Homework Set 3

Problem 1: Electrons in graphene

Graphene is one-atom-thick crystal of carbon atoms densely packed into a honeycomb lattice. Its surprising discovery (according to Mermin-Wagner theorem of statistical mechanics, two-dimensional crystals do not exist!) in 2004 has led to Nobel Prize in Physics 2010. The band structure of graphene close to the Fermi energy is such that electrons behave as the so-called massless Dirac fermions with energy-momentum relationship:

${\epsilon }_{\pm }(\mathbf{𝐤})=\pm \hslash v_F|\mathbf{𝐤}|$

akin to photons or neutrinos of high energy physics, except that instead of the velocity of light $c$ is replaced by the Fermi velocity is $v_F\approx c/300$.

(a) For any fermionic system at chemical potential $\mu$, show that the probability of finding an occupied state of energy $\mu +\delta$ is the same as that of finding an unoccupied state of energy $\mu -\delta$ where $\delta$ is any constant energy. HINT: According to Fermi-Dirac statistics, the probability of finding $n_{\epsilon }$ particles in a single particle state of energy $\epsilon$ is

$P[n_{\epsilon }]=\frac{e^{-\beta (\epsilon -\mu )n_{\epsilon }}}{1+e^{-\beta (\epsilon -\mu )}}$

(b) At zero temperature, all negative energy states (the so-called holes) in graphene are occupied and all positive energy states are empty, so that $\mu (T=0)=0$. Using the result in (a), find the chemical potential at finite temperature $T$.

(c) Show that the mean excitation energy of this system at finite temperature satisfies:

$E(T)-E(0)=4A\int \frac{d^2\mathbf{𝐤}}{(2\pi {)}^2}\frac{{\epsilon }_{+}(\mathbf{𝐤})}{\exp [\beta {\epsilon }_{+}(\mathbf{𝐤})]+1}$

where A is the surface are of graphene. HINT: You can find $E(T)-E(0)$ from the following formula:

$E(T)-E(0)=\sum _{\mathbf{𝐤},s_z}[⟨n_{+}(\mathbf{𝐤})⟩{\epsilon }_{+}(\mathbf{𝐤})-(1-⟨n_{-}(\mathbf{𝐤})⟩){\epsilon }_{-}(\mathbf{𝐤})]$

where $s_z$ is sum over the for spin degree of freedom (which gives factor 2 in the formula that you have to prove).

(d) Give a closed form answer for the excitation energy by evaluating integral in (c).

(e) Using (d), calculate the heat capacity $C_V$ of massless Dirac fermions in graphene as a function of temperature.

Note that your final results can be expressed using Riemann zeta function which specifies the values of $f_m^{-}(z)$ for $z=e^{\beta \mu }=1$ in the case of Dirac fermions.

Problem 2: Pauli paramagnetism

Calculate the contribution of electron spin to its magnetic susceptibility as follows. Consider non-interacting electrons where each is subject to a Hamiltonian operator:

${\hat{H}}_1=\frac{{\widehat{\mathbf{𝐩}}}^2}{2m}-{\mu }_0\overrightarrow{\sigma }\cdot \mathbf{𝐁}$

where ${\mu }_0$ is the Bohr magneton and we ignore orbital effects of magnetic field (if they are taken into account then, $\mathbf{𝐩}\to \mathbf{𝐩}-e\mathbf{𝐀}$ for vector potential $\mathbf{𝐀}$).

(a) Calculate the grand potential $\mathrm{\Phi }$ at a chemical potential $\mu$. HINT: The energy of electron gas is given by $E=\sum _p{E}_p(n_p^{+},n_p^{-})$ where $n_p^{\pm }$ (=0 or 1 as in the case of any fermionic system) denotes the number of particles having $\pm$ spins and momentum p, and

$E_p(n_p^{+},n_p^{-})=(\frac{p^2}{2m}-{\mu }_{0}B)n_p^{+}+(\frac{p^2}{2m}+{\mu }_{0}B)n_p^{-}=(n_p^{+}+n_p^{-})\frac{p^2}{2m}-(n_p^{+}-n_p^{-}){\mu }_{0}B$.

(b) Find the densities $n_{+}=N_{+}/V$ and $n_{-}=N_{-}/V$ of electrons pointing parallel and antiparallel to the magnetic field, respectively.

(c) Using result in (b), find the magnetization $M={\mu }_0(N_{+}-N_{-})$, and expand the result for small B.

(d) Sketch the zero-field susceptibility $\chi (T)=\partial M/{\partial }_B{|}_{B=0}$, and indicate its behavior at low and high temperatures. HINT: In the low temperature limit $\ln z=\beta \mu \to \mathrm{\infty }$ you will find useful the following identity:

$f_n^{-}(z)\approx \frac{[\ln (z){]}^n}{n\mathrm{\Gamma }(n)}$.

(e) Estimate the magnitude of $\chi /N$ for a typical metal at room temperature. HINT: Since room temperature is always smaller that $T_F\sim 10^4$ K of typical metals, you can take low temperature limit $T\to 0$ of your result in (d).

Problem 3: Neutron star

A neutron star can be considered to be a collection of non-interacting neutrons, which are spin 1/2 fermions. A typical neutron star has a mass $M$ close to one solar mass $M_{⨀}\approx 2\times 10^{30}$ kg. The mass of a neutron is about $m=1.67\times 10^{-27}$ kg. In the following, we will estimate the radius $R$ of the neutron star.

(a) Find the energy of a neutron star at $T=0$ as a function of $R$, $M$, and $m$ assuming that the star can be treated as an ideal non-relativistic Fermi gas.

(b) Assume that the density of the star is uniform and show that its gravitational energy is given by $E_G=-3G{M}^2/5R$, where the gravitational constant $G=6.67\times 10^{-11}N{m}^2/k{g}^2$. HINT: From classical mechanics find the gravitational potential energy between an existing sphere of radius $r$ and a shell of volume $4\pi r^{2}dr$ coming from infinity to radius $r$. Then integrate that expression from $r=0$ to $R$.

(c) Assume that gravitational equilibrium occurs when the total energy is minimized and find an expression for the radius $R$.

(d) Estimate the actual value of $R$ in kilometers. Estimate the mass density and compare it with the density of material on the surface of Earth such as water.

(e) Determine the Fermi energy and Fermi temperature of a neutron star. A typical internal temperature for a neutron star is $T=10^8$ K. Compare this value with the Fermi temperature and determine if the zero temperature approximation that we assumed is applicable.

(f) Compare the rest energy $m{c}^2$ of a neutron star with Fermi energy of a neutron star. Is the non-relativistic approximation justified?