1. Chapter 7 The electronic theory of metal Objectives At the end of this Chapter, you should: 1. Understand the physical meaning of Fermi statistical distribution function. 2. Be able to calculate the Fermi level at 0 K. 3. Understand the concept of work function and contact potential. 4. Be able to calculate the heat capacity of electrons for metals

2. 7.1 Fermi statistics and the heat capacity of electron For the equilibrium state of one system, the basic principle of Fermi statistics can be Which the probability that the eigenstate at energy E will be occupied by an electron. The function of Fermi distribution

3. The function of Fermi distribution f (E) has the form The function of Fermi distribution shows that the function f (E) changes from value 1 to value 0 within the scope of several k B T up and down E F. such eigenstate is almost empty ； 3 ） When E is lower than E F by several k B T ， 2 ） When E is higher than E F by several k B T ， 1）1）

4. The image left is the situation at 0 K, and the right is the situation at T K. Fermi distribution specifically shows the situation of thermal excitation. the change of Fermi distribution from 0K to TK shows that some electrons at energy below transfer to the higher state at energy above after getting the energy with the magnitude being k B T. Fermi surface and thermal excitation

5. Density of States N(E) We often need to know the density of electron states, which is the number of states per unit energy, so we can quickly calculate it: Now using the general relation: we get: The differential number of electron states in a range of energy dE or wavevector dk is:

6. Utility of the Density of States We can simplify by using the relation: With N(E) we can immediately calculate the average energy per electron in the 3-D FEG system: Why the factor 3/5? A look at the density of states curve should give the answer: N(E) E EFEF

7. The density of energy state and the electron distribution in accordance with energy we often need to know the distribution of the electronic energy, then we can get the statistical average number of electrons according to the Fermi distribution function is :

8. Determination of E F In the system, the total number of electron is given by Introducing a new function Because Q(E) =0 when E=0, and f (E)=0 when E=∞, The first is zero. So Q (E) means the total number of the quantum states with energy above E. Then we can obtain

9. Function has the characteristic of δ function, and its contribution to the integral mainly comes from E nearby E F. We can perform Taylor expansion on Q(E) near E F, E F electrons can be approximately given for nearly free by It is obvious that the energy at Fermi surface has slight decrease (about 1/10000) when the temperature is increased, but the slight decline has a decisive effect on the metal properties. the characteristic of

10. So the electronic contribution to the molar heat capacity would be expected to be This is half of the 3R we found for the lattice heat capacity at high T. But experiments show that the total C for metals is only slightly higher than for insulators—which conflicts with the classical theory! 19 th century puzzle: each monatomic gas molecule in sample at temperature T has energy, so if the N free electrons in a metal make up a classical “gas” they should behave similarly. The heat capacity of electron

11. The total energy of electron: Introducing a new function Then we can obtain The heat capacity of electron The ratio of its value to the value of classic theory is about. Therefore, the value of quantum theory is much smaller than the value of classic theory. C el  T !

12. 1. The energy of most electrons is far lower than. Because of the limit of Pauli principle, they don't nearly participate in the thermal excitation, and only those electrons with energy within the scope of several k B T nearby can contribute to the heat capacity. In the condition of general temperature, the heat capacity of lattice is far greater than the heat capacity of electron. At low temperature, the heat capacity of lattice drops rapidly in a way of T 3, the heat capacity of electron is proportional to T and it drops linearly with the temperature. Important points for understanding 2. According to the above analysis, the heat capacity of electron is proportional to. Thus we can get the density of energy state near Fermi face by the heat capacity data.

13. A Rough and Ready Estimate electrons that can absorb thermal energy N(E) E EFEF N(E)f(E)  2kT thermal energy of electrons FEG heat capacity But this linear dependence is impossible to measure directly, since the heat capacity of a metal has two contributions. : Assuming we can measure C(T) for a metal, how can we test this relationship?

14. The experimental value of coefficient r which is the electronic heat capacity of some metal

15. Heat Capacity of Metals: Theory vs. Expt. at low T Very low temperature measurements reveal: Meta l  expt  FEG  expt /  FEG = m*/m Li1.630.7492.18 Na1.381.0941.26 K2.081.6681.25 Cu0.6950.5051.38 Ag0.6460.6451.00 Au0.7290.6421.14 Al1.350.9121.48 Results for simple metals show that the FEG values are in reasonable agreement with experiment, but are always too high: The discrepancy is “accounted for” by defining an effective electron mass m* that is due to the neglected electron-ion interactions

16. 7.2 Work function and contact potential For the homogeneous surface of a metal, its work function is defined as the potential difference between Fermi energy level and vacuum energy level. In fact, the electrons that may be excited to escape from metal are near the Fermi face. therefore, there is Here V 0 is vacuum energy level ， and it can also be seen as the depth of the potential well. Work function The potential energy of an electron in the surface of metal Explanation of the work function

17. The work function of part metal, its unit is eV.

18. Thermal emission Experiments show that the current density of emission of thermionic electrons is given by Here A is constant. And W is the work function, and it is also the energy that electrons need to overcome the barrier when escaping from the metal. Richardson － Dushman formula the work function can be expressed empirically To interpret the formula, we have known that

19. In the k space, the number of states in the volume dk 3 is when is replaced by, we can get the number of states in the volume Thus, the number of electron in the unit volume is when the electron speed is distributed in the range of

20. We set that the direction x is perpendicular to the surface of metal. The electrons with velocity greater than a particular value (Their kinetic energy is greater than particular value) in the direction x can escape from the metal. the current density of thermionic electrons emission

21. Here (work function) Thus we can obtain Different metals have different woke function. Due to thermal expansion, W is a function of temperature.

22. When two different pieces of metal A and B are in contact or connected by the wire, they will be charged. They have different potential VA and VB, and this potential is called contact potential The contact potential Contact potential Explanation of the contact potential

23. Vacuum energy level Chemical potential is equal The chart of energy level and the woke function of metal The difference of contact potential and the woke function e

24. The potential energy of metal A is － eV A ( 0), which enhance its energy. When their Fermi energy is equal, the electrons stop to move from A to B. The energy of metal A is decreased by eV A, the energy of metal B is increased by eV B. This makes their chemical potential equal and the electrons stop to move. The difference of these two pieces of metal is If we set W A (E F ) B. When two pieces of metal is connected, the electrons will move from the metal A to the metal B, So the metal A has positive charge and the metal B has negative charge Process