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Unformatted text preview: ECE 3080  Dr. Alan Doolittle Georgia Tech Lecture 4 Density of States and Fermi Energy Concepts Reading: (Contd) Notes and Anderson 2 sections 2.82.13 ECE 3080  Dr. Alan Doolittle Georgia Tech How do electrons and holes populate the bands? Density of States Concept In lower level courses, we state that Quantum Mechanics tells us that the number of available states in a cubic cm per unit of energy, the density of states, is given by: eV cm States of Number unit E E E E m m E g E E E E m m E g v v p p v c c n n c 3 3 2 * * 3 2 * * , ) ( 2 ) ( , ) ( 2 ) ( ECE 3080  Dr. Alan Doolittle Georgia Tech How do electrons and holes populate the bands? Density of States Concept Thus, the number of states per cubic centimeter between energy E and E+dE is otherwise and and , E E if )dE (E g , E E if )dE (E g v v c c But where does it come from? ECE 3080  Dr. Alan Doolittle Georgia Tech How do electrons and holes populate the bands? Where Does the Density of States Concept come from? Approach: 1. Find the smallest volume of kspace that can hold an electron. This will turn out to be related to the largest volume of real space that can confine the electron. 2. Next assume that the average energy of the free electrons (free to move), the fermi energy E f , corresponds to a wave number k f . This value of k f , defines a volume in kspace for which all the electrons must be within. 3. We then simply take the ratio of the total volume needed to account for the average energy of the system to the smallest volume able to hold an electron ant that tells us the number of electrons. 4. From this we can differentiate to get the density of states distribution. ECE 3080  Dr. Alan Doolittle Georgia Tech How do electrons and holes populate the bands? Derivation of Density of States Concept First a needed tool: Consider an electron trapped in an energy well with infinite potential barriers. Recall that the reflection coefficient for infinite potential was 1 so the electron can not penetrate the barrier. After Neudeck and Pierret Figure 2.4a 2 2 2 2 n 2 2 2 2 2 2 2 2 2 2 2 2 E and sin ) ( 3... , 2 1, n for a n k ka Asin (a) B ) ( : Conditions Boundary 2 or E 2 2 k where cos sin ) ( : Solution General 2 2 ma n a x n A x m k mE kx B kx A x k x E x m E V m n n ECE 3080  Dr. Alan Doolittle Georgia Tech How do electrons and holes populate the bands?...
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This note was uploaded on 08/23/2011 for the course ECE 3080 taught by Professor Staff during the Spring '08 term at Georgia Institute of Technology.
 Spring '08
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