**3. Dynamic characteristics of each stage of epitaxial growth**

#### **3.1 The process from vapor to adatoms**

In this section, the factors controlling the early growth of thin films on the substrate are described from the perspective of atomism. This process starts with a clean surface of the substrate, which at a temperature of *Ts* is exposed to the vapor of the chemically compatible film material, which is at a temperature of *Tv*. In order to form a single-crystal film, the film material atoms in the vapor must reach the substrate surface, adhere to it, and locate a possible equilibrium position before the structural defects remain in the growth front. On the other hand, in order to form amorphous films, it is necessary to prevent atoms from reaching the growth surface to obtain a stable equilibrium position. In both cases, this must occur in a more or less identical way over a very large area of the substrate surface for structural development. At first glance, the result seems unlikely, but such films are made as usual. The atoms in the vapor touch the surface of the substrate, where they form chemical bonds with the atoms in the substrate. The idea that the temperature of the substrate must be sufficiently low so that the vapor phase is in a sense supersaturated with respect to the substrate will make the following more specific. In the process of adhesion, energy is reduced due to the formation of bond. As shown in the diagram in **Figure 3**, if the energy generated due to thermal fluctuation is enough to overcome the adhesion energy occasionally, some parts of the adhesion atom (called the adsorption atom) can be returned to vapor by evaporation.

**Figure 3.** *Schematic showing the atomistics of film formation on substrates.*

In order to make the discussion more specific, we assume a simple hexagonal closepacked crystal structure to facilitate the calculation of bonding. It has been recognized that in order to make the film growth possible, the vapor contacting the growth surface must be supersaturated with respect to the substrate at the substrate temperature *Ts*. For a uniform crystal at a certain temperature in contact with its own vapor at the same temperature, the equilibrium vapor pressure *pe* of the system is defined as the pressure under which the vapor atoms condense on the solid surface and the atoms evaporate from the surface at the same rate. In equilibrium, the entropy free energy of each atom in the vapor is equal to the free energy of each atom in the crystal. Due to the effect of chemical bond, the lower internal energy of atoms in the crystal is offset by the lower entropy energy in the crystal. For the net deposition of the substrate surface, the pressure *p* in the vapor must be higher than the equilibrium vapor pressure *pe* at the substrate temperature, that is, the vapor must be supersaturated. For pressure *p*, the entropy free energy of each atom in the vapor, which is higher than the pressure *pe*, is estimated as the work of each atom required to increase the vapor pressure from *pe* to *p* at a constant temperature. According to the ideal gas law, *kTvlnp=pe* where *<sup>k</sup>* = 1.38 � <sup>10</sup>�<sup>23</sup> J/K = 8.617 � <sup>10</sup>�<sup>5</sup> eV/K is Boltzmann constant, *Tv* is the absolute temperature of the vapor. If the vapor becomes supersaturated, there is a difference of free energy between the vapor and the crystal, which provides the chemical potential of the interface driving the interface toward the vapor. As the interface progresses in a self-similar way, some distant layers of vapor of the same mass are transformed into inner layers of the same mass. When these energies are the same, the interface will not move forward or backward.

In thin film deposition, because the vapor phase and the substrate are not the same material phase, and the temperature of the substrate is usually lower than that of the vapor phase, the situation is often complex. In this case, the definition of equilibrium vapor pressure is not clear. However, in most cases, when the vapor pressure is lower than the equilibrium vapor pressure, the film material will not deposit on the growth surface, which is an operational definition of *pe* in the film growth process. Once atoms are attached to the substrate, their entropy free energy will be reduced from that of the vapor. These atoms form a two-dimensional vapor distribution on the substrate surface. The deposited material will be rapidly heated to the substrate temperature *Ts*. On the surface of the crystal, there are some atoms whose density *ρad* is balanced with a straight surface step or wall frame adjacent to some monolayer atoms, that is, the step does not move forward or backward. Step is the boundary between two phases in a homogeneous material system. Therefore, the concept of equilibrium vapor pressure or equilibrium vapor pressure density *ρ<sup>e</sup> ad* can be called in a similar way. In order to realize the film growth by condensation, the free energy of each atom in the two-dimensional gas must exceed the free energy of the fully entrained surface atom, the amount of which is:

$$
\varepsilon\_{phase} = kT \ln \left( \rho\_{ad} / \rho\_{ad}^{\epsilon} \right) \tag{2}
$$

Condensation is just a special case of adsorption in which the substrate composition is the same as that of the adsorbate. This is sometimes the case in thin film deposition and sometimes not. In either case, the molecule accelerates down the curve of the potential well until it passes the bottom and is repelled by the steeply rising portion, which is caused by mutual repulsion of the nuclei. If enough of the molecule's perpendicular component of momentum is dissipated into the surface during this interaction, the molecule will not be able to escape the potential well after being repelled, though it will still be able to migrate along the surface. This molecule is trapped in a weakly adsorbed state known as physical adsorption or physisorption. The fraction of approaching molecules so adsorbed is called the trapping probability, δ, and the fraction escaping (reflecting) is (1 � δ) as shown in **Figure 4**. The quantity δ is different from the thermal accommodation coefficient, *η*, which was defined by

*Molecular potential energy diagram for evaporation and condensation.*

*Adsorption processes and quantities. a, is used only for condensation (adsorption of a material onto itself). A*

*<sup>η</sup>* <sup>¼</sup> *Trs* � *Trh Trs* � *Th*

(3)

Eq. (3):

**9**

**Figure 5.**

**Figure 4.**

*vertical connecting bar denotes a chemical bond.*

*Growth Kinetics of Thin Film Epitaxy*

*DOI: http://dx.doi.org/10.5772/intechopen.91224*

Consider a molecule approaching a surface from the vapor phase, as shown in **Figure 4**. Upon arriving within a few atomic distances of the surface, it will begin to feel an attraction due to interaction with the surface molecules. This happens even with symmetrical molecules and with inert gases, neither of which has dipole moments. It happens because even these molecules and atoms act as oscillating dipoles, and this behavior creates the dipole-induced-dipole interaction known as the Van der Waals force or London dispersion force. Polar molecules, having permanent dipoles, are attracted more strongly. The approaching molecule is being attracted into a potential well like the one that was illustrated in **Figure 5** for condensation.

#### **Figure 4.**

In order to make the discussion more specific, we assume a simple hexagonal closepacked crystal structure to facilitate the calculation of bonding. It has been recognized that in order to make the film growth possible, the vapor contacting the growth surface must be supersaturated with respect to the substrate at the substrate temperature *Ts*. For a uniform crystal at a certain temperature in contact with its own vapor at the same temperature, the equilibrium vapor pressure *pe* of the system is defined as the pressure under which the vapor atoms condense on the solid surface and the atoms evaporate from the surface at the same rate. In equilibrium, the entropy free energy of each atom in the vapor is equal to the free energy of each atom in the crystal. Due to the effect of chemical bond, the lower internal energy of atoms in the crystal is offset by the lower entropy energy in the crystal. For the net deposition of the substrate surface, the pressure *p* in the vapor must be higher than the equilibrium vapor pressure *pe* at the substrate temperature, that is, the vapor must be supersaturated. For pressure *p*, the entropy free energy of each atom in the vapor, which is higher than the pressure *pe*, is estimated as the work of each atom required to increase the vapor pressure from *pe* to *p* at a constant temperature. According to the ideal gas law, *kTvlnp=pe* where *<sup>k</sup>* = 1.38 � <sup>10</sup>�<sup>23</sup> J/K = 8.617 � <sup>10</sup>�<sup>5</sup> eV/K is Boltzmann constant, *Tv* is the absolute temperature of the vapor. If the vapor becomes supersaturated, there is a difference of free energy between the vapor and the crystal, which provides the chemical potential of the interface driving the interface toward the vapor. As the interface progresses in a self-similar way, some distant layers of vapor of the same mass are transformed into inner layers of the same mass. When these energies are the same, the interface will not move

In thin film deposition, because the vapor phase and the substrate are not the same material phase, and the temperature of the substrate is usually lower than that of the vapor phase, the situation is often complex. In this case, the definition of equilibrium vapor pressure is not clear. However, in most cases, when the vapor pressure is lower than the equilibrium vapor pressure, the film material will not deposit on the growth surface, which is an operational definition of *pe* in the film growth process. Once atoms are attached to the substrate, their entropy free energy will be reduced from that of the vapor. These atoms form a two-dimensional vapor distribution on the substrate surface. The deposited material will be rapidly heated to the substrate temperature *Ts*. On the surface of the crystal, there are some atoms whose density *ρad* is balanced with a straight surface step or wall frame adjacent to some monolayer atoms, that is, the step does not move forward or backward. Step is the boundary between two phases in a homogeneous material system. Therefore, the concept of equilibrium vapor pressure or equilibrium vapor pressure density *ρ<sup>e</sup>*

can be called in a similar way. In order to realize the film growth by condensation, the free energy of each atom in the two-dimensional gas must exceed the free

*<sup>ε</sup>phase* <sup>¼</sup> *kTln <sup>ρ</sup>ad=ρ<sup>e</sup>*

Consider a molecule approaching a surface from the vapor phase, as shown in **Figure 4**. Upon arriving within a few atomic distances of the surface, it will begin to feel an attraction due to interaction with the surface molecules. This happens even with symmetrical molecules and with inert gases, neither of which has dipole moments. It happens because even these molecules and atoms act as oscillating dipoles, and this behavior creates the dipole-induced-dipole interaction known as the Van der Waals force or London dispersion force. Polar molecules, having permanent dipoles, are attracted more strongly. The approaching molecule is being attracted into a potential well like the one that was illustrated in **Figure 5** for condensation.

*ad*

(2)

energy of the fully entrained surface atom, the amount of which is:

forward or backward.

*21st Century Surface Science - a Handbook*

**8**

*Adsorption processes and quantities. a, is used only for condensation (adsorption of a material onto itself). A vertical connecting bar denotes a chemical bond.*

#### **Figure 5.**

*ad*

*Molecular potential energy diagram for evaporation and condensation.*

Condensation is just a special case of adsorption in which the substrate composition is the same as that of the adsorbate. This is sometimes the case in thin film deposition and sometimes not. In either case, the molecule accelerates down the curve of the potential well until it passes the bottom and is repelled by the steeply rising portion, which is caused by mutual repulsion of the nuclei. If enough of the molecule's perpendicular component of momentum is dissipated into the surface during this interaction, the molecule will not be able to escape the potential well after being repelled, though it will still be able to migrate along the surface. This molecule is trapped in a weakly adsorbed state known as physical adsorption or physisorption. The fraction of approaching molecules so adsorbed is called the trapping probability, δ, and the fraction escaping (reflecting) is (1 � δ) as shown in **Figure 4**. The quantity δ is different from the thermal accommodation coefficient, *η*, which was defined by Eq. (3):

$$\eta = \frac{T\_{\rm rs} - T\_{\rm rh}}{T\_{\rm rs} - T\_{\rm h}} \tag{3}$$

*η* represents the degree to which the molecule accommodates itself to the temperature *Th* of the surface from which it is reflected. Gas-conductive heat transfer is shown in **Figure 6**.

In general, a molecule is at least partially accommodated thermally to the surface temperature, *Ts*, even when it is reflected without having been trapped.

In addition to the low temperature *T*, the physical adsorption molecules are mobile on the surface; as shown in **Figure 4**, the adsorption molecules hop (diffuse) between the surface atomic sites. Adsorption molecules can also be desorbed by obtaining enough energy at the tail of thermal energy distribution, or they can form chemical bonds through further interaction with surface atoms, namely, chemical adsorption. If both adsorption states exist, the physical adsorption state is called precursor state. Chemical adsorption involves the sharing of electrons on the new molecular orbital, which is much stronger than physical adsorption. Physical adsorption only involves dipole interaction. These two types of adsorption can be distinguished in almost all gas phase surface combinations, so they constitute a valuable model for analyzing any surface process. This model [4] has long been applied to heterogeneous catalysis, thin film deposition, and condensation of molecular vapors. Recent theory indicates that even the condensation of a monatomic vapor such as *Al* can involve both adsorption states, the precursor state in that case being an *Al* � *Al* dimer whose bonding to the bulk *Al* is inhibited by the existence of the dimer bond [5]. In such a case, and in the case of condensing molecular vapors such as As4, the vapor would not be considered actually condensed until it had become fully incorporated into the solid phase by chemisorption. Thus, the condensation coefficient, *αc*, defined by *Jc* ¼ *αcJi* [*Ji*, impinging flux; *Jc*, the portion of *Ji* that condenses; *αc*, corresponding condensation coefficient] is that fraction of the arriving vapor that becomes not only trapped but also chemisorbed, as indicated in **Figure 4**. However, the term *α<sup>c</sup>* is not used in the case of chemisorption on a foreign substrate. Then, the chemisorption reaction probability, ξ, will be derived later. The precursor model may also be applied to cases where both of the adsorption states involve chemical bonding, but where the bonding in one state is weaker than in the other.

chemisorbed state (c). By convention, the zero of *Ep* is set at the *Ep* of the element Y in its thermodynamic standard state, which we specify for this element to be the diatomic molecule in the gas phase. In fact, all gaseous elements except the inert gases have diatomic standard states. Note that lifting atomic Y out of its potential well along curve c results in a much higher *Ep* in the gas phase, which corresponds roughly to the heat of formation, Δ*fH*, of 2Y(g) from Y2(g). [Δ*fH* usually can be found in thermodynamic tables] [6–8]. The result of this high *Ep* for Y(g) is that curves a and c intersect at positive *Ep*, meaning that there is an activation energy, *Ea*, to be overcome for Y2(g) to become dissociatively chemisorbed. For the deeper precursor well, b, chemisorption is not "activated," though there still is a barrier, as shown. The level of *Era* or *Erb* and hence of *Es* is determined by the degree to which the bonds within both the precursor and the surface must be strained from their relaxed condition before new bonds can be formed between the precursor and

*Energetics of the precursor adsorption model. Energy scale is typical only.*

*Growth Kinetics of Thin Film Epitaxy*

*DOI: http://dx.doi.org/10.5772/intechopen.91224*

There are two ways in which vapor can arrive at the surface having an *Ep* > 0. Gaseous molecules have their *Ep* raised by becoming dissociated. Solids and liquids have it raised by evaporating. If the *Ep* of the arriving vapor is high enough, curve c is followed, and direct chemisorption can occur without involving the precursor state. In the language of surface chemistry, direct reaction between an incoming species and a surface site or adsorbate is called the Eley-Rideal mechanism, whereas reaction among surface species is called the Langmuir-Hinshelwood mechanism. A principal advantage of the energy-enhanced deposition processes is that they can provide enough energy so that the arriving molecules can surmount the *Ea* barrier and adsorb directly into the chemisorbed state. In other words, the arriving molecules immediately react with the surface to deposit the film. In sputter deposition, species arrive having kinetic energies of around 1000 kJ/mol as well as having *Ep* > 0 by having been vaporized. In plasma-enhanced deposition, vapor molecules become dissociated in the plasma and thus arrive along curve c, above the *Ea* barrier. Thus, an energy-enhanced process can supply *Ea* to the arriving species either as kinetic energy

of accelerated molecules or as potential energy of dissociated ones.

Conversely, in thermally controlled deposition processes such as evaporation and CVD, the vapor often adsorbs first into the precursor state, that is, it falls to the bottom of the well on curve a or b. Thence, it may either chemisorb by overcoming the barrier *Er a*ð Þ ,*<sup>b</sup>* shown in **Figure 7** or it may desorb by overcoming the heat of

the surface.

**11**

**Figure 7.**

These examples will be revisited after a more detailed study of the energetics of the precursor adsorption model.

Consider a hypothetical diatomic gas phase molecule Y2(g) adsorbing and then dissociatively chemisorbing as two Y atoms. **Figure 7** shows a diagram of the potential energy versus molecular distance, *z*, from the surface. This is similar to **Figure 5** for condensation except that we have changed from the molecular (*εp*) to the molar (*Ep*) quantities of potential energy which are more conventional in chemistry. The energy scales shown represent typical bond strengths. Three curves are shown: two alternate ones for the precursor state (a and b) and one for the

**Figure 6.** *Gas-conductive heat transfer between parallel plates at (a) low and (b) high Knudsen numbers, K.*

*η* represents the degree to which the molecule accommodates itself to the temperature *Th* of the surface from which it is reflected. Gas-conductive heat

temperature, *Ts*, even when it is reflected without having been trapped.

In general, a molecule is at least partially accommodated thermally to the surface

These examples will be revisited after a more detailed study of the energetics of

Consider a hypothetical diatomic gas phase molecule Y2(g) adsorbing and then

dissociatively chemisorbing as two Y atoms. **Figure 7** shows a diagram of the potential energy versus molecular distance, *z*, from the surface. This is similar to **Figure 5** for condensation except that we have changed from the molecular (*εp*) to the molar (*Ep*) quantities of potential energy which are more conventional in chemistry. The energy scales shown represent typical bond strengths. Three curves are shown: two alternate ones for the precursor state (a and b) and one for the

*Gas-conductive heat transfer between parallel plates at (a) low and (b) high Knudsen numbers, K.*

In addition to the low temperature *T*, the physical adsorption molecules are mobile on the surface; as shown in **Figure 4**, the adsorption molecules hop (diffuse) between the surface atomic sites. Adsorption molecules can also be desorbed by obtaining enough energy at the tail of thermal energy distribution, or they can form chemical bonds through further interaction with surface atoms, namely, chemical adsorption. If both adsorption states exist, the physical adsorption state is called precursor state. Chemical adsorption involves the sharing of electrons on the new molecular orbital, which is much stronger than physical adsorption. Physical adsorption only involves dipole interaction. These two types of adsorption can be distinguished in almost all gas phase surface combinations, so they constitute a valuable model for analyzing any surface process. This model [4] has long been applied to heterogeneous catalysis, thin film deposition, and condensation of molecular vapors. Recent theory indicates that even the condensation of a monatomic vapor such as *Al* can involve both adsorption states, the precursor state in that case being an *Al* � *Al* dimer whose bonding to the bulk *Al* is inhibited by the existence of the dimer bond [5]. In such a case, and in the case of condensing molecular vapors such as As4, the vapor would not be considered actually condensed until it had become fully incorporated into the solid phase by chemisorption. Thus, the condensation coefficient, *αc*, defined by *Jc* ¼ *αcJi* [*Ji*, impinging flux; *Jc*, the portion of *Ji* that condenses; *αc*, corresponding condensation coefficient] is that fraction of the arriving vapor that becomes not only trapped but also chemisorbed, as indicated in **Figure 4**. However, the term *α<sup>c</sup>* is not used in the case of chemisorption on a foreign substrate. Then, the chemisorption reaction probability, ξ, will be derived later. The precursor model may also be applied to cases where both of the adsorption states involve chemical bonding, but where the bonding in one state

transfer is shown in **Figure 6**.

*21st Century Surface Science - a Handbook*

is weaker than in the other.

**Figure 6.**

**10**

the precursor adsorption model.

**Figure 7.** *Energetics of the precursor adsorption model. Energy scale is typical only.*

chemisorbed state (c). By convention, the zero of *Ep* is set at the *Ep* of the element Y in its thermodynamic standard state, which we specify for this element to be the diatomic molecule in the gas phase. In fact, all gaseous elements except the inert gases have diatomic standard states. Note that lifting atomic Y out of its potential well along curve c results in a much higher *Ep* in the gas phase, which corresponds roughly to the heat of formation, Δ*fH*, of 2Y(g) from Y2(g). [Δ*fH* usually can be found in thermodynamic tables] [6–8]. The result of this high *Ep* for Y(g) is that curves a and c intersect at positive *Ep*, meaning that there is an activation energy, *Ea*, to be overcome for Y2(g) to become dissociatively chemisorbed. For the deeper precursor well, b, chemisorption is not "activated," though there still is a barrier, as shown. The level of *Era* or *Erb* and hence of *Es* is determined by the degree to which the bonds within both the precursor and the surface must be strained from their relaxed condition before new bonds can be formed between the precursor and the surface.

There are two ways in which vapor can arrive at the surface having an *Ep* > 0. Gaseous molecules have their *Ep* raised by becoming dissociated. Solids and liquids have it raised by evaporating. If the *Ep* of the arriving vapor is high enough, curve c is followed, and direct chemisorption can occur without involving the precursor state. In the language of surface chemistry, direct reaction between an incoming species and a surface site or adsorbate is called the Eley-Rideal mechanism, whereas reaction among surface species is called the Langmuir-Hinshelwood mechanism.

A principal advantage of the energy-enhanced deposition processes is that they can provide enough energy so that the arriving molecules can surmount the *Ea* barrier and adsorb directly into the chemisorbed state. In other words, the arriving molecules immediately react with the surface to deposit the film. In sputter deposition, species arrive having kinetic energies of around 1000 kJ/mol as well as having *Ep* > 0 by having been vaporized. In plasma-enhanced deposition, vapor molecules become dissociated in the plasma and thus arrive along curve c, above the *Ea* barrier. Thus, an energy-enhanced process can supply *Ea* to the arriving species either as kinetic energy of accelerated molecules or as potential energy of dissociated ones.

Conversely, in thermally controlled deposition processes such as evaporation and CVD, the vapor often adsorbs first into the precursor state, that is, it falls to the bottom of the well on curve a or b. Thence, it may either chemisorb by overcoming the barrier *Er a*ð Þ ,*<sup>b</sup>* shown in **Figure 7** or it may desorb by overcoming the heat of

physisorption, which is roughly *Ed a*ð Þ ,*<sup>b</sup>* . The competition between these two reactions results in a net rate of chemisorption whose behavior we would like to describe, since it is the basic film-forming reaction. We start with the conventional expression for the rate of a first-order chemical reaction, first-order meaning that rate is proportional to the concentration of one reactant; thus, *Rk* ¼ *kkns* ¼ *kkns*0Θ, where *Rk* = rate of the *k*th surface reaction per unit surface area, mc/cm<sup>2</sup> s; *kk* = rate constant, s�<sup>1</sup> ; *ns* = surface concentration of reactant, mc/cm<sup>2</sup> ; *ns*<sup>0</sup> = monolayer surface concentration, mc/cm<sup>2</sup> ; Θ = fractional surface coverage by reactant.

the language of reaction rate theory. **Figure 8b** illustrates a typical adsorbate situation leading to this corrugation. It is a hexagonally close-packed surface lattice on which the adsorption sites are the centers of the triangles of surface atoms and the transition state is the "saddle point" between them. Other bonding situations can lead to the adsorption sites being other points, such as the centers of the surface atoms. In the process of surface diffusion, the bond between adsorbate and surface should be partially destroyed, so that the adsorbate can migrate to the adjacent surface and form a new bond there. This process can be regarded as the basic form of chemical reaction, because any reaction involves the partial fracture of reactant bond and the formation of product bond when the atom moves through the transition state. Therefore, the principles discussed below apply to any chemical reaction,

There will be some flux, *Js* (mc/cm s), of adsorbate across the *Es* barrier between sites 1 and 2 in the *x* direction of **Figure 8b**. The flux here is in surface units, which are per linear cm of crosswise distance, *y*, instead of the previously encountered volume flux units, which are per cm<sup>2</sup> of cross-sectional area. If the distance between sites is a, then the rate of barrier crossing by transition state molecules, per unit area

Considering the adsorbate to be a two-dimensional gas at thermal equilibrium,

the numerical proportionality factors that arise in going from a three-dimensional to a two-dimensional situation. (It turns out that these factors cancel each other, anyway.) Inserting these equations into Eq. *Rs* <sup>¼</sup> *Js=<sup>a</sup>* mc*=*cm2 ð Þ � <sup>s</sup> , we have:

r

being the concentration of molecules in adsorption sites. At thermal equilibrium, statistical mechanics says that the concentration of molecules in a given state is proportional to the total number of ways of distributing the available thermal energy around a large system of molecules in that state. For each type of kinetic energy contributing to the thermal energy, the number of ways, *Z*, is equal to the sum over all of the quantized energy levels, *εj*, of the following products: the Boltzmann factor for each energy level times the number of ways of distributing

> *<sup>Z</sup>* <sup>¼</sup> <sup>X</sup> *j gj e*

To understand nucleation, the concept of surface energy needs to be introduced. The familiar experiment of drawing a liquid membrane out of soapy water on a wire

for the mean speed. Here, for simplicity, we ignore the small changes in

ffiffiffiffiffiffiffiffiffiffi *RT* 2*πM*

the Maxwell-Boltzmann distribution applies to these translating molecules.

*<sup>s</sup> <sup>c</sup>=<sup>a</sup>* <sup>¼</sup> *<sup>n</sup>*<sup>∗</sup> *s a*

the transition state. Now we must find the relation between *n*<sup>∗</sup>

energy at that level (the degeneracy of the level, *g*). Thus:

*Rs* <sup>¼</sup> <sup>1</sup> 4 *n*∗

*Rs* <sup>¼</sup> *Js=<sup>a</sup>* mc*=*cm2 � <sup>s</sup> � � (4)

<sup>4</sup> *nc* for the flux of molecules impinging on the barrier and

¼ *n*∗ *s a*

) denotes the surface concentration of adsorbate residing in

ffiffiffiffiffiffiffiffiffi *kBT* 2*πm*

�*εj=kBT* (6)

(5)

*<sup>s</sup>* and *ns*, the latter

r

including those occurring in CVD.

*Growth Kinetics of Thin Film Epitaxy*

*DOI: http://dx.doi.org/10.5772/intechopen.91224*

of surface, is:

*c* ¼

Thus, we may use *Ji* <sup>¼</sup> <sup>1</sup>

ffiffiffiffiffiffi 8*RT πM* q

where *n*<sup>∗</sup>

**3.3 Nucleation**

**13**

*3.3.1 Surface energy*

*<sup>s</sup>* (mc/cm<sup>2</sup>

#### **3.2 Diffusion of adsorbed atoms on substrate surface**

Surface diffusion is one of the most important determinants of film structure because it allows the adsorbing species to find each other, find the most active sites, or find epitaxial sites. Various methods have been applied to measure surface diffusion rates of adsorbed molecules. The role of surface diffusion in thin films has mainly been inferred from observations of film structure. Scanning tunneling microscope (STM) gives us the extraordinary power to directly observe individual atoms on surfaces in relation to the entire array of available atomic surface sites. STM observation of the diffusion of these atoms should ultimately provide a wealth of data relevant to thin film deposition.

The expression of the surface diffusion rate will be derived using the absolute reaction rate theory [9]. Although this approach cannot provide a quantitative estimate of the diffusion rate, it will provide valuable insight into what factors determine this rate. **Figure 7** showed that adsorbed atoms or molecules reside in potential wells on the surface, but it did not consider the variation in well depth with position, *x*, along the surface. **Figure 8a** shows that this depth is periodic, or corrugated, with a potential energy barrier of height *Es* between surface sites. The top of the barrier is considered to be the "transition state" between surface sites, in

#### **Figure 8.**

*Surface diffusion: (a) potential energy vs. position x along the surface and (b) typical adsorption sites on a surface lattice.*

physisorption, which is roughly *Ed a*ð Þ ,*<sup>b</sup>* . The competition between these two reactions results in a net rate of chemisorption whose behavior we would like to describe, since it is the basic film-forming reaction. We start with the conventional expression for the rate of a first-order chemical reaction, first-order meaning that rate is proportional to the concentration of one reactant; thus, *Rk* ¼ *kkns* ¼ *kkns*0Θ, where *Rk* = rate of the *k*th surface reaction per unit surface area, mc/cm<sup>2</sup> s; *kk* = rate

Surface diffusion is one of the most important determinants of film structure because it allows the adsorbing species to find each other, find the most active sites, or find epitaxial sites. Various methods have been applied to measure surface diffusion rates of adsorbed molecules. The role of surface diffusion in thin films has mainly been inferred from observations of film structure. Scanning tunneling microscope (STM) gives us the extraordinary power to directly observe individual atoms on surfaces in relation to the entire array of available atomic surface sites. STM observation of the diffusion of these atoms should ultimately provide a wealth

The expression of the surface diffusion rate will be derived using the absolute reaction rate theory [9]. Although this approach cannot provide a quantitative estimate of the diffusion rate, it will provide valuable insight into what factors determine this rate. **Figure 7** showed that adsorbed atoms or molecules reside in potential wells on the surface, but it did not consider the variation in well depth with position, *x*, along the surface. **Figure 8a** shows that this depth is periodic, or corrugated, with a potential energy barrier of height *Es* between surface sites. The top of the barrier is considered to be the "transition state" between surface sites, in

*Surface diffusion: (a) potential energy vs. position x along the surface and (b) typical adsorption sites on a*

; Θ = fractional surface coverage by reactant.

; *ns*<sup>0</sup> = monolayer

; *ns* = surface concentration of reactant, mc/cm<sup>2</sup>

**3.2 Diffusion of adsorbed atoms on substrate surface**

constant, s�<sup>1</sup>

**Figure 8.**

**12**

*surface lattice.*

surface concentration, mc/cm<sup>2</sup>

*21st Century Surface Science - a Handbook*

of data relevant to thin film deposition.

the language of reaction rate theory. **Figure 8b** illustrates a typical adsorbate situation leading to this corrugation. It is a hexagonally close-packed surface lattice on which the adsorption sites are the centers of the triangles of surface atoms and the transition state is the "saddle point" between them. Other bonding situations can lead to the adsorption sites being other points, such as the centers of the surface atoms. In the process of surface diffusion, the bond between adsorbate and surface should be partially destroyed, so that the adsorbate can migrate to the adjacent surface and form a new bond there. This process can be regarded as the basic form of chemical reaction, because any reaction involves the partial fracture of reactant bond and the formation of product bond when the atom moves through the transition state. Therefore, the principles discussed below apply to any chemical reaction, including those occurring in CVD.

There will be some flux, *Js* (mc/cm s), of adsorbate across the *Es* barrier between sites 1 and 2 in the *x* direction of **Figure 8b**. The flux here is in surface units, which are per linear cm of crosswise distance, *y*, instead of the previously encountered volume flux units, which are per cm<sup>2</sup> of cross-sectional area. If the distance between sites is a, then the rate of barrier crossing by transition state molecules, per unit area of surface, is:

$$R\_t = J\_s/\mathfrak{a} \text{ (mc/cm}^2 \cdot \text{s)}\tag{4}$$

Considering the adsorbate to be a two-dimensional gas at thermal equilibrium, the Maxwell-Boltzmann distribution applies to these translating molecules. Thus, we may use *Ji* <sup>¼</sup> <sup>1</sup> <sup>4</sup> *nc* for the flux of molecules impinging on the barrier and *c* ¼ ffiffiffiffiffiffi 8*RT πM* q for the mean speed. Here, for simplicity, we ignore the small changes in the numerical proportionality factors that arise in going from a three-dimensional to a two-dimensional situation. (It turns out that these factors cancel each other, anyway.) Inserting these equations into Eq. *Rs* <sup>¼</sup> *Js=<sup>a</sup>* mc*=*cm2 ð Þ � <sup>s</sup> , we have:

$$R\_s = \frac{1}{4} n\_s^\* \overline{c} / a = \frac{n\_s^\*}{a} \sqrt{\frac{RT}{2\pi M}} = \frac{n\_s^\*}{a} \sqrt{\frac{k\_B T}{2\pi m}}\tag{5}$$

where *n*<sup>∗</sup> *<sup>s</sup>* (mc/cm<sup>2</sup> ) denotes the surface concentration of adsorbate residing in the transition state. Now we must find the relation between *n*<sup>∗</sup> *<sup>s</sup>* and *ns*, the latter being the concentration of molecules in adsorption sites. At thermal equilibrium, statistical mechanics says that the concentration of molecules in a given state is proportional to the total number of ways of distributing the available thermal energy around a large system of molecules in that state. For each type of kinetic energy contributing to the thermal energy, the number of ways, *Z*, is equal to the sum over all of the quantized energy levels, *εj*, of the following products: the Boltzmann factor for each energy level times the number of ways of distributing energy at that level (the degeneracy of the level, *g*). Thus:

$$Z = \sum\_{j} \mathbf{g}\_{j} \mathbf{e}^{-\varepsilon\_{j}/k\_{\mathrm{B}}T} \tag{6}$$

#### **3.3 Nucleation**

#### *3.3.1 Surface energy*

To understand nucleation, the concept of surface energy needs to be introduced. The familiar experiment of drawing a liquid membrane out of soapy water on a wire ring is illustrated in **Figure 9**. The force required to support the membrane per unit width of membrane surface is known as the surface tension, *γ*, expressed as N/m in SI units, or more commonly as dynes/cm in cgs units. For a wire of circumference *b*, the width of surface is 2*b*, since the membrane has both an inner and an outer surface. Thus, the total force required to support the membrane is *F* ¼ 2*bγ*. As the membrane is extended upward in the *x* direction, work *F*ð Þ Δ*x* (N m or J) is done to create the new surface, and the surface area created is *A* ¼ 2*b*Δ*x*, assuming for simplicity a constant membrane circumference. The work is stored as surface energy (as in stretching a spring), so the surface energy per unit area of surface is:

$$F\Delta\mathbf{x}/A\left(\mathbf{N}\cdot\mathbf{m}/\mathbf{m}^2\right) = (2b\chi)\Delta\mathbf{x}/2b\Delta\mathbf{x} = \chi\left(\mathbf{N}/\mathbf{m}\text{ or }\mathbf{J}/\mathbf{m}^2\right) \tag{7}$$

on crystal orientation, passivation, and other factors. It is assumed that there is sufficient surface diffusion to enable the deposited materials to rearrange to minimize *γ*, i.e., it is assumed that nucleation is not limited by dynamics and can approach equilibrium. For this reason, we must have Λ ≫ *a* (Λ is the diffusion length). On the contrary, if Λ < *a*, every atom will stick where it falls, and the growth behavior is "quenched." According to our hypothesis Λ ≫ *a*, there are two kinds of nucleation on the bare substrate, as shown in the **Figure 10a** and **b**. In a,

In other words, the total surface energy of the wetted substrate is lower than that of the bare substrate. This leads to the smooth growth of the atomic layer, which is the Frank-van der Merwe growth mode. To achieve this mode, there must be a strong enough bond between the film and the substrate to reduce the *γ<sup>i</sup>* in Eq. (8). If there is no such bond, we will get *γ<sup>i</sup>* ¼ *γ<sup>f</sup>* þ *γs*, so spreading the film on the substrate will always increase the total surface energy 2*γ<sup>f</sup>* , as shown in the freestanding liquid film in **Figure 9**. Therefore, in the case of insufficient substrate bonding, Eq. (8) cannot be maintained, and the film does not wet the substrate, but forms a threedimensional (3D) island, as shown in **Figure 10b**, which is called the Volmer-Weber growth mode. There is a third growth mode, Stranski-Krastanov growth mode, as shown in **Figure 10c**. In this mode, due to the change of the energy situation of the continuous single layer, the growth changes from one layer to an island after one or two layers. For liquids contacting solids, the degree of wetting is most easily observed by the rise or depression of a liquid column in a narrow tube (a capillary). Thus, film nucleation analysis in terms of degree of wetting is known as the "capillarity" model. Three-dimensional nucleation is usually undesirable, since

Different crystal shapes imply that underlying substrates critically influence the

vapor phase growth mode. The substrate-dependent growth characteristics of various low-dimensional nanocrystals in both solution and vapor phase growth have

*Film growth modes: (a) Frank-Van der Merwe (layer), (b) Volmer-Weber (island), and*

*γ<sup>f</sup>* þ *γ<sup>i</sup>* < *γ<sup>s</sup>* (8)

the film diffuses or "wets" the substrate on the substrate because:

it leads to rough, nonuniform films.

*Growth Kinetics of Thin Film Epitaxy*

*DOI: http://dx.doi.org/10.5772/intechopen.91224*

**Figure 10.**

**15**

*(c) Stranski-Krastanov.*

been discussed for their growth mechanisms [10, 11].

Thus, surface tension (N/m) and surface energy per unit area (J/m2 ) are identical, at least for liquids. For solids at *T* > 0 K, surface Gibbs free energy is reduced by an entropy factor [*G* ¼ ð Þ� *U* þ *pV TS* ¼ *H* � *TS*] which depends on the degree of surface disorder. For solids, there is also a quantity called surface stress, which differs from surface energy by a surface elastic strain term. Liquids cannot support such strain, because the atoms just rearrange to relax it.

The surface energy exists because the molecules in the condensed phase attract each other, which is the reason for condensation. The generation of a surface involves the removal of molecular contact (bond breaking) from above the surface, thus involving energy input. Therefore, the movement in the condensed phase can occur within a certain range, and this movement will continue to minimize the total surface energy, *γA*. In the liquid membrane case, where *γ* is fixed, this means minimizing *A*. Thus, when the wire is lifted far enough, the membrane snaps taut into the plane of the ring, and when a bubble is blown, it becomes spherical. In the case of solids, surface energy can be minimized by surface diffusion, which is the basis of the development of film structure. In the film growth, *A* and *γ* are varying. Area *A* depends on the surface morphology, and *γ* depends on many characteristics of the exposed surface, including chemical composition, crystal orientation, atomic reconstruction, and atomic-scale roughness.

For the deposition on foreign substrates, the substrate *γ* strongly affects the nucleation behavior. Here, the surface energy *γ<sup>s</sup>* of the free surface of the substrate, the surface energy *γ<sup>i</sup>* of the substrate film interface, and the surface energy *γ<sup>f</sup>* of the free surface of the film should be considered. These three *γ* values generally depend

**Figure 9.** *Surface tension of a liquid membrane.*

ring is illustrated in **Figure 9**. The force required to support the membrane per unit width of membrane surface is known as the surface tension, *γ*, expressed as N/m in SI units, or more commonly as dynes/cm in cgs units. For a wire of circumference *b*, the width of surface is 2*b*, since the membrane has both an inner and an outer surface. Thus, the total force required to support the membrane is *F* ¼ 2*bγ*. As the membrane is extended upward in the *x* direction, work *F*ð Þ Δ*x* (N m or J) is done to create the new surface, and the surface area created is *A* ¼ 2*b*Δ*x*, assuming for simplicity a constant membrane circumference. The work is stored as surface energy (as in stretching a spring), so the surface energy per unit area of surface is:

Thus, surface tension (N/m) and surface energy per unit area (J/m2

such strain, because the atoms just rearrange to relax it.

*21st Century Surface Science - a Handbook*

reconstruction, and atomic-scale roughness.

**Figure 9.**

**14**

*Surface tension of a liquid membrane.*

cal, at least for liquids. For solids at *T* > 0 K, surface Gibbs free energy is reduced by an entropy factor [*G* ¼ ð Þ� *U* þ *pV TS* ¼ *H* � *TS*] which depends on the degree of surface disorder. For solids, there is also a quantity called surface stress, which differs from surface energy by a surface elastic strain term. Liquids cannot support

The surface energy exists because the molecules in the condensed phase attract

For the deposition on foreign substrates, the substrate *γ* strongly affects the nucleation behavior. Here, the surface energy *γ<sup>s</sup>* of the free surface of the substrate, the surface energy *γ<sup>i</sup>* of the substrate film interface, and the surface energy *γ<sup>f</sup>* of the free surface of the film should be considered. These three *γ* values generally depend

each other, which is the reason for condensation. The generation of a surface involves the removal of molecular contact (bond breaking) from above the surface, thus involving energy input. Therefore, the movement in the condensed phase can occur within a certain range, and this movement will continue to minimize the total surface energy, *γA*. In the liquid membrane case, where *γ* is fixed, this means minimizing *A*. Thus, when the wire is lifted far enough, the membrane snaps taut into the plane of the ring, and when a bubble is blown, it becomes spherical. In the case of solids, surface energy can be minimized by surface diffusion, which is the basis of the development of film structure. In the film growth, *A* and *γ* are varying. Area *A* depends on the surface morphology, and *γ* depends on many characteristics of the exposed surface, including chemical composition, crystal orientation, atomic

*<sup>F</sup>*Δ*x=<sup>A</sup>* <sup>N</sup> � <sup>m</sup>*=*m2 <sup>¼</sup> ð Þ <sup>2</sup>*b<sup>γ</sup>* <sup>Δ</sup>*x=*2*b*Δ*<sup>x</sup>* <sup>¼</sup> *<sup>γ</sup>* <sup>N</sup>*=*m or J*=*m<sup>2</sup> (7)

) are identi-

on crystal orientation, passivation, and other factors. It is assumed that there is sufficient surface diffusion to enable the deposited materials to rearrange to minimize *γ*, i.e., it is assumed that nucleation is not limited by dynamics and can approach equilibrium. For this reason, we must have Λ ≫ *a* (Λ is the diffusion length). On the contrary, if Λ < *a*, every atom will stick where it falls, and the growth behavior is "quenched." According to our hypothesis Λ ≫ *a*, there are two kinds of nucleation on the bare substrate, as shown in the **Figure 10a** and **b**. In a, the film diffuses or "wets" the substrate on the substrate because:

$$
\chi\_f + \chi\_i < \chi\_s \tag{8}
$$

In other words, the total surface energy of the wetted substrate is lower than that of the bare substrate. This leads to the smooth growth of the atomic layer, which is the Frank-van der Merwe growth mode. To achieve this mode, there must be a strong enough bond between the film and the substrate to reduce the *γ<sup>i</sup>* in Eq. (8). If there is no such bond, we will get *γ<sup>i</sup>* ¼ *γ<sup>f</sup>* þ *γs*, so spreading the film on the substrate will always increase the total surface energy 2*γ<sup>f</sup>* , as shown in the freestanding liquid film in **Figure 9**. Therefore, in the case of insufficient substrate bonding, Eq. (8) cannot be maintained, and the film does not wet the substrate, but forms a threedimensional (3D) island, as shown in **Figure 10b**, which is called the Volmer-Weber growth mode. There is a third growth mode, Stranski-Krastanov growth mode, as shown in **Figure 10c**. In this mode, due to the change of the energy situation of the continuous single layer, the growth changes from one layer to an island after one or two layers. For liquids contacting solids, the degree of wetting is most easily observed by the rise or depression of a liquid column in a narrow tube (a capillary). Thus, film nucleation analysis in terms of degree of wetting is known as the "capillarity" model. Three-dimensional nucleation is usually undesirable, since it leads to rough, nonuniform films.

Different crystal shapes imply that underlying substrates critically influence the vapor phase growth mode. The substrate-dependent growth characteristics of various low-dimensional nanocrystals in both solution and vapor phase growth have been discussed for their growth mechanisms [10, 11].

#### **Figure 10.**

*Film growth modes: (a) Frank-Van der Merwe (layer), (b) Volmer-Weber (island), and (c) Stranski-Krastanov.*

#### *3.3.2 Kinetics vs. thermodynamics*

In general, within the framework of the nucleation kinetics model [12], a gas phase growth reaction can be divided into two steps: (1) adsorption of vaporized precursors onto substrates and diffusion to the preferential growth sites and (2) incorporation of precursors into existing nuclei. The rate-limiting step in vapor phase crystal growth can be determined as either the diffusion-limited step or the reaction-limited step.

One way to achieve smooth growth is to reduce substrate temperature, *T*, to inhibit surface diffusion, thus "freezing" the nucleation and coalescence process. If the arriving species do not have enough heat energy to desorb or diffuse, they will stay where they land, leading to the aforementioned quenching growth. In this case, the nucleation process is kinetically inhibited by the surface diffusion activation energy barrier, *Es*, in **Figure 8(a)**. This is also the case for ion-bombardment dissipation of 3D nuclei; the nuclei do not have time to reassemble themselves by surface diffusion before they are buried by depositing material.

The question of whether a process is approaching equilibrium or is instead limited by kinetics is an important one, and it arises often in thin film deposition. Process behavior and film properties are profoundly affected by the degree to which one or the other situation dominates. The answer is not always apparent in a given process, and this often leads to confusion and to misinterpretation of observed phenomena. Therefore, to elaborate briefly, the generalized mathematical representation of this dichotomy is embodied in Eq. (9):

$$R\_{\sharp} = R\_{-\sharp} \text{ or } \mathfrak{n}\_{\sharp}\mathfrak{k}\_{\sharp} = \mathfrak{n}\_{\sharp}^{\*}\mathfrak{k}\_{-\sharp} \tag{9}$$

The difficulty of answering the question of kinetics versus thermodynamics arises from the fact that the applicable rate constants, *kk*, are often unknown or not known accurately enough. The measurement of *kk* is much more difficult than just measuring equilibrium concentration, both because it is a dynamic measurement

When wetting is complete and Eq. (8) holds, the adsorbing atoms do not accumulate into 3D islands but, instead, spread out on the surface in a partial monolayer

> *pv*

�

ð Þ <sup>4</sup>*=*<sup>3</sup> *<sup>π</sup>r*<sup>3</sup> *Vmc*

þ *γ<sup>f</sup>* 4*πr*

<sup>2</sup> (13)

as shown in **Figure 10a**. Because total surface energy is reduced rather than increased by this process, there is no nucleation barrier in going from the vapor state to the adsorbed state, that is, the term in Eq. (13) is negative when the

<sup>þ</sup> *<sup>γ</sup><sup>f</sup> Af* ¼ � *RTln <sup>p</sup>*

where *pv* is saturation vapor pressure and *Vmc* is the molar volume of the

This means that deposition can proceed even in undersaturated conditions. Assuming, as we did for 3D nucleation, that there is sufficient surface diffusion for equilibration, the partial monolayer of adsorbed atoms will behave as a 2D gas. By analogy to a 3D gas condensing into 3D nuclei, the 2D gas then condenses into 2D nuclei as illustrated in **Figure 11**. Here, only the top monolayer of atoms is drawn. The "atomic terrace" to the left represents a monolayer which is one atomic step (a) higher than the surface to the right. But unlike the 3D nucleation case, 2D nucleation from a 2D gas involves no change in any of the *γ* values, so one might expect there to be no nucleation barrier. However, the chemical potential, *μ*, of a 2D nucleus is higher than that of a continuous monolayer because of the exposed edge. This situation may be viewed in terms of an excess edge energy, *β* (J/m), which is analogous to the surface energy, *γ*, of the 3D case. The surface concentration of the 2D gas for which its *μ* is the same as that on the straight terrace edge of a continuous

and because it must be made in the absence of the reverse reaction.

*3.3.3 Two-dimensional nucleation*

*Growth Kinetics of Thin Film Epitaxy*

*DOI: http://dx.doi.org/10.5772/intechopen.91224*

interfacial area is included:

condensate.

**Figure 11.**

**17**

*Geometry of 3D nucleation, looking down at the surface.*

*<sup>Δ</sup><sup>G</sup>* ¼ � *<sup>μ</sup><sup>v</sup>* � *<sup>μ</sup><sup>c</sup>* ð Þ *<sup>V</sup>*

*Vmc*

where �*s* denotes the reverse reaction from the transition state back to adsorption site.

Eq. (9) describes the rate balance of a reversible reaction, and Eq. (10) defines its equilibrium constant:

$$K = \frac{k\_\sharp}{k\_{-\sharp}} = \frac{n\_\sharp^\*}{n\_\sharp} e^{-\Delta\_r G^0 / RT} \tag{10}$$

Approach to equilibrium requires the forward and reverse rates to be fast enough so that they become balanced within the applicable time scale, which may be the time for deposition of one monolayer, for example. Then, the concentrations of reactant and product species are related by the difference in their free energies, Δ*rG*<sup>0</sup>*:* If, on the other hand, the forward rate is so slow that the product concentration does not have time to build up to its equilibrium level within this time scale, then the product concentration is determined not by Δ*rG*<sup>0</sup> but, instead, by the forward rate. This rate is governed by Eqs. (11) and (12):

$$R\_k = k\_k n\_s = k\_k n\_{s0} \Theta \tag{11}$$

where *Rk* = rate of the kth surface reaction per unit surface area, mc*=*cm<sup>2</sup> s; *kk* = rate constant, s�1; *ns* = surface concentration of reactant, mc*=*cm2; *ns*<sup>0</sup> = monolayer surface concentration, mc*=*cm2; Θ = fractional surface coverage by reactant.

$$k\_k = \nu\_{0k} e^{-E\_k/RT} \tag{12}$$

where *υ*0*<sup>k</sup>* = frequency factor or pre-exponential factor; *Ek* = reaction activation energy, kJ/mol in which *Ek=T* plays the dominant role. So it is that reactions can be frozen out and equilibration avoided if so desired, by lowering the *T*.

#### *Growth Kinetics of Thin Film Epitaxy DOI: http://dx.doi.org/10.5772/intechopen.91224*

The difficulty of answering the question of kinetics versus thermodynamics arises from the fact that the applicable rate constants, *kk*, are often unknown or not known accurately enough. The measurement of *kk* is much more difficult than just measuring equilibrium concentration, both because it is a dynamic measurement and because it must be made in the absence of the reverse reaction.

## *3.3.3 Two-dimensional nucleation*

*3.3.2 Kinetics vs. thermodynamics*

*21st Century Surface Science - a Handbook*

reaction-limited step.

adsorption site.

**16**

its equilibrium constant:

In general, within the framework of the nucleation kinetics model [12], a gas phase growth reaction can be divided into two steps: (1) adsorption of vaporized precursors onto substrates and diffusion to the preferential growth sites and (2) incorporation of precursors into existing nuclei. The rate-limiting step in vapor phase crystal growth can be determined as either the diffusion-limited step or the

One way to achieve smooth growth is to reduce substrate temperature, *T*, to inhibit surface diffusion, thus "freezing" the nucleation and coalescence process. If the arriving species do not have enough heat energy to desorb or diffuse, they will stay where they land, leading to the aforementioned quenching growth. In this case, the nucleation process is kinetically inhibited by the surface diffusion activation energy barrier, *Es*, in **Figure 8(a)**. This is also the case for ion-bombardment dissipation of 3D nuclei; the nuclei do not have time to reassemble themselves by

The question of whether a process is approaching equilibrium or is instead limited by kinetics is an important one, and it arises often in thin film deposition. Process behavior and film properties are profoundly affected by the degree to which one or the other situation dominates. The answer is not always apparent in a given process, and this often leads to confusion and to misinterpretation of observed phenomena. Therefore, to elaborate briefly, the generalized mathematical repre-

*Rs* <sup>¼</sup> *<sup>R</sup>*�*<sup>s</sup>* or *nsks* <sup>¼</sup> *<sup>n</sup>*<sup>∗</sup>

Eq. (9) describes the rate balance of a reversible reaction, and Eq. (10) defines

¼ *n*∗ *s ns e*

Approach to equilibrium requires the forward and reverse rates to be fast enough so that they become balanced within the applicable time scale, which may be the time for deposition of one monolayer, for example. Then, the concentrations of reactant and product species are related by the difference in their free energies, Δ*rG*<sup>0</sup>*:* If, on the other hand, the forward rate is so slow that the product concentration does not have time to build up to its equilibrium level within this time scale, then the product concentration is determined not by Δ*rG*<sup>0</sup> but, instead, by the

where *Rk* = rate of the kth surface reaction per unit surface area, mc*=*cm<sup>2</sup> s; *kk* = rate constant, s�1; *ns* = surface concentration of reactant, mc*=*cm2; *ns*<sup>0</sup> = monolayer

where *υ*0*<sup>k</sup>* = frequency factor or pre-exponential factor; *Ek* = reaction activation energy, kJ/mol in which *Ek=T* plays the dominant role. So it is that reactions can be

surface concentration, mc*=*cm2; Θ = fractional surface coverage by reactant.

*kk* ¼ *υ*0*ke*

frozen out and equilibration avoided if so desired, by lowering the *T*.

where �*s* denotes the reverse reaction from the transition state back to

*<sup>K</sup>* <sup>¼</sup> *ks k*�*s*

forward rate. This rate is governed by Eqs. (11) and (12):

*<sup>s</sup> k*�*<sup>s</sup>* (9)

�*ΔrG*<sup>0</sup>*=RT* (10)

*Rk* ¼ *kkns* ¼ *kkns*0Θ (11)

�*Ek=RT* (12)

surface diffusion before they are buried by depositing material.

sentation of this dichotomy is embodied in Eq. (9):

When wetting is complete and Eq. (8) holds, the adsorbing atoms do not accumulate into 3D islands but, instead, spread out on the surface in a partial monolayer as shown in **Figure 10a**. Because total surface energy is reduced rather than increased by this process, there is no nucleation barrier in going from the vapor state to the adsorbed state, that is, the term in Eq. (13) is negative when the interfacial area is included:

$$
\Delta \mathbf{G} = - (\mu\_v - \mu\_c) \frac{V}{V\_{mc}} + \gamma\_f A\_f = - \left( RT \ln \frac{p}{p\_v} \right) \cdot \frac{(4/3) \pi r^3}{V\_{mc}} + \gamma\_f 4 \pi r^2 \tag{13}
$$

where *pv* is saturation vapor pressure and *Vmc* is the molar volume of the condensate.

This means that deposition can proceed even in undersaturated conditions.

Assuming, as we did for 3D nucleation, that there is sufficient surface diffusion for equilibration, the partial monolayer of adsorbed atoms will behave as a 2D gas. By analogy to a 3D gas condensing into 3D nuclei, the 2D gas then condenses into 2D nuclei as illustrated in **Figure 11**. Here, only the top monolayer of atoms is drawn. The "atomic terrace" to the left represents a monolayer which is one atomic step (a) higher than the surface to the right. But unlike the 3D nucleation case, 2D nucleation from a 2D gas involves no change in any of the *γ* values, so one might expect there to be no nucleation barrier. However, the chemical potential, *μ*, of a 2D nucleus is higher than that of a continuous monolayer because of the exposed edge. This situation may be viewed in terms of an excess edge energy, *β* (J/m), which is analogous to the surface energy, *γ*, of the 3D case. The surface concentration of the 2D gas for which its *μ* is the same as that on the straight terrace edge of a continuous

**Figure 11.** *Geometry of 3D nucleation, looking down at the surface.*

monolayer (*μc*) may be thought of as the 2D saturation vapor concentration, *nv* (mc*=*m<sup>2</sup> ). If *ns* is the actual concentration of the 2D gas, then (*ns=nv*) becomes the 2D supersaturation ratio. By the same procedures as in the 3D case, we may then derive expressions for the critical nucleus as:

$$r^\* = \frac{\beta}{a\left(\frac{RT}{V\_{mc}}\right)\ln\left(\frac{n\_r}{n\_v}\right)}\tag{14}$$

wetting requires low *γ<sup>i</sup>* and therefore requires strong adsorption. As a result, it will not always be possible to achieve strong enough adsorption for wetting without immobilizing the adsorbate and preventing grain growth. Even so, small-diameter

The texturing described here refers to the crystal structure rather than the surface morphology, although they are often correlated. The degree of texturing is the degree to which the crystallites in a polycrystalline film are similarly oriented. In one limit, there is random orientation (no texturing), and in the other limit, there is the single crystal. A material in which the crystallites are nearly aligned in all three dimensions is called a "mosaic," and the limit of a perfect mosaic is a single crystal. The degree of texturing is best measured by X-ray techniques. Texturing can occur in one, two, or three dimensions. Epitaxy is the best way to achieve perfect threedimensional texturing. Epitaxy occurs when the bonds of the film crystal align with the bonds of the substrate surface, making the interfacial energy, *γi*, very low, zero in the case of homoepitaxy; this is when the film material is the same as the

substrate material. In other cases, when there is no such arrangement to operate, the most common form of film texture is two-dimensional texture, in which the crystal plane is arranged relative to the rotation of two axes on the substrate plane. This means that the film has a preferred growth plane parallel to the substrate, but has a random orientation relative to the rotation of the axis (i.e., azimuth) perpendicular to the substrate plane. It is often desirable to deposit the film on a substrate that cannot be crystallographically aligned, such as an amorphous substrate (such as glass) or a substrate with crystal symmetry or lattice size very different from the film. In this case, it is very ideal to realize two-dimensional texture when the

Because of the importance of atomically abrupt interfaces, we will focus next on physical and chemical vapor deposition processes which operate far from equilibrium in the sense that *Jv* ≪ *Rr*. This is achieved by reducing the deposition *T*, *Ts*, until *Jv* becomes negligible. It does not mean, however, that all of the steps in the deposition process are operating far from equilibrium. Recall from Section 3.2 that deposition is a series of steps: adsorption, surface diffusion, reaction, nucleation, structure development, and interdiffusion. To obtain good deposition rate control, it is only important that either the adsorption or the reaction step be far from equilibrium. To prevent the broadening of interfaces after they are formed, it is important also that *Ts* be low enough so that interdiffusion is negligible during the total time of structure deposition. However, if *Ts* is too low, surface diffusion will become negligible, and structural equilibration will not occur. This is the "quenched growth" regime, and the crystallographic quality of epilayers is poorer in this regime than at higher *Ts*. Fortunately, there is often a *Ts* "window" within which good crystallography and sharp interfaces can both be obtained. Much of the development work in epitaxy has involved modifying processes to widen this

In addition to non-equilibrium growth, one must also have chemical compatibility and reasonably good lattice match between layers to obtain good heteroepitaxy. Now let us move on to chemical interactions. Epitaxy is particularly sensitive to degradation by impurities and defects. Moreover, complete disruption of epitaxy can occur if even a fraction of a monolayer of disordered contaminant

grains can become wider as the film grows thicker.

*Growth Kinetics of Thin Film Epitaxy*

*DOI: http://dx.doi.org/10.5772/intechopen.91224*

required film properties are also crystal anisotropy.

**3.5 Submonolayer and lattice mismatch**

**3.4 Texturing**

window.

**19**

and

$$
\Delta G^\* = \frac{\pi \beta^2}{a \left(\frac{RT}{V\_{mc}}\right) \ln \left(\frac{n\_r}{n\_v}\right)}\tag{15}
$$

Here, *a* is the monolayer thickness. Once supercritical nuclei form, the 2D gas continues to attach to their edges until coalescence occurs and the monolayer is complete. Meanwhile, the next monolayer is beginning to form, and the film continues to build up in this way, atomic layer by layer. In the special case of singlecrystal film deposition (epitaxy), the surface may contain many atomic terraces with straight edges as shown in **Figure 11**. The "kink" sites shown in **Figure 11** are also important surface features. Attachment of a 2D gas atom to a random site on the straight edge involves an increase in total edge energy, because it increases the length of the edge. Conversely, attachment to the kink site makes no change in the length of the edge; this is therefore an energetically preferred site, and edge growth can most easily occur by attachment-driven motion of these kink sites along the edge.

It can be seen from the above that the surface energy depends not only on the facet direction discussed in Section 3.3.1 but also on the density of steps and kinks (Williams, 1994). The equilibrium densities of these two features increase with *T* because of their associated entropy (disorder), *S*. That is, when the *Ts* term for the Gibbs free energy, *G*, becomes larger; the internal energy term, *U*, also becomes larger to minimize *G*; and *U* here mostly consists of the potential energy of step and kink formation. This is the same *T*-driven tendency toward disorder that causes vapor pressure to rise with *T*.

During film deposition, if the surface diffusion rate is high enough and *ns* is low enough so that the 2D gas atoms are more likely to attach to an edge than to form a critical nucleus within an atomic terrace, then edge attachment becomes the dominant growth mode, that is, we have Λ > *L*, where Λ is the surface diffusion length and *L* is the distance between terraces. This is called the "continuous" growth mode, as opposed to the nucleated mode. The continuous mode of 2D growth is analogous to the type of 3D nucleation in which nucleation is more likely to occur at active surface sites than by spontaneous nucleation elsewhere on the surface. Active sites and step edges, especially kinked edges, break the nucleation barrier by providing wetting at those sites.

Two-dimensional nucleation is usually preferred to 3D because it leads to smooth growth. In nonepitaxial growth, large grain size (coarse nucleation) may be desired in addition to smoothness. Unlike in the 3D nucleation case, here large grain size and smoothness are not incompatible. That is, if adatom mobility on the substrate is sufficient, large 2D nuclei will form before the first monolayer coalesces, and then subsequent monolayers will grow epitaxially on those nuclei. But there is another problem. High adatom mobility requires a low surface diffusion activation energy, *Es*, in accordance with **Figure 8**, but *Es* tends to increase with the strength of the adsorption, *Ed* or *Ec*, as suggested in **Figure 7b**. At the same time, good

wetting requires low *γ<sup>i</sup>* and therefore requires strong adsorption. As a result, it will not always be possible to achieve strong enough adsorption for wetting without immobilizing the adsorbate and preventing grain growth. Even so, small-diameter grains can become wider as the film grows thicker.
