Research Collection Doctoral Thesis Linear and non-linear spectroscopy of semiconductors using synchrotron infrared Author(s): Friedli, Peter Publication Date: 2013 Permanent Link: https://doi.org/10.3929/ethz-a-009904545 Rights / License: In Copyright - Non-Commercial Use Permitted This page was generated automatically upon download from the ETH Zurich Research Collection . For more information please consult the Terms of use . ETH Library
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Research Collection
Doctoral Thesis
Linear and non-linear spectroscopy of semiconductors usingsynchrotron infrared
This chapter describes optical gain and loss studies of strained and doped germanium
(Ge) layers directly grown on silicon (Si) substrates. This investigation represents the first
31
32 Chapter 3. Direct-Bandgap Gain and Optical Absorption in InGaAs and Germanium
application of the broadband infrared pump-probe (IRPP) spectroscopy using the setup
described in the previous chapter. Here, the IRPP system was utilised to follow direct-
bandgap (DG) gain and optical absorption in InGaAs and Ge as a function of injected
carrier density, doping and strain. InGaAs serves as a reference system benchmarking
the new approach. The mid-IR reflection is used to identify the plasma frequency of the
(photo-excited and/or doping) carriers, and enables the determination of their density. The
mid- and near-IR transmission spectra are used to study the absorption and population
inversion processes. It is shown that although conditions for direct-gap inversion (and
so, optical gain) in Ge can be reached, the optical amplification needed for lasing does
not occur because of the competing effect of valence intraband (VIB) absorption. The
obtained results were published recently in [37] and are discussed and expanded upon in
this chapter.
3.1 Introduction
Nowadays, despite the impressive increase in component density in complementary metal-
oxide-semiconductor (CMOS) circuits, recent increases in computational power are delivered
with increasing difficulty. [38]. Apart from the advances in computational methods, such as
parallelism and smart caching, the main potential for increasing computer performance in
the future lies in the realisation of fast and low-latency on-chip communication methods.
Currently, this so-called inter-core communication is done using electrical signals carried
over sophisticated wire-grids connecting the cores. This type of transfer consumes large
parts of the total processor electrical and cooling power and is likely to increase as future
consumer level multicore architectures go beyond their current multi-core configurations.
These drawbacks can be largely avoided by the use of an all-optical inter-core data transfer
technology: it runs on low-power, provides easy multiplexing of signals at different frequen-
cies, is not susceptible to electronic noise, and removes the need for insulation between
wires as present in the electronic grid setup [39].
This is where the field of Si-photonics comes into play. Today, with the sophisticated tech-
3.1. Introduction 33
nology of Si, two out of the three basic elements for Si-photonics have reached maturity.
Ge-on-Si photodetectors provide high responsivity throughout the visible and near-IR spec-
tral range and offer high bandwidths [40]. Si photo-modulators based on Mach-Zehnder
interferometers1 or based on electro-absorption effects in Ge/SiGe multi quantum wells
provide high performance [41]. The main missing piece for a complete assemblage, however,
is an efficient, cost-effective light-emitter. Ideally, this emitter should be monolithically
integrable with Si to enable high component density, low-costs, and full scalability [42].
This difficulty of light emission arrises from a fundamental characteristics of Si - it is an
indirect-bandgap material, as illustrated in Fig. 3.1(b). In contrast to DG materials, see
Fig. 3.1(a), radiative recombination of carriers is a low probability process, because it
requires an interaction with a phonon to balance the momentum mismatch between the
holes populating the valence bands (Γ-valley) and the electrons populating the L-valley
conduction band. In Si, the main source of recombination is of the non-radiative type,
such as Auger recombination and free-carrier absorption, which is introduced later in this
chapter.
Different approaches may be employed to achieve coherent light emission directly in Si or in
Si compatible material platforms [39]: Raman-lasing in Si, defect implantation and the rare-
earth-doping of Si, zone folding in SiGe superlattices, the epitaxial growth of laser materials
onto Si, and Si-based quantum cascade emitters. The different approaches are discussed in
the following. The current state-of-the-art technique, however, is a hybrid Si laser consisting
of III-V laser materials wafer-bonded onto Si. This represents an effective workaround
solution. Drawbacks concern the high costs, complicated process flows and the lack of an
efficient thermal management at the wafer-interfaces [43].
Stimulated Raman scattering (RS)2, benefiting from the high RS cross-section in Si, can
provide optical gain and lasing in Si as demonstrated first in [44,45]. Following the first
demonstration in pulsed operation, the Si Raman laser was shown to run also in continuous-
1Mach-Zender interferometers split light into two channels and recombine the signals with a phase-shift to
either achieve constructive (1) or destructive (0) interference.2Raman scattering denotes the inelastic scattering of a photon, hence the subsequent emission of a photon
occurs at a slightly different energy
34 Chapter 3. Direct-Bandgap Gain and Optical Absorption in InGaAs and Germanium
EΓ#EL#
Energy# Γ*valley#
L*valley#
Wave#vector#HH#
LH#
SO#
(a) The momentum of electrons and holes
match in direct-bandgap semiconductors,
such as InP and InGaAs. Efficient light
absorption and emission is possible.
EΓ# EL#
Energy#
L*valley#
Wave#vector#HH#
LH#
SO#
Γ*valley#
(b) In indirect-bandgap semiconductors,
such as Si or Ge, electrons and hole popu-
lations arise at different momentums. Non-
radiative recombination prevails emission
of photons.
Figure 3.1: Typical energy band diagrams for two types of semiconductors, showing the conduction
(blue) and valence (green) bands. The latter consists of light-hole (LH), heavy-hole (HH)
and Split-off (SO) bands.
wave (CW) [46]. This became possible because of the significant reduction of free-carrier
absorption by applying a strong electrical field [46]. However, this principle of light
generation requires an external pump, which may conflict with the need for miniaturisation
and simplicity.
Ng et al. [47] were able to show light-emission at room-temperature from a Si-based diode,
which had been implanted by boron to form a p-n junction. The implantation also gives
rise to dislocations, which modify the local band structure and spatially confine the charge
carriers. In this way, the carrier diffusion, and hence, the non-radiative recombination
path of carriers at point defects in Si is suppressed, and the fraction of radiative emission
increases. Implantation of Si with rare-earths, such as erbium, reveals luminescence at the
telecommunications-wavelength 1.55µm due to transitions within the erbium shown by
Ennen et al. under optical [48] and electrical [49] carrier injection. Emission from such
diodes was first achieved at room temperature by Zheng et al. [50]. Erbium-doped silicon
3.1. Introduction 35
microdisk lasers were demonstrated by Polman et al. [51] and Kippenberg [52] under optical
pumping. Most promising towards an electrically pumped diode laser seems the use of silicon
nanocrystals as sensitisers in a silicon-rich oxide [53,54].
The epitaxial growth on Si of chemically compatible group IV elements such as germanium
(Ge) and tin (Sn) also represents a promising alternative to fabricate a Si based laser
source. One of the approaches investigated in the 1980s was to create DG material from
superlattices obtained by alternating layers of Si and Ge deposited onto a Si substrate [55].
Such Si/Ge monolayer sequences were predicted to feature interband transitions with high
oscillator strengths, emerging from the bulk electron states at the conduction band-minima
at the L- or X-point of Ge and Si, folded back to the centre (Γ-valley) of the Brillouin zone.
The effect of folding has been demonstrated in photo reflection [56]. This concept was
recently revisited by Avezac et al. [57]. Their genetic algorithm predicts SL designs with
strongly enhanced (50x) DG matrix elements. However, no experimental verification is yet
available.
Reinforced by the steady advances in epitaxial growth of Si and Ge by molecular beam
epitaxy (MBE) [58], the quantum cascade laser (QCL) scheme [59] has become a viable
option for making a Si-based laser. The QCL approach indeed resolves the drawback of
the indirect bandgap as it relays on unipolar charge transport without any recombination
across the bandgap. However, it was found that for p-doped structures designed for emission
in the mid-IR (> 100 meV), the lifetime of the relevant states, the current transport, and
waveguide loss parameters are unfavourable compared to the traditional InAs or GaAs
based systems, due to the strong mixing between the diverse states in the valence band,
the low carrier mobility and strong free carrier absorption, respectively [60]. Si-based QCL
structures for the THz were shown to exhibit long lifetimes (> 20 ps) [61], but transport and
waveguide concerns may be severe as well. In fact, neither gain nor lasing structures have
been reported, in spite of intense research conducted in the past years by several groups
inclusive ours.
A plausible and natural approach, however, would be to make use of the DG transition at
an energy of 0.8 eV in Ge, which is close to the C-band telecommunications wavelength of
36 Chapter 3. Direct-Bandgap Gain and Optical Absorption in InGaAs and Germanium
1.55µm. Due to the recent report of optical gain in Ge mesas by Liu et al. [62], scientific
interest has been raised again for the feasibility of a Ge-on-Si laser. They previously modelled
that by applying tensile strain and simultaneous n-type doping, population inversion in the
direct bandgap of Ge is achievable [63]. Tensile strain decreases the initial energy separation
of 136 meV between the L- and Γ-valleys. At the same time, due to the strong n-type doping
(> 6 · 1019 cm−3), the L-valley is pre-populated by electrons. With strong optical pumping,
efficient injection of electrons in the DG Γ-valley then becomes possible. In this way,
optically [64] and electrically [65] pumped lasing was reported in Ge. Those results seem
ambiguous, however, as will be explained in the results section of this chapter. Optical gain
in biaxially tensile-strained (0.4%) and n-type doped (2 · 1019 cm−3) germanium photonic
wires was reported independently [66]. The direct bandgap in Ge, however, is predicted to
be achieved with very large strain values only, which are 4.2% in the uniaxial [67] and ∼2%
in the biaxial [63] case, which is both technologically challenging. Micromachining-based
technologies to achieve such large values are currently being developed, and uniaxial strain of
0.98% and biaxial strain of 0.82% were recently reported [68].
Nevertheless, gain and optical absorption features in Ge have not been fully characterised
experimentally, despite a number of theoretical models and approximations being put
forward. The reported doping levels in the optically pumped Ge structures (1 · 1019 cm−3) as
well as the pumping-intensities were much lower than predicted by the model, demonstrating
the need for improved understanding of the mechanisms playing a role in the emission, and
experimental investigation of the relevant parameters, such as the balance between gain
and loss. This is addressed below.
3.2 Experimental Overview
The IRPP system is uniquely suited for studying the dynamics of gain and loss in optically
pumped samples. Its large spectral range gives access to and correlates the physics of the
carrier population inversion (at ∼1 eV), the free-carrier absorption of the injected carriers
(at ∼100 meV), and also the broadband absorption effects between 0.1 - 1 eV. In this way, as
3.2. Experimental Overview 37
shown in Fig. 2.14, physical phenomena such as Auger recombination, carrier diffusion, and
electron/hole recombination can be investigated simultaneously, under precisely the same
conditions of e.g. pumping, doping and strain.
3.2.1 Investigated Samples
Using low energy plasma-enhanced chemical vapour deposition, 1 to 2µm thick Ge layers
was grown on Si substrates [69]. This series of samples were n-type doped using phospho-
rous, and strain enhanced by rapid thermal annealing [70].
The In0.53Ga0.47As samples were grown on [001]-InP substrates by molecular beam epitaxy
(MBE), with elemental Si supplied by a effusion cell providing n-type doping [71]. Doping
densities from the not-intentionally-doped (NID) limit of 2.1 · 1015 cm−3 to 2.1 · 1019 cm−3
were prepared. Material composition and doping were determined by high-resolution x-
ray diffraction (XRD), Hall and secondary ion mass spectrometry (SIMS) measurements,
respectively [71].
A summary of the investigated samples is given in Table 3.1.
Table 3.1: Characteristics of the Ge and InGaAs samples. For not-intentionally-doped (NID) samples,
the maximum background doping is < 1 ·1015 cm−3 for Ge and < 1 ·1016 cm−3 for InGaAs.
For not-intentionally-strained (NIS) samples, the biaxial tensile strain is < 0.05%.
Sample Thickness (µm) Doping ( cm−3) Strain Growth Number
Ge#1 1.0 NID NIS 56426
Ge#2 1.0 2.5 · 1019 cm−3 NIS 56429
Ge#3 1.9 NID 0.25% 8300
InGaAs#1 4.1 NID NIS Ep939
InGaAs#2 1.0 2.1 · 1019 cm−3 NIS Ep962
InGaAs#2 1.0 5.3 · 1018 cm−3 NIS Ep963
38 Chapter 3. Direct-Bandgap Gain and Optical Absorption in InGaAs and Germanium
3.3 The Case of a III-V Laser Material (InGaAs)
3.3.1 Direct-Gap Band Structure
When InGaAs is pumped optically, electrons are vertically excited across the DG, which
results in a non-equilibrium distribution of electrons in the Γ-valley of the conduction band
(CB), and a distribution of the holes in the valence bands (VB). The co-existence of electrons
and holes in the Γ-valley leads to a population inversion that changes the DG absorption
αDG into DG gain (i.e. αDG < 0), see Fig. 3.2.
HH"LH"
SO"
EFe"
EFh"
Γ"
αDG"<"0"αDG">"0"
αVIB"
EDG"
0"
Figure 3.2: Direct-bandgap gain or absorption is achieved in InGaAs, depending on the photon
transition energy. Intraband absorption in the valence band is present at the same time.
This is described by the Bernard-Duraffourg (BD) condition [72], which states that the quasi-
Fermi energy levels of the conduction and valence band have to be separated by more than
the direct-bandgap energy EDG in order to achieve optical gain, and reads
EFc − EF v > EDG, (3.1)
where EFc and EF v are the quasi-Fermi energy levels for the two bands. The BD condition
is discussed in detail in Sec. 3.5. Gain is achieved for all transition energies E in the range
EDG < E < EFc − EF v.
In addition to recombination with electrons (a bipolar process), the holes may undergo
unipolar vertical transitions between the light-hole (LH), heavy-hole (HH) and split-off
3.3. The Case of a III-V Laser Material (InGaAs) 39
(SO) bands of the VB driven by photo-excitation. The high absorption cross-section of
these unipolar, momentum-conserving transitions results in a strong broadband valence
intraband (VIB) absorption, that can be comparable in strength to DG absorption. The
measurement of the transmission through such an optically-excited sample reveals the
contributions from the DG absorption and/or gain, and the hole-induced broadband MIR
absorption. Finally, the charge carriers (induced by doping and/or photo-injection) give rise
to a plasma, whose frequency depends on the density and effective masses of the carriers.
The plasma response of the carriers is described by the complex dielectric permittivity of
the material, and manifests itself as a well-defined minimum in the reflection spectra, at the
characteristic plasma frequency ωp.
3.3.2 Transmission Measurement
Figure Fig. 3.3(a) shows the normalised broadband transmission spectra of the InGaAs#1
sample using the parabolic-mirror setup (c.f. Fig. 2.8(a)), when pump and probe pulses
are in full overlap. The unpumped spectrum shows a 725 meV DG absorption edge, the
expected Fresnel losses of ≈57%, and thin-film interference effects with an oscillation period
matching the InGaAs layer thickness. The pumped spectrum for a peak pump intensity of
150 MW cm−2 using a 200µm spot is considerably richer in terms of features, and exhibits
(i) a strong absorption band below 300 meV, (ii) a constant broadband absorption between
200 - 650 meV, and (iii) a recovery of the reflection-limited transmission above 650 meV,
with only a small net gain at 740 meV.
However, this gain is more clearly visible in Fig. 3.3(b), where the experiment is repeated
using a micro-focus setup (as shown in Fig. 2.8(b)). Here, the smaller illuminated sample
area (35µm pump) minimises carrier recombination by reducing the possibility of stimulated
emission. It was this effect that resulted in early gain-clamping in the material illuminated
by the bigger pump spot in the parabolic-mirror setup.
The reduced transmission below 300 meV originates from both the Drude-type free carrier
absorption (FCA) of the optically injected carriers and the increase in refractive index, as
40 Chapter 3. Direct-Bandgap Gain and Optical Absorption in InGaAs and Germanium
100
80
60
40
20
0
Tran
smiss
ion
(%)
1.00.80.60.40.2Energy (eV)
pum
p-in
d. a
bsor
ptio
n
FCA Absbandgap
renormalization
InGaAs#1 Unpumped Pumped
(a) With the parabolic mirror setup (pump spot of
200µm), only a marginal gain is achieved upon
pumping with an intensity of 150 MW cm−2.
100
80
60
40
20
0
Tran
smiss
ion
(%)
1.11.00.90.80.70.60.5Energy (eV)
tota
l gai
n
pum
p-in
d. a
bsor
ptio
nne
t ga
in
InGaAs#1 unpumped
-50ps 250ps 0ps 450ps 100ps 950ps
(b) With the micro-focus setup (pump spot of 35µm),
clear optical amplification exceeding the reflection-
limited transmission is achieved.
Figure 3.3: Normalised IR transmission spectra for sample InGaAs#1.
discussed below.
The constant broadband absorption spanning between 300 meV up to higher energies is
consistent with VIB absorption [73]. The broadness of VIB absorption originates from the
superposition of the three LH-HH, LH-SO, and HH-SO transitions, and is further enhanced
by impurity and carrier-carrier scattering [74]. As shown in [73] [74], this VIB absorption
extends up to the DG energy.
The increase in transmission at energies above 650 meV is due to the growing DG inver-
sion, which is superimposed on the VIB absorption. The onset of this increase indicates a
bandgap renormalisation of up to 75 meV with respect to the unpumped absorption edge,
as shown in Fig. 3.3(a). Renormalisation effects of this order have been measured in
doped InGaAs [75] and optically excited materials [76]. When the DG gain exceeds the
pump-induced VIB absorption, the InGaAs material provides optical amplification, as shown
in Fig. 3.3(b). Here, the transmission rises up to ≈90% when pump and probe overlap
maximally. If the optical amplification is high enough to match mirror loses, then the system
lases.
3.3. The Case of a III-V Laser Material (InGaAs) 41
3.3.3 Determination of Carrier Density from Reflection Measurement
Free carriers of a given density and effective mass in a semiconductor have a well-defined
plasma frequency ωP , given by the following expression
EP = ~ωP = ~p
NT e2/mPε, (3.2)
as described in [77]. Here, NT is the total number of carriers, mP is the reduced mass (which
is taken as the weighted average of the masses of electron and different types of holes) and e
is the elementary charge. This follows from a simple Drude model of the motion of electrons
in the semiconductor material. The plasma response of the carriers enters into the (complex)
dielectric permittivity εr of the material
εr(~ω) = ε∞ ·
1−τS(ωP)2
τSω2+ iω
(3.3)
where ε∞ is the DC permittivity of the material and τS is the scattering time [78]. The
permittivity undergoes a phase change close to ωP . Hence, the optical properties of the
material differ above and below ωP . While above ωP , the semiconductor acts as a dielectric
material, below ωP its behaviour follows that of a metallic conductor. The transition from
one state to the other is accompanied by a minimum in the reflection spectrum, which is
used to measure and identify ωP .
The reflectivity R of a semiconductor surface exposed to air (vacuum) is given by
R=(n− 1)2+κ2
(n+ 1)2+κ2 , (3.4)
where n is the real index of refraction and κ is the extinction coefficient [77]. These parame-
ters are derived from the complex relative permittivity ε = ε1+ iε2 by
n=
r
1
2
p
ε21+ ε
22+ ε1
κ=
r
1
2
p
ε21+ ε
22− ε1
,
(3.5)
where ε1 = Re[ε] and ε1 = Im[ε] are the real and imaginary part of ε, respectively.
The ratio of pumped and unpumped reflection is then defined as differential reflection. A
characteristic blue-shift of the reflection minima with increasing carrier density is shown
42 Chapter 3. Direct-Bandgap Gain and Optical Absorption in InGaAs and Germanium
in Fig. 3.4 for an InGaAs bulk material for different carrier densities. Here, ε∞ = 11.56,
mp = me = 0.041 ·m0, and tS= 20 fs were used.
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
Pum
ped
/ Un
pum
ped
1.41.21.00.80.60.40.20.0Energy [eV]
Carrier densities [ cm-3 ] 1e19 2e19 3e19
4e19 5e19 1e20 2e20 4e20Energy Blueshift
Figure 3.4: The ratio of pumped and unpumped reflection (differential reflection) for bulk InGaAs
shows a characteristic blue-shift of the reflection minima with increasing carrier density.
In the case of thin-film layers grown on substrates, however, multiple-path reflections and co-
herent superposition have to be taken into account, c.f. Fig. 3.5.
Thin%film))))))))
Substrate)rSA)
tLS)
rLS)
rAL)
Coherent)Superposi7on) tLA)
tSL)tLA)×A)Probe)
Figure 3.5: Sketch of the multiple-path reflection. In the thin-film layer, coherent interaction takes
place, while the reflection on the back of the substrate is added incoherently.
The total reflectivity Rtotal is modelled by the FP model
Rtotal = A ·
tAL t LS rSA tSL t LA
2+
rAL +tAL rLS t LA · exp (2iδL l)1− rLA rLS · exp (2iδL l)
2
, (3.6)
where A accounts for the spread, i.e. divergence, of the beam on its path through the
substrate, t x y and rx y are the amplitude transmission and reflection coefficients at the
different interfaces (Air, Layer for the active material, and Substrate), respectively, and
3.3. The Case of a III-V Laser Material (InGaAs) 43
δL takes into account the attenuation and phase shift during the path length l in the
active layer. In summary, the first part represents the non-coherent summation fully down
to the substrate-air interface and back, and the second part accounts for the multiple-
path coherent interaction in the active layer. The propagation constant δL is defined
as
δL =ωpεr
c. (3.7)
Fig. 3.6(a) shows an unpumped and pumped normalised broadband reflection spectra
from the n-doped InGaAs#2 thin-film layer grown in InP (n = 3.1). The energy of the
reflection minima clearly blue shifts upon increasing pumping. The ratio of pumped and
unpumped reflection spectra at different pumping intensities is shown in Fig. 3.6(b) (dots).
The double feature in the ratio RP/RU originates from the two ωP conditions, where the
first is the unpumped case and depends only on the dopant density ND , while the second
depends on the sum ND + NP . This series of reflection measurements for each sample
allows for the fit of the plasma-frequencies for the doped (ND) and photoexcited NP carrier
densities as the only two fitting parameters. The model yields excellent fit to the measured
ratio, and later on enables the correlation of carrier density with the determined gain and
losses.
44 Chapter 3. Direct-Bandgap Gain and Optical Absorption in InGaAs and Germanium
1.0
0.8
0.6
0.4
0.2
0.0
Norm
aliz
ed R
eflec
tanc
e
0.400.350.300.250.200.150.10Energy [eV]
InGaAs#2 Unpumped Pumped
(a) An example of the absolute reflection spectra that
are used to generate the ratio spectra.
3.0
2.5
2.0
1.5
1.0
0.5
0.0
R p /
Ru
0.450.400.350.300.250.200.150.10Energy [eV]
InGaAs#2Pump Intensity ( MW/cm2 )
3 13 63 284 Drude Reflection Model
(b) The differential reflection spectra are fitted with
a Drude-model.
Figure 3.6: The IR reflection spectra, here for the InGaAs#2 sample, allow for the determination of
the optically-injected carrier densities.
3.4 Optically Pumped Germanium
3.4.1 Indirect-Gap Band Structure
Contrary to the DG III-V material as described in the previous section, Ge is of indirect-
bandgap (IG) type. This is illustrated in Fig. 3.7. The relevant section of the conduction-band
(CB, red) consists of two distinct minima valleys, which are the Γ- and the L-valley. The
structure of the valence band is analogous to that of InGaAs. N-type doping and tensile
strain act to bring the quasi-Fermi energy level of the CB closer to the minimum of the
Γ-valley.
The physics of direct-gap inversion and VIB absorption, however, is for Ge very similar
as for InGaAs. What differs, is that optical injection of carriers does not automatically
lead to inversion, because the L-valley in the CB has to be filled first. The high (2 ·
1020 cm−3) injected densities used to generate a population inversion at the direct-gap
simultaneously provide a proportionately high density of holes, creating a strong competing
VIB absorption.
3.4. Optically Pumped Germanium 45
HH"LH"SO"
EIG"
EFe"
EFh"
Γ"
αDG"<"0"αDG">"0"
αVIB"
L"
EDG"
0"
Figure 3.7: Upon pumping, the lower L-valley is filled first, which does not lead to population
inversion at the direct-bandgap. Only for very strong pumping, the Γ-valley is being
populated. However, also strong Intraband absorption in the valence band is then present.
3.4.2 Normal-Incidence Measurements
Normal-incidence (NI) broadband transmission and reflection spectra of sample Ge#1 at a
pump intensity of approximately 1 GW cm−2, as published in [22], are shown in Fig. 3.8(a)
and Fig. 3.8(b), respectively. To access the full bandwidth, two detectors are used and shown
in combination.
The pump-induced VIB absorption extends linearly from 350 meV up to the DG at 775 meV.
The strength of the VIB absorption is about 3500 cm−1. Above 830 meV, bleaching of the
DG absorption builds up, and thus the normalised transmission starts to exceed 100%.
Obviously, such a high VIB absorption suppresses any population inversion related gain
and therefore no optical amplification is observed. Due to the NI configuration, thin-film
interference effects create strong FP oscillations with a period of 150 meV. These oscillations
make the identification of a gain feature difficult and ambiguous, and motivate the move to
a Brewster-angle setup, described below.
The analysis of the reflection spectra follows Eq. 3.6. The injected carrier density is deter-
mined to be NP =2.5 ·1020 cm−3. The model gives a good fit of the measured data, as shown
in Fig. 3.8(b).
46 Chapter 3. Direct-Bandgap Gain and Optical Absorption in InGaAs and Germanium
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
Nor
mal
ized
Tra
nsm
issi
on
0.90.80.70.60.50.40.30.2
Energy [eV]
Vigo-MCT Teledyne-Judson InSb
(a) Normalised transmission spectrum.
2.0
1.5
1.0
0.5
0.0
Nor
mal
ized
Ref
lect
ion
0.90.80.70.60.50.40.30.20.1
Energy [eV]
Measurement Vigo-MCT Teledyne-Judson InSb
Model
(b) Normalised reflection spectrum.
Figure 3.8: Broadband transmission- and reflection-spectra of sample Ge#1. The spectra are com-
bined of data from the measurements with two detectors with different spectral coverage.
The transmission shows the strong VIB absorption, which extends from 350 meV up to
the DG [22].
3.4.3 Brewster-Angle Measurements
For an accurate determination of DG gain and absorption, samples Ge#1 to #3 are measured
in Brewster geometry. Here, the p-polarised light, i.e. the light whose polarisation lies in
the plane defined by the surface normal of the interface and the light propagation direction,
is not reflected. Hence, the angle between light propagation direction and the sample is
optimised to suppress reflection from the second interface, i.e. the boundary between the
thin-film and the substrate. Such a measurement for the case of sample Ge#1 is shown
in Fig. 3.9. The flat part of the unpumped spectrum below 740 meV is normalised to
100% transmission. As clearly evident, FP oscillations are efficiently removed from the
measurement.
The spectra differ significantly from the above shown InGaAs ones. First, the increase
of the transmission above the reflection-limited transmission during pumping is not even
achieved for maximum pump-probe overlap (∆ t = 0 ps). This means that no net gain,
i.e. optical amplification, is present in the structures. This can be directly attributed to
strong pump-induced VIB absorption. In fact, the cross-sections for the VIB absorption in
3.5. Absorption and Gain 47
100
90
80
70
60
50
40
Tran
smiss
ion
[%]
1.00.90.80.70.60.5Energy [eV]
Ge#1Measurement
P U
-150ps
-100ps
-50ps
0ps
Figure 3.9: Normalised IR transmission spectra for sample Ge#1 at an intensity of 285 MW cm−2
measured under Brewster geometry. The time-dependence reflects the overlap of optical
pump and probe, which results in measurements at different effective carrier densities.
[37]
Ge and InGaAs are practically identical, given the similar effective masses of the holes in
both systems. However, the absorption in the Ge samples is much stronger, because they are
examined under conditions of higher electron/hole densities needed to establish inversion of
the carriers at the Γ-point. In contrast, in photoexcited InGaAs, the gain from the inversion
rises faster than the induced VIB absorption, and optical amplification is achieved readily.
In Ge, the VIB absorption loss always exceeds the gain that can be generated from the
inversion.
3.5 Absorption and Gain
3.5.1 Model
For a quantitative assessment of the absorption and the gain, the total absorption α(ω)
needs to be modelled:
α(~ω) = αDG(~ω)
fv(~ω)− fc(~ω)
, (3.8)
48 Chapter 3. Direct-Bandgap Gain and Optical Absorption in InGaAs and Germanium
where ~ω is the energy of the photon, fv and fc are the Fermi-Dirac (FD) distributions for
valence band (VB) and conduction band (CB), respectively. The optical absorption in the
parabolic approximation of DG transitions reads
αDG(~ω) = Kabs
p
~ω− EDG, (3.9)
where Kabs is defined as
Kabs =q2 x2
vc
λ0ε0hnop
2mr
h
3/2
, (3.10)
where q is the electric charge, xvc is the optical matrix element, and nop is the refractive
index [79].
As a direct consequence, optical gain, which corresponds to negative optical absorption, is
achieved when
fc(~ω)− fv(~ω)> 0. (3.11)
The FD distributions are defined as
fc(~ω) =1
1+ exp
(EC(~ω)− EFc)/kB T
fv(~ω) =1
1+ exp
(EV (~ω)− EF v)/kB T
(3.12)
where EFc and EF v are the quasi-Fermi energy levels for the conduction and valence
band, respectively. They are calculated from the number of carriers in a fermi-sphere
as
EFc(N) =~2
2me(3π2Ne)
2/3
EF v(N) =~2
2mh(3π2Nh)
2/3,
(3.13)
where the factor 2 arises from spin degeneracy. The number of carriers is Ne = NP + ND for
electrons and Nh = NP for holes, where NP is pumping and ND doping density, and me and
mh are the effective masses of electrons and holes, respectively. The latter is a reduced mass
of the light- and heavy-hole masses, weighted under the parabolic band approximation [80],
which reads
mh =m3/2
HH +m3/2LH
m1/2HH +m1/2
LH
. (3.14)
3.5. Absorption and Gain 49
The position of the energy levels EC(~ω) and EV (~ω) are defined as
Ec(~ω) = EDG +mr
mc(~ω− EDG)
Ev(~ω) =−mr
mv(~ω− EDG).
(3.15)
The reduced effective mass mr is calculated as
mr =
1
me+
1
mh
−1
, (3.16)
It follows, that Eq. 3.11 can also be expressed in terms of the position of the quasi-Fermi
energy levels, i.e.
EFc − EF v > EDG, (3.17)
where EFc and EF v are the quasi-Fermi energy levels for the two bands. This is the earlier
introduced Bernard-Duraffourg condition [72].
3.5.2 Measurement Fit
For the determination of the loss and gain created at the DG transition, the transmission
spectra shown in Fig. 3.3(b) and Fig. 3.9 are fitted to Eq. 3.8. The fitted parameters are
sample temperature T , direct-gap energy EDG, VIB absorption, and the variation of electron
mass m∗ with carrier density N .
Like in the case of the Drude-model for the reflection spectra, the individual reflection coeffi-
cients R at the interfaces are derived from the complex permittivity. Thin-film interference,
i.e. FP oscillations, are accounted for as well. The elevated temperature is deduced from
the absorption edge of the transmission spectra with an unpumped probe pulse during
the pump/probe measurement. It is found to be ∼ 400 K and is subsequently used in the
Fermi-Dirac distribution. The VIB absorption in Ge is found to depend linearly on ~ω. In
the case of InGaAs (with much lower carrier densities), it appears spectrally flat, consistent
with the behaviour observed in low intensity pumped Ge samples. It is the stronger Auger
recombination in InGaAs, which hampers the achievement of similarly high injected densities
50 Chapter 3. Direct-Bandgap Gain and Optical Absorption in InGaAs and Germanium
as in the strongly pumped Ge. The VIB contribution to the spectra is fitted by looking at the
low-energy end of the spectrum.
The measured InGaAs#1 transmission (as introduced in Fig. 3.3(b)) and fit are shown in
the left panel of Fig. 3.10.
10000
8000
6000
4000
2000
0
-2000
Abso
rptio
n Co
effic
ient
[cm
-1]
1.00.80.60.4Energy [eV]
InGaAs#1, Δτ = 0ps Optical Amplification Direct-Gap Gain
Unpumped αDG (U)
Pumped αtotal (P) αVIB (P) αDG (P)
100
80
60
40
20
0
Tran
smiss
ion
[%]
0.5 0.6 0.7 0.8 0.9 1.0 1.1Energy [eV]
Meas. P U
Model P U
Figure 3.10: Left Panel: The JDOS model yields an excellent fit to the transmission spectra of sample
InGaAs#1. Right panel: The differential direct-gap (DG) gain is highlighted in yellow.
Due to the much lower pump-induced VIB absorption, optical amplification is achieved
(highlighted in orange) [37].
The unpumped and pumped transmission are well reproduced both below and above the
direct-gap energy. The FP oscillations are well matched with the sample thickness of 4µm.
In the right panel of Fig. 3.10, the contributions to the overall gain and absorption curve
are sketched individually. The VIB absorption is modelled as constant absorption with a
value of αV IB =1000 cm−1. The direct-gap gain −αDG reaches approximately 1750 cm−1 at
0.85 eV. The maximum net gain is the given by the sum of αV IB and −αDG to approximately
750 cm−1. The ranges of optical amplification and gain are highlighted as by orange and
yellow colors, respectively.
The transmission of sample Ge#1 (as introduced in Fig. 3.9) and fit are shown in the
left panel of Fig. 3.11. The conversion in the centre panel to absorption coefficient re-
veals a maximum value of up to 10000 cm−1, due to VIB absorption. Note that this value
exceeds even the DG absorption, showing that VIB is a significant process that must be
3.5. Absorption and Gain 51
accounted for in any full treatment of optical gain and loss analysis. As evident, inver-
sion, and hence, also gain, is reached in the direct-gap transition, which is highlighted
in yellow. However, due to the strong pump-induced VIB absorption there is no optical
amplification.
10000
8000
6000
4000
2000
0
-2000
Abso
rptio
n Co
effic
ient
[cm
-1]
1.00.80.6Energy [eV]
Meas. Model αDG (U) αVIB (P) αDG (P)
Ge#1, Δτ = 0ps10000
8000
6000
4000
2000
0
-2000
Abso
rptio
n Co
effic
ient
[cm
-1]
1.00.90.80.70.60.5Energy [eV]
Meas. P U
Model P U
-150ps
-100ps
-50ps
0ps100
90
80
70
60
50
40
Tran
smiss
ion
[%]
1.00.90.80.70.60.5Energy [eV]
Ge#1Meas.
P U
Model P U
-150ps
-100ps
-50ps
0ps
Figure 3.11: Left Panel: The JDOS model yields an excellent fit to the transmission spectra of sample
Ge#1. Center Panel: The conversion yields absorption coefficients up to 10000 cm−1.
Right panel: The differential direct-gap (DG) gain is highlighted in yellow. However,
due to the strong pump-induced VIB absorption there is no optical amplification [37].
3.5.3 Discussion
The relation between pumping intensity and photoexcited carrier density is given in Fig. 3.12.
The vertical offset in carrier densities between the three InGaAs samples is due to the different
intrinsic doping levels. The measured non-linear behaviour between photoexcited carrier
density and pumping intensity is due to Auger recombination and is described further in
detail in [37]. The considerably higher carrier density in Ge compared to InGaAs for a
given pumping level, is due to the significantly different Auger recombination parameters,
which are rAug,Ge = 1 · 10−30 cm6 s−1 for Ge and rAug,InGaAs = 1 · 10−28 cm6 s−1 for InGaAs,
respectively [81].
The measured VIB absorption can be expressed as a function of determined carrier densities,
see Fig. 3.13. The observed VIB absorption is well described by a linear absorption cross-
52 Chapter 3. Direct-Bandgap Gain and Optical Absorption in InGaAs and Germanium
1.5x1020
1.0
0.5
0.0Tota
l Car
rier D
ensit
y N T
[ c
m-3
]
150100500Pumping Intensity [MW/cm2]
InGaAs#1 InGaAs#2 InGaAs#3 Ge#1 Ge#2
Figure 3.12: The determined non-linear relation between photoexcited carrier density and pumping
intensity is due to Auger recombination. Partly shown in [37].
section model
αV IB(EDG) = σeNe +σhNh, (3.18)
where σe and σh are the cross-section of the electrons and holes, respectively. In the case
of the undoped samples, the electron and hole densities correspond to the photoexcited
carrier density, i.e. Ne = Nh = NP . For the n-type doped samples, the electron density
has to be taken as the total carrier density NT . From this figure, it is clear that σh σe,
because the absorption scales linearly with NP and not NT . Fitting to the experimental data
gives σh = 3.8 · 10−3 nm2 and σh/σe > 10, which is in good agreement to values reported
elsewhere [82]. The hole absorption cross-section is much higher, because it is related to
vertical transitions between the valence subbands, while this is not the case for the electrons.
As can be seen in Fig. 3.13, the absorption cross-section in the case of strained Ge is slightly
reduced. It should be noted that the higher absorption observed in the Ge samples is not
due to any inherently higher absorption cross-section in the Ge, but rather is due to the far
higher hole density that is reached in these samples, as they are optically pumped to the
high levels needed to reach the inversion condition. This is an unwanted consequence of the
present scheme to create a high density of electrons at the Γvalley-in the CB. As a far more
attractive alternative, high doping and/or strained structures would increase this density
without the parasitic high hole density.
3.5. Absorption and Gain 53
0.60.0
5000
4000
3000
2000
1000
0Pum
p-in
duce
d VI
B Ab
sorp
tion
[ cm
-1 ]
1.61.20.80.40.0NT x 1020 [ cm-3 ]
ND
ND
Ge#1 Ge#2 Ge#3
InGaAs#1 InGaAs#2 InGaAs#3
Linear Fits
Figure 3.13: The measured pumped-induced valence intraband (VIB) absorption is linear with respect
to the injected carrier densities. Part of the data is shown in [37].
The measured peak gain is shown as a function of determined carrier densities, c.f. Fig. 3.14.
Both n-type doping and strain decrease the onset threshold of the peak gain, as visible
from the measurement, and also confirmed by the JDOS model. However, even though
a high gain of up to (850±50) cm−1 is achieved in for Ge, the simultaneously created
VIB absorption is always much stronger. For injected carrier densities of approximately
1 · 1020 cm−3, the VIB absorption is typically x10 higher than the gain. This demonstrates,
that optical amplification is not possible in any of the three investigated samples. In fact,
the VIB absorption is already more than > 500 cm−1 at the predicted gain onset densities of
around 0.3 · 1020 cm−3.
Hence, in Ge-on-Si, carriers excited by optical pumping lead to a strong direct-gap gain.
However, optical amplification is suppressed by the much stronger pump-induced VIB
absorption. This contradicts the straight-forward explanation of lasing in Ge-on-Si put
forward in [64]. In that article, the described pumping conditions correspond to an optically
injected carrier density of around 1 · 1019 cm−3, which is well below the reported density
needed for the onset of gain. It is therefore unclear, how optical transparency (as suggested
by the Fabry-Pérot oscillations) can be achieved, given that the pumped-induced absorption
is at least an order of magnitude stronger. Also, the slope of their reported emission versus
pump power only changes by a factor of ∼ 4 above threshold. In a standard laser system,
54 Chapter 3. Direct-Bandgap Gain and Optical Absorption in InGaAs and Germanium
1000
800
600
400
200
0
Peak
Gai
n [
cm-1
]
2.01.51.00.50.0
NP x 1020 [ cm-3 ]
Measurement Ge#1 Ge#2 Ge#3
Model Ge#1 Ge#2 Ge#3
Figure 3.14: The measured pumped-induced VIB absorption is linear with respect to the injected
carrier densities. Part of the data is shown in [37].
this ratio is normally well-above 104. It is only with a much larger strain than the 0.24%
reported, that peak gain could possibly overcome the pumped-induced absorption. This
could be due to the applied processing, which uses a selective regrowth of micron-sized Ge
waveguides onto Si. As pointed out in [83], a high point defect density in epitaxially grown
Ge close to the Si substrate interface gives rise to increased non-radiative recombination
between electrons from the L-valley and holes, which is due to the energetically closely
placed trap states below the L-valley. This favourably increases the ratio of electrons in the Γ-
with respect to the L−valley, and was observed to lead to an increased photoluminescence
(PL). Due to high surface-to-volume ratio in the Ge micro-cavities reported in [64], this
effect is expected to play a major role as well, which could be a further explanation for the
observed lasing.
In summary, the current understanding of the direct-bandgap physics in Ge is improved by
the experimental data and their interpretation introduced above, and will help the quest for
a light-emitting structure directly implemented on Si.
4.4.3 Quantum Cascade Structures based on Quantum Dashes . . . . . . . 81
This chapter is devoted to the gain and loss analysis of quantum cascade lasers (QCLs).
Here, synchrotron infrared is used to characterise active material embedded in micron-
sized waveguide structures. The measurements are then used to test two widely-used
models. These results were recently published in [84]. In addition, the investigation of
polarisation-resolved losses in typical mid-IR waveguides is added. A short discussion of gain
55
56 Chapter 4. Gain and Loss of mid-IR Quantum Cascade Structures
investigations in advanced QCL structures with quantum dashes, as well as multi-section
broadband QCL devices, concludes this chapter.
Despite the impressive advances made in QCL research over the last decade in terms of
wavelength coverage, gain bandwidth, emission efficiency, as well as operation temperature,
the extent of quantitative methods to assess the device’s main optical properties – gain and
loss – are still underdeveloped. There is, therefore, a need for methods that deliver significant
parameters for benchmarking models and simulation tools, such as Monte Carlo, density
matrix formalism and non-equilibrium Green’s function theory.
4.1 Quantum Cascade Laser
QCLs are semiconductor heterostructures emitting light in the mid- to far-IR. Lasing is
achieved from transitions between electron subbands which are defined by the the confining
potential from quantum wells. They were first realised in 1994 by Faist et al. [59], emitting at
4.2µm. The electron states and subbands are engineered by carefully designing the quantum
well and barrier widths, their order, and the respective materials in these heterostructures.
The transition energy between the states then is purely a result of design, and opens up
the path for flexible emission throughout the mid-IR region [85]. QCLs are fully unipolar
devices, because only a single type of carrier, the electrons, undergoes transitions. As
a consequence, the band curvatures of the initial and final states is the same, as shown
in Fig. 4.1. This results in the joint density of states of QCLs being comparable to an
atomic transition. QCLs are essentially transparent below and above the transition energy
range.
In contrast to a QCL, the emission of an interband laser is mainly defined by the direct-gap
energy of the material, and thus cannot easily be engineered to specific energies [86].
Light generation in interband systems is due to the recombination of electrons and holes
in a forward biased bipolar junction. Fundamental changes of the transition energy in
an interband laser are still possible, but require laborious efforts, such as the change of
the material alloy, or the engineering of strain. Luckily, the use of a heterostructure offers
4.1. Quantum Cascade Laser 57
the possibility to slightly engineer the transition energy as well [87]. Nevertheless, the
availability of such sources in the mid-IR is limited. Most of these diode laser sources cover
the visible and near-IR range due to interband emission over the bandgap, where they have
proven energy- and cost-efficient operation. It is only the lead-salt material system, which
provides emission across the bandgap between 3 - 20µm [88], at however limited power and
reliability. On the other side, QCLs cannot reach-up to the NIR, because the quantum wells
are not deep enough to confine such high-energy transitions.
E12$+$E&$+$E1fe$$
E12$
Efe$
E&$
E$ E$
k//$ k//$
E$ E$E12$ E12$
Intersubband$ Interband$
Figure 4.1: Left panel: In the intersubband case, the electron undergoes a transition between con-
duction band states, which have the same in-plane dispersion. The joint density of states
is comparable to a typical atomic transition. Right panel: In the interband case, the
transition occurs between electrons in the conduction band and holes in the valence band
with opposite curvature of the dispersion. Adapted from [78].
The theoretical foundation of light amplification in semiconductor heterostructures dates
back to Kazarinov and Suris in the 1970s [85,89]. They described the transitions between
electron states in quantum wells of an electrically biased superlattice. A further milestone
towards the realisation of a QCL, was the first observation of an infrared transition between
quantum well states in GaAs/AlGaAs heterostructures in absorption measurements by
West and Eglash [90]. They demonstrated a large oscillator strength f of 12.2, defined
58 Chapter 4. Gain and Loss of mid-IR Quantum Cascade Structures
as
f =2me
~
zi j
2 E j − Ei
~, (4.1)
where me is the free-electron mass, ~ is the reduced Planck constant, zi j is the dipole matrix
element, and E j and Ei are the energies of the upper (state number j) and lower (i) quantum
well states, respectively. The dipole matrix element zi j for the case of an infinite quantum
well is defined as
zi j =Lw
π2
8i j
(i2− j2)2∝Ç
me
m∗, (4.2)
where Lw is the width of the quantum well, and m∗ is the effective mass. In the design
by West and Eglash, zi j was found to reach almost 2 nm. It is the beneficial ratio of me
m∗
in semiconductors (e.g. me
m∗u 15 for GaAs [90]) which leads to these large dipole matrix
elements and oscillator strengths. Furthermore, the oscillator strength f increases mono-
tonically for transitions between higher states in the quantum well. As shown in Chap. 5, a
direct consequence of the large dipoles are the strong non-linear coefficients in QCLs. In
addition, West and Eglash also showed that the transition energy is directly depending on the
quantum well width, which confirmed the possibility of engineering such transitions. These
characteristics, together with the proposed design of Kazarinov and Suris, were therefore
expected to lead to an efficient light-emitting device.
However, Choi et al. [91] demonstrated that the main condition for lasing, which is popula-
tion inversion between the upper and lower laser states, could only be achieved in an unstable
electrical regime along with high electric fields in such a superlattice. Intersubband light
emission between 10 - 20 meV from a superlattice was demonstrated shortly after by Helm
et al. [92], however, still at the lack of population inversion.
It was then the inclusion of graded regions with n-type doing outside of the active quantum
wells, shown in Fig. 4.2, which enabled the achievement of a population inversion and led
to the first QCL realisation in the InGaAs/AlInAs system [59]. Here, the carrier injection
and relaxation regions were properly engineered, and allowed the structure to align the
quantum well levels in an electrically stable region.
Furthermore, the technical achievement of a proper molecular beam epitaxy (MBE), shortly
shown before [93], with which accurate nm-sized layers could repeatedly be grown with
4.1. Quantum Cascade Laser 59
7.2. ACTIVE REGION: FUNDAMENTAL CONCEPTS 119
roughly speaking, each cell can be divided in a gain region and an injection/relaxation re-gion. The gain region is the structure that will create and maintain a population inversion
3
2
1
!32
!2!32 > !2
Active region Relaxation/Injection
a)
b)
one period
Figure 7.2: a) Schematic conduction band diagram of a quantum cascade laser [20]. Each stageof the structure consists of an active region and a relaxation/injection region. Electrons can emitup to one photon per stage. b) General philosophy of the design: The active region is a three-levelsystem. The lifetime of the 3! 2 transition has to be longer than the lifetime of level 2 to obtainpopulation inversion.
between the two levels of the laser transition. As it will be shown later, this result can beobtained using various designs. In general, the active region contains a ladder of at leastthree states (or continuum of states), such that electrons are injected in the n = 3 state andthe population inversion is maintained between the states n = 3 and n = 2. Assuming thatelectrons are exclusively injected in the n = 3 state, the population inversion requirementtranslates into the following requirement on the lifetimes
32 > 2, (7.2.1)
i.e. the total lower state lifetime 2 is shorter than the electron scattering time from the n=3to the n=2 levels.
3"
2"
1" 3"
2"
1" 3"
One"Period"
Light"Emission"
Relax8"a9on/"
Injec9on"Region"
Ac9ve"Region"
Figure 4.2: Schematic of the conduction band diagram of a QCL featuring the important wave
functions. Upon injection into the upper laser state 3, the electron undergoes a radiative
transmission down to the lower laser state 2, is extracted via state 1 and then raised in
energy for subsequent injection. Adapted from [78].
different compositions and doping levels, opened the way for the QCL realisation. Dozens of
these periodic regions are grown on top of each other (typically 35), in order to increase the
electron "recycling" and to match the height of the active region to the typical optical mode
sizes of the mid-IR. Depending on the targeted wavelength, various material compositions
are used, which mainly differ by their conduction band discontinuity ∆EC . In the following
years, various material systems, such as GaAs/AlGaAs [94], were tackled for QCL. The
lattice-matched InGaAs/AlInAs (∆EC =520 meV) on InP is widely used for emission above
5µm. When aiming for wavelengths below 5µm, thermal activation of electrons from the
the upper laser state is increased, which hampers operation at room temperature. However,
the same material system (when grown in a strain-compensated way, i.e. compressively-
strained wells and tensile-strained barriers) can provide up to ∆EC =1.2 eV, and lasing
has been reported down to 3µm [95,96]. Also, InAs/AlSb grown on InAs substrates is a
promising candidate for emission down to 3µm [97] and at 2.6µm [98]. The avoidance of
aluminum-free barriers, as shown by Nobile et al. [99], is expected to provide larger matrix
element, and thus enhanced optical gain.
Two main level designs provide the basis of most QCL high-performance operation, the
bound-to-continuum (BTC) and the two-phonon-resonance (2Ph) designs [100–103]. In
both cases, the electron injection into the upper laser state is performed by resonant
60 Chapter 4. Gain and Loss of mid-IR Quantum Cascade Structures
tunnelling injection [104]. However, these designs differ in the extraction technique of the
carriers from the lower subband. For the BTC, the extraction is facilitated by a series of
lower states, which define a continuum-of-states, a so-called miniband. In the second case,
the series of three lower levels, separated by the optical phonon energy, lead to a effective
carrier extraction.
In general, when defining the radiative transition in the design, there is a trade-off between
vertical and or diagonal transitions. In the first case, a maximum overlap between upper and
lower states, and hence, large oscillator strengths, a narrower gain spectrum, and a lower
threshold is achieved at the expense of an unfavourable ratio of the lifetimes. In the case of
the diagonal transition, upper and lower laser levels are separated by a thicker barrier. This
increases the lifetime of the upper laser state and results in a better population inversion, at
the expense of a reduced oscillator strength.
The design of the waveguide (WG) is important in terms of thermal optimisation. The
waveguides must allow for efficient removal of the heat created in the devices, and they
are the limiting design factor for maximum operation temperature. The best example of an
efficient waveguide, featuring low losses and efficient heat extraction, is that of the buried
heterostructure waveguide [105]. Here, after definition (i.e. etching) of the active material
to form a ridge waveguide, the side and top of the waveguide are covered with regrown InP,
which offers a high thermal conductivity. With a high doping of the dielectric layers close to
the top contact, an efficient decoupling from the lossy surface plasmon mode, the so-called
plasmon-enhanced waveguide, is achieved [106,107].
QCLs emit light with electric field polarised parallel to the growth direction of the structure,
which is termed transverse-magnetic (TM) polarisation. The polarisation condition is due to
selection rules of the intersubband transition, and was shown to be strongly obeyed [108].
For the investigation of QCLs, this means that only light of this polarisation will interact
with the intersubband transitions. This offers a convenient means of distinguishing between
intersubband and waveguide losses.
An important characteristic for efficient operation and practicability is the so-called wall plug
4.1. Quantum Cascade Laser 61
efficiency (WPE), which is defined as the ratio of injected electrical and extracted optical
power. Up to now, most QCLs still operate at a rather low WPE around 10-20%, which is
mainly due to the efficient non-radiative electron transition in connection with the emission
of a phonon. It was shown, that the a priori maximally achievable WPE is strongly reduced
with increasing wavelength [109]. Recently, two different approaches were reported with
higher WPEs [110]. In the first approach, presented by Bai et al. [111], the voltage drop
across the injector was minimised. Furthermore, an increased number of periods overlapped
with the optical mode due to the shorter period length and yielded a higher differential gain,
resulting in a WPE of 50%. It seems, however, ambiguous to optimise a design, which will
only ever operate at cryogenic temperatures due to heavy thermal backfilling with increased
temperature. A second approach, reported by Liu et al. [112], consisted of increasing the
coupling strength between injector and upper laser states. In this way, WPEs over 40%
have been achieved. For a real quantitation, however, the (cryogenic) cooling power should
be considered as well, which drastically lowers the above reported WPEs. Hence, a WPE
of 27% reported at room temperature in a shallow-well design by Bai et al. [113] is very
remarkable.
Power output of more than 5 W was achieved by minimising carrier losses in unwanted energy
states and in the continuum, which was achieved by implementing shallow wells and taller
barriers [113]. To allow for dense packaging and maximum portability, QCLs can be designed
to run at low power dissipation (below 1 W), which was achieved by the use of a genetic algo-
rithm for the QCL design and a low doping density [114,115].
QCls are shown to cover a spectral range throughout the mid-IR to the THz from 2.9 -
250µm [116]. In the mid-IR, they operate at room-temperature in most of that range,
which is mainly achieved by a rigorous optimisation of the waveguide structure for thermal
management, e.g. burying the waveguide in the high thermal conductivity material InP
and mounting the QCL epitaxial-side-down [105]. The operation at room-temperature in
the THz, however, is limited. After the first THz QCL by Köhler et al. [117], which ran up
to 50 K, the maximum operation temperatures of contemporary THz QCLs, which mainly
rely on a resonant-tunneling injection scheme, seems limited by Tmax ≈ ~ω/kB, as reviewed
62 Chapter 4. Gain and Loss of mid-IR Quantum Cascade Structures
in [118]. However, newer schemes, such as the scattering-assisted injection mechanism, are
promising for higher operation temperatures [119].
To allow for broadband spectroscopy, a wide tuneability is requested. This can be achieved
by the combination of a QCL and an external cavity setup, as shown by Luo et al. [120].
The tuning range is further extended by the use of broadband QCL design, such as the BTC
[121]. Recently, tuning over 430 cm−1 in the wavelength range between 7.6 - 11.4µm was
achieved by a QCL consisting of several periods designed for different emitting wavelengths
[122].
An easier way to achieve wavelength-tuning, at the cost of a reduced range, is the use of
a distributed feedback (DFB) laser. This type of QCL operates in single longitudinal mode
at a fixed frequency, with a high side-mode suppression ratio (SMSR). The principle of
DFB was first shown and in a laser source by Kogelnik et al. [123], who investigated laser
oscillations in a dye laser pumped by the interference pattern of two UV laser sources. The
first realisation of a semiconductor laser was soon demonstrated by Nakamura et al. [124]
for the case of an optically pumped GaAs interband laser. The first DFB-QCL was realised by
Faist et. al [125], and was soon followed by high average power DFB-QCLs [101]. With the
on-chip integration of 24 DFB-QCLs into an array, tuneability was recently demonstrated
over 220 cm−1 from 8.0 to 9.8µm. This setup however demands for a complex electrical
pumping scheme.
Using photonics crystals (PhCs) is a further means of frequency-selection. In PhCs, the
refractive index is periodically modulated, which a modulation period close to the optical
wavelength [126]. In this way, a QCL incorporating 2D optical mirrors with PhCs was
shown by Dunbar et al. [127]. Benz et al. [128] then enhanced the concept by merg-
ing the gain and PhC structure. This resulted in a further miniaturisation of the whole
structure.
Recently, QCLs were also shown to operate as a frequency comb generator in the mid-IR,
which is due to the phase-locking in a broadband QCL through the process of four-wave
mixing [12], as further introduced and explained in Chap. 5.
4.2. Overview on Measurement Techniques 63
In summary, QCL technology has achieved a mature state [116,129,130]. Today’s and future
QCLs exceed and are fully able to replace other methods for the creation of coherent mid-IR
radiation, such as the lead-salt laser introduced above, or lasers based on the discharge of
an electric current inside a gas.
4.2 Overview on Measurement Techniques
The methods for measuring the key parameters of laser sources, which are their optical gain
and loss, can be divided loosely in two main groups.
The first group comprises all the techniques that use the actual device as the only source,
i.e. by looking at threshold current densities, or internally created light. In such cases, the
determination of gain and waveguide losses is made using the measurement of the lasing
threshold current densities for samples with identical design and processing, but different
length, which is the so-called 1/L-measurement, as applied in [131]. In a similar way, these
key parameters can also be determined by measuring the threshold current densities for a
sample before and after evaporation of a coating onto the facets. These methods, however,
are susceptible to sample variations and uncertainties in facet reflectivities in the later case.
Also, they stop short of providing spectral information. The measurement of the photolu-
minescence (PL) under optical pumping, or the electroluminescence (EL) under applied
electrical bias yields information on the energy position and the width of the radiative transi-
tion(s), and on the achievable efficiency. Measurement of the EL from waveguide cavities can
be further analysed for the modulation depth of Fabry-Pérot oscillations in sub-threshold op-
eration [132], which provides a direct feedback on gain and loss in the luminescent spectral
region. However, outside that region no information can be accessed this way. Furthermore,
the use of a high-resolution spectrometer is necessary.
The second category of measurements techniques makes uses of an external light source,
whose transmission through the waveguide of a QCL is determined. The measurement of
the transmission of a Fabry-Pérot waveguide can be performed with a narrow-band light, e.g.
a HeNe laser [133]. Here, the temperature of the waveguide under investigation is changed.
64 Chapter 4. Gain and Loss of mid-IR Quantum Cascade Structures
The introduced variation in cavity length lead to FP oscillations. In an analogue way, the
use of the frequency chirp of a DFB-QCL can be used [134]. Here, the waveguide cavity
is kept at constant temperature, while the increase in temperature of the DFB during the
electrical pulse leads to a frequency-sweep (typ. in the order of a few cm−1) of the latter.
As a consequence, the waveguide is sampled across several FP resonances. Note, however,
that these two methods only delivers gain and loss information at the narrow-band light
energy. Furthermore, stray light has to be carefully suppressed, as it results in a decrease of
the modulation depth, and, hence, artificially high losses.
Measurements using a multi-section cavity technique [135] overcome this limitation,
but require specially processed devices, where the waveguide is cut in transverse direc-
tion to the light propagation. This cutting obviously introduces absorption and coupling
losses.
Gain and loss studies based on the transmission of externally-created broadband IR light
through broad-area waveguide QCL devices have been introduced recently. The source
employed here consists of either a globar (mid-IR) or He-lamp (NIR) [136]. Information
over the full broad spectral range (of the source) is accessible. In addition, passive structures,
such as waveguides without any embedded quantum cascade materials, can be characterised.
Furthermore, due to the fact that intersubband transition only interact with TM-polarised
light, TE-polarised light can be used for normalisation, and to characterise the material
and waveguide losses independently of the intersubband losses. Nevertheless, this method
only works reasonably well with rather broad (>20µm) waveguides to achieve sufficient
coupling for a practical transmission signal. The alignment of the source with respect to
the waveguide and their spatial stability is experimentally challenging, and quantitative
results are ambiguous, because the effective coupling efficiency of the external light into
the waveguide is difficult to estimate, and stray light introduces artifactual offsets into the
measurement. So far, the characterisation of active QC structures has only been shown
to work in CW operation. Structures under development, which are usually operated
in pulsed mode, cannot be assessed. A further variant of this method also aims for the
interpretation of the Fabry-Pérot fringes in the transmission spectrum [137]. Contrary to the
4.2. Overview on Measurement Techniques 65
above described narrow-band FP method, however, the broadband source naturally yields
broadband information. Through the analysis of the ratio of the higher-harmonics in the
Fourier transform of the spectrum, the propagation losses and the refractive index may be
independently determined.
Alternatively, time domain spectroscopy introduced recently [138,139] has been shown to
be able to resolve spectral features. Here, ultrashort IR pulses are transmitted through the
active medium. The broadband IR pulses are created by phase-matched difference frequency
(DFG) generation in a non-linear medium (e.g. a GaSe) by Ti:sapphire laser pulses with
lengths < 100 fs. The phase resolved transmission is then measured using electro-optic
sampling. This coherent detection scheme allows to measure transient phenomena with a
time-resolution below 10 fs.
The use of a synchrotron source for transmission measurement is one of the most di-
rect way to characterise high performance QCL devices, which typically involve single
transversal-mode waveguides, see Fig. 4.3. Because such waveguides feature widths well
below 10µm (which is at the diffraction limit in the mid-IR) the external IR probe source
should also be diffraction-limited, in order to maximise modal coupling, and this is one
of the key features of synchrotron IR. The technical aspects of this method were de-
scribed in detail in Chap. 2. In the following sections, its application in different cases
is given.
Transverse magnetic (TM)
Figure 4.3: Facet view on a typical mid-IR waveguide with dimensions 2 x 5µm. The active polarisa-
tion (TM) is shown as well.
66 Chapter 4. Gain and Loss of mid-IR Quantum Cascade Structures
4.3 High-Performance Quantum Cascade Laser
4.3.1 Device Design and Processing
In this chapter, the performed investigations of a high-performance QCL based on the
two-phonon resonance (2Ph) [140] are presented. The modelled band structure together
with the electron states are given in Fig. 4.4. Those and the dipole-matrix elements were
calculated with the self-consistent solution of the Poisson and Schrödinger equations. An
energy-dependent effective mass was employed, thereby accounting for non-parabolicity.
Light emission for the sub-threshold conditions takes place from the upper laser state
12 down to states 8-10, before concentrating for increased biases on the designed lasing
transition 12 to 10 at 8.4µm.
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
Pote
ntia
l [eV
]
706050403020100
Distance [nm]
active region injector
12
13
1198
10
g
Figure 4.4: Moduli squared of the relevant wave functions and the conduction band diagram of a
period at an average field of 33 kV/cm for the QCL. The designed lasing transition is from
level 12 to 10.
The devices were processed into narrow-ridge low-loss heterostructure waveguides with
a width of 13.9µm). The waveguides were buried in iron-doped InP in order to improve
the thermal behaviour of the device, while at the same time ensuring electrical blocking
to suppress leakage currents. The measurement was performed on devices with a length
of 3 mm, which was determined to be the best trade-off between the suppression of stray
light while still guaranteeing a measurable transmission for the device without any ap-
4.3. High-Performance Quantum Cascade Laser 67
plied bias, when the optical absorption in transverse-magnetic (TM) polarisation is very
strong.
4.3.2 Measurement
The relative transmission spectra t = TT0
, which is the determined transmission T through
the device normalised with respect to the transmission T0 through a 15×8µm2 aperture with
entrance and exit optics aligned to each other, are given for TM- and TE-polarised light in
Fig. 4.5(a). The effective device temperature was 349K, and the data spans from zero bias
up to a current of I = 0.97 · Ith, with Ith being the lasing threshold current. The effective
device temperature is held constant by keeping the transmission cutoff at around 700 meV