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Electron dynamics in a biased quantum well dos

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    Electron dynamics in a biased quantum well   dos Electron dynamics in a biased quantum well dos Document Transcript

    • Electron Dynamics In A Biased Quantum Well: Physics of a Biased Quantum Well: 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 Quantum Well
    • 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
    • Biased Quantum Well Characteristics  Density of States ◦ Continuum of States - Tunneling Energy ◦ Spreading in States Around Energy Level ◦ 3-Dimensional DOS ◦ Virtual Bound States  Carrier Velocity ◦ Local Velocity - position and energy ◦ Average Velocity ◦ Not Free Particle Velocity  Wavefunction ◦ Drastically changes as a function of the bias  Physics Change with Bias ◦ Electron Escape Time ◦ Electron-Phonon Interaction ◦ Absorption Coefficient ◦ Dark Currents
    • Results of an Electric Field  Application of an Electric Field onto a Quantum Well Tilts the Quantum Well ◦ Creates an Open System ◦ Changes the Density of States, Group Velocity and Wavefunction of the Carriers
    • Solving Schoedinger’s Eq.  Starting with Schoedinger’s Equation   2 d  1 d( z, E z )     V z  E z ( z, E z )  0 2 dz  m( z)  dz   Rewriting Schroedinger’s Equation: dX( z, E z )  m( z)   X( z, E z )2  V z  E z  4   j dz  2    1 d( zi )  X( zi , E z )  ( 2 j m( zi ))   X R ( z, Ez)  jXI (z, E z )  ( zi ) dzi   where X(z,Ez) is the logarithmic derivative of the wavefunction.  Using transmission lines theory: X( z i1 )cosh(  i z )  Xoi sinh(  i z ) X  ( z i , E z )  Xoi X oi cosh(  i z )  X( z i1 )sinh(  i z )  X  ( z, E z )  jXI ( z, E z ) R X oi  2 i  j m i ;  i  j ( 2m( z i )  )( E  V( z i ))    Descretize in position and energy to calculate X for all z and Ez
    • Density of States (DoS)  The Method of Logarithmic Derivative Yields: ◦ The 1D Density of State as a Function of Position and Energy.   g ( z, E )  Im   i 8  1D z  +       X z, E z  X z, E z     ◦ Integration over the Quantum Well Width Results in the Effective 1D Density of States. Lw g ( E )  0 g ( z, E )dz 1D z 1D z   Lw Im   0 i 8  dz  +      X z, E z  X z, E z     ◦ Convolution Yield 3D Density of States m  E g ( E )   dE t  g1D ( E z ) dE z ( E  E t  E z ) 3D 2 0 0 m E   g ( E z ) dE z  2 0 1d
    • 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.
    • Sample1-D Density of States for 30kV/cm
    • 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
    • Density of States Concluding Remarks  Applied Electric Field Tilts the Quantum Well ◦ 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