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Band nonparabolicity effect on eigenstates

Brief description and algorithm for estimating band nonparabolicity effect on eigenstates

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Band nonparabolicity effect on eigenstates

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  1. Course: Quantum Electronics Arpan Deyasi Quantum Topic: Effect of Band nonparabolicity on Eigenstates Electronics Arpan Deyasi Arpan Deyasi, RCCIIT 5/11/2020 1

  2. DQWTB structure Arpan Deyasi Quantum Electronics Z 5/11/2020 Arpan Deyasi, RCCIIT 2

  3. Graphical representation of Transmission Coefficient Arpan Deyasi T(E) Quantum Electronics E E1 E2 E3 E0 5/11/2020 Arpan Deyasi, RCCIIT 3

  4. 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 2 2 dz 5/11/2020 Arpan Deyasi, RCCIIT 4

  5. Parabolic Dispersion Relation Arpan Deyasi E  2 2 = E Quantum * 2 m Electronics k Ideal; most of the materials do not obey the relationship 5/11/2020 Arpan Deyasi, RCCIIT 5

  6. Nonparabolic Dispersion Relation Arpan Deyasi E  Quantum 2 2 +  +  + = 2 (1 ...) E E E * 2 m Electronics k 5/11/2020 Arpan Deyasi, RCCIIT 6

  7. α, β,…. are nonparabolic coefficients Arpan Deyasi Expressions of κ1& κ2will change Quantum Magnitudes of band nonparabolic coefficients control the energy states Electronics 5/11/2020 Arpan Deyasi, RCCIIT 7

  8. Modified wave-vectors Arpan Deyasi Consider first-order band nonparabolicity  Quantum * +  2 ( ) (1 z E ) m E = w for well region 2 2 Electronics * +  − 2 ( )[ (1 z E ) ] m E V  = 0 b for barrier region 1 2 5/11/2020 Arpan Deyasi, RCCIIT 8

  9. Graphical representation of Transmission Coefficient Arpan Deyasi T(E) Quantum N Electronics P ΔE32n ΔE32p ΔE21n ΔE21p ΔE31p ΔE31n E4n E1n E2n E3n E E2p E3p E4p E1p 5/11/2020 Arpan Deyasi, RCCIIT 9

  10. Effect of nonparabolic coefficients Arpan Deyasi 1. Energy values will be lowered Quantum 2. The effect will be more significant for higher order states Electronics 3. Corresponding intersubband transition energy will be reduced 4. Operating wavelength of the photodetectorwill be enhanced 5/11/2020 Arpan Deyasi, RCCIIT 10

  11. Algorithm to calculate transmission coefficient using TMT Arpan Deyasi S1: Input parameters: ‘a’, ‘b’, ‘mb*’, ‘mw*’, ‘V0’ Quantum S2: Consider energy range of interest (E<V0) Electronics S3: Calculate wave vectors ‘κ1’ and ‘κ2’ S4: Calculate interface matrices ‘Mi’ S5: Calculate composite interface matrix ‘M’ S6: Calculate transmission coefficient from ‘M11’ 5/11/2020 Arpan Deyasi, RCCIIT 11

  12. Algorithm to calculate transmission coefficient using PMM Arpan Deyasi S1: Input parameters: ‘a’, ‘b’, ‘mj*’, ‘Vj’ Quantum S2: Consider energy range of interest (E<V0) Electronics S3: For energy ‘Ej’, calculate wave number ‘κj’for each position in the potential ‘Vj’ S4: Calculate junction matrix ‘Pjunc(j)’ between any two consecutive points inside the structure S5: Consider generalized length ‘Lj’ between two consecutive points 5/11/2020 Arpan Deyasi, RCCIIT 12

  13. Algorithm to calculate transmission coefficient using PMM Arpan Deyasi S6: Calculate step matrix ‘Pstep(j)’ between any two consecutive points inside the structure S7: Calculate total propagation matrix ‘Pprop(j)’ as a Cartesian product of junction matrix and step matrix Quantum S8: Repeat it for every points inside the structure Electronics S9: Calculate transmission coefficient from ‘P11’ 5/11/2020 Arpan Deyasi, RCCIIT 13

  14. Quasi-peak Arpan Deyasi As per the transmission coefficient profile, eigenenergy states are obtained from sharp peaks Quantum This implies are energy states are very thin levels Electronics Practically all energy bands consist of multiple energy levels Henceforth, there should exist finite width of energy peak 5/11/2020 Arpan Deyasi, RCCIIT 14

  15. Quasi-peak Practically, one small peak is associated with every principle peak with a small finite energy difference Arpan Deyasi The second peak is called quasi-peak Quantum Electronics 5/11/2020 Arpan Deyasi, RCCIIT 15

  16. Quasi-peak in Transmission Coefficient profile Arpan Deyasi T(E) Quantum Electronics E4qp E2qp E3qp E1qp E E4 E1 E3 E2 5/11/2020 Arpan Deyasi, RCCIIT 16

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