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| == Problem 1: Electrons in graphene ==
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| Graphene is one-atom-thick crystal of carbon atoms densely packed into a honeycomb lattice. Its surprising discovery (according to [http://link.aps.org/doi/10.1103/PhysRev.176.250 Mermin-Wagner theorem] of statistical mechanics, two-dimensional crystals do not exist!) in 2004 has led to [http://nobelprize.org/nobel_prizes/physics/laureates/2010/ Nobel Prize in Physics 2010]. The band structure of graphene
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| close to the Fermi energy is such that electrons behave as the so-called massless Dirac fermions with energy-momentum relationship:
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| <math> \varepsilon_{\pm} (\mathbf{k}) = \pm \hbar v_F |\mathbf{k}| </math>
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| akin to photons or neutrinos of high energy physics, except that instead of the velocity of light <math> c </math> is replaced by the Fermi velocity is <math> v_F \approx c/300 </math>.
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| (a) For any fermionic system at chemical potential <math> \mu </math>, show that the probability of finding an occupied state of energy <math> \mu + \delta </math> is the same as that of finding an unoccupied state of energy <math> \mu - \delta </math> where <math> \delta </math> is any constant energy. HINT: According the Fermi-Dirac statistics, the probability of occupation of a single particle state of energy <math> \varepsilon_\mathbf{k} </math> is
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| <math> P_\mathbf{k} = \frac{e^{-\beta(\varepsilon_\mathbf{k} - \mu)n_\mathbf{k}}}{1+e^{-\beta(\varepsilon_\mathbf{k} - \mu)}} </math>
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| (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 <math> \mu (T=0) =0 </math>. Using the result in (a), find the chemical potential at finite temperature <math> T </math>.
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| (c) Show that the mean excitation energy of this system at finite temperature satisfies:
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| <math> E(T) - E(0) = 4 A \int \frac{d^2 \mathbf{k}}{(2 \pi)^2} \frac{\varepsilon_+(\mathbf{k})}{\exp[\beta \varepsilon_+(\mathbf{k})] + 1 } </math>
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| where A is the surface are of graphene.
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| (d) Give a closed form answer for the excitation energy by evaluating integral in (c).
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| (e) Using (d), calculate the heat capacity <math> C_V </math> of massless Dirac fermions in graphene as a function of temperature.
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| ==Problem 2: Pauli paramagnetism ==
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| Calculate the contribution of electron spin to its magnetic susceptibility as follows. Consider non-interacting electrons where each is subject to a Hamiltonian operator:
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| <math> \hat{H}_1 = \frac{\hat{\mathbf{p}}^2}{2m} - \mu_0 \vec{\sigma} \cdot \mathbf{B} </math>
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| where <math> \mu_0 </math> is the Bohr magneton and we ''ignore'' orbital effects of magnetic field (if they are taken into account then, <math> \mathbf{p} \rightarrow \mathbf{p} - e\mathbf{A} </math> for vector potential <math> \mathbf{A} </math>).
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| (a) Calculate the grand potential <math> \Phi </math> at a chemical potential <math> \mu </math>. HINT: The energy of electron gas is given by <math> E = \sum_p E_p (n_p^+,n_p^-) </math>
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| where <math> n_p^\pm </math> (=0 or 1 as in the case of any fermionic system) denotes the number of particles having <math> \pm </math> spins and momentum p, and
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| <math> E_p(n_p^+,n_p^-) = \left(\frac{p^2}{2m} - \mu_0 B \right) n_p^+ + \left(\frac{p^2}{2m} + \mu_0 B \right) n_p^- = (n_p^+ + n_p^-) \frac{p^2}{2m} - (n_p^+ - n_p^-) \mu_0 B </math>.
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| (b) Find the densities <math> n_+ = N_+ /V </math> and <math> n_- = N_- /V </math> of electrons pointing parallel and antiparallel to the magnetic field, respectively.
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| (c) Using result in (b), find the magnetization <math> M = \mu_0 (N_+ - N_-) </math>, and expand the result for small B.
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| (d) Sketch the zero-field susceptibility <math> \chi(T) = \partial M/\partial_B |_{B=0} </math>, and indicate its behavior at low and high temperatures.
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| (e) Estimate the magnitude of <math> \chi/N </math> for a typical metal at room temperature. HINT: Since room temperature is always smaller that <math> T_F \sim 10^4 </math> K of typical metals, you can take low temperature limit <math> T \rightarrow 0 </math> of your result in (d).
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| == Problem 3: Stoner ferromagnetism ==
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| The conduction electrons in a metal can be treated as a gas of fermions of spin 1/2 and density <math> n=N/V </math>. 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:
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| <math> U = \alpha \frac{N_+N_-}{V} </math>
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| where <math> N_+ </math> and <math> N_- = N - N_+ </math> are the numbers of electrons with up and down spins, and <math> V </math> is the volume. The parameter <math> \alpha </math> is related to the scattering length <math> a </math> by <math> \alpha = 4 \pi \hbar^2 a/m </math>.
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| (a) The ground state has two Fermi seas filled with spin-up and spin-down electrons. Express the corresponding Fermi wavevector <math> k_{F\pm} </math> in terms of densities <math> n_\pm = N_\pm /V </math>.
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| (b) Calculate the kinetic energy density of the ground state as a function of densities <math> n_\pm </math> and the fundamental constants.
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| (c) Assuming small deviations <math> n_\pm = n/2 \pm \delta </math> from the symmetric state, expand the kinetic energy to
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| ''fourth order'' in <math> \delta </math>.
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| (d) Express the spin-spin interaction density <math> U/V </math> in terms of <math> n </math> and <math> \delta </math>. Find the critical value of <math> \alpha_c </math> such that <math> \alpha > \alpha_c </math> the electron gas can lower its total energy by spontaneously developing magnetization. This is known as the '''Stoner instability'''.
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| (e) Explain qualitatively the behavior of spontaneous magnetization as a function of <math> \alpha </math> and sketch the corresponding graph.
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