1. Energy Bands and Charge Carriers in Semiconductors
Lecture-5
Energy Bands and Charge Carriers in Semiconductors
Metals, Semiconductors, and Insulators in Energy Band Structure ?
For electrons to experience an acceleration in an applied electric field, they must be able to move into new energy states. This implies that there must be empty states (allowed energy states which are not already occupied by electrons) available to the electrons. For example, if relatively few electrons reside in an otherwise empty band, ample unoccupied states are available into which the electrons can move. On the other hand, the Si band structure is such that at 0K, the valence band is completely filled with electrons and the conduction band is empty. Thus, there can be no charge transport within the valence band because no empty states are available into which electrons can move. Also, there are no electrons in the conduction band so that no charge transport can take place there either. This is why a pure Si has a high resistivity typical of insulators.

2. - At 0K, semiconductors have basically the same structure as insulators: a filled valence band is separated from an empty conduction band by a band gap containing no allowed energy states (See Fig. 3-4).
Difference lies in the size of the band gap (Eg), which is much smaller in semiconductors than in insulators. [Si: Eg = 1.1 eV & Diamond : Eg = 5 eV]
This relatively small energy band gaps of semiconductors allow for excitation of electrons from lower (valence) band to the upper (conduction) band by reasonable amount of thermal or optical energy.
At room temperature, Si with 1.1 eV band gap will have a significant number of electrons excited thermally across the energy band gap into the conduction band, whereas an insulator with 10 eV band gap will have a negligible number of such excitations. Unlike insulators, in Si, the number of electrons available for conduction greatly increases by thermal and/or optical energy.
In metals, the bands either overlap or are partially filled. Thus, electrons and empty energy states are intermixed within the bands so that electrons can move freely under the influence of an electric field. This is why metals have a high conductivity.

3. Typical Energy Bands (Insulator, Semiconductor, Metal)
Figure 3—4
Typical band structures at 0 K.

4. Direct and indirect energy band
GaAs has a minimum in the conduction band and a maximum in the valence band for the same k value (k = 0)  Direct Energy Band : an electron can transit from C.B. to V.B. without a change in k value.
Si has its valence band maximum at different value of k than its conduction band minimum  Indirect Energy Band. Thus, a transition of electron from the minimum in the C.B. to the maximum in the V.B. requires some change in k value.
(a) Direct Semiconductor (GaAs) : an electron in the conduction band can fall to an empty state in the valence band, giving off the energy difference (Eg) as a photon of light. Thus, this semiconductor material is used for light emitters and lasers.
(b) Indirect Semiconductor (Si) : an electron in the conduction band minimum in Si can not fall directly to the valence band maximum. Instead, it must undergo a momentum change as well as changing its energy. The indirect transition involves a change in k, and the energy is generally given up as heat to the lattice rather than as an emitted photon.

5. Figure 3—5
Direct and indirect electron transitions in semiconductors:
(a) direct transition with accompanying photon emission;
(b) indirect transition via a defect level.

6. Energy Bands Variation vs. Composition (x)
Energy bands vary with their compositions in compound semiconductors :
(Ex) GaAs : direct band gap of 1.43 eV at conduction band  minimum at RT
AlAs : indirect band gap of 2.16 eV at conduction band X minimum at RT
AlxGa1-xAs :Lowest-lying conduction band for x < 0.38 : direct  minimum
Lowest-lying conduction band for x > 0.38 : indirect X minimum
Overall, the ternary alloy AlxGa1-xAs has three conduction band minima (L, , X) and these minima move up relative to the valence band as the composition x varies : from x = 0 (GaAs) to x = 1 (AlAs). Thus, AlxGa1-xAs is a direct semiconductor for Al compositions on the column III sublattice up to 38% and is an indirect semiconductor for higher mole fractions (See Fig. 3-6).
GaAs1-x Px: direct for x < 0.45, indirect for x > 0.45 :
Light emission is most efficient for direct semiconductors where electrons can drop from conduction band to valence band without changing k or momentum (ħk).

7. Figure 3—6
Variation of direct and indirect conduction bands in AIGaAs as a function of composition: (a) the ( E,k) diagram for GaAs, showing three minima in the conduction band; (b) AIAs band diagram; (c) positions of the three conduction band minima in AIx Ga1- x As as x varies over the range of compositions from GaAs ( x = 0) to AIAs ( x = 1). The smallest band gap, Eg (shown in color), follows the direct band to x = 0.38, and then follows the indirect X band.

8. Charge Carriers in Semiconductors
• Consideration of Current Conductions :
Metals : Metal atoms are imbedded in a “sea” of free electrons, and these electrons can move as a group under the influence of an electric field.
Semiconductors : Since the semiconductors has a filled valence band and an empty conduction band at 0K, the increase of electrons in conduction band by thermal excitations across the band gap must be considered as the temp. is raised. In addition, after electrons are excited to the conduction band, the empty states left in the valence band can contribute to the conduction process. Also, the introduction of impurities is considered to have an important effect on the energy band structure and on the availability of charge carriers.
• Electrons & Holes : As the temp. is raised from 0K, some electrons in the V.B. receive enough thermal energy to be excited across the band gap to the C.B., and this results in a semiconductor with some electrons in an otherwise empty C.B. and some unoccupied states (called holes) in an otherwise filled V.B.

9. Figure 3—7
Electron-hole pairs in a semiconductor.

10. Electron-Hole Pair (EHP)
• EHP = A pair of conduction band electron and valence band hole created by the excitation of a valence band electron to the conduction band.
- Equilibrium number of EHPs in pure Si at RT = 1010 EHP/cm3
Si atom density = 5 x 1022 atoms/cm3
 Very few electrons are free to move about via the many available empty states

11. E – k Energy Band Diagram vs. Simplified Band Diagram
Hole energy increases oppositely to electron energy because the two carriers have opposite charge. In E-k diagram, hole energy increases downward while the electron energy increases upward. The holes, seeking the lowest energy state available, are generally found at the top of the valence band while the conduction band electrons are found at the bottom of the conduction band.
Since E-k diagram is a plot of the total electron energy (PE +KE) as a function of the crystal-direction-dependent electron wave vector k (the momentum and therefore the velocity) at some point in space. Hence, the bottom of the C.B. corresponds to zero electron velocity (or KE), simply indicating the PE level at that point in space. For holes, the top of the valence band corresponds to zero kinetic energy.
For simplified band diagrams, we plot the edges of C.B. and V.B. (i.e., potential energy) as a function of position. Energies higher in the band correspond to additional kinetic energy of the electron. The fact that the band edge corresponds to the PE of the electron implies that the variation of the band edge in space is related to the electric field at the different points in the semiconductor.

12. Superposition of E-k Diagram on E-x Simplified Band Diagram
(a) In Fig. 3-9, an electron at point A sees an electric field given by the slope of the band edge, corresponding to potential energy (PE), and gains kinetic energy (KE) at the expense of PE by moving to point B. In E-k diagram, the electron starts at k = 0, but moves to a non-zero wave vector kB.
(b) The electron loses KE to heat by scattering mechanisms and returns to the bottom of the band at point B. The slopes of the E–x band edges at different points in space reflect the local electric fields at those points.
(c) Practically, the electron may lose its KE in stages by a series of scattering events by the dashed lines.

13. Figure 3—9
Superimposition of the (E, k) band structure on the E-versus-position (x) simplified band diagram for a semiconductor in an electric field. Electron energies increase going up, while hole energies increase going down. Similarly, electron and hole wave vectors point in opposite directions and these charge carriers move opposite to each other, as shown.