Quantum Cascade Laser Code Prediction
Dr. Christopher Baird Submillimeter-Wave Technology Laboratory University of Massachusetts Lowell February, 2009
OUTLINE
I. Current Work A. Introduction & Code Structure Overview B. Finding the Wave Functions C. Finding the Scattering Times D. Finding the Populations and Gain II. Future Work
WHAT IS A QCL?
A Quantum Cascade Laser (QCL) is a next-generation device that uses electron transitions between man-made quantum well levels, as opposed to traditional lasers that use transitions between atomic levels. WHAT IS A QCL?
Quantum wells are built by stacking up alternating layers of semiconductors with thicknesses of only a few atoms.
WHAT IS A QCL?
Molecular Beam Epitaxy (MBE) system used at UMass to build QCL's layer by layer.
HOW DOES A QCL WORK?
E AlGaAs GaAs AlGaAs E (z) c Ψ (z) Conduction V(z) Φ(z) n Band E E (z) n F Φ(z) E (z) Φ(z) E (z) g v
Valence Band -eΔΦ app z
ΔΦ : Externally applied voltage drop E : Energies of possible quantum states applied n Φ(z): Built-in potential E (z) : Fermi energies F E (z) : Conduction band edge energy in bulk E (z) : Valence band edge energy in bulk c v V(z) : Effective conduction band edge energy E (z) : Band-gap energies in bulk g Ψ (z): Wave functions n HOW DOES A QCL WORK?
By properly adjusting the layer thicknesses, the quantum states are fine-tuned until lasing occurs. The electrons cascade down through the quantum states like a waterfall, emitting laser radiation at each drop. WHY TERAHERTZ?
Terahertz (THz) radiation is used at UMass Lowell's STL and in the military for radar imaging of scaled targets. Expensive, room-sized gas lasers must be currently used to generate THz radiation. QCL's are much more compact and potentially cheaper at providing THz radiation. WHY TERAHERTZ?
Terahertz (THz) radiation is used to image vehicles and to pursue EM scattering, chemical sensor, and medical research.
STUDENT INVOLVEMENT
Our QCL teams rely on student involvement, providing them with hands-on experience in high-tech research. We are continually looking for additional students to join our team. Current student involvement includes: ● High School Students through the SOS program ● Undergraduate Students working part-time ● Graduate Students working as research assistants ● Graduate Students working on dissertation research
Our Team's Efforts
Our QCL team includes groups at STL, the Photonics Center, in collaboration with groups at Sandia, etc.
● Theoretically model, predict, and design QCL performance ● Grow QCL's using MBE ● Process the QCL's ● Test and characterize the QCL's ● Use results to improve future QCL's
EXPERIMENTAL SUCCESS
Our team successfully built a 2.4 THz quantum cascade laser based on the 2.9 THz Barbieri structure shown previously. CODE STRUCTURE OVERVIEW
Repeat for Different Temperatures, Biases, Scaling Factors
Build QCL Find Wavefunctions and Energy Levels Structure
Calculate Load Material Number of Parameters Free Electrons & Ionized Donors Find Scattering Times
Read User Inputs Calculate Initial Fermi Levels Find Populations
Find Gain, Intensity, Current, etc.
FINDING THE WAVE FUNCTIONS
repeat until converges repeat until converges
repeat until converges Poisson Equation
Built in Voltage Steady State Equation Schroedinger Equation Fermi Levels Charge Density Wavefunctions Equation Charge Density Equation Charge Distribution
Charge Distribution
FINDING THE WAVE FUNCTIONS
d d z Poisson − z =z Equation dz dz
d 1 d 2 Schrödinger [ ]z=− E−V z z Equation dz m* z dz ℏ2
− * Charge em z k T − / = B ∑∣ ∣2 E n E F z k BT Density z z ln 1 e ℏ2 n Equation n
d d Steady-State [zz E z]=0 Equation dz dz F
Differential Equations are solved numerically using Convergent Finite Difference Methods BARBIERI STRUCTURE RESULTS
Barbieri Experimental Results: 2.90 THz
STL Code Results, Barbieri Code Results: Schrödinger Eq. Only 2.660 THz 2.602 THz BARBIERI STRUCTURE RESULTS
Barbieri Experimental Results: 2.90 THz
STL Code Results, STL Code Results, Schrödinger-Poisson Eqs. Only Schrödinger-Poisson-Steady State 2.898 THz 2.906 THz WORRAL STRUCTURE RESULTS
Worral Experimental Results: 2.06 THz
STL Code Results, Worral Code Results: Schrödinger Eq. Only 1.934 THz 1.854 THz WORRAL STRUCTURE RESULTS
Worral Experimental Results: 2.06 THz
STL Code Results, STL Code Results, Schrödinger-Poisson Eqs. Only Schrödinger-Poisson-Steady State 2.122 THz 2.004 THz FINDING THE TRANSITION TIMES n 2 τ τ τ 21 2 laser n 1 τ 1
(all other lower levels)
Lasing Requires Population Inversion:
τ and τ must be slow transitions (high transition times) 21 2
τ and τ must be fast transitions 1 laser FINDING THE TRANSITION TIMES
● Types of non-laser transitions – Optical Phonon Scattering – Acoustic Phonon Scattering – Spontaneous Photon Emission – Electron-Electron Scattering – Impurity Scattering – Interface Roughness Scattering
LO PHONON TRANSITION TIMES
1. Apply quantized phonon field Hamiltonian (Froelich Interaction) to QCL geometry
2. Integrate over all possible final electron momenta
* 2 2 m e E − ∣ − ∣ 1 = LO 1 − 1 / ± / ∫ ∫ ∫ 1 q z z ' nLO 1 2 1 2 d d z d z ' i z f z f z ' i z ' e ems,abs ℏ3 ∞ i f ki 2 s 0 q * 2 2 2m 2= 2 2 − and k =k E 0−E 0∓E where q ki k f 2ki k f cos f i ℏ2 i f LO
3. Average over all possible initial electron momenta
∞ ∫ 1 d Ek f D Ek i i 1 0 i f Ek = 1 = i where f E − / ∞ D ki E E k T 1e k i F B i f ∫ d E f E k i D k i 0
4. All 4 integrals are approximated by sums over a fine grid and calculated numerically
5. Repeat calculations to find transition times for all possible combinations of levels RESULTS
LO phonon scattering times between lasing levels for the Williams FL175C structure
Williams' Results STL Results
FINDING THE POPULATIONS
Use rate equations to find the electron population in each level. In equilibrium, the rate of electrons transitioning into a given level equals the rate transitioning out. For generality, consider every possible level and every possible transition to that level. n a τ τ τ τ τ τ n ab ba ai ia af fa i τ τ τ if fi st n f τ τ τ τ fb bf ib bi n b FINDING THE GAIN
1. Apply quantized photon field Hamiltonian to QCL Hamiltonian
2. Replace the Dirac delta in Energy with a more realistic Lorentzian lineshape function 2 st,band = e if W i f f i f mif if 4 m* V
* zN = 2m if ∑ − 2 where the oscillator strength is: f i f | zi zi−1 f zi zi i zi | ℏ = zi 0
3. Replace the number of inducing photons present m with the intensity of the if incident EM wave.
4. Define the gain in terms of the power added to the incident wave.
n −n e2 g = i j ij f ij * ij ij 4m c n 0
FUTURE WORK
● Finish implementing the gain calculations ● Test the gain calculations against published works ● Further test and fine-tune all calculations ● Finish implementing non-laser transitions (e-e) ● Finish Implementing Waveguide predictions