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E-raamat: Advances in Imaging and Electron Physics

Edited by (Founder-President of the European Microscopy Society and Fellow, Microscopy and Optical Societies of America; member of the editorial boards of several microscopy journals and Serial Editor, Advances in Electron Optics, France)
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Advances in Imaging and Electron Physics, Volume 210, merges two long-running serials, Advances in Electronics and Electron Physics and Advances in Optical and Electron Microscopy. The series features extended articles on the physics of electron devices (especially semiconductor devices), particle optics at high and low energies, microlithography, image science, digital image processing, electromagnetic wave propagation, electron microscopy and the computing methods used in all these domains. Sections in this new release cover Electron energy loss spectroscopy at high energy losses, Examination of 2D Hexagonal Band Structure from a Nanoscale Perspective for use in Electronic Transport Devices, and more.

  • Contains contributions from leading authorities on the subject matter
  • Informs and updates on the latest developments in the field of imaging and electron physics
  • Provides practitioners interested in microscopy, optics, image processing, mathematical morphology, electromagnetic fields, electrons and ion emission with a valuable resource
  • Features extended articles on the physics of electron devices (especially semiconductor devices), particle optics at high and low energies, microlithography, image science and digital image processing
Contributors ix
Preface xi
1 Introduction to examination of 2D hexagonal band structure from a nanoscale perspective for use in electronic transport devices
1(6)
Clifford M. Krowne
References
5(1)
Further reading
6(1)
2 Determination of reciprocal lattice from direct space in 3D and 2D- Examination of hexagonal band structure
7(16)
Clifford M. Krowne
2.1 3D analysis- direct and indirect space vectors
7(4)
2.2 2D analysis-direct and indirect space vectors
11(3)
2.3 2D analysis- first Brillouin zone vertex points
14(2)
2.4 2D analysis- uniqueness properties of crystallographic K points
16(1)
2.5 2D analysis- closest atoms for tight-binding method
17(3)
References
20(1)
Further reading
21(2)
3 Tight-binding formulation of electronic band structure of 2D hexagonal materials
23(24)
Clifford M. Krowne
3.1 π and σ orbitals in graphene
23(1)
3.2 Relationship between atomic orbitals and crystalline wavefunction
24(6)
3.3 Assessment of overlap between atomic orbitals
30(7)
3.4 Reduction of the spatially varying Schrodinger equation into a solvable system
37(6)
References
43(1)
Further reading
43(4)
4 Evaluation of the matrix elements for the tight-binding formulation of 2D hexagonal materials
47(20)
Clifford M. Krowne
4.1 Determination of an arbitrary Hamiltonian and self-matrix elements
47(7)
4.2 Secular equation of the system using the Hamiltonian
54(1)
4.3 Nearest neighbor hopping and overlap integrals
55(5)
4.4 Next nearest neighbor hopping and overlap integrals
60(4)
References
64(1)
Further reading
64(3)
5 Solving the secular equation of the system for eigenenergy- 2D hexagonal materials
67(10)
Clifford M. Krowne
5.1 Exact solution based upon the Hamiltonian matrix elements
67(3)
5.2 Approximate solution viewing various parameters as possessing order
70(1)
5.3 Unnormalizing the parameters in eigenenergy
71(1)
5.4 Unnormalizing the eigenenergy
72(1)
References
73(1)
Further reading
74(3)
6 Properties of the bare shifted eigenenergy determined as a function of k vector- 2D hexagonal metarials
77(14)
Clifford M. Krowne
6.1 Eigenenergy found as an explicit function of k vector
77(2)
6.2 Examination of the bands of graphene
79(2)
6.3 Determination of the Dirac reciprocal space k points
81(6)
6.4 Symmetry property of the eigenenergy
87(1)
References
88(1)
Further reading
89(2)
7 Hamiltonian of the two atom sublattice system- 2D hexagonal materials
91(70)
Clifford M. Krowne
7.1 Reduced Hamiltonian of the system- identical atoms in sublattices
91(2)
7.2 General solution for eigenenergy and eigenvector
93(5)
7.3 Specialized solution- dropping next nearest neighbor hopping term
98(3)
7.4 Low energy, small momentum deviations about the Dirac points
101(1)
7.5 Phase factor used in Hamiltonian and eigenfunction at Dirac points
102(3)
7.6 Hamiltonian and eigenenergy about Dirac points
105(8)
7.7 Band and Dirac point symmetry breaking due to second order q momentum effects
113(2)
7.8 Density of states near Dirac points
115(7)
7.9 Density of states in vicinity of electron group velocity approaching zero
122(8)
7.10 Density of states in vicinity of electron group velocity zero for finite k
130(7)
7.11 Eigenenergy at 1BZ edge at the Van Hove singularity point
137(20)
References
157(1)
Further reading
157(4)
8 2-Spinor and 4-spinor wavefunctions and Hamiltonians- 2D hexagonal materials
161(18)
Clifford M. Krowne
8.1 Review of 2-spinor construction
161(2)
8.2 4-Spinor: obtaining a reordered wavefunction and restructured Hamiltonian
163(3)
8.3 4-Spinoreigenenergiesand eigenfunctions under an approximation
166(3)
8.4 4-Spinor eigenstates: eigenfunctions and eigenenergies using reordered wavefunction and restructured Hamiltonian with the approximation
169(7)
Further reading
176(3)
9 Examination of the relativistic Dirac equation and its implications for 2D hexagonal materials
179(72)
Clifford M. Krowne
9.1 The relativistic energy
179(2)
9.2 The Dirac conditions obtained from linear energy representation and its Hamiltonian
181(4)
9.3 Allowable α- and β matrices
185(3)
9.4 Choosing a specific a,- and p set and their satisfaction of Dirac conditions
188(3)
9.5 Non-uniqueness of the Dirac matrices and their transformations
191(2)
9.6 The Dirac equation
193(4)
9.7 Plane wave form of the Dirac equation
197(2)
9.8 Eigenvalues of the plane wave Dirac equation
199(5)
9.9 Eigenvectors of the plane wave Dirac equation and comparison to graphene
204(11)
9.10 Spinor eigenvectors with transverse momentum plane wave Dirac equation
215(10)
9.11 Transforming from one Dirac matrix set to another
225(11)
9.12 Transformed plane wave Dirac equation for transverse momentum
236(11)
References
247(1)
Further reading
247(4)
10 Different onsite energies for the two atom problem- 2D hexagonal materials
251(40)
Clifford M. Krowne
10.1 Governing equation when onsite sublattice energies differ
251(5)
10.2 Examination of the Hamiltonian for the two atom system
256(3)
10.3 Evaluating the normalized BB self-energy matrix element
259(2)
10.4 Hamiltonian and governing equation with pivoting element unity
261(6)
10.5 Solving the governing equation for eigenvectors and eigenenergies
267(6)
10.6 Eigenenergy solution extracted from its governing equation form
273(14)
Further reading
287(4)
11 Overall conclusion for 2D hexagonal materials
291(8)
Clifford M. Krowne
Acknowledgments
294(1)
References
294(2)
Further reading
296(3)
12 Performing EELS at higher energy losses at both 80 and 200 kV
299(58)
Ian MacLaren
Rebecca B. Cummings
Fraser Gordon
Enrique Frutos-Myro
Sam McFadzean
Andrew P. Brown
Alan J. Craven
1 Introduction
300(5)
2 Technical challenges in performing EELS at higher energy losses
305(17)
3 High loss EELS in practice using optimized camera lengths
322(26)
4 Conclusions
348(1)
Acknowledgments
349(1)
References
349(8)
Index 357
Peter Hawkes obtained his M.A. and Ph.D (and later, Sc.D.) from the University of Cambridge, where he subsequently held Fellowships of Peterhouse and of Churchill College. From 1959 1975, he worked in the electron microscope section of the Cavendish Laboratory in Cambridge, after which he joined the CNRS Laboratory of Electron Optics in Toulouse, of which he was Director in 1987. He was Founder-President of the European Microscopy Society and is a Fellow of the Microscopy and Optical Societies of America. He is a member of the editorial boards of several microscopy journals and serial editor of Advances in Electron Optics.
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