FERMI’S GOLDEN RULE IN THESCHRODINGER AND WIGNER

PICTURE

1

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 mecha-

nism (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 Schrodinger 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 sys-

tems,” 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.

3

P. Argyres: Quantum kinetic equations for electrons in high

electric and phonon fields, Phys. Lett. A 171 North Holland

(1992).

N. Ashcroft, M. Mermin: Solid State Physics, Holt - Saun-

ders, New York (1976).

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 thecrystal lattice (pseudo - particles, phonons)

∂tf(p, t) = Q[f ](p, t)

=∫

[S(p, p′)f(p′, t)− S(p′, p)f(p, t)]dp′

f(p, t): density of electrons with momentum p,

S(p, p′): scattering cross section; FGR states specified amount/ loss of energy in scattering event

S(p, p′) =∑

ν=±1

sν(p, p′)δ(ε(p)− ε(p′) + ν~Ω)

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 mo-mentum p and phonons with momentum q under some equi-librium 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 me-

chanics ⇒ 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; argu-

ment about relevance

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The semi - classical Boltzmann equation

∂tfc(x, k, t) + [V (x) + ε(k), fc]c = QFGR[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 Schrodinger 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 := Heψm +Hpψm +∑nHepmnψn

H: Frohlich Hamiltonian

He: Free particle.

Hp: Free phonon.

Hepmn Interaction of electron with phonon (generation, de-

struction).

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Hamiltonians

Free electron Hamiltonian, acts only on r; V : mean field

potential

He = ε(−i∇r) + V (r), E(r, k) =~2

2m|k|2 + V (r)

Free phonon Hamiltonian (acts only on km = (k1, .., km)

Hpψm(r,km) =∫Hp(km, sm)ψm(r, sm)dsm

He and Hp are diagonal elements in the Frohlich Hamiltonian

H

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Electron phonon interaction Hamiltonian

Hepmn = Hep+

mn +Hep−mn

Creation of a phonon:

Hep+mn ψn(r,k

m) = δm−n−1~m∑l=1

F (kl)eir·klψn(r,kml−)

kml− = (k1, .., kl−1, kl+1, .., km)

Destruction of a phonon: H self adjoint

∞∑m=0

∫φm(r,km)∗(HΨ)m(r,km)drdkm

=∞∑

m=0

∫ψm(r,km)∗(HΦ)m(r,km)drdkm

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Hep−mn is the adjoint of Hep+

nm

Hep−mn ψn(r,k

m) = δm−n+1~n∑l=1

∫F (s)∗e−ir·sψn(r,kml+(s))ds

kml+(s) = (k1, .., kl−1, s, kl, .., km)

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THE PHONON TRACE

The density matrix corresponding to Ψ

ρmn(r,km, r′,k

′n) = ψm(r,km)∗ψn(r′,k′n)

Find equation for phonon trace:

(Trpρ)(r, r′) =

∞∑m=0

∫ρmm(r,km, r′,km)dkm

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 bathof harmonic oscillators (Caldeira, Legget); mathematicaljustification 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, Ar-gyres, Barker & Ferry); mathematical justification in-complete (Frommlet, Markowich & Ringhofer); arrive atLevinson (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 − [He +Hp, ρmn] =

Hepm,m−1ρm−1,n +H

epm,m+1ρm+1,n

−ρm,n−1Hepn−1,n − ρm,n+1H

epn+1,n

Notation:

∂tρmn −Aρmn = (Bρ)mn

A diagonal in m,n, ⇒ commutes with taking the phonon

trace∞∑

m=0

∫(Aρ)mmdk

m = [He, (Trpρ)]

→ free streaming operator acting on Trpρ

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Fixed point argument:

Assume B small and iterate, SA(t) semigroup operator

ρj+1 = SA(t)(ρI) +∫ t

0SA(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 equiv-

alent to the density function in the Boltzmann equation)

f(x, k, t) =∫Trpρ(x−

1

2y, x+

1

2y, t)eiy·kdy

Remark: Since we iterated the semigroup operator the L-B-

F equation will be non-local in time.

∂tf(x, k, t) + [ε+ V, f ]w =∫ t

0dt′

∫dk′[S(k, k′, t− t′)f(x, k′, t′)− S(k′, k, t− t′)f(x, k, t′)]

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, p′, t) =

∑ν=±1

aν cos[1

~

∫ t

0(ε(q(p, τ))− ε(q(p′, τ)) + ν~Ω)dτ ]

q(k, τ) := p− τ∇xV, ε(p) =|p|2

2mΩ: 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, p′, t) =∑

ν=±1

aν cos[t

~(ε(p)− ε(p′) + ν~ω)]

THE SPATIALLY HOMOGENEOUS ZERO - FIELD

LEVINSON EQUATION

Scaling and dimensionless formulation:

ε(p0) = KT, t0 =1

Ωλ, λ = O(~)

∂tf(p, t) = Qλ[f ](p, t) =∫ t

0dt′

∫dp′×

1

λ[S(p, p′,

t− t′

λ)f(p′, t′)− S(p′, p,

t− t′

λ)f(p, t′)]

dimensionless cross section

S(p, p′, t) =∑

ν=±1

aν cos[t(ε(p)− ε(p′) + νΩ)]

aν = O(1) and Ω (scaled emission energy) = O(1).

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FERMI’S GOLDEN RULE - A HEURISTIC ARGU-

MENT

Qλ[f ](p, t) =∫ t

0dt′

∫dp′×

1

λ[S(p, p′,

t− t′

λ)f(p′, t′)− S(p′, p,

t− t′

λ)f(p, t′)]

Assume: limt→∞ S(p, p′, t) = 0

Qλ[f ](p, t) →∫dp′[S0(p, p

′)f(p′, t)− S0(p′, p)f(p, t)]

S0(p, p′) =

∫ ∞

0S(p, p′, t)dt

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cross section:

S(p, p′, t) =∑

ν=±1

aν cos[t(ε(p)− ε(p′) + νΩ)]

∫∞0 cos(zt)dt = δ(z) yields FGR.

Since limt→∞ S(p, p′, t) 6= 0 this becomes a weak limit.

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THEOREM 1

Define the functional

Yλ(φ, f) =∫ ∞

0dt

∫dpφ(p, t)Qλ[f ](p, t)

for smooth, compactly supported test functions φ. Then

limλ→0

Yλ(φ, f) =∫ ∞

0dt

∫dpφ(p, t)Q0[f ](p, t)

holds for fixed φ with

S0(p, p′) =

∑ν=±1

aνδ(ε(p)− ε(p′) + νΩ)

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THEOREM 2

(first order perturbation)

Yλ(φ, f) =∫ ∞

0dt

∫dpφ(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 ]:∫φ(p)Q1[∂tf ](p)dp =

−∑

ν=±1

aν

∫ ∫ln(|ε− ε′ + νΩ|)

d

dε

d

dε′[(φ′ − φ)∂tf ]dpdp

′

ddε: directional derivative perpendicular to surfaces of equal

energy.

Corresponds (formally) to a first order scattering cross sec-

tion

S = S0 + λS1

S0(p, p′) =

∑ν=±1

aνδ(ε(p)− ε(p′) + νΩ)

S1(p, p′, ∂t) =

∑ν=±1

aν

(ε(p)− ε(p′) + νΩ)2∂t

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REMARK: Parabolic bands ε(p) = |p|22 : polar coordinates and

radial derivatives

NUMERICAL VERIFICATION

THEOREM 3: Let Γ be a time - direction smoothing oper-

ator

(Γf)(t) =∫ ∞

0γ(t− t′)f(t′)dt′.

Then

ΓQλ[f ] = Q0(Γf) + λQ1[Γ∂tf ] + o(λ)

pointwise in t weakly in p.

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Computations

• Parabolic bands ε = |p|22m

• an = const: δ− function interaction potential

• Symmetry: f(p, t) = f(ε(p), t)

• Compare smoothed energies:∫εΓQλ[f ]dp

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0

1

2

3

4

0

2

4

6

80

0.5

1

1.5

2

energy

FIGURE 1

time

f

28

0 1 2 3 4 5 6 7 8−200

−100

0

100

200

300

400FIGURE 2

time

ener

gy

29

0

1

2

3

4

0

2

4

6

80

0.2

0.4

0.6

0.8

1

energy

FIGURE 3

time

f (fil

tere

d)

30

0 1 2 3 4 5 6 7 8−5

0

5

10

15

20

25

30

35

40FIGURE 4

time

ener

gyK*Q[f] Q

0[K*f]

Qfgr

[K*f]

31

0 1 2 3 4 5 6 7 80

10

20

30

40

50

60

70

80FIGURE 5

time

ener

gyK*Q[f] Q

0[K*f]

Qfgr

[K*f]

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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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0 50 100 150 200 2500

0.2

0.4

0.6

0.8

1

1.2

1.4

energy (meV)

initi

al c

ondi

tion

FIGURE 6

35

050

100150

200250

0

1

2

3

4−0.2

0

0.2

0.4

0.6

0.8

1

1.2

energy (meV)

FIGURE 7

time (ps)

f

36

050

100150

200250

0

1

2

3

4−0.2

0

0.2

0.4

0.6

0.8

1

1.2

energy (meV)

FIGURE 8

time (ps)

f filt

ered

37

050

100150

200250

0

1

2

3

4−0.2

0

0.2

0.4

0.6

0.8

1

1.2

energy (meV)

FIGURE 9

time (ps)

f 0+la

mbd

a*f 1

38

0 50 100 150 200 2500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

energy (meV)

appr

oxim

atio

n at

t=0.

8541

1ps

FIGURE10

f0

f0+lam*f

1f

0 50 100 150 200 2500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

energy (meV)

appr

oxim

atio

n at

t=1.

7082

ps

FIGURE11

f0

f0+lam*f

1f

0 50 100 150 200 2500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

energy (meV)

appr

oxim

atio

n at

t=2.

5623

ps

FIGURE12

f0

f0+lam*f

1f

0 50 100 150 200 2500

0.1

0.2

0.3

0.4

0.5

energy (meV)

appr

oxim

atio

n at

t=3.

4164

ps

FIGURE13

f0

f0+lam*f

1f

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CONCLUSIONS

• The first order correction produces a significant changein the transient behavior.

• This can explain collisional broadening in actual devices.

Open Problems

• The non-zero field case (→ the intra - collisional fieldeffect).

• Numerics for non - radially symmetric solutions (→ MC).

• How to solve the first order approximation using weightedparticle methods.

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