Diffusion and Interdiffusion in the Synthesis of Semiconductor Nanostructures

Semiconductor nanostructures have been studied extensively over the last two decades. Under proper processing conditions, the fabrication of heterogeneous junctions between different semiconductor materials results into three dimensional nanostructures with lateral dimensions in the 1-100 nm length scale. A notable example is the case of Group IV semiconductors such as silicon (Si) and germanium (Ge).

The deposition of Ge on a Si substrate just a few atomic layers thick induces self organization of a high density nanostructure with physical and chemical properties different to their neighboring environment. For instance, one peculiar trait of their electrical behavior is the ability to trap discrete amounts of opposite charge (electrons and electron holes), similarly to the case of natural atoms. As a result, these nanostructures are often referred to as 'quantum dots' (QDs) and 'artificial atoms'. Likewise, mutual interactions within functional architectures of QDs may give rise to artificial analogues of molecules and crystals, leading to a gamut of new opportunities.

The potential applications of QDs are enormous. Technological fields where the use of QDs may exert the highest impact include light emitting diode (LED) and laser technologies, single photon sources, new transistors, cellular automata and quantum computers, advanced catalysts, photovoltaic devices, environmental and biomedical diagnostics, imaging and therapeutics, biosensing, etc. In particular, the development of processes compatible with silicon technology holds potential for immediate integration of QDs in state of the art semiconductor fabrication processes.

The fabrication of germanium / silicon nanostructures using the bottom-up approach could become a viable option to realize arrays of epitaxially grown QDs. The prototypical experiment consists involves the slow deposition of germanium atoms on a silicon substrate (e.g. fractions of monatomic layers per second), which may be realized by a variety of chemical and physical methods already in use in semiconductor processing.

At high temperatures, Ge atoms replicate the crystal lattice geometry of the Si substrate, due to similarities between these elements. However the lattice parameter of Ge is about 4% larger than that of Si, which causes excessive strain accumulation at the heterogeneous interface.

Beyond a certain thickness, spontaneous mechanisms intervene to accomplish partial relaxation of this strain. One of these mechanisms is the creation of roughness, which ultimately leads to the emergence of three dimensional nanostructures. Other mechanisms include the nucleation of misfit dislocations and intermixing of Ge and Si atoms, which reduces the effective lattice mismatch at the interface. The geometrical, strain and elemental profile within and around the three dimensional nanostructures governs fundamental characteristics of these QDs.

While the principal concept at the origin of the self organization of semiconductor nanostructures is a thermodynamic instability, over recent years a novel paradigm has been proposed, which refers to a leading role of kinetic parameters and energy barriers against atomic diffusion. Thermodynamic stability, which is one of the most ubiquitous concepts in physics, does not explain a number of physical and chemical properties observed under typical experimental conditions, including e.g. strain and elemental profiles.²

To achieve thermodynamic stability all material within and around the three dimensional nanostructures should sustain massive rearrangements and great multitude of competitive configurations. However this is obstructed by energy barriers against atomic diffusion and exchanges. Under typical experimental conditions, there is a large unbalance between the probability of surface diffusion and bulk diffusion.²

In practice, surface diffusion proves extremely rapid and is essentially governed by Brownian motion (random movement) and only partially directed by the thermodynamic landscape of the surface.

In contrast bulk diffusion is negligible, i.e. atoms below the topmost atomic layer are frozen as soon as overlaid by new atoms. Moreover when temperature is quenched soon after deposition, the overall configuration of the sample comprising e.g. size and shape statistics, strain and elemental profiles and mutual separations of the QDs cannot undergo significant evolution, which gives the highest importance to the dynamic processes realized during growth.

An important feature which is defined early in the deposition process is the mutual positions of the resulting three dimensional nanostructures. The probability of nucleation of an individual nanostructure increases with the local concentration of available atoms, whose diffusion and clustering may generate stable nuclei.

This probability suddenly drops as soon as one nucleus appears and begins to enlarge by capture of nearby atoms essentially driven by Brownian motion.³ This explains why nuclei tend to keep a certain distance apart, which correlates with the atomic surface diffusion length.⁴ Surface diffusion also mediates nanostructure growth, size and shape by capture of mobile atoms.³

Under the simplest assumption of Brownian motion, this is an intuitive competitive process between coexisting nanostructures, whereby the closer their mutual proximity the smaller their relative size.⁴ The correlation between size and shape is a solid concept.¹ Finally, surface diffusion determines the elemental profile within the nanostructures, whose principal features may be explained in terms of Brownian motion once again, the different mobility of germanium and silicon and its dependence on temperature.²

At moderate temperatures for example (say approximately 500 Celsius) the mobility of germanium is much higher than that of silicon, which causes Si atoms to accumulate at nanostructure edges and perimeters,^5,6 whereas thermodynamic stability would require the opposite, i.e. Si rich cores and Ge rich peripheries.

While oversimplified, the picture described above is a reasonable start to understand individual and collective properties of semiconductor nanostructures as observed in experimental data. A variety of additional thermodynamic components, including e.g. strain interactions between nanostructures and substrate, between coexisting nanostructures and within individual nanostructures, may induce effective perturbations in the definition of preferential nucleation sites, the transfer of mass and modulation of size and shape (see e.g. Ostwald ripening which promotes the growth of large over small nanostructures), and the exchange of germanium and silicon atoms.

Both diffusive dynamics and additional thermodynamic components may become modulated by integration of suitable top-down interventions, which may be designed and implemented in keeping with the spontaneous behavior described above, which cannot be suppressed. This is a hybrid approach to achieve semiconductor nanostructures with enhanced control over their position, size, shape and elemental composition.

In this context the notion of 'surface cues' is a powerful concept,⁷ whereby a preliminary modification of the substrate alters the kinetic and thermodynamic landscape at the surface, thus guiding adsorption and diffusion of atoms and molecules. Examples of 'surface cues' may be arrays of steps,⁸ dislocations and chemical inhomogeneities introduced onto the silicon substrate prior to germanium deposition.

In conclusion, over recent years there has been significant progress in the fundamental understanding and fabrication of QDs based on semiconductor nanostructures. While there are still many critical issues ahead, the potential for radical innovation behind these concepts provides strong motivation for future investigations of semiconductor nanostructures.

References

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5. G. Katsaros, G. Costantini, M. Stoffel, R. Esteban, A.M. Bittner, A. Rastelli, U. Denker, O.G. Schmidt, K. Kern, Phys. Rev. B 72, 195320 (2005).
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