Full Wave Electromagnetics for Simulating Terahertz ... · John Albrecht, Robert Bedford, Sarah...

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Full Wave Electromagnetics for Simulating Terahertz Quantum Well Laser Diode Modulation

Matt Grupen, Paul Sotirelis, & Steve WongHPTi/PET

John Albrecht, Robert Bedford, Sarah Maley, Tom Nelson, & Bill Siskaninetz

US Air Force Research Lab, Wright Patterson AFB

7th International Conference on Numerical Simulation of Optoelectronic DevicesSeptember 26, 2007


Outline

Brief backgroundcarrier heating bottleneck to direct current modulationBloch equation analysis of plasma heating by radiation

New self-consistent simulation of plasma heating effectfull wave electromagneticsFermi gas thermodynamics

Delaunay/Voronoi Surface Integration (DVSI)curl operators in Ampere’s and Faraday’s lawsdivergences in electrostatics, charge, and energy conservation (i.e. box integration)

Simulate plasma heating modulation of single QW laser structureelectrical current pumping to achieve lasingpatch antenna like structure to inject high frequency radiation

Summary


Current Injection Gain Saturation by Dual Modulation

Both Fermi level & temperature fluctuateGorfinkel & LuryiGrupen & Hess

Quantum mechanical tunneling injectiondecreases heatingdecreases pumping efficiency Bhattacharya & Ghosh

Plasma heatingBloch equation analysis• carrier density• energy density• over estimates heat capacity of degenerate Fermi gas• modulation depth saturation with increasing field intensity• Ning et al.; Li & Ning; Chow et al.self-consistent EM & nonlinear charge transport• Maxwell’s full wave vector field theory• Boltzmann’s equation solved for a Fermi gas

M. Grupen and K. Hess, IEEE JQE 34 120, 1998.


Classical Theory of Electromagnetics

( )−+ −+−=⋅∇ AD NNnpqEεelectrostatic Gauss’s Law

0=⋅∇ Hμmagnetostatic Gauss’s Law

( )t

q np ∂∂

+−=×∇EJJH ε

Ampere’s Law

t∂∂

−=×∇HE μ

Faraday’s Law

SRHn Utn

+⋅∇=∂∂

− Jelectron continuity

SRHp Utp

+⋅∇=∂∂

− Jhole continuity

⎟⎟⎠

⎞⎜⎜⎝

⎛ −++⋅∇+⋅=

∂∂

−n

latnSRH

kinn

totnn

n kTkTFFnUEq

tE

τ2/1

2/3SJEelectron energy conservation

⎟⎟⎠

⎞⎜⎜⎝

⎛ −++⋅∇+⋅−=

∂

∂−

p

latpSRH

kinp

totpp

p kTkTFFpUEq

tE

τ2/1

2/3SJEhole energy conservation

)( npappliedlatlat

p qTt

TC JJE −⋅−∇⋅∇=∂∂

− κρlattice energy conservation


Defining Field Components

( )irrrottEEAE +=⎟

⎠⎞

⎜⎝⎛ ∇+∂∂

−= εψεε

rotttEAAE ×∇=

∂∂

×−∇=⎟⎠⎞

⎜⎝⎛ ∇+∂∂

×−∇=×∇ ψ

Two vector fields and four field equationsDecompose field flux densities into orthogonal functionals

irrtEAE εψεψεε ⋅∇=∇⋅−∇=⎟

⎠⎞

⎜⎝⎛ ∇+∂∂

⋅−∇=⋅∇equivalent to

Coulomb gaugefor 0=∇ε

from Faraday’s Lawand the vector potential AH ×∇=μ

( ) 0=×∇⋅∇=⋅∇ AHμ

rotHAH ×∇=⎟⎟⎠

⎞⎜⎜⎝

⎛×∇×∇=×∇

μ1


Defining Charge & Energy Densities

( ) ( )nCn

E n

Cn FkTNde

kTmdE

kTFEEEmn

n

C

ηεεππ ηε 2/1

2/3

0

2/332

2/3

32

2/3

)(1

2

1exp

2=

+=

+⎟⎠⎞

⎜⎝⎛ −

−= ∫∫

∞

−

∞

hh

( ) ( ) ( )nCE

n

E n

Cnn FkTNd

ekTmdE

kTFE

EEmEC

n

C

ηεεππ ηε 2/3

2/52/3

2/532

2/32/3

32

2/3

)(1

2

1exp

2=

+=

+⎟⎠⎞

⎜⎝⎛ −−

= ∫∫∞

−

∞

hh

( ) εεη ηε dekT

EFFFn

j

n

Cnjnj ∫

∞

− +=⎟⎟

⎠

⎞⎜⎜⎝

⎛ −=

0 1

Michele Goano, “Algorithm 745: Computation of the Complete and IncompleteFermi-Dirac Integral,” ACM Trans. Math. Software, vol. 21, no. 3, Sept. 1995,pp. 221-232


Bulk Charge Fluxes: 1st Moment of Boltzmann Equation

( ) kvBvEv k dffqtff

⎭⎬⎫

⎩⎨⎧ ∇⋅−∇⋅×+=

∂∂

+∫∞

∞−0034

1 τττπ h

( ) ( ) ( ) ⎥⎦

⎤⎢⎣

⎡∇⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛∇+

+−=

∂∂

+ 2/12/3

2/1

2/3

2/1

2/32/1

2/32/1

2/322 3

232

1FkT

qkT

FF

qkT

FFFkTFkTN

mq

t nC

n

EB

MJJμ

ττ

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+++−+−++++−+

=2222

2222

2222

11

1

znzynxnzxnyn

zynxnynyxnzn

zxnynyxnznxn

BBBBBBBBBBBBBB

BBBBBBB

μμμμμμμμμμμμμμμ

M

[ ]1221

22/12/3

212/12/3

1222/1

2/321

,21, )())(()())((1

132

→→

→→

−=

−−+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

∂∂

+

nn

Cn

n

ave

netnnetn

JJ

FkTBFkTBL

Nq

kTFF

tJ

J ηξηξμμτ

BM

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛Δ+⎟⎟

⎠

⎞⎜⎜⎝

⎛= → q

kTFF

LE

kTq

FFL

ave 2/1

2/321

2/3

2/1

321

23ξ

1)exp()(

−=

ξξξB

Scharfetter-Gummel discretization


Bulk Energy Fluxes: 3rd Moment of Boltzmann Equation

( ) kvBvEv k dffqtffE

⎭⎬⎫

⎩⎨⎧ ∇⋅−∇⋅×+=

∂∂

+∫∞

∞−0034

1 τττπ h

( ) ( )⎥⎦

⎤⎢⎣

⎡∇+

+−=

∂+∂

++ 2/5

2/7

2/32/5

22

)(32)(

35

1F

qkTFkTN

t nCn

workn

kinnwork

nkinn E

BMSSSSμ

μτ

( ) kvBvEv k dffqtffCV

⎭⎬⎫

⎩⎨⎧ ∇⋅−∇⋅×+=

∂∂

+∫∞

∞−0034

1 τττπ h

( ))(kTd

dFEC nV −≡

heat capacity

kinetic energy and chemical work

( ) ( ) ( )[ ]( ) ( )[ ]{ }2

20

1/exp/exp

)( +−−

⎟⎠⎞

⎜⎝⎛ −

=−kTFE

kTFEkT

FEkTd

dfFEn

nnn


Discretized Energy Fluxes

Scharfetter-Gummel kinetic energy & work

[ ]1221

22/32/5

212/32/5

1222/3

2/521

,21, )())(()())((1

132

→→

→→

−=

−−+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

∂∂

+

kwkw

Cn

n

ave

netkwnetkw

SS

FkTBFkTBL

Nq

kTFF

tS

S ηξηξμμτ

BM

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛Δ+⎟⎟

⎠

⎞⎜⎜⎝

⎛= → q

kTFF

LE

kTq

FFL

ave 2/3

2/521

2/5

2/3

321

35

23ξ

1)exp()(

−=

ξξξB

heat exchange

( ) ( )( )[ ] ( )[ ]12

,12

2,12

,1221

,21

1,21

,21

122,

12211,

2112211221

1122

1

2

2

1

1

2

2

1

→→→→→→→→

→→→→→→→→

−−−−−−−=

⎥⎦⎤

⎢⎣⎡ −−−−=⎥⎦

⎤⎢⎣⎡ −−=− ∫∫∫∫

TnnnTkwkwTnnnTkwkw

T

T nnkw

T

T nnkw

T

T V

T

T Vhh

JJFSSJJFSS

dTJFSdTddTJFS

dTddTCdTCSS


Heterojunction Flux

highCE

lownF

lowCE

lhJ → hlJ →

( ) ( ) ⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ −−⎟⎟⎠

⎞⎜⎜⎝

⎛ −=−= →→ low

n

highC

lownlow

nhighn

highC

highnhigh

nhllhnet kTEFFkT

kTEFFkTmJJJ 1

21

2

3221

hπ

highnF

thermionic emission


Quantum Well

( ) TEtrans JP−1

TEJ TEtrans JP

2-Dsub-bands

wellE

meanQW LLtrans eP /−=

*/2 mEL wellscatmean τ=

conduction sub-bands

g 2D(m

-2)

E-EV(eV)

pk ⋅

effective masses

( )∑

∑ ∫

⎟⎟⎠

⎞⎜⎜⎝

⎛ −=

+⎟⎟⎠

⎞⎜⎜⎝

⎛ −=

∞

j n

jnn

Dj

j E

n

n

jD

TEF

FTN

dE

TFE

mn

j

02

2

*

2

1exp

1hπ


Optical Photon Emission

( ) mvcredsponm

mvcspon EffggBU Δ−=∑ → 1 ( ) ( )[ ] mcvvcred

stimm

mmvcstim EffffggSBU Δ−−−=∑ → 11

( )m

mstimm

sponm

m SdVUUdt

dSτ

−+= ∫

∫=Δ

dVEg m

stimm 2

2

φφ

⎟⎟⎠

⎞⎜⎜⎝

⎛

+=∑ *

,*,

*,

*,

2

21

ivic

ivic

ired mm

mmg

hπ2

20

20

2

M⎟⎟⎠

⎞⎜⎜⎝

⎛=→ nm

qB vc ωεπ

⎪⎩

⎪⎨

⎧

=⋅+⋅

=Δ+Δ+

=

−−222

2*

20

2

2ˆˆ3/21

/112

blhchhc

bG

G

c

G

eeE

EmEm

MMM

MM

bulk

QW

⎪⎩

⎪⎨⎧

Δ

Δ=Δ

mstimm

mm

msponm

Eg

EcE

Eg 332

2

hπPlanck density over estimate

coherent density under estimate

( ) 0222 =−+∇ φβωεφ ωt


Collision Broadening

( ) ( )[ ] icvvcredstimi

ivcim

stimm EffffggBSU Δ−−−= ∑ → 11ρ

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡ −−−∝ 2

2,,exp

broad

ivicmi E

EEEρ

energy (eV)

broa

d fa

ctor

(arb

. uni

ts)


Delaunay/Voronoi Surface Integration (DVSI): Ampere’s Law

( ) ∑=∂∂

+j

jjii

ii LHAtEAJ ε

( ) HEJ ×∇=∂

∂+

tε

Ei

Hj

Ai


Delaunay/Voronoi Surface Integration (DVSI): Faraday’s Law

( )∑=∂

∂−

iiij

j LEAtHμ

( ) EH×∇=

∂∂

−tμ

Ei

Hj

Aj


DVSI: Compatible with Box Integration Method

Ei

Hj

Ai


Divergences: Electrostatics & Charge Conservation

Aij

vi

vj

ijV S j

ij ADddVi i

∫ ∫ ∑=⋅=⋅∇ SDD

divergence theorem


Solving the Discretized Equations with Newton Method

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−=

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

ΔΔΔΔΔΔΔΔ

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

l

p

n

p

n

l

p

n

p

n

rot

rot

l

l

p

l

n

l

p

l

n

l

rot

l

rot

ll

l

p

p

p

n

p

p

p

n

p

rot

p

rot

pp

l

n

p

n

n

n

p

n

n

n

rot

n

rot

nn

l

p

p

p

n

p

p

p

n

p

rot

p

rot

pp

l

n

p

n

n

n

p

n

n

n

rot

n

rot

nn

rotrot

lpnpnrotrot

pnpn

TTTFF

HE

TTTFFHE

TTTFFHE

TTTFFHE

TTTFFHE

TTTFFHE

HE

TTTFFHE

TTFF

energy

energy

energy

cont

cont

Far

Amp

Gauss

RRRRRRRR

energyenergyenergyenergyenergyenergyenergyenergy



contcontcontcontcontcontcontcont

contcontcontcontcontcontcontcont

00000FarFar0

AmpAmpAmpAmpAmpAmpAmpAmp

0GaussGaussGaussGauss00Gauss

ψ

ψ

ψ

ψ

ψ

ψ

ψ

ψDDDDD


Microwave Heating Structure

injectedcurrent

heatingvoltage signal

p-type

n-typeinsulator insulator

current (mA)

outp

ut p

ower

(mW

)

single facet L-I characteristics


200 mA Bias, 0.5 V Heating Voltage Stepinjectedcurrent

heatingvoltage signal

applied electric field change in QW electron temperature


Current & Plasma Heating Modulation Schemes

ΔP

(mW

)

ΔP

(mW

)

time (signal periods) time (signal periods)

0.2 mA sinusoidal current signal

200 mA quiescent bias current

1 V sinusoidal voltage heating signal

5 GHz

20 GHz

50 GHz

200 GHz

50 GHz

200 GHz

500 GHz1 THz


QW Carrier Temperature Variations Affect Optical Gainte

mpe

ratu

re (K

)

time (ps) time (ps)

QW electron temperature conduction band lasing state

occu

patio

n pr

obab

ility

200 mA dc current bias2 V amplitude, 500 GHz heating signaldistance measured from the injected radiation boundary

2 μm

3 μm4 μm

2 μm

3 μm

4 μm


Relative Variations in QW Electron GasΔ

F n/F

n,0

(x 1

0-3 )

time (ps) time (ps)

chemical potential temperature

ΔT n

/Tn,

0(x

10-

3 )

200 mA dc current bias1 V amplitude, 500 GHz heating signal


Approximately Linear Scaling with Heating Voltage

200 mA dc current bias500 GHz voltage heating signal

radiation energy goes as square of field strengthlinear scaling may be due to coupling efficiency

ΔP

(mW

)

time (ps)

10 V

4 V2 V

1 V


Summary

Gain saturation from carrier heating limits laser diode current modulation

increasing injected current raises QW chemical potentialdissipating injected carriers’ excess energies increases QW temperaturegain increase from chemical potential offset by temperature increase

New simulation techniques used to study dual modulation effect full wave electromagneticsFermi gas dynamics with full thermal effectsnew treatment of heat flow using heat capacity of Fermi gasesnew vector field discretization scheme allows self-consistent solutions

Terahertz modulation through QW carrier heating by injected radiationappears permissible by Maxwell’s vector field theory and kinetictheory of Fermi gasespractical laser diode structures• will exhibit significant lattice heating• likely adversely affect this high frequency modulation scheme


Acknowledgements

This work was supported by the Department of Defense’s High PerformanceComputing Modernization Program under the User Productivity Enhancementand Technology Transfer contract #GS04T01BFC0061.

Full Wave Electromagnetics for Simulating Terahertz ... · John Albrecht, Robert Bedford, Sarah...

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Transcript of Full Wave Electromagnetics for Simulating Terahertz ... · John Albrecht, Robert Bedford, Sarah...