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First-Principles Methods

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Conduction in Carbon Nanotube Networks

Part of the book series: Springer Theses ((Springer Theses))

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Abstract

In the previous chapter, we demonstrated how the conductance through a device can be calculated from a quantum-mechanical description of the system.

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Notes

  1. 1.

    Here and throughout this thesis we make the adiabatic approximation [1], also known as the Born-Oppenheimer approximation, that neglects the quantum-mechanical nature of the ionic nuclei and treats them instead as stationary, charged classical particles. This is justified by the large difference in mass between electrons and the nuclei, a factor of \(\approx 2000\) for Hydrogen and larger for heavier nuclei, which results in the electronic structure relaxing rapidly to changes in the nuclear positions. In terms of the system Hamiltonian, this approximation neglects the kinetic energy of the nuclei and interactions between electrons and the lattice vibrations (electron-phonon interactions). The Coulombic interactions between pairs of nuclei are included simply as a classical electrostatic energy.

  2. 2.

    We adopt atomic units: \(e=1\), \(\hbar =1\), \(m_e=1\).

  3. 3.

    For systems with degenerate ground-states, the upper bound on the ground-state energy still holds; the corresponding wavefunction is contained within the subspace of degenerate ground-state wavefunctions.

  4. 4.

    Somewhat coincidentally, the memory footprint for carbon in this estimation scheme corresponds, roughly, to the maximum memory address space that can be stored within a 64-bit computer architecture.

  5. 5.

    Note that equality is not allowed as \(\vert \Psi ^{\prime }_0\rangle \) cannot be the ground-state of \(\hat{H}\) by the first part of the first Hohenberg-Kohn theorem. The wavefunctions are assumed normalised \(\langle \Psi _0\vert \Psi _0\rangle = 1\).

  6. 6.

    A small caveat exists with this approach as the minimisation must proceed only over those densities that can be created by an external potential, the so-called v-representability problem; examples have been found of trial densities for which there exists no external potential [8–10]. This problem can be solved by an alternative approach to the one presented here, the constrained search formulation, [7] that removes this constraint, and instead imposes much weaker constraints on the trial densities which are more generally satisfied.

  7. 7.

    The name exchange-correlation is historic due to the Hartree-Fock method for electronic structure, which preceded DFT; see Sect. 4.3.3.

  8. 8.

    Here we assume that the Brillouin zone is sampled at the \(\Gamma \) point only; see Sect. 4.4.3.

  9. 9.

    Strictly speaking, only the valence orbitals are solved for in the pseudo-atom; the orbitals from core electrons are removed by the pseudopotential approximation. See Sect. 4.4.2.

  10. 10.

    For instance, computing the Hartree potential can be performed most efficiently in reciprocal space:

    $$\begin{aligned} V_{\mathrm {H}}(\mathbf {r}) = \int \mathrm {d}^3{\mathbf {r}^{\prime }}~ \frac{n(\mathbf {r}^{\prime })}{|\mathbf {r} - \mathbf {r}^{\prime }|} \quad \rightarrow \quad \widetilde{V}_{\mathrm {H}}(\mathbf {G}) = \frac{4\pi }{\Omega _{\mathrm {cell}}}\frac{\widetilde{n}(\mathbf {G})}{|\mathbf {G}|^2}, \end{aligned}$$
    (4.44)

    where \(\widetilde{V}\), \(\widetilde{n}\) are the Fourier transforms of the Hartree potential and electronic density, and \(\Omega _{\mathrm {cell}}\) is the unit cell volume. The Hartree potential in real-space is then obtained by inverse Fourier transform. This procedure replaces integrals over the whole simulation cell for each \(\mathbf {r}\) with the considerably computationally faster Fourier transform, multiplication and inverse transform.

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  1. Cavendish Laboratory, Theory of Condensed Matter Group, University of Cambridge, Cambridge, UK

    Robert A. Bell

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Bell, R.A. (2015). First-Principles Methods. In: Conduction in Carbon Nanotube Networks. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-19965-8_4

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