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1
Lecture 2
Semiconductor Device Physics
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2 0 1 3
2
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
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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)
14
Chapter 2 Carrier Modeling
Effect of Temperature on f(E)
15
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):
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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
28
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
30
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
31
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.
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