This is a brief review of the emerging role of quantum dots as biological markers. The main theoretical approaches (including statistical ensemble calculations) are discussed and compared against, after which current research promising their productive application to biological analysis techniques is presented. The current and arising challenges faced in this developing field are then discussed.

FIG. 1:  Colloidal quantum dots irradiated with a UV light. Different sized  quantum dots emit different color light due to quantum confinement. Reproduced from nature.com.

Quantum dots (QD) are nanoscale semiconductor particles characterised by the confinement of the electron, electon-hole or exciton wavefunction in three spatial dimensions; systems of which exhibit optoelectronic properties that are greatly differing to those of comparable systems consisting of mesoscopic or macroscopic particles. There has been great interest in QDs since it was shown by Chan (1998) that highly luminescent semiconductor QDs could be coupled via covalent bonding mechanisms to biological molecules. The development of highly sensitive biological detection techniques utilising such QDs is thought within the scientific community to offer great promise given their potential to replace the radiological alternatives that are presently in place as accepted practise. It is well known that such techniques can pose serious health hazards to patients through long term exposure (Picano, 2004) as well as the fact that treatment specialists face logistical challenges due to the short life-times of these currently adopted biomarkers; the result of which is to increase radiological exposure (hence added risk) to patients.

I. Theory

A. Overview of the main theoretical approaches

The optoelectronic properties of QDs are influenced in large part by quantum-mechanical phase coherence, and vary as a function of size and shape (Murray et al.,2000). Mathematical modelling of QD systems is often carried out in the statistically formulated regime of electrons interacting with a random matrix, explained in detail by Beenakker (1997), or by interaction with a pseudopotential, investigated by Wang and Zunger (1996); Franceschetti and Zunger (1997). A surprising link to the distribution of the zeros of the Riemann zeta function is accessed through spectral theory (Keating, 1993) via the Hilbert-Pólya conjecture (later refined by Berry). 

Semiclassical approaches have also been developed, notably by Shpatakovskaya (2006) who used a one-dimensional quantum dot as a primitive model for developing a consistent semiclassical method, which has scope to be applied to systems of higher dimension, that yield to separation of variables.

B. Comparison of the main theoretical approaches

The advantage of semiclassical and pseudopotential approaches over a random matrix approach is that they reduce the complexity of problems that would otherwise be analytically intractable in the stochastic random matrix theory. For example Puangmali et al. (2008) employed an atomistic psuedopotential approach to investigate the optoelectronic properties of spherical InAs nanocrystals, with their calculated interband absorption spectra replicating the spectra observed in experiment via STM imaging. Such approaches are often more practical in highly complex systems offering experimenters a powerful tool of predictive power, which can be used as a productive means of guiding experimental investigation.

Within random matrix theory, however, efforts have also been made to establish simplified means of calculation, most notably by Dyson (1962) who introduced the circular ensemble of scattering matrices as an alternative to the often intractable Gaussian ensemble. As an example, we consider an approach in the regime of a Gaussian ensemble of Hamiltonians describing chaotic transport through a cavity. The Hamiltonian of the cavity is given by (Mahaux and Weidenmuller, 1979)

where the set {a_i} (a = 1,2,...N, with N = N_1 + N_2 the total number of propagating modes) forms a basis of scattering states in the leads at the Fermi energy, E_F. Blmel and Smilansky (1990) later found that for chaotic scattering, the correlations of the phase shifts φ_n are accurately modelled by the distribution function

The major advantage offered by the random matrix approach is that it constitutes a powerful tool in the computational modelling of the behaviour of stochastic QD systems, in cases where one can construct an analytically tractable framework, either by direct solution or numerical approximation. One such example of this is the identification of a new universality class which is distinct from the ubiquitous Wigner-Dyson class by Altland and Zirnbauer (1996), realised via a random matrix description of a chaotic Andreev QD.

II. Application

It was shown by Leatherdale et al. (2002) that QDs have a high extinction coefficient and operate in a manner that can be thought analogous to a single electron transistor. For this reason, coupled with evidence that they exhibit the Coulomb blockade effect (Beenakker, 1991), great promise is held for a wealth of optical applications. Meir et al. (1993) showed that the density of states of QDs is far more sharply peaked than in materials of a lower degree of spatial confinement (such as quantum wires, for example). It is this characteristic to which their remarkable quantum transport properties are owed, and their potential application as biological markers is further validated.

A. Application to biological analysis

The high extinction coefficient of QDs has been linked to their high degree of luminescence (Chan et al., 2002) which has led to their emergence in the field of biological analysis where there is great demand for the development of organic dyes used in vivo; in particular, in the targeting of cancer cells (Gao et al., 2004). QDs exhibit high levels of photo-stability (Chan, 1998) therefore there exists the possibility of high sensitive, ‘real-time’ cellular imaging and tracking of biological matter in a time-evolving system.

B. Toxicity

There exist concerns regarding the toxicity of QDs, given their potential for biomedical applications. Hauck et al. (2010) stated ‘most toxicology data is derived from in vitro studies and may not reflect in vivo responses’, but there is still a great deal of investigation left to be carried out until a conclusive answer is agreed upon. Hardman (2006) pointed out that ‘not all QDs are alike’, and further investigation is necessary in discerning the risks associated with each category of engineered QDs.

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