1. Conduction of Electricity in Solids
Chapter 41
Conduction of Electricity in Solids
Key contents
Electric properties of metals
Semiconductors and doping
The p-n junction and transistors

2. The Electrical Properties of Solids:

3. The Electrical Properties of Solids:

4. Energy Levels in a Crystalline Solid:
If we bring the atoms of Fig. 41-2a close together, their wave functions will overlap, beginning with those of the outermost electrons. Then we have a single two-atom system; here the system contains 2 x29 =58 electrons.
If we bring up more atoms, we gradually assemble a lattice of solid copper. If the lattice contains N atoms, then each level of an isolated copper atom must split into N levels in the solid.
Thus, the individual energy levels of the solid form energy bands, adjacent bands being separated by an energy gap, with the gap representing a range of energies that no electron can possess.

5. Energy Levels in a Crystalline Solid:

6. Insulators:
In insulators, the highest occupied band is full. The gap energy to the higher band is large, e.g., 5.5 eV for diamond. The room temperature is only about eV.

7. Example, Probability of electron excitation in an insulator:

8. Metals:
In metals, the highest occupied band is partially filled.
If the electric potential energy U of a conduction electron is uniform throughout the lattice, let’s set U = 0, so that the mechanical energy E is entirely kinetic. Then the level at the bottom of the partially filled band of Fig corresponds to E =0. The highest occupied level in this band at absolute zero (T =0 K) is called the Fermi level, and the energy corresponding to it is called the Fermi energy EF ; for copper, EF =7.0 eV.
The electron speed corresponding to the Fermi energy is called the Fermi speed vF . For copper the Fermi speed is=1.6 x106 m/s. All motion does not cease at absolute zero; at that temperature the conduction electrons are stacked up in the partially filled band of Fig with energies that range from zero to the Fermi energy.

9. Metals: How Many Conduction Electrons Are There?

10. Example, Number of conduction electrons in a metal:

11. Metals: How Many Quantum States Are There?
From slide 11 to slide 17, what are discussed are the number density of states, occupation number (Fermi-Dirac distribution), and the computation of Fermi energy. This part can be skipped without affecting the understanding of the slides that follow.

12. Example, Number of states per electron volt in a metal:

13. Metals: The Occupancy Probability P(E):

14. Example, Probability of occupancy of an energy state in a metal :

15. Metals: How Many Occupied States Are There?
Fig (a) The density of occupied states No(E) for copper at absolute zero. The area under the curve is the number density of electrons n. Note that all states with energies up to the Fermi energy EF =7 eV are occupied, and all those with energies above the Fermi energy are vacant. (b) The same for copper at T =1000 K. Note that only electrons with energies near the Fermi energy have been affected and redistributed.

16. Example, Number of occupied states in an energy range in a metal:

17. (Here m is the electron mass)
Metals: Calculating the Fermi Energy:
(The number of occupied states per unit volume at T = 0 K for all energies between E=0 and E=EF).
(Here m is the electron mass)

18. Semiconductors:
Fig (a) The band–gap pattern for a semiconductor. It resembles that of an insulator except that here the energy gap Eg is much smaller; thus electrons, because of their thermal agitation, have some reasonable probability of being able to jump the gap. (b) Thermal agitation has caused a few electrons to jump the gap from the valence band to the conduction band, leaving an equal number of holes in the valence band.

19. (Here r is the resistivity)
Semiconductors: Temperature Coefficient of Resistivity:
(Here r is the resistivity)
The resistivity of copper increases with temperature (that is, dr/dT >0) because collisions of copper’s charge carriers occur more frequently at higher temperatures. This makes a positive for copper. The collision frequency also increases with temperature for silicon.
In contrast, the resistivity of silicon actually decreases with temperature (dr/dT <0) since the number of charge carriers n (electrons in the conduction band and holes in the valence band) increases so rapidly with temperature. (More electrons jump the gap from the valence band to the conduction band.) Thus, the fractional change a is negative for silicon.

20. Doped Semiconductors:

21. Doped Semiconductors:

22. Doped Semiconductors:

23. Example, Doping silicon with phosphorus:

24. The p-n junction: diffusing current, majority carriers
drifting current,
minority carriers

25. The Junction Rectifier

26. The Junction Rectifier:
If a potential difference is applied across a p-n junction in one direction (here labeled and “Forward bias”), there will be a current through the junction.
However, if the direction of the potential difference is reversed, there will be approximately zero current through the junction.

27. The Junction Rectifier

28. The Light-Emitting Diode (LED):
In some semiconductors, including gallium arsenide, the energy can be emitted as a photon of energy hf at wavelength

29. Check also photoconductors, Charge Coupled Devices (CCDs),
More Applications :
Shining light on a suitably arranged p-n junction can produce a current in a circuit that includes the junction. This is the basis for the photodiode.
Photodiode
A p-n junction, where there is population inversion, can act as a junction laser, where its light output being highly coherent and much more sharply defined in wavelength than light from an LED.
The Junction Laser
Check also photoconductors, Charge Coupled Devices (CCDs),
Complimentary Metal-Oxide-Semiconductor (CMOS) sensors,
Avalanche Photo Diodes (APDs), etc.

30. Example, Light Emitting Diode (LED):

31. The Transistor:
A transistor is a three-terminal semiconducting device that can be used to amplify (or quench) input signals.

32. The Transistor: an example
When the gate terminal sets up a negative potential, which tends to repel electrons from the n-channel down to the substrate, the depletion zone (at the p-n junction interface, not shown in the figure) widens and the n-channel width decreases. The number of charge carriers in the n-channel therefore drops and the current IDS decreases.

33. Key contents
Electric properties of metals
Semiconductors and doping
The p-n junction and transistors