MICROEI.~CTROI¢IC ENGINEERING ELSEVIER
Microelectronic Engineering 41/42 (1998) 515-518
Monte Carlo simulation of the growth of metallic quantum dots M. Boero a, P.A. Mulheranb and J.C. Inksona aDepartment of Physics, University of Exeter, Stocker Road, Exeter, EX4 4QL, UK b Department of Physics, The University of Reading, Whitenights, Reading RG6 6AF, UK
The phenomenon of Coulomb Blockade allows to control the flow of individual electrons and has raised hopes of building electronics devices based on such phenomenon. In order to observe Coulomb Blockade, islands, or dots, on the nanometre scale must be fabricated in order to obtain a small enough capacitance for the electrostatic energy associated with adding a single electron to the island to exceed thermal fluctuations. Coulomb Blockade has been observed both in semiconductor and in metallic structures. In particular irregular 2D arrays of metallic dots of typical size less than 10rim have been fabricated by means of Electron Beam Lithography combined with granular film deposition. In this paper we present a theoretical study of the grain deposition and subsequent island formation based on a Monte Carlo technique. The model allows to account for a variety of important parameters such as temperature, diffusion vs. deposition rate, and in particular the effect of contacts. Optimum parameters to obtain reproducible Coulomb Blockade devices are suggested. building principle underpinning future electronics 1. INTRODUCTION devices. In this paper we address one of the techniques that Coulomb Blockade is a phenomenon that impedes have been proven to yield Coulomb Blockade electric conduction through small islands. It devices, namely the deposition of granular metallic manifests itself in the IV characteristics where below particles (typically gold) on insulating substrates a threshold voltage no-current flows [see e.g.1 and (typically silicon oxide) [1,2,3]. references therein ]. When an extra electron attempts This technique has successfully been combined with to occupy the island, it causes an increase in the electron beam lithography (EBL) to produce electrostatic energy of the island which is inversely disordered arrays of metallic dots which show an IV proportional to the capacitance of the island. characteristic that presents the typical non-linearity It follows that two key conditions must be met in due to Coulomb Blockade [1,2,31. order to observe Coulomb Blockade. Firstly the In this work we simulate the deposition technique by islands must be very small in order for the means of a Monte Carlo technique. We allow for the capacitance C to be small and the electrostatic energy variation of deposition temperature, deposition and associated to the addition of a single electron to be diffusion rate, and for the presence of metallic considerably larger than the thermal energy. contacts on the surface where the islands are Secondly the islands must be sufficiently well deposited. The latter in particular prove to have an isolated so that quantum fluctuations do not affect important effect on the position of the islands. The the occupancy of the island. In, practice this contacts act as sinks for the metallic particles condition translates in the requirement that the deposited in their neighbourhood so that the tunnelling resistance between the island and the probability of forming dots in the vicinity of the contacts or the neighbouring islands must be much contacts is considerably reduced. This effect can be greater than the resistance quantum . used to control the position where the islands are The two previous conditions pose a strong challenge formed. to fabrication techniques. Several methods to fabricate Coulomb Blockade devices have been 2. METHOD successfully developed. However several problems such as temperature and reproducibility must still be The simulation process is based on a Monte Carlo solved if Coulomb Blockade is to be adopted as the technique. This consists in choosing a 2D mesh of points on which atoms are deposited. Once the 0167-9317/98/$19.00 © Elsevier Science B.V. All rights reserved. Pll: SO 16%9317(98)00120-8
M. Boero et al./Microelectronic Engineering 41/42 (1998) 515-518
atoms are dropped on the surface they move on the mesh in a random fashion. The motion of the atoms on the surface continues until one of the following things happens. Either an atom encounters an existing island and merges with it, or an atom meets at a point on the mesh a sufficient number of other atoms to form a new island . Two key parameters must be chosen in order to perform the simulation. Firstly the minimum number of atoms which is necessary to form a new island must be specified. Secondly the ratio D - - b e t w e e n the diffusion rate and the deposition rate F must be chosen. As for the first parameter, it is often referred to as icrit, the minimum number of atoms needed to form an island minus one. Its value, typically 1,2,3,4, depends on several parameters such as temperature, conditions of the surface and type of atoms that are deposited on the surface. Broadly speaking, the number of atoms needed to form an island depends on the binding energy of the cluster of atoms versus the temperature at which the growth process is performed. Clearly the smaller icrit the lower the temperature at which the g o w t h occurs. D The diffusion-deposition ratio F depends also on a series of parameters such as the actual rate at which atoms are deposited on the surface, the mobility of atoms on the surface, temperature, etc. Once an island is formed at a mesh site, its size depends on the number of atoms contained on it. The precise relation between size of the island and number of atoms contained in it depends on the geometry of the island . In this work we have a s s u m e d the shape o f the islands to be h e m i s p h e r i c a l . T h e r e f o r e the radius of the hemisphere is proportional to the ctibic root of the number of islands. In this simulations we have assumed that once an atom becomes part of an island it can no longer move, i.e. we have neglected the process of evaporation of atoms from already formed islands , and we have assumed that once the islands are formed they cannot change their position on the mesh.
3. RESULTS We have applied the previous method to simulate the growth of islands by means of atomic deposition on a surface. For the purpose of the simulation file 2D mesh representing the surface is assumed to be perfectly clean, i.e. no attempt is made to include the presence of impurities and defects on the surface of the island . Our results have been compared with the experimental data  for what concerns the average density of islands and their average size. The comparison shows good agreement  between theory and experiment. In this work we concentrate on the simulations accounting for the presence of metallic contacts which are present on the surface before the grain deposition process occurs. These are actually the conditions which apply to the experimental situations [1,2,31. Fig. la shows a snap-shot of the growth simulation taken when the total surface covered by the metallic grains is 5% of the total surface. The simulation of fig. 1 is performed with a critical parameter icrit = 1, in other words it is assumed that the simultaneous presence of two atoms on the same point of the 2D mesh is enough to create a new island.. This corresponds to a low temperature situation in which the temperature of the substrate is lower than the binding energy of two atoms on the surface. As fig.1 clearly shows, even with only 5% of the surface covered there are already several islands which are formed. By comparison figs. lb and lc show the situation when 15% and 25% respectively of the total surface is covered. A close analysis of figures la, lb and lc shows that the great majority of islands is already formed in the first stages of the growth process, i.e. by the time that a 5% coverage of the surface is obtained the majority of the islands is already formed. Further deposition of material tends to increase the size of existing islands rather than create new ones. This is due to the fact that the increasing presence of islands on the surface renders the probability of two atoms to meet at the same point on the surface less likely than the probability of an atom to meet an existing island during its diffusion on the surface. This fact has important consequences for the fabrication of Coulomb Blockade devices because it indicates that the initial stages of the deposition process are paramount to determine the type of structure produced by the deposition process.
M. Boero et al. / Microelectronic Engineering 41/42 (1998) 515-518
Figs. 2a, 2b, 2c refer to a the simulation of a similar deposition process as simulated in figs. la but this time the parameter icrit=2, i.e. at least three atoms must converge on the same point for a new island to be created. This simulation refers to the experimental situation where the temperature of the substrate is higher than the binding energy of a two-atom complex but lower than that of a three-atoms complex. As before figs 2a refers to a 5% coverage of the surface, 2b to 15% and 2c to 25 %. A comparison between the set of figures 1 and 2 shows that performing the deposition at higher temperature yields a lower number of islands. This is expected because with i=2 the formation of new islands is statistically less favourable. As in the case of icrit=l, the majority of islands are formed in the initial stages of the growth process, while the deposition of material after 5% of the surface is covered increases the size of new islands with a much reduced probability of creating new ones. A closer analysis of figs. 2 indicates that the formation of islands is depressed in the vicinity of the contacts. This is due to the fact that the contacts are fabricated prior to the deposition process. Those atoms which are deposited in the vicinity of the contacts have a large probability of being captured by them, so that formation of islands in their vicinity is considerably suppressed. This fact is further emphasised by fig. 3a, 3b, 3c which refer to the growth simulation with icrit=2
but with a - - ratio ten times larger than in figures 1 and 2. Clearly the extent of the suppression of islands in the vicinity of the contacts depends on the actual size of the contacts. We will address this point in detail in a later publication. As can be seen, with this choice of parameters only one island is present in the region between the contacts. This is a totally different situation from the experimental structures grown so far [1,2,3]. In experimental structures obtained by means of atomic deposition, a large number of dots is always present between the two contacts [1,2,3]. The possibility of tailoring the growth conditions so that only a single dot is present between the contacts will allow to study Coulomb Blockade in a more controlled way and more importantly will allow to tailor the size of the dot so as to obtain devices showing Coulomb Blockade at higher temperatures. For example once the number of islands present between the contacts
can be predetermined, the growth process can be suitably interrupted so as to obtain dots of the optimum size for Coulomb Blockade devices. 4. CONCLUSIONS In this work we have undertaken a Monte Carlo simulation of the growth process of Coulomb Blockade devices by means of atomic deposition of metallic particles on an insulating surface. The model allovrs to account for different growth conditions such as variation of temperature, deposition rate, and presence of metallic contacts on the surface. The results have been compared with the experimental findings and have shown good agreement in terms of overall density and size of the islands formed. In this work we have concentrated on finding the optimum growth conditions to control the deposition process and obtain reliable Coulomb Blockade devices. The growth conditions for which only a single dot is produced in the region between the contacts have been found. This will allow to study Coulomb Blockade in a cleaner environment and to optimise the growth parameters to obtain devices showing Coulomb Blockade at higher temperatures. The presence of metallic contacts on the surface has been shown to hinder the formation of dots in their vicinity. This could be used as a means to control the deposition process and predetermine the position of the islands. Future work will concentrate on optimising the size of the contacts and on the presence of impurities and defects on the surface. These may well play a crucial role on islands formation, especially in terms of the position where the islands are formed.
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M. Boer• et aL/Microelectronic Engineering 41/42 (1998) 515-518
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