Semiconductor Physics Overview

Semiconductor Physics Overview

Day 7-8

Jeff Davis

ECE3030

Online Reserve Reading: Streetman, Chapter 3 Energy Bands and Charge Carriers in

Semiconductors, Solid State Electronic Devices, 1990.

Barrier Model for Computation

A B

Modulate Barrier

pass

block

Billiard Balls

Switches

General Barrier

Barrier Model for Computation

Tox

SOURCE DRAIN

Leff

GATE

Vg

source drain

gate

Goal

To understand this idea we must first

understand some key ideas about how

electrons move in a semiconductor!

5

What is a semiconductor?

6

A high level description could be …

• Metals -- Highly conductive to current flow!

• Insulators -- Highly resistive to current flow!

• Semiconductors -- Somewhere in between!

Classifications of Electronic Materials

Doping is a good thing!

7


Another could be based on bond strengths…

8

Metal Crystal

Semiconductor Crystal

Strong Bonds Metal atoms are ionized..

Electrons free to roam (1022 cm-3)

Classifications of Electronic MaterialsClassifications of Electronic Materials

Results in how conductive a material is…

9

†Michael Shur, GaAs Devices and Circuits, 1987 Spring Publisher

“The partially heteropolar [ionic&covalent] bonds in GaAs are stronger than the homopolar [covalent] bonds in Si and Ge. It leads to a smaller amplitude of the lattice vibrations (and as a consequence to higher mobility), higher melting point, and wider energy gaps”†


The strength of the bonds determines material

properties

10

Origin of Allowable Energy Bands…

‘N’ for this figure is the number of

atoms with overlapping electron

wavefunctions

sp3 hybrid orbitals

Carbon (C) 1s22s22p2

Note - at zero temp all

electrons are in the valence

band!

Reference:

Energy split due to Pauli Exclusion Principle once

electron wave functions start to overlap.

Energy Band Theory of Solids

Hybrid Orbitals

sp3 Hybrid Orbitals

13


Ev

Ec

Insulators Semiconductors

e- e- e- e- e-

8.8eV

SiO2

Ev

Ec

e- e- e-

e-

e-

1.12eV

Si

Energy necessary to break the electron from its bond is the bandgap energy.

Ev

Ec

e- e- e-

e-

e-

0.17eV

InSb

Indium

Antimonide“Silicon”

Remember: Metals have no forbidden region!

14

What is an electron volt?

• It is an ENERGY!

• 1eV = 1.6e-19 J

Amount of potential energy given to a electron held at an electrostatic potential

of 1 volt

W = F ⋅ ∆x

W = (−qE) ⋅ ∆x

W = −qV

∆x∆x = −qV


Ec

0.17eV

InSb

15

Why do we like semiconductors?

16

Si

Si

Si

Si

Si

Si

Si

SiSi Si

Si

SiSi Si

Si

Si

Si

Si

Si

Si

Si

Si

Si

SiSi

Si

Si

Si

SiSi

Si

Si

Si

Si

Si

Si

Si

Si

Si

Si

Si

SiSi

Si

Si

Si

SiSi

Si

Si

Si

Si

Si Si Si

SiSiSi

Si

Doping SemiconductorsOCCASIONALLY add in other elements to substitute for silicon in x-tal to change

conductivity of material!

“OCCASIONALLY” typically means 1 dopant atom in 1000 Silicon atoms to 1 dopant atom in

100M Silicon atoms

17

Doping Semiconductors

Si

Si

Si

Si

SiSi

silicon has

4 valence electrons

4 covalent bonds

P

phosphorus has

5 valence electrons

Si

Si

SiSi P

free electron

n-type dopant

B

boron has

3 valence electrons

Si

Si

SiSi B

“hole”

p-type dopant

18

How do we describe x-tal structure?

19

X-tal properties

1. Si thin film transistors

2. Amorphous Si Solar Cells

1. Gate material of MOSFET

2. Polycrystalline solar cells

Most high-quality devices

are made of pure x-tal!

20

Importance of Crystal (x-tal) Structure

• Example – Carbon

– Graphite (conductor)~ 7.837 μ∧-cm

– Diamond (great insulator) ~ 5.45 eV

• We must have a working vocabulary to

describe different x-tal structures

21

Unit Cell DefinitionA portion of the crystal that could be used to reproduce the entire x-tal… kind of like

a “rubber stamp”

Examples of unit

cells

22

Unit Cell Definition

A unit cell can be used to recreate the entire x-tal lattice!

23

Unit Cell Definition

A unit cell can be used to recreate the entire x-tal lattice!

Is this a unit cell?

24

Primitive Unit Cell

The smallest portion of the crystal that could be used to reproduce the entire x-tal.

25

3-D Cubic X-tal Cells

(BCC) (FCC)

a = “lattice constant”

26

Number of Atoms Per Unit Cell

(BCC) (FCC)

1 2 4

Answer is

covered!

a = “lattice constant”

27

Diamond Lattice

One-quarter of diagonal length

28

Zinc Blende

Show using rasmol - intro to crystal planes!

29

Atomic Density

3

333

224.1

2241.1)82.5(

2

cm

atomse

cellunitofVolume

cellunitperAtomsofNumberDensity

cmecmeacellunitofVolume

cellunitperAtomsofNumber

==

−=−==

=

What is the atomic density of a bcc material with lattice constant 5.2

angstroms?

30

X-tal planes

a

a

2a

31

a

a

a

X-tal planes

32

How do we specify x-tal planes?

Answer: Miller Indices!

I will not go into depth here, but I would like for you to know these

basic x-tal planes in the cubic lattice.

33

How do we produce silicon

x-tals?

34

start with silica

(impure SiO2)

heat silica with

carbon to remove

oxygen

very impure

silicon

Chlorinated

impure siliconliquefied SiCl4

distillation

purification

process

ultra-pure

SiCl4

heat SiCl4 in H2

atmosphere

(SiCl4+2H2�

4HCL+ Si )

ultra-pure

polycrystalline

Silicon

Producing ultra-pure silicon

“silicon tetrachloride”

35

Ingot

Wafer

36300mm wafer

Wafer Boat

37

Czochralski Crystal Growth

40

What type of carriers do we have in

a semiconductor?

Conduction Band Completely

Empty

Valence Band Completely full

Band Occupation at T=0K

Ec

Ev

+

Electron free to move in

conduction band

“Hole” free to move in valence

band

Band Occupation at Higher Temperature (T>0 Kelvin)

For (Ethermal=kT)>0

Ec

Ev

+

For (Ethermal=kT)>0

Carrier Movement Under Bias

Direction of

Current Flow

Direction of

Current Flow

Ec

Ev


conduction band

“Hole” movement in

valence band

+

Ec

Ev


conduction band


valence band


Direction of

Current Flow

Direction of

Current Flow

For (Ethermal=kT)>0

+

Ec

Ev



conduction band


valence band

Direction of

Current Flow

Direction of

Current Flow

For (Ethermal=kT)>0

Material Classification based on Size of Bandgap:

Ease of achieving thermal population of conduction band determines

whether a material is an insulator, semiconductor, or metal

47

How many carriers do we have in an

UNDOPED semiconductor (i.e.

“intrinsic material”)?

Intrinsic Carrier Concentration

•Intrinsic carrier concentration is the number of electron (=holes)

per cubic centimeter populating the conduction band (or valence

band) is called the intrinsic carrier concentration, ni

•ni= f(T) that increases with increasing T (more thermal energy)

ni~2e6 cm-3 for GaAs with Eg=1.42 eV,

ni~1e10 cm-3 for Si with Eg=1.1 eV,

ni~2e13 cm-3 for Ge with Eg=0.66 eV,

ni~1e-14 cm-3 for GaN with Eg=3.4 eV

At Room Temperature (T=300 K)

Ec

Ev

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+08

1.E+09

1.E+10

1.E+11

1.E+12

1.E+13

1.E+14

1.E+15

1.E+16

1.E+17

1.E+18

100 1000 10000

Temperature (K)

Temperature Dependence of Intrinsic Carrier Concentration

ni= N

cN

ve−

Eg

2kT

k = boltzmann constant = 1.38e-23 J/K

Definition of “extrinsic semiconductor”

Example: P,

As, Sb in Si

Extrinsic, (or doped material):

Concept of a Donor “adding extra” electrons

Concept of a Donor “adding extra” electrons: Band diagram

equivalent view

Example: B,

Al, In in Si

Extrinsic, (or doped material):

Concept of an acceptor “adding extra” holes

Concept of an Acceptor“adding extra hole”: Band diagram

equivalent view

Hole Movement

Empty state is located next to the Acceptor

All regions of

material are

neutrally

charged.

+

Hole Movement

Another valence electron can fill the empty state located next to the Acceptor leaving

behind a positively charged “hole”.

+

Hole Movement

The positively charged “hole” can move throughout the crystal (really it is the valance

electrons jumping from atom to atom that creates the hole motion).

+

Hole Movement

Region around

the acceptor

has one extra

electron and

thus is

negatively

charged.

Region

around the

“hole” has

one less

electron and

thus is

positively

charged.

The positively charged “hole” can move throughout the crystal (really it is the valance

electrons jumping from atom to atom that creates the hole motion).

Summary of Important terms and symbols

Bandgap Energy: Energy required to remove a valence electron and allow it to freely conduct.

Intrinsic Semiconductor: A “native semiconductor” with no dopants. Electrons in the conduction

band equal holes in the valence band. The concentration of electrons (=holes) is the intrinsic

concentration, ni.

Extrinsic Semiconductor: A doped semiconductor. Many electrical properties controlled by the

dopants, not the intrinsic semiconductor.

Donor: An impurity added to a semiconductor that adds an additional electron not found in the

native semiconductor.

Acceptor: An impurity added to a semiconductor that adds an additional hole not found in the

native semiconductor.

Dopant: Either an acceptor or donor.

N-type material: When electron concentrations (n=number of electrons/cm3) exceed the hole

concentration (normally through doping with donors).

P-type material: When hole concentrations (p=number of holes/cm3) exceed the electron

concentration (normally through doping with acceptors).

Majority carrier: The carrier that exists in higher population (ie n if n>p, p if p>n)

Minority carrier: The carrier that exists in lower population (ie n if n<p, p if p<n)

Other important terms (among others): Insulator, semiconductor, metal, amorphous, polycrystalline,

crystalline (or single crystal), lattice, unit cell, primitive unit cell, zincblende, lattice constant,

elemental semiconductor, compound semiconductor, binary, ternary, quaternary, atomic density,

Miller indices, various notations, etc...

How do we calculate the electron or hole

concentration in equilibrium for

EXTRINSIC MATERIALS?

Parking Lot Analogy

If we have a lot with 100 spaces and the probability of a

single space being occupied is 25%, on average how many

parking spaces should be occupied.

Change percentage

Change # of parking spaces







Modified Football Stadium Analogy

Valence Seats

Conduction Seats

Forbidden Seats

Valence Seats

Forbidden Seats

Conduction Seats

H If heat energy > Egap

then move to

conduction seats

•Sell enough tickets for valence seats only!

•Hot plate under each seat that is randomly activated in the valence seats!

•What is the number of “Conduction Seats” occupied?

G[H] = Density of seats

G[H] ΔH = # of seats between H and H+ΔH

F[H] = probability that seat at level H is occupied

Modified Football Stadium Analogy

# of fans between H and H+⊗H

= (# seats)*(prob seat is occupied)

=(G[H] ⊗H) F[H]

Conduction Band

Valence Band

E

Forbidden Region

Conduction Seats

Valence Seats

H

Forbidden Region

gc(E) = Density of states (i.e. energy states)

gc(E) ΔE = # energy levels per vol. from E to E+ΔE

f(E) = probability electron at Energy E

electrons conc. between E and E+⊗E

= (# E states density)*(prob E is occupied)

= (gc(E) ΔE ) f(E)

Electron Statistics in Semiconductor

Quantum Mechanics tells us that the number of available states in a cubic cm

per unit of energy, the density of states, is given by:

eVcm

StatesofNumber

unit

EEEEmm

Eg

EEEEmm

Eg

v

vpp

v

c

cnn

c

≡

≤−

=

≥−

=

3

32

**

32

**

,)(2

)(

,)(2

)(

h

h

π

π

Density of States Concept

h = planck’s constant= 6.63e-34 [J-sec]

h = reduced planck’s constant (pronounced “h-bar”)= h/2π

Effective Mass for Different Estimations

We also need to know the

probability that an energy level is

occupied at a given temperature!

Here we combine statistical

thermodynamics and quantum ideas!

Fermi-Dirac Function Origins

Energy Levels

E1

1 2 3 4 g1

n1

number of “particles” at energy E1

positions at energy E1

E2

1 2 3 4 g2

n2

1 2 3 4 g3

n3E3

1 2 3 4 gk

nkEk

W1=

g1g1−1( ) g1 − 2( )...(g1 − (n

1−1))

n1!

# of arrangements at each level

assuming order of balls doesn’t matter!

W2=

g2g2−1( ) g2 − 2( )...(g2 − (n2 −1))

n2!

W3=

g3g3−1( ) g3 − 2( )...(g3 − (n3 −1))

n3!

Wk=

gkgk−1( ) gk − 2( )...(gk − (nk −1))

nk!

S = lnW = ln(W1*W

2*W

3...*W

k)


Wk=

gkgk−1( ) gk − 2( )...(gk − (nk −1))

nk!

S = lnW = ln(W1*W

2*W

3...*W

k)

Here is the question…..

What values of g1, g2, g3,… gk and n1, n2, n3, … nk MAXIMIZE the entropy

under the following constraints:

ni= N

total

i=1

i=k

∑

niEi= E

total

i=1

i=k

∑


Energy Levels

E1

1 2 3 4 g1

n1

E2

1 2 3 4 g2

n2

1 2 3 4 g3

n3E3

1 2 3 4 gk

nkEk ni

gi

=1

1+ e

(Ei−α )

β

The result of this optimization is the

following fundamental relationship.

f (Ei) =

1

1+ e

(Ei−E

F)

kT

“Occupation Probability”

Probability of Occupation (Fermi Function) Concept

The probability of an electron at the Fermi energy is 0.5

Source: Pierre Textbook (Need to include full source)

f (Ei) =

1

1+ e

(Ei−E

F)

kT

At higher temperatures, higher energy states can be occupied, leaving more

lower energy states unoccupied (1-f(E)).

Probability of Occupation (Fermi Function) Concept


Expected Electron Concentration

Thus, the density of electrons (or holes) occupying the states in energy between

E and E+dE is:

otherwise

and

and

0

,EE if dEf(E)]-(E)[1g

,EE if dEf(E)(E)g

vv

cc

≤

≥Electrons/cm3 in the conduction

band between Energy E and E+dE

Holes/cm3 in the valence band

between Energy E and E+dE

G[H] = Density of seats

G[H] ΔH = # of seats between H and H+ΔH

F[H] = probability that seat at level H is occupied

# of fans between H and H+⊗H

= (# seats)*(prob seat is occupied)

=(G[H] ⊗H) F[H]

remember….

Expected Electron Concentration

Decreasing (Ec-

Ef) increases

electron

concentration

Decreasing (Ef-

Ev) increases

electron

concentration


Intrinsic Energy (or Intrinsic Level)

…Equal numbers of

electrons and holes

Efis said to equal E

i

(intrinsic energy) when

material is intrinsic

NOTE: Eiis approximately mid-bandgap BUT not quite!


Additional Dopant States: Changing Ef

Intrinsic:

Equal number

of electrons

and holes

n-type: more

electrons than

holes

p-type: more

holes than

electrons


Developing the mathematical model for electrons and holes

The density of electrons is:

∫=bandconductionofTop

bandconductionofBottom

E

Ec

dEEfEgn )()(

∫ −=

bandvalenceofTop

bandvalenceofBottom

E

Ev

dEEfEgp )](1)[(

The density of holes is:

Probability the state is filled

Probability the state is empty

Number of states per cm-3 in energy range dE

Number of states per cm-3 in energy range dE

Note: units of n and p are #/cm3


The result of this integration gives:

kTEE

i

kTEE

i

fi

if

enp

and

enn

/)(

/)(

−

−

=

=Ec

Ev

EF

Ei

n-type material

Ec

Ev

EF

Ei

p-type material

Ec

Ev

Ei=EF

intrinsic material


Other useful Relationships: n - p product

2

/)(/)(

i

kTEE

i

kTEE

i

nnp

enpandennSince fiif

=

==

−−

Known as the Law of Mass Action

Calculating Electron and Hole

Concentrations in Semiconductors


If ND>>N

Aand N

D>>n

i

D

i

D

N

npandNn

2

≅≅

If NA>>N

Dand N

A>>n

i

A

i

A

N

nnandNp

2

≅≅

np = ni

2


Example:

An intrinsic Silicon wafer has 1e10 cm-3 holes. When 1e18 cm-3

donors are added, what is the new hole concentration?

n ≅ ND= 10

18 cm−3

p =

ni

2

n=

1010( )

2

1018

cm−3= 100 cm−3

Concept of a Donor “adding extra” electrons: Band diagram

equivalent view

Concept of an Acceptor“adding extra hole”: Band diagram

equivalent view

Temperature Depedance of Carrier Concentration

Extrinsic Temperature Region

n/ND

100 200 300 400 500 600

T(K)

0.5

1.0

1.5

2.0

Freeze Out

Intrinsic T-Region

Semiconductor Physics Overview

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