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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 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.
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| (b) At zero temperature all negative energy states 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 \matbf{k}}{(2 \pi)^2} \frac{\varepsilon_+(\mathbf{k}}{\exp(\beta \varepsilon_+(\mathbf{k})) +1}
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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) 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:
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| <math> \hat{H}_1 = \frac{\hat{\mathbf{p}^2}{2m} - \mu_B \vec{\sigma} \cdot \mathbf{B} </math>
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| where <math> \mu_B </math> is the Bohr magneton and we ignore orbital effects of magnetic field (if they are taken into account than, <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>.
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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_B (N_+ - N_-), 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: Black hole entropy ==
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