modal density distribution in QW and varies from 2 to 0
depending on the relative position of the QW and the mode
field. The lasers used here are designed with the QWs at the
center of the l cavity and we calculate
j
'1.8. The
b
bulk
is
calculated from classical electromagnetic theory as one half
of the inverse value of the total number of modes in the
emission spectrum:
17
b
bulk
5
e
l
4
4
p
n
eff
3
VDl
, ~7!
where
e
is the power confinement factor of a mode, V is the
waveguide volume, and Dl is the full width half maximum
~FWHM! of the spontaneous emission spectrum.
The calculated
b
QW
is plotted in Fig. 3 ~solid line! as a
function of the cavity width. Since for the smallest of our
devices ~333
m
m! the cavity width a'11l/n
eff
, we expect
our experimental data to follow the straight line. Indeed, for
a.5
m
m, the agreement is evident. However, for smaller
sizes, the measured
b
is almost a factor of 4 higher than that
predicted by theory. This disagreement needs to be ex-
plained. For the smallest lasers it is difficult to assess the
cavity size and waveguide parameters with sufficient accu-
racy. In the calculation we assumed the cavity size to be
equal to that of the oxide aperture. We also assumed a step-
like index variation due to the confining oxide. The actual
index profile can be affected by the edge diffraction and
self-heating, especially in small devices. A ;1
m
m overesti-
mate in the cavity size would place the small device data
points back on the calculated line.
We also observe a clear dependence of the spontaneous
emission factor on the threshold carrier density. For the two
devices of the same size the lower threshold laser always
exhibits higher
b
~see Fig. 3!. Since the measured transpar-
ency current is almost constant for a given device size, the
difference in threshold is assumed to be mostly due to the
difference in optical loss. The decrease in
b
with increasing
carrier density can then be attributed to the broadening of the
spontaneous emission spectrum as well as increased optical
loss.
To compare the two methods of determining the sponta-
neous emission factor, we fit rate equations to measured L-I
characteristics for selected devices. This measurement is
done for the zeroth-order transverse mode in the vicinity of
threshold. Data points obtained in this way are plotted as
triangles in Fig. 3. For mid-size devices ~5–10
m
m cavity
width! the best fit to the L-I curve gives approximately the
same
b
value as those obtained by harmonic distortion analy-
sis.
In conclusion, we report the results of spontaneous emis-
sion factor measurement for oxide-confined InGaAs vertical
cavity lasers emitting at 980 nm. The
b
factor is determined
from the measurement of the small-signal harmonic distor-
tion at threshold. For a 333 oxide aperture device we obtain
b
5 4.2• 10
2 2
which is, to our knowledge, the highest value
reported for any VCSELs to date.
The authors thank J. Banas for technical assistance. The
work at Sandia was supported in part by the United States
Department of Energy under Contract DE-AC04-
94AL85000, work at Texas Tech is supported by the Na-
tional Science Foundation and the Maddox Foundation.
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FIG. 3. Spontaneous emission factor dependence on the cavity width: solid
circles—experimental data obtained from harmonic distortion measure-
ments, open triangles—experimental data obtained from the fit to the mea-
sured L-I curve, solid line—theoretical result.
15Appl. Phys. Lett., Vol. 70, No. 1, 6 January 1997 Kuksenkov
et al.
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