Quantum mechanics course Microsoft Power Point Time Dependent Perturbation Theory

8/14/2019 Quantum mechanics course Microsoft Power Point Time Dependent Perturbation Theory

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Quantum Transition

* Time-Dependent Perturbation Theory

* Fermi-Golden Rule

*Impurity Scattering


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Time Evolution of Quantum StatesIn quantum mechanics, one in general deals with two kinds of problems. One is todetermine all possible states of a system. This is possible only if the Hamiltonian ofthe system is time independent, that is, the potentials or forces do not vary from

time to time. The basic procedure is to solve time-independent Schrdingerequation with all kinds of approximations, as what we have already seen.When the Hamiltonian is time dependent, the state or the wavefunction

of the system will be also time dependent. In other words, an electron will have aprobability to transfer from one state (molecular orbital) to another. Thetransition probability can be obtained from the time-dependent Schrdinger Eq.

)()(

tHt

ti =

(23.1)

Equation 1 says once the initial wavefunction, (0), is known, the wavefunction at a

given later time can be determined. If H is time independent, we can easily found thatn

n

tiE

n

neat = /)( (23.2)In this case, it is easy to see that 22 )0()( = t

so the probability density does not change.

(23.3)


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Time-Dependent Perturbation TheoryNow we determine the transition rate according to the above definition. We assumethat the initial state of the system is

(23.7)k= )0(

the external perturbation, H, is switched on at t=0. The time dependent SchrdingerEq. is

)()'()(

0 tHHt

ti +=

(23.8)

For simplicity, we can rewrite Eq. 23.5 as

n

tiE

n

nk

netCt /

)(')(= (23.9)

Note that Cnk(t) in Eq. 23.9 is different from Cnk(t) that in Eq. 23.5, but|Cnk(t)|

2=| Cnk(t)|2 and we can omit the prime in equation 10. Substituting Eq. 23.9

into Eq. 23.8, we have (can you show it?)

.'//n

n

tiE

nkn

n

tiEnkHeCe

dt

dCi nn =

(23.10)


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Time-Dependent Perturbation TheoryMultiplying Eq. 23.10 by k and integrate, we obtain

(23.10)nk

n

tEEikkCnHke

dt

dCi nk

>


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Time-Dependent Perturbation Theory

>< kHk |'|'

,

1 /)(

0

'

''

' dteHi

CtEEi

t

kkkk

kk

=

We replace Cnk on the right hand side of Eq. 23.10 with Cnk(0) and obtain the first

order correction

in the above equation is often denoted as Hkk and it measured thecoupling strength between the k and k states. Solving Eq. 23.13, we have

(23.13)

(23.14)

One important case is that H is fixed once switched on. In this case, Eq. 23.14becomes

/)(11)(

'

/)('

''

'

kk

tEEi

kkkkEEi

eHi

tC

kk

=

(23.15)


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Fermi-Golden Rule

)(||2

]/)[(

)/)((sin||

1|)(| '

2'

'22

'

'

22'

'2

2

' kkkk

t

kk

kk

kkkkEEH

t

EE

tEEHtC

=

)(||2

'

2'

'2' kkkkkkEEHw =

From Eq. 23.15, we can obtain

So the transition rate is

(23.16)

(23.17)

We can conclude from Eq. 23.17 that (1) the transition rate is independent of time,(2) the transition can occur only if the final state has the same energy as theinitial state. The later one reflects energy conservation. In the case when theenergy levels are continuous band, the number of states near Ek for an interval ofdE

k

is In the case when the energy levels are continuous band, the number ofstates near Ek for an interval of dEk is (Ek)dEk , where is the density of states.The transition rate from k state to the states near Ek is then

)(||2

)( 2' '2''' kkkkkkk EHdEwEw

== This is Fermi Golden rule,

(23.18)


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Time dependent perturbation theory - Revisited

Assume the Hamiltonian may be decomposed as H=H0+Vs,

where H0 is the Hamiltonian of the perfect crystal (described by

Bloch states), Vs(r,t) is a small random potential. IfVs


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If the initial wave packet is centered around ko, so that

In the limit at t, the probability of finding the particle inanother state ko is

Define the transition rate

Solve for using the S.E. and the previous expansion

( ) ( ) 01 00 tctc kkk

( )2

000

tcP kt

kk

= lim 0k0k

sV

( )t

tck

tkk

2

0

00

= lim

{ } ( ) ( ) ( ) ( ) = +

k

tiEkk

k

tiEkks

kk etct

ietcVH // rr0

0kc


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Assume sufficiently weak scattering that cko1, and ckko0 for

all time. The dominant term in the sum is:

which integrates to

Suppose V(r,t) may be Fourier decomposed, so that

Note that this form of V(r,t) may correspond to interaction with

lattice vibrations or with optical excitation.

( )( ) ( )

/tEEi

sk

k kkekVktct

tci 00

0

0

00

=

( ) ( ) ( )01

0

00

0

0

00 k

ttEEi

sk cekVktd

i

tc kk

+ =

/

( ) ( )

ti

ss eVtV

=

rr,


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Then substituting

and integrating this last expression leads to

Since the probability of being in k0 is given by

( ) ( )

/; = =

000

000

1kk

tti

sk EEetdkVki

tc

( )

=

i

eV

itc

ti

skk

k

1100

0

( ) ( ) tt

teVi

tc tikk

sk

= sin

/ 200

0

1

( )2

000

tcP kt

kk

= lim


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Substituting forcand taking the magnitude squared gives

where asymptotically

This gives the famous Fermis Golden Rule (droping 0s index)

Assumptions made:(1) Long time between scattering (no multiple scattering events)

(2) Neglect contribution of othercs (Collision broadening ignored)

( ) 2222

00

00

1t

t

tVP

kk

st

kk

=

sinlim

( ) ( ) ( ) tEEtt

tkk

t//sinlim ==

00

222

( )

==

kk

kk

skk

kk EEVt

P 22


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A.3 Scattering Theory

What contributes to ?kk

Scattering Mechanisms

Defect Scattering Carrier-Carrier Scattering Lattice Scattering

CrystalDefects

Impurity Alloy

Neutral Ionized

Intravalley Intervalley

Acoustic OpticalAcoustic Optical

Nonpolar Polar Deformation

potential

Piezo-

electric

Scattering Mechanisms

Defect Scattering Carrier-Carrier Scattering Lattice Scattering

CrystalDefects

Impurity Alloy

Neutral Ionized

Intravalley Intervalley

Acoustic OpticalAcoustic OpticalAcoustic Optical

Nonpolar Polar Nonpolar Polar Deformation

potential

Piezo-

electric


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Ionized Impurities Scattering

(Ionized donors/acceptors, substitutional impurities, charged

surface states, etc.) The potential due to a single ionized impurity with net charge

Ze is:

In the one electron picture, the actual potential seen by

electrons is screenedby the other electrons in the system.

( ) unitsmks

r

ZeVi

=

4

r2

0


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What is Screening?

+

-

-

-

-

-

-

--

-

D - Debye screening length

Ways of treating screening:

Thomas-Fermi Method

static potentials + slowly varying in space

Mean-Field Approximation (Random Phase Approximation)

time-dependent and not slowly varying in space

r

3D:1

r

1

rexp

r

D

screeningcloud

Example:


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Considering the induced charge caused by the change in the

electron gas by the impurity, the net potential seen is

In the above expression, q is the wavevector associated withFourier transforming the potential (and Poissons equation),

Vi(q) is the total potential seen by an electron due to an

impurity, and (q,) is the dielectric function characterizing thepolarization of the electron gas to the impurity potential.

In linear response theory, this may be calculated in the random

phase approximation (RPA) to give the Lindhard dielectric

function

( )( )

++

=

+

+

k qk

k0qk0

2

2

1qiEE

EfEf

q

e

kscs lim,

( )( )

( )=

,q

qq

0i

i

VV


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Assuming low frequencies, and assuming long wavelengths, the

Thomas-Fermi function is obtained to be of the form:

where the inverse screening length 2

is given as (3D):

In here, n is the carrier density and EF is the Fermi energy.

Assuming the Fermi Thomas form, inverse Fourier transforming

gives the form of the screened potential in real space as:

KTE

neetemperaturhigh

Tk

ne

FscBsc

02

32

22

2 =

=

= ;;

( ) ri er

ZqV

=4

r2

( )2

2

01q

qq

+

,lim

,


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The usual argument is that since the us are normalized within a

unit cell (i.e. equal to 1), the Bloch overlap integral I, isapproximately 1 forn=n [interband(valley)]. Therefore, for

impurity scattering, the matrix element for scattering is

approximately

where the scattered wavevector is:

This is the scattering rate for a single impurity. If we assume that

there are Ni impurities in the whole crystal, and that scattering is

completely uncorrelated between impurities:

where ni is the impurity density (per unit volume).

kkq =

( ) ( )( )

volumeVqV

eZVVsc

ii =+= ;

2222

4222qkrk

( ) ( ) 22242

2222

42

sc

i

sc

ikki

qVeZn

qVeZNV

+=

+


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The total scattering rate from k to k is given from Fermis golden

rule as:

If is the angle between k and k, then:

Comments on the behavior of this scattering mechanism:

- Increases linearly with impurity concentration

- Decreases with increasing energy (k2), favors lowerT

- Favors small angle scattering

- Ionized Impurity-Dominates at low temperature, or room

temperature in impure samples (highly doped regions)

Integration over all k gives the total scattering rate k :

( )( )kk222

422EE

qV

eZn

sc

iikk +

=

( )=+== coscos 122kk 222 kkkkkq

( )=

+= /;

*1

4

4

8 222

2

332

42

D

DDsc

iik q

qkq

k

k

meZn


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Total Electron Scattering Rate Versus Energy:

Intrinsic Si GaAs

In both cases the electron scattering rates were calculated

by assuming non-parabolic energy bands.

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Quantum mechanics course Microsoft Power Point Time Dependent Perturbation Theory