NonequilibriumGreen’sfunctiontheoryforgainand

transportpropertiesofquantumcascadestructures

ToniS.-C.LeeInst.furTheoretischePhysik,TechnischeUniversitatBerlin,10623Berlin,Germany

Overview

•Quantumcascadelaserstructures

•Theoreticalformalism:nonequilibriumstationarystate→Green’sfunctions

•Physicalproperties

•Applicationtoexamplestructures

Acknowledgment:A.Wacker,UniversityofLund,Sweden

•unipolar,intraband•intersubband,interminiband•midinfrared:3.5–24µm

farinfrared:67–100µm

activeregion

injectoractive

region

1 period

GaAs/Al1−xGaxAs

Sirtorietal.Appl.Phys.Lett.731722(1998)—33%Al

Pageetal.Appl.Phys.Lett.783529(2001)—45%Al

NONEQUILIBRIUMSTATIONARYSTATE

correlationfunction,G<(E)

retardedGreen’sfunction,Gret

(E)

Extractphysicalproperties

•spectralfunction,densityofstates

•levelbroadening,energyrenormalisation

•populations

•gainspectra

•transportproperties:I−Vcharacteristic

Hamiltonian

H=Ho+Hscatt

kineticenergy,Hke

superlatticepotential,HSL

appliedbiasHE

scatteringprocesses︷︸︸︷

interfaceroughnessimpurityscatteringelectron-phonon—acoustic,LO

electron-electron

Self-energiesΣ(E)

Mean-fieldPoisson’sequation

Quantumtransportequations

Dysonequation:EGretα1α2,k(E)−

X

β

»

(Ho+HMF)α1β,k+Σretα1β,k(E)

–

Gretβα2,k(E)

=δα1α2

Keldyshrelation:G<α1α2,k(E)=

X

ββ′

Gretα1β,k(E)Σ

<ββ′,k(E)G

advβ′α

2,k(E)

Self-energies

Interfaceroughness/impurityscattering:

Σ<,rough/impα1α1,k(E)=

X

β,k′

〈|Vrough/imp

α1β(k−k′)|

2〉G

<ββ,k′(E)

Optical/acousticphononscattering:

Σ<,phα1α1,k(E)=

X

β,k′

|Vphon

α1β(k,k′)|

2

»

fB(Eph)G<ββ,k′(E−Eph)+[fB(Eph)+1]G

<ββ,k′(E+Eph)

–

AndequationsforΣret

.Electron-electronscatteringMeanfieldapprox,Poisson’sequation.

Self-consistentLoopInitialguess:G

ret(E),G

<(E)

Evaluate:mean-fieldΣ

ret(E,k),Σ

<(E,k)

Evaluate:G

ret/new(E),G

</new(E)

nnew

=P

k

R

dEG</new

(E,k)

Test:|G

new(E)−G(E)|<ε

|nnew

−n0|<ε

Evaluatephysicalproperties,e.g,currentdensities,populations

NewguessforG

ret(E),G

<(E)

NO

YES

choiceofbasisstates

•Blochfunctions:eigenstatesofHke,z+HSL,spatially-extended

•Wannier-Starkstates:energyeigenstatesofHke,z+HSL+HE,spatially-extendedatlowbias

•Wannierstates:spatially-localisedstates

computationalbenefit:

•independentofbiasevaluatescatteringmatrixelementsatzerobiasonly

•possibletoreducenumberofself-energymatrixelements?

disadvantage:notenergyeigenstates,physicalinterpretation?

injector

activeregion

GaAs/Al0.33Ga0.67As ε

d = 45.3 nm

= 55 kV/cm

approximations

Self-energies(generalcase)

Σα1α2,k(E)=∑

ββ′,k′

〈Vα1β(k,k′)Vβ′α2(k,k

′)〉Gββ′,k′(E)

neglectoff-diagonalelementsofself-energy

Σα1α1,k(E)=∑

β,k′

|Vα1β(k,k′)|

2Gβ,k′(E)

k-independentmatrixelements

∑

k′

|Vαβ(k,k′)|

2Gβ,k′(E)|Vαβ(kfix,k

′fix)|

2∑

k′

Gβ,k′(E)

Approx:evaluate|Vαβ(k,k′)|atfixedmomentakfixandk

′fix.

InformationcontainedinGretk(E)

SpectralfunctionIm[G

retνν,k=0(E)]Energyeigenstates

–renormalisation–broadening:lifetime,Γi

-0.0500.050.1E (eV)

0

20

40

60

Im[G

νν ret(k = 0, E)] (a. u.)

36

Wannier states

ν

Sir98 0.2 V/per 77 K

-0.0500.050.1E (eV)

0

20

40

60

80

100

Im[G

νν ret(k = 0, E)] (a. u.)1

36

Wannier-Stark states

νSir98 0.2 V/per 77 K

Densityofstates

Im[Gretνν(E)]=

∫dkIm[G

retνν(k,E)]

∑

ν

Im[Gretνν(E)]

-0.1-0.0500.050.10.150.2E (eV)

0

0.2

0.4

0.6

0.8

1

Im[G

νν ret(E)] (a. u.)3

6

Wannier states

ν

Sir98 0.2 V/per 77 K

00.20.4E (eV)

0

2

4

6

8

Σν Im[G

νν ret(E)] (a. u.)

Sir98 0.2 V/per 77K

DensityofstatesTotaldensityofstatesinνthWannierstateperperiod

Basis-independent

G<k(E):Energetically-andspatially-resolvedpopulations

n(z,E)=∑

αβ,k

G<αβ,k(E)ψα(z)ψβ(z)

0.1V/per,J<<Jth0.2V/per,J∼Jth

-0.1

-0.05

0

0.05

0.1

0.15

0.2

-60-40-20020406080100

E (eV)

position (nm)

activeregion

injector

-0.1

0

0.1

0.2

0.3

0.4

0.5

-40-20020406080

E (eV)

position (nm)

active region

injector

Currentdensities

J=e

V〈dz

dt〉=

e

V

i

~〈[Ho,z]〉

︸︷︷︸Jo

+e

V

i

~〈[Hscatt,z]〉

︸︷︷︸Jscatt

Jo=2e

~V

∑

αβ,k

∫dE

2π[Ho,z]αβG

<βα,k(E).

Jscatt=2e

~V

∑

αβγ,k

∫dE

2π[G

<βγ,k(E)Σ

adv(α)γγ,k(E)+G

retβγ,k(E)Σ

<(α)γγ,k(E)]zγβ

−zαγ[Σ<(β)γγ,k(E)G

advγα,k(E)+Σ

ret(β)γγ,k(E)G

<γα,k(E)].

Sir98,I-V:expvstheoryExp.datafromSirtorietal.APL73

1722(1998)

02468Current (A)

0

2

4

6

8

Voltage (V

)

TheorySir98 77 K

Exp:Ith∼4Aseriesresistance(cladding)∼1Ω

NegativeDifferentialResistivity(NDR)

0510152025J (kA/cm

2)

0

5

10

15

20

Voltage (V

)

77 K233 K

Theory

a

b

c

Pageetal.APL783529(2001)

0.2

0.3

0.4

energy (eV)

0.2

0.3

0.4

energy (eV)

-20-1001020position (nm)

0.1

0.2

0.3

0.4

0.5

energy (eV)

10 V, 15 kAcm-2

13 V, 23 kAcm-2

16.5 V, 15 kAcm-2

52%

10%

28%

20%

51%

8%

∆E = 32 meV

∆E = 6 meV

∆E = 14 meV

(a)

(b)

(c)

iu

i

i

u

u

77 K

Gain:linearresponsetoanappliedopticalfield

Gaincoefficient:g(ω)=−ω

nBcIm[χ(ω)]

LinearresponsetoappliedopticalfieldE(r,t):

δG</ret

,δΣ</ret

⇒δJ⇒δP(ω)[inducedpolarisation]

DefineχNGF(ω)=δP(ω)/εoE(ω)gNGF(ω).

0.080.10.120.140.160.18Ephoton (eV)

-150

-100

-50

0

50

gain (cm-1)

0.12, 0.480.16, 2.20.18, 4.10.2, 6.5

V/period, J (kA/cm2)

Sir98 77 K

Gain:Simplifiedtwo-levelmodel

Wannier-Starkbasis

G<ii:populations,niIm[G

retii(E)]:energiesEi,broadeningΓi

Im[χWS(ω)]=2π

εoV

∑

ij,k

|dij|2(fik−fjk)δ(~ω+Eik−Ejk)

=π

εoLp

∑

ij

|dij|2(ni−nj)Lij(ω)

withLij(ω)=(Γij/2π)/[(~ω−∆Eij)2+(Γij/2)

2].Im[χWS(ω)]gWS(ω)

Drawbacks

•notapplicableatlowbias

•neglectsquantum-mechanicaleffects

(off-diagonalelementsofdensitymatrix)

ComparisonofgNGF(ω)andgWS(ω)

0.080.10.120.140.160.18Ephoton (eV)

-150

-100

-50

0

50

gain (cm-1)

0.12, 0.480.16, 2.20.18, 4.10.2, 6.5

V/period, J (kA/cm2)

Sir98 77 K

dashed: gWS(ω)solid: gNGF(ω)

0.060.080.10.120.140.160.18Ephoton (eV)

-200

-100

0

100

gain (cm-1)

gWS total spectra

Sir98 77 K0.22 V/period

A

B

Separatecontributionsfromdifferenttransitionstototalspectrum.

-40-200204060position (nm)

0

0.1

0.2

0.3

0.4

E (eV

)

1

22’

33’ A

B

0.50.7

714

1.2

rel. pop. (%)

Sir98 77 K0.2 V/period

0

Maintransitions:(A)upperlaserleveltocontinuum(B)upper–lowerlaserlevel

‘3.4-THzquantumcascadelaserbasedonlongitudinal-optical-phononscatteringfordepopulation’

Williamsetal.,Appl.Phys.Lett821015(2003)

-40-2002040position (nm)

0

0.1

0.2

E (eV

)

1

2

345

1’

2’

64 mV/period30 K

Lp= 52.4 nm

lasing transition3.4 THz

LO-phononemission

ne = 2.8 x 1010

cm-2

G<k(E):Energetically-andspatially-resolvedpopulations

n(z,E)=∑

αβ,k

G<αβ,k(E)ψα(z)ψβ(z)

−40−200204060

−0.1

−0.05

0

0.05

0.1

0.15

position (nm)

E (eV

)

active region

injector

TheoreticalGainSpectra

00.0050.010.0150.02E (eV)

-60

-40

-20

0

20

40

60

80gain (cm

-1)

646668

30 K

mV/period

64mV/period—maincontributionstogainspectra

00.0050.010.0150.02E (eV)

0

20

40

60

80

gain (cm-1)

00.010.02E (eV)

-40

0

40

gain (cm-1) 30K

1’ - 4

2’ - 564 mV/per

total gain spectra

n5 = 0.077

n4 = 0.08 n1’ = 0.354

51’

2’

64 mV/per 30 K

E54 = 14.2 meV

n2’ = 0.33

E1’4 = 12.5 meVE2’5 = 5.5 meV

τ5 = 0.5 ps

τ4 = 0.44 ps

τ2’ = 1 ps

τ1’ = 0.94 ps

z54 = 5.14 nm

z2’5 = 6.1 nmz1’4 = 3.2 nm

•above10meV:maingaincontributionfrom1’–4transition,i.e.,lowercollectorleveltolowerlaserlevel.NOT(designed)5–4transition.

•at'5meV,substantialgainfrom2’–5transition.

•low-energyabsorptionfeature(∼2.5meV)from1’–5

66mV/period—maincontributionstogainspectra

00.0050.010.0150.02E (eV)

0

20

40

60

80

gain (cm-1)

2’ - 5

2’ - 1’

1’ - 4

5 - 4

00.010.02E (eV)

-40

0

40

gain (cm-1) 30K

66 mV/period

total gain spectra

5

41’

2’

n5 = 0.12τ5 = 0.55 ps

τ4 = 0.44 psn4 = 0.096

n1’ = 0.26τ1’ = 0.71 ps

n2’ = 0.36τ2’ = 0.81 ps

E54 = 15.1 meV

30 K66 mV/perz54 = 4.6 nmz1’4 = 3.8 nmz2’5 = 5.9 nm

E1’4 = 13.4 meVE2’5 = 6 meV

•again,maincontributionstogainfrom1’–4and2’-5transitions.

•additionalcontributionsfrom5–4transition(>10meV)and2’–1’(<10meV)transition

•low-energyabsorptionfeature(∼2.5meV)from1’–5

•forhigherbias(&68mV/period)1’–5featurebecomesgain

Effectofvaryingparameters:

•broadeningparametersΓi

–includesbothintrasubbandandintersubbandscattering–alternativeapproachtogaincalculation:

linearresponseofstationarystatetotime-dependent(optical)perturbation,doesnotrequireΓi

•numberoflevels

•temperature:5–110K

•conductionbandoffset:

–I-V∗66%offset(Vurgaftmanetal.JAP89,5818(2001)):2×exp.I-V∗80%offset(Williamsetal.):∼exp.I-V

–gainspectra

Gain:linearresponsetoanappliedopticalfield

Gaincoefficient:g(ω)=−ω

nBcIm[χ(ω)]

LinearresponsetoappliedopticalfieldE(r,t):

δG</ret

,δΣ</ret

⇒δJ⇒δP(ω)[inducedpolarisation]

DefineχNGF(ω)=δP(ω)/εoE(ω)gNGF(ω).

00.0050.010.0150.02E (eV)

-50

0

50

gain (cm-1)

gWSgNGF

30 K64 mV/period

Changingnumberoflevels

00.0050.010.0150.02E (eV)

-60

-40

-20

0

20

40

60gain (cm

-1)

56

5 K64 mV/period

no of levels

Varyingtemperature

00.0050.010.0150.02E (eV)

-60

-40

-20

0

20

40gain (cm

-1)

53050657790110

64 mV/period

temp (K)

Conductionbandoffset

00.0050.010.0150.02E (eV)

-60

-40

-20

0

20

40

60gain (cm

-1)

66%80%

30 K64 mV/period

conduction band offset

Summary

•NonequilibriumGreen’sfunctiontheory:descriptionofnonequilibriumstationarystateinQCstructures

•Physicalproperties:

–spectralfunction:renormalisedenergies,broadening,densityofstates–populations,distributionfunctions

•Applicationtoquantumcascadestructures:

–calculationofI−Vcharacteristic–evaluationandanalysisofgain/absorptionspectra,

predictionofgainat1THz

Workinprogress/Futurework

•validityofapproximations

–Wannierstates,sufficientlylocalised?–off-diagonalelementsofself-energies?

•electron-electronscattering,beyondmean-field