Carrier Transport Modeling in
Q
Quantum
Cascade
C
Lasers
C. Jirauschek
Institute for Nanoelectronics,
Nanoelectronics TU München
München,
Germany
Overview
•Methods overview
•Open
p two-level model
•Monte
Monte Carlo simulation
•Quantum transport
Technische Universität München
2
Institute for Nanoelectronics
Typical QCL Structure
V
G
GaAs/Al
/ 0.15Ga
G 0.85As
Technische Universität München
3
z
Institute for Nanoelectronics
Carrier Transport Simulation Methods
Quantum transport
(semiclassical)
(e.g., density matrix, NEGF)
Ekin
Monte Carlo method
Ekin
Rate equations
Complexity, accuracy
Technische Universität München
4
Institute for Nanoelectronics
Overview
•Methods overview
•Open
p two-level model
•Monte
Monte Carlo simulation
•Quantum transport
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5
Institute for Nanoelectronics
Op
ptical Pow
wer
Quantum Cascade Ring Laser
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Propagation coordinate
6
Institute for Nanoelectronics
Extended Maxwell-Bloch Equations
Gain
n
Los
ss
Gain recovery
Propagation coordinate
Propagation coordinate
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7
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Rate Equations
Δp/T1
T1
T 0
T→0
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8
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Coherent Effects
T2
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9
E
Institute for Nanoelectronics
Coherent Effects
Standard amplification: Rate equations
Coherent effects (Maxwell-Bloch equations)
In
nversion
Pulse
u
on
Rabi frequency
Rabi
R
bi oscillations:
ill i
•Quantum beat between two states
•Oscillatory driving field
Time
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10
Institute for Nanoelectronics
Coherent instability = Rabi splitting
Gain
Coherent instability
[Spatial hole burning]
+ΩRabi
37
2.0
2xRabi frequency, theory
Spectrum splitting, experiment
Frequ
uency (TH
Hz)
Optical
Opt
ca Frequency
eque cy (THz)
(
)
35
36
15
1.5
Opttical Powe
er
1110
Mode locking
Q switching
F
Frequency
-ΩRabi
34
Slow
gain
Fast
gain
1.5
0.5
1150
1190
Wavenumber (cm-1)
Technische Universität München
1230
00
0.0
11
0
2
Power1/2
4
6
1/2
(mW )
8
Institute for Nanoelectronics
Overview
•Methods overview
•Open
p two-level model
•Monte
Monte Carlo simulation
•Quantum transport
Technische Universität München
12
Institute for Nanoelectronics
Simulation
V
Schrödinger-Poisson solver
Monte Carlo simulation
z
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13
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k⊥, Ekin
k⊥, Ekin
Monte Carlo Solver
5
4
3
2
1
Carrier distribution
function f(k)
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14
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Boltzmann equation
carrier distribution function f(r,k,t)
∂ t f = − v∇r f − (E + v × B)∇k f + ∂ t f |scatt
qc
=
Acoustic phonons
Optical phonon absorption/emission Carrier-carrier scattering
(polar LO / non-polar TO phonons)
E
Optical phonon
E
E
E
k
k
k
O ti l phonon
Optical
h
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k
15
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m
⁞ ⁞
⁞ ⁞
⁞
⁞
2ΔE
1 ΔE
0
⁞ ⁞
.
.
.
3ΔE
2ΔE
ΔE
5 ΔE
0
⁞ ⁞
2ΔE
1 ΔE
1
0
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16
5
⁞
1
5
⁞
1
5
⁞
1
⁞
5
⁞
1
5
⁞
1
5
⁞
1
⁞
5
⁞
1
5
⁞
1
5
⁞
1
Institute for Nanoelectronics
Rando
om selecction
2
nE
2ΔE
LO
O phonon
n
sscattering
g
.
.
.
3ΔE
2ΔE
ΔE
Tabulate
e
sca
attering rates
.
.
.
.
3ΔE
.
2ΔE
ΔE
.
.
5 2ΔE
.
ΔE
.
4
ΔE
3
Acousstic phono
on…
sca
attering
k, Ekin
k
Evaluation of Scattering Rates
Ensemble Monte Carlo Method
Optical phonon Acoustic phonon
scattering
scattering
Carrier-carrier
scattering
Particle
N
.
.
.
3
2
1
Subinterval Δt
Time t
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Ensemble Monte Carlo Method
State
m Carrier N
.
.
.
.
.
.
C i 3
Carrier
Carrier 2
3
2
C i 1
1 Carrier
Subinterval Δt
Time t
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Ensemble Monte Carlo Method
State
m Carrier N
.
.
.
.
.
.
C i 3
Carrier
Carrier 2
3
2
C i 1
1 Carrier
Subinterval Δt
Time t
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Monte Carlo Solver
Start
Import parameters
and wavefunctions
Wavefunctions,
eigenenergies
Calculate
scattering rate
Schrödinger-Poisson
solver
(effective mass)
Initialize carrier
distribution
Next subinterval
Next particle
No
New free flight
Stationary
state?
Scattering process
Update state of
next particle
No
End of
subinterval?
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Yes
End
No
Yes
Last
particle?
O t t off scattering
Output
tt i
rates, carrier
distribution, etc.
Yes
20
Evaluation of
subinterval update
subinterval,
of carrier
distribution
Institute for Nanoelectronics
Monte Carlo Simulation - Example THz QCLs
0.2
0.06
0.04
Kinetiic carrier dis
stribution fu
ucntion
T=200 K
L
0.1
T=300 K
L
0
0.020
20
40
60
Position (nm)
0.02
Upper laser level
4
T =200 K
3
L
80
2
1
100
x 10-3
6
Lower laser level
5
T =100 K
L
0.01
T=100 K
L
T =300 K
L
2
1
0
0
0
0.05 0.1 0.15 0
0.05 0.1 0.15
Kinetic energy (eV) Kinetic energy (eV)
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Current d
C
dens. (kA/cm2)
Ene
ergy (eV)
0.08
In
nversion
2'
5
1'
4
3
2
1
0
9
10
11
12
13
14
Applied bias (kV/cm)
21
Institute for Nanoelectronics
Overview
•Methods overview
•Open
p two-level model
•Monte
Monte Carlo simulation
•Quantum transport
Technische Universität München
22
Institute for Nanoelectronics
Carrier Transport Simulation Methods
E
kin
n
Non-equilibrium Green's
function method (NEGF)
kin
n
E
Monte Carlo method
semiclassical; no q
quantum
correlations (e.g., dephasing)
most g
general scheme for
incoherent quantum transport
modest computational effort
huge computational effort
⇒ neglect e-e scattering,…
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23
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Eigene
energies (eV)
2
Current density (kA
C
A/cm )
Current Density for 3.4 THz Structure (100 K)
2.5
NEGF
Monte Carlo
2
1.5
1
0.5
0
9
0.14
10
11
12
10
11
12
13
14
15
16
17
18
19
13
14
15
16
Applied field (kV/cm)
17
18
19
0 12
0.12
0.1
0.08
0.06
9
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24
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Electron Resolved Density of States
Energy resolved electron density [1018 cm-3eV-1]
2
0.2
0 02
0.02
T. Kubis et al., phys. stat. sol. (c) 5, 232 (2008)
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25
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Optical Gain at Current Peak
Absorption coefficient α [1/cm]
300
175
50
-75
75
-200
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26
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Results (Summary)
Open two-level model
• Includes rate equation elements and quantum effects
• Description of optical instabilities/mode-locking in QCLs
C Y
C.
Y. Wang et al.,
al Phys.
Phys Rev
Rev. A 75,
75 031802(R) (2007)
(2007).
Monte Carlo simulation of QCLs
• Takes into account kinetic electron distribution
• Analysis of experimental results
C. Jirauschek et al
C
al., JJ. Appl
Appl. Phys
Phys. 101,
101 086109 (2007);
C. Jirauschek and P. Lugli, phys. stat. sol. (c) 5, 221 (2008);
C. Jirauschek and P. Lugli, J. Comput. Electron. 7, 436 (2008).
Quantum transport
• Includes quantum correlations/dephasing
• Allows for simulation of spectral gain
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Acknowledgment
Collaborations
• Monte Carlo simulations:
¾ Prof. P. Lugli, TUM
¾ Scamarcio group,
group Bari (experimental)
• Maxwell-Bloch model:
¾ Kärtner group, MIT
¾ Capasso g
group, Harvard
• Further collaborations:
¾ Vogl group/Tillmann Kubis
Kubis, TUM
Financial support
pp