Chapter 6 Free Electron Fermi Gas Chapter 6 Free Electron Fermi Gas Free Electron Fermi Gas as Particles in a Box (1D) � As particles in a box (in 1D) the energy looks like a free particle of waveleng[.]
Chapter 6: Free Electron Fermi Gas Free Electron Fermi Gas as Particles in a Box (1D) • • As particles in a box (in 1D) the energy looks like a free particle of wavelength λn = 2L/n Therefore, the energy is: p h k h 2π = = εn = 2m 2m 2m λn 2 2 h 2π h nπ εn = = 2m L / n 2m L • The energy goes like (and it is quantized) n2 n=3 ε1=9(ħπ)2/(2mL2) = Ψ(x) ε1=4(ħπ)2/(2mL2) n=2 ε1=(ħπ)2/(2mL2) n=1 L Free Electron Fermi Gas • So, we have a quantized system of energy levels • How are these levels filled? • According to the Pauli Exclusion Principle, no two electrons can be in the same energy level (because they are fermions and follow Fermi-Dirac statistics) • We know that electrons can exist in the same energy level only if their spins are in opposite directions: mz = +1/2 or -1/2 (and therefore they have slightly different energy levels, but these differences are so small that we usually just say that they have the same energy) • We call these energy levels orbitals Energy level filling • So, what we have is that the energy levels will fill up, with electrons in each level, up to a maximum energy level which is nf = N/2 (at T = K) Energy levels fill up until N/2 No of electrons • • The number of orbitals with the same level is called the degeneracy This maximum energy level is often called the Fermi energy h nfπ εf = 2m L 2 h Nπ = 2 m L (in one dimension) 2 electrons/energy level Effect of Temperature • • • • We said on the last slide that this was the distribution at T = K Why? If we increase the temperature, there will be thermal energy which can excite electrons to higher energy levels What is the distribution at higher temperatures? This is determined by the Fermi-Dirac distribution: f (ε ) = Probability that an orbital of energy ε will be occupied exp[(ε − µ ) / kT ] + Chemical potential (~ amount of energy needed to add an electron to the system) Fermi-Dirac Distribution • • • • What does this mean? Chemical potential: this is rougly the Fermi energy εf (if you need to add an electron, it must go into the next occupied state which is at the fermi energy) The chemical potential changes a bit as the temperature changes, but it is usually ~ Fermi energy What does this distribution function look like? f (ε ) = • • exp[(ε − µ ) / kT ] + Maximum value: exp part goes to zero (as the energy becomes small, for example) f = (lower energy levels are usually filled) Minimum value: exp part becomes large (as the energy becomes large, for example) f = (higher energy levels are usually not filled) Fermi-Dirac Distribution ~ Fermi energy • • • • This is what the Fermi-Dirac distribution looks like at different temperatures (for a real 3D system) In this case εf ~ 50 000 K As T → K, this becomes a step function Note that the lower energy levels are usually filled, and as you raise the temperature, you increase the no of electrons at higher energy levels Technically, the Fermi energy changes as the temperature changes because it is defined as: µ = Fn+1 – Fn where F is the Helmholtz free energy (F = U – TS) (n = no of particles in system) 1.0 T = 500 K T = 5000 K T = 10000 K 0.8 0.6 f(ε) • 0.4 0.2 0.0 ε/k (in units of 10 K) Energy of system Entropy This goes down as T increases 10 Free Electron Gas in 3D • • Now we have to take this idea in 1D and change it so that we have a 3D solid (a particle in a box of dimensions L x L x L) The free-particle Schrödinger equation in 3D is: r r r h2 ∂2 ∂2 ∂2 + + ψ k (r ) = ε kψ k (r ) Hψ k (r ) = − 2m ∂x ∂y ∂z • The solution to this is (where nx, nY, and nz are integers): r nxπx n yπy nzπz sin ψ n (r ) = A sin sin L L L Periodic Boundary Conditions • Just like we did for phonons, we are going to insist that this wavefunction satisfies periodic boundary conditions: ψ ( x + L, y , z ) = ψ ( x, y , z ) • and similarly for the y and z coordinates This means that our k values in the x, y and z-direction are quantized: 2π 4π k x = 0,± ,± , etc L L • and similarly for kY and kz Just like for phonons, only certain wavelengths are possible for these electrons confined to the “box” Proof • • Our electrons moving throughout the lattice are moving like free particles in a box Therefore, let’s use the free-particle wavefunction which assumes that they have a well-defined momentum k: r r r Ψk (r ) = exp(ik • r ) • If we impose the boundary condition on this for the x-direction say (it is equivalent in the y or z-directions) Ψk ( x) = exp(ik x x) = Ψ ( x + L) = exp(ik x ( x + L)) = exp(ik x x) exp(ik x L) • We therefore get the result that: exp(ik L) = x ⇒ k x L = 2πn ⇒ k x = 2πn L 2π 4π ⇒ k x = 0,± ,± , L L (n is an integer) Energies of the electrons • The energy of an electron in state k is therefore: p h 2k h 2 2 εk = = = (k x + k y + k z ) 2m 2m 2m • • And the electron’s velocity is v = p/m = ħk/m The Fermi energy is therefore defined as: εf = • How we figure out what kf is? h kf 2m Back to k-space • Each value of k occupies a volume of (2π/L)3 in k-space KZ Kf KY KX How many k states (orbitals) are in a sphere of radius kf? Total no of states Unit of K-space (N/2)= (Volume of sphere)/(2π/L)3 (N/2)= (4πkf3/3)/(2π/L)3 So : kf = (3π2N/V)1/3 (each dot is separated by 2π/L, and represents a wave with wavevectors (Kx, Ky, Kz)) Fermi Energy • Our Fermi energy (at T = K) is therefore: εf = • h 2k f 2m = h 3π N 2m V 2 1/ h 3π N vf = = m m V • This is just related to the density of conduction electrons And we can define a velocity at the Fermi surface to be: hk f • 2/3 (~ 106 m/s!) This is just an electron at the highest occupied energy level, which can have a k-vector pointing in any direction (so it is on a sphere for the free-electron gas) We can also define a Fermi temperature to be Tf = εf/kB (Boltzmann’s constant) Meaning of the Fermi Temperature • The Fermi temperature is not the temperature of the electron gas! • It is a measure of where the Fermi energy is at (typically on the order of ~ 10000 K) • So, for most metals say at room temperature, not many electrons are excited above the Fermi energy • Problem of Chapter 6: Fermi Gases in Astrophysics uses the physics we just talked about to find the Fermi Energy of White Dwarf Stars (an enormous amount of electrons in a small space – high density, so high Fermi energy), and why pulsars are thought to be made of neutrons (and not electrons) Density of States of a Fermi Gas • • What is the density of states for a Fermi gas in 3D? Using our definition of DOS in terms of energy instead of frequency: dN D(ε ) = dε • And expressing the number of orbitals in terms of the energy V 2mε N= 2 3π h • We have: 3/ 3/ 3/ dN d V 2mε V 2m d / (ε ) = 2 = D(ε ) = dε dε 3π h 3π h dε 3/ V 2m 1/ D(ε ) = ε 2π h 3D Density of States • • • What does this look like? This represents how many energy values are occupied as a function of energy in the 3D k-sphere Why does this look differently than the FD distribution function? There can be many states occupied at the same energy, but they have different kvalues The larger your sphere is in kspace, the more k-values you have (and therefore, the more degenerate energy states you have) Width of ~ kT Density of states • • Density of states 1/2 (T > 0K) ε 1 0 εf 10 Energy ε The red curve is the ε1/2 function The black is the ε1/2 x FD distribution Note that the electrons that were once in section at T = K are now in section at some higher temperature