Download presentation

Presentation is loading. Please wait.

Published byAlysa Thorman Modified over 3 years ago

1
The Quantized Free Electron Theory ++++++++ Energy E Spatial coordinate x Nucleus with localized core electrons Jellium model: electrons shield potential to a large extent Electron “sees” effective smeared potential

2
Electron in a box In one dimension: In three dimensions: where and

3
Fixed boundary conditions: ++++++++ x 0 L + + + + + + + + Periodic boundary conditions: and kxkx “free electron parabola” density of states Remember the concept of # of states in

4
1. approach use the technique already applied for phonon density of states where Density of states per unit volume

5
1/ Volume occupied by a state in k-space kxkx kyky kzkz Volume( )

6
Free electron gas: Independent from and Independent from and 2 Each k-state can be occupied with 2 electrons of spin up/down k2k2 dk

7
2. approach calculate the volume in k-space enclosed by the spheres and kxkx kyky # of states between spheres with k and k+dk : with 2 2 spin states

8
E D(E) E’E’+dE D(E)dE =# of states in dE / Volume

9
Statistics of the electrons (fermions) Fermions are indistinguishable particles which obey the Pauli exclusion principle T=0 E n=1 E n=2 E n=3 E n=4 E n=5 Let us distribute 4 electrons spin E n=6 Occupation number 0 for state Occupation number 1 for states of a given spin E f(E,T=0) Probability that a qm state is occupied EFEF 1 x

10
Fermi Dirac distribution function at T>0 With accuracy sufficient for many estimations: f(E,T) linearized at E F here chemical potential Fermi energy

11
More detailed approach to Fermi statistics The grand canonical ensemble Heat Reservoir R T=const. Particle reservoir System Average energy Average particle # Normalized probabilities Now we consider independent particles Total energy of N fermion system occupation # n i =0,1 of single particle state i with energy i where

12
average occupation of state j is given by Chemical potential For details see& additional info see where the summation means Repeat this step

13
The Fermi gas at T=0 E f(E,T=0) EFEF 1 E D(E) EF0EF0 Electron density #of states in [E,E+dE]/volume Fermi energy depends on T Probability that state is occupied T=0

14
Energy of the electron gas @ T=0: there is an average energy of per electron without thermal stimulation with electron densitywe obtain Click for a table of Fermi energies, Fermi temperatures and Fermi velocities

15
Specific Heat of a Degenerate Electron Gas here: strong deviation from classical value only a few electrons in the vicinity of E F can be scattered by thermal energy into free states Specific heat much smaller than expected from classical consideration D(E) Density of occupied states E EFEF energy of electron state #states in [E,E+dE] probability of occupation, average occupation # 2k B T Before we calculate U let us estimate: These # of electrons increase energy from to

16
π2π2 3 subsequent more precise calculation Calculation of C el from Trick: Significant contributions only in the vicinity of E F

17
with and E D(E) EFEF decreases rapidly to zero for

18
withand in comparison with for relevant temperatures W.H. Lien and N.E. Phillips, Phys. Rev. 133, A1370 (1964) Heat capacity of a metal: electronic contribution lattice contribution @ T<<Ө D

19
Selected phenomena which don’t require detailed knowledge of the band structure Temperature dependence of the electrical resistance T Scattering of electrons: deviations from a perfect periodic potential Impurities: temperature independent imperfection scattering phonon scattering Matthiessen’s rule:

20
Simple approach to understand for T>>Ө D Remember Drude expression: scattering rate #of scattering centers/volume scattering cross section Fermi velocity of electrons: compare lecture notes: Thermal Properties of Crystal Lattices

21
with and Let us consider the high temperature limit: T>>Ө D Note: T 5 –low temperature dependence not described by this simple approach Lindemann melting temperature T M : where emperical value average atomic spacing

22
Thermionic Emission Finite barrier height of the potential E x EFEF work function E vac Current density for homogeneous velocity generalized Current density for k-dependent velocity

23
Again: Occupied and Spin degeneracy Since Fermi distribution approximated by Maxwell Boltzmann distribution approximated

24
Let us investigate the integral Remember integrals of the type: Richardson-Dushman

25
Richardson Constant Typical value of Tungsten: Nobel prizeNobel prize in 1928 "for his work on the thermionic phenomenon and especially for the discovery of the law named after him". Owen Willans Richardson Universal constant: A=1.2 X 10 6 A/m 2 K Reflection at the potential step A (1-r)=0.72 X 10 6 A/m 2 K Vacuum tube

26
Field-Aided Emission EFEF E vac E x Image potential Electric field in x-direction @ very high electric fields of tunneling through thin barrier cold emission

Similar presentations

Presentation is loading. Please wait....

OK

Free Electron Fermi Gas

Free Electron Fermi Gas

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on p/e ratio Ppt on uses of synthetic fibres Ppt on alternative sources of energy and energy crisis View ppt on mac Ppt on business communication process Ppt on learning styles Ppt on history of australia penal colony Ppt on chapter why do we fall ill for class 9 Ppt on online shopping procedure Ppt on human resource recruitment