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

1. 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

2. Motivation NEGF Formulation Calculation Results Conclusion
Speed of Light Motivates Research on Electron-Photon
Interaction 1
1
Images courtesy of IBM.

3. Motivation NEGF Formulation Calculation Results Conclusion
Speed of Light Motivates Research on Electron-Photon
Interaction 2
2
Images courtesy of Intel.

4. Motivation NEGF Formulation Calculation Results Conclusion
Speed of Light Motivates Research on Electron-Photon
Interaction 2
Six Building blocks
2
Images courtesy of Intel.

5. 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.

6. 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.

7. 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.

8. 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.

9. 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.

10. 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.

11. Motivation NEGF Formulation Calculation Results Conclusion
We Study Optical Absorption in Quantum Well Infrared
Photodetector
Zero bias with a terminating Biased and no terminating barrier
barrier on the right. at the contacts.
Henrickson, JAP, (91) 6273,
2002.

12. Motivation NEGF Formulation Calculation Results Conclusion
NEGF Framework with Electron-Photon Interaction

13. 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

14. Motivation NEGF Formulation Calculation Results Conclusion
Self-Enery Matrix for Electron-Photon Interaction
Σ< (E ) =
rs
< <
Mrp Mqs [NGpq (E − ω) + (N + 1)Gpq (E + ω)]
pq
1 N is the number of photon.
2 G < is the less-than Green’s function, giving us the electron
distribution.
3 Mij is the coupling matrix obtained from the Interaction
Hamiltonian, and is a function of photon flux.

15. Motivation NEGF Formulation Calculation Results Conclusion
Calculation Steps

16. Motivation NEGF Formulation Calculation Results Conclusion
Photocurrent Calculation
q < <
I = t(Gp,q (E ) − Gq,p (E ))dE
π
and
I
RI =
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.

17. Motivation NEGF Formulation Calculation Results Conclusion
Our Calculation Agrees Well with Published Result
Photocurrent Response, RI (nm2/photon)
0
10
Our Simulation
10
-1 Henrickson’s
10-2
10-3
10-4
-5
10
-6
10
10-7
10-8
0 0.5 1 1.5 2 2.5
Photon Energy (eV)
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.

18. Motivation NEGF Formulation Calculation Results Conclusion
Effect of Bias on Photocurrent Spectral Response Peak
Locations is not Significant
Photocurrent Response, RI (nm2/photon)
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 1.5 2 2.5
Photon Energy (eV)
1 Peak Locations do not change significantly.
2 Magnitude seems to be affected.

19. Motivation NEGF Formulation Calculation Results Conclusion
Plot of Transmission Curves Under Various Bias
100
-1
10
-2
10
-3
Transmission
10
-4
10
-5
10
-6
10
-7
10 Vb = 0.05 V
10-8 Vb = 0.10 V
-9
Vb = 0.20 V
10
0 0.5 1 1.5 2 2.5
Energy (eV)
1 Resonant peak locations are shifted to the left for higher bias.
2 Distance between resonant peaks, however, does not change
significantly.

20. Motivation NEGF Formulation Calculation Results Conclusion
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.

21. 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
Nc
Iω ≡ √ (1)
V µr r
Since the photocurrent response is normalized
I
RI = (2)
qIω
hence, we can set Iω = 1.

22. Derivation of Self-Energy Matrices Device Simulator Approach Photocurrent Response from Absorption Coefficient
Interaction Hamiltonian
The vector potential is given by
A(r, t) = ˆ
a (be −ıωt + b † e ıωt ) exp(ık · r) (3)
2ω V
We also assume dipole approximation, i.e. e k·r ≈ 1.
The interaction Hamiltonian in the second quantized form is
†
H1 = r |H 1 |s ar as (4)
rs
q
r |H 1 |s = r |A · p|s (5)
m0

23. 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 direction. Therefore,
z
the interaction Hamiltonian can be shown to be
iq
H1 = (zr − zs ) (be −iωt + b † e iωt ) × ˆzr r H 0 s ar as
a †
(6)
rs
If we use finite difference, it can be shown that
H1 = Mrs be −ıωt + b † e ıωt (7)
rs
where
∗

√  +1/ms , s = r + 1
q µr r Prs = ∗
−1/ms , s = r − 1
Mrs = Iω Prs
ı2a 2Nω c 
0 , else

24. Derivation of Self-Energy Matrices Device Simulator Approach Photocurrent Response from Absorption Coefficient
Self-Energy Matrices
And the self-energy matrices is given by
Σrs (t1 , t2 ) = Gpq (t1 , t2 )Drp;qs (t1 , t2 ) (8)
pq
and
> 1 1
Drp;qs (t1 , t2 ) ≡ Hrp (t1 )Hqs (t2 ) (9)
< 1 1
Drp;qs (t1 , t2 ) ≡ Hqs (t2 )Hrp (t1 ) (10)
Hence, we can write the self-energy matrices as
Σ< (E ) =
rs
< <
Mrp Mqs [NGpq (E − ω) + (N + 1)Gpq (E + ω)]
pq

25. 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 (11)
∂t q
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.

26. 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)