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Lecture 2

Semiconductor Device Physics

http://zitompul.wordpress.com

2 0 1 3

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Manipulation of Carrier Numbers – Doping

Donors: P, As, Sb

Phosphorus, Arsenic, Antimony

Acceptors: B, Ga, In, Al

Boron, Gallium Indium, Aluminum

Chapter 2 Carrier Modeling

By substituting a Si atom with a special impurity atom (elements from Group III or Group V), a hole or conduction electron can be created.

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Doping Silicon with Acceptors

Al– is immobile

Chapter 2 Carrier Modeling

Example: Aluminum atom is doped into the Si crystal.

The Al atom accepts an electron from a neighboring Si atom, resulting in a missing bonding electron, or “hole”.

The hole is free to roam around the Si lattice, and as a moving positive charge, the hole carries current.

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Doping Silicon with Donors

P+ is immobile

Chapter 2 Carrier Modeling

Example: Phosphorus atom is doped into the Si crystal.

The loosely bounded fifth valence electron of the P atom can “break free” easily and becomes a mobile conducting electron.

This electron contributes in current conduction.

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Ec

Donor LevelED

Donor ionization energy

Ev

Acceptor LevelEA

Acceptor ionization energy

+

+

Ionization energy of selected donors and acceptors

in Silicon (EG = 1.12 eV)

Acceptors

Ionization energy of dopant Sb P As B Al In

EC – ED or EA – EV (meV) 39 45 54 45 67 160

Donors

Chapter 2 Carrier Modeling

Donor / Acceptor Levels (Band Model)

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Chapter 2 Carrier Modeling

Dopant Ionization (Band Model)

Donor atoms

Acceptor atoms

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Chapter 2 Carrier Modeling

Carrier-Related Terminology

Donor: impurity atom that increases n (conducting electron).Acceptor: impurity atom that increases p (hole).

n-type material: contains more electrons than holes.p-type material: contains more holes than electrons.

Majority carrier: the most abundant carrier.Minority carrier: the least abundant carrier.

Intrinsic semiconductor: undoped semiconductor n = p = ni.Extrinsic semiconductor: doped semiconductor.

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Ec

Ev

DE

Chapter 2 Carrier Modeling

Density of States

E

Ec

Ev

gc(E)

gv(E)density of states g(E)

g(E) is the number of states per cm3 per eV.

g(E)dE is the number of states per cm3 in the energy range between E and E+dE).

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Ec

Ev

DE

* *

n n c

c 2 3

2( )

m m E Eg E

h

* *

p p v

v 2 3

2( )

m m E Eg E

h

E Ec

E Ev

Density of States

*

n

*

n o*

p o

31

o

: effective mass of electron

For Silicon at 300 K,

1.18

0.81

9.1 10 kg

m

m m

m m

m

mo: electron rest mass

Chapter 2 Carrier Modeling

Near the band edges:

E

Ec

Ev

gc(E)

gv(E)density of states g(E)

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Electrons as Moving Particles

Chapter 2 Carrier Modeling

In free space In semiconductor

0F qE m a *

nF qE m a

mo: electron rest mass mn*: effective mass of electron

Si Ge GaAs

m n*/m 0 1.18 0.55 0.066

m p*/m 0 0.81 0.36 0.52

Effective masses at 300 K

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F( ) /

1( )

1 E E kTf E

e

+

Fermi Function

F

F

F

If , ( ) 0

If , ( ) 1

If , ( ) 1 2

E E f E

E E f E

E E f E

k : Boltzmann constant

T : temperature in Kelvin

Chapter 2 Carrier Modeling

The probability that an available state at an energy E will be occupied by an electron is specified by the following probability distribution function:

EF is called the Fermi energy or the Fermi level.

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F( ) /

1( )

1 E E kTf E

e

+

Chapter 2 Carrier Modeling

Boltzmann Approximation of Fermi FunctionThe Fermi Function that describes the probability that a state

at energy E is filled with an electron, under equilibrium conditions, is already given as:

Fermi Function can be approximated, using Boltzmann Approximation, as:

F( ) /( ) E E kTf E e

F( ) /1 ( ) E E kTf E e

if E – EF > 3kT

if EF – E > 3kT

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Chapter 2 Carrier Modeling

Effect of Temperature on f(E)

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Chapter 2 Carrier Modeling

Effect of Temperature on f(E)

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Energy banddiagram

Density ofstates

Probabilityof occupancy

Carrier distribution

Chapter 2 Carrier Modeling

Equilibrium Distribution of Carriersn(E) is obtained by multiplying gc(E) and f(E),

p(E) is obtained by multiplying gv(E) and 1–f(E).

Intrinsic semiconductor material

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Energy banddiagram

Density ofStates

Probabilityof occupancy

Carrier distribution

Chapter 2 Carrier Modeling

Equilibrium Distribution of Carriers

n-type semiconductor material

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Energy banddiagram

Density ofStates

Probabilityof occupancy

Carrier distribution

Chapter 2 Carrier Modeling

Equilibrium Distribution of Carriers

p-type semiconductor material

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Chapter 2 Carrier Modeling

Important Constants

Electronic charge, q = 1.610–19 C

Permittivity of free space, εo = 8.85410–12 F/m

Boltzmann constant, k = 8.6210–5 eV/K

Planck constant, h = 4.1410–15 eVs

Free electron mass, mo = 9.110–31 kg

Thermal energy, kT = 0.02586 eV (at 300 K)

Thermal voltage, kT/q = 0.02586 V (at 300 K)

2

hh

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v F c3 3E kT E E kT+

Ec

Ev

3kT

3kT

EF in this range

Chapter 2 Carrier Modeling

Nondegenerately Doped SemiconductorThe expressions for n and p will now be derived in the range

where the Boltzmann approximation can be applied:

The semiconductor is said to be nondegenerately doped(lightly doped) in this case.

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Chapter 2 Carrier Modeling

Degenerately Doped Semiconductor If a semiconductor is very heavily doped, the Boltzmann

approximation is not valid.

For Si at T = 300 K,EcEF > 3kT if ND > 1.6 1018 cm–3

EFEv > 3kT if NA > 9.1 1017 cm–3

The semiconductor is said to be degenerately doped (heavily doped) in this case.

• ND = total number of donor atoms/cm3

• NA = total number of acceptor atoms/cm3

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Chapter 2 Carrier Modeling

Equilibrium Carrier Concentrations Integrating n(E) over all the energies in the conduction band to

obtain n (conduction electron concentration):

top

c

c ( ) ( )

E

E

n g E f E dE

By using the Boltzmann approximation, and extending the integration limit to ,

F c

3/ 2*

( ) nC C 2

where 22

E E kT m kTn N e N

h

• NC = “effective” density of conduction band states

• For Si at 300 K, NC = 3.22 1019 cm–3

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Chapter 2 Carrier Modeling

Equilibrium Carrier Concentrations Integrating p(E) over all the energies in the conduction band to

obtain p (hole concentration):

V

bottom

v ( ) 1 ( )

E

E

p g E f E dE

By using the Boltzmann approximation, and extending the integration limit to ,

v F

3/ 2*

p( )

V V 2 where 2

2

E E kTm kT

p N e Nh

• NV = “effective” density of valence band states

• For Si at 300 K, NV = 1.83 1019 cm–3

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Chapter 2 Carrier Modeling

Intrinsic Carrier ConcentrationRelationship between EF and n, p :

For intrinsic semiconductors, where n = p = ni,

v F( )

V

E E kTp N e

F c( )

C

E E kTn N e

G

2

i

2

i C V E kT

np n

n N N e

• EG : band gap energy

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F c v F( ) ( )

C V( ) ( ) E E kT E E kT

np N e N e

Chapter 2 Carrier Modeling

Intrinsic Carrier Concentration

v c( )

C V

E E kTN N e

G

C V

E kTN N e

2

i C VGE kT

n N N e

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In an intrinsic semiconductor, n = p = ni and EF = Ei, where Ei denotes the intrinsic Fermi level.

v i v F( ) ( )

i

E E kT E E kTp n e e

i c F c( ) ( )

i

E E kT E E kTn n e e

Chapter 2 Carrier Modeling

Alternative Expressions: n(ni, Ei) and p(ni, Ei)

F c( )

C

E E kTn N e

i F( )

i

E E kTp n e

v F( )

V

E E kTp N e

F i( )

i

E E kTn n e

i c( )

C i

E E kTN n e

i c( )

i C

E E kTn N e

F i

i

lnn

E E kTn

+

F i

i

lnp

E E kTn

v i( )

i V

E E kTp N e

v i( )

V i

E E kTN n e

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i c v i( ) ( )

C V

E E kT E E kTN e N e

Intrinsic Fermi Level, Ei

c v Vi

C

ln2 2

E E NkTE

N

+ +

Ec

Ev

EG = 1.12 eV

Si

Ei

*

pc vi *

n

3ln

2 4

mE E kTE

m

+ +

Chapter 2 Carrier Modeling

To find EF for an intrinsic semiconductor, we use the fact that n = p.

c vi

2

E EE

+ • Ei lies (almost) in the middle

between Ec and Ev

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Example: Energy-Band Diagram

175

10

100.56 8.62 10 300 ln eV

10

+

F i

i

lnn

E E kTn

+

Chapter 2 Carrier Modeling

For Silicon at 300 K, where is EF if n = 1017 cm–3 ?

Silicon at 300 K, ni = 1010 cm–3

0.56 0.417 eV +

0.977 eV

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Charge Neutrality and Carrier Concentration

2

iD A 0,

np n N N p

n +

Chapter 2 Carrier Modeling

ND: concentration of ionized donor (cm–3)

NA: concentration of ionized acceptor (cm–3)?

Charge neutrality condition:

2

iD A 0

nn N N

n +

2 2

D A i( ) 0n n N N n • Ei quadratic equation in n

+

D A 0p n N N +

+

Assumptions: nondegeneracy (np product relationship applies) and total ionization.

Setting ND = ND and NA = NA, as at room temperature almost all of donor and acceptor sites are ionized,

+ –

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Charge-Carrier Concentrations

1 22

2A D A Di

2 2

N N N Np n

+ +

1 22

2D A D Ai

2 2

N N N Nn n

+ +

Chapter 2 Carrier Modeling

The solution of the previous quadratic equation for n is:

New quadratic equation can be constructed and the solution for p is:

• Carrier concentrations depend on net dopant concentration ND–NA or NA–ND

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Dependence of EF on Temperature

F i

i

F i

i

ln , donor-doped

ln , acceptor-doped

nE E kT

n

pE E kT

n

+

1013 1014 1015 1016 1017 1018 1019 1020

Net dopant concentration

(cm–3)

Ec

Ev

Ei

Chapter 2 Carrier Modeling

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Carrier Concentration vs. TemperaturePhosphorus-doped Si

ND = 1015 cm–3

Chapter 2 Carrier Modeling

• n : number of majority carrier

• ND : number of donor electron

• ni : number of intrinsic conductive electron

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Chapter 2 Carrier Modeling

Homework 1

1. (Problem 2.6 in Pierret) (4.2)

a) Under equilibrium condition at T > 0 K, what is the probability of an

electron state being occupied if it is located at the Fermi level?

b) If EF is positioned at Ec, determine (numerical answer required) the

probability of finding electrons in states at Ec + kT.

c) The probability a state is filled at Ec + kT is equal to the probability a

state is empty at Ec + kT. Where is the Fermi level located?

2. (2.36)

a) Determine for what energy above EF (in terms of kT) the Boltzmann

approximation is within 1 percent of the exact Fermi probability function .

b) Give the value of the exact probability function at this energy.

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Chapter 2 Carrier Modeling

Homework 1

3.a. (4.2)

Calculate the equilibrium hole concentration in silicon at T = 400 K if the Fermi energy level is 0.27 eV above the valence band energy.

3.b. (E4.3)

Find the intrinsic carrier concentration in silicon at:(i) T = 200 K and (ii) T = 400 K.

3.c. (4.13)

Silicon at T = 300 K contains an acceptor impurity concentration of NA = 1016 cm–3. Determine the concentration of donor impurity atoms that must be added so that the silicon is n-type and the Fermi energy level is 0.20 eV below the conduction band edge.