Temporary HW

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:

$\varepsilon _{\pm }(\mathbf {k} )=\pm \hbar v_{F}|\mathbf {k} |$

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.

(b) At zero temperature all negative energy states 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:

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

where A is the surface are of graphene.

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

(e) Calculate the heat capacity $C_{V}$ of massless Dirac fermions in graphene as a function of temperature.

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:

Failed to parse (syntax error): {\displaystyle \hat{H}_1 = \frac{\hat{\mathbf{p}^2}{2m} - \mu_B \vec{\sigma} \cdot \mathbf{B} }

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

(a) Calculate the grand potential $\Phi$ at a chemical potential $\mu$.

(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 $M=\mu _{B}(N_{+}-N_{-})$, 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.

(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 $T\rightarrow 0$ of your result in (d).