Quantum Cascade Laser Code Prediction

Dr. Christopher BairdSubmillimeter-Wave Technology Laboratory

University of Massachusetts LowellFebruary, 2009

OUTLINE

I. Current WorkA. Introduction & Code Structure OverviewB. Finding the Wave FunctionsC. Finding the Scattering TimesD. 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?

EF(z)

V(z)

Eg(z)

-eΔΦapp

ΔΦapplied

: Externally applied voltage drop

Φ(z): Built-in potential

Ec(z) : Conduction band edge energy in bulk

V(z) : Effective conduction band edge energy

Ψn(z): Wave functions

En : Energies of possible quantum states

EF(z) : Fermi energies

Eg(z) : Band-gap energies in bulk

Ev(z) : Valence band edge energy in bulk

Ev(z)

Ec(z)

Φ(z)

Φ(z)Φ(z)

En

Ψn(z)

GaAsAlGaAs

AlGaAs

z

E

ValenceBand

ConductionBand

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

Find Scattering Times

Find Populations

Find Gain, Intensity,Current, etc.

Find Wavefunctionsand Energy Levels

Load MaterialParameters

Build QCLStructure

CalculateNumber of

Free Electrons& Ionized Donors

Read UserInputs

Repeat for Different Temperatures, Biases, Scaling Factors

Calculate InitialFermi Levels

FINDING THE WAVE FUNCTIONS

PoissonEquation

SchroedingerEquation

Charge DensityEquation

Steady StateEquation

Charge DensityEquation

Built in Voltage

Wavefunctions

Fermi Levels

Charge Distribution

Charge Distribution

repeat until converges

repeat until converges

repeat until converges

FINDING THE WAVE FUNCTIONS

SchrödingerEquation

PoissonEquation

ChargeDensity

Equation

Steady-StateEquation

ddz [ 1

m* z ddz ] z =−

2ℏ2 E−V z z

−ddz z d z

dz = z

z =−em* z k BT

ℏ2 ∑n

∣ n z ∣2 ln 1e−E nE Fz / k BT

ddz [ z z d

dzEF z ]=0

Differential Equations are solved numerically using Convergent Finite Difference Methods

BARBIERI STRUCTURE RESULTS

Barbieri Experimental Results:2.90 THz

Barbieri Code Results:2.660 THz

STL Code Results,Schrödinger Eq. Only

2.602 THz

BARBIERI STRUCTURE RESULTS

Barbieri Experimental Results:2.90 THz

STL Code Results,Schrödinger-Poisson Eqs. Only

2.898 THz

STL Code Results,Schrödinger-Poisson-Steady State

2.906 THz

WORRAL STRUCTURE RESULTS

Worral Experimental Results:2.06 THz

Worral Code Results:1.934 THz

STL Code Results,Schrödinger Eq. Only

1.854 THz

WORRAL STRUCTURE RESULTS

Worral Experimental Results:2.06 THz

STL Code Results,Schrödinger-Poisson Eqs. Only

2.122 THz

STL Code Results,Schrödinger-Poisson-Steady State

2.004 THz

FINDING THE TRANSITION TIMES

n2

n1

τ21

τ1

τ2

τlaser

Lasing Requires Population Inversion:

(all other lower levels)

τ21

and τ

2 must be slow transitions (high transition times)

τ1

and τlaser

must be fast transitions

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

1i f

=∫

0

∞

d Ek i

1 i f Ek i

f DEk i

∫0

∞

d E k if DEk i

f D Ek i=

11eEk i

−E F / k BT

1i f

ems,absk i=

m* e2 E LO

2ℏ3 1∞

−1s nLO1/2±1/2∫

0

2

d ∫ d z∫ d z ' i z f z f z 'i z ' 1q

e−q∣z−z '∣

1. Apply quantized phonon field Hamiltonian (Froelich Interaction) to QCL geometry

2. Integrate over all possible final electron momenta

q2=k i2k f

2 −2k i k f cos k f2 =k i

22 m*

ℏ2 E i0−E f 0∓ELOwhere and

3. Average over all possible initial electron momenta

where

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

RESULTSLO phonon scattering times between lasing levels for the Williams FL175C structure

Williams' Results STL Results

FINDING THE POPULATIONS

ni

nf

na

nb

τif

τfi

τfb

τbf

τib

τbi

τfa

τaf

τia

τai

τbaτ

ab

τst

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.

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

f i f =2m*if

ℏ|∑

zi=0

zN

z i−z i−1 f z i zi i z i |2

where the oscillator strength is:

W i fst,band =

e2 if

4 m* V f i f mif if

3. Replace the number of inducing photons present mif with the intensity of the

incident EM wave.

4. Define the gain in terms of the power added to the incident wave.

g ij =ni−n je

2 ij

4 m* c n0 f ij ij

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