Silicon VLSI Technology Fundamentals, Practice and Modeling © 1999

by Plummer, Deal and Griffin

This is the appropriate expression for electrons and holes in a semiconductor crystal. The quantity
EF in this equation is called the Fermi level and is defined by this equation to be simply the energy
level at which the probability of finding an electron is exactly 0.5. 1 - F(E) is the probability of
not finding an electron at energy E, or in other words, the probability of finding a hole there.
Fig. 1.22 illustrates the concept of the Fermi level. The left hand case represents an undoped
semiconductor. In that case, n = p and typically there will be a very small number of electrons in
the conduction band and holes in the valence band. Thus the probability of a valence band energy
level being occupied by an electron is essentially 1; the probability of finding an electron in the
conduction band is essentially 0. Therefore, EF must be located somewhere in the forbidden band
between EV and EC. In fact, it is almost exactly in the middle of the bandgap. EF is usually called
the intrinsic Fermi level EFi or Ei in this case. If we add N type doping (middle case in Fig 1.22),
we know that the number of electrons in the conduction band increases, so the probability of
finding an electron above EC also increases. The Fermi level will move up in the bandgap to reflect
this. In fact, a little experience with these sorts of band diagrams will allow the reader to quickly
estimate the electron or hole populations in a given situation from a picture, which shows EF. The
higher EF is in the bandgap, the higher n is.

F ree El ectrons

EC
ED
EF
EF Ei
EF
EV EA

H oles
Fig. 1.22: Fermi level position in an undoped (left), N type (center) and P type (right)
semiconductor. The dots represent free electrons; the open circles represent mobile
holes.

Note also in Fig. 1.22 that we are now showing (schematically) only the holes in the valence
band rather than all the bound electrons as we have in earlier figures. The holes are the mobile
charge carriers in the valence band and they are what we care about when we describe device
operation or process physics issues. From this point forward we will show only the mobile carriers
in these diagrams - holes in the valence band and electrons in the conduction band.
P type doping is analogous (right side of Fig 1.22). Introducing P type dopants increases the
hole population in the valence band and decreases the free electron population in the conduction
band. EF moves down in the bandgap to reflect the lower free electron population. The closer EF
is to EV, the higher the hole population is. Thus, while EF is a well-defined mathematical quantity,
perhaps its greatest use in semiconductors is in visualizing the electronic properties (carrier
concentrations and type) through the band diagram concept.
To actually calculate n and p in a given situation, we need to know not only the probability
of finding them at an energy level E (Eqn. 1.6), but also the number and position of allowed energy
levels. Quantum mechanics tells us how the electrons fill the energy levels in an atom or in a
system of atoms like a crystal. In such a system the energy levels are not continuous (no two

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Silicon VLSI Technology Fundamentals, Practice and Modeling © 1999
by Plummer, Deal and Griffin

electrons can have exactly the same set of quantum numbers). From these concepts comes an
approximate expression for the allowed electron energy levels at an energy E

3 1
4π
()
NE = ( )
* 2
me (E − EC ) 2 for E > EC (1.7)
3
h

3 1
()
NE =
4π
( )
mh* 2
(EV − E) 2 for E < EV (1.8)
h3

where the first equation describes the allowed energy levels for electrons in the conduction band
and the second, the allowed levels for holes in the valence band. In this equation m*e and m*h are
the density of states effective masses of the electron and hole respectively in the crystal and are
different from the electron rest mass. m*e and m*h account for the fact that the electrons and holes
are located in a crystal rather than in free space. N(E) is of course zero within the bandgap of a
pure semiconductor.
Given these expressions for the allowed energy levels and Eqn. 1.6 which describes the
probability that an allowed level will actually be occupied, we can now determine how many free
electrons and holes are actually present in the crystal. Fig. 1.23 illustrates graphically how we do
this.

F r ee E l ectr ons

EC

EF

EV

H oles

N(E ) 0.5 1.0 n, p

F(E )
Fig. 1.23: Density of allowed states, probability function, and resulting electron and hole
populations in a semiconductor crystal.

F(E) is the probability distribution for electrons, which, for an undoped crystal, has a value
of 0.5 at the middle of the forbidden band. N(E), the density of allowed states, is plotted from
Eqns. 1.7 and 1.8. Note that there are no allowed energy levels in the forbidden band. Above EC
or below EV, the densities increase as the square root of the energy. The multiple lines in this part
of the figure are meant to schematically represent the discrete allowed states that exist at any
energy E. For a crystal with an appreciable number of atoms, the discrete energy levels are so
close together and so numerous they appear to be a continuous distribution as shown in Eqns. 1.7
and 1.8. The product of the F(E) and N(E) curves on the right represents the electron population
at any energy. Since we are interested only in the free electrons, we have shown this population
only in the conduction band. In an analogous way, the product of N(E) and 1 - F(E) represents the
hole population, which we have shown only in the valence band since this is where holes are
mobile. Note that most of the free electrons are located fairly close to EC and most of the free

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Silicon VLSI Technology Fundamentals, Practice and Modeling © 1999
by Plummer, Deal and Griffin

holes close to EV, because the respective probability functions rapidly fall towards zero further
away from the band edges. This is in spite of the fact that the number of allowed energy levels
increases away from EC or EV.
Usually in device physics or in process physics, we are interested in the total number of
electrons in the conduction band and the total number of holes in the valence band. These
quantities are the areas under the right hand curves in Fig. 1.23 and can be calculated as follows.
∞ ⎛ E − EF ⎞ (1.9)
n = ∫ F (E )N (E )dE ≅ N C exp⎝ − C
EC kT ⎠

E
V ⎛ E − EV ⎞ (1.10)
p = ∫ [1− F (E )]N( E )dE ≅ N V exp⎝ − F
−∞ kT ⎠

$/&
2πm∗( kT $/& 2πm∗+ kT
where N' = 2 - 2 and N * = 2 ? @ (1.11)
h& h&

NC and NV are often called the effective densities of states in the conduction and valence bands.
They have values of 2.8 x 1019 cm-3 and 1.04 x 1019 cm-3 respectively in silicon at room temperature.
In integrating Eqns. 1.9 and 1.10, we have made use of the fact that the Fermi Dirac
probability function in Eqn. 1.6, can be approximated by the Boltzmann distribution when the
Fermi level is at least a few kT away from the conduction and valence bands and from any other
allowed energy levels in the bandgap. When this is true, the E - EF term in Eqn. 1.6 is much larger
than kT and the 1 in the denominator may be dropped, resulting in the Boltzmann distribution
function. Eqns. 1.9 and 1.10 only apply as long as this is true. When EF approaches EC, EV, or
other allowed energy levels in the bandgap, the full Fermi Dirac distribution function must be used
to describe the electrons populating the various energy levels. This may be required at low
temperatures when not all donors or acceptors are ionized and EF can approach ED or EA. Fermi
Dirac statistics are also generally required when doping levels exceed 1019 cm-3 in silicon, since
then EF moves up into the conduction band or down into the valence band, and allowed energy
levels exist near EF. Such heavily doped semiconductors are often called degenerate and act more
like metals than semiconductors. We will in general use the simple results in Eqns. 1.9 and 1.10
in this text. However there will be cases where this is not valid. Generally when this is occurs,
computer techniques are used to calculate n, p and EF. Such programs are widely available.
By combining Eqn. 1.5 with 1.9 and 1.10 above, we arrive at the result that

⎛ EG ⎞ ⎛ E
np = n2
i = N C N V exp⎝ − = KT3exp⎝ − G ⎞⎠ (1.12)
kT ⎠ kT

where EG = EC - EV. Note the similarity of this result to Eqn. 1.4. The exponential behavior of ni
with temperature is also apparent in Fig. 1.16.
By combining Eqns. 1.9, 1.10 and 1.12, we obtain the following expressions

⎛ E − Ei ⎞ (1.13)
n = ni exp⎝ F
kT ⎠

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