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k.p model

Derivation of Kronig-Penny model for understanding formation of Brillouin zone

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k.p model

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  1. Arpan Deyasi Kṙonig-Penney Model Electron Device Arpan Deyasi Dept. of ECE, RCCIIT, Kolkata

  2. What the model is all about? Simplified one-dimensional quantum mechanical model of a crystal with periodic potential Arpan Deyasi Result provides electronic band structure Electron Device Assumption: [i] Potential varies only in one direction [ii] only one electron is present in the crystal 03-08-2025 Arpan Deyasi, RCCIIT 2

  3. Mathematical Formulation Schrodinger equation in one-dimension is given by Arpan Deyasi Electron Device 2  * d dz 2 m   + E V z −  = V0 ( ) 0 2 2 Where potential is defined as z a 0 -b = = −  0  a V V 0 for for b  z 0 0  z 03-08-2025 Arpan Deyasi, RCCIIT 3

  4. Mathematical Formulation Arpan Deyasi 2  * d dz 2 m   + E V −  = 0 In barrier region 0 Electron Device 2 2 ( ) E V − * 2m  = 0 2 2  d dz d dz +   = 2 2 0 2 2  * 2 mE +  = 0 In well region 2 2 * 2m E  = 2 2  d dz +   = 2 2 0 2 03-08-2025 Arpan Deyasi, RCCIIT 4

  5. Mathematical Formulation For periodic potential, the wave function solution to the Schrodinger’s equation is modulated by a function with the same periodicity. Arpan Deyasi Therefore solution may be considered as Bloch function Electron Device  = jkz ( ) z ( ) u z e where = + ( ) u z u a ( z ) 03-08-2025 Arpan Deyasi, RCCIIT 5

  6. Mathematical Formulation Modified equation for well region Arpan Deyasi Electron Device 2 d u dz du dz     + − −  = 2 2 2 jk k u z ( ) 0 2 Modified equation for barrier region 2 d u dz du dz      + − + = 2 2 2 jk k ( ) u z 0 2 03-08-2025 Arpan Deyasi, RCCIIT 6

  7. Mathematical Formulation Arpan Deyasi Solution for well region 1( ) u z Electron Device ( ) ( )  − −  + j k z j k z = + Ae Be Solution for barrier region ( ) ( )  − −  + jk z jk z = + 2( ) u z Ce De 03-08-2025 Arpan Deyasi, RCCIIT 7

  8. Mathematical Formulation Final solution may be put into the following form: Arpan Deyasi Electron Device  −  2 2 ( ) ( ) ( ) ( ) ( )   +   = +     sinh b sin a cosh b cos a cos k a b  2  =  j where 03-08-2025 Arpan Deyasi, RCCIIT 8

  9. Mathematical Formulation To obtain a more convenient solution, potential barriers are represented as Delta functions: Arpan Deyasi → Electron Device → = V b 0 but V b finite 0 0 Modified equation becomes ( )  sin a ( ) ( ) +  = p cos a cos ka  a * m V ab p = 0 2 where 03-08-2025 Arpan Deyasi, RCCIIT 9

  10. Band diagram Arpan Deyasi Electron Device Allowed Energy Bands Forbidden Region 4π/a 2π/a -π/a π/a 3π/a -2π/a 0 -3π/a -4π/a 03-08-2025 10 Arpan Deyasi, RCCIIT

  11. Outcome The free electron solutions are also solutions to the periodic potentials for certain energies Arpan Deyasi Electron Device The energy discontinuity at the boundary of a Brillouin zone follows from the fact that the limiting values of k corresponds to standing waves rather than traveling waves. When k=±π/a, as we have seen, the waves are Bragg-reflected back and forth, and so the only solutions of Schrödinger’s equation consist of standing waves whose wavelength is equal to the periodicity of the lattice. 03-08-2025 Arpan Deyasi, RCCIIT 11

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