Advanced Solid State Physics

7.1.3 Sommerfeld Expansion When you calculate the thermodynamic quantities the normal way you get stuck analytically at the dens ...

where every odd term is zero (x,x^3 ,x^5 ) and the other ones are just numbers. If you put in these terms you get n= ∫∞ −∞ H(E)f ...

Figure 21: Plot of the specific heat to get the constants Sommerfeld Expansion: Entropy, Free Energy, Pressure We can also calcu ...

1-D, free particle 2-D, free particle 3-D, free particle i~ddtΨ=−~ 2 2 m∆Ψ i~ dΨ dt =− ~^2 2 m∆Ψ i~ dΨ dt =− ~^2 2 m∆Ψ Eigenfunc ...

Figure 23: Shifting the electrons as the density of states over~ω(energy of photons) 7.2 Electronic Band Structure Calculations ...

7.2.1 Empty Lattice Approximation The empty lattice approximation is the easiest one for most materials. We know from previous e ...

Figure 25: Dispersion relationship for a simple cubic lattice for the empty lattice approximation If you make a calculation you ...

Figure 26: Dispersion relationship for a fcc for the empty lattice approximation 7.2.2 Plane Wave Method - Central Equations Now ...

matrix for everykin the first Brillouin zone: ( ~^2 k^2 2 m −E ) Ck+ ∑ G UGCk−G= 0 (49) By solving these equations we get both, ...

Now back to the Central equations (1 dim). These are the algebraic equations we get when we put the Fourier- series in the Schrö ...

Example: Simple Cubic At first we have to figure out what the Fourier- series is for simple cubic. Just take the nearest neighbo ...

Figure 29: Dispersion relation and the density of states over the energy. 7.2.3 Tight Binding Model The tight binding model is a ...

Figure 30: Tight binding model.     〈Ψ 1 , 1 |H|Ψ 1 , 1 〉 〈Ψ 1 , 1 |H|Ψ 1 , 2 〉 ... 〈Φ 1 , 1 |H|ΦM,N〉 〈Ψ 1 , 2 |H|Ψ 1 , 1 ...

       −t 0 0 −t −t  −t 0 ··· 0 0 −t  −t 0 0 0 −t  .. . .. . ... −t −t 0 0 ··· −t             ...

Example: Square Well Potential A square well potential can be solved by the tight binding model by looking for a solution for a ...

Easier way to do the calculation: Back to the linear chain model we used for phonons (see chapter 6.1). Now we are talking about ...

elements) the phase factor cancels out. = ∫ d^3 rΨ∗(r)HΨ(r) 〈Ψk|Ψk〉 Now to themnearest neighbors. They are all equivalent in th ...

Fig. 33 shows the dispersion relationship of a simple cubic. It starts fromΓto X because in the x-direction there is a nearest n ...

In fig. 34 the dispersion relationship for an fcc crystal is shown. It is just plotted fromΓ(centre) to L (111-direction) and to ...

Figure 36: Dispersion relationship for a bcc crystal. Figure 37: Surfaces of constant energy for a bcc crystal calculated with t ...