- 3.3.1 Strain-induced Energy Splitting

- 3.3.2 Valence Band Splitting
- 3.3.3 Different Stress Configurations
- 3.3.3.1 Biaxial Stress: Tetragonal Distortion
- 3.3.3.2 Stress along [111]: Trigonal Distortion
- 3.3.3.3 Stress along [110]: Orthorhombic Distortion
- 3.3.3.4 Stress along [123]: Triclinic Distortion

- 3.3.4 Stress-induced Degeneracy Lifting

3.3 Effect of Strain

(3.39) |

where denote the strain tensor components. The effect of strain can be incorporated into band structure calculations by introducing an additional perturbation term into the unstrained potential [Manku93b]. For the empirical pseudopotential method, an interpolation of the pseudopotential form factors is required. Moreover, in the presence of an arbitrary strain condition, there occurs a movement of the vertex atoms as well as the central atom, which results in an ambiguity in the exact location of the central atom in the bulk strained primitive unit cell. This effect can be captured by taking into account an additional internal displacement parameter [de Walle86].

The effect of strain on the conductivity of Si was first investigated by Smith [Smith54]. The principal finding of his experimental work was the observation of a change in the Si resistivity on applying uniaxial tensile stress. This change occurs due to a modification of the electronic band structure. Microscopically, the modification stems from a reduction in the number of symmetry operations allowed, which in turn depends on the way the crystal is stressed. This breaking of the symmetry of the fcc lattice of Si can result in a shift in the energy levels of the different conduction and valence bands, their distortion, removal of degeneracy, or any combination. In the following sections these effects are discussed in detail.

3.3.1.1 Deformation Potential Theory

Within the framework of this theory, the energy shift of a band extremum is expanded in terms of the components of the strain tensor .

(3.40) |

The tensor quantity is called the deformation potential. The symmetry of the strain tensor is also reflected in that of the deformation potential tensor, giving

(3.41) |

The maximum number of independent components of this tensor is six which is reduced to two or three for a cubic lattice. These are usually denoted by , the uniaxial deformation potential constant, and , the dilatation deformation potential constant. The deformation potential constants can be calculated using theoretical techniques such as density functional theory [de Walle86], the non-local empirical pseudopotential method [Fischetti96], or ab-initio calculations. However, a final adjustment of the potentials is obtained only after comparing the calculated values with those obtained from measurement techniques.

The general form of the strain-induced energy shifts of the conduction band valleys for an arbitrary strain tensor can be written as

where is a unit vector of the valley minimum for the () valley type. The first term, in (3.42) shifts the energy level of all the valleys equally and is proportional to the hydrostatic strain, . The difference in the energy levels of the valleys arises from the second term in (3.42). The analytical expressions for this term for different stress directions is listed in Table 3.4.

Direction of Stress | |||

Valleys | |||

(3.43) |

(3.44) |

The parameters are related to the Luttinger parameters [Yu03] while denote valence band deformation potentials. The effect of spin-orbit coupling is taken into account by introducing a spin-orbit interaction term, , dependent on the unstrained energy level , of the split off band, as an additional perturbation into the Hamiltonian,

(3.45) |

After performing a unitary transformation on followed by some mathematical manipulations, Manku [Manku93b] arrived at the following form to describe the valence band spectrum.

Here the and the are related to the matrices and through the relation

(3.47) |

Equation (3.47) can be simplified to

which is a cubic equation in . Its solutions give the energies for the HH, LH and the SO bands for a particular

where the coefficients are given as

(3.50) | |

(3.51) | |

(3.52) | |

(3.53) | |

(3.54) | |

(3.55) |

3.3.3 Different Stress Configurations

In this section stress configurations are discussed in which the stress is applied along the , and directions, as well as an academic case of stress along the direction.

3.3.3.1 Biaxial Stress: Tetragonal Distortion

For the case of biaxial stress in the (001) plane, the 6-fold degenerate -valleys in Si are split into a 2-fold degenerate valley pair (located along the [001] direction) and a 4-fold degenerate valleys pair. In terms of symmetry considerations, this stress condition is equivalent to applying a uniaxial stress along the [001] direction. The cubic lattice of Si gets distorted to a tetragonal crystal system (right parallelopiped with a square base). The number of symmetry operations for this system is reduced by a factor of 3 compared to the unstrained case.

Biaxial tensile strain obtained by epitaxially growing Si on relaxed SiGe results in a lowering of the valleys in energy while the valleys move up in energy. As a result, the following effects become important: a) electron transfer from the high energy valleys to the low energy valleys resulting in increased population of the valleys. This is indicated by the increased size of the lobes in Fig. 3.8a. b) Reduced probability of electron scattering from to , and c) the lowered valleys experience a smaller effective mass, , in the (001) plane.

For the valence bands, the degeneracy of the HH and LH bands at the point is lifted. The top band moves to a lower hole energy and is HH like, while the other band moves higher in energy. A schematic of the band splitting is shown in Fig. 3.8b where the in-plane direction is denoted as and the out-of-plane direction as . The curvature of the top band is higher in the out-of-plane direction as compared to the direction.

3.3.4 Stress-induced Degeneracy Lifting

The first and second conduction bands are degenerate at the X-point. This
coupling of the two bands at the X-point was understood in terms of the X-ray
scattering results obtained on the *diamond*
lattice [Bouckaert36]. The effect of strain on the degeneracy of the bands
at the X-point was first examined in the theoretical study performed by Bir and
Pikus [Bir74] and later verified experimentally by Hensel [Hensel65]
and Laude [Laude71].

For any stress condition which causes the strain tensor to have non-diagonal components, there is a distortion of the band structure and the degeneracy at some of the X-points is lifted. This leads to a change in the electron effective mass which has been detected using cyclotron resonance experiments [Hensel65]. We consider the case in which a uniaxial stress is applied along the [110] direction.

The band splitting at the X-point can be calculated from the solution of the eigenvalue problem stated in [Hensel65].

where

(3.57) | |

(3.58) |

The constant in (3.59) is a new deformation potential ascribed to the degeneracy lifting at X-point. Two different values of have been suggested: eV predicted from cyclotron resonance experiments [Hensel65] and eV from indirect exciton spectrum measurements [Laude71]. The energy levels of the two conduction bands are thus given by

(3.59) | |

(3.60) |

The energy dispersion of the first () and second ( ) conduction band can be determined from the eigenvalues of the Hamiltonian suggested by Bir and Pikus [Bir74],

where describes the dispersion of and that of , and

(3.62) |

The parameters to have been obtained in [Ungersboeck07] and are given as

where

(3.64) |

and denotes the distance of the conduction band minimum of unstrained Si measured from the point. By adopting a new primed coordinate system that is rotated by 45 with respect to the crystallographic coordinate system,

the energy dispersion (3.62) can be written as

It should be noted that is invariant under the transformation described by (3.66). The effective masses in the and direction can be obtained using the relations

while the longitudinal mass is given by

Here denotes the minimum of the conduction band and can be obtained from (3.62). Substituting the values of to from (3.64) into (3.62) and setting , the dispersion relations becomes

From , the position of the conduction band minimum is obtained [Ungersboeck07]

where . The expression (3.72) reveals that the minimum in the [001] direction moves closer to the X-point. In Fig. 3.11, the impact of shear strain on the shape of the and conduction bands is plotted. For the position of the minimum is located at the point , thus , and remains fixed.

Evaluating the derivatives in (3.68) to (3.70), the strain dependence of the transversal and longitudinal masses is obtained as

in the direction,

in direction, and

for the longitudinal mass along the direction. Here, denotes the signum function and . This modification of the band structure translates into a change in the shape of the constant energy surfaces. The constant energy surfaces of unstrained Si having a prolate ellipsoidal shape are now deformed to a scalene ellipsoidal shape, characterized by the masses and . As can be seen from Fig. 3.12, the mass along the stress direction is reduced whereas perpendicular to the stress direction is increased.

where can be calculated from (3.71) and (3.72).

In Chapter 5 it is shown how the change in the effective masses contributes to the mobility enhancement. While the strain-induced deformation of the valence band structure leading to direction-dependent effective masses has been well known, a similar attention was not received by conduction band and the information of shear stress-induced electron effective mass change was lost, despite its discovery several decades back.

S. Dhar: Analytical Mobility Modeling for Strained Silicon-Based Devices