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CHARGE CARRIERS IN SEMICONDUCTORS
Objectives:
• Discovery of the nature of charge carriers in intrinsic and extrinsic
semiconductors
• Finding on what, how and why densities of charge carriers in semiconductors
depend
• Calculation methods of of charge carrier densities in semiconductors
Content:
Charge carriers in intrinsic semiconductors
Nature of charge carriers
Fermi level in an intrinsic semiconductor
Densities of carriers
Charge carriers in extrinsic semiconductors
n-type semiconductors
p-type semiconductors
Compensation doping
Excess carriers and lifetime
ELEKTRONIKOS PAGRINDAI 2008
VGTU EF ESK [email protected]
1
Nature of charge carriers in intrinsic semiconductors
Carefully refined semiconductors are called intrinsic semiconductors. .
In a silicon crystal each atom is surrounded by four neighbour atoms. At 0 K
all valence electrons take part in covalent bonding and none are free to move
through the crystal.
As the temperature increases, the lattice vibrations arise. Some of the energy
of the lattice vibrations is transferred to the valence electrons. If sufficient
energy is given to an electron, it leaves a bond and becomes free.
At very low temperatures the
valence band is full (filled to
capacity) and the conduction band
is completely empty.
Thus, at a very low temperature
(close to 0 K) the crystal behaves
as an insulator.
ELEKTRONIKOS PAGRINDAI 2008
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The jump of an electron from the
valence band to the conduction
band corresponds to the release
of an electron from covalent
bonding. The minimum energy
required for that is equal to the
width of the forbidden band.
If the width of the forbidden band is less, electrons are released at lower
temperature.
When an electron is released, a positively charged vacancy appears. This
vacancy may be considered as a positive hole.
According to the energy band diagram an uncompleted allowed energy level in
the valence band corresponds to a hole.
Nature of charge carriers in intrinsic semiconductors
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Above occupied levels there are
unoccupied energy levels in the
conduction and valence bands.
The situation is similar to that in
conductors: electrons are able to accept
energy, change their velocity and
participate in conductivity.
… a hole behaves as a positive charge
carrier having a positive charge equal in
magnitude to the electronic charge.
... In the intrinsic semiconductor the number of electrons in the
conduction band equals to the number of holes in the valence band.
Charge carriers appear as a result of charge carrier generation.
Positive holes attract negative electrons. If an electron is drawn into the bond, it
recombines with a hole.
The jump of an electron from the conduction band to the valence band
corresponds to the recombination process.
Nature of charge carriers in intrinsic semiconductors
ELEKTRONIKOS PAGRINDAI 2008
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4
Fermi level in an intrinsic semiconductor
n
pcvF lnk
4
3
2 m
mT
WWW +
+=
If the Fermi level is below the bottom of the conduction
band, it is possible to use the simplified formula
... The Maxwell-Boltzmann distribution function can be used for calculation of the
probability of occupation of energy levels in the conduction band.
The same conclusion can be made for holes in the valence band.
TWWWf
k/)(F
Fe)(−−
≅
The Fermi level in an intrinsic
semiconductor lays at the middle
of the forbidden band.
ELEKTRONIKOS PAGRINDAI 2008
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5
Densities of charge carriers in intrinsic semiconductors
We have derived the expression for density of electrons in a non-generate system.
Now it is necessary to take into account that conduction electrons exist in a crystal.
We must consider the effective mass:
( ) TWWTWWN
Tmn
k/)(c
k/)(
3
2/3n FcFc eeh
kπ22 −−−−==
Acting in a similar way we can find density or holes in the valence band.
( )TWWTWW
NTm
pk/)(
vk/)(
3
2/3p
vFvF eeh
kπ22−−−−
==
TWTWWNNNNnp k/vc
k/)(vc ee vc ∆−−−
==
2i
2iii pnpnnp ===
TWNNnppn k2/vcii e ∆−===
... The densities of electrons and
holes in an intrinsic semiconductor
are strongly dependent on
temperature and gap energy.
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The number of conduction electrons and holes increases rapidly with the
increase of temperature and decreases with the increase of the gap
energy.
( )Tba
T
WNNpn
11
k2ln2
1lnln vcii −≅−==
∆
k2tan
W∆α ≅
... Using the expressions for the densities of electrons and holes and taking into
account the condition n= p, it is possible to derive the formula for the Fermi
level in an intrinsic semiconductor.
Densities of charge carriers in intrinsic semiconductors
TWNNnppn k2/vcii e ∆−===
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Densities of charge carriers in intrinsic semiconductors. Problems
1. Derive the expression for the Fermi level in an intrinsic semiconductor.
2. Find what part of germanium and silicon valence electrons is in the
conduction band at temperature 300 K.
3. Find the ratio of carrier densities in germanium and silicon at room
temperature (T = 300 K).
4. Find how many times carrier density in the intrinsic germanium increases, if
the temperature increases from 20 to 1000C. Repeat the calculations for the
intrinsic silicon. Comment on the results.
ELEKTRONIKOS PAGRINDAI 2008
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8
Extrinsic (doped) semiconductors
The intrinsic carrier densities are very small and depend strongly on
temperature. In order to fabricate devices such as diodes or transistors, it is
necessary to increase the free electron or hole population. This is done
intentionally doping the semiconductor, i. e. adding specific impurities in
controlled amounts.
Doped semiconductor are called extrinsic semiconductors.
... The percentage of impurity in non-degenerate semiconductors must be
small (for example, about 10-5 % in the substrates for integrated circuits). Then
impurity atoms are isolated from each other by semiconductor atoms.
In order to have necessary conductivity type, donor and acceptor impurities are
used.
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Charge carriers in the semiconductors doped with donor impurities
The elements from the column V of the periodic table, e. g. phosphorus (P),
arsenic (As) and antimony (Sb) are added to an intrinsic elemental
semiconductor to modify the semiconductor into an n-type semiconductor.
If the fifth electron gets such energy, it leaves the
impurity atom and becomes free.
The impurity atom becomes ionized.
... An electron and a positive ion appear as a result
of ionization of a donor impurity atom.
Let us consider silicon doped with phosphorus.
Each pentavalent atom occupies a site usually occupied by a silicon atom. Four
valence electrons are in covalent bonds, the fifth electron rotates round the
positively charged impurity atom.
The binding energy of this extra electron is small. In the case of phosphorus in
silicon it is only about 0.044 eV.
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The free electron is in the conduction band. So
donor energy levels are in the forbidden band near
the bottom of the conduction band
At room temperature each donor impurity atom
donates an additional charge carrier – a negative
conduction electron. According to this
semiconductors that contain donor impurities are
called donor-type, electronic or n-type
semiconductors.
Because usually impurity density is small, donor
levels are isolated and are shown by the dashed line.
Charge carriers in the semiconductors doped with donor impurities
ELEKTRONIKOS PAGRINDAI 2008
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Charge carrier densities and Fermi level in extrinsic semiconductors strongly
depend on temperature and impurity density.
At 0 K all allowed energy levels in the valence band are filled by electrons. All
donor levels are filled by unbound electrons. The conduction band is free. So
charge carriers do not exist, and the semiconductor behaves as an insulator.
At 0 K the Fermi level is between the donor levels and the bottom of the
conduction band.
If the temperature increases, unbound
electrons obtain energy and jump to the
conduction band. So, in the low
temperature or impurity ionisation
range, density of conduction electrons
increases.
Charge carriers in the semiconductors doped with donor impurities
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At some temperature, that is sufficiently lower than 300 K, all impurity atoms
become ionised and the density of conduction electrons becomes equal to the
donor density Nd.
If the temperature further increases, in the wide
range the density of conduction electrons
remains constant. This temperature range is
called the extrinsic range.
dn Nn ≅ iin
in pp
n
np <<=
... In the n-type semiconductor electrons are the majority carriers and
holes are the minority carriers.
Charge carriers in the semiconductors doped with donor impurities
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In the extrinsic range
( )[ ] dFccn k/exp NTWWNn =−−=
d
ccF lnk
N
NTWW −=
... The Fermi level in the extrinsic range falls as the temperature
increases.
As the density of the intrinsic charge carriers increases with temperature, at
some temperature that is sufficiently higher than 300 K it becomes equal to Nd.
At higher temperatures thermally exited intrinsic carriers predominate. Then the
semiconductor obtains the properties of an intrinsic semiconductor.
In the high temperature or intrinsic range the Fermi level approaches the mid-
gap position.
Charge carriers in the semiconductors doped with donor impurities
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... In the impurity ionisation (low temperature) range the density of electrons increases with
temperature.
In the extrinsic range (at middle temperatures) the density of electrons is almost constant.
At last in the intrinsic range (at high temperatures) the intrinsic carriers predominate and their
density increases with temperature.
Plotting the variable 1/T along the x-axis and lnn and lnp along the y-axis, we obtain curves
that may be approximated by the segments of a straight line.
( ) 2/1dF =Wf 2/dNn ≅( )dc
dcs
/2lnk NN
WWT
−≅
di Nn =)/ln(k 2
dvc
iNNN
∆WT =
Charge carriers in the semiconductors doped with donor impurities
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... In the vicinity of room temperature the conduction electron density in the n-
type semiconductor is approximately constant and equals to the density of donor
impurity atoms. The hole density is much lower than the intrinsic density and is
strongly dependent upon temperature.
At normal temperature carrier densities and Fermi level
depend on impurity density.
... In the n-type semiconductor the Fermi level is always over the middle of the
forbidden band.
If the donor density is higher, the Fermi level is higher, closer to the bottom of
the conduction band; the electron density is higher, the hole density is lower.
If the donor concentration is lower, the Fermi level is lower, closer to the middle
of the forbidden band.
Charge carriers in the semiconductors doped with donor impurities
dn Nn ≅ iin
in pp
n
np <<=
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1. How many charges and how many charge carriers appear as a result of
ionization of semiconductor atom?
2. How many charges and how many charge carriers appear as a result of
donor atom ionization?
3. A germanium specimen is doped with phosphorus. Its density is 1016 cm–3.
Find densities of electrons and holes and position of the Fermi level at
temperature 300 K.
4. Silicon is doped with phosphorus. The Fermi level is in the forbidden band
at the distance of 0,044 eV from the bottom of the conduction band.
Impurity density is 1016 cm–3. Find the position of the Fermi level at
temperature 300 K.
Charge carriers in the semiconductors doped with donor impurities.
Problems
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Semiconductors with majority charge carrier density of holes are called p-type.
They can be produced adding impurities from column III of the periodic table, e. g.
boron (B), aluminium (Al), gallium (Ga) or indium (In) to intrinsic silicon or
germanium.
These p-type impurities are characterized by three valence electrons in their outer
shell.
Let us consider silicon doped with
boron.
The vacancy thus created by the
impurity is not a hole, since it is
bound to the atom.
At some temperature above 0 K the electron from a bond of a neighbouring parent
atom can fill the vacant site leaving a hole.
So dopants from column III accept electrons to create holes for conduction.
Therefore they are called acceptors.
Charge carriers in the semiconductors doped with acceptor impurities
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The energy required by a valence electron to fill the vacancy created by an
impurity atom and thus to create a hole is of the similar magnitude to the
ionisation energy of a donor atom. So acceptors create discrete acceptor levels
just above the top of the valence band.
At 0 K acceptor levels are free. Electrons occupy the valence band of the
semiconductor. There are no charge carriers. Semiconductor has properties of
insulator.
If the temperature increases, electrons jump from the valence band to the
acceptor levels leaving holes in the valence band. So in the impurity ionisation
range the density of holes increases with temperature.
Charge carriers in the semiconductors doped with acceptor impurities
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In the extrinsic range :
Density of electrons is small and increases with temperature:
ap Np ≅
a
2i
p
2i
pN
n
p
nn ≅=
As the temperature in the extrinsic range is
raised, the Fermi level increases.
a
vvF lnk
N
NTWW =−
At high temperatures the material becomes intrinsic and the Fermi level
approaches midway between the conduction and valence bands.
Charge carriers in the semiconductors doped with acceptor impurities
... Positive holes are the majority carriers and electrons are the minority carriers
in the p-type semiconductor.
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ap Np ≅a
2i
p
2i
pN
n
p
nn ≅=
In the extrinsic range:
a
vvF lnk
N
NTWW =−
Charge carriers in the semiconductors doped with acceptor impurities
1 – low temperature, impurity ionization range;
2 – middle temperature, extrinsic conductivity range;
3 – high temperature, intrinsic conductivity range
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At normal temperature, Fermi level and carrier densities
depend on the impurity density.
ap Np ≅ a2ip / Nnn ≅
... The Fermi level below the middle of the forbidden band is the
characteristic feature of the band model of the p-type semiconductor.
Charge carriers in the semiconductors doped with acceptor impurities
If acceptor density is higher, the Fermi level is lower,
closer to the top of the valence band; the hole density
is higher, the electron density is lower.
If the acceptor density is lower, the Fermi level is
higher, closer to the middle of the forbidden band.
a
vvF lnk
N
NTWW =−
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1. How many charges and how many charge carriers appear as a result of
acceptor ionization?
2. Silicon plate is doped by boron. Its density is 1016 cm–3. Find carrier
densities and Fermi level at temperature 300 K.
Charge carriers in the semiconductors doped with acceptor impurities.
Problems
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Compensation doping
When both acceptor and donor impurities are added simultaneously to an
intrinsic semiconductor, the compensation takes place.
At higher donor density the crystal is n-type
semiconductor since n-type impurity predominates.
The free carriers supplied by the less concentrated
dopant recombine with an equal number of carriers
of the opposite type. So some of the donor states
are cancelled by acceptor states..
The process is called compensation.
'' addn NNNn −≅= d2in / Nnp =
'' daap NNNp −≅= a2ip / Nnn =
At higher donor impurity density when ,'' AD NN >>
If
,'' DA NN >>If
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Compensation doping is widely used in manufacturing of semiconductor devices
and integrated circuits.
The compensation is possible, if impurity density is not very high. Then the
distance between impurity atoms is relatively great and impurity atoms cannot
interact.
If impurity density increases, the distance between impurity atoms decreases.
When impurity density in silicon becomes approximately 1019 cm-3, degeneration
of the semiconductor arises. Then impurity levels split and allowed bands
appear.
p-type material can be converted to the n-type and vice-versa, by the addition of
excess dopant atoms of the appropriate type.
If , the intrinsic charge carriers predominate. The semiconductor
has properties of the intrinsic compensated or near-fully compensated material.
The Fermi level lies near the middle of the forbidden gap.
ADi NNn −>
Compensation doping
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In the degenerate n-type semiconductor, conduction
and donor bands overlap and form the hybrid
conduction band. Energy levels at the bottom of the
hybrid conduction band are occupied by electrons.
The Fermi level is above the bottom of the
conduction band.
In the degenerate p-type semiconductor, we have
overlapping of the valence and acceptor bands.
Then energy levels at the top of the hybrid valence
band are not occupied by electrons. The Fermi level
is below the top of the valence band
W
Wv
Wc
WF
W
Wv
Wc
WF
Compensation doping
In both cases there are unoccupied allowed energy
levels over the levels that are filled by electrons. The
situation is very similar to that in conductors.
Therefore conduction is possible even at 0 K in
degenerate semiconductors and they are sometimes
called semimetals.
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Germanium is doped with phosphorus and boron. Their densities are
1017 cm–3 and 1016 cm-3, respectively. Find carrier densities and position of
the Fermi level at 300 K.
Compensation doping. Problem
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Excess carriers and lifetime
Under thermal equilibrium the generation rate and the recombination rate are
equal and the carrier densities and remain constant.
The equilibrium may be disturbed by light or carrier injection.
Excess electrons and holes are always equal in number:
nnn ∆+= 0 ppp ∆+= 0 .pn ∆=∆ .2inpn >
If the external excitation (activation) stops, the density of the excess carriers
reduces exponentially:
)/exp()/exp()()( 00 τ−∆=τ−∆=∆=∆ tptntptn
The recombination of the excess carriers
may be radiative or nonradiative, direct
or indirect, band-to-band or through
recombination centres and traps.
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.
,
0n0
0n0
ppp
nnn
∆∆+=
+=
τ
∆∆ n
t
n
t
n−== ...
d
)d(
d
d
τ
∆∆ p
t
p
t
p−== ...
d
)d(
d
d
In practical device design small lifetime may be desirable for high-speed
switching applications.
Nanosecond switching speed is realised using gold doping.
)/exp()/exp()()( 00 τ−∆=τ−∆=∆=∆ tptntptn
The lifetime τ represents the average time a carrier remains free before it
recombines. During the lifetime the number of excess carriers reduces e times.
The recombination rate of the excess carriers is dependent on the lifetime:
Excess carriers and lifetime
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29