# Continuing Reduction Of Silicon Integrated Circuits Biology Essay

With the continuing reduction of silicon integrated circuits, new engineering solutions and innovative techniques are required to improve bipolar transistors performance, and to overcome the physical limitations of the device scaling. Therefore, strained-silicon technology has become a strong competitor in search for alternatives to transistor scaling and new materials for improved devices and circuits performances. Strained-Si technology enables improvements in electronic devices performance and functionality via replacement of the bulk, crystalline-Si substrate with a strained-Si substrate. The improved performance comes from the properties of strained-Si itself through changing the nature of the wafer by stretching and/or compressing the placement of the atoms. This chapter gives an overview of the elasticity theory of solids , physics behind strain, different strain types, and application techniques of strain.

## Theory of Elasticity

Elasticity is the ability of a solid to recover its shape when the deforming forces are removed. The deformation of an elastic material obeys Hooke's law, which states that deformation is proportional to the applied stress up to a certain point. This point is called the elastic limit. Beyond this point additional stresses will cause permanent deformation [1]. The main law governing elasticity of materials is the theory of stress, strain and their interdependence will be discussed.

## The Stress Tensor

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Stress is defined as the force per unit area. When a deforming force is applied to a body, the stress is defined as the ratio of the force to the area over which it is applied. There are two basic types of stress. If the force is perpendicular (normal) to the surface over which it is acting, then the stress is termed a normal stress; if it is tangential to the surface, it is called a shear stress. Usually, the force is neither entirely normal nor tangential, but is at some arbitrary intermediate angle. In this case it can be resolved into components which are both normal and tangential to the surface; so the stress is composed of both normal and shearing components. The sign convention is that tensional stresses are positive and compression stresses are negative.

Let's take an arbitrary solid body oriented in a Cartesian coordinate system. With a number of forces are acting on it in different directions, such that the net force (the vector sum of the forces) on the body is zero. Conceptually, we slice the body on a plane normal to the x-direction (parallel to the YZ-plane). A small area on this plane can be defined as:

The total force acting on this small area is given by :

We can define three scalar quantities:

## The first subscript refers to the plane and the second refers to the force direction. Similarly considering slices orthogonal to the Y and Z -directions, we obtain

For static equilibrium ,, and , resulting in six independent scalar quantities. These scalar quantities can be arranged in a matrix form to yield the stress tensor [2]:

## The Strain Tensor

Strain is defined as the change in length in a given direction divided by the initial length in that direction. If a force is applied to a solid object, that may simultaneously translating, rotating and deforming the object [3]. If we consider the two arbitrary neighboring points P and Q are marked at initial position x and respectively. after deformation these points move to position and respectively. The absolute squared distance between the deformed points can be written as

For small displacement , a Taylor expansion about the point x gives the absolute squared distance as

The squared distance between the original points can be written as

The change in the squared distance can be written as

Where is the strain tensor components, and are defined as

For , the second term in equation (2.11) can be neglected, and the resultant tensor is

Therefore, the strain tensor is analogous to the stress tensor and can be written as

The diagonal terms are the normal strains in the directions X, Y, and Z respectively. While the off-diagonal terms are equal to one half of the engineering shear strain.

The strain components in three dimensions can be written as

Where u, v, and w are the displacements in the X, Y, and Z directions, respectively [4].

## Stress-Strain Relationship

Stress and strain are linked in elastic media by a stress-strain or constitutive relationship. This relation between stress and strain was first identified by Robert Hook. For Hookean elastic solid, the stress tensor is linearly proportional to the strain tensor over a specific range of deformation. The most general linear relationship between the stress and strain tensors can be written as

Where is a fourth-order elastic stiffness tensor with 81 () elements.

However, due to the symmetries involved for the stress and strain tensors under equilibrium, is reduced to a tensor of 36 elements. To simplify the notations , the stress and strain tensors can be written as vectors using the contracted notations. First the off-diagonal strain terms are converted to engineering shear strains (The off-diagonal terms are equal to one-half of the engineering shear strain).

Where is the engineering shear strain.

The resulting matrix in no longer a tensor because it doesn't follow the coordinate-transformation rules. Then the elements are renumbered as the following

## The relationship between the stress vector and the strain vector can be written as

The material property matrix with all of the elastic tensor constants (C's) is known as the stiffness matrix. The inverse of the stiffness matrix is called compliance matrix, S, where [5]-[6]. The compliance matrix is written as

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for linear elastic isotropic materials where the physical properties are independent of direction.

Therefore, Hooke's law takes on a simple form involving only two independent variables [7]. In stiffness form, Hooke's law for the isotropic medium is

where E is the Young's modulus and v is the Poisson ratio.

For anisotropic materials such as cubic crystals (i.e. Si, and Ge crystals), in which their elastic properties are direction dependent. It is possible to simplify Hook's law by considerations of cubic symmetry. If the X, Y, and Z axes coincide with the [100], [010], and [001] directions in the cubic crystal, respectively, then Hooke's law in stiffness form can be written as

for cubic crystals, the compliance-stiffness constants relationships are given by

The stiffness coefficients and compliance coefficients for Si and Ge are listed in Table.2.1.

## C11

## C12

## C44

## S11

## S12

## S44

## Si

165.64

63.94

79.51

0.7691

-0.2142

1.2577

## Ge

128.7

47.7

66.7

0.9718

-0.2628

1.499

Table.2.1: Elastic stiffness coefficients Cij in GPa and elastic compliance coefficients Sij in (10âˆ’12 m2.N-1).

## Young's Modulus

Young's Modulus, E, is defined as the ratio of elastic stress to strain. It is a measures of the material's resistance to elastic deformation. The value of modulus, E, depends on the direction of the applied force (anisotropic). For an arbitrary crystallographic direction, E, can be written as:

where are the elastic compliance constants. , , and are the direction cosines of the applied force with respect to the crystallographic axis [8].

The following are the measured values for E in silicon at room temperature for different directions of the applied force [9]-[10].

## Miller Indices (hkl)

The orientations and properties of the surface crystal planes are important. Since semiconductor devices are built on or near the semiconductor surface. A convenient method of defining the various planes in a crystal is to use Miller indices [11]. Miller Indices are a symbolic vector representation in crystallography for the orientation of an atomic plane in a crystal lattice and are defined as the reciprocals of the fractional intercepts which the plane makes with the crystallographic axes, and denoted as h,k, and l. The direction [hkl] defines a vector direction normal to surface of a particular plane or facet. shows the Miller indices of three important planes in a cubic crystal [12].

Fig. 1: Miller indices of three important planes in a cubic crystal.

## Coordinate transformation

It is often useful to know the stress tensor in the crystallographic coordinate system for a stress applied along a general direction with respect to the crystallographic coordinate system [13]. A stress applied in a generalized direction [] can be transformed to stress in the crystallographic coordinate system using the following transformation matrix, U

Where is the polar angel, and is the azimuthal angel of the applied stress direction relative to the crystallographic coordinate system.

The stress in the crystallographic coordinate system is given by

Where is the stress applied in a generalized coordinate system.

## Piezoresistivity

Piezoresistance is defined as the change in electrical resistance of a solid when subjected to stress. The piezoresistance coefficients (Ï€) that relate the piezoresistivity and stress are defined by

Where R is the original resistance that is related to semiconductor sample dimension by where is the resistivity, l, w, and h are the length, the width and the height of the sample respectively. Î”R signifies the change of resistance, and is the applied mechanical stress.

The ratio of Î”R to R can be expressed as the following

The first three terms of the of equation (2.35) represent the geometrical change of the sample under stress, and the last term Î”Ï/Ï is the resistivity dependence on stress. For most semiconductors, the stress-induced resistivity change is much larger than the geometrical change-induced resistance change, therefore, the resistivity change by stress is the determinant factor of the piezorestivity.

The resistivity change, Î”Ï, is connected to stress by a fourth-rank tensor , and is given by

In the vector form we can rewrite as , where i=1,2,â€¦,6. Therefore, equation 2.36 can be written as

Where is a 66 matrix.

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For a cubic crystals such as Si, has only three independent elements due to the cubic symmetry.

Where describes the piezoresistive effect for stress along the principal crystal axis (longitudinal piezoresistive effect), describes the piezoresistive effect for stress directed perpendicular to the principal crystal axis (transverse piezoresistive effect), and describes the piezoresistive effect on an out-of-plane electric field by the change of the in-plane current induced by in-plane shear stress [14]-[15].

## Element of bulk Si and Ge

Si and Ge are elements of group IV with four electrons in the outermost shell, and they have diamond lattice structure, where each atom is surrounded by four equidistant nearest neighbors which lie at the corners of a tetrahedron. The unit cell can be considered as two interpenetrating face-centered cubic (fcc) lattices separated by a/4 along each axis of the cell, where a is the lattice constant as shown in Fig. 2.a. At 300K, the lattice constants of Si and Ge are 5.431 Å and 5.6575 Å, respectively [9].

The first Brillouin zone represents the central (Wigner-Seitz) cell of the reciprocal lattice. It contains all points nearest to the enclosed reciprocal lattice point. The first Brillouin zone for cubic semiconductors is a truncated octahedron. It has fourteen plane faces; six square faces along the <100> directions and eight hexagonal faces along the <111> directions. The coordinate axes of the Brillouin zone are the wave vectors of the plane waves corresponding to the Bloch states (electrons) or vibration modes (phonons). The points and directions of symmetry are conventionally denoted by Greek letters, as shown in Fig. 2.b. The zone center is called the point (k=0), the directions <100>, <110> and <111> are called, respectively,, , and directions and their intersections with the zone boundaries are called, X, K and L points respectively [12].

(b)

## Fig. 2Â : a) Structure of the fcc crystal lattice b) The first Brillouin zone of the fcc lattice.

## Energy Band Structure

Band structure is one of the most important concepts in solid state physics, it describes the variation of energy, E, with the wave vector, k, The band of filled or bonding states is called the valence band. The band of empty or anti-bonding states is called the conduction band. The highest energy occupied states are separated from the lowest energy unoccupied states by an energy region containing no states known as the bandgap. The energy difference between the top of the valence band and the bottom of the conduction band is, Eg, the bandgap energy.

Si and Ge are an indirect gap semiconductor materials. The conduction band minima of silicon is a six-fold degenerate, and located close to the X point at in the <100> direction. The valence band maximum is located at G-point and it consists of light hole (LH), heavy hole (HH) and spin-orbit (SO) hole bands. The LH and HH bands are degenerate at G-point while the SO has 44 meV split from the others bands. In contrast, Germanium has a smaller band gap than silicon and a higher atomic mass. The Ge conduction band minima are four-fold degenerated and located along at L-point along the <111> direction on the first Brillouin zone boundary. The energy band diagram of Si and Ge are shown in . At 300 K, The indirect bandgap energy for Si and Ge are 1.12 eV and 0.664 eV, respectively [16]-[17].

## Calculation of Energy Bands

A wide range of techniques have been employed to calculate the energy band dispersion curves of semiconductor materials. The most frequently used methods are the orthogonalized plane-wave method (OPW), the pseudopotential method, and the k.p method.

## Fig. 3Â : Electronic band-structure of Si and Ge calculated by Pseudopotential method.

From quantum mechanics, Schrödinger equation can be solved by expending the eigenfunction in terms of a complete basis function and developing a matrix eigenvalue equation. A plane wave basis can used to do so, but it has a difficulty because many plane waves are needed to describe the problem adequately. The OPW method has been proposed by Herring in 1940 [18]. It is an approach to avoid having to deal with a very large number of plane wave states. The basic idea is that the valence and conduction band states are orthogonal to the core states of the crystal, and this fact should be utilized in the selection of the plane wave, resulting in a reduced number of plane wave states used in solving the problem.

The pseudopotential method, and the k.p method for calculating the band structure will be discussed in details in the following sections.

## The Pseudopotential Method

The pseudopotential method is a technique to solve for band structures of semiconductors. This method makes use of the information that the valence and conduction band states are orthogonal to the core states. In addition to that, this method uses an empirical parameters known as pseudopotentials to solve Schrödinger equation in the one-electron approximation [19]:

Assuming that the electron wavefunctions of the core states and their energies are given by and respectively. We then have

Where

the orthogonality condition is given by

this equation is called the orthogonalized plane wave.

The orthogonality condition is satisfied when we choose the wave function given by

by substituting this equation into equation 2.39 we have

Then, the following relation is obtained

Equation 2.44 can be written as the following

or

Where

There exists an inequality relation between the energies of the core states and the energies of the valence and conduction bands , which is given by

thus,

Therefore equation 2.45 can be written as

Where is called the pseudopotential, which periodic and can be expanded by Fourier series as the following

## where are the Fourier coefficients, and they are given by

maybe chosen so that the potential is expressed with a small number of the Fourier coefficients , and therefore, the small values of can be neglected.

By using the empirical pseudopotential method, the Fourier coefficients of are empirically chosen so that the shape of the critical points and their energies are in good agreement with experimental observation. The energy band calculations based on the empirical pseudopotential method takes into account as few the pseudopotentials as possible, and use the Bloch functions of the free-electron bands for the wave functions .

The energy bands are obtained by solving the equation:

Where

## Substituting equation 2.53 and equation 2.50 in equation 2.52, then the energy band structures are given by the following eigenvalue equation:

To obtain the eigenvalues and eigenfunctions of the above equation, we define the following

Equation 2.54 becomes:

Where

Then the solutions are obtained by solving the determinate

## The k.p Method

The k.p method starts with the known form of the band structure problem at the edges, and using the perturbation theory to study wave functions according to the crystal symmetry, so that band structures away from the highly symmetry points in k space can be obtained. Additionally, using this method one can obtain analytic expressions for band dispersion and effective masses around high-symmetry points [20]-[21].

Assuming that the eigenvalues and Bloch functions are know for a semiconductor with a band edge at k0 (k0 is at position -point (=[000] in the Brillouin zone). The Schrödinger equation for a one-electron system is given by

Where is potential energy with the lattice periodicity, is the wave function, and is the total energy.

The solution of equation 2.39 is given by the Bloch function

Where is a function of the lattice periodicity for band index n.

Substituting the Bloch function into equation (2.39) and using the following relations

By using the relation for the momentum operator, equation (2.39) becomes

The secular equation is represented by

Therefore, the eigenvalue determinant becomes

Where is the momentum matrix element between the different bandedges states, and is given by

is non zero only for certain symmetries of and , hence reducing the number of independent parameters.

The k.p description of the non degenerate bands (i.e., conduction bandedge or the split-off band in the valence band for the case of large spin-orbit coupling) can be done using the perturbation theory to obtain the energy wave functions away from k0.

For k0=0, the Schrödinger equation for the perturbation Hamiltonian is given by

Where

## and is the central part of the Bloch functions . In the perturbation approach H0 is a zero order term in , H1 is a first order term in , and H2 is a second order term in .

## To zero order, we have

To first order perturbation we have

For crystals with inversion symmetry such as Si and Ge, the states or will have inversion symmetry, therefore, the first order matrix elements vanish because P has an odd parity.

For the second order, the energy is given by

Equation 5.53 can be expressed in terms of the effective mass as the following

## Where

This equation is valid for conduction bandedge and the split-off bands. For the conduction band, keeping only the valance bandedge bands in the summation, then the energy eigenvalue can be expressed as

with

## Where is the energy gap at the zone center, and is the HH-SO (Split-Orbit) band separation.

For the Split-off band the energy eigenvalue is given by

Where

The HH valence band structure is that of a free-electron, therefore, the effective mass is the same as free-electron mass, and is given by

and the light-hole band structure is given by

Where

## Impact of Strain

## 4.1 Crystal Symmetry

Due to the communication between symmetry operations and the crystal Hamiltonian, crystal symmetry determines the symmetry of the band structure. Therefore, straining the silicon lattice will reduce the crystal symmetry and changes the inter-atomic spacing. The breaking of the crystal symmetry also causes band warping from symmetry restrictions. When the band structure of a material is changed, many material properties are altered including band gap, effective mass, carrier scattering, and mobility. Associated modifications in the electronic band structure and density of states contribute to changes in carrier mobility through modulated effective transport masses [22].

## 4.2 Band Structure and Band Alignment

The impact of strain on the band structure of a semiconductor can be discussed in two parts; biaxial strain which can be decomposed into a hydrostatic contribution and uniaxial strain. The hydrostatic component leads to an energy band shift and change of the bandgap. while the uniaxial strain component splits the degeneracy of the conduction and valance bands, but it has no effects on the average band energy. The effects of the hydrostatic and biaxial stress on the energy bands in Si for both tensile and compressive stresses is schematically illustrated in Fig. 4. The biaxial tensile stress corresponds to hydrostatic tension plus uniaxial compression in the z-direction. For this case, a uniform hydrostatic stretching of the lattice shifts the conduction and valence band edges to lower energies relative to their equilibrium positions, while the uniaxial compression component induces a splitting that does not affect the average band energy. The six-fold degenerate Si conduction band energy (âˆ†6) is splitted. The âˆ†2 and âˆ†4 bands move in opposite directions and the âˆ†2 level moves by twice the amount as the âˆ†4 level. The energies of the two out-of-plane valleys âˆ†2 are lowered in comparison to the four in-plane valleys âˆ†4.

## Fig. 4Â : Schematic representation of the effects of hydrostatic and biaxial stress on the energy bands in Si for tensile and compressive stresses.

In addition to that, biaxial tensile strain splits the valence band degeneracy. In the valence band, biaxial strain reduces the heavy hole (HH) and split of band energies with respect to the light hole (LH) band as shown in Fig. 4. In the case of biaxial compressive stress, a uniform hydrostatic compressing of the lattice shifts the conduction and valence band edges to higher energies relative to their equilibrium positions, while the uniaxial tensile component induces a splitting of the energy band. The six-fold degenerate Si conduction band energy (âˆ†6) is splitted. The âˆ†2 and âˆ†4 bands move in opposite directions and the âˆ†2 level moves by twice the amount as the âˆ†4 level. The energies of the two out-of-plane valleys âˆ†2 are higher in comparison with the four in-plane valleys âˆ†4. In addition to that, biaxial tensile strain splits the valence band degeneracy. In the valence band, biaxial strain increase the heavy hole (HH) and split of band energies with respect to the light hole (LH) band [23].

## The k.p method incorporated with Bir-Pikus strain Hamiltonian is used to calculated strain effect on band structures by introducing an additional perturbation term into the unstrained potential [24]. Therefore, the total Hamiltonian is given by

Where

and

Where and are the dilation and uniaxial deformation potentials at the Si conduction bandedge required for symmetry considerations, is the trace strain tensor, is the longitudinal (transverse) strain component (along [001], , and ), and is the longitudinal (transverse) effective mass.

The general form of the strain-induced energy change in the energy of carrier bands in silicon is given by

where, a, b, and d are deformation potentials that correspond to the model, i corresponds to the carrier band number, and are the components of the strain tensor in the crystal coordinate system. The final value of the change in the energy band can be calculated by averaging the energy changes in all the sub-bands. The expression for the change in energy can be summarized as:

where and are the number of subvalleys considered in the conduction and valence bands, respectively, and =300K [25].

## 4.3 Mobility Enhancement

To understand the effect of strain on mobility, the simple qualitative Drude model of electrical conduction which explain the transport properties of electrons in materials dictates that

Where is the carrier mobility, is the scattering time, and is the conductivity effective mass. Therefore, the mobility improvement in strained silicon takes place mainly due to the reduction of the carrier conductivity effective mass, and the reduction in the intervalley phonon scattering rates.

The conduction band of unstrained bulk silicon has six equivalent valleys along the <100> direction of the Brilloun zone, and the constant energy surface is ellipsoidal with the transverse effective mass, mt = 0.19m0, and the longitudinal effective mass, ml = 0.916m0, where m0 is the free electron mass [26]. If biaxial tensile strain is applied, the degeneracy between the four in-plane valleys (Î”4) and the two out-of-plane valleys (Î”2) is broken as shown in Fig. 4. As a consequence, the electrons prefer to populate the lower valleys, which are energetically favored. This result in an increased electron mobility via a reduced in-plane and increased out-of-plane electron conductivity mass . In addition to that, electron scattering also reduced due to the conduction valleys splitting into two sets of energy levels, which lowers the rate of intervalley phonon scattering. Therefore, if the optimum strain is applied, both reduction in scattering rate and in effective mass will contribute to the electron mobility enhancement. The stress-induced electron mobility enhancement is given by

Where is electron mobility without the strain, and are the electron longitudinal and transverse masses in the subvalley, respectively, and are the change in the energy of the unstrained and the strained carrier sub-valleys, is the quasi-Fermi level of electrons. The index i corresponds to a direction (for example, is the electron mobility in the direction of the x-axis of the crystal system and, therefore, should correspond to the two-fold subvalley along the x-axis) [27].

For holes, the valence band structure of silicon is more complex than the conduction band. For unstrained silicon at room temperature, holes occupy the top two bands; the heavy and light hole bands. Applying strain, the hole effective mass becomes highly anisotropic due to band warping, and the energy levels become mixtures of the pure heavy, light, and split-off bands. Thus, the light and heavy hole bands lose their meaning, and holes increasingly occupy the top band at higher strain due to the energy splitting. To achieve high hole mobility, a low in-plane conductivity mass for the top band is required, in addition to that, a high density of states in the top band and a sufficient band splitting to populate the top band are also required [25].

## Strain Application Techniques

In the previous sections It has been shown that the introduction of a compressive and/or tensile strain in the Si substrate can improve the mobility of both carrier types. Therefore, this provides a very important way to modify and enhance the electric properties of Si through proper design, implementation, and control of strain in the active layers. Consequently, various methods and approaches have been proposed to induce the desired strain in electronic devices, such as "Global strain" through SiGe epitaxial processes [28]-[29], "Local strain" using specially engineered high tensile films [29], and "Mechanical strain" by mechanically pending the wafer post fabrication[30]-[31]. The different strain generation methods will be discussed in details.

## Global Strain Approach

Global strain on wafer level is mostly induced by the epitaxial growth of Si1-x Gex and Si layers. Because the lattice parameter of Si1-xGex (0 â‰¤ x â‰¤ 1) alloys varies between 0.5431nm for Si (x=0) and 0.5657nm for Ge (x=1), tensile strain is induced in a silicon layer epitaxially grown on top of the SiGe layer. And compressive strain is induced in the SiGe layer epitaxially grown on top of a Si layer as shown in Fig. 5. In this technology the degree of strain is controlled by changing the content of Ge in the Si1âˆ’xGex layer, or by changing the thickness of the strained Si layer. In both cases the strain is in the plane of the layer (), but this strain also produces a perpendicular strain, resulting in a tetragonal distortion to the lattice.

The strains are connected through the isotropic elasticity theory as

Where v is the Poisson's ratio.

The tetragonal distortion produced by the perpendicular strain results in a parallel lattice constant, and is given by

Where is the Si lattice constant, is the SiGe lattice constant, f is the misfit between the two layers, is the Si layer thickness, is the SiGe layer thickness, and , are the shear moduli of Si and SiGe respectively.

(b)

## Fig. 5: (a) A schematic diagram of the bulk lattice of a thin Si1âˆ’xGex film to be grown on top of a thin bulk silicon layer with the top Si1âˆ’xGex film being compressively strained. (b) A schematic diagram of the bulk lattice of Si film to be grown on top of a bulk Si1âˆ’xGex film with the top Si film being tensile strained.

The misfit between the two layers , f, is defined as

In equilibrium, the in-plane strain in the Si layer and SiGe layer are related together by the relation

Under appropriate growth conditions, good quality layers of crystalline Si1-xGex alloys on Si substrates can be grown. If the SiGe thickness below a critical thickness (hC), which depends on the alloy composition and the growth temperature, a Si1-xGex film pseudomorphically grown on top of Si substrate, can be grown without the introduction of extended defects. If the Si1- xGex thickness exceeds the critical thickness. Or the substrate is exposed to sufficiently high temperatures for long period of time, at which the pseudomorphically grown layer is no longer thermodynamically stable, the lattice constant relaxes to its original value. This means that the strain in the Si1- xGex layer will be relaxed and misfit dislocations will generate at the Si/ Si1- xGex interface. Thus, the Si1- xGex/Si strained heterostructures are limited in thickness and stability.

Various models have been developed to predict the critical thickness for which the epitaxial strain layer can be grown. van der Merwe produced a thermodynamic equilibrium model by minimizing the total energy of a system with the generation of a periodic array of dislocations. In his model the critical thickness is when the strain energy equals the interface energy, and is given by

Where b is the magnitude of the Burger's vector. For a bulk Si substrate b=0.4 nm, and in general , where a is the lattice constant of the relaxed substrate.

Matthews and Blackeslee in their model have proposed that the critical thickness is when the misfit stress on an existing threading dislocation equals the line tension of the dislocation, or equivalently, when a dislocation half-loop is stable against the misfit stress. The critical thickness according to Matthews and Blackeslee model is given by

Where is the angel between the dislocation line and the Burgers vector, is the angel between the Burgers vector and the direction in the interface normal to the dislocation line, v is the Poisson's ratio, and is the in-plane strain.

However, it was verified that van der Merwe model and Matthews and Blackeslee model calculations were not consistent with the experimental data of the critical thickness, and if epitaxy conditions are carefully controlled, then a Si1-xGex layers with thickness above (hC) could be grown. The simplicity of these models, as well, not taking in consideration the nucleation, propagation, and interaction of dislocations in their calculation were the reasons for the models failure. Afterwards, more accurate results were proposed by People and Bean model. In their model they tried to explain the metastable critical thickness (hc,MS) through a nonequilibrium model approach. According to their model, the metastable critical thickness is defined as the film thickness at which its strain energy density becomes greater than the self-energy of an isolated screw dislocation, and is given by

In Fig. 6 the equilibrium (stable) and metastable values of critical thicknesses are plotted versus the Ge content, x, of a Si1-xGex epitaxial layer grown on a Si substrate. As shown in the figure, increasing the Ge content, will increase the strain in the SiGe layer, and thus the critical thickness decreases.

Even though the global strain approach described above has the advantage that it is wafer-level and the transistor fabrication process requires little or no change. It suffers from several process integration issues. The presence of Ge modifies dopants diffusion and change thermal conductivity of the substrate. The growth of a thick SiGe strain-relaxed buffer can be costly. Also the relaxation of SiGe via misfit dislocation formation and thermal processing during the fabrication steps [32].

## Fig. 6Â : The equilibrium and metastable critical thickness versus Ge content for pseudomorphic Si1âˆ’xGex layers grown on bulk silicon substrate.

## Local Strain Approach

A second technique for introducing strain in semiconductor devices is the use of a tensile and/or compressive strain layers. In this approach, either uniaxial or biaxial strain is created through the device fabrication process using strain layers such as silicon dioxide (SiO2), and silicon nitride (Si3N4). In this technique, strain develops primarily during the deposition process and consists of two components: the intrinsic strain and the extrinsic strain. The intrinsic strain is the component of strain in the layer caused by the deposition process itself. Processing conditions such as temperature, thickness, pressure, deposition power, reactant and impurity concentrations, are important factors in determining the magnitude and strain type (i.e. compressive or tensile).

The extrinsic stress is the component of strain caused by a change in the external conditions on the wafer. The thermal expansion coefficient of materials like SiO2, and Si3N4 layers are different from the silicon substrate thermal coefficient, therefore, when the temperature changes, the layer and substrate try to expand or contract by different amounts. Because the substrate and the stress layer are bound together, a stress will develop in both the layer and the substrate. Since layers are typically deposited above room temperature, the process of cooling after deposition will introduce a thermal component of strain. So, after deposition, the film tends to back to its initial state by shrinking if it was stretched earlier, thus creating compressive intrinsic stress, and similarly tensile intrinsic stress if it was compressed during deposition.

The thermal expansion coefficient, , is defined as the rate of change of strain with temperature, and is given by

## Therefore, the thermal strain, induced by a variation in temperature is given by

The intrinsic stress generated can be quantified by Stoney's equation by relating the stress to the substrate curvature as

where ESi and Ï…Si are Young's modulus and Poisson's ratio of Silicon, hSi and hf are substrate and film thickness, and R is the radius of curvature of the substrate [33]-[37].

The local strain approach through using tensile and/or compressive strain nitride layer has been used to optimize NMOSFET and PMOSFET devices on the same wafer independently by applying different levels of strain as shown in Fig. 7 [38]. More than 2 GPa of tensile stress and more than 2.5 GPa of compressive stress have been developed by Applied Materials through controlling the growth conditions of Si3N4 layers [39].

## Fig. 7Â : TEM micrographs of 45-nm n-type MOSFET with nitride-capping film with a large tensile stress

As well, IBM, AMD and Fujitsu [40]-[41] have reported a CMOS architecture in which longitudinal uniaxial tensile and compressive stress in the Si channel have been created. In this approach, the process flow consist of a uniform deposition of a highly tensile Si3N4 liner post silicidation over the entire wafer, followed by patterning and etching the film off p-channel transistors. Next, a highly compressive Si3N4 layer is deposited, and this film is patterned and etched from n-channel regions. The advantages of this technique over the epitaxial SiGe technique, that the DSL approach reduces the process complexity and integration issues. In addition to simultaneously improve both n- and p-channel transistors. The local strain approach through using SiO2 and Si3N4 strain layers in the collector region will be presented in chapter 4.

## Mechanical Strain Approach

The third technique of introducing strain into the transistors is through external mechanical stress post fabrication. In this approach, the strain is engendered into the Si either through direct mechanical bending of the Si wafer, or by bending a packaged substrate with a Si chip glued firmly onto its surface. two methods used to apply external mechanical strain on the Si wafer are shown in Error: Reference source not found [30]-[31]. This technique is an extremely low-cost technique, and it allows the reversible application of either compressive or tensile strain.

## Fig. 8Â : Schematic diagram of the externally applied mechanical stress on the Si (100) wafer.

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