

In [1]:

    
import numpy as np
from scipy import linalg
import plotly.plotly as py
import plotly.graph_objs as go
from plotly import tools
from plotly.offline import init_notebook_mode, iplot
from ipywidgets import interactive

init_notebook_mode()

def fermi_gas_1D(n=1):
    n_down = (n + 1) // 2
    n_up = n // 2
    momenta = np.linspace(-10, 10, 101)
    energies = momenta**2
    data = [go.Scatter(x=momenta, y=energies, hoverinfo='none', line={'color':'black'})]
    discrete_momenta = np.arange(-19, 20)
    discrete_energies = discrete_momenta**2
    sort_order = np.argsort(discrete_energies)
    discrete_momenta = discrete_momenta[sort_order]
    discrete_energies = discrete_energies[sort_order]
    data.append(go.Scatter(x=discrete_momenta[:n_down], y=discrete_energies[:n_down] - 2,
                           marker=go.Marker(color='red', symbol='circle'), line={'width':0}))
    data.append(go.Scatter(x=discrete_momenta[:n_up], y=discrete_energies[:n_up] + 3,
                           marker=go.Marker(color='blue', symbol='circle'), line={'width':0}))
    layout = go.Layout(
            showlegend=False,
            autosize=False,
            width=600,
            height=300,
            xaxis={'title':'k'},
            yaxis={'title':'E'})
    fig = go.Figure(data=data, layout=layout)
    iplot(fig, show_link=False)

fermi_gas_1D = interactive(fermi_gas_1D, n=(1, 39))

def plot_dos(dim=1, mu=1., T=0.001):
    E = np.linspace(0, 2, 300)
    n = E**((dim - 2) / 2) / (np.exp((E - mu) / T) + 1)
    data = [go.Scatter(x=E, y=n, hoverinfo='none', line={'color':'black'})]
    layout = go.Layout(
            showlegend=False,
            autosize=False,
            width=600,
            height=300,
            xaxis={'title':'E'},
            yaxis={'title':'n'})
    iplot(go.Figure(data=data, layout=layout), show_link=False)

dos = interactive((lambda dim=1., mu=1.: plot_dos(dim, mu)), dim=(1, 3), mu=(0., 2.))

dos_finite = interactive(plot_dos, dim=(1, 3), mu=(0., 2.), T=(0.001, .1, 0.01))

Free electron model

By Anton Akhmerov (also it's my very first lecture ever today!)

This lecture:

Fermi surface
Fermi energy
Fermi velocity
Electron heat capacitance

Next lecture:
Scattering and magnetic field

Electrons

Q: In which ways are electrons different from phonons?

They have electric charge.
You cannot create new electrons, while you can create phonons
(So they obey different statistics, fermions, not bosons)
They have a mass:
$E = p^2/2m$ for electrons instead of $E = v|p|$ for (acoustic) phonons

NB: Electrons are my favorite particle

Q: What is similar between electrons and phonons?

Both interact with lattice
Both can store extra energy when the system is heated

Focus on differences first, forget the lattice exists (imagine the nuclei form an average potential for electrons).

Density of states

Plan: count states in a big box

Electrons obey Schrödinger equation:

$$ H = -\frac{\hbar^2}{2m}\left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}\right) $$

Take $L\times L \times L$ box with periodic boundary conditions: $\psi(L, y, z) = \psi(0, y, z)$, same for $y$ and $z$.

Wave vectors are now quantized: $k_{x,y,z} = 2\pi n_{x, y, z}/ L$

...and energies as well: $E(n_x, n_y, n_z) = \frac{(2\pi\hbar)^2}{mL^2}(n_x^2 + n_y^2 + n_z^2)$

Lowest energy state of many electrons

Let's start with a 1D illustration:



In [2]:

    
fermi_gas_1D

Fermi-everything

The lowest energy state is called Fermi sea, and the top of Fermi sea is Fermi surface
All the states below Fermi energy ($E_\text{F}$) are filled, those above are empty
This means $|k| < k_\text{F}$, the Fermi momentum

Counting the states

Number of states with $|\vec{k}| < k$:

1D: $N = \frac{2 \times 2 k}{2\pi/L} = \frac{2 k V}{\pi}$
2D: $N = \frac{2 \pi k^2}{(2\pi/L)^2} = \frac{k^2 V}{2\pi}$
3D: $N = \frac{2 (4/3) \pi k^3 }{ (2\pi/L)^3} = \frac{k^3 V}{3\pi^2}$

Density of states

We need to get:

$$g(E) \equiv \frac{dN}{dE} = \frac{dN}{dk}\frac{dk}{dE}.$$

From Schrödinger equation:

$$k = \hbar^{-1}\sqrt{2mE};\quad \frac{dk}{dE}=\hbar^{-1}\sqrt{m/2E}$$

So:

1D: $g(E) = \frac{2 V}{\pi} \frac{dk}{dE} \sim E^{-1/2}$
2D: $g(E) = \frac{k V}{2\pi} \frac{dk}{dE} \sim \text{const}$
3D: $g(E) = \frac{k^2 V}{\pi^2} \frac{dk}{dE} \sim E^{1/2}$

Number of states with each energy



In [3]:

    
dos

Effects of finite temperature

Fermi-Dirac distribution (check wiki: WP:Fermi Distribution):

$ f(E, T) = \frac{1}{e^{(E - \mu)/k_\text{B}T} + 1}$ ($\mu$ is chemical potential, energy cost for adding an extra electron)

At finite temperature $n(E) = g(E)\times f(E, T)$



In [4]:

    
dos_finite

Heat capacitance:

$n \approx (1/2) g(E_F) \times k_B T$ electrons have their energy increased by $\delta E \sim k_B T$

$$E(T) - E(0) \approx g(E_F) (k_b T)^2$$

So: $$ C_{v, e} \approx g(E_F) k_b^2 T = \ldots = \frac{3}{2} N k_B \frac{T}{T_F}$$

($T_F = E_F/ k_B$ is Fermi temperature)

At room temperature $C_{v, f} = 3Nk_B \gg C_{v, e}$.

At $T \to 0$, $C_{v, f} \sim T^3$, while $C_{v, f} \sim T$.