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Fermi-Dirac distribution and the Fermi-level.
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Fermi-Dirac distribution and the Fermi-level Density of states tells us how many states exist at a given energy E. The Fermi function f(E) specifies how many of the existing states at the energy E will be filled with electrons. The function f(E) specifies, under equilibrium conditions, the probability that an available state at an energy E will be occupied by an electron. It is a probability distribution function. EF = Fermi energy or Fermi level k = Boltzmann constant = 1.38 1023 J/K = 8.6 105 eV/K T = absolute temperature in K
Fermi-Dirac distribution: Consider T 0 K For E > EF : For E < EF : E EF 0 1 f(E)
Fermi-Dirac distribution: ConsiderT > 0 K If E = EF then f(EF) = ½ If then Thus the following approximation is valid: i.e., most states at energies 3kT above EF are empty. If then Thus the following approximation is valid: So, 1f(E) = Probability that a state is empty, decays to zero. So, most states will be filled. kT (at 300 K) = 0.025eV, Eg(Si) = 1.1eV, so 3kT is very small in comparison.
Equilibrium distribution of carriers Distribution of carriers = DOS probability of occupancy = g(E) f(E) (where DOS = Density of states) Total number of electrons in CB (conduction band) = Total number of holes in VB (valence band) =
Integration by parts (I) • In calculus, integration by parts is a rule that transforms the integral of products of functions into other, hopefully simpler, integrals. The rule arises from the product rule of differentiation. • If u = f(x), v = g(x), and the differentials du = f '(x) dx and dv = g'(x) dx; then in its simplest form the product rule is
Integration by parts (II) • In the traditional calculus curriculum, this rule is often stated using indefinite integrals in the form • As a simple example, consider • Since ln x simplifies to 1/x when differentiated, we make this part of ƒ; since 1/x2 simplifies to −1/x when integrated, we make this part of g. The formula now yields
At T = 0, U = (3/5)NεF , this energy is large because all the electrons must occupy the lowest energy states up to the Fermi level. • The average energy of a free electron in silver at T = 0 is • The mean kinetic energy of an electron, even at absolute zero, is two orders of magnitude greater than the mean kinetic energy of an ordinary gas molecule at room temperature.
Heat capacity • The electronic heat capacity Ce can be found by taking the derivative of Equation (19.18): • For temperatures that are small compared with the Fermi temperature, we can neglect the second term in the expansion compared with the first and obtain
Thus the electronic specific heat capacity is 2.2 x 10-2 R. This small value explains why metals have a specific heat capacity of about 3R, the same as for other solids. • It was originally believed that their free electrons should contribute an additional (3/2) R associated with their three translational degrees of freedom. Our last calculation shows that the contribution is negligible. • The energy of the electrons changes only slightly with temperature (dU/dT is small) because only those electrons near the Fermi level can increase their energies as the temperature is raised, and there are precious few of them.
At very low temperatures the picture is different. From the Debye theory, Cv is proportional to T3 and so the heat capacity of a metal takes the form Cv = AT + BT3, • where the first term is the electronic contribution and the second is associated with the crystal lattice. • At sufficiently low temperatures, the AT term can dominate, as the sketch of Figure 19.9 indicates. Figure 19.9 Sketch of the heat capacity of a metal as a function of temperature showing the electronic and lattice contributions.
The fermion gas pressure is found from S = 0 at T = 0, as it must be. The Helmholtz function F = U -TS is
For silver we find that • N/V = 5.9 x1028 m-3 and TF= 65,000K . • Thus • P = 2/5 *5.9*1028 *(1.38*10-23) (6.5*104) • = 2.1*1010 Pa = 2.1*105 atm. • Given this tremendous pressure, we can appreciate the role of the surface potential barrier in keeping the electrons from evaporating from the metal.
19.5 Applications to White Dwarf Stars The temperature inside the core of a typical star is at the order of 107K. The atoms are completely ionized at such a high T, which creates a hugh electron gas The loss of gravitational energy balances with an increase in the kinetic energy of the electrons and ions, which prevent the collapse of star!
Example: The pressure of the electron gas in Sirius B can be calculated with the formula Using the following numbers Mass M = 2.09 × 1030 kg Radius R= 5.57 × 106 m Volume V= 7.23 × 1020 m3
Assuming that nuclear fusion has ceased after all the core hydrogen has been converted to helium! The number nucleons = Since the ratio of nucleons and electrons is 2:1 there are electrons
Therefore, T(=107 K) is much smaller then TF . i.e. is a valid assumption ! Thus: P can be calculated as
A white dwarf is stable when its total energy is minimum For Since can be expressed as Where
For gravitational energy of a solid With In summary To find the minimum U with respect to R
19.7 a) Calculate Fermi energy for Aluminum assuming three electrons per Aluminum atom.
19.7b) Show that the aluminum at T=100 K, μ differs from εFby less than 0.01%. (The density of aluminum is 2.69 x 103 kg m-3 and its atomic weight is 27.)
19.7c) Calculate the electronic contribution to the specific heat capacity of aluminum at room temperature and compare it to 3R. • Using the following equation
19.13. Consider the collapse of the sun into a white dwarf. For the sun, M= 2 x 1030 kg, R = 7 x 108 m, V= 1.4 x 1027 m3. Calculate the Fermi energy of the Sun’s electrons.
(b) What is the Fermi temperature? • (c) What is the average speed of the electrons in the fermion gas (see problem 19-4). Compare your answer with the speed of light.
