Solid State Physics• Bonding in solids (metals, isolators, semiconductors)• Classical free electron theory of metals• Quantum theory of metals• Band theory of solids• Semiconductors •LasersPhysics 215Winter 2002Prof. Ioan KosztinLecture #24Introduction to Modern PhysicsQuantum free electron theory of metals• Quantum free electron model of metals:metal = an ideal gas of conduction electronsmoving through thefixed lattice of positive ion cores+ the electrons obey (quantum) Fermi-Dirac statistics222282221,322.45 10 WΩ/K33FFFBne ne LKTvLmmvEKkTne eτσγγπσ−== =⇒= = =ו Determine σ and K by replacing in the classical formulas3/223/2232,,382,3rmsFFVV BFvvnnDEmCCT DkE Dhππγγ→→≈→= = =Lorentz numbergood agreement with experimentFermi sphere (sea) in electric fieldAvxvxvyvy2FFEvm=AA’E!dv!()0()Ffp p pθ=−!stepfunction:1for 0()0for 0xxxθ>≡<0() () ()fp fp fpδ=+!! !()() ()pEp fpfp e pmδτε⋅∂≈−∂!!!!!relaxation time approximationThe net effect of E is a small displacement of the Fermi surfaceTemperature dependence of σ and K• From the Boltzmann kinetic equation in the relaxation time approximation213/202(,)() () (,)3eD f TTT dTmεσρ ετεεε−∂== −∂∫transport relaxation time depends on the scattering mechanism of the electrons• At low temperature the electron-impurity scattering is dominant• At high temperature the electron-phonon scattering is dominanta similar relation holds for K(T)"""25()ieeephimpTaTbTρρ−−=+ +() ,TTKconstρα==(Matthiessen’s rule)Resistivity of Ag and Na[D.K.C. MacDonald and K. Mendelssohn, Proc. Roy. Soc. (London) A202:103, 1950]higher impurityconcentrationBand Theory of Solids• The energy spectrum of crystalline solids can be calculated by using the Self-Consistent-Field (SCF) method from atomic physics• Consider that each electron moves in the static, periodic potential created by the lattice of nuclei (Born-Oppenheimer approximation) and the rest of the electrons• The eigenvalues of the corresponding Schrödinger equation form energy bands εs(k); s = band index, 0<ki<bi(k=quasi wave vector, and b=reciprocal lattice vector)• The eigenfunctions of the corresponding Schrödinger equation have the generic form() (), ( ) ()ikrsk sk sk skr eur ura urψ=+=lattice vector()()sk skkb kεε+=Isolated-atom approach to band theorySplitting of the 3s level when 2 Na atoms are brought togetherSplitting of the 3s level when 6 Na atoms are brought togetherFormation of a 3s energy band when Na atoms form a crystalline solidEnergy bands in sodiumFor N atoms, the capacity of each band is 2(2ℓ+1)NEnergy bands in metals(Fermi energy)conduction band• In metals the conduction band is partially filled with (conduction) electrons with E<EF• States with E>EFin the conduction band are empty and can be easily occupied (at the cost of a tiny amount of energy) by the electrons near the Fermi surface, i.e., conduction electrons can move freelyin a perfect metallic crystalEnergy bands in insulataors• The valence (conduction) band is completely filled (empty) • The conduction and valence bands are separated by and energy gap Eg~10eV• The Fermi energy (chemical potential) falls inside the energy gap• A valence electron requires ∆E>Egto become a conduction electron, i.e., the density of conduction electrons is ~ exp(-Eg/kBT)Energy bands in semiconductorsThere are two types of charge carriers in an intrinsic semiconductor (i.e., #electrons = #holes):electronsand holes• similar band structure to insulators but with much smaller energy gap (Eg~1eV)• poor (good) conductor (insulator) at T=0• conductivity increases rapidly with