Ionization energy – Bohr model2

centrifugala

m* vF

r

Coulomb centrifugalF F 2 2

2aa

e m* v

r4 r

2

Coulomb 2a

eF

4 r

angular momentum is quantized am* r v n

radius is quantized 2 2

a 2

n 4r

m* e

2 2 22a

a

m* r ve

4 m* r

2

a

n

m* r

Bohr radius 2

0 20

4a

m e

2a 0

0

r n m

a m*

2a

a

m* r v

m* r

+ -

ar

Ionization energy – Bohr model

21m*

2v

a

2

r

1 nm*

2 m*

2

2 2

2

1 nm*

2 n 4m*

m* e

am* r v n 2

a

2

2

n 4

er

m*

2

2 2

2

e

n 44

m* e

2

2 2

2

e

n 44

m* e

total energy = kinetic energy

a

2e

4 r

+ potential energy

Ionization energy – Bohr model2 2

2 2 2 2

2 2

1 n em*

2 n 4 n 4m* 4

m* e m* e

4

2

m* e

n 4

3 ) ( .37

4

2

1 m* e

2 n 4

cE

vE

4

2

1 m* e

2 n 4

Happy Hearts Day!

Happy Hearts Day!

Fermi-Dirac distribution

E EFTB

1f ( E )

e 1

E EF

TB

11 f ( E ) 1

e 1

E EFTB

E EFTB

e 1 1

e 1

E EFTB

E EFTB

e

e 1

Distribution of states & electronsFermi function

c FermiEcn g ( E ) f ( E )dE

Evv Fermi0

p g ( E ) 1 f ( E ) dE

cE

vE

cE

vE

..

Fermif ( E )1

vE cE

FermiE

Figure 3-1c Fermin( E ) g ( E ) f ( E )

a

Figure 3-1 v Fermip( E ) g ( E ) 1 f ( E )

a

Problem 3.5The magnitude of the product g(E) & Fermi(E) in the conduction band is a function of energy as shown in Figure 3.1. Assume the Boltzmann approximation is valid. Determine the energy with respect to Ec at which the maximum occurs.

c Fermin( E ) g ( E ) f ( E )

Fc

B

E EE E exp

T

c c Fc

B B

E E E EE E exp exp

T T

Problem 3.5

c c Fc

B B

E E E En( E ) E E exp exp

T T

cc

B

E Ed E E exp

Td n( E )

dE dE0

cx E E B

xd x exp

T0

dx

Problem 3.5

B

cT

E E2

B

xd x exp

T0

dx

B B

1 x xexp 0

T T2 x

cx E E

Typical numbers

Silicon 1.08 0.56

Normalized electroneffective mass

Normalized holeeffective mass

Gallium arsenide 0.067 0.48

Germanium 0.55 0.37

Normalized mass equals effective mass/rest mass of the electron

n type extrinsic semiconductor

*p

Fi c v B *n

m1 3E E E T ln

2 4 m

cE

vE

c Fermin( E ) g ( E ) f ( E )

v Fermip( E ) g ( E ) 1 f ( E )

donor energy levelFermi energy between donor energy

band and conduction band

p type extrinsic semiconductor

*p

Fi c v B *n

m1 3E E E T ln

2 4 m

cE

vE

c Fermin( E ) g ( E ) f ( E )

v Fermip( E ) g ( E ) 1 f ( E )

densities may differ by 1010

acceptor energy levelFermi energy between acceptor

energy band and conduction band

Ionization energies in silicon and germanium in electron volts

Impurity Si GeDonors Phosphorus 0.045 0.012

Arsenic 0.05 0.0127

Acceptors

Boron 0.045 0.0104Aluminum 0.06 0.0102

Conduction band

Valence band

Current density – charges in motion

2A

mdriftJ v

electron electron driftJ env

equation of motion for a hole in an electric field

pF m * a eE

19e 1.602 10 C

mobility & electron drift electron hole drift holev E v E

n nv E p pv E

electron nJ en( E ) hole pJ ep E

+- hole hole driftJ epv

p nJ e p n E

collisions

equation of motion for a hole in an electric field

pdv

F m * eEdt

+

cp

starting velocity after each collision is 0

average time between collisions

p

eEv t

m *

collisions

+

cp

starting velocity after each collision is 0

average time between collisions

p

eEv t

m *

cp

average velocity

p

eEv

2m *

cp

statistical analysis yields

p

eEv

m *

collisions

+

cp

statistical analysis yields

p

eEv

m * cp

hole mobility

pp

e

m *

cnelectron mobility nn

e

m *

Lattice structure will also vibrate due to T > 0

“lattice scattering”

lattice mobility 3

2

lattice1

T

Conductivity and resistivity

p nJ e p n E

J E

mhos or mp nSe p n m

p n

1 1m

e p n

Conductivity and resistivity

VE

L

J E

I

VIJ

A

You enter the laboratory and see an experiment. How will you know which class is it?

If it's green and wiggles, it's biology.

If it stinks, it's chemistry.

If it doesn't work, it's physics.

Teachers' remarks that changed the history of physics Copernicus, when will you understand that you are not the center of the world?Galileo, if you will drop stones from the top of the tower one more time, you will be dismissed forever.Kepler, till when will you stare at the sky?Newton, will you please stop idling away under the apple tree?Ohm, must you resist Ampere's opinions on current events?Nikola Tesla, I see that everyone is attracted to your magnetic personality.Einstein, a crocodile is greener or is it wider? Schrödinger, stop abusing cats!Heisenberg, when will you be sure of yourself?

Simple three-dimensional unit cell

a

b

c