理学院 物理系 沈嵘 Chapter 6 Free Electron Fermi Gas 6.1 Electron Gas Model and its Ground State 6.2 Thermal Properties of Electron Gas 6.3 Free Electrons in Electric Fields 6.4 Hall Effect 6.5 Thermal Conductivity of Metals 6.6 Failures of the free electron gas model 1 6.1 Electron Gas Model and its Ground State 6.1 Electron Gas Model and its Ground State I. Basic Assumptions of Electron Gas Model Metal: valence electrons → conduction electrons (moving freely) ü The simplest metals are the alkali metals—lithium, sodium, 2 potassium, cesium, and rubidium. 6.1 Electron Gas Model and its Ground State density of electrons: Zr n = N m A A where Z is # of conduction electrons per atom, A is relative atomic mass, rm is the density of mass in the metal. The spherical volume of each electron is, 1 3 1 V 4 3 æ 3 ö = = p rs rs = ç ÷ n N 3 è 4p nø Free electron gas model: Suppose, except the confining potential near surfaces of metals, conduction electrons are completely free. The conduction electrons thus behave just like gas atoms in an ideal gas --- free electron gas. 3 6.1 Electron Gas Model and its Ground State Basic Properties： ü Ignore interactions of electron-ion type (free electron approx.) ü And electron-eletron type (independent electron approx). Total energy are of kinetic type, ignore potential energy contribution. ü The classical theory had several conspicuous successes 4 6.1 Electron Gas Model and its Ground State Long Mean Free Path: ü From many types of experiments it is clear that a conduction electron in a metal can move freely in a straight path over many atomic distances. ü In a very pure specimen at low temperatures, the mean free path may be as long as 108 interatomic spacings (more than 1 cm). ü Condensed matter so transparent to conduction electrons: l Due to periodic lattice structure. l Due to quantum nature of electrons: Pauli exclusive principle. 5 6.1 Electron Gas Model and its Ground State II. Quantum Free Electron Model -- single electron state and eigenenergy 1. Schrödinger Equation Þ Single electron problem in a 3D space Equation of wave for each electron (within the potential well)： 2 " 2 - Ñ y (r!) = ey (r!) 2m !2 æ ¶ 2 ¶ 2 ¶ 2 ö - ç 2 + 2 + 2 ÷y (x, y, z) = ey (x, y, z) 2m è ¶ x ¶ y ¶ z ø 6 6.1 Electron Gas Model and its Ground State 2. Boundary Condition ——periodic B.C. y (x + L, y, z,t) =y (x, y, z,t) y (x, y + L, z,t) =y (x, y, z,t) y (x, y, z + L,t) =y (x, y, z,t) 3. Solution of Schrödinger's equation ! ! 1 ik×r! ! ! y ! (r ) = e k --wave vector, r --position vector k V ! k = (k x ,k y ,kz ) energy eigenvalues： 2 2 2 2 2 2 " ! k ! (kx + k y + kz ) e (k) = = 2m 2m 7 6.1 Electron Gas Model and its Ground State ! 4. physical meaning of k vector ! ! ! ! ! - i"Ñy k (r ) = "ky k (r ) eigen value pˆ = -i!Ñ --momentum operator ! ! Electron in state y k ( r ) has a certain momentum proportional to wave vector ! : ! ! k p = "k ! ! ! p "k velocity: v = = m m p2 1 energy: e = = mv 2 2m 2 2p wavelength： l = k 8 6.1 Electron Gas Model and its Ground State Require the plain wave solutions to fullfil PBC: 2p ik x L ik yL ikz L 2p 2p e = e = e = 1 kx = nx k = n k = n L y L y z L z nx ,ny ,nz —— a series of integer numbers. Free electron's eigen energy levels are discontineous, and are pretty close to each other. 5. k-space and density of states (DOS) ! In the k-space, every k point represents a possible state of electron (orbital). Neigboring representative points are equally spaced by 2 p L in three dimensions. So the volume each point occupies is: 3 ! æ 2p ö 8p 3 Dk = ç ÷ = 9 è L ø V 6.1 Electron Gas Model and its Ground State 3 ! æ 2p ö 8p 3 Dk = ç ÷ = è L ø V Unit k-space volume ! contains # of k points: 1 V = Dk 8p 3 ! For each k state (orbital), electrons have two different spin ! 3 states: unit k space volthus can ume contain at most V 4p individual spinful electronic states： 10 6.1 Electron Gas Model and its Ground State ! ! DOS in k space: g(k ) ! V !! dddNgk==( ) ttkk3 4p Densisy of states in energy scale g ( e )： unit volume, unit energy invterval, spinful electronic states: dN = Vg(e )de ! e = !2k 2 2m determines the isosurface in k space, which turns out to be spherical surfaces with radius 2 m e ! . Electronic states between energy shells e ~ e + d e corresponds to those with wave vectors between k ~ k + dk V 1 1 2 d4dNkk= p 2 g(e ) = (2m 3e ) 4p 3 p 2!3 11 6.1 Electron Gas Model and its Ground State 12 6.1 Electron Gas Model and its Ground State III. Ground state properties of free electron gas Ground state of free electron gas means its property at the absolutely zero temperature. Fermi sphere: obeying Pauli principle, electrons fills the orbitals from low energy to high levels, and form a sphere in k space. Fermi surface: Interface betwee occupied and unoccupied states ! Fermi wave vector: kF 1 3 k = 3p 2n F ( ) 13 6.1 Electron Gas Model and its Ground State Fermi energy: Single electron energy on the Fermi surface !2k 2 e = F F 2m Fermi velocity: The electron velocity vF at the Fermi surface Fermi temperature: TF is defined as ƐF/kB Has nothing to do with the real temperature of the electron gas! 14 6.1 Electron Gas Model and its Ground State Excise: mean energy of electrons： 1 eF eeee= g ( )d N ò0 ee = FFeeegg( )dd( ee) òò00 3 3 e = e = k T 5 F 5 B F Even the ground state has some finite kinetic energy --- due to Pauli principle 15 6.2 Thermal properties of free electron gas 6.2 Thermal Properties of Free Electron Gas I. Fermi Distribution Distribution of free electrons in energy levels: Fermi-Dirac statistics At temperature ,T the probability that energy level e isi occupied in thermal equilibrium: 1 fi = (e -µ ) k T e i B +1 µ--chemical potential: in the condition of fixed temperature and volume, the increasement of free energy by adding a singlet electron to the system. ¥ N = fi =V g(e )f (e )de å ò0 16 i 6.2 Thermal properties of free electron gas 1. T ® 0 lim f (e i ) =1, e i < µ T®0 lim f (e i ) = 0, e i > µ T®0 At absolutely zero temperature, levels with energy < µare all occupied, and the levels above it are all empty. Chemical potential (Fermi energy) is the highest energy level electrons takes at T=0; 2. T > 0 1 e = µ f (e ) = 2 µ level have equal probability to be filled or to ramain empty. T rises, the step becomes e << µ f (e ) = 1 more and more blurred. e >> µ f (e ) = 0 17 6.2 Thermal properties of free electron gas II. Chemical Potential 1. ground state T = 0, e F 2 2 2 2 ! k ! 2 µ(0) = e = F = (3np )3 F 2m 2m 2. thermal excitation T > 0 kBT << µ Electrons near Fermi surface can be excited to states outside Fermi surface, leaving some empty states (holes) inside. ¥ 1 3 1 2 N = V g(e )f (e )de g(e ) = 2 3 (2m e ) ò0 p ! 2 é p 2 æ k T ö ù µ = e ê1- ç B ÷ ú generally, k T << e µ » e F 12 e B F F ëê è F ø ûú 18 6.2 Thermal properties of free electron gas Fermi-Dirac distribution function 1/2 ü Fermi temperature is high: TF= 50,000 K. ü The chemical potential at each temperature may be read off the graph as the energy at which f = 0.5. 19 6.2 Thermal properties of free electron gas 3. Density of States 6.2 Thermal properties of free electron gas III.electron heat capacity contant volume æ ¶u ö U c = u = specific heat： V ç ÷ è¶T øV V ! U—energy： ! ! U = 2 e (k) fk N = 2 fk å! å! k k ¥ ¥ ugf= ee( ) ( ee)d ngf= (eee) ( )d ò0 ò0 u0 --average energy at ground state æ ¶u ö p 2 T c = ç ÷ = nk V ¶T 2 B T è øV F 21 6.2 Thermal properties of free electron gas Analysis：classically, mean energy of 1 mol electron gas æ 3 ö 3 e = N ç k T ÷ = RT mol A è 2 B ø 2 ¶e 3 monovalent metal： C e = mol = R V ,mol ¶T 2 high T total specific heat (constant volume) of metal: 3 C = C Ph + C e = 3R + R » 37.40J / mol× K V V ,mol V ,mol 2 e actual CV，mol is smaller than this classical limit e T quantum：CV ~ TF At room temperatures, electrons' contribution to total specific heat is negligible. 22 6.2 Thermal properties of free electron gas e CV = gT g is elctron specific heat constant 3 Ph 12 4 æ T ö low T：CV ,mol = p Rç ÷ 5 è QD ø e Ph CV = CV ,mol + CV ,mol = gT + bT 3 C V = g + bT 2 T C V ~ T 2 ——quadratic T 23 6.2 Thermal properties of free electron gas Thermal Effective Mass Thermal effective mass mth to the electron mass m: Due to 3 facts: interaction with periodic potential, with phonon, with electrons themselves.