Download Handout 26 2D Nanostructures: Semiconductor Quantum Wells

Handout 26
2D Nanostructures: Semiconductor Quantum Wells
In this lecture you will learn:
• Effective mass equation for heterojunctions
• Electron reflection and transmission at interfaces
• Semiconductor quantum wells
• Density of states in semiconductor quantum wells
Leo Esaki (1925-)
Nobel Prize
Nick Holonyak Jr. (1928-)
Charles H. Henry (1937-)
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Transmission and Reflection at Heterojunctions
r
t
Ec 2
Ec
Ec 1
x
0
The solution is:
t
Where:
2
1  m x 1k x 2 m x 2k x 1
r 
1  m x 1k x 2 m x 2k x 1
1  m x 1k x 2 m x 2k x 1
 2k y2  1
 2 k x2 2  2 k x21
1   2 k z2  1
1 


 E c 






2m x 2 2m x 1
2  m y 2 m y 1 
2  m z 2 m z 1 

 2k x2 2  2 k x21

 Veff k y , k z 
2m x 2 2m x 1
Special case: If the RHS in the above equation is negative, then kx2 becomes imaginary
and the wavefunction decays exponentially for x>0 (in semiconductor 2). In this case:
r 1
and the electron is completely reflected from the hetero-interface
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
1
Semiconductor Quantum Wells
Ec 2
A thin (~1-10 nm) narrow
bandgap material
sandwiched between two
wide bandgap materials
Ec 2
Ec
Ec 1
AlGaAs
GaAs
AlGaAs
Ev 1
Ev 2
Ev 2
Semiconductor quantum wells can be composed of pretty much any semiconductor
from the groups II, III, IV, V, and VI of the periodic table
TEM micrograph
GaAs
GaAs
InGaAs
InGaAs
quantum well
(1-10 nm)
GaAs
GaAs
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Semiconductor Quantum Well: Conduction Band Solution
L
Ec 2
Ec 2
Ec
Ec 1
x
0
Assumptions and solutions:

 2k 2
Ec 2 k  Ec 2 
2me


 2k 2
Ec 1 k  Ec 1 
2me

 Eˆc1 i   1r   E 1r 
 Eˆc 2  i   2 r   E 2 r 
  2 2




 Ec1 1r   E 1r 
m
2
e


  2 2




 Ec 2  2 r   E 2 r 
m
2
e


Symmetric
cosk x e i k y y  k z z 

x
1r   A
i k y  k z z 
 sink x x e y
Anti-symmetric

e   x  L 2  e i k y y  k z z 
i k y  k z z 
e   x  L 2  e y
2 r   B 

 e  x  L 2  e i k y y  k z z 
i k y  k z z 
  e  x  L 2  e y
2 r   B 
xL2
x  L 2
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
2
Semiconductor Quantum Well: Conduction Band Solution
L
Ec 2
Ec 2
Ec
Ec 1
x
0
Energy conservation condition:
E  Ec 1 
 

 2 k x2  k||2
2me
2me
2
 E
c2


 2   2  k||2

k||2  k y2  k z2
2me
Ec  k x2
The two unknowns A and B can be found by imposing the continuity of the
wavefunction condition and the probability current continuity condition to get the
following conditions for the wavevector kx:

2me
Ec  k x2

2
kxL  


 tan



kx
 2  kx

2
m
e

Ec  k x2
2

 kxL  

  cot  2   k 
kx

x
Wavevector kx cannot
be arbitrary!
Its value must satisfy
these transcendental
equations
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Semiconductor Quantum Well: Conduction Band Solution
 r 
Ec 2
Ec 2
Ec
Ec 1
Graphical solution:
L
x
0

2me
Ec  k x2

2
 kxL  

 tan



kx
 2  kx

m
2
e

Ec  k x2
2

 kxL  

  cot  2   k 
kx

x
In the limit Ec  ∞ the values of kx
are:
k x  p  L ( p = 1,2,3……..
Different red curves for Increasing Ec values
0


2
3
2
2
5
2
kxL 2
• Values of kx are quantized
• Only a finite number of solutions are possible – depending on the value of Ec
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
3
Electrons in Quantum Wells: A 2D Fermi Gas
Ec 2
Ec 2
E2
E1
Ec
Ec 1
L
x
0
Since values of kx are quantized, the energy dispersion can be written as:
E  Ec 1 
2 2
2k x2  k||

2me 2me
 Ec 1  E p 
k||2  k y2  k z2
2k||2
p = 1,2,3……..
2me
In the limit Ec  ∞ the values of Ep are: E p 
 2  p 


2me  L 
2
p = 1,2,3……..
• We say that the motion in the x-direction is quantized (the energy associated with
that motion can only take a discrete set of values)
• The freedom of motion is now available only in the y and z directions (i.e. in
directions that are in the plane of the quantum well)
• Electrons in the quantum well are essentially a two dimensional Fermi gas!
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Energy Subbands in Quantum Wells
Ec 2
Ec 2
E2
E1
Ec
Ec 1
E
E
L
x

2k||2
Ec p, k||  Ec1  E p 
2me

0

p =1,2,3……..
k||2  k y2  k z2
The energy dispersion for
Ec1  E3 electrons in the quantum
Ec 1  E 2
kz
ky
wells can be plotted as
shown
It consists of energy
Ec1  E1 subbands (i.e. subbands of
the conduction band)
Ec 1

k||
Electrons in each subband
constitute a 2D Fermi gas
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
4
Density of States in Quantum Wells
Ec 2
Ec 2
E2
E1
Ec
Ec 1
L
x
0
Suppose, given a Fermi level position Ef , we need to find the electron density:
We can add the electron present in each subband as follows:
n  2 
p

d 2k||
 


f Ec p, k||  Ef
2 
2

If we want to write the above as:
Ec 1  E 3
Ef
Ec 1  E 2

n   dE gQW E  f E  Ef 
Ec1  E1
Ec 1
Ec 1

k||
Then the question is what is the density of states gQW(E ) ?
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Density of States in Quantum Wells

2k||2
Ec p, k||  Ec1  E p 
2me


Start from:
n  2 
p

d 2k||
2 2
 


f Ec p, k||  Ef

Ec 1  E 3
Ef
Ec 1  E 2
Ec1  E1
And convert the k-space integral to energy space:
Ec 1
 m 
n    dE  e2  f E  Ef 
p Ec 1  E p    


  dE
Ec 1

k||
 m 
  e2   E  Ec1  E p f E  Ef 
p   
gQW E 


This implies:
3
 m 
gQW E     e2   E  Ec1  E p
p   


2
me
me
me
 2
 2
 2
Ec1
Ec1  E1
Ec1  E2 Ec1  E3
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
5
Density of States: From Bulk (3D) to QW (2D)
E
E
E
E
Ec1  E3
Ec1  E2
Ec1  E1
Ec1
Ec1

k||
g3D E 

k
gQW
g2DE 
me
 2
2
me
 2
3
me
 2
The modification of the density of states by quantum confinement in nanostructures
can be used to:
i) Control and design custom energy levels for laser and optoelectronic applications
ii) Control and design carrier scattering rates, recombination rates, mobilities, for
electronic applications
iii) Achieve ultra low-power electronic and optoelectronic devices
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Semiconductor Quantum Well: Valence Band Solution
Ev 1
Ev 2
0

 2k 2
Ev 1 k  Ev 1 
2 mh


Eˆ v 1 i  1r   E 1r 




i k y k z

Anti-symmetric

  




 Ev 2  2 r   E 2 r 
m
2
h


2 2

  2 2



 Ev 2  2 r    E 2 r 
m
2
h


  2 2




 Ev 1 1r    E 1r 
2
m
h


z
cosk x e y

x
1r   A
i k y y  k z z 
 sink x x e

 2k 2
Ev 2 k  Ev 2 
2mv


ˆ
Ev 2  i  2 r   E 2 r 

 2 2



 
 Ev 1 1r   E 1r 
2
m
h


Symmetric
Ev 2
x
Assumptions and solutions:

Ev
L

e   x  L 2  e i k y y  k z z 
i k y  k z z 
e   x  L 2  e y
2 r   B 

 e  x  L 2  e i k y y  k z z 
i k y  k z z 
  e  x  L 2  e y
2 r   B 
xL2
x  L 2
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
6
Semiconductor Quantum Well: Valence Band Solution
Ev 1
Ev 2
Ev
Ev 2
L
x
0
Energy conservation condition:
E  Ev 1 
 

 2 k x2  k||2
2mh
2mh
2
 E

v2

 2   2  k||2

2me
Ev  k x2
The two unknowns A and B can be found by imposing the continuity of the
wavefunction condition and the probability current conservation condition to get the
following conditions for the wavevector kx:

2mh
Ev  k x2

2
kxL  


 tan



kx
 2  kx

2
m
h

Ev  k x2
2

 kxL  

  cot  2   k 
kx

x
Wavevector kx cannot
be arbitrary!
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Semiconductor Quantum Well: Valence Band Solution
Ev 1
L
Ev
Ev 2
Ev 2
x
Graphical solution:
0

2mh
Ev  k x2

2
kxL  


 tan


kx

 2  kx

2
m
h

Ev  k x2
2

 kxL  

  cot  2   k 
kx

x
In the limit Ev  ∞ the values of kx
are:
k x  p  L ( p = 1,2,3……..
Different red curves for Increasing Ev values
0


2
3
2
2
5
2
kxL 2
• Values of kx are quantized
• Only a finite number of solutions are possible – depending on the value of Ev
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
7
Semiconductor Quantum Wells: A 2D Fermi Gas
L
Ev 1
E1
Ev
E2
Ev 2
Ev 2
x
0
Since values of kx are quantized, the energy dispersion can be written as:
E  Ev 1 
2 2
2k x2  k||

2mh 2mh
 Ev 1  E p 
Light-hole/heavy-hole
degeneracy breaks!
 2k||2
p = 1,2,3……..
2mh
In the limit Ev  ∞ the values of Ep are: E p 
 2  p 


2mh  L 
2
p = 1,2,3……..
• We say that the motion in the x-direction is quantized (the energy associated with that
motion can only take a discrete set of values)
• The freedom of motion is now available only in the y and z directions (i.e. in directions
that are in the plane of the quantum well)
• Electrons (or holes) in the quantum well are essentially a two dimensional Fermi gas!
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Density of States in Quantum Wells: Valence Band

 2k||2
Ev p, k||  Ev 1  E p 
2 mh


Start from:
p  2 
p

k||

d 2k||
Ev 1
1  f Ev p, k||   Ef 
2
Ev 1  E1
2 
p
Ev 1  E2
Ef
And convert the k-space integral to energy space:
Ev 1  E3
Ev 1  E p
p
 mh 
 1  f E  Ef 
 dE 
2

  
Ev 1
  dE

 m 
  h2   Ev 1  E p  E 1  f E  Ef 
p   


3
This implies:
 m 
gQW E     h2   Ev 1  E p  E
p   

gQW E 
mh
 2
2

mh
 2
mh
 2
Ev 1  E3
Ev 1  E2
Ev 1  E1
Ev 1
E
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
8
Example (Photonics): Semiconductor Quantum Well Lasers
A quantum well laser (band diagram)
electrons
A ridge waveguide laser structure
stimulated and
spontaneous
emission
non-radiative
recombination
N-doped
metal
photon
P-doped
holes
Some advantages of quantum wells for
laser applications:
• Low laser threshold currents due to
reduced density of states
• High speed laser current modulation
due to large differential gain
• Ability to control emission
wavelength via quantum size effect
All lasers used in
fiber optical
communication
systems are
semiconductor
quantum well
lasers
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Compound Semiconductors and their Alloys: Groups IV, III-V, II-VI
CdS
850 nm
980 nm
InP
1300 nm
1550 nm
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
9
Example (Electronics): Silicon MOSFET
x
NMOS Band diagram
SiO2
Si
Ec  E 2
Ef
Ec  E1
Ec
Inversion layer
(2D Electron gas)
y
100 nm
x
z
x
A 50 nm gate MOS transistor
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Example (Electronics): Silicon MOSFET
For minima 1 and 2:
NMOS
  

 Band diagram
2  2
2  2
2  2


 Ec  U r   r   E  r 

2
2
2
Si
2mt y
2mt z

 2m x

ik z  ik z z
Ef
 r   f  x e y
SiO2
f(x)
Ec  Et
Ec  E 
Ec

 2  2
2k y2

 2k z2 





U
r
f
x
E
E
f x 








c
2

2mt
2mt 
 2m x



 E  Ec  E  
2k y2
2mt

2k z2
2mt
x
For minima 3 and 4:

 r   g  x e
3
ik y z  ik z z

  
 2k y2

 2k z2 
gx 
 U r  g  x    E  Ec 


2

2m
2mt 
 2mt x



2
2
 E  Ec  Et 
 2k y2
2 m
5 2
1

 2k z2
2mt
Et  E
6
4
y
x
z
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
10
Example (Electronics): Silicon MOSFET
For minima 5 and 6:

 r   g  x  e
ik y z  ik z z
3

  

 U r  g  x    E  Ec 

2

2
2mt
m
t x



2
5 2
1
 2k y2
2
 E  Ec  E t 
 2k y2
2mt

 2k z2

 gx 
2m 

 2k z2

6
x
Et  E
2m
4
y
z
Advantage of Quantum Confinement and Quantization:
SiO2
NMOS Band diagram
• As a result of quantum confinement the degeneracy
among the states in the 6 valleys or pockets is lifted
• Most of the electrons (at least at low temperatures)
occupy the two valleys (1 & 2) with the lower quantized
energy (i.e. Eℓ )
Si
f(x)
Ec  Et
Ef
Ec  E 
Ec
x
• Electrons in the lower energy valleys have a lighter
mass (i.e. mt) in the directions parallel to the interface
(y-z plane) and, therefore, a higher mobility
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Example (Electronics): HEMTs (High Electron Mobility Transistors)
Source
Gate
Drain
AlGaAs
InGaAs (Quantum Well)
GaAs (Substrate)
The HEMT operates like a MOS
transistor:
The application of a positive or
negative bias on the gate can
increase or decrease the electron
density in the quantum well
channel thereby changing the
current density
InGaAs
QW
InGaAs
QW
GaAs
GaAs
AlGaAs
AlGaAs
Metal
Modulation doping
Metal
Band diagram in
equilibrium
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
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