Lundstrom ECE 305 S15

ECE-305: Spring 2015

Carrier Properties: II

Professor Mark Lundstrom Electrical and Computer Engineering

Purdue University, West Lafayette, IN USA lundstro@purdue.edu

1/23/15

Pierret, Semiconductor Device Fundamentals (SDF) pp. 32-49

Lundstrom ECE 305 S15 2

announcements

1.  Exam 1: Friday, Jan. 30 in class see the class homepage for info https://nanohub.org/groups/ece305lundstrom

2.  Do the homework!

3.  Review the quizzes.

4. Ask questions on Piazza

3

vocabulary 1.  Miller indices 2.  Diamond and Zinc blende lattices 3.  Energy bands 4.  Conduction band, valence band, band gap 5.  Energy band diagrams 6.  Intrinsic carriers, intrinsic carrier concentration 7.  Effective mass and band structure 8.  Doping 9.  Intrinsic semiconductor 10. Extrinsic semiconductor 11.  Density of States 12. Fermi function, Fermi level 13. Non-degenerate semiconductor

carrier concentration vs. temperature

4 4 Fig. 2.22 from R.F. Pierret, Semiconductor Device Fundamentals

extrinsic

intrinsic freeze-out

Lundstrom ECE 305 S15 5

outline 1.  Density of States

2.  Fermi function

3.  Carrier distributions

4.  Carrier concentrations -given Fermi level -given doping densities

DOS

6

conduction “band”

valence “band”

4Na states / band

Na = 5 x 1022 /cm3 • • • • • • • • •

• • • • • • • • • How are the energy levels distributed with the bands?

density-of-states

Number of states per unit energy per unit volume. Units: (J-m3)-1

g E( )dE

Number of states in an energy range, dE, per m3.

DOS

7

E

g E( )

EC

EV

ECtop

EVbot

gC E( )dE

gV E( )dE

gC E( )EC

ECtop

∫ dE = 4Na

gV E( )EVbot

EV

∫ dE = 4Na

density of states near the band edge

8

E

D E( )

EC

EV

ECtop

EVbot

gC E( ) = mn

* 2mn* E − EC( )

π 2!3

gV E( ) =

mp* 2mp

* EV − E( )π 2!3

Lundstrom ECE 305 S15 9

outline

1.  Density of States

2.  Fermi function

3.  Carrier distributions

4.  Carrier concentrations -given Fermi level -given doping densities

✓

Occupation of states

10

1S2

2S2

2P6

3S2

3P2

4S0

Si atom (At. no. 14)

ener

gy

States below this energy have a high probability of being occupied.

States way above have very little probability of being occupied.

Fermi level

11

E

D E( )

EC

EV

ECtop

EVbot

E = EF

(electrochemical potential)

f E( ) = 11+ e E−EF( ) kBT

(Fermi function)

occupying the bands

12

f E( )

E

0 1

small probability of being empty.

small probability of being filled.

Fermi function

f E( ) = 11+ e E−EF( ) kBT

0.5

EF f EF( ) = 12

Probability that a state at energy, E, is occupied in equilibrium.

Fermi function

13

• • • • • • • • •

f E( )

E0

1

EF

Fermi level

kBT = 0.026 eV

f E( ) = 11+ e E−EF( ) kBT

1 2

effect of temperature

14

• • • • • • • • •

f E( )

E0

1

EF

kBT0

f E( ) = 11+ e E−EF( ) kBT

T1 > T0

T2 < T0

electrons and holes

15

EC

EV

EG = 1.1eV

These states are way above the Fermi level.

These states are way below the Fermi level.

Typically, we will find the Fermi level somewhere inside the bandgap.

conduction band

16

• • • • • • • • •

f E( )

E0

1

EF

f E( ) = 11+ e E−EF( ) kBT

small probability of being full

f E( ) ≈ e EF−E( ) kBT

non-degenerate semiconductor E >> EF

EC

valence band

17

• • • • • • • • •

f E( )

E0

1

EF

f E( ) = 11+ e E−EF( ) kBT

small probability of being empty

non-degenerate semiconductor

fh E( ) = 1− f E( ) = 11+ e EF −E( ) kBT

E << EF fh E( ) ≈ e E−EF( ) kBT

Non-degenerate semiconductors

18

EC

EV

EG = 1.1eV

≈ 3kBT

f EC( ) = 11+ e EC−EF( ) kBT

f EC( ) ≈ EF−EC( ) kBT

f EV( ) = 11+ e EV −EF( ) kBT

1− f EV( ) ≈ e EV −EF( ) kBT

energy band diagram of an intrinsic semiconductor

19

EC

EV

EG = 1.1eV f E( ) = 11+ e E−EF( ) kBT

n = ni

EF = Ei

p = ni

temperature dependence of intrinsic density

20

EC

EV

EG = 1.1eV

f E( )

E

0 1

small probability of being empty.

Fermi function

1 2

EF

T2

T1 > T2n = ni T( )

p = ni T( )

carrier concentration vs. temperature

21 21 Fig. 2.22 from R.F. Pierret, Semiconductor Device Fundamentals

intrinsic

energy band diagram of an n-type semiconductor

22

EC

EV

EG = 1.1eV f E( ) = 11+ e E−EF( ) kBT

n >> ni

EF

n ∝ e EF−EC( ) kBT

Expect:

n = NCeEF−EC( ) kBT

carrier concentration vs. temperature

23 23 Fig. 2.22 from R.F. Pierret, Semiconductor Device Fundamentals

extrinsic

energy band diagram of an p-type semiconductor

24

EC

EV

EG = 1.1eV f E( ) = 11+ e E−EF( ) kBT

EF

p >> ni

p ∝ e EV −EF( ) kBT

Expect:

p = NVeEV −EF( ) kBT

Lundstrom ECE 305 S15 25

outline

1.  Density of States

2.  Fermi function

3.  Carrier distributions

4.  Carrier concentrations -given Fermi level -given doping densities

✓

✓

distribution of electrons within a band

26

E

g E( )

EC

EV

ECtop

EVbot

gC E( )∝ E − EC( )

EF

n E( )dE = gC E( )dE × f E( )

p E( )dE = gV E( )dE × 1− f E( )( )

gV E( )∝ EV − E( )

carrier distribution

27

from R.F. Pierret, Semiconductor Device Fundamentals

more carrier distributions

28 from R.F. Pierret, Semiconductor Device Fundamentals

question

29

1) At T = 0 K, what is the density of holes in the valence band of a pure semiconductor?

a) the atomic density of the material. b) Avogadro’s number. c) The density of dopants. d) The packing fraction of the material. e) Zero

another question

30

2) At T = 0 K, where is the Fermi level located in a pure semiconductor?

a) Deep inside the conduction band b) Near EC. c) Near the middle of the bandgap. d) Near EV. e) Deep inside the valence band.

electrons and holes

31

EC

EV

EG = 1.1eV

How many electrons in the conduction band?

How many holes in the valence band?

gC E( )dE

n E( )dE = f E( )gC E( )dEEF

n E( )dEEC

∞

∫ = f E( )gC E( )dEEC

∞

∫

final result (electrons)

32

n = NCeEF −EC( ) kBT

NC = 2

mn*kBT( )2π!2

⎡

⎣⎢⎢

⎤

⎦⎥⎥

3/2

“effective density of states”

For Si at T = 300K:

mn* = 1.182 (DOS effective mass)

NC = 3.23×1019 cm-3

E

x

EC

EV

EF

EC − EF > 3kBT

final result (holes)

33

p = NVeEV −EF( ) kBT

NV = 2mp*kBT( )2π!2

⎡

⎣⎢⎢

⎤

⎦⎥⎥

3/2

“effective density of states”

For Si at T = 300K:

mn* = 0.81 (DOS effective mass)

NV = 1.83×1019 cm-3

E

x

EC

EV

EF

EF − EV > 3kBT

summary

34

p = NVeEV −EF( ) kBT

NV = 2mp*kBT( )2π!2

⎡

⎣⎢⎢

⎤

⎦⎥⎥

3/2

E

x

EC

EV

EF

n = NCeEF −EC( ) kBT

NC = 2

mn*kBT( )2π!2

⎡

⎣⎢⎢

⎤

⎦⎥⎥

3/2

Fermi level should be at least 3kBT away from a band edge.

summary

35

1) Semiconductor devices are made by controllably putting a few electrons in the conduction band and a few holes in the valence band.

2) In equilibrium, the Fermi function gives the probability that a state at energy, E, is occupied by an electron.

3) The two parameters in the Fermi function are the Fermi level and the temperature.

4) The density of states, g(E), tells how the states are distributed in energy.

5) From the DOS and the Fermi function, we can relate the electron and hole densities to the location of the Fermi level.