Carrier capture delay and modulation bandwidth in

Copyright by the AIP Publishing. Asryan, Levon V.; Wu, Yuchang; Suris, Robert A., "Carrier capture delay and modulation
bandwidth in an edge-emitting quantum dot laser," Appl. Phys. Lett. 98, 131108 (2011); http://dx.doi.org/10.1063/1.3571295
Carrier capture delay and modulation bandwidth in an edge-emitting quantum dot
laser
Levon V. Asryan, Yuchang Wu, and Robert A. Suris
Citation: Applied Physics Letters 98, 131108 (2011); doi: 10.1063/1.3571295
View online: http://dx.doi.org/10.1063/1.3571295
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APPLIED PHYSICS LETTERS 98, 131108 共2011兲
Carrier capture delay and modulation bandwidth in an edge-emitting
quantum dot laser
Levon V. Asryan,1,a兲 Yuchang Wu,1,b兲 and Robert A. Suris2,c兲
1
Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061, USA
Ioffe Physico-Technical Institute, Saint Petersburg 194021, Russia
2
共Received 1 March 2011; accepted 8 March 2011; published online 30 March 2011兲
We show that the carrier capture from the optical confinement layer into quantum dots 共QDs兲 can
strongly limit the modulation bandwidth ␻−3 dB of a QD laser. As a function of the cross-section ␴n
of carrier capture into a QD, ␻−3 dB asymptotically approaches its highest value when ␴n → ⬁ 共the
case of instantaneous capture兲. With reducing ␴n, ␻−3 dB decreases and becomes zero at a certain
nonvanishing ␴min
n . The use of multiple-layers with QDs significantly improves the laser modulation
response—␻−3 dB is considerably higher in a multilayer structure as compared to a single-layer
structure at the same dc current. © 2011 American Institute of Physics. 关doi:10.1063/1.3571295兴
Due to the quantum-size effect, reducing dimensionality
of the active region has been a key to developing lowthreshold semiconductor lasers.1,2 In commercial diode lasers, a two-dimensional 共2D兲 active region 关quantum well
共QW兲兴 is used.3,4 In quantum dot 共QD兲 lasers, an ultimate
case of a zero-dimensional active region is realized.5,6 The
interesting physics involved and the potential for wide range
of applications have motivated extensive studies of QD lasers. However, in contrast to the steady-state characteristics,
the dynamic properties of QD lasers need to be further scrutinized. In particular, the potential of QD lasers for highspeed direct modulation of the output optical power by injection current should be clarified.
In Ref. 7, the highest modulation bandwidth attainable
in QD lasers was estimated. For this purpose, an idealized
situation of instantaneous carrier capture into QDs was assumed. In actual semiconductor lasers, carriers are not directly injected into the quantum-confined active region—
they are first injected into the optical confinement layer
共OCL兲 and then captured into the active region 共Fig. 1兲.
Indirect injection adversely affects the laser operating
characteristics—the threshold current is increased8 and more
temperature-sensitive,9 and the output optical power is
decreased.10,11 Due to a transport delay across the OCL and a
capture delay from the OCL into the active region, the bandwidth of direct modulation of the output power by injection
current is also reduced 共see, e.g., Ref. 12 for QW lasers兲.
In this letter, we briefly report on the effect of noninstantaneous capture of carriers into QDs on the modulation bandwidth of an edge-emitting QD laser. Our model is based on
the following set of three coupled rate equations for free
carriers in the OCL, carriers confined in QDs, and photons:
冉冊
⳵ NS
NS
NS
2 f n = ␴nvn 共1 − f n兲nOCL − ␴nvnn1 f n
⳵t
b
b
b
−
NS f 2n
− vggmax共2f n − 1兲nph ,
b ␶QD
⳵ nph
= vggmax共2f n − 1兲nph − vg␤nph ,
⳵t
共2兲
共3兲
where nOCL is the free carrier density in the OCL, j is the
injection current density, b is the OCL thickness, ␴n is the
cross-section of carrier capture into a QD, vn is the carrier
thermal velocity, NS is the surface density of QDs, f n is the
occupancy of the energy-level of a carrier confined in a QD,
B is the spontaneous radiative recombination constant for the
OCL, ␶QD is the spontaneous radiative time in a QD, vg is the
group velocity of light, gmax is the maximum modal gain,8
nph is the photon density 共per unit volume of the OCL兲 in the
lasing mode, ␤ = 共1 / L兲ln共1 / R兲 is the mirror loss, L is the
cavity length, and R is the facet reflectivity.
In Eqs. 共1兲 and 共2兲, the quantity n1 = N3D
c exp共−En / T兲
characterizes the carrier thermal escape from a QD to the
OCL, where N3D
c is the effective density of states in the OCL,
En is the carrier thermal excitation energy from a QD, and T
is the temperature 共in units of energy兲.
Strictly speaking, ␴n is the only parameter adequately
describing the capture/escape into/from a QD. Using ␴n, two
distinct characteristic times can be introduced—the capture
time into an unoccupied QD ensemble11 and the thermal escape time from an individual QD,8,11
⳵ nOCL
NS
NS
j
=
− ␴nvn 共1 − f n兲nOCL + ␴nvnn1 f n
⳵t
eb
b
b
2
− BnOCL
,
共1兲
a兲
Electronic mail: [email protected].
Electronic mail: [email protected].
c兲
Electronic mail: [email protected].
b兲
FIG. 1. Indirect injection into the active region of a QD laser.
0003-6951/2011/98共13兲/131108/3/$30.00
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Appl. Phys. Lett. 98, 131108 共2011兲
Asryan, Wu, and Suris
NS
b
冊
−1
,
␶esc = 共␴nvnn1兲−1 .
共4兲
In a specific structure considered below, ␶capt,0 = 1.63 ps and
␶esc = 0.07 ps at ␴n = 10−11 cm2.
We consider the spontaneous radiative recombination as
the only mechanism of nonstimulated recombination in the
OCL and QDs. The inclusion of the nonradiative Auger recombination will increase the threshold current density and
the steady-state carrier density in the OCL, and, within the
framework of the small-signal analysis, will decrease the differential nonstimulated recombination time, while not otherwise affecting the main derivations of this letter.
Our model does not include the wetting layer 共WL兲,
which is inherently present in self-assembled Stranski–
Krastanow grown QD structures. The WL can affect the carrier capture into QDs. In addition to the direct capture from
the bulk OCL into QDs, carriers will also be captured from
the OCL into the 2D WL and then from the WL into QDs.
The inclusion of the WL will thus require a careful consideration of all these capture processes.
Due to inhomogeneous broadening of the transition energy in a QD-ensemble in an edge-emitting laser, we do not
also consider the optical mode resonance with the QDtransition. This resonance in the context of QD nanocavity
lasers was considered in Ref. 13.
Applying the small-signal analysis of rate equations, we
consider the injection current density in Eq. 共1兲 in the form
of j = j0 + 共␦ jm兲exp共i␻t兲, where j0 is the dc component and
the amplitude ␦ jm of the time-harmonic ac component is
small 共␦ jm Ⰶ j0 − jth, where jth is the threshold current density兲. We correspondingly look for nOCL, f n, and nph in Eqs.
共1兲–共3兲 in the form of nOCL = nOCL,0 + 共␦nOCL−m兲exp共i␻t兲, f n
= f n,0 + 共␦ f n−m兲exp共i␻t兲, and nph = nph,0 + 共␦nph−m兲exp共i␻t兲,
where nOCL,0, f n,0, and nph,0 are the solutions of the steadystate rate equations at j = j0.10,11 In particular,
f n,0 =
冉
冊冉
冊
1
1
1
␤
1 + max = 1 +
,
2
2
g
␶phvggmax
共5兲
where the photon lifetime in the cavity is
␶ph =
1
L
.
=
vg␤ vg ln共1/R兲
共6兲
20
(GHz)
冉
␶capt,0 = ␴nvn
j0 = 63.8 kA/cm
ω-3 dB / 2π
131108-2
10
2
2
10 kA/cm
2
1 kA/cm
0
0
5
10
σn (x10-11cm2)
FIG. 2. Modulation bandwidth vs capture cross-section into a QD at different values of the dc component j0 of the injection current density in a
single-layer structure. The horizontal dashed lines show ␻−3 dB for the case
of instantaneous capture 关Eq. 9 of Ref. 7兴. 63.8 kA/ cm2 is the optimum
value of j0 maximizing ␻−3 dB for the case of instantaneous capture and,
max
correspondingly, the top horizontal line shows ␻−3
dB for that case 关Eq.
共11兲兴. T = 300 K and L = 1.1 mm.
= 6.11⫻ 1010 cm−2, and an ideal overlap between the electron and hole wave functions in a QD. At these parameters,
the maximum modal gain in a single-QD-layer structure
gmax = 29.52 cm−1. The OCL thickness b = 0.28 ␮m and the
cavity length L = 1.1 mm 共at this L and the as-cleaved facet
reflectivity R = 0.32, the mirror loss ␤ = 10 cm−1兲.
The modulation bandwidth depends strongly on the capture cross-section ␴n. At a fixed j0, with making slower the
capture into QDs 共reducing ␴n兲, ␻−3 dB decreases and finally
becomes zero 共Figs. 2 and 3兲.
As seen from Fig. 3, ␻−3 dB = 0 at a certain nonvanishing
value ␴min
n . This is due to the fact that, at a given j 0, no lasing
is attainable in the structure if ␴n ⬍ ␴min
n . Indeed, while j 0 is
fixed, the threshold current density increases with decreasing
␴n 共the curve corresponding to the left axis兲,
jth =
冉
2
f n0
1
f n0
eNS 2
f n0 + ebB n1
+
␶QD
1 − f n0 ␴nvn␶QD 1 − f n0
冊
2
,
共7兲
where f n,0 is given by Eq. 共5兲. In order for the lasing to
occur, j0 should be higher than jth. At a certain ␴min
n , j th
becomes equal to j0 共Fig. 3兲. At ␴n ⱕ ␴min
,
j
ⱖ
j
,
th
0 which
n
means that there can be no lasing and hence no direct modulation in the structure 共the shaded region in Fig. 3兲.
The minimum tolerable ␴n for the lasing to occur at j0 is
found from the condition jth = j0 and is given by
冑
j0 −
eNS 2
f +
␶QD n0
冑
eq
jth
−
eNS 2
f
␶QD n0
As seen from Eq. 共5兲, the confined-carrier level1
冑ebB
␴min
,
n 共j 0兲 =
eq
occupancy f n,0 in a QD at the steady-state is pinned at its
vn␶QD 1 − f n0
j0 − jth
threshold value and does not change with j0 above the lasing
共8兲
threshold. In contrast to f n,0, the steady-state free-carrier deneq
eq
where jth is jth for the case of instantaneous capture 关jth is
sity nOCL,0 in the OCL is not pinned—it rises with j0 above
obtained using ␴n = ⬁ in Eq. 共7兲兴.
the lasing threshold. It should be emphasized that it is the
noninstantaneous capture of carriers from the OCL into QDs
that causes this rise in nOCL,0 above the lasing threshold.10,11
We obtain from Eqs. 共1兲–共3兲 a set of algebraic equations
in the frequency-dependent small amplitudes ␦nOCL−m,
␦ f n−m, and ␦nph−m, the solution of which yields the modulation response function H共␻兲 = 兩␦nph−m共␻兲 / ␦nph−m共0兲兩2. Finally, we arrive at a cubic equation for the square of the
modulation bandwidth ␻−3 dB—the frequency, at which
H共␻兲 has fallen to half its dc 共␻ = 0兲 value.
For an illustration of our results, room-temperature operation of a GaInAsP heterostructure lasing near 1.55 ␮m
共Ref. 8兲 is considered here. We assume 10% QD-size flucFIG. 3. Modulation bandwidth 共at a very low j0兲 and threshold current
tuations,
the surface
density
of article.
QDs in
a single-layer
density
vs terms
captureat:cross-section
into a QD in a single-layer structure.
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131108-3
Appl. Phys. Lett. 98, 131108 共2011兲
Asryan, Wu, and Suris
(GHz)
20
ω-3 dB / 2π
2
10
共␻−3
j0 (kA/cm ) (5 QD-layers)
10
0
20
⬀
where ␻−3
5 QD-layers
Single QD-layer
50
100
j0 (kA/cm ) (Single QD-layer)
2
FIG. 4. Modulation bandwidth vs dc component of the injection current
density in single- and 5-QD-layer-structures. A plausible value ␴n
= 10−11 cm2 共Refs. 14 and 15兲 of the capture cross-section is used.
As seen from Eq. 共8兲, when j0 decreases and approaches
min
,
increases infinitely, i.e., no lasing is attainable at
jeq
th ␴n
eq
j0 ⱕ jth even if the carrier capture into QDs is instantaneous.
With increasing j0, ␴min
n becomes smaller, i.e., the lasing
can occur and hence the direct modulation of the output
power is possible at a slower capture. At high j0 共when
␴min
n → 0兲, the asymptotic expression for ␻−3 dB for ␴n in the
min
关共␴n − ␴min
vicinity of ␴min
n
n 兲 / ␴n Ⰶ 1兴 is
dB ⬇
2冑r − 1vggmax
f n0共1 − f n0兲 ␴n − ␴min
n
,
2 − f n0
␴min
n
共9兲
/ 2π
ω -3max
dB
(GHz)
where the numerical parameter r = 100.3 ⬇ 1.995 originates
from the definition of the ⫺3 dB bandwidth,
10 log10 H共␻−3 dB兲 = −3.
As a function of the dc component j0 of the injection
current density, ␻−3 dB has a maximum 共Fig. 4兲. In a singleQD-layer structure 共the dotted curve兲, the optimum value jopt
max
max
of j0, at which ␻−3
dB is attained, is very high, i.e., ␻−3 dB is
unattainable. As seen from the figure, there are the following
two advantages in a multi-QD-layer structure 共the solid
curve兲 as compared to a single-layer structure: 共i兲 ␻−3 dB is
considerably higher at the same j0 and 共ii兲 jopt is considerably
max
reduced, which means that ␻−3
dB is practically attainable.
At large ␴n, when ␶capt,0 / ␶ph Ⰶ 1 关␶capt,0 and ␶ph are given
by Eqs. 共4兲 and 共6兲, respectively兴, both ␻−3 dB at a given j0
max
共Fig. 2兲 and ␻−3
dB 共Fig. 5兲 asymptotically approach their
saturation values 共the horizontal dashed lines兲 corresponding
to the case of instantaneous capture into QDs,
20
5 QD-layers
Single
QD-layer
10
0 -14
10
10
-12
σ n (cm
10
2
− ␻−3
-10
)
FIG. 5. Maximum modulation bandwidth vs capture cross-section into a
max
QD. The horizontal dashed line shows ␻−3
dB for the case of instantaneous
capture into QDs 关Eq. 共11兲兴.
dB兲,
max
max
共␻−3
dB兩␴n=⬁ − ␻−3 dB兲
␶capt,0 1
⬀ ,
␶ph
␴n
dB 兩␴n=⬁
max
␻−3
dB兩␴n=⬁ ⬇
0
0
␻−3
dB兩␴n=⬁
共10兲
is given by Eq. 9 of Ref. 7 and
冑2
␶ph
.
共11兲
max
As seen from Fig. 5, while the saturation value of ␻−3
dB
at ␴n → ⬁ and at a fixed L 关Eq. 共11兲兴 does not depend on the
max
number of QD-layers, ␻−3
dB at a given finite ␴n is higher in
a multilayer structure as compared to a single-layer structure.
In conclusion, we have shown that the carrier capture
from the OCL into QDs can strongly limit the modulation
bandwidth ␻−3 dB of a QD laser. ␻−3 dB is highest in the case
of instantaneous capture into QDs, when the cross-section of
carrier capture into a QD ␴n = ⬁. With reducing ␴n, ␻−3 dB
decreases and becomes zero at a certain nonvanishing ␴min
n .
This ␴min
n presents the minimum tolerable ␴n for the lasing to
occur at a given dc component j0 of the injection current
density. The use of multiple-layers with QDs has been shown
to significantly improve the modulation response of the
laser—␻−3 dB is considerably higher in a multilayer structure
as compared to a single-layer structure at the same j0. At a
plausible cross-section ␴n = 10−11 cm2,14,15 ␻−3 dB as high
as 19 GHz can be obtained in a 5-QD-layer structure with
the cavity length L = 1.1 mm at a practical value of j0
= 7 kA/ cm2. Our analysis provides a basis for optimizing
the QD laser design for high-speed operation.
L.V.A. and Y.W. acknowledge the U.S. Army Research
Office 共Grant No. W911-NF-08-1-0462兲, Y.W. also acknowledges the China Scholarship Council, and R.A.S. acknowledges the Russian Foundation for Basic Research 共Grant No.
08-02-01337兲 and the Program “Fundamental Research in
Nanotechnology and Nanomaterials” of the Presidium of the
Russian Academy of Sciences for support of this work.
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