ABSTRACT
A quantum dot can be de†ned as any solid material in the form of a particle with a diameter comparable to the wavelength of an electron. The de Broglie wavelength, λB, of a particle, with mass m, in motion, is given by
(8.1)
where ℏ is Planck’s constant, and p is the linear momentum of the particle. From the Pauli uncertainty principle, the uncertainty in the linear momentum, Δpx, of the particle is given by
p x
x ≈
(8.2)
where Δx is the positional uncertainty. Con†nement of the particle gives rise to an additional kinetic energy, Econf, of
= ≈E p
m m x
( ) 2 2 ( )conf
2 (8.3)
The additional energy of con†nement will be signi†cant if it is greater than the energy of the thermal motion of the particle, that is, if
≥E k T1 2conf B
(8.4)
where kB is the Boltzman constant. Thus, size quantization will occur if the range of con†nement, Δx, is comparable to the de Broglie wavelength of the particle. For instance, the effective mass, m*, of an electron in a semiconductor is typically 0.1m0 (where m0 is the free electron rest mass), which corresponds to Δx ≈ 5 nm. It is thus possible to think of a quantum dot as a rather large atom, with the consequence that atom-like, rather than band-like, electronic structure, should be found. The example is for con†nement in one dimension, where the other two dimensions may be macroscopic, but the argument can be extended readily to 2-D or 3-D con†nement.
