Solid State Device Fundamentals ENS 345 Lecture Course by Alexander M. Zaitsev alexander.zaitsev@csi.cuny.edu Tel: 718 982 2812 Office 4N101b 1
The free electron model of metals The free electron model of metals assumes that electrons are free to move within the metal but are confined to the metal by potential barriers. The minimum energy needed to extract an electron from the metal equals qf M, where F M is the work function. F M may vary from 2.1 V (for Cs) to 5.9 V (Pt). This model ignores the periodic potential due to atoms and as such it does not work well for semiconductors. 5.3 ev 4.1 ev 2
Electrons in periodic potential When two identical atoms, e.g. two Si atoms, are brought together, the orbitals of the two outer electrons form two combined orbitals: bonding orbital with lower energy and antibonding orbital of higher energy. The electron probability density of these orbitals is the highest in the region between the atoms and it serves as directional covalent bond. Due to this directionality, lagre number of atoms brought together form regular crystal structure. 3
Electrons in periodic potential (a) Energy of electron versus wavevector k (linear momentum p=ħk) for free electron. (b) Energy of electron versus wavevector k in monoatomic linear crystal lattice of lattice constant a. The energy gap E g is associated with the first Bragg reflection at k=±π/a. 4
Formation of energy bands of electrons in crystals 5
Electron energy bands in crystals Possible energy band diagrams of a crystal. Shown are: a) a half filled band, b) two overlapping bands, c) an almost full band separated by a small bandgap from an almost empty band and d) a full band and an empty band separated by a large bandgap. 6
Electron energy bands in silicon and SiO 2 Totally filled bands and totally empty bands do not allow current flow (just as there is no motion of liquid in a totally filled, or totally empty bottle). Semiconductors have lower E g s than insulators. 7
Measuring the Band Gap Energy by Light Absorption electron photons photon energy: h v > E g E g E c E v hole E g can be determined from the minimum energy (hn) of photons that are absorbed by the semiconductor. Bandgap energies of selected semiconductors Semi-conductor InSb Ge Si GaAs GaP ZnSe Diamond Eg (ev) 0.18 0.67 1.12 1.42 2.25 2.7 6 8
Temperature dependence of the energy bandgap The temperature dependence of the energy bandgap, E g, has been experimentally determined yielding the following expression for E g as a function of the temperature, T: 9
Doping dependence of the energy bandgap E g (N) = 3e2 16πε s e 2 N ε s kt E g Si N = 22.5 N 10 18 (cm 3 ) mev A = πr 2 High doping densities cause the bandgap to shrink. This effect is explained by the fact that the wavefunctions of the electrons bound to the impurity atoms start to overlap as the density of the impurities increase. For instance, at a doping density of 10 18 cm -3, the average distance between two impurities is only 10 nm. This overlap forces the energies to form an energy band rather than a discreet level. 10