Pierret, Chapter 2
“Carrier Modeling”
Jeff Davis
FALL 2007
ECE3040
References
• Prof. Alan Doolittle’s Notes
– users.ece.gatech.edu/~alan/index_files/ECE3040index.htm
• Prof. Farrokh Ayazi’s Notes
– users.ece.gatech.edu/~ayazi/ece3040/
• Figures for Require Textbooks
– (Pierret and Jaeger)
Important Early Dates Leading
to Quantum Mechanics
•
•
•
•
•
•
•
•
1901 - Planck models blackbody radiation assume that energy
release from atoms is quantized
1905 - Einstein defines photons to describe photoelectric effect
1910 - Rutherford experiment to determine atom core
1913 - The Bohr atom is proposed (semi-classical)
1920s - Wave particle duality of EM radiation is broadly
accepted
1925 - DeBroglie postulates that wave-particle duality applies to
matter
1926 - Schrodinger proposes wave mechanics with his famous
equation … Wave functions are born!
1927 - Davisson and Germer experiment with diffracting e-beam
from nickel crystal to support DeBroglie’s hypothesis
Energy Quantization - READING APPENDIX A
“The Bohr Atom (1913)”
Energy Hydrogen electron
mo q 4
13.6 eV
=!
=
!
2( 4" # 0 ! n) 2
n2
where mo = electron mass, ! = planks cons tan t / 2" = h / 2"
q = electron ch arg e, and n = 1,2,3...
n is the principle quantum number
Quantum Numbers
Designates shell
N = principle quantum number (N = 1,2,3,..) Size of orbital
N>=(L+1)
L = angular quantum number (L = 0,1,2,3,4,5,6,7 OR s,p,d,f,g,h, i, k)
Designates subshell
M = magnetic quantum number ( orientation of the orbitals)
M can be negative!
|M|<=L
S = Angular momentum of electron spin (+1/2 or -1/2)
Pauli Exclusion Principle
NO TWO ELECTRONS CAN HAVE THE SAME
4 QUANTUM NUMBERS!!!
(If overlapping wave functions)
Examples
1.
2.
3.
4.
5.
6.
Hydrogen (H) 1s
Helium (He) 1s2
Lithium (Li) 1s22s
Beryllium (Be) 1s22s2
Boron (B) 1s22s22p
Carbon (C) 1s22s22p2
Carbon has six electrons.. its electrons could have the
following set of quantum numbers
electron 1: (N=1,L=0(s),M=0,S=+1/2)
electron 2: (N=1,L=0(s),M=0,S=-1/2)
electron 3: (N=2,L=0(s),M=0,S=+1/2)
electron 4: (N=2,L=0(s),M=0,S=-1/2)
electron 5: (N=2,L=1(p),M=0,S=+1/2)
electron 6: (N=2, L=1(p), M=0,S=-1/2)
Pauli Exclusion Principle
Splitting Energy Levels
Carbon (C) 1s22s22p2
‘N’ for this figure is the number of
atoms with overlapping electron
wavefunctions
sp3
hybridation
Note - at zero temp
all electrons are in
the valence band!
3
sp
Hybrid Orbitals
Different orientation of orbitals
What about silicon?
4 empty states
n=2: Complete Shell
2 “2s electrons”
6 “2p electrons”
Silicon
4 Valence
Shell
Electrons
n=1: Complete Shell
2 “s electrons”
n=3:
2 “3s electrons”
Only 2 of 6 “3p electrons”
T=0K
“Band Gap”
where ‘no’ states
exist
EC or conduction
band
EV or valence
band
Band Occupation at T=0K
Conduction Band
Completely Empty
Ec
Ev
Valence Band
Completely full
Band Occupation at Higher Temperature (T>0 Kelvin)
For (Ethermal=kT)>0
Electron free to move
in conduction band
Ec
+
Ev
“Hole” free to move in
valence band
Carrier Movement Under Bias
For (Ethermal=kT)>0
Electron free to move in
conduction band
Ec
+
Ev
“Hole” movement in
valence band
Direction of
Current Flow
Direction of
Current Flow
Carrier Movement Under Bias
For (Ethermal=kT)>0
Electron free to move in
conduction band
Ec
+
Ev
“Hole” movement in
valence band
Direction of
Current Flow
Direction of
Current Flow
Carrier Movement Under Bias
For (Ethermal=kT)>0
Electron free to move in
conduction band
Ec
+
Ev
“Hole” movement in
valence band
Direction of
Current Flow
Direction of
Current Flow
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
Energy Dependence of Bandgap Energy
GaAs
Eg(T=0) =1.519 [eV]
α= 5.41x10-4 (eV/K)
β= 204 [K]
Silicon
Eg(T=0) =1.166 [eV]
α = 4.73x10-4 (eV/K)
β = 636 [K]
"T 2
E g (T ) = E g (T = 0) !
T +#
Germanium
Eg(T=0) =0.7437 [eV]
α = 4.77x10-4 (eV/K)
β = 235 [K]
Physically why does this occur?
Energy Dependence of Bandgap Energy - Example
GaN
Ramirez-Flores, G., H. Navarro-Contreras, A. Lastrae-Martinez, R.C. Powell, J.E. Greene,
Temperature-dependent optical band gap of the metastable zinc-blende structure
beta -GaN, Phys. Rev. B 50(12) (1994), 8433-8438.
Definition of Intrinsic Carrier
Concentration, ni
Intrinsic Carrier Concentration
Ec
Ev
•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)
At Room Temperature (T=300 K)
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
Temperature Dependence of Intrinsic Carrier Concentration
ni = N c N v e
!
Eg
2 kT
k = boltzmann constant = 1.38e-23 J/K
Definition of “effective mass” of a
charge carrier
Carrier Movement in Free Space
Newton’s second law of motion!
F=ma
dv
F = "qE = mo
dt
F ! force, v ! velocity, t ! time,
q ! electronic ch arg e, mo ! electron mass
Carrier Movement Within the Crystal
Electron sees a periodic potential due to the atomic cores
Carrier Movement Within the Crystal
dv
F = "qE = m
dt
F ! force, v ! velocity, t ! time,
q ! electronic ch arg e,
dv
dt
F ! force, v ! velocity, t ! time,
q ! electronic ch arg e,
mn* ! electron effective mass
m*p ! hole effective mass
F = qE = m*p
*
n
Effective Mass used to estimate “mobility” of carrier in the lattice
electron
holes
mn*= 0.33mo
mh*= 0.5mo
Germanium
mn*= 0.22mo
mh*= 0.31mo
GaAs
mn*= 0.063mo
mh*= 0.5mo
Silicon
A.K.A. Conductivity Effective Mass
Effective Mass for Different Estimations
Definition of “extrinsic
semiconductor” of a charge carrier
Extrinsic, (or doped material):
Concept of a Donor “adding extra” electrons
Example:
P, As, Sb
in Si
Concept of a Donor “adding extra” electrons:
Band diagram equivalent view
Extrinsic, (or doped material):
Concept of an acceptor “adding extra” holes
Example:
B, Al, In
in Si
Concept of an Acceptor“adding extra hole”:
Band diagram equivalent view
Hole Movement
All regions of
material are
neutrally
charged.
Empty state is located next to the Acceptor
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
+
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
+
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 density in
equilibrium?
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
If we have a lot with 100 spaces and the
probability of a single space being
occupied is 50%, on average how many
parking spaces should be occupied.
Change # of parking spaces
If we have a lot with 200 spaces and the
probability of a single space being
occupied is 25%, on average how many
parking spaces should be occupied.
Modified Football Stadium Analogy
Conduction
Conduction
Seats
Seats
Forbidden
Seats
Valence Seats
H
If heat energy >
Egap then move to
conduction seats
Forbidden
Seats
Valence 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?
Modified Football Stadium Analogy
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 level H and H+ΔH
= (# seats)*(prob seat is occupied)
=(G[H] ΔH) F[H]
How do electrons and holes populate the bands?
Density of States Concept
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:
mn* 2mn* ( E ) Ec )
gc (E) =
, E * Ec
2 3
+ !
gv (E) =
m*p 2m*p ( Ev ) E )
+ !
2
3
, E ( Ev
& Number of States #
$
!
3
cm
"
unit ' %
eV
Remember Plancks Constant!
h = planck’s constant= 6.63e-34 [J-sec]
h = reduced planck’s constant (pronounced “h-bar”)= h/2π
Ephoton = hν = hω
Photon frequency
Photon angular frequency
Density of States Concept
Thus, the number of states per cubic centimeter between
energy E’ and E’+dE is
g c (E’ )dE if E’ " E c
and ,
g v (E’ )dE if E’ ! E v
and ,
0 otherwise
Probability of Occupation (Fermi Function)
Concept
Now that we know the number of available states at each energy, how
do the electrons occupy these states?
We need to know how the electrons are “distributed in energy”.
Again, Quantum Mechanics tells us that the electrons follow the
“Fermi-distribution function”.
f (E) =
1
(E ! EF )
where k " Boltzman constant, T " Temperature in Kelvin
kT
1+ e
and EF " Fermi energy
f(E) is the probability that a state at energy E is occupied
1-f(E) is the probability that a state at energy E is unoccupied
Probability of Occupation (Fermi Function)
Concept
At T=0K, occupancy is “digital”: No occupation of states above EF and
complete occupation of states below EF
At T>0K, occupation probability is reduced with increasing energy.
f(E=EF) = 1/2 regardless of temperature.
The probability of an electron at the Fermi energy is 0.5
Probability of Occupation (Fermi Function)
Concept
At higher temperatures, higher energy states can be occupied, leaving more
lower energy states unoccupied (1-f(E)).
Expected Electron Concentration
Thus, the density of electrons (or holes) occupying the
states in energy between E and E+dE is:
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 c (E) f(E) dE
if E " E c
and ,
g v (E)[1 - f(E)] dE if E ! E v
0 otherwise
and ,
Expected Electron Concentration
Decreasing
(Ec -Ef)
increases
electron
concentration
Decreasing
(Ef-Ev)
increases
electron
concentration
Intrinsic Energy (or Intrinsic Level)
Ef is said to equal Ei
(intrinsic energy) when
material is intrinsic
…Equal numbers of
electrons and holes
NOTE: Ei is 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
Probability the state is filled
The density of electrons is:
n=!
ETop of
conduction band
E Bottom of
conduction band
g c ( E ) f ( E )dE
Number of states per cm-3 in energy range dE
The density of holes is:
Probability the state is empty
p=!
ETop of
valence band
E Bottom of
valence band
g v ( E )[1 " f ( E )]dE
Number of states per cm-3 in energy range dE
Note: units of n and p are #/cm3
Developing the mathematical model for electrons and holes
n=
*
n
m
*
n
2m
# 2! 3
!
ETop of
1+ e
Ec
E # Ec
Letting $ =
kT
when E = Ec , $ = 0
( E " E f ) / kT
and $c =
dE
E f # Ec
kT
ETop of conduction band " !
Let
n=
E " Ec
conduction band
*
n
m
*
n
2 3
2m (kT )
% !
3/ 2
!
"
0
$
d$
($ #$c )
1+ e
1/ 2
This is known as the Fermi-dirac integral of order 1/2 or, F1/2(ηc)
Developing the mathematical model for electrons and holes
We can further define:
& m (kT ) #
N c = 2$
2 !
2
'
!
%
"
and
*
n
3/ 2
& m*p (kT ) #
N v = 2$
2 !
2
'
!
$%
!"
the effective density of states in the conduction band
3/ 2
the effective density of states in the valence band
This is a general relationship holding for all materials and
3
results in:
*
2
' mn $
"" cm !3 at 300 K
N c = 2.51x10 %%
& mo #
19
3
2
'
$
m
"" cm !3 at 300 K
N v = 2.51x1019 %%
& mo #
*
v
Developing the mathematical model for electrons and holes
n = Nc
2
F1/ 2 ("c )
#
and
p = Nv
2
F1/ 2 ("v ) where "v = ( Ev ! E f ) / kT
#
Fermi-dirac integrals
can be numerically
determined or read
from tables or...
Developing the mathematical model for electrons and holes
Useful approximations to the Fermi-dirac integral
If Ef < Ec-3kT
n
e
g
e
D
3kT
Nondegenerate
1
" (! "!c )
#e
! "!c
1+ e
e
t
a
er
" ( E f ! Ec ) / kT
F1/ 2 (#c ) =
e
2
Similarly when Ef > Ev+3kT
" ( Ev ! E f ) / kT
F1/ 2 (#v ) =
e
2
De
Ef
3kT
gen
era
te
Ec
Ev
Developing the mathematical model for electrons and holes
Useful approximations to the Fermi-dirac integral:
Non-degenerate Case
n = Nce
( E f ! Ec ) / kT
and
p = Nve
( Ev ! E f ) / kT
Developing the mathematical model for electrons and holes
When n=ni, Ef=Ei (the intrinsic energy), then
( Ei ! Ec ) / kT
or
N c = ni e
ni = N v e ( Ev ! Ei ) / kT
or
N v = ni e ( Ei ! Ev ) / kT
ni = N c e
( Ec ! Ei ) / kT
and
n = ni e
( E f ! Ei ) / kT
and
p = ni e
( Ei ! E f ) / kT
Developing the mathematical model for electrons and holes
Other useful Relationships: n - p product
ni = N c e
2
( Ei ! Ec ) / kT
ni = N c N v e
and
! ( Ec ! Ev ) / kT
ni = N c N v e
ni = N v e
= Nc Nve
! EG / 2 kT
( Ev ! Ei ) / kT
! EG / kT
Developing the mathematical model for electrons and holes
Other useful Relationships: n - p product
Since n = ni e
( E f ! Ei ) / kT
and
p = ni e
np = ni2
Known as the Law of mass Action
( Ei ! E f ) / kT
Developing the mathematical model for electrons and holes
Charge Neutrality
• Must maintain charge neutrality
•Thus, all charges within the semiconductor must cancel.
+
D
Immobile - charge
Mobile - charge
Immobile + charge
Mobile + charge
p+N =n+N
!
A
Developing the mathematical model for electrons and holes
Charge Neutrality: Total Ionization Case
N-A = Concentration of “ionized” acceptors ~ = NA
N+D = Concentration of “ionized” donors ~ = ND
(p ! N A )+ (N D ! n ) = 0
Developing the mathematical model for electrons and holes
Charge Neutrality: Total Ionization Case
(p ! N A )+ (N D ! n ) = 0
' ni2
$
%% ! N A "" + (N D ! n ) = 0
& n
#
n 2 ! n(N D ! N A )! ni2 = 0
2
2
ND ' NA
NA ' ND
& ND ' NA #
& NA ' ND #
2
2
n=
+ $
+ $
! + ni or p =
! + ni
2
2
2
2
%
"
%
"
and
pn = ni2
Developing the mathematical model for electrons and holes
If ND>>NA and ND>>ni
n ! ND
and
2
i
n
p!
ND
If NA>>ND and NA>>ni
p ! NA
and
ni2
n!
NA
Developing the mathematical model for electrons and holes
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 = 1018 cm"3
(
10
)
2
10
n
"3
"3
p=
=
cm = 100 cm
18
n
10
2
i
Developing the mathematical model for electrons and holes
Example:
An intrinsic Silicon wafer has 1e10 cm-3 holes. When 1e18 cm-3
acceptors and 8e17 cm-3 donors are added, what is the new hole
concentration?
2
" 1x10 ! 8x10 %
1x10 ! 8x10
10
p=
+ $
+
1x10
'
2
2
#
&
18
17
18
17
p = 2x1017 cm!3 = NA ! ND
(
)
2
Developing the mathematical model for electrons and holes
Example:
An intrinsic Silicon wafer at 470K has 1e14 cm-3 holes. When 1e14
cm-3 acceptors are added, what is the new electron and hole
concentrations?
ND=0
2
! 1x10 $
1x10
14
p=
+ #
+
1x10
&
2
" 2 %
14
14
(
p = 1.62x1014 cm'3 ( NA ' ND
1x10 )
(
n=
14
2
1.62x1014
= 6.2x1013 cm'3
)
2
Developing the mathematical model for electrons and holes
Example:
An intrinsic Silicon wafer at 600K has 4e15 cm-3 holes. When 1e14
cm-3 acceptors are added, what is the new electron and hole
concentrations?
ND=0
2
! 1x10 $
1x10
15
p=
+ #
+
4x10
&
2
" 2 %
14
14
(
)
2
p = 4x1015 cm'3 = ni ( NA ' ND
4x10 )
(
n=
15
4x1015
2
= 4x1015 cm'3 = ni
)
Intrinsic Material at High Temperature
Where is Ei?
Since we started with descriptions of intrinsic materials then it makes
sense to reference energies from the intrinsic energy, Ei.
Intrinsic Material:
n = Nce
( E f ! Ec ) / kT
Nce
( Ei ! Ec ) / kT
= Nve
( Ev ! E f ) / kT
= Nve
( Ev ! Ei ) / kT
E c + E v kT & N v #
!!
Ei =
+
ln$$
2
2 % Nc "
=p
Where is Ei?
Intrinsic Material:
But,
N v &$ m #!
=
N c $% mn* !"
*
p
3/ 2
*
&
m
E c + E v 3kT $ p
Ei =
+
ln *
$m
2
4
% n
Letting Ev=0,
this is Eg / 2 or
“Midgap”
#
!
!
"
-0.007 eV for Si @
300K ( 0.6% of EG )
Where is Ei?
Extrinsic Material:
n = ni e
( E f ! Ei ) / kT
p = ni e
( Ei ! E f ) / kT
Solving for (Ef-Ei)
&n
E f ' Ei = kT ln$$
% ni
or for N D >> N A
#
& p#
!! = 'kT ln$$ !!
"
% ni "
and N D >> ni
& ND #
!!
E f ' Ei = kT ln$$
% ni "
or for N A >> N D and N A >> ni
& NA #
!!
E f ' Ei = 'kT ln$$
% ni "