Book contents
- Frontmatter
- Contents
- Preface
- Introduction
- 1 FOUNDATIONS
- 2 ELECTRONS AND PHONONS IN CRYSTALS
- 3 HETEROSTRUCTURES
- 4 QUANTUM WELLS AND LOW-DIMENSIONAL SYSTEMS
- 5 TUNNELLING TRANSPORT
- 6 ELECTRIC AND MAGNETIC FIELDS
- 7 APPROXIMATE METHODS
- 8 SCATTERING RATES: THE GOLDEN RULE
- 9 THE TWO-DIMENSIONAL ELECTRON GAS
- 10 OPTICAL PROPERTIES OF QUANTUM WELLS
- A1 TABLE OF PHYSICAL CONSTANTS
- A2 PROPERTIES OF IMPORTANT SEMICONDUCTORS
- A3 PROPERTIES OF GaAs–AlAs ALLOYS AT ROOM TEMPERATURE
- A4 HERMITE'S EQUATION: HARMONIC OSCILLATOR
- A5 AIRY FUNCTIONS: TRIANGULAR WELL
- A6 KRAMERS–KRONIG RELATIONS AND RESPONSE FUNCTIONS
- Bibliography
- Index
2 - ELECTRONS AND PHONONS IN CRYSTALS
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface
- Introduction
- 1 FOUNDATIONS
- 2 ELECTRONS AND PHONONS IN CRYSTALS
- 3 HETEROSTRUCTURES
- 4 QUANTUM WELLS AND LOW-DIMENSIONAL SYSTEMS
- 5 TUNNELLING TRANSPORT
- 6 ELECTRIC AND MAGNETIC FIELDS
- 7 APPROXIMATE METHODS
- 8 SCATTERING RATES: THE GOLDEN RULE
- 9 THE TWO-DIMENSIONAL ELECTRON GAS
- 10 OPTICAL PROPERTIES OF QUANTUM WELLS
- A1 TABLE OF PHYSICAL CONSTANTS
- A2 PROPERTIES OF IMPORTANT SEMICONDUCTORS
- A3 PROPERTIES OF GaAs–AlAs ALLOYS AT ROOM TEMPERATURE
- A4 HERMITE'S EQUATION: HARMONIC OSCILLATOR
- A5 AIRY FUNCTIONS: TRIANGULAR WELL
- A6 KRAMERS–KRONIG RELATIONS AND RESPONSE FUNCTIONS
- Bibliography
- Index
Summary
Few low-dimensional systems are periodic (superlattices provide an obvious exception), but they all consist of relatively large scale structures superposed on the structure of a host. This may be a true crystal such as GaAs or a random alloy such as (Al,Ga)As; we shall ignore the complications introduced by the alloy and treat it as a crystal ‘on average’. We must understand the electronic behaviour of the host before treating that of the superposed structure.
This chapter deals first with one-dimensional crystals, followed by three-dimensional materials. The final section is devoted to phonons, lattice waves rather than electron waves, which also have a band structure imposed by the periodic nature of the crystal. Photons are the third kind of wave that we shall encounter, and structures that display band structure for light have recently been demonstrated. Their behaviour can be described with a similar theory but we shall not pursue this.
Band Structure in One Dimension
The potential energy in a real crystal is clearly far more complicated than the systems that we have studied in the previous chapter. In Section 5.6 we shall solve the simple example of a square-wave potential in detail, but the most important results follow from the qualitative feature that the potential is periodic. In one dimension this means that V(x + a) = V(x), where a is the lattice constant, the size of each unit cell of the crystal.
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- The Physics of Low-dimensional SemiconductorsAn Introduction, pp. 45 - 79Publisher: Cambridge University PressPrint publication year: 1997