1. Semiconductor Materials Revision
1.1 Introduction
Our understanding of materials, in particular conductors and
semiconductors, relies on the Band Theory of Solids, which means that
solids contain energy bands, each being a continuum of energy levels,
which may be occupied by an electron orbiting the nucleus of an atom.
Of primary interest are the outermost conduction band and the valence
band located a small energy gap above this.
1.2 Fermi Dirac Distribution and Fermi Energy Level
The behaviour of electron and hole concentrations is complex but is
described in simplified terms by the Fermi-Dirac Probability Function.
This function indicates the probability that any particular energy state
is occupied by an electron. The probability function applies to allowable
energy levels in the case of energy bands. The Fermi Dirac function is
given as:
f(E) 
1
1  e(E-EF )/kT
where E is the energy of a particular state, T is temperature in degrees
Kelvin and k = 1.38 x 10-23 J/K is Boltzmann’s Constant.
The value of energy EF is known as the Fermi Energy, or Fermi Level,
and by definition is given for a conducting metal as the highest energy
level which is occupied by an electron at absolute zero temperature of
0K.
The Fermi-Dirac function is shown plotted in Fig. 1.1. At absolute
zero temperature of 0K, no electrons have any energy due to thermal
agitation and consequently occupy all of the lowest energy levels
available as shown for T=0K. All of the energy levels below the Fermi
level are occupied and so the probability function has a value of 1 here.
By the same token, all of the available energy levels above the Fermi
level are unoccupied or empty and consequently the probability function
has a value of 0 here. Therefore the function has the rectangular shape
shown in Fig 1.1 for T=0K.
When the temperature is raised above 0K, some electrons will gain
thermal energy and will be able to occupy higher energy states greater
than EF. These electrons will leave vacant energy levels behind below
the Fermi level. Consequently, the curve spreads symmetrically on
either side of the Fermi level as shown in Fig. 1.1 for T>0K.
1
Prob
1
T = 0K
1- f(E)
1/2
T > 0K
f(E)
0
EF
Fig. 1.1
E
The Fermi-Dirac Probability Function
This function can also be applied to semiconductor materials in order to
calculate the concentration of charge carriers present in the energy
bands. In the case of intrinsic Silicon the free electrons and holes are
generated in pairs when electrons jump from states in the valence band
to states in the conduction band. For each electron in the conduction
band above the Fermi energy, there is a vacated state in the valence
band an equal amount below the Fermi energy, which has the associated
positive charge of the hole. Hence, the curve of the distribution function
is symmetrical on either side of the Fermi level with f(EF) = ½. The
concentration of free mobile electrons in the conduction band can be
established from the Fermi Dirac function as:
f(E) =
1
for
(E-EF )/kT
1+ e
EC ≤ E ≤ ∞
which is shown in blue in Fig. 1.1.
In addition, the concentration of holes in the valence band can equally
be established as:
1 - f(E) = 1 -
1
(E-EF )/kT
1+ e
for
0 ≤ E ≤ EV
which is shown in red in Fig. 1.1 and is the difference between unity and
the probability function for electron occupancy. Hence for intrinsic
Silicon, the Fermi level is located mid-way between the conduction and
valence bands as can be seen in Fig. 1.2, which shows only the hole and
electron free mobile charge carriers in the energy bands.
2
1.3 Intrinsic Silicon
Intrinsic Silicon material is un-doped Silicon with a homogeneous
crystalline structure. At absolute zero temperature 0K, all of the
covalent bonds formed in the valence band are intact and there are no
electrons present in the conduction band. At room temperature of 300K
(27OC), atoms undergo thermal agitation and a significant number of
electrons in the valence band gain sufficient energy to jump the energy
gap to the conduction band. These electrons can move around easily in
the conduction band as there are many vacant energy levels for them to
transfer between, i.e. they become free mobile charge carriers.
Electrons which have jumped to the conduction band leave behind
empty energy states in the valence band. These empty energy states are
associated with atoms which now have one more proton in the nucleus
than the number of electrons closely bound to it in the atomic structure.
This means that there is a net positive charge associated with the empty
state known as a hole in the valence band for each atom in which a
covalent bond has been broken and an electron freed.
Electrons in the conduction band can move with ease between the
plentiful vacant states available here. This gives rise to an electron
current. In the valence band, on the other hand, it is possible for an
electron to move from a neighbouring atom to occupy the hole or vacant
state in an atom. This is in effect the transfer of positive charge in the
opposite direction. This means the current flowing in the valence band
is composed essentially of positive charge carriers and is therefore a
hole current. The total current then consists of both positive and
negative charges and the technology is referred to as ‘bipolar’.
In intrinsic or un-doped Silicon at room temperature, holes and
free electrons are created in pairs. Some of the free electrons in the
conduction band will collide with each other and with other atoms in the
lattice and consequently lose some of their energy and return to the
valence band. When this occurs they recombine with a vacant state here
and neutralise the charge of a proton in reforming a covalent bond.
There is always the same number of holes in the valence band as there
is of free electrons in the conduction band. That is, they have the same
concentrations in Intrinsic Silicon. Thermal equilibrium is reached when
the rate of generation of free mobile electrons is equal to the rate of
recombination. If the concentration of electrons is denoted n0 and that
of holes as p0, then:
10
n0 =p0 =ni =1.5 x 10
cm-3
and the Law of Mass Action gives:
20
n0 =p0 =n2
i =2.25 x 10
3
E
Intrinsic
Silicon
conduction band
e- e- e-
eE c conduction band edge
energy
gap
E F Fermi Energy level
Eg
+
h
+
h
-
e
-
e
+
h
E v valence band edge
h+
valence band
0
1/2
1
f(E)
E
n-type
Silicon
conduction band
Ec
Ed
Donor level
Fermi level
EF
EV
e- h+
0
valence band
1
1/2
f(E)
E
conduction band
p-type
Silicon
EC
EF
Ea
EV
Fermi level
Acceptor level
-
e h+
0
Fig. 1.2
1/2
valence band
1
f(E)
Mobile Carrier Concentrations in Intrinsic and Doped Silicon
4
1.4 n-Type Doped Silicon
Intrinsic Silicon can be made n-type by doping the Silicon with a
Group V material such as Phosphorous which has an atomic number of
15 and has 5 electrons in its outer sub-shell, one more than Silicon. The
concentration of dopant atoms is usually between 1013 and 1018 cm-3,
which is much lower than the density of the Silicon atoms themselves
of 5 × 1022 cm-3.
In n-type material the dopant atoms readily enter into covalent
bonding with the neighbouring Silicon atoms but have an additional
electron available that cannot form a covalent bond. The extra free
electron originally occupies the donor energy level, Ed, in the dopant
atom and this energy level is located just below the conduction band,
EC, of the Silicon. At room temperature, practically all of the surplus
electrons of the impurity atoms gain sufficient energy to leave the donor
level and enter the conduction band of the Silicon lattice.
The Law of Mass Action still applies to the n-type doped material
and n0p0=ni2. However, the number of electrons in the conduction band
in n-type Silicon due to doping far exceeds those generated by thermal
agitation so that the electron concentration in the conduction band can
be taken as equal to the doping concentration of n-type impurities
present, i.e. n0 ≈ Nd . Moreover, the increased concentration of electrons
in the conduction band also increases the rate of carrier recombination
with the thermally generated holes in the valence band of the doped
material. Hence, the concentration of holes in the valence band of ntype material falls and is much lower than is the case for un-doped
material. The carrier concentrations in n-type material are given as:
n0 p0 = n2i
n0 ≈ Nd >> ni
p0 =
n2i
<< ni
Nd
majority
carriers
minority
carriers
When the Fermi-Dirac distribution is applied to doped, n-type
material the electron concentration is higher in the conduction band so
the distribution function, f(E), can be seen to shift upwards as can be
seen in Fig. 1.2, where the electron concentration is shown in blue.
Consequently, the function 1–f(E) becomes reduced in the valence band,
as can be seen in red in Fig. 1.2. The result of this is that the Fermi level
in n-type Silicon shifts upwards from its position in intrinsic Silicon and
appears close to the donor energy level.
5
1.5 p-type Doped Silicon
Intrinsic Silicon material can be made p-type by doping the Silicon
with a Group III material such as Aluminium having an atomic number
of 13 and has only 3 electrons in its outer sub-shell, one less than
Silicon. The concentration of dopant atoms lies between 1013 and 1018
cm-3, much lower than the density of the Silicon atoms of 5 × 1022 cm-1.
In p-type material the dopant atoms readily enter into covalent
bonding with the neighbouring Silicon atoms but find themselves with
one electron less than the number needed to form a complete set of
covalent bonds in its outer sub-shell. The dopant atom has one vacant
energy level in its outer shell, referred to as an acceptor level, Ea, and
this energy level is located just above the valance band, EV, of the
Silicon. At room temperature, electrons in the valence band of the
Silicon easily gain sufficient energy to leave the valence band and
occupy practically all of the dopant acceptor levels. Hence many
covalent bonds in the Silicon lattice are broken and electrons move
freely between vacant states in the valence band. Hence, positively
charged holes are generated in the valence band and become mobile.
The Law of Mass Action also applies to the p-type doped material
and n0p0=ni2. The number of holes in the valence band in p-type Silicon
due to doping far exceeds those generated thermally so that the hole
concentration in the valence band can be taken as the concentration of
p-type impurities present, i.e. p0 ≈ Na . The increased concentration of
holes in the valence band also increases the rate of carrier
recombination with the thermally generated electrons in the conduction
band. Hence, the concentration of electrons in the conduction band of
p-type material is much lower than is the case for un-doped material.
The carrier concentrations for p-type material are given as:
n0 p0 = n2i
p0 ≈ Na >> ni
n0 =
n2i
<< ni
Na
majority
carriers
minority
carriers
When the Fermi-Dirac distribution is applied to p-type material the
electron concentration is lower in the conduction band so the
distribution function, f(E), shifts downwards as seen in blue in Fig. 1.2.
Consequently, the function 1–f(E) reflects the increased concentration
of holes in the valence band, as seen in red in Fig. 1.2. The result of this
is that the Fermi level in p-type Silicon shifts downwards from its
position in intrinsic Silicon and appears close to the acceptor energy
level.
6
1.6 Drift Current
When an electric field is applied across a semiconductor, by way
of a battery or power supply for example, an electrostatic force is
exerted on the charge carriers in the material. The carriers, as a result
of the force they experience in the field, gain electrostatic energy. This
gives rise to a gradient in the energy bands occupied by the carriers in
the direction of the electric field as shown in Fig. 1.3 as well as a
gradient of potential across the material.
conduction band
V
eE
qV
C
h+
semiconductor
+
E
EV
+
-
direction of electric field
Fig. 1.3
valence band
E
direction of electric field
-
Effect of an Electric Field on Energy Bands
Since electrons seek to occupy the lowest energy status available,
they will tend to move from locations of higher electrostatic energy to
locations of lower energy. This gives rise to a net movement of carriers
through the material and hence sets up a flow of current. Electrons
move in a direction opposite to the electric field while holes move in the
direction of the field. When charge carriers drift in a uniform manner
through a piece of homogeneous semiconductor material, there is an
average sustained uniform flux of charge through the material. The
average charge flow considered per unit area normal to the direction of
flow is referred to as the Charge Flux Density as seen in Fig. 1.4 below.
electric
E
field
Jn drift  nqμnE
A
charge
flux
density
Jp drift  pqμpE
for electrons
for holes
J
Fig. 1.4
Charge Flux Density in an Electric Field
7
Electrons and holes drift in opposite directions in any given electric field.
However, since the charge on the electron is negative, both electrons
and holes make positive contributions to conventional current specified
as being positive in the direction of the electric field. When both flux
components are integrated over the area, A, of the semiconductor, the
Total Drift Current is obtained as:
Idrift  Jn drift A  Jp drift A  AEq(nμn  pμp )
where:
n and p are the free charge carrier concentrations,
q is the magnitude of the charge on the carriers,
μn , μp are the respective carrier mobilities and
E is the electric field strength.
1.7 Diffusion Current
If the concentration of free charge carriers at a particular location
in a semiconductor is higher than at other surrounding locations, then
electrostatic forces cause carriers to migrate away from the regions of
higher concentration towards regions of lower concentration. This
means that the concentration at the higher point falls while in the
surrounding regions it rises, thereby establishing a concentration
gradient as it does so. This process is known as diffusion and gives rise
to a diffusion current. Fig. 1.5 shows the corresponding change in the
concentration gradient with distance from the source and its variation
with time. Note that the movement of carriers is always in the direction
of decreasing concentration.
n
n
t = t 2 > t1
t = t1
Carrier flow
Carrier flow
x
Fig. 1.5
x
Change in Concentration Gradient with Diffusion
8
The diffusion of carriers will continue as long as the concentration
gradient is maintained and will cease if the gradient eventually becomes
zero. The rate at which diffusion takes place determines the associated
charge flux density which depends on:
electrons
holes
dn
dx
dp
dx
the carrier concentration gradients,
q
q
the charge on the carriers,
Dn
Dp
the diffusion coefficients.
The charge flux density for diffusion is then:
Jndiff  -q .Dn . -
dn

dx
Jpdiff   q .Dp . -
dp
dp
 -qDp
dx
dx
qDn
dn
dx
for electrons
for holes
The negative signs in front of the concentration gradients above
account for the fact that the carriers move in the direction of decreasing
concentration. Note that if the concentration gradients for both holes
and electrons are in the same direction, then both types of carrier will
be travelling in the same direction and the charge flows will tend to
neutralize each other. If, on the other hand, the gradients are in
opposite directions, then the flows of charge of opposite polarity will
combine and add together.
For a piece of homogeneous semiconductor of uniform crosssectional area, A, the Total Diffusion Current due to both electrons and
holes in the positive x direction is given as:
Idiff  Jn diff A  Jp diff A  Aq(D n
dn
dp
 Dp
)
dx
dx
The polarity of the gradients will determine the direction of flow of
individual carriers and hence the direction of the overall diffusion
current.
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1.8 Carrier Injection, Diffusion and Concentration Gradient
Consider a length of homogeneous p-type semiconductor of uniform
cross-sectional area, A, as shown in Fig. 1.6. Under equilibrium
conditions, the concentration of majority carriers is determined by the
doping concentration, p0 = Na, and the concentration of minority carriers
governed by the Law of Mass Action, n0 = ni2/Na. If now, additional or
‘excess’ minority charge carriers are injected into the semiconductor
from an external source, this equilibrium is disturbed as the excess
minority carriers injected begin to recombine with the majority carriers
present within the material. This gives rise to changes in the charge
concentration within the material. The rate of change of carrier
concentration depends on the rate of injection of excess carriers and the
rate of recombination in the volume. For minority carrier electrons:
Total Carrier Concentration: n
Equilibrium Carrier Concentration: n0
Excess Carrier Concentration: n’
Excess Carrier Concentration is:
excess minority
carriers recombine
with majority
carriers inside the
volume of the
semiconductor
n’ = n – n0
A
Jn (x)
uniform
cross
sectional
area
charge flux density
becomes a function
of distance into the
material
excess minority
carriers injected
from an external
source
Fig 1.6
Excess Charge Carriers Injected into a Semiconductor
10
If there is no electric field present across the semiconductor and the
material is homogeneous or uniform, then the injected minority carriers
will move into the material in the x direction by diffusion. As minority
carriers diffuse into the doped semiconductor some will recombine with
the majority carriers in the material. If the supply of excess minority
carriers into the material can be maintained by some external means
and the majority carriers which recombine can also be replaced, then
this will support the recombination process and eventually a new
steady-state condition will become established and the concentration of
charge within the volume will reach a stable value. The excess minority
concentration profile with distance x into the semiconductor is then
described as:
 Lx
n'(x)  n'(x  0)e
n
where n’(x = 0) is the quantity of excess minority carriers injected into
the material at x = 0.
The excess injected minority carriers will penetrate a distance Ln
on average into the material before recombining. This distance is
equivalent to the depth of penetration which would take place if the
concentration gradient at the boundary face of the material persisted
into it. The diffusion length is related to the average carrier lifetime
through the diffusion coefficient so that for electrons:
Dn  n  L n
2
The profile of the minority carrier concentration across the material into
which excess carriers are injected therefore has an exponential form as
shown in Fig. 1.7.
n (x)
n(x = 0)
excess
carriers
n’(x = 0)
thermal n
carriers 0
0
Ln
distance x
Fig. 1.7 Profile of Minority Carrier Concentration across Semiconductor
11