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Electron dynamics in a biased quantum well dos

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Electron dynamics in a biased quantum well dos

  1. 1. Electron Dynamics In A Biased Quantum Well: Physics of a Biased QuantumWell: Redistribution of the Density of States Kevin R. Lefebvre, Ph.D. kevin@lefebvres.com This presentation is a subset of several presentations that will describe the Electron Dynamics in a Biased QuantumWell
  2. 2. Motivation  Biased Quantum Well Devices ◦ SEEDs ◦ QWIPs ◦ Modulators ◦ Solar Cells ◦ MQW Avalanche Photodiode  Device Characteristics is a Function of the: ◦ Confinement of Electrons and Holes within the Biased Quantum Well  Function of the Bias and the Quantum Well System ◦ Interaction between the Carriers, Photons, Phonons and the Quantum Well  Function of the Bias and the Quantum Well System
  3. 3. Biased Quantum Well Characteristics  Density of States ◦ Continuum of States - Tunneling Energy ◦ Spreading in States Around Energy Level ◦ 3-Dimensional DOS ◦ Virtual Bound States  CarrierVelocity ◦ LocalVelocity - position and energy ◦ AverageVelocity ◦ Not Free ParticleVelocity  Wavefunction ◦ Drastically changes as a function of the bias  Physics Change with Bias ◦ Electron Escape Time ◦ Electron-Phonon Interaction ◦ Absorption Coefficient ◦ Dark Currents
  4. 4. Results of an Electric Field  Application of an Electric Field onto a Quantum Well Tilts the QuantumWell ◦ Creates an Open System ◦ Changes the Density of States, Group Velocity and Wavefunction of the Carriers
  5. 5. Solving Schoedinger’s Eq.  Starting with Schoedinger’s Equation  Rewriting Schroedinger’s Equation: where X(z,Ez) is the logarithmic derivative of the wavefunction.  Using transmission lines theory:  Descretize in position and energy to calculate X for all z and Ez             2 2 1 0 d dz m z d z E dz V z E z Ez z z  ( ) ( , ) ( , )   X X X( , ) (  ( )) ( ) ( ) ( , )z E m z z d z dz z Ezi z i i i R I        2 1  j ji   (z,E )z   d z E dz m z z E V z Ez z z X X ( , )  ( ) ( , )       j 2 42   X X X  X  X  X  X X X            ( , ) ( )cosh( ) sinh( ) cosh( ) ( )sinh( ) ( , ) ( , )  (  ( ) )( ( )) z E z z z z z z z E z E m m z E V z i z oi i i oi i oi i i i R z I z oi i i i i i 1 1 2 2           j j j;
  6. 6. Density of States (DoS)  The Method of Logarithmic DerivativeYields: ◦ The 1D Density of State as a Function of Position and Energy. ◦ Integration over the Quantum WellWidth Results in the Effective 1D Density of States. ◦ ConvolutionYield 3D Density of States     g 1D i8 + ( , ) , , z E z Im z E z z E z               X X     g 1D z ( ) ( , )dz , , E z E z Lw z E z z E z dzLw                 g 1D Im i8 + 0 0  X X g D E m dEt g D Ez dEz E E Et Ez m g d Ez dEz E 3 2 0 10 2 10 ( ) ( ) ( ) ( )                  
  7. 7. Density of States Results  1-D Density of States Become Continuous ◦ Spreading of the DOS around the Energy Level ◦ Spreading increases with Applied Field Kevin R. Lefebvre and A. F. M. Anwar, “Redistribution of the Quantum Well Density of States”, Semi. Science and Tech., vol. 12, p. 1226, , 1997.
  8. 8. Sample1-D Density of States for 30kV/cm
  9. 9. 3-D Density of States  3-D DOS can be found by convoluting the 1-D DOS with the 2-d DOS  Unbiased Step-like 3-D DOS Converts Towards Bulk DOS as a Bias Increases
  10. 10. Density of States Concluding Remarks  Applied Electric Field Tilts the QuantumWell ◦ Redistributes the Density of States ◦ 1-D DOS Spread Around the Energy Level ◦ 3-D DOS Step-like Function Smooths and approaches the 3- D DOS as the Applied Field Increases ◦ Redistribution of the DOS will Change the Thermionic Emission and Scattering of Electrons by Electrons,Photons and Phonons

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