Failures and successes of the free electron model

Transcription

1 Failures and successes of the free electron model

2 Temperature dependence on the electric conduc1vity j e = E = ne2 m electron gas electron- phonon coupling Drude s model predicts that σ is independent of temperature unless we put in by hand that = (T ) electrical conductivity is about momentum transfer Superconductor R 1

3 Temperature dependence on the thermal conduc1vity electron gas phonons j Q = rt = 2 3 k 2 B Tn m METAL Trea<ng the electron gas as non- interac<ng fermions gives (only) the correct low temperature behaviour for metals. thermal conductivity is about energy transfer NON- METAL

4 Thermal conduc1vity of diamond

5 Temperature dependence on the specific heat c v = u T V 3nk B T T 3 c v = 2 2 k B T F nk B Diamond Again, only the low temperature behaviour for metals is fairly well described. For high temperatures one needs to consider heat carried by the ion cores

6 Hall coefficient Free electron theory predicts: - Constant Hall coefficient R H =- (nec) Hall voltage independent of H The free electron result works well for alkali metals (Na, Li, K) but fails for Al, Mg and many more (see AM Tab. 1.4). The Hall effect is a powerful way to discover anomalous proper<es of materials! Normalised Hall coefficient for two types of high temperature super- conductors. No<ce the sign shig.

7 More problems Direc1on of DC current: Does not need to be parallel to the applied electric field! Too long mean free path: v F = p 2 F /m Å and mysteries Why are some materials metals and others insulators?

8 Main success: Widemann- Franz law T = 2 3 kb e 2 = [W Ohm K 2 ] Metal 273K 373K Mo Pb Pt Sn W Zn Metal 273K 373K Ag Au Cd Cu Ir Lorentz number in 10-8 [W Ohm K - 2 ] Note: outside this temperature range the Lorentz number can depend strongly on temperature.

9 Review of main assump1ons 1. Free electron approxima1on The ion cores play no other role than source of collisions. Between the collisions they do not affect the electron s mo<on. Resolu1on: we will let the electrons move in a sta<c (periodic) poten<al due to a fixed array of ions rather than in free space. 2. Independent electron approxima1on electron- electron interac<ons are ignored. Resolu1on: ignore. 3. Sta1c ions The ions are also dynamics en<<es which can contribute to physical phenomena. One prominent example is heat transfer. Resolu1on: we will let the ions vibrate around their equilibrium posi<ons due to thermal fluctua<ons

10 CRYSTALLOGRAPHY

11 Defini1on of a Bravais lasce (i) Infinite array of discrete points with an arrangement and orienta<on that appears exactly the same from whichever of the points the array is viewed. (ii) All points with posi<on vectors R on the form R = n 1 a 1 + n 2 a 2 + n 3 a 3 (3D) n 1,n 2,n 3 integers not unique!

12 Five unique Bravais lakces in 2D Oblique Hexagonal (triangular) Rectangular Square Centred rectangular

13 14 unique Bravais lakces in 3D

14 Example: BCC (e.g. Cesium, Sodium) Primi<ve (Bravais) lakce vectors: a 1 = aˆx a 2 = aŷ a 3 = a (ˆx +ŷ +ẑ) 2 ˆx a Cs = 6.05 Å a Na = 4.23 Å ŷ ẑ

15 Example: FCC (e.g. Gold, Argon) Primi<ve (Bravais) lakce vectors: a 1 = a 2 (ŷ +ẑ) a 2 = a (ˆx +ẑ) 2 a 3 = a (ŷ +ˆx) 2 aau ˆx ŷ ẑ = 4.08 Å a Ar = 5.26 Å

16 Wigner- Seitz primi<ve cell AM: One can always choose a primi6ve cell with the full symmetry of the Bravais la;ce. By far the most common such choice is the Wigner- Seitz cell. FCC The Wigner- Seitz cell is the region of space around a la;ce point that is closer to that la;ce point than to any other la;ce point

17 Lakce with a basis 2D BCC Bravais primi1ve lasce vectors a 1 = aˆx, Two- point basis 0, a (ˆx +ŷ) 2 a 2 = aŷ

18 WAVES IN SOLIDS Experimental setup for X- ray diffrac1on Outgoing wave e ik r k = 2 ˆn0 Incoming wave e ik 0 r k 0 = 2 ˆn θ Bragg angle sample (fcc) Bragg s law gives the angles for coherent and incoherent scaqering from a crystal lakce

19 Reciprocal lakce AM: Consider a set of points R cons6tu6ng a Brvais la;ce and a plane wave exp(i k.r). For general k such a plane wave will, of course, not have the periodicity of the la;ce, but for special choices it will. The set of all wave vectors K that yield plane waves with the periodicity of the given Bravais la;ce is known as the reciprocal la;ce. e ik R =1 R = n 1 a 1 + n 2 a 2 + n 3 a 3

20 Laue condi1on (AM): Construc6ve interference will occur if the change in wave vector equals a reciprocal la;ce vector. K = k 0 k 0 = k k k K = 1 2 K Bragg plane! A geometrical interpreta<on of the Laue condi<on

21 Theorem (AM): For any family of la;ce planes there are reciprocal la;ce vectors that are (i) perpendicular to the planes (ii) the shortest of which have length 2π/d. Miller indices (AM): The Miller indices of a plane are the coordinates (h, k, l) of the shortest reciprocal la;ce vector to that plane.