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Propagation Matrix Method applied on quantum well

calculation of transmission coefficient and eigenstates of double quantum well structure

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Propagation Matrix Method applied on quantum well

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  1. Course: Quantum Electronics Arpan Deyasi Quantum Transmission Coefficient using PMM Electronics Calculation of Arpan Deyasi Arpan Deyasi, RCCIIT 14/10/2020 1

  2. Graphical representation of Transmission Coefficient Arpan Deyasi T(E) Quantum Electronics E E1 E2 E3 E0 14/10/2020 Arpan Deyasi, RCCIIT 2

  3. Drawbacks of Transfer Matrix Technique Arpan Deyasi Step potential well: Ideal case Quantum Real quantum well can’t have step potential profile, in fact, different complex potential profiles are considered for application purpose Electronics 14/10/2020 Arpan Deyasi, RCCIIT 3

  4. Solution Arpan Deyasi We have to consider a mathematical technique where variation of potential at each point of both barrier and well layers can be incorporated Quantum Electronics V = V0 V = 0 14/10/2020 Arpan Deyasi, RCCIIT 4

  5. Change required in Mathematical Formulation Arpan Deyasi  = Quantum * − 2 ( )( z E V ) m * 2 ( ) z E m  = 0 b w 1 2 2 2 Electronics for well for barrier 14/10/2020 Arpan Deyasi, RCCIIT 5

  6. Change required in Mathematical Formulation We have to make a generalized wave-vector for both barrier and well Arpan Deyasi Quantum Electronics Why? Then we will be able to incorporate simultaneous variation of barrier and well potentials 14/10/2020 Arpan Deyasi, RCCIIT 6

  7. Generalized wave-vector Arpan Deyasi * − 2 ( )( z E V ) m Quantum j j  = j 2 calculated at jthpoint Electronics 14/10/2020 Arpan Deyasi, RCCIIT 7

  8. DQWTB structure Arpan Deyasi Quantum Electronics j j+1 Z 14/10/2020 Arpan Deyasi, RCCIIT 8

  9. Generalized wave equations Arpan Deyasi  =  + −  exp( ) exp( ) A i z B i z Quantum j j j j j = Electronics   + −  exp( ) exp( ) C i z D i z + + + + + 1 1 1 1 1 j j j j j 14/10/2020 Arpan Deyasi, RCCIIT 9

  10. Generalized boundary conditions Arpan Deyasi  =  Quantum j+ 1 j Electronics   d d + 1 j j = dz dz 14/10/2020 Arpan Deyasi, RCCIIT 10

  11. At interface Arpan Deyasi  =  Quantum j+ 1 j  + −  exp( ) exp( + ) A = i z  B i z Electronics j C j j j −  exp( ) exp( ) i z D i z + + + + 1 1 1 1 j j j j 14/10/2020 Arpan Deyasi, RCCIIT 11

  12. At interface Arpan Deyasi  =  ' ' j+ 1 j  Quantum  −  −  exp( ) exp(  − ) i = Electronics A i z i B i z j j j j z j j   −  exp( ) exp( ) i C i i D i z + + + + + + 1 1 1 1 1 1 j j j j j j  =  −  −  exp( ) exp(  − ) A i z B i z j j j j j j   −  exp( ) exp( ) C i z D i z + + + + + + 1 1 1 1 1 1 j j j j j j 14/10/2020 Arpan Deyasi, RCCIIT 12

  13. At interface Arpan Deyasi Quantum  − −  exp( ) exp( ) A i z B i z j        j j j           + 1 j =  exp( ) C i z Electronics + + 1 1 j j j   + 1 j − −  exp( ) D i z + + 1 1 j j j 14/10/2020 Arpan Deyasi, RCCIIT 13

  14. At interface  + −  exp( ) exp( + ) A = Arpan Deyasi i z  B i z j C j j j −  exp( ) exp( ) i z D i z + + + + 1 1 1 1 j j j j Quantum  − −  exp( ) exp( ) A i z B i z j     j j j                 + Electronics + 1 1 j j =  − −  exp( ) exp( ) C i z D i z + + + + 1 1 1 1 j j j j j j         1 1            A B C D       1 1 1 − In matrix notation + 1       j j = + + 1 1 j j − 1 + 1 j j j j 14/10/2020 Arpan Deyasi, RCCIIT 14

  15. At interface Arpan Deyasi         1 1            A B C D       1 1 1 − + 1       j j = Quantum + + 1 1 j j − 1 + 1 j j j j Electronics       1 1 − 1             A B C D         1 1 1 − + 1       j j = + + 1 1 j j − 1 + 1 j j j j 14/10/2020 Arpan Deyasi, RCCIIT 15

  16. At interface Arpan Deyasi       1 1    Quantum          A B C D       1 1 1 − 1 2 + 1       j j = + + 1 1 j j − 1 + 1 j j j j Electronics                                                     + + 1 1 j j + − 1 1             A B C D 1 2 j j + 1 j j =     + 1 j j + + 1 1 j j − + 1 1 j j 14/10/2020 Arpan Deyasi, RCCIIT 16

  17. Arpan Deyasi At interface                                                     + + Quantum 1 1 j j + − 1 1 1 2 j j = P Let j     Electronics + + 1 1 j j − + 1 1 j j Pj: junction matrix 14/10/2020 Arpan Deyasi, RCCIIT 17

  18. DQWTB structure Arpan Deyasi Quantum Electronics Lj j j+1 Z 14/10/2020 Arpan Deyasi, RCCIIT 18

  19. Significance of Lj Arpan Deyasi travelling from ‘j’ to ‘j+1’ with a distance ‘Lj’ matches the positive coefficients as well as negative coefficients Quantum Electronics 14/10/2020 Arpan Deyasi, RCCIIT 19

  20. For Lj Arpan Deyasi  = exp( ) A i L C + 1 j j j j Quantum −  = exp( ) B i L D + 1 j j j j Electronics                 exp( ) 0 i i L A B C D + 1 j j j j In matrix notation =  −  0 exp( ) L + 1 j j j j 14/10/2020 Arpan Deyasi, RCCIIT 20

  21. For Lj Arpan Deyasi −     Quantum             exp( ) 0 i  A B i L C D + 1 j j j j =  0 exp( ) L + 1 j j j j Let Electronics −      exp( ) 0 i  i L j j = P L 0 exp( ) L  j j PL : step matrix 14/10/2020 Arpan Deyasi, RCCIIT 21

  22. Q: How to calculate propagation matrix? Arpan Deyasi It is the Cartesian product of Quantum Junction matrix with Step matrix Electronics 14/10/2020 Arpan Deyasi, RCCIIT 22

  23. Propagation Matrix calculation Arpan Deyasi         Electronics                                             + + 1 1 j j + − 1 1 Quantum 1 2 j j = P j     + + 1 1 j j − + 1 1 j j −      exp( ) 0 i  i L j j = P L 0 exp( ) L  j j 14/10/2020 Arpan Deyasi, RCCIIT 23

  24. Propagation Matrix calculation Arpan Deyasi                         −                                + + 1 1 j j + − 1 1 Quantum 1 2 j j = P     Electronics + + 1 1 j j − + 1 1 =  P P P j L j j   exp( ) 0 i  i L j j  0 exp( ) L  j j 14/10/2020 Arpan Deyasi, RCCIIT 24

  25. Propagation Matrix calculation Arpan Deyasi                               exp( Quantum + + 1 1 j j −  + −  − ) 1        exp( ) 1        i L i L j j j j 1 2 j j = P exp( Electronics             + + 1 1 j j  −  + ) 1 exp( ) 1 i L i L j j j j j j      P P P P 11 12 = P 21 22 14/10/2020 Arpan Deyasi, RCCIIT 25

  26. Propagation Matrix calculation Arpan Deyasi             A B C D + 1 j j = P Quantum + 1 j j Electronics             A B C D       P P P P + 1 j j 11 12 = + 1 j j 21 22 14/10/2020 Arpan Deyasi, RCCIIT 26

  27. Propagation Matrix calculation Arpan Deyasi             A B C D       P P P P + 1 j j 11 12 = Quantum + 1 j j 21 22 Electronics = + A P C P D + + 11 1 12 1 j j j = + B P C P D + + 21 1 22 1 j j j 14/10/2020 Arpan Deyasi, RCCIIT 27

  28. Arpan Deyasi Cj+1 Aj P11 P12 Quantum P21 P22 Electronics Dj+1 Bj P12is the transmission coefficient when the wave is traversing from port 2 to port 1 and port 1 is terminated by matched load 14/10/2020 Arpan Deyasi, RCCIIT 28

  29. P12 = 0 for practical device Arpan Deyasi = + A P C P D + + 11 1 12 1 j j j Quantum A j = P 11 C+ Electronics 1 j 2         C 1 ( ) + 1 j = = T E * A 11 11 P P j 14/10/2020 Arpan Deyasi, RCCIIT 29

  30. Graphical representation of Transmission Coefficient Arpan Deyasi T(E) Quantum Electronics E E1 E2 E3 E0 14/10/2020 Arpan Deyasi, RCCIIT 30

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