Quantum Transport in Nanostructures and Molecules

Name: Quantum Transport in Nanostructures and Molecules | An introduction to molecular electronics
Brand: Institute of Physics Publishing
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An introduction to molecular electronics

Colin John Lambert(Autor*in)

Institute of Physics Publishing

Erschienen am 23. März 2021

446 Seiten

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Inhalt

Intro
Preface
Acknowledgements
Author biography
Colin John Lambert
Chapter 1 Introduction to molecular-scale electronics
1.1 Background
1.2 The theoretical challenges addressed by this book
1.3 De Broglie wave patterns
1.4 The Landauer formula
1.5 Resonant transport
1.6 Thermoelectricity
1.7 Off-resonance transport
1.8 Intuitive picture of Green's functions
1.9 Magic ratio theory
1.10 Controlling quantum interference
1.11 Summary
References
Chapter 2 Connectivity theory and noisy neighbour equations for quantum transport
2.1 Introduction
2.2 Green's functions for beginners
2.3 The principle of superposition
2.4 Destructive versus constructive interference
2.5 Green's functions produced by noisy neighbours
2.6 NNEs for transport through more complex coupled quantum structures
2.7 Identifying quantum interference within the five-component NNE
2.8 Magic ratio theory
2.9 Green's function of a molecular core
2.10 Mid-gap energies versus E=0
2.11 Summary
Appendix A Basic properties of matrices
A.1 Matrix addition and subtraction
A.2 Matrix multiplication
A.3 Multiplication of a matrix by a number
A.4 Expressions for individual matrix elements of a product
A.5 The unit matrix
A.6 The determinant of a matrix
A.7 Inverse of a matrix
A.8 Book keeping the Green's function of a structure composed of two substructures
A.9 Multiplying matrices sub-matrix by sub-matrix
A.10 Products of three matrices
A.11 Products of inverses of matrices
A.12 Proof of the equivalence of the NNEs G11H12g22=g11H12G22
A.13 Proof that the equation G11=g11-1-S1-1 implies G11=g11+g11H12G22H21g11
A.14 Dealing with large (possibly infinite) matrices
A.15 Proof that the NNEs imply G11G12G21G22=g11-1-H12-H21g22-1-1
A.16 Note regarding the notation G11G12G21G22=E-H11-H12-H21E-H22-1
References
Chapter 3 A beginner's guide to solving the Schrödinger equation
3.1 Introduction
3.2 Mathematical aim: an introduction to linear algebra
3.3 A simple harmonic oscillator
3.4 The equations of motion of two harmonic oscillators
3.5 Matrix form of the equations of motion of two harmonic oscillators
3.6 Combining linear independence with the best choice of basis to solve equation (3.7)
3.7 Eigenvalues and eigenvectors
3.8 Solution to the general N×N dynamical problem
3.9 Solution to the general N×N Schrödinger equation
3.10 How many linearly-independent eigenvectors does a dynamical matrix H possess?
3.11 A two-level quantum system
3.12 The Hamiltonian of a two-level quantum system
3.13 Eigenvalues and eigenvectors of N×N Hermitian matrices
3.14 Proof that a N×N Hermitian matrix possesses N linearly independent eigenvectors
3.15 Completeness and initial conditions revisited
3.16 The eigenstates of a two-level quantum system
3.17 Summary
Appendix A Basic properties of matrices and complex numbers
A.1 Functions of matrices
A.2 Complex numbers
A.3 The fundamental theorem of algebra
Chapter 4 Quantum properties of linear chains and simple molecules
4.1 Introduction
4.2 The eigenvalues and eigenvectors of a two-level molecule
4.3 The eigenvalues and eigenvectors of a doubly infinite chain of identical atoms
4.4 Effective mass and group velocity
4.5 Degeneracies and the allowed values of k
4.6 The most general solution to the Schrödinger equation for a linear chain
4.7 A finite chain of N sites with periodic boundary conditions
4.8 Group velocity revisited
4.9 The eigenvalues and eigenvectors of a linear chain of N identical atoms
4.10 The two-level system revisited
4.11 A three-level system
4.12 Normal modes of a vibrating atomic chain
4.13 Radicals and Huckel's rule
4.14 Summary of results for the eigenstates and eigenvalues of one-dimensional chains
4.15 A quantum spider's web
4.16 Eigenstates of a semi-infinite one-dimensional chain
4.17 Summary
Appendix A Relationship to Fourier analysis
Appendix B Continuity equations
B.1 Forward continuity
B.2 Backward continuity
B.3 Egalitarian continuity
Chapter 5 Quantum properties of electrodes in higher dimensions
5.1 Introduction
5.2 Peierl's distortion and oligoynes
5.3 Conduction bands versus valence bands
5.4 Flexibility in labelling states
5.5 A finite-width electrodes formed from a linear chain of cells
5.6 A linear chain of cells, each containing two sites
5.7 A linear chain of cells, each containing N sites, with an intra-cell Hamiltonian H1 proportional to the unit matrix
5.8 A linear chain of cells, each containing N sites, with an intra-cell Hamiltonian H1 proportional to the unit matrix and an intra-cell Hamiltonian H0 describing a linear chain with free ends
5.9 A linear chain of cells, each containing N sites, with an intra-cell Hamiltonian H1 proportional to the unit matrix and an intra-cell Hamiltonian H0 describing a linear chain with periodic boundary conditions
5.10 A two-dimensional crystal on a square lattice
5.11 The most general solution to the Schrödinger equation for the periodic structure of figure 5.7, for which H1 is proportional to the unit matrix
5.12 The equivalence between a finite-width lead and many one-dimensional leads
5.13 Summary
Chapter 6 Scattering theory of electrical conductance and thermopower
6.1 Introduction
6.2 A single impurity in one dimension
6.3 The scattering matrix
6.4 Bond currents and conservation of probability
6.5 Current carried by counter-propagating plane waves
6.6 Unitarity of the scattering matrix
6.7 Physical meaning of scattering matrix elements
6.8 Consequences of unitarity of the scattering matrix
6.9 Bound states
6.10 The S matrix in higher dimensions
6.11 The Landauer formula
6.12 Derivation of the Landauer formula and formulae for thermoelectric properties of a quantum object connected to external electrodes
6.13 Summary
Appendix A A general expression for bond currents
Appendix B Unitarity of the multi-channel S matrix
Chapter 7 Thermoelectricity in nanostructures and molecules
7.1 Introduction
7.2 A qualitative description of thermoelectricity
7.3 Linear response formulae for thermoelectric coefficients
7.4 An expressions for thermoelectric efficiency
7.5 Relationship between thermoelectric efficiency and ZT
7.5.1 Efficiency at maximum power
7.5.2 Maximum efficiency
7.6 Expressions for thermoelectric coefficients
7.7 Proof that ZTe is positive
7.8 Strategies for maximising thermoelectric performance
7.9 Summary
Chapter 8 A very useful formula (VUF) for the transmission coefficient of an arbitrary scatterer connected to one-dimensional leads and a magic ratio theory for intra-molecular currents
8.1 Introduction
8.2 The Schrödinger equation for a scatterer connected to two one-dimensional leads
8.3 Note about the choice of wave vectors
8.4 Solution to the Schrödinger equation
8.5 Expression for the transmission amplitude in terms of the full Green's function G
8.6 Expression for the transmission amplitude in terms of the Green's function of the isolated scatterer
8.7 Expression for the reflection amplitude
8.8 Expression for the wave function inside the scatterer
8.9 Expressions for bond currents
8.10 Magic ratio theory for intra-molecular currents
8.11 Properties of the very useful formula for the transmission coefficient
8.12 The relationship between magic ratio theory and the VUF
8.13 The relationship between the Breit-Wigner formula, on-resonance transport and the VUF
8.14 Summary
References
Chapter 9 A quantum system connected to many scattering channels
9.1 Introduction
9.2 Solving the Schrödinger equation
9.3 Unitarity of the many-channel scattering matrix
9.4 Expressions for the transmission coefficients
9.5 Expressions for the reflection coefficients
9.6 The self-energy matrix
9.7 Dyson's equation
9.8 Example: a scatterer connected to two 1-dimensional leads
9.9 A scatterer attached to M leads, in which each lead connects to only one site of the scatterer
9.10 Scattering theory in the presence of an open scatterer
9.11 The effect of closed channels
9.12 A scatterer connected to finite-dimensional leads
9.13 Summary
Chapter 10 Relationship between Green's functions, wave functions and scattering amplitudes
10.1 Introduction
10.2 Green's functions of closed systems
10.3 Approximations to Green's functions of closed systems
10.4 Green's functions of a doubly infinite linear chain
10.5 Relationship between a Green's function of a linear chain and the wave function of a T-shaped junction
10.6 Green's functions of a semi-infinite linear chain
10.7 The relationship between Green's functions and wave functions in the presence of a scatterer
10.8 Green's function of a linear chain with periodic boundary conditions
10.9 Green's functions of a linear chain with free-end boundary conditions
10.10 Green's functions of a finite-width nanoribbon
10.11 Relationships between Green's functions and transmission amplitudes
10.12 Summary
Chapter 11 Connectivity theory revisited: heteroatom substitution, decimation and the Breit-Wigner formula
11.1 Introduction
11.2 Recursively describing the effect of changes to a molecular junction
11.3 Decimation and the effect of pendant groups
11.4 Illustration of destructive quantum interference due to pendant atoms
11.5 The effect of a local perturbation
11.6 The effect of introducing a single heteroatom
11.7 The effect of introducing two heteroatoms
11.8 The Green's function of a semi-infinite lead revisited
11.9 Fusion of substructures via Dyson's equation
11.10 Constructing a lead-scatterer-lead system using Dyson's equation
11.11 From Green's functions to transmission functions
11.12 Simplification of the Green's function of a scatterer in the presence of electrodes
11.13 Summary of Green's function equations for the transmission coefficient
11.14 The Breit-Wigner formula for an arbitrary scatterer connected to finite-width leads
11.15 Derivation of the five-component NNE
11.16 Summary
Appendix A Similarity transformations
Appendix B An alternative derivation of the core Green's function
Chapter 12 Linear molecules
12.1 Introduction
12.2 Single-site decimation and the variation of conductance with length
12.3 A negative eigenvalue theorem
12.4 Recurrence relations via the noisy neighbour equations
12.5 Decay of conductance with length of an oligoyne and a diatomic chain
12.6 Quantum circuit rules for the conductance and Seebeck coefficient of linear molecules
12.7 Summary
References
Chapter 13 Quantum interference in molecules with parallel paths and pendant groups
13.1 Introduction
13.2 The effect of pendant groups
13.3 The Green's function of an isolated molecule containing a bridging unit
13.4 A molecular backbone coupled to two different atoms of a bridging unit
13.5 A molecule with three moieties in series
13.6 A molecule with two bridging groups in parallel
13.7 The electrical conductance of a molecule with weakly coupled bridges
13.8 The electrical conductance of a molecule with many parallel bridges
13.9 Summary
Chapter 14 Connectivity theory and equations of motion
14.1 Introduction
14.2 Stationary states and Green's functions
14.3 Green's function of a 2-component system revisited
14.4 Connectivity equations from combinations of active and passive viewpoints
14.5 Stationary states of vibrating structures
14.6 Connectivity theory, equality of currents and unitarity of the scattering matrix
14.7 The limit ?0
14.8 The case ??0
14.9 Comparison between advanced and retarded Green's functions
14.10 Summary
Appendix A Explicit integration of equation (14.4)
Appendix B Green's functions of open systems
Appendix C Green's functions of a double infinite tight-binding chain revisited
Appendix D Iterative analysis of the 3-component structure of figure 14.3
Appendix E The Green's function of a 5-component system
Chapter 15 Relationship between Green's functions, molecular orbitals and densities of states
15.1 Introduction
15.2 The Coulson-Rushbrooke theorem
15.3 An orbital symmetry rule
15.4 Densities of states
15.5 van Hove singularities
15.6 A more precise form of the density of states
15.7 Properties of delta functions
15.8 Examples of delta functions
15.9 Relationship between Green's functions and densities of states
15.10 Summary
References
Chapter 16 Solving the time-dependent Schrödinger equation and the theory of representations
16.1 Introduction
16.2 The theory of representations
16.3 Representations involving orthonormal basis functions
16.4 Representations involving non-orthogonal basis functions
16.5 Transformations between representations
16.6 A more compact notation for the transformation equation
16.7 Atomic orbitals versus pseudo-atomic orbitals
16.8 Contribution of overlap integrals to nearest-neighbour couplings
16.9 Dyson's equation in a non-orthogonal basis
16.10 Summary
Reference
Chapter 17 Scattering theory in the presence of material-specific leads
17.1 Introduction
17.2 The most general solution to the Schrödinger equation for an arbitrary crystalline lead
17.3 Matrix elements between channel eigenvectors
17.4 Cancellation of off-diagonal contributions to currents
17.5 Note about degeneracies
17.6 Analysis of a general scatterer connected to a periodic structure
17.7 Analysis of currents
17.8 Expressions for transmission amplitudes
17.9 Determining the allowed wave vectors for a given energy E
17.10 The Green's function of an arbitrary material-specific, finite-width, doubly infinite lead
17.11 The Green's function of an arbitrary finite-width, semi-infinite lead
17.12 The surface Green's function of an arbitrary finite-width, semi-infinite lead
17.13 A simple derivation of the surface Green's function of an arbitrary finite-width, semi-infinite lead
17.14 Dyson's equation for the surface Green's function of an arbitrary finite-width, semi-infinite lead
17.15 Expressions for transmission and reflection amplitudes of a scatterer connected to arbitrary, material-specific leads
17.16 Summary
Appendix A Solving the H1 problem
A.1 Decomposing a singular matrix
A.2 Solving an eigenvalue problem involving the singular matrix A
A.3 Solving the Schrödinger equation for a finite-width lead with a singular H1
A.4 An expression for inter-cell currents in terms of renormalized inter-cell couplings
Appendix B Translation operators, velocity operators, Green's functions, time reversal symmetry and the transmission coefficient
B.1. The translation operator ?
B.2. Book keeping for wave vectors and four useful translation operators
B.3. Translation operators for eigenstates
B.4. Left and right, retarded and advanced, surface Green's functions of semi-infinite leads
B.5. Identification of retarded and advanced Green's functions
B.6. A generalised group velocity
B.7. Relationships between surface Green's functions
B.8. Retarded and advanced Green's functions
B.9. Time reversal
B.10. The velocity operator
B.11. A general expression for the transmission coefficient
B.12. A general expression for the transmission coefficient in the presence of two arbitrary leads
B.13. Summary of the conceptual framework for solving a general scattering problem
Appendix C A note about degeneracies
References

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