Electronic conduction in metals
Prof. Stanislav S. Fedotov, Prof. Dmitry Aksyonov
Materials Chemistry
Center for Energy Science and Technology
October 6th, 2025
Why conductivity is important?
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Our technological civilization relies on electricity, which is made possible by the electrical conductivity of materials:
What is electrical conductivity?
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Why do metals conduct electric current?
Electronic structure of a metal vs insulators and semi
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Electronic structure of a metal
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d
Copper band structure
The electrons are easily excited into empty state by E
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Room temperature conductivity
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Ohm’s law
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I = U/R
I is current in Amperes
U is voltage across two points in Volts
R is resistance across two points
j = I/A
A is area, j is current density
R = ρL/A
Georg Simon Ohm 1789 - 1854, German
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Why does current depend linearly on voltage?
Drude free electron model (1900) of electrical conduction
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Paul Drude, 1863-1906, German
In 1894 he was responsible for introducing the symbol "c" for the speed of light in a perfect vacuum.
Why do electrons has quantum behaviour in metal?
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Δp is uncertainty in momentum
Δx is uncertainty in position
h is Planck constant
Reasons: High density and low mass of electrons
Fermi-Dirac statistics
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F(E) is probability to occupy a state with energy E
Ei is the energy
μ is the chemical potential of the electron, e.g. Fermi level
k is Boltzmann's constant
T is temperature
Distribution of particles, classical vs quantum
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VF = ℏ/m (3π2n)⅓ ~ 1.56x106 m/s As a result the mean three path is up to several mm, contradicting Drude approximations
How electrons can move such a long way without interacting with atomic cores?
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Felix Bloch
Nobel prize in 1952
Physical reasons of resistivity in metals (different τ)
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Temperature dependence of conductivity and defects
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Matthiessen’s rule the resistivity arises from independent scattering processes
which are additive, i.e.,
linear dep is valid only at t above Debye T, below it the dependence is T2
<n> = kT/ℏ𝛚 - concentration of phonons
λ ~ 1/<n> is mean free path
Electroconductivity for alloys
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Electroconductivity for single-phase alloys (Nordheim’s rule)
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XA, XB are component fractions and C is a materials constant. Nordheim’s rule does not take into consideration the changes in the density of states with composition. This is particularly true
for alloys containing a transition metal.
For two-phase alloys the volume fractions are used:
Electron-electron scattering
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Has very low probability of collision, how is it possible, the average distance between electrons is less than 2A.
The main reason is exclusion principle,
the cross-section is reduced by (kBT/Ef)2 the second reason is screening
Jellium model
e-
e-
e-
Smeared positive charge
Exchange-correlation hole
Important only at very low temperatures
Conductivity of different metals from QM
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D
Resistors
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Wattage is highly important not to broke the resistor
Specific electronic heat
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Cel = π2/3 N(EF)k2T,
Thermal electronic conductivity
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Wiedemann-Franz law
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Kel/σ = LT,
where L is Lorentz number
the ratio of the electronic contribution of the thermal conductivity (Kel) to the electrical conductivity (σ) of a metal is proportional to the temperature
Thermal conductivity vs Electrical conductivity
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Summary
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Task 1
The room-temperature electrical resistivities of pure lead and pure tin are 2.06x10-7 and 1.11x10-7 Ohm-m, respectively.
(a) Make a schematic graph of the 100 C electrical resistivity versus composition for all compositions between pure lead and pure tin.
(b) On this same graph schematically plot electrical resistivity versus composition at 150C.
(c) Explain the shapes of these two curves, as well as any differences between them.
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Individual studies:
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Thank you for your attention!