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 the Fermi-Dirac statistics, the probability of occupation of a single particle state of energy ${\epsilon }_{\mathbf{𝐤}}$ is

$P_{\mathbf{𝐤}}=\frac{e^{-\beta ({\epsilon }_{\mathbf{𝐤}}-\mu )n_{\mathbf{𝐤}}}}{1+e^{-\beta ({\epsilon }_{\mathbf{𝐤}}-\mu )}}$

(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.

(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.

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.

(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: Stoner ferromagnetism

The conduction electrons in a metal can be treated as a gas of fermions of spin 1/2 and density $n=N/V$. The Coulomb repulsion favors wave functions that are antisymmetric in position coordinates, thus keeping the electrons apart. Because of the full (position and spin) anti-symmetry of fermionic wave functions, this interaction may be approximated by an effective spin-spin coupling that favors states with parallel spins. In this simple approximation, the net effect is described by an interaction energy:

$U=\alpha \frac{N_{+}{N}_{-}}{V}$

where $N_{+}$ and $N_{-}=N-N_{+}$ are the numbers of electrons with up and down spins, and $V$ is the volume. The parameter $\alpha$ is related to the scattering length $a$ by $\alpha =4\pi {\hslash }^{2}a/m$.

(a) The ground state has two Fermi seas filled with spin-up and spin-down electrons. Express the corresponding Fermi wavevector $k_{F\pm }$ in terms of densities $n_{\pm }=N_{\pm }/V$.

(b) Calculate the kinetic energy density of the ground state as a function of densities $n_{\pm }$ and the fundamental constants.

(c) Assuming small deviations $n_{\pm }=n/2\pm \delta$ from the symmetric state, expand the kinetic energy to fourth order in $\delta$.

(d) Express the spin-spin interaction density $U/V$ in terms of $n$ and $\delta$. Find the critical value of ${\alpha }_c$ such that $\alpha >{\alpha }_c$ the electron gas can lower its total energy by spontaneously developing a magnetization. This is known as the Stoner instability.

(e) Explain qualitatively the behavior of spontaneous magnetization as a function of $\alpha$ and sketch the corresponding graph.