Part 5
14-Dec-18
Semiconductors
1
OUTLINE
Semiconductors Models
Bonding model of semiconductor crystals
Energy band Model of semiconductor crystals
Semiconductor Density of States
Fermi-Dirac Distribution Function
Electron and Hole Concentrations
Intrinsic Carrier Concentration
Charge Neutrality Equation
Position of Fermi Energy Level
Thermal Equilibrium in Electronic Systems
14-Dec-18
Semiconductors
2
Bonding Model of Semiconductor Crystals
To understand the properties of semiconductor materials we need to consider how
atoms combine to form crystal structures.
• We begin by considering the atomic structure of SILICON which is known to
contain fourteen electrons that in turn occupy three different energy levels.
The first of these two levels are completely filled with electrons and form a fairly
inert CORE that is tightly bound to the central nucleus of the atom.
The remaining four valence electrons only partially fill the third energy level and
these electrons participate strongly in chemical bonding.
Si : 1s 2 2 s 2 2 p 6 3s 2 3 p 2
• In the silicon atom the first two energy levels hold
their full compliments of two- and eight-electrons
respectively.
• In the third energy level the shell 3s are filled
completely but the shell 3p hold only two electrons
so that four states of this orbital are left unfilled.
14-Dec-18
Semiconductors
3
Bonding Model of Semiconductor Crystals
When silicon atoms combine to form a crystal structure they share their valence
electrons covalently with four other atoms in the crystal. This way, each atom in the
crystal structure appears to fill its outermost orbitals completely.
SIMPLE BONDING MODEL FOR A SILICON
CRYSTAL
Si Atom
•
Each solid circle represents the filled core levels
of the silicon atom.
• Each line represents a shared valence electron.
• Remember that in a silicon crystal the bonds are
actually tetrahedral coordinated so this model
here should be considered as schematic.
Si Crystal
• This basic model is strictly valid for an entire
semiconductor at T = 0 k (insulator).
• At T > 0 k we visualize the breaking
atom to atom bond and the associated
release or freeing of an electron.
14-Dec-18
Semiconductors
-
4
Bonding Model of Semiconductor Crystals
Electrons are not the only charge carriers in a semiconductor at finite temperatures !!!
In the Bonding Model we picture the hole as carrying an electrical current when an
electron hops from a nearby bond to fill the original hole.
By completing the initially broken bond in this manner ANOTHER bond is broken
and we can think that the positive hole has moved to a new position.
1
1
2
2
This completes the originally broken
bond but at the same time breaks a
bond in region 2.
Initially a bond is broken in region 1
but then an electron hops from
region 2 to 1.
In INTRINSIC ( pure) semiconductors the number of electrons (n)
EXACTLY equals the number of holes (p)
n = p = ni
14-Dec-18
where ni is the intrinsic carrier concentration
Semiconductors
5
Bonding Model of Semiconductor Crystals
Extrinsic Semiconductors
The great advantage of semiconductors is that we can modify the electron and hole
densities by the controlled addition of IMPURITIES known as DOPANTS
n-type
In the specific case of silicon, Group V elements (pentavalent) of the periodic
table serve as donors (ND) to increase the electron density.
Since a Group V element has five valence electrons, an
extra electron is left over after the donor completes its
bonding and this electron is loosely bound.
e
At room temperature, the extra electron may escape from
the donor and wander through the crystal.
Electrons called the majority carriers.
Holes called the minority carriers.
14-Dec-18
Semiconductors
e
6
Bonding Model of Semiconductor Crystals
p-type
In contrast to the previous case, if we dope silicon with GROUP III elements of the
periodic table, these serve as acceptors (NA) to increase the hole density.
Since a Group III element has three valence electrons, a
bond is left incomplete when the acceptor is incorporated
into the crystal structure.
h
Now at room temperature, electrons hop from nearby sites
to complete the broken bond so that the acceptor has
essentially added an extra hole to the material.
Holes called the majority carriers.
Electrons called the minority carriers.
14-Dec-18
Semiconductors
h
7
Energy Band Model of Semiconductor Crystals
The energy band model shows how the ENERGY of charge carriers varies as a function
of their position within the crystal. Basics of the energy band model was discussed in
part 3.
Increasing
electron
energy.
Model Features
•
In terms of the energy band model, the energy required to break the
bond is simply equal to the gap energy Eg.
•
The electron energy increases upward while the holes energy increases
downward.
Electron
Eg
Hole
Ev
Increasing
hole
energy.
According to the predictions of the band theory, the velocity of
electrons or holes is zero at the conduction or valence band EDGES.
•
Ec
Only at energies AWAY from the band edges, is the kinetic energy
of the electrons or holes greater than zero.
Electron KE.
Ec
•
In the absence of an applied voltage or electric field, the potential
energy of the charge carriers is independent of position and the
energy bands are consequently FLAT.
All electrons in conduction band have the same potential energy.
All holes in valence band have the same potential energy.
T.E.
P.E.
Ev
Hole KE.
Eref is any convenient
reference energy.
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8
Eref
Energy Band Model of Semiconductor Crystals
In the following sections, we will introduce more information related to intrinsic
and extrinsic semiconductors through the energy band model.
Intrinsic Semiconductors (n = p = ni)
The next table shows the intrinsic carrier concentration and energy band gaps for most
important semiconductors at room temperature.
Clearly the value of the band gap has a very
strong effect on the number of carriers
present.
In a silicon crystal, the density of atoms is
5×1022 cm-3, so we see that the carrier density
(n or p) represents a small fraction of the total
number of atoms.
Semiconductor
GaAs
Si
Ge
Eg (eV)
ni @ 300 K
1.42
1.12
0.66
2×106 cm-3
1×1010 cm-3
2×1013 cm-3
It should be noted that the energy gap and the
intrinsic carrier concentration are functions of
Temperature.
14-Dec-18
Semiconductors
9
Energy Band Model of Semiconductor Crystals
15
10
Ge
13
10
Energy Band gap (eV)
-3
INTRINSIC CARRIER CONCENTRATION (cm )
Dependence of the Intrinsic Concentration and Energy Bandgap on Temperature
Si
11
10
GaAs
9
10
7
10
5
10
200
300
400
500
600
TEMPERATURE (K)
TEMPERATURE (K)
14-Dec-18
Semiconductors
10
Energy Band Model of Semiconductor Crystals
Extrinsic Semiconductors
The energy required for the extra electron in a donor atom (or for the extra hole in an
acceptor atom) to escape from the atom is called the Binding (Ionization) Energy .
The donor and the acceptor binding energies are expected to be relatively smaller than
the band gap energy.
Host
Material
Binding Energy for Donors (eV)
P
As
Sb
Phosphorus
Arsenic
Antimony
Binding Energy for Acceptors (eV)
B
Al
Ga
In
Boron
Aluminum
Gallium
Indium
Si
.045
.049
.039
.045
.057
.065
.016
Ge
.012
.013
.010
.010
.010
.011
.011
The effect of the dopants is therefore to
create impurity levels (dotted lines in the
next figure) that lie very close to the
conduction or valence band edge.
Donor level
Acceptor level
Ev
14-Dec-18
Semiconductors
Ec
Ec
Ev
11
Energy Band Model of Semiconductor Crystals
Effect of Temperature Variation
At zero temperature none of the dopants are ionized.
For n-type, with increasing temperature electrons are excited into the conduction band
from the donor levels. Ionization of the dopants is usually complete at room temperature .
At higher temperatures, more carriers are excited across the energy gap from the host
material and these eventually swamp the carriers provided by the dopants.
The extrinsic semiconductor then looks like
an intrinsic one.
Electron Concentration, n (cm-3)
n-type Semiconductor
Ec
Ev
Zero
Temperature
Low
Temperatures
Room
Temperature
3×1015
2×1015
1×1015
High
Temperatures
0
14-Dec-18
Semiconductors
12
Semiconductor Density of States
• We would now like to provide a quantitative description of the occupation of electron and
hole states in semiconductors.
• The first quantity we need to introduce is the Density of States in the valence and the
conduction band .
• The density of states is defined as the number of states per unit energy per unit
volume that is available at a certain energy E. (review part 3 for more details)
Z (E ) =
1
4π 2
 2m 
 ℏ 2 
3/ 2
E 1/ 2
• Now for an electron in a certain band of a semiconductor material, if we regard the
electron as free electron with an effective mass m*, the semiconductor density of
states can be defined as follows.
14-Dec-18
Semiconductors
13
Semiconductor Density of States
Nsc(E)dE is the number of conduction band states per unit volume lying in the energy
range between E and E+dE.
Nsv(E)dE is the number of valence band states per unit volume lying in the energy range
between E and E+dE.
Z (E ) =
E
1
4π 2
 2m 
 ℏ 2 
Ec
N sc ( E ≥ E c ) = 2×
ENERGY
GAP Eg
3/2
1
4π 2
E 1/2
FREE ELECTRON within an infinite well
 2m n* 
 ℏ2 


3/2
( E − E c )1/2
(1)
CONDUCTION BAND
Ev
N sv ( E ≤ Ev ) = 2 ×
1
4π 2
Ns(E)
 2m p* 
 2 
 ℏ 
3/2
( Ev − E )1/2
(2)
VALENCE BAND
• Note that the positive energy axis for the electrons in the conduction band is directed upward while for the
holes in the valence band is directed downward.
• Also note that, the density of states here is twice that in part III. The reason is that, one energy state
accommodates two electrons with opposite spin. The electron spin represents the direction of rotation of the
electron about its axis of symmetry.
14-Dec-18
Semiconductors
14
Fermi-Dirac Distribution Function
While the density of states gives information of the number of states that exist in a given
energy range, it does not tell us how many of these states are occupied by carriers.
To determine the number of occupied states in a given energy range, we must first
introduce the Fermi-Dirac Function f(E).
This function gives the probability that a state at given energy E is occupied by an
electron at a particular temperature T .
f (E) =
1
1 + exp [ ( E − EF ) / k T ]
(3)
EF is called the Fermi energy or the Fermi level
k is the Boltzmann constant = 1.38×10-23 J/K
T is the temperature in K
By definition, 1– f(E) gives the probability that a state at a given energy E is
not occupied by an electron. In other words 1– f(E) gives the probability that
a state at given energy E is occupied by a hole.
14-Dec-18
Semiconductors
15
Fermi-Dirac Distribution Function
• Since the Fermi function represents a probability function it takes values
between 0 and 1.
• The significance of Ef (Fermi energy or Fermi level ) is revealed by
considering the energy dependence of the Fermi function at absolute zero.
f (E) =
1
1 + exp [ ( E − EF ) / k T ]
f(E)
1
T = 0 & E < EF : exp[( E − EF ) / k T ] = 0 ∴ f ( E ) = 1
T = 0 & E > EF : exp[( E − EF ) / k T ] = ∞ ∴ f ( E ) = 0
0
EF
E
From this figure we easily see that the Fermi level represents the boundary between
filled and empty electron states at absolute zero.
14-Dec-18
Semiconductors
16
Fermi-Dirac Distribution Function
• In the next figure the form of the Fermi function is plotted for three different
temperatures.
• For T > 0 K, the function shows that
initially occupied states just below the
Fermi level will be empty as electrons
move to occupy states located just
above the Fermi level.
The graph of f(E) is symmetric
about Ef
by definition of the Fermi function
f(Ef ) = ½ for all T > 0 K
• With increasing temperature, the value of f(E) below the Fermi level decreases while
the value above it increases indicating that electrons move to occupy higher energy
states.
• Typically we may assume that at energies below Ef - 3kT all electron states are
occupied, while at energies greater than Ef + 3kT all electron states are empty.
14-Dec-18
Semiconductors
17
Fermi-Dirac Distribution Function
Example:
In a system with a Fermi energy of 8 eV, what is the probability that an electron will
occupy a state at an energy of 8.1 eV at temperatures of 0 K, 300 K and 600 K?
f (8.1 eV, 0 K) = 0 since
f (8.1 eV, 300 K) =
f (8.1 eV, 600 K) =
8.1 eV > EF
1
exp (8.1 − 8.0) ×1.6 ×10−19 / (1.38 ×10−23 × 300)  + 1
1
exp (8.1 − 8.0) ×1.6 ×10−19 / (1.38 ×10−23 × 600)  + 1
≈ 0.02
≈ 0.13
Note how doubling the temperature from 300 k to 600 k increases the probability
of occupation of the state at 8.1 eV by a factor of six. This is a consequence of the
exponential dependence of f(E) on temperature.
ASSIGNEMENT
Prove that the probability of finding an electron at an energy level E that is ઢ۳
below ࡱࢌ equals the probability of finding a hole at an energy level E that is ઢ۳
above ࡱࢌ .
14-Dec-18
Semiconductors
18
Electron and Hole Concentrations
• The density of states gives the number of available states at an energy level E, while the
Fermi function gives the probability that these states are occupied at a temperature T.
•
By multiplying these two quantities, we obtain the number of states at energy E that are
actually occupied by electrons at that particular temperature.
• These distributions depend on the position of the Fermi level in the semiconductor. The
next figure depicts intrinsic semiconductor case.
⇒ the number of free electrons and holes is identical in such materials.
E
E
E
HOLES
Ec
ELECTRONS
Ec
Nsc(E)
f(E)
EF
Energy band
structure
14-Dec-18
ELECTRONS
Ev
Nsv(E)
Electron and hole
densities of states
Electron and hole
Fermi-Dirac functions
Semiconductors
EF
Ev
HOLES
Electron and hole
state occupation
19
Electron and Hole Concentrations
• For n-type semiconductors, electrons are provided by the donors and by the host
material, while holes are only provided by the host.
This implies that the effect of doping with donors is to push the Fermi level up,
towards the conduction band.
E
HOLES
Ec
EF
Energy band
structure
14-Dec-18
ELECTRONS
Ev
f(E)
Nsc(E)
Nsv(E)
Electron and hole
densities of states
Electron and hole
Fermi-Dirac functions
Semiconductors
Ec
EF
Ev
Electron and hole
state occupation
20
Electron and Hole Concentrations
• For p-type semiconductors, electrons are provided by the acceptors and by the host
material while electrons are only provided by the host.
This implies that the effect of doping with acceptors is to push the Fermi level
down, towards the valence band.
E
HOLES
Ec
Ec
NSc(E)
Energy band
structure
14-Dec-18
f(E)
NSv(E)
ELECTRONS
EF
Ev
Electron and hole
densities of states
Electron and hole
Fermi-Dirac functions
Semiconductors
EF
Ev
Electron and hole
state occupation
21
Electron and Hole Concentrations
The number of electrons per cubic centimeter in the entire conduction band is
n=
∫
Ectop
Ec
N sc ( E ) f ( E )dE
(4)
1  2 m e* 
n =


2π 2  ℏ 2 
From Eqs. (1), (3) and (4):
3/ 2
∫
∞
Ec
E − Ec
1 + e[
( E − E F ) / kT ]
dE
(5)
Resetting the upper limit of integration to infinity is acceptable because of the rapid
fall of f(E) with increasing E.
In order to integrate Eq.(5) and obtain a closed form for electron density, we are often
forced to make some approximations:
We assume that the Fermi level is positioned such that for all energies in the conduction
band E - EF > 3kT i.e. exp[(E – EF)/3kT] >> 1
(Nondegenerate Semiconductor)
1  2me* 
n=


2π 2  ℏ 2 
3/2
1  2me* 
= 2 2 
2π  ℏ 
3/2
14-Dec-18
∫
e
∞
Ec
E − Ec e
−[( Ec − EF )/ kT ]
∫
−[ ( E − EF )/ kT ]
∞
Ec
dE
E − Ec e
−[( E − Ec )/ kT ]
Semiconductors
dE
(6)
22
Electron and Hole Concentrations
By making the substitution x = (E - Ec)/kT the integral in Eq.(6) then reduces to
1  2me* 
n= 2 2 
2π  ℏ 
n = Nc e
3/2
e
−[ ( Ec − EF )/ kT ]
−[( Ec − EF )/ kT ]
,
∫
∞
0
x e − x dx
 2π mn* kT 
Nc = 2 

2
h


∞
Gama function ∫0
vbottom
−[( EF − Ev )/ kT ]
,
3/2
The factors Nc and Nv in Eqs
7 and 8 are referred to as the
Effective Density of States in
the conduction and valence
bands respectively.
(8)
• Eqs. (7)&(8) are valid for intrinsic and extrinsic semiconductors.
Ec
3kT
• Eqs. (7)&(8) are Only valid if the Fermi level lies more
than 3kT away from either band edge.
When this condition is violated however the semiconductor is
said to be DEGENERATE and Eqs. (7) & (8) No Longer hold.
14-Dec-18
2
(7)
v
p = Nv e
π
3/2
A similar analysis may also be performed for the hole
E
concentration in the valence band using: p = ∫E Nsv(E)[1− f (E)] dE
 2π m*p kT 
Nv = 2 

2
h


xe − x dx =
Semiconductors
DEGENERATE
NON-DEGENERATE
REGION
3kT
DEGENERATE
Ev
23
Electron and Hole Concentrations
In an Intrinsic semiconductor (p = n = ni), use Ei to denote the position of the Fermi
Level in the intrinsic semiconductor. We may reorganize Eqs. (7) & (8) to be:
ni = N c e −[( Ec − Ei )/ kT ]
⇒
N c = ni e[( Ec − Ei )/ kT ]
(9)
ni = N v e −[( Ei − Ev / kT ]
⇒
N v = ni e −[( Ei − Ev / kT ]
(10)
By substituting Eqs. (9)&(10) into Eqs. (7)&(8), we obtain two Very Important relations
n = ni e[( EF − Ei )/ kT ]
(11)
p = ni e[( Ei − EF )/ kT ]
(12)
Equations (11) and (12) like Equations (7)
and (8) are valid for both intrinsic and
nondegenerate extrinsic semiconductors.
Multiplying Eqs. (11)&(12) leads to
np = ni2
14-Dec-18
(13)
Mass-Action Law
Semiconductors
24
Energy Band Model of Semiconductor Crystals
ASSIGNMENT
The next figure depicts the
electron concentration for two ntype materials 1 and 2.
1. Estimate
the
donor
concentration
for
both
materials.
2. If the two materials are Si and
Ge, material 2 is ………
3. Estimate the minority carrier
concertation for material 1 at
300 K and 500 K.
14-Dec-18
2
1
Semiconductors
25