4.2  The critical thickness criterion

The system shown in Figure 4.1 is composed of a thin film with a finite thickness in the region 0 < z < h, and a semi-infinite substrate filling the half space z < 0. The dislocation is along the interface z = 0 aligned along the y-axis. The critical thickness criterion compares the dislocation energy per unit length, d∕dy, with the work, d∕dy, done by the misfit stress, σ_ij^m, in forming the unit length of a dislocation along the film–substrate interface. As a consequence, the critical thickness criterion is given by

(4.1)

4.2.1  Work done by the misfit strain

Since the film is much thinner than the semi-infinite substrate, the strain in the substrate is assumed to be zero while the complete lattice mismatch is accommodated in the film (i.e., any mismatch strain relaxation by substrate bending is neglected). According to the geometry shown in Figure 4.1, the top surface z = h of the film is traction free and therefore σ_xz^m(x,h) = σ _zz^m(x,h) = σ _yz^m(x,h) = 0. Assuming all directions within the xy-plane (the hexagonal plane) are equivalent, a planar stress state is obtained and thus the mismatch strain components are ε_xx^m = ε _yy^m = ε _m, where ε_m is the misfit strain, defined as

(4.2)

considering that the thin film’s in-plane lattice constant a adjusts to the rigid substrate’s lattice constant a^*.

Hooke’s law (2.10) for isotropic structures reads

(4.3)

where c₄₄ = ∕2. In the case of cubic and hexagonal symmetries, the shape of the stiffness matrix c_ij changes to

(4.4)

respectively, where c₆₆ = (c₁₁ - c₁₂)∕2 for the hexagonal symmetry (the right matrix above).

Considering for example the hexagonal symmetry, Hooke’s law gives

(4.5)

which results in

(4.6)

Finally, Hooke’s law yields for the mismatch stress

(4.7)

In the chosen Cartesian coordinate system, the Burgers vector b, the tensor of the misfit stress σ^m, and the outer normal to the cut surface Γ = S ₂ + S₄ (see Figure 3.1) denoted by n take the forms

=	∫ 0h∕cosϕ ∑ i,jbiσijmn jdz = ∫ 0h∕cosϕ ∑ i dz
=	bσmh sin θ sin ϕ.	(4.9)

4.2.2  Dislocation energy

The formula for the dislocation energy d_d∕dy for a straight dislocation inside an infinite isotropic medium reads  [19,23,32]:

(4.10)

where μ and ν are the shear modulus and the Poisson ratio, respectively, φ is the angle between the Burgers vector b and the dislocation line, and R is the outer cut-off radius. Here it is taken to be equal to the film thickness.

Chapter 3.2 describes how to calculate the dislocation energy d_d∕dy inside an infinite anisotropic medium according to the procedure developed by Steeds  [72] and described in Section 3.4:

(4.11)

where K is the pre-logarithmic factor and it is a function of the crystal structure and the elastic properties (see Section 3.4). Holec  [25] derived a simplified expression of K from the Steeds procedure specifically for a dislocation along the y-axis inside a hexagonal crystal. The expression is easily adapted to the cubic symmetry by simply modifying the stiffness tensor.

If we want to calculate the dislocation energy at the interface between the film and the substrate of a heteroepitaxial structure, it is necessary to evaluate the effect of the free surface of the film and the different elastic properties of the two materials. Willis, Jain and Bullough  [82] have evaluated both effects on the dislocation energy within isotropic elasticity.

The adaptation of the Willis, Jain and Bullough model to the hexagonal symmetry can be found in the works of Holec  [24] and Coppeta  [8]. This procedure is reported in details in the next Paragraph 4.2.3 for a straight dislocation at the interface between a finite hexagonal film and a semi-infinite hexagonal substrate with different elastic constants.

In conclusion, four approaches (summarized in Table 4.1) for calculating the dislocation energy were defined above, which will be used to evaluate the impact of different approximations.

4.2.3  A straight dislocation at the interface of anisotropic materials

The evaluation of the integrals along the surfaces S₂ and S₄ (see Figure 3.1) deals with a straight dislocation at the interface between a finite anisotropic film and a semi-infinite anisotropic substrate with different elastic properties:

(4.12)

Considering Figure 4.2, the previous equation becomes

(4.13)

If the extremes of the second integral are inverted, one obtains

(4.14)

where u_i(x → 0⁺) - u _i(x → 0^-) = b _i.
The evaluation of the stress components in (4.14) is performed through the treatment suggested by the Willis, Jain and Bullough model  [82] for hexagonal and cubic symmetries following the Steeds procedure  [72].

The dislocation of interest is a misfit dislocation along the c-plane of the wurtzite structure or the closed packed plane of the diamond structure. The z-axis is perpendicular to the c-plane or to the closed packed plane, respectively (See Figure 4.3). The dislocation is considered straight and extended to infinity along the y-axis. This assumption simplifies the problem to a plane strain problem where no quantity depends on the y-coordinate, so

(4.15)

The Burgers vector of the dislocation is b. Displacements are given by the functions u_x, u_y and u_z. The displacements u_x and u_z correspond to the edge component of the considered dislocation whereas u_y corresponds to the screw component. The strain components are

	εxx = ,εxy = ,
	εyy = 0,εxz = ,	(4.16)
	εzz = ,εyz = .

	+ = 2,	(4.17a)
	- = 0.	(4.17b)

The fact that ε_yy = 0 yields a relation between particular stress components:

(4.18)

The compliances reflecting the proper hexagonal symmetry have been used to obtain:

(4.19)

where

(4.20)

The shape of the tensor for cubic symmetry (in both Si(100) and Si(110)) is the same as for the hexagonal one. The subsequent treatment is valid for the cubic symmetry with s₆₆ = s₄₄ and s₁₃ = s₁₂.

According to the Willis, Jain and Bullough model, it is useful now to have the jumps in displacements occur over a surface which is perpendicular to the free surface instead of across the slip plane. All quantities in the substrate have a * superscript to distinguish them from quantities related to the thin film. The Fourier transform of a function f is denoted by . It is possible to split the problem into two independent parts, resolving the edge and the screw components separately.

Edge component

The displacement field in the thin film can be decomposed into

(4.21)

where be = and the function v is continuous for all x. The strain field (and thus also the stress field) inside the thin layer is determined only by the v part as (except at the cut surface x = 0) the derivatives of the sgn(x) function are equal to zero everywhere. To solve this problem, Fourier-transformed variables are employed:

(4.22)

The Fourier transform of the ∂∕∂x operator is -iξ. Using (4.21) and (4.16), the Fourier components of the strain tensor are:

	xx(ξ,z) = -iξ x(ξ,z),	(4.23a)
	zz(ξ,z) = (ξ,z),	(4.23b)
	xz(ξ,z) = .	(4.23c)

The equilibrium conditions  [72] take the form

+ = 0,		(4.24a)
+ = 0.		(4.24b)

Transforming these equations into their Fourier equivalents and using Hooke’s law with the stiffness tensor gives the following system of partial differential equations for the Fourier-transformed components of the displacements _x and _z:

(4.25)

In order to simplify the notation, the quantities A = -c₁₁ , B = c₄₄, C = -i(c₁₃ + c₄₄), D = -c₄₄, E = c₃₃ are introduced. From the second equation of the system one obtains

(4.26)

Using this equation and its twice differentiated form with respect to z and substituting them into the first equation of the system (4.25), which is once differentiated with respect to z, yields

(4.27)

The solution has the exponential form e^λz. The corresponding proper λs are solutions of the characteristic equation

(4.28)

The solutions are λ₁ = κ₁, λ₂ = -κ₁, λ₃ = κ₂ and λ₄ = κ₂, where

(4.29)

Therefore, the general solution to (4.27) has the form  [72]

(4.30)

where A_i are functions of ξ to be determined. The component _x can then be obtained from (4.26). Its general solution is

(4.31)

Evidently

(4.32)

The Fourier-transformed components of the strain tensor are now easy to obtain from combining Equations (4.30), (4.31), and (4.23):

xx	= ∑ i=14A iξ2eλiz,	(4.33a)
xz	= - ∑ i=14iA iξλieλiz,	(4.33b)
zz	= ∑ i=14A iξ2λ i2eλiz.	(4.33c)

Using Hooke’s law, which has the same form when Fourier transformed, one obtains:

xx	= c11 xx + c13 zz = -∑ i=14A iξ2λ i2c 44eλiz,	(4.34a)
xz	= 2c44 xz = -∑ i=14iA iξλic44eλiz,	(4.34b)
zz	= c13 xx + c33 zz = ∑ i=14A iξ2c 44eλiz,	(4.34c)

where the identity c₁₁c₄₄ + λ_i² + c ₄₄c₃₃λ_i⁴ = 0 was used in the expression for _xx.

The general solution for the substrate region takes the same form (except that all variables have a star superscript). It is assumed that the substrate is not influenced by the thin layer as z →-∞. To fulfill this condition, the constants A₂^* and A ₄^* must be identically zero. Boundary conditions must be employed now in order to determine the constants A₁, A₂, A₃, A₄, A₁^*, and A ₃^*.

The Fourier transform of 1∕2sgn(x) is i∕. The continuity of displacements across the interface z = 0 is expressed by the following equations:

x + = x* ,		(4.35a)
z + = z* ,		(4.35b)

which result in the equations

	Γ1 + Γ2 + = A1*Γ 1* + A 3*Γ 2*,	(4.36a)
	κ1 + κ2 + = A1*κ 1* + A 3*κ 2*,	(4.36b)

where the notation

Γi = ,Γi* = ,i = 1, 2.

(4.37)

The second set of equations is obtained from the requirement that tractions must be continuous across the interface z = 0:

xz = xz* ,		(4.38a)
zz = zz* .		(4.38b)

Combining Equations (4.34) and (4.38) gives

	κ1Λ1 + κ2Λ2 = A1*κ 1*Λ 1* + A 3*κ 2*Λ 2*	(4.39a)
	Λ1 + Λ2 = A1*Λ 1* + A 3*Λ 2*,	(4.39b)

where the simplifying notation

Λi = ,Λi* = ,i = 1, 2,

(4.40)

The condition _xz = 0 is expressed as

κ 1Λ1 + κ 2Λ2 = 0

(4.41)

Λ 1 + Λ 2 = 0.

(4.42)

Screw component

The procedure for the screw component is similar but simpler. The Burgers vector of a screw dislocation has only one non-zero component, b_y, and thus the only non-zero component of the displacement is u_y. Moreover, it is again a plane strain problem, i.e., ∂∕∂y = 0. As a consequence, the only non-zero strain components are ε_xy and ε_yz. Similarly to the case of the edge dislocation, it is more convenient to deal with the Fourier-transformed component _y rather than with u_y itself. The Fourier-transformed components of the strain are

xy = -iξ y,		(4.43a)
yz = ,		(4.43b)

and the FT stress components obtained from Hooke’s law are

	xy = 2c66 xy = -iξc66 y,	(4.44a)
	yz = 2c44 yz = c44.	(4.44b)

The equilibrium condition  [72]

+ = 0

(4.45)

- ξ2c 66 y + c44 = 0,

(4.46)

y = A1e z + A2e-z .

(4.47)

	A1 + A2 + = A1*,	(4.48a)
	= A1*,	(4.48b)
	A1e h - A2e-h = 0.	(4.48c)

Values of A₁ and A₂ fulfilling these equations are

	A1 = ,	(4.49a)
	A2 = .	(4.49b)

The stress components can be obtained by the inverse Fourier transform of _xy and _xz with the substituted A₁ and A₂ from the last two equations.

After that, the components of the stress and the Burgers vector can be substituted in (4.14) to calculate the dislocation energy.

4.2.4  The critical thickness models

According to equation (4.1), equating the work done by the misfit strain with each of the four formulas for dislocation energy yields four different models with which to calculate the equilibrium critical thickness . These models are summarized with the respective hypotheses in Table 4.1 and their results are discussed in the subsequent Paragraphs. In particular, the critical thickness model developed by Freund  [17], which neglects the differences between film and substrate, considers only isotropic elasticity, and in its simplified (often used) formula also neglects the effect of the free surface. The Freund model is corrected for the elastic anisotropy using the Steeds treatment for the dislocation energy  [72]. The fully isotropic model, but with the free surface and different elastic constants in the film and the substrate, employs the Willis, Jain and Bullough formula  [82]. Finally, a model addressing all effects  [8,24] is reported in Paragraph 4.2.3.