**3.5 Submonolayer and lattice mismatch**

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 window.

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

exists on the substrate surface or accumulates on the film surface during deposition. This is because the depositing atoms need to sense the crystallographic order of the underlying material and chemical forces extend only one or two atomic distances. An island of surface contaminant becomes the nucleus for the growth of nonepitaxial material, and this region often spreads with further deposition, as shown in **Figure 12**, rather than being overgrown by the surrounding epilayer. Contamination can enter at any step in the thin film process. Removal of substrate contamination to improve adhesion is not discussed here. The additional substrate requirements that must be met to achieve epitaxy are of great concern. These include crystallographic order, submonolayer surface cleanliness, and chemical inertness toward the depositing species. Any crystallographic disorder at the substrate surface will be propagated into the depositing film. A few materials can be obtained as prepolished wafers with excellent surface crystallography. In other cases, careful preparation is necessary to remove the disorder introduced by wafer sawing and mechanical polishing. The crystallographic damage produced by polishing-grit abrasion extends into the crystal beneath the surface scratches, to a distance of many times the grit diameter, as shown by the dislocation line networks in **Figure 13a**. This damaged region must be removed by chemical etching. To promote uniform etching and prevent pitting, the "chemical polishing" technique is used. In this technique, the etchant is applied to a soft, porous, flat pad which is wiped across the wafer. If the depth of etching is insufficient, some damage will remain, as shown in **Figure 13b**, even though the surface may appear absolutely flat and smooth under careful scrutiny by Nomarski microscopy. However, these defects can be revealed by dipping the wafer in a "dislocation" etchant [13] that preferentially attacks them and thereby decorates the surface with identifying pits and lines. The crystallographic disorder at these defects, consisting of strained and broken bonds, raises the local free energy and thereby increases reactivity toward the etchant. After sufficient chemical polishing, the only remaining defects will be those grown into the bulk crystal, as shown at the etch pits in **Figure 13c**.

Finally, the lattice mismatch is discussed. The expression of lattice mismatch

Having now dealt with avoiding precipitates and controlling point defects, we can proceed to the problem of minimizing other crystallographic defects. It is useful

≈ð Þ ae � as *=*as (16)

<sup>f</sup> <sup>¼</sup> ð Þ ae � as ð Þ ae þ as *=*2

*Modes of accommodating epilayer lattice (solid circles) to substrate lattice (white circles).*

*Crystallographic damage due to wafer sawing and mechanical polishing.*

to think of defects in terms of their dimensionality. Point defects are zerodimensional (0D), while precipitates or disordered regions are 3D. Planar (2D) defects include grain boundaries, twin planes, stacking faults, and antiphase domain boundaries. Dislocations are line (1D) defects. We will see below how dislocations arise from the fractional lattice mismatch, f, at heteroepitaxial interfaces. For this purpose, we consider the simple square symmetry of cubic material growing in (001) orientation on a (001)-oriented substrate, although the same principles apply to other symmetries. **Figure 14** shows the various modes of

factor is as follows:

*Growth Kinetics of Thin Film Epitaxy*

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

**Figure 13.**

**Figure 14.**

**21**

After crystallographic preparation of the substrate, surface contamination must be removed. In the final chemical cleaning step prior to wafer installation in the deposition chamber, one seeks to minimize residual surface contamination and also to select its composition so that it is more easily removed by the techniques available in the chamber.

**Figure 12.**

*Effect of submonolayer surface contamination on epitaxy.*

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

exists on the substrate surface or accumulates on the film surface during deposition. This is because the depositing atoms need to sense the crystallographic order of the underlying material and chemical forces extend only one or two atomic distances.

An island of surface contaminant becomes the nucleus for the growth of nonepitaxial material, and this region often spreads with further deposition, as shown in **Figure 12**, rather than being overgrown by the surrounding epilayer. Contamination can enter at any step in the thin film process. Removal of substrate contamination to improve adhesion is not discussed here. The additional substrate requirements that must be met to achieve epitaxy are of great concern. These include crystallographic order, submonolayer surface cleanliness, and chemical inertness toward the depositing species. Any crystallographic disorder at the substrate surface will be propagated into the depositing film. A few materials can be obtained as prepolished wafers with excellent surface crystallography. In other cases, careful preparation is necessary to remove the disorder introduced by wafer sawing and mechanical polishing. The crystallographic damage produced by polishing-grit abrasion extends into the crystal beneath the surface scratches, to a distance of many times the grit diameter, as shown by the dislocation line networks in **Figure 13a**. This damaged region must be removed by chemical etching. To promote uniform etching and prevent pitting, the "chemical polishing" technique is used. In this technique, the etchant is applied to a soft, porous, flat pad which is wiped across the wafer. If the depth of etching is insufficient, some damage will remain, as shown in **Figure 13b**, even though the surface may appear absolutely flat and smooth under careful scrutiny by Nomarski microscopy. However, these defects can be revealed by dipping the wafer in a "dislocation" etchant [13] that preferentially attacks them and thereby decorates the surface with identifying pits and lines. The crystallographic disorder at these defects, consisting of strained and broken bonds, raises the local free energy and thereby increases reactivity toward the etchant. After sufficient chemical polishing, the only remaining defects will be

*21st Century Surface Science - a Handbook*

those grown into the bulk crystal, as shown at the etch pits in **Figure 13c**.

able in the chamber.

**Figure 12.**

**20**

*Effect of submonolayer surface contamination on epitaxy.*

After crystallographic preparation of the substrate, surface contamination must be removed. In the final chemical cleaning step prior to wafer installation in the deposition chamber, one seeks to minimize residual surface contamination and also to select its composition so that it is more easily removed by the techniques avail-

**Figure 13.** *Crystallographic damage due to wafer sawing and mechanical polishing.*

Finally, the lattice mismatch is discussed. The expression of lattice mismatch factor is as follows:

$$\mathbf{f} = \frac{(\mathbf{a\_e} - \mathbf{a\_s})}{(\mathbf{a\_e} + \mathbf{a\_s})/2} \approx (\mathbf{a\_e} - \mathbf{a\_s})/\mathbf{a\_s} \tag{16}$$

Having now dealt with avoiding precipitates and controlling point defects, we can proceed to the problem of minimizing other crystallographic defects. It is useful to think of defects in terms of their dimensionality. Point defects are zerodimensional (0D), while precipitates or disordered regions are 3D. Planar (2D) defects include grain boundaries, twin planes, stacking faults, and antiphase domain boundaries. Dislocations are line (1D) defects. We will see below how dislocations arise from the fractional lattice mismatch, f, at heteroepitaxial interfaces. For this purpose, we consider the simple square symmetry of cubic material growing in (001) orientation on a (001)-oriented substrate, although the same principles apply to other symmetries. **Figure 14** shows the various modes of

**Figure 14.** *Modes of accommodating epilayer lattice (solid circles) to substrate lattice (white circles).*

mismatch accommodation. In the special case of perfect match (a), the lattices are naturally aligned, and the growth is therefore "commensurate" without requiring lattice strain. In (b–d), the atomic spacing of the epilayer, ae, is larger than that of the substrate, as. In fact, f has been made quite large (0.14) here so that it may be readily observed, but it is much smaller in most heteroepitaxial systems of interest. coherently strained epilayer U<sup>ϵ</sup> is obtained by integrating force over distance as the film is compressed toward a fit to the substrate, starting from the relaxed state shown in **Figure 14b**. The force to maintain the compression is supplied from the rigid substrate by bonding across the interface. The integration can be done in one direction and then doubled to account for the orthogonal direction. The force, F, in the x (or y) direction, per unit width of film in y (or x), is given by Eq. (20):

> Y 1 � υ � �

where Ff, Fs; σf, σs; and hf, hs are force, stress, and thickness of film and substrate, respectively, and the distance in x is the same as the strain Ex if we use a

The force of compression creates shear stresses in crystal planes that are not perpendicular to it, and along certain of these planes, the film will "slip" to relieve stress by breaking and then reforming bonds. After slippage, there will be extra rows of substrate atoms which are not bonded to the film, such as the one shown along y in **Figure 14d**. These features are known as misfit dislocations. Film stress is relieved by the development of a grid of such dislocations in the interface, the grid periodicity being determined by energy minimization, as we will see below. This growth mode is known as "discommensurate." In addition to the bulk film strain energy, Uε, there is an interface energy, γi, as discussed in Section 3.3.1, but since it

Usually, defects of any dimensionality (0D through 3D) are undesirable within a

The above discussion has examined the factors determining epitaxy film structure, topography, interfacial properties, and stress. The kinetic mechanism of atom adsorption, diffusion, reaction, nucleation, and texture is given. The kinetic characteristics and related technological conditions of two-dimensional nucleation and layered ordered growth are described. A new optimized denotation index (a *Ts* "window" within which good crystallography and sharp interfaces can both be obtained) for epitaxy growth is proposed. Much of the development work in epitaxy has involved modifying processes to widen this window. Finally, two main factors in epitaxial growth are proposed. Two principal factors are the degree of interaction of the depositing vapor with the substrate and with itself and the amount of energy input to the deposition surface. When the energy input is ther-

film unless they are introduced for a specific purpose such as doping. Films in electronic applications are particularly sensitive to degradation by defects. They disturb the lattice periodicity and thus locally alter the band structure of a semiconductor crystal, often producing charge carrier traps or charge recombination centers within the band gap. Defects of 1D and 2D also provide paths for electrical leakage and impurity diffusion. Thus, in heteroepitaxial growth, it is important to know what conditions have to be met to avoid the generation of misfit dislocations. This situation needs to be analyzed based on the discussion of the properties of

f

Y 1 � υ

<sup>ε</sup>fhf <sup>¼</sup> <sup>Y</sup>

� �εxhdε<sup>x</sup> <sup>¼</sup> <sup>Y</sup>

1 � υ � �

fs

1 � υ � �ε<sup>2</sup>

ε<sup>s</sup> (20)

xh (21)

Ff ¼ Fs or σfhf ¼ σshs or

ðx 0 Fx dx Lx ¼ 2 ð<sup>ε</sup><sup>x</sup> 0

normalized film length of Lx ¼ 1. Thus:

¼ 2Ux ¼ 2

does not depend on h, it will be neglected below.

dislocations. It is not discussed here because of the space.

mal, care must be taken to achieve good substrate T control.

Uϵ

J m2 or

**4. Conclusion**

**23**

N m � �

*Growth Kinetics of Thin Film Epitaxy*

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

There are several ways in which lattice mismatch can be accommodated. In **Figure 14b**, bonding across the interface is weak, so that the epilayer "floats" on top of the substrate and is therefore "incommensurate" with it. This mode occurs, for example, with materials having a 2D, layered structure, such as graphite and MoS2 [14]. In such compounds, there is no chemical bonding perpendicular to the hexagonally close-packed and tightly bonded basal plane, so that interaction of such a film with the substrate is only by Van der Waals forces. These weak forces are often strong enough to maintain rotational alignment with the substrate and to produce a small periodic compression and expansion in the epilayer lattice, but they are not strong enough to strain the epilayer so that it fits that of the substrate. There is a small periodic distortion in ae as the lattices fall in and out of alignment periodically across the interface, and this produces a beautiful Moire pattern in STM images of the epilayer surface. Incommensurate growth can also occur when chemical bonding is weak because of a difference in bonding character between film and substrate. Chemical bonding can also be blocked by passivating the substrate surface.

In the more common situation, the epilayer is chemically bonded to the substrate, thus forming a unit called a "bicrystal." A thin epilayer with small f is likely to become strained to fit the substrate in *x* as shown in **Figure 14c** and similarly in *y*. This is sometimes referred to as "pseudomorphic" growth, but it really is not, because no change in crystal structure has occurred. It is properly termed "commensurate growth" or "coherent epitaxy." In **Figure 14c**, it is assumed that the substrate is much thicker than the epilayer, so that the substrate is rigid and all of the strain is in the epilayer. This "coherency" strain is then just Ex ¼ Ey ¼ Ex,y ¼ �f, and the corresponding biaxial stress, σxy, is given by Eq. (18). The biaxial stress produces a strain in z, perpendicular to the growth plane, which is given by the three-dimensional form of Hooke's law:

$$\mathfrak{e}\_{\mathbf{z}} = \frac{1}{\mathbf{Y}} \left( \mathfrak{o}\_{\mathbf{z}} - \mathfrak{u}\mathfrak{o}\_{\mathbf{x}} - \mathfrak{u}\mathfrak{o}\_{\mathbf{y}} \right) = \frac{-2\mathfrak{u}\mathfrak{o}\_{\mathbf{x},\mathbf{y}}}{\mathbf{Y}} = \frac{-2\mathfrak{u}\mathfrak{e}\_{\mathbf{x},\mathbf{y}}}{\mathbf{1} - \mathfrak{u}} \tag{17}$$

Here, the second equality was obtained by setting σ<sup>z</sup> as it must be for the unconstrained direction, and the third was obtained using Eq. (18):

$$
\mathbf{e}\_{\mathbf{x}} = \mathbf{e}\_{\mathbf{y}} = \mathbf{e}\_{\mathbf{x},\mathbf{y}} = \frac{(\mathbf{1} - \mathbf{o})}{\mathbf{Y}} \sigma\_{\mathbf{x},\mathbf{y}} \frac{\sigma\_{\mathbf{x},\mathbf{y}}}{\mathbf{Y}'} \tag{18}
$$

(where Y<sup>0</sup> is sometimes known as the biaxial elastic modulus. Poisson's ratio).

In **Figure 14c**, the epilayer is shown compressed in x and y and expanded in z in accordance with the above formula. This lattice is said to be "tetragonally" distorted, and the tetragonal strain is defined as:

$$
\varepsilon\mathbf{r} = \mathbf{e}\_{\mathbf{z}} - \mathbf{e}\_{\mathbf{x}, \mathbf{y}} = -\left(\frac{\mathbf{1} + \mathbf{v}}{\mathbf{1} - \mathbf{v}}\right) \mathbf{e}\_{\mathbf{x}, \mathbf{y}} \tag{19}
$$

X-ray diffraction measurement of the expanded atomic plane spacing a<sup>0</sup> in z can be used with Eq. (17) to determine the fraction by which the epilayer lattice has compressed to fit the substrate in x and y. Electron diffraction can be used only when the change in a is larger than a few percent, because the peaks are much broader than in X-ray diffraction. The strain energy stored per unit area in the

mismatch accommodation. In the special case of perfect match (a), the lattices are naturally aligned, and the growth is therefore "commensurate" without requiring lattice strain. In (b–d), the atomic spacing of the epilayer, ae, is larger than that of the substrate, as. In fact, f has been made quite large (0.14) here so that it may be readily observed, but it is much smaller in most heteroepitaxial systems of interest. There are several ways in which lattice mismatch can be accommodated. In **Figure 14b**, bonding across the interface is weak, so that the epilayer "floats" on top of the substrate and is therefore "incommensurate" with it. This mode occurs, for example, with materials having a 2D, layered structure, such as graphite and MoS2 [14]. In such compounds, there is no chemical bonding perpendicular to the hexagonally close-packed and tightly bonded basal plane, so that interaction of such a film with the substrate is only by Van der Waals forces. These weak forces are often strong enough to maintain rotational alignment with the substrate and to produce a small periodic compression and expansion in the epilayer lattice, but they are not strong enough to strain the epilayer so that it fits that of the substrate. There is a small periodic distortion in ae as the lattices fall in and out of alignment periodically across the interface, and this produces a beautiful Moire pattern in STM images of the epilayer surface. Incommensurate growth can also occur when chemical bonding is weak because of a difference in bonding character between film and substrate.

Chemical bonding can also be blocked by passivating the substrate surface.

three-dimensional form of Hooke's law:

*21st Century Surface Science - a Handbook*

<sup>ε</sup><sup>z</sup> <sup>¼</sup> <sup>1</sup>

distorted, and the tetragonal strain is defined as:

**22**

<sup>Y</sup> <sup>σ</sup><sup>z</sup> � υσ<sup>x</sup> � υσ<sup>y</sup>

unconstrained direction, and the third was obtained using Eq. (18):

<sup>¼</sup> �2υσx,y

Here, the second equality was obtained by setting σ<sup>z</sup> as it must be for the

Y

(where Y<sup>0</sup> is sometimes known as the biaxial elastic modulus. Poisson's ratio). In **Figure 14c**, the epilayer is shown compressed in x and y and expanded in z in

X-ray diffraction measurement of the expanded atomic plane spacing a<sup>0</sup> in z can be used with Eq. (17) to determine the fraction by which the epilayer lattice has compressed to fit the substrate in x and y. Electron diffraction can be used only when the change in a is larger than a few percent, because the peaks are much broader than in X-ray diffraction. The strain energy stored per unit area in the

σx,y σx,y

1 � υ 

<sup>ε</sup><sup>x</sup> <sup>¼</sup> <sup>ε</sup><sup>y</sup> <sup>¼</sup> <sup>ε</sup>x,y <sup>¼</sup> ð Þ <sup>1</sup> � <sup>υ</sup>

accordance with the above formula. This lattice is said to be "tetragonally"

<sup>ε</sup><sup>T</sup> <sup>¼</sup> <sup>ε</sup><sup>z</sup> � <sup>ε</sup>x,y ¼ � <sup>1</sup> <sup>þ</sup> <sup>υ</sup>

<sup>Y</sup> <sup>¼</sup> �2υεx,y 1 � υ

(17)

<sup>Y</sup><sup>0</sup> (18)

εx,y (19)

In the more common situation, the epilayer is chemically bonded to the substrate, thus forming a unit called a "bicrystal." A thin epilayer with small f is likely to become strained to fit the substrate in *x* as shown in **Figure 14c** and similarly in *y*. This is sometimes referred to as "pseudomorphic" growth, but it really is not, because no change in crystal structure has occurred. It is properly termed "commensurate growth" or "coherent epitaxy." In **Figure 14c**, it is assumed that the substrate is much thicker than the epilayer, so that the substrate is rigid and all of the strain is in the epilayer. This "coherency" strain is then just Ex ¼ Ey ¼ Ex,y ¼ �f, and the corresponding biaxial stress, σxy, is given by Eq. (18). The biaxial stress produces a strain in z, perpendicular to the growth plane, which is given by the

coherently strained epilayer U<sup>ϵ</sup> is obtained by integrating force over distance as the film is compressed toward a fit to the substrate, starting from the relaxed state shown in **Figure 14b**. The force to maintain the compression is supplied from the rigid substrate by bonding across the interface. The integration can be done in one direction and then doubled to account for the orthogonal direction. The force, F, in the x (or y) direction, per unit width of film in y (or x), is given by Eq. (20):

$$\mathbf{F}\_{\mathbf{f}} = \mathbf{F}\_{\mathbf{s}} \text{ or } \sigma\_{\mathbf{f}} \mathbf{h}\_{\mathbf{f}} = \sigma\_{\mathbf{s}} \mathbf{h}\_{\mathbf{s}} \text{ or } \left(\frac{\mathbf{Y}}{\mathbf{1} - \mathbf{u}}\right)\_{\mathbf{f}} \mathbf{e}\_{\mathbf{f}} \mathbf{h}\_{\mathbf{f}} = \left(\frac{\mathbf{Y}}{\mathbf{1} - \mathbf{u}}\right)\_{\mathbf{f} \mathbf{s}} \mathbf{e}\_{\mathbf{s}} \tag{20}$$

where Ff, Fs; σf, σs; and hf, hs are force, stress, and thickness of film and substrate, respectively, and the distance in x is the same as the strain Ex if we use a normalized film length of Lx ¼ 1. Thus:

$$\mathbf{U\_e \left(\frac{\mathbf{J}}{\mathbf{m}^2} \text{or} \frac{\mathbf{N}}{\mathbf{m}}\right)} = 2\mathbf{U\_x} = 2\int\_0^\mathbf{x} \mathbf{F\_x} \frac{d\mathbf{x}}{\mathbf{L\_x}} = 2\int\_0^{\mathbf{e\_x}} \left(\frac{\mathbf{Y}}{\mathbf{1} - \mathbf{u}}\right) \mathbf{e\_x} \mathbf{h} d\mathbf{e\_x} = \left(\frac{\mathbf{Y}}{\mathbf{1} - \mathbf{u}}\right) \mathbf{e\_x^2} \mathbf{h} \tag{21}$$

The force of compression creates shear stresses in crystal planes that are not perpendicular to it, and along certain of these planes, the film will "slip" to relieve stress by breaking and then reforming bonds. After slippage, there will be extra rows of substrate atoms which are not bonded to the film, such as the one shown along y in **Figure 14d**. These features are known as misfit dislocations. Film stress is relieved by the development of a grid of such dislocations in the interface, the grid periodicity being determined by energy minimization, as we will see below. This growth mode is known as "discommensurate." In addition to the bulk film strain energy, Uε, there is an interface energy, γi, as discussed in Section 3.3.1, but since it does not depend on h, it will be neglected below.

Usually, defects of any dimensionality (0D through 3D) are undesirable within a film unless they are introduced for a specific purpose such as doping. Films in electronic applications are particularly sensitive to degradation by defects. They disturb the lattice periodicity and thus locally alter the band structure of a semiconductor crystal, often producing charge carrier traps or charge recombination centers within the band gap. Defects of 1D and 2D also provide paths for electrical leakage and impurity diffusion. Thus, in heteroepitaxial growth, it is important to know what conditions have to be met to avoid the generation of misfit dislocations. This situation needs to be analyzed based on the discussion of the properties of dislocations. It is not discussed here because of the space.

### **4. Conclusion**

The above discussion has examined the factors determining epitaxy film structure, topography, interfacial properties, and stress. The kinetic mechanism of atom adsorption, diffusion, reaction, nucleation, and texture is given. The kinetic characteristics and related technological conditions of two-dimensional nucleation and layered ordered growth are described. A new optimized denotation index (a *Ts* "window" within which good crystallography and sharp interfaces can both be obtained) for epitaxy growth is proposed. Much of the development work in epitaxy has involved modifying processes to widen this window. Finally, two main factors in epitaxial growth are proposed. Two principal factors are the degree of interaction of the depositing vapor with the substrate and with itself and the amount of energy input to the deposition surface. When the energy input is thermal, care must be taken to achieve good substrate T control.

*21st Century Surface Science - a Handbook*
