Intermolecular forces - Silicon Needle-like Surfaces for Room Temperature Silicon-Silicon Bondi

The adhesion between two interacting bodies without any intermediate layer at room temperature is generally due to intermolecular forces (e.g., VdW forces, hydrogen bridging, electrostatic forces, and capillary forces) acting between them [32, 108]. In general, VdW forces are dominant over other forces when the environment is dry and the distance between involved bodies is in the range of nanometer [28].

However, in humid environments, capillary forces contribute to the adhesion with a considerable influence and dominate other forces [23]. The following sections briefly introduce these two forces since they are the main intermolecular forces responsible for adhesions in micro- and nano-scales interactions.

A. Van der Waals forces

Van der Waals forces are essentially weak attractive forces existing between nearby electrically neutral molecules. They are initiated by the attraction between electron rich region of one molecule and electron poor region of the other one. Van der Waals forces (𝐹_𝑉𝑑𝑊) between atoms or molecules are the combination of Keesom force (𝐹_{𝐾𝑒𝑒𝑠𝑜𝑚}), Debye force (𝐹_{𝐷𝑒𝑏𝑦𝑒}), and London dispersion forces (𝐹_𝐿𝐷) [191].

𝐹𝑉𝑑𝑊 ≈ 𝐹𝐾𝑒𝑒𝑠𝑜𝑚+ 𝐹𝐷𝑒𝑏𝑦𝑒+ 𝐹𝐿𝐷 (14) The Keesom force is a result of interacting of electric fields of two permanent dipole moments and highly depends on temperature [110]. Certain molecules can develop permanent dipoles moment due to a nonuniform distribution of electrons within their atoms. When two such molecules (polar molecules) are brought close to each other, there will be a dipole-dipole interaction between them (similar to alignment of two magnets) [192]. The Keesom force can be either attractive or repulsive depending on configuration of dipoles. When dipoles are antiparallel, the force is attractive, and when they are parallel, it is repulsive. The force is decreasing with increasing of temperature since thermally induced motions of permanent dipoles disturb their own mutual alignments [192].

The Debye force is arising from interactions between rotating permanent dipoles and from the polarizability of atoms and molecules (induced dipoles). It is always attractive and does not vanish by a temperature increase. The interaction between a

polar molecule and a nonpolar molecule is due to the polarization field, which comes from a polar molecule polarizing another atom or nonpolar molecule [192].

The London dispersion forces are arising due to the non-zero instantaneous dipole moments of all atoms and molecules (fluctuating dipole–induced dipole). They are exhibited by nonpolar molecules resulting from the presence of correlated movements of electrons in the interacting molecules. Dipoles create an electric field, which polarizes nearby neutral atom including temporary dipole moment in it [193]. This polarization forces molecules to attract each other. The London dispersion forces have the largest contribution to van der Waals forces since they are always present in contrast to the other types of forces [194]. They are also the key intermolecular forces, which cause molecules to adhere in liquid or solid, and are responsible for surface tensions and capillary actions. Some of their main features can be described as follows [108]:

a. They are relatively long-range forces and can be effective from large distances (>10 nm) down two interatomic spacing (~ 0.2 nm).

b. They can be repulsive or attractive.

c. Besides bringing molecules together, they can tend to mutually align or orient them as well.

d. They are not additive. It means that the force between two bodies can be affected by the presence of the other nearby bodies.

The van der Waals interaction potential (𝑈) between two molecules is usually presented through the Lennard-Jones potential equation as [108]:

𝑈 = 𝐶_𝑛 𝑟¹²−𝐶_𝑚

𝑟⁶ (15) where 𝐶𝑛 = 4𝜁𝜂¹² and 𝐶𝑚 = 4𝜁𝜂⁶ are interaction potential constants with 𝜁 as depth of the potential well and 𝜂 as the infinite distance at which the inter-particle potential is zero, and 𝑟 is distance between two molecules.

The first term (^𝐶^𝑛

𝑟¹²) represents the repulsion, and the second term (−^𝐶_𝑟^𝑚₆) represents the attraction between two molecules separated by a distance 𝑟. The attractive van der Waals forces are then obtained by taking first derivative of the attraction term of 𝑈 in respect to 𝑟 as:

47 𝐹𝑉𝑑𝑊 = −𝑑𝑈

𝑑𝑟 (16) For macroscopic bodies with known volumes and numbers of atoms or molecules per unit volume, the total attractive van der Waals potential is often computed by summation of all interacting pairs as [195]:

𝑈𝑚𝑎𝑐𝑟𝑜𝑠𝑐𝑜𝑝𝑖𝑐= − ∭𝐶_𝑚𝜌₁𝜌₂

𝑟⁶ 𝑑𝑉 (17) where 𝜌1 and 𝜌2 are molecular densities of two interacting bodies.

B. Capillary forces

Capillary forces are pervasive in nature and are responsible for many macroscopic and microscopic phenomena, such as ascension of liquids through slim tubes [196]

and self-assembly of micro-particles [197]. Capillary forces are fundamentally originated from an interfacial tension. In a finite volume of condensed matter, molecules in the bulk undergo attractive forces from other molecules in all directions.

However, molecules or atoms at the solid-vapor-, liquid-vapor-, or solid-liquid- interface are in an environment different than that in the bulk [198]. This environmental difference results in a surplus energy associated with creation of a unit interfacial area, indicated as interfacial tension or energy. The interfacial energy can also be expressed as an energy by surface unit or as a force by length unit. In a system consisting of solid objects and a liquid medium, it is the tendency of the system to minimize the amount of its interfacial energies. The interfacial energy (𝛾_𝐿𝑉) is defined at the interface between a liquid and a solid (𝛾_𝑆𝐿), and a solid and a vapor (𝛾_𝑆𝑉) through the Young-Dupré equation as [199]:

𝛾𝐿𝑉𝑐𝑜𝑠𝜓 + 𝛾𝑆𝐿 = 𝛾𝑆𝑉 (18) where ψ is the contact angle that liquid forms with a solid.

Eq. (18) itself can be also derived by minimizing sum of interfacial energies for a fixed droplet volume of a liquid and a surface of solid as it is shown in Figure 7.

Figure 7. A liquid droplet on a solid surface to illustrate the Young-Dupré equation (retrieved from [200]).

Although the basis of all capillary forces is minimization of total interfacial tensions, their appearances can be easily varied depending on the system geometry and surrounding environment. As the surface tension, the Laplace equation, the Kelvin equation, and the capillary condensation are relevant to the adhesion based on capillary forces, a brief introduction of them is presented below.

The “surface tension” is related to the concept that the surface is under a tension [195]. In this way, it is macroscopically comparable to a rubber balloon in which a force is needed to increase surface area of its rubber membrane against the tension. In the molecular level, a molecule energetically desires to be surrounded by other molecules and attract other molecules through various interactions (e.g., VdW forces or hydrogen bonds). At the surface, molecules are only partially surrounded by other molecules, and number of neighboring molecules is less than the ones in the bulk (see Figure 8).

This is not energetically desirable for molecules; therefore, to bring a molecule from the bulk to the surface, a work needs to be done. In this prospect, the surface tension can be interpreted as the energy, which brings molecules from inside of the liquid to the surface and creates a new surface area.

49 Figure 8. Schematic molecular structure of a liquid-vapor interface (retrieved from [195]).

In general, due to the surface tension, a pressure difference across the interface between a liquid and a gas is arisen. This pressure difference is the so-called Laplace pressure after Pierre Laplace, who studied it in the beginning of the 19^th century. This pressure difference is associated with curvature of the liquid interface and the surface tension according to the Young-Laplace equation as [201]:

∆𝑃 = 𝛾_𝐿𝑉( 1

𝑅𝑝1+ 1

𝑅𝑝2) (19) where ∆𝑃 is the pressure difference or Laplace pressure over the interface between the liquid and the gas, and 𝑅𝑝1 and 𝑅𝑝2 are principal radii of curvatures (in two normal planes that cut the interface along two principal curvature sections) of the interface between the liquid and the gas.

The Kelvin equation is one of the fundamental equations in the surface science and is based on a thermodynamic principle like the Young-Laplace equation. However, it does not refer to any special materials or conditions. When a liquid surface is curved, the vapor pressure increases due to the Laplace pressure. The elevated Laplace pressure in a droplet causes its molecules to evaporate easily. However, in a liquid that surrounds a bubble, the pressure reduces with respect to inner part of the bubble and causes molecules to evaporate [195]. William Tomson named as Lord Kelvin was the first who realized that the vapor pressure of a liquid governs strongly by the

curvature of its surfaces [202]. His equation describes the actual vapor pressure (𝑃_𝑣) as a function of curvature of the surface of condensed liquid as [203]:

𝑋𝑇𝑙𝑛 (𝑃𝑣

A significant application of the Kelvin equation is the representation of the capillary condensation (condensation of vapor into capillaries or porous mediums even for vapor pressures below the saturation vapor pressure). Once condensation occurs, a meniscus is formed at the liquid-vapor interface, which permits an equilibrium below the saturation vapor pressure [204]. This meniscus formation mainly depends on surface tension of the liquid and shape of the capillary explained by the Young-Laplace equation. Since the liquid-vapor interface involves a meniscus, the Kelvin equation provides a relation for difference between the equilibrium and the saturation vapor pressures as [195]:

𝑋𝑇𝑙𝑛 (𝑃𝑣

𝑃𝑜) = − (2𝛾𝑉𝑚

𝑟𝑐 ) (21) where 𝑟𝑐 is capillary radius at the point where the meniscus is in the equilibrium, and curvature of the liquid surface is constant everywhere (radius of curvature of the

In general, two common approaches are used to calculate attractive capillary forces between macroscopic objects [127]. The first approach is based on the thermodynamic equilibrium in which capillary forces are directly computed from the meniscus geometry obtained by a numerical solution of the Young-Laplace equation

51 or approximated by predefined geometrical profiles, such as circle or parabola [128].

The second approach is based on the thermodynamic non-equilibrium in which capillary potentials are minimized for a given fixed liquid volume [129]. Nevertheless, both approaches have been shown equivalent results in terms of total capillary adhesion forces [126, 128]. In both approaches, due to presence of unknown variables such as radius of curvature of the meniscus, filling angle, and volume of the liquid enclosed, additional expressions have been proposed considering a circular approximation for the curvature of the liquid vapor interface [130–133] and for the numerical computation of the curvature [134, 135]. The circular approximation has been shown a strong validity, where adhering objects and their corresponding liquid volumes and bond numbers were very small. Additionally, in several approaches, a symmetric configuration has been assumed for the liquid bridge by considering equal contact angles [108, 136–138].

3.3 Summary

This chapter provides the theoretical background relevant to anodic etching of silicon, which is the basis for generation of needle-like surfaces using the anodization technique. It also covers the common electrochemical etching setups used for anodization of Si and describes the chemical reactions occurring during this process.

Additionally, it reviews the pore formation models, which are the bases for modelling the formation mechanisms of needles during the anodization.

In addition, it provides the theoretical background and the fundamental equations relevant to van der Waals forces and capillary forces which are the bases for modelling the adhesion between single objects and surfaces at room temperature.

4 | Generation of silicon needle-like surfaces by anodic etching of silicon

This chapter introduces a process to generate silicon needle-like surfaces based on anodic etching of surfaces of lowly doped p-type silicon wafers in an aqueous HF solution in the transition region (region between the pore formation and the electropolishing). A simple model based on pore formation models is also presented to describe formation of needles in the transition region for lowly doped <100> p-type Si wafers anodized in a 7.2 wt.% aqueous HF solution. Additionally, impacts of anodization parameters and substrate properties on morphology of needle-like surfaces and geometry of needles are discussed.

4.1 Deriving of required anodic etching parameters

Physical and chemical processes such as wet and dry etching processes can be used to generate silicon needle-like surfaces. Dry etching processes such as the RIE (black silicon) has a disadvantage of redeposition of used chemical on surface of needles (a thin layer of SiOxFy) over wet etching processes. This thin layer can change the nature of the Si or SiO2 surface, and consequently the direct fusion bonding mechanism between these surfaces. Therefore, in order to generate pure Si or SiO2

needles suitable for the room temperature direct fusion bonding, the anodic etching technique has used to generate silicon needle-like surfaces.

Pore morphology of porous silicon is largely dependent on applied current density, HF concentration of the solution, and substrate type and its doping concentration [206]. Morphology of pores can be varied from sponge-type (porous) to needle-type by employing proper anodization parameters and appropriate substrate materials. In the transition region, self-organized needle-like surfaces can be created through a special dissolution process of lowly doped p-type Si wafers in an aqueous HF solution by widening pores and dissolving side walls (interpore spacing) of macropores in which remaining bulk Si between pores (islets) results in needles (see Figure 9).

Figure 9. Formation of silicon islets: a) top view of randomly distributed macropores with different sizes and b) top view of overlapped and widened macropores where the remaining bulk Si between them results in islets.

Pore diameters and interpore spacings of porous silicon are generally dependent on two groups of factors [27]: i) Ones, which affect carrier density on the surface of a pore bottom (e.g., substrate doping type and concentration, potential, and illumination direction (for n-type only)), and ii) Ones, which only affect distribution of the reactions (e.g., current density, HF concentration, and illumination frequency and intensity (for n-type only)). In addition, both pore diameters and interpore spacings are largely dependent on thickness of the space charge region, except for macro-pores formed on lowly doped p-type Si [169]. In general, pore diameters have the same order of magnitude as thickness of the space charge region, whereas interpore spacings are less than twice the space charge region thickness. Due to overlapping of two space charge regions, interpore spacing regions (walls) are mostly depleted of carriers and are not conductive. If a wall thickness is greater than twice the space charge region, the wall is not depleted of carriers, and thus dissolution will occur to form new pores in the wall. Pore diameters generally increase by increasing current density and by decreasing HF concentration [27, 189]. However, variation of interpore spacings is more complicated than pore diameters. In particular, interpore spacings are dependent on potential and increase by increasing potential at small currents. However, at certain currents, they begin to decrease by increasing potential [27]. Additionally, the growth of individual pores or the dissolution at pore tips is anisotropic and depends mainly on the orientation of the substrate and the direction of the current [207].

Macro-pores on lowly doped p-type Si can be only formed in medium to low HF concentrated aqueous solutions at current densities lower than 𝐽_𝑃𝑆𝑖 [176]. However, at current densities, slightly above 𝐽_𝑃𝑆𝑖 (the transition region), macro-pores with very thin walls in comparison to size of pores will be formed since pores are consistent with

55 respect to size and depth from area to area, as the coverage of PSi is not uniform on the surface [169]. In organic and aqueous solutions, p-type macro-pores grow preferentially along <100> direction and towards the source of holes, regardless of crystal orientation of the substrate (investigated crystalline orientations: (100), (111), (511), and (5 5 12)), only in (111) samples, <100> and <113> oriented macro-pores have been observed [208–210]. Typically, macro-pores are well aligned perpendicular to the surface of (100) substrates but may have a less than 90° angle to the surface of substrates with other orientations [169]. The degree of the orientation dependence of p-type macro-pores is dependent on the electrolyte mixture [208, 209]. Aqueous HF electrolytes or more generally, electrolytes with relatively large amount of oxygen result in heavily branched and wavy pores. Whereas, electrolytes with reduced oxygen and increased hydrogen availability for the interface reactions result in extremely straight and smooth macro-pores. Formation of this kind of macro-pores is viewed as a dimensional transition from pores with small radius of curvatures in an exponential region to a flat surface, which has an infinite radius of curvature. Formation of these large pores is mainly related to the extended coverage of oxides on pore bottoms from the exponential region to the electropolishing region [211]. When current density is larger than a certain value (required current density to start formation of oxide at tip of pores), increasing current density will expands coverage of oxides on pore bottoms.

In this case, according to Eq. 13, the relative change of current density at a pore tip (^∆𝐽^𝑡

𝐽𝑡) with an increase in the current density in the pore (∆𝐽) is less than the one at sides of the pore bottom (^∆𝐽^𝑠

𝐽_𝑠) [207]. Therefore, curvature of the pore bottom increases and results in a larger pore with thinner walls. However, when the current density is relatively small in such a way that the current density at the pore tip is still much less than the required current density for oxide formation, an increase in the applied current density causes a relatively larger increase in the current density at tip than on sides of the pore bottom (^∆𝐽^𝑡

𝐽_𝑡 >^∆𝐽_𝐽^𝑠

𝑠). As a result, the pore bottom becomes sharper by increasing of the current density.

Although most of researchers agree on an Arrhenius type of dependency between J_PS_i and HF concentration and temperature (see Eq. 23) [106, 212–215], there are still few with seemingly contractive data. Lehmann [186] found an exponential relation between HF concentration and J_PS_i for lowly doped p-type Si (< 10¹⁵cm^-3 [177]) in

aqueous solutions, whereas van den Meerakker [216] reported a linear dependency, as are shown in Figure 10.

Figure 10. Critical current density as a function of HF concentrations for lowly doped p-type Si in aqueous electrolytes. Parameters described by Lehmann and by van den Meerakker.

(retrieved from [176]).

While formation and dissolution of oxide depend strongly on the amount of water in the electrolyte, condition for occurrence of the transition region and formation of macro-pores with very thin walls or no walls on p-type Si can be significantly differed by type of the electrolyte. Hence, to form straight needles (perpendicular to the surface of the wafer), the anodization parameters given by van den Meerakker (see Figure 10) and <100> p-type Si wafers were used to derive suitable anodization parameters for varying pore morphology of macro-pores from sponge-type to needle-type. For this purpose, a quarter of a 4-inch <100> Czochralski (CZ) 12 - 17 Ωcm p-type Si wafer with backside sheet resistances of 16 ± 0.3 Ω/□ (achieved by the thermal diffusion of boron dopants) and a 7.2 wt.% aqueous HF solution at a constant temperature of 21

°C were chosen as the starting point of this investigation. To find the transition region for this specific condition, anodic current density-voltage (J-V) characteristic of Si-electrolyte of the Si wafer (see Figure 11) with a circular anodization aperture with an area of 5.7 cm² was obtained through the linear sweep voltammetry method using a potentiostat (Bio Logic SP-150) with a scanning rate of 100 mV/s. For this purpose, a tinny cylindrical platinum (Pt) reference electrode with a radius of 250 µm and a length of 5 mm was positioned 2 mm away from the Si surface. The process was carried out in a double tank electrochemical etching cell (MPSB 100 with electrolyte circulation and cooling capabilities from AMMT GmbH) containing 1.7 liters of the electrolyte. A 4-inch <100> CZ 12 - 17 Ωcm p-type Si wafer with a backside sheet resistance of 16.2

± 0.3 Ω/□, similar to the one which had used to obtain the J-V curve, was then used to experimentally find the pore formation region, the transition region, and the electropolishing region in respect to the obtained J-V curve. The wafer was cut into

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