What is a 'quantum dot'?
A quantum dot is a very small chunk of semiconductor material with quantum-like properties. These are any effects that the bulk form of the same material does not possess. This phenomenon is called quantum confinement. You may have also heard the terms 'nanocrystal' or 'nanoparticle.' Nanoparticles can be just about anything whose dimensions are on the nanometer scale, while nanocrystals are usually nanometer-sized inorganic solids such as metals, insulators or semiconductors. 'Quantum dot' is a term usually applied to semiconductor nanocrystals in a size limit whose volume is smaller than the volume defined by the bohr radius of that particular semiconductor. A note on fabrication: 'Colloid' is usually used to distinguish the method of preparing quantum dots, the predominant methods of preparation being growth by molecular beam epitaxy (MBE) or organometallic synthesis (colloidally-prepared). Our quantum dots are all colloidally-prepared.
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What is quantum confinement in semiconductor nanocrystals?
Imagine that as you chisel away at a uniform red stone, all the flecks and bits and pieces of the stone turn blue. This is quantum confinement. The optical properties of a material depend not on its composition - but its size!

The term quantum confinement, when applied to low-dimensional semiconductors, describes the confinement of the exciton within the physical boundaries of the semiconductor. This is an inherently quantum phenomenon - hence the names, "quantum well", "quantum wire", and "quantum dot", which describe confinement in 1, 2 and 3 dimensions, respectively. The exciton bohr radius (a_B) is often used as a meter-stick to judge the extent of confinement in a low-dimensional structure. The confinement regimes describe a size range in semiconductor quantum dots that compare the bohr radius to the diameter of the nanocrystal (D),

Strongly-confined regime: D < 2a_B

Intermediate confinement regime: D ~ 2a_B

Weakly-confined regime: D > 2a_B

It is in the strongly-confined regime that the optical properties of these quantum dots are most affected. For example, in the CdSe system, as the dimensions of the nanocrystal are reduced below the exciton bohr radius (a_B~5.0-5.5nm), the optical transitions shift toward the blue (higher energy).
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Brus [1983] arrived at an intuitive description of the lowest electronic transition (1S_3/2-1S_e), and described it primarily as the confinement of the exciton to the boundaries of the quantum dot (confinement term) and a perturbation that resulted from the coulombic interaction between the electron and the hole (coulombic term). Thus, the size-dependent optical properties of the nanocrystal can be described as:


(click for a larger picture)

Note that the confinement term follows a 1/R² dependence - that is the energy is dependent upon the boundaries just as in the particle-in-a-box. Recall that the particle-in-a-box energy levels are: E=n²h²/8ml² which is identical to the confinement term. Moreover, the coulombic term arises from electrostatics: V=-q₁q₂/4pee₀r. One may also include further terms that describe the interaction for precisely; for example, the exchange interaction can be written as a ~1/R³ dependence, which becomes important in understanding exciton fine structure [Bawendi, the "dark exciton" papers].
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What are the selection rules for electronic transitions?
Perhaps it is more accurate to describe the selection rules for excitonic creation/annihilation coupled to a radiation field, since the transitions in quantum dots involve not only electron motion, but hole motion as well. To understand the optical selection rules of quantum dots, we can start with the rules of the bulk semiconductor - which necessarily leads us to consider the electron and hole energy levels. CdSe is the most well-studied system, and so this will be used as a model, as Efros and Bawendi have done.
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Electron and Hole Structure (in CdSe)

We first turn our attention to the electron levels.  The conduction band of CdSe is composed primarily of the Cd²⁺ molecular orbitals.  Cd²⁺ has an electronic configuration of [Kr]4d¹⁰5s⁰ (basically, Kr). Therefore, in the excited state, the electron must populate the 5s derived conduction band. Since the s-orbitals are singly degenerate (l=0 --> L=0), this makes any spin-orbit-coupling also singly degenerate (J = L+S = 0+1/2 = 1/2).

We then turn our attention to the hole levels.  The valence band of CdSe is composed, primarily of the Se^2- molecular orbitals.  Se^2- has an electronic configuration of [Ar]4s²3d¹⁰4p⁶.  Therefore, the hole will populate the 4p levels (thus removing one electron).  Since the p-orbitals are doubly degenerate (l=1 --> L=1, 0) then spin-orbit-coupling produces doubly degenerate hole levels (J = L+S = 0+1/2, 1+1/2 = 1/2, 3/2).
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Exciton States - Absorption (E1)

Therefore, even in the bulk two excitons are observed, each one derived from a different hole level coupled to the same electron level. These have been referred to as the A and B excitons. It turns out that the A excitons are electric-dipole allowed (E1) only with polarizations perpendicular to the c-axis of CdSe, while the B excitons are E1-allowed only with polarizations parallel to the c-axis.

We can now turn our attention to a quantum-confined system, where it will be convenient to describe the electron and hole with distinct term symbols with the following components:

a label that describes the ordering of the energy shells, which is similar to the atomic case with principal quantum number (n);

a label describing the overall shape of the wavefunction in the nanocrystal that is similar to the atomic case with the spherical harmonics (L);

and finally the spin-orbit-coupling term that will describe the interaction of the electron/hole orbit within the degeneracy of the conduction/valence band (J).

By convention, this has been written down as: n L_J.

Since the electron in CdSe is singly degenerate, J=1/2 is replaced with J=e. The exciton states are then described as a coupling between distinct electron and hole states. For example, the lowest energy transition in CdSe is the 1S_3/2-1S_e state which is an exciton whose electron is in the first energy shell with spherical symmetry and a hole in the first energy shell with spherical symmetry, derived from the upper J=3/2 valence band.

As you will see later, another term, F, is used to describe the exciton fine structure as a coupling of the hole spin-orbit-coupling (J_h) to the electron spin projection (m_e,z); F=F=J_h+m_e,z.
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