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
ELECTRICAL CONDUCTION
• Ohm's Law:
voltage drop (volts)
DV = I R
current (amps)
resistance (Ohms)
• Resistivity, r and Conductivity, s:
--geometry-independent forms of Ohm's Law
E: electric
field
intensity
DV I
 r
L
A
rL
L
• Resistance: R 

A As
resistivity
(Ohm-m)
J: current density
conductivity
I
s
r
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Resistivity and Conductivity as charged
particles
mobility, m =
V
E
Where V
is the average
velocity
is the average distance between
collisions,
divided by the average time between
collisions,
V
d
V 
t
d
t
t
d
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CONDUCTION IN TERMS OF ELECTRON AND
HOLE MIGRATION
• Concept of electrons and holes:
• Electrical Conductivity given by:
s  ne m e  p e m h
# electrons/m3
# holes/m3
electron mobility
hole mobility
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CONDUCTIVITY: COMPARISON
• Room T values (Ohm-m)
-1
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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.
CONDUCTION & ELECTRON TRANSPORT
• Metals:
-- Thermal energy puts
many electrons into
a higher energy state.
• Energy States:
-- the cases below
for metals show
that nearby
energy states
are accessible
by thermal
fluctuations.
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ENERGY STATES: INSULATORS AND
SEMICONDUCTORS
• Insulators:
--Higher energy states not
accessible due to gap.
• Semiconductors:
--Higher energy states
separated by a smaller gap.
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PURE SEMICONDUCTORS: CONDUCTIVITY VS T
• Data for Pure Silicon:
--s increases with T
--opposite to metals
s undoped  e
 E gap / 2 kT
electrons
can cross
gap at
higher T
material
Si
Ge
GaP
CdS
band gap (eV)
1.11
0.67
2.25
2.40
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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.
INTRINSIC VS EXTRINSIC CONDUCTION
• Intrinsic:
# electrons = # holes (n = p)
--case for pure Si
• Extrinsic:
--n ≠ p
--occurs when impurities are added with a different
# valence electrons than the host (e.g., Si atoms)
• N-type Extrinsic: (n >> p)
• P-type Extrinsic: (p >> n)
Phosphorus atom
Boron atom
hole
4+ 4+ 4+ 4+
s  n e me
4+ 5+ 4+ 4+
4+ 4+ 4+ 4+
no applied
electric field
conduction
electron
valence
electron
Si atom
4+ 4+ 4+ 4+
4+ 3+ 4+ 4+
s  p e mh
4+ 4+ 4+ 4+
no applied
electric field
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Equations describing Intrinsic and
Extrinsic conduction
Using the Fermi-Dirac equation, we can find the number of charge carrier per
unit volume as:
Ne = Noexp(-Eg/2kT)
No is a preexponential function,
Eg is the band-gap energy and
k is Boltzman’s constant (8.62 x 10-5 eV/K)
If
If
Eg > ~2.5 eV
0 < Eg < ~2.5 eV
the material is an insulator
the material is a semi-conductor
Semi-conductor conductivity can be expressed by:
s(T) = so exp(-E*/nkT)
E* is the relevant gap energy (Eg, Ec-Ed or Ea)
n is 2 for intrinsic semi-conductivity and 1 for extrinsic semiMSE-512
conductivity
DOPED SEMICON: CONDUCTIVITY VS T
• Data for Doped Silicon:
--s increases doping
--reason: imperfection sites
lower the activation energy to
produce mobile electrons.
• Comparison: intrinsic vs
extrinsic conduction...
--extrinsic doping level:
1021/m3 of a n-type donor
impurity (such as P).
--for T < 100K: "freeze-out"
thermal energy insufficient to
excite electrons.
--for 150K < T < 450K: "extrinsic"
--for T >> 450K: "intrinsic"
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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,
while 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 
P-N RECTIFYING JUNCTION
• Allows flow of electrons in one direction only (e.g., useful
to convert alternating current to direct current.
• Processing: diffuse P into one side of a B-doped crystal.
• Results:
--No applied potential:
no net current flow.
--Forward bias: carrier
flow through p-type and
n-type regions; holes and
electrons recombine at
p-n junction; current flows.
--Reverse bias: carrier
flow away from p-n junction;
carrier conc. greatly reduced
at junction; little current flow.
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Piezoelectrics
Field
produced
by stress:
  gs
Strain
produced
by field:
  d
Elastic
modulus:
1
E
gd
 = electric field
s = applied
stress
E=Elastic
modulus
d = piezoelectric
constant
g = constant
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APPLIED MAGNETIC FIELD
• Created by current through a coil:
• Relation for the applied magnetic field, H:
NI
H
L
current
applied magnetic field
units = (ampere-turns/m)
2
Bo  mo H
In
Air:
mo1.257  106 Wb/(Am)
With
Magntic
cor:
Bo  mo H (1  c m )
MAGNETIC SUSCEPTIBILITY
• Measures the response of electrons to a magnetic
field.
• Electrons produce magnetic moments:
magnetic moments
electron
nucleus
electron
spin
Adapted from Fig.
20.4, Callister 6e.
• Net magnetic moment:
--sum of moments from all electrons.
• Three types of response...
4
Magntic domains align in
prsnc of magntic fild, H
Hysteresis Loop
Soft and Hard Magnetic Materials
Typical proprtis
of soft and hard
magntic matrials
MAGNETIC
STORAGE
• Information is stored by magnetizing material.
• Head can...
--apply magnetic field H &
align domains (i.e.,
magnetize the medium).
--detect a change in the
magnetization of the
medium.
recording medium
recording head
• Two media types:
--Particulate: needle-shaped
g-Fe2O3. +/- mag. moment
along axis. (tape, floppy)
--Thin film: CoPtCr or CoCrTa
alloy. Domains are ~ 10-30nm!
(hard drive)
~2.5mm
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Magntic Forcs
Sheet Resistivity
V
EL
=
R=
JA
I
L
= r
=
A
rs is the sheet resistivity
L
r L
L
= rs
w
t w
Sheet resistivity is the
resistivity divided by
the thickness of the
doped region, and is
denoted W/□
w
If we know the area per square, the
resistance is r s  n squares  area/square
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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, 
Vd = -mn 
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 = nqmn  s
n is the number of charges, and
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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) 
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