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Density of States of bulk and quantum confined structures

brief analysis of density of states for bulk, quantum well and quantum wire

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Density of States of bulk and quantum confined structures

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  1. Course: Quantum Electronics Arpan Deyasi Quantum Topic: Density of states of Bulk and Quantum Structures Electronics Arpan Deyasi Arpan Deyasi, RCCIIT 6/10/2020 1

  2. What is DoS? Arpan Deyasi E Number of available energy states Quantum per unit energy interval Electronics EC+ dEC dEC per unit dimension EC in real space k EV dEV EV+ dEV 6/10/2020 Arpan Deyasi, RCCIIT 2

  3. What do we mean by ‘dimension’? Arpan Deyasi For ‘bulk’, it is ‘volume’ Quantum For ‘quantum well’, it is ‘area/surface’ Electronics For ‘quantum wire’, it is ‘line/length’ For ‘quantum dot’, it is a ‘point/dot’ 6/10/2020 Arpan Deyasi, RCCIIT 3

  4. Energy band diagram is drawn in E-k plane Arpan Deyasi ‘k’ is wave-vector, not a physical quantity Quantum No of electrons is measured by magnitude of current Electronics So we must know the density of electrons in real space instead of k-space 6/10/2020 Arpan Deyasi, RCCIIT 4

  5. Fermi sphere Arpan Deyasi Quantum Electronics 6/10/2020 Arpan Deyasi, RCCIIT 5

  6. Fermi surface Arpan Deyasi Quantum Electronics 6/10/2020 Arpan Deyasi, RCCIIT 6

  7. DoS for bulk semiconductor Arpan Deyasi Let’s start with Bloch theorem Quantum ( , , ) x y z  Electronics Consider a 3D semiconductor with dimensions Lx, Ly, Lz =  + + + ( , , ) x L y L z L x y z 6/10/2020 Arpan Deyasi, RCCIIT 7

  8. DoS for bulk semiconductor Arpan Deyasi For validity of wave function Quantum = = =    2 2 2 k L k L n n x x x Electronics y y y k L n z z z (2 )  3 n n n x y z = k k k Volume in k-space x y z L L L x y z 6/10/2020 Arpan Deyasi, RCCIIT 8

  9. DoS for bulk semiconductor Arpan Deyasi Let Quantum = = = 1 n n n x y z = = = L L L L Electronics x y z  3 (2 ) L = k V Volume of unit cell in k-space 3 6/10/2020 Arpan Deyasi, RCCIIT 9

  10. DoS for bulk semiconductor Arpan Deyasi Volume of Fermi sphere in k-space Quantum 4 3 =  3 V k F Volume of semiconductor in real space Electronics = 3 V L R 6/10/2020 Arpan Deyasi, RCCIIT 10

  11. DoS for bulk semiconductor Arpan Deyasi number of energy states in real space Quantum 1 =  N V V F V number of energy states in real space Electronics k 1 1 =   N V R F V V k R 6/10/2020 Arpan Deyasi, RCCIIT 11

  12. DoS for bulk semiconductor Arpan Deyasi Quantum 3 4 3 1 L L  =    3 N k 3 3 8 Electronics 3 k  = = ( ) N N k 2 6 6/10/2020 Arpan Deyasi, RCCIIT 12

  13. DoS for bulk semiconductor Arpan Deyasi Introducing Pauli’s exclusion principle Quantum 3 3 k  k  =  = ( ) 2 N k 2 2 6 3 Electronics   2 k  = ( ) N k 2 k 6/10/2020 Arpan Deyasi, RCCIIT 13

  14. DoS for bulk semiconductor Arpan Deyasi From parabolic dispersion relation Quantum 2 2 k = E * 2 m Electronics   2 E k k = * m   2 2 * m E m 2 m E k E = = * * 6/10/2020 Arpan Deyasi, RCCIIT 14

  15. DoS for bulk semiconductor Arpan Deyasi   * 1 k E m = Quantum 2 E Electronics       N E N k k E =    2 * 1 N E k  m =  2 2 E 6/10/2020 Arpan Deyasi, RCCIIT 15

  16. DoS for bulk semiconductor Arpan Deyasi ρ(E)   * 2 * m E  1 N E m =  Quantum 2 2 2 E Electronics 3/2        * 1  2 N E m =  E 2 2 2 E   N E  E For a particular material 6/10/2020 Arpan Deyasi, RCCIIT 16

  17. DoS for Quantum Well Arpan Deyasi  2 (2 ) L = k A Area in k-space Quantum 2 Area of Fermi circle in k-space Electronics =  2 A k F Area of semiconductor in real space = 2 A L R 6/10/2020 Arpan Deyasi, RCCIIT 17

  18. DoS for Quantum Well Arpan Deyasi number of energy states in real space Quantum 1 A 1 A =   N A F k R Electronics 2 1 L L  =    2 N k 2 2 4 2 k = = ( ) N N k  4 6/10/2020 Arpan Deyasi, RCCIIT 18

  19. DoS for Quantum Well Arpan Deyasi Introducing Pauli’s exclusion principle Quantum 2 2 k k =  = ( ) 2 N k Electronics   4 2   k  = ( ) N k k 6/10/2020 Arpan Deyasi, RCCIIT 19

  20. DoS for Quantum Well Arpan Deyasi From parabolic dispersion relation Quantum   * 1 k E m = 2 E Electronics       N E N k k E =    * 1 N E k  m =  2 E 6/10/2020 Arpan Deyasi, RCCIIT 20

  21. DoS for Quantum Well Arpan Deyasi   * * 2 1 N E m E  m =  Quantum 2 E Electronics   * N E m =  2 DoS is independent of energy? 6/10/2020 Arpan Deyasi, RCCIIT 21

  22. DoS for Quantum Well Arpan Deyasi The result is obtained for a particular sub-band Quantum ρ(E) Considering all the sub-bands Electronics   * n N E m  =  − ( ) E E i  2 = 1 i E E1 E2 E3Ei-1EiEi+1 6/10/2020 Arpan Deyasi, RCCIIT 22

  23. DoS for Quantum Wire Arpan Deyasi  2 = k L Length in k-space Quantum L Length of Fermi line in k-space Electronics = 2 L k F Length of semiconductor in real space = L L R 6/10/2020 Arpan Deyasi, RCCIIT 23

  24. DoS for Quantum Wire Arpan Deyasi number of energy states in real space Quantum 1 L 1 L =   N L F k R Electronics 1 L L  =   2 N k 2 k  = N 6/10/2020 Arpan Deyasi, RCCIIT 24

  25. DoS for Quantum Wire Arpan Deyasi Introducing Pauli’s exclusion principle Quantum 2k  = N Electronics   2  = ( ) N k k 6/10/2020 Arpan Deyasi, RCCIIT 25

  26. DoS for Quantum Wire Arpan Deyasi From parabolic dispersion relation Quantum   * 1 k E m = 2 E Electronics       N E N k k E =    * 2  1 N E m =  2 E 6/10/2020 Arpan Deyasi, RCCIIT 26

  27. DoS for Quantum Wire Arpan Deyasi         2 * m 1 N E = Quantum ρ(E)  E Electronics E E1E2E3E4E5 6/10/2020 Arpan Deyasi, RCCIIT 27

  28. DoS for Quantum Dot Arpan Deyasi ρ(E) ρ(E) Quantum Electronics E E E1 E2 E3 E4 E1 E2 E3 E4 6/10/2020 Arpan Deyasi, RCCIIT 28

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