Slide set 4

Electrons and Holes For T>0

some electrons in the valence band receive
enough thermal energy to be excited
across the band gap to the conduction
band.
The result is a material with some electrons
in an otherwise empty conduction band and
some unoccupied states in an otherwise
filled valence band.
An empty state in the valence band is
referred to as a hole.
Electron-hole pairs in a semiconductor. If the conduction band electron and the
The bottom of the conduction band hole are created by the excitation of a
denotes as Ec and the top of the valence valence band electron to the conduction
band denotes as Ev.
band, they are called an electron-hole pair
(EHP).
•
•
Transport in valence band
Transport in valence band
Effective Mass
 Intrinsic Material
A perfect semiconductor crystal with no impurities or lattice defects is called an
intrinsic semiconductor.

At T=0 K – At T>0
No charge carriers Electron-hole pairs are generated
Valence band is filled with electrons EHPs are the only charge carriers in
Conduction band is empty intrinsic material

Since electron and holes are created in

pairs – the electron concentration in
conduction band, n (electron/cm3) is
equal to the concentration of holes in the
valence band, p (holes/cm3).
Each of these intrinsic carrier
concentrations is denoted ni.
Thus for intrinsic materials n=p=ni

Electron-hole pairs in the covalent bonding

model in the Si crystal.
 Intrinsic Material
• At a given temperature there is a certain concentration of electron-hole pairs ni. If
a steady state carrier concentration is maintained, there must be recombination of
EHPs at the same rate at which they are generated. Recombination occurs when
an electron in the conduction band makes a transition to an empty state (hole) in
the valence band, thus annihilating the pair. If we denote the generation rate of
EHPs as gi (EHP/cm3·s) and the recombination rate as ri, equilibrium requires that
r i = gi
• Each of these rates is temperature dependent. For example, gi(T) increases when
the temperature is raised, and a new carrier concentration ni is established such
that the higher recombination rate ri (T) just balances generation. At any
temperature, we can predict that the rate of recombination of electrons and holes ri,
is proportional to the equilibrium concentration of electrons n0 and the
concentration of holes p0:
r i = α r n 0 p 0 = α r n i2 = g i

• The factor αr is a constant of proportionality which depends on the particular

mechanism by which recombination takes place.
 Increasing conductivity
• The conductivity of the semiconductor material increases when the
temperature increases.
• This is because the application of heat makes it possible for some
electrons in the valence band to move to the conduction band.
• Obviously the more heat applied the higher the number of electrons that
can gain the required energy to make the conduction band transition and
become available as charge carriers.
• This is how temperature affects the carrier concentration.

• Another way to increase the number of charge carriers is to add them in

from an external source.
• Doping or implant is the term given to a process whereby one element is
injected with atoms of another element in order to change its properties.
• Semiconductors (Si or Ge) are typically doped with elements such as
Boron, Arsenic and Phosphorous to change and enhance their electrical
properties.
Increasing conductivity by temperature
As temperature increases, the number of free electrons and holes
created increases exponentially.

Therefore the conductivity of a semiconductor is influenced by

temperature
 Extrinsic Material
By doping, a crystal can be altered so that it
has a predominance of either electrons or
holes. Thus there are two types of doped
semiconductors, n-type (mostly electrons)
and p-type (mostly holes). When a crystal is
doped such that the equilibrium carrier
concentrations n0 and po are different from
the intrinsic carrier concentration ni, the
material is said to be extrinsic.
Extrinsic Material – donation of electrons
An impurity from column V introduces an
n-type material energy level very near the conduction band
in Ge or Si. This level is filled with electrons
at 0 K, and very little thermal energy is
required to excite these electrons to the
conduction band. Thus, at about 50-100 K
nearly all of the electrons in the impurity
level are "donated" to the conduction band.
Such an impurity level is called a donor level,
and the column V impurities in Ge or Si are
called donor impurities. From figure we note
that the material doped with donor
impurities can have a considerable
concentration of electrons in the conduction
band, even when the temperature is too low
for the intrinsic EHP concentration to be
Donation of electrons from
appreciable. Thus semiconductors doped
a donor level to the
with a significant number of donor atoms will
conduction band
have n0>>(ni,p0) at room temperature. This
is n-type material.
Extrinsic Material – acceptance of electrons
Atoms from column III (B, Al, Ga,
P-type material and In) introduce impurity levels in Ge
or Si near the valence band. These
levels are empty of electrons at 0 K.
At low temperatures, enough thermal
energy is available to excite
electrons from the valence band into
the impurity level, leaving behind holes
in the valence band. Since this type of
impurity level "accepts" electrons
from the valence band, it is called an
acceptor level, and the column III
impurities are acceptor impurities in
Acceptance of valence band Ge and Si. As figure indicates, doping
electrons by an acceptor level, and with acceptor impurities can create a
the resulting creation of holes. semiconductor with a hole
concentration p0 much greater than
the conduction band electron
concentration n0 (this is p-type
material).
 Calculation of binding energy
We can calculate rather simply the approximate energy required to
excite the fifth electron of a donor atom into the conduction band
(the donor binding energy) based on the Bohr model results:

• where m n* is the effective mass typical of semiconductors

• (m0 = 9.11x10-31 kg is the electronic rest mass) ,
• ), is a reduced Planck’s constant and

where εr is the relative dielectric constant of the semiconductor

material and ε0 = 8.85x10-12 F/m is the permittivity of free space.
 The Fermi level
Electrons in solids obey Fermi - Dirac statistics: The following consideration
are used in the development
of this statistics:
(4.6)
1. indistinguishability of the
electrons,

where k is Boltzmann’s constant 2. electron wave nature,

k=8.62ּ10-5 eV/K=1.38 10-23 J/K. 3. the Pauli exclusion principle.

The function f(E) called the Fermi-Dirac distribution function gives the probability that
an available energy state at E will be occupied by an electron at absolute temperature T.
The quantity EF is called the Fermi level, and it represents an important quantity in the
analysis of semiconductor behavior. For an energy E = EF the occupation probability is

(4.7)

This is the probability for electrons to occupy the Fermi level.

The Fermi – Dirac distribution function
At T=0K f(E) has rectangular shape
the denominator of the exponent is
1/(1+0)=1 when (E<Ef), exp. negative
1/(1+∞)-0 when (E>Ef), exp. positive

At 0 К every available energy state up to EF is filled

with electrons, and all states above EF are empty.
At temperatures higher than 0 K, some probability
f(E) exists for states above the Fermi level to be
filled with electrons and there is a corresponding
The Fermi – Dirac distribution
function for different temperatures
probability [1 - f(E)] that states below EF are empty.
The Fermi function is symmetrical about EF for all
temperatures. The probability exists for
state ΔE above EF is filled – f(EF+ ΔE)
state ΔE below EF is filled – [1- f(EF - ΔE)]

The symmetry of the distribution of empty and filled states about EF makes the Fermi level a
natural reference point in calculations of electron and hole concentrations in semiconductors.
In applying the Fermi-Dirac distribution to semiconductors, we must recall that f(E) is the
probability of occupancy of an available state at E. Thus if there is no available state at E (e.g.,
in the band gap of a semiconductor), there is no possibility of finding an electron there.
 Relation between f(E) and the band structure
Electron probability
tail f(E)

Hole probability
tail [1-f(E)]

In intrinsic material the Fermi level EF must lie at the middle of the band gap.
In n-type material the distribution function f(E) must lie above its intrinsic position on the energy scale. The energy
difference (Ec – EF) gives a measure of n.
For p-type material the Fermi level lies near the valence band such that the [1-f(E)] tail below Ev is larger than the
f(E) tail above Ec. The value of (EF – Ev) indicates how strongly p-type the material is.
The distribution function has values within the band gap between Eν and Ec, but there are no energy states
available, and no electron occupancy results from f(E) in this range.
 Electron and Hole Concentrations at Equilibrium
The Fermi distribution function can be used to calculate the concentrations of electrons
and holes in a semiconductor if the densities of available states in the valence and
conduction bands are known. The concentration of electrons in the conduction band is

(4.8)

where N(E)dE is the density of states (cm-3) in the energy range dE. The subscript 0 used
for the electron and hole concentration symbols (n0, p0) indicates equilibrium conditions.

The number of electrons per unit volume in the energy range dE is the product of the density of
states and the probability of occupancy f(E). Thus the total electron concentration is the integral
over the entire conduction band. The function N(E) can be calculated by using quantum
mechanics and the Pauli exclusion principle.
N(E) is proportional to E1/2, so the density of states in the conduction band increases with
electron energy. On the other hand, the Fermi function becomes extremely small for large
energies. The result is that the product f(E)N(E) decreases rapidly above Ec, and very few
electrons occupy energy states far above the conduction band edge.
Similarly, the probability of finding an empty state (hole) in the valence band [1 - f(E)]
decreases rapidly below Ev, and most holes occupy states near the top of the valence band.
Band diagram, density of states, Fermi-Dirac distribution,
and the carrier concentrations at thermal equilibrium

Intrinsic
semiconductor

n-type
semiconductor

p-type
semiconductor
The conduction band electron concentration is simply the effective density of states at Ec
times the probability of occupancy at Ec:
(4-9)
In this expression we assume the Fermi level EF lies at least several kT below the
conduction band. Then the exponential term is large compared with unity, and the
Fermi function f(Ec) can be simplified as
(4-10)

Since kT at room temperature is only 0.026 eV, this is generally a good approximation.
For this condition the concentration of electrons in the conduction band is
(4-11)

It can be shown that the effective density of states Nc is

(4-12)

Values of Nc can be tabulated as a function of temperature. As Eq. (4-11) indicates, the

electron concentration increases as EF moves closer to the conduction band.
By similar arguments, the concentration of holes in the valence band is
(4-13)
where Nv is the effective density of states in the valence band.
The probability of finding an empty state at Ev, is

(4-14)

for EF larger than Ev by several kT. From these equations, the concentration of holes
in the valence band is
(4-15)

The effective density of states in the valence band reduced to the band edge is
(4-16)

Eq. (4-15) predicts that the hole concentration increases as EF moves closer to the
valence band.
The electron and hole concentrations predicted by Eqs. (4-11) and (4-15) are valid
whether the material is intrinsic or doped, provided thermal equilibrium is maintained.
Thus for intrinsic material, EF lies at some intrinsic level Ei near the middle of the band
gap, and the intrinsic electron and hole concentrations are
(4-17)

,
 The product of n0 and p0 at equilibrium is a constant for a particular material and
temperature, even if the doping is varied:

(4-18a)

(4-18b)

In Eqns. (4-18a) and (4-18b) Eg = Ec – Ev. The intrinsic electron and hole
concentrations are equal (since the carriers are created in pairs), ni = pi ; thus the
intrinsic concentration is
(4-19)

The constant product of electron and hole concentrations in Eq. (4-18) can be
written conveniently as
(4-20)
This is an important relation, and we shall use it extensively in later calculations. The
intrinsic concentration for Si at room temperature is approximately ni = 1.5 x 1010 cm-3.
 Comparing Eqs. (4-17) and (4-19), we note that the intrinsic level Ei is the middle
of the band gap (Ec - Ei= Eg/2), if the effective densities of states Nc and Nv are equal.
There is usually some difference in effective mass for electrons and holes (e.g. for
Si – mn*=0.26m0, mn*=0.39m0), however, and, therefore, Nc and Nν are slightly different
as Eqs. (4-12) and (4-16) indicate.
Another convenient way of writing Eqs. (4-11) and (4-15) is

(4-21)

(4-22)

obtained by the application of Eq. (4-17). This form of the equations indicates
directly that the electron concentration is ni, when EF is at the intrinsic level Ei, and
that n0 increases exponentially as the Fermi level moves away from Ei toward the
conduction band. Similarly, the hole concentration p0 varies from ni, to larger values
as EF moves from Ei toward the valence band. Since these equations reveal the
qualitative features of carrier concentration so directly, they are particularly
convenient to remember.
Conductivity of Intrinsic and Extrinsic Semiconductors

For Si μn = 0.135 m2/Vs, μp = 0.048 m2/Vs;

for Ge μn = 0.39 m2/Vs, μp = 0.19 m2/Vs.
 Conductivity of Extrinsic Semiconductors
Typical carrier densities in intrinsic & extrinsic semiconductors
Si at 300K, intrinsic carrier density ni = 1.5 x 1016/m3
Extrinsic Si doped with As → typical concentration 1021atoms/m3:
Majority carriers n0 = 1021 e/m3 ; Mass action law: ni2 = n0p0
Minority carriers: p0 = (1.5×1016)2/1021 = 2.25 x 1011 holes/m3
Conductivity:
Majority carriers: σn = 1021x0.135x1.6x10-19 (e/m3 ) (m2 /Vs) (A⋅s C) =0.216 (Ω
cm)-1
Minority carriers: σp = 2.25x10-11 x 0.048 x1.6x10-19 = 0.173x10-10 (Ω cm)-1
Conductivity total σtotal = σn + σp ≈ 0.216 (Ω cm)-1
----------------------------------------------------------------------------------------------------------------------------
Conductivity of Intrinsic and Extrinsic Semiconductors:
Effect of Temperature

Illustrative Problem: calculate σ

of Si at room temperature (20 oC
→293 K) and at 150 oC →423 K).
 Modulators of conductivity

Just reviewed how conductivity of a semiconductor is affected by:

Temperature – Increasing temperature causes conductivity to
increase
• Dopants – Increasing the number of dopant atoms (implant dose)
cause conductivity to increase.
• Holes are slower than electrons therefore n-type material is
more conductive than p-type material.
• These parameters are in addition to those normally affecting
conducting material,

Cross sectional area 🡹 Resistance 🡹

Length 🡹 Resistance 🡹
 Silicon Resistivity Versus Dopant Concentration

1021

1020

Dopant Concentration (atoms/cm3)

1019

1018

1017
n-type p-type
16
10

1015

1014

101310-3 10-2 10-1 100 101 102 103

Electrical Resistivity (ohm-cm)

Redrawn from VLSI Fabrication Principles, Silicon and Gallium Arsenide, John Wiley & Sons, Inc.
 Summary
Intrinsic semiconductors

Doped semiconductors
n-type p-type
Carrier concentration vs inverse
Temperature
Compensated
semiconductors
pn Junction
The interface separating the n and p
regions is referred to as the
metallurgical junction.

For simplicity we will consider a step

junction in which the doping
concentration is uniform in each
region and there is an abrupt change
in doping at the junction.

Initially there is a very large density

gradient in both the electron and
hole concentrations. Majority carrier
electrons in the n region will begin
diffusing into the p region and
majority carrier holes in the p region
will begin diffusing into the n
region. If we assume there are no
external connections to the
semiconductor, then this diffusion
process cannot continue
indefinitely.
 pn Junction
This cannot occur in the case of the charged particles
in a p-n junction because of the development of space
charge and the electric field ε. As electrons diffuse from the n region,
positively charged donor atoms are left
behind. Similarly, as holes diffuse from the p
region, they uncover negatively charged
acceptor atoms. These are minority carriers.
The net positive and negative charges in the
n and p regions induce an electric field in the
region near the metallurgical junction, in the
direction from the positive to the negative
charge, or from the n to the p region.

The net positively and negatively charged regions

are shown in Figure. These two regions are
referred to as the space charge region (SCR).
Essentially all electrons and holes are swept out of
the space charge region by the electric field. Since
the space charge region is depleted of any mobile
charge, this region is also referred to as the
depletion region

Density gradients still exist in the majority carrier concentrations at each edge of the space charge
region. This produce a "diffusion force" that acts the electrons and holes at the edges of the space
charge region. The electric field in the SCR produces another force on the electrons and holes
which is in the opposite direction to the diffusion force for each type of particle. In thermal equilib
rium, the diffusion force and the E-field (ε) force exactly balance each other.
 pn Junction – built-in potential barrier
No applied voltage across pn-junction
The junction is in thermal equilibrium
—the Fermi energy level is constant
throughout the entire system. The
conduction and valence band energies
must bend as we go through the space
charge region, since the relative
position of the conduction and valence
bands with respect to the Fermi energy
changes between p and n regions.

Electrons in the conduction band of the n region see a potential barrier in trying to move
into the conduction band of the p region. This potential barrier is referred to as the built-in
potential barrier and is denoted by Vbi (or V0). The built-in potential barrier maintains
equilibrium between majority carrier electrons in the n region and minority carrier electrons
in the p region, and also between majority carrier holes in the p region and minority carrier
holes in the n region. The potential Vbi maintains equilibrium, so no current is produced by
this voltage.

The intrinsic Fermi level is equidistant from the conduction band edge through the junction,
thus the built-in potential barrier can be determined as the difference between the intrinsic
Fermi levels in the p and n regions.
pn Junction An applied voltage bias V appears across the
transition region of the junction rather than in the
neutral n and p region. Of course, there will be some
voltage drop in the neutral material, if a current flows
through it. But in most p-n junction devices, the
length of each region is small compared with its area,
and the doping is usually moderate to heavy; thus the
resistance is small in each neutral region, and only a
small voltage drop can be maintained outside the
space charge (transition) region. V consider to be
positive when the external bias is positive on the p
side relative to the n side.

The electrostatic potential barrier at the junction is

lowered by a forward bias Vf from the equilibrium
contact potential V0 to the smaller value V0-Vf. This
lowering of the potential barrier occurs because a
forward bias (p positive with respect to n) raises the
electrostatic potential on the p side relative to the n
side. For a reverse bias (V=-Vr ) the opposite occurs;
the electrostatic potential of the p side is depressed
relative to the n side, and the potential barrier at the
junction becomes larger (V0 + Vr ).
The electric field within the transition region can be
deduced from the potential barrier. We notice that the
field decreases with forward bias, since the applied
electric field opposes the buid-in field. With reverse
bias the field at the junction is increased by the
applied field, which is in the same direction as the
equilibrium field.
 Physics of the Depletion
Region
• When n and p type material are placed in contact with
each other, the electrons diffuses into the p-type region in
order to equalise the Fermi levels.

• This loss of electrons from the n-type material leaves the

surface layer positively charged.

• Similarly the p-type material will have a negatively

charged surface layer.

• Thus an electric field is established which opposes the

diffusion of electrons when the Fermi levels are equal
(dynamic equilibrium is established)
Compensated semiconductors