3.23 Electrical, Optical, and Magnetic Properties of Materials

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1 MIT OpenCourseWare Electrical, Optical, and Magnetic Properties of Materials Fall 2007 For information about citing these materials or our Terms of Use, visit:

2 3.23 Fall 2007 Lecture 13 THE LAW OF MASS ACTION Last time 1. Band structure of oides (perovskites), semiconductors (silicon, and compared with lead), late (fcc) transition metals (same period, or same group), graphene and nanotubes 2. Independent electron gas: states, energy, density, DOS 3. DOS of massive and massless bands in 1, 2 and 3 dimensions 4. Statistics of classical and quantum particles, Fermi-Dirac distribution, chemical potential 1

3 Chap 6 Singleton, or, much better, Study Chap 28 (Homogeneous semiconductors) Ashcroft-Mermin (to be posted) Sb-doped Germanium Courtesy of Elsevier, Inc., Used with permission. 2

4 Semiconductors Please see any graph of semiconductor band gaps vs. lattice constants, such as Valence+conduction bands in Si Please see: Fig. 6.1 in Singleton, John. Band Theory and Electronic Properties of Solids. Oford, England: Oford University Press,

5 Band structure of Si, Ge, GaAs Please see any image of Si, Ge, and GaAs energy bands, such as Conduction band minima (in 3d) Please see Fig in Marder, Michael P. Condensed Matter Physics. New York, NY: Wiley-Interscience,

6 Optical absorption in Ge Please see: Fig. 6.3 and 6.4 in Singleton, John. Band Theory and Electronic Properties of Solids. Oford, England: Oford University Press, Impress your eaminers (orals) Tet removed due to copyright restrictions. Please see Table 19.1 in Marder, Michael P. Condensed Matter Physics. New York, NY: Wiley-Interscience,

7 Number of carriers at thermal equilibrum Please see Fig in Pierret, Robert F. Semiconductor Device Fundamentals. Reading, MA: Addison-Wesley, Conduction and valence DOS (non-degenerate sc, isotropic effective mass) Please see: Fig. 18 in Kittel, Charles. "Introduction to Solid State Physics." Chapter 8 in Semiconductor Crystals. New York, NY: John Wiley & Sons,

8 Density of available states Miracle! Law of Mass Action 3/2 m 3/2 c T 19 N T = / c m 3 c ( ) m 300K m 3/2 3/2 v T 19 3 Pv ( T ) = / cm m 300K 7

9 n c Intrinsic case ( T) = p T n T ) ( ) n i ( T ) v Intrinsic case 8

10 Etrinsic case n T p T = c ( ) ( ) n v Etrinsic case 9

11 Impurity levels Adding impurities can lead to controlled domination of one carrier type n-type is dominated by electrons p-type is dominated by holes Adding other impurities can degrade electrical properties Impurities with close electronic structure to host p-type n-type Impurities that create deep levels deep level Al - As + - Au Si Si E c E E c D E c E DEEP E A E v E v E v Impurity states as embedded hydrogen atoms Consider the weakly bound 5 th electron in Phosphorus as a modified hydrogen atom For hydrogenic donors or acceptors, we can think of the electron or hole, respectively, as an orbiting electron around a net fied charge We can estimate the energy to free the carrier into the conduction band or valence band by using a modified epression for the energy of an electron in the H atom me E n = = =e 8ε h n n (ev) o me 4 e 2 m * e m * 1 E n = ε r = ε o h n 8ε o h n ε r n m ε Thus, for the ground state n=1, we can see already that since e is on the order of 10, the binding energy of the carrier to the impurity atom is <0.1eV Epect that many carriers are then ionized at room T B acceptor in Si: ev P donor in Si: ev As donor in Si:

12 Temperature dependence of majority carriers Please see Fig in Pierret, Robert F. Semiconductor Device Fundamentals. Reading, MA: Addison-Wesley, (Inverse temperature plot) Please see: Fig in Singleton, John. Band Theory and Electronic Properties of Solids. Oford, England: Oford University Press,

3.23 Electrical, Optical, and Magnetic Properties of Materials

3.23 Electrical, Optical, and Magnetic Properties of Materials MIT OpenCourseWare http://ocw.mit.edu 3.23 Electrical, Optical, and Magnetic Properties of Materials Fall 2007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.

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