Electrons and Holes in a 3D “Quantum Dot”

Equations 2.11-2.16 and 3.15 are applicable to the electron and hole states in semiconductor “quantum dots,” which are used in biological research as color – coded fluorescent markers. Typical semiconductors for this application are CdSe and CdTe.

A “hole” (missing electron) in a full-energy band behaves very much like an electron, except that it has a positive charge, and tends to float to the top of the band. That is, the energy of the hole increases opposite to the energy of an electron.

The rules of quantum mechanics that have been developed so far are also applicable to holes in semiconductors. To create an electron-hole pair in a semi­conductor requires an energy at least equal to the energy bandgap, Eg, of the semiconductor.

We found earlier that the wavefunction for a particle in a three-dimensional infinite trap of volume L3 with impenetrable walls are given as

ipn(x, y, z) — (2/L)3/2sin (nxPx/L)sin (nyPy/L)sin (nzPz/L), where nx — 1, 2 …, etc.,

and

En — [h2/8mL2] (n2x + nj + nj).

The conceptual leap that is required here is that instead of a free electron in a vacuum box of side L with infinite potential walls, we have a conduction electron free to roam about in a cube of intrinsic semiconductor of side L, with the electron being confined to the conductor by the work function barrier. The moving carrier will be endowed with an effective mass by its immersion in the semiconductor, and it will be affected also by the permittivity of the semiconductor medium.

(Different wavefunction and energy equations will apply to other geometries than a cube, for example, the quantum dots grown by molecular beam epitaxy (MBE) are often pyramids. The wavefunctions in these cases will be different, but the differ­ences in the end do not matter very much. Sharply defined energies inversely proportional to the square of the container size will result.)

This application to semiconductor quantum dots requires L in the range of 3-5 nm, the mass m must be interpreted as an effective mass m*, which may be as small as 0.1me. The electron and hole particles are generated by light of energy

he/1 = ЕП;electron T En, hole T Eg. (8-3)

Here, the first two terms are strongly dependent on particle size L, as L~2, which allows the color ofthe light to be adjusted by adjusting the particle size. The bandgap energy Eg is the minimum energy to create an electron and a hole in a pure semiconductor. The electron and hole generated by light in a bulk semiconductor may form a bound state along the lines of the Bohr model, described above, called an exciton. However, as the size of the sample is reduced, the Bohr orbit becomes inappropriate and the states of the particle in the 3D trap are a more correct description.

In this context, Figure 8.6 shows levels in a quantum dot as an element of absorber in a solar cell. The process shown is one of absorption of a high-energy photon

Electrons and Holes in a 3D “Quantum Dot”

Figure 8.6 Multiple exciton generation in a between the first energy levels for electrons and

quantum dot [105]. Because of quantum holes in the quantum dot can create three

confinement, the energy levels for electrons and excitons, tripling the charge in the external holes are discrete. A single absorbed photon of circuit. energy at least three times the energy difference

Electrons and Holes in a 3D “Quantum Dot”

Figure 8.7 Predicted solar cell efficiency as enhanced by impact ionization. Evidence for such processes has been advanced for nanocrystals of the small bandgap semiconductor PbSe [106].

hc/l = Eg + 6Д, where Д is the energy spacing of succeeding size-quantized states in the quantum dot. The process depicted leads to three electrons compared to one electron eligible to flow through the load resistance, but requires an energy threshold E > Eg + 8Д, and the further condition

6Д > 2Eg + 4Д, or Д > Eg.

The lower threshold to produce two electron-hole pairs would be E > Eg + 4Д, with 4Д > Eg + 2Д or 2Д > Eg.

The minimum excess energy released by exciton decay to give charge multipli­cation is Eg + 2 Д. This is a high threshold: since 2 Д > Eg, the initial energy has to be more than 3Eg. The further assumption is that the charge appearing at the lowest electron and hole energies will efficiently leave the quantum dot to flow in the external circuit. This may be possible but apparently has not been demonstrated in situations other than the Ge detector.

If such processes are important, the optimum solar cell would have smaller values of Eg, as shown in Figures 8.7 and 8.8.

8.3

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