effective mass

7/27/2019 Effective Mass

1/23

The Quasi-Free Electron and

Electron Effective Mass, m*

ECE G201(Partly adapted from Prof. Hopwood)


2/23

Quasi-Free Electron:

This is the BIG approximation in looking at electrons in

crystals.

What is a Quasi-Free Electron?

Under some conditions (often found in devices)electrons behave like free particles with an effective mass

that is different than the mass in vacuum.

We want to understand this approximation.

We also want to understand when behavior beyond this

approximation occurs in devices.


3/23

Copy!


4/23

Copy!


5/23


6/23

Goal: show that an electron

behaves like a particle with mass

m* = 2(d2E/dK2)-1

Recall that the electron energy is related to

the frequency of the electron waveE =

and the group velocity of the wave is the

velocity of the electronvg = d/dK = 1/ dE/dK (as in text)


7/23

The acceleration of a particle is given by

the time-derivative of its velocity:

a = dvg/dt =d/dt(

d/dK)

=

d

/dK(

d

/dK)

dK

/dt= (1/

2) d/dK(d/dK)(

d (K)/dt)

= (1/2)(d2E/dK2)(d (K)/dt)

This is the term we are looking to show is:

(1/2)(d2E/dK2) = 1/m*


8/23

What is d (K)/dt?

If we apply an external force on the electron, forexample an electric field (Fext=qE), then we will do

work on the electron:

dWe = Fextdx = Fext(vgdt) since vg = dx/dt = d/dK

= Fext(d/dK)dt


9/23

Doing work on the electron

increases its energydWe = Fext(

d/dK)dt = dE

= (dE/dK)dK

= [d()/dK]dK

= (d/dK)dK

therefore: Fextdt = dK

or Fext = d (K)/dtnote: since F=d(mv)/dt,

K is called the crystal momentum


10/23

Finally

a = (1/2)(d2E/dK2)(d (K)/dt)

and

Fext = d(K)/dt

gives us

a = (1/m*)Fext or Fext = m*a

Where m* = [(1/2)(d2E/dK2)]-1 = 2 (d2E/dK2)-1


11/23


12/23


13/23


14/23


15/23

Without the crystal lattice, thehole cannot exist. It is an artifact

of the periodic potential (Ep)created by the crystal.


16/23

E(K) and E(x)

p/a

E(K)

K

conduction band

valence band

EC

EV +

-

x

E(x)

Eg


17/23

Generation and Recombination

of electron-hole pairs

conduction band

valence band

EC

EV +

-

x

E(x)

+

-


18/23


19/23

Non-cubic lattices:

(FCC, BCC, diamond, etc.)

a

b

x

y

p/a

E(Kx)

Kx

p/b

E(Ky)

Ky

Different lattice spacings lead to different curvatures for E(K)

and effective masses that depend on the direction of motion.


20/23

(from S.M. Sze, 1981)


21/23

Memory Aid

a hairpin is lighter than a frying pan

light m*

(larger d2E/dK2)

heavy m*

(smaller d2E/dK2)


22/23

Notes on E(K)

The extrema for the conduction and valence

bands are at different values of K for silicon

and germaniumthese are called indirectbandgap semiconductors

The conduction band minimum and valence

band maximum both occur at K=0 for GaAsthis is called a directbandgap semiconductor


23/23

effective mass

Documents