Electrons in a periodic potential: Free electron approximation

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1 Dr. A. Sapelin, Jan 01 Electrons in a periodic potential: ree electron approximation ree electron ermi gas - gas of non-interacting electrons subject to Pauli principle Wealy bound electrons move freely through metal Assume valence electrons conduction electrons Neglect electron ion core interaction Valence electron= immobile electron involved in the bonding Conduction electron= mobile electron able to move within the solid Good model for metals; can explain: () K Electrical conductivity Optical properties Thermal conductivity C v Specific heat capacity Quite a few of the properties of interest (e.g. C V ) are defined by or dependent on the total energy of the system, so we need a way of calculation the total energy based on a microscopic consideration. One way of doing it is to obtain information about the energy of each particle in the system and then add up all the energies. We now from QM that one has to solve the Schrödinger for a quantum mechanical system in order to obtain all available energy states. The general form of the equation is: where in general ( r, t). However, for the majority of problems in solid state physics we can safely assume (r). In our case the system is a periodic lattice and the energy term would in general include the inetic and the potential parts: Solving such an equation for a simple system is a reasonably involved tas and even more so for a large system of atoms and electrons. So, as a first approximation we assume a free electron approximation.

2 Dr. A. Sapelin, Jan 01 We can then consider D Schrödinger equation for a free electron, ( r ) ( r ) m x y z Consider metal cube of side L=V 1/ Periodic boundary conditions for box dimensions L ( x, y, z L) ( x, y, z) ( x, y L, z) ( x, y, z) ( x L, y, z) ( x, y, z) Solution: r 1 i.r ( ) e - a plane wave with energy V ( ) m (we have used the normalisation condition dr ( r) 1) ( ) m x i L y izl Boundary conditions permit only discrete values of since: e x e e 1 i L x n x, L y n y, L z n z, n x, n y, L n z integers y In -space (D representation) In d dimensions, the volume per point is (π/l) d x /L

3 Dr. A. Sapelin, Jan 01 V - allowed values of ( / L) 8 In practice: L is very large allowed -space points form continuum m x y z x Now, counting all the electron energies in the system one by one is a bit of a tedious tas, especially for a large system hence it would be good to find an alternative. or a sufficiently large system we can consider -space continuum rather than discreet states. Under these conditions we hope to move from finding a sum of a sequence to integration. The ground state of N free electrons described by -space sphere of a radius (ermi wavevector), energy at the surface (ermi energy): m z x y Total number of allowed values within sphere is 4 V 8 V 6 Each -value leads to two one-electron levels (one for each spin value Pauli Exclusion Principle) N = V 6 Electronic density, n

4 Dr. A. Sapelin, Jan 01 Pauli Exclusion Principle: no two electrons can have all quantum numbers identical Density of states (in energy); Number of states < ɛ, N V m Now we are almost ready to calculate total electron energy of a system. Classically: resulting in But we now from experiments that this is not correct. So, where is the problem? The problem is in that we have to factor in the Pauli Exclusion Principle and the ermi-dirac distribution f(ɛ). With these in mind Where is the small energy range around and is the density of states dn 1 m D ( ) d 1 THERMAL PROPERTIES O THE REE ELECTRON ERMI GAS: THE ERMI-DIRAC DISTRIBUTION N wealy interacting ermions in thermal equilibrium at temperature T have occupancy probability f(ɛ) given by the ermi-dirac distribution 1 f ( ) ( ) / e T B 1 Where: ɛ is the ermi energy and B is the Boltzman constant

5 Dr. A. Sapelin, Jan 01 f( B T f( 1 T=0 K T T T > T 1

6 Dr. A. Sapelin, Jan 01 At all temperatures f(ɛ) =1/ when ɛ=ɛ Some textboos will use the chemical potential μ instead of ɛ in the ermi Dirac equation. This strictly correct but we will use ɛ because ɛ =μ to good approximation for most purposes Df inite T D( T=0 K Actual occupation 1 1/ f( N D( ) f (, T) d 0 = shaded area or real metals, V N is very high so that ɛ >> B T at RT and below (typically ɛ 5 ev, while B T =1/40 RT)

7 Dr. A. Sapelin, Jan 01 HEAT CAPACITY O A REE ELECTRON GAS inite T >> B T A T=0 K Df B As T increases from 0 K, points in shaded area A become unoccupied and points in B become occupied increase in energy U U(number of electrons that have increased energy) x (energy increase for each one). We can also see that electrons near ermi level are the ones contributing to the energy increase. Then for ɛ >> B T : Measured from ɛ = 0. And for the heat capacity: and since we obtain: [ ] [ ] [ ]

8 Dr. A. Sapelin, Jan 01 which leads to [ ] In real metals, C v AT D( ) B T Heat capacity of lattice C V T Electronic contribution dominates at low T Measures D(ɛ ) T ELECTRICAL CONDUCTIVITY O REE ELECTRON GAS Remember that for free electron, ( ) m p m Momentum related to wavevector by: m v In electric field E and magnetic field B the force on electron, e ( E v B) When B=0,

9 Dr. A. Sapelin, Jan 01 ee Equation of motion, dv d m dt dt ermi sphere is displaced at constant rate as the value of the state occupied by each electron changes uniformly and equally, eet ( t) (0) t=0 t>0 y y x x E, d/dt Sphere reaches steady state due to scattering: τ = relaxation time = scattering time = time to change state by collision with impurities/ lattice imperfection/lattice vibrations dv d m dt dt ee for each electron

10 Dr. A. Sapelin, Jan 01 If n electrons per unit volume and v inc =incremental (drift) velocity increase: j n( e) v inc E e ne n( e) E E m m NB Kittel uses v for actual velocity and also for incremental velocity Electrical conductivity defined as j=e Drude equation ne m Resistivity is defined as 1/ so m ne At RT, τ ~10-14 s l=mean free path between collisions=vτ or electrons at the ermi surface, l= v τ At RT, l~0.1 μm, i.e. > lattice spacing

11 Dr. A. Sapelin, Jan 01 ree electron approximation Successes Temperature dependence of Heat Capacity paramagnetic (Pauli) susceptibility Ratio of thermal and electrical conductivities (Lorentz number) Magnitudes of heat capacities and Hall effect in simple metals ailures Heat capacities and Hall effect of many metals are wrong Hall effect can be positive Does not explain why mean free paths can be so long Does not explain why some materials are metals, some insulators and some are semiconductors

Unit III Free Electron Theory Engineering Physics

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