第五讲+半导体量子阱

5
Growth and structure of semiconductor quantum wells
Epitaxial crystal growth techniques: molecular beam epitaxy (MBE, 分子束外延) and metal-organic chemical vapour deposition (MOCVD，金属有机物化学气相沉积)
7
Separation of the variables
The electrons and holes in a quantum well layer are free to move in the x, y plane but are confined in the z direction. This allows us to write the wave functions in the form:
px 2m
2
~
2 2m x
2
This confinement energy will be significant only if it is comparable to or greater
than the kinetic energy of the particle ects criterion
1
Quantum confined structures
•The optical properties of solids do not usually depend on their size. Ruby! • For very small crystals, then the optical properties do in fact depend on the size. Semiconductor doped glasses. •The size dependence of the optical properties in very small crystals is a consequence of the quantum confinement effect.
px x
If the particle is free and has a mass m, the confinement in the x direction gives it an additional kinetic energy of magnitude ：
E ~ confinement
Econfinement
~
2 2m x
2

1 2 kBT
It means quantum size effects will be important if:
x ~ 2 deB mkBT px 2
x must be of the same order of magnitude as the de Broglie wavelength deB for the thermal motion in order to obtain obvious quantum effects.
3
Three basic types of quantum confined structures
Quantum Well
Quantum Wire
Quantum Dot
1D confinement
2D confinement
3D confinement
Bulk: 3-D crystals; Quantum wells: 2-D crystals; Quantum wires: 1-D crystals; Quantum dots: 0-D crystals.
第六讲 Semiconductors quantum wells
Quantum confined structures Growth and structure of semiconductor quantum wells Electric levels Optical absorption and excitons Optical emission Quantum wires and dots
(x, y, z) (x, y)(z)
Free motion in the x, y plane, the states can by described by the wave vector k:
(x, y) 1 eikr
A
2k 2 Ek 2m*
Quantized energy in the z direction described by the quantum number n:
2
)
1. The energy of the levels is inversely proportional to the effective mass and the square of the well width.
2. the electrons, heavy holes and light holes will all have different quantization energies. In the valence band, the heavy holes are dominant in most situations because they form the ground state level.
Preparation techniques:
Quantum Wells: advanced epitaxial crystal growth Quantum Wires: lithographic techinique or by epitaxial growth Quantum Dots: lithographic patterning or by spontaneous growth techniques
The infinite well model overestimates the quantization energy.
The quantization energies of holes are smaller than those of electrons (smaller
mass)
11
第六讲 Semiconductors quantum wells
Quantum confined structures Growth and structure of semiconductor quantum wells Electric levels Optical absorption and excitons Optical emission Quantum wires and dots
multiple quantum wells (MQWs): Have larger b values; isolated Superlattices: Have much thinner barbers; coupled and new extended states are formed in the z direction; additional properties.
The Schrodinger equation:

2 2m*
d 2(z)
dz2

E ( z )
The boundary condition:
(d / 2) (d / 2) 0
The wave function:
n (z)
2
n
d sin(kn z 2 )
This form of wave function describes a standing wave inside the well:
3. The wave functions can be identified by their number of nodes, the nth level has (n - 1) nodes.
4. States of odd n have even parity, and vice versa. φ(-z) = + φ(z) (even parity) or φ(-z) = - φ(z) (odd parity).
4
第六讲 Semiconductors quantum wells
Quantum confined structures Growth and structure of semiconductor quantum wells Electric levels Optical absorption and excitons Optical emission Quantum wires and dots
En
n (z)
The total energy for an electron or hole in the nth quantum level is therefore
given by:
2k 2
E total (n, k ) En 2m*
8
Infinite potential wells
6
第六讲 Semiconductors quantum wells
Quantum confined structures Growth and structure of semiconductor quantum wells Electric levels Optical absorption and excitons Optical emission Quantum wires and dots
•d is chosen to be close to: m*kBT
•quantized motion in the z direction, and free motion in the x, y plane.