Transcript Conduction and Semiconductors

MSE-630 Week 2
Conductivity, Energy Bands and
Charge Carriers in
Semiconductors
Objectives:
• To understand conduction, valence energy
bands and how bandgaps are formed
• To understand the effects of doping in
semiconductors
• To use Fermi-Dirac statistics to calculate
conductivity and carrier concentrations
• To understand carrier mobility and how it is
influenced by scattering
• To introduce the idea of “effective mass”
• To see how we can use Hall effect to determine
carrier concentration and mobility
Conductivity
Charge carriers follow a
random path unless an
external field is applied.
Then, they acquire a drift
velocity that is dependent
upon their mobility, mn and the
strength of the field, x
Vd = -mn x
The average drift velocity, vav is dependent
Upon the mean time between collisions, 2t
Charge Flow and Current Density
Current density, J, is the rate at
which charges, cross any plane
perpendicular to the flow direction.
J = -nqvd = nqmnx = sx
n is the number of charges, and
-19
q is the charge (1.6 x 10
C)
The total current density depends upon the total charge
carriers, which can be ions, electrons, or holes
J = q(nmn + pmp) x
OHM’s Law:
V = IR
Resistance, R(W) is an extrinsic quantity. Resistivity, r(Wm), is the
corresponding intrinsic property.
r = R*A/l
Conductivity, s, is the reciprocal of resistivity: s(Wm)-1 = 1/r
As the distance between
atoms decreases, the
energy of each orbital
must split, since
according to Quantum
Mechanics we cannot
have two orbitals with
the same energy.
The splitting results in “bands” of
electrons. The energy difference
between the conduction and valence
bands is the “gap energy” We must
supply this much energy to elevate an
electron from the valence band to the
conduction band. If Eg is < 2eV, the
material is a semiconductor.
Simple representation of silicon atoms bonded in a crystal.
The dotted areas are covalent or shared electron bonds.
The electronic structure of a single Si atom is shown
conceptually on the right. The four outermost electrons are
the valence electrons that participate in covalent bonds.
Electron (-) and hold (+) pair
generation represented b a broken
bond in the crystal. Both carriers are
mobile and can carry current.
Portion of the periodic table relevant
to semiconductor materials and
doping. Elemental semiconductors
are in column IV. Compound
semiconductors are combinations of
elements from columns III and V, or
II and VI.
Doping of group IV semiconductors
using elements from arsenic (As, V)
or boron (B, III)
Intrinsic carrier concentration vs.
temperature.
Dopant designations and
concentrations
Resistivity as a function of
charge mobility and number
When we add carriers by doping, the number of additional carrers, Nd, far
exceeds those in an intrinsic semiconductor, and we can treat conductivity as
s = 1/r = qmdNd
Simple band and bond representations of pure
silicon. Bonded electrons lie at energy levels
below Ev; free electrons are above Ec. The
process of intrinsic carrier generation is
illustrated in each model.
Simple band and bond representations of doped
silicon. EA and ED represent acceptor and donor
energy levels, respectively. P- and N-type
doping are illustrated in each model, using As as
the donor and B as the acceptor
Behavior of free carrier concentration
versus temperature. Arsenic in silicon is
qualitatively illustrated as a specific
example (ND = 1015 cm-3). Note that at high
temperatures ni becomes larger than 1015
doping and n≈ni. Devices are normally
operated where n = ND+. Fabrication occurs
as temperatures where n≈ni
Probability of an electron occupying
a state. Fermi energy represents the
energy at which the probability of
occupancy is exactly ½.
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.
The density of allowed states at an
energy E.
Integrating the product of the probability of occupancy with the density of
allowed states gives the electron and hole populations in a
semiconductor crystal.
Effective Mass
In general, the curve of Energy vs. k is nonlinear, with E increasing as k increases.
E = ½ mv2 = ½ p2/m = h2/4pm k2
We can see that energy varies inversely with
mass. Differentiating E wrt k twice, and
solving for mass gives:
2
h
m =
2
d E
2p
2
dk
*
Effective mass is significant because it
affects charge carrier mobility, and
must be considered when calculating
carrier concentrations or momentum
Effective mass and other semiconductor properties may be found in
Appendix A-4
Substituting the results from the previous slide into the expression for the
product of the number of holes and electrons gives us the equation above.
Writing NC and NV as a function of ni and substituting gives the equation
below for the number of holes and electrons:
In general, the number of electron
donors plus holes must equal the
number of electron acceptors plus
electrons
The energy band gap gets smaller with
increasing temperature.
Fermi level position in the forbidden band for a
given doping level as a function of temperature.
In reality, band structures are highly
dependent upon crystal orientation. This
image shows us that the lowest band gap
in Si occurs along the [100] directions, whil
for GaAs, it occurs in the [111]. This is why
crystals are grown with specific
orientations.
The diagram showing the
constant energy surface
(3.10 (b)), shows us that
the effective mass varies
with direction. We can
calculate average effective
mass from:
1 1 1
2 
=   
*
mn 3  ml mt 