BR
Wiley/Razavi/
Fundamentals of Microelectronics
[Razavi.cls v. 2006]
June 30, 2007 at 13:42
24 (1)
24
Chap. 2
Basic Physics of Semiconductors
dislodge an electron from a covalent bond. Called the “bandgap energy” and denoted by
, this
minimum is a fundamental property of the material. For silicon,
eV.
The second question relates to the conductivity of the material and is as follows. How
many
free electrons are created at a given temperature? From our observations thus far, we postulate
that the number of electrons depends on both
and
: a greater
translates to fewer electrons,
but a higher
yields more electrons. To simplify future derivations, we consider the
density
(or
concentration) of electrons, i.e., the number of electrons per unit volume,
, and write for silicon:
(2.1)
where
J/K is called the Boltzmann constant. The derivation can be found in
books on semiconductor physics, e.g., [1]. As expected, materials having a larger
exhibit a
smaller
. Also, as
, so do
and
, thereby bringing
toward zero.
The exponential dependence of
upon
reveals the effect of the bandgap energy on the
conductivity of the material. Insulators display a high
; for example,
eV for dia-
mond. Conductors, on the other hand, have a small bandgap. Finally,
semi
conductors exhibit a
moderate
, typically ranging from 1 eV to 1.5 eV.
Example
2.1
Determine the density of electrons in silicon at
K (room temperature) and
K.
Solution
Since
eV
J, we have
(2.2)
(2.3)
Since for each free electron, a hole is left behind, the density of holes is also given by (2.2) and
(2.3).
Exercise
Repeat the above exercise for a material having a bandgap of 1.5 eV.
The
values obtained in the above example may appear quite high, but, noting that silicon
has
, we recognize that only one in
atoms benefit from a free
electron at room temperature. In other words, silicon still seems a very poor conductor. But, do
not despair! We next introduce a means of making silicon more useful.
2.1.2 Modification of Carrier Densities
Intrinsic and Extrinsic Semiconductors
The “pure” type of silicon studied thus far is an
example of “intrinsic semiconductors,” suffering from a very high resistance. Fortunately, it is
possible to modify the resistivity of silicon by replacing some of the atoms in the crystal with
atoms of another material. In an intrinsic semiconductor, the electron density,
, is equal
The unit eV (electron volt) represents the energy necessary to move one electron across a potential difference of 1
V. Note that 1 eV
J.