1ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityHandout 2Sommerfeld Model for Metals – Free Fermion GasIn 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 principleTo account for these shortcomings Sommerfeld in 1927 developed a model for electrons in metals that took into consideration the Fermi-Dirac statistics of electronsNote 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tvmBxtvEetvmFdttvdmdttpd2ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityQuantum Mechanics and the Schrodinger EquationThe quantum state of an electron is described by the Schrodinger equation:ttritrH,,ˆ   rErrVrm222 rVmPPPrVmPHzyxˆ2ˆˆˆˆ2ˆˆ2222 tEiertr,Suppose:Where the Hamiltonian operator is:then we get:rErHˆ(Time independent form)The momentum operator is:iPˆTherefore:222222222222.212ˆ.ˆ2ˆzyxmmiimmPPmP The time independent form of the Schrodinger equation is:ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversitySchrodinger Equation for a Free Electron   rErrVrm222The time independent form of the Schrodinger equation is:For a free-electron:0rV rErm222We have:Solution is a plane wave (i.e. plane wave is an energy eigenstate):zkykxkirkikzyxeVeVr11.Energy:The energy of the free-electron state is:mkmkkkEzyx22222222Note: The energy is entirely kinetic (due to motion)Momentum:ipˆ  rkrirpkkkˆThe energy eigenstates are also momentum eigenstates:123rrdk3ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityElectrons in Metals: The Free Electron ModelThe quantum state of an electron is described by the time-independent Schrodinger equation:   rErrVrm222Consider a large metal box of volume V = LxLyLz: xLyLzLIn 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 forbox the inside for0rrVrrVfree electrons (experience no potential when inside the box)zyxLLLVECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityElectrons in Metals: The Free Electron ModelxLyLzL rErm222Need to solve:With the boundary condition that the wavefunction iszero at the boundary of the boxrSolution is: zkykxkVrzyxksinsinsin8Where:zzyyxxLpkLmkLnkAnd n, m, and p are non-zero positive integers taking values 1, 2, 3, 4, …….zyxLLLVxyzNormalization:The wavefunction is properly normalized:123rrdkEnergy:The energy of the electron states is:mkmkkkEzyx22222222Note: The energy is entirely kinetic (due to motion)4ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityxLyLzLElectrons in Metals: The Free Electron ModelxkykzkLabeling Scheme:All electron states and energies can be labeled by the corresponding k-vectorzkykxkVrzyxksinsinsin8mkkE222k-space Visualization:The allowed quantum states can be visualized as a 3D grid of points in the first quadrant of the “k-space”zzyyxxLpkLmkLnkProblems:• The “sine” solutions are difficult to work with – need to choose better solutions• The “sine” solutions come from the boundary conditions – and most of the electrons inside the metal hardly ever see the boundaryn, m, p = 1, 2, 3, 4, …….ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityBorn Von Karman Periodic Boundary ConditionsxLyLzLxyz rErm222Solve:Instead of using the boundary condition:0boundaryrUse periodic boundary conditions:zyxLzyxzyxzLyxzyxzyLxzyx,,,,,,,,,,,,These imply that each facet of the box is folded and joined to the opposite facetSolution is:zkykxkirkikzyxeVeVr11.The boundary conditions dictate that the allowed values of kx , ky , and kz, are such that:      zzLkizkiLzkiyyLkiykiLykixxLkixkiLxkiLpkeeeLmkeeeLnkeeezzzzzyyyyyxxxxx212121n = 0, ±1, ±2,….m = 0, ±1, ±2,….p = 0, ±1, ±2,….5ECE 407 – Spring 2009 – Farhan Rana – Cornell UniversityBorn Von Karman Periodic Boundary ConditionsLabeling Scheme:All electron states and energies can be labeled by the corresponding k-vectormkkE222rkikeVr.1Momentum Eigenstates:Another advantage of using the plane-wave energy eigenstates (as opposed to the “sine” energy eigenstates) is that the plane-wave states are also momentum eigenstatesMomentum operator:ipˆ  rkrirpkkkˆNormalization: The wavefunction is properly normalized:123rrdkOrthogonality: Wavefunctions of two different states are orthogonal: kkrkkikkVerdrrrd,'.'3*'3Velocity:Velocity of eigenstates