Strain Effects on the Silicon Band Structure

The deformation potential theory to describe the influence of strain on the band structure was developed by Bardeen and Shockley(2 ) and later generalized by Herring and Vogt(67 ). Within this theory the energy is represented as a Taylor series in powers

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Strain Effects on the Silicon Band Structure

8.1 Linear Deformation Potential Theory 8.1.1 Conduction Band The deformation potential theory to describe the influence of strain on the band structure was developed by Bardeen and Shockley [2] and later generalized by Herring and Vogt [7]. Within this theory the energy is represented as a Taylor series in powers of lattice strain, and the expansion is truncated after the terms linear in strain. The theory thus relates the shifts of the energy bands to small deformations of the crystal as: X .k/ Dij "ji : (8.1) E.k/ D ij

It follows from (8.1) that the band shift is linear in strain. The coefficients of proportionality Dij.k/ form a second rank tensor. This tensor called the deformation potential tensor is a characteristic of a given non-degenerate band at a chosen point k. The tensor is symmetric and therefore has only six independent components. In cubic semiconductors the number of components is further reduced to three [7]. Based on the linear deformation potential theory, the shift of the conduction band minima with stress in silicon and germanium can be evaluated. The shift depends on the magnitude of forces applied and their directions with respect to the valley orientations. For arbitrary stress conditions, the degenerate minima in silicon are split. The value of the valley splitting, which is linear in strain within the linear deformation potential theory [7] is completely determined by the only two deformation potentials Dd and Du [1]. The general form of the linear energy shift (8.1) for one of the six degenerate valleys i D 1; 2; ::; 6 in silicon for an arbitrary homogeneous deformation can be written in the following form Ec.i / D Dd Tr.O"/ C Du aTi "Oai ;

(8.2)

where ai is a unit vector parallel to the K0 vector determining the minimum position of the valley i . It follows from (8.2) that the shift of the mean energy of the V. Sverdlov, Strain-Induced Effects in Advanced MOSFETs, Computational c Springer-Verlag/Wien 2011 Microelectronics, DOI 10.1007/978-3-7091-0382-1 8, 

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8 Strain Effects on the Silicon Band Structure

conduction band depends on the hydrostatic pressure: 1 .k/ Ec;av D .Dd C Du /Tr.O"/: 3

(8.3)

Because the deformation potentials Dd and Du have different values at different locations in the Brillouin zone, the average energy shifts are different for different valley types. This difference has to be taken into account, when more than one type of valleys is considered, since the relative shift of the mean energy causes a repopulation of carriers between these types of valleys. For example, the energy shifts of the conduction valleys at the L-point is determined by the same expression (8.2), with the deformation potentials DdL and DuL computed at the L-point [1]. However, if the valley were at the  symmetry point, it is enough to know a single deformation potential to determine the valley shift: ıE0 D D  Tr.O"/:

(8.4)

Using the above relations the valley splitting due to stress along any direction can be obtained onc