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Free Electron Theory Many solids conduct electricity. There are electrons that are not bound to atoms but are able to move through the whole crystal. Conducting solids fall into two main classes; metals and semiconductors. ( RT ) metals ;106 108 m and increases by the addition of small amounts of impurity. The resistivity normally decreases monotonically with decreasing temperature. ( RT ) pure semiconductor ( RT )metal and can be reduced by the addition of small amounts of impurity. Semiconductors tend to become insulators at low T. Why mobile electrons appear in some solids and others? When the interactions between electrons are considered this becomes a very difficult question to answer. The common physical properties of metals; • Great physical strength • High density • Good electrical and thermal conductivity, etc. This chapter will calculate these common properties of metals using the assumption that conduction electrons exist and consist of all valence electrons from all the metals; thus metallic Na, Mg and Al will be assumed to have 1, 2 and 3 mobile electrons per atom respectively. A simple theory of ‘ free electron model’ which works remarkably well will be described to explain these properties of metals. Why mobile electrons appear in some solids and not others? According to free electron model (FEM), the valance electrons are responsible for the conduction of electricity, and for this reason these electrons are termed conduction electrons. Na11 ￫ 1s2 2s2 2p6 3s1 Valance electron (loosely bound) Core electrons This valance electron, which occupies the third atomic shell, is the electron which is responsible chemical properties of Na. When we bring Na atoms together to form a Na metal, Na metal Na has a BCC structure and the distance between nearest neighbours is 3.7 A˚ The radius of the third shell in Na is 1.9 A˚ Solid state of Na atoms overlap slightly. From this observation it follows that a valance electron is no longer attached to a particular ion, but belongs to both neighbouring ions at the same time. A valance electron really belongs to the whole crystal, since it can move readily from one ion to its neighbour, and then the neighbour’s neighbour, and so on. This mobile electron becomes a conduction electron in a solid. The removal of the valance electrons leaves a positively charged ion. + + + The charge density associated the positive + + + ion cores is spread uniformly throughout the metal so that the electrons move in a constant electrostatic potential. All the details of the crystal structure is lost when this assunption is made. According to FEM this potential is taken as zero and the repulsive force between conduction electrons are also ignored. Therefore, these conduction electrons can be considered as moving independently in a square well of finite depth and the edges of well corresponds to the edges of the sample. Consider a metal with a shape of cube with edge length of L, Ψ and E can be found by solving Schrödinger equation V 2 2 E Since, V 0 2m L/2 0 L/2 • By means of periodic boundary conditions Ψ’s are running waves. ( x L, y L, z L) ( x, y, z) The solutions of Schrödinger equations are plane waves, 1 ik r 1 i ( kx x k y y kz z ) ( x, y , z ) e e V V Normalization constant where V is the volume of the cube, V=L3 2 2 2 2 Na p Na p where, k k p p k Na L So the wave vector must satisfy 2 ; k 2 q ; k 2 r kx p y L L z L where p, q, r taking any integer values; +ve, -ve or zero. The wave function Ψ(x,y,z) corresponds to an energy of 2 2 2 k E (k x 2 k y 2 k z 2 ) E 2m 2m the momentum of p (k x , k y , k z ) Energy is completely kinetic p k 2 2 1 2 mv k m2v 2 2 k2 2 2m Typical values may be obtained by using monovalent potassium metal as an example; for potassium the atomic density and hence the valance electron density N/V is 1.402x1028 m-3 so that EF 3.40 10 19 J 2.12eV k F 0.746 A1 Fermi (degeneracy) Temperature TF by EF k BTF EF TF 2.46 104 K kB Typical values of monovalent potassium metal; 2/3 2 3 N 2 EF 2.12eV 2m V 1/ 3 3 N 2 kF 0.746 A1 V PF 1 VF 0.86 10 ms 6 me EF TF 2.46 104 K kB The free electron gas at finite temperature At a temperature T the probability of occupation of an electron state of energy E is given by the Fermi distribution function 1 f FD 1 e( E EF ) / kBT Fermi distribution function determines the probability of finding an electron at the energy E. Fermi Function at T=0 and at a finite temperature 1 fFD=? At 0°K f FD ( E EF ) / k BT 1 e fFD(E,T) i. E<EF 1 f FD ( E EF ) / k BT 1 1 e 0.5 ii. E>EF 1 f FD ( E EF ) / k B T 0 1 e E E<EF EF E>EF The Electrical Conductivty In the presence of DC field only, eq.(*) has the steady state solution e e v E e Mobility for me electron me a constant of proportionality (mobility) Mobility determines how fast the charge carriers move with an E. Electrical current density, J e N J n(e)v v E n me V Where n is the electron density and v is drift velocity. Hence ne 2 ne 2 J E Electrical conductivity me me Ohm’s law Electrical Resistivity and Resistance 1 L J E R A Collisions In a perfect crystal; the collisions of electrons are with thermally excited lattice vibrations (scattering of an electron by a phonon). This electron-phonon scattering gives a temperature dependent ph (T ) collision time which tends to infinity as T 0. In real metal, the electrons also collide with impurity atoms, vacancies and other imperfections, this result in a finite scattering time 0 even at T=0. The total scattering rate for a slightly imperfect crystal at finite temperature; 1 1 1 ph (T ) 0 Due to phonon Due to imperfections So the total resistivity ρ, me me me I (T ) 0 ne 2 ne ph (T ) ne 0 2 2 Ideal resistivity Residual resistivity This is known as Mattheisen’s rule and illustrated in following figure for sodium specimen of different purity. Residual resistance ratio Residual resistance ratio = room temp. resistivity/ residual resistivity and it can be as high as 106 for highly purified single crystals. impure pure Temperature Thermal conductivity, K Due to the heat tranport by the conduction electrons Kmetals Knon metals Electrons coming from a hotter region of the metal carry more thermal energy than those from a cooler region, resulting in a net flow of heat. The thermal conductivity 1 K CV vF l where CV is the specific heat per unit volume 3 vF is the mean speed of electrons responsible for thermal conductivity since only electron states within about k BT of F change their occupation as the temperature varies. l is the mean free path; l vF and Fermi energy F 1 mevF 2 2 1 12 N T 2 2 nk BT 2 2 T K CV vF 2 kB ( ) F where Cv Nk B 3 3 2 V TF me 3me 2 TF Wiedemann-Franz law ne 2 2 nkBT 2 K me 3me The ratio of the electrical and thermal conductivities is independent of the electron gas parameters; 2 kB 2 K Lorentz 2.45 x108W K 2 number T 3 e K L 2.23x108W K 2 For copper at 0 C T

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