1. Metals I: Free Electron Model
Physics 355
The free electron model allows us to visualize and explain some properties of metals, but not all. Every model has its limitations. So, let see where this one takes us.

2. Free Electron Model
Schematic model of metallic crystal, such as Na, Li, K, etc.
The equilibrium positions of the atomic cores are positioned on the crystal lattice and surrounded by a sea of conduction electrons.
For Na, the conduction electrons are from the 3s valence electrons of the free atoms. The atomic cores contain 10 electrons in the configuration: 1s22s2p6. + + +
In this model, electrons are completely free to move about, free from collisions, except for a surface potential that keeps the electrons inside the metal.
In the alkali metals, with a bcc structure, the cores take up about 15% of the volume of the crystal, but in the noble metals (Cu, Ag, Au), with an fcc structure, the atomic cores are relatively larger and maybe close to contacting or in contact with each other. + +

3. Free Electrons? How do we know there are free electrons?
You apply an electric field across a metal piece and you can measure a current – a number of electrons passing through a unit area in unit time.
But not all metals have the same current for a given electric potential. Why not?

4. Paul Drude
resistivity ranges from 108 m (Ag) to 1020 m (polystyrene)
Drude (circa 1900) was asking why? He was working prior to the development of quantum mechanics, so he began with a classical model:
positive ion cores within an electron gas that follows Maxwell-Boltzmann statistics
following the kinetic theory of gases- the electrons in the gas move in straight lines and make collisions only with the ion cores – no electron-electron interactions.
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5. Paul Drude
He envisioned instantaneous collisions in which electrons lose any energy gained from the electric field.
The mean free path was approximately the inter-ionic core spacing.
Model successfully determined the form of Ohm’s law in terms of free electrons and a relation between electrical and thermal conduction, but failed to explain electron heat capacity and the magnetic susceptibility of conduction electrons.
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The failures of the model are the result of the limitations of the classical model and Maxwell-Boltzmann statistics in particular.

6. Ohm’s Law E
Experimental observation:

7. Ohm’s Law: Free Electron Model
Conventional current
The electric field accelerates each electron for an average time  before it collides with an ion core.

8. Ohm’s Law: Free Electron Model

9. Ohm’s Law: Free Electron Model
If electrons behave like a gas…
The mean free time is related to this average speed…
typical value
About 1014 s
So, this predicts that the resistivity varies as the square root of the temperature.
Then,

10. Ohm’s Law: Free Electron Model
Predicted
behavior
High T: Resistivity limited by lattice thermal motion.
Low T: Resistivity limited by lattice defects.
The mean free path is actually many times the lattice spacing – due to the wave properties of electrons.

11. Wiedemann-Franz Law (1853)
Electrical
Thermal
Conductivities
where
The Lorentz number predicted by this treatment gives approximately 1.6x10-8 W-W/K2.
This is wrong!
Lorentz number (Incorrect!!)

12. Wiedemann-Franz Law (1853)
(Ludwig) Lorenz Number
(derived via quantum mechanical treatment)

13. Free Electron Model: QM Treatment
Assume N electrons (1 for each ion) in a cubic solid with sides of length L – particle in a box problem.
These electrons are free to move about without any influence of the ion cores, except when a collision occurs.
These electrons do not interact with one another.
What would the possible energies of these electrons be?
We’ll do the one-dimensional case first. L

14. Free Electron Model: QM Treatment
At x = 0 and at L, the wavefunction must be zero, since the electron is confined to the box.
One solution is:
The energy is proportional to n2 and it is quantized.
How are these levels occupied with electrons?
According to the Pauli Exclusion Principle, no two electrons can be in the same energy level because they are fermions and follow Fermi-Dirac statistics.
We know that 2 electrons can exist in the same energy level only if their spins are in opposite directions: mz= +1/2 or -1/2 (and therefore they have slightly different energy levels, but these differences are so small that we usually just say that they have the same energy)
We call these energy levels orbitals.

15. Free Electron Model: QM Treatment

16. Free Electron Model: QM Treatment
Chemical potential: this is roughly the Fermi energy εF (if you need to add an electron, it must go into the next occupied state which is at the Fermi energy)
The chemical potential changes a bit as the temperature changes, but it is usually ~ Fermi energy

17. Free Electron Model: QM Treatment
Chemical Potential
If an electron is added, it goes into the next available energy level, which is at the Fermi energy. It has little temperature dependence. m
Fermi-Dirac Distribution
If f goes to zero, it means that few states are occupied. If f goes to 1, it means that nearly all the states are full.
For lower energies,
f goes to 1.
For higher energies,
f goes to 0.

18. Free Electron Model: QM Treatment
From thermodynamics,
the chemical potential, and thus the Fermi Energy, is related to the Helmholz Free Energy:
where
This is what the Fermi-Dirac distribution looks like for a real 3D system at various temperatures.
In this case, εF is about K.
As T →0 K, this becomes a step function.
Note that the lower energy levels are usually filled, and as you raise the temperature, you increase the no of electrons at higher energy levels
The free energy is for a system of N particles. The product of the temperature and entropy decreases with temperature and this is represented in the graph of the Fermi-Dirac distribution.

19. Free Electron Model: QM Treatment
When an electron is confined to a cube-shaped box, the wavefunction is a standing wave.
Since we’ll also need this wavefunction to satisfy periodic boundary conditions, like we did for phonons, the wavefunctions will also be periodic in x, y, and z with period L.
Of course, we want to deal with free-particles, so Mr. Schrodinger with periodic conditions gives us wavefunctions that are traveling plane waves...
where nx, ny, and nz are integers

20. Free Electron Model: QM Treatment
and similarly for y and z, as well
The traveling plane wave solution is valid as long as the components of the wavevector satisfy these relations...any component of k is of the form 2n/L where n is our positive or negative quantum number. This arises because when you satisfy the periodic boundary conditions, you find
Just as in the case of phonons...only certain wavelengths are possible for these electrons confined to the box.

21. Free Electron Model: QM Treatment
Energy
Fermi Energy
The momentum i quantum mechanics is an operator
In the ground state of a system of N particles,
the occupied orbitals may be represented as points inside a sphere in k space...
The energy at the surface of the sphere is the Fermi energy and the wavevectors have a maximum value kF.
Velocity

22. Free Electron Model: QM Treatment
Each value of k exists within a volume
The number of states inside the sphere of radius kF is
This successfully relates the Fermi energy to the electron density.

23. million meters per second
Free Electron Model: QM Treatment
million meters per second
We can define a velocity at the Fermi surface...
This is for an electron at the highest occupied energy level, which can have a k-vector pointing in any direction.
We can also define what is called a Fermi temperature, but this is not a temperature of the electron gas.
It is a measure of where the Fermi energy is at (typically on the order of ~ K)
So, for most metals say at room temperature, not many electrons are excited above the Fermi energy.
Fermi Temperature

24. Free Electron Model: QM Treatment
Density of States

25. Free Electron Model: QM Treatment
The number of orbitals per unit energy range at the Fermi energy is approximately the total number of conduction electrons divided by the Fermi energy.

26. Free Electron Model: QM Treatment
This represents how many energies are occupied as a function of energy in the 3D
k-sphere.
As the temperature increases above T = 0 K, electrons from region 1 are excited into region 2.
The red curve is the square root of the energy.
The black curve is the red curve multiplied by the Fermi-Dirac distribution.
The blue line represents the Fermi energy, filled levels at 0 K.