Table of Contents
Preface xii
Author biography xiv
1 Introduction 1
1.1 A brief history of embedding 2
1.2 Overview 5
1.3 A note on Green functions 7
1.4 Units 8
References 8
2 The variational embedding method 1
2.1 The variational principle 1
2.2 The embedded Schrödinger equation 5
2.3 A first application 6
2.4 The embedded Green function 10
2.5 Application to continuum states 12
2.6 Resonances and complex eigenvalues 14
2.6.1 Resonances in the spherical square well 17
References 22
3 Embedding at surfaces 1
3.1 Surface embedding and the embedding surface 3
3.1.1 Embedding and the reflection matrix 4
3.1.2 Shifting the embedding surface 5
3.1.3 Testing the shifted embedding plane 7
3.1.4 Embedding on to vacuum 9
3.2 Embedded surface calculations 10
3.2.1 LAPW basis functions 10
3.2.2 Calculating the Green function 13
3.2.3 Self-consistency-the electron density and potential 14
3.3 First results 18
3.3.1 Electron density and work-function 18
3.3.2 Surface states and bulk states at the surface 19
3.3.3 Problems 21
3.4 Sub-volume embedding 22
3.4.1 Adding an atomic layer 22
3.4.2 Bulk embedding potential 23
3.4.3 Tests 24
3.4.4 Applications 27
3.5 Embedding with buffer regions 28
3.5.1 Applications 30
3.6 The transfer matrix and embedding 31
3.6.1 Transferring the transfer matrix 35
3.6.2 The transfer matrix and Green functions 36
3.6.3 Model calculation 37
3.7 Embedding an isolated adsorbate 40
3.7.1 The adsorbate embedding potential 42
3.7.2 Adsorbate calculations: density of states and energy 43
3.7.3 Adsorption results 45
References 46
4 Electrons at surfaces 1
4.1 Surface states and surface resonances 1
4.2 Image states 3
4.2.1 Threshold behaviour 4
4.2.2 Magnetic image states 7
4.3 Screening of an external field 9
4.3.1 Screening at the Ag(001) surface 10
4.3.2 Screening at stepped surfaces 13
4.4 Adsorbates 14
4.4.1 Alkali metal overlayers on Cu(111) quantum-well systems 15
4.4.2 Alkali and alkaline-earth atoms adsorbed on Cu(111) 17
4.4.3 Silicene on Ag(111) 19
References 21
5 Confined electrons and embedding 1
5.1 Variational principle for confined systems 1
5.2 Confined H atom 5
5.3 Surface state confinement by islands on Ag(111) 6
5.4 Electron transport through nanostructures 7
5.4.1 Eigenstates 8
5.4.2 Transmission 10
5.5 Mixed boundary conditions 12
5.5.1 Kinks revisited 14
5.5.2 Kink density of states 16
5.6 Linear dependence 18
References 20
6 Tight-binding and the embedding self-energy 1
6.1 LCAO embedding 2
6.1.1 The embedded Hamiltonian matrix 2
6.1.2 Matrix variational method 3
6.1.3 The Green function and Dyson's equation 6
6.1.4 Baraff-Schlüter approach 6
6.2 The Grimley-Newns chemisorption model 7
6.3 Finite differences and tight-binding 12
6.3.1 Discretizing the continuum 12
6.3.2 Embedding muffin-tins into a finite-difference grid 14
6.4 LCAO codes for the self-energy 19
6.4.1 Recursion 21
6.4.2 Summing over bulk states 22
6.4.3 Fixing the energy 24
6.4.4 Applications 25
References 26
7 Electron transport 1
7.1 The embedding potential and transport 1
7.1.1 Channel functions 2
7.1.2 Left and right spectral functions 4
7.1.3 Total transmission 8
7.1.4 Applications 9
7.2 Transport with localized basis functions 12
7.2.1 Current in LCAO 13
7.2.2 Spectral functions in LCAO 14
7.2.3 Transmission in LCAO 17
7.3 LCAO transport calculations 18
7.3.1 Zero-bias calculations 18
7.3.2 Finite-bias calculations 21
References 24
8 Relativistic embedding 1
8.1 Embedding the Dirac equation 2
8.1.1 Dirac variational principle 2
8.1.2 Solving the embedded Dirac equation 5
8.1.3 H-atom in a spherical cavity 7
8.1.4 The Dirac Green function and continuum states 10
8.2 Embedded surface calculations with the Dirac equation 13
8.2.1 The Dirac transfer matrix and embedding potential 14
8.2.2 Rashba surface states on Au(111) 16
8.3 The scalar-relativistic equation + spin-orbit coupling 19
8.3.1 Spin-orbit derivation 19
8.3.2 Application to surface states 21
References 25
9 Embedding in electromagnetism 1
9.1 Embedding Maxwell's equations 2
9.1.1 Variational method for the Helmholtz equation 2
9.1.2 First tests, and eliminating Laplace solutions 5
9.1.3 Leaky electromagnetic waves and the continuum 8
9.1.4 Working with the magnetic field 10
9.2 Embedding dielectric spheres 14
9.2.1 Matrix elements of the embedding operator 14
9.2.2 Photonic band structure of dielectric spheres 17
9.3 Plasmonics of metal cylinders 19
9.3.1 Embedding with ε(ω) 19
9.3.2 Embedding a line of cylinders 20
9.3.3 Spectral density and photonic band structure 23
9.3.4 Eigenmodes 25
9.4 Good conductors 28
9.5 Conclusions 33
References 34
10 Time-dependent embedding 1
10.1 Time-dependent embedding formalism 2
10.2 Model atomic problem 5
10.2.1 Time-dependent embedding potential 6
10.2.2 Time evolution in region I 7
10.3 Time evolution of extended states 9
10.4 Excitation of electrons at the Cu(111) surface 12
10.4.1 Cu(111) surface electronic structure 12
10.4.2 Time-dependent embedding potentials for Cu(111) 15
10.4.3 Excitation of the Cu(111) surface state 17
10.4.4 Excitation of a Cu(111) bulk state 21
10.4.5 Outlook 22
10.5 Time-dependent embedding in a localized basis 22
10.5.1 Time-dependent self-energy 24
10.5.2 Discretized model atomic problem 26
10.5.3 Time-dependent calculations for molecular transport 29
10.6 Conclusions 33
References 33
11 Connections 1
11.1 Embedding and R-matrix theory 1
11.1.1 Electron energy-loss spectroscopy from NiO 5
11.2 Resonant states 8
References 12
