free electron fermi gas_supp_123456789abcdefghij.pptx

free electron fermi gas

1. Free Electron Fermi Gas

2. CONTENTS Energy Levels in One Dimension Effect of Temperature on the Fermi-Dirac Distribution Free Electron Gas in Three Dimensions Heat Capacity of the Electron Gas Electrical Conductivity and Ohm’s Law Motion in Magnetic Fields Thermal Conductivity of Metals Nanostructures .

3. INTRODUCTION Free electron model : Works best for alkali metals (Group I: Li, Na, K, Cs, Rb ) Na: ionic radius ~ .98A, n.n . dist ~ 1.83A.

4. Successes of classical model: Ohm’s law. σ / κ Failures of classical model: Heat capacity. Magnetic susceptibility. Mean free path . Quantum model ~ Drude model

5. Energy Levels in One Dimension Orbital: solution of a 1-e Schrodinger equation Boundary conditions: Particle in a box

6. Pauli-exclusion principle: No two electrons can occupy the same quantum state. Quantum numbers for free electrons: ( n , m s ) Degeneracy: number of orbitals having the same energy. Fermi energy ε F = energy of topmost filled orbital when system is in ground state. N free electrons:

7. Effect of Temperature on the Fermi-Dirac Distribution Fermi-Dirac distribution : Chemical potential μ = μ ( T ) is determined by   (Boltzmann distribution)

8. Free Electron Gas in Three Dimensions Particle in a box ( fixed ) boundary conditions: Periodic boundary conditions: Standing waves → → Traveling waves

9. → ψ k is a momentum eigenstate with eigenvalue  k . N free electrons:

10. Density of states: →

11. Heat Capacity of the Electron Gas N free electrons: ( 2 orders of magnitude too large at room temp) Pauli exclusion principle → T F ~ 10 4 K for metal free electrons Using the Sommerfeld expansion formula

12. → 3-D e-gas for 3-D e-gas              

13. for 3-D e-gas for 1-D e-gas  

14. Experimental Heat Capacity of Metals For T <<  and T << T F : el + ph Deviation from e-gas value is described by m th :      

15. Possible causes: e-ph interaction e-e interaction Heavy fermion: m th ~ 1000 m UBe 3 , CeAl 3 , CeCu 2 Si 2 .

16. Electrical Conductivity and Ohm’s Law Lorentz force on free electron: No collision: Collision time  : Ohm’s law Heisenberg picture: Free particle in constant E field

17. Experimental Electrical Resistivity of Metals Dominant mechanisms high T : e -ph collision. low T : e -impurity collision. phonon impurity Matthiessen’s rule : Sample dependent Sample independent Residual resistivity : Resistivity ratio :  imp ~ 1  ohm-cm per atomic percent of impurity K  imp indep of T ( collision freq additive )

18. Consider Cu with resistivity ratio of 1000: Impurity concentration: = 17 ppm Very pure Cu sample:  For T >  : See App.J From Table 3, we have   imp ~ 1  ohm-cm per atomic percent of impurity            

19. Umklapp Scattering Normal: Umklapp: Large scattering angle ( ~  ) possible Number of phonon available for U -process  exp (  U / T ) For Fermi sphere completely inside BZ, U -processes are possible only for q > q q = 0.267 k F for 1e /atom Fermi sphere inside a bcc BZ. For K ,  U = 23K,  = 91K  U -process negligible for T < 2K

20. Motion in Magnetic Fields Equation of motion with relaxation time  : be a right-handed orthogonal basis Let Steady state: = cyclotron frequency q = – e for electrons

21. Hall Effect →  Hall coefficient: electrons

22. Thermal Conductivity of Metals From Chap 5: Fermi gas: In pure metal, K el >> K ph for all T. Wiedemann-Franz Law : Lorenz number : for free electrons

23. for free electrons