3.1 Diffusion

The diffusion coefficient of a solute impurity in the solid and liquid phases is a fundamental quantity required to characterize mass transport rates. The mechanisms of impurity diffusion in the solid phase are well established. Diffusivity depends exponentially on the diffusion temperature and is generally defined by the Arrhenius equation:

$D=D_0\text{exp}\left[-\frac{E}{k_{B}T}\right]$ (1.1)

(1.1)

Here D is the diffusivity in cm²/s; D₀ is the temperature-independent preexponential factor in cm²/s; E is the diffusion activation energy in kJ/mol; k_B is the Boltzmann constant; and T is the absolute temperature in degrees Kelvin.

By contrast, the mechanisms for diffusion in the liquid phase are not well understood and several theories have been proposed to explain the process that involve local density fluctuations in the liquid, viscosity of the liquid, molecular friction, free volume theory, hard sphere model, vibration models, and entropy correlation [56–59]. However, there are numerous difficulties in accurately measuring diffusion coefficients in liquid metals, primarily attributable to mass transport by bulk motion of the liquid due to natural convection. This is driven by buoyancy forces produced by any temperature or concentration gradients, which can lead to decrease of liquid density with depth. The relative dearth of experimental data for self-diffusion and solute diffusion in liquid metals and alloys compounds the problem of developing a theory that is universally applicable, which can accurately represent existing experimental data and which can predict diffusion coefficients when no experimental data are available.

In terms of the temperature dependence, the Arrhenius equation is still widely used for solute diffusion in liquids. However, the preexponential factor and the activation energy are determined by fitting available experimental data, or are estimated from first-principle simulations and semiempirical correlations [60–75]. Several modified forms of the Arrhenius equation have been proposed to estimate solute diffusivity in liquid metals. One equation is based on numerical combination of equations derived from the fluctuation mechanism and the hard sphere mechanism, which yields the following equation [58,70]:

$D=D_0\frac{d_B}{d_A}\text{exp}\left[-\frac{0.17T_m(16\text{+}{K}_0)}{T}\right]$ (1.2)

(1.2)

Here d_A and d_B are the Goldschmidt diameters of the atoms A and B, respectively; T_m is the melting point of the solute species in degrees Kelvin; K₀ is the atomic valence of the solute species, defined as 1, 2, and 3 for body-centered cubic, hexagonal close-packed, and face-centered cubic structures, respectively.

Another predictive equation given below is based on combining the modified Stokes–Einstein–Sutherland formula with the melting point viscosity model [60,61,72–74].

$D=D_0\exp \left[-\frac{16T_m^{1.07}}{T}\right]$ (1.3)

(1.3)

and

$D_0=2.19\text{×}{10}^{-16}\left[\frac{{\xi }_T{T}_m}{M}\right]\frac{T_m}{V_m^{1/3}{C}_0{\gamma }_m}\exp \left[\frac{16T_m^{1.07}}{T}\right]$ (1.4)

(1.4)

Here ξ_T is a correction factor introduced by the authors; V_m is the molar volume; C₀ is a constant that is approximately the same for all metals (≈0.369 kg^−1/2 m⁻¹ s K^1/2 mol^1/2); γ_m is the surface tension.

These equations have been used to estimate impurity diffusion coefficients in the liquid phase for silicon as a function of temperature for several solute species for which experimental data are sparse or nonexistent [76].

Figs. 1.1–1.3 show the temperature dependence of the diffusivities of selected impurities in solid and liquid silicon and in solid germanium, plotted on a logarithmic scale as a function of the reciprocal temperature using critically assessed data and first-principle calculations to validate the experimental data [32,76–127]. Often the reported experimental values vary widely. For example, the reported activation energies vary from 12.54 to 409.1 kJ/mol (0.13–4.24 eV), while the preexponential factor ranges from 10⁻¹³ to 0.5 cm²/s for the diffusivity of interstitial nickel in crystalline silicon. Similarly, cobalt diffusion in silicon also shows large deviations in the preexponential factor (10⁻⁸ to 10⁻¹³) and the activation energy (35–269 kJ/mol) among the reported data using different measurement techniques [107].

Diffusivity defines the time that is required for a solute impurity to reach thermal equilibrium. At low diffusivities, the solutes are prevented from reaching thermal equilibrium and will exist as quenched, metastable defects at room temperature. High diffusivities prevent the impurity solutes from forming such metastable defects, but can form precipitates on cooling. The transition metals, Co, Ni, Cu, and Pd, are among the fastest-diffusing solute impurities in silicon, with diffusion coefficients as high as 10⁻⁴–10⁻⁵ cm²/s at 800°C. At lower temperature the diffusivity of copper in silicon is even higher compared to other transition metals. Penetration of these impurities in a wafer of 500 μm thickness will take less than 10 seconds. During heat treatment, these metals form the respective silicide precipitates very rapidly on cooling, which then act as a diffusion source. For example, nickel diffuses outward so rapidly that more than half the surface of a 100-mm diameter wafer can be contaminated with haze after heat treatment for 20 minutes at 1050°C. Rhodium is known to form haze on the surface, and although its diffusivity in silicon is unknown, it is likely to be on the order of 10⁻⁶ cm²/s at 800°C, similar to Pd, which also forms haze [9,85]. This is a reason why these metal impurities are detrimental during wafer processing.

3.2 Solubility

The solubility of a solute impurity is the maximum concentration of the solute that can be dissolved at thermal equilibrium in the sample at a given temperature without forming a second phase. It is temperature dependent and is represented by the solidus or liquidus boundaries in the respective phase diagrams. The phase diagrams provide information on the solubility and composition of the phases present in the system and the temperature at which these phases form [128]. In the case of solid silicon, the solubility of impurity elements ranges from parts per billion for the transition metals to few percent for the light elements (Fig. 1.4).

Below the eutectic temperature, the solubility S can be represented by the Arrhenius equation.

$S=S_0\text{exp}\left[S_s-\frac{H_s}{k_B{T}_{eu}}\right]$ (1.5)

(1.5)

Here S₀ is the temperature-independent preexponential factor; S_s and H_s are the solution entropy and enthalpy, respectively; T_eu is the eutectic temperature.

The solubility of an impurity atom differs by several orders of magnitude between room temperature and higher temperatures, such as a typical silicon process diffusion temperature of 1100°C, as well as the solubilities among the elements at the same temperature, as shown in Fig. 1.5 for the transition metals in silicon [9,111,115]. The very steep slopes of the Arrhenius plots indicate drastic reduction in the solubility at room temperature, resulting in precipitation of the impurity within the bulk, or rapid diffusion to the surface of impurities with high diffusion coefficients, where they can form silicide precipitates and haze. Of the 18 transition metals that have been investigated, Fe, Co, Ni, Cu, Pd, and Rh form haze [9,84,85]. If the impurity atoms remain within the bulk volume due to low diffusivity, they can form electrically active complexes with other impurities, resulting in deterioration of the device performance. Fig. 1.6 shows the impact of the concentration of various impurities on p-type silicon solar cell efficiency [95,101,118,129–131]. For example, Cu and Ni can be tolerated in concentrations at the sub-ppm level (~10¹⁴–10¹⁷ atoms/cm³), while heavier metals such as Mo, Ti, or W can degrade cell performance in concentrations as low as 10¹¹–10¹² atoms/cm³ and should be present at sub-ppb levels.

Above the eutectic temperature but below the melting temperature, the solubility is retrograde and describes the change in impurity concentration in the solid. The impurities tend to precipitate on cooling. Review of the phase diagrams shows that metatectic or catatectic phase reactions (solid (S1)→solid (S2)+liquid) occur in more than 50 binary metallic systems of technical interest, including Fe-Mn, Fe-S, Fe-Pu, Fe-Zr, Ag-In, Al-Li, and Mg-Bi, as well as several rare-earth and Si-metal systems [132–144]. Even organic systems exhibit retrograde melting [145–152]. As an example of a binary metallic system, consider the Gd-rich end of the phase diagram of the Gd+Au system shown in Fig. 1.7 [128,135]. This system exhibits a catatectic invariant at 1220°C. An alloy containing 2 at.% Au, when heated, will first melt at 845°C, will resolidify completely above 1220°C, and will begin to melt again at around 1242°C. The reverse process occurs on cooling.

In most common silicon-impurity systems, retrograde melting cannot occur by this mechanism, since these systems do not exhibit a catatectic point [128,142,143]. An alternate pathway is illustrated in the generic silicon-metal phase diagram of Fig. 1.8. The solid solubility of an impurity within the crystal structure increases with temperature and reaches the maximum concentration well above the invariant reaction temperature (eutectic or peritectic), leading to supersaturated solutions. The metal solutes tend to form second-phase precipitates that are distributed throughout the solid matrix upon cooling. Since the supersaturated solutions exist at temperatures above the eutectic temperature, the precipitates form in the liquid phase causing retrograde melting, where local melting occurs upon cooling. The solubility of metal solutes in the liquid phase is much higher than in the solid phase, which results in solid-to-liquid gettering and reduces the concentration of impurities in the bulk.

Many dissolved elements in silicon exhibit such invariant behavior (Table 1.3), although other elements do not (Table 1.4). If supersaturation occurs above the eutectic temperature, second-phase particles can precipitate and can lead to retrograde melting [142,143]. The driving force for precipitation is the reduction in free energy per metal atom [136,137,143] and is given by

$\Delta F_c=k_B{T}_r\left[\frac{C_{max}}{C_{inv}}\right]$ (1.6)

(1.6)

where ∆F_c is the driving force, C_max is the actual solute concentration at the temperature of maximum solubility, and C_inv is the concentration of the solute at the invariant temperature.

As shown in Table 1.3, several silicon-metal systems have large driving forces, especially at low invariant temperature, and are likely to exhibit retrograde melting. A good example is the Si-Ni system shown in Fig. 1.9. It has a large driving force and the invariant temperature (993°C) is 44 degrees higher than the eutectic temperature (949°C).

This phenomenon has been used for purification of metallurgical-grade silicon for use in solar cell applications [153–159], as well as for fabricating silicon nanowires and predictive control of the concentration and distribution of metal impurities in semiconductors [138,139,141–143,160–162]. However, retrograde melting can also lead to deleterious effects such as hot shortness or solidification cracking of welds in stainless steels [134,163–166], as well as diffusion-driven liquid metal embrittlement in susceptible systems [167–171].

4.1 Metal Impurities and Particles

In silicon processing, metal impurities may enter the starting materials from an external source that is directly or indirectly in contact with the melt or the cooling crystal, including furnace parts, growth surfaces, or feedstock [103]. Cu, Cr, Fe, Hf, Mn, Mo, Ni, and Ti have been observed. Metals can also diffuse from crucible walls directly into the Si ingot after crystallization [172–179]. Impurity concentrations can vary in the ingot and between different ingots processed in the same equipment. Therefore, it is critical to control all process parameters and impurity sources to ensure high-quality consistently reproducible material.

Trace metals can be introduced into a sample from containers that have been washed and rinsed in a conventional siphon-type washer as compared with washing in a high-efficiency washer designed to minimize trace metal contamination by first flushing the container with high-purity nitric acid, followed by rinsing with ASTM Type I water [180–182]. As Table 1.5 shows, conventional washing can introduce nearly two orders of magnitude higher levels of trace metal contamination in a process liquid compared with processing in a high-efficiency washer.

Common borosilicate glass has been found to release trace elements such as Na, Al, and B, especially at both extremes of pH, and is unsuitable for use in cleanroom manufacturing and processing activities. A much better but more expensive option is fused silica prepared from the vapor-phase hydrolysis of silicon tetrachloride or tetrafluoride. Fused silica can be cleaned effectively with acids and it can also be used at elevated temperatures. The trace metal contents of different kinds of silica are given in Table 1.6 [182–184].

Polymers are widely used in semiconductor manufacturing. They can also be a source of trace metal contamination and should be carefully assessed for their suitability, especially contamination arising from residual catalysts and additives used in manufacture. The trace element content of some plastics is summarized in Table 1.7 [182].

Particles are generated by human or industrial activities from expended energy that may be mechanical, chemical, thermal, electrical, or radiological in nature. Table 1.8 shows typical sources and sizes of particles from common activities. If the materials involved in these activities are metals or have a large metal content, the generated particles become a source of metal contaminants.

Many modern high-technology processes demand cleanliness of the product, the production environment, and the assembly process. Specifically, they demand an absence of particulate contamination. For example, in semiconductor manufacturing, the size of the contaminant airborne particles that must be filtered is equal to and larger than the feature size on the ICs; particles smaller than the feature size are not big enough to cause a short circuit or other particle-related failure. ICs are multilayered devices, with each layer being extremely thin. The density of IC surface areas compounds the likelihood that stray particles could destroy the entire chip. Controlling or eliminating particle contamination within the production environment is the primary concern of a semiconductor manufacturer.

The pharmaceutical industry commonly manufactures parenteral drugs. Parenteral (injectable) drugs must be free of particles (metallic and nonmetallic) and impurities that could infect the body—either human or animal [185]. Particles that can negatively affect the body tend to be larger than 2.0 or 3.0 μm and the pharmaceutical industry, like the semiconductor manufacturer, must manage the production environment to eliminate particle contamination. Typically, pharmaceutical companies determine process cleanliness by monitoring 0.5-μm particles and determine product sterility by monitoring 5-μm particles. In contrast, semiconductor manufacturing tends to concentrate on particles from 0.05 to 0.3 μm.

A major source of contaminant particles is the manufacturing process. Even in the highest quality cleanroom environment in nonoperational mode, particles will always be present, particularly nanosize particles. The current cleanliness specification for an ISO Class 1 cleanroom allows 10 particles of 0.1 µm (100 nm) size per cubic meter of air and 1200 particles of 10 nm size per cubic meter of air in the facility, regardless of the chemical nature of the particles [46,47]. Manufacturing operations performed in the cleanroom will change the environment relative to the number, size, range, and the physical and chemical characteristics of the particles. For example, the tools used to assemble a product are fabricated from materials that will wear with use. Also, the presence of people in the assembly process in a cleanroom or other controlled environment is itself a source of contamination. Particles are generated from human activity by shedding skin cells, emitting perfume/colognes/hairsprays, losing hair, breathing, and sneezing, among other activities. These particles contain elemental chemicals (Table 1.9) that can cause deleterious effects such as corrosion of the product [39].

Incoming parts used to assemble the final product can contribute significantly to particle contamination. Typical surface defects that are sources of particles include burrs, roughness, flaking, porosity, inclusions, and loose debris. There is an ongoing effort throughout the industry to eliminate the sources of these defects from incoming parts, but in many applications it is physically or financially impossible to eliminate these sources entirely.

Moving parts in a product create a tribological system that can generate metal particles due to friction and wear. Both these phenomena depend on many different factors, including surface and intrinsic properties of the contacting surfaces, mechanical and physical properties, and microstructural characteristics. At the macroscale, the laws of friction and wear are well established. Typical mechanisms generating particles are abrasive, erosive, adhesive, fatigue, plastic flow, corrosive, diffusive and melting wear, several of which can occur simultaneously in the contact region. At the nanoscale, friction and wear behavior of materials is very different. For example, the frictional force between two nanoscopically smooth surfaces depends on the area of contact of the surfaces. If the area of contact is sufficiently small, the load can be increased without increasing friction, resulting in less wear and reduced particle generation. This phenomenon can also be facilitated by nanoscale additives [186–188]. This is significant for many interfaces of technological importance such as microelectromechanical systems (MEMS) and hard disk drives with flying heights of 1–5 nm. The tribological implications of particles have been reviewed recently [189].

Another source of metal contamination is the growth of whiskers, which affects the reliability of a wide range of products from cardiac pacemakers and watches to missile systems and satellites, and have even resulted in the shutdown of a nuclear reactor [190–195]. The general reliability risks are short circuits, low-pressure-induced metal vapor arcing, and debris contamination, which can interfere with the performance of sensitive components. Whiskers are hairlike, single-crystal structures of various metals (such as Al, Au, Ag, Cd, In, Mo, Sb, Sn, Ta, W, and Zn), alloys, and metal oxides, which grow spontaneously from the surface of the material (Fig. 1.10). They can grow as kinked, bent or striated needles, or may take odd shapes. Whiskers are typically 1 nm to 30 μm in diameter, but they can grow from 100 μm to 10 mm in length. The primary mechanism for whisker formation is attributed to mechanical stresses on the surface that are relieved by whisker growth. Recently, an alternative mechanism has been proposed in which whisker formation is attributed to the energy gain due to electrostatic polarization of metal filaments in the electric field. The field is induced by surface imperfections such as contamination, oxidation states, and grain boundaries [196].

4.2 Ionic Contamination

Ionic contamination refers to the presence of undesirable cationic (such as Na⁺ and K⁺) and anionic (such as Cl⁻, F⁻, Br⁻, ${\text{SO}}_3{}^{-}$, ${\text{BO}}_{\text{3}}{}^{3-}$, and ${\text{PO}}_4{}^{3-}$) ions from production processes (e.g., fluxes used during soldering, plating, chemical mechanical polishing, and photolithography), human activity, environment, and materials and chemicals that come in contact with the manufactured product. The ions can cause chemical, electrochemical, or galvanic corrosion of a part.

The sources of ionic contamination are varied. In general, ionic contamination can be classified as coming from the processes, people and objects that come in contact with the manufactured product. Ionic contamination includes ions in chemicals, ions deposited during handling, ions from the air, ions from cleaning products and equipment, and ions from packaging materials. For example, isopropyl alcohol used for cleaning may contain enough water to transfer the ionic contaminant to the part from the medium used for cleaning.

Some of the most common sources of ionic residue in the manufacture of printed wiring boards include plating chemicals, flux activators, human perspiration and fingerprints, component packaging materials, ionic surfactants and alkaline cleaning solutions, and ethanolamines used in cleaning products. In the presence of moisture, ionic residues can dissociate into either negatively or positively charged species and increase the overall conductivity of the solution, resulting in corrosion or metal deposition in the form of dendrites [197–200].

6.1 Metal Particles and Trace Metal Impurities

From a contamination perspective, nanosize metal particles can have major impact on the performance of precision and other products. For example, the presence of undesirable trace metal impurities such as nickel or iron in the parts per billion range can result in irrecoverable loss of the entire production lot of semiconductor wafers or degraded performance of the device fabricated from the wafers.

To develop mitigation and remediation strategies for small particle contamination, it is necessary to resolve the particles by their sizes [202]. Discrete particles can be generally classified by size according to Table 1.13.

The challenges associated with accurately determining atomic positions at the nanoscale are formidable because atom positions must be known to very high precision, of the order of 0.01–0.001 Å (1 Å=10⁻¹⁰ m) needed for theoretical calculations of electronic structure and the functional properties of nanostructured materials [278–280]. A major advance in this direction has been to demonstrate measurement of atomic positions at a precision of 0.04–0.06 Å in aberration-corrected scanning transmission electron microscopes [281–284], which can be combined with statistical parameter estimation theory to determine atomic structural information for atoms and crystalline nanoparticles [285–287]. Similar levels of precision of atomic positions have been demonstrated by neutron holography [288].

In referring to very small particles, the size of the particles can be discussed in terms of various physical phenomena. For example, the interactions of particles much larger than 1 μm diameter are dominated by gravitational forces, while van der Waals and other forces tend to dominate their interactions below that size. Particles with diameters of 0.3–0.7 μm are of the same size as the wavelength of visible light, which is the limit of resolution in optical microscopic observation of particles in that size range. Particles in the size range 20–100 nm are referred to as ultrafine, while nanosize particles have diameters smaller than 20 nm. Due to the need to understand aerosol behavior, two additional classes of particle sizes have been defined. Very small particles refer to particles smaller than 5 nm, while molecular size defines particles with diameters smaller than 1 nm [289].

The physical nature of very small particles cannot be thought of in terms of classical surface or volume continua. Rather, the molecules statistically associated with the particle will tend to define their interactions. As Table 1.14 shows, the number of molecules associated with a particle increases with increasing particle size, but the fraction of the molecules at the surface decreases with increasing particle size. For a particle of 20 nm diameter, the number of molecules at the surface is only about 12%. Consequently, the overall behavior of nanosize particles is governed by the surface and binding energies of the molecules in the particle. These particles are neither solid nor liquid, and do not behave like individual molecules. The particle may be regarded as a complex structure whose behavior depends on the positions of the individual molecules and the combined electronic charge distribution [289].

Only simple interactions of nanosize particles are presently understood. Due to the technological importance of very small particles in advanced materials and nanotechnology, advances in kinetic theory, solid state chemistry, quantum mechanics, and aerosol dynamics will make it possible to predict the behavior of very small particles in complex systems in the future. In turn, it will provide the basis for understanding the transport, adhesion, and detachment of these particles in real systems, as well as making it possible to design materials with specific properties.

The understanding of gas-phase reactions in the formation of particles has advanced considerably, but important chemical and physical mechanisms are still unresolved, particularly for particles in the size range 1–10 nm. As noted earlier, such particles have to be considered as complex structures whose properties and interactions are determined by the number and position of the molecules on the surface. In fact, the interactions of these particles will be largely between the molecules at the surface, which are, in turn, determined by atomic, electronic, and nuclear motions. The application of advanced techniques such as ultrafast characterization using femtosecond or even attosecond laser pulses [290–294] will help resolve many of the gas-to-particle conversion questions. Other techniques for characterizing small particles have been reviewed recently [295].

6.2 Ionic Contamination

Devices such as printed circuit boards, semiconductor wafers, MEMS, data storage components, medical devices, automotive and aerospace electronics, and flat-panel displays are the most susceptible to ionic contamination as they have electronic circuitry that in the presence of ions and moisture can develop conductance paths resulting in a variety of electronic failure modes. These failure modes can degrade the performance of the device or cause the device to fail completely.

Fig. 1.11 shows an example of dendrite formation on a printed circuit board and a base metal electrode capacitor [198,200].

As mentioned previously, dendrites can grow on electronic parts such as printed circuit boards and capacitors (Fig. 1.11) and cause failures by short circuits [197,198,296]. Dendrites can be silver, copper, tin, lead, palladium, or a combination of metals, and their growth can be very rapid; failures have been known to occur in less than 30 minutes, but can take several months or more [198]. Corrosion of the interconnect lines can also lead to failures of the devices. The amount of contamination required for dendrites to form can be extremely small and stringent cleanliness requirements must be met (Table 1.15) [297]. Migration of metallic ions and dendrite growth has been observed at voltages as low as 0.5 V and 40% relative humidity [197].

This type of failure mechanism has resulted in pacemaker failures and subsequent extensive recalls for more than one manufacturer. For those electronic medical devices that operate in a high-humidity environment, such as implants, it is especially important that all ionic contaminants are removed. Ionic contamination can occur through cleaning, plating, etching, handling, fluxing, or soldering operations, as well as from airborne sources [298].

Data storage products such as disk drive components are susceptible to ionic contamination [197]. Microcontamination and subsequent corrosion of the recording head is of particular concern. The heads are composed of nickel/iron alloys that are highly susceptible to corrosive attack by anions and cations such as chloride, sulfate, and sodium. The primary mechanism is gas-phase transfer of volatile inorganic species. As in semiconductor processing, the source of the ions may be materials that are used during fabrication, ionic surfactants, aqueous rinse solutions, coatings that are used on the disk surface, packaging materials, human contact, the component assembly environment, solvents, adhesives, and lubricants.

Similar phenomena may be encountered in the manufacture of nonelectronic medical devices that are cleaned using alcohol, fluorinated, and/or chlorinated cleaning processes. Such parts include bone implants, catheters, syringes, disposable blood filters, oxygenators, dental tools, surgical tools, optical lenses, and heart valves.