2.6: Conductors (Summary)
 Page ID
 2578
What counts are the specific quantities:

\[\left[\sigma\right]=(\Omega m)^{1}=\text{S/m;}\,S=1/ \Omega=\text{"Siemens"}\\\left[\rho\right]=\Omega m\] 
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).
Nonmetallic 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: MoSi_{2} for heating elements in corrosive environments (dishwasher!). 
Organic conductors (and semiconductors)  The future HighTech 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:
\[j=A\cdot T^2\cdot\text{exp}\frac{E_\text{A}}{\text{k}T}\]
 A_{theo} = 120 A · cm^{–2} · K^{–2} for free electron gas model
A_{exp} ≈ (20  160) A · cm^{–2} · K^{–2}  E_{A} = work function≈ (2  >6) eV
 Materials of choice: W, LaB_{6} 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 nonequilibrium.
Seebeck effect: Thermovoltage develops if a metal Ametal 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 (∝ I^{2}) is small. 
Electrical current can conducted by ions in

Challenge: Find / design a material with a "good" ion conductivity at room temperature

Basic principle
Diffusion current j_{diff}driven by concentration gradients grad(c) of the charged particles (= ions here) equilibrates with the  \[j_\text{diff}=D\cdot\text{grad(c)}\] 
Field current j_{field}caused 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 V_{ec}:  \[V_\text{ec}=\text{const.}=\text{e}\cdot U(x)+\text{k}T\cdot\text{ln}c(x)\] 
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 nonzero on a length scale given by the Debye length d_{Debye}  \[d_\text{Debye}=\left(\frac{\varepsilon\cdot\varepsilon_0\cdot\text{k}T}{\text{e}^2\cdot c_0}\right)^{1/2}\] 
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 c_{0} 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 c_{i} =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 ZrO_{2}  based O_{2} 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.