제 4 장 Metals I: The Free Electron Model Outline 4.1 Introduction 4.2 Conduction electrons 4.3 the free-electron gas 4.4 Electrical conductivity 4.5 Electrical resistivity (T) 4.6 heat capacity of conduction electrons 4.7 Fermi surface 4.8 Electrical conductivity: effects of the Fermi surface 4.9 Thermal conductivity in metals 4.10 Motion in a magnetic field; cyclotron resonance and the Hall effect 4.11 AC conductivity and optical properties 4.12 Thermionic emission 4.13 failure of the free-electron model

4.1 Introduction Metals in life from ancient to future. Example: Duralumin (Cu, Mn, and Mg) in Car, Cu in electrical wire, Ag and Au as jewelry Common properties: physical strength, density, electrical and thermal conductivity Optical (visible) reflectivity Explanation of many physical properties by assuming cloud of free-electrons all over sample Most of optical properties can be explained but with a limit.

4.2 Conduction electrons Na gas; a simple collection of free atoms. Each atom has 11 electrons orbiting around the nucleus. 11 Na: 1s 2 2s 2 2p 6 3s 1 1 valence el. loosely bound to the rest of the system(=Na 1+ ). can explain ordinary chemical reaction. Na atoms to form a metal bcc structure (section 1.7) d = 3.7  Å Core electrons Valence electron d

4.2 Conduction electrons (p138) Na atoms to form a metal. bcc structure (section 1.7) two atoms overlap slightly  valence el. can hop from one to the neighboring atom  move around freely all over the crystal.  conduction electrons in a crystal !! Naming since they carriers electrical current in an electric field. Cf) core electrons do not give electrical current. N=concentration M’=atomic weight, Z = atomic valence (1+, 2+,..),  m =density of the substance Fig. 4.1 overlap of the 3s in solid sodium

4.3 Free electron gas (p140) e cond. is completely free, except for a potential at the surface. Free motion except occasional reflection from the surface which confines e cond.. Like gas!! So called free electron gas! How about interaction between 1) e cond. and e core.? a)Strong coulomb attraction between e cond. and core ion is compensated by repulsive potential due to array of core ion by quantum effect. b)e cond. spend tiny time near core ion. c)Short-range screened coulomb potential rather than a long-range pure coulomb potential.

Dipole field and potential Quadrapole field and potential

interaction between 2) conduction electrons themselves? Pauli exclusion principle:  =  1 (a)+  2 (b) 1)electrons of parallel  or  spins tend to stay away. 2)electron pair having opposite spin  can stay together Each electron is surrounded by a hole” The hole with a radius of about 1 Å. The hole move with each electron. Interaction between two specific electron is always screened by the other electrons.  very low interaction between the two electrons. Free electron gas vs ordinary gas 1)charged, i.e. like plasma 2)N is very large N~10 23 cm -3 while ordinary gas ~ cm -3 Jellium model : ions form a uniform jelly into which e cond. move around

4.4 electrical conductivity (p142) Law of electrical conduction in metal : Ohm’s law J: current density per area, E=electrical field Substitute into (4.2) Substitute J and E (4.3) and (4.4) Electric field accelerates e cond. (not ions in the crystal) thus electric current.

Newton’ equation with friction force with  = collision time. Collision and friction tends to reduce the velocity to zero. Final (or steady) state solution. Terminal velocity (drift velocity) vs. random velocity 4.4 electrical conductivity (p142)

Newton’ equation with friction (collision) force with  = collision time. Collision and friction tends to reduce the (average vector)velocity to zero. Terminal velocity (drift velocity) vs. random velocity J: current density per area is proportional to E field,

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 = collision time, mean free lifetime?  ~ sec

metal semiconductor

 = relaxation time?  ~ sec  vs. mean free path and random velocity

Microscopic expression for Joule heat = power per volume absorbed by the electron system from the electric field. (i.e. current), Maybe Skip

Origin of collision time l ≈ m ≈ 10 2 Å Quantum mechanics, deBroglie relation: A light wave traveling in a crystal is not scattered at all. The ions form a perfect lattice, no collision at all, that is l =∞ and hence τ≈∞, leads to infinite conductivity. In reality, l ≈ 10 2 Å. d≈ 4Å 25x unit cell Well trained soldiers at ground, Movie (hero)

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4.5 Electrical resistivity versus temperature varies with the metal’s temperature, so do resistivity Fig. 4.5 The normalized resistivity  (T)/  (290 K) versus T for Na at low Temp. (a), and high Temp. (b).  (290 K) ≈ 2.10x10 -8  m. Deviation from perfect lattice: a.Lattice vibration (phonon) of the ions around their equilibrium position due to thermal excitation of the ions. b.All static imperfections (impurities or defect). Electron undergoes a collision only because the lattice is not perfectly regular. probability of the electron scatter per unit area

τ ph : collision time due to phonon τ i : collision time due to impurities Probability of electron being scattered by phonon and impurities are additive, since these 2 mechanisms are assumed to act independently. Substitute (4.16) into (4.15)  splits into 2 terms: 1.Due to scattering by impurities (independent of T)  residual resistivity 2.Due to scattering by phonon (dependent of T)  ideal (intrinsic?) resistivity.

τ ph : collision time due to phonon τ i : collision time due to impurities  splits into 2 terms: 1.Due to scattering by impurities (independent of T)  residual resistivity 2.Due to scattering by phonon (dependent of T)  intrinsic low T: scattering by phonon is negligible (amplitude of oscillation very small) τ ph  ∞ ;  ph  0 hence  =  i  constant As T   scattering by phonons more effective  ph (T)   high T:  ph (T)  linearly with T

Assumption: collision is of the hard sphere l i : mean free path collision impurity  i : scattering cross section of an impurity Substitute (4.18) and (4.19) into (4.17)  i proportional to N i Expectation: (impurity concentration) For small N i,  i proportional to N i   ph >>  i, T.

4.6 Heat capacity of conduction electrons Drude-Lorentz model  kinetic theory of gasses that a free particle in has average energy Average energy per mole: N A : Avogadro number R= N A k Total specific heat in metal C ph : specific heat phonon C e : specific heat electrons Experiment C in metal T Contribution C e <<

Fig. 4.6 (a) occupation of energy levels according to the Pauli exclusion principle. (b) the distribution function f(E) versus E, at T=0 0 K and T>0 0 K. E in metal is quantized. The electrons fill the energy level from the lowest level to the highest. The highest energy occupied level is called Fermi energy or Fermi level. Distribution function of electrons: f(E): Probability that the energy level is occupied by electrons. f(E)=0  level certainly empty f(E)=1  level certainly

Only electron in the range kT are excited of E f only fraction are affected. Total number of electron excited per mole= Each electron absorb energy kT Fig. 4.6 (b) the distribution function f(E) versus E, at T=0 0 K and T>0 0 K.

T f : Fermi temperature  The solid must be heated by T f, but impossible. Solid would long since have melted and evaporated, therefore specific heat of electrons far below its classical value. linear function of temperature. Unlike specific heat of lattice which is constant at high temperature and proportional to T 3.

4.7 The Fermi Surface Electrons in metal are in continuous state random of motion. v:speed of particle velocity of space (v x, v y, v z )  magnitude and direction of electron. Electron in FS move very fast. Fig. 4.7 The Fermi surface and the Fermi sphere Fig. 4.6 (b) The distribution function f(E) vs E f at t=0K and T>0K

4.8 Electrical conductivity; effect of the Fermi surface Fig. 4.8 (a) The Fermi surface at equilibrium. (b) Displacement of the Fermi sphere due to an electric field. Absence of an electric field: FS is centered at origin. Electron moving at high speed and carry currents. Total current system is zero. Electric field (+) is applied: electron acquired v d. FS is displaced to the left.

Current density: The fraction electron which remain uncompensated ≈ Concentration  and velocity Current density : 