Lectures

What is nanophysics: Survey of course topics

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• Video lectures on AFM and STM from nanohub.org

Additional references

• Foa Torres et al. textbook Chapters 1 and 3.
• M. Gibertini, M. Koperski, A. F. Morpurgo, and K. S. Novoselov, Magnetic 2D materials and heterostructures, Nat. Nanotech. 14, (2019). [PDF]
• A. Fert, 2D magnets: From fundamentals to spintronic devices
• Y. Ando, Topological insulator materials, J. Phys. Soc. Jpn. 82, 102001 (2013). [PDF]

Survey of quantum statistical tools

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• Density of states on square tight-binding lattice via Green functions

Additional references

• Dirac notation and mathematical prerequisites for quantum mechanics (Chapter 1 from L. E. Ballentine: Quantum Mechanics - A modern development, second edition).
• Probability current and current operators in quantum mechanics
• B. K. Nikolić, L. P. Zarbo, and S. Souma, Imaging mesoscopic spin Hall fow: Spatial distribution of local spin currents and spin densities in and out of multiterminal spin-orbit coupled semiconductor nanostructures, Phys. Rev. B 73, 075303 (2006). [PDF]
• M. M. Odashima, B. G. Prado, and E. Vernek, Pedagogical introduction to equilibrium Green's functions: Condensed matter examples with numerical implementations, Rev. Bras. Ens. Fis. 39, e1303 (2017). [PDF]
• W. J. Herrera and H. Vinck-Posada, and S. Gómez Páez, Green's functions in quantum mechanics courses, Am. J. Phys. 90, 763 (2022). [PDF]

From atoms to 1D nanowires: Tight-binding Hamiltonian

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• Discretization of 1D continuous Hamiltonian
• Discretization of continuous Hamiltonian for any device shape and dimensionality in KWANT
• How to put magnetic field into tight-binding Hamiltonian
• Wannier functions for 1D nanowires

Additional references

• Ryndyk textbook Chapter 3.
• J. G. Analytis, S. J. Blundell, and A. Ardavan, Landau levels, molecular orbitals, and the Hofstadter butterfly in finite systems, Am. J. Phys. 72, 5 (2004)]. [PDF]
• E. Canadell, M.-L. Doublet, and C. Iung, Orbital Approach to the Electronic Structure of Solids (Oxford University Press, Oxford, 2012).
• D. C. Ralph, Berry curvature, semiclassical electron dynamics, and topological materials: Lecture notes for Introduction to Solid State Physics, arXiv:2001.04797.

Band structure of graphene via tight-binding Hamiltonian

• P. B. Allen and B. K. Nikolic, Band structure of graphene, massless Dirac fermions as low-energy quasiparticles, Berry phase, and all that
• Visualization of graphene electronic energy dispersion using Mathematica
• How to construct matrix representation of tight-binding Hamiltonian of graphene for numerical calculations
• Effective mass of electrons in graphene
• Hofstadter butterfly using graphene in magnetic field

Additional references

• Foa Torres et al. textbook Chapter 2.
• B. A. McKinnon and T. C. Choy, A tight-binding model for the density of states of graphite-like structures calculated using Green's functions, Aust. J. Phys. 46, 601 (1993). [PDF]
• A. Matulis and F. M. Peeters, Analogy between one-dimensional chain models and graphene, Am. J. Phys. 77, 595 (2009). [PDF]
• Tight-binding Hamiltonian of other materials using physical intuition:
  • S. Mao, A. Yamakage, and Y. Kuramoto, Tight-binding model for topological insulators: Analysis of helical surface modes over the whole Brillouin zone, Phys. Rev. B 84, 115413 (2011). [PDF]
  • T. M. McCormick, I. Kimchi, and N. Trivedi, Minimal models for topological Weyl semimetals, Phys. Rev. B 95, 075133 (2017). [PDF]

Density functional theory for first-principles band structure calculations

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• Delta benchmark of DFT codes

Additional references

• Foa Torres et al. textbook Appendix A.
• Chapter 6 in C. Fiolhais, F. Nogueira, and M. A. L. Marques, A Primer in Density Functional Theory (Springer-Verlag, Berlin, 2003). [PDF]
• Tight-binding Hamiltonian via fitting of density functional theory calculations:

Textbook tight-binding Hamiltonians are created by assuming the shape of the orbitals---for instance s, p or d orbitals centered around a particular atom---and then using symmetry to calculate orbital-orbital hopping up to a particular range. In a second step the parameters associated with the degrees of freedom are determined by fitting to experimental data or first-principles calculations.

• • T. B. Boykin, M. Luisier, G. Klimeck, X. Jiang, N. Kharche, Yu. Zhou, and S. K. Nayak, Accurate six-band nearest-neighbor tight-binding model for the p-bands of bulk graphene and graphene nanoribbons, J. Appl. Phys. 109, 104304 (2011). [PDF]
  • J. M. Marmolejo-Tejada, J. H. García, M. Petrović, P.-H. Chang, X.-L. Sheng, A. Cresti, P. Plecháč, S. Roche, and B. K. Nikolić, Deciphering the origin of nonlocal resistance in multiterminal graphene on hexagonal-boron-nitride with ab initio quantum transport: Fermi surface edge currents rather than Fermi sea topological valley currents, J. Phys.: Mater. 1, 0150061 (2018). [PDF]
  • E. Ridolfi, D. Le, T. S. Rahman, E. R. Mucciolo, and C. H. Lewenkopf, A tight-binding model for MoS₂ monolayers, J. Phys.: Condens. Matter 27, 365501 (2015). [PDF]

• Tight-binding Hamiltonian via Wannierization of density functional theory calculations:

Wannierization of density functional theory (DFT) calculations starts from the diagonal Kohn-Sham Hamiltonian in the Bloch state basis and transforms into a basis of maximally localized Wannier functions (typically via Wannier90 package). The first-principles Wannier tight-binding Hamiltonian preserves the phase and the orbital information from the DFT calculations.

• • J. Kuneš, Wannier functions and construction of model Hamiltonians
  • S. Fang and E. Kaxiras, Electronic structure theory of weakly interacting bilayers, Phys. Rev. B 93, 235153 (2016). [PDF]
  • S. Carr, S. Fang, H. Chun Po, A. Vishwanath, and E. Kaxiras, Derivation of Wannier orbitals and minimal-basis tight-binding Hamiltonians for twisted bilayer graphene: First-principles approach, Phys. Rev. Res. 1, 033072 (2019). [PDF]

• Wannierization vs. fitting:

• • A. C. Jacko, Deriving ab initio model Hamiltonians for molecular crystals, arXiv:1508.07735.
  • J. Sifuna, P. García-Fernández, G. S. Manyali, G. Amolo, and J. Junquera, Comparison of band-fitting and Wannier-based model construction for WSe₂, arXiv:2001.05959.

Landauer formula for ballistic quasi-1D nanowires with application to edge state transport in 2D topological insulators

• PDF
• Experiment on the electronic energy distribution along nanowire
• Crash course on topology in condensed matter
• Graphene goes topological -> Haldane model (introduced in video by Haldane himself)

Additional references

• Ryndyk textbook Chapter 2.2.
• M. Payne, Electrostatic and electrochemical potentials in quantum transport, J. Phys.: Condens. Matter 1, 4931 (1989). [PDF]
• U. Bajpai, M. J. H. Ku, and B. K. Nikolić, Robustness of quantized transport through edge states of finite length: Imaging current density in Floquet topological versus quantum spin and anomalous Hall insulators, Phys. Rev. Res. 2, 033438 (2020). [PDF]
• X.-L. Sheng and B. K. Nikolić, Monolayer of the 5d transition metal trichloride OsCl₃: A playground for two-dimensional magnetism, room-temperature quantum anomalous Hall effect, and topological phase transitions, Phys. Rev. B 95, 201402(R) (2017). [PDF]

Graphene nanoribbons and carbon nanotubes

• PDF
• Subband structure of zigzag and armchair graphene nanoribbons
• From Brillouin zone of graphene to that of single-wall carbon nanotube
• Subband structure of a single-wall carbon nanotube via the zone-folding method
• Subband structure of single-wall CNT via the Wannier tight-binding Hamiltonian

Additional references

• Foa Torres et al. textbook Chapter 10

Landauer-Büttiker formula for two-terminal and multi-terminal quantum-coherent nanostructures

• PDF

Additional references

• Ryndyk textbook Chapters 2.3 and 2.4.
• J. Walker and J. Gathright, Exploring one-dimensional quantum mechanics with transfer matrices, Am. J. Phys. 62, 408 (1994)]. [PDF]

Application of Landauer-Büttiker formula to quantum interference effects in electronic transport

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Additional references

• G. B. Lesovik and I. A. Sadovskyy, Scattering matrix approach to the description of quantum electron transport, Physics Uspekhi 54, 1007 (2011). [PDF]

Quantum transport via Nonequilibrium Green function (NEGF) formalism

• PDF
• Self-energy for semi-infinite electrodes modeled on a cubic tight-binding lattice
• How to use NEGF matrix formulas: Step-by-step tutorial in pictures

Additional references

• Ryndyk textbook Chapter 3
• S. Datta, Nanoscale device modeling: The Green's function method
• R. Golizadeh-Mojarad and S. Datta, Nonequilibrium Green’s function based models for dephasing in quantum transport, Phys. Rev. B 75, 081301(R) (2007). [PDF]
• C.-L. Chen, C.-R. Chang, and B. K. Nikolić, Quantum coherence and its dephasing in the giant spin Hall effect and nonlocal voltage generated by magnetotransport through multiterminal graphene bars, Phys. Rev. B 85, 155414 (2012). [PDF]

Application of NEGF and NEGF+DFT to magnetic tunnel junctions

• PDF

Additional references

• NEGF+DFT:
  • Foa Torres et al. textbook Appendix C.
  • S. Sanvito, Electron transport theory for large systems.
  • D. A. Areshkin and B. K. Nikolić, Electron density and transport in top-gated graphene nanoribbon devices: First-principles Green function algorithms for systems containing a large number of atoms, Phys. Rev. B 81, 155450 (2010). [PDF]
• MTJs:
  • W. H. Butler, Tunneling magnetoresistance from a symmetry filtering effect, Sci. Technol. Adv. Mater. 9, 014106 (2008). [PDF]
  • K. K. Saha, A. Blom, K. S. Thygesen, and B. K. Nikolić, Magnetoresistance and negative differential resistance in Ni/Graphene/Ni vertical heterostructures driven by finite bias voltage: A first-principles study, Phys. Rev. B 85, 184426 (2012). [PDF]
  • M. Piquemal-Banci, R. Galceran, M.-B. Martin, F. Godel, A. Anane, F. Petroff, B. Dlubak, and P. Seneor, 2D-MTJs: Introducing 2D materials in magnetic tunnel junctions, J. Phys. D: Appl. Phys. 50, 203002 (2017). [PDF]

Application of NEGF and NEGF+DFT to spin torque and spin pumping

• PDF

Additional references

• D. C. Ralph and M. D. Stiles, Tutorial on spin transfer torque [NOTE: arXiv:0711.4608 version linked here is corrected and contains additional material compared to the officially published J. Magn. Magn. Mater. 320, 1190 (2008)].
• N. Locatelli, V. Cros, and J. Grollier, Spin-torque building blocks, Nat. Mater. 13, 11 (2014). [PDF]
• B. K. Nikolić, K. Dolui, M. Petrović, P. Plecháč, T. Markussen, and K. Stokbro, First-principles quantum transport modeling of spin-transfer and spin-orbit torques in magnetic multilayers (Chapter of Handbook of Materials Modeling, Volume 2 Applications: Current and Emerging Materials (Springer, Cham, 2018). [PDF]
• S.-H. Chen, C.-R. Chang, J. Q. Xiao, and B. K. Nikolić, Spin and charge pumping in magnetic tunnel junctions with precessing magnetization: A nonequilibrium Green function approach, Phys. Rev. B. 79, 054424 (2009). [PDF]

Application of NEGF and NEGF+DFT to nanoscale thermoelectrics

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Additional references

• B. K. Nikolić, K. K. Saha, T. Markussen, and K. S. Thygesen, First-principles quantum transport modeling of thermoelectricity in single-molecule nanojunctions with graphene nanoribbon electrodes, J. Comp. Electronics 11, 78 (2012). [PDF]

Coulomb blockade

Additional references

• Ryndyk textbook Chapter 5.