Chapter 6 Free Electron Fermi Gas – Electrical Conductivity Chapter 6 Free Electron Fermi Gas � Electrical Conductivity Free Electron Fermi Gas � Last lecture, we showed that the free electron fermi g[.]
Chapter 6: Free Electron Fermi Gas – Electrical Conductivity Free Electron Fermi Gas • • • • • • Last lecture, we showed that the free electron fermi gas was successful in describing why Cel = γT This lecture, we will be look at electrical conductivity Can the free electron model reproduce experimental resistivities? The momentum of a free electron is mv = ħk What happens when we apply an electromagnetic field to a system of free electrons? Electric charge They feel a Lorentz force: (CGS units) r r r 1r r r dv dk F =m =h = − e( E + v × B ) dt dt c Speed of light Shift in k-vectors • For now, let’s look at the case where B = (we’ll look at what happens when B ≠ later) r • So we then have: r r r dk e r F =h = −eE → dk = − Edt dt h • When we apply an electric field from some time t = to some time t, what happens to the k vectors? r k (t ) t r r r e r e r ∫r dk = −∫0 h Edt → k (t ) − k (0) = − h Et k (0) k-vector shifts in the direction of E Fermi Spheres • • What is happening? Say the field is applied in the x-direction ky ky F Net momentum in kx direction kx kx Before a field is applied, there is no net momentum of the Fermi Gas After the field is applied, the gas feels a net momentum in the x-direction (all the states are shifted slightly) Electrical Conductivity • The electrons all feel a shift in their k-values of: • Now, according to this simple theory, the longer we leave the field on, the faster and faster the electrons start to move (the k-values, which are proportional to the momenta, keep on increasing in the x-direction) Is this observed in the real world? This would mean that if you apply a field to a copper wire, and create an electrical current (movement of electrons), the current would grow as a function of time, apparently without a limit What stops the electrons from moving faster and faster in this electric field? (they are accelerating under this force) • • • r r r Ft − eEt ∆k = = h h Collisions • • • • The reason why we don’t see this experimentally is that the electrons suffer collisions which decrease the velocity Collisions typically occur with Impurities in the lattice Lattice imperfections (ie Dislocations, point defects) Phonons The simplest approximation we can make is that the electrons lose all of their kinetic energy after each collision In this model, the electrons have a typical collision time τ, which is the time that they are accelerated from zero velocity to some maximum velocity, v, and after another collision, v = again τ τ τ τ F Back to the Fermi Sphere • Another way of saying this is that in the steady state, the Fermi sphere is displaced according to the equation we had before with t = τ ky F The incremental (additional) velocity of the electrons is then: v = -eτE/m kx and for a concentration of n electrons, the current density J (electrons per unit area per second) is: J = nqv = ne2τE/m Ohm’s law • • • • We have actually just derived Ohm’s law The electrical conductivity σ is defined by Ohm’s law to be J=σE On the last slide we showed that J = ne2τE/m So we can now say that the electrical conductivity is simply: σ = ne2τ/m = (ne) (e/m) τ Collision time Charge density is ne • e/m factor from the acceleration in electric field And the resistivity is the inverse of the conductivity: ρ = m/(ne2τ) (note: units are in Ohm cm) Other definition: ρ = E/ J Mean Free Path • • • • • • • How we determine the collision time? Electrons have a mean free path that they travel before a collision occurs l = v τ At low temperatures, most of the mobile electrons are right at the Fermi surface, so v = vf (Fermi velocity) At these temperatures, one can have mean free paths on the order of ~ cm (!) for very pure crystals (even up to 10 cm for some extremely pure metals!) Eg Copper has l (4K) = 0.3 cm Compare to the high temperature value: l (300 K) = x 10-6 cm More collisions at high temperatures (as expected) This leads to shorter collision times, and therefore higher resistivities (ρ tends to grow as you increase the temperature) Experimental Resistivities • • • At room temperature (300 K), the electrical resistivity is dominated by electron collisions with phonons At low temperatures (~ K), it is determined by collisions with impurities (there aren’t very many phonons around) The rates of these collisions are pretty much independent of one another, so we have: Matthiessen’s rule: ρ = ρL + ρi Phonon resistivity (related to the concentration of phonons, so it is temp dependent) • Imperfection resistivity (temp independent) Note: this implies that the collision times are related by: 1/τ = 1/τL + 1/τi Breakdown of Fermi Electron Gas Theory • • • • • At extremely low temperatures for some metals, the resistivity undergoes a remarkable change Metals such as Zn, Ti, and V superconduct! (at Tc = 0.875 K, 0.39 K, and 5.38 K respectively) This means that the resistivity drops to zero – the electron free paths become infinite! Kammerlingh Ones was the first one to notice this for Hg at ~ 4.153 K What is happening here? Some residual resistivity at T = K Superconductivity • • • In a superconductor, the currents effectively run forever – there are no collisions to slow them down (measurements by File and Mills suggest that the decay time of a supercurrent through a solenoid is no less than 100 000 years) Another odd property of superconductors: The Meissner Effect If a superconducting sample is cooled in a small magnetic field, the magnetic field lines will be expelled from the sample (due to the supercurrents forming in a direction to oppose the field, and therefore the field inside the superconductor is zero) Magnetic Levitation • • This is what causes the levitation of magnets above superconducting samples (the supercurrents form to counterbalance the magnetic force, and when the forces are equal and opposite, the magnet floats) Potential application: levitation of magnetic trains (no friction) Electrons in magnet, which create a fixed magnetic field Superconducting electrons in sample (in a direction which counters the magnet to expel the magnetic field) Type I and Type II Superconductors • • Type I superconductor: A field can be applied to some maximum value before it becomes “normal” The field does not penetrate the superconductor (Meissner effect) Most metals belong to this class (eg Zn, V, Ti) Type II superconductor: A field can be applied up to a critical value, HC1, where the field lines penetrate the sample This is known as the vortex state After the field is increased to HC2, the material is no longer superconducting These are the “high-Tc” superconductors, like YBa2Cu3O7 Penetration of magnetic field lines in a type II superconductor Explanation using statistical physics Total electron spin = • • • • • • • • What happens to our free electron theory? The whole reason why this theory works is that you can only have electrons in each state (because electrons have s=1/2 and they are fermions) But at low enough temperatures, the electrons are moving slow enough for other effects to become important (free electron theory breaks down) In BCS theory (Bardeen-Cooper-Schrieffer), phonons cause a temporary build up of positive charge, which attracts electrons So, the electrons, which move much faster than the phonons, are attracted to one another for brief periods of time They can “pair up” into integer states (eg S = “singlet” or S = “triplet”) If they have integer spin, they are bosons They can all be in the same energy state – this is often called Bose Condensation So, they can all be at the same energy – they can all move in the same way when an electric field is applied (a boson, no longer a fermion) In BCS theory, phonons can cause electrons to become attracted to one another e