We review basic concepts on semiconductors physics that will help us to understand future posts. Detailed accounts can be found in several textbooks [1] and websites.
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Figure 1: Atoms arranged in a
two-dimensional lattice. The unit
cell of the crystal is in yellow. |
Semiconductors belong the class of materials known as crystal, where the atoms are well arranged in a lattice. The lattice can be decomposed into unit cells, see Fig. 1. According to the theory of crystals, the wave-function of an electron living in the lattice is a product of envelope [eikr] and microscopic/periodic [ubk(r)] parts
\begin{eqnarray}
\large
\varphi_{bk} ({\bf{r}})= L^{-3/2} e^{i \bf{k} \bf{r}} u_{bk}(\bf{r}), ~~~~~~ (1)
\end{eqnarray}
with L the linear size of the lattice. An electron can exist in a particular state specified by the quasi-momentum k and the energy band b.
Semiconductor differ from other crystals, such as metals, because all electrons occupy the valence band b=v, and this band is separated from the next more energetically one by an energy gap Eg (in the electron-Volt range). In Fig. 2 we see a simplified two-band model, commonly used in theoretical studies.
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| Figure 2: In an unexcited semiconductor all electrons are in the valence band. |
In addition to their energy and linear momentum, electrons carry spin and band angular momenta. In many systems, the valence band has angular momentum equal to one, and the conduction band has angular momentum equal to zero. Then, the band+spin angular momentum for the former is J=3/2 (heavy hole) or J=1/2 (split-off), while for the latter is J=1/2.
Electrons in the valence band can be promoted to the conduction band, for example, by optical excitation. The conservation of angular momentum in an optical transition determines which states are coupled to each other. To be more specific, circularly polarized light with photon spin equal to −1 induces transition between the states in the valence band with {J=3/2, Jz=3/2} and states in the conduction band with {J=1/2, Jz=1/2}.
Nanostructures
In the case of man-make nanometric structures - such as quantum dots, quantum rings, nanowires, quantum wells, etc - we obtain an approximate electron wave-function substituting the envelope eikr in Eq. (1) by the appropriate function having the symmetry of the nano-system. [2] For instance, the complete wave-function of a thin quantum ring is
\begin{eqnarray} \large \varphi_{bk}({\bf{r}})= (2\pi)^{-1/2} e^{i m \theta} u_{b0}(\bf{r}), \end{eqnarray}
where the normalization was also corrected.
References
[1] Neil W. Ashcroft and N. David Mermin, Solid State Physics, Cengage Learning; 1 edition (January 2, 1976).
[2] Gerald Bastard, Wave mechanics applied to semiconductor heterostructures (Chapter 3), Wiley-Interscience; 1 edition (January 29, 1991).
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