Chapter 6 Free Electron Fermi Gas The Hall Effect Chapter 6 Free Electron Fermi Gas The Hall Effect and Thermal Conductivity Motion in Magnetic Field • Last time, we looked at what happens when you ap[.]
Chapter 6: Free Electron Fermi Gas: The Hall Effect and Thermal Conductivity Motion in Magnetic Field • • • Last time, we looked at what happens when you apply an electric field to a Fermi gas Now, we are going to look at what happens when you apply a magnetic field For the application of an electric field, we found that the Fermi sphere is displaced by ∆k r r r d r F = ma = h ∆k = −eE dt ky F kx (for a field in the x-direction) Effect of Collisions • We also showed that the electrons undergo collisions in a material after a collision time τ This means that the equation we wrote above has to be modified: ( ) r ⎛ d 1⎞ r F = h⎜ + ⎟ ∆k ⎝ dt τ ⎠ • • This second term, which takes collisions into account, acts like a frictional force Since there is no derivative in this form, it says that the force is proportional to ∆k/τ, which is like saying it is proportional to the velocity of the particles (k ~ momentum) Other frictional forces act this way, like friction due to air resistance (this is proportional to the velocity, and explains why a terminal velocity exists Similarily, it explains why the electrons’ velocity doesn’t grow without limit when you apply an electric field) Electric and Magnetic Fields • Using this expression for the force, let’s go back to our original equation (which included both electric and magnetic fields): ( ) ) r ⎛ d 1⎞ r ⎛ r r r⎞ F = h⎜ + ⎟ ∆k = −e⎜ E + v × B ⎟ c ⎝ dt τ ⎠ ⎝ ⎠ r r r = −e E + v × B ( • (CGS units) (SI units) Now, assuming that mv = ħ∆k (ie the electrons velocities are shifted by this amount due to the electric/magnetic field) we have: r ⎛ r r r⎞ ⎛ d ⎞r F = m⎜ + ⎟v = −e⎜ E + v × B ⎟ c ⎝ ⎠ ⎝ dt τ ⎠ An example: The Hall Effect • To illustrate how this equation works, lets take a metal and apply a static magnetic field B in the z-direction, and a static electric field Ex in the x-direction B z y Ex x • How the electrons move? (the result of this experiment is called the Hall Effect) The Hall Effect • • • Consider the motion of the electrons in the x, y, and z direction Initially: v = (vx,vY,vz), E = (Ex, 0, 0), B = (0, 0, B) Apply the fields, and observe what happens r ⎞ ⎛ ⎛ d 1⎞ ⎛ r r r⎞ ⎛ d 1⎞ F = m⎜ + ⎟v = −e⎜ E + v × B ⎟ → m⎜ + ⎟v x = −e⎜ E x + vY B ⎟ c c ⎠ ⎝ ⎝ dt τ ⎠ ⎠ ⎝ ⎝ dt τ ⎠ (net force in –x direction) y Ex (electrons build up down here) B x Initial state: electrons start to flow in the –y direction This creates a field EY ⎛1 ⎞ ⎛ d 1⎞ m⎜ + ⎟vY = e⎜ v x B ⎟ ⎝c ⎠ ⎝ dt τ ⎠ (net force in the –y direction) We need to consider what happens after this initial electron drift to the bottom part of the metal slab The Hall Effect • • The electrons will build up in the lower part of the metal, generating an electric field EY This will occur until the motion reaches a steady state – that is, when the forces are balanced in the y-direction (the Lorentz force from the magnetic field, and the Electric force from the build-up of the electrons will be equal and cancel eachother out) Steady-state conditions: d/dt part = → (forces are balanced) (from Lorentz Force) y EY + + + EY - + Ex - - - (electrons build up down here) Steady state equations: x ⎞ ⎛ ⎛ d 1⎞ m⎜ + ⎟v x = −e⎜ E x + vY B ⎟ c ⎠ ⎝ ⎝ dt τ ⎠ ⎞ ⎛ ⎛ d 1⎞ m⎜ + ⎟vY = −e⎜ EY − v x B ⎟ c ⎠ ⎝ ⎝ dt τ ⎠ mvx ⎞ ⎛ → = −e⎜ E x + vY B ⎟ τ c ⎠ ⎝ mvY ⎞ ⎛ → = −e⎜ EY − v x B ⎟ τ c ⎠ ⎝ The Hall Effect • These equations of motion are often expressed as: eτ eτ vx = − E x − ωcτvY ; vY = − EY + ωcτvx m m • • • For electrons moving in the z-direction, we have: vz = (they don’t feel the magnetic field), unless there is an electric field and then we have: vz = -(eτ/m)Ez The constant ωC = (eB/mc) is called the cyclotron frequency (see the next assignment to find out why) The Hall Effect occurs when we apply an electric field in the xdirection and a magnetic field in the z-direction (which generates, as well, an electric field in the y-direction EY) What happens? The Hall Effect • • If the current can’t flow out of the metal in the y-direction, then in the steady state, the velocity of the electrons in the y-direction must be zero (vY = 0) Using the equations we just derived, we then have: • And: eτ vY = − EY + ωcτv x = m ωcτv x ωc v x → EY = m =m eτ e eτ vx = − Ex m vx → Ex = −m eτ so putting it together: eBτ Ex EY = − mc The Hall Effect • So, what happens is that we get a net flow of electrons in the x-direction, but we also get an electric field set up in the y-direction Hall Effect measurements compare the ratio of the field created in the ydirection, to the current in the x-direction, and the magnetic field in the zdirection The Hall coefficient is defined by: • • (from steady state solution) eBτE x / mc EY =− =− RH = jx B ne τE x B / m nec y EY + + + EY - (from last lecture) + Ex - - (CGS units) - (in steady state, no of e- is constant at bottom) x jx RH = − ne (SI units) (electric charge) (density of conduction electrons) The Hall Effect • • • • • Why is this an important measurement? It relates simple, easily measured quantities (the current density, the electric field in y-direction, and the magnetic field in the zdirection) to the density of conduction electrons (which we can calculate) Also, it gives some strange results! In particular, RH changes signs for some materials! Why? Metal RH (exp) No of RH carriers (theory) /atom Li -1.89 el -1.48 Na -2.619 el -2.603 K -4.946 el -4.944 Rb -5.6 el -6.04 Cu -0.6 el -0.82 Ag -1.0 el -1.19 Au -0.8 el -1.18 Al 1.136 hole 1.135 In 1.774 hole 1.780 Electrons and holes • The Hall Effect was an important experiment historically because it suggested that a carrier could have a positive charge • These carriers are “holes” in the electron sea (and thus, being the absence of an electron, they have a net positive charge) These were first explained by Heisenburg • We can’t explain why this would happen with our free electron theory (but it arises naturally in band theory) • Note: the conditions we derived for the steady state can be invalid for several conditions (ie when there is a distribution of collision times) But in general, it is a very powerful tool for looking at properties of materials Thermal Conductivity • • • • Last chapter, we found that for phonons, the thermal conductivity is κ = 1/3 Cvl (heat capacity/unit volume = C, v is the velocity of phonons, l is the mean free path) The exact same theory can be applied for free electrons (just like phonons, they move with a certain velocity, and have a mean free path) Most of the mobile electrons are at the Fermi energy, so εF = ½ mvF2 We also have, using Cel = 1/2 π2Nk2T/εf and l = vf τ ⎞ 1 ⎛⎜ N T ⎟v l κ el = Cvl = π k ⎟ f ⎜ 3 V mv f ⎠ ⎝ π nk 2Tτ (so, κel ~ T at low temperatures) → κ el = 3m (note: n = electron density) Thermal Conductivity • What carries the heat in a metal? The electrons or phonons? • In pure metals, the electronic part is much greater than the phonons (density of free electrons is high) • If the metal is impure, the mean free path of the electrons will be lower, so the phonon part may dominate • Note that the thermal conductivity of a metal should look like the specific heat at low temperatures: C ~ γT + βT3 κ ~ AT + BT3 (different constants, but same functional form at low temperatures) Wiedemann-Franz Law • • In the early days, scientists did not know if what carried heat and what carried electrical current in metals was the same particle (ie the electron) One test of this was to take the ratio of the thermal conductivity to the electrical conductivity: κ π k Tnτ / 3m π = = ne τ / m σ • • 2 ⎛k⎞ ⎜ ⎟ T = LT ⎝e⎠ (Wiedemann-Franz law L = Lorentz number) Note that the Lorentz number is just a constant – the terms of m and n cancel It should be the same for all metals and equal to 2.45 x 10-8 WΩ/K2 (using the Free Electron Fermi Gas model in SI units) How well does this hold up? Wiedemann-Franz Law • • • • This model works incredibly well for most metals (L ~ 2.45 W Ω/K2) Therefore, it appears that electrons carry both heat and current (which should be expected) At low temperatures, the Lorentz number decreases slightly (our model starts to break down) Over all though, it is very successful L L Metal (O deg C) (100 deg C) Ag 2.31 2.37 Au 2.35 2.40 Cd 2.42 2.43 Cu 2.23 2.33 Mo 2.61 2.79 Pb 2.47 2.56 Pt 2.51 2.60 Sn 2.52 2.49 W 3.04 3.20 Zn 2.31 2.33 (L in units of 108 W Ω/K2)