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Drudes Classical Model of Metallic Conduction Contrasted with Sommerfelds Quantum Mechanical Theory Slide 2 Drudes Classical Model of Metals (Beautifully explained in depth in Ashcroft and Mermin, Ch. 1) Modern condensed matter physics was BORN with the discovery of the electron by J.J. Thompson in 1897. Soon afterwards (1900) Drude used the new concept to postulate A theory of metallic conductivity Slide 3 3 Drudes Assumptions 1. Matter consists of light negatively charged electrons which are mobile, & heavy, static, positively charged ions. 2. The only interactions are electron-ion collisions, which take place in a very short time t. The neglect of the electron-electron interactions is The Independent Electron Approximation. The neglect of the electron-ion interactions is The Free Electron Approximation. Slide 4 4 The mean time between collisions is Trajectory of a mobile electron Ion Drudes Assumptions Continued 3. Electron-ion collisions are assumed to dominate. These will abruptly alter the electron velocity & maintain thermal equilibrium. 4. The probability of an electron suffering a collision in a short time dt is dt/, where 1/ The Electron Scattering Rate. Electrons emerge from each collision with both the direction & magnitude of their velocity changed; the magnitude is changed due to the local temperature at the collision point. 1/ is often an adjustable parameter. See the figure. Slide 5 Drude Conductivity: Ohms Law V = IR The Resistance R is a property of a conductor (e.g. a wire) which depends on its dimensions, V is a voltage drop & I is a current. In microscopic physics, it is more common to express Ohms Law in terms of a dimension-independent conductivity (or resistivity) which is intrinsic to the material the wire is made from. In this notation, Ohms Law is written E = j or j = E (1) E = Electric Field, j = Current Density, The resistivity & The conductivity of the material. Consider a wire of cross sectional area A, with a current flowing in it. The current consists of n electrons per unit volume, all moving in the direction of the current with velocity v: The number of electrons crossing area A in time dt is nAvdt A j Slide 6 The charge crossing A in dt is -nevAdt, so j = -nev. (2) In the real material, we expect the electrons to be moving randomly even in zero electric field due to thermal energy. However, they will have an average, or drift velocity along the field direction. v drift = -eE m (3) This comes from integrating Newtons 2 nd Law over time . This is the velocity that must be related to j. Combining (2) & (3) gives j = (ne 2 m)E. Comparison of this with j = E gives the Drude conductivity: Slide 7 7 As is often true for physics models, this result for has been obtained using some very simple assumptions, which surely cannot be correct in reality! How can this result be tested? First ask: Does the Drude assumption of scattering from ions seem reasonable? Check it experimentally by measuring for a series of known metals, and, using sensible estimates for n, e and m, estimate . Results show that,at least for simple metals s at room temperature. Slide 8 8 For many cases, instead of a an average scattering time , its often necessary to formulate a theory of conductivity in terms of An average distance between collisions. This distance is called The mean free path between collisions. To do this, we have to consider the average electron velocity. This should not be v drift, which is the electron velocity due to the electric field. Instead, it should be v random, the velocity associated with the intrinsic thermal energy of the electrons. Slide 9 Lets estimate v random by treating the electrons as a classical ideal gas and using the result from classical statistical physics: The Equipartition Theorem (m)v 2 random = ( 3 / 2 )(k B )T (For a simple derivation, see any Physics I textbook!) Results: The mean free path is = v random This is of the order of interatomic distances, so it is reasonable! Slide 10 A Very Important Experimental Result for Metals: The Wiedemann-Franz Law Since 1853, it has been known that one of the most universal properties of metals is an experimentally well verified relationship between the thermal and the electrical conductivities. As we just discussed, in the presence of an external electric field E, the current density j is given by Ohms Law: j = E jqjq j V -T In simple one dimensional geometry as in the figure, j = E = dV/dx The thermal analogue of this is j q = - dT/dx. Thermal Conductivity. Slide 11 Drudes assumption was that in metals is dominated by the electronic contribution. A result from elementary kinetic theory is: = ( 1 / 3 )v random c el (4) where c el is the electronic specific heat per unit volume. Each electron has an energy ( 3 / 2 )k B T so, for n electrons per unit volume: E tot = ( 3 / 2 )k B T, c el = dE tot /dT = ( 3 / 2 ) nk B T Recall that = v random , & divide by = (ne 2 m) giving: Slide 12 Dividing by T gives the simple result that This is a very nice result!! All the parameters that might be regarded in some way as poorly known have dropped out, leaving what looks like it might be a universal quantity. Experimentally, this is indeed the case. The measured number is a factor of two different from this Drude result, but in his original work, a numerical error made the agreement appear to be exact! Slide 13 13 So, Drudes model appeared to be reasonably self-consistent in identifying electron-ion collisions as the main scattering mechanism, and had a triumph regarding the most universal known property of metals. This was enough to set it up as the main theory of metals for two decades. However, fundamental problems began to emerge: 1. It could not explain the observation of positive Hall coefficients in many metals (discussed later). 2. As more became known about metals at low temperatures, it was obvious that since the conductivity increased sharply, was far too long to be explained by simple electron-ion scattering. 3. A vital part of the thermal conductivity analysis is the use of the kinetic theory value of 3 / 2 nk B for the electronic specific heat. Measurements gave no evidence for a contribution of this size. Slide 14 14 Advanced general interest reading on this issue (probably more suitable some time later in the year unless you have already read quite a bit about quantum mechanics): The theory of everything, R.B. Laughlin and D. Pines, Proc. Nat. Acad. Sci. 97, 28 (2000). The solution to these issues and the other unpleasant difficulties like the existence of insulators would not have come in a thousand years of hard work, if we had been restricted to classical physics. The physics of solids is deeply quantum mechanical; indeed condensed matter is arguably the best laboratory for studying subtle quantum mechanical effects in the 21st century. Slide 15 15 The Free Electron Gas A Non-trivial Quantum Fluid Bohr, de Broglie, Schrdinger, Heisenberg, Pauli, Fermi, Dirac.. With the development of quantum mechanics, a natural step was to formulate a quantum theory of electrons in metals. This was first done by Sommerfeld. Assumptions Most are very similar to those of Drude. Free & independent electrons, but no assumptions about the nature of the scattering. Starting point: time-independent Schrdinger equation Summerfelds Quantum Mechanical Model of Electron Conduction in Metals Slide 16 16 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 the Schrdinger equation we need appropriate boundary conditions for a metal. Standard particle in a box: set = 0 at boundaries. This is not a good representation of a solid, however. 1. It says that the surface is important in determining the physical properties, which is known not to be the case. 2. 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. Slide 17 17 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: Here V = L 3 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 integer, etc. (9) The most appropriate boundary condition for solid state physics is the periodic boundary condition : Slide 18 18 What is the physical meaning of these eigenstates? Then, note that k is also an eigenstate of the momentum operator, with eigenvalue p = k. 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: (11) (10) First, note energy eigenvalues: Slide 19 19 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. 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. Sommerfelds great contribution: to apply Paulis exclusion principle to the states of this system, not just to an individual atom. Slide 20 20 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.... kyky kzkz kxkx State separation 2 L State volume (2 L) 3 Filled states Empty states Fermi surface Fermi wavenumber k F Slide 21 21 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 - doesnt k F appear to depend on the arbitrary cube size L? Quantities of interest depend on the carrier number per unit volume; the sample dimensions drop out neatly. No - (12) Slide 22 22 How can we scale these quantum mechanical effects against something we are more familiar with? Calculate numerical values for the parameters. Use potassium Result: k F 0.75 -1 v F 1 x 10 6 ms -1 F 2 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! T F 25000 K ( recall k B T at room T 1/40 eV) Slide 23 23 k kFkF A couple of much-used graphs relating to the Sommerfeld model: a) The free electron dispersion Probability of state occupation 1 0 k F or k F b) The T = 0 state occupation function. Slide 24 24 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: where the chemical potential F. (13) At finite T: f()f() F ~ 2k B T As expected, T is a minor player when it comes to changing things. Slide 25 25 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. Our next task, then, is to derive a quantity of high and general importance, the density of states g( ). and the specific heat c el from dE tot /dT as before. (14) Then internal energy E tot (T) can be calculated from (15) Slide 26 26... kyky kzkz kxkx State separation 2 L State volume (2 L) 3 dk Number of allowed states per unit volume per shell thickness dk: spin Slide 27 27 Convert to density of states per unit volume per unit (the quantity usually meant by the loose term density of states): Very important result, but note that dependence is different for different dimension (see tutorial question 5). (16a, b) (17) Slide 28 28 n( ,T) F g( Movement of electrons in energy at finite T 2kBT2kBT 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 >> k B T: [E tot (T) - E tot (0)]/V 1/2g( F ). k B T.2k B T = g( F ). (k B T) 2 (18) Slide 29 29 Differentiating with respect to T gives our estimate of the specific heat capacity: c el = 2g( F ). k B 2 T (19) The exact calculation gives the important general result that c el = ( g( F ). k B 2 T (20) c.f. Drude: 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 (do this for yourselves as an exercise): (21) Slide 30 30 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 k B T. The quantum fermion gas is in some senses like a rigid fluid, and its thermal properties are defined by the behavior of its excitations. Slide 31 31 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 centered 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 = d /dk = -1 d /dk = k/m for free particles like those we are concerned with at present. Slide 32 32 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 k F ). 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 This equation would produce continuous acceleration, which we know cannot occur in the presence of scattering. Slide 33 33 Include scattering by modifying (22) to m(dv/dt + v -eE - ev B 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 (3) when we discussed the Drude model! It is no surprise, then, that it leads to the same expression for the electrical conductivity: Slide 34 34 Set B to zero and stress that the relevant velocity is v drift ; (23) becomes m(dv drift /dt + v drift -eE Steady state solution (dv drift /dt = 0) is just v drift = -(e /m)E Following the procedure from Kittel gives us the Drude expression (3): If you give this some thought, it should concern you. What happened to our new quantum picture? Slide 35 35 kyky kzkz kxkx To understand, consider physical meaning of the process: Fermi surface is shifted along the k x axis by an E field along x. The quasi-Drude derivation assumes that every electron state in the sphere is shifted by dk. This is mathematically correct, but physically entirely the wrong picture. kyky kzkz kxkx dk = -1 mv drift = -eE/ E = 0 Slide 36 36 Which states can interact with the outside world? kyky kzkz kxkx dkdk 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? Consider amount of extra velocity/momentum acquired in equilibrium: Drude- like picture: In the quantum model, only those within k B T of F, i.e. those very near the Fermi surface. # of states mom. gain # of states (1/2 FS area) mom. gain x comp. only Quantum picture: (24) Slide 37 37 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. Slide 38 38 What about thermal conductivity? Recall (4) from Drude model: = 1 / 3 v random c el Here, v random can clearly be identified with v F, and = v F . Provided that 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: (25) Slide 39 39 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? Slide 40 40 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). dkdk kyky kzkz dkdk Thermal excitation: All particles with k k F, but sum over k = 0. kxkx kyky kzkz Electrical excitation: All particles with k k F, but sum over k = 2 k F /3. kxkx Slide 41 41 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 dont ask about the microscopic origins of the scattering time - 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. I asked my tutor why this was so - and he told me that it was not understood. It was good to know the limits of knowledge at the time Quote from Sir Nevill Mott, writing about his time as a student in Cambridge, around 1925. Slide 42 42 Classical Drude gas Random velocity purely thermal: Specific heat c el = Large number of particles moving slowly. Quantum Sommerfeld gas: do wave mechanics and then think in an equivalent particle picture Random velocity dominantly quantum (due to Pauli principle): Small effective number of particles moving very fast, due to special quantum mechanical constraints.