Temperature effects and transportphenomena in terahertz quantum cascade

lasers

Philip Slingerland

Submillimeter-Wave Technology LaboratoryUniversity of Massachusetts Lowell

November 29, 2011

Outline

IntroductionTerahertz BackgroundQCL DescriptionMotivationComputational Model

QCL OptimizationFrequency TargetingTemperature Optimization

Conclusions

Outline

IntroductionTerahertz BackgroundQCL DescriptionMotivationComputational Model

QCL OptimizationFrequency TargetingTemperature Optimization

Conclusions

THz Spectrum

I THz radiation is part of EM radiation spectrum: Between IRand Microwave

I Many potential industries: medical imaging, security,spectroscopy, communication, manufacturing

I What can produce coherent THz radiation?

THz Gap

I Absence of sources in region of interestI Most applications require powerful sourceI Graph does not address portability

Are QCL’s the solution?

I (Semi-)portablecoherent THz emitters

I Do they have enoughpower for applications?

I Can they operate abovecryogenictemperatures?

What is a QCL?

I Stacked layers ofsemiconductor material formconduction band quantum well

I Material doped to provideexcess electrons

I Quantized electronic statesexist within well

Electron Energy

k

E

Eki

Ei

0

I Each state (subband) hasassociated quantized and freeenergy

I Can define potential (Ei ) andkinetic energy (Ek

i )I Ek

i related to electrontemperature

Electron Dynamics

k

E

k

E I Scattering between statesdetermines how electrons andtheir energy distribute

I Dictates how device willperform - establishespopulation inversion

Population Inversion

upper lasingstate

lower lasingstate

~ω

I Inversion: more electrons inupper state

I Allows stimulated emission todominate transition

I Requires fast depopulation oflower state

QCL Description

I Many repeated layers withbias applied results in complexelectronic structure

I Spacing of electron energylevels and wavefunctionshapes controlled by design(thickness and alloyconcentration)

I Lasing can occur by tailoringenergy levels andwavefunctions

I Single electron can emit manyphotons

QCL Design Issues

I First successful THz QCL grown in 2002 [Köhler, 2002]I But need to be further tailored for applications

I Need to target certain frequenciesI Need to increase output powerI Need to operate at higher temperatures

I Making devices is time consuming and expensiveI Need efficient way to predict device performance

Computational Methods

I Methods exist which describe QCL performance in detail(MC, NEGF)

I Need technique which will allow for high productionI Model devices over broad ranges of many parametersI Self-consistent energy balance method implemented

(SCEB)

Waveguide Effects

input

waveguide

SchrödingerPoisson

?

transition rates

state densities

?

electrontemperatures

?

?

output

I QCL dimensions: slab model w/ TM modesI In each layer, Hy has the form

Hy (x) = Aeikx x + Be−ikx x

I Gives E-field components

Ex (x) =kz

ωεHy (x) Ez(x) =

iωε

∂

∂xHy (x)

wherekx =

√εµω2 − k2

z

Schrödinger-Poisson

input

waveguide

SchrödingerPoisson

?

transition rates

state densities

?

electrontemperatures

?

?

output

I Poisson eq. for space charge

− ddz

(ε(z)

dΦ(z)

dz

)= ρ(z)

I Schrödinger eq. for electron states

ddz

(1

m∗(z)

dψ(z)

dz

)= − 2

~2 (E−V (z,Φ))ψ(z)

I Equations solved self-consistently untilstates converge

Transition Rates

input

waveguide

SchrödingerPoisson

?

transition rates

state densities

?

electrontemperatures

?

?

output

I Scattering rates from Fermi’s “golden rule”:

Wi→f =2π~∣∣〈f |H ′|i〉∣∣2 δ(∆E)

I Every type described by Hamiltonian:

H ′e-e =e2

4πε∣∣rj − ri

∣∣H ′photon = − e

m∗A · p

H ′phonon =∑

q

[α(q)(eiq·rbq + e−iq·rb†q)

]

Electron Populations

input

waveguide

SchrödingerPoisson

?

transition rates

state densities

?

electrontemperatures

?

?

output

I Set of rate equations solved for everyelectron and photon state

ni =

∑Ni nj

[Wji + mijVW sp

ij

]∑N

i

[Wij + mijVW sp

ij

]mij =

ni

W pij /(ΓW sp

ij )− V (ni − nj)

I Every subband population determinediteratively until convergence is reached

I Based on steady-state condition

Electron Temperatures

input

waveguide

SchrödingerPoisson

?

transition rates

state densities

?

electrontemperatures

?

?

output

I Set of energy rate equations solved forevery electron state

dEemi

dt+

dEabsi

dt+

dEe-ei

dt= 0

I Every subband temperature determinediteratively until convergence is reached

I Based on steady-state condition

Outline

IntroductionTerahertz BackgroundQCL DescriptionMotivationComputational Model

QCL OptimizationFrequency TargetingTemperature Optimization

Conclusions

Does the model work?

I The SCEB model has been employed in collaboration withUML Photonics Center

I Need for focused growth campaigns to provide devices forexperimental applications

I Example follows of frequency scaling to meet real-worlddemand

Water Absorption

0

0.2

0.4

0.6

0.8

1

1 1.5 2 2.5 3

trans

mis

sion

frequency (THz)

I Water absorption is issue in THzI Need to use absorption “windows”I Tailor device to emit within frequency window

Scaled 2.83 THz QCL

2

2.2

2.4

2.6

2.8

3

0 0.2 0.4 0.6 0.8 1

frequ

ency

(TH

z)

transmission

Scaled 2.83 THz QCL

2

2.2

2.4

2.6

2.8

3

0 0.2 0.4 0.6 0.8 1

frequ

ency

(TH

z)

transmission

2

2.2

2.4

2.6

2.8

3

-2 0 2 4 6 8 10 12scale %

2.53 THz

Scaled 2.83 THz QCL

2

2.2

2.4

2.6

2.8

3

0 0.2 0.4 0.6 0.8 1

frequ

ency

(TH

z)

transmission

2

2.2

2.4

2.6

2.8

3

-2 0 2 4 6 8 10 12scale %

2.53 THz

Scaled 1.9 THz QCL

1.9

1.95

2

2.05

2.1

2.15

2.2

0 0.2 0.4 0.6 0.8 1

frequ

ency

(TH

z)

transmission

Scaled 1.9 THz QCL

1.9

1.95

2

2.05

2.1

2.15

2.2

0 0.2 0.4 0.6 0.8 1

frequ

ency

(TH

z)

transmission

1.9

1.95

2

2.05

2.1

2.15

2.2

-3 -2 -1 0 1 2 3 4scale %

1.97 THz

Scaled 1.9 THz QCL

1.9

1.95

2

2.05

2.1

2.15

2.2

0 0.2 0.4 0.6 0.8 1

frequ

ency

(TH

z)

transmission

1.9

1.95

2

2.05

2.1

2.15

2.2

-3 -2 -1 0 1 2 3 4scale %

1.97 THz

Modeling Scaled Structures

-5

0

5

10

12

3

0

5

10

15

output power (mW)

scale %

bias (×105 V/m)

I Simulate structure atvarious scaling factors

I Requires studyingmultiple parameters

I Optimum voltagechanges with scaling

Temperature Degradation

I Problem: Performance of any QCL degrades astemperature increases

I This limits the potential application of QCL’sI Helium or nitrogen cooling is impractical in field

Understanding Temperature

I Experiment [Vitiello,2005], MC studies showelectron temperaturesdeviate from latticetemperature

I Variation also seenamong electronsubband temperatures

I Electron dynamicstemperature dependent

I scattering ratesI populations (gain)I current

How does temperature affect performance?

k

Eupper laser

level

lower laserlevel

~ω

I Close energy spacing of electron states leads to potentialissues

I Thermal back-filling occurs when electrons scatter up anenergy level

I Harmful to population inversion

How does temperature affect performance?

k

E

k

E

~ω ELO

I Phonon issuesI Electron-phonon scattering

highly probableI Phonon emission can easily

replace photon emissionI Electrons can escape into

“continuum”I Upper lasing state electrons

can leave wellI Both effects are product of one

parameter: T ue

Energy Transition Rate

ki

EiI Every electron transition

transfers energy betweensubbands

I Removes energy from oneand adds energy to another

Energy Transition Rate

ki

Ei

kf

Ef

ELO

I Every electron transitiontransfers energy betweensubbands

I Removes energy from oneand adds energy to another

Energy Transition Rate

ki

Ei

kf

Ef

ELO

−(Eki )W em

if

(Ekf )W em

if

I Every electron transitiontransfers energy betweensubbands

I Removes energy from oneand adds energy to another

Subband Energy Balance Formula

I Kinetic energy is average thermal energy of electron

Eki = βi = kBT i

e

Ekf = Ei + βi − Ef + δE

I For phonon emission:

dEemi

dt=∑

f

nf (Ekf )W em

fi − ni∑

f

(βi)W emif

I All relevant scattering types must balance

dEemi

dt+

dEabsi

dt+

dEe-ei

dt= 0

Electron Temperature

input

waveguide

SchrödingerPoisson

?

transition rates

state densities

?

electrontemperatures

?

?

output

I Energy balance condition providesadditional level of consistency

I Provides another set of parameters tooptimize device

Determining temperature

1

2

3

4

5

-4

-2

0

2

4

6

40 80 120 160 200 240en

ergy

trans

ition

rate

(×10

12J/

s)electron temp. (K)

12345

Influence On Scattering Rates

I All scattering processes temperature dependentI Screening of electron-electron scattering also affectedI Multi-subband, temperature-dependent screening model

employed

ε(q,Te) = 1 +e2

2εsq

∑i

Πii(q,Te)Aiiii(q)

I Scattering rates influence other output parametersI populations (gain)I current

Testing Te predictions

80

100

120

140

Ti e

(K)

(a)

T 5e T inj

e

40 50 60 70

1

2

3

electric field (mV/module)

n 5/n

4

(b)

Optimizing Temperature

+ −

u

l

I Picked structure withexcellent temperatureperformance [Kumar,2010]

I Examined effect of shiftingbarrier thickness to altertransition rates

Optimizing Temperature

+ −

u

l

180190200210

-3 -2 -1 0 1 2 3elec

tron

tem

p.(K

)

TL = 150 K

0.1

0.15

0.2

0.25

-3 -2 -1 0 1 2 3

pow

er(W

)

barrier shift (Å)

T ue

Outline

IntroductionTerahertz BackgroundQCL DescriptionMotivationComputational Model

QCL OptimizationFrequency TargetingTemperature Optimization

Conclusions

Conclusions

I QCL’s offer a viable source for the terahertz frequencyrange, provided they can be improved

I Further computational modeling is essential part of thisoptimization

I Demonstrated success of SCEB model to predictperformance of THz QCL’s

I Many parameters available to optimize, but subbandtemperature seems promising

References

I Tonouchi Nature Photonics 1, 97 (2007)I Köhler et al. Nature 417, 156 (2002)I Vitiello et al. Appl. Phys. Lett. 86, 111115 (2005)I Vitiello et al. Appl. Phys. Lett. 90, 191115 (2007)I Freeman et al. Appl. Phys. Lett. 93, 191119 (2008)I Kumar et al. Nature Physics 7, 166 (2010)

Thank you

I Christopher BairdI Robert Giles, Viktor Podolskiy and William GoodhueI Students and staff at STL and POD

I Andriy Danylov and Neelima ChandrayanI Xifeng Qian and Shiva Vangala

I Many friends and family