handout2 - Handout 2 Sommerfeld Model for Metals Free...

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1 ECE 407 – Spring 2009 – Farhan Rana – Cornell University Handout 2 Sommerfeld Model for Metals – Free Fermion Gas In this lecture you will learn: • Sommerfeld theory of metals Arnold Sommerfeld (1868-1951) ECE 407 – Spring 2009 – Farhan Rana – Cornell University • Does not say anything about the electron energy distribution in metals - Are all electrons moving around with about the same energy? • Does not take into account Pauli’s exclusion principle To account for these shortcomings Sommerfeld in 1927 developed a model for electrons in metals that took into consideration the Fermi-Dirac statistics of electrons Note added: Six of Sommerfeld’s students - Werner Heisenberg, Wolfgang Pauli, Peter Debye, Hans Bethe, Linus Pauling, and Isidor I. Rabi - went on to win Nobel prize in Physics. Sommerfeld himself was nominated 81 times (more than any other person) but was never awarded the Nobel prize. Problems with the Drude Theory
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2 ECE 407 – Spring 2009 – Farhan Rana – Cornell University Quantum Mechanics and the Schrodinger Equation The quantum state of an electron is described by the Schrodinger equation: () ( ) t t r i t r H = , , ˆ r h r ψ () () () () r E r r V r m r r r r h = + 2 2 2 r V m P P P r V m P H z y x ˆ 2 ˆ ˆ ˆ ˆ 2 ˆ ˆ 2 2 2 2 r r + + + = + = () ( ) t E i e r t r h r r = , Suppose: Where the Hamiltonian operator is: then we get: ( )( ) r E r H r r = ˆ (Time independent form) The momentum operator is: = i P h r ˆ Therefore: + + = = = = 2 2 2 2 2 2 2 2 2 2 2 2 . 2 1 2 ˆ . ˆ 2 ˆ z y x m m i i m m P P m P h h h h r r The time independent form of the Schrodinger equation is: ECE 407 – Spring 2009 – Farhan Rana – Cornell University Electrons in Metals: The Free Electron Model The quantum state of an electron is described by the time-independent Schrodinger equation: r E r r V r m r r r r h = + 2 2 2 Consider a large metal box of volume V = L x L y L z : x L y L z L In the Sommerfeld model: • The electrons inside the box are confined in a three-dimensional infinite potential well with zero potential inside the box and infinite potential outside the box • The electron states inside the box are given by the Schrodinger equation box the outside for box the inside for 0 r r V r r V r r r r = = free electrons (experience no potential when inside the box) z y x L L L V =
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3 ECE 407 – Spring 2009 – Farhan Rana – Cornell University Electrons in Metals: The Free Electron Model x L y L z L () () r E r m r r h ψ = 2 2 2 Need to solve: With the boundary condition that the wavefunction is zero at the boundary of the box ( ) r r Solution is: () ( ) z k y k x k V r z y x sin sin sin 8 = r Where: z z y y x x L p k L m k L n k π = = = And n , m , and p are non-zero positive integers taking values 1, 2, 3, 4, …….
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This note was uploaded on 04/04/2009 for the course ECE 4070 taught by Professor Rana during the Spring '08 term at Cornell.

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handout2 - Handout 2 Sommerfeld Model for Metals Free...

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