固体物理ppt6.ppt

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1、Chapter 6Free electron Fermi gasKey points:Free electron gas Heat capacity of the electron gas the equation of motion in fields electrical conductivity Hall effectWhat is the physical properties of metals?Electrical conductivitygoodThermal conductivitygoodHeat capacityC=T+AT3 T 0Magnetic susceptibil

2、ityWe can understand many physical properties of metals in terms of the free electron model.According to this model,the valence electrons of the constituent atoms become conduction electrons and move about freely though the volume of the metal.Different models:Classical free electron gas model(Drude

3、 model):treat the free electrons gas as classical ideal gas.success:Ohms lawfailure:heat capacity,magnetic susceptibilityFree electron Fermi gas model(Sommerfeld model):treat the free electrons gas as Fermi gas following Fermi-Dirac distribution.success:heat capacityfailure:what is the different fro

4、m metals and isolators.Near free electron gas model:Consider the free electrons Fermi gas in a periodical potential V(r+R)=V(r)0.success:energy bandsOther models:Other corrections of the periodical potential V(r).success:details of the energy bandsEnergy levels in one dimensionConsider a free gas co

5、nfined to a length L by infinite barriers,taking account of quantum theory and of the Pauli principle.The Hamiltonian H=p2/2mThe Schrdinger equationThe fixed boundary conditions:Then the wave functionAnd the energyAccording to the Pauli principle no two electrons can have all their quantum numbers i

6、dentical.The quantum numbers of a conduction electron orbital:n(is any positive integer)ms(=1/2)A pair of orbitals labeled by the quantum number n can accommodate two electrons,one with spin up(ms=1/2)and one with spin down(ms=1/2).The Fermi energy:the Fermi energy is defined as the energy of the to

7、pmost filled level in the ground state of N electron system.The topmost filled energy level in the ground statenF=N/2Then in one dimension,the Fermi energy isThe ground state is the state of the N electron system at absolute zero.The Fermi-Dirac distributionThe ground state is the state of the N ele

8、ctron system at absolute zero.The kinetic energy of the electron gas increases as the temperature is increased:some energy levels are occupied which were vacant at T=0K,and some levels are vacant which were occupied at T=0K.The Fermi-Dirac distribution gives the probability that an orbital at energy

9、 will be occupied in an ideal electron gas in thermal equilibrium:Generally,the chemical potential is a function of temperature.is to be chosen in the way that the total number of particles in the system equals to N.For two systems in diffusive and thermal contact,the entropy will be maximum with re

10、spect to the transfer of particles as well as to the transfer of energy.The chemical potential is introduced by(Appendix D,page 645)At T=0K,=F,because in the limit T 0 the function f()changes discontinuously from 1(filled)to 0(empty)at =F=.For a small temperature range at T 0,is approximately consta

11、nt,F.The high energy tail of the distribution is that part for which kBT,where f()exp()/kBT (Boltzmann or Maxwell distribution)Free electron gas in three dimensionsConsider the free electron Fermi gas is confined to a cube of edge L.The Schrdinger equation is:Due to the periodicity condition:the wav

12、efunction is where the wavevector k satisfy Substituting the wavefunction in the Schrdinger equation,we have the energy of the orbitalThe eigenvalue of the momentum:The particle velocity:The Fermi sphere:in the ground state(T=0),the occupied orbitals may be represented inside a sphere in k space.The

13、 Fermi energy:the energy at the surface of Fermi sphere.The Fermi surface:the surface in k space where the energy equals to Fermi energy.In free electron gas,the Fermi surface is the surface of the Fermi sphere.According to the potential from lattice,the Fermi surface will deviate from the surface o

14、f sphere.The total number of the electrons inside the Fermi sphere should be N,i.e.Thus we have the radius of the Fermi sphere(Fermi wavevector)where n is the particle concentration.There is one allowed wavevector per volume(2/L)3 in k space.For each wavevector,two electrons(spin up and spin down)ar

15、e allowed in each orbital.The Fermi energy:The Fermi temperature:The Fermi velocity:The density of state D():the number of orbitals per unit energy range.Strictly,D()is the density of one-particle states,or density of orbitals.Where the total number of orbitals of energy :Then i.e.Thus Heat capacity

16、 of the electron gasClassical statistical prediction:the heat capacity of electron Cel=3kBN/2.Experiments:the capacity contributed from electron is less than 0.01 of the predicted value.Question:how can the electrons participate in electrical conduction processes as if they were mobile,while not con

17、tributing to the heat capacity?Answer:the Pauli exclusion principle and the Fermi-Dirac distributionDue to the Pauli exclusion principle,not every electron gains an energy kBT,but only those electrons in orbitals within an energy range kBT of the Fermi level can be excited.Estimate the heat capacity

18、:The total number of electron is N;The number of the electrons within an energy range of the order of kBT of the top of the energy distribution is in the order of NT/TF.The thermal energy of these electrons contributed to the heat capacity is U (NT/TF)kBT.The electronic heat capacityFor room tempera

19、ture T/TF 0.01.Derive a quantitative expression for the heat capacity:For normal temperature kBT F,and the second term gives the energy needed to bring the electrons to F from orbitals below F.For normal temperature kBT F,F.Then/kB(104K)For normal temperature kBT .Therefore T for high T().Umklapp sc

20、atteringUmklapp scattering of electrons by phonons accounts for most of the electrical lattice resistivity of metals at low temperatures.These are electron-phonons scattering processes in which a reciprocal lattice vector G is involved,so that electron momentum change in the process may be much larg

21、er than in a normal electron-phonon scattering process at low temperatures.a normal electron-phonon scattering processk=k+qan umklapp electron-phonon scattering processk=k+q+GAt very low temperature,the number of umklapp processes is negligible and the lattice resistivity is caused only by small ang

22、le scattering.Motion in magnetic fieldsConsider the collision,the equation of the motion for displacement of a Fermi sphere of particles:ForLet a static magnetic field B lie along the z axis.Then In the steady state in a static electric field,Then we havewhere is the cyclotron frequency.Hall effectT

23、he Hall field is the electric field developed across two faces of a conductor,in the direction jB,when a current j flows across a magnetic field B.Consider a rod-shaped specimen,j=(jx,0,0),B=(0,0,Bz).There is no current flow in y direction,we have vy=0.i.e.The Hall coefficient(CGS)(SI)The lower the

24、carrier concentration,the greater the magnitude of the Hall coefficient.Measuring RH is an important way of measuring the carrier concentration.The positive Hall coefficient indicates the motion of the carriers of apparent positive sign,hole.Thermal conductivity of metalsThermal conductivity of part

25、icles:The electric heat capacity per unit volume:The electric thermal conductivity:In pure metals the electric thermal conductivity is dominant at all temperature.Ratio of thermal to electrical conductivityThe thermal conductivity:The electrical conductivity:The Wiedemann-Franz law:The Lorenz number:nanostructuresThe term nanostructure denotes a condensed matter structure have a minimum dimension approximately in nanometer scale,i.e.between 1 nm and 100 nm.These structures may be fine particles,fine wires,or thin films.