Simulation of Resonant Tunneling in Semiconductors

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Simulation of Resonant Tunneling in Semiconductors
Jeff Noel UW-Madison Dr. A.-B. Chen Auburn
University
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What is Tunneling?

• Tunneling is the occurrence of a particle in a
  region which classically is energetically
  forbidden.

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A Tunneling Semiconductor Device

• Semiconductors are grown up layer by layer of
  atoms
• Very accurate boundaries and widths are possible
• Different materials have dissimilar potential
  differences
• A tunneling device is on the nano-scale.

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Semiconductor Band Structure (GaAs)

• Effective Mass Approximation
• Treat electrons as free particles with some
  effective mass

• Band Gap Energies
• Si 1.12 eV
• GaAs 1.42 eV
• AlAs 2.16 eV

• Effective Masses
• Si 1.08 me
• GaAs 0.067 me
• AlAs 0.15 me

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Semiconductor Heterojunctions

• Electrons traveling through the heterojunction
  experience a square potential barrier or
  potential well

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Tunnel (Esaki) Diode

• Heavily doped P-N junction which displays
  negative differential resistance and tunneling
• Used in microwave production

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Metal-Oxide-Semiconductor Transistor

• Tunneling limits the width of the oxide
  dielectric
• Width too small and current drains into the metal
  instead of the semiconductor drain

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Double Barrier Resonance Tunneling Diode

• Square well has quasi-bound state energies.
• If energy of incident wave differs with resonant
  energy, T0.
• If energy of incident wave coincides with
  resonant energy, T1.
• Able to align quasi-bound state energy with a
  voltage drop over device.

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Spin Selection with RTD

• Ferromagnetic dopant (ex. Mn) causes splitting of
  the double barrier resonant energy due to
  electron spin
• Able to control transmitted ratio of spin-up vs.
• spin-down

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What did I do?
Tunneling is Important
Goal Understand current transport (tunneling)
properties of a device

• Calculated T(k) and transmitted current for
  arbitrary potential barrier
• Solved the time dependent Schrodinger equation
  for a Gaussian wave packet

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Calculate T(k) for arbitrary potential barrier

• First ratio
  grid width

• Recurse from right to left until you have
  
  
  which gives you the ratio B/A

• B/A 2 R T 1-R

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Time Dependent Problem
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Double Barrier Resonant Tunneling
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Calculating T(k) for Double Barrier Resonant
System
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First Resonant Energy Level .11eV
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Second Resonant Energy Level .41eV
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Double Barrier Resonant Tunneling Diode
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T(k) for DBRTD
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DBRTD No Bias Voltage
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DBRTD Resonant Bias Voltage
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How to connect time-independent calculation with
time-dependent simulation?

• Fourier analyze your initial wave packet gt g(k)
• Integrate over all plane waves with eachs
  respective transmission coefficient T(k)

Theory -gt
Simulation -gt
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IV curve .25eV
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Nanostructures

• Similar numerical methods can easily be applied
  to nanostructures such as carbon nanotubes and
  quantum wires
• Grids replaced by atoms
• Accurate representation of realistic system

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Summary

• Tunneling is important in semiconductor devices
• Calculation of transmission coefficient for any
  barrier is very easy computationally
• Many applications of resonance tunneling
• Allow next generation transistors to shrink
• Emitters/detectors for electron microscopy
• Spin selection