1 / 37

Transfer Matrix Technique applied on quantum well

calculation of transmission coefficient and eigenstates using transfer matrix technique for DQWTB structure

Arpan3
Télécharger la présentation

Transfer Matrix Technique applied on quantum well

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Course: Quantum Electronics Arpan Deyasi Quantum Electronics Calculation of Transmission Coefficient using Transfer Matrix Technique Arpan Deyasi Arpan Deyasi, RCCIIT 9/10/2020 1

  2. Single Quantum Well Arpan Deyasi Quantum Electronics 9/10/2020 Arpan Deyasi, RCCIIT 2

  3. Properties to be evaluated: Arpan Deyasi Electronic Properties Quantum 1. Transmission Coefficient 2. Eigen Energy 3. Density of States Electronics Optical Properties 1. Absorption Coefficient 2. Oscillator Strength 9/10/2020 Arpan Deyasi, RCCIIT 3

  4. Numerical Techniques may be considered for Calculation Arpan Deyasi Transfer Matrix Technique (TMT) Quantum Propagation Matrix Method (PMM) Electronics Perturbation Method WKB Approximation Finite Element Method (FEM) Finite Difference Time Domain Method (FDTD) 9/10/2020 Arpan Deyasi, RCCIIT 4

  5. Q. Which have better accuracy? Arpan Deyasi Quantum Q. Which are faster for calculation? Electronics We have to optimize between them 9/10/2020 Arpan Deyasi, RCCIIT 5

  6. FDTD and FEM are most accurate as per the literatures Arpan Deyasi TMT & PMM are faster which incorporate fast principle Quantum Electronics 9/10/2020 Arpan Deyasi, RCCIIT 6

  7. Arpan Deyasi Today we will start the calculation of Electronic Properties using Transfer Matrix Technique Quantum Electronics We will consider Double Quantum Well structure for our theoretical work 9/10/2020 Arpan Deyasi, RCCIIT 7

  8. Arpan Deyasi DQWTB structure A1 A2 A3 A4 A5 Quantum a a b Electronics Z=a+b Z=0 Z=a Z Z=2a+b B5 B1 B2 B3 B4 I II III IV V 9/10/2020 Arpan Deyasi, RCCIIT 8

  9. Schrödinger Equation for well region Arpan Deyasi for V=0 Quantum * 2 ( ) z E m 2  ( ) z d  = w 2 +  ( ) ( ) z  = 0 z 2 2 2 2 dz Schrödinger Equation for barrier region Electronics for V=V0 ( 2 ) * E V − 2 ( ) z m 2  ( ) z d  = 0 b 2 +  ( ) ( ) z  = 0 z 1 1 2 dz 9/10/2020 Arpan Deyasi, RCCIIT 9

  10. Solution of Schrödinger Equation in different regions Arpan Deyasi  =  + i z  − exp( Quantum ) exp( ) A i z B 1 1 1 1 I  =  + −  exp( ) exp( ) A i z B i z 2 2 2 2 II Electronics  =  + i z  − exp( ) exp( ) A i z B 3 1 3 1 III  =  + −  exp( ) exp( ) A i z B i z 4 2 4 2 IV  =  + i z  − exp( ) exp( ) A i z B 5 1 5 1 V 9/10/2020 Arpan Deyasi, RCCIIT 10

  11. Ben-Daniel Duke Boundary Conditions Arpan Deyasi  =  Quantum I II Electronics   1 1 d d = I II * * dz dz m m I II A little modification is required in second boundary condition. Why? 9/10/2020 Arpan Deyasi, RCCIIT 11

  12. Both κ1and κ2are functions of m* Arpan Deyasi So to avoid dual effect of m*, we will modify the 2ndcondition as Quantum Electronics   d d = I II dz dz 9/10/2020 Arpan Deyasi, RCCIIT 12

  13. at Z = 0 (1stinterface) Arpan Deyasi  =  Quantum I II  + i z  − − exp( A ) exp( exp( B ) A = i z B + 1 1 i  1 1 i Electronics  exp( ) ) z z 2 2 2 2 + = + A B A B 1 1 2 2 9/10/2020 Arpan Deyasi, RCCIIT 13

  14. at Z = 0 (1stinterface) Arpan Deyasi  =  ' '  Quantum I II  −  i z  − − exp( exp( A ) exp( exp( B ) i = Electronics A i z i B i − 1  1 1  1  1 1 i  ) ) i i z z 2 2 2 2 2 2  −  =  −  A B A B 1 1 1 1 2 2 2 2 9/10/2020 Arpan Deyasi, RCCIIT 14

  15. at Z = 0 (1stinterface) Arpan Deyasi + = + A B A B Quantum 1 1 2 2  −  =  −  A B A B 1 1 1 1 2 2 2 2 Electronics                 1 1  − 1 1  − A B A B 1 2 = In matrix notation       1 1 1 2 2 2 9/10/2020 Arpan Deyasi, RCCIIT 15

  16. at Z = 0 (1stinterface) Arpan Deyasi Quantum                 1 1  − 1 1  − A B A B 1 2 =       1 1 1 2 2 2 Electronics             A B A B 1 2 = M M 1 2 1 2 9/10/2020 Arpan Deyasi, RCCIIT 16

  17. at Z = a (2ndinterface) Arpan Deyasi  =  Quantum II III  + −  exp( exp( A ) exp( exp( B ) A = i z B + i z 2 2 i  2 2 i z  − Electronics ) ) z 3 1 3 1  + −  exp( exp( A ) exp( exp( B ) A = i a B + i a 2 2 i  2 2 i a  − ) ) a 3 1 3 1 9/10/2020 Arpan Deyasi, RCCIIT 17

  18. at Z = a (2ndinterface) Arpan Deyasi  Quantum =  ' ' II III   −  − −  exp( exp( A ) exp( exp( i B  ) i = Electronics A i z i B i z 2 i  2 2 i 2 2 2 i z  −  ) ) z 1 3 1 1 3 1  =  −  − −  exp( exp( A ) exp( exp( B  ) A i a  B i a 2  2 2 i 2 2 2 i a  − ) ) a 1 3 1 1 3 1 9/10/2020 Arpan Deyasi, RCCIIT 18

  19. at Z = a (2ndinterface) Arpan Deyasi  + −  =  + i a  − exp( ) exp( ) exp( ) exp( ) A i a B i a A i a B 2 2 Quantum 2 2 3 1 3 1  =  −  − −  exp( exp( A ) exp( exp( B  ) A i a  B i a 2  2 2 i 2 2 2 i a  − ) ) a 1 3 1 1 3 1 Electronics  −  −         exp( exp(    =   ) a  exp( exp(  − ) i a i a   − A B 2 2 i 2   − ) ) i a In matrix notation 2 2 i a i a  − 2 2 2         exp( exp(  ) exp( exp(  − ) A B i a 3 1 1 i a  − ) ) 3 1 1 1 1 9/10/2020 Arpan Deyasi, RCCIIT 19

  20. at Z = a (2ndinterface) Arpan Deyasi  −  −         exp( exp(    =   Electronics ) a  exp( exp(  − ) i a i a   − A B 2 2 i 2  Quantum  − ) ) i a 2 2 i a i a  − 2 2 2         exp( exp(  ) exp( exp(  − ) A B i a 3 1 1 i a  − ) ) 3 1 1 1 1          =   A B A B 3 2 M M 3 4 3 2 9/10/2020 Arpan Deyasi, RCCIIT 20

  21. at Z = (a+b) (3rdinterface) Arpan Deyasi  =  III IV Quantum  + i z  − exp( exp( A ) exp( exp( ) A = i z B + 3 1 i  3 B 1 i  − ) ) z z 4 2 4 2 Electronics  + + −  + exp( exp( A (  ) exp( B ( i )) + A = i a b a b B i a b  3 1 i 3 + 1 − + ( )) exp( ( )) a b 4 2 4 2 9/10/2020 Arpan Deyasi, RCCIIT 21

  22. at Z = (a+b) (3rdinterface)  Arpan Deyasi =  ' ' III IV Quantum   −  i z  − exp( exp( ) z exp( exp( ) i = A i z i B i − 1  3 A 1  1  3 B 1 i  − ) ) i i z Electronics 2 4 2 2 4 2  =  + −  − −  − + exp( exp( ( )) exp( B ( )) + A i a b a b B  i a b  1  3 A 1  1 3 1 i + ( )) exp( ( )) i a b 2 4 2 2 4 2 9/10/2020 Arpan Deyasi, RCCIIT 22

  23. at Z = (a+b) (3rdinterface) Arpan Deyasi  + + −  − a b + exp( exp( A Quantum (  )) + exp( B (  )) + A = i a b a b B + i 3 1 i 3 1 ( )) exp( ( )) i a b 4 2 4 2  =  + −  − −  − + exp( exp( ( )) exp( B ( )) + A i a b a b B  i a b  1  3 A 1  1 3 1 i 2 Electronics + ( )) exp( ( )) i a b 4 2 2 4 2  + −  − +           exp( exp(    =   ( )) exp( exp(  − ( )) A B i a b a b i a b a b In matrix notation 3 1   1 i   − − + + + + ( )) ( )) i 3 1 1 1 1       exp( exp(  ( )) exp( exp(  − ( )) + i a b a b i a b a b A B 2 2 i  4   + ( )) ( )) i 2 2 2 2 4 9/10/2020 Arpan Deyasi, RCCIIT 23

  24. at Z = (a+b) (3rdinterface) Arpan Deyasi  + −  − +           exp( exp(    =   Electronics ( )) exp( exp(  − ( )) A B i a b a b i a b a b 3 1   1 i   − − + + + + ( )) ( )) i 1 Quantum 3 1 1 1       exp( exp(  ( )) exp( exp(  − ( )) + i a b a b i a b a b A B 2 2 i  4   + ( )) ( )) i 2 2 2 2 4             A B A B 3 4 = M M 5 6 3 4 9/10/2020 Arpan Deyasi, RCCIIT 24

  25. at Z = (2a+b) (4thinterface) Arpan Deyasi  =  IV V Quantum  + −  exp( exp( A ) exp( exp( B ) A = i z B + i z 4 2 i  4 2 i z  − ) ) z 5 1 5 1 Electronics  + − −  + exp( exp( A (2 )) exp( exp( (2 )) A = i a b a b B + i − a b a b 4 2 i  4 B 2  + + (2 )) (2 )) i 5 1 5 1 9/10/2020 Arpan Deyasi, RCCIIT 25

  26. at Z = (2a+b) (3rdinterface) Arpan Deyasi  =  ' ' IV V i = Quantum  Electronics  −  − −  exp( exp( A ) exp( exp( i B  ) A i z  i B i z 2 i  4 2 i 2 4 2 i z  − ) ) z 1 5 1 1 5 1  =  + −  − −   + exp( exp( (2 )) exp( exp( (2 (2 )) )) A i a b a b B i − a b a b + 2  4 A 2  2 4 B 2 +  (2 )) i i 1 5 1 1 5 1 9/10/2020 Arpan Deyasi, RCCIIT 26

  27. at Z = (2a+b) (3rdinterface) Arpan Deyasi ( ) ( ) )  a b + − −  a b + exp Quantum (2 ) exp (2 ) A = i B + i − 4 2 i  4 B 2  ( ) ( ) a b + a b + exp (2 ) exp (2 A i 5 1 5 1  =  + −  − −   + exp( exp( (2 )) exp( exp( (2 (2 )) )) A i a b a b B i − a b a b + 2  4 A 2  2 4 B 2 5 Electronics +  (2 )) i i 1 1 1 5 1 In matrix notation  + −  − − +               exp( exp(    =   (2 )) exp( exp(  − (2 )) + i a b a b + i a b a b + A B 2    2 i    − 4  + (2 )) (2 )) i 2 2 (2 2 exp( exp(  − 2 4   exp( exp(  )) (2 )) A B i a b a b i a b a b 5 1 1 i + + (2 )) (2 )) i 5 1 1 1 1 9/10/2020 Arpan Deyasi, RCCIIT 27

  28. at Z = (2a+b) (3rdinterface) Arpan Deyasi  + −  − − +               exp( exp(    =   Electronics (2 )) exp( exp(  − (2 )) + i a b a b + i a b a b + A B 2    2 i    − 4  Quantum + (2 )) (2 )) i 2 2 (2 2 exp( exp(  − 2 4   exp( exp(  )) (2 )) A B i a b a b i a b a b 5 1 1 i + + (2 )) (2 )) i 5 1 1 1 1          =   A B A B 5 4 M M 7 8 5 4 9/10/2020 Arpan Deyasi, RCCIIT 28

  29. Arpan Deyasi          =   A B A B 5 4 M M 7 8 Quantum 5 4          =   A B A B − Electronics 1 5 4 M M 7 8 5 4 9/10/2020 Arpan Deyasi, RCCIIT 29

  30. Arpan Deyasi             A B A B 3 4 = M M 5 6 3 4 Quantum             A B A B − 1 3 4 = M M Electronics 5 6 3 4             A B A B − − 1 1 3 5 = M M M M 5 6 7 8 3 5 9/10/2020 Arpan Deyasi, RCCIIT 30

  31. Arpan Deyasi          =   A B A B 3 2 M M 3 4 3 2 Quantum          =   A B A B − 1 3 2 M M Electronics 3 4 3 2          =   A B A B − − − 1 1 1 5 2 M M M M M M 3 4 5 6 7 8 5 2 9/10/2020 Arpan Deyasi, RCCIIT 31

  32. Arpan Deyasi             A B A B 1 2 = M M 1 2 1 2 Quantum             A B A B − 1 1 2 = M M Electronics 1 2 1 2          =   A B A B − − − − 1 1 1 1 5 1 M M M M M M M M 1 2 3 4 5 6 7 8 5 1 9/10/2020 Arpan Deyasi, RCCIIT 32

  33.         Quantum =   A B A B Arpan Deyasi − − − − 1 1 1 1 5 1 M M M M M M M M 1 2 3 4 5 6 7 8 5 1          =   A B A B 5 1 M Electronics 5 1                  A B A B M M M M 5 1 11 12 = 5 1 21 22 9/10/2020 Arpan Deyasi, RCCIIT 33

  34. Arpan Deyasi                  A B A B M M M M 5 1 11 12 = Quantum 5 1 21 22 = + A M A M B Electronics 1 11 5 12 5 = + B M A M B 1 21 5 22 5 9/10/2020 Arpan Deyasi, RCCIIT 34

  35. Arpan Deyasi A5 A1 M11 M12 Quantum M21 M22 Electronics B5 B1 M12is the transmission coefficient when the wave is traversing from port 2 to port 1 and port 1 is terminated by matched load 9/10/2020 Arpan Deyasi, RCCIIT 35

  36. M12 = 0 for practical device Arpan Deyasi = + A M A M B 1 11 5 12 5 Quantum A A = 1 M 11 Electronics 5 2       1 A A ( ) = = 5 T E * M M 1 11 11 9/10/2020 Arpan Deyasi, RCCIIT 36

  37. Graphical representation of Transmission Coefficient Arpan Deyasi T(E) Quantum Electronics E E1 E2 E3 E0 9/10/2020 Arpan Deyasi, RCCIIT 37

More Related