Download Quantum-mechanical wavepacket transport in Quantum Cascade Lasers

Quantum mechanical wavepacket
transport in QCLs
Andreas Wacker
in collaboration with:
S.-C. Lee, F. Banit, M. Woerner
– p.1
Concepts for Transport
– p.2
Hopping transport
Evaluate
energy-eigenstates
– p.3
Hopping transport
Evaluate
energy-eigenstates
k
population of states with
in-plane momentum k
– p.3
Hopping transport
Evaluate
energy-eigenstates
population of states with
in-plane momentum k
k
k’
scattering transitions
scatt
Rαk→βk
0
– p.3
Hopping transport
Evaluate
energy-eigenstates
population of states with
in-plane momentum k
k
k’
scattering transitions
scatt
Rαk→βk
0
Current by scattering between states with different
center of gravity
– p.3
Hopping transport
Evaluate
energy-eigenstates
population of states with
in-plane momentum k
k
k’
scattering transitions
scatt
Rαk→βk
0
Current by scattering between states with different
center of gravity
Concept of rate equation and Monte-Carlo models
– p.3
States in different
quantum wells
Energy
Kazarinov and Suris
z
– p.4
States in different
quantum wells
Tunnel-coupling Ω
provides nondiagonal
density matrix
ρ12 (k) = ha†2k a1k i
⇔ wave function
Ψ(z) = aΨ1 (z) + bΨ2 (z)
Energy
Kazarinov and Suris
1
Ω
2
z
– p.4
States in different
quantum wells
Tunnel-coupling Ω
provides nondiagonal
density matrix
ρ12 (k) = ha†2k a1k i
⇔ wave function
Ψ(z) = aΨ1 (z) + bΨ2 (z)
P
Current ∝ Ω k ρ12 (k)
Energy
Kazarinov and Suris
1
Current
2
z
– p.4
States in different
quantum wells
Tunnel-coupling Ω
provides nondiagonal
density matrix
ρ12 (k) = ha†2k a1k i
⇔ wave function
Ψ(z) = aΨ1 (z) + bΨ2 (z)
P
Current ∝ Ω k ρ12 (k)
Energy
Kazarinov and Suris
1
Current
2
z
Current by coherent superposition of localized states
– p.4
Bulk-transport
Bloch functions eikz ukν (z)
– p.5
Bulk-transport
Bloch functions eikz ukν (z)
carry current by velocity v(k) = ~k/meff
– p.5
Bulk-transport
Bloch functions eikz ukν (z)
carry current by velocity v(k) = ~k/meff
Boltzmann equation
∂
e ∂
∂
f (r, k, t) + v(k) f (r, k, t) + F f (r, k, t)
∂t
∂r
~ ∂k
X
Wk0 →k f (r, k0 , t) − Wk→k0 f (r, k, t)
=
k0
Scattering does not change particle position
– p.5
Bulk-transport
Bloch functions eikz ukν (z)
carry current by velocity v(k) = ~k/meff
Boltzmann equation
∂
e ∂
∂
f (r, k, t) + v(k) f (r, k, t) + F f (r, k, t)
∂t
∂r
~ ∂k
X
Wk0 →k f (r, k0 , t) − Wk→k0 f (r, k, t)
=
k0
Scattering does not change particle position
Current by coherent propagation of wavepackets
– p.5
Formulation of the problem
Spatial propagation of carriers by:
Coherent evolution or scattering?
– p.6
Formulation of the problem
Spatial propagation of carriers by:
Coherent evolution or scattering?
Relevant for
Conceptual understanding
– p.6
Formulation of the problem
Spatial propagation of carriers by:
Coherent evolution or scattering?
Relevant for
Conceptual understanding
Quantitative simulation (?)
– p.6
Formulation of the problem
Spatial propagation of carriers by:
Coherent evolution or scattering?
Relevant for
Conceptual understanding
Quantitative simulation (?)
Ultrafast phenomena
– p.6
Quantum transport model
– p.7
Modeling of Transport
Start with (single particle) basis Ψα (z)ei(kx x+ky y)
o
Ĥ = Ĥ + Ĥscatt
with
o
Ĥ =
X
o
Hαβ
(k)a†αk aβk
α,β,k
Ĥscatt contains nondiagonal terms a†αk aβk0
– p.8
Modeling of Transport
Start with (single particle) basis Ψα (z)ei(kx x+ky y)
o
Ĥ = Ĥ + Ĥscatt
with
o
Ĥ =
X
o
Hαβ
(k)a†αk aβk
α,β,k
Ĥscatt contains nondiagonal terms a†αk aβk0
Current:
ie
J=
h[Ĥ, ẑ]i =
~V
ie
h[Ĥo , ẑ]i
|~V {z
}
= Jo
diagonal in k
terms ρβα (k) = ha†αk aβk i
– p.8
Modeling of Transport
Start with (single particle) basis Ψα (z)ei(kx x+ky y)
o
Ĥ = Ĥ + Ĥscatt
with
o
Ĥ =
X
o
Hαβ
(k)a†αk aβk
α,β,k
Ĥscatt contains nondiagonal terms a†αk aβk0
Current:
ie
J=
h[Ĥ, ẑ]i =
~V
ie
ie
h[Ĥo , ẑ]i
h[Ĥscatt , ẑ]i
+
|~V {z
|~V {z
}
}
= Jo
= Jscatt
diagonal in k
with terms ha†αk aα0 k0 i
terms ρβα (k) = ha†αk aβk i
– p.8
Modeling of Transport
Start with (single particle) basis Ψα (z)ei(kx x+ky y)
o
Ĥ = Ĥ + Ĥscatt
with
o
Ĥ =
X
o
Hαβ
(k)a†αk aβk
α,β,k
Ĥscatt contains nondiagonal terms a†αk aβk0
Current:
ie
J=
h[Ĥ, ẑ]i =
~V
ie
ie
h[Ĥo , ẑ]i
h[Ĥscatt , ẑ]i
+
|~V {z
|~V {z
}
}
= Jo
= Jscatt
diagonal in k
with terms ha†αk aα0 k0 i
terms ρβα (k) = ha†αk aβk i
Evaluation using nonequilibrium Green functions
– p.8
J0 and coherence
ie X
J0 =
[Ĥo , ẑ]αβ ρβα (k)
~V αβk
For many basis sets (energy-eigenstates, Wannier. . . ):
[Ĥo , ẑ]αα = 0
J0 results from nondiagonal elements ρβα (k) (coherences)
– p.9
Interpretation of Jscatt
Semiclassical rate equations for energy-eigenstates Ψ α (z):
small correlations hâ†αk âα0 k i ≈ δαα0 fα (k)
– p.10
Interpretation of Jscatt
Semiclassical rate equations for energy-eigenstates Ψ α (z):
small correlations hâ†αk âα0 k i ≈ δαα0 fα (k)
⇒ J0 = 0
and
k
k’
Jscatt
e X scatt
=
Rαk→βk0 (zβ − zα )
V α,kβ,k0
– p.10
Interpretation of Jscatt
Semiclassical rate equations for energy-eigenstates Ψ α (z):
small correlations hâ†αk âα0 k i ≈ δαα0 fα (k)
⇒ J0 = 0
and
k
k’
Jscatt
e X scatt
=
Rαk→βk0 (zβ − zα )
V α,kβ,k0
Jscatt is electron transfer
in space by scattering
events
– p.10
Results
– p.11
Importance of full self-energy
Approx. in
Σimp
αα0 ,k
=
P
ββ 0 ,k0
imp
0
hVαβ
(k − k0 )Vβimp
0 α0 (k − k)iGββ 0 ,k0
– p.12
Importance of full self-energy
Approx. in
Σimp
αα0 ,k
=
P
ββ 0 ,k0
imp
0
hVαβ
(k − k0 )Vβimp
0 α0 (k − k)iGββ 0 ,k0
3000
Diagonal self-energy (DG)
α = α0 , β = β 0
2500
J [A/cm]
2000
T=78 K
J (DG)
J0 (DG)
Jscatt (DG)
1500
1000
500
0
0
0.01
0.02
0.03
0.04
eFd [eV]
0.05
0.06
0.07
– p.12
Importance of full self-energy
Approx. in
Σimp
αα0 ,k
=
P
ββ 0 ,k0
imp
0
hVαβ
(k − k0 )Vβimp
0 α0 (k − k)iGββ 0 ,k0
Diagonal self-energy (DG)
α = α0 , β = β 0
3000
Full self-energy (ND)
2000
J [A/cm]
2500
1500
T=78 K
J (DG)
J0 (DG)
Jscatt (DG)
J (ND)
J0 (ND)
Jscatt (ND)
1000
500
0
0
0.01
0.02
0.03
0.04
eFd [eV]
0.05
0.06
0.07
– p.12
Importance of full self-energy
Approx. in
Σimp
αα0 ,k
=
P
ββ 0 ,k0
imp
0
hVαβ
(k − k0 )Vβimp
0 α0 (k − k)iGββ 0 ,k0
Diagonal self-energy (DG)
α = α0 , β = β 0
3000
Full self-energy (ND)
2000
J [A/cm]
Jscatt vanishes
in all bases considered
2500
1500
T=78 K
J (DG)
J0 (DG)
Jscatt (DG)
J (ND)
J0 (ND)
Jscatt (ND)
1000
500
0
0
0.01
0.02
0.03
0.04
eFd [eV]
0.05
0.06
0.07
– p.12
Importance of full self-energy
Approx. in
Σimp
αα0 ,k
=
P
ββ 0 ,k0
imp
0
hVαβ
(k − k0 )Vβimp
0 α0 (k − k)iGββ 0 ,k0
Diagonal self-energy (DG)
α = α0 , β = β 0
3000
Full self-energy (ND)
2000
J [A/cm]
Jscatt vanishes
in all bases considered
2500
1500
T=78 K
J (DG)
J0 (DG)
Jscatt (DG)
J (ND)
J0 (ND)
Jscatt (ND)
1000
500
0
0
12
0.01
0.02
0.03
0.04
eFd [eV]
0.05
0.06
0.07
Bias (V)
10
J (DG)
J (ND)
Exp
8
6
Full self energy agrees
with experiment
4
2
0
0
THz QCL by Kumar 2004
0.5
Current (A)
T = 78 K
1
1.5
– p.12
Position Eigenstates
E (eV)
(a)
Wannier
(b)
0.1
0
position eigenstate
0.1
0
20
40
60
position (nm)
80
Eigenstates of Position operator ẑ
imp
(k − k0 ) small for α 6= β
Vαβ
0
0
20
40
60
position (nm)
80
– p.13
Current in Position Eigenstates
Wan ND
Pos ND
Pos DG
Pos NDlocal
-2
Jo (kA cm )
1
0.5
0
0
0.02
0.04
V/per
0.06
imp
Full current with local scattering matrix elements Vαα
(k − k0 )
– p.14
Current in Position Eigenstates
Wan ND
Pos ND
Pos DG
Pos NDlocal
-2
Jo (kA cm )
1
0.5
0
0
0.02
0.04
V/per
0.06
imp
Full current with local scattering matrix elements Vαα
(k − k0 )
Only local scattering of relevance
– p.14
Resolution of Current
X
e
~ ∂ψβ (z) <
∗
Gβα,k (E)
Rewrite Jo (E, z) = −
Re ψα (z)
πA αβ,k
m(z) ∂z
– p.15
Resolution of Current
X
e
~ ∂ψβ (z) <
∗
Gβα,k (E)
Rewrite Jo (E, z) = −
Re ψα (z)
πA αβ,k
m(z) ∂z
Coherent
evolution plus
local scattering
– p.15
Current carrying states
Diagonalize G<
βα,k (E)
Eigenvalues fnk (E)
Eigenfunctions φnk (z)
– p.16
Current carrying states
Diagonalize G<
βα,k (E)
Eigenvalues fnk (E)
Eigenfunctions φnk (z)
complex φnk (z) carry current
~
∂φnk (z)
φnk (z)
J(z) ∝ fnk (E)Re
i
∂z
– p.16
Current carrying states
complex φnk (z) carry current
~
∂φnk (z)
φnk (z)
J(z) ∝ fnk (E)Re
i
∂z
Diagonalize G<
βα,k (E)
Eigenvalues fnk (E)
Eigenfunctions φnk (z)
200
WS at -7.3 meV
WS at -6.6 meV
50 mV/period
-100 0
20
z (nm)
300
2
(c)
|φ (z)|
E (meV)
-4
n=2
-6
10
0
n=4
10 -15
-10
40
300
Re{-iφ*10φ’10/m}
100
(c)
0
20
z (nm)
40
n=3
-8
10
200
-100
(b) n=1
10
100
0
-2
10
fn0(E) (1/meV)
(a)
E (meV)
E (meV)
300
200
0
-5
E (meV)
2
|φ20(z)|
Re{-iφ*
φ’ /m}
20 20
5
(d)
100
0
-100
(d)
0
20
z (nm)
40
– p.16
Relation to hopping models
Standard hopping models:
neglect coherences, i.e., ραα0 (k) = δαα0 fα (k).
Current P
by scattering rates between different levels
scatt
Jhopp = αk,βk0 [fα (k) − fβ (k 0 )]Rαk→βk
0 (zβ − zα )
– p.17
Relation to hopping models
Standard hopping models:
neglect coherences, i.e., ραα0 (k) = δαα0 fα (k).
Current P
by scattering rates between different levels
scatt
Jhopp = αk,βk0 [fα (k) − fβ (k 0 )]Rαk→βk
0 (zβ − zα )
Our findings:
Current entirely given by nondiagonal elements of ραβ (k).
Scattering only redistributes energy and momentum, but
does not propel carriers through the structure
– p.17
Relation to hopping models
Standard hopping models:
neglect coherences, i.e., ραα0 (k) = δαα0 fα (k).
Current P
by scattering rates between different levels
scatt
Jhopp = αk,βk0 [fα (k) − fβ (k 0 )]Rαk→βk
0 (zβ − zα )
Our findings:
Current entirely given by nondiagonal elements of ραβ (k).
Scattering only redistributes energy and momentum, but
does not propel carriers through the structure
Reconciliation:
Approximate coherences in J0 and obtain Jhopp
see Wacker, Phys. Rep 357, 1 (2002) for superlattices
– p.17
Relation to hopping models
Standard hopping models:
neglect coherences, i.e., ραα0 (k) = δαα0 fα (k).
Current P
by scattering rates between different levels
scatt
Jhopp = αk,βk0 [fα (k) − fβ (k 0 )]Rαk→βk
0 (zβ − zα )
Our findings:
Current entirely given by nondiagonal elements of ραβ (k).
Scattering only redistributes energy and momentum, but
does not propel carriers through the structure
Reconciliation:
Approximate coherences in J0 and obtain Jhopp
see Wacker, Phys. Rep 357, 1 (2002) for superlattices
Hopping model is an approximation for coherences!
– p.17
Conclusion
Nature of transport in quantum cascade lasers
– p.18
Conclusion
Nature of transport in quantum cascade lasers
Spatial transport by coherent
evolution
– p.18
Conclusion
Nature of transport in quantum cascade lasers
Spatial transport by coherent
evolution
Jscatt vanishes
– p.18
Conclusion
Nature of transport in quantum cascade lasers
Spatial transport by coherent
evolution
Jscatt vanishes
Only diagonal scattering
matrix elements Vαα of
importance in position
basis
– p.18
Conclusion
Nature of transport in quantum cascade lasers
Spatial transport by coherent
evolution
Jscatt vanishes
Only diagonal scattering
matrix elements Vαα of
importance in position
basis
Current carried by wave
functions φnk (z)
– p.18
Conclusion
Nature of transport in quantum cascade lasers
Spatial transport by coherent
evolution
Jscatt vanishes
Only diagonal scattering
matrix elements Vαα of
importance in position
basis
Current carried by wave Coherent evolution in space
functions φnk (z)
plus local scattering
Lee, Banit, Woerner, and Wacker, PRB 73, 245320 (2006)
– p.18
Supplements
– p.19
Nonequilibrium Green Functions
Concept: Information in Green functions
0
G<
(k;
t,
t
)
α,β
=
iha†βk (t0 )aαk (t)i
steady
state
=
Z
dE <
−iE(t−t0 )/~
Gα,β (k, E)e
2π
– p.20
Nonequilibrium Green Functions
Concept: Information in Green functions
0
G<
(k;
t,
t
)
α,β
=
Keldysh relation
Z
dE <
−iE(t−t0 )/~
Gα,β (k, E)e
=
2π
ret
o
ret
E − H (k) − Σ (k, E) G (k, E) = 1
iha†βk (t0 )aαk (t)i
Dyson equation
steady
state
Σ< (k, E)Gadv (k, E)
G< (k, E) = Gret (k, E)Σ
– p.20
Nonequilibrium Green Functions
Concept: Information in Green functions
0
G<
(k;
t,
t
)
α,β
=
Keldysh relation
Z
dE <
−iE(t−t0 )/~
Gα,β (k, E)e
=
2π
ret
o
ret
E − H (k) − Σ (k, E) G (k, E) = 1
iha†βk (t0 )aαk (t)i
Dyson equation
steady
state
Σ< (k, E)Gadv (k, E)
G< (k, E) = Gret (k, E)Σ
Self-energies (in self-consistent Born approx.)
X
Σ ret,imp (k, E) =
hV(k, k 0 )Gret (k 0 , E)V(k 0 , k)iimp
Σ < (k, E) = . . .
k0
have to be solved self-consistently
– p.20
Current carrying states for QCL
– p.21
Current versus density
– p.22
IR-QCL from Sirtori 1998
– p.23
Vanishing of Jscatt
Jscatt
ie
=
h[Ĥscatt , ẑ]i
~V
Ĥscatt is a microscopic function of r̂, ẑ, due to local fluctuating
potentials.
Thus: [Ĥscatt , ẑ] = 0
– p.24