## 2.6 Summary: Conductors

| What counts are the specific quantities: Conductivity s (or the specific resistivity r = 1/ s current density j (Electrical) field strength · E |  |
[ s] = ( Wm)–1 = S/m; S = 1/ W = "Siemens" [ r] = Wm
|  | The basic equation for s is: n = concentration of carriers µ = mobility of carriers |  |
s = |q| · n · µ
|  | Ohm's law states: It is valid for metals, but not for all materials |  |
j = s · E
| s (of conductors / metals) obeys (more or less) several rules; all understandable by looking at n and particularly µ. |  |
|  | Matthiesen rule Reason: Scattering of electrons at defects (including phonons) decreases µ. |  |
r = rLattice(T) + rdefect(N)
|  | "r(T) rule": about 0,04 % increase in resistivity per K Reason: Scattering of electrons at phonons decreases µ |  |
D r | = | ar · r · D T | » | 0,4% oC
--- | --- | --- | --- | ---
|  | Nordheim's rule: Reason: Scattering of electrons at B atoms decreases µ |  |
r » rA + const. · [B]
| Major consequence: You can't beat the conductivity of pure Ag by "tricks" like alloying or by using other materials. (Not considering superconductors). |  |
| Non-metallic conductors are extremely important. |  |
|  | Transparent conductors (TCO's) ("ITO", typically oxides) |  |
No flat panels displays = no notebooks etc. without ITO!
Batteries, fuel cells, sensors, ...
Example: MoSi2 for heating elements in corrosive environments (dishwasher!).
The future High-Tech key materials?
|  | Ionic conductors (liquid and solid) |
|  | Conductors for high temperature applications; corrosive environments, .. (Graphite, Silicides, Nitrides, ...) |
|  | Organic conductors (and semiconductors) |
| Numbers to know (order of magnitude accuracy sufficient) |  |
r(decent metals) about 2 m Wcm r(technical semiconductors) around 1 Wcm r(insulators) > 1 GWcm

| No electrical engineering without conductors! |  |
Money, Chemistry (try Na!), Mechanical and Thermal properties, Compatibility with other materials, Compatibility with production technologies, ...
|  | Hundreds of specialized metal alloys exist just for "wires" because besides s, other demands must be met, too: |
|  | Example for unexpected conductors being "best" compromise: |  |
Poly Si, Silicides, TiN, W in integrated circuits
| Don't forget Special Applications: |  |
Contacts (switches, plugs, ...); Resistors; Heating elements; ...

| Thermionic emission provides electron beams. The electron beam current (density) is given by the Richardson equation: |  |
j = A · T 2 · exp – | EA kT
--- | ---
|  | Atheo = 120 A · cm–2 · K–2 for free electron gas model Aexp » (20 - 160) A · cm–2 · K–2 |
|  | EA = work function » (2 - >6) eV |
|  | Materials of choice: W, LaB6 single crystal |
| High field effects (tunneling, barrier lowering) allow large currents at low T from small (nm) size emitter |  | Needs UHV!
| There are several thermoelectric effects for metal junctions; always encountered in non-equilibrium. |  |
|  | 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 |  |
Essential for measuring (high) temperatures with a "thermoelement" Future use for efficient conversion of heat to electricity ???
|  | 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". |  |
Used for electrical cooling of (relatively small) devices. Only big effect if electrical heating (µ I2) is small.

| 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". |  |
Challenge: Find / design a material with a "good" ion conductivity at room temperature
| Basic principle |  |
|  | Diffusion current jdiff driven by concentration gradients grad(c) of the charged particles (= ions here) equilibrates with the |  |
jdiff = – D · grad(c)
jfield = s · E = q · c · µ · E
|  | Field current jfield caused by the internal field always associated to concentration gradients of charged particles plus the field coming from the outside |
|  | 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. |  |
µ = eD/kT
– | d2U dx2 | = | dE dx | = | e · c(x) e e0
--- | --- | --- | --- | --- | ---
| Immediate results of the equations from above are: |  |
|  | In equilibrium we find a preserved quantity, i.e. a quantity independent of x - the electrochemical potential Vec: |  |
Vec | = const. = | e · U(x) + | kT | · ln c(x)
--- | --- | --- | --- | ---
|  | 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: |  |
c(x) = exp – | (V x) – Vec kT
--- | ---
|  | Electrical field gradients and concentration gradients at "contacts" are coupled and non-zero on a length scale given by the Debye length dDebye |  |
dDebye = | æ ç è | e · e0 · kT e2 · c0 | ö ÷ ø | 1/2
|  | 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. |
|  | The Debye length is not an important material parameter in metals since it is so small that it doesn't matter much. |
| The potential difference between two materials (her ionic conductors) in close contact thus... |  |
|  | ... extends over a length given (approximately) by : |  |
dDebye(1) + dDebye(2)
|  | ... is directly given by the Boltzmann distribution written for the energy: (with the ci =equilibrium conc. far away from the contact. |  |
c1 c2 | = exp – | e · D U kT |  | Boltz- mann
--- | --- | --- | --- | ---
D U = – | kT e | · ln | c1 c2 |  | Nernst's equation
--- | --- | --- | --- | --- | ---
|  | The famous Nernst equation, fundamental to ionics, is thus just the Boltzmann distribution in disguise! |
| "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. |  |
Questionaire
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© H. Föll (Advanced Materials B, part 1 - script)