Macropore formation in p-type silicon: toward the modeling of morphology
© Slimani et al.; licensee Springer. 2014
Received: 25 April 2014
Accepted: 2 October 2014
Published: 21 October 2014
The formation of macropores in silicon during electrochemical etching processes has attracted much interest. Experimental evidences indicate that charge transport in silicon and in the electrolyte should realistically be taken into account in order to be able to describe the macropore morphology. However, up to now, none of the existing models has the requested degree of sophistication to reach such a goal. Therefore, we have undertaken the development of a mathematical model (phase-field model) to describe the motion and shape of the silicon/electrolyte interface during anodic dissolution. It is formulated in terms of the fundamental expression for the electrochemical potential and contains terms which describe the process of silicon dissolution during electrochemical attack in a hydrofluoric acid (HF) solution. It should allow us to explore the influence of the physical parameters on the etching process and to obtain the spatial profiles across the interface of various quantities of interest, such as the hole concentration, the current density, or the electrostatic potential. As a first step, we find that this model correctly describes the space charge region formed at the silicon side of the interface.
Macropores have first been obtained upon electrochemical dissolution of n-type silicon in hydrofluoric acid (HF)-based electrolytes [1–5], in conditions where the current is limited by the supply of photogenerated holes to the electrochemical interface. For this reason, it has been initially thought that no macropores could be obtained on p-type silicon. However, several experimental groups subsequently proved that such a feeling was erroneous and reported on the growth of macropores on p-Si, either in aqueous or non-aqueous conditions [6–17]. Accounting for these observations became a mandatory task for the theoretical models. A noticeable contribution has been performed by Lehmann and Rönnebeck  who tried to extend to the p-Si case the generally admitted model accounting for macropore formation on n-Si, the so-called Lehmann's model . These authors assumed that in the case of p-Si, the electrode is under depletion conditions and that the silicon/electrolyte interface behaves as a Schottky diode. Under these conditions, the hole supply to the interface is limited by the transport across the space charge region (SCR), making the analogy with the n-Si case rather straightforward. In this picture, the holes are focused on the pore tips where the space charge is somewhat thinner due to the 3D configuration of the electric field. Other authors highlight that the coupling of hole transport in the semiconductor and ion transport in the electrolyte could make the semiconductor/electrolyte interface unstable upon dissolution, accounting for the initial formation of pores [20, 21]. Since the approach is based on linear stability analysis, it has some intrinsic limitations in describing the pore development and propagation. It has however been refined by Chazalviel et al., who described the practical case where the electrochemical dissolution is under both charge transport and reaction rate control . From this short overview, it appears that taking into account charge transport in the semiconductor and in the electrolyte is needed for accounting for pore formation but that new modeling tools need to be developed in order to account for the pore development and morphology.
The present work is devoted to developing a model which is able to account for macropore formation. We will first recall the experimental results suggesting that such a model should be able to perform a realistic description of charge transport in conditions where silicon is under depletion conditions. Then, we will present first steps toward the development of a phase-field model for macropore formation. The phase-field method has emerged over the last 30 years as a robust method for the solution of moving boundary problems and has been applied to a wide range of phenomena [23–25], including electrochemical growth [26–29]. However, to the best of our knowledge, no phase-field model for silicon dissolution is presently available. The key ingredient for the phase-field approach is to transform the sharp interface into a spatially diffuse one by the use of an order parameter, which is a scalar field Φ(r) that distinguishes between phases and is hence called phase field. We couple the phase field with the concentrations of two mobile carrier species (holes and ions) and the electric potential in order to construct a minimal model for silicon dissolution. We suppose that the conduction takes place by holes in the semiconductor and by positive ions (cations) in the electrolyte. Both species exist in the whole space, as is the case in any phase-field model; however, their respective concentrations are consistent with the change in conduction mechanism (negligible hole concentration in the electrolyte and negligible ion concentration in the semiconductor).
We present a mathematical analysis as well as numerical simulations of our model and show that it exhibits two new key features that are important for the description of pore formation: at equilibrium, a large space charge region is present on the semiconductor side of the interface, and the application of an external potential yields a nonlinear current-voltage characteristic of Schottky type. Finally, further developments that are needed for a full model of pore formation are discussed.
A (100) electrode was cut from a p-Si single-crystal of 10 Ω cm resistivity. The electrochemical setup consisted of a polystyrene cell and a classical three-electrode arrangement, including a rotating working electrode, a saturated calomel (SCE) reference electrode, and a Pt-wire counter electrode. The electrolyte was prepared from deionized water, ammonium fluoride, and hydrofluoric acid (0.05 M HF, 0.05 M NH4F, and 0.9 M NH4Cl, i.e., total fluoride concentration cF = 0.1 M and pH = 3), which provides slow etching conditions favorable for the elaboration of large-sized macropores .
The macropore sizes found here are somewhat larger than those found in [14, 16, 32] for p-Si electrodes of similar doping, which indeed motivated the choice of using a low fluoride concentration in the electrolyte. As for any kind of porous silicon structure, macropores are observed in the rising part of the I-V characteristics only. This part of the I-V characteristics has been shown to correspond to a situation where the surface is at least partly hydrogen-covered, in contrast to the situation beyond the current peak where a continuous oxide film covers the surface. From an electrical viewpoint, it means that the interface is not yet under blocking conditions for the charge carriers (unlike in the electropolishing regime) and that the main limitation to carrier transport is likely to be the interfacial Schottky barrier, in agreement with the assumptions of Lehmann and Rönnebeck. However, the macropores more easily form in the region close to the potential corresponding to JPS, and their shape exhibits a strong dependency on the exact potential value. It suggests that a realistic model for the charge transport at the interface is a mandatory condition for capturing the important features governing the morphology of the macropores. Such is the major goal of the modeling effort presented in the following.
Our goal is to develop a model that is capable of simulating the formation of macropores. This is a highly complex problem, and in the present contribution, we will limit ourselves to the analysis of a stationary planar interface, which can be treated as a one-dimensional problem. The future developments needed to incorporate the anisotropy of the etching are shortly discussed below.
The problem of macropore formation contains multiple length scales. The electrochemical interface exhibits a Helmholtz double-charge layer of atomic width, a diffuse Gouy-Chapman layer on the electrolyte side of the interface, and an extended space charge region in the semiconductor. For the conditions of our experiments, the space charge region typically has a width of 100 to 500 nm, whereas the width of the Gouy-Chapman layer is a few nanometers. Finally, typical pores are a few micrometers in size.
Clearly, it would be highly demanding to resolve all these dramatically different length scales within the same model. Our main goal is to understand the interplay between hole transport in the semiconductor, the geometry of the space charge region, and macropore formation. Therefore, we will set up a simple phenomenological model that works on the scale of the space charge region but neglects all the microscopic details of the Helmholtz and Gouy-Chapman layers.
Experimentally, is the final stable product for silicon dissolution in HF solution. In the semiconductor, the electrical current which drives the reaction is transported by holes, whereas in the electrolyte, both positive and negative ions contribute to the charge transport. In the present work, we will replace the electrolyte by a simpler ‘effective medium’. That is, we assume that there are only two mobile species: holes (h + ) and the mobile cations (H+). Indeed, if the negatively charged and neutral species are excluded from reaction (1), it can be seen that the net charge transfer from the semiconductor to the electrolyte can be formally described by these two species. The negative charges are supposed to form a fixed ‘background charge’, which is correct for the semiconductor but is an approximation for the electrolyte.
where KB is the Boltzmann constant, T the temperature, C the concentration, V the electrostatic potential, and e the elementary charge. The first two terms in the above expression represent the contribution of the carrier statistics and the electrostatic energy, respectively. The last terms, eWh and eWi, represent a chemical energy difference between the two media (semiconductor and electrolyte), where Wh and Wi are constants with the dimension of an electrostatic potential. These energies can roughly be interpreted as standard levels for each species and determine the concentrations of each type of carrier in the two bulk phases, as will be shown below.
where and are the acceptor concentration in the semiconductor and the counterion concentration in the electrolyte, respectively. The form of this interpolation is motivated by the fact that for carrier concentrations that follow Equation 6, the first and last terms in Equations 4 and 5 both vary linearly with Φ across the interface.
where ϵSi and ϵel are the relative dielectric constants of silicon and the electrolyte, respectively.
where μh,i is the mobility of holes and ions, respectively.
where μr is a rate constant of dimension of cm3/(eV s): the reaction rate is proportional to the product of the concentration of the involved species and to the difference in their electrochemical potentials.
The dopant concentration in the semiconductor . To guarantee the electric neutrality of bulk silicon, the concentrations far from the interface on the silicon side, where Φ = 1, must satisfy .
The concentration of negative ions in the electrolyte. Again, to guarantee neutrality of the electrolyte, we must have .
Since there are no currents at equilibrium, the electrochemical potentials must be constant; in particular, they must be the same in the two bulk media for each species. For example, , where V Si and V el are the electric potentials in the semiconductor and electrolyte, respectively.
Since there is also no reaction at equilibrium, the electrochemical potentials for ions and holes must be the same.
This relation shows that the equilibrium potential at the interface is reached when the energy variation of electrostatic origin compensates exactly the energy variation of chemical origin.
Numerical method and parameters
4 × 10−3 cm2 V−1 s−1
400 cm2 V−1 s−1
6 × 1017 cm−3
Experimental observations strongly suggest that a proper modeling of the pore morphology should carefully take into account the space charge effects (responsible for the lateral size of the pore/wall structures) at the same time as the chemical dissolution aspects, which represents a real challenge. As a preliminary step toward the modeling of pore formation, we have presented here a phenomenological phase-field model that exhibits a space charge region at a planar interface and correctly reproduces both equilibrium and non-equilibrium properties. In the future, the model will be extended to include interface motion in two dimensions. For this purpose, Equation 3 has to be replaced by an evolution equation for the phase field, with a rate of transformation that is proportional to the electrochemical reaction. Anisotropy can then be included by making the rate constant μr depend on the interface orientation following the standard procedures for phase-field models . While this procedure is straightforward in principle, we expect the numerical calculations (in particular, the solution of the Poisson equation) to become quite heavy. Nevertheless, we believe that, in the long term, this type of model provides a realistic perspective for a modeling of the interplay between the geometry of the space charge region and the macropore formation.
AS thanks the Ministry of Science and Higher Education of Algeria for financing her research stay at the PMC laboratory at Ecole Polytechnique (Palaiseau, France) with the goal of realizing this research work.
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