Subsections

• 3.5.1.1 Strain Induced Conduction Band Splitting
• 3.5.1.2 Strain Induced Degeneracy Lifting at the $X$ Point
• 3.5.1.3 Strain Induced Valence Band Splitting

3.5.1 Deformation Potential Theory

Bardeen and Shockley [165] originally developed the deformation potential theory. Herring and Vogt [166] generalized this theory. Bir and Pikus [161] studied various semiconductors via group theory and showed how to calculate strain effects on the band structure with deformation potentials. A short introduction into the deformation potential theory is given subsequently.

The deformation potential theory introduces an additional Hamiltonian $\mathcal{H}(\bar{\varepsilon})$, that is attributed to strain and its effects on the band structure. This Hamiltonian is based on first order perturbation theory and its matrix elements are defined by

$$\left\{ \mathcal{H}\left(\bar{\varepsilon}\right) \right\}_{ij} =... ...,\beta=1}^{3} \mathcal{D}_{ij}^{\alpha \beta} \varepsilon_{\alpha \beta} \quad,$$ (3.15)

$\mathcal{D}^{\alpha \beta}$ denotes the deformation potential operator which transforms under symmetry operations as second rank tensor [167] and $\varepsilon_{\alpha \beta}$ describes the $\left(\alpha \beta\right)$ strain tensor component. The subscripts $\left( ij \right)$ in $\mathcal{D}^{\alpha \beta}_{ij}$ denote the matrix element of the operator $\mathcal{D}^{\alpha \beta}$. Due to the symmetry of the strain tensor with respect to $\alpha$ and $\beta$, also the deformation potential operator has to obey this symmetry $\mathcal{D}^{\alpha\beta}=\mathcal{D}^{\beta\alpha}$ and thus limits the number of independent deformation potential operators to six.

In the case of cubic semiconductors the edges of the conduction band and the valence band are located on symmetry lines. These symmetries are reproduced in the energy band structure and in the basis states. Furthermore, the symmetry of the basis states allows to describe the deformation potential operator of a particular band via two or three deformation potential constants [166].

Although, theoretically the deformation potential constants can be calculated via the empirical pseudo potential method or by ab initio methods, it is more convenient to fit the deformation potentials to experimental results obtained by electrical, optical, microwave techniques, or by analyzing stress induced absorption edges. Even though, theoretical predictions and measurements match quite well, deformation potentials in literature and found by different methods deviate from each other [168].

3.5.1.1 Strain Induced Conduction Band Splitting

Cubic crystalls exhibit a strain induced energy shift for the non-degenerate energy levels of the conduction band. Along the $\Delta$ symmetry line it is sufficient to describe the deformation potential operators $\mathcal{D}^{\alpha \beta}$ as scalars by one or two independent constants. The energy shifts of the conduction band edge of valleys along the $\langle111\rangle$ and $\langle100\rangle$ directions is determined by two independent deformation potential constants ^3.1 [169]:

$$\delta E_{0}^{v_{i}}=\Xi_{\mathrm{d}}^{v}\: \mathrm{Tr}\left(\bar... ...{v}_{\mathrm{u}}\; \vec{a}_{i}^{\mathrm{T}}\bar{\varepsilon}\,\vec{a}_{i}\quad.$$ (3.16)

$\Xi_{\mathrm{u}}^{v}$ describes the uniaxial- and $\Xi_{v}^{\mathrm{d}}$ the dilatation deformation potential constants for valleys of the type $v=L,\,\Delta$. $\vec{a}_{i}$ denotes the unit vector parallel to the $\vec{k}$ vector of valley $i$. The $\Gamma^{'}_{2}$ conduction band minimum valley shift can be determined from a single deformation potential constant

$$\delta E_{0}^{\Gamma}=\Xi_{\mathrm{u}}^{\Gamma}\: \mathrm{Tr}\left(\bar{\varepsilon}\right) \quad.$$ (3.17)

Via the two relations from above the valley splitting from uniaxial stress along arbitrary directions can be calculated.

3.5.1.2 Strain Induced Degeneracy Lifting at the $X$ Point

Additionally to strain induced energy shifts of energy levels of the conduction band edges, there can also be a partially or complete lifting of degeneracy for degenerate bands, caused by the reduction of symmetry. Due to the special symmetry of the diamond structure (three glide reflection planes at $x=a_{0}/8$, $y=a_{0}/8$ and $z=a_{0}/8$), the lowest two conduction bands $\Delta _{1}$ and $\Delta _{2'}$ touch at the zone boundary $X$. Shear strain $\varepsilon _{xy}$ due to stress along $\left[110\right]$ reduces the symmetry of the diamond crystal structure and produces an orthorhombic crystal. The glide reflection plane $z=a_{0}/8$ is removed by the shear strain component and thus the degeneracy of the two lowest conduction bands $\Delta _{1}$ and $\Delta _{2'}$ at the symmetry points $X=\frac{2\pi}{a_{0}}\left(0,0,\pm1\right)$ is lifted [161,170]. It should be mentioned that in biaxially strained $Si$ layers grown on $\left\{001\right\}$ $Si_{1-x}Ge_{x}$ substrates and for uniaxially strained/stressed $Si$ along a fourfold rotation axis $\left\langle100\right\rangle$ the glide reflection symmetry is preserved.

Bir and Pikus found from k.p theory, that when the degeneracy at the zone boundary $X$ is lifted, a relatively large change in the energy dispersion of the conduction band minimum located close to this $X$ point arises [161]. This effect was experimentally proved for $Si$ by Hensel and Hasegawa [170], who measured the change in effective mass for stress along $\langle110\rangle$, and by Laude [171], who showed the effect via the indirect exciton spectrum.

Therefore, in order to take the lifting of the degeneracy of the two lowest conduction bands $\Delta _{1}$ and $\Delta _{2'}$ at the $X$ points $\frac{2\pi}{a_{0}}\left(0,0,\pm1\right)$ into account, (3.16) has to be adapted [170]

$$\left( \begin{array}{cc} \delta E_{0}&\delta E_{1}\\ \delta E_{1}... ... = \delta E \left( \begin{array}{c} \xi \\ \hat{\xi} \end{array} \right) \quad,$$ (3.18)

where $\Xi_{\text{u}'}$ denotes a new deformation potential,

$$\begin{array}{cc} \delta E_{0}=& \Xi_{\mathrm{d}}^{\Delta}\: ... ...e_{xy} = 2 \Xi_{\mathrm{u}'} \varepsilon_{xy} \quad.\end{array}$$ (3.19)

The solutions of the eigenvalue problem look like:

$$\delta E = \delta E_{0} \pm \delta E_{1} \qquad \mathrm{for} \qquad \hat{\xi}=\pm \xi \quad,$$ (3.20)

which shows that at the $X$ points $\frac{2\pi}{a_{0}}\left(0,0,\pm1\right)$ the band shifts by an amount of $\delta E_{0}$ (like before in (3.16)) plus an additional splitting of $2\, \delta E_{1}$, which lifts the degeneracy. (3.19) shows the proportional dependence on shear strain $\varepsilon _{xy}$ for the splitting

$$\left(E_{\Delta_{1}}- E_{\Delta_{2'}}\right) \Big\vert _{X_{\left[001\right]}}=2 \delta E_{1} = 4 \Xi_{\mathrm{u}'} \varepsilon_{xy} \quad.$$ (3.21)

A value of $5.7\pm1\,$eV has been predicted by Hensel for the shear deformation potential $\Xi_{\text{u}'}$ [170]. Laude [171] confirmed this value by his measurement of $7.5\pm2\,$eV via the indirect exciton spectrum of $Si$.

The splitting is already strongly pronounced for shear strain $<1\%$. Due to the lifting of the degeneracy the $\Delta _{1}$ conduction band is deformed close to the symmetry points $X=\frac{2\pi}{a_{0}}\left(0,0,\pm1\right)$ (Fig. 3.2).

Figure 3.2: Energy dispersion of the conduction bands $\Delta _{1}$ and $\Delta _{2'}$ near the zone boundary $X$ point along $\left [001\right ]$. For $\varepsilon _{xy}=0\%$ the conduction bands are degenerate at the zone boundary. Introduction of shear strain $\varepsilon_{xy}\neq 0$ lifts this degeneracy and opens up a gap. The energy separation $2\,\delta\!E_{1}$ between the bands becomes larger with increasing strain $\varepsilon _{xy}$. At the same time the two minima of the lower conduction band $\Delta _{1}$ move closer to the zone boundary with rising strain $\varepsilon _{xy}$, until they merge at the zone boundary and stay there for further increasing strain.

i

A non-vanishing shear strain component $\varepsilon _{xy}$ has the following effects on the energy dispersion of the lowest conduction band:

• The band edge energy of the valley pair along $\left [001\right ]$ direction shifts down with respect to the other four valleys along $\left [100\right ]$ and $\left [010\right ]$.
• The effective mass of the valley pair along $\left [001\right ]$ changes with increasing $\varepsilon _{xy}$.
• The conduction band minima along $\left [001\right ]$ move to the zone boundary $X$ points at $\frac{2\pi}{a_{0}}\left(0,0,\pm1\right)$ with increasing $\varepsilon _{xy}$.

Figure:3.3 Energy dispersion of the two lowest conduction bands at the zone boundaries $X=\frac {2\pi }{a_{0}}\left (1,0,0\right )$   and$\: \frac{2\pi}{a_{0}}\left(0,1,0\right)$. The band separation of unstrained $Si$ at the conduction band edge $\vec{k}_{\mathrm{min}}=\frac{2\pi}{a_{0}}\left(0,0,0.85\right)$ is denoted by $\Delta$. Contrary to the conduction bands along $\left [001\right ]$ the conduction bands along $\left [100\right ]$ and $\left [010\right ]$ are not affected by shear strain $\varepsilon _{xy}$.

For differing strains ( $\varepsilon_{xy} \neq \varepsilon_{xz} \neq \varepsilon_{yz}$), the conduction band minima along the $\left\langle001\right\rangle$ axes are different in their energies, causing a repopulation between the six conduction band valleys. This kind of effect is not covered with (3.16), due to the negligence of possible degeneracy liftings by shear strain and by ignoring a possible repopulation of energy states.

The model presented shows no change in the conduction bands near the zone boundaries $X=\frac{2\pi}{a_{0}}\left(\pm1,0,0\right)$ and $X=\frac{2\pi}{a_{0}}\left(0,\pm1,0\right)$ for a shear component $\varepsilon _{xy}$ (Fig. 3.3). However, shear components like $\varepsilon_{xz}$ or $\varepsilon_{yz}$ lift the degeneracy at $X=\frac{2\pi}{a_{0}}\left(\pm1,0,0\right)$ or $X=\frac{2\pi}{a_{0}}\left(0,\pm1,0\right)$.

Applying a degenerate k.p theory at the zone boundary $X$ point [161,170] enables an analytical description for the valley shift along the $\Delta$ direction. Shear strain $\varepsilon _{xy}$ causes an energy shift between the conduction band valleys along $\left [100\right ]$/ $\left [010\right ]$ and the valleys along $\left [001\right ]$. This shift is described by

$$\delta E_{\mathrm{shear}}=\left\{\begin{array}{l@{,}c} -\frac{\De... ...t)\quad&\quad \left\vert\varepsilon_{xy}\right\vert>1/\kappa \end{array}\right.$$ (3.22)

$\kappa=\left(4 \Xi_{u'}\right)/\Delta$ is a dimensionless parameter and $\Delta$ denotes the band separation between the lowest two conduction bands at the conduction band edge

$$\Delta = \left(E_{\Delta_{2'}}-E_{\Delta_{1}} \right) \Big\vert _{\vec{k}=\vec{k}_{\mathrm{min}}}\quad.$$ (3.23)

$\vec{k}_{\mathrm{min}}=\frac{2\pi}{a_{0}}\left(0,0,0.85\right)$ denotes the position of the band edge in the unstrained lattice.

3.5.1.3 Strain Induced Valence Band Splitting

Caused by the degeneracy at the maximum of the valence bands the deformation potential is different than that of the conduction bands. The deformation potential operators $\mathcal{D}^{\alpha \beta}$ are no longer scalars and have to be expressed as $3\times3$ matrices. Using symmetries the six independent operators can be described via three independent entries, commonly named $l,m,n$ or $a,b,d$, related to the applied set of eigenfunctions [172]. For the basis $\vert x,s\rangle$, $\vert y,s\rangle$, $\vert z,s\rangle$, with $s=\uparrow, \downarrow$ denoting the spin state, the perturbation Hamiltonian takes the following form:

$$\bar{H}_{\; strain}= \left(\begin{array}{cc} \bar{H}& \bar{0}_{\:... ...egin{array}{c} \vert\uparrow\rangle \\ Vert\downarrow\rangle \end{array} \quad,$$ (3.24)

$\bar{H}$ denotes the $3\times3$ matrix

$$\bar{H} = \left( \begin{array}{ccc} l \varepsilon_{xx} + m \left(... ...gin{array}{c} \vert x\rangle\\ Vert y\rangle\\ Vert z\rangle \end{array} \quad.$$ (3.25)

In the case of the valence band the description of the strain induced shifts of the heavy-hole, light-hole, and the split-off band are more complex[169].

Footnotes

... constants^3.1

neglecting strain induced splitting of the degenerate conduction bands $\Delta _{1}$ and $\Delta _{2'}$ at the $X$ point