CHAPTER 4 FREE ELECTRON THEORY Classification of Solids Electrical Properties Some solids conduct current at all temperatures and, generally, the resistivity of such solids increases with temperature. These are METALS Other solids stop conducting at low temperatures and their resistivity falls with increasing temperature. These INSULATORS and SEMICONDUCTORS 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. V ( RT ) metals ;10 10 8 ; m and increases by the addition of small amounts of impurity. The resistivity normally decreases monotonically with decreasing temperature. V(RT)puresemiconductor ? V(RT)metal and can be reduced by the addition of small amounts of impurity. Semiconductors tend to become insulators at low T. Free Electron Theory 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 µ free electron model¶ which works remarkably well will be described to explain these properties of metals. Free Electron Theory 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 Core electrons Valance electron (loosely bound) 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 AÛ The radius of the third shell in Na is 1.9 AÛ 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. + + + Free Electron Theory Classical and Quantum Free Electron Models of Electrical Conductivity Electron Theory of Metals(DrundeLorentz) Postulates of CFEM ** In an atom electrons revolue around the nucleus and a metal is composed Of such atoms 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. + + + + + + CFEM These free electrons move in random directions and collide with either positive ions fixed to the lattice or other free electrons. All the collision are elastic. i.e. there is no loss of energy. The movements of free electrons obey the classical free electron theory of gases CFEM The electron velocities in a metal obey the Classical Statistics. i.e Maxwell-Boltzmann Distribution of velocities . When an electric field is applied to the metal ,the free electrons are accelerated in the direction opposite to the direction of applied electric field. Electrical Conductivity In metals free electrons roam freely through the crystal lattice . In the absence of applied external field the net current due to the movement of electrons is zero since they are randomly in all directions . In between two collisions the electron move with uniform velocity. Electrical Conductivity + + + During every collision both the direction And the magnitude of velocity change. + + + Electrical Conductivity According to the Ohm¶s Law I=V/R R resistance of the wire ** The current is due to the motion of the conduction electrons under the influence of the electric field. ** The field E exerts a force ±eE on the electron When an external field applied the electrons accelerated. Here we consider the frictional force acting on the electron due to the collision. IfY is the velocity of the electron and X is the time between two consective collisions The frictional force can be written as Y F ! m X useNewton ' sLaw dY Y ! eE m dt X underStead yStateCond ition m This is the steady-state dY !0 dt velocity eX Y ! E m Drift Velocity And Mean Free Path In the absence of the field the electrons have random motion ,just as gas molecules move randomly in a gas container ,the randomly moving electrons undergo scattering and the change the direction. This random motion contributes zero current and the corresponding velocity is called the random velocity Drift Velocity And Mean Free Path In the presence of a field, in addition to random velocity ,there is an additional net velocity associated with electrons called drift velocity due to applied electric field . Due to drift velocity (vd )electrons with negative charge move opposite to the field direction Drift Velocity And Mean Free Path If n is the number of conduction per unit volume ,then the charge per unit volume is (±ne).The amount of charge crossing a unit area per unit time is given by the current density J J ! ( ne) ne 2X ! E m J ! WE ne 2X W! m ! ne( eX E) m d Drift Velocity And Mean Free Path Mean Free Path: The average distance traveled by an electron between two successive collision in the presence of applied field is known as Mean Free Path Relaxation Time Relaxation Time can be defined as the time taken for the drift velocity to decay to 1/e of its initial value. Let assume that the applied field is cut off after the drift velocity of the electron has reached its steady value.Drift velocity after this instant is governed by dY d Y ! dt X dY d dt ! Yd X Y d (t ) ! Y d (0) exp( t / X ) Vd(0)is the stedy state drift velocity Relaxation Time Vd(0) Let t=T Vd(t)=vd(0)/e vd t Mobility Mobility of the electron is defined as the steady state drift velocity<vd> per unit electric field. Q! ! Yd E ne 2X ! eX eX ! ne. ! neQ ! neQ ne 2X Where ( V ) resistivit y V! 1 ! The electrical conductivity depends on two factors ,the charge density n and their mobility . These two quantities depend on temperature. In metals n is constant and decreases slightly with temperature and hence with increase of temperature ,the conductivity decreases. In semiconductors the exponential increase of n with temperature is responsible for increase of conductivity with temperature In insulator n remains constant and above certain temperature increase exponentially resulting in dielectric breakdown Success of classical free electron theory ** It verifies Ohm¶s Law ** It explains the electrical and thermal conductivities of metals ** It derives Wiedemann-Franz Law ** It explains Optical Properties of metals. Drawbacks of Classical Free Electron Theory The phenomena such as photoelectric Effect, Compton Effect and the Black Body Radiation couldn¶t be explained by free electron theory. According to CFET the value of specific heat of metals is given by 4.5Ru.Where as the experimental value nearly equal to 3Ru. Electrical conductivity of Semiconductor or Insulators Couldn¶t be explained using this model Quantum Theory of Free Electrons This theory is proposed by Sommerfeld in 1928,with help of quantum of statistics (FermiDirac) explained QFET. The difficulty of classical FET arises M-B it permits all the free electrons to gain energy. But in Quantum Statistics turn out that only about one percent of the free electron to gain a energy Free electron moving in uniform potential within in a metal .potential field inside the metal not uniform. But instead ,the field experienced by a moving free electrons varies periodically with the periodicity of the crystal. To determine the restriction imposed by quantum mechanics the energies that free electron can have inside the metal It is assumed that valence electrons are traped in constant potential well V(x) 2 J 2 n2h2 k E(n) ! ! 2 2m 8mL X=0 X=L Quantum Theory of Free Electrons When an external electric field E is applied the force exerted on the electron is ±eE. Since force is also rate of change of momentum dp eE ! dt h h 2T v !J p! ! k P 2T P d dk ( eE ! (J) ! J ) k dt dt eE dk ! dt J Quantum Theory of Free Electrons This Means that origin of the k space moves through a distance dk in time dt on application of external field. Because of collision with imperfection, displacement of k space becomes steady k and dt is then the average collision time Quantum Theory of Free Electrons (k ! eEX J p ! m ! J, incrementalVelocity( k ( ! J J eEX (k ! ( )( ) m m J eEX ( ! m No.of e-s per unit volume is n,then J is J ! n( e ) ( ne 2 EX J! m J ! WE ne 2X W! m This treatment tell us that current carried out by very few electrons Free Electron Models Classical Model: Metal is an array of positive ions with electrons that are free to roam through the ionic array Electrons are treated as an ideal neutral gas, and their total energy depends on the temperature and applied field In the absence of an electrical field, electrons move with randomly distributed thermal velocities When an electric field is applied, electrons acquire a net drift velocity in the direction opposite to the field Quantum Model: Electrons are in a potential well with infinite barriers: They do not leave metal, but free to roam inside Electron energy levels are discrete (quantized) and well defined, so average energy of electron is not equal to (3/2)kBT Electrons occupy energy levels according to Pauli¶s exclusion principle Electrons acquire additional energy when electric field is applied Fermi-Dirac distribution . The Fermi function f(E) specifies how many of the existing states at the energy E will be filled with electrons. The function f(E) specifies, under equilibrium conditions, the probability that an available state at an energy E will be occupied by an electron. It is a probability distribution function. EF = Fermi energy or Fermi level k = Boltzmann constant = 1.38v 1023 J/K = 8.6 v 105 eV/K T = absolute temperature in K Fermi-Dirac distribution function At a temperature T the probability of occupation of an electron state of energy E is given by the Fermi distribution function f FD ! 1 1 e ( E EF ) / kBT Fermi distribution function determines the probability of finding an electron at the energy E. Fermi-Dirac distribution: Consider T p 0 K 1 f ( E " EF ) ! ! 0 1 exp (g) For E > EF : For E < EF : E f (E 1 EF ) ! ! 1 1 exp (g) EF 0 1 f(E) Temperature dependence of FermiDirac distribution Fermi Function at T=0 and at a finite temperature f FD ! 1 1 e ( E E F ) / kBT fFD=? At 0°K fFD(E,T) i. E<EF f FD ! 1 e 1 ( E EF ) / k B !1 0.5 ii. E>EF f FD ! E E<EF EF E>EF 1 ( E EF ) / kB 1 e !0 Fermi-Dirac distribution function at various temperatures, Fermi-Dirac distribution function At any temperature other than 0k,if E=Ef F(E)=1/2 Fermi level is that state at which the probability of electron occupation is ½ at any temperature above 0k and also it is the highest level of the filled energy state at 0k Fermi-Dirac distribution function Fermi energy is the energy of the state at which the probability of electron occupation is ½ at any temperature above 0k Electrons with Fermi energy move with Fermi velocity and the same is related to the Fermi temperature by the relation 1 2 mY F ! kT 2 Thank u