Download 1 Energy bands in semiconductors

From MSc Thesis of Lars Gimmestad Johansen
1
Energy bands in semiconductors
Electrons in an isolated atom can only have discrete energy levels, but when
atoms are brought together as in crystalline solids, these degenerate energy
levels will split into many separated levels due to the atomic interaction. Because the levels are so closely separated, they may be treated as a continuous
band of allowed energy states. The two highest energy bands are the valence
band and the conduction band. These bands are separated by a region which
designates energies that the electrons in the solid cannot possess. This region
is called the forbidden gap, or bandgap Eg . This is the energy difference between the maximum valence band energy EV and the mimimum conduction
band energy EC .
For insulators the valence electrons form strong bonds between neighbouring
atoms. These bonds are difficult to break, and consequently there are no free
electrons to participate in current conduction.
Bonds between neighbouring atoms in a semiconductor are only moderately
strong. Therefore thermal vibrations may break some bonds. When a bond
is broken, an electron is injected from the valence band into the conduction
band. This is now a mobile negative charge carrier, and the atom from which
the electron emerges is left with a negative charge deficiency, i.e. a positive
net charge, also called a hole. The bandgap for Si at 300 K is 1.12 eV. A
characteristic property of semiconductors is that the bandgaps have a negative temperature coefficient. The bandgap in Si and GaAs as a function of
temperature can be described by:
4.73 × 10−4 T 2
T + 636
5.4 × 10−4 T 2
Eg (T ) = 1.52 −
T + 204
Eg (T ) = 1.17 −
for Si
(1)
for GaAs
(2)
as seen in figure 1, where the temperature T is expressed in Kelvin. At room
temperature (300 K) and under normal atmosphere, the values of the bangap
are 1.12 eV for Si and 1.42 eV for gallium arsenide.
In conductors such as metals, the conduction band is either partially filled
Figure 1: Bandgaps of Si and GaAs as a function of temperature.
(the Fermi energy level EF is located in the middle of the conduction band)
or overlaps the valence band so that there is no bandgap. As a consequence,
the uppermost electrons in the partially filled band or electrons at the top of
the valence band can move to the next-higher available energy level when they
gain kinetic energy (e.g. from an applied electric field). Therefore, current
conduction can readily occur in conductors.
2
The Effective Mass Approximation
For conduction electrons in a semiconductor, the electrons are relatively free
to move, but bound by the periodic potential of the lattice nuclei. Thus the
effective mass is different from that of a free electron. The energy-momentum
relationship of a conduction electron can be written as:
E=
p̄2
2mn
(3)
where p̄2 is the crystal momentum and mn is the electron effective mass. The
crystal momentum is analogous to the particle momentum, i.e. the particle
momentum along a given crystal direction as defined previously by the Miller
index.
For both Si and GaAs the maximum valence band energy of an electron occurs
at p̄ = 0. The minimum energy of the conduction band electron in GaAs also
occurs for p̄ = 0, hence a transition across the bandgap only requires an Eg
energy absorbtion or emission. Because of this, GaAs is called a direct semiconductor. Si, on the other hand has its minimum conduction band electron
energy along the [100] direction where p̄ 6= 0, and hence an electron transition
across the bandgap not only requires the exchange of the energy quantum Eg ,
but also a change in the crystal momentum. Therefore Si is called an indirect
semiconductor. As mentioned earlier, Si is not a suitable material for building
light-emitting diodes and semiconductor lasers, and the reason is that these
devices require direct semiconductors for efficient generation of photons.
With a known E − p̄ relationship, one can obtain the effective mass from the
second derivative of E with respect to p̄ from equation (3):
mn =
d2 E
dp̄2
!−1
(4)
Values for the effective mass are 0.19m0 for Si and 0.07m0 for GaAs, where
m0 is the free electron mass. The effective-mass approximation is useful, since
the electrons and holes then may be treated essentially as classical charged
particles.
Because of the mutual attraction between the negative electron and the positive
atom nucleus, the potential energy of an electron increases as the distance
increases. Therefore, if a hole is treated as a classical charged particle, it has a
potential energy oppositely directed to that of an electron due to its opposite
electric charge.
2.0.1
Intrinsic carrier concentration and Fermi energies
The electron density of a semiconductor can be obtained by first finding the
density in an incremental energy range dE. This density n(E) is given by the
product of the density of allowed energy states per unit volume N (E) and the
probability of occupying that energy range F (E). Thus, the conduction band
electron density is given by integrating the product from the bottom of the
conduction band EC to the top of the conduction band Etop :
n=
Z Etop
n(E) dE =
EC
Z Etop
N (E)F (E) dE
(5)
EC
It can be shown by using equation (3) that the density of allowed energy states
per unit energy in the phase space is:
2mn
N (E) = 4π
h2
3/2
E 1/2
(6)
where h is Planck’s constant. Furthermore F (E) is given by the Fermi-Dirac
distribution function:
F (E) =
1
1 + e(E−EF )/kT
(7)
where k is the Boltzmann constant, T is the absolute temperature, and EF is
the Fermi energy level. The Fermi level is the energy at which the probability
of occupation by an electron is exactly one half. By assuming that F (E) '
exp[−(E −EF )/kT ] for (E −EF ) > 3kT , the electron density of the conduction
band in equation (5) can be shown to be:
n = NC e
−
EC −EF
kT
where
2πmn kT
NC ≡ 2
h2
!3/2
(8)
where NC is the effective density of states in the conduction band and mn
is the electron effective mass. In a similar manner, the hole density p in the
valence band may be obtained:
p = NV e
−
EF −EV
kT
where
2πmp kT
NV ≡ 2
h2
!3/2
(9)
where NV is the effective density of states in the valence band and mp is the
hole effective mass. For an undoped semiconductor, the number of electrons
per unit volume in the conduction band equals the number of holes per unit
volume in the valence band, i.e. n = p = ni , where ni is the intrinsic carrier
density (intrinsic and extrinsic semiconductors are defined below). The Fermi
level for an intrinsic semiconductor is obtained by equating equations 8 and 9:
EC + EV
3kT
mp
+
ln
2
4
mn
EF = Ei =
(10)
At room temperature, the second term is much smaller than the bandgap.
Hence the intrinsic Fermilevel Ei of a semiconductor generally lies very close
to the middle of the bandgap. The intrinsic carrier density ni is also obtained
directly from equations 8 and 9:
np = n2i
(11)
n2i = NC NV e
ni =
q
E −E
− VkT C
NC NV e
E
g
− 2kT
(12)
The relation in equation (11) is called the mass action law, and is independent
of the Fermi energy. The relation is valid for both intrinsic and extrinsic (to be
explained in section 3) semiconductors under a thermal equilibrium condition,
and therefore central for studying the generation and recombination currents
in section 4.5 where the increase of one type of carrier tends to reduce the
number of the other.
3
Doping of semiconductors
Pure semiconductors, such as a Si crystal without any impurities, are called
intrinsic semiconductors. When a semiconductor is doped with impurities,
it becomes extrinsic and impurity energy levels are introduced. The doping
occurs when some atoms in the lattice are replaced with foreign atoms, altering
the lattice structure.
Figure 2a shows a Si lattice where one of the atoms have been replaced by
a type-V atom, e.g. phosphor. Four of the phosphor valence electrons form
covalent bonds to the nearest neighbouring Si-atoms, while the fifth becomes
a conduction electron that is ’donated’ to the conduction band. Thus, the
material is called an n-type donator because of the additional negative charge
carrier, or simply an n-type material. A complementary situation is seen in
figure 2b, where a Si atom is replaced by a type-III atom, e.g. boron. Here the
local lattice deficiency of one electron can receive an electron from one of the
neighbouring atoms, or a free conduction band electron. Thus, the material is
called a p-type acceptor because of the positive charge released by accepting
an electron. The convention of viewing this occurrence as a movement of a
positive charge, rather than a negative electron, will become clear in section 4.
+4
Si
+4
Si
+4
Si
+4
Si
+4
Si
CONDUCTION
ELECTRON
+4
Si
HOLE
-q
+q
+4
Si
+5
P
+4
Si
+4
Si
+3
B
+4
Si
+4
Si
+4
Si
+4
Si
+4
Si
+4
Si
+4
Si
(a)
(b)
Figure 2: Schematic picture of bonds between atoms in an (a)
n-type Si lattice with phosphor doping, and a (b) p-type Si lattice
with Boron doping.
The energies associated with shallow dopants in a semiconductor are shown
schematically in figure 3.
EC
ED
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CONDUCTION
ELECTRONS
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DONOR
IONS
Eg
Ei
Ep
En
ACCEPTOR
IONS
EA
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Figure 3: Schematic picture of energies in doped lattices. A
donator is ionized by accepting a valence electron, thus ’injecting’
a conduction hole to the valence band.
EV
CONDUCTION
HOLES
Conduction electrons in an n-doped semiconductor are called majority carriers
and conduction holes are called minority carriers. For a p-doped semiconductor, the conduction electrons become minority carriers and conduction holes
become majority carriers.
4
4.1
Current conduction in semiconductors
Intrinsic charge carriers
Electric currents may occur in intrinsic silicon. For intrinsic silicon at room
temperature, ni (the intrinsic carrier density) is about 1.45 × 1010 cm−3 . The
free electric charges are created as the thermal ionization of atoms where covalent bonds are broken. The free electrons may either move around in the
crystal, or recombine with positive Si ions, and under thermal equilibrium, the
ionization rate equals the recombination rate.
The current is seen as the net movement of electrons through the periodic
potential of the silicon lattice. As the positive charges are all fixed within the
atomic nuclei of the lattice, there is no positive electric current in the same
sense as for the electrons. As described in the previous section an electron of
a neighbouring atom may be captured by this positive charge, resulting in an
ionized atom with a positive net charge. If this step is successively repeated
in one direction, only electrons have moved, but it is still regarded a positive
current of electron deficiencies, i.e. a positive current of holes.
4.2
Extrinsic charge carriers
The intrinsic conductivity of Si is very low. By doping the material, the
conductivity can be increased by many orders of magnitude, even by small
impurity concentrations. Elements utilized in semiconductor doping should
preferrably introduce energy levels into the forbidden bandgap. A shallow
donor will have an energy level ED in the band gap close to the lowest conduction band energy EC , and a shallow acceptors state EA is similarly located
in the bandgap close to the highest valence band energy EV . Phosphor is a
shallow dopant in silicon, and its dopant level is 0.045eV below EC (where the
band gap Eg in Si is 1.12 eV as mentioned earlier). Similarly, boron introduces
a shallow acceptor state in silicon, and the acceptor level of boron is 0.045 eV
above EV in silicon.
For shallow dopants in Si and GaAs, there usually is enough thermal energy
to supply the energy ED or EA to ionize all donor or acceptor impurities at
room temperature, and thus provide an equal number of electrons or holes
in the conduction band or valence band respectively. This condition is called
complete ionization, and leads to:
n = ND
p = NA
(13)
where n is the electron density, ND is the donor concentration, p is the hole
density and NA is the acceptor concentration. The charge carrier density
for Si is in a large temperature range (typically 100 - 500 K) approximately
constant and dominated by complete ionization of the impurity states (called
the extrinsic region). Below this range, the density approaches 0, and above
this range, the charge carrier density is dominated by intrinsic electrons, hence
it is called the intrinsic region. Therefore, for given applications, the number
of free charge carriers may be tailored to the specific requirements by the
impurity atom doping concentration for a wide temperature range.
Assuming complete ionization of an n-doped material at room temperature,
the Fermi level may be calculated by combining equations 8 and 13:
EC − EF = kT ln
NC
ND
(14)
Similarly for a p-doped material it is easily shown by combining equations 9
and 13 that:
EF − EV = kT ln
NV
NA
(15)
It is seen that for high donor concentrations, the difference (EC − EF ) is
reduced, hence the Fermi level will move closer to the bottom of the conduction
band. Similarly, a high acceptor concentration will move the Fermi level closer
to the valence band. This means that for extrinsic semiconductors, the Fermi
level is different from the intrinsic Fermi level, i.e. EF 6= Ei .
4.3
4.3.1
Drift current
Mobility
Under thermal equilibrium, the average thermal energy of a conduction electron may be obtained from the theorem of equipartition of energy. The electrons in a semiconductor have three degrees of freedom, since they can move
about in three dimensional space. Therefore, the kinetic energy of the electrons
is given by:
1
3
2
mn vth
= kT
2
2
(16)
where mn is the effective electron mass, vth is the average thermal velocity,
k is the Boltzmann’s constant and T is the absolute temperature. At room
temperature the thermal velocity in equaiton 16 is about 107 cm/s for Si and
GaAs. Without the presence of either an electric field, or a charge gradient
in the material, the random motion leads to a zero net displacement of an
electron over a sufficiently long period of time. The Drude model, introduced
in 1900 by P. Drude, assumes the conduction electrons to move around freely
in a static lattice of positive ions, forming a ’gas’ of conduction electrons which
is then treated using the method of kinetic theory.
When an electric field is applied, each electron is accelerated in the field direction until it undergoes a collision (considered by Drude to be a collision with
the lattice), after which its velocity is randomized again. But during the mean
free path, the momentum applied to an electron is given by −qEτc , and the
momentum gained is mn vn , where vn is called the drift velocity. We have:
− qEτc = mn vn
⇒
vn = −
qτc
E
mn
(17)
which states the proportionality of the drift velocity to the applied electric
field. The proportionality factor is called the electron mobility µn in units of
cm2 /Vs, or:
µn ≡
qτc
mn
⇒
vn = −µn E
(18)
A similar expression can be written for holes in the valence band (with sign
inversion due to the positive charge):
v p = µp E
(19)
The mobility in equation (18) is related directly to the mean free time between
collisions, which in turn is determined by the various scattering mechanisms,
the two most important mechanisms being lattice scattering and impurity
scattering.
Lattice scattering results from thermal vibrations of the lattice atoms at any
temperature above zero. These vibrations disturb the lattice periodic potential and allow energy to be transferred between the carriers and the lattice.
Since lattice vibration increases with increasing temperature, lattice scattering
becomes dominating at high temperatures, hence the mobility decreases with
increasing temperature. Theoretical analysis shows that the mobility due to
lattice scattering µL will decrease in proportion to T −3/2 .
Impurity scattering results when a charge carrier travels past an ionized dopant
impurity (donor or acceptor). The charge carrier path will be deflected due to
Coulomb force interaction. The probability of impurity scattering depends on
the total concentration of ionized impurities, i.e. the sum of the concentration
of the negatively and positively charged ions. However, impurity scatterting
becomes less significant at higher temperatures, since the carriers then remain
near the impurity atom for a shorter time due to higher carrier velocity. Hence,
they are less effectively scattered. The mobility due to impurity scattering µI
can be shown to vary as T 3/2 /NT , where NT is the total impurity concentration.
Impurity scattering not only predicts the mobility reduction for high temperatures, it also explains the fact that semiconductor mobility has a maximum
value for low doping concentrations. Increasing the doping concentration above
this limit will increase the concentration of ionized impurities and thus decrease
the mobility.
4.3.2
Resistivity
The transport of carriers under the influence of an applied electric field produces a current called drift current. Considering free electrons in a semiconductor, they will experience a force −qE which is equal to the negative gradient
of the potential energy:
− qE = −
dEC
dx
⇒
E=
1 dEC
1 dEi
=
q dx
q dx
(20)
and if the material is homogeneous, E is constant throughout the sample. A
conduction electron is accelerated in this field, gaining kinetic energy at the
same rate as the potential energy loss. When lattice scattering occurs, the
electron loses all, or parts of its kinetic energy, and the total energy drops
towards its thermal equilibrium position. This process is repeated as long as
the field exists. Conduction by holes may be described in a similar manner,
but in the opposite direction.
Considering a sample with a homogeneous electric field and a cross-sectional
area A, the electron current density Jn may be found by summing the product
of the charge on each electron multiplied by the electron’s velocity over all
electrons per unit volume n:
Jn =
n
In X
= (−qvi ) = −qnvn = qnµn E
A
i=1
(21)
where In is the electron current, and equation (18) is used for the relationship
between vn and E. A similar argument applies to holes yielding Jp = qpµp E,
thus the total current density due to the applied field may be written as:
J = Jn + Jp = (qnµn + qpµp)E = σE
(22)
The quantity σ is known as the conductivity. The corresponding resistivity is
defined as the reciprocal of σ:
ρ≡
1
1
=
σ
q(nµn + pµp )
(23)
Generally, in extrinsic semiconductors, only one of the current components
is significant due to the large difference between the two carrier densities.
Therefore, equation (23) reduces to:
1
for an n-type semiconductor
qnµn
1
for a p-type semiconductor
ρ '
qpµp
ρ '
4.4
(24)
(25)
Diffusion current
Diffusion currents occur in a semiconductor at locations with a nonuniform
charge distribution, i.e. where the spatial charge distribution gradient is
nonzero. In effect, the charge carriers tend to move away from a high concentration of charge carriers of the same sign and towards a direction with
lower concentration. Diffusion currents of carriers of the opposite sign move
in the opposite direction.
In regions with a nonzero spatial charge distribution gradient, drift currents
will also exist due to the electric field produced by the nonuniformity. But since
the field originates from the electrically interacting charge carriers themselves,
the field will change continuously, unlike the case for an externally applied
static field.
Drift and diffusion currents are superimposed components, and the diffusion
may be studied in more detail by a one dimensional first order approximation.
Considering a nonuniform electron distribution along the x-axis at the points
x = −l, x = 0 and x = l, where l is the mean free path of a particle with
thermal velocity vth and mean free time τ c, i.e. l = vth τ c. Electrons at x = −l
have equal probability of moving a distance l in either direction within a time
period of τc , hence the average rate of electron flow per unit area F1 crossing
x = 0 from the x = −l is:
F1 = n(−l)
1 l
1
= n(−l) vth
2 τc
2
(26)
where n(x) is the electron density at x. Similarly, the average rate of electron
flow per unit area F2 from x = l is:
1
F1 = n(l) vth
2
(27)
The net rate of carrier flow F from left to right (across x = 0) is then
1
F = F1 − F2 = vth [n(−l) − n(l)]
2
(28)
An approximation of n(x) at x = ±l by a first order Taylor series expansion
gives:
("
#
"
dn
dn
n(0) − l
− n(0) + l
dx
dx
dn
dn
= −vth l
≡ −Dn
dx
dx
1
F =
vth
2
#)
(29)
where Dn ≡ vth l is called the diffusivity. The flow of electrons results in a
current:
Jn = −qF = qDn
dn
dx
(30)
and it is seen that for an increasing electron concentration in positive xdirection, there is a (positive) current in the same direction, i.e. the electrons
flow in the opposite direction. The same relations hold for diffusion of holes,
and the total drift and diffusion current for holes and electrons are simply
additive, giving the total conduction current density Jcond as:
dn
dx
dp
= qµp pE − qDp
dx
⇓
= Jn + Jp
Jn = qµn nE + qDn
Jp
Jcond
(31)
4.5
Generation and recombination currents
In thermal equilibrium the relationship (equation (11)) pn = n2i is valid. If
exess carriers are introduced to a semiconductor so that pn > n2i , we have a
nonequilibrium situation, and the process of introducing exess carriers is called
carrier injection. Excess carriers may be injected by various means, including optical excitation and ionizing particles. Carrier injection by ionization
requires energies larger than the band gap Eg in order to inject electrons from
the valence to the conduction band.
Carrier injection by ionization produces an equal amount of excess electrons
and holes ∆n = ∆p. For extrinsic semiconductors, e.g. n-doped, the injected
electron density may be many orders of magnitude smaller than the ionized
donor concentration ∆n ND , but the injected hole concentration may be
comparable or larger than the minority carrier concentration. This condition is
referred to as low-level injection. High-level injection is the situation where the
injected carrier concentration is comparable, or much larger than the majority
carrier concentration.
4.5.1
Direct semiconductors
In thermal equilibrium, the electron and hole concentration in an n-doped
material is nn0 and pn0 respectively. In a nonequilibrium carrier injection
situation, the rate at which conduction electrons are generated by injection to
the conduction band is called the generation rate G. This is the sum of the
thermally generated electrons from the valence band Gth and any unspecified
mechanism that generates conduction electrons GM . The recombination rate
R is proportional to the density of conduction electrons e and recombination
sites p (holes):
G = Gth + GM
R = βnn pn = β(nn0 + ∆n)(pn0 + ∆p)
(32)
(33)
where β is the proportionality constant. Recombination is the process where a
conduction electron is captured by an ionized atom, releasing energy either as
a photon or as thermal lattice energy. The net recombination rate is defined
as U ≡ R − Gth = GM , and it is easily verified that for low-level injection in
steady state that:
U ' β(nn0 + pn0 + ∆p)∆p ' βnn0 ∆p
(34)
This shows that the net recombination rate is proportional to the excess minority carrier concentration. Also, because of this proportionality, an exponential
decay is expected. Considering a steady state nonequilibrium situation where
GM 6= 0 and then suddenly set to G = 0 at t = 0. The solution to the
differential equation dpn /dt = −U = −βnn0 ∆p is:
pn (t) = pn0 + τp GM e−t/τp
4.5.2
where
τp ≡
1
βnn0
(35)
Indirect semiconductors
Since the Eg transition of an electron in an indirect semiconductor requires a
nonzero crystal momentum exchange, a direct generation or recombination is
very unlikely to occur. Rather an indirect process is more likely where electrons step via localized intermediate energy states in the forbidden bandgap.
These intermediate states are called generation-recombination centers, or sim-
ply recombination centers.
The generation and recombination currents have a far more complex description than for direct semiconductors. Electrons captured by recombination
centers may either go to the conduction or to the valence band. This leads
to four different currents, and the probability of these transitions depend on
the electron and hole capture crossections σn and σp as well as the location of
the recombination center Ei in the forbidden bandgap and the density of the
centers Nt . The net recombination rate U in this case may be written as:
U ≡ Ra − Rb =
vth σn σp Nt (pn nn − n2i )
σp [pn + ni e(Ei −Et )/kT ] + σn [nn + ni e(Et −Ei )/kT ]
(36)
where Ra and Rb are the electron capture and electron emission rates respectively, and Et is the recombination energy level. However, by applying similar
assumtions as for the direct semiconductor case above, and also assuming that
the recombination centers are located near the middle of the bandgap, equation
36 may be reduced to:
U ' vth σp Nt (pn − pn0 )
(37)
which has the same form as for the direct semiconductor in equation 34.
4.6
High-Field operation
One assumption so far has been that the mean free time τc is constant, which
holds for low values of E. But for high electric fields, as vn becomes comparable
to vth , this additional velocity component will result in a shorter average time
between lattice scatterings, thus the mobility is not longer constant. The
mobility will finally saturate for high electric fields, and may be approximated
by the empirical expression:
vn , v p =
vs
[1 + (E0 /E)]1/γ
(38)
where vs is the saturation velocity (107 cm/s for Si at 300 K), and E0 is a
constant equal to 7 × 103 V/cm for electrons and 2 × 104 V/cm for holes, and γ
is 2 for electrons and 1 for holes.
Another current component that may occur for high voltage operation of semiconductors results from an avalanche process. This is governed by conduction
electrons with sufficient energy to ionize atoms in the lattice. These electrons
gain kinetic energy from the strong electric field between each interaction with
the lattice, and the ionized electron-hole pair may gain sufficient kinetic energy
in the field to ionize other atoms. This chain reaction gives rise to a very large
current increase for a small increase of the electric field around a critical value.
5
The p-n junction
The basic constituent of the ATLAS semiconductor tracker is the detector
element which utilizes some important properties of the p-n junction. The p-n
junction, usually called a diode, is a nonlinear element that is most commonly
used in electronics as a rectifying, current limiting or voltage stabilizing device.
5.1
Steady state thermal equilibrium operation
The diode is a two-terminal device with one n-doped and one p-doped region.
During fabrication the diode is produced on a single piece of crystal, either by
epitaxial growth, diffusion or ion implantation. However, it may be useful to
visualize the junction as the face of two elements brought in close contact with
each other.
ε
1111
0000
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I
0000
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0000
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I
0000
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drift
p
p
n
n
diff
Drift
EC
EF
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EV 11111111111
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EC
EF
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V
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(a)
EC
Diffusion
EC
E
F
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EF
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E
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V
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Diffusion
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(b)
Figure 4: The p-n junction in thermal equilibrium. (a) Uniformly doped samples and their energy levels when separated. (b)
Energy levels and currents in a p-n junction.
Considering the n-material initially, before the two samples are brought together (fig. 4a), the n-side is electrically neutral, since an equal amount of
positive ionized atoms and negative conduction electrons exists. After the two
samples are joined, there will be a large charge concentration gradient at the
junction due to all the free electrons that exist at the n-side of the junction,
which results in a diffusion current across the junction of free electrons into
the p-region. The positively charged ionized dopant atoms at the n-side, however, are bound by the crystal lattice and may not diffuse. As more and more
electrons diffuse across the junction, a positive net charge is created at the
n-side, and the positive ions are said to be uncovered.
The same process occurs in the p-material, where positive holes diffuse into the
n-region, leaving behind uncovered bound negatively charged acceptor ions in
the p-lattice. The uncovered charges on both sides govern an electric field E
across the junction region directed from n to p as seen in figure 4b. This field
will produce a drift current of free electrons and holes, which is oppositely
directed to the initial diffusion currents; electrons will drift from the p-side
across the junction and holes will drift from the n-side towards the p-side.
In thermal equilibrium and with no bias applied, the current density of holes
and electrons across the p-n junction must be zero. For the hole current density
this yields:
Jp = Jp (drift) + Jp (diffusion)
dp
= qµp pE − qDp
!dx
1 dEi
dp
=0
= qµp p
− kT µp
q dx
dx
(39)
where equation (20) and the Einstein relation Dp = kT µp /q have been used.
It can be verified from equation (9) that:
p = ni e(Ei −EF )/kT
⇓
!
dp
p dEi dEF
=
−
dx
kT dx
dx
Inserted into equation (39) yields the net hole current density:
(40)
(41)
J p = µp p
dEF
=0
dx
or
dEF
=0
dx
(42)
Thus, the Fermi level must be constant throughout the sample as illustrated
in figure 4b. Close to the junction on each side, there are now two regions
where all bound excited states are left uncompensated. Going outwards one
reaches the transition region where the ions are partially compensated by free
charge carriers. The region from the junction and outwards on both sides,
including the transition region, is called the space charge region, or depletion
region, since it is depleted for free charge carriers (see figure 5). Outside the
transition region charge neutrality is maintained.
The transition region is usually small compared to the total space charge region, thus the space charge distribution may be approximated by a rectangular
where the n-side depletion region extends uniformly up to xn , and at the p-side
up to xp .
The electrostatic potential ψ is defined in order that its negative gradient equals
the electric field:
Depletion Region
ND -N A
Neutral n
region
Neutral p
region
111111111111
000000000000
000000000000
111111111111
000000000000
111111111111
000000000000
111111111111
x
000000000000
111111111111
0000000000
1111111111
0
x
0000000000
1111111111
Transition
0000000000
1111111111
Region 1111111111
0000000000
0000000000
1111111111
0000000000
1111111111
0000000000
1111111111
0000000000
1111111111
p
n
x
Figure 5: Space charge distribution of a p-n junction with
abrupt doping changes at the metallurgical junction.
E ≡−
dψ
dx
⇒
ψ=−
Ei
q
(43)
according to equation (20). In the neutral p-region the electrostatic potential
calculated with respect to the Fermi level ψp will be:
1
kT NA
ψp ≡ − (Ei − EF )
=−
ln
x≤−x
p
q
q
ni
(44)
where equation (40) has been used by first setting p = NA . This is valid since
Poisson’s equation yields (for the neutral region):
dE
ρs
q
=
= (ND − NA + p − n)
dx
εS
εS
(45)
assuming all donors and acceptors to be ionized. And due to the p-type material; ND = 0 and p n. εS is the permittivity of Si.
A similar expression may be obtained for the electrostatic potential at x > xn
in an n-type material with respect to the Fermi level:
1
ND
ψn ≡ − (Ei − EF )
= kT ln
x≥xn
q
ni
(46)
Hence the total electrostatic potential difference between the p-side and the
n-side neutral regions at thermal equilibrium is called the built-in potential
Vbi :
Vbi = ψn − ψp =
5.2
kT NA ND
ln
q
n2i
(47)
Highly doped abrupt junctions
As will be described in another chapter, the p − n junction of the ATLAS SCT
tracker is of a special design with a large difference in doping concentration
between the p and the n region. The p-material is heavily doped, in the order
of 1019 cm−3 , whereas the n-material is lightly doped, in the order of 1016 cm−3 .
To indicate the heavy doping concentration, the p region is often designated
as p+ -type, and the lightly doped n-region simply as n-type.
In unbiassed thermal equilibrium, the total space charge must be neutral. And
since there are no free charge carriers in the depleted regions, the total p and
n space charge on each side must be equal:
Depletion Region
ND -N A
Neutral p
region
Lightly doped
n region
111111111111
000000000000
000000000000
111111111111
000000000000
111111111111
000000000000
x 111111111111
000000000000
111111111111
0
1
0
x
0
1
0
1
doped
0 Heavily
1
p region
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
Neutral n
region
p
n
x
Figure 6: Space charge distribution of a highly doped abrupt p-n
junction with abrupt doping changes at the metallurgical junction.
NA x p = ND x n
(48)
where the space charge regions are approximated by rectangular regions, also
called an abrupt junction, with −xp and xn as the lower and upper limits
respectively. The total depletion region width d is then given by d = xn + xp .
d is easily found by first calculating the electric field of the space charge region
from Poisson’s equation:
A
E = − qN
(xp + x)
εS
E=
qND
(x
εS
− xn )
, −xp ≤ x ≤ 0
(49)
, 0 ≤ x ≤ xn
(50)
and then the built-in potential Vbi :
Vbi = −
=
Z xn
−xp
qNA x2p
2 + εS
E(x) dx = −
Z 0
−xp
+
qND x2n
2εS
E(x) dx −
Z xn
E(x) dx
0
(51)
and from this, the depletion width d is given by:
s
d=
2εS
q
NA + ND
Vbi
NA ND
(52)
For approximate abrupt space charge regions where one impurity type is dominant (as for the p+ –n junction mentioned above), the depletion width will
almost entirely depend on the light dopant in accordance with the above equations. These junctions are called one-sided abrupt junctions, and equation (52)
may be approximated by:
s
d≡
2εS Vbi
qND
for a one-sided abrupt p+ -n junction where NA ND .
(53)
5.3
Biassed p-n junction in thermal equilibrium
If an additional electric field is introduced to the juntion by an external voltage,
this will shift the depletion width depending on the sign of the bias voltage. If
the applied potential has the opposite sign as that of the built-in electrostatic
potential, the total electrostatic potential will be reduced by |VF | to Vbi + VF ,
where VF is negative. The junction is now forward biassed. The reduction of
the electrostatic potential across the junction reduces the drift current, and
there is no longer a balance between the drift and diffusion current; there will
be a nonzero net diffusion current across the junction. Since p-n junctions of
a Si microstrip detector are reverse biassed, the forward bias properties will
not be discussed any further here.
If the junction is biassed with opposite polarity, the electrostatic potential will
be additional to the built-in potential, and thus increasing the total electrostatic potential. The junction is said to be reverse biassed. As a result the
depletion width will increase accordingly for a one-sided abrupt junction as:
d≡
v
u
u 2εS (Vbi
t
+V)
qNef f
(54)
where Nef f is the effective doping concentration of the lightly doped bulk, and
V is positive for reverse bias voltages. A method for measuring this value is
outlined in a later section. The space charge region that results from a reverse
bias voltage is important when using the p-n junction as a particle detector,
as will be explained in a later chapter.