Non-equilibrium Green's Function Calculation Of Optical Absorption In Nano Optoelectronic Devices
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Motivation
NEGF Formulation
Calculation Results
Conclusion
Non-equilibrium Green’s Function Calculation of Optical Absorption in Nano Optoelectronic Devices Oka Kurniawan, Ping Bai, Er Ping Li Computational Electronics and Photonics Institute of High Performance Computing Singapore
28th May 2009
Motivation
NEGF Formulation
Calculation Results
Conclusion
Speed of Light Motivates Research on Electron-Photon Interaction 1
1
Images courtesy of IBM.
Motivation
NEGF Formulation
Calculation Results
Conclusion
Speed of Light Motivates Research on Electron-Photon Interaction 2
2
Images courtesy of Intel.
Motivation
NEGF Formulation
Calculation Results
Conclusion
Speed of Light Motivates Research on Electron-Photon Interaction 2
Six Building blocks
2
Images courtesy of Intel.
Motivation
NEGF Formulation
Calculation Results
Conclusion
Motivation Studying Electron-Photon Interaction with Non-equilibrium Green’s Function (NEGF) Framework
1
Commonly used for nanoscale transport with phase-breaking phenomena.
2
Electron-photon interaction is important for optoelectronics.
3
Takes into account open systems with complex potentials and geometries.
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 with Non-equilibrium Green’s Function (NEGF) Framework
1
Commonly used for nanoscale transport with phase-breaking phenomena.
2
Electron-photon interaction is important for optoelectronics.
3
Takes into account open systems with complex potentials and geometries.
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 with Non-equilibrium Green’s Function (NEGF) Framework
1
Commonly used for nanoscale transport with phase-breaking phenomena.
2
Electron-photon interaction is important for optoelectronics.
3
Takes into account open systems with complex potentials and geometries.
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 with Non-equilibrium Green’s Function (NEGF) Framework
1
Commonly used for nanoscale transport with phase-breaking phenomena.
2
Electron-photon interaction is important for optoelectronics.
3
Takes into account open systems with complex potentials and geometries.
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 with Non-equilibrium Green’s Function (NEGF) Framework
1
Commonly used for nanoscale transport with phase-breaking phenomena.
2
Electron-photon interaction is important for optoelectronics.
3
Takes into account open systems with complex potentials and geometries.
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 Infrared Photodetector
Zero bias with a terminating barrier on the right. Henrickson, JAP, (91) 6273, 2002.
Motivation
NEGF Formulation
Calculation Results
Conclusion
We Study Optical Absorption in Quantum Well Infrared Photodetector
Zero bias with a terminating barrier on the right. Henrickson, JAP, (91) 6273, 2002.
Biased and no terminating barrier at the contacts.
Motivation
NEGF Formulation
Calculation Results
NEGF Framework with Electron-Photon Interaction
Conclusion
Motivation
NEGF Formulation
Calculation Results
Conclusion
The Device is Represented by its Hamiltonian, and the Interaction by its Self-Energy Matrices G (E ) = [ES + ıη − H0 − diag(U) − Σ1 − Σ2 − Σph ]−1
Motivation
NEGF Formulation
Calculation Results
Self-Enery Matrix for Electron-Photon Interaction
Σ< rs (E ) =
X
< < Mrp Mqs [NGpq (E − ~ω) + (N + 1)Gpq (E + ~ω)]
pq
1 2
3
N is the number of photon. G < is the less-than Green’s function, giving us the electron distribution. Mij is the coupling matrix obtained from the Interaction Hamiltonian, and is a function of photon flux.
Conclusion
Motivation
NEGF Formulation
Calculation Steps
Calculation Results
Conclusion
Motivation
NEGF Formulation
Calculation Results
Photocurrent Calculation
I =
q π~
Z
< < t(Gp,q (E ) − Gq,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.
Conclusion
Motivation
NEGF Formulation
Calculation Results
Conclusion
Photocurrent Response, RI (nm2/photon)
Our Calculation Agrees Well with Published Result 100 10
Our Simulation Henrickson’s
-1
10-2 10-3 10-4 10
-5
10-6 10
-7
10-8
0
0.5
1 1.5 Photon Energy (eV)
2
2.5
1
LE = LC = 2 nm and LW = 5nm.
2
Barrier height is 2.0 eV, and terminating barrier height on the right 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 from Henrickson, JAP, (91) 6273, 2002.
Motivation
NEGF Formulation
Calculation Results
Conclusion
Photocurrent Response, RI (nm2/photon)
Effect of Bias on Photocurrent Spectral Response Peak Locations is not Significant
10-1
Vb = 0.05 V Vb = 0.10 V Vb = 0.20 V
10-2
10-3
10-4
0.4
1.9 1.1
10-5
0
0.5
1
Peak Locations do not change significantly.
2
Magnitude seems to be affected.
1 1.5 Photon Energy (eV)
2
2.5
Motivation
NEGF Formulation
Calculation Results
Conclusion
Plot of Transmission Curves Under Various Bias
100 10-1 Transmission
10-2 10-3 10-4 10-5 10-6 10-7
Vb = 0.05 V Vb = 0.10 V Vb = 0.20 V
10-8 -9
10
0
0.5
1 1.5 Energy (eV)
2
2.5
1
Resonant peak locations are shifted to the left for higher bias.
2
Distance between resonant peaks, however, does not change significantly.
Motivation
NEGF Formulation
Conclusion
1
We study electron-photon interaction using the NEGF framework.
2
Our calculation agrees with the previously published result.
3
Peak locations of photocurrent spectral response under various bias does not change significantly.
4
Transmission curves show the shift in the peaks of the resonant energies.
Calculation Results
Conclusion
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 r
(1)
Since the photocurrent response is normalized RI = hence, we can set Iω = 1.
I qIω
(2)
Derivation of Self-Energy Matrices
Device Simulator Approach
Photocurrent Response from Absorption Coefficient
Interaction Hamiltonian The vector potential is given by r ~ (be −ıωt + b † e ıωt ) exp(ık · r) A(r, t) = ˆ a 2ωV We also assume dipole approximation, i.e. e k·r ≈ 1. The interaction Hamiltonian in the second quantized form is X H1 = hr |H 1 |siar† as
(3)
(4)
rs
hr |H 1 |si =
q hr |A · p|si m0
(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 ˆ z direction. Therefore, the interaction Hamiltonian can be shown to be H1 =
X
iq (zr − zs ) (be −iωt + b † e iωt ) × ˆazr r H 0 s ar† as ~ rs
If we use finite difference, it can be shown that X Mrs be −ıωt + b † e ıωt H1 =
(6)
(7)
rs
where r √ q~ ~ µr r Mrs = Iω Prs ı2a 2Nωc
+1/ms∗ , s = r + 1 −1/ms∗ , s = r − 1 Prs = 0 , 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 X ≷ ≷ Σ≷ Gpq (t1 , t2 )Drp;qs (t1 , t2 ) rs (t1 , t2 ) =
(8)
pq
and 1 1 > Drp;qs (t1 , t2 ) ≡ hHrp (t1 )Hqs (t2 )i < Drp;qs (t1 , t2 )
≡
1 1 hHqs (t2 )Hrp (t1 )i
(9) (10)
Hence, we can write the self-energy matrices as X < < Σ< (E ) = Mrp Mqs [NGpq (E − ~ω) + (N + 1)Gpq (E + ~ω)] rs pq
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 the continuity equations. ∂n 1 = ∇Jn + Gn − Rn ∂t q
(11)
where Jn is the electron current density, Gn is the generation rate and Rn is the recombination rate. The generation is calculated from Pλ G = η0 α exp (αy ) (12) hc where η0 is the internal quantum efficiency, P is the intensity, α is the 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 the current from the Drift-Diffusion equation. Jn = qnµn En + qDn ∇n
(13)
NEGF Formulation
Calculation Results
Conclusion
Non-equilibrium Green’s Function Calculation of Optical Absorption in Nano Optoelectronic Devices Oka Kurniawan, Ping Bai, Er Ping Li Computational Electronics and Photonics Institute of High Performance Computing Singapore
28th May 2009
Motivation
NEGF Formulation
Calculation Results
Conclusion
Speed of Light Motivates Research on Electron-Photon Interaction 1
1
Images courtesy of IBM.
Motivation
NEGF Formulation
Calculation Results
Conclusion
Speed of Light Motivates Research on Electron-Photon Interaction 2
2
Images courtesy of Intel.
Motivation
NEGF Formulation
Calculation Results
Conclusion
Speed of Light Motivates Research on Electron-Photon Interaction 2
Six Building blocks
2
Images courtesy of Intel.
Motivation
NEGF Formulation
Calculation Results
Conclusion
Motivation Studying Electron-Photon Interaction with Non-equilibrium Green’s Function (NEGF) Framework
1
Commonly used for nanoscale transport with phase-breaking phenomena.
2
Electron-photon interaction is important for optoelectronics.
3
Takes into account open systems with complex potentials and geometries.
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 with Non-equilibrium Green’s Function (NEGF) Framework
1
Commonly used for nanoscale transport with phase-breaking phenomena.
2
Electron-photon interaction is important for optoelectronics.
3
Takes into account open systems with complex potentials and geometries.
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 with Non-equilibrium Green’s Function (NEGF) Framework
1
Commonly used for nanoscale transport with phase-breaking phenomena.
2
Electron-photon interaction is important for optoelectronics.
3
Takes into account open systems with complex potentials and geometries.
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 with Non-equilibrium Green’s Function (NEGF) Framework
1
Commonly used for nanoscale transport with phase-breaking phenomena.
2
Electron-photon interaction is important for optoelectronics.
3
Takes into account open systems with complex potentials and geometries.
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 with Non-equilibrium Green’s Function (NEGF) Framework
1
Commonly used for nanoscale transport with phase-breaking phenomena.
2
Electron-photon interaction is important for optoelectronics.
3
Takes into account open systems with complex potentials and geometries.
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 Infrared Photodetector
Zero bias with a terminating barrier on the right. Henrickson, JAP, (91) 6273, 2002.
Motivation
NEGF Formulation
Calculation Results
Conclusion
We Study Optical Absorption in Quantum Well Infrared Photodetector
Zero bias with a terminating barrier on the right. Henrickson, JAP, (91) 6273, 2002.
Biased and no terminating barrier at the contacts.
Motivation
NEGF Formulation
Calculation Results
NEGF Framework with Electron-Photon Interaction
Conclusion
Motivation
NEGF Formulation
Calculation Results
Conclusion
The Device is Represented by its Hamiltonian, and the Interaction by its Self-Energy Matrices G (E ) = [ES + ıη − H0 − diag(U) − Σ1 − Σ2 − Σph ]−1
Motivation
NEGF Formulation
Calculation Results
Self-Enery Matrix for Electron-Photon Interaction
Σ< rs (E ) =
X
< < Mrp Mqs [NGpq (E − ~ω) + (N + 1)Gpq (E + ~ω)]
pq
1 2
3
N is the number of photon. G < is the less-than Green’s function, giving us the electron distribution. Mij is the coupling matrix obtained from the Interaction Hamiltonian, and is a function of photon flux.
Conclusion
Motivation
NEGF Formulation
Calculation Steps
Calculation Results
Conclusion
Motivation
NEGF Formulation
Calculation Results
Photocurrent Calculation
I =
q π~
Z
< < t(Gp,q (E ) − Gq,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.
Conclusion
Motivation
NEGF Formulation
Calculation Results
Conclusion
Photocurrent Response, RI (nm2/photon)
Our Calculation Agrees Well with Published Result 100 10
Our Simulation Henrickson’s
-1
10-2 10-3 10-4 10
-5
10-6 10
-7
10-8
0
0.5
1 1.5 Photon Energy (eV)
2
2.5
1
LE = LC = 2 nm and LW = 5nm.
2
Barrier height is 2.0 eV, and terminating barrier height on the right 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 from Henrickson, JAP, (91) 6273, 2002.
Motivation
NEGF Formulation
Calculation Results
Conclusion
Photocurrent Response, RI (nm2/photon)
Effect of Bias on Photocurrent Spectral Response Peak Locations is not Significant
10-1
Vb = 0.05 V Vb = 0.10 V Vb = 0.20 V
10-2
10-3
10-4
0.4
1.9 1.1
10-5
0
0.5
1
Peak Locations do not change significantly.
2
Magnitude seems to be affected.
1 1.5 Photon Energy (eV)
2
2.5
Motivation
NEGF Formulation
Calculation Results
Conclusion
Plot of Transmission Curves Under Various Bias
100 10-1 Transmission
10-2 10-3 10-4 10-5 10-6 10-7
Vb = 0.05 V Vb = 0.10 V Vb = 0.20 V
10-8 -9
10
0
0.5
1 1.5 Energy (eV)
2
2.5
1
Resonant peak locations are shifted to the left for higher bias.
2
Distance between resonant peaks, however, does not change significantly.
Motivation
NEGF Formulation
Conclusion
1
We study electron-photon interaction using the NEGF framework.
2
Our calculation agrees with the previously published result.
3
Peak locations of photocurrent spectral response under various bias does not change significantly.
4
Transmission curves show the shift in the peaks of the resonant energies.
Calculation Results
Conclusion
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 r
(1)
Since the photocurrent response is normalized RI = hence, we can set Iω = 1.
I qIω
(2)
Derivation of Self-Energy Matrices
Device Simulator Approach
Photocurrent Response from Absorption Coefficient
Interaction Hamiltonian The vector potential is given by r ~ (be −ıωt + b † e ıωt ) exp(ık · r) A(r, t) = ˆ a 2ωV We also assume dipole approximation, i.e. e k·r ≈ 1. The interaction Hamiltonian in the second quantized form is X H1 = hr |H 1 |siar† as
(3)
(4)
rs
hr |H 1 |si =
q hr |A · p|si m0
(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 ˆ z direction. Therefore, the interaction Hamiltonian can be shown to be H1 =
X
iq (zr − zs ) (be −iωt + b † e iωt ) × ˆazr r H 0 s ar† as ~ rs
If we use finite difference, it can be shown that X Mrs be −ıωt + b † e ıωt H1 =
(6)
(7)
rs
where r √ q~ ~ µr r Mrs = Iω Prs ı2a 2Nωc
+1/ms∗ , s = r + 1 −1/ms∗ , s = r − 1 Prs = 0 , 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 X ≷ ≷ Σ≷ Gpq (t1 , t2 )Drp;qs (t1 , t2 ) rs (t1 , t2 ) =
(8)
pq
and 1 1 > Drp;qs (t1 , t2 ) ≡ hHrp (t1 )Hqs (t2 )i < Drp;qs (t1 , t2 )
≡
1 1 hHqs (t2 )Hrp (t1 )i
(9) (10)
Hence, we can write the self-energy matrices as X < < Σ< (E ) = Mrp Mqs [NGpq (E − ~ω) + (N + 1)Gpq (E + ~ω)] rs pq
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 the continuity equations. ∂n 1 = ∇Jn + Gn − Rn ∂t q
(11)
where Jn is the electron current density, Gn is the generation rate and Rn is the recombination rate. The generation is calculated from Pλ G = η0 α exp (αy ) (12) hc where η0 is the internal quantum efficiency, P is the intensity, α is the 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 the current from the Drift-Diffusion equation. Jn = qnµn En + qDn ∇n
(13)
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