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What counts are the specific quantities:
- Conductivity s (or the specific
resistivity r = 1/
s
- current density j
- (Electrical) field strength · E
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[ s] =
( Wm)–1 = S/m; S = 1/ W = "Siemens"
[ r] = Wm |
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The basic equation for
s is:
n = concentration of carriers
µ = mobility of carriers |
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Ohm's law states:
It is valid for metals, but not for all materials |
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s
(of conductors / metals) obeys (more or less) several rules; all understandable
by looking at n and particularly µ. |
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Matthiesen rule
Reason: Scattering of electrons at defects (including phonons) decreases
µ. |
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| r = rLattice(T) + rdefect(N) |
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"r(T) rule":
about 0,04 % increase in resistivity per K
Reason: Scattering of electrons at phonons decreases µ |
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| Dr |
= |
ar ·
r · DT |
» |
0,4%
oC |
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Nordheim's rule:
Reason: Scattering of electrons at B atoms decreases µ |
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Major consequence: You can't beat the
conductivity of pure Ag by "tricks" like alloying or by using
other materials.
(Not considering superconductors). |
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Non-metallic conductors are extremely important. |
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Transparent conductors (TCO's)
("ITO", typically oxides) |
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No flat panels displays = no
notebooks etc. without ITO! |
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Ionic conductors (liquid and solid) |
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Batteries, fuel cells, sensors,
... |
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Conductors for high temperature applications;
corrosive environments, ..
(Graphite, Silicides, Nitrides, ...) |
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Example: MoSi2 for
heating elements in corrosive environments (dishwasher!) |
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Organic conductors (and semiconductors) |
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The future High-Tech key
materials? |
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Numbers to know (order of magnitude
accuracy sufficient) |
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r(decent metals) about 2 mWcm
r(technical semiconductors) around 1
Wcm
r(insulators) > 1 GWcm |
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No electrical engineering without
conductors! Hundreds of specialized metal alloys exist just for
"wires" because besides s, other
demands must be met, too: |
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Money, Chemistry (try Na!),
Mechanical and Thermal properties, Compatibility with other materials,
Compatibility with production technologies, ... |
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Example for unexpected conductors being
"best" compromise: |
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Poly Si, Silicides, TiN,
W in integrated circuits |
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Don't forget Special
Applications: |
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Contacts (switches, plugs, ...);
Resistors;
Heating elements; ... |
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Thermionic emission provides electron
beams.
The electron beam current (density) is given by the Richardson equation: |
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| j = A
· T 2 · exp – |
EA
kT |
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Atheo = 120 A ·
cm–2 · K–2 for free electron gas model
Aexp » (20 - 160) A ·
cm–2 · K–2 |
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EA = work function
» (2 - >6) eV |
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Materials of choice: W,
LaB6 single crystal |
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High field effects (tunneling,
barrier lowering) allow large currents at low T from small (nm)
size emitter |
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Needs UHV! |
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There are several thermoelectric
effects for metal junctions; always encountered in non-equilibrium. |
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Seebeck
effect:
Thermovoltage develops if a metal A-metal B junction is at a temperature
different form the "rest", i.e. if there is a temperature
gradeient |
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Essential for measuring (high)
temperatures with a "thermoelement"
Future use for efficient conversion of heat to electricity ??? |
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Peltier
effect:
Electrical current I through a metal - metal (or metal -
semiconductor) junction induces a temperature gradient µ I, i.e. one of the junction may
"cool down". |
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Used for electrical cooling of
(relatively small) devices. Only big effect if electrical heating (µ I2) is small. |
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Electrical current can conducted by
ions in
- Liquid electrolytes (like H2SO4 in your
"lead - acid" car battery); including gels
- Solid electrolytes (= ion-conducting crystals). Mandatory for fuel cells
and sensors
- Ion beams. Used in (expensive) machinery for "nanoprocessing".
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Challenge: Find / design a
material with a "good" ion conductivity at room temperature |
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Basic principle |
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Diffusion
current jdiff driven by concentration
gradients grad(c) of the charged particles (= ions here)
equilibrates with the |
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| jfield = s · E = q · c · µ
· E |
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Field current
jfield caused by the internal field always associated
to concentration gradients of charged particles plus the field coming from the
outside |
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Diffusion coefficient D and
mobility µ are linked via theEinstein relation;
concentration c(x) and potential U(x)
or field E(x) =
–dU/dxby the Poisson equation. |
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| – |
d2U
dx2 |
= |
dE
dx
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= |
e ·
c(x)
ee0
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Immediate results of the equations
from above are: |
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In equilibrium we find a preserved quantity, i.e.
a quantity independent of x - the electrochemical potential
Vec: |
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| Vec |
= const. = |
e · U(x) + |
kT |
· ln c(x) |
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If you rewrite the equaiton for
c(x), it simply asserts that the particles are distributed
on the energy scale according to the Boltzmann distrubution: |
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| c(x) = exp – |
(Vx) – Vec
kT |
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Electrical field gradients and concentration gradients at "contacts" are coupled and
non-zero on a length scale given by the Debye length
dDebye |
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dDebye = |
æ
ç
è |
e ·
e0 · kT
e2 · c0 |
ö
÷
ø |
1/2 |
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The Debye length is an extremely important
material parameter in "ionics" (akin to
the space charge region width in semiconductors); it depends on temperature
T and in particular on the (bulk) concentration
c0 of the (ionic) carriers. |
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The Debye length is not an important material
parameter in metals since it is so small that it doesn't matter much. |
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The potential difference between two
materials (her ionic conductors) in close contact thus... |
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... extends over a length given (approximately)
by : |
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... is directly given by the Boltzmann
distribution written for the energy:
(with the ci =equilibrium conc. far away from the
contact. |
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c1
c2 |
= exp – |
e · DU
kT |
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Boltz-
mann |
| DU = – |
kT
e |
· ln |
c1
c2 |
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Nernst's
equation |
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The famous Nernst
equation, fundamental to ionics, is thus just the Boltzmann
distribution in disguise! |
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"Ionic" sensors (most
famous the ZrO2 - based O2 sensor in your
car exhaust system) produce a voltage according to the Nernst equation because
the concentration of ions on the exposed side depends somehow on the
concentration of the species to be measured. |
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© H. Föll
