2.6: Conductors (Summary)
- Page ID
- 2578
What counts are the specific quantities:
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\[\left[\sigma\right]=(\Omega m)^{-1}=\text{S/m;}\,S=1/ \Omega=\text{"Siemens"}\\\left[\rho\right]=\Omega m\] |
The basic equation for σ is: n = concentration of carriers µ = mobility of carriers |
\[\sigma=|q|\cdot n\cdot\mu\] |
Ohm's law states: It is valid for metals, but not for all materials |
\[\underline{j}=\sigma\cdot {\color{purple}\underline{E}}\] |
σ (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. |
\[\rho=\rho_\text{Lattice}(T)+ \rho_\text{defect}(N)\] |
"ρ(T) rule": about 0,04 % increase in resistivity per K Reason: Scattering of electrons at phonons decreases µ |
\[\Delta\rho=\alpha_\rho\cdot\rho\cdot\Delta T\approx\frac{0.4\%}{^\circ C}\] |
Nordheim's rule: Reason: Scattering of electrons at B atoms decreases µ |
\[\rho\approx\rho_\text{A} +\text{const.}\cdot [B]\] |
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! |
Ionic conductors (liquid and solid) | Batteries, fuel cells, sensors, ... |
Conductors for high temperature applications; corrosive environments, .. (Graphite, Silicides, Nitrides, ...) |
Example: MoSi2 for heating elements in corrosive environments (dishwasher!). |
Organic conductors (and semiconductors) | The future High-Tech key materials? |
Numbers to know (order of magnitude accuracy sufficient) | ρ(decent metals) about 2 μΩcm ρ(technical semiconductors) around 1 Ωcm ρ(insulators) > 1 GΩcm |
No electrical engineering without conductors!
Hundreds of specialized metal alloys exist just for "wires" because besides σ, other demands must be met, too: | Money, Chemistry (try Na!), Mechanical and Thermal properties, Compatibility with other materials, Compatibility with production technologies, ... |
Example for unexpected conductors being "best" compromise: | Poly Si, Silicides, TiN, W in integrated circuits |
Don't forget Special Applications:
- Contacts (switched, plugs, ...);
- Resistors;
- Heating elements; ...
Thermionic emission provides electron beams.
The electron beam current (density) is given by the Richardson equation:
\[j=A\cdot T^2\cdot\text{exp}-\frac{E_\text{A}}{\text{k}T}\]
- 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 ∝ I, i.e. one of the junction may "cool down". |
Used for electrical cooling of (relatively small) devices. Only big effect if electrical heating (∝ I2) is small. |
Electrical current can conducted by ions in
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Challenge: Find / design a material with a "good" ion conductivity at room temperature
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Basic principle
Diffusion current jdiffdriven by concentration gradients grad(c) of the charged particles (= ions here) equilibrates with the | \[j_\text{diff}=-D\cdot\text{grad(c)}\] |
Field current jfieldcaused by the internal field always associated to concentration gradients of charged particles plus the field coming from the outside | \[j_\text{field}=\sigma\cdot{\color{purple}E}=q\cdot c\cdot\mu\cdot{\color{purple}E}\] |
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. |
\[\mu=\text{e}D/\text{k}T\\-\frac{\text{d}^2U}{\text{d}x^2}=\frac{\text{d}{\color{purple}E}}{\text{d}x}=\frac{\text{e}\cdot c(x)}{\varepsilon\varepsilon_0}\] |
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: | \[V_\text{ec}=\text{const.}=\text{e}\cdot U(x)+\text{k}T\cdot\text{ln}c(x)\] |
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: | \[c(x)=\text{exp}\left(\frac{(Vx)-V_\text{ec}}{\text{k}T}\right)\] |
Electrical field gradients and concentration gradients at "contacts" are coupled and non-zero on a length scale given by the Debye length dDebye | \[d_\text{Debye}=\left(\frac{\varepsilon\cdot\varepsilon_0\cdot\text{k}T}{\text{e}^2\cdot c_0}\right)^{1/2}\] |
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 : | \[d_\text{Debye}(1)+d_\text{Debye}(2)\] | |
... is directly given by the Boltzmann distribution written for the energy: (with the ci =equilibrium conc. far away from the contact. |
\[\frac{c_1}{c_2}=\text{exp}\left(\frac{\text{e}\cdot\Delta U}{\text{k}T}\right)\;\text{Boltzmann's Equation}\] | |
The famous Nernst equation, fundamental to ionics, is thus just the Boltzmann distribution in disguise! |
\[\Delta U=-\frac{\text{k}T}{\text{e}}\cdot\text{ln}\frac{c_1}{c_2}\;\text{Nernst's Equation}\] |
"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.