固体物理第四章

Chapter IVEssential Theories for Electrons in Solids4.1 Drude-Lorentz Theories for Free Electrons- Drude’s Hypothesis (at the turn of 20th century):The electrons in metals were hypothesized to behave like free molecules in an ideal gas. The electron density may be written as:M N z n oρ=Its typical magnitude varies from 1022~1023/cm 3. They move randomly and collide with one another apart from with their parental ionic cores . The directional movement of these electrons in electric field or temperature field leads to the electric conductivity and the partial contribution to heat conduction of materials:qE v dtdv m e −=+γ, γ-damping coefficient ⎥⎦⎤⎢⎣⎡⎟⎠⎞⎜⎝⎛−−=τt v v e exp 1, where qE v m e e −=τ is called relaxation time constant . Andwe have: ev qE −=γ, E m nq q nv j ee τ2== Electric conductivity: em nq E j τσ2== Free path-length:τe v l = - Lorentz’s Theory, Maxwell-Boltzmann distribution- Widemann-Franz-Lorentz law, a relationship between electric conduction and heat conduction:LT e =σλ where L is called Lorentz number . This formula may be theoretically verified if the heat conductivity is written as:l v c e e e 31=λ Shadows cost over the Drude model!Problem 1 - According to the homogeneous partition of energy by the degrees of freedom, the predicted thermal capacity (specific heat) per electron should be:B e k c 23=, It is much higher than the experimental value, which is just accountable for only a few percent of it.Problem 2 - the real free path-length is much larger than the expected by Drude’s model;Problem 3 – no way to access to insulators and semiconductors;4.2 Sommerfeld’s Free Electrons by Quantum Mechanics4.2.1 Energy states of free electrons 1) Schrödinger Eq. for an electron in free space)()(222r E r mΨ=Ψ∇−h It has a general solution:ikr ikr Be Ae r −+=Ψ)(and the eigenvalue of the Hamiltonian operator is:mk k E 2)(22h = where k may take any positive numbers.2) Electrons in a limited space with boundary conditions - Stationary waves boundary conditions0),,(),,0(==z y L z y x ψψ0),,(),0,(==z L x z x y ψψ0),,()0,,(==z L y x y x ψψSubstituting the above conditions into the general equation, it follows that (0)0,,(),0,(),,0(===y x z x z y ψψψ, stationary waves):z k y k x k A r z y x sin sin sin )(=ψ,and (0),,(),,(),,(===z y x L y x z L x z y L ψψψ)i i i n L k π=, (i=x,y,z)where L i stands for the edge lengths of crystal and n i isallowed to take any positive integers . According to the normalization condition:1)(2=∫dr r V ψ one may arrives at:V L L L A zy x 88== and mk k E i i 2)(22h = It should be noted that with this boundary condition, a stationary wave is formed and, therefore, the average velocity and kinetic momentum are given:0=k h and 0==mk v h - Periodic (Born-von Karman) boundary conditionsNote: A wave function ikr Ae r =Ψ)(should be adopted when it is required to be the eigenstate of both Hamiltonian operator ⎟⎟⎠⎞⎜⎜⎝⎛∇−=222m H h and momentum ⎟⎠⎞⎜⎝⎛∇=i p h operator , where k can take any real numbers .Taking into account the periodic conditions:),,(),,(z y L x z y x x +=ψψ),,(),,(z L y x z y x y +=ψψ),,(),,(z L z y x z y x +=ψψit comes that:)1(,1)(z y x ikr L L L A e V r ==ψmk k E i i 2)(22h = ),,(,2z y x i n L k i i i ==πwhere n i takes any real integers (positive and negative).4.2.2 Electrons state density()()∫∫∇=ωωπωρq dSV q 32)(, for phonons ()()∫∫∇=E kk E dS V E 32)(πρ, for electrons Substituting into the above equation the energy dispersion relationship, it comes out: ()∫∫=⎟⎟⎠⎞⎜⎜⎝⎛=m k E m k dS V E 223222)(h h πρ ()E m k k m d k d V =∫∫=220022322sin 2h h ππθθϕπ E m V 232224⎟⎠⎞⎜⎝⎛=h πAs each energy level can accommodate 2 spin states, the state density for free electrons can be rewritten as:()()∫∫∇=E kk E dS V E 322)(πρE m V 232222⎟⎠⎞⎜⎝⎛=h π4.3 Fermi-Dirac Distribution and Fermi energy for Electrons gas 1) Fermi-Dirac distributionT k E E B F e E f /)(11)(−+=where E F is the Fermi energy of electrons. It means that⎩⎨⎧>≤==F FT E E E E E f 01)(0Fig. 4-1 The schematic of Fermi-Dirac distributionIn the case where T=0K, the electrons would occupy all the levels of f E E ≤and the levels of f E E >remain entirely empty. The equi-energy surface at F E E =is usually called Fermi surface, varying as a function of temperature. In addition, when T~0K, EE f ∂∂−)( behaves like a )(f E E −δ, and has the following integration with regard to any E Ffunction )(E g :...)()(6)()()(''220++≈∂∂−=∫∞f B f E g T k E g dE E E f E g I π This approximation is actually suitable for most of the frequently encountered metals since F B E T k <<holds for them at not very high temperatures.2) Number density of free electronsTotal number of free electrons:∫∞=0)()(dE E f E N ρdE E f E m V )(2202322∫∞⎟⎠⎞⎜⎝⎛=h π⎭⎬⎫⎩⎨⎧∂∂−⎟⎠⎞⎜⎝⎛=∫∞∞023232322)()(230dE E E f E E E f m V h π ⎥⎥⎦⎤⎢⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛+⎟⎠⎞⎜⎝⎛=222/323228123F B F E T k E m V ππh (A) Therefore, the number density of free electrons:VN n = when T=0K, one has:232322231o f E m n ⎟⎠⎞⎜⎝⎛=h π (B) and ()322232n m E of πh =Since n can not change with the temperature, the above equations (A) and (B) may further produce:3/22281⎥⎥⎦⎤⎢⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛+=F B F o F E T k E E πWhen F B E T k <<, E F may be derived from the above equation:⎥⎥⎦⎤⎢⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛−≈22121o F B o F F E T k E E π 3) Average energy of free electronsTotal energy:∫∞=0)()(dE E f E E E T ρ ()⎥⎦⎤⎢⎣⎡+⎟⎠⎞⎜⎝⎛=F B F E T k E m V 222523228525ππh So, the average one: N E E T =Making use of the condition: F B E T k <<, the average energy per electron is:⎥⎥⎦⎤⎢⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛+=222153F B F E T k E E π or ⎥⎥⎦⎤⎢⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛+≈22125153o F B o E T k E E F π When T=0K,o F o E E 53=4) Specific heat from free electrons By the definition of thermal capacity, one may arrive at:⎟⎟⎠⎞⎜⎜⎝⎛==⎟⎟⎠⎞⎜⎜⎝⎛∂∂=o F B o F B V e VT T k E T k T E C 22222ππ T γ= where 2010−≈FT T at room temperature. It means that only those electrons around the Fermi surface by T k B may be excited up to the higher energy levels and contribute to the thermal capacity.The total heat capacity of a metal:3bT T C C C L V e V V +=+=γ, at low temperatures; and 2bT TC v +=γFig. 4-2 The C v /T ~T 2 relationship for zinc5) Electric conductivity of metalsFig. 4-3 Schematic diagram of Fermi sphere displacementdriven by external electric field'N qv j F −=, E E N F Δ=)('ρWhen the steady state is reached after time length τ, the change in the momentum should be:ττE q F p −==Δ,Considering m p mv E 2/2/22==, one arrives atm p p E /=⎟⎟⎠⎞⎜⎜⎝⎛∂∂ and ττE qv E q m p p p E E F −=−=Δ⎟⎟⎠⎞⎜⎜⎝⎛∂∂=Δ)( Therefore:E E v q jF F τρ)(22−=orτρσ)(22F F E v q −= ( In comparison with the classic model: em nq E j τσ2==)ESolid State Physics, by Prof.Zeng An Introductionary Course for Postgraduate Students in Materials Science- p. 11 - 4.4 Work function and contact potentials- work function , the minimum energy φ needed to movean electron at Fermi surface into the outside of the metal.Fig. 4-4 Schematic diagram of the potential for electronsin metals- contact potential difference , when two pieces of differentmetals are brought into contact, the electrons in the metal with higher Fermi energy will spontaneously move into the metal with lower Fermi energy until an equilibrium is reached, resulting in a potential difference:)(12112φφ−=eV - thermal emission of electrons – the electrons inside ametal may escape away from the surface of the metal to form an electric current:T k B B e h T k m e j /32)(4φπ−= Vacuumaffinity。