3/21/2017 3/20/2017 FREE ELECTRON FERMI GAS Outlines Free ElectronGas 1. Energy levels in one dimension 2. Effect of temp on the Fermi-Dirac function 3. Free electron gas in 3D 4. Heat capacity of the electron gas 5. Thermal conductivity of metal Physics UCF 2 In this unit, we study electrons in solid, in particular Inside a real metal, the electron in the conduction band electrons inside a metal. There are some overlap between this unit and unit#1, where we study the Drude model. 1. Will interact with other electron. (correlation, exchange) 3/20/2017 2. Will interact with the atomic potential (Coulomb force) 3/20/2017 In a simple metal, the atom at each lattice site usually has both core electrons and valence electrons. The core Free ElectronGas Free ElectronGas electrons are very similar to those electrons in a free When we ignore atom. However, the valence electrons are quite different from the outer electrons of the free atom. 1. >>>>>>>> Independent electron model 2. >>>>>>>> Free electron model Of course, we can also treat electrons as classical particles or as quantum particles. In this unit, we will examine the success and the failure of this model, and 3 try to understand why. 4 Electron Energy Levels in one dimension One dimensional infinite square well For a particle in a 1D box, the boundary conditions are Let’s consider a free electron gas in one dimension. WE 3/20/2017 ; . The solution can be found 3/20/2017 take into account the fact that electrons are quantum particles satisfy the Schrodinger equation, Free ElectronGas (3) Free ElectronGas (1) where is the wave-function of the electron, is where & Hamiltonian operator and is the eigenvalue of the Hamiltonian operator. Substitute (3) into (1) we obtain (2) (4) In this unit, we are dealing with free electrons, so the 5 6 potential is zero. 1 3/21/2017 So you can see from the figure below the first few energy For each of the energy levels associated with a quantum levels and the wavefunctions look like. number n, there can have only two electrons. For N electrons, the last filled level will be , and we define Since this is a one dimensional solid, so only one principle 3/20/2017 3/20/2017 quantum number n, but electron has an intrinsic Fermi energy as the energy of the topmost filled level in the quantum number called spin. So for 1D electron gas, ground state of the N electron system. there are two degree of freedom. Now electrons are Free ElectronGas Free ElectronGas fermions so they satisfy the Fermi-Dirac distribution. (5) For 6 electrons in a 1D infinite square well the ground state is given by: The Fermi-Dirac Distribution n Occupancy 1 ↑ 1 The Fermi-Dirac distribution gives the probability that an 1 ↓ 1 orbital at energy will be occupied in an ideal electron gas 2 ↑ 1 at thermal equilibrium 2 ↓ 1 3 ↑ 1 (6) 7 / 8 3 ↓ 1 4 ↑ 0 Free electron gas in 3D The free electron in 3D satisfies the Schrodinger eq. 3/20/2017 3/20/2017 (7) Free ElectronGas Free ElectronGas The solution is · (8) , , ,⋯ , , ,⋯ The quantity µ is a function of temperature. At , . At other temperature, is equal to ½ when , , ,⋯ 9 10 . The energy of the system is given by The total number of electrons is just (volume density) (9) · · 3/20/2017 3/20/2017 In the reciprocal space, as the states are occupied by N electrons, when ≫, the occupied states form a sphere Free ElectronGas (11) Free ElectronGas The surface of this sphere is Substitute back into Eq. (10), we end up with called Fermi surface and the energy at the surface at T=0 (12) is called Fermi energy. The electron velocity at the Fermi surface is given by (10) (13) 11 12 2 3/21/2017 Re-arrange Eq. (12), we express N in terms of If we assume each electron occupies a volume of , (14) where is the Wigner-Seitz radius, 3/20/2017 3/20/2017 The density of states is · · (15) Free ElectronGas Free ElectronGas ≡ · · · · ≅. (16) 13 14 For free electron, the band structure is dominated by Heat capacity of the electron gas parabolic function since One of the early difficulties encountered in the Drude model was the experimental heat capacity of the electron gas was 3/20/2017 much smaller than that expected from the classic model. 3/20/2017 (17) Classical theory Free ElectronGas Each electron in 3D has energy, so the heat capacity Free ElectronGas is Experimental data showed %of The answer to this question was given by Fermi through the Fermi- Dirac distribution of 15 Fermion. 16 At temp T only a fraction of all electrons been excited , Quantitative calculation of The energy required to raise the temperature by 1 °C is 3/20/2017 3/20/2017 · · (18) ·· ·· (20) Free ElectronGas Free ElectronGas ~ ~ (19) ·· (21) At room temperature, Since · · (22) % 17 18 Combine Eq. (21) and (22), 3 3/21/2017 In 1D or 2D electron gas model, For ≪ 3/20/2017 3/20/2017 (23) Experimental heat capacity / Free ElectronGas At temperature below Debye temp and Fermi temp, the Free ElectronGas (24) heat capacity of metals can be written as the sum of / electron and lattice contributions (Eq. 15-1(#6) & eq. 26) / (25) (27) / (26) 19 20 The departure from simple free electron model come from From Eq. (26) we can see that (1) interaction with atomic potential, (2) interaction with phonon, (3) interaction with other electrons. (28) 3/20/2017 3/20/2017 Take into account Eq. (12) & (15), we can see that Free ElectronGas Free ElectronGas is proportional to the mass of the electron. Since experimental measurement of is typically different from the simple , so we usually redefine the mass of electron to force fit the equation (28), we can the fitted mass as effective mass, (29) 21 22 Thermal conductivity of metals Wiedemann-Franz Law At moderate temperature, the ratio between and σ In Drude model, the thermal conductivity K is expressed as depends on T linearly, the factor is called Lorentz number, L 3/20/2017 3/20/2017 (30) / Free ElectronGas (31) Free ElectronGas At low T, the head capacity dominated by the electronic / contribution, therefore we can substitute (26) into (30) ·· (31) 23 24 4.