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Slides presentation for 13th International Workshop on Computational Electronics 27-29 May 2009 in Beijing
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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)