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Mics. print

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Mics. print

  1. 1. 1 3-1-4. Direct & Indirect Semiconductors • A single electron is assumed to travel through a perfectly periodic lattice. • The wave function of the electron is assumed to be in the form of a plane wave moving. xjk xk x exkUx ),()(   x : Direction of propagation  k : Propagation constant / Wave vector   : The space-dependent wave function for the electron
  2. 2. 2 3-1-4. Direct & Indirect Semiconductors  U(kx,x): The function that modulates the wave function according to the periodically of the lattice.  Since the periodicity of most lattice is different in various directions, the (E,k) diagram must be plotted for the various crystal directions, and the full relationship between E and k is a complex surface which should be visualized in there dimensions.
  3. 3. 3 3-1-4. Direct & Indirect Semiconductors Eg=hν Eg Et k k EE Direct Indirect Example 3-1
  4. 4. 4 3-1-4. Direct & Indirect Semiconductors Example 3-1: Assuming that U is constant in for an essentially free electron, show that the x-component of the electron momentum in the crystal is given by xx khP  Example 3-2 ),()( xkUx xk  xjkx e
  5. 5. 5 3-1-4. Direct & Indirect Semiconductors x x xjkxjk x kh dxU dxUkh dxU dxe xj h eU P xx                  2 2 2 2 )(Answer: The result implies that (E,k) diagrams such as shown in previous figure can be considered plots of electron energy vs. momentum, with a scaling factor . h
  6. 6. 6 3-2-2. Effective Mass • The electrons in a crystal are not free, but instead interact with the periodic potential of the lattice. • In applying the usual equations of electrodynamics to charge carriers in a solid, we must use altered values of particle mass. We named it Effective Mass.
  7. 7. 7 3-2-2. Effective Mass Example 3-2: Find the (E,k) relationship for a free electron and relate it to the electron mass. E k
  8. 8. 8 3-2-2. Effective Mass khmvp  2 22 2 22 1 2 1 k m h m p mvE  Answer: From Example 3-1, the electron momentum is: m h dk Ed 2 2 2 
  9. 9. 9 3-2-2. Effective Mass Answer (Continue): Most energy bands are close to parabolic at their minima (for conduction bands) or maxima (for valence bands). EC EV
  10. 10. 10 3-2-2. Effective Mass • The effective mass of an electron in a band with a given (E,k) relationship is given by 2 2 2 * dk Ed h m   X L k E 1.43eV )()( or ** LXmm  Remember that in GaAs:
  11. 11. 11 3-2-2. Effective Mass • At k=0, the (E,k) relationship near the minimum is usually parabolic: gEk m h E  2 * 2 2 In a parabolic band, is constant. So, effective mass is constant. Effective mass is a tensor quantity. 2 2 dk Ed 2 2 2 * dk Ed h m 
  12. 12. 12 3-2-2. Effective Mass EV EC 02 2  dk Ed 02 2  dk Ed 0* m 0* m 2 2 2 * dk Ed h m  Ge Si GaAs † m0 is the free electron rest mass. Table 3-1. Effective mass values for Ge, Si and GaAs. mn * mp * 055.0 m 01.1 m 0067.0 m 037.0 m 056.0 m 048.0 m
  13. 13. 13 3-2-5. Electrons and Holes in Quantum Wells • One of most useful applications of MBE or OMVPE growth of multilayer compou-nd semiconductors is the fact that a continuous single crystal can be grown in which adjacent layer have different band gaps. • A consequence of confining electrons and holes in a very thin layer is that
  14. 14. 14 3-2-5. Electrons and Holes in Quantum Wells these particles behave according to the particle in a potential well problem. GaAs Al0.3Ga0.7AsAl0.3Ga0.7As 50Å E1 Eh 1.43eV1.85eV 0.28eV 0.14eV 1.43eV
  15. 15. 15 3-2-5. Electrons and Holes in Quantum Wells • Instead of having the continuum of states as described by ,modified for effective mass and finite barrier height. • Similarly, the states in the valence band available for holes are restricted to discrete levels in the quantum well. 2 222 2mL hn En  
  16. 16. 16 3-2-5. Electrons and Holes in Quantum Wells • An electron on one of the discrete condu-ction band states (E1) can make a transition to an empty discrete valance band state in the GaAs quantum well (such as Eh), giving off a photon of energy Eg+E1+Eh, greater than the GaAs band gap.

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