Motivation NEGF Formulation Calculation Results Conclusion

Non-equilibrium Green’s Function Calculation ofOptical Absorption in Nano Optoelectronic Devices

Oka Kurniawan, Ping Bai, Er Ping Li

Computational Electronics and PhotonicsInstitute of High Performance Computing

Singapore

28th May 2009

Motivation NEGF Formulation Calculation Results Conclusion

Speed of Light Motivates Research on Electron-PhotonInteraction 1

1Images courtesy of IBM.

Motivation NEGF Formulation Calculation Results Conclusion

Speed of Light Motivates Research on Electron-PhotonInteraction 2

2Images courtesy of Intel.

Motivation NEGF Formulation Calculation Results Conclusion

Speed of Light Motivates Research on Electron-PhotonInteraction 2

Six Building blocks

2Images courtesy of Intel.

Motivation NEGF Formulation Calculation Results Conclusion

Motivation Studying Electron-Photon Interaction withNon-equilibrium Green’s Function (NEGF) Framework

1 Commonly used for nanoscale transport with phase-breakingphenomena.

2 Electron-photon interaction is important for optoelectronics.

3 Takes into account open systems with complex potentials andgeometries.

4 no prior assumptions on the nature of the transitions.

5 Other interaction can be included, such as electron-phonon.

Motivation NEGF Formulation Calculation Results Conclusion

Motivation Studying Electron-Photon Interaction withNon-equilibrium Green’s Function (NEGF) Framework

1 Commonly used for nanoscale transport with phase-breakingphenomena.

2 Electron-photon interaction is important for optoelectronics.

3 Takes into account open systems with complex potentials andgeometries.

4 no prior assumptions on the nature of the transitions.

5 Other interaction can be included, such as electron-phonon.

Motivation NEGF Formulation Calculation Results Conclusion

Motivation Studying Electron-Photon Interaction withNon-equilibrium Green’s Function (NEGF) Framework

1 Commonly used for nanoscale transport with phase-breakingphenomena.

2 Electron-photon interaction is important for optoelectronics.

3 Takes into account open systems with complex potentials andgeometries.

4 no prior assumptions on the nature of the transitions.

5 Other interaction can be included, such as electron-phonon.

Motivation NEGF Formulation Calculation Results Conclusion

Motivation Studying Electron-Photon Interaction withNon-equilibrium Green’s Function (NEGF) Framework

1 Commonly used for nanoscale transport with phase-breakingphenomena.

2 Electron-photon interaction is important for optoelectronics.

3 Takes into account open systems with complex potentials andgeometries.

4 no prior assumptions on the nature of the transitions.

5 Other interaction can be included, such as electron-phonon.

Motivation NEGF Formulation Calculation Results Conclusion

Motivation Studying Electron-Photon Interaction withNon-equilibrium Green’s Function (NEGF) Framework

1 Commonly used for nanoscale transport with phase-breakingphenomena.

2 Electron-photon interaction is important for optoelectronics.

3 Takes into account open systems with complex potentials andgeometries.

4 no prior assumptions on the nature of the transitions.

5 Other interaction can be included, such as electron-phonon.

Motivation NEGF Formulation Calculation Results Conclusion

We Study Optical Absorption in Quantum Well InfraredPhotodetector

Zero bias with a terminatingbarrier on the right.Henrickson, JAP, (91) 6273,2002.

Motivation NEGF Formulation Calculation Results Conclusion

We Study Optical Absorption in Quantum Well InfraredPhotodetector

Zero bias with a terminatingbarrier on the right.Henrickson, JAP, (91) 6273,2002.

Biased and no terminating barrierat the contacts.

Motivation NEGF Formulation Calculation Results Conclusion

NEGF Framework with Electron-Photon Interaction

Motivation NEGF Formulation Calculation Results Conclusion

The Device is Represented by its Hamiltonian, and theInteraction by its Self-Energy Matrices

G (E ) = [ES + ıη − H0 − diag(U)− Σ1 − Σ2 − Σph]−1

Motivation NEGF Formulation Calculation Results Conclusion

Self-Enery Matrix for Electron-Photon Interaction

Σ<rs(E ) =

∑pq

MrpMqs [NG<pq(E − ~ω) + (N + 1)G<

pq(E + ~ω)]

1 N is the number of photon.

2 G< is the less-than Green’s function, giving us the electrondistribution.

3 Mij is the coupling matrix obtained from the InteractionHamiltonian, and is a function of photon flux.

Motivation NEGF Formulation Calculation Results Conclusion

Calculation Steps

Motivation NEGF Formulation Calculation Results Conclusion

Photocurrent Calculation

I =q

π~

∫t(G<

p,q(E )− G<q,p(E ))dE

and

RI =I

qIω

1 t is the off-diagonal coupling element of the Hamiltonian.

2 Iω is the photon flux at energy ~ω.

3 RI is the photocurrent response.

Motivation NEGF Formulation Calculation Results Conclusion

Our Calculation Agrees Well with Published Result

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

0 0.5 1 1.5 2 2.5

Pho

tocu

rren

t Res

pons

e, R

I (nm

2 /pho

ton)

Photon Energy (eV)

Our SimulationHenrickson’s

1 LE = LC = 2 nm and LW = 5nm.

2 Barrier height is 2.0 eV, and terminating barrier height on theright is 0.2 eV.

3 We use a uniform GaAs effective mass for all region.

4 First peak location agrees pretty well with the result fromHenrickson, JAP, (91) 6273, 2002.

Motivation NEGF Formulation Calculation Results Conclusion

Effect of Bias on Photocurrent Spectral Response PeakLocations is not Significant

10-5

10-4

10-3

10-2

10-1

0 0.5 1 1.5 2 2.5

Pho

tocu

rren

t Res

pons

e, R

I (nm

2 /pho

ton)

Photon Energy (eV)

0.4

1.1

1.9

Vb = 0.05 VVb = 0.10 VVb = 0.20 V

1 Peak Locations do not change significantly.

2 Magnitude seems to be affected.

Motivation NEGF Formulation Calculation Results Conclusion

Plot of Transmission Curves Under Various Bias

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

0 0.5 1 1.5 2 2.5

Tra

nsm

issi

on

Energy (eV)

Vb = 0.05 VVb = 0.10 VVb = 0.20 V

1 Resonant peak locations are shifted to the left for higher bias.

2 Distance between resonant peaks, however, does not changesignificantly.

Motivation NEGF Formulation Calculation Results Conclusion

Conclusion

1 We study electron-photoninteraction using the NEGFframework.

2 Our calculation agrees with thepreviously published result.

3 Peak locations of photocurrentspectral response under variousbias does not change significantly.

4 Transmission curves show the shiftin the peaks of the resonantenergies.

Derivation of Self-Energy Matrices Device Simulator Approach Photocurrent Response from Absorption Coefficient

Photon Flux

We assume that the photon flux is a constant and is given by

Iω ≡Nc

V√µr εr

(1)

Since the photocurrent response is normalized

RI =I

qIω(2)

hence, we can set Iω = 1.

Derivation of Self-Energy Matrices Device Simulator Approach Photocurrent Response from Absorption Coefficient

Interaction Hamiltonian

The vector potential is given by

A(r, t) = a

√~

2ωεV(be−ıωt + b†eıωt) exp(ık · r) (3)

We also assume dipole approximation, i.e. ek·r ≈ 1.The interaction Hamiltonian in the second quantized form is

H1 =∑rs

〈r |H1|s〉a†ras (4)

〈r |H1|s〉 =q

m0〈r |A · p|s〉 (5)

Derivation of Self-Energy Matrices Device Simulator Approach Photocurrent Response from Absorption Coefficient

Interaction Hamiltonian

We assume that the field is polarized in the z direction. Therefore,the interaction Hamiltonian can be shown to be

H1 =∑rs

(zr − zs)iq

~(be−iωt + b†e iωt)× azr

⟨r∣∣H0∣∣ s⟩ a†ras (6)

If we use finite difference, it can be shown that

H1 =∑rs

Mrs

(be−ıωt + b†eıωt

)(7)

where

Mrs =q~ı2a

√~√µr εr

2NωεcIωPrs

Prs =

+1/m∗s , s = r + 1−1/m∗s , s = r − 10 , else

Derivation of Self-Energy Matrices Device Simulator Approach Photocurrent Response from Absorption Coefficient

Self-Energy Matrices

And the self-energy matrices is given by

Σ≷rs(t1, t2) =

∑pq

G≷pq(t1, t2)D≷

rp;qs(t1, t2) (8)

and

D>rp;qs(t1, t2) ≡ 〈H1

rp(t1)H1qs(t2)〉 (9)

D<rp;qs(t1, t2) ≡ 〈H1

qs(t2)H1rp(t1)〉 (10)

Hence, we can write the self-energy matrices as

Σ<rs(E ) =

∑pq

MrpMqs [NG<pq(E − ~ω) + (N + 1)G<

pq(E + ~ω)]

Derivation of Self-Energy Matrices Device Simulator Approach Photocurrent Response from Absorption Coefficient

Device Simulator Approach to Photogeneration

Simulator calculate the change in carrier density from thecontinuity equations.

∂n

∂t=

1

q∇Jn + Gn − Rn (11)

where Jn is the electron current density, Gn is the generation rateand Rn is the recombination rate. The generation is calculatedfrom

G = η0Pλ

hcα exp (αy) (12)

where η0 is the internal quantum efficiency, P is the intensity, α isthe absorption coefficient, and y is distance.

Derivation of Self-Energy Matrices Device Simulator Approach Photocurrent Response from Absorption Coefficient

From Photogeneration to Photocurrent

Once we know the change in carrier density, we can calculate thecurrent from the Drift-Diffusion equation.

Jn = qnµnEn + qDn∇n (13)