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1 Sommerfield Model for Free Electron Theory Prof.P. Ravindran, Department of Physics, Central University of Tamil Nadu, India http://folk.uio.no/ravi/CMP2013 P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory Quantum free electron theory deBroglie wave concepts The universe is made of Radiation(light) and matter(Particles).The light exhibits the dual nature(i.e.,) it can behave s both as a wave [interference, diffraction phenomenon] and as a particle[Compton effect, photo-electric effect etc.,]. Since the nature loves symmetry was suggested by Louis deBroglie. He also suggests an electron or any other material particle must exhibit wave like properties in addition to particle nature P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory In mechanics, the principle of least action states” that a moving particle always chooses its path for which the action is a minimum”. This is very much analogous to Fermat’s principle of optics, which states that light always chooses a path for which the time of transit is a minimum. de Broglie suggested that an electron or any other material particle must exhibit wave like properties in addition to particle nature. The waves associated with a moving material particle are called matter waves, pilot waves or de Broglie waves. P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory Wave function A variable quantity which characterizes de-Broglie waves is known as Wave function and is denoted by the symbol . The value of the wave function associated with a moving particle at a point (x, y, z) and at a time ‘t’ gives the probability of finding the particle at that time and at that point. de Broglie wavelength deBroglie formulated an equation relating the momentum (p) of the electron and the wavelength () associated with it, called de-Broglie wave equation. hp where h - is the planck’s constant. P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory Schrödinger Wave Equation Schrödinger describes the wave nature of a particle in mathematical form and is known as Schrödinger wave equation. They are , 1. Time dependent wave equation and 2. Time independent wave equation. To obtain these two equations, Schrödinger connected the expression of deBroglie wavelength into classical wave equation for a moving particle. The obtained equations are applicable microscopic and macroscopic particles. P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory for both 6 Schrödinger Time Independent Wave Equation The Schrödinger's time independent wave equation is given by 2 8 m 2 ( E V ) 0 2 h For one-dimensional motion, the above equation becomes d 8 2 m 2 ( E V ) 0 2 dx h 2 P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory 7 Introducing, h 2 In the above equation d 2 2m ( E V ) 0 2 2 dx For three dimension, 2m 2 ( E V ) 0 2 P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory 8 Schrödinger time dependent wave equation The Schrödinger time dependent wave equation is 2 2 V i 2m t 2 2 V 2 m i t (or) H E where H = 2 2 V 2m E = i t = Hamiltonian operator = Energy operator P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory 9 The salient features of quantum free electron theory Sommerfeld proposed this theory in 1928 retaining the concept of free electrons moving in a uniform potential within the metal as in the classical theory, but treated the electrons as obeying the laws of quantum mechanics. Based on the deBroglie wave concept, he assumed that a moving electron behaves as if it were a system of waves. (called matter waves-waves associated with a moving particle). According to quantum mechanics, the energy of an electron in a metal is quantized.The electrons are filled in a given energy level according to Pauli’s exclusion principle. (i.e. No two electrons will have the same set of four quantum numbers.) P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory 10 Each Energy level can provide only two states namely, one with spin up and other with spin down and hence only two electrons can be occupied in a given energy level. So, it is assumed that the permissible energy levels of a free electron are determined. It is assumed that the valance electrons travel in constant potential inside the metal but they are prevented from escaping the crystal by very high potential barriers at the ends of the crystal. In this theory, though the energy levels of the electrons are discrete, the spacing between consecutive energy levels is very less and thus the distribution of energy levels seems to be continuous. P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory 11 Success of quantum free electron theory According to classical theory, which follows MaxwellBoltzmann statistics, all the free electrons gain energy. So it leads to much larger predicted quantities than that is actually observed. But according to quantum mechanics only one percent of the free electrons can absorb energy. So the resulting specific heat and paramagnetic susceptibility values are in much better agreement with experimental values. According to quantum free electron theory, both experimental and theoretical values of Lorentz number are in good agreement with each other. P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory 12 Drawbacks of quantum free electron theory It is incapable of explaining why some crystals have metallic properties and others do not have. It fails to explain why the atomic arrays in crystals including metals should prefer certain structures and not others P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory Summerfeld’s Quantum Mechanical Model of Electron Conduction in Metals The Free Electron Gas: A Non-trivial Quantum Fluid Bohr, de Broglie, Schrödinger, Heisenberg, Pauli, Fermi, Dirac….. The development of the new theory of quantum mechanics. A natural step was to formulate a quantum theory of electrons in metals. First done by Sommerfeld. Assumptions Most are very similar to those of Drude. Free and independent electrons, but no assumptions about the nature of the scattering. Starting point: time-independent Schrödinger equation 2 2 2m P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory (7) 13 Note that no other potential terms are included; hence we can solve for a single, independent electron and then investigate the consequences of putting in many electrons. To solve (7), we need appropriate boundary conditions for a metal. Standard ‘particle in a box’: set ψ = 0 at boundaries. This is not a good representation of a solid, however. a) It says that the surface is important in determining the physical properties, which is known not to be the case. b) It implies that the surfaces of a large but not infinite sample are perfectly reflecting for electrons, which would make it impossible to probe the metallic state by, for example, passing a current through it. P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory 14 Most appropriate boundary condition for solid state physics: the periodic boundary condition first introduced by Born and von Karman: x L, y L, z L x, y, z (8) (We consider a cube of side L for mathematical convenience; a different choice of sample shape would have no physical consequence at the end of the calculation.) Solving then gives allowed wavefunctions: k x, y , z 1 V 1/ 2 e i (kx xk y y kz z ) 2 p , , kx L p integer, etc. (9) Here V = L3 and the V-1/2 factor ensures that normalisation is correct, i.e. that the probability of finding the electron somewhere in the cube is 1. P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory 15 What is the physical meaning of these eigenstates? First, note energy eigenvalues: 2k 2 k 2m (10) Then, note that k is also an eigenstate of the momentum operator , with eigenvalue p = k. pˆ i The state k is just the de Broglie formulation of a free particle! It has a definite momentum k. Then we see the close analogy with a well-known classical result: 2k 2 p 2 k 2m 2m P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory (11) It thus also has a velocity v = k/m. How does the spectrum of allowed states look? Cubic grid of points in k-space, separated by 2/L; volume per point (2/L)3. So, why have we come anywhere here? We have just done a quantum calculation of a free particle spectrum, and seen close analogies with that of classical free particles. Answer: now we have to consider how to populate these states with a macroscopic number of electrons, subject to the rules of quantum mechanics. Sommerfeld’s great contribution: to apply Pauli’s exclusion principle to the states of this system, not just to an individual atom. P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory Each k state can hold only two electrons (spin up and down). Make up the ground (T = 0) state by filling the grid so as to minimise its total energy. Result: At T = 0, get a sudden demarkation between filled and empty states, which (for large N), has the geometry of a sphere. Fermi wavenumber kF ky Filled states Fermi surface . . . . . . . . . . . . kz State volume (2L)3 kx Empty states State separation 2L P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory We set out to do a quantum Drude model, and did not explicitly include any direct interactions due to the Coulomb force, but we ended up with something very different. The Pauli principle plays the role of a quantum mechanical particle-particle interaction. The quantum-mechanical ‘free electron gas’ is a non-trivial quantum fluid! Is everything OK here - doesn’t kF appear to depend on the arbitrary cube size L? No - 4 3 N 2 k F 3 2 L 3 1/ 3 2 N k F 3 V (12) Quantities of interest depend on the carrier number per unit volume; the sample dimensions drop out neatly. P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory 19 How can we scale these quantum mechanical effects against something we are more familiar with? Calculate numerical values for the parameters. Use potassium (tutorial question 4). Result: kF 0.75 Å-1 vF 1 x 106 ms-1 F 2 eV TF 25000 K ( recall kBT at room T 1/40 eV) This is a huge effect: zero point motion so large that a Drude gas of electrons would have to be at 25000 K for the electrons to have this much energy! P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory 20 A couple of much-used graphs relating to the Sommerfeld model: a) The free electron dispersion b) The T = 0 state occupation function. Probability of state occupation 1 kF k 0 , k F or kF P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory 21 The specific heat of the quantum fermion gas The T=0 occupation discussed previously is a limit of the Fermi-Dirac distribution function for fermions: f (, T ) 1 e ( ) / k BT 1 where the chemical potential F. (13) At finite T: ~ 2kBT As expected, T is a minor player when it comes to changing things. f() F P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory 22 The Fermi function gives us the probability of a state of energy being occupied. To proceed to a calculation of the specific heat, we need to know the number of states per unit volume of a given energy that are occupied per unit energy range at a given T. n(, T ) g () f (, T ) (14) Then internal energy Etot(T) can be calculated from Etot (T ) n( , T )d (15) 0 and the specific heat cel from dEtot/dT as before. Our next task, then, is to derive a quantity of high and general importance, the density of states g(). P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory 23 ky dk . . . . . . . . . . . . kz State volume (2L)3 kx State separation 2L Number of allowed states per unit volume per shell thickness dk: 2 Vol. of shell at k 2 4k 2 dk g (k )dk 3 3 L Vol. per k L 2 3 spin L P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory 24 Convert to density of states per unit volume per unit (the quantity usually meant by the loose term ‘density of states’): md 2m dk 2 ; k 2 k 1/ 2 1/ 2 2m m 4 2 2 2m g ()d 2 (2)3 2 (16a, b) d 2m g () 2 2 3 3 / 2 1/ 2 (17) Very important result, but note that dependence is different for different dimension . P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory 25 Evaluating integral (15) is complicated due to the slight movement of the chemical potential with T (see Hook and Hall and for details Ashcroft and Mermin). However, we can ignore the subtleties and give an approximate treatment for F >> kBT: g( n(,T) Movement of electrons in energy at finite T F 2kBT [Etot(T) - Etot(0)]/V 1/2g(F). kBT.2kBT = g(F). (kBT)2 P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory (18) 26 Differentiating with respect to T gives our estimate of the specific heat capacity: cel = 2g(F). kB2T (19) The exact calculation gives the important general result that cel = (23g(F). kB2T (20) How does this compare with the classical prediction of the Drude model? Combining g(F) from (17) with the expression for F derived in tutorial question 4 gives, after a little rearrangement : k BT 2 cel nkB 2 F (21) c.f. Drude: P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory 3 nkB 2 27 A remarkable result: Even though our quantum mechanical interaction leads to highly energetic states at F, it also gives a system that is easy to heat, because you can only excite a highly restricted number of states by applying energy kBT. The quantum fermion gas is in some senses like a rigid fluid, and its thermal properties are defined by the behaviour of its excitations. P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory 28 What about the response to external fields or temperature gradients? To treat these simply, should introduce another vital and wide-ranging concept, the Semi-Classical Effective Model. Faced with wave-particle duality and a natural tendency to be more comfortable thinking of particles, physicists often adopt effective models in which quantum behaviour is conceptualised in terms of ‘classical’ particles obeying rules modified by the true quantum situation. In this case, the procedure is to think in terms of wave packets centred on each k state as particles. Each particle is classified by a k label and a velocity v. Velocity is given by the group velocity of the wave packet: v = dw/dk = -1d/dk = k/m for free particles like those we are concerned with at present. 29 P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory Assumption of the above: we cannot localise our ‘particles’ to better than about 10 lattice spacings. The uncertainty principle tells us that if we try to do that, we would have to use states more than 10% of our full available range (defined roughly by kF). Not, however, a particularly heavy restriction, since it is unlikely that we would want to apply external fields which vary on such a short length scale. In the absence of scattering, we then use the following ‘classical’ equation of motion in applied E and/or B fields: mdv/dt = dk/dt= -eE - ev B (22) This equation would produce continuous acceleration, which we know cannot occur in the presence of scattering. P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory 30 Include scattering by modifying (22) to m(dv/dt + vt -eE - ev B (23) This is just the equation of motion for classical particles subject to ‘damped acceleration’. If the fields are turned off, the velocity that they have acquired will decay away exponentially to zero. This reveals their ‘conjuring trick’. The physical meaning of v in (23) must therefore be the ‘extra’ or ‘drift’ velocity that the particles acquire due to the external fields, not the group velocity that they introduced in their (3.22). In fact, this is formally identical to the process that we discussed in deriving equation when we discussed the Drude model! It is no surprise, then, that it leads to the same expression for the electrical conductivity: P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory Set B to zero and stress that the relevant velocity is vdrift ; (23) becomes m(dvdrift/dt + vdriftt -eE Steady state solution (dvdrift/dt = 0) is just vdrift = -(et/m)E Following the procedure from Kittel gives us the Drude expression (3): ne 2t m If you give this some thought, it should concern you. What happened to our new quantum picture? P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory 32 To understand, consider physical meaning of the process: ky ky kz kz kx E=0 kx dk = -1mvdrift= -eEτ/ Fermi surface is shifted along the kx axis by an E field along x. The ‘quasiDrude’ derivation assumes that every electron state in the sphere is shifted by dk. This is ‘mathematically correct’, but physically entirely the wrong picture. P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory 33 Which states can ‘interact with the outside world’? ky kz In the quantum model, only those within kBT of F, i.e. those very near the Fermi surface. kx Pauli principle: only those states can scatter, so only processes involving them can relax the Fermi surface. So how does the ‘wrong’ picture work out? dk Consider amount of extra velocity/momentum acquired in equilibrium: Drudelike picture: 3 2 4 3 k F .dk L 3 # of states mom. gain Quantum picture: 3 2 2 4 k F dk 2 . kF 2 3 L # of states (1/2 FS area) mom. gain P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory x comp. only 34 (24) So the two pictures, one of which is conceptually incorrect, give the same answer, because of a cancellation between a large number of particles acquiring a small extra velocity and a small number of particles acquiring a large extra velocity. However, this is only the case for a sphere. As we shall see later, Fermi surfaces in solids are not always spherical. In this case, the Drude-like picture is simply wrong, and the conductivity must be calculated using a Fermi surface integral. P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory 35 What about thermal conductivity? Recall (4) from Drude model: k = 1/3vrandomlcel Here, vrandom can clearly be identified with vF, and l = vFt. Provided that t is the same for both electrical and thermal conduction (basically true at low temperatures but not at high temperatures; see Hook and Hall Ch. 3 after we have covered phonons), we can now revisit the Wiedemann-Franz law using (21) for the specific heat: k BT k 1 m 1 2 vF t nkB 2 T T ne t 3 2 F 2 3 2 kB e P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory 2 (25) The approximate factor of two error from the Drude model has been corrected (2/3 in quantum model cf. 3/2 in Drude model). Real question - how on earth was the Drude model so close? Answer: Because a severe overestimate of the electronic specific heat was cancelled by a severe underestimate of the characteristic random velocity. Thinking for the more committed (i.e. non-examinable): Would all quantum gas models give the same result for the Wiedemann-Franz law as the quantum fermion gas? P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory The modern conceptualisation of the quantum free electron gas: Make an analogy with quantum electrodynamics (QED). Filled Fermi sea at T = 0 is inert, so it is the vacuum. Temperature and / or external fields excite special particle-antiparticle pairs. The role of the positron is played by the holes (vacancies in the filled sea with an effective positive charge). Thermal excitation: All particles with k kF, but sum over k = 0. ky Electrical excitation: All particles with k kF, but sum over k = 2 kF/3. ky kz kz kx kx dk P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory dk 38 Scorecard so far; achievements and failures of the quantum Fermi gas model 1. Successful prediction of basic thermal properties of metals. 2. Successful prediction of conductivity, as long as we don’t ask about the microscopic origins of the scattering time t - why is the mean free path so long in metals at low temperatures? What happened to electron-ion and electron-electron scattering? 3. Failure to predict a positive Hall coefficient. 4. No understanding whatever of insulators. ‘… So insulators, which cannot carry a current, must contain electrons too. In a metal they must be free to move, and in an insulator they must be stuck. P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory Classical Drude gas Quantum Sommerfeld gas: do wave mechanics and then think in an ‘equivalent particle’ picture Random velocity purely thermal: Random velocity dominantly quantum (due to Pauli principle): 3k BT / m 1/ 3 Specific heat cel = 3 nkB 2 Large number of particles moving slowly. 2 N vF k F / m 3 V k BT 2 cel nkB 2 F Small effective number of particles moving very fast, due to special quantum mechanical constraints. P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory 40 /m 41 P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory The Sommerfeld Model Electrons are fermions. - Ground state: Fermi sphere, k F (3 2 n)1 / 3 - Distribution function m mv 2 ) 3 / 2 exp( ) 2 k B T 2k B T (m / ) 3 1 f FD (v) 3 1 4 exp[( mv 2 E F ) / k B T ] 1 2 f MB (v) n( Modification of the Drude model 3k B T 1 / 2 3 2 k BT v : vT ( ) v F , cv n k B ( )nk B m 2 2 EF - the mean free path l vT t v Ft - the Wiedemann-Frantz law - the thermopower Q k 3 k 2 kB 2 ( B )2 ( ) T 2 e 3 e kB 2 k B k BT ( )( ) 2e 6 e EF P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory 43 The Sommerfeld theory of metals the Drude model: electronic velocity distribution is given by the classical Maxwell-Boltzmann distribution m f MB ( v ) n 2 k T B the Sommerfeld model: electronic velocity distributionf FD is given by the quantum Fermi-Dirac distribution m / ( v) 3 4 3 3/ 2 mv 2 exp 2 k T B 1 1 2 mv k T B 0 exp 2 1 k BT Pauli exclusion principle: at most one electron can occupy any single electron level n dvf ( v ) normalization condition P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory T0 44 consider noninteracting electrons electron wave function associated with a level of energy E satisfies the Schrodinger equation 2 2 2 2 2 2 2 ( r ) E ( r ) 2m x y z x, y , z L x, y , z periodic boundary conditions x, y L, z x, y, z x L, y, z x, y, z a solution neglecting the boundary conditions normalization constant: probability of finding the electron somewhere in the whole volume V is unity energy momentum velocity wave vector de Broglie wavelength k (r ) 1 ikr e V 1 dr (r ) 2k 2 E (k ) 2m p k k v m k 2 k 3D: 1D: L 2 p2 1 2 E mv 2m 2 P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory 45 x, y , z L x, y , z x, y L, z x, y, z 1 ikr k (r ) e V x L, y, z x, y, z e e 1 2 2 2 kx nx , k y ny , kz nz L L L nx, ny, nz integers apply the boundary conditions components of k must be e ik x L ik y L ik z L V 2 / L 3 2 3 V the number of states per unit volume of k-space, 2 3 a region of k-space of volume contains the area per point 2 L 2 the volume 2 3 2 3 per point V L states i.e. allowed values of k k-space density of states P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory k-space consider T=0 the Pauli exclusion principle postulates that only one electron can occupy a single state therefore, as electrons are added to a system, they will fill the states in a system like water fills a bucket – first the lower energy states and then the higher energy states 46 the ground state of the N-electron system is formed by occupying all single-particle levels with k < kF state of the lowest energy volume density of states the number of allowed values of k within the sphere of radius kF 4 k F 3 V kF 3 2V 3 3 6 2 to accommodate N electrons 2 electrons per k-level due to spin kF 3 N 2 2V 6 kF 3 n 2 3 Fermi wave vector k F Fermi energy ~108 cm-1 E F 2 k F / 2m 2 Fermi temperature TF EF k B Fermi momentum pF k F ~1-10 eV ~104-105 K ky Fermi sphere kF kx Fermi surface at energy EF k F 3 2n 13 23 2 EF 3 2n 2m 13 vF 3 2n m vF k F / m Fermi velocity ~108 cm/s 1/ 2 compare to the vthermal 3k BT / m ~ 107 cm/s at T=300K classical P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: 0 Sommerfield Model for Free Electron Theory thermal velocity at T=0 47 Density of states V 3 k 3 2 Total number of states with wave vector < k N Total number of states with energy < E V 2mE N 2 2 3 E 32 The density of states – number of states per unit energyD ( E ) dN V 2m dE 2 2 2 The density of states per unit volume or the density of states P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory k 2m 32 dn 1 2m D( E ) 2 2 dE 2 k-space density of states – the number of states per unit volume of k-space 2 2 E 32 E V 2 3 48 Ground state energy of N electrons 2 2 E 2 k Add up the energies of all electron k k F 2m states inside the Fermi sphere volume of k-space per state smooth F(k) The energy density The energy per electron in the ground state k 8 3 V F (k ) k V V k 0i . e.V F ( k ) k F ( k )dk 3 3 8 k 8 5 E 1 2k 2 1 2k F 3 dk 2 V 4 k k F 2m 10m 2 E 3 2k F 3 EF N 10 m 5 2k 2 F (k ) 2m dk 4k 2 dk kF 3 N 2V 3 P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory Remarks on statistics I In quantum mechanics particles are indistinguishable 49 systems where particles are exchanged are identical exchange of identical particles can lead to changing ia , e 2 , 1 1 2 of the system wave function by a phase factor only 1, 2 2 , 1 repeated particle exchange → e2ia 1 system of N=2 particles 1, 2 - coordinates and spins for each of the particles Antisymmetric wavefunction with respect to the exchange of particles 1 1, 2 p1 1 p2 2 p1 2 p2 1 2 symmetric wavefunction with respect to the exchange of particles 1 1, 2 p1 1 p2 2 p1 2 p2 1 2 Fermions are particles which have half-integer spin the wavefunction which describes a collection of Fermions must be antisymmetric with respect to the exchange of identical particles Bosons are particles which have integer spin the wavefunction which describes a collection of bosons must be symmetric with respect to the exchange of identical particles Fermions: electron, proton, neutron Bosons: photon, Cooper pair, H atom, exciton p1, p2 – single particle states if p1 = p2 0 → at most one fermion can occupy any single particle state – Pauli principle Unlimited number of bosons can occupy a single particle state obey Fermi-Dirac statistics Obey Bose-Einstein statistics P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory Distribution function f(E) → probability that a state at energy E will be occupied at thermal equilibrium 50 fermions particles with half-integer spins Fermi-Dirac distribution function f FD ( E ) 1 E exp 1 k BT degenerate Fermi gas fFD(k) < 1 bosons particles with integer spins Bose-Einstein distribution function f BE ( E ) 1 E exp 1 k T B degenerate Bose gas fBE(k) can be any both fermions and bosons at high T when E k BT Maxwell-Boltzmann E f ( E ) exp distribution MB k BT function n dEn( E ) dED( E ) f ( E ) =(n,T) – chemical potential P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory classical gas fMB(k) << 1 remarks on statistics II BE and FD distributions differ from the classical MB distribution because the particles they describe are indistinguishable. Particles are considered to be indistinguishable if their wave packets overlap significantly. Two particles can be considered to be distinguishable if their separation is large compared to their de Broglie wavelength. m v (x) 12 2 h dB ~ p mk BT Particles become indistinguishable when dB ~ d n 2 1 3 2 2 2 3 n i.e. at temperatures below TdB mk B At T < TdB fBE and fFD are strongly different from fMB At T >> TdB fBE ≈ fFD ≈ fMB Electron gas in metals: n = 1022 cm-3, m = me → TdB ~ 3×104 K Gas of Rb atoms: n = 1015 cm-3, matom = 105me → TdB ~ 5×10-6 K Excitons in GaAs QW n = 1010 cm-2, mexciton= 0.2 me → TdB ~ 1 K x vg=v x x k g(k’) Thermal de Broglie wavelength 51 k’ k0 k '2 (r, t ) g (k ') exp i k ' r t 2 m k' A particle is represented by a wave group or wave packets of limited spatial extent, which is a superposition of many matter waves with a spread of wavelengths centered on 0=h/p The wave group moves with a speed vg – the group speed, which is identical to the classical particle speed Heisenberg uncertainty principle, 1927: If a measurement of position is made with precision x and a simultaneous measurement of momentum in the x direction is made with precision px, then p x x Theory P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron 2 T≠0 the Fermi-Dirac distribution 3D dn 1 2m Density of states D( E ) 2 2 dE 2 1 Distribution function f ( E ) E exp 1 k T B n dED ( E ) f ( E ) 32 E lim f ( E ) 1, E T 0 0 E lim EF T 0 1 [the number of states in the energy range from E to E + dE] V 1 D( E ) f ( E )dE [the number of filled states in the energy range from E to E + dE] V D( E )dE Density of states D(E) per unit volume Density of filled states D(E)f(E,T) shaded area – filled states at T=0 P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory EF E 52 Specific heat of the degenerate electron gas, estimate 1 U u V T V T V U U – thermal u kinetic energy V Specific heat 53 T ~ 300 K for typical metallic densities T=0 cv f(E) at T ≠ 0 differs from f(E) at T=0 only in a region of order kBT about because electrons just below EF have been excited to levels just above EF 1 3 Classical gas u n mv 2 nkBT 2 2 3 cv nkB The observed electronic 2 contribution at room T is usually 0.01 of this value Classical gas: with increasing T all electron gain an energy ~ kBT Fermi gas: with increasing T only those electrons in states within an energy range kBT of the Fermi level gain an energy ~ kBT k T Number of electrons which gain energy with increasing temperature ~ N B EF k BT The total electronic thermal kinetic energy U ~ N k BT E F EF/kB ~ 104 – 105 K 1 U k BT kBTroom / EF ~ 0.01 The electronic specific heat cv ~ nk B V T V Model for Free EF Electron Theory P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Specific heat of the degenerate electron gas dk u 3 E (k ) f E (k ) dED( E ) Ef ( E ) 4 0 u and u c v dk T V n 3 f E ( k ) dED( E ) f ( E ) 4 0 54 The way in which integrals of the form H ( E ) f ( E )dE differ from their zero T values H ( E )dE is determined by the form of H(E) near E= ( E )n d n Replace H(E) by its Taylor expansion about E= H ( E ) H ( E ) E n n ! dE n 0 EF H ( E )dE k T The Sommerfeld expansion H ( E ) f ( E )dE n 1 B d 2 n 1 an H ( E ) E 2 n 1 dE 7 4 k BT H ( ) kBT 4 H ( ) O kBT H (E )E 6 360 Successive terms are smaller by O(kBT/)2 2 2n u ED ( E )dE 2 0 For kBT/ << 1 n D ( E )dE 0 6 2 6 k BT 2 D( ) D( ) O (T 4 ) k BT 2 D( ) O (T 4 ) EF H ( E )dE H ( E )dE ( E F ) H ( EF ) P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory 0 0 6 2 Replace by T0 = EF correctly to order T2 Specific heat of the degenerate electron gas 1 k T 2 EF 1 B 3 2 E F u u0 1 2m D( E ) 2 2 2 EF 2 3 n 2m 2 23 32 E 6 k BT 2 D( EF ) u 2 2 cv k B TD ( E F ) T 3 3 n D( EF ) 2 EF cv (1) 2 2 k BT 2 EF cclassical nk B 3 nk B 2 FD statistics depress k BT cv by a factor of 3 EF 2 (2) cv T P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory 55 Thermal conductivity 56 thermal current density jq – a vector parallel to the direction of heat flow whose magnitude gives the thermal energy per unit time jq kT crossing a unite area perpendicular to the flow 1 1 k v 2tcv lvcv 3 3 ne2t m k cv mv 2 3 k B T 3ne2 2 e Drude: application of classical ideal gas laws cv 2 3 nkB 2 1 2 3 mv k BT 2 2 Wiedemann-Franz law (1853) Lorenz number ~ 2×10-8 watt-ohm/K2 success of the Drude model is due to the cancellation of two errors: at room T the actual electronic cv is 100 times smaller than the classical prediction, but v is 100 times larger 2 k BT For cv nkB cv cv classical ~ k BT / EF ~ 0.01 at room T the correct degenerate 2 EF Fermi gas of 2 2 the correct estimate of v2 is vF2 vF vclassical ~ EF / k BT ~ 100 at room T electrons k 2 kB T 3 e 2 P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory Thermopower Seebeck effect: a T gradient in a long, thin bar should be accompanied by an electric field directed opposite to the T gradient E QT high T low T thermoelectric field gradT E Drude: application of classical ideal gas laws For degenerate Fermi gas of electrons the correct Thermopower cv 3 nkB 2 cv 2 k BT 2 EF Q 57 cv 3ne Q kB 2e nkB cv cv classical ~ k BT / EF ~ 0.01 at room T Q/Qclassical ~ 0.01 at room T Q 2 k B k BT 6 e EF P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory Electrical conductivity and Ohm’s law Equation of motion dv dk Newton’s law m eE dt dt In the absence of collisions the Fermi sphere in k-space is displaced as a whole at a uniform rate k (t ) k (0) eE t by a constant applied electric field eE Because of collisions the displaced Fermi sphere k t avg is maintained in a steady state in an electric field ky k eE F v avg Fermi sphere kx avg m m 58 dp ( t ) p( t ) f (t ) 0 dt t p ft eEt Ohm’s law t j nev avg kavg the mean free path l = vFt because all collisions involve only electrons near the Fermi surface vF ~ 108 cm/s for pure Cu: at T=300 K t ~ 10-14 s l ~ 10-6 cm = 100 Å at T=4 K t ~ 10-9 s l ~ 0.1 cm ne2t E j m ne2t m 1 m 2 ne t kavg << kF for n = 1022 cm-3 and j = 1 A/mm2 vavg = j/ne ~ 0.1 cm/s << vF ~ 108 cm/s P.Ravindran, PHY075- Condensed Matter Physics, Spring 2013 16 July: Sommerfield Model for Free Electron Theory