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Classical and Quantum Free Electron Models of Electrical ...

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**1. **Classical and Quantum Free Electron Models of Electrical Conductivity (Garcia Chapter 23)

**2. **Classification of Solids Electrical Properties Some solids conduct current at all temperatures and, generally, the resistivity of such solids increases with temperature. These are METALS
Other solids stop conducting at low temperatures and their resistivity falls with increasing temperature. These INSULATORS and SEMICONDUCTORS

**3. **Classical Model for Electrical Conduction

**4. **Ohm’s Law for Electrical Resistance

**5. **Ohm’s Law

**6. **Resistance, Resistivity, Conductance, Conductivity Resistance is R = V/I
Conductance is G = I/V = 1/R
Resistance is higher for longer wires and for wires with smaller cross section
R ~ L/A ? R = ? (L/A)
G ~ A/L? G = s (A/L)
? is called resistivity. The units are [O·m]
“Conductivity” is inverse of resistivity s = 1/?
Conductivity and Resistivity are material properties, and do not depend on the size or shape of the material

**7. **Examples of Resistivity (?) Ag (Silver): 1.59×10-8 O·m
Cu (Copper): 1.68×10-8 O·m
Graphite (C): (3 to 60)×10-5 O·m
Diamond (C): ~1014 O·m
Glass: ~1010 - 1014 O·m
Pure Germanium: ~ 0.5 O·m
Pure Silicon: ~ 2300 O·m

**8. **Classical Model for Electrical Conduction The average thermal velocity for ideal gas depend on temperature, T

**9. **Microscopic Model of Current For current carried by electrons, |q| = e
Estimate drift speed

**10. **Drift Speed and Current While charges move chaotically and very fast, their average motion is slow
However, the onset of current moves along a wire with the speed of light, since the electric field moves with the speed of light and electrons are already “inside” the wire

**11. **Quantum Free Electron Theory of Metals (Garcia Chapter 23)

**12. **3D Cubic Infinite Potential Well
1-D Well
3-D “Cubic” Well (with sides length L)

**13. **3D Infinite Potential Well : Degeneracy Consider three different wavefunctions (quantum states) for a particle in the 3-D Well:
i, j, and k are integers
Although the states are different, the energy of these states are the same, i.e. these are degenerate

**14. **Electrons in 3D Infinite Potential Well Each electron is described by the wavefunction of a particle in the infinite well, i.e. the electron state is defined by three quantum number n1, n2, n3; however, in addition, we have to include the spin quantum number, ms
The electron states are thus determined by four quantum numbers: n1, n2, n3, ms
The energy, of course, still depends only on n1, n2, n3!
It is convenient to use the following notations:
for ms = ˝, we shall call it the spin “up” (?)
for ms = -˝, we shall call it the spin “down” (?)
Example:

**15. **Electrons in 3D Infinite Potential Well: Pauli’s Principle What is the ground state configuration of many electrons in the 3D infinite potential well?
Electrons cannot be in the lowest energy state, since this would violate the Pauli Exclusion Principle.
consider case of solid with 34 electrons

**16. **34 Electrons in 3D Infinite Well (n1, n1, n1)
The lowest energy for this system is 3E0, which corresponds to n1 = n2 = n3 = 1
Thus only 2 (two) electrons can have this energy: one with spin ? and one with spin ?
Next energy level (6E0), for which one of n’s is 2
Thus total of 6 (six) electrons can have this energy
Next energy level (9E0) can also accommodate 6 electrons
What are the combinations of n’s for this energy level?
Next energy level (11E0) also accommodates 6 electrons
What are the n’s?
Next energy level (12E0) is two-fold degenerate

**17. **34 Electrons in 3D Infinite Well So far we have placed 22 electrons, so we need to add another 12 electrons
What is the next energy level?
The next energy level is 14E0
What are n1, n2, and n3?
What is the degeneracy?
This energy can be had by 12 electrons
We have placed all 34 electrons!!

**18. **34 Electrons in 3D Infinite Well Can demonstrate with diagram
Energy is plotted in terms of E0
Each arrow represents an electron with “up” or “down” spin
Numbers in parenthesis show the set of n’s for a given energy level

**19. **34 Electrons in 3D Infinite Well In this configuration,
What is the probability at T =0 that a level with energy 14E0 or less will be occupied?
It is 1!
What is the probability that the level with energy above 14E0 will be occupied?
It is 0!

**20. **Fermi Energy Generally:
The highest filled energy is called the Fermi Energy
It is often denoted as EF
In our case: EF = 14E0
An electron with E = 14E0 is said to be at the Fermi level

**21. **Density of States of Metal Consider a free electron gas in a macroscopic sample
Density of states of free electrons confined within a metal by reflecting walls can be calculated in the same way as the density of states of EM modes in a cavity with reflecting walls
Recall…

**22. **Einstein’s Photon Interpretation of Blackbody Radiation

**23. **Density of EM Modes, 1

**24. **The number of modes between f and (f+df) is the number of points in number space with radii between n and (n+dn) multiplied by (1) 1/8 and (2) by 2,
The first factor arises because modes with positive and negative n correspond to the same modes, so take n1, n2, n3,> 0
The second factor arises because there are two modes with perpendicular polarization (directions of oscillation of E) for each value of f
Since the density of points is 1 (one point per unit volume) in number space, the number of modes between f and (f+df) is the number of points dN in number space in the positive octant of a shell with inner radius n and outer radius (n+dn) multiplied by 2
The volume of a complete shell is the area of the shell multiplied by its thickness, 4? n2dn
The number of modes with associated radii in number space between n and (n+dn) is, therefore, dN = (2)(1/8)4? n2dn = ? n2dn Density of EM Modes, 2

**25. **Density of EM and Electron States

**26. **Velocity Distribution Find the velocity distribution – the probability per unit velocity of finding a state with velocity v

**27. **Electron Density per unit Energy The density of states dN/dE is often written g(E)
The number of electrons per unit energy, N(E) also depends upon the probability that a given state is occupied, F(E). It is,
N(E) = g(E)F(E)

**28. **Fermi-Dirac Distribution Function We introduce the probability distribution function, F(E), which describes the probability that a state with energy E is occupied
For electrons this function is the Fermi-Dirac Distribution Function
At T = 0

**29. **Fermi-Dirac Distribution Function At T > 0 K

**30. **Free Electron Models Classical Model:
Metal is an array of positive ions with electrons that are free to roam through the ionic array
Electrons are treated as an ideal neutral gas, and their total energy depends on the temperature and applied field
In the absence of an electrical field, electrons move with randomly distributed thermal velocities
When an electric field is applied, electrons acquire a net drift velocity in the direction opposite to the field Quantum Mechanical Model:
Electrons are in a potential well with infinite barriers: They do not leave metal, but free to roam inside
Electron energy levels are discrete (quantized) and well defined, so average energy of electron is not equal to (3/2)kBT
Electrons occupy energy levels according to Pauli’s exclusion principle
Electrons acquire additional energy when electric field is applied

**31. **Consequences for Metal Theories At T = 0 K
N is the electron density, i.e. the number of electrons per unit volume of metal
Calculations show that
Thermal energy at room temperature:
kBT ~ 0.025 eV ? kBT << EF (0)

**32. **Consequences for Metal Theories Only electrons occupying levels close to the Fermi Energy will participate in the conduction, since only these electrons can be excited into the higher energy states by the electric field
From QM point of view, energy supplied by the electric field excites electrons into higher lying energy levels

**33. **Temperature Dependence of Metals Resistivity Let’s introduce the Fermi velocity:
Then:

**34. **Conductivity of Metals There is still a problem, since according to our definition of the free mean path, the conductivity is temperature independent
QM resolves the problem:
Electrons inside the metal have de Broglie wavelength:
But wave in a crystal lattice undergoes Bragg scattering when the condition, 2dsin? = n?, is satisfied. If ? > 2d, Bragg scattering cannot occur at any angle. In copper, ?F = 4.65 A, while, d = 2.09 A. So Bragg scattering cannot occur.

**35. **Conductivity of Metals Thus, the scattering mechanism is not collisions of electrons with ions, but rather scattering of electron wave from deviations form perfection in the crystal that may arise from imperfections in the lattice or vibrations of the ions in the lattice
In a high quality metal the latter mechanism dominates
When the atoms vibrate the lattice is no longer ideal and presents an effective cross sectional area for scattering of ?r2, where r is the amplitude of vibration.
The electron mean free path l is inversely proportional to the scattering cross section
But the energy of a vibrating atom, E ~ r2 ~ kT
So l ~ 1/T
Hence as is found experimentally