Research Projects

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Guidelines

The idea of a Research Project is to tackle a single topic (which could be composed of several intertwined problems) borrowed from recent research literature and spend time on researching references, doing computation, and writing a Report. In this way a student will gain experience in the same activities practiced by research scientists which require different way of thinking than involved in solving simple homework problems. When you have enough results to tell a coherent story, you should end the Research Project by writing (in a clear writing style, obeying the rules of grammar and spelling) and submitting a Report. The Report should be understandable to a person who has not done the assignment.

Format of the report for the midterm project

The midterm project should be finalized as a paper similar to research articles dealing with Mesoscale and Nanoscale Physics posted every day on arxiv.org. The format of the paper mimicking this is:

  • Title, Name of the person and affiliation, Abstract, Introduction, Methods, Results, Dicussion, Conclusion, and References.
  • Paper should be typed in two column style. For this you can use:
    • LaTeX in the form of RevTeX style for Physical Review journals, as implemented by PHYS 824 template and the embedded EPS figure for this example. You can also find more examples of typing mathematical formulas in Math into LaTeX: How to Beautify Equations (and the embedded EPS figure).
    • Open Office version of Microsoft Word (Word itself is not advisable since you need additional programs, such as MathType, on the top of it to be able to type equations).
  • The final report should be produced as a PDF file and emailed to the instructor before the deadline.

Format of the report for the final project

The final project will be reported through a Poster Session, during the final exam time, and it will also include peer reviewing. To make a poster, you can use this PowerPoint Template. Poster printing is available in Smith Hall and its cost will be covered by the Department.

Midterm Research Project: Electronic structure of graphene nanoribbons (deadline: 11/23 at midnight)

The project explores recently discovered graphene nanoribbons (GNRs) by computing their electronic structure as equilibrium property using simple tight-binding method and more involved density functional codes:


a) Using the nearest-neighbor tight-binding Hamiltonian with single 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_z } orbital per carbon atom, compute the subband structure of three armchair GNRs whose width 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 N_a=4,5, 30} . The expected result is shown in Lecture 9. Pay attention to select proper range 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 k_x } values within the first 1D Brillouin zone.


b) Using the nearest-neighbor (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_1=2.7 } eV) tight-binding Hamiltonian with single 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_z } orbital per carbon atom, compute the subband structure of three armchair GNRs whose width 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 N_z=4,5, 30} . The expected result is shown in Lecture 9. Plot the amplitude squared 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(y)|^2 } across 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_z=30} ZGNR as the transverse part of a selected eigenfunction (i.e., conducting channel) whose eigenergy is close to the Dirac point 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=10^{-3} \gamma } . This plot should show that probability to find electron peaks around the nanoribbons edges.


c) Repeat subband structure calculations for 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_a=5 } AGNR 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_z=5 } ZGNR using the tight-binding Hamiltonian which includes up to third-nearest neighbour hoppings whose values are: 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_1=2.7 } eV, 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_2 = 0.20 } eV, 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 t_3 = 0.18} eV.


d) At low temperatures, the energy band gap 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 \approx 0.5 K } due to intrinsic spin-orbit coupling, as well as the chiral spin-filtered edge states whose subbands pass through the gap, should become visible in experiments. This systems, termed topological insulator where time-reversal invariance ensures the crossing of the energy levels at special points in the Brillouin zone so that their energy spectrum cannot be adiabatically deformed into topologically trivial insulator without such states, can be studied using the following tight-binding model:

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}_{\mathrm{TI}} = - t_1 \sum_{\langle ij \rangle} \hat{c}_{i}^\dagger \hat{c}_{j} + \frac{2i}{\sqrt{3}} t_{\mathrm{SO}} \sum_{\langle \langle ij \rangle \rangle} \hat{c}_i^\dagger \hat{\mathbf{\sigma}} \cdot (\mathbf{d}_{kj} \times \mathbf{d}_{ik})\hat{c}_j } (1)

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 \hat{c}^\dagger_i = (\hat{c}^\dagger_{i\uparrow}, \hat{c}^\dagger_{i\downarrow}) } are electron creation operators 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 \hat{\mathbf{\sigma}} } is the vector of the Pauli matrices. The third sum in Eq.~(1) introduces the intrinsic SO coupling compatible with the symmetries of the honeycomb lattice. The SO coupling acts as spin-dependent next-nearest neighbor hopping 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 i } 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 j } are two next-nearest neighbor sites, 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 k } is the only common nearest neighbor 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 i } 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 j } , 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 \mathbf{d}_{ik} } is a vector pointing from 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 k } to 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 i } .

Compute the band structure 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 N_z=30} ZGNR with SO coupling described by Hamiltonian (1) assuming 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_1 = 2.7 } eV 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 t_{\mathrm{SO}}=0.03 t_1 } . The value for 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_{\mathrm{SO}} } is selected to be much larger than the realistic one in order to see clearly opening of the band gap 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 \Delta_{\rm SO}=6\sqrt{3} t_{\rm SO} } in your figure.


e) Using DFT code SIESTA (installed of fermi), compute subband structure for 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_a=5 } AGNR 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_z=5 } ZGNR and compare this with your result in c).


MAIN REFERENCE: A. Cresti, N. Nemec, B. Biel, G. Niebler, F. Triozon, G. Cuniberti, and S. Roche, Charge transport in disordered graphene-based low-dimensional materials, Nano Research 1, 361 (2008). [PDF].


REFERENCE FOR GNR AS TOPOLOGICAL INSULATOR: C. K. Kane and E. J. Mele, Quantum spin Hall effect in graphene, Phys. Rev. Lett. 95, 226801 (2005).

Final Research Project

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