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. 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. 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. 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. 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. 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. 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. Sample1-D Density of
States for 30kV/cm

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. 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