Chapter 5 The Drude Theory of Metals • Basic assumption of Drude model • DC electrical conductivity of a metal • Hall effect • Thermal conductivity in a metal 1 Basic assumptions of Drude model * A “ gas of conduction electrons of mass m, which move against a background of heavy immobile ions Zρ Electron density n = .0 6022 ×10 24 m A .0 6022 ×10 24 Avogadro’s number ρm Mass density in g/cm 3 A Atomic mass in g/mole Z Number of electron each atom contribute rs Radius of a sphere whose volume is equal to the volume per conduction electron V 1 4 3 3/1 = = πr 3 r = N n 3 s s 4πn r s ~ 2 − 3 in typical metal a0 Bohr radius The density is typically 10 3 times greater than those of a classical gas at normal T and P. 2 * Between collisions the interaction of a given electron, both with others and with the ions, is neglected. * Coliisons in the Drude model are instantaneous events that abruptly alter the velocity of an electron. Drude attributed them to the electrons bouncing off the impenetrable ion cores. 1 * We shall assume that an electron experiences a collision with a probability per unit time τ Probability dt during time interval dt τ τ : relaxation time * Electrons are assumed to achieve thermal equilibrium with their surroundings only through collisons 3 DC Electrical Conductivity of a Metal r n electrons per unit volume all move with velocity v . n(vdt )A electrons will cross an area A perpendicular to the direction of flow. Charge crossing A in time dt : − nevAdt − nevAdt r r j = = −nev j = −ne v r Adt v the average electronic velocity r r When E = 0 , v = 0 r In a electric field E t: time elapse since last collision r r v0 randomly oriented, and does not contribute to average v r r eE r eE 2 Acquired velocity: − t v = − t v ~ 1.0 cm / s at 1A/mm m m r r ne 2t r j = −ne v = E The average t is relaxation time τ m 2 r ne 2τ r r r ne τ j = E j = σE σ = m m 4 m τ = ~ 10 -14 to 10 -15 sec at RT ρne 2 τ 1 3 mv 2 = k T Mean free path l = v0τ 2 0 2 B v 7 0 ~ 10 cm/sec l ~ 1 –10 A at RT Estimate of v0 is an order of magnitude too small Actual l ~ 10 3 A at low temperature, a thousand times the spacing between ions • Use Drude model without any precise understanding of the cause of collisions. •τ -independent quantities yield more reliable information m •τ calculated using τ = 2 is accurate ρne • Be cautious about quantities such as average electron velocity v, and electron specific heat cv 5 r r r r ne p(t) p(t) j At any time t, average electronic velocity v = = − m m r Momentum p ( t ) at time t r t+dt p(t + dt ) at time dt Fraction of electrons without suffering a collision from t to ( t+dt ) 1− τ r r Each of these electrons acquire an additional momentum under the influence of an external force f ( t ) : f (t)dt dt Fraction of electrons that undergo a collision: τ After a collision, the electronic velocity is randomly directed, and the average velocity is 0. 6 r The acquired velocity for each of these electrons after dt: ~ f (t)dt r f (t) 2 The contribution to momentum per electron: ~ ()dt neglected τ r r r dt r 2 r dt r 2 p(t + dt ) = 1− [p(t) + f (t)dt + O(dt ) ]= p(t) − p(t) + f (t)dt + O()dt τ τ r r r r r r dt r 2 dp(t) p(t) p(t + dt ) − p(t) = − p(t) + f (t)dt + O()dt = − + f (t) τ dt τ equivalent to a damping term Hall Effect and Magnetoresistance z y H x Ex + + + + + + + + + + + + + Ey --------------- jx -e vx r r 7 − ve × H e r r Lorentz force − v × H deflects electrons in –y direction c Electric field build up in y direction that oppose electron motion in y direction. In equilibrium the traverse field (Hall field) Ey balance the Lorentz force Ey Ex Two important parameters: resistivity ρ(H ) = Hall coefficient RH = jx H jx negative value for electrons, and positive value for positive charge R To calculate ρ ( H ) and H , consider the current density j x and j y in the presence of an electric field with arbitrary components , and in the presence of magnetic field Ex Ey H z r r r r H f = −e E + v × c r r r d p r p r p The momentum per electron = −e E + × H − dt mc τ p 0 = −eE −ω p − x x c y τ eH In steady state the current is independent of time where ω = p c 0 = −eE + ω p − x mc y c x τ r σ E = ω τ j + j ne 2τ ne τ r 0 x c y x Where σ = Multiply by − and using j = ne v 0 m m σ 0 Ey = −ωcτ jx + jy In steady state ω τ H 1 jy = 0 So c Ey = − jx = − jx RH = − σ 0 nec nec 8 9 10 11 12.