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1.3: Crystals

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    We used silicon in the preceding example on purpose. The fact that it has a halffilled valence shell with four electrons puts it in a special place. As is, it's neither a great conductor nor a superior insulator. With some attention to detail, it will become a semiconductor. Silicon is not the only material that can be used for semiconductors. In fact, many of the earliest semiconductors were made from germanium and currently we make semiconductors from other materials. Silicon, however, remains the source of most semiconductors today.

    It is possible for pure silicon to be arranged in a mono crystaline structure. That is, all of the silicon atoms align in a very specific, well-ordered manner, without any voids or breaks in the pattern. As silicon has only four electrons in its valence shell, four more electrons would be needed to fill the shell. In the crystal, any given atom of silicon effectively “shares” an electron from its four closest neighbors through a covalent bond (meaning “with or among the valence”). Each atom does this, therefore each atom is tightly bound to its neighbors. This is illustrated in Figure \(\PageIndex{1}\) using simplified Bohr models. Note the color coding that indicates the sharing.

    clipboard_e5ac43d4449b33781c9264b46a6528675.png

    Figure \(\PageIndex{1}\): Silicon crystal covalent bonding.

    Remember, this is an energy diagram. We are not trying to indicate that individual valence electrons are zipping between atoms in figure eight patterns. Indeed, this diagram is drawn flat whereas a real crystal is not a simple sheet, but is three dimensional with varying thickness.

    While it would be a practical impossibility to draw a highly realistic representation of atoms in the crystal, given a few liberties we can draw something that at least comes a little closer to reality. We start by representing each silicon atom as a ball and the covalent bond as a connecting tube. Recalling that each atom must be bound to four others in a regular, equal pattern, we come up with the drawing of Figure \(\PageIndex{2}\). Notice that the overall structure is essentially that of a cube. Further, at the center of each face of the cube there exists an atom of silicon. Therefore, we say that the crystal structure is face centered cubic.

    clipboard_e9cdca4a6d4b18e1392b82f8517616b53.png

    Figure \(\PageIndex{2}\): Representation of silicon crystal structure. Image source

    An interesting thing happens in the crystal when we examine the energy levels. With a single atom we would expect to see an energy diagram like that of Figure 1.2.7. That is, discrete, permissible steps. Within a crystal, though, each atom is affected by those around it. This causes slight changes in the energy levels. Taken as a whole, all of these individual variations cause the discrete levels to blur into broader bands. If we were to examine the valence and conduction energy levels, instead of discrete, thin lines we'd see the thicker bands as illustrated in Figure \(\PageIndex{3}\). These bands still represent permissible electron energy levels, it's just that now there is a continuum rather than a discrete level. There will still be non-permissible or forbidden zones between these regions. A forbidden zone is referred to as a band gap.

    Associated with this idea is the concept of the Fermi level, named after physicist Enrico Fermi. Basically, the Fermi level is the energy level in a given material at which there is a 50% probability that it is filled with electrons. In other words, levels below this value tend to be filled with electrons and levels above tend to be empty. If the Fermi level lies within a band, the material will be good a conductor. On the other hand, if the Fermi level lies between two widely separated bands, the material will be a good insulator. If the Fermi level is between bands that are relatively close, the material is a semiconductor.

    clipboard_e187e759ef266989490907ec3ad526183.png

    Figure \(\PageIndex{3}\): Energy diagram for an intrinsic semiconductor.

    Figure \(\PageIndex{3}\) shows the energy bands for an intrinsic semiconductor, such as an ideal silicon crystal. The term intrinsic simply means that there are no impurities in the crystal. Between the valence band and conduction band is an impermissible or forbidden region. This is a band gap. In practical terms you can think of the band gap as the amount of energy that needs to be applied to an electron in order to move it from the valence band to the conduction band. The value of the band gap will depend on a variety of factors, the precise material being used for the semiconductor is of particular importance.

    Without any external energy applied (i.e., isolated and at absolute zero), the crystal lattice is stable and there is no electron movement through the crystal. As we add thermal energy, it is possible for valence electrons to jump up to the conduction band. At this point, the electron can “wander” through the crystal in the manner depicted in Figure \(\PageIndex{4}\).

    clipboard_eb2da6d39c3b026b66bc15a4ccea0f266.png

    Figure \(\PageIndex{4}\): Electron movement in a crystal

    clipboard_e20c3ed55ba7e275314d2c3c5ec87a8c3.png

    Figure \(\PageIndex{5}\): Electron versus hole flow.

    Here is how this happens: Because the thermal energy causes the electron to jump to the higher energy level of the conduction band, it leaves behind a “hole”, that is, a place devoid of an electron. Now that the hole exists, it provides a place for another electron to “fall into”. The higher the temperature, the greater the number of freed electrons and the greater the number of corresponding holes. We now have thermally-induced electron movement. We can also look at this from the opposing perspective, namely that we have an equal magnitude but opposite direction “hole flow”. If you find this idea hard to grasp, simply look at Figure \(\PageIndex{5}\). Each horizontal bar contains four dots representing electrons. In the topmost bar there is an empty space (a hole) to the extreme left. When the leftmost electron moves into this hole it fills it in a process called electron-hole recombination, which of course, sounds much more impressive than it really is. The result is the second bar. We repeat this process of moving an electron right to left as we traverse down the diagram. Eventually we end up with the four electrons packed together toward the left. Finally, instead of focusing on the dots, focus instead on the negative space (the empty white bit). Moving from top to bottom, the hole moves left to right, in the opposing direction.

    Just as we think of the movement of electrons as a movement of negative charge, then the movement of holes can be thought of as a movement of positive charge. We can say that the electron is the carrier of negative charge while the hole is the carrier of positive charge.

    Before moving on to the next section, it is important to remember that in an intrinsic (pure) semiconductor, the number of thermally produced electrons and holes will be equal. Also, even at room temperature the total number will also be quite small compared to the number of electrons in the crystal.


    This page titled 1.3: Crystals is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by James M. Fiore via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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