Sommerfield Model for Free Electron Theory. Prof.P. Ravindran, Department of Physics, Central University of Tamil Nadu, India

Transcription

1 Soerfield Model for ree lectron Theory Prof.P. Ravindran, Departent of Physics, Central University of Tail Nadu, India P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory

2 Quantu free electron theory deroglie wave concepts The universe is ade of Radiationlight and atterparticles.the light exhibits the dual naturei.e., it can behave s both as a wave [interference, diffraction phenoenon] and as a particle[copton effect, photo-electric effect etc.,]. Since the nature loves syetry was suggested by Louis deroglie. He also suggests an electron or any other aterial particle ust exhibit wave lie properties in addition to particle nature P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory

3 In echanics, the principle of least action states that a oving particle always chooses its path for which the action is a iniu. This is very uch analogous to erat s principle of optics, which states that light always chooses a path for which the tie of transit is a iniu. de roglie suggested that an electron or any other aterial particle ust exhibit wave lie properties in addition to particle nature. The waves associated with a oving aterial particle are called atter waves, pilot waves or de roglie waves. P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory

4 Wave function A variable quantity which characterizes de-roglie waves is nown as Wave function and is denoted by the sybol. The value of the wave function associated with a oving particle at a point x, y, z and at a tie t gives the probability of finding the particle at that tie and at that point. de roglie wavelength deroglie forulated an equation relating the oentu p of the electron and the wavelength associated with it, called de-roglie wave equation. h p where h - is the planc s constant. P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory

5 Schrödinger Wave quation Schrödinger describes the wave nature of a particle in atheatical for and is nown as Schrödinger wave equation. They are,. Tie dependent wave equation and. Tie independent wave equation. To obtain these two equations, Schrödinger connected the expression of deroglie wavelength into classical wave equation for a oving particle. The obtained equations are applicable for both icroscopic and acroscopic particles. P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory

6 P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory 6 Schrödinger Tie Independent Wave quation The Schrödinger's tie independent wave equation is given by 0 8 V h or one-diensional otion, the above equation becoes 0 8 V h dx d

7 P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory 7 Introducing, h In the above equation 0 V dx d or three diension, 0 V

8 P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory 8 Schrödinger tie dependent wave equation The Schrödinger tie dependent wave equation is t i V t i V or H where H = V = Hailtonian operator t i = nergy operator =

9 9 The salient features of quantu free electron theory Soerfeld proposed this theory in 98 retaining the concept of free electrons oving in a unifor potential within the etal as in the classical theory, but treated the electrons as obeying the laws of quantu echanics. ased on the deroglie wave concept, he assued that a oving electron behaves as if it were a syste of waves. called atter waves-waves associated with a oving particle. According to quantu echanics, the energy of an electron in a etal 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 sae set of four quantu nubers. P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory

10 0 ach nergy level can provide only two states naely, 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 assued that the perissible energy levels of a free electron are deterined. It is assued that the valance electrons travel in constant potential inside the etal but they are prevented fro 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 sees to be continuous. P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory

11 Success of quantu free electron theory According to classical theory, which follows Maxwell- oltzann statistics, all the free electrons gain energy. So it leads to uch larger predicted quantities than that is actually observed. ut according to quantu echanics only one percent of the free electrons can absorb energy. So the resulting specific heat and paraagnetic susceptibility values are in uch better agreeent with experiental values. According to quantu free electron theory, both experiental and theoretical values of Lorentz nuber are in good agreeent with each other. P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory

12 Drawbacs of quantu free electron theory It is incapable of explaining why soe crystals have etallic properties and others do not have. It fails to explain why the atoic arrays in crystals including etals should prefer certain structures and not others P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory

13 Suerfeld s Quantu Mechanical Model of lectron Conduction in Metals The ree lectron Gas: A Non-trivial Quantu luid ohr, de roglie, Schrödinger, Heisenberg, Pauli, eri, Dirac.. The developent of the new theory of quantu echanics. A natural step was to forulate a quantu theory of electrons in etals. irst done by Soerfeld. Assuptions Most are very siilar to those of Drude. ree and independent electrons, but no assuptions about the nature of the scattering. Starting point: tie-independent Schrödinger equation P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory 7

14 Note that no other potential ters are included; hence we can solve for a single, independent electron and then investigate the consequences of putting in any electrons. To solve 7, we need appropriate boundary conditions for a etal. 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 iportant in deterining the physical properties, which is nown not to be the case. b It iplies that the surfaces of a large but not infinite saple are perfectly reflecting for electrons, which would ae it ipossible to probe the etallic state by, for exaple, passing a current through it. P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory 4

15 Most appropriate boundary condition for solid state physics: the periodic boundary condition first introduced by orn and von Karan: x L, y L, z L x, y, z 8 We consider a cube of side L for atheatical convenience; a different choice of saple shape would have no physical consequence at the end of the calculation. Solving then gives allowed wavefunctions: x, y, z V / e i x x y yz z, x p L, p integer, etc. 9 Here V = L and the V -/ factor ensures that noralisation is correct, i.e. that the probability of finding the electron soewhere in the cube is. P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory 5

16 What is the physical eaning of these eigenstates? irst, note energy eigenvalues: 0 Then, note that is also an eigenstate of the oentu operator pˆ i, with eigenvalue p =. The state is just the de roglie forulation of a free particle! It has a definite oentu. Then we see the close analogy with a well-nown classical result: p P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory

17 It thus also has a velocity v = /. How does the spectru of allowed states loo? Cubic grid of points in -space, separated by /L; volue per point /L. So, why have we coe anywhere here? We have just done a quantu calculation of a free particle spectru, and seen close analogies with that of classical free particles. Answer: now we have to consider how to populate these states with a acroscopic nuber of electrons, subject to the rules of quantu echanics. Soerfeld s great contribution: to apply Pauli s exclusion principle to the states of this syste, not just to an individual ato. P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory

18 ach state can hold only two electrons spin up and down. Mae up the ground T = 0 state by filling the grid so as to iniise its total energy. Result: At T = 0, get a sudden dearation between filled and epty states, which for large N, has the geoetry of a sphere. eri wavenuber y z State volue L eri surface illed states P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory x pty states State separation L

19 We set out to do a quantu Drude odel, and did not explicitly include any direct interactions due to the Coulob force, but we ended up with soething very different. The Pauli principle plays the role of a quantu echanical particle-particle interaction. The quantu-echanical free electron gas is a non-trivial quantu fluid! Is everything OK here - doesn t appear to depend on the arbitrary cube size L? No - 4 N L N V / Quantities of interest depend on the carrier nuber per unit volue; the saple diensions drop out neatly. P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory 9

20 How can we scale these quantu echanical effects against soething we are ore failiar with? Calculate nuerical values for the paraeters. Use potassiu tutorial question 4. Result: 0.75 Å - v x 0 6 s - ev T 5000 K recall T at roo T /40 ev This is a huge effect: zero point otion so large that a Drude gas of electrons would have to be at 5000 K for the electrons to have this uch energy! P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory 0

21 A couple of uch-used graphs relating to the Soerfeld odel: a The free electron dispersion b The T = 0 state occupation function. Probability of state occupation 0 or, P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory

22 The specific heat of the quantu ferion gas The T=0 occupation discussed previously is a liit of the eri-dirac distribution function for ferions: f, T / e T where the cheical potential. At finite T: f ~ T As expected, T is a inor player when it coes to changing things. P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory

23 The eri function gives us the probability of a state of energy being occupied. To proceed to a calculation of the specific heat, we need to now the nuber of states per unit volue of a given energy that are occupied per unit energy range at a given T. n, T g f, T 4 Then internal energy tott can be calculated fro tot T n, T d 0 5 and the specific heat c el fro d tot /dt as before. Our next tas, then, is to derive a quantity of high and general iportance, the density of states g. P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory

24 d y z State volue L x State separation L Nuber of allowed states per unit volue per shell thicness d: g d spin L Vol. of shell at Vol. per P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory 4 L 4 d L

25 P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory 5 Convert to density of states per unit volue per unit the quantity usually eant by the loose ter density of states : / / / 4 g d d g Very iportant result, but note that dependence is different for different diension. / ; d d 6a, b 7

26 valuating integral 5 is coplicated due to the slight oveent of the cheical potential with T see Hoo and Hall and for details Ashcroft and Merin. However, we can ignore the subtleties and give an approxiate treatent for >> T: n,t g Moveent of electrons in energy at finite T T [ tot T - tot 0]/V /g. T. T = g. T 8 P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory 6

27 Differentiating with respect to T gives our estiate of the specific heat capacity: c el = g. T 9 The exact calculation gives the iportant general result that c el = g. T 0 How does this copare with the classical prediction of the Drude odel? Cobining g fro 7 with the expression for derived in tutorial question 4 gives, after a little rearrangeent : c el n T c.f. Drude: n P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory 7

28 A rearable result: ven though our quantu echanical interaction leads to highly energetic states at, it also gives a syste that is easy to heat, because you can only excite a highly restricted nuber of states by applying energy T. The quantu ferion gas is in soe senses lie a rigid fluid, and its theral properties are defined by the behaviour of its excitations. P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory 8

29 What about the response to external fields or teperature gradients? To treat these siply, should introduce another vital and wide-ranging concept, the Sei-Classical ffective Model. aced with wave-particle duality and a natural tendency to be ore cofortable thining of particles, physicists often adopt effective odels in which quantu behaviour is conceptualised in ters of classical particles obeying rules odified by the true quantu situation. In this case, the procedure is to thin in ters of wave pacets centred on each state as particles. ach particle is classified by a label and a velocity v. Velocity is given by the group velocity of the wave pacet: v = dw/d = - d/d = / for free particles lie those we are concerned with at present. P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory 9

30 Assuption of the above: we cannot localise our particles to better than about 0 lattice spacings. The uncertainty principle tells us that if we try to do that, we would have to use states ore than 0% of our full available range defined roughly by. Not, however, a particularly heavy restriction, since it is unliely 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 otion in applied and/or fields: dv/dt = d/dt= -e - ev This equation would produce continuous acceleration, which we now cannot occur in the presence of scattering. P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory 0

31 Include scattering by odifying to dv/dt + vt -e - ev This is just the equation of otion for classical particles subject to daped acceleration. If the fields are turned off, the velocity that they have acquired will decay away exponentially to zero. This reveals their conjuring tric. The physical eaning of v in ust 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.. In fact, this is forally identical to the process that we discussed in deriving equation when we discussed the Drude odel! It is no surprise, then, that it leads to the sae expression for the electrical conductivity: P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory

32 Set to zero and stress that the relevant velocity is v drift ; becoes dv drift /dt + v drift t -e Steady state solution dv drift /dt = 0 is just v drift = -et/ ollowing the procedure fro Kittel gives us the Drude expression : ne t If you give this soe thought, it should concern you. What happened to our new quantu picture? P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory

33 To understand, consider physical eaning of the process: y y z z x x = 0 d = - v drift = -eτ/ eri surface is shifted along the x axis by an field along x. The quasi- Drude derivation assues that every electron state in the sphere is shifted by d. This is atheatically correct, but physically entirely the wrong picture. P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory

34 y Which states can interact with the outside world? z In the quantu odel, only those within T of, i.e. those very near the eri surface. d x Pauli principle: only those states can scatter, so only processes involving the can relax the eri surface. So how does the wrong picture wor out? Consider aount of extra velocity/oentu acquired in equilibriu: Drudelie picture: L 4. d Quantu picture: 4 d. L 4 # of states o. gain # of states / S area o. gain x cop. only P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory 4

35 So the two pictures, one of which is conceptually incorrect, give the sae answer, because of a cancellation between a large nuber of particles acquiring a sall extra velocity and a sall nuber of particles acquiring a large extra velocity. However, this is only the case for a sphere. As we shall see later, eri surfaces in solids are not always spherical. In this case, the Drude-lie picture is siply wrong, and the conductivity ust be calculated using a eri surface integral. P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory 5

36 P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory What about theral conductivity? Recall 4 fro Drude odel: = /v rando lc el Here, v rando can clearly be identified with v, and l = v t. Provided that t is the sae for both electrical and theral conduction basically true at low teperatures but not at high teperatures; see Hoo and Hall Ch. after we have covered phonons, we can now revisit the Wiedeann-ranz law using for the specific heat: t t e T n v ne T T 5

37 The approxiate factor of two error fro the Drude odel has been corrected / in quantu odel cf. / in Drude odel. Real question - how on earth was the Drude odel so close? Answer: ecause a severe overestiate of the electronic specific heat was cancelled by a severe underestiate of the characteristic rando velocity. Thining for the ore coitted i.e. non-exainable: Would all quantu gas odels give the sae result for the Wiedeann-ranz law as the quantu ferion gas? P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory

38 The odern conceptualisation of the quantu free electron gas: Mae an analogy with quantu electrodynaics QD. illed eri sea at T = 0 is inert, so it is the vacuu. Teperature 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. Theral excitation: All particles with, but su over = 0. lectrical excitation: All particles with, but su over = /. y y z z x x d P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory 8 d

39 Scorecard so far; achieveents and failures of the quantu eri gas odel. Successful prediction of basic theral properties of etals.. Successful prediction of conductivity, as long as we don t as about the icroscopic origins of the scattering tie t - why is the ean free path so long in etals at low teperatures? What happened to electron-ion and electron-electron scattering?. ailure to predict a positive Hall coefficient. 4. No understanding whatever of insulators. So insulators, which cannot carry a current, ust contain electrons too. In a etal they ust be free to ove, and in an insulator they ust be stuc. P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory

40 Classical Drude gas Quantu Soerfeld gas: do wave echanics and then thin in an equivalent particle picture Rando velocity purely theral: T / Specific heat cel = Large nuber of particles oving slowly. n Rando velocity doinantly quantu due to Pauli principle: / N v / / V c el n T Sall effective nuber of particles oving very fast, due to special quantu echanical constraints. P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory 40

41 P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory 4

42 P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory The Soerfeld Model lectrons are ferions. - Ground state: eri sphere, - Distribution function Modification of the Drude odel - the ean free path - the Wiedeann-rantz law - the theropower / n t t v T v l e e T 6 T e e Q exp / T v T n v f M ] / exp[ 4 / T v v f D v T n T n c v T v v, : /

43 The Soerfeld theory of etals the Drude odel: electronic velocity distribution is given by the classical Maxwell-oltzann distribution the Soerfeld odel: electronic velocity distribution f is given by the quantu eri-dirac distribution Pauli exclusion principle: at ost one electron can occupy any single electron level f M D / v v n exp T T v n dvf v / 4 v T0 exp T noralization condition T 0 4 P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory

44 consider noninteracting electrons 44 electron wave function associated with a level of energy satisfies the Schrodinger equation periodic boundary conditions a solution neglecting the boundary conditions noralization constant: probability of finding the electron soewhere in the whole volue V is unity energy oentu velocity wave vector de roglie wavelength r P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory x y x, y, z L x, y, z x, y L, z x, y, z x L, y, z x, y, z r p v dr r V e ir z D: D: p v L r

45 45 r V e ir x, y, z L x, y, z x, y L, z x, y, z x L, y, z x, y, z ixl y izl e e e apply the boundary conditions n, n, n coponents of ust be L L L i L x x y y z z nx, ny, nz integers the area per point the volue per point L L V a region of -space of volue contains the nuber of states per unit volue of -space, -space density of states / L V V states i.e. allowed values of -space P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory

46 consider T=0 the Pauli exclusion principle postulates that only one electron can occupy a single state therefore, as electrons are added to a syste, they will fill the states in a syste lie water fills a bucet first the lower energy states and then the higher energy states 46 the ground state of the N-electron syste is fored by occupying all single-particle levels with < state of the lowest energy volue the nuber of allowed values of within the sphere of radius to accoodate N electrons electrons per -level due to spin eri wave vector eri energy eri teperature eri oentu eri velocity T p v / / 4 V V 6 N V 6 n ~08 c- ~-0 ev ~04-05 K ~08 c/s density of states copare to the v / ~ 07 c/s at T=00K theral T / classical theral velocity 0 at T=0 eri sphere eri surface at energy P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory v n y n n x

47 P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory Total nuber of states with energy < The density of states nuber of states per unit energy V d dn D V N Density of states V -space density of states the nuber of states per unit volue of -space The density of states per unit volue or the density of states d dn D V N Total nuber of states with wave vector < 47

48 P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory e V i N d V d V V V d d 4 N V Ground state energy of N electrons Add up the energies of all electron states inside the eri sphere volue of -space per state sooth The energy density The energy per electron in the ground state 48

49 Rears on statistics I In quantu echanics particles are indistinguishable systes where particles are exchanged are identical exchange of identical particles can lead to changing of the syste wave function by a phase factor only repeated particle exchange e ia ia, e,,, 49 syste of N= particles, - coordinates and spins for each of the particles Antisyetric wavefunction with respect to the exchange of particles p p p p, erions are particles which have half-integer spin the wavefunction which describes a collection of erions ust be antisyetric with respect to the exchange of identical particles erions: electron, proton, neutron p, p single particle states syetric wavefunction with respect to the exchange of particles, p p p p osons are particles which have integer spin the wavefunction which describes a collection of bosons ust be syetric with respect to the exchange of identical particles osons: photon, Cooper pair, H ato, exciton if p = p 0 at ost one ferion can occupy any single particle state Pauli principle Unliited nuber of bosons can occupy a single particle state obey eri-dirac statistics Obey ose-instein statistics P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory

50 Distribution function f probability that a state at energy will be occupied at theral equilibriu 50 ferions particles with half-integer spins bosons particles with integer spins both ferions and bosons at high T when T eri-dirac distribution function ose-instein distribution function Maxwell-oltzann distribution function f f f D M exp T exp T exp T n dn dd f degenerate eri gas f D < degenerate ose gas f can be any classical gas f M << =n,t cheical potential P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory

51 and D distributions differ fro the classical M distribution because the particles they describe are indistinguishable. Particles are considered to be indistinguishable if their wave pacets overlap significantly. Two particles can be considered to be distinguishable if their separation is large copared to their de roglie wavelength. Theral de roglie wavelength Particles becoe indistinguishable when i.e. at teperatures below rears on statistics II ~ d At T < Td f and fd are strongly different fro fm At T >> Td f fd fm lectron gas in etals: n = 0 c -, = e Td ~ 0 4 K Gas of Rb atos: n = 0 5 c -, ato = 0 5 e Td ~ K xcitons in GaAs QW n = 0 0 c -, exciton = 0. e Td ~ K T d d d T n P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron x Theory n ~ h p A particle is represented by a wave group or wave pacets of liited spatial extent, which is a superposition of any atter waves with a spread of wavelengths centered on 0=h/p The wave group oves with a speed vg the group speed, which is identical to the classical particle speed Heisenberg uncertainty principle, 97: If a easureent of position is ade with precision x and a siultaneous easureent of oentu in the x direction is ade with precision px, then p x g 0 ' r, t g 'exp i ' r t ' x x v vg=v x x 5

52 Density of states Distribution function T 0 D f n the eri-dirac distribution D dn d exp T dd f li f, T 0 0 li T 0 D d [the nuber of states in the energy range fro to + d] V D f d [the nuber of filled states in the energy range fro to + d] V Density of states D 5 per unit volue shaded area filled states at T=0 Density of filled states Df,T P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory

53 Specific heat of the degenerate electron gas, estiate 5 Specific heat U theral inetic energy Classical gas U u cv V T V T U u V u n v cv n n T The electronic specific heat cv ~ n V T P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield V Model for ree lectron Theory V The observed electronic contribution at roo T is usually 0.0 of this value f at T 0 differs fro f at T=0 only in a region of order T about because electrons just below have been excited to levels just above Classical gas: with increasing T all electron gain an energy ~ T eri gas: with increasing T only those electrons in states within an energy range T of the eri level gain an energy ~ T Nuber of electrons which gain energy with increasing teperature ~ The total electronic theral inetic energy T U ~ N T U T T ~ 00 K for typical etallic densities T = 0 T N / ~ K Troo / ~ 0.0

54 P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory f dd f d n f dd f d u H d H d H T O D T d D n T O D D T d D u correctly to order T Specific heat of the degenerate electron gas The way in which integrals of the for differ fro their zero T values is deterined by the for of H near = n n n n H d d n H! 0 Replace H by its Taylor expansion about = The Soerfeld expansion T O H T H T H H d d a T d H d f H n n n n n d f H d H Successive ters are saller by O T/ or T/ << Replace by T0 = and v V u u c T 54

55 Specific heat of the degenerate electron gas 55 D D statistics depress cv by a factor of n u u0 T D 6 u cv TD T n D c c v classical c v T T T T n n P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory

56 Theral conductivity 56 theral current density j q ne t j q Drude: application of classical ideal gas laws T v tc v c v v lvc v cvv ne n T a vector parallel to the direction of heat flow whose agnitude gives the theral energy per unit tie crossing a unite area perpendicular to the flow e T Wiedeann-ranz law 85 Lorenz nuber ~ 0-8 watt-oh/k success of the Drude odel is due to the cancellation of two errors: at roo T the actual electronic cv is 00 ties saller than the classical prediction, but v is 00 ties larger or degenerate eri gas of electrons the correct the correct estiate of v is v c v T n v c v c vclassical v classical ~ ~ T / / T ~ 0.0 ~ 00 at roo T at roo T T e P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory

57 Theropower Seebec effect: a T gradient in a long, thin bar should be accopanied by an electric field directed opposite to the T gradient high T low T QT theroelectric field 57 gradt Drude: application of classical ideal gas laws Theropower c v n Q e Q cv ne or degenerate eri gas of electrons the correct c v T n c v c vclassical ~ T / ~ 0.0 at roo T Q/Q classical ~ 0.0 at roo T Q 6 e T P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory

58 lectrical conductivity and Oh s law quation of otion Newton s law In the absence of collisions the eri sphere in -space is displaced as a whole at a unifor rate by a constant applied electric field ecause of collisions the displaced eri sphere is aintained in a steady state in an electric field eri sphere y x avg dv d e dt dt e t 0 t v avg avg e t avg e t j nev avg the ean free path l = v t because all collisions involve only electrons near the eri surface v ~ 0 8 c/s for pure Cu: at T=00 K t ~ 0-4 s l ~ 0-6 c = 00 Å at T=4 K t ~ 0-9 s l ~ 0. c avg << for n = 0 c - and j = A/ v avg = j/ne ~ 0. c/s << v ~ 0 8 c/s dp t p t f t 0 dt t p ft et Oh s law 58 ne t j ne t ne t P.Ravindran, PHY075- Condensed Matter Physics, Spring 0 6 July: Soerfield Model for ree lectron Theory