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. HINT: According the Fermi-Dirac statistics, the probability of occupation of a single particle state of energy $\varepsilon _{\mathbf {k} }$ is

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle P_\mathbf{k} = \frac{e^{-\beta(\varepsilon_\mathbf{k} - \mu)n_\mathbf{k}}}{1+e^{-\beta(\varepsilon_\mathbf{k} - \mu)n_\mathbf{k}}} }

(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 Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mu (T=0) =0 } . Using the result in (a), find the chemical potential at finite temperature Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle T } .

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

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 } }

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 Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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:

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{H}_1 = \frac{\hat{\mathbf{p}}^2}{2m} - \mu_0 \vec{\sigma} \cdot \mathbf{B} }

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mu_0 } is the Bohr magneton and we ignore orbital effects of magnetic field (if they are taken into account then, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbf{p} \rightarrow \mathbf{p} - e\mathbf{A} } for vector potential Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbf{A} } ).

(a) Calculate the grand potential Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Phi } at a chemical potential Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mu } . HINT: The energy of electron gas is given by Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E = \sum_p E_p (n_p^+,n_p^-) } where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n_p^\pm } (=0 or 1 as in the case of any fermionic system) denotes the number of particles having Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \pm } spins and momentum p, and

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 } .

(b) Find the densities Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n_+ = N_+ /V } and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n_- = N_- /V } of electrons pointing parallel and antiparallel to the magnetic field, respectively.

(c) Using result in (b), find the magnetization Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle M = \mu_0 (N_+ - N_-) } , and expand the result for small B.

(d) Sketch the zero-field susceptibility Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \chi(T) = \partial M/\partial_B |_{B=0} } , and indicate its behavior at low and high temperatures.

(e) Estimate the magnitude of Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \chi/N } for a typical metal at room temperature. HINT: Since room temperature is always smaller that Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle T_F \sim 10^4 } K of typical metals, you can take low temperature limit Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle T \rightarrow 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 Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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:

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle U = \alpha \frac{N_+N_-}{V} }

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle N_+ } and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle N_- = N - N_+ } are the numbers of electrons with up and down spins, and V is the volume. The parameter Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \alpha } is related to the scattering length Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle a } by Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \alpha = 4 \pi \hbar^2 a/m } .