Appunti lezione 6 di Misure

«soft zone» of Fermi statistics at T≠0

the «soft zone»

The fact that even a small # of impurities have a strong impact

on the semiconductor properties is the reason why it was difficult

to understand semiconductors

Metals and insulators / semiconductors

Metal insulators / semiconductors

Density of states finite at E=E Density of states=0 at E=E

F F

conduction

band

valence band figure 7.1a

we have defined the difference between a metal and a non metal now we

still have to define the difference between a semiconductor and an insulator

5 definition of bands

#4 T=0K T>0K figure 7.1b

«spilling» of electrons from VB to CB

#5 m

The meaning of ( )

By Ashcroft/Mermin

We are in a «gran canonical ensemble»

so the

at T≠0 corresponding potential

can be F or G. This means that they

Depends only on thermodinamical variables

T,P,V and not from the path of the transformation. m

Charge neutrality determine the position of

#6 Qualitative deduction

#7 * *

➔positively charged solid

Figure 7.1c

* m

This is because electrons with energy KT would easily overcome but very few of

them would reach the CB because the gap is much bigger than KT

8 m

Equivalent argument for just below the CB minimum

*

#8 m ➔neutrally

➔negatively charged solid

charged solid (charge conservation respected)

Figure 7.1d

* m m

This is because electrons with energy KT close to would easily overcome and reach the CB

instead very few electrons in the VB would leave it because the gap is much bigger than KT and

they remains in the VB.

9 m

( in the middle of the gap)

#9

CB and VB partially filled will both contribute to the conduction

#10 Example of conduction of partially filled VB

10 Example of conduction in partially filled VB

#11 *

filling the previous vacancy and leaving a new vacancy

*

#12 The conduction can be viewed by the movement of the hole e

equation of motion t

➔dk=(e/h)

remember also dk →D E≠0

Not filled band dk

finite

no field

field dk →D E=0

filled band

fully occupied bands do not conduct electricity

12 The Fermi-Dirac distribution for a

semiconductor

• For a metal, the Fermi energy is the highest occupied

energy at 0 K. The chemical potential is temperature-

dependent (but not much for metals) and so the two are

essentially the same.

• For a semiconductor, the definition of the Fermi energy is

not so clear because it is located in the gap, where

electrons cannot be. We better use the chemical potential.

• Some (many) people also use the term “Fermi energy” for

semiconductors but then it is considered temperature-

dependent.

For metals, it really does not make much of a difference. For semiconductors it does

because the chem. pot. can change a lot with temperature. We use the chemical

potential rather than a T-dep Fermi energy to emphasize this difference.

(both e- and h)

(density of electrons in the CB)

e

Notice: the 2 statistical distribution: f (E,T) for electrons and

h e e h

f (E,T)=1-f (E,T) for holes, is due to f (E,T)+ f (E,T)=1

(density of total probability definition (charge neutrality)

➔

holes in

the VB)

Approximation to parabolic shape for VB and CB

#13 h and e- close to VB and CB

#14

#15 Figure 7.3a

#16 (The motion of electrons/holes can be

described using the effective mass)

#17

#18 e- conduction in an electric field E

#19 m* >0

e

#20 In other words we can think of VB as a CB for holes

h conduction in an electric field E

#21

#22

Interpretation eq. of motion

VB maximum

as E=0 effective mass

conduction band

A negatively charged particle

with a positive mass (”electron”)

valence band

VB: negative charge and negative mass

or: positive charge and positive mass: the or

effective masses for hole and electron

are positive “hole”

E>E

g

E<0

free electrons

Simplified band structure

VB maximum

as E=0 conduction band

valence band *

* 1/2

Notice that (-E) is well defined because

in the VB E<0

21 Temperature dependence of the carrier density

(we approximate f(E,T)) p

n

electrons in the conduction band (CB)

missing electrons (holes) in the valence band (VB)

Simplified Fermi-Dirac Distribution

for the conduction band m

for the valence band -E>>K T

B

remember: 1/(1-x)1+x

where x=

m>E

<<1 so the exponent is negative

Both are Boltzmann distributions!

This is called the non-degenerate case.

The conduction band: occupation

substitution