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Title THz Quantum Cascade Lasers: Simulation of GaN-Based Active Regions and Fabrication of Integrated Waveguide Probes
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Author Naghibi Mahmoudabadi, Partia
Publication Date 2013
Peer reviewed|Thesis/dissertation
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THz Quantum Cascade Lasers: Simulation of GaN-Based Active Regions and Fabrication of Integrated Waveguide Probes
A thesis submitted in partial satisfaction
of the requirements for the degree
Master of Science in Electrical Engineering
by
Partia Naghibi Mahmoudabadi
2013
© Copyright by
Partia Naghibi Mahmoudabadi
2013
ABSTRACT OF THE THESIS THz Quantum Cascade Lasers: Simulation of GaN-Based Active Regions and Fabrication of Integrated Waveguide Probes
by
Partia Naghibi Mahmoudabadi
Master of Science in Electrical Engineering
University of California, Los Angeles, 2013
Professor Benjamin S. Williams, Chair
The terahertz region of electromagnetic wave spectrum spanning from 300 GHz to 10 THz (the transition from electronics to photonics) has recently attracted considerable attention based on its numerous applications in different fields. Quantum cascade lasers (QCLs) have been developed in the past ten years as important sources of terahertz radiation. However, there still remain challenges to overcome such as room-temperature operation, efficient out-coupled power and directive radiative beam pattern. Achieving these goals will make THz QCLs very special sources of light to be utilized in broad range of applications in astronomy, biology, medical imaging, etc. This research tackles the aforementioned challenges in two sections. First, it targets the room-temperature issue by focusing on active region engineering and utilizing GaN/AlGaN quantum wells instead of the conventional
GaAs/AlGaAs heterostructure QCLs. Second, it introduces a novel approach in waveguide engineering by harnessing the transmission line theory in order to improve the out-coupled power and radiation beam pattern.
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The thesis of Partia Naghibi Mahmoudabadi is approved.
Oscar M. Stafsudd
Robert N. Candler
Benjamin S. Williams, Committee Chair
University of California, Los Angeles
2013
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Table of Contents
1 Introduction ...... 1
1.1 Background ...... 1
1.2 Motivations and Challenges ...... 3
2 Simulation of 2D Quantum Structures ...... 8
2.1 Electron’s Particle-Wave Duality and Wave Mechanics ...... 8
2.2 Effective Mass Approximation ...... 11
2.3 Electronic States in Heterostructures ...... 12
2.4 Shooting Method (A Numerical Approach to Solve Schrödinger’s Equation) . 15
2.4.1 Implementation Conditions ...... 18
2.4.2 Extension to Variable Effective Mass ...... 19
2.5 Fermi Level Calculations ...... 22
2.6 Poisson’s Equation ...... 23
2.7 Self-Consistent Schrödinger-Poisson’s Solver ...... 26
2.8 Simulation Results ...... 27
2.8.1 One Quantum Well ...... 27
2.8.2 Multi Quantum Wells (FL178C-M7) ...... 30
3 Simulation and Design of GaN/AlGaN Terahertz Quantum Cascade Laser . . 32
3.1 Introduction ...... 32
3.2 Spontaneous Polarization in III-Nitrides ...... 33
3.3 Properties of GaN, AlN and AlxGa1-xN ...... 35
3.3.1 Bowing Parameters ...... 35
3.3.2 Structural Parameters ...... 36
3.3.3 Elastic Constants ...... 36 iv
3.3.4 Piezoelectric Coefficients ...... 37
3.3.5 Spontaneous Polarizations ...... 38
3.3.6 Piezoelectric Polarizations ...... 39
3.3.7 Bandgap Energy ...... 41
3.3.8 Electron’s Effective Mass ...... 42
3.3.9 Dielectric Constants ...... 42
3.4 Piezoelectric and Pyroelectric Effects in III-V Nitride Nanostructures . . . . 42
3.5 Resonant Tunneling and Anticrossings ...... 44
3.6 Oscillator Strength, Intersubband Gain and Threshold Current Density . . . . 46
3.7 Polar Longitudinal Optical (LO) Phonon Scattering ...... 48
3.8 Simulation Results ...... 53
3.8.1 Design at 3.93 THz ...... 53
3.8.2 Design at 5.4 THz ...... 57
3.9 Conclusion ...... 59
4 Fabrication of GaAs/AlGaAs Terahertz Quantum Cascade Laser with . . . . 61
Integrated Radial Probe
4.1 Introduction ...... 61
4.2 Waveguide Simulation and Design ...... 62
4.3 Micro-Fabrication of QCL with a Radial Probe ...... 68
4.4 Conclusion ...... 77
A Four-Level Resonant-Phonon QCL Rate Equations ...... 82
B Processing Recipes ...... 84
References ...... 88
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Acknowledgements
I would like to sincerely thank Benjamin Williams for his truly systematic and meaningful guidance during my presence at UCLA terahertz lab. That certainly played an essential role in developing my technical understanding and skills in order to accomplish this thesis.
I would like to thank Amir Ali Tavallaee for his patience and very detailed training during the fabrication process. I would also like to thank Luyao Xu and Pradeep Senanayake from
UCLA, and Choonsup Lee from JPL for assisting me in different parts of this research.
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Chapter 1
Introduction
1.1 Background
The Terahertz (THz) frequency range spanning roughly from 300 GHz to 10 THz is considered the gap between electronics and photonics in the electromagnetic frequency spectrum
(Fig. 1.1). Unlike its neighboring frequency ranges, it has remained relatively less developed mainly due to lack of a mature technology for generating THz radiation.
Figure 1.1: The electromagnetic wave spectrum. Figure is taken from Williams [1].
Surprisingly, it has variety of important applications such as terahertz imaging, spectroscopy, astronomy, biology, biomedical imaging, explosive detection and homeland security [2,3,4].
Many molecules such as carbon monoxide, carbon dioxide, water, oxygen, etc have THz signature since their vibrational and/or rotational resonances fall in the THz range. Cancer tumor, bones and other biological samples can be detected using non-invasive THz imaging compared to the X-ray imaging which is dangerous [4]. Materials such as clothing, that are opaque in
1 optical range, show transparency in the THz range (depending on a frequency) [3]. Therefore, terahertz imaging can be used in fields required security such as airport, to discover concealed illegal items such as weapons and drugs.
To operate in the THz region, various efforts and techniques have been made using either electronics or photonics. In the electronics region, sources such as transistors, Gunn oscillators, and Schottky diode frequency multipliers are capable of producing power on the order of milliwatts in the gigahertz frequency range using frequency up-conversion techniques. However, their output power will be reduced drastically near the THz region to about 10 μW at 1 THz [5] since electronics are limited by transport transit time and parasitic capacitance effects. In the photonics region, lower frequencies (near THz range) can be produced using electron-hole recombination scheme in semiconductor lasers, but the emission is strongly bandgap dependent, and cannot span the whole THz range (e.g. lead-salt materials could go as low as 15 THz which is still in the far-IR region [6]). As another technique, THz frequency is created using nonlinear down-conversion of visible or infrared light, but the power is on the order of microwatts [7, 8, 9,
10, 11]. Moreover, there exist other THz sources such as gas lasers and free-electron beam lasers that are application-limited due to their huge size, complexity and cost [1].
Quantum cascade lasers (QCLs) are the semiconductor-based sources of terahertz which have had impressive progress since two decades ago. The first QCL was invented at Bell lab in
1994 [12] and operated in infrared region. In addition, the first THz QCL was demonstrated in
2001 [13]. Unlike diode lasers which operate based on electron-hole recombination, the QCL operation is based on intersubband (ISB) transition of electron carriers in conduction band. The
2 gain of QCL is supplied electrically due to the current flowing through many numbers of identical quantum-well modules in a cascading manner.
1.2 Motivations and Challenges
Quantum cascade lasers as recently developed sources of terahertz operate based on intersubband (ISB) transition of conduction band electrons in semiconductor quantum wells, see
Fig. 1.2. GaAs/AlGaAs quantum wells have been the materials of interest in QCL active region which resulted structures capable of producing a broad frequency range of 0.6 – 5 THz (λ ~ 60 –
500 μm).
Figure 1.2: Two cascaded modules of a resonant-phonon active region with the squared magnitude of the wavefunctions for the various subband states. The upper- and lower-radiative states are shown in red and blue, respectively. Figure is taken from Williams [2]
However, there are limitations which only allow operation of GaAs/AlGaAs THz QCLs under cryogenic conditions. So far, the maximum temperatures of 199.5 K pulsed [14] and 117 K
CW [15] have been reported. These limitations are due to the fact that longitudinal optical
3 phonon (LO-phonon) energy LO 36 meV in GaAs, which is comparable to the room temperature thermal energy kBT of 26 meV.
Figure 1.3: Upper-state depopulation scheme. hv and LO correspond to the radiative photon and LO- phonon energies respectively.
This would cause two essential problems in QCL’s operation. One is called the upper-state depopulation due to non-radiative relaxation of electrons by spontaneously emitting LO phonons
(see Fig 1.3). At low electron temperatures, electrons residing in the upper radiative state have insufficient energy to emit an LO-phonon, since the subband separation (~4-20 meV) is less than
LO . However, at higher temperatures, electrons residing at the bottom of the upper-state subband | u gain sufficient thermal energy and are excited to the higher-energy states within the subband. As a result, they can scatter into the lower-state subband | l via LO-phonons emission.
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Another mechanism is called lower-state thermal backfilling. In this process, Fig. 1.4, at
Figure 1.4: Lower-state thermal backfilling scheme. hv and LO correspond to the radiative photon and LO-phonon energies respectively.
higher temperatures, electron residing at the bottom to the ground-state subband | g gain sufficient thermal energy to get excited to higher-energy states within the subband. As a result, they can scatter into the lower-level subband | l via LO-phonon absorption.
One promising system that has drawn much attention for ISB-transition-based devices is the
GaN/AlGaN quantum wells material system. GaN has LO-phonon energy of ~90 meV at the - point; therefore, it is considered a good candidate to avoid the aforementioned issues for GaAs- based THz QC-lasers. Previous theoretical studies have suggested that lasing is feasible using
III-Nitride QWs even at higher temperatures. Jovanovich [16] showed that laser action is possible by calculating the population inversion of 30% to 40% of the total sheet carrier density for a GaN /Al0.2Ga0.8N system at temperatures up to 155 K. Also Bellotti [17] used Monte Carlo simulation to demonstrate that population inversion of GaN /Al0.15Ga0.85N THz QCL is three time more than GaAs /Al0.15Ga0.85As one at the room temperature. Moreover, Sun [18] showed
5 relatively low threshold current density of 832 A/cm2 can provide a threshold optical gain of
50/cm at the room temperature.
However, fabricating intersubband devices based on group III-Nitride materials, e.g. GaN and related alloys, seems to be challenging in terms of epitaxial growth, design and device processing [19-24]. For fabrication of short-wavelength intersubband devices where ultra-thin layers are needed, the best suited techniques are plasma assisted molecular beam epitaxy or ammonia-source molecular beam epitaxy [25-27]. Molecular beam epitaxy, because of its inherent low growth temperature and slow growth rate, allows accurate control of layer thickness and interface abruptness. The growth of 1-1.5 nm thick layers with sharp interfaces is very challenging with metal-organic vapor phase epitaxy, because of interface instabilities induced by the high growth temperature and built-in strain [28-30]. However, this technique is perfectly suited for intersubband devices operating in mid- and far- infrared.
Substrates, such as sapphire and SiC, are commonly employed in nitride epitaxy due to the lack of commercially available nitride substrates that have both a sufficiently high structural quality and a large area. The heteroepitaxy process on dissimilar substrates results in a high defect density in the GaN/AlGaN structures due to the large lattice mismatch between substrate and the grown layers. This high defect density has a negative impact on the device performance.
However, ongoing efforts are also been made to achieve a high quality GaN substrate [31-33].
It is also very difficult to achieve a short wavelength emission for intersubband nitride structures such as quantum cascade laser since precise control of layer thicknesses must be maintained throughout the whole structure which normally includes many modules.
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Additionally, the intrinsic internal electric fields across heterostructures grown along the c- direction represent an important challenge since these electric fields may affect both electron confinement and efficient current injection. Finally, achieving efficient n-type doping of AlxGa1- xN alloys with a high Al content is yet another challenge [34].
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Chapter 2
Simulation of 2D Quantum Structures
2.1 Electron’s Particle-Wave Duality and Wave Mechanics
In 1926, Louis De Broglie extended the idea of coexistence of waves and particles (it was previously discovered by Albert Einstein for light waves and photons in 1905) for all particles.
He stated that for any particle with momentum p, there is a wavelength λ associated with which is given by the following:
h (2.1) p
Therefore, an electron at a position r can be described by a state function which has the wave form, i.e.
ei(krt) (2.2) where t is the time, ω is the angular frequency and the modulus of the wave vector is given by:
2 k (2.3)
The quantum mechanical translational (linear) momentum operator is defined as:
pˆ i (2.4)
8 where is the reduced Planck’s constant and operator is defined in three-dimensional (3D)
Cartesian coordinate system as:
iˆ ˆj kˆ (2.5) x y z
When pˆ acts upon , the momentum p arises as the eigenvalue solution:
i p (2.6)
Using the expression (2.2) representing electron vacuum wavefunction, (2.6) can be written as:
ie(krt) pe(krt) (2.7) where p is deduced as:
ˆ ˆ ˆ p (kxi k y j kz k) k (2.8)
Based on the classical mechanics, the kinetic energy of a particle with mass m can be written as:
1 (mv)2 p 2 E mv2 (2.9) 2 2m 2m
Therefore, it would be a reasonable guess to expect the same form in quantum mechanical analogy:
1 2 (i)2 2 E (2.10) 2m 2m where E is the kinetic energy eigenvalue and 2 can be written using (2.5):
9
2 2 2 2 (2.11) x 2 y 2 z 2
Using expression (2.2), (2.10) can be rewritten as:
2 2e(krt) Ee (krt) (2.12) 2m
2 (i 2k 2 i 2k 2 i 2k 2 )e(krt) Ee (krt) (2.13) 2m x y z where E is deduced as:
2 k 2 E (2.14) 2m
The result (2.14) is in accordance with the classical case. Eq. (2.14) says that for a free electron in the vacuum away from any electromagnetic interaction, the kinetic energy
(equivalent to the total energy) is related to the momentum (proportional to the wavevector by ) by a parabolic relationship, Fig 2.1.
Figure 2.1: The energy versus wavevector curve for an electron in the vacuum
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Eq. (2.10) represents the time-independent Schrödinger equation which describes the energy of a particle in the vacuum. There is also a time-dependent Schrödinger equation, i.e.
i e(krt) i(i)e(krt) (2.15) t
i (2.16) t where is called the energy eigenvalue. Eqs. (2.10) and (2.16) are two complimentary descriptions associated with wave-particle duality.
2.2 Effective Mass Approximation
The free electron assumption will be rather complex in case of solid crystals. In crystals, electrons interact with periodic lattice potentials instead of non-interacting vacuum. However,
(2.10) can still be applicable considering an assumption known as effective mass approximation.
In this approximation, all the information related to electron’s interaction with crystal’s potential are lumped into a new defined mass called effective mass, m*.
2 k 2 E (2.17) 2m*
However, (2.17) is only valid near a band minimum where a parabolic band is a good approximation. Depending on crystal’s orientation, m* would have different values. The anisotropy in m* has also been confirmed via experimental measurements (e.g. GaAs has an isotropic m* near conduction band minimum but a more anisotropic m* in valance band). Adachi
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* [35] has reported the effective mass of GaAs and its alloys. The m has a value of about 0.067m0
(m0 is the free electron mass) for GaAs at the -point, Fig 2.2.
Figure 2.2: Energy versus wavevector (proportional to momentum) curves for an electron in GaAs at the -point compared to that in a vacuum.
2.3 Electronic States in Heterostructures
When two different materials are brought adjacent, the effective mass is no longer a constant value but varying as a function of position. The bandgap also varies from one material to another,
Fig 2.3.
Figure 2.3: two dissimilar semiconductors with different bandgaps joined to form a heterojunction. Black curves represent freedom of motion in the direction parallel to the interface. Figure is taken from Harrison [36].
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At the heterojunction between two different materials, there exists a discontinuity in the band energies. When the layers thicknesses are so thin, i.e. on the order of De Broglie wavelength, i.e.
Eq (2.1), the electron motion is restricted in the growth direction, e.g. zˆ direction, and the energy is quantized.
In general the wavefunction for an electron in a solid is given by:
(r) F(r)U n,0 (r) (2.18)
where Un,0(r) is the Bloch state wavefunction at the band minimum and F(r) is the envelope function that satisfies the effective mass equation (2.19). Effective mass variation is represented
* by m (z). Conduction band edge profile is represented by Ec(z) which also includes external electric field (E-field) and any variations due to space charge. Therefore, the effective mass equation is rewritten as:
2 2 2 || 1 * * Ec (z)F(r) EF (r) (2.19) 2m (z) 2 z m (z) z
where || is the in-plane differential operator [37]. The solution for the wavefunction is given by:
1 ik|| r|| F(r) e n (k|| , z) (2.20) S||
where n (k|| , z) satisfies
2 2 2 1 k|| * Ec (z) * n (k|| , z) En (k|| ) n (k|| , z) (2.21) 2 z m (z) z 2m (z)
13 and k|| is the in-plane wavevector, n is the subband index, and S|| is the normalization area. The spatially varying effective mass causes a coupling between the in-plane and z directions. This coupling is usually neglected, and equation (2.21) simplifies to the 1D Schrödinger equation:
2 1 * Ec (z) n (z) En (z) n (z) (2.22) 2 z m (z) z
Where the total energy is given by
2k 2 E (k ) E || (2.23) n || n 2m*
Where m* is the effective mass of the electron in the well. Here the total energy of the electron is the summation of energies in both in-plane and z directions, (See Fig. 2.4).
Figure 2.4: Example of AlxGa1-xAs /GaAs/ AlxGa1-xAs quantum well and in-plane energy dispersion of a trapped electron. Figures are taken from Harrison [38].
The coupling mentioned above only can be neglected if the in-plane kinetic energy is modest compared to the barrier height and effective masses of the barrier and well don’t differ too much.
Otherwise, the barrier height would effectively change by: 14
2k 2 m* || 1 (2.24) * * 2m m (z)
By introducing free carriers to the structure, the conduction band edge profile is perturbed due to the induced electrostatic potential. To account for this effect, Poisson’s equation, i.e.
d d (z) (z) (z) (2.25) dz dz has to be solved, where Φ(z) is the electrostatic potential, ε(z) is the spatially varying dielectric
constant, and ρ(z) is the charge density. This gives Ec (z) Ec,0 (z) e(z), where Ec,0 (z) is the intrinsic conduction band profile. In order to find the end result for the conduction band profile,
Poisson’s and Schrödinger’s equations need to be solved iteratively until a self-consistent solution yields.
2.4 Shooting Method (A Numerical Approach to Solve
Schrödinger’s Equation)
In this section, I will explain how to solve Schrödinger’s equation numerically for 1D confined structure (i.e. quantum well). The method I will explain is called the shooting method and it’s a powerful method to solve for the energy eigenfunctions and eigenstates of a system composed of many adjacently grown quantum wells. Most of the derivations in this section are taken from Harrison [39].
Let’s start with the time-independent Schrödinger’s equation:
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2 2 (z) V (z) (z) E (z) (2.26) 2m* z 2 where V(z) is an undefined 1D potential. Eq. (2.26) can be rewritten as:
2 2 (z) [V (z) E] (z) 0 (2.27) 2m* z 2
The goal is to solve Eq. (2.26) numerically for eigenenergy E and eigenstate ψ(z). At the first step, let’s derive the second-order derivative in terms of finite differences. The first derivative of a function is defined as the following:
f df lim (2.28) z0 z dz
Using the Fig. 2.5,
Figure 2.5: The first derivative of a function. Figure is taken from Harrison [39].
Eq. (2.28) can also be written in the approximate form:
df f f (z z) f (z z) (2.29) dz z 2z 16
Following the same procedure, the second-order derivative can be derived as:
df df d 2 f dz dz zz zz (2.30) dz 2 2z
f (z 2z) f (z) f (z) f (z 2z) 2 d f 2z 2z f (z 2z) 2 f (z) f (z 2z) (2.31) dz 2 2z (2z) 2
Considering 2δz to be some small value, it can mathematically be called as a new δz. This simplifies Eq. (2.31) to:
d 2 f f (z z) 2 f (z) f (z z) (2.32) dz 2 (z)2
Applying Eq. (2.32) result upon Eq. (2.27) and assuming δz to be very small (i.e. going from
‘ ’to ‘=’), we get:
2 (z z) 2 (z) (z z) * 2 V (z) E (z) 0 (2.33) 2m (z)
2m* (z z) 2 (z) (z z) (z)2 V (z) E (z) (2.34) 2 which would finally have the form:
* 2m 2 (z z) 2 (z) (V (z) E) 2 (z) (z z) (2.35)
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Eq. (2.35) expresses that if the wavefunction ψ values at two points (z - δz) and z are known; value of ψ can be calculated numerically for point (z + δz). Eventually, values of ψ can be evaluated using the same procedure for all other points along the z-axis. This procedure forms the basis of solving differential equations numerically, and is known as shooting method.
2.4.1 Implementation Conditions
Assuming that the potentials of interest are all confining potentials, then all the standard boundary conditions can apply for wavefunction:
(z) 0 and (z) 0 as z (2.36) z
Also, wave can exponentially decay or grow in barrier and well respectively based on the evaluated wavevector in any regions:
2m* (V E) (2.37)
Hence, after selecting the value of wavefunction’s starting point (z z) , the value of (z) is calculated using the boundary condition of exponential growth:
(z z) (z)exp( z ) (2.38)
However, these exponential boundary conditions won’t be reliable enough since the boundary conditions at either ends of the structure, i.e. Eq. (2.36), also need to be satisfied. The simple but bizarre initial value conditions are chosen as:
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(z1 ) (z2 z) 0 and (z2 ) 1 (2.39) because of their generality [39] and the fact that they are applicable for any potential profiles
such as symmetric, anti-symmetric or random. In (2.39), z1 and z2 correspond to the first and second points.
In practice, it is very critical how thick the end barriers’ thicknesses are selected. Even though it still converges to an energy eigenvalue, choosing too thick or too thin barrier would cause the wavefunction to diverge to . Finally, the wavefunctions obtained by this method are not normalized, i.e. they don’t satisfy:
* (z) (z)dz 1 (2.40) space
This can be done by the following transformation:
(z) (z) (2.41) * (z) (z)dz space
2.4.2 Extension to Variable Effective Mass
The numerical solutions to the Schrödinger’s equation, by far, have been based on the constant effective mass approximation. More realistically, the mass is varying as a function of position (as it’s the case for any heterostructures material systems). If so, previously derived equations require corresponding revision. Therefore, Eq. (2.26) would have the new form:
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2 1 (z) V (z) (z) E (z) (2.42) 2 z m* (z) z which can also be written as:
1 2 (z) V (z) E (z) (2.43) z m* (z) z 2
The variable kinetic energy operator can be expanded to give:
1 1 2 2 * (z) * 2 (z) 2 V (z) E (z) (2.44) z m (z) z m (z) z or equivalently:
2 1 * 1 2 2 m (z) (z) * 2 (z) 2 V (z) E (z) (2.45) m* (z) z z m (z) z
However, applying the previous method to convert Eq. (2.45) to finite difference form would create high inaccuracy in the case of large effective mass discontinuity (e.g. GaAs and
AlAs). This occurs due to δ-function nature of m* (z) / z . A more robust result can be derived by expanding the left-hand derivative of (2.43):
1 (z) 1 (z) m* (z z) z m* (z z) z 2 zz zz V (z) E (z) (2.46) 2z 2
1 (z) 1 (z) 22z * * 2 V (z) E (z) (2.47) m (z z) z zz m (z z) z zz
Taking Advantage of Eq. (2.29), the Eq. (2.47) can be written as:
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1 (z 2z) (z) 1 (z) (z 2z) * * m (z z) 2z m (z z) 2z
22z V (z) E (z) (2.48) 2
(z 2z) (z) (z) (z 2z) 22z2 * * 2 V (z) E (z) (2.49) m (z z) m (z z)
By simplifying the result:
(z 2z) (z 2z) 22z2 1 1 * * 2 V (z) E * * (z) (2.50) m (z z) m (z z) m (z z) m (z z) and performing the transformation, 2z z , we get:
(z z) 2z2 1 1 * 2 V (z) E * * (z) m (z z / 2) m (z z / 2) m (z z / 2)
(z z) (2.51) m* (z z / 2)
Eq (2.51) is called the shooting method equation for varying effective mass. The intermediate points’ masses z z / 2can be calculated by taking the mean of z and z z . Clearly, Eq.
(2.51) collapses back to Eq. (2.35) in the case of the constant effective mass.
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2.5 Fermi Level Calculations
Hereafter, I will explain how to calculate the Fermi level for 2D quantum structures (e.g. quantum wells) numerically. It then will be used to solve the Poisson’s equation for a doped structure in order to find the induced electrostatic potential Φ(z). Finally Poisson’s and
Schrödinger’s equations will be solved iteratively as many times as it converges to a solution.
Most of the derivations in this chapter are based on those explained in Harrison [39].
Electrons in a solid are distributed thermally base on the Fermi-Dirac distribution:
1 f (E) EEF (2.52) e kBT 1
where EF is the Fermi energy level of the system and kB is the Boltzmann’s constant. The density of states in a two-dimensional (2D) structure (e.g. quantum well) follows:
m* N(E) (E E ) (2.53) 2 i i
Where θ is the step function and Ei is the discrete energy eigenvalue. Assuming that 100% of the dopants in the structure are ionized then the total 2D free carrier concentration can be calculated via the following:
n N(E)f (E)dE (2.54) 2D,total Bound States
Knowing n3D,total then n2D,total can easily be calculated using:
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n n dz (2.55) 2D,total 3D,total Structure Length
EF can then be determined numerically and with a high accuracy whenever right- and left-hand sides of Eq. (2.54) happen to be equal.
2.6 Poisson’s Equation
The 2D free carrier concentration in the i-th subband with energy Ei is given by:
n N(E) f (E)dE (2.56) 2D,i Ei
Moreover, the total 3D carrier concentration is calculated as:
Top Occupied State 1 2 1 1 n (z) (z) n (2.57) 3D,total 3 i 2D,i 2 m iGround m m State
2 where i (z) is the probability function describing electron probabilistic distribution in a
quantum confinement (Note that within a subband, i (z)have different distributions since each subband contains a spectrum of eigenstates). Then the net 3D charge density distribution along the growth z-direction is given by:
(z) q(N D (z) n3D,total (z)) (2.58)
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where q is the elementary charge and ND (z) is the dopant ion concentration. Since ρ(z) is a discrete quantity, it can be imagined as sheet career density σ(z) at each point in the z-direction simply by the following conversion:
(z) (z)z (2.59)
The E-field which results from σ(z) in each infinitesimal length, δz, is shown in Fig. 2.6.
Figure 2.6: E-field for an infinite plane of charge. Figure is taken from Harrison [14].
The magnitude of the E-field in Fig. 2.6 can be derived by the Maxwell’s first equation (i.e.
Gauss’s law) which gives:
(z) E (2.60) 2 (z) where ε(z) is the dielectric constant as a function of z. Finally, to calculate the net E-field at each point in z-axis, E-fields contributed from all the sheets (Fig. 2.7) have to be summed up:
(z ' ) E(z) sign(z z ' ) (2.61) ' z' 2(z )
24 where:
sign(z) 1 z 0
sign(z) 1 z 0 (2.62)
Figure 2.7: Example of infinite sheets of charge along the structure. Each sheet contributes an E-field in each point of z-axis. Figure is taken from Harrison [39].
After the E-field was calculated along the structure, electric potential is given by:
E V (2.63)
To numerically solve Eq. (2.61), it can be expanded to:
zl V V E z (2.64) B A z0 where l is the length of the structure along z-axis. Subsequently, the potential energy for an electron (a negative sign factor is multiplied) is given by:
E (q)V (2.65)
25
2.7 Self-Consistent Schrödinger-Poisson’s Solver
The derived potential energy, Eq. (2.65), has to be added to the band edge potential profile to give the new band edge potential:
Ec (z) Ec0 (z) E (2.66)
Where Ec is the new band edge profile, Ec0 is the initial band edge profile and Eρ is the potential energy introduced by the dopants. The Schrödinger’s and Poisson’s equations have to be utilized in an iterative manner in order that the end result converges to a self-consistent solution.
The systematic method of this approach is illustrated in Fig. 2.8:
Figure 2.8: Block diagram showing the self-consistent Schrödinger’s-Poisson solver procedure
26
2.8 Simulation Results
2.8.1 One Quantum Well
In this section, the validity of my code is verified using a one-well structure. The structure is composed of one 100 Å thick GaAs well sandwiched between two 200 Å thick Al0.2Ga0.8As barriers. The well is n-doped to 21018 cm-3. Fig. 2.9 illustrates the band diagram of an undoped structure under no bias at 0 K.
Figure 2.9: Band diagram of a single well Al0.2Ga0.8 As/GaAs/Al0.2Ga0.8As at 0 K and under no bias. There exist only first (ground) and second states with ~80 meV energy separation.
Now, let’s assume that our structure is doped. Here, it’s being investigated how the initial band edge profile will change due to induced potential caused by the dopants. Let’s have an assumption that all the introduced carriers will only occupy the ground state. It’s a reasonable assumption since the Fermi level is located at 99.67 meV, therefore all the carriers will freeze at the ground state based on Fermi-Dirac statistics. Fig. 2.10 illustrates the areal charge density for this structure:
27
Figure 2.10: Areal charge density σ for a doped 100 Å GaAs well at 0 K and under no bias. Electrons are distributed in the growth direction based on the ground state probability function.
Obviously, σ is zero at far ends of the structure since there exists no dopants and the wavefunction is also zero. It also shows a positive σ at the both junctions of GaAs/AlxGa1-xAs and a negative σ in the middle of the well. Those are very much expected due to the ground state wavefunction probability (z) 2 symmetry.
The E-field can also be calculated using Eq. (2.61). Fig 2.11 illustrates the E-field strength distribution along the structure:
Figure 2.11: E-field strength for a doped 100 Å GaAs well at 0 K and under no bias. 28
Clearly, the E-field is zero at far ends based on the charge neutrality principle. It also has the value of zero in the middle due to the symmetric distribution of the charges. Finally, the potential energy due to the introduced charges can be derived using Eqs. (2.64) and (2.65). The result is illustrated in Fig. 2.12.
Figure 2.12: Potential energy for a doped 100 Å GaAs well at 0 K and under no bias.
In order to find the true existing band edge potential profile, the effect of dopants has to be accounted for. In other words, the self-consistent Schrödinger-Poisson solver explained in Fig.
2.8 need to be employed. Iterations can be continued until negligible changes (this criterion is optional depending on application and required accuracy) are observed in the values of energy eigenstates. Fig. 2.13 shows the final result after twenty iterations.
29
Figure 2.13: Resultant band edge potential energy profile for a doped 100 Å GaAs well at 0 K and under no bias.
The results presented in the section very accurately match the results done by Harrison [39].
Therefore, the validity of my code has been verified, and this code can confidently be used for any other complicated structures with more wells, random dopings and even different materials.
2.8.2 Multi Quantum Wells (FL178C-M7)
To visualize the effect of doping in a more complex structure, I also applied my code to one period of a resonant-phonon THz QC-laser design structure [40], named as FL178C-M7.
A single period is composed of four GaAs/Al0.15Ga0.8As quantum wells/barriers. The layers’ thicknesses are 29/9/24/15/57/12/33ML, that boldfaced numbers correspond to the barriers and each GaAs monolayer (ML) is 2.825 Å. The 57 ML well is doped to 1.51016 cm-3. Fig. 2.14 and Fig. 2.15 illustrate the band edge profiles and energy eigenstates without and with the effect of dopants respectively. Also, both simulations are done at 0 K and under no bias.
30
Figure 2.14: Undoped FL178C-M7 structure’s band edge profile and bound energy eigenstates at 0 K and under no bias.
Figure 2.15: Doped FL178C-M7 structure’s band edge profile and bound energy eigenstates at 0 K and under no bias.
Here again, the valid assumption of free carrier distribution in the ground state only is made
(Fermi level is calculated to be 13.45 meV which is slightly above the ground state). Also the number of iterations is chosen to be twenty. As the result, Fig. 2.15 just slightly differs from Fig.
2.14 mainly due to low dopant concentration.
31
Chapter 3
Simulation and Design of GaN/AlGaN Terahertz
Quantum Cascade Laser
3.1 Introduction
Serious efforts began to fabricate nitride devices four decades ago. In 1971, Pankove et al. reported the first GaN-based light-emitting diode [41]. However, most of these research programs were abandoned due to fundamental materials problems. Lack of a suitable technology for growing bulk-crystal GaN substrates forced epitaxy to be done on highly lattice-mismatched substrate which caused a high defect density and poor surface morphology.
It was not until mid-1980s that the growth difficulties started to be overcome. The use of Al or GaN nucleation layers facilitated the growth of high-quality GaN films on sapphire substrates by metalorganic chemical vapor deposition (MOCVD). The first n-GaN/AlGaN transistor was demonstrated by Khan et al. in 1993 [42]. Then, Nakamura et al. in 1995, manufactured the first nitride-based laser diode with continuous-wave room-temperature emission of 417 nm wavelength [43].
Now, blue and green nitride-based LEDs are wide used in full-color displays and traffic light.
Nitride laser diodes are the key components of high-definition DVD players. They also have possible applications in areas such as sensors, communications and medical equipments.
32
However, despite of all the worldwide research and development, there still remains challenges in understanding the microscopic transport process of nitride devices. Numerical simulations are important tools which establish quantitative links between the material properties and device performance in order to overcome those challenges.
3.2 Spontaneous Polarization in III-Nitrides
In the nature, there exist two types of materials carrying spontaneous polarizations Psp, ferroelectrics and pyroelectrics. In the ferroelectrics, electric field can be inverted by applying an external electric field. Therefore Psp can be measured accurately. Conversely, in the pyroelectrics, electric field direction and magnitude cannot be altered and is always parallel to the low symmetry axis of the crystal.
Among tetrahedral coordinated solids, the most common pyroelectrics such as GaN and AlN have wurtzite structure. In wurtzite crystals, the pyroelectric axis or Psp orientation is parallel to
[0001] direction (generally, [hkil]convention is used for wurtzite crystal orientation, and i (h k) ). This permanent polarization is essentially related to the bonding nature of these materials, see Fig. 3.1.
33
Figure 3.1: A unit cell of III-N wurtzite crystal. Nitride atoms are represented by silver spheres. Figure is taken from Wikipedia [44].
The main reason behind this permanent polarization is that center of negative charges
(electrons) and positive charges (ions) do not coincide at the same point in the space. An intuitive way of understanding this phenomenon is that bonds connecting an atom to its neighbors are not equivalent (e.g. bonds along [0001] direction in Fig. 3.1). This is not a case for other tetrahedrally coordinated semiconductor crystals such as Si and GaAs which have zinc- blende structure; see Fig. 3.2.
Figure 3.2: A unit cell of zinc-blende crystal. Figure is taken from Wikipedia [44].
34
3.3 Properties of GaN, AlN and AlxGa1-xN
There exist two types of parameters or material properties for any materials, macroscopic and microscopic. The macroscopic parameters are those that can be seen in larger scale such as lattice constants a and c. Conversely, the microscopic parameters cannot be extracted as easy, and more rigorous methods might be needed. Example of microscopic parameters includes bond length and electronic properties such as band gap and polarizations.
3.3.1 Bowing Parameters
AlxGa1-xN alloys can be made by mixing binary compounds GaN and AlN (x and 1-x represent the percentage of AlN and GaN in the alloy respectively). Normally, properties of a compound can be calculated using a linear interpolation called Vegard’s law. For instance, property M for compound AxB1-x can be interpolated as follows:
M AB xM A (1 x)M B (3.1)
However, Vegard’s law cannot be exactly followed in the case of AlxGa1-xN, primarily due to atomic size mismatch in group-III elements. This gives rise to a parabolic third term in Eq. (3.1), which is known as the bowing term. Therefore, Eq. (3.1) turns into:
M AB xM A (1 x)M B bAB x(1 x) (3.2)
where bAB is the bowing parameter.
35
3.3.2 Structural Parameters
There exist structural parameters a and c for GaN and AlN [45], see Tab. 3.1.
Table 3.1: Structural parameters of GaN and AlN.
a and c parameters for disordered (where group-III elements are distributed randomly on cation sites) AlxGa1-xN alloys are the followings:
a xa (1 x)a b x(1 x) (3.3) Al xGa1x N AlN GaN Al xGa1x N
c xc (1 x)c b x(1 x) (3.4) Al xGa1x N AlN GaN Al xGa1x N
where bAlxGa1-xN has values of -0.002 Å and 0.028 Å for a and c structural parameters respectively.
3.3.3 Elastic Constants
The required elastic constants [46] for GaN and AlN are listed in Tab. 3.2.
36
Table 3.2: Elastic constants of GaN and AlN
Also, the elastic constants for AlxGa1-xN can be interpolated using the Vegard’s law.
3.3.4 Piezoelectric coefficients
The required piezoelectric coefficients [45] for GaN and AlN are e31 and e33. It’s important to know that there exist two piezoelectric coefficients e31 for GaN and AlN, which are known as proper and improper e31. The improper e31 is used for computation of electric field in nanostructures. On the other hand, the proper e31 is used when polarization is used to calculate the flow of electric current due to piezoelectric field in an experimental setup such as that used to measure Psp in ferroelectrics. Both of these values are listed in Tab. 3.3.
Table 3.3: Piezoelectric coefficients for GaN and AlN.
Also, the piezoelectric coefficients for AlxGa1-xN can be interpolated using Vegard’s law.
37
3.3.5 Spontaneous Polarizations
Speaking of GaN, there can be grown two types of GaN named Ga-face and N-face, Fig 3.3.
Figure 3.3: Schematic drawing of crystal structure of wurtzite Ga-face and N-face GaN. Figures are taken from Ambacher et al [47].
By Ga-faced we mean that Ga is on the top position of [0001] bilayer, corresponding to [0001] polarity (by convention, the [0001] direction is given by the vector pointing from a Ga atom to the nearest-neighbor N atom). The same definitions are true for AlN.
Psp [45] has a negative (positive) value for Ga-face/Al-face (N-face) GaN/AlN, meaning that it is antiparallel to the conventional [0001] direction, see Tab. 3.4.
Table 3.4: Spontaneous polarization for GaN and AlN
38
The spontaneous polarization for disordered AlxGa1-xN alloys is given by considering the bowing effect:
P xP (1 x)P b x(1 x) (3.5) sp,Al xGa1x N sp,AlN sp,GaN Al xGa1x N
2 where bAlxGa1-xN has value of 0.0191 Cm . This value is to some extent comparable with Psp for
GaN and AlN. Therefore, the effect of spontaneous polarization bowing factor is sensible for
AlxGa1-xN alloys.
3.3.6 Piezoelectric Polarizations
Piezoelectric polarization is an induced polarization due to mechanical deformation (tensile or compression) of the crystal symmetry. It can be computed in two forms for III-Nitride compounds, linear and nonlinear (more accurate). The linear form is the following:
Ppz e33z e31(x y ) (3.6)
where x , y and z are the strains due to the bulk substrate on the top-grown layer, and are given as:
csubs c0 asubs a0 z , x y (3.7) c0 a0
where c0 and a0 are the structural parameters of the unstrained top-grown layer. Likewise, csubs and asubs are the bulk substrate structural parameters. By substituting Eq. (3.7) into Eq. (3.6), we obtain:
39
a a C subs 0 13 Ppz 2 e31 e33 (3.8) aa C33
Moreover, Bernardini [48] showed that at the typical strain values considered in usual III-N based MQW systems, Ppz of the binaries has quite a large nonlinear component. This nonlinearity is modeled by a second-order polynomial:
AlN 2 Ppz 1.808 x 5.624x for x 0
AlN 2 Ppz 1.808 x 7.888x for x 0
GaN 2 Ppz 0.918x 9.541x (3.9)
The nonlinear piezoelectric polarization for AlxGa1-xN alloys is also given by Vegard’s law:
Ppz (Al xGa1x N,x ) xPpz (AlN,x ) (1 x)Ppz (GaN,x ) (3.10)
For the case of MQW grown on a substrate, there exists a buffer alloy substrate that minimizes the global strain. Lattice constant of the buffer material, and consequently its composition are determined (using Vegard’s formula and solve for unknown x composition) once the well and barrier widths and their compositions are known [49]:
C j w j j a a j (3.11) C j w j j 2 a j
2 where aj and wj are the lattice constant and width of each layer, and C j C11 C12 2C13 /C33 is the elastic parameter for each layer.
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3.3.7 Bandgap Energy
Bandgaps of GaN and AlN are listed in Tab. 3.5.
Table 3.5: Bandgaps of GaN and AlN at different temperatures
The compositional and temperature dependent bandgap is defined as:
(x)T 2 E (x,T) E (x,0) (3.12) g g (x) T
Where T is the temperature and Eg(x,0) is the compositional dependence bandgap at 0 K, and can be calculated using Vegard’s law:
E (x,0) xE AlN (1 x)E GaN b x(1 x) (3.13) g g g ALxGa1x N
where bAlxGa1-xN = 1eV is the bandgap bowing factor parameter [50]. In Eq. (3.12),(x)and (x) are the Varshni parameters [51] and are given by:
(x) x AlN (1 x)GaN cx(1 x) (3.14)
(x) x AlN (1 x)GaN dx(1 x) (3.15)
where AlN 2.63 meV/K,GaN 0.94 meV/K, AlN 2082 K, GaN 791 K, c 2.15 meV/K and d 1561K.
41
3.3.8 Electron’s Effective Mass
The value of the electron’s effective mass at the -point is extracted from [52] for different compositions of GaN and AlN in AlxGa1-xN:
* m (x) [0.33x 0.22(1 x)]m0 (3.16)
3.3.9 Dielectric Constants
The values of zero and high frequency dielectric constants [53] are given in the Tab. 3.6.
Table 3.6: Dielectric constants of GaN and AlN at zero (static) and high frequency
The values of zero and high frequency dielectric constants for AlxGa1-xN can also be computed using Vegard’s law.
3.4 Piezoelectric and Pyroelectric Effects in III-Nitride
Nanostructures
For an inhomogeneous medium, Gauss’s law states:
42
D free (3.17)
where D is the electric field displacement and σfree is the free charge density. Similarly, Eq. (3.17) can be used to calculate the polarization-induced charge density, σpol, localized at the interface between two materials carrying polarization fields:
T P .(Psp Ppz ) pol (3.18) where PT is called the transverse polarization which is the summation of spontaneous and piezoelectric polarizations. Here, the negative sign behind σpol results from the convention that polarization goes from negative to positive charge, while electric field does the opposite.
By considering the built-in polarization of pyroelectric materials, the electric field displacement, D, can be written as:
D E Psp Ppz (3.19) where ε is the dielectric constant. More explicitly, D is composed of the external applied electric field, Dext, and the electric field due to the free carriers (electrons and holes), Dfree. Also, Dfree can be split into Dbulk due to the carriers distributed inside the volume of the sample, and Dsurf which arises from the screening effect at the outer surface of the sample. Therefore, D can be rewritten as:
(3.20) where χ is the electric susceptibility, and colors indicating the equivalent physical terms in each segment.
43
Eq. (3.20) is explaining the electric field displacement for a massive homogeneous sample. In
T a multilayered nanostructure made of n layers of thickness lk , transverse polarization Pk and dielectric constant of εk , the expression for surface screening is more complicated:
l PT / D surf k k k k (3.21) l / k k k
Plugging Eq. (3.21) back into Eq. (3.19) and assuming that free carriers screening in the bulk is negligible, we obtain:
l PT / PT l / E k k k k j k k k (3.22) j l / j k k k
This is the electric field expression in each layer of a generic nanostructure (e.g. a generic MQW) where the interfaces between the layers are oriented in [0001] direction.
In the case of nanostructures with a continuous change of composition, a simple rule cannot be given and the self-consistent solution of the differential equation (3.23) is necessary in order to compute the electric field.
d d d (z) E(z) E(z) (z) (D(z) PT (z)) (3.23) dz dz dz
3.5 Resonant Tunneling and Anticrossings
Before we address our overall goal of designing GaN-based QC-laser structures, it is useful to establish some fundamental concepts and terminology. We first consider the phenomena of 44 resonant tunneling, which is used to control the selective injection and extraction of carriers from the active region, but also as a tool to design electronic states with wavefunctions that span multiple wells with varying degrees of overlap. . Let’s first consider
Figure 3.7: (a) Two isolated quantum wells. (b) Two coupled quantum wells. Figures are taken from Williams [1].
two isolated quantum wells shown in Fig. 3.7(a). They both have identical ground energy eigenstates, and wavefunctions | L and | R are confined to their wells. Now, let’s assume the case that the isolating barrier in-between two wells becomes so thin that states | L and | R are no longer isolated, Fig 3.8(b). Consequently, the degeneracy of states and is broken.
These states hybridize to form two new non-degenerate eigenstates; one is symmetric | S with lower energy and the other is anti-symmetric | A with higher energy (as an analogy, one can think of chemical bonding between two hydrogen atoms which results in bonding and anti- bonding states). This phenomenon is called resonant tunneling (RT) as both degenerate states
| L and | R tunneled through the thin barrier and now (newly-formed as non-degenerate states
| S and| A ) have presence in both wells. The minimum energy difference between states
and is called the anti-crossing gap and is represented by 0 EA Es . The bigger 0 represents the stronger coupling between two wells and more effective tunneling. By detuning
45 the initial energies of the two states (for example by varying the electric field), the states can be moved out of resonance, and the eigenstates regain their localized character.
3.6 Oscillator Strength, Intersubband Gain and Threshold
Current Density
Wavefunctions of different states can interfere with each other and cause radiative transition of an electron from one to another due to formation of an electric dipole moment.
Oscillator strength is a dimensionless quantity that expresses the strength of this transition, and is given by:
2 * * m 2m (E f Ei ) zi f fi f fi f ,unscaled 2 (3.24) m0
where fi f is the scaled oscillator strength, Ei the energy of the initial state that electron
transitions from, Ef is the energy of the final state that electron transitions to and zi f is called the dipole matrix element which is defined as:
zi f f | z | i (3.25)
The scaled oscillator strength will also obey the Thomas-Reiche-Kuhn sum rule:
fi f 1 (3.26) i f
46
Eq. (3.26) means that the summation of the probabilities of transitions of an electron from the state i to all available states (excluding state i) will be equal to one.
The small signal bulk gain per unit length for a transition with the population inversion per unit volume ΔN is:
(3D) 2 2 (3D) 2 N e 0 | zi f | N e fi f g( ) ( ) * ( ) (3.27) cn 0 4m cn 0
(3D) (2D) (2D) The three-dimensional population density is written as N N / Lmod , where N
is the two-dimensional population inversion density, and Lmod is the length of the module.
Assuming a homogeneous broadened transition, lineshape ( ) has the form of Lorentzian:
( / 2 ) ( ) 2 2 (3.28) ( 0 ) ( / 2) where
1 1 1 1 1 (3.29) * T 2 i 2 f T
is the full-width half maximum linewidth of the transition centered about ν0. In Eq (3.29), T
* is the total phase breaking time, i and f are the initial and final state lifetimes., and T is the pure dephasing time. Note that in Eq (3.27), ΔN is in units of cm-3, and n is the effective refractive index of the mode of interest. Eq (3.27) relates the growth of the inducing wave intensity to the traveled distance z by:
47
dI g( )I (3.30) dz
In general, a laser starts to operate once the condition
g( ) / (3.31) is satisfied; where is the total loss of the waveguide and is the confinement factor of the
propagating waveguide mode. At this condition g( ) is called the threshold gain, gth ( ) . Also
can generally be separated into two parts: waveguide loss w and cavity or mirror loss m . The waveguide loss (can be calculated using the Drude model) is a contribution of the metal cladding loss and bulk free carrier loss. The mirror loss is defined as:
m ln(R1R2 ) / 2L (3.32) where R1 and R2 are the facet reflectivities and L is the length of the waveguide. Finally, the threshold current density Jth can be expressed using the expression for population inversion
between the upper and lower radiative states N43 (see Appendix A), Eqs (3.31) and (3.27):
4m*cn L J 0 mod (3.33) th 1 1 1 1 1 1 1 1 1 1 ef ( ) i f 43 42 41 43 42 31 21
3.7 Polar Longitudinal Optical (LO) Phonon Scattering
In this section, the electron-LO-phonon scattering will be discussed as the main scattering mechanism in polar heterostructure devices (most of the derivations are taken from Williams
[1]). The phonon dispersion is assumed to be the same as the one in bulk at equilibrium.
Assuming the non-dispersive phonon branch at -point, the scattering rate for an electron 48 initially in state | i,ki (subband i, in-plane wavevector ki) to the final state | f ,k f through an interaction potential H’ is evaluated using Fermi’s golden rule:
2 2 W (k ,k ) f ,k | H'| i,k (E (k ) E (k ) ) (3.34) i f i f f i f f i i LO
The electron-phonon interaction Hamiltonian takes the form of:
' iqr iqr † H (q)(e bq e bq ) (3.35) q
where α(q) is the electron-phonon interaction and bq and bq are the creation and annihilation operators for a phonon in mode q. The Frӧhlich interaction strength for electron-polar-optical- phonon scattering is given in SI unites by:
2 2 LO e 1 1 (q) 2 ( ) (3.36) 2 q dc
-2 where dc and are the static and high frequency permittivities. The q dependence of this term results in reduced scattering rates when large in-plane momentum transfers are required.
This reduces scattering between subbands with a large energy separation.
The matrix element is given by:
2 2 e 1 1 1 2 f ,k | H ' | i,k LO A (q ) (n 1/ 2 1/ 2) (3.37) f i 2 2 i f z ki ,k f q|| LO 2V qz q||
where q|| and qz are the components of the phonon wavevector that are perpendicular (in-plane) and parallel to the growth direction zˆ respectively. n is the Bose-Einstein occupation, and LO
49 the upper and lower signs correspond to phonon absorption and emission respectively. The delta function ensures in-plane momentum conservation, and the form factor:
* * qz z Ai f (qz ) dz f (z) i (z)e (3.38)
is related to the qz momentum uncertainty due to the spatially localized envelope wavefunctions ψi(z) and ψf(z).
This expression can then be integrated over the phonon modes q and final states kf to find the scattering rate from an initial wavevector W(ki). Assuming parabolic subband dispersion, the final states lie on a circle with radius kf determined by conservation of energy:
2m* (E (0) E (0) ) k 2 k 2 f i LO (3.39) f i 2
Energy conservation and the in-plane momentum conservation rule allows us to write the phonon wavevector q|| in terms of ki and kf :
2 2 2 2 q|| ki k f ki k f 2ki k f cos (3.40)
where θ is the angle between the in-plane ki and kf . This is illustrated in Fig. 3.8.
50
Figure 3.8: (a) Electron-LO-phonon scattering scheme. (b) In-plane reciprocal lattice space diagram illustrating the relation between initial and final electron wavevectors ki and kf and in-plane phonon wavevector q||. Pictures are taken from Williams [22].
After summation over the phonon modes q, Eq. (3.34) can be integrated over these final states kf to yield the total scattering rate, Eqs (3.41) and (3.42) for phonon absorption and emission respectively, from an initial wavevector:
m*e2 1 1 2 W abs (k ) LO n dB (q ) (3.41) i f i 2 LO i f || 8 dc 0
m*e2 1 1 2 W em (k ) LO (n 1) dB (q ) (3.42) i f i 2 LO i f || 8 dc 0
where Bi f is given by:
1 q zz' B dz dz ' * (z) (z) * (z') (z') e || (3.43) i f f i i f q||
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Similarly, intrasubband scattering times can also be calculated using the same formulism, simply by setting i=f and ψi (z)= ψf (z).
The total scattering time between subbands i f can then be obtained by averaging over all possible initial states in the subband:
dE f [E ]W (E ) 1 k i k i f k 0 (3.44) i f dE f [E ] k k 0
2 2 * where Ek k|| / 2m is the in-plane kinetic energy in the initial subband, and fi [Ek] is the corresponding Fermi function.
When the transition energy Efi < ELO, rather than performing the full average described in Eq.
(3.44), the thermally activated expression
1 E E (hot) fi LO Wi f exp (3.45) i f kBT
(hot) where Wi f is the scattering rate at the lowest energy Ek in the subband where LO-phonon scattering is energetically allowed.
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3.8 Simulation Results
3.8.1 Design at 3.93 THz
Based on all the information presented in this chapter, I designed and simulated two 4-level
GaN/AlGaN QCLs candidate designs at the operating design biases. The first QCL is designed at
3.93 THz (corresponding to the energy difference between upper-state 4 to lower-state 3), see
Fig 3.9. One period of this QCL is made of alternating layers of GaN/Al0.15Ga0.85N with layer
Figure 3.9: GaN/Al0.15Ga0.85N quantum cascade laser designed at 3.93 THz. The conduction band offset is taken to be 0.2 eV [32]. Electrons are injected from state 1’ to state 4 through tunneling. Then they lose their energy as photon emission and relaxe into state 3. Finally, after they tunnel through into state 2, they relax into state 1 via electron-LO-phonon scattering.
thickness of 27/53/16/30/26/75 Å (bold-faced numbers correspond to the barriers). At the design bias, electrons are injected from state 1’ (magenta) into upper-state 4 (cyan) through tunneling.
53
Then, they relax into lower-state 3 (black) and emit a photon with an energy equal to the energy difference of state 4 and 3. After that, they tunnel though into state 2. Finally, they relax into ground-state 1 (injecting state of the next module) via electron-LO-phonon scattering. This mechanism repeats in a cascading manner for the next modules, and at each step a photon will be emitted. The scheme shown in Fig. 3.9 is designed at the design bias of 62.6 kV/cm. Fig. 3.10 illustrates the energy difference among different states versus the applied bias.
Figure 3.10: Anticrossing gaps Δ (graphs’ minima) among different states. In an ideal design, minima of 3-2 and 1’-4 has to align at the same bias.
At the design bias of 62.6 kV/cm anticrossing gaps between states 3 and 2, and states 1’and 4 happen to be nearly aligned. At this condition, tunneling mechanism is allowed for these states.
In other words, at the design bias, electron population can be built up in the upper state while electrons are extracted from the lower state. Therefore, the population inversion which is critical for the laser operation can be maintained.
54
In this design, the energy difference between states 2 and 1 is 135 meV which is higher than the LO-phonon energy of ~ 90 meV in GaN. It turned out to be very difficult to come up with a design near GaN LO-phonon energy. This is mainly due to the huge built-in electric field in
GaN/AlGaN heterostructures which produces a saw-tooth band edge profile (see Fig. 3.9).
Here, I also performed the electron-LO-phonon scattering time calculation based on Eq. 3.42.
For simplicity, I assumed that all the electrons are at the bottom of a subband (ki=0). Using the temperature of 300 K (same as the one used for the design), I computed the scattering time from state 2 to 1, τ2-1 = 0.0443 ps, which is very fast compared to the case of GaAs/AlGaAs QCL (~17 ps for an energy difference of ~37 meV at 0 K).
Fig. 3.11 also shows the oscillator strength (f) calculation for states 4-3, 1’-3 and 2-P (P is the
parasitic state). In order to operate at a single frequency, it is desired that f 4,3 to be high relative
to the other states’ oscillator strength. However, f 4,3 and f 2,P seem to be very close in this design, which was a challenge to overcome. Near the design bias, oscillator strength f
Oscillator Strength versus bias Electric Field 0.5
0.4
0.3
0.2 Oscillator Strength(f) Oscillator 0.1 43 P2 1'3 0 61 61.5 62 62.5 63 63.5 64 64.5 65 Electric Field(KV/cm)
Figure 3.11: Oscillator strength (f) for states 4-3, 2-P and 1’-3.
55 between the upper and lower radiative states is approximately equivalent to the summation of f4,3 and f1’,3 due to the tunneling phenomenon. Therefore, f has a value of ~ 0.45 near the design bias.
Lifetime of each state with an energy separation less than the LO-phonon with respect to another state can be calculated using Eq. (3.45). In this calculation, it’s assumed that sufficiently excited electrons within the initial subband would scatter to the bottom of the final subband
(kf=0). For simplicity, Jth can be estimated using Eq. (3.33) at the design bias 62.6 kV/cm with the following approximations:
1 gth m w (3.45)
1 1 1 1 * 4 , 3 T (3.46) T 2 4 2 3
Taking into account all the relevant transition lifetimes (see Appendix A), Eq. (3.46) can be rewritten as:
1 1 1 1 1 1 1 (3.47) T 2 41 42 43 31 21
-1 It’s been assumed that w 10 cm (taken from [1] assuming the bulk free carrier density of
51015cm-3). The reflectivity at GaN/air interface is R= 0.28 and the length of the Fabry-Pérot
-1 ridge laser is assumed to be L=2 mm; which results in m 6.4cm . Therefore, the total loss is
-1 16.4cm . Having computed 43= 0.5 ps, 42 = 1.7 ps, 41=49.7 ps, 31= 0.16 ps, 21= 0.044 ps result in = 5 THz. Subsequently, the lineshape at the peak frequency of 3.93 THz will be
( 0 ) 127.4 fs. Ultimately, Jth at the peak frequency can be estimated using Eq. (3.33):
2 Jth = 8100 A/cm
56
3.8.2 Design at 5.4 THz
The second QCL is designed at 5.40 THz, see Fig 3.12. One period of this QCL is made of
Figure 3.12: GaN/Al0.15Ga0.85N quantum cascade laser designed at 5.4 THz. The conduction band offset is taken to be 0.2 eV [32]. Electrons are injected from state 1’ to state 4 through tunneling. Then, they lose their energy as photon emission, and relax into state 3. Finally, after they tunnel through into state 2, they relax into state 1 via electron-LO-phonon scattering.
alternating layers of GaN/Al0.15Ga0.85N with layer thickness of 27/53/14/27/26/74 (bold-faced numbers correspond to the barriers). The scheme shown in Fig. 3.12 is designed at the design bias of 67.4 kV/cm. Fig. 3.13 illustrates the energy difference among different states versus the applied bias.
57
Figure 3.13: Anticrossing gaps Δ (graphs’ minima) among different states. In an ideal design, minima of 3-2 and 1’-4 has to align at the same bias.
In this design, the energy difference between states 2 and 1 is 135.46 meV. I also performed the electron-LO-phonon scattering time calculation based on Eq. 3.42. Again, I assumed that all the electrons are at the bottom of a subband. Using the temperature of 300 K (same as the one used for the design), I computed the scattering time of τ2-1 = 0.0385 ps.
Fig. 3.14 also shows the oscillator strength (f) calculation for states 4 and 3, and states 2 and
P. However in this design, compared to the previous one, f3,4 is much higher than f 2,P which is a much more desirable result in order to achieve a single frequency operation. Near the design bias (as it was explained in the first design), the oscillator strength can be approximated ~ 0.55.
58
Oscillator Strength versus bias Electric Field 0.7
0.6
0.5
0.4
0.3
0.2 Oscillator Strength(f) Oscillator
0.1 43 P2 1'3 0 65 66 67 68 69 70 Electric Field(KV/cm)
Figure 3.14: Oscillator strength (f) for states 4-3, 2-P and 1’-3.
Having computed 43= 0.3 ps, 42 = 3 ps, 41= 58 ps, 31= 0.27 ps, 21= 0.039 ps result in
= 5.33 THz. Subsequently, the lineshape at the peak frequency of 5.4 THz will be ( 0 )
119.4 fs. Ultimately, Jth at the peak frequency can be estimated using Eq. (3.33):
2 Jth = 10608 A/cm
3.9 Conclusion
Although QCLs based on GaN/AlGaN heterostructure active region have been proven very promising for the next generation of QCLs, the design part seems very challenging. In the mentioned designs, the energy difference between states 1 and 2 was ~135 meV which is much higher compared to GaN LO-phonon energy of ~90 meV. This effect, which is mainly due to the huge built-in electric field in GaN/AlGaN QWs, can be reduced possibly by using different
59 layers of GaN and AlN alloys with different composition. This would allow more ideal resonant
LO-phonon depopulation.
Additionally, it was seen that anticrossing tunneling occurs more efficiently and less bias sensitive in the case of 5.4 THz QCL compared to the 3.93 THz one. It also shows much better oscillator strength in the case of 5.4THz QCL. Having said all, it brings us to the conclusion that designing GaN-based QCL in higher frequencies could perform more ideally than those in lower frequencies.
Finally in both designs, a much smaller scattering time (~ 0.04 ps) between the state 2 and state 1 (ground state) was observed. This would widen the linewidth and consequently increase the threshold current density (as it was estimated for both designs), which is not desirable.
2 Considering a smaller linewidth of = 2 THz in both cases would result in Jth = 3233 A/cm
2 and Jth = 4000 A/cm in the first and second designs respectively, which is much more improvement (Note that is related to states’ lifetimes. However, in the following assumption only the decrease of is considered. Taking into account the corresponding changes in the lifetimes would result even better Jth).
60
Chapter 4
Fabrication of GaAs/AlGaAs Terahertz Quantum
Cascade Laser with Integrated Radial Probe
4.1 Introduction
Despite having impressive output power, a divergent output radiation beam pattern has been one of the important challenges for THz QCLs. As a solution, we have proposed a QCL with slightly different design. We’ve designed and fabricated a metal-metal QCL with addition of an integrated coupling radial probe on one facet using micro-fabrication techniques. The laser is designed to be mounted in a micro-scale full-height rectangular waveguide. The radial probe would help transitioning the laser’s output into a waveguide TE10 mode. The waveguide will terminate in a diagonal horn antenna which will improve the laser’s highly divergent output radiation beam. Achievement of a quality radiation beam pattern would make QCLs able to be employed in various terahertz instruments. Furthermore, it would allow for integration of THz
QC-lasers into the widely used split-block rectangular waveguide technology. Split block waveguide is the standard for housing THz electronic devices (such as mixers, multipliers, and phase shifters) in the sub-1-THz range, but increasingly used at higher frequencies (particularly in the astrophysics and space science community). This project was performed as part of a collaboration with Dr. Goutam Chattopadhyay at NASA Jet Propulsion Laboratory (JPL). Our role was to develop the laser with radial probe, where JPL would fabricate the rectangular waveguide and horn block. 61
4.2 Waveguide Simulation and Design
Terahertz QC-lasers currently require nontrivial waveguide designs for simultaneous single- mode low-divergence emission with efficient power out-coupling. The highest temperature performance has been achieved using so-called double-metal waveguide (Fig. 4.1), where the active gain material is sandwiched between metal cladding layers fabricated using metallic wafer bonding. This yields a waveguide scheme very similar to microwave microstrip transmission-line.
Figure 4.1: A Metal-Metal QCL and the corresponding active region’s layered structure. It also shows the 2D first fundamental mode profile at λ=80 μm for a 10 μm tall and 50 μm wide structure. Figure of the Metal-Metal QCL is taken from Kumar [54].
Metal-metal waveguide provides a very good confinement factor (i.e. lower loss factor and higher gain) which consequently leads to a smaller lasing threshold current. This is due to the fact that propagating THz mode is tightly confined in-between the metal claddings, which causes a large mode mismatch between the waveguide mode and the free-space propagating mode.
However, this leads to a poor output-coupling of the light from the facets (high facet reflectivity), a highly divergent radiation beam pattern from the edge emitting lasers (Fig. 4.2), and easy excitation of the lateral modes (along the width of the waveguide) which makes the laser multi- moded even for narrow waveguides. 62
Figure 4.2: Far-Field radiation beam pattern for a metal-metal QCL (length=670 μm, width=25 μm, height=10 μm) at λ=102.7 μm. A highly divergent beam pattern is shown in two different angles. Figures are taken from [55].
Various works have been done in terms of waveguide engineering to overcome these THz
QCLs’ challenges. Low-loss and low-threshold current THz QC-lasers with mode selectivity can also be achieved by embedding QC-lasers into an engineered photonic crystal structure [56].
Moreover, Kumar et al [57] demonstrated that single mode frequency can be achieved using a combination of second-order distributed feedback (DFB) and metallization of laser ridge side walls which suppresses the lateral modes. Also harnessing third-order DFB [58] has proved to be successful in terms of efficient out-coupling. Besides, engineering radiation beam pattern has shown impressive progress. Metamaterial-based gradient (GRIN) lenses are used to focus the
THz QC-laser’s output radiation beam [59]. Alternatively, horn antenna has been utilized as a good option either monolithically [60] or attached externally [61] to the waveguide to direct the divergent beam pattern. Plasmonic collimators (e.g. second-order gratings and metasurfaces) [62] 63 sculpted on the facet of the laser’s substrate are also demonstrated to help the beam directionality by reradiating the coupled output power from the laser’s facet edge (i.e. mimicking antenna functionality). Additionally, steering of output propagating beam from the laser has been shown using one-dimensional metamaterial-based waveguide which acts as a leaky antenna [63].
As the main goal of this research, a QCL with excellent beam pattern and high efficiency, suitable for use as a local oscillator in a heterodyne spectrometer for astrophysical and space science instruments is to achieve. Therefore as a solution to the aforementioned problems, we proposed our own method of waveguide implementation by introducing a radial probe to one facet of the laser which is mounted on a full-height rectangular waveguide.
In submillimeter and THz technology such as superconductor-insulator-superconductor (SIS) mixers, extended-into-waveguide planar radial probe [64] has been proven to efficiently couple the signal from thin-film microstrip line to full-height rectangular waveguide (see Fig. 4.3), resulting in broad RF bandwidth. In other words, it helps smooth transitioning of the signal from one medium to another via impedance matching. It also has a convenience at which the active device can be biased and the IF signal extracted.
Figure 4.3: 90o radial probe on a thin-film substrate mounted on a split block waveguide via SMA connector. The orientation of the probe is parallel to E-field of the TE10 waveguide mode. Picture is taken from Kooi [64].
64
Waveguide blocks, Fig. 4.4, have been used as a standard technique with the most possible simplicity in terms of configuration and fabrication for submillimeter-wave and terahertz waveguide circuits (e.g. in conjunction with thin-film diode mixers such as MOMED (monolithic membrane diode mixers) [65].
Figure 4.4: A 180-300 GHz sideband-separating mixer block includes an eight-branch quadrature hybrid coupler, two-branch LO directional couplers, and an in-phase LO power splitter. Picture is taken from Bruneau [66].
Likewise, in our method, a probe is designed to enhance the out-coupling from the laser and to excite the fundamental TE10 mode in the rectangular waveguide (Fig. 4.5a). This way, output beam divergence is suppressed by propagating through the waveguide. The coupling (smooth transition) from the waveguide to the free space is done by a horn antenna (Fig. 4.5b) compensating for the impedance mismatch between the waveguide and the air, and creating a much directive radiating beam pattern.
65
(a) (b)
Figure 4.5. (a) Mounted QCL into the waveguide and the TE10 mode’s E-field profile. The color arrows describe the TE10 mode profile as red and blue correspond to the max and the min respectively. (b) Micro- machined half waveguide block by JPL.
Optimum coupling from the QC-laser to the waveguide requires impedance matching. The
Probe’s impedance depends on different parameters [64], see Table. 4.1.
Table 4.1: Probe’s impedance’s dependence on different parameters. εr=12.9 is the dielectric constant of o o GaAs and Δθ|90 is the probe’s neck’s deviation angle from 90
0.5 Re[Z ] Probe Radius, , | o probe r 90
Im[Z probe] Substrate Size, Waveguide Height
Based on the rectangular waveguide’s impedance in the waveguide block, a 14 μm probe is designed on one facet of the laser (Fig. 4.6) in order to most efficiently out-couple the energy from the laser into the waveguide. As the simulation below shows the best result occurs at frequency of about 2.77 THz (Fig. 4.7).
66
(a) (b)
Figure 4.6: QCL’s design and attributes; (a) Top view, (b) 3D view. The bottom metal is extended 2 μm from each side relative to the top metal. There is also a 50 μm extension, also known as the beam lead, from the laser’s top metal to serve as the bias point.
Figure 4.7: Simulated S-Parameters by Jet Propulsion Laboratory (JPL), S11(reflectivity, dark blue) and
S21(transmittivity, light blue) for the radiation of the laser with a 14 μm radial probe to the waveguide.
67
4.3 Micro-Fabrication of QCL with a Radial Probe
The primary challenge of this research is to development a process to microfabricate the THz
QC-laser ridge waveguide with an integrated waveguide probe. A large fraction of this work took place within the microfabrication facilities at the UCLA Nanolab and the CNSI Integrated
Nanomaterials Lab.
The steps can be summarized:
1. Photolithography and top metal evaporation
2. Ridge etching
3. Wafer bonding
4. Substrate removal
5. Backside processing
6. Release
The process began with cleaved wafer pieces of size about 1cm1.5cm (Fig. 4.8) containing
10 μm of GaAs/AlGaAs quantum wells that make the laser’s active region (AR). It also included a 0.7 μm n doped GaAs layer and a .15 μm etch-stopper between the substrate and AR.
Figure 4.8: Growth layers from top to bottom are AR, doped GaAs layer (not shown), etch-stopper and GaAs substrate.
68
The very first step of the process was image reversal [Appendix B.1] photolithography (Fig.
4.9) required for the top metal evaporation. AZ nLOF 2020 was used as negative photoresist.
After development step, it was seen that features as small as 3 μm were well developed (the smallest feature of our mask was the neck of the probe which was at least 3 μm considering all the neck size variations on the mask).
Figure 4.9: AZ nLOF 2020 photoresist after development. The inward slope of photoresist’s side walls after image reversal step is shown.
A two-minute O2 plasma ashing needed following the photolithography step to remove any residual photoresist.
Pieces should be dipped 10 seconds in BOE prior to metal evaporation to remove the oxide formed on the surface. This step would enhance the surface cleanliness and the quality of metal/semiconductor non-alloyed Ohmic contact which is a very important factor for the metal evaporation step. The metal evaporation involved three different metals (Fig. 4.10); 20 nm Cr was the adhesion layer, 200 nm Au was the top metal of the waveguide and 550 nm Ni was the etch mask during the dry etching.
69
Figure 4.10: Different evaporated metal layers (20 nm Cr, 200 nm Au and 550 nm Ni)
Following this step, the pieces were left in acetone overnight for the metal lift-off. Acetone dissolves the photoresist and metal will be patterned only on the desired areas (Fig. 4.11).
Figure 4.11: The piece after metal lift-off; the cross section (left) and the top view (right).
After that, the pieces were ready for the etching step to create the laser ridges. The etching was done by the Chlorine Plasma Etcher (Appendix B.2). Since the pieces were small, they had to be mounted on a 4” carrier wafer individually by cool grease (a good heat conductor) in order to be loaded into the chamber. A silicon wafer was used as the carrier wafer. It was covered by a
few microns of SiO2 to produce a consistent etching rate (the etching rate of ~0.5 μm per minute was already characterized using the mask). Some pieces were etched past the etch-stopper and some before (Fig. 4.12). The etch time was approximately 20 min for etching 10 μm in each piece.
70
Figure 4.12: The ridge etched past the etch-stopper (left) and before the etch-stopper (right). Ni residues are already removed by TFG solution.
The pieces were also dipped 2 minutes in nickel etchant TFG solution (sulfuric acid and sulfonate) to remove the Ni residues (since wire bonding on Ni is more difficult than Au). The
SEM images below show how the ridges would look like after etching. We can see that the side walls look pretty smooth (Fig. 4.13).
Figure 4.13: SEM images of the ridges after the dry etching by Chlorine Plasma Etcher; the ridge’s end (left) and ridge’s probe (right)
Next, a 1 μm PECVD SiO2 layer was deposited (Appendix B.3) on the ridges (Fig. 4.14). As we can see, the conformality looks pretty good.
71
Figure 4.14: PECVD oxide deposited on the ridge (left); the SEM image (right).
Following that step, a bonding agent called BCB (benzocyclobutene) was spun (Appendix
B.4) on the pieces (Fig. 4.15). BCB is a polymer which becomes hard at elevated temperature. It
has a very good adhesion to SiO2 and Si but not to III-V materials. It also planarizes well to form a flat surface over the ridges.
Figure 4.15: To cover the ridge fully, more than 10 μm of thick (high viscous) BCB is spun on the PECVD-oxide-deposited ridge.
Then, the pieces were bonded to Si substrate (Fig. 4.16) by the Karl-Suss bonder under thermal bonding conditions (Appendix B.5). Additionally, a piece of graphite was placed on top of the Si substrate to help equal distribution of the tool force and relieve the stress on the sample.
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Figure 4.16: Bonded piece to the Si substrate via BCB
After the bonding was done, GaAs back substrate had to be removed. This was done by mechanical lapping.1 μm slurry (aluminum oxide) was used as the lapping material. The lapping was continued until reaching to about 70 μm thickness of the GaAs piece. Then the rest of 60 μm (excluding the AR thickness) was removed by chemical etching. The etchant solution’s ratio was
NH 4OH : H 2O2 1:19. The chemical etching almost took 40 minutes for each piece. Fig. 4.17 shows the pieces after the back-substrate removal.
Figure 4.17: The pieces after the back-substrate removal; the one etched past the etch-stopper (left) and the one etched before the etch-stopper (right).
Next, the etch-stopper (AlxGa1-xAs) and 0.7 μm n doped GaAs layer had to be removed by
HF and the etchant solution made by NH 4OH : H 2O2 : H 2O 10:6:480 (etching rate of 200 nm/min) respectively. These two steps were done differently for two types of etched ridges. For the one which was etched before the etch-stopper, it was done first by etching the etch-stopper and then by etching the 0.7 doped GaAs layer (Fig. 4.18, 4.19).
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Figure 4.18: The ridge etched before the etch-stopper; the etch-stopper is removed by HF (left) and 0.7 μm n doped GaAs layer is removed by NH 4OH : H 2O2 : H 2O solution (right).
Figure 4.19: The SEM image of a 5μm-active-region test piece after the etch stopper and doped GaAs layer removal.
But the ridges which were etched past the etch-stopper, first, required a photolithography
step to protect the sacrificial SiO2 from HF contact during etch-stopper removal. The photolithography needed for this step was merged with the photolithography step required for the back metal evaporation. As it’s shown in Fig. 4.20, after the photolithography step (identical to the one for the top metal evaporation), both etch-stopper and 0.7 μm doped GaAs layer were removed by the corresponding solutions which were already described.
Figure 4.20: The ridge etched past the etch-stopper. Etch-stopper and 0.7 m doped GaAs layer are removed by HF and solution respectively.
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Then, the process could proceed to the metal evaporation step; Note that we also had to do the photolithography step for the pieces which were etched before the etch-stopper. The same metals and thicknesses were chosen for the top metal evaporation (Fig. 4.21).
Figure 4.21: Bottom metal evaporation (20 nm Cr, 200 nm Au and 550 nm Ni)
It then was followed by an overnight lift-off in acetone (Fig. 4.22).
Figure 4.22: The piece after acetone lift-off (left) and SEM image of the ridges (right); It is seen that metal still exists on the tail and probe side of some of the ridges.
Next, the GaAs above the tail and probe was removed (Fig. 4.23) by dry etching (same as the dry etching for the ridges) using Ni as a mask.
Figure 4.23: The cross section view of the piece after etching the GaAs above the tail and probe (in reality, some GaAs would still be left on the tail and the probe to avoid etching through the top gold). 75
Finally, the laser ridges were released (Fig. 4.24) by soaking the pieces in BOE solution to etch 1 μm sacrificial SiO2 layer in between the ridges and BCB. This step took about 30 to 40 min using 1:10 ratio BOE.
Figure 4.24: The released device (intended) from the substrate by BOE etching; cross-section view (left) and 3D view (right).
The released devices were collected on a filter after the BOE solution, which is diluted by water, is completely drained. Then, they are mounted into the waveguide block (Fig. 4.25). This part of the work was done by a specialist in JPL.
(a) (b)
Figure 4.25: (a) The mounted laser into the waveguide block. (b) A color-contrasted zoomed-in version of (a).
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4.4 Conclusion
Although the proposed devices were successfully fabricated, the final yield of the good devices was very little. That was mainly due to an imperfect bonding process which resulted in spider-shape cracks for bonded pieces. The most likely reason for the cracks is the very non- uniform thick BCB spun on the pieces. Since our ridges are 10 μm tall, to cover them fully, the thick and very viscous BCB have to be spun with a quite slow speed of 2500 rpm, which produces huge edge beads. The bonding, however, was strong enough to survive the lapping step.
Consequently, it resulted in obtaining a very low yield of laser ridges.
Also as it was already mentioned, despite leaving the pieces in acetone overnight and using
Ultra-Sonic, the bottom evaporated metal didn’t come off from the tail and probe areas in many ridges (Fig. 4.26). This happened more likely because the ridges’ heights were about 1 m lower than its surrounding BCB and oxide, see Fig.4.20. Therefore, it might have required extra effort to lift off the metal from those areas, which would have possibly incurred more damages to the ridges and especially to the narrow-necked probes.
Figure 4.26: The remained metal on the tail and probe (left); the zoomed-in look (right).
77
Unfortunately, the lasing did occur on the mounted device shown in Fig. 4.25 due to the mentioned issues.
I also did a second batch of fabrication in order to improve the bonding step and avoid getting cracks. This time, instead of spinning the BCB at low rpm (resulting thicker and less uniform layer) on the piece only, it was spun at higher rpm (resulting thinner and more uniform layer) on both the piece and the handling substrate (selected as GaAs to avoid the thermal expansion mismatch between GaAs and Si as one of the probable reasons of creating cracks).
Also, the bonding was done on a hot plate (see Appendix B.6) in order to manually control the process. Although, first, I ran into the problem of moving the piece and the substrate due to reflowing of BCB, I finally kept them fix by putting relatively heavy weight around the bonding setup. Fortunately, no cracks were observed after removing the substrate. However, I ran into a problem of misaligning during the back-metal-evaporation photolithography step. It was observed that the mask and pieces were misaligned in an increasing order on the right and left of an aligned ridge. This would imply some bowing effect which occurred due to either stress or thermal expansion during the process. Therefore, it decreased the yield of the fabricated devices drastically. After I attempted the release of these devices, they looked damaged and many of them did not have the top metal, see Fig. (4.27)
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Figure 4.27: SEM images of the release devices. It is shown how metals are come off. In overall, this fabrication process has been found very challenging. The somewhat unsuccessful results of both fabrication batches are strong reasons to rethink the whole process in a scrutinizing manner and improve some steps or consider other possible approaches.
Deformation of the laser ridge was the fact which was observed as an important issue. Stress plays a very critical role in this process. The ridges experience the stress in steps like bonding, lapping and also during the dry etching the GaAs underneath the probe and tail. Moreover, the
200 nm thin probe with a narrow neck seems to be very sensitive to any mechanical force variation (as we were later informed that the probe fell off of the only mounted device, see Fig.
4.25). In order to better evaluate the stress tolerance of our structure, precise MEMS simulations need to be preformed. As a possible solution to this issue, bonding on a hot plate should be well characterized using a lighter bonding weight and at a lower temperature. This would help to alleviate the stress and thermal expansion on very fragile GaAs pieces. Also, characterizing the needed Ni mask thickness would be very important during both top and bottom metallization step since a thicker Ni layer would increase the stress on the ridges.
One additional and quite important step in the process was to remove the extra 700 nm doped
GaAs layer (Fig. 4.18) in the pieces. As it was already explained, the pieces that were not etched
79 through etch-stopper saved us one extra photolithography step compared to the pieces which were. However, 700 nm seems relatively thick as it produced a dip (Fig. 4.20) on the piece’s back surface, which consequently caused problems during lift-off step (Fig. 4.26). Therefore, a thinner layer might be a better option since it takes less effort to remove it.
Although hard-cured BCB seems to invulnerable to acetone, in practice I found that spray bottle acetone during standard cleaning would cause BCB to delaminate. It is very critical to keep away from excessive use of acetone. Unfortunately, I lost a small fraction of my devices due to this unforeseen effect.
Additionally, I think that release of the devices in BOE can also be tried using HF vapor. HF vapor molecules have more freedom to penetrate in the gap between the BCB and bonded device, which seems to be more efficient in terms of etching time and etching profile. Dealing with the gas-phase HF vapor also seems to be less messy compared to the BOE solution.
Finally, more precise RF-based electromagnetic simulations should be done considering the effect of the residue GaAs underneath the probe. In reality, it’ll be almost impossible to remove all the residue since it might damage the probe as the GaAs gets etched more unless a very well characterized etching scheme is followed. This residue will definitely play a non-negligible role in the impedance matching calculations in order to determine the probe’s geometry.
Considering all the issues I mentioned, I believe that this is a doable project. However, to accomplish this novel approach, it might require more work and funding. In terms of THz waveguide radiation beam pattern engineering, there exists more developed and somewhat less complicated methods with well proven results such as DFB waveguides, integrated horn antennas and plasmonic collimators. In terms of JPL’s goal from this project (a QCL with 80 excellent beam pattern and high efficiency, suitable for use as a local oscillator in a heterodyne spectrometer for astrophysical and space science instruments), they are capable of producing frequency up to 3 THz using planar GaAs Schottky diodes multiplier chains and the monolithic membrane (MOMED) Diode technology [60]. Although Schottky diodes can likely produce enough power up to 3 THz to pump one mixer, future detector concepts involve arrays of heterodyne mixers. These arrays will need correspondingly more power, which is a motivation for QC-lasers as local oscillators even in the 2-3 THz range. Moreover, for frequencies higher than 3 THz, QCLs might be the only available sources of light. Therefore, it seems the idea of using QCLs in conjunction with full-height waveguide blocks is promising if and only if every step of this project is analyzed carefully, and in the presence of ample time and sufficient funds.
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Appendix A
Four-Level Resonant- Phonon QCL Rate Equations
In this section, I derive the rate equations for a four-level laser. The results are then used to calculate the population inversion density between the upper and lower states. In Fig. A.1, the population densities in states 3 and 2 are assumed to be equal due to the tunneling between two states. So hereafter, I name the upper and lower radiative states state 4 and state 3 respectively.
(2D) The 2D population inversion density between states 4 and 3 is represented by N 43 .
Figure A.1: The schematic of a four-level laser. Lifetime between two states is shown by τ. Radiative photon and non-radiative phonon emissions are shown by red and blue respectively.
dN (2D) J N (2D) N (2D) N (2D) 4 4 4 4 dt e 43 42 41 (A.1) dN (2D) N (2D) N (2D) N (2D) N (2D) 3 4 4 3 3 dt 43 42 31 21 82
Solving (A.1) in steady-state condition results in:
J 1 1 1 (2D) N 4 e 43 42 41 (A.2) 1 1 1 1 (2D) (2D) N 3 N 4 31 21 43 42
After some simplifications, (A.2) can be rewritten as:
1 J 1 1 1 (2D) N 4 e 43 42 41 (A.3) 1 1 J 1 1 1 1 1 1 1 (2D) N3 e 43 42 41 43 42 31 21
(2D) Ultimately, N 43 can be written as:
1 1 J 1 1 1 1 1 1 1 N (2D) N (2D) N (2D) 1 43 4 3 e (A.4) 43 42 41 43 42 31 21
(3D) (2D) Note that to compute the 3D population inversion density; it follows as N N / Lmod , where Lmod is the thickness of one QCL module. Therefore, Eq. (A.4) can be rewritten as:
1 1 J 1 1 1 1 1 1 1 N (3D) (N (2D) N (2D) ) / L 1 43 4 3 mod L e (A.5) mod 43 42 41 43 42 31 21
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Appendix B
Processing recipes
B.1 AZ nLOF 2020 Photolithography
(1) Wafer prebake at 150 oC for 3 minutes on hotplate, then cool to ambient.
(2) Soak in HMDS vapor for 15 minutes.
(3) Deposit AZ nLOF 2020, spin at 4000 rpm for 30 seconds.
(4) Wafer softbake at 110 oC for 60 seconds on hotplate.
(5) Clean backside of the wafer with a swab.
(6) Karl-Suss exposure for 12 seconds (8 mW/cm2 power, 365 nm wavelength).
(7) Wafer postbake at 110 oC for 60 seconds on hotplate.
(8) Develop in AZ 300 MIF for 75 seconds, DI water rinse and blow dry with N2.
B.2 GaAs dry etching
Dry etching was performed in the Unaxis SLR770 ICP-RIE using the ATGAAS.RCP recipe as listed below. Etch rate is 0.5 μm/min.
BCL3 flow rate: 50 sccm.
Pressure: 10 mTorr.
RF1 (RIE): power 100 W.
RF2 (ICP): power 800 W.
Temperature: 21 oC. 84
B.3 High-Deposition Rate SiO2 PECVD
PECVD SiO2 deposition was performed in the STS using HIDROXID.RCP recipe as listed below. Calibration runs on blank Si wafer(s) before performing the deposition is essential, since the deposition rate varies from run to run and/or may not be linear in the first few minutes.
Deposition rate is ~3000 Å/min.
N2O flow rate: 2000 sccm.
SiH4 flow rate: 50 sccm.
Pressure: 400 mTorr.
LF power: 140 W.
Temperature: 300 oC.
B.4 3022-63 BCB Spinning
(1) Standard cleaning using acetone, methanol, IPA and water; N2 blow to dry.
(2) Dehydration for 3 min at 150o C on hotplate.
(3) Deposit AP3000 adhesion promoter. Spin at 3000 rpm for 30 seconds.
(4) Wafer bake at 100 oC for 1 min (adhesion enhancement of AP3000) on hotplate.
(5) Deposit BCB, spin at 2500 rpm for 30 seconds.
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B.5 Thermal Bonding in Machine
Bonding was done in the Karl-Suss bonder. Prior to bonding, surfaces of blank substrate and
BCB-deposited wafer should be free of any undesired particles. The table below shows the recipe used for thermal bonding of BCB-deposited GaAs piece on Si substrate.
Table B.1: The Karl-Suss bonder recipe for BCB bonding.
B.6 Thermal Bonding on Hotplate
Bonding was done on a hotplate under air ambient at 250 oC for 2 hours. A weight was used for bonding on top of the stack (see Fig. B.1). Three fixed weights were used to prevent the whole stack to drift around during BCB reflowing. The bonding setup is shown below.
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Figure B.1: The bonding setup using weights on a hotplate; Left (top view), Right (side view).
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