quantum transport :atom to transistor,phonons, emission & absorption

8/8/2019 Quantum Transport :Atom to Transistor,Phonons, emission & Absorption

1/10

Quantum Transport:Quantum Transport:Atom to Transistor

P rof. Supriyo DattaECE 659Purdue University

Network for Computational Nanotechnology

04.23.2003

Lecture 37: Phonons, emission & AbsorptionRef. Chapter 10.2 & 10.4


2/10

Interband transitions are a lotlike atomic transitions

Intraband transitions,otherwise known as Cerenkovradiation, can occur with bothphonons and photons

Intraband transitions couldtake picoseconds whereasinterband transitions usuallytake nanoseconds

In the first part of this lecture we discussintraband transitions. Up till now we havecovered only interband transitions

E

k v

intraband

interband

Transitions

Introduction00:00


3/10

However, silicon interbandtransitions take as long as amillisecond

From the diagram on the left one can seethat for emission,

and absorption

where the photon wave vector requiredis very large. Since is often very smallfor light the coupling between non-verticaltransitions is weak and hence is large

Note: vertical transitions are notpermitted in silicon because there is notany empty state available in the valenceband if the electron is to drop vertically.

Interband Silicon Transitions

E

k v

interband transition

vvv

= k k f

vvv

+= k k f

v

v

Silicon InterbandTransitions

00:50


4/10

Just as with interband transitions, intraband transition lifetimes are defined as:Emission

Absorption

Importantly, for intraband transitions the final state retains the initial state wavefunction. Therefore, and for emission we requirelikewise for absorption

( ) ( )[ ] +== vv

vh

vvvh

)()()(212

k k K N

( ) ( )[ ] ++==

vv

vh

vvvh )()()(2

2k k K N

vvv

= k k f ( ( 0=+

vh

vvvwk k

( ) 0cos22222

22

2

=+ ck k m

k m

hhh

k mk cm

k mc

22

cos 2

2

+=

+

= hh

hh)()(

rvrv k k

) ) ) = 0 vhvvv

wk k

0= v

hvvv

wk k

Intraband Formalism05:30


5/10

So, we need to have a phonon or photon with direction and an electronwith direction

Note: for phonons c s is usuallyused in place of (Velocity of sound instead of velocity of light)

The term cos gives theCerenkov cone for phonons/photons, this effect is exactly the

same as the sonic boom producedby jets

We can work out the same resultfor absorption, only a sign changeresults Inspecting the equality

we see that since cos 1 emission isonly possible if the electron velocity,k/m, exceeds the photon/phononvelocity

v

k v

v

k v

k mk c

2cos += h

c

c

Coupling13:28


6/10

Similar to a classical harmonicosscilator this mass-spring system hasa resonant frequency defined by

with discrete energies. We definephonons as the quantized energy

exchanged between vibrating masses

Next : An introduction to phononsand their dispersion curves

Basic Definition : phonons are thelattice vibrations (sound) whichpropagate through a solid or molecule

For example, hydrogen moleculesvibrate about an equilibrium bonddistance, R eq , with an intensityproportional to temperature muchlike a spring mass system

H2 Harmonic Oscillator

H H

m m

Req

M K =

Phonons20:52


7/10

Now, assuming sine or cosinesolutions to the equation we can write

or

The above translates easily into matrixform:

We can make a similar extensionto the infinite 1-D lattice producing amass-spring structure of the form

where m is the mass of each atomand U m(t) is the displacement of them th atom from equilibrium its

equilibrium position. Thus, according to Newtons lawthe force exerted on the m th atom is

Um-1 (t)

a

Um(t) Um+1 (t)

( ) ( )1122

+ = mmmmm U U K U U K dt U d m

( ) mm U dt t U d 2

2

2

=

( )112 2 + += mmmm U U U m K U

=

MO

OO

M

3

2

1

3

2

1

2

1

1

121112

U

U U

m K

U

U U

1-Dimensional Lattice23:00


8/10

This gives rise to a dispersion relation whichlooks like

Likewise, takingadvantage of matrixperiodicity, for phonon wemay use the ansatz

such that

v

aim eU U

=0

( )

( )

=

=

=

2sin2

2sin22

cos12

2

2

amk

amk

amk

v

- /a /a

and around the origin the curve is oftenapproximated linearly by the relation

amk =

1-D Dispersion Relation28:37


9/10

Acousticand OpticalBranches

Acoustic phonons propagate in onedirection

Optical phonons squeeze or pushtwo atoms together

In semiconductors another phonon vs. branch often occurs,otherwise known as optical phonons.This results when two types of atoms,and hence two different masses,exist in a unit cell. For example,GaAs:

In the dispersion relation we denotethe upper branch optical branch(has nothing to do with light) and thelower branch acoustic branch. Thereason we call the upper one opticalis because if you shine light on thesolid you can excite this branch.

Ga As Ga As

v

- /a - /a

GaAsGa acoustic phonon

AsGaoptical phonon

Optical Phonons32:40


10/10

The electric field, , of a photon is uniform on an atomic scaleand induces opposite force on the +ve and -ve bonding pair

Physical intuition on the flatness of the upper branch & strongdependence of lower branch on

Lets ask ourselves why does the frequency generally go up with ? Frequency goes up asthe spring gets stiffer is a certain deformation is caused. is like the wavelength. When youhave a 1-D series of atoms and you excite the first one, if the wavelength is relatively large,the first atom moves a lot, the second moves by almost the same amount, the third movesalmost the same and so on. So the spring itself is not distorted a lot. However , for shorter

wavelengths the displacement in the atom 1 and atom 2 is considerably different such thatthe spring feels a lot stiffer and so for short wavelengths the frequency goes up. This does

not happen for the optical branch. It doesnt matter if the wavelength is lone or short. Within aunit cell, the two atoms move opposite to each other and thats what determines the stiffness(they are moving against each other, its very stiff anyway). For short or long wavelength the

spring is distorted. So - curve for the optical branch tends to be relatively flat.

The final coupling constant for any phonon as found in is a product of the phonon strain andthe induced deformation potential. The deformation potential is the actual shift in theconduction band of a solid due to the passage of a phonon. It is derived experimentally

v

AsGa

+ve -ve E v

Closing Comments38:00

quantum transport :atom to transistor,phonons, emission & absorption

Documents