The effect of dispersion of the refractive index on the performance of ...

Journal <strong>of</strong> Crystal Growth 227–228 (2001) 313–318
<strong>The</strong> <strong>effect</strong> <strong>of</strong> <strong>dispersion</strong> <strong>of</strong> <strong>the</strong> <strong>refractive</strong> <strong>index</strong> on <strong>the</strong>
performance <strong>of</strong> mid-infrared quantum cascade lasers
A.Z. Li*, J.X. Chen, Q.K. Yang, Y.G. Zhang, C. Lin
State Key Laboratory <strong>of</strong> Functional Materials for Informatics, Shanghai Institute <strong>of</strong> Metallurgy, Chinese Academy <strong>of</strong> Science,
Shanghai 200050, China
Abstract
In this paper, we present <strong>the</strong> results <strong>of</strong> <strong>the</strong> calculation and design for waveguide and <strong>the</strong> dependence on <strong>the</strong>
performance <strong>of</strong> mid-infrared InAlAs/InGaAs/InP quantum cascade (QC) lasers grown by gas source molecular beam
epitaxy. <strong>The</strong> <strong>dispersion</strong> <strong>of</strong> <strong>refractive</strong> indices are calculated by taking into account <strong>the</strong> contribution <strong>of</strong> <strong>the</strong> free-carrier
concentration absorption. We also report <strong>the</strong> first GSMBE grown InAlAs/InGaAs/InP QC lasers emitting at 5.1 mm,
operated in pulse mode up to 130 K with T 0 <strong>of</strong> 208 K and J 0 <strong>of</strong> 2.6 kA/cm 2 . For <strong>the</strong> QC laser designed with lower
waveguide cladding InP concentration <strong>of</strong> 1 10 18 cm 3 , it does not lase. Presented <strong>the</strong>oretical and experimental results
indicate that <strong>the</strong> free-carrier absorption in both upper InGaAs contact layer and lower InP waveguide cladding does
play an important role in <strong>the</strong> <strong>dispersion</strong> <strong>of</strong> <strong>the</strong> <strong>index</strong> <strong>of</strong> refraction <strong>of</strong> semiconductors at mid–far-infrared wavelengths
and influence <strong>the</strong> performance <strong>of</strong> QC lasers. # 2001 Elsevier Science B.V. All rights reserved.
PACS: 85.60.Bt; 85.30.De; 78.30.Fs
Keywords: A1. Characterization; A3. Molecular beam epitaxy; B1. Phosphides; B3. Infrared devices; B3. Laser diode
1. Introduction
Quantum cascade (QC) lasers based on an
InGaAs/InAlAs/InP material system are new light
sources working in <strong>the</strong> important mid–far-infrared
wavelength atmospheric window, and thus <strong>the</strong> QC
lasers have potential for applications in <strong>the</strong> air
pollution detection and gas remote monitoring
[1–7].
<strong>The</strong> emission <strong>of</strong> photons in InGaAs/InAlAs QC
lasers are based on electron transitions between
*Corresponding author. Tel.: +86-21-62511070-8201; fax:
+86-21-62513510.
E-mail address: azli@itsvr.sim.ac.cn (A.Z. Li).
energy conduction sub bands created by <strong>the</strong>
quantum confinement. Thus <strong>the</strong> emission wavelength
is completely determined by <strong>the</strong> layer
thickness but not <strong>the</strong> band gap <strong>of</strong> <strong>the</strong> materials,
and <strong>the</strong> lasers can work at <strong>the</strong> mid–far-infrared
wavelength region.
As a result <strong>of</strong> <strong>the</strong> mid–far-infrared operation <strong>of</strong>
<strong>the</strong> QC lasers, <strong>the</strong> intraband free-carrier absorption
becomes essential in <strong>the</strong> design and fabrication
<strong>of</strong> <strong>the</strong> QC lasers. As we have pointed out [8,9],
<strong>the</strong> <strong>effect</strong> <strong>of</strong> intraband transitions on <strong>the</strong> optical
properties <strong>of</strong> semiconductors is becoming increasingly
pronounced towards longer wavelength. For
longer wavelength, <strong>the</strong> contribution <strong>of</strong> intraband
free carrier absorption plays an important role
0022-0248/01/$ - see front matter # 2001 Elsevier Science B.V. All rights reserved.
PII: S 0022-0248(01)00712-6

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A.Z. Li et al. / Journal <strong>of</strong> Crystal Growth 227–228 (2001) 313–318
that cannot be ignored. This problem becomes
essentially serious when <strong>the</strong> fundamental mode <strong>of</strong>
<strong>the</strong> waveguide couples with <strong>the</strong> high-loss plasmon
mode (approximates to a wavelength <strong>of</strong> 9.5 mm in
<strong>the</strong> QC laser case) propagating along <strong>the</strong> metal
contact-semiconductor interface.
<strong>The</strong> free carrier absorption in <strong>the</strong> substrate,
which at <strong>the</strong> meantime serves as a lower waveguide
in <strong>the</strong> QC lasers, have been experimentally shown
to be a key feature <strong>of</strong> <strong>the</strong> QC lasers, especially for
those working in <strong>the</strong> longer wavelength. When <strong>the</strong>
operation wavelength gets longer, <strong>the</strong> doping
concentration in <strong>the</strong> substrate should be decreased,
o<strong>the</strong>rwise <strong>the</strong> light output can be hardly obtained.
<strong>The</strong> doping concentration in <strong>the</strong> substrate has
attracted remarkable attention during <strong>the</strong> past few
years only because <strong>the</strong> problem is not essential for
<strong>the</strong> traditional semiconductor lasers, including <strong>the</strong>
heterojunction lasers and <strong>the</strong> multiple quantum
well lasers.
Although it has been experimentally demonstrated
by ano<strong>the</strong>r group and our group that a farinfrared
QC laser deserves a lowerly doped
substrate than a mid-infrared QC laser does [1–
7,10], <strong>the</strong>re is lack <strong>of</strong> systematically <strong>the</strong>oretical
research on <strong>the</strong> <strong>effect</strong> <strong>of</strong> doping concentration in
<strong>the</strong> substrate on <strong>the</strong> behavior <strong>of</strong> QC lasers.
<strong>The</strong>refore, in this paper, we present <strong>the</strong> results <strong>of</strong>
<strong>the</strong> calculation and design for waveguide and <strong>the</strong>
dependence on <strong>the</strong> performance <strong>of</strong> mid-infrared
InAlAs/InGaAs/InP quantum cascade lasers.
2. <strong>The</strong>oretical results
As we know, <strong>the</strong> realization <strong>of</strong> optical confinement
is acquired by <strong>the</strong> <strong>refractive</strong> <strong>index</strong> difference
between <strong>the</strong> waveguide core and <strong>the</strong> waveguide
claddings. Taking into account both <strong>the</strong> interband
transitions and intraband transitions, which have
been shown to yield results in excellent agreement
with experiments, <strong>the</strong> <strong>refractive</strong> <strong>index</strong> nðoÞ can be
expressed as [8,11]
p
nðoÞ ¼
ffiffiffiffiffiffiffiffiffiffiffi
e 1 ðoÞ ðho4E 0 Þ;
ð1aÞ
n
o
e 1 ðoÞ ¼A Fðx o Þþ0:5½E 0 =ðE 0 þ D 0 ÞŠ 3=2 Fðx os Þ
þ B=ðhoÞ 2 þ C;
ð1bÞ
where e 1 ðoÞ is <strong>the</strong> real part <strong>of</strong> <strong>the</strong> complex
dielectric constant, E 0 <strong>the</strong> bandgap, D 0 <strong>the</strong> spinorbit
splitting energy, and <strong>the</strong> subscript os
represents this term is brought by orbit-spin
split-<strong>of</strong>f band absorption, where
h pffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffi
i
FðxÞ ¼x 2 2 1 þ x 1 xHðxÞ
; ð2Þ
x o ¼ ho ; x os ¼ ho
ð3Þ
E 0 E 0 þ D 0
(
1; x41
HðxÞ ¼
ð4Þ
0; x > 1
and A, B, C are parameters corresponding to <strong>the</strong>
strength parameters <strong>of</strong> <strong>the</strong> E 0 =ðE 0 þ D 0 Þ transitions,
<strong>the</strong> free carrier absorption, and <strong>the</strong> nondispersive
contributions <strong>of</strong> <strong>the</strong> higher-lying band
and <strong>the</strong> lattice, respectively.
As being pointed out in Refs. [8,9,11], <strong>the</strong>
parameters A, B can be expressed as
A ¼ 4
3 3=2
3 2 m P 2 ; ð5aÞ
B ¼ h 2 n e e 2
;
ð5bÞ
m * e 0 e r
where m is <strong>the</strong> combined density-<strong>of</strong>-states mass, P
is <strong>the</strong> momentum matrix element, and n e is <strong>the</strong>
electron concentration, e 0 e r is <strong>the</strong> dielectric constant,
and m * is <strong>the</strong> electron <strong>effect</strong>ive mass. From
Eq. (5b) one knows that <strong>the</strong> electron concentration
affects <strong>the</strong> <strong>refractive</strong> <strong>index</strong> especially in <strong>the</strong> heavily
doping case.
Having a narrow bandgap, small <strong>the</strong>rmal
resistivity, and easy to get an Ohmic contact,
GaInAs is generally used as <strong>the</strong> contact layer for
QC lasers. <strong>The</strong> <strong>refractive</strong> <strong>index</strong> versus wavelength
for different materials and for GaInAs doped with
different concentrations are shown in Fig. 1. From
Fig. 1, one knows that for QC lasers working
at l 9 mm, <strong>the</strong> doping concentration n ¼ 1
10 19 cm 3 is required for <strong>the</strong> contact layer. While if
l ¼ 5 mm, <strong>the</strong> doping concentration n 1 10 20
cm 3 is required for <strong>the</strong> contact layer.
Using this kind <strong>of</strong> heavily doped contact layer,
<strong>the</strong> coupling between <strong>the</strong> fundamental mode <strong>of</strong> <strong>the</strong>
waveguide and <strong>the</strong> high-loss plasmon mode
propagating along <strong>the</strong> metal contact-semiconduc-

A.Z. Li et al. / Journal <strong>of</strong> Crystal Growth 227–228 (2001) 313–318 315
tor interface can be <strong>effect</strong>ively suppressed, because
<strong>the</strong> <strong>refractive</strong> <strong>index</strong> in <strong>the</strong> contact layer is so small.
By solving <strong>the</strong> Maxwell’s equation, we are able to
get <strong>the</strong> mode distribution for a QC laser. A
schematic structure showing <strong>the</strong> waveguide <strong>of</strong> a
quantum cascade laser working at l 8:5 mm is
shown in Fig. 2.
<strong>The</strong> free-carrier absorption can be obtained
from <strong>the</strong> permittivity <strong>of</strong> semiconductors,
Fig. 1. (a) <strong>The</strong> <strong>dispersion</strong> <strong>of</strong> <strong>refractive</strong> <strong>index</strong> versus wavelength
for GaInAs, AlInAs, and InP. <strong>The</strong> points in <strong>the</strong> lines are
distinction for different materials. (b) <strong>The</strong> relationship <strong>of</strong>
<strong>refractive</strong> <strong>index</strong> versus wavelength for GaInAs doped with
different concentrations.
a ¼ oe 2
n r c ;
ð6Þ
where e 2 is <strong>the</strong> imaginary part <strong>of</strong> <strong>the</strong> permittivity,
n r is <strong>the</strong> <strong>effect</strong>ive refractivity, c is <strong>the</strong> speed <strong>of</strong> light
in free space.
Let us start with <strong>the</strong> semi-classical <strong>the</strong>ory <strong>of</strong>
dynamic electro-magnetic field. <strong>The</strong> behavior <strong>of</strong> a
particle with mass m, charge e in an electricmagnetic
field E 0 e iot (optical field) with a
resistance g can be described by <strong>the</strong> equation,
m d2 x
dt 2 þ mgdx dt þ mo2 0 x ¼ eE 0e iot ;
ð7Þ
where o 0 is <strong>the</strong> intrinsic frequency <strong>of</strong> <strong>the</strong> particle.
When we are interested in <strong>the</strong> behavior <strong>of</strong> <strong>the</strong> free
carriers around <strong>the</strong> optical frequency, <strong>the</strong> last term
in <strong>the</strong> left side on Eq. (7) can be neglected, with m
being substituted by an <strong>effect</strong>ive mass m * . One
Fig. 2. Schematic structure with InP buffer layer <strong>of</strong> a quantum cascade laser working at l 8:0 mm.

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A.Z. Li et al. / Journal <strong>of</strong> Crystal Growth 227–228 (2001) 313–318
gets
If we are interested in <strong>the</strong> mid–far-infrared
m * d2 x
dt 2 þ m * g dx
wavelength, l 6 mm, we have o 0:5 10 14 Hz,
dt ¼ eE 0e iot :
ð8Þ and ðotÞ 2 2500. Thus we get
Solving this equation, one gets
aðoÞ 4pn 0e 2 1 1
x ¼
eE 0e iot
n
1
r m * c o 2 t
m * o 2 þ iog :
ð9Þ
¼ 4pn 0e 2 1
Under <strong>the</strong> dipole approximation, <strong>the</strong> polarization n r m * c l2 t ;
ð16Þ
is
where l is <strong>the</strong> emission wavelength.
P ¼ n 0 ex;
ð10Þ Eq. (16) shows that in <strong>the</strong> mid–far-infrared
where n 0 is <strong>the</strong> number <strong>of</strong> electrons per unit volume,
i.e. <strong>the</strong> doping concentration in <strong>the</strong> substrate (1) proportional to <strong>the</strong> doping concentration, and
wavelength range, <strong>the</strong> free-carrier absorption is
for <strong>the</strong> problem we are discussing. <strong>The</strong>refore, <strong>the</strong> (2) proportional to <strong>the</strong> square <strong>of</strong> <strong>the</strong> wavelength.
polarization ratio Z can be expressed as:
When <strong>the</strong> emission wavelength in QC lasers gets
P
Z ¼
eE 0 e iot ¼ n 0e 2
longer, <strong>the</strong> doping concentration in <strong>the</strong> substrate
1
m * o 2 ð11Þ (<strong>the</strong> lower waveguide) should be decreased, in
þ iog
order to lower <strong>the</strong> free-carrier absorption; or a
and <strong>the</strong> permittivity
lowly doped buffer layer should be inserted
4pn 0 e 2 1
between <strong>the</strong> substrate and <strong>the</strong> active region.
eðoÞ ¼1 þ 4pZ ¼ 1
m * o 2 þ iog : ð12Þ
We can get <strong>the</strong> real part and <strong>the</strong> imaginary part <strong>of</strong> 3. Experimental results
<strong>the</strong> permittivity:
4pn 0 e 2 1
e 1 ðoÞ ¼1
m * o 2 þ g 2;
ð13aÞ <strong>The</strong> wafers were grown on (0 0 1) oriented S-
doped InP substrates with a VG Semicon V80 H
e 2 ðoÞ ¼ 4pn gas source molecular beam epitaxy system [10,12].
0e 2 g
m * oðo 2 þ g 2 Þ :
ð13bÞ <strong>The</strong> growth temperature was 5208C. During <strong>the</strong>
growth <strong>of</strong> <strong>the</strong> AlInAs/GaInAs heterostructure,
Thus <strong>the</strong> free carrier absorption
<strong>the</strong> Ga shutter switched on at <strong>the</strong> same time when
aðoÞ ¼ oe <strong>the</strong> Al shutter switched <strong>of</strong>f, no interruption has
2ðoÞ
been adopted during <strong>the</strong> growth <strong>of</strong> <strong>the</strong> multiplequantum
wells.
n r c
¼ o
n r c 4pn 0e 2 g
<strong>The</strong> parameters for all samples were preadjusted
with single epilayers and checked by
m * oðo 2 þ g 2 Þ
reflection high-energy electron diffraction
¼ 4pn 0e 2 g
n r m * c o 2 þ g 2:
ð14Þ (RHEED), high-resolution XRD, and high-resolution
simulation based on dynamical <strong>the</strong>ory <strong>of</strong>
XRD. <strong>The</strong> growth rate was determined by optical
<strong>The</strong> resistance factor g can be approximately
microscopy to be 1.50 mm per hour. <strong>The</strong> lattice
obtained by <strong>the</strong> intraband relaxation time (phonon
mismatch between epilayers and substrates was
assisted relaxation time) with <strong>the</strong> relation ship:
controlled to be no more than 8 10 4 .
gt ¼ 1
<strong>The</strong> X-ray diffraction was performed on a
and <strong>the</strong> intraband relaxation time t1 ps. <strong>The</strong>refore
Eq. (14) can be rewritten as
folded Ge (2 2 0) monochromator and a resolution
Philips X’Pert MRD system, which had a four-
aðoÞ ¼ 4pn 0e 2
<strong>of</strong> 12 arcsec. During <strong>the</strong> measurements, <strong>the</strong> signalto-noise
ratio was controlled to be 2 10 4 . <strong>The</strong>
1 1 1
n r m * c o 2 t 1 þ 1=o 2 t 2: ð15Þ optical measurements were performed on a Nicolet

A.Z. Li et al. / Journal <strong>of</strong> Crystal Growth 227–228 (2001) 313–318 317
driven by pulsed current under various heat-sink
temperatures.
Considering <strong>the</strong> free-carrier absorption in <strong>the</strong>
mid–far-infrared wavelength range, we grew different
doping concentration <strong>of</strong> 1 10 17 ,5 10 17
and 1 10 18 cm 3 for InP lower waveguide cladding.
Figs. 3 and 4 show <strong>the</strong> pulse emission spectra
and <strong>the</strong> dependence <strong>of</strong> threshold current density
on <strong>the</strong> operating temperature. For a QC laser
operating at 8.2 mm with lower waveguide cladding
InP concentration <strong>of</strong> 2 10 17 cm 3 , for a QC laser
working at 5.1 mm with lower waveguide cladding
InP concentration <strong>of</strong> 5 10 17 cm 3 . As we previously
demonstrated, our one-step GSMBE
Fig. 3. Pulse emission spectra at several temperatures for a
quantum cascade laser <strong>of</strong> 2 mm length and 12 mm width. (a) For
5 mm QC lasers with lower waveguide cladding InP concentration
<strong>of</strong> 5 10 17 cm 3 . (b) For 8 mm QC lasers with lower
waveguide cladding InP concentration <strong>of</strong> 2 10 17 cm 3 .
760 Fourier transform infrared spectrometer with
an Ar ion (514.5 nm) laser as <strong>the</strong> excitation source
and a liquid-nitrogen cooled InSb detector for
receiving signal.
<strong>The</strong> laser samples were processed into mesa
etched ridge waveguides <strong>of</strong> width 8–14 mm. <strong>The</strong>
lasers were cleaved into 2 mm long bard and <strong>the</strong>
facets were left uncoated. <strong>The</strong> lasers were <strong>the</strong>n
Fig. 4. Experimental plot threshold current density versus
temperature for 2 mm cavity length QC lasers. (a) 5 mm QCL.
(b) 8 mm QCL.

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A.Z. Li et al. / Journal <strong>of</strong> Crystal Growth 227–228 (2001) 313–318
grown QC lasers emitting at 8.16 mm. from Figs. 3
and 4 we can see <strong>the</strong> 8.16 mm lasers operating at
250 K and with a characteristic temperature (T 0 )
<strong>of</strong> 229 K and a J 0 <strong>of</strong> 1.27 KA/cm 2 . In this paper,
we report <strong>the</strong> first one-step GSMBE grown
InAlAs/InGaAs/InP QC lasers emitting at
5.1 mm, operated in pulse mode up to 130 K with
T 0 <strong>of</strong> 208 K and J 0 <strong>of</strong> 2.6 KA/cm 2 . For <strong>the</strong> QC
laser designed with lower waveguide cladding InP
concentration <strong>of</strong> 1 10 18 cm 3 , it does not lase.
4. Conclusion
In summary, we have presented <strong>the</strong>oretical and
experimental results on <strong>the</strong> <strong>effect</strong> <strong>of</strong> <strong>dispersion</strong> <strong>of</strong>
<strong>the</strong> <strong>refractive</strong> <strong>index</strong> on <strong>the</strong> performance <strong>of</strong> midinfrared
quantum cascade lasers. <strong>The</strong> results
indicate that <strong>the</strong> free-carrier absorption in both
upper InGaAs contact layer and lower InP
waveguide cladding does play an important role
in <strong>the</strong> <strong>dispersion</strong> <strong>of</strong> <strong>the</strong> <strong>index</strong> <strong>of</strong> refraction <strong>of</strong>
semiconductors at mid–far-infrared wavelengths
and influence <strong>the</strong> performance <strong>of</strong> QC lasers. It
must <strong>the</strong>refore taken into account in order to
properly interpret <strong>the</strong> <strong>dispersion</strong> <strong>of</strong> <strong>the</strong> <strong>refractive</strong>
<strong>index</strong> over this range <strong>of</strong> wavelengths.
References
[1] J. Faist, F. Capasso, D.L. Sivco, C. Sirtori, A.L.
Hutchinson, A.Y. Cho, Science 264 (1994) 553.
[2] G. Scamarcio, F. Capasso, C. Sirtori, J. Faist, A.L.
Hutchinson, D.L. Sivco, A.Y. Cho, Science 276 (1997)
5313.
[3] J. Faist, F. Capasso, C. Sirtori, D.L. Sivco, A.L.
Hutchinson, A.Y. Cho, Nature 387 (1997) 777.
[4] C. Gmachl, A. Tredicucci, D.L. Sivco, A.L. Hutchinson,
F. Capasso, A.Y. Cho, Science 286 (1999) 749.
[5] G. Scamarcio, F. Capasso, C. Sirtori, D.L. Sivco, A.L.
Hutchinson, A.Y. Cho, Science 276 (1997) 773.
[6] S. Slivken, C. Jelen, A. Rybaltowski, J. Diaz, M. Razeghi,
Appl. Phys. Lett. 71 (1997) 2593.
[7] C. Gmachl, A. Tredicucci, F. Capasso, A.L. Hutchinson,
D.L. Sivco, J.N. Bailargeon, A.Y. Cho, Appl. Phys. Lett.,
72 (1998) 3130.
[8] W.G. Bi, A.Z. Li, J. Appl. Phys. 71 (1992) 2826.
[9] Q.K. Yang, A.Z. Li, Chin. Phys. Lett. 16 (1999) 610.
[10] A.Z. Li, J.X. Chen, Q.K. Yang, C. Lin, Y.C. Ren, J. Cryst.
Growth 201/202 (1999) 901.
[11] S. Adachi, J. Appl. Phys. 53 (1982) 5863.
[12] Q.K. Yang, J.X. Chen, A.Z. Li, J. Crystal Growth 209
(2000) 8.
Acknowledgements
This work is supported in part by <strong>the</strong> Chinese
Academy <strong>of</strong> Sciences under Contract No. KJ951-
B1-706-01.