Electronic States in Crystals of Finite Size

by Ren, Shang Yuan

• ISBN13: 9780387263038
• ISBN10: 0387263039
• Format: Hardcover
• Copyright: 2005-10-21
• Publisher: Springer Verlag

Summary

This book presents an analytical theory on the electronic states in low-dimensional systems and finite crystals recently developed by the author based on a differential equation theory approach. This theory gives some exact and general fundamental understandings on the electronic states in ideal low-dimensional systems and finite crystals such as the numbers and the behaviors of electronic states of different types; It provides new insights on some fundamental problems in low-dimensional systems such as surface states, quantum confinement effects etc, some of them are quite different from what are traditionally believed in the solid state physics community.

Author Biography

The author has been working in various areas in the field of theoretical condensed matter physics for more than twenty years and has published more than ninety research papers. Some of them are well cited.

Table of Contents

Part I Why a Theory of Electronic States in Crystals of Finite Size is Needed
Introduction	3	(20)
Electronic States Based on Translational Invariance	4	(2)
Energy Band Structure of Several Typical Crystals	6	(2)
Fundamental Difficulties of the Theory of the Electronic States in Traditional Solid State Physics	8	(2)
The Effective Mass Approximation	10	(2)
Some Numerical Results	12	(2)
Subject of the Book and Main Findings	14	(9)
References	18	(5)
Part II One-Dimensional Semi-infinite Crystals and Finite Crystals
Mathematical Basis	23	(26)
Elementary Theory and Two Basic Theorems	24	(2)
Floquet Theory	26	(4)
Discriminant and Linearly Independent Solutions	30	(3)
Basic Theory of the Schrodinger Equation in One-Dimensional Crystals	33	(6)
Two Different Eigenvalue Problems	34	(1)
The Function D(λ)	35	(4)
Energy Band Structure of One-Dimensional Crystals	39	(4)
Zeros of Solutions	43	(6)
References	47	(2)
Surface States in One-Dimensional Semi-infinite Crystals	49	(16)
Basic Considerations	50	(2)
Two Relevant Theorems	52	(2)
Surface States in Ideal Semi-infinite Crystals	54	(3)
Cases Where Vout Is Finite	57	(4)
Comparisons with Previous Work and Discussions	61	(4)
References	62	(3)
Electronic States in Ideal One-Dimensional Crystals of Finite Length	65	(24)
Basic Considerations	65	(1)
Two Types of Electronic States	66	(6)
τ-Dependent States	72	(3)
Electronic States in One-Dimensional Finite Symmetric Crystals	75	(3)
Comments on the Effective Mass Approximation	78	(1)
Comments on the Surface States	79	(4)
Two Other Comments	83	(1)
A Comment on the Formation of the Energy Bands	83	(1)
A Comment on the Boundary Locations	84	(1)
Summary	84	(5)
References	85	(4)
Part III Low-Dimensional Systems and Finite Crystals
Electronic States in Ideal Quantum Films	89	(28)
A Basic Theorem	90	(4)
Consequences of the Theorem	94	(1)
Basic Considerations on the Electronic States in an Ideal Quantum Film	95	(1)
Stationary Bloch States	96	(5)
The Simplest Cases	96	(2)
More General Cases	98	(3)
τ3-Dependent States	101	(1)
Several Practically More Interesting Films	102	(4)
(001) Films with a fcc Bravais Lattice	102	(1)
(110) Films with a fcc Bravais Lattice	103	(2)
(001) Films with a bcc Bravais Lattice	105	(1)
(110) Films with a bcc Bravais Lattice	105	(1)
Comparisons with Previous Numerical Results	106	(4)
Si (001) Films	106	(2)
Si (110) Films and GaAs (110) Films	108	(2)
Further Discussions	110	(7)
References	115	(2)
Electronic States in Ideal Quantum Wires	117	(26)
Basic Considerations	118	(1)
Further Quantum Confinement of ψn (k, x; τ3)	119	(4)
Further Quantum Confinement of ψn,j3 (k, x; τ3)	123	(4)
Quantum Wires of Crystals with a sc, tetr, or ortho Bravais Lattice	127	(2)
fcc Quantum Wires with (110) and (001) Surfaces	129	(8)
fcc Quantum Wires Obtained from (001) Films Further Confined by Two (110) Surfaces	130	(2)
fcc Quantum Wires Obtained from (110) Films Further Confined by Two (001) Surfaces	132	(2)
Results Obtained by Combining Sections 6.5.1 and 6.5.2	134	(3)
fcc Quantum Wires with (110) and (110) Surfaces	137	(1)
bcc Quantum Wires with (001) and (010) Surfaces	138	(1)
Summary and Discussions	139	(4)
References	142	(1)
Electronic States in Ideal Finite Crystals or Quantum Dots	143	(30)
Basic Considerations	144	(1)
Further Quantum Confinement of ψn (k, x; τ2,τ3)	144	(4)
Further Quantum Confinement of ψn,j3 (k, x; τ2,τ3)	148	(3)
Further Quantum Confinement of ψn,j2 (k, x; τ,τ3)	151	(3)
Further Quantum Confinement of ψn,j2,j3 (k, x; τ2,τ3)	154	(4)
Finite Crystals or Quantum Dots with a sc, tetr, or ortho Bravais Lattice	158	(2)
fcc Finite Crystals with (001), (110), and (110) Surfaces	160	(3)
bcc Finite Crystals with (100), (010), and (001) Surfaces	163	(3)
Summary and Discussions	166	(7)
References	170	(3)
Part IV Epilogue
Concluding Remarks	173	(30)
Summary and Brief Discussions	173	(5)
Some Relevant Systems	178	(3)
Electronic States in Ideal Cavity Structures	178	(1)
Other Finite Periodic Systems, such as Finite Photonic Crystals	179	(2)
Could a More General Theory Be Possible?	181	(4)
References	182	(3)
Part V Appendices
A. Electronic States in One-Dimensional Symmetric Finite Crystals with a Finite Vout	185	(6)
References	190	(1)
B. Electronic States in Ideal Cavity Structures	191	(12)
B.1 Electronic States in Ideal Cavity Structures of One-Dimensional Crystals	191	(1)
B.2 Electronic States in Ideal Two-Dimensional Cavity Structures of Three-Dimensional Crystals	192	(2)
B.3 Electronic States in Ideal One-Dimensional Cavity Structures of Three-Dimensional Crystals	194	(1)
B.3.1 Wire Cavities in Crystals with a sc, tetr, or ortho Bravais Lattice	195	(1)
B.3.2 Wire Cavities with (001) and (110) Surfaces in fcc Crystals	196	(1)
B.3.3 Wire Cavities with (110) and (110) Surfaces in fcc Crystals	197	(1)
B.3.4 Wire Cavities with (010) and (001) Surfaces in bcc Crystals	197	(1)
B.4 Electronic States in Ideal Zero-Dimensional Cavity Structures of Three-Dimensional Crystals	198	(1)
B.4.1 BDot Cavities in Crystals with a sc, tetr, or ortho Bravais Lattice	199	(1)
B.4.2 Dot Cavities with (110), (110), and (001) Surfaces in fcc Crystals	200	(1)
B.4.3 Dot Cavities with (100), (010), and (001) Surfaces in bcc Crystals	201	(2)
Index	203