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Transcript
Fermi-Dirac Statistics • At finite temperature the electrons are no longer entirely in their ground state configuration – Fermi-Dirac distribution describes the probability that an energy level E will be occupied in a system of fermions at temperature T. f (E) = 1 exp[(E ! EF ) / k BT ] + 1 States unoccupied at T=0 have electrons in them at finite temperature Definitions • E is the energy we are interested in. • T is the temperature at equilibrium. Fermi-Dirac function at Zero Temperature • Plot the function on the board in the limit of T→0. Show the step-wise nature of the transition when E=ΕF. • Note how the step-wise transition at T=0 occurs because E-EF is divided by a number approaching zero, which is equivalent to multiplying by infinity. Thus if E-EF is less than zero, the exponential is zero, and f(E)=1, while if E-EF is greater than zero, then then exponential is infinity, and f(E)=0. At the transition temperature the Fermi-Dirac function has a width of order kT. Specifically, the 10%-90% width is 4.4 kT, which for T=300K corresponds to 0.11 eV. 1 Electronic Heat Capacity • Heat Capacity: Coefficient relating the energy added to a system and the resultant increase in temperature "U C! "T – The product D(E)f(E)dE gives the number of occupied states at a given temperature T. Only electrons near EF can gain energy – Energy at a given temperature found by evaluating " U(T ) = # E ! D(E)! f (E) dE 0 Classical Statistics: U = 32 NkT C= !U 3 = Nk !T 2 Quantum Statistics: The product D(E)×f(E) gives the number of occupied states at temperature T. The observed value of the heat capacity is reduced from the classical value by order T/TF (factor of 0.01). !2 T2 U= Nk B 4 TF "U ! 2 T C= = NkB "T 2 TF A two-dimensional model provides a useful means to compute the integral for U(T) graphically, since D(E)=constant in 2D. We consider that all electrons within kBT of EF have their energy increased by exactly kBT . All other electrons do not change their energy, as their occupancy remains unity. Thus, for electrons which can increase their energy, we have the following • Energy increase per electron is kT • Number of electrons increasing their energy is N × kBT/EF=NT/TF • Total energy change is: energy per electron × number of electrons = kT × NT/TF=NkT × T/TF (which is the same as the analytic result bar a factor of π 2/4) 2 Comparison with Experiment • Phonon contribution ∝ T3 • Electron contribution ∝ T γexpt:γfree C = !T + AT 3 " C = ! + AT 2 T Li Na K Rb Cs Be 2.18 1.26 1.25 1.26 1.43 0.34 For small temperatures the heat capacity is a sum of linear and cubic terms in the temperature - the linear behaviour here is due to the Fermionic nature of the heat conduction electrons, while the cubic behaviour arises from the phonons (which are Bosons). The linear term is expressed in the y-intercept, and the cubic term in the slope. See that the free electron gas model provides an excellent starting point for understanding metals. The coefficient of the linearity is denoted γ and is known as the Sommerfeld parameter. For Potassium: γexpt=2.08 mJ/mol/K2 ! free " 2 Nk B ! = 2 TF 23 $2 3 " 2 (6.02 # 10 )(1.38 # 10 ) = = 1.67 mJ/mol/K 2 2 24,600 Departures from unity due to the interaction of the conduction electrons with 1) the periodic potential the rigid crystal lattice. 2) the quanta of lattice vibrations (phonons) 3) other conduction electrons. 3