Subsections

• 3.5.2.1 Effective Electron Mass in Unstrained Silicon
• 3.5.2.2 The Silicon Conduction Band Minimum's Dependence on Strain
• 3.5.2.3 The Conduction Band Minimum of Silicon and its Energy Dispersion under Strain

3.5.2 The k.p Method

The k.p method allows to derive analytical expressions for the energy dispersion and the effective masses [161]. It enables the extrapolation of the band structure over the entire Brillouin zone from the energy gaps and matrix elements at the zone center. In addition to the common use of the k.p method to model the valence band of semiconductors, it is also well suited to describe the influence of strain on the conduction band minimum.

The k.p method can be derived from the one-electron Schrödinger equation as follows:

$$\mathcal{H}\Psi_{n}(\vec{r})=\left(\frac{\vec{p}^{\phantom{1}2}}{2m}+V(\vec{r})\right)\Psi_{n}(\vec{r})=E_{n} \Psi_{n}(\vec{r})$$ (3.26)

$V(\vec{r})$ denotes the periodic lattice potential and $\mathcal{H}$ the one-electron Hamilton operator. $\Psi_{n}$ describes the one-electron wave function in an eigenstate $n$ and $E_{n}$ the eigenenergy for the eigenstate $n$. Due to the periodicity of the lattice potential (3.26) the Bloch theorem is applicable and the solution can be written in the form of:

$$\Psi_{n\vec{k}}(\vec{r})=e^{i\vec{k}\cdot\vec{r}} u_{n\vec{k}}(\vec{r}) \quad,$$ (3.27)

The wave function $\Psi_{n \vec{k}}(\vec{r})$ can be expressed as the product of a plane wave and the function $e^{i \vec{k}_{0}\cdot \vec{r}} u_{n \vec{k}}(\vec{r})$, which reflects the periodicity of the lattice. $n$ denotes the band index and $\vec{k}$ represents a wave vector. If the given potential $V(\vec{r})$ only depends on one spatial coordinate (also called local), (3.27) can be substituted in (3.26).

Luttinger [173] showed that it is possible to use the eigenfunctions of the ground states as a complete set of eigenfunctions and that the wave function can be expanded by

$$\chi_{n \vec{k}}=e^{i \Delta \vec{k}}\,\Psi_{n \vec{k}_{0}} = e^{i\left(\vec{k}_{0}+\Delta \vec{k} \right)}\,u_{n \vec{k}_{0}}$$ (3.28)

for $k_{0}\neq0$. Inserting (3.28) into (3.26) yields:

$$\left(\frac{\left(\hbar\,\Delta\vec{k}+\vec{p}\right)^{2}}{2m} + ... ...eft(\vec{k}_{0}+\Delta\vec{k}\right)^{2}}{2m}\right)u_{n\vec{k}}(\vec{r})\quad.$$ (3.29)

This way, for any fixed wave vector $\vec{k}=\vec{k}_{0}$, (3.29) for the unperturbed system, delivers a complete set of eigenfunctions $u_{n\vec{k}_{0}}$, which completely cover the space of the lattice periodic functions in real space. Therefore, the wave function $\Psi_{n \vec{k}}(\vec{r})$ at $\vec{k}$, for the full system, can be expressed via $u_{n\vec{k}_{0}}$

$$\Psi_{n\vec{k}}(\vec{r})=\sum_{n'}C_{n,n'}(\vec{k},\vec{k}_{0}) e^{i\left(\vec{k}_{0}+\Delta \vec{k}\right)\cdot\vec{r}} u_{n'\vec{k}_{0}} \quad.$$ (3.30)

As soon as the eigenenergy $E_{n\vec{k}_{0}}$ and the $u_{n\vec{k}_{0}}$ of the unperturbed system are determined, the eigenfunctions $\Psi_{n \vec{k}}(\vec{r})$ and eigenenergies $E_{n\vec{k}}$ can be calculated for any $\vec{k}=\vec{k}_{0}+\Delta \vec{k}$ in the vicinity of $\vec{k}_{0}$ by accounting the $\frac{\hbar\vec{k}\cdot\vec{p}}{m}$ term in (3.29) as a perturbation. This method has been introduced by Seitz [174] and extended by [172,173,175] to study the band structure of semiconductors.

Due to the $\frac{\hbar\vec{k}\cdot\vec{p}}{m}$ term in (3.29) this method is also known as the k.p method. Provided that the energies at $\vec{k}_{0}$ and that the matrix elements of $\vec{p}$ between the wave functions, or the wave functions themselves, are known, the band structure for small $\Delta\vec{k}$'s around $\vec{k}_{0}$ can be calculated. The entire first Brillouin zone can be calculated by diagonalizing (3.29) numerically, provided a sufficiently large set of $u_{n\vec{k}_{0}}$ to approximate the complete set of basis functions is used [172].

The following subsections will explain the effective masses for the non-degenerate conduction band of silicon and the energy dispersion utilizing a non-degenerate k.p theory. In order to analyze the effects of shear strain on the two lowest conduction bands $\Delta _{1}$ and $\Delta _{2'}$, the k.p method is adapted to enable degeneracy, due to the coincidence of the $\Delta _{1}$ and $\Delta _{2'}$ bands at the $X$ point.

3.5.2.1 Effective Electron Mass in Unstrained Silicon

The conduction band minima of silicon reside on the $\langle001\rangle$ axes at a distance of $0.15\frac{2\pi}{a_{0}}$ from the $X$ symmetry points. By means of non-degenerate perturbation theory and the knowledge of the eigenenergies $E_{n\vec{k}_{0}}$ and the wave functions $u_{n\vec{k}_{0}}$ at the conduction band minima $\vec{k}_{0}$, the eigenvalues $E_{n\vec{k}}$ at neighboring points $\vec{k}$ can be expanded to second order terms in $k_{i}$.

$$E_{n\vec{k}}=E_{n\vec{k}_{0}}+\frac{\hbar^{2}}{2m_{\text{0}}}\sum... ...t u_{n\vec{k}_{0}}\right\rangle}{E_{n\vec{k}_{0}}-E_{n'\vec{k}_{0}}}k_{j}\quad.$$ (3.31)

Scalar products $\vec{k}\cdot\vec{p}$ are expressed via index notation $\sum_{i}k_{i}p_{i}$ and the matrix elements with Dirac's notation

$$\left\langle u_{n\vec{k}_{0}} \right\vert p_{j} \left\vert u_{n'\... ...frac{\hbar}{i} \frac{\partial}{\partial x_{j}} u_{n'\vec{k}_{0}}d\vec{r} \quad.$$ (3.32)

The linear terms in $k_{i}$ can be set to zero under the assumption that $E_{n,\vec{k}_{0}}$ is a minimum. The expression for the effective mass tensor $m_{n,ij}^{*}$ can be derived from the dispersion relation (3.31)

$$\frac{1}{m^{*}_{n,ij}}=\frac{1}{m_{\text{0}}}+\frac{2}{m_{\text{0... ...t\vert u_{n\vec{k}_{0}}\right\rangle}{E_{n\vec{k}_{0}}-E_{n'\vec{k}_{0}}}\quad.$$ (3.33)

The effective mass tensor for the lowest conduction band $\Delta _{1}$ in diamond crystal structures is characterized by two masses. In the principal coordinate system for the $\left [001\right ]$ valley the effective masses can be written as

$$\frac{1}{m_{l}}=\frac{1}{m_{\text{0}}}+\frac{2}{m_{\text{0}}^{2}}... ..._{0}}\left\rangle \right\vert^{2}}{E_{\Delta_{1}\vec{k}_{0}}-E_{n'\vec{k}_{0}}}$$ (3.34)

and

$$\frac{1}{m_{t}}=\frac{1}{m_{\text{0}}}+\frac{2}{m_{\text{0}}^{2}}... ...left\rangle \right\vert^{2}}{E_{\Delta_{1}\vec{k}_{0}}-E_{n'\vec{k}_{0}}}\quad.$$ (3.35)

$\Delta _{1}$ denotes the band index $n$ of the lowest conduction band. Therefore, the energy dispersion can be formulated as:

$$E(\vec{k})=\frac{\hbar^{2}\left(k_{z}-k_{0}\right)^{2}}{2m_{\text{l}}}+\frac{\hbar^{2}\left(k_{x}^{2}+k_{y}^{2}\right)}{2m_{\text{t}}} \quad.$$ (3.36)

From the derived equations follows that due to the coupling between electronic states in different bands (via k.p term), an electron in a solid has a different mass than a free electron. The coupling terms are related to the following criteria:

• The bigger the energetic gap between two bands, the smaller is the effect on the effective mass. The relative importance of a band $n'$ to the effective mass of band $n$ is controlled by the energy gap between the two bands.
• All bands $n'$ with non-zero matrix elements $\langle u_{n\vec{k}_{0}}\vert\vec{k}\cdot\vec{p}\vert u_{n'\vec{k}_{0}}\rangle$ can be found via the matrix element theorem [176] by group theoretical considerations checking all possible symmetries for $u_{n'\vec{k}_{0}}$.

It is possible to calculate numerically all matrix elements and subsequently the effective masses from (3.33) via the empirical pseudo potential method [177].

3.5.2.2 The Silicon Conduction Band Minimum's Dependence on Strain

(3.55) only requires the direction of the $\vec{k}$ vector, indicating the location of the valley, to describe the shift of the valley minima. Hence, the valley shift is independent of the exact value of the wave vector $\vec{k}$ and all $\vec{k}$ points belonging to a particular valley experience the same shift. Since the effective mass is given by the second derivative of the energy dispersion $\frac{1}{m^{*}_{ij}}=\frac{1}{\hbar^{2}}\frac{\partial^{2}E}{\partial k_{i}\partial k_{j}}$ and (3.16) does not change the curvature of the energy band, the formula predicts no change in the effective electron mass due to strain.

However, there is a clear experimental proof that shear strain changes the effective masses of electrons in the lowest conduction band[170] and the exciton spectrum of silicon[171]. In order to explain this behavior one has to take the splitting of the lowest two conduction bands at the $X$ symmetry point by shear strain into account. The lifting of the degeneracy can be calculated with the deformation potential constant $\Xi_{u'}$ via (3.20). (3.20) is only valid at the $X$ symmetry point and cannot be used to predict the effect of strain on the valley minima $\vec{k}_{\text{min}}$. In order to circumvent this obstacle a degenerate k.p theory has to be applied around the $X$ symmetry point.

A different approach was adapted in [161]. The Hamiltonian at the $X=\frac{2\pi}{a_{0}}\left(0,0,\pm1\right)$ points can be described via the theory of invariants:

$$\mathcal{H}\left(\bar{\varepsilon},\vec{k}\right) = \lambda\, \ma... ...} k_{x} k_{y} + D_{3} \varepsilon_{xy}\right) + A_{4}\, \sigma_{z} k_{z} \quad.$$ (3.37)

where,

$$\lambda=A_{1} k_{z}^{2}+A_{2}\left(k_{x}^{2}+k_{y}^{2}\right)+D_{1}\varepsilon_{zz}+D_{2}(\varepsilon_{xx}+\varepsilon_{yy}) \quad,$$ (3.38)

$\sigma_{x}$ and $\sigma_{z}$ are the Pauli's matrices and $A_{1}$ and $A_{2}$ denote scalar constants

$$\sigma_{x}=\left(\begin{array}{cc} 0&1\\ 1&0\end{array}\right) \quad\quad\sigma_{z}=\left(\begin{array}{cc} 1&0\\ 0&-1\end{array}\right)$$ (3.39)

The scalar constants $D_{1}$, $D_{2}$, and $D_{3}$ are connected to the deformation potential constants $\Xi_{u}$, $\Xi_{d}$, and $\Xi_{u'}$ through

$$\begin{array}{cl} D_{1}=&\Xi_{u}+\Xi_{d}\quad,\\ D_{2}=&\Xi_{d}\quad,\\ D_{3}=&2\,\Xi_{u'}\quad. \end{array}$$ (3.40)

From (3.37) eigenvalues can be calculated which represent the energy dispersion for the first and second conduction band

$$E_{\pm}(\hat{\varepsilon},\vec{k}) = \lambda\pm\sqrt{A_{4}^{2}k_{z}^{2}+\left(2\, \Xi_{u'}\varepsilon_{xy}+A_{3}k_{x}k_{y}\right)^{2}}\quad,$$ (3.41)

where $E_{-}$ denotes the energy dispersion of $\Delta _{1}$ and $E_{+}$ that of $\Delta _{2'}$. Under the assumption that this description is valid around the $X$ point up to the minimum of the lowest conduction band at $\vec{k}_{\text{min}}=\frac{2\pi}{a_{0}}\left(0,0,\pm0.85\right)$, $A_{4}$ and $A_{1}$ can be related to each other via

$$\frac{\partial E_{-}(\hat{\varepsilon}=0,\vec{k})}{\partial k_{z}... ...min}}}=2\,A_{1} k_{0}+\frac{A_{4}^{2} k_{0}}{\sqrt{A_{4}^{2}k_{0}^{2}}}=0\quad.$$ (3.42)

$\vec{k}_{0}=0.15\frac{2\pi}{a_{0}}$ describes the distance of the conduction band minimum of unstrained silicon to the $X$ point. $A_{4}$ can be determined from (3.42)

$$\vert A_{4}\vert=2\, A_{1} k_{0}\quad.$$ (3.43)

The effect of shear strain on the shape of the lowest conduction band is examined in the following section.

3.5.2.3 The Conduction Band Minimum of Silicon and its Energy Dispersion under Strain

Up to now it has been assumed that the conduction band minima are located at $\vec{k}_{\text{min}}=\frac{2\pi}{a_{0}}\left(0,0,\pm0.85\right)$. This is only valid for small shear strain. The minimum of the conduction band moves towards the $X$ point in conjunction with an increasing splitting between the conduction bands, when the shear strain rises (as can be seen in Fig. 3.2). This causes a change in the shape of the conduction bands and the assumption that the minima lie fixed at $\vec{k}_{\text{min}}=\frac{2\pi}{a_{0}}\left(0,0,\pm0.85\right)$ does not hold anymore.

Therefore, a model which is able to cover the effects of shear strain on the effective masses has to take the movement of the conduction band as a function of strain into account. In the following a model will be derived that takes this movement of $\vec{k}_{\text{min}}(\varepsilon_{xy})$ into account.

Starting with (3.41) and setting $k_{x}=k_{y}=0$ the minimum can be found from the dispersion relation

$$E_{-}=\frac{\hbar^{2}k_{z}^{2}}{2 m_{\text{l}}}-\sqrt{\frac{\hbar^{4} k_{0}^{2}k_{z}^{2}}{m_{\text{l}}}+4 \Xi_{u'}^{4}\varepsilon_{xy}^{2}} \quad.$$ (3.44)

The constants $A_{1}$ and $A_{4}$ are replaced with the relations (3.43) and (3.36), and $\vec{k}_{0}=0.15\,\frac{2\pi}{a_{0}}$ describes the position of the conduction band minimum measured from the zone boundary $X$. Setting the first derivative of (3.44) to zero, $\frac{\partial E_{-}}{\partial k_{z}}=0$, and solving for $k_{z}$ results in the desired relation between $\vec{k}_{z,\text{min}}$ and shear strain.

$$k_{z,\text{min}}=\left\{ \begin{array}{cc} k_{0}\sqrt{1-\kappa^{2... ...ppa \\ 0 &,\quad \vert\varepsilon_{xy}\vert>1/\kappa \end{array} \right. \quad.$$ (3.45)

Here $\kappa=\frac{4\Xi_{u'}}{\Delta}=\frac{2\Xi_{u'}m_{\text{l}}}{\hbar^{2}k_{0}^{2}}$ is introduced and represents the ratio between the shear deformation potential $\Xi_{u'}$ and the band separation between the two lowest conduction bands $\Delta$ at zero shear strain (Fig. 3.3). (3.45) shows that for strain smaller than $1/\kappa$, the minimum position shifts towards the $X$point. At $\vert\varepsilon_{xy}\vert=1/\kappa$, the minimum is located at the $X$ point ( $k_{z,\text{min}}$). Increasing shear strain above $\varepsilon_{xy}>1/\kappa$ does not shift $k_{z,\text{min}}$ anymore. The change of shape of the two lowest conduction bands $\Delta _{1}$ and $\Delta _{2'}$ and accordingly the position change of the minimum with increasing shear strain can be seen in Fig. 3.2.

The strain dependent longitudinal mass $m_{\text{l}}(\varepsilon_{xy})$ can be calculated from (3.44) with

$$\frac{1}{m_{\text{l}}( \varepsilon_{xy} )}=\frac{1}{\hbar^{2}}\, ... ...2} E_{-}}{\partial k_{z}^{2}}\Big\vert _{\vec{k}=(0,0,k_{z,\text{min}})} \quad.$$ (3.46)

After some algebraic manipulations the strain dependent mass $m_{\text{l}}(\varepsilon_{xy})$ can be expressed as

$$m_{\text{l}}(\varepsilon_{xy})=\left\{ \begin{array}{cc} {m_{\tex... ...ight)^{-1}&,\quad \vert\varepsilon_{xy}\vert>1/\kappa \end{array} \right.\quad.$$ (3.47)

Accordingly to (3.45) the dependence of the longitudinal masses is different for a strain level above or below $1/\kappa$. For the derivation of the transversal masses we rotate the principal coordinate system by $45^{\circ}$ around the z-axis with the following transformation:

$$k_{x'}=\frac{k_{x}+k_{y}}{\sqrt{2}}\quad\quad k_{y'}=\frac{k_{x}-k_{y}}{\sqrt{2}}\quad\quad k_{z'}=k_{z} \quad.$$ (3.48)

The energy dispersion in the rotated coordinate system is

$$E_{\pm}(\hat{\varepsilon},\vec{k})=\lambda\pm\sqrt{A_{4}^{2}k_{z}... ...silon_{xy} + \frac{A_{3}}{2}\left(k_{x}^{2}-k_{y}^{2}\right)\right)^{2}} \quad.$$ (3.49)

The effective mass in the $\left[110\right]$ and $\left[ 1\bar{1}0\right]$ directions is defined by

$$\frac{1}{m_{\text{t},x'}(\varepsilon_{xy})}=\frac{1}{\hbar^{2}}\f... ...{2}E_{-}}{\partial k_{x'}^{2}}\Big\vert _{\vec{k}=(0,0,k_{z,\text{min}})}\quad,$$ (3.50)

and

$$\frac{1}{m_{\text{t},y'}(\varepsilon_{xy})}=\frac{1}{\hbar^{2}}\f... ...{2}E_{-}}{\partial k_{y'}^{2}}\Big\vert _{\vec{k}=(0,0,k_{z,\text{min}})}\quad.$$ (3.51)

Applying (3.50) and (3.51) to (3.49) gives for the $\left[110\right]$ direction

$$m_{\text{t},x'}(\varepsilon_{xy})=\left\{ \begin{array}{cc} \frac... ...on_{xy})}&,\quad \vert\varepsilon_{xy}\vert>1/\kappa \end{array} \right. \quad,$$ (3.52)

and

$$m_{\text{t},y'}(\varepsilon_{xy})=\left\{ \begin{array}{cc} \frac... ...on_{xy})}&,\quad \vert\varepsilon_{xy}\vert>1/\kappa \end{array} \right. \quad,$$ (3.53)

for the $\left[ 1\bar{1}0\right]$ direction with the parameter $\eta=\frac{m_{t}}{2m'}$ and $m'$ is defined by

$$\frac{1}{m'}=\frac{2}{m_{\text{0}}^{2}}\,\sum_{ n \neq \Delta_{1}... ...eft\vert p_{y}\right\vert\Delta_{2'}\right\rangle }{E_{n}-E_{\Delta_{1}}}\quad.$$ (3.54)

As can be seen along the $\left[110\right]$ direction the effective mass is reduced (mobility is enhanced) for $\varepsilon_{xy>0}$, while for the $\left[ 1\bar{1}0\right]$ direction the effective mass is increased (the mobility is reduced) for increasing shear strain ( $\varepsilon_{xy}<a/\kappa$). For shear strain above $\varepsilon_{xy}>1/\kappa$ the effective mass is a constant which depends on the sign of the strain.

The analytical valley shift induced by shear strain $\varepsilon _{xy}$ (given in (3.22)) can now be calculated. Substituting the expression for $k_{z,\text{min}}$ from (3.45) into equation (3.44) delivers the equation for shear strain.The shift between the valley pair along $\left [001\right ]$ and the valley pairs $\left [100\right ]$ or $\left [010\right ]$ due to $\varepsilon _{xy}$ can be obtained in the form of

$$\delta E_{\text{shear}} =E(\varepsilon_{xy},\vec{k}_{\text{min}})... ...}\vert-1)&,\quad \vert\varepsilon_{xy}\vert>1/\kappa \end{array} \right. \quad.$$ (3.55)