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Commit 0e78cd78 authored by Anton Akhmerov's avatar Anton Akhmerov
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2 merge requests!65Crystal structure changes,!16WIP: exercises lecture 7 (Tight-Binding)
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src/4_sommerfeld_model.md
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......@@ -33,7 +33,32 @@ $$
k_x=\frac{2\pi p}{L},\ k_y=\frac{2\pi q}{L},\ k_z=\frac{2\pi r}{L}\ \Rightarrow\ \varepsilon=\frac{2\pi^2\hbar^2}{mL^2}\left( p^2+q^2+r^2 \right)
$$
![](figures/free_electron.svg)
```python
kf = 3;
extrapol = 1.1;
ks = np.arange(-kf, kf+1);
kcont = np.linspace(-extrapol*kf, extrapol*kf, 200);
Edis = ks**2;
Econt = kcont**2;
fig = pyplot.figure();
ax = fig.add_subplot(111);
ax.plot(kcont, Econt);
ax.plot(ks, Edis, 'k.', markersize=10);
for i in range(2*kf + 1):
ax.plot([ks[i], ks[i]], [0.0, Edis[i]], 'k:');
ax.set_xlim(-3.75, 3.75);
ax.set_ylim(0.0, 11);
ax.set_xlabel(r"$k \enspace \left[ \frac{2 \pi}{L} \right]$");
ax.set_ylabel(r"$\varepsilon$");
ax.set_xticklabels([""] + ks.tolist() + [""]);
ax.set_yticks([]);
draw_classic_axes(ax, xlabeloffset=.6);
```
Comparable to phonons, but: electrons are _fermions_.
......@@ -97,7 +122,35 @@ For copper, the Fermi energy is ~7 eV. It would take a temperature of $\sim 70 0
The total number of electrons can be expressed as $N=\frac{2}{3}\varepsilon_{\rm F}g(\varepsilon_{\rm F})$.
![](figures/fermi_level.svg)
```python
kf = 3.0;
extrapol = 4.0/3.0;
kfilled = np.linspace(-kf, kf, 500);
kstates = np.linspace(-extrapol*kf, extrapol*kf, 500);
Efilled = kfilled**2;
Estates = kstates**2;
fig = pyplot.figure();
ax = fig.add_subplot(111);
ax.fill_between(kfilled, Efilled, kf*kf, alpha=0.5);
ax.plot([kf, kf], [0.0, kf*kf], 'k:');
ax.plot(kstates, Estates, 'k--');
ax.plot(kfilled, Efilled, linewidth=4);
ax.axhline(kf*kf, linestyle="dotted", color='k');
ax.set_xticks([kf]);
ax.set_yticks([kf*kf + 0.4]);
ax.set_xticklabels([r"$k_F$"]);
ax.set_yticklabels([r"$\varepsilon_F$"]);
ax.set_xlabel(r"$k$");
ax.set_ylabel(r"$\varepsilon$");
ax.set_xlim(-kf*extrapol, kf*extrapol);
ax.set_ylim(0.0, kf*kf*extrapol);
draw_classic_axes(ax, xlabeloffset=.6);
```
The bold line represents all filled states at $T=0$. This is called the _Fermi sea_. Conduction takes place only at the _Fermi surface_: everything below $\varepsilon_{\rm F}-\frac{eV}{2}$ is compensated.
......
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