Anodic etching of a single crystalline silicon (c-Si) in a HF based solution can be resulted in a sponge-like material so-called porous silicon (PSi), a needle-like surface, or electropolishing of the surface depending on the used materials and conditions [169]. The accidental discovery of PSi was reported for the first time in 1956 by Uhlir as a result of investigation of electropolishing of silicon in HF solutions [170]. Etching of silicon in a HF based solution, and thereby growing of pores with different sizes can be classified into three groups in respect to their pores diameters according to the International Union of Pure and Applied Chemistry (IUPAC) as [171]:
• Macro-pores (> 50 nm)
• Meso-pores (2 - 50 nm)
• Micro-pores (< 2 nm)
In the dissolution process, a silicon wafer serves as an anode while a HF resistant conducting material (e.g., platinum or gold) is used as a cathode. Although there are still some debates concerning the surface dissolution mechanism during the electrochemical etching process, it is commonly agreed that holes are needed to initiate the dissolution mechanism [172]. The necessary holes are already present in p-type crystalline Si since they are majority carriers; however, they need to be supplied
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for n-type Si since they are minority carries. Illumination of samples during anodization is the most common approach to generate additional electron hole pairs in n-type Si substrates [173].
Anodic etching of Si is typically carried out in a chemical etching cell made of Teflon (highly HF resistive material) in either single tank cell arrangement or double tank cell arrangement [174] as they are shown in Figure 2. In both arrangements, the Si wafer is used as an anode to acquire electrons from a HF based solution while a platinum or a gold plate is served as a cathode to supply electrons to the solution [175]. In the single tank cell arrangement (see Figure 2a), only front side of silicon wafer is in contact with the solution, while in the double tank cell arrangement (see Figure 2b), both sides of silicon wafer are exposed to the solution.
Figure 2. Typical electrochemical etching cells for anodic etching of Si. a) single tank cell arrangement and b) double tank cell arrangement.
In double tank cells, entire solution is polarized by applying a voltage or a current to platinum or gold electrodes. The rear side of Si wafer plays as a secondary cathode while its front side plays as a secondary anode. The etching occurs at its anodic side with the negatively polarized HF solution. However, in single tank cells, entire Si wafer plays as anode and is connected to the positive terminal of a galvanostat or a potentiostat. In both arrangements, the current flow between the cathode and the silicon releases free holes from the surface to the electrolyte. Arrival of a hole at Si-electrolyte interface leads to breaking of a Si-Si bond and injection of an electron under certain circumstances, e.g., by ripping off a second bond [176]. In the case of anodic
a) b)
37 etching of Si in an aqueous HF solution, following reactions are possible to occur [27, 177]:
Si + 2Fad− + 𝑥ℎ+ → SiF2+ (2 − 𝑥)𝑒− (𝑥 < 2) (6) Si + 4OHad− + 𝑥ℎ+ → SiO2+ 2H2O + (4 − 𝑥)𝑒− (𝑥 < 4) (7) SiO2+ 6HF → H2SiF6+ 2H2O (8) SiF4+ 2HF → H2SiF6 (9) where OHad− and Fad− are adsorbed ions out of the electrolyte, ℎ+ and 𝑒− are representing a hole and an electron respectively, and x is a coefficient (valance number).
All above reactions may proceed on the Si-surface and may compete in rate and coverage of the surface area. The reaction described by Eq. (6) leads to direct (divalent) dissolution of silicon and requires two electrons transfer (valence number of 2). However, the reaction described by Eq. (7) leads to oxide formation and involves four electrons transfer (valence number of 4). The formed silicon oxide is not stable in HF-based solutions and will dissolve according to the reaction described by Eq. (8).
This reaction is purely a chemical reaction process and does not involve any electron transfer. The combination of reactions described by Eqs. (7 and 8) explain the indirect (tetravalent) dissolution of silicon via oxidation [178]. Oxide free surfaces are fluorine terminated and tend to be totally covered by hydrogen (hydrogen termination (the reaction based on Eq. (9)) if no other reaction take place). Hydrogen-terminated Si surfaces are then passivated against further dissolution in the same way.
Pore formation during anodic etching of Si is best characterized by current density-voltage (J-V) curve of the process. A typical J-V curve for a p-type doped Si in a HF solution is illustrated in Figure 3. Once a potential is applied to the Si in a HF based solution, an external current will flow through the system. For any current to pass through the Si/electrolyte interface, the electric current must be transformed into an ionic current. This means that a specific redox reaction needs to occur at the silicon interface. The J-V curve of Si in a HF-based solution is very similar to current-voltage (I-V) curve of a normal Schottky diode with few differences [179]. First, the chemical reaction at the Si/electrolyte interface remains the same, irrespective of the change in sign of majority carriers between n- and p-type. Second, reverse bias dark currents
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are at least three times larger than in the normal Schottky diodes. Under a cathodic polarization (reverse bias), Si is stable, and the main cathodic reaction is the reduction of water at the Si/electrolyte interface with contemporary formation of hydrogen gas (hydrogen evaluation). This happens only at high cathodic overpotentials or at reverse bias according to the Schottky diode terminology. Dissolution of silicon only occurs under the anodic polarization (forward bias) and goes toward the electropolishing at high anodic overpotentials. Pore formation arises during preliminary increasing part of the J-V curve, where the applied potential is below the potential of the first current density peak (𝐽𝑃𝑆𝑖). Whereas, electropolishing occurs where the applied potential is above the potential of the 𝐽𝑃𝑆𝑖 and the entire surface starts to oxidize and goes beyond the second current density peak (𝐽𝑂𝑥𝑖𝑑𝑒). Between the pore formation and the electropolishing regions, there is a very narrow region, which is called the transition region [180]. In this region, both pore formation and electropolishing can occur at the same time and compete for control over morphology of the surface.
Figure 3. The current density-voltage (J-V) curve of a p-type silicon wafer in a HF based solution and positions of current density peaks.(Retrieved from [181]).
Various models have been suggested to describe formation of porous silicon and its pore morphology in different length scales and at different growth stages. These models are generally divided in three groups [182]: i) models for pore nucleation, ii)
39 models for stationary pore growth, and iii) models explaining whole pore growth process.
A mathematical representative of the pore nucleation model is proposed by Knag and Jorne [106, 183] where nucleation of pores at a silicon substrate is mathematically treated as an occurrence of instability of a planar surface towards small perturbations.
The predication of the model is that the distance between pores varies by square root of the applied potential similar to width of the solid-state junction (space charge region length (𝑆𝐶𝑅𝐿)) as [183]:
𝑆𝐶𝑅𝐿 = √2𝐾𝜀(𝑉𝑏− 𝑉𝑎)
𝑒𝑁𝐴 (10) where 𝑉𝑏 and 𝑉𝑎 are built-in and applied potential at zero current, respectively, 𝑁𝐴 is acceptor concentration (1 × 1015 atoms/cm³ for lowly doped p-type Si), K is dielectric constant (11.8), 𝜀 is permittivity of free space (8.85 × 10−14 farad/cm), and 𝑒 is elementary charge (1.602 × 10−19 C). Assuming the electrolyte as a heavily-doped silicon and acting like a metal, the built-in potential (𝑉𝑏) can obtained from [184]:
𝑉𝑏 = −𝐸𝐺 186] is the most accepted one in the porous silicon community. The Lehmann model is centered on the 𝑆𝐶𝑅𝐿 and considers the critical current density (JPSi) as the major parameter to govern pore geometry and morphology. The model relates the pore
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The model assumes that pore walls are passivated against holes due to presence of a space charge region (SCR). However, pores are freely adjusting themselves to diameters and distances to obtain 𝐽𝑃𝑆𝑖 in all active areas (pore tips) during anodic etching. If pores are pre-structured on a Si substrate, all of them will do the same. If not, randomization will occur, where the average distance between pores (interspacing) is determined by twice the 𝑆𝐶𝑅𝐿, and pores diameters are adjusted accordingly [177]. According to the model, in lowly doped p-type Si, SCR is not fully depleted of holes at zero bias and under forward conditions. Hence, a diffusion current (𝐼𝑑𝑖𝑓𝑓) exists at pore tip and walls resulting from concentration gradient of holes, according to the Schottky’s theory [186]. At a thermal equilibrium and zero applied bias, 𝐼𝑑𝑖𝑓𝑓 is compensated by a field current (𝐼𝑓𝑖𝑒𝑙𝑑) as it is shown in Figure 4. Absolute values of 𝐼𝑑𝑖𝑓𝑓 and 𝐼𝑓𝑖𝑒𝑙𝑑 at the pore tip are larger compared with the ones at pore walls since the concentration gradient of holes and the electric field strength increase by decreasing of 𝑆𝐶𝑅𝐿. A SCR is thinner at curved surfaces in comparison to a flat surface. When a forward bias voltage is applied, 𝐼𝑓𝑖𝑒𝑙𝑑 decreases while 𝐼𝑑𝑖𝑓𝑓 increases.
Hence, higher absolute current values at the pore tip become decisive, and total current at the pore tip (𝐼𝑡𝑖𝑝 = (𝐼𝑑𝑖𝑓𝑓 – 𝐼𝑓𝑖𝑒𝑙𝑑)𝑡𝑖𝑝) becomes larger than the total current at the pore walls (𝐼𝑤𝑎𝑙𝑙 = (𝐼𝑑𝑖𝑓𝑓 – 𝐼𝑓𝑖𝑒𝑙𝑑)𝑤𝑎𝑙𝑙). Consequently, a depression in the Si surface grows faster than a planar area; a pore tip develops. If the distance between two neighboring pores becomes smaller than 2 × 𝑆𝐶𝑅𝐿, the pore wall then becomes passivated due to the depletion of holes in the SCR [186].
41 Figure 4. Left: equilibrium (V = 0), field and diffusion currents across the SCR on a macro-pore tip and wall regions in a p-type Si. Right: field and diffusion currents under forward bias (V > 0). Note that due to geometric field enhancement around the pore tip, tip currents are always larger than the pore walls currents (retrieved from [186]).
The model proposed by Beale et. al [187, 188], is one of the pioneer models in providing a quantitative estimation of pore geometry of meso-pores porous silicon. The model considers that a Schottky-type barrier at the Si-interface detained at reverse conditions can be overcome by a breakdown. It also considers that the electron transport through the barrier is performed by the tunneling in heavily doped Si and by the Schottky emission in moderately and lowly doped Si. The Beale model assumes that pore tips are semispherical, and the current preferentially flows at pore tips due to concentration of the electric field since the barrier height is lower at pore tips.
Distribution of the electric field at different regions of a pore and explanation of localized dissolution of silicon are described by the Zhang model [27, 169, 189]. The model describes the pore growth through two competing processes (the indirect dissolution and the direct dissolution of silicon) where the probability of occurrence of one or another process is highly dependent on the applied current density and the potential. The model states that for a steady growth of porous silicon, the reactions described by Eqs. (6 - 9) and the dissolution rates are different at pore walls and pore bottoms (see Figure 4). The reactions and dissolution rates are also different at every position of a pore bottom due to the difference in radius of the curvature. The current at the pore tip is the largest (since radius of the curvature is the smallest at tip) and decreases from the pore tip to the pore wall as radius of the curvature increases. Since
I
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the reactions are dependent on the current density, for a given condition, the direct dissolution of Si (Eq. 6) dominates at relatively low current densities, whereas the indirect dissolution (Eqs. 7 and 8) dominates at higher current densities. Therefore, the indirect dissolution tends to occur at pore tips at a lower potential than at the side of the pore bottom. For a pore to propagate under a steady state, the current density at sides of the pore bottom (𝐽𝑠), and the current density at the pore tip (𝐽𝑡) have to follow the following relation [169]:
𝐽𝑠 = 𝐽𝑡𝑐𝑜𝑠 𝜑 + 𝐽𝑏 (13) where 𝐽𝑏 is extra current density exists at side of a pore bottom and is responsible for formation of side pores, and 𝜑 is a polar angle between the axis of the pore and the radius vector placed to some points at the pore surface [106].
The current density at different sites on the pore bottom is mainly dependent on the angle φ (see Fig. 5). The current density is the largest at the tip (φ =0) and is the smallest at the boundary of the bottom (φ = 90°). For porous silicon formed by a controlled 𝑆𝐶𝑅𝐿, the actual wall thickness is dependent on the relative dissolution rates between edges of the pore bottom due to 𝐽𝑏 and the tip due to 𝐽𝑡. If 𝐽𝑏 is very small compared to 𝐽𝑡, the pore tip propagates relatively fast, and edges of the pore bottom turn into the wall regions due to lack of carriers and stopping of the dissolution before much dissolution occurs at edges of the pore bottom. However, if 𝐽𝑏 is comparable to 𝐽𝑡, substantial dissolution occurs at edges of the pore bottom before the pore tip propagates further and results in a thin walls or even no walls at all (overlapped or widened pores) [189].
43 Figure 5. Current variation and coverage of silicon oxide on surface of a pore bottom (retrieved from [169]).
A further developed pore growth model considering stochastic nature of pores geometries and morphologies (the current burst model (CBM)) is proposed by Föll et al. [176]. The model assumes that there are definite correlations between two different processes (the direct dissolution and the indirect dissolution of silicon) during anodic etching of silicon. The model assumes that [176]:
- Charge (electron) transfer, and thereby current flow is always inhomogeneous in space (x, y) and time (t), meaning that there are times when no charge is transferred in some areas. A charge transfer process nucleates at (x, y, t) on the Si surface (or through a thin oxide) with a certain probability, which depends mainly on the surface state at (x, y, t). The sequence of events started in this way is called “a current burst” (CB).
- The sequence of events in a CB is logically determined and started with the direct dissolution, followed by oxidation, followed by oxide dissolution, and if a new CB does not immediately nucleate at the same spot, followed by hydrogen passivation.
- Individual CBs may interact in space and time. This means the nucleation probability of a current burst is not only a function of the surface state S (x, y, t). It may also depend on what has happened before at (x, y) (interaction in time) or on what is going on in the neighborhood at (t) (interaction in space).
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The CB model describes formation of micro-pores due to anti-correlation of current bursts in time (interaction in time) and formation of meso- and macro-pores due to positive correlation of current bursts in time (clustering of current bursts (interaction in space)).
According to the model, under conditions where current is relatively large (near to
JPSi), and the direct dissolution is dominant, each individual CB generates a nm-sized pore by the direct dissolution and oxide removal [178]. Anti-correlation of CBs in time is then resulting in formation of micro-pores (see Figure 6a). This means wherever a CB is occurred, it is less likely that a new CB nucleates on the same spot in a short time after. On the other hand, a new CB nucleates somewhere else (most likely between some former CBs where the oxide was thinnest) [178]. When the current exceeds JPSi, CBs begin to correlate positively in time. This means that the likelihood of nucleation of a new CB in the place where an old one had occurred, is increased, and CBs begin to cluster. The clustered CBs are then resulting in formation of either meso-pores or macro-pores (see Figure 6b and c) depending on density of CB events in a particular area, correlation in space [190].
a) Micro-pores b) Meso-pores c) Macro-pores Anodization time
Figure 6. Formation of pores due to correlation of current bursts in time. Time is increasing from left to right, black dot: active current bursts and hallow circles: inactive current bursts, which dissolve oxide bumps (retrieved from [178]).
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