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Free electron theory of Metals Introduction • The electrons in the outermost orbital's of the atoms determine its electrical properties. • The electron theory of solids aims to explain the electrical, thermal and magnetic properties of solids. • This theory has been developed in three main stages. 1.Classical free electron theory 2.Quantum free electron theory 3.Band theory of solids. • Classical free electron theory: Drude and Lorentz developed this theory in 1900. According to this theory the metals containing free electrons obey the laws of Classical Mechanics. • Quantum free electron theory: Somerfield developed this theory during 1928. According to this theory free electrons obey the Quantum laws. • Band theory of solids or Zone theory: Bloch stated this theory in 1928. According to this theory, the free electrons move in a Periodic field provided by the lattice and the theory is also called Band theory of solids. Classical free electron theory o In an atom electrons revolve around the nucleus and every metallic substance is composed of such like atoms. o The valance electrons of atoms are free to move about the whole volume of metals like the molecules of a perfect gas in a container. o These free electrons moves in random directions and collide with either positive ions fixed to the lattice or other free electrons. o These all collisions are elastic that is there is no loss of energy. o The electron velocities in a metal obey the laws of Classical Maxwell - Boltzmann distribution. • The free electrons move in a completely uniform potential field due to ions fixed in the lattice. • When an electric field is applied to the metal, the free electrons are accelerated in the direction opposite the direction of applied electric field. • When an electric field E is applied on an electron of charge e then it moves in opposite the direction of applied electric field with a velocity Vd. Electric field Fixed Positive metallic ions The Lorentz force acting on the electron FL eE The frictional force or opposing force can be expressed as Fr mvd r Where vd is a drift velo city & r is a relaxation time When the system is in a steady state total force F{FL Fr } 0 m mv dv eE d dt r eE vd ( mvd r e r m 0 )E • If n is the number of conduction electrons 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. i J A (ne)vd (ne)( e E) m ne 2 E E m ne 2 where Electrical Conductivi ty m Drift velocity: The amount of velocity gained by the electron by the application of unit electric field is known drift velocity. • • • • Relaxation time: The duration of a time required for an electron to decay its drift velocity to 1/e times of its initial velocity. When electric field is applied on an electron then it drifts in opposite direction to the field applied. After removal of electric field the drift velocity decays exponentially. Let us assume that the applied field is cut off after the drift velocity of the electron has reached its steady value. dvd v m d dt dvd v d dt dvd dt vd m t t dvd dt 0 vd 0 {log vd }t0 t log vd (t ) log vd (0) Relaxation time can be defined as the v (t ) t log d time taken for the drift velocity to decay v d ( 0) to 1/e of its initial value. t vd (t ) vd (0) exp( ) ift vd (t ) v d ( 0) e t Mean free Path λ: The average distance travelled by an electron between two successive collisions in the presence of applied field is known as mean free path. Mobility µ: Mobility of the electron is defined as the steady state drift velocity per unit electric field. e E vd e m E E m ne 2 e conductivity ne. ne m m 1 m 1 resistivit y 2 ne ne Drawbacks of Classical free electron theory 1. The phenomena such as photoelectric effect, Compton effect and black body radiation couldn't be explained by classical free electron theory. 2. According to the classical free electron theory the value of specific heat and electronic specific heat of metals is given by 4.5Ru and 3/2Ru while the actual values are about 3Ru and 0.015Ru. 3. Electrical conductivity of semiconductor or insulators couldn’t be explained using this model. 4. Ferromagnetism couldn’t be explained by this theory. 5. Wiedemann-franz law ( K / σT =L ) according to the classical free electron theory, it is not constant at low temperature. • • • • • Quantum theory of free electrons Somerfield proposed the quantum free electron theory and he assumed that the valance electron are free in a metal piece and they obey quantum laws . According to quantum theory the free electrons occupy different energy levels present in the metal. According to this theory only Fermi level electrons are responsible for electrical conductivity and other properties of metals. By this theory Somerfield exactly predicted the values of specific heat of metals ,electrical conductivity and thermal conductivity of metals. If free electrons are moving with very higher velocities, the potential energy becomes zero. Total energy of outer most electron E= KE + PE= p2/2m + V → p2/2m From de Broglie Matter wave equation h p 2 2 p p k h h ( ) p 2 p K p2 fromeq (1) E 2m 2k 2 E 2m n k L n 2 2 ( ) 2 2 2 2 2 n h n L E E E 2m 2mL2 8mL2 n When an external electric field E is applied the force exerted on the electron is F eE dp eE dt d (k ) eE dt dk eE eE dk dt dt dv dp F m dt dt dv d (k ) m dv dk dt dt m eE eE dv ( dt ) dt m m If the no of electrons per unit volume is n, then the current density J is given by…. J ne( dv ) eEdt J ne( ) m ne 2 dt J E m J E ne 2 dt where m Draw backs •It does not explain temperature variation of electrical conductivity •It does not explain why metals prefer only certain structures. •Distribution means how the electrons in materials distribute among the different possible energy states. •A small metal piece contains very large number of electrons •Each electron posses quantized energy states . •The electrons obey Pouli’s exclusion principle. •Hence the suitable statistics to find the distribution of electrons in a metal is Fermi Dirac distribution. According to this law, the probability F(E) or P(E) of an electron occupying that a given energy state ‘ E ’ at temperature ‘ T ’ is given by F(E) = P(E) = 1 --------- (1) 1 + e (E –EF) / KBT Where, F(E) = P(E) = Fermi – Dirac probability function KB = Boltzmann constant Explanation(F –D statistics) E E At T=0K EF Where, E3 > E 2 > E1 > E F T3 > T 2 > T 1 > T E3 E2 E1 At T=0K EF O 1.0 T1 T2 T3 P(E) Figure-1 O 1.0 Figure-2 P(E) From equation (1) we may discuss the following 2 cases. CASE:1 (1) Let T=0K for E < EF , e(E-E F/KT) = 0 F(E)=1 This means all energy states below EF are completely filled. (2) Let T=0K for E > EF , e(E-E F/KT) = F(E)=0 then the exponential term becomes infinite and F(E) = 0 i.e. there is no probability of finding an occupied state of energy greater than EF at absolute zero. Hence, This means all energy states EF are empty. That means EF is the Fermi energy level is the energy level of the Highest occupied states at 0k. CASE:2 Let T> 0K and E = EF , Then F(E) = P(E) = ½ at any temperature. Thus, Fermi level is that energy level for which the probability of occupation is (½) at any temperature above 0K. The effect of temperature on Fermi Dirac Distribution As the temperature increases, thermal agitations grows up. As a result more and more electrons jump up to the levels above EF from the levels below EF leaving vacancies as shown EF EF T=0 T>0 When T > 0 K , some of levels below EF are not completely filled up and some of the levels above EF are not completely empty. They are partially filled and partially empty. Both the electrons in the levels above EF and vacancies in the levels below EF contribute for conduction in semi conductors. But the metals are good conductors even at room temperatures. If the temperature raised, the resistance of the metals increases due to increase of the collisions or decrease of the mobility. Use : By using this we can explain specific heat of metals. Free electron theory of Metals Introduction • The electrons in the outermost orbital's of the atoms determine its electrical properties. • The electron theory of solids aims to explain the electrical, thermal and magnetic properties of solids. • This theory has been developed in three main stages. 1.Classical free electron theory 2.Quantum free electron theory 3.Band theory of solids. • Classical free electron theory Drude and Lorentz developed this theory in 1900. According to this theory the metals containing free electrons obey the laws of Classical mechanics. • Quantum free electron theory Somerfield developed this theory during 1928. According to this theory free electrons obey the Quantum laws. • Band theory of solids or Zone theory Bloch stated this theory in 1928. According to this theory, the free electrons move in a Periodic field provided by the lattice and the theory is also called Band theory of solids. Classical free electron theory o In an atom electrons revolve around the nucleus and every metallic substance is composed of such like atoms. o The valance electrons of atoms are free to move about the whole volume of metals like the molecules of a perfect gas in a container. o These free electrons moves in random directions and collide with either positive ions fixed to the lattice or other free electrons. o These all collisions are elastic that is there is no loss of energy. o The electron velocities in a metal obey the laws of Classical Maxwell - Boltzmann distribution. • The free electrons move in a completely uniform potential field due to ions fixed in the lattice. • When an electric field is applied to the metal, the free electrons are accelerated in the direction opposite the direction of applied electric field. • When an electric field E is applied on an electron of charge e then it moves in opposite the direction of applied electric field with a velocity Vd. Positive ion cores _ _ _ _ _ _ + + + + + + + + + + + + + + + + Electron clouds _ _ _ _ _ _ The Lorentz force acting on the electron FL eE The frictional force or opposing force can be expressed as Fr mvd r Where vd is a drift velo city & r is a relaxation time When the system is in a steady state total force F{FL Fr } 0 m mv dv eE d dt r eE vd ( mvd r e r m 0 )E • If n is the number of conduction electrons 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. i J A (ne)vd (ne)( e E) m ne 2 E E m ne 2 where Electrical Conductivi ty m Drift velocity: The amount of velocity gained by the electron by the application of unit electric field is known drift velocity. • • • • Relaxation time: The duration of a time required for an electron to decay its drift velocity to 1/e times of its initial velocity. When electric field is applied on an electron then it drifts in opposite direction to the field applied. After removal of electric field the drift velocity decays exponentially. Let us assume that the applied field is cut off after the drift velocity of the electron has reached its steady value. dvd v m d dt dvd v d dt dvd dt vd m t t dvd dt 0 vd 0 {log vd }t0 t log vd (t ) log vd (0) Relaxation time can be defined as the time taken for the drift velocity to decay to 1/e of its initial value. log vd (t ) t v d ( 0) t vd (t ) vd (0) exp( ) ift vd (t ) v d ( 0) e t Mean free Path λ: The average distance travelled by an electron between two successive collisions in the presence of applied field is known as mean free path. Mobility µ: Mobility of the electron is defined as the steady state drift velocity per unit electric field. e E vd e m E E m ne 2 e conductivity ne. ne m m 1 m 1 resistivit y 2 ne ne Drawbacks of classical free electron theory 1. The phenomena such as photoelectric effect, Compton effect and black body radiation couldn't be explained by classical free electron theory. 2. According to the classical free electron theory the value of specific heat and electronic specific heat of metals is given by 4.5Ru and 3/2Ru while the actual values are about 3Ru and 0.015Ru. 3. Electrical conductivity of semiconductor or insulators couldn’t be explained using this model. 4. Ferromagnetism couldn’t be explained by this theory. 5. Wiedemann-franz law ( K/œT =L ) according to the classical free electron theory, it is not constant at low temperature. Quantum theory of free electrons • Somerfield proposed the quantum free electron theory and he assumed that the valance electron are free in a metal piece and they obey quantum laws . • According to quantum theory the free electrons occupy different energy levels present in the metal. • According to this theory only Fermi level electrons are responsible for electrical conductivity and other properties of metals. • By this theory Somerfield exactly predicted the values of specific heat of metals ,electrical conductivity and thermal conductivity of metals. • If free electrons are moving with very higher velocities, the potential energy becomes zero. Total energy of outer most electron E= KE + PE= p2/2m + V → p2/2m From de Broglie matter wave equation…. h p 2 2 p p k h h ( ) p 2 p K p2 fromeq (1) E 2m 2k 2 E 2m n k L n 2 2 ( ) 2 2 2 2 2 n h n L E E E 2m 2mL2 8mL2 n • When an external electric field E is applied the force exerted on the electron is F eE dp eE dt d (k ) eE dt dk eE eE dk dt dt dv dp F m dt dt dv d (k ) m dv dk dt dt m eE eE dv ( dt ) dt m m If the no of electrons per unit volume is n, then the current density J is given by…. J ne( dv ) eEdt ) m ne 2 dt J E m J E J ne( ne 2 dt where m Draw backs •It does not explain temperature variation of electrical conductivity •It does not explain why metals prefer only certain structures.