skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Quantum mechanics of graphene with a one-dimensional potential

Abstract

Electron states in graphene with a one-dimensional potential have been studied. An approximate solution has been obtained for a small angle between vectors of the incident electron momentum and potential gradient. Exactly solvable problems with a potential of the smoothened step type U(x) Utanh(x/a) and a potential with a singularity U(x) = -U/(|x| + d) are considered. The transmission/reflection coefficients and phases for various potential barriers are determined. A quasi-classical solution is obtained.

Authors:
 [1];  [2]
  1. Novosibirsk State University (Russian Federation)
  2. Russian Academy of Sciences, Institute of Semiconductor Physics, Siberian Branch (Russian Federation)
Publication Date:
OSTI Identifier:
22069303
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Experimental and Theoretical Physics; Journal Volume: 115; Journal Issue: 4; Other Information: Copyright (c) 2012 Pleiades Publishing, Ltd.; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 77 NANOSCIENCE AND NANOTECHNOLOGY; APPROXIMATIONS; ELECTRONS; EXACT SOLUTIONS; GRAPHITE; NANOSTRUCTURES; ONE-DIMENSIONAL CALCULATIONS; POTENTIALS; QUANTUM MECHANICS; REFLECTION; SINGULARITY; TRANSMISSION

Citation Formats

Miserev, D. S., and Entin, M. V., E-mail: entin@isp.nsc.ru. Quantum mechanics of graphene with a one-dimensional potential. United States: N. p., 2012. Web. doi:10.1134/S1063776112090087.
Miserev, D. S., & Entin, M. V., E-mail: entin@isp.nsc.ru. Quantum mechanics of graphene with a one-dimensional potential. United States. doi:10.1134/S1063776112090087.
Miserev, D. S., and Entin, M. V., E-mail: entin@isp.nsc.ru. 2012. "Quantum mechanics of graphene with a one-dimensional potential". United States. doi:10.1134/S1063776112090087.
@article{osti_22069303,
title = {Quantum mechanics of graphene with a one-dimensional potential},
author = {Miserev, D. S. and Entin, M. V., E-mail: entin@isp.nsc.ru},
abstractNote = {Electron states in graphene with a one-dimensional potential have been studied. An approximate solution has been obtained for a small angle between vectors of the incident electron momentum and potential gradient. Exactly solvable problems with a potential of the smoothened step type U(x) Utanh(x/a) and a potential with a singularity U(x) = -U/(|x| + d) are considered. The transmission/reflection coefficients and phases for various potential barriers are determined. A quasi-classical solution is obtained.},
doi = {10.1134/S1063776112090087},
journal = {Journal of Experimental and Theoretical Physics},
number = 4,
volume = 115,
place = {United States},
year = 2012,
month =
}
  • A two-dimensional model of relativistic quantum mechanics is suggested which permits any number of particles and a wide class of potentials and is as general as the two-dimensional nonrelativistic quantum mechanics. This proposed model is an example of the application of given axioms. (JFP)
  • To accurately determine the reaction path and its energetics for enzymatic and solution-phase reactions, we present a sequential sampling and optimization approach that greatly enhances the efficiency of the ab initio quantum mechanics/molecular mechanics minimum free-energy path (QM/MM-MFEP) method. In the QM/MM-MFEP method, the thermodynamics of a complex reaction system is described by the potential of mean force (PMF) surface of the quantum mechanical (QM) subsystem with a small number of degrees of freedom, somewhat like describing a reaction process in the gas phase. The main computational cost of the QM/MM-MFEP method comes from the statistical sampling of conformations ofmore » the molecular mechanical (MM) subsystem required for the calculation of the QM PMF and its gradient. In our new sequential sampling and optimization approach, we aim to reduce the amount of MM sampling while still retaining the accuracy of the results by first carrying out MM phase-space sampling and then optimizing the QM subsystem in the fixed-size ensemble of MM conformations. The resulting QM optimized structures are then used to obtain more accurate sampling of the MM subsystem. This process of sequential MM sampling and QM optimization is iterated until convergence. The use of a fixed-size, finite MM conformational ensemble enables the precise evaluation of the QM potential of mean force and its gradient within the ensemble, thus circumventing the challenges associated with statistical averaging and significantly speeding up the convergence of the optimization process. To further improve the accuracy of the QM/MM-MFEP method, the reaction path potential method developed by Lu and Yang [Z. Lu and W. Yang, J. Chem. Phys. 121, 89 (2004)] is employed to describe the QM/MM electrostatic interactions in an approximate yet accurate way with a computational cost that is comparable to classical MM simulations. The new method was successfully applied to two example reaction processes, the classical S{sub N}2 reaction of Cl{sup -}+CH{sub 3}Cl in solution and the second proton transfer step of the reaction catalyzed by the enzyme 4-oxalocrotonate tautomerase. The activation free energies calculated with this new sequential sampling and optimization approach to the QM/MM-MFEP method agree well with results from other simulation approaches such as the umbrella sampling technique with direct QM/MM dynamics sampling, demonstrating the accuracy of the iterative QM/MM-MFEP method.« less
  • The general problem involving an infinite potential barrier is treated by first constructing a pseudopotential H' that is shown to reproduce the effect of the barrier. A new procedure is then developed to handle the perturbative effect of H' since the standard formulae become invalid. The case of a finite potential well with large height V/sub o/ can the be solved by reducing it to that of an equivalent infinite barrier. The perturbation parameter turns out to be proportional 1/V/sub o/-E)/sup 1/2/ where E is the energy of the unperturbed state, defying, therefore, the conventional perturbation series treatment that dependsmore » on a split-off Hamiltonian for its expansion parameter. These methods are first illustrated with simple examples and then compared to more complex cases. Recently, they have extended this method to the case of degenerate perturbation. The calculation of the hydrogen-like impurity states in quantum well is in progress. Their result should also furnish a check for any specific problem involving a barrier solved by other approximate means such as the variational method.« less
  • The review presents a constructive investigation of exactly integrable one- and two-dimensional quantum systems that possess a nontrivial internal symmetry group.
  • We report on local measurements of the surface potential and quantum capacitance in single layer graphene as well as multilayers thereof as a function of the carrier density by using frequency-modulated Kelvin probe force microscopy. We find excellent agreement to tight-binding calculations reported for graphene monolayers and extract the minimum quantum capacitance from density sweeps at room temperature. The surface potential of graphene multilayers is found to depend linearly on the carrier density, which suggests treating them as two-dimensional electron gases. In addition, we demonstrate that the simultaneously detected second harmonic of the Kelvin modulation, proportional to |∂{sup 2}C/∂z{sup 2}|,more » is directly sensitive to local changes in the quantum capacitance of graphene.« less