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3.4.1 Band Structure of Relaxed Si

The band structure describes the variation of the energy $ E$ with the wave-vector $ \mathitbf{k}$. The valence bands contain the last filled energy levels at $ T=0$K, whereas the conduction bands are empty at $ T=0$K. The band gap $ E_{\mathrm gap}$ separates the conduction band from the valence band. The band structure is usually visualized by plotting $ E_n({\ensuremath{\mathitbf{k}}})$ on symmetry lines, where $ n$ denotes the band index. In Figure 3.6 the band structure of Si is plotted on the symmetry lines given in (3.27).

The band structure close to the conduction band edge can be approximated by ellipsoidal energy surfaces and a parabolic energy dispersion $ E_n({\ensuremath{\mathitbf{k}}})$. In Si the conduction band edge is located near the zone boundary $ X$ points along the $ \Delta $ symmetry lines. For the conduction band valley at $ \mathitbf{k}$ $ _{\mathrm{min}} = \frac{2\pi}{a_0}(0,0,0.85) = (0,0,k_\mathrm{min})$ the energy dispersion reads

$\displaystyle E({\ensuremath{\mathitbf{k}}}) = \frac{\hbar^2 (k_z - k_\mathrm{m...
...m_\mathrm{l}}} + \frac{\hbar^2 (k_x^2 + k_y^2)}{2 \ensuremath{m_\mathrm{t}}}\ ,$ (3.28)

where $ \ensuremath{m_\mathrm{l}}= 0.916 \ensuremath{\mathrm{m}}_0$ is the longitudinal and $ \ensuremath{m_\mathrm{t}}= 0.19 \ensuremath{\mathrm{m}}_0$ the transverse mass of Si [Singh93].

Due to the point symmetry of the fcc lattice the six directions $ [100]$, $ [\bar{1}00]$,$ [010]$, $ [0\bar{1}0]$,$ [001]$, and $ [00\bar{1}]$ are equivalent. Consequently, there are six conduction band valleys. The constant energy surfaces of all six equivalent valleys along the principal axes $ \langle
100 \rangle$ are shown in Figure 3.7. Since electron transport in unstrained Si involves the electrons moving in all of the six valleys, it shows little anisotropy, even though there is strong anisotropy in each valley.

Figure 3.6: (a) Band structure of Si calculated using the pseudopotential method. The edges of the valence bands are located at the $ \Gamma $ point, the minimum of the lowest conduction band lies on the $ \Delta $ symmetry line close to the $ X$ point. (b) Valence band structure with heavy hole (HH), light hole (LH) and split-off (SO) band along the $ \Lambda $ and the $ \Delta $ direction.
[a]\includegraphics[width=2.6in]{xmgrace-files/Si_bandstructure.eps} [b]\includegraphics[width=2.6in]{xmgrace-files/Si_Vbandstructure.eps}

Figure 3.7: Constant energy surfaces for the lowest conduction band of Si. In unstrained Si the six valleys along the three principal axes are equivalent.
\includegraphics[scale=1.0, clip]{inkscape/bzEle.eps}

The valence band edge is located at the $ \Gamma $ point, where the heavy hole (HH) and light hole (LH) band are degenerate. The split-off band (SO) is very close, since the split-off energy is only 44 meV in Si. For very small energies constant energy surfaces can be approximated by

$\displaystyle E({\ensuremath{\mathitbf{k}}}) = - A k^2 \mp \sqrt{B^2 k^4 + C^2 (k_x^2k_y^2 + k_y^2k_z^2 + k_z^2k_x^2)}\ .$ (3.29)

This is the warped band approximation [Yu03]. The warping occurs along the [100] and [111] directions because of the cubic symmetry of the crystal.

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E. Ungersboeck: Advanced Modelling Aspects of Modern Strained CMOS Technology