Advanced Solid State Physics

Theentropy(density)sis s=− ∂f ∂T =− 4 kB^4 T^3 3 ~^4 c^3 π^2 ∫∞ 0 x^3 ex− 1 dx and theinternal energy(density)uis u=f+Ts=− 3 f=− ...

0 1 2 3 4 5 6 7 8 x 10−6 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 wavelength / m internal energy density u / J/m 3 T ...

So it is∝T^4 .σ=^2 π (^5) k (^4) B 15 h^3 c^2 = 5.^67 ·^10 − (^8) Wm− (^2) K− (^4) is the Stefan-Boltzmann constant. TheentropyS ...

5 Photonic Crystals 5.1 Intruduction: Plane Waves in Crystals A crystal lattice is a periodic arrangement of lattice points. If ...

Now we expand the electron densityn(r)in a fourier series (this is possible because the periodic- ity ofn(r)is the same as for t ...

O P 2 P 1 G 2 2 k 2 G 1 2 k 1 Figure 8: Brillouin zone Now we take the inner product ofGCwith some vectork 1. If this product eq ...

(a) (b) Figure 9: a) Dispersion relation for photons in vacuum (potential is zero!); b) Density of states in terms ofωfor photon ...

5.3 Central Equations We can also take thecentral equationsto do this numerically. Now we have the wave equation with the speed ...

Figure 10: Dispersion relation for photons in a material. of the matrix andω^2 are the eigenvalues of the matrix. We just need t ...

solved to a two-by-two matrix, which means that we only take one fourier coefficient into account. ( (k−G 0 )^2 U 0 k^2 U 1 (k−G ...

(a) (b) Figure 11: a) Density of states for photons in one dimension in a material in terms ofω; b) Density of states of photons ...

6 Phonons In the long wavelength limit (at low frequencies), sound obeys the wave equation: c^2 ∇^2 u(x,t) = ∂^2 u(x,t) ∂t^2 (42 ...

(^00) omega_max 1 2 3 4 5 6 ω D(ω ) (a) (^00) pi/(2a) pi/a 0.2 0.4 0.6 0.8 1 k ω (b) Figure 13: a) Dispersion relation of a 1-di ...

−ω^2 M 2 v=C(u(1 +eika)− 2 v) These equations can be written in a matrix ( ω^2 M 1 − 2 C C(1 +e−ika) C(1 +eika) ω^2 M 1 − 2 C )( ...

−2pi/a^0 −pi/a (^0) k pi/a 2pi/a ω Figure 16: Dispersion relation of a 1-dimensional chain of two different types of atoms with ...

m ∂^2 zp,q,r ∂t^2 =C 1 (zp,q,r− 1 − 2 zp,q,r+zp,q,r+1) Once again, we assume plane wave solutions like rp,q,r=rkei(pka^1 +qka^2 ...

(a) (b) Figure 18: a)ωas a function ofkin the 11-direction; b)ωas a function ofkx Plottingωin the 10-direction (kx-direction), w ...

little amount of light, that is scattered inelastically, submits part of its energy to phonons. Therefore, by measuring the chan ...

7 Electrons 7.1 Free Electron Fermi Gas The density of statesD(E)of an electron gas isD(E) =D(k)dEdk, as stated previously. The ...

(^001020304050) 1 D(E) E F E/EF (a) (^0012345) 0.2 0.4 0.6 0.8 1 1.2 E/EF 2 E D(E)/F π (b) (^0020406080100) 1 2 3 4 5 6 7 8 9 10 ...