Download Lecture 4

FERMI’S GOLDEN RULE IN THE
SCHRÖDINGER AND WIGNER
PICTURE
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INTRODUCTION
Particle trajectories are not only bent by the presence of the
crystal. They are changed by collisions with the vibrating
crystal lattice.
In these collisions energy between the particles and the lattice
is exchanged.
This is modeled by the creation and destruction of pseudo
particles (phonons).
In crystals this is by far the most important collision mechanism (more frequent than particle - particle collisions).
The energy exchange is governed by Fermi’s Golden Rule,
stating that the amount of particle energy gained / lost in
the process is given by the dominant frequency of the lattice.
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The exact description of phonon collisions is given by an
infinite system of many body Schrödinger for a non-constant
number of particles.
The Fermi Golden Rule can be derived (not rigorously) from
a long time average of this system.
LITERATURE
I. Levinson, “Translational invariance in uniform fields and the
equation for the density matrix in the Wigner representation,”
Soviet Phys.JETP, vol. 30, no. 2, pp. 362–367, 1970.
A. Bertoni, P. Bordone, R. Brunetti, and C. Jacoboni, “The
Wigner function for electron transport in mesoscopic systems,” J.Phys.:Condensed Matter, vol. 11, pp. 5999–6012,
1999.
M. Nedjalkov, R. Kosik, H. Kosina, and S. Selberherr, “A
Wigner Equation for Nanometer and Femtosecond Transport
Regime,” in Proceedings of the 2001 First IEEE Conference
on Nanotechnology, (Maui, Hawaii), pp. 277–281, IEEE, Oct.
2001.
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P. Argyres: Quantum kinetic equations for electrons in high
electric and phonon fields, Phys. Lett. A 171 North Holland
(1992).
N. Ashcroft, M. Mermin:
ders, New York (1976).
Solid State Physics, Holt - Saun-
J. Barker, D. Ferry: Phys. Rev. Lett. 42 (1997).
F. Fromlet,P. Markowich,C. Ringhofer: A Wignerfunction
Approach to Phonon Scattering, VLSI Design 9 pp.339-350
(1999).
OVERVIEW
FGR electron - phonon scattering in a crystal.
Discrepancy to experiment, the intra - collisional field effect.
Derivation of the FGR from the Fock - space many body
picture.
Infinite collision times; the Levinson and the Barker - Ferry
equation.
Local time asymptotics → first order corrections to the FGR.
Numerical results.
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FERMI’S GOLDEN RULE (FGR)
Models the interaction of electrons with vibrations of the
crystal lattice (pseudo - particles, phonons)
∂tf (p, t) = Q[f ](p, t)
=
Z
[S(p, p0)f (p0, t) − S(p0, p)f (p, t)]dp0
f (p, t): density of electrons with momentum p,
S(p, p0): scattering cross section; FGR states specified amount
/ loss of energy in scattering event
S(p, p0) =
X
sν (p, p0)δ(ε(p) − ε(p0) + ν ~Ω)
ν=±1
ε(p): energy corresponding to momentum p;
Ω: dominant eigen - frequency of the phonon lattice.
Derived from a joint density fep(p, q, t) for electrons with momentum p and phonons with momentum q under some equilibrium assumptions on the phonons.
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COLLISIONAL BROADENING
Experiment: Apply electric field to a piece of bulk silicon.
Switch it off instantenously: FGR predicts time constants
(relaxation times).
Keep it constant: FGR predicts steady state.
E · ∇pf = Q[f ]
For large fields and small scales both predictions are not quite
correct.
Problem: Collisions in FGR are instantenous; quantum mechanics ⇒ collisions take a finite time.
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THE INTRA - COLLISIONAL FIELD EFFECT
The spatially inhomogeneous case, the Boltzmann equation:
∂tf + ∇pε · ∇xf − ∇xV · ∇pf = Q[f ]
Remark: Extremely difficult to observe experimentally; argument about relevance
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The semi - classical Boltzmann equation
∂tfc(x, k, t) + [V (x) + ε(k), fc]c = QF GR [fc]
The quantum Boltzmann equation for the Wigner function
∂tf (x, k, t) + [V (x) + ε(k), f ]w = Q[f ]
Quantum phonon operator Q non-local in x, k, t.
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DERIVATION OF THE FGR
The Fock-space Schrödinger equation:
ψm(r, k1, .., km, t)
ψm: wave function for one electron with position r and m
phonons with wave vectors k1, .., km
i~∂tΨ = HΨ,
Ψ = (ψ0, ψ1, ....)
i~∂tψm = (HΨ)m := H eψm + H pψm +
X
n
ep ψ
Hmn
n
H: Fröhlich Hamiltonian
H e: Free particle.
H p: Free phonon.
ep
Hmn Interaction of electron with phonon (generation, destruction).
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Hamiltonians
Free electron Hamiltonian, acts only on r; V : mean field
potential
H e = ε(−i∇r ) + V (r),
~2 2
|k| + V (r)
E(r, k) =
2m
Free phonon Hamiltonian (acts only on km = (k1, .., km)
H pψm(r, km) =
Z
H p(km, sm)ψm(r, sm)dsm
H e and H p are diagonal elements in the Fröhlich Hamiltonian
H
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Electron phonon interaction Hamiltonian
ep = H ep+ + H ep−
Hmn
mn
mn
Creation of a phonon:
ep+ ψ (r, km) = δ
Hmn
n
m−n−1 ~
m
X
F (kl )eir·kl ψn(r, km
l− )
l=1
km
l− = (k1 , .., kl−1 , kl+1 , .., km)
Destruction of a phonon: H self adjoint
∞ Z
X
φm(r, km)∗(HΨ)m(r, km)drdkm
m=0
=
∞ Z
X
ψm(r, km)∗(HΦ)m(r, km)drdkm
m=0
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ep−
ep+
Hmn is the adjoint of Hnm
ep− ψ (r, km) = δ
Hmn
n
m−n+1 ~
n Z
X
F (s)∗e−ir·sψn(r, km
l+ (s))ds
l=1
km
l+ (s) = (k1 , .., kl−1 , s, kl , .., km)
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THE PHONON TRACE
The density matrix corresponding to Ψ
0n
0n
m
0
m
∗
0
ρmn(r, k , r , k ) = ψm(r, k ) ψn(r , k )
Find equation for phonon trace:
(T rpρ)(r, r 0) =
∞ Z
X
ρmm(r, km, r 0, km)dkm
m=0
using asymptotics → quantum Boltzmann equation for single
electron density matrix.
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Time Reversibility
PROBLEM:
Quantum mechanics is time reversible
G(Ψ)m(r, km, t) = ψm(r, −km, −t)∗
G(i~∂tΨ) = i~∂tGΨ,
G(HΨ) = HG(Ψ)
⇒ know Ψ(t = T ); solve same equation backward in time to
compute Ψ(t = 0).
FGR not time reversible; loss of information by building the
phonon trace.
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APPROACHES
1. Start from simplified physical model; phonons as bath
of harmonic oscillators (Caldeira, Legget); mathematical
justification incomplete (Erdos); derive quantum Fokker
- Planck operator (Arnold, Markowich)
2. Start with randomized interaction potential (Papanicolao,
Shi Jin et al); derive FGR; weak theory.
3. Direct weak e-p interaction asymptotics (Levinson, Argyres, Barker & Ferry); mathematical justification incomplete (Frommlet, Markowich & Ringhofer); arrive at
Levinson (Barker - Ferry) equation → FGR.
4. Direct weak e-p interaction asymptotics theory complete;
arrive at Pauli master equation (Castella, Degond)
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WEAK ELECTRON - PHONON INTERACTIONS
i~∂tρmn − [H e + H p, ρmn] =
ep
ep
Hm,m−1ρm−1,n + Hm,m+1ρm+1,n
ep
ep
−ρm,n−1Hn−1,n − ρm,n+1Hn+1,n
Notation:
∂tρmn − Aρmn = (Bρ)mn
A diagonal in m, n, ⇒ commutes with taking the phonon
trace
∞ Z
X
(Aρ)mmdkm = [H e, (T rpρ)]
m=0
→ free streaming operator acting on T rpρ
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Fixed point argument:
Assume B small and iterate, SA(t) semigroup operator
ρj+1 = SA(t)(ρI ) +
Z t
0
SA(t − τ )(Bρj )(τ )dτ,
j = 0, 1, ...
Not rigorous:
Formulate in Wigner picture and assume classical transport
operator.
Assume parabolic bands.
Corresponds to assuming quadratic potentials.
Use characteristics.
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THE LEVINSON (BARKER-FERRY) EQUATION
Formulated in terms of the Wigner function (classical equivalent to the density function in the Boltzmann equation)
1
1
T rpρ(x − y, x + y, t)eiy·k dy
2
2
Remark: Since we iterated the semigroup operator the L-BF equation will be non-local in time.
f (x, k, t) =
Z
∂tf (x, k, t) + [ε + V, f ]w =
Z t
0
dt0
Z
dk0[S(k, k0, t − t0)f (x, k0, t0) − S(k0, k, t − t0)f (x, k, t0)]
Remark: The semigroup operator is expressed in terms of
tracing back characteristics in the Wigner picture. (Only true
for linear potentials V .)
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The scattering cross section
Write everything in terms of momentum p = ~k
S(p, p0, t) =
1 t
aν cos[
(ε(q(p, τ )) − ε(q(p0, τ )) + ν ~Ω)dτ ]
~ 0
ν=±1
X
Z
q(k, τ ) := p − τ ∇xV,
|p|2
ε(p) =
2m
Ω: eigen - frequency of the lattice
Remark: The collisions are never completed! The integral
kernel is highly oscillatory for ~ → 0.
Sano (1995): Numerical solution of the B-F equation
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The zero field case (V = const)
S(p, p0, t) =
t
aν cos[ (ε(p) − ε(p0) + ν ~ω)]
~
ν=±1
X
THE SPATIALLY HOMOGENEOUS ZERO - FIELD
LEVINSON EQUATION
Scaling and dimensionless formulation:
ε(p0) = KT,
t0 =
1
,
Ωλ
∂tf (p, t) = Qλ[f ](p, t) =
λ = O(~)
Z t
0
dt0
Z
dp0×
1
t − t0
t − t0
0
0
0
0
[S(p, p ,
)f (p , t ) − S(p , p,
)f (p, t0)]
λ
λ
λ
dimensionless cross section
S(p, p0, t) =
X
aν cos[t(ε(p) − ε(p0) + νΩ)]
ν=±1
aν = O(1) and Ω (scaled emission energy) = O(1).
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FERMI’S GOLDEN RULE - A HEURISTIC ARGUMENT
Qλ[f ](p, t) =
Z t
0
dt0
Z
dp0×
0
0
1
t
−
t
t
−
t
[S(p, p0,
)f (p0, t0) − S(p0, p,
)f (p, t0)]
λ
λ
λ
Assume: limt→∞ S(p, p0, t) = 0
Qλ[f ](p, t) →
Z
dp0[S0(p, p0)f (p0, t) − S0(p0, p)f (p, t)]
S0(p, p0) =
Z ∞
0
S(p, p0, t)dt
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cross section:
S(p, p0, t) =
X
aν cos[t(ε(p) − ε(p0) + νΩ)]
ν=±1
R∞
0 cos(zt)dt = δ(z) yields FGR.
Since limt→∞ S(p, p0, t) 6= 0 this becomes a weak limit.
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THEOREM 1
Define the functional
Yλ(φ, f ) =
Z ∞
0
dt
Z
dp{φ(p, t)Qλ[f ](p, t)}
for smooth, compactly supported test functions φ. Then
lim Yλ(φ, f ) =
λ→0
Z ∞
0
dt
Z
dp{φ(p, t)Q0[f ](p, t)}
holds for fixed φ with
S0(p, p0) =
X
aν δ(ε(p) − ε(p0) + νΩ)
ν=±1
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THEOREM 2
(first order perturbation)
Yλ(φ, f ) =
Z ∞
0
dt
Z
dp{φ(p, t)(Q0[f ] + λQ1[∂tf ])(p, t)} + o(λ)
Remark:
• Q1 acts on ∂tf (non-local in time);
• Q1 only defined weakly in p
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Weak formulation of Q1[∂tf ]:
Z
X
−
aν
Z Z
φ(p)Q1[∂tf ](p)dp =
ln(|ε − ε0 + νΩ|)
ν=±1
d d
0 − φ)∂ f ]dpdp0
[(φ
t
dε dε0
d : directional derivative perpendicular to surfaces of equal
dε
energy.
Corresponds (formally) to a first order scattering cross section
S = S0 + λS1
S0(p, p0) =
X
aν δ(ε(p) − ε(p0) + νΩ)
ν=±1
S1(p, p0, ∂t) =
aν
∂
0 ) + νΩ)2 t
(ε(p)
−
ε(p
ν=±1
X
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|p|2
REMARK: Parabolic bands ε(p) = 2 : polar coordinates and
radial derivatives
NUMERICAL VERIFICATION
THEOREM 3: Let Γ be a time - direction smoothing operator
(Γf )(t) =
Z ∞
0
γ(t − t0)f (t0)dt0.
Then
ΓQλ[f ] = Q0(Γf ) + λQ1[Γ∂tf ] + o(λ)
pointwise in t weakly in p.
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Computations
|p|2
• Parabolic bands ε = 2m
• an = const: δ− function interaction potential
• Symmetry: f (p, t) = f (ε(p), t)
• Compare smoothed energies:
R
εΓQλ[f ]dp
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FIGURE 1
2
f
1.5
1
0.5
0
8
4
6
3
4
2
2
time
1
0
0
energy
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FIGURE 2
400
300
energy
200
100
0
−100
−200
0
1
2
3
4
time
5
6
7
8
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FIGURE 3
1
f (filtered)
0.8
0.6
0.4
0.2
0
8
4
6
3
4
2
2
time
1
0
0
energy
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FIGURE 4
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K*Q[f]
Q0[K*f]
Qfgr[K*f]
35
30
energy
25
20
15
10
5
0
−5
0
1
2
3
4
time
5
6
7
8
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FIGURE 5
80
K*Q[f]
Q0[K*f]
Qfgr[K*f]
70
60
energy
50
40
30
20
10
0
0
1
2
3
4
time
5
6
7
8
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SOLVING THE LEVINSON EQUATION
∂tfλ = Qλ[fλ]
∂tf01 = Q0[f01] + λQ1[∂tf01]
REMARK:
• fλ → f01 not covered by the theory.
• Extremely difficult to solve; implicit; Q1 singular integral
kernel
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Example:
∂tf01 = Q0[f01] + λQ1[Q0[f01]]
yields an ill posed equation.
Solve for expansion terms directly.
f01 = f0 + λf1
∂tf0 = Q0[f0]
∂tf1 = Q0[f1] + Q1[∂tf0] = Q0[f1] + Q1[Q0[f0]]
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FIGURE 6
1.4
1.2
initial condition
1
0.8
0.6
0.4
0.2
0
0
50
100
150
200
250
energy (meV)
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FIGURE 7
1.2
1
0.8
f
0.6
0.4
0.2
0
−0.2
4
250
3
200
2
150
100
1
50
time (ps)
0
0
energy (meV)
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FIGURE 8
1.2
1
f filtered
0.8
0.6
0.4
0.2
0
−0.2
4
250
3
200
2
150
100
1
50
time (ps)
0
0
energy (meV)
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FIGURE 9
1.2
1
f0+lambda*f1
0.8
0.6
0.4
0.2
0
−0.2
4
250
3
200
2
150
100
1
50
time (ps)
0
0
energy (meV)
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FIGURE10
FIGURE11
0.7
0.6
f0
f +lam*f
0.5
f
0
approximation at t=1.7082ps
approximation at t=0.85411ps
0.7
1
0.4
0.3
0.2
0.1
0
0
50
100
150
energy (meV)
200
0.6
f0
f +lam*f
0.5
f
0
0.4
0.3
0.2
0.1
0
250
0
50
FIGURE12
200
250
0.5
0.6
f0
f0+lam*f1
0.5
f
approximation at t=3.4164ps
approximation at t=2.5623ps
100
150
energy (meV)
FIGURE13
0.7
0.4
0.3
0.2
0.1
0
1
0
50
100
150
energy (meV)
200
250
f0
f0+lam*f1
0.4
f
0.3
0.2
0.1
0
0
50
100
150
energy (meV)
200
250
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CONCLUSIONS
• The first order correction produces a significant change
in the transient behavior.
• This can explain collisional broadening in actual devices.
Open Problems
• The non-zero field case (→ the intra - collisional field
effect).
• Numerics for non - radially symmetric solutions (→ MC).
• How to solve the first order approximation using weighted
particle methods.
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