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ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Chapter 2
Semiconductor Heterostructures
In this lecture you will learn:
• Energy band diagrams in real space• Semiconductor heterostructures and heterojunctions• Electron affinity and work function• Heterojunctions in equilibrium• Electrons at Heterojunctions• Semiconductor Quantum wells Herbert Kroemer
(1920-)Nobel Prize 2000 for the Semiconductor Heterostructure Laser
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Band Diagrams in Real Space - I
For devices, it is useful to draw the conduction and valence band edges in real space:
cE
vE
fE
x
cE
vEfE
x
N-type semiconductor
Energy
k
fE
cE
vE
Energy
k
fEcE
vE
P-type semiconductor
KTEEc
fceNn
KTEEv
vfeNp
2
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Band Diagrams in Real Space - II
Electrostatic potential and electric field:
cE
vE
fE
xN-type semiconductor
An electrostatic potential (and an electric field) can be present in a crystal:
The total energy of an electron in a crystal is then given not just by the energy band dispersion but also includes the potential energy coming from the potential:
Therefore, the conduction and valence band edges also become position dependent:
rrEr
and
kEn
rekEkE nn
reEEreEE vvcc
Example: Uniform x-directed electric field
xeExExE
xExr
xErE
xcc
x
x
0
0
ˆ
xErE x ˆ
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Electron Affinity and Work Function
Electron affinity “” is the energy required to remove an electron from the bottom of the conduction band to outside the crystal, i.e. to the vacuum level
0 x
0
Vacuum level
Potential in a crystal
Conduction band
Energy
V
cE
vE
fE
x
W
Work function “W ” is the energy required to remove an electron from the Fermi level to the vacuum level
• Work function changes with doping but affinity is a constant for a given material
3
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Semiconductor N-N Heterostructure: Electron Affinity Rule
Heterostructure: A semiconductor structure in which more than one semiconductor material is used and the structure contains interfaces or junctions between two different semiconductors
Consider the following heterostructure interface between a wide bandgap and a narrow bandgap semiconductor (both n-type):
1gE 2gE
1 2
1cE
1vE
1fE 2cE
2vE
2fE
21
V
1gE2gE
The electron affinity ruletells how the energy band edges of the two semiconductors line up at a hetero-interface
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Semiconductor N-N Heterojunction
1cE
1vE
1fE 2cE
2vE
2fE
21
V
1gE2gE
Something is wrong here:the Fermi level (the chemical potential) has to be the same everywhere in equilibrium (i.e. a flat line)
• Once a junction is made, electrons will flow from the side with higher Fermi level (1) to the side with lower Fermi level (2)
Electrons
4
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
• Electrons will flow from the side with higher Fermi level (1) to the side with lower Fermi level (2)
• Electron flow away from semiconductor (1) will result in a region at the interface which is depleted of electrons (depletion region). Because of positively charged donor atoms, the depletion region has net positive charge density
• Electron flow into semiconductor (2) will result in a region at the interface which has an accumulation of electrons (accumulation region). The accumulation region has net negative charge density
1cE
1vE
1fE2cE
2vE
2fE
2
1
V
1gE2gE
Depletion region Accumulation
region
1gE 2gE
1 2+++++++++++++++
---------------
Note: the vacuum level follows the electrostatic potential:
00 xxexVxV
Semiconductor N-N Heterojunction: Equilibrium
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
1cE
1vE
1fE2cE
2vE
2fE
2
1
V
1gE2gE
• Electron flow from semiconductor (1) to semiconductor (2) continues until the electric field due to the formation of depletion and accumulation regions becomes so large that the Fermi levels on both sides become the same
• In equilibrium, because of the electric field at the interface, there is a potential difference between the two sides – called the built-in voltage
• The built-in voltage is related to the difference in the Fermi levels before the equilibrium was established:
Depletion region Accumulation
region
1cE
1vE
1fE 2cE
2vE
2fE
21
V
1gE2gE
beV
21 ffb EEeV
beV
Semiconductor N-N Heterojunction: Equilibrium
5
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
1cE
1vE
1fE 2cE
2vE2fE
21
V
1gE 2gE
Once a junction is made:
• Electrons will flow from the side with higher Fermi level (1) to the side with lower Fermi level (2)
• Holes will flow from the side with lower Fermi level (2) to the side with higher Fermi level (1)
beV
Electrons
Holes
Semiconductor P-N Heterojunction
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
1cE
1vE
1fE
2cE
2vE2fE
2
1
V
1gE
2gE
Depletion region Depletion
region
beV• Electron flow away from semiconductor (1) will result in a region at the interface which is depleted of electrons (depletion region). Because of positively charged donor atoms, the depletion region has net positive charge density
• Hole flow away from semiconductor (2) will result in a region at the interface which is depleted of holes (depletion region). Because of negatively charged acceptor atoms, the depletion region has net negative charge density
1gE 2gE
1 2+++++++++++++++
---------------
Note: the vacuum level follows the electrostatic potential:
00 xxexVxV
Semiconductor P-N Heterojunction: Equilibrium
6
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
1cE
1vE
1fE
2cE
2vE2fE
2
1
V
1gE
2gE
Depletion region Depletion
region
1cE
1vE
1fE2cE
2vE2fE
21
V
1gE 2gE
beV
beV
• Electron flow from semiconductor (1) to semiconductor (2) and hole flow from semiconductor (2) to semiconductor (1) continues until the electric field due to the formation of depletion regions becomes so large that the Fermi levels on both sides become the same
• The built-in voltage is related to the difference in the Fermi levels before the equilibrium was established:
21 ffb EEeV
Semiconductor P-N Heterojunction: Equilibrium
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Types of Semiconductor Heterojunctions
Type-I: Straddling gap
Type-II: Staggered gap
1cE
1vE
2cE
2vE
21
V
1gE 2gE
1cE
1vE
2cE
2vE
21V
1gE
2gE
1cE
1vE
2cE
2vE
2
1
V
1gE
2gE
Type-III: Broken gap
7
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Band Offsets in Heterojunctions
1cE
1vE
2cE
2vE
21
V
1gE2gE
cE
vE
The conduction and valence band offsets are determined as follows:
cggcgv
c
EEEEEE
E
21
12
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
A PN Heterojunction
Ec1
Ef1
Ev1
Ec2Ef2
Ev2
q1q2
Eg1 Eg2
Vacuum level
Ec
Ev
x
x
1 (p-doped) 2 (n-doped)
+ +
+ +
+ +
+ +
- -
- -
- -
- -
-xpxn
12 ffbi EEqV
222 ln
c
dcf N
NKTEE
111 ln
v
afv N
NKTEE
.ln12
2
vc
davgbi NN
NNKTEEqV
a
ipo N
nnn
21
apo Np dno Nn
d
ino N
npp
22
8
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
A PN Heterojunction in Equilibrium
Ec1
Ef
Ev1
Ec2
Ef
Ev2
q1
q2Eg1
Eg2
Vacuum level
Ec
Ev
xxn-xp
x
+
-
-xp
xn
(x)
+qNd
-qNa
The Depletion Approximation:
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
A PN Heterojunction in Equilibrium
x
-xp
xn
E(x)
elsewhere0
0
0
1
2
xxxxqN
xxxxqN
xE ppa
nnd
xxExdxd
Electric Field:
x-xp
xn
(x)
VbiElectrostatic Potential:
2
2
2nd xN
q
1
2
2paxN
q
pand xqNxqN
1
2
2
2
22 pand
bixN
qxN
qV
21
2121
21
2121
2
2
da
bi
d
an
da
bi
a
dp
NN
V
N
N
qx
NN
V
N
N
qx
QxqNxqN pand
Charge per unit area:
9
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Ec1
Ef1Ev1 Ec2
Ef2
Ev2
q1
q2
Eg1
Eg2
Vacuum level
Ec
Ev
xxn-xp
-qV
A PN Heterojunction in Reverse Bias
x-xp xn
1 (p-doped) 2 (n-doped)+ ++ ++ +
- -- -- -
V+ -
Wn-Wp
V<0
Quasi Fermi Levels and their Splitting:
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
A PN Heterojunction in Reverse Bias
x-xpxn
1 (p-doped) 2 (n-doped)+ ++ ++ +
- -- -- -
Wn-Wp
V<0
21
2121
21
2121
2
2
da
bi
d
ap
da
bi
a
dn
NN
VV
N
N
qVx
NN
VV
N
N
qVx
Depletion regions grow in width:
V+ -
10
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
A PN Heterojunction in Forward Bias
x-xpxn
1 (p-doped) 2 (n-doped)+ +-
V+ -
Wn-Wp
V>0
Ec1
Ef1Ev1
Ec2Ef2
Ev2
q1
q2
Eg1
Eg2
Vacuum level
Ec
Ev
xxn-xp
qV
--
+ ++ +
--
-
Now diffusion exceeds drift!!Minority carrier injection………
Electrons
Holes
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
A PN Heterojunction in Forward Bias
Ec1
Ef1Ev1
Ec2Ef2
Ev2
q1
q2
Eg1
Eg2
Vacuum level
Ec
Ev
xxn-xp
qV
Electrons
Holes
Assumption:
Main bottleneck for current flow are the quasineutral regions and not the depletion regions
a
KTxExEvp
KTqVpo
KTqV
a
i
KTxExEKTxExEc
KTxExEcp
NeNxp
eneNn
eeNeNxn
pfpv
pfpfpcpfpcpf
)()(1
21
)()()()(1
)()(1
1
1212
)(
)(
Electron concentration on the p-side:
11
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
A PN Heterojunction in Forward Bias
Ec1
Ef1Ev1
Ec2Ef2
Ev2
q1
q2
Eg1
Eg2
Vacuum level
Ec
Ev
xxn-xp
qV
Electrons
Holes
Assumption:
Main bottleneck for current flow are the quasineutral regions and not the depletion regions
Hole concentration on the n-side:
KTqVno
KTqV
d
i
KTxExEKTxExEv
KTxExEvn
dKTxExE
cn
epeNn
eeNeNxp
NeNxn
nfnfnfnvnfnv
ncnf
22
)()()()(2
)()(2
)()(2
1221
2
)(
)(
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
A PN Heterojunction in Forward Bias
x-xp xn Wn-Wp
p(x)
n(x)
Minority carrier concentrations:
Electrons on the p-side:
1e
poee
nxnxGxR
Excess electrons injected in the p-side will recombine with the holes
xxn
DqxJ ee
1 Diffusion current
xRxGxJxqt
neee
1
Need to solve: 0
1
2
2
1e
poe
nxn
x
xnD
12
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
A PN Heterojunction in Forward BiasMinority carrier concentrations:
111
21
2
2
12
2
1
eee
e
po
e
poe
DL
L
nxn
x
xn
nxn
x
xnD
222
22
2
2
hhh
h
no
DL
L
pxp
x
xp
P-side: N-side:
Boundary conditions:KTqV
pop enxn )(
pop nWn )(
Boundary conditions:KTqV
non epxp )(
non pWp )(?? ??
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
A PN Heterojunction in Forward BiasMinority carrier concentrations:
ppKT
qV
e
pp
e
p
popo xxWe
L
xW
L
xW
nnxn
1
sinh
sinh
1
1
nnKT
qV
h
nn
h
n
nono Wxxe
LxW
LxW
ppxp
1
sinh
sinh
2
2
13
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
A PN Heterojunction in Forward Bias
Majority carrier concentrations and charge neutrality:
P-side:
One must have: poa nxnxnNxpxp
Excess majority carrier density must balance the excess minority carrier density
N-side:
One must have: nod pxpxpNxnxn
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
A PN Heterojunction in Forward BiasMinority carrier current:
ppKTqv
e
pp
e
p
e
e
a
i
eee
xxWe
L
xW
L
xW
LD
Nn
q
xxn
DqxEqnxJ
1
sinh
cosh
1
1
1
121
11
P-side:~0
nnKT
qv
n
nn
n
n
h
h
d
i
nhhh
Wxxe
LxW
LxW
LD
Nn
q
xxp
DqxEqpxJ
1
sinh
cosh
2
2
2
222
2
N-side:~0
14
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
x-xp xn Wn-Wp
Jh(x)Je(x)
A PN Heterojunction in Forward Bias
Since there is no obstacle to current flow in the depletion regions, and if we ignore electron-hole recombination in the depletion region, we must have:
x-xp xn Wn-Wp
Jh(x)Je(x)
Total current:
xJxJJ heT Must be constant throughout the device
JT
Minority carrier current:
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
A PN Heterojunction in Forward Bias
x-xp xn Wn-Wp
Jh(x)Je(x)
JT
Total current:
1cothcoth
22
222
11
121 KT
qv
h
nn
h
h
d
i
e
pp
e
e
a
iT e
LxW
LD
Nn
L
xW
L
D
Nn
qJ
1KT
qv
oT eIAJI
22
222
11
121 cothcoth
h
nn
h
h
d
i
e
pp
e
e
a
io L
xWLD
Nn
L
xW
L
D
Nn
qAI
V+ -
I
A
15
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
A PN Heterojunction in Forward Bias
x-xp xn Wn-Wp
Jh(x)Je(x)
JT
Majority carrier current:
P-side:
N-side:
xJJxJ eTh
xJJxJ hTe
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
A PN Heterojunction in Forward Bias
Ec1
Ef1
Ev1
Ec2
Ef2
Ev2
q1
q2
Eg1
Eg2
Vacuum level
Ec
Ev
xxn-xp
qV
Quasi Fermi Levels:
1KT
qv
oT eIAJI
16
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
A PN Heterojunction in Reverse Bias
Ec1
Ef1Ev1 Ec2
Ef2
Ev2
q1
q2
Eg1
Eg2
Vacuum level
Ec
Ev
xxn-xp
-qV
1KT
qv
oT eIAJI
Reverse bias current: I -Io
Quasi Fermi Levels:
Why?
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
A PN Heterojunction in Forward Bias
Electron-Hole Recombination in the Depletion Region:
Ec1
Ef1
Ev1
Ec2
Ef2
Ev2
q1
q2
Eg1
Eg2
Vacuum level
Ec
Ev
xxn-xp
qV
xRxGxJxq eee
1
dxxGxRqxJxJn
p
x
xeepene
17
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
A PN Heterojunction in Forward Bias
Electron-Hole Recombination in the Depletion Region:
x-xp xn Wn-Wp
Jh(x)Je(x)
JT
dxxGxRqxJxJn
p
x
xeepene
xRxGxJxq eee
1
dxxGxRqxJxJn
p
x
xhhnhph
Similarly:
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
A PN Heterojunction in Forward Bias
Electron-Hole Recombination in the Depletion Region:
x-xp xn Wn-Wp
Jh(x)Je(x)
JT
dxxGxRqeL
xWLD
Nn
L
xW
L
D
Nn
qJn
p
x
xee
KTqv
h
nn
h
h
d
i
e
pp
e
e
a
iT
1cothcoth
22
222
11
121
2iee nnpxRxG
KTEEi
ffennp 122
1KT
qv
ee exRxG
18
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Effective Mass Schrodinger Equation
Consider a semiconductor with energy band dispersion:
oeocc kkMkkEkE
..2
12
Energy
k
fEcE
vE
The Bloch functions are solutions of the equation:
rkErrVm kcckcLattice
,,
22
2
ruV
er kc
rki
kc
,
.
,
What if one needs to solve the equation:
rErrUrVm Lattice
2
22
Some extra potential (perhaps due to some crystal impurity, defect, or external electric field)
ok
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Effective Mass Schrodinger EquationEnergy
k
fEcE
vE
One can in most cases write the solution as:
rrrokc
,
Envelope function
Where the envelope function satisfies the “effective mass Schrodinger equation”:
rErrUikE oc
ˆ
ok
rErrUrVm Lattice
2
22
19
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
The Envelope Function
rErrUikE oc
ˆ
Energy
k
rrrokc
,
r
rokc
,
Slowly varying envelope function Bloch function
ok
Electron wavefunction
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
The Effective Mass Schrodinger Equation: An Example Energy
k
ok
Consider a conduction energy band with the dispersion:
zz
ozz
yy
oyy
xx
oxxcc m
kkm
kk
mkk
EkE222
222222
rErrUikE oc
ˆ
Note that one has to make the following replacements in the energy dispersion relation:
z
ikky
ikkx
ikkikEkE ozzoyyoxxocc
ˆ
The operator is then: ikE oc
ˆ
2
22
2
22
2
22
222 zmymxmEikE
zzyyxxcoc
The effective mass Schrodinger equation becomes:
rErrUEzmymxm c
zzyyxx
2
22
2
22
2
22
222
What is this equation:
20
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Electrons at Heterojunctions
1cE
1vE
2cE
2vE
1gE 2gE
cE
vE
Question: What happens to the electron that approaches the interface (as shown)? How does it see the band offset? Does it bounce back? Does it go on the under side?
The effective mass equation can be used to answer all the above questions
In semiconductor 1:
In semiconductor 2:
rErrUikE oc
111ˆ
rrrokc
,111
rErrUikE oc
222ˆ
rrrokc
,222
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Electrons at Heterojunctions; Effect of Band Offsets
1cE
1vE
2cE
2vE
1gE 2gE
cE
vE
1
22
11 2 ecc m
kEkE
rrr
okc
0,111
Assume for the electron in the conduction band of semiconductor 1:
rErEm c
e
111
2
1
2
2
2
22
22 2 ecc m
kEkE
rrr
okc
0,222
And for the electron in semiconductor 2:
rErEm c
e
222
2
2
2
2
Notice that the conduction band edge energy (i.e. Ec1 or Ec2) appears as a constant potential in the effective mass Schrodinger equation
Conduction band offset at the heterojunction therefore appears like a potential step to the electron
0rU
21
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Electrons at Heterojunctions: Boundary Conditions
(1) Continuity of the wavefunction at the boundary:
0201 xx rr
(2) Continuity of the normal component of the probability current at the boundary:
In text book quantum mechanics the probability current is defined as:
rim
rrim
rccrim
rrJ ***
22..
2
..2
* ccrim
rrJ
Or in shorter component notation:
Probability current is always continuous across a boundaryWe need an expression for the probability current in terms of the envelope function
0201 xx rr
If one assumes: rr
oo kckc
,2,1
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Electrons at Heterojunctions: Boundary Conditions
Probability Current: In a material with energy band dispersion given by:
oonoonn kkkkm
EkkMkkEkE
,
21
2
2..
2
The expression for the electron probability current (in terms of the envelope function) is:
..2
* ccrim
rrJ
Continuity of the probability current:The continuity of the normal component of the probability current across a heterojunction gives another boundary condition for the envelope function:
02
2011
11xxxx
rm
rm
For:
zz
yy
xx
m
m
m
M
1
1
11
0
2
20
1
1
11
xxxxxx xr
mxr
m
22
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Electrons at Heterojunctions: Boundary Conditions
(1) Continuity of the envelope function at the boundary:
(2) Continuity of the normal component of the probability current at the boundary:
0x x
Semiconductor 1 Semiconductor 2
0201 xx rr
02
2011
11xxxx
rm
rm
zz
yy
xx
m
m
m
M
1
1
11
0
2
20
1
1
11
xxxxxx xr
mxr
m
If in both the materials the inverse effective mass matrix is diagonal then this boundary condition becomes:
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
The Effective Mass Theory for Heterojunctions
1cE2cE
cE
x0
Assume in semiconductor (1):
rErEzmymxm
rEriE
rErrUikE
czyx
c
oc
1112
2
1
2
2
2
1
2
2
2
1
2
111
111
222
ˆ
ˆ
1
22
1
22
1
22
11 222 z
z
y
y
x
xcc m
km
k
mk
EkE
Assume in semiconductor (2):
2
22
2
22
2
22
22 222 z
z
y
y
x
xcc m
km
k
mk
EkE
In semiconductor (1):
0ok
0ok
23
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
The Effective Mass Theory for Heterojunctions
1cE2cE
cE
x0
rErEzmymxm c
zyx
1112
2
1
2
2
2
1
2
2
2
1
2
222
Assume a plane wave solution: zkykxki zyxer 1
1
Plug it in to get:1
22
1
22
1
21
2
1 222 z
z
y
y
x
xc m
km
k
mk
EE
We expect a reflected wave also so we write the total solution in semiconductor (1) as:
zkykxkizkykxki zyxzyx erer 11
1
A plane wave solution works
In semiconductor (1):
r t r
1 r
2
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
The Effective Mass Theory for Heterojunctions
rErEzmymxm c
zyx
2222
2
2
2
2
2
2
2
2
2
2
2
222
Assume a plane wave solution: zkykxki zyxetr 2
2
Plug it in to get:2
22
2
22
2
22
2
2 222 z
z
y
y
x
xc m
km
k
mk
EE
A plane wave solution works here also
In semiconductor (2):
1cE2cE
cE
x0
r t r
1 r
2
24
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Boundary Conditions at Heterojunctions
zkykxkizkykxki zyxzyx erer
111
zkykxki zyxetr
22
(1) Envelope functions must be continuous at the interface:
1
22
1
22
1
21
2
1 222 z
z
y
y
x
xc m
km
k
mk
EE
2
22
2
22
2
22
2
2 222 z
z
y
y
x
xc m
km
k
mk
EE
tr
etere
xxzkykizkykizkyki zyzyzy
1
00 21
Note that this boundary condition can only be satisfied if the components of the wavevector parallel to the interface are the same on both sides
1cE2cE
cE
x0
r t r
1 r
2
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Boundary Conditions at Heterojunctions
zkykxkizkykxki zyxzyx erer
111
zkykxki zyxetr
22
Energy conservation:
1
22
1
22
1
21
2
1 222 z
z
y
y
x
xc m
km
k
mk
EE
2
22
2
22
2
22
2
2 222 z
z
y
y
x
xc m
km
k
mk
EE
zyeffx
x
x
x
zz
z
yy
yc
x
x
x
x
z
z
y
y
x
xc
z
z
y
y
x
xc
kkVmk
mk
mmk
mm
kE
mk
mk
mk
m
k
mk
Emk
m
k
mk
EE
,22
112
11222
222222
1
21
2
2
22
2
12
22
12
22
1
21
2
2
22
2
2
22
2
22
2
22
2
21
22
1
22
1
21
2
1
Note that the effective barrier height depends on the band offset as well as the parallel components of the wavevector
1cE2cE
cE
x0
r t r
1 r
2
25
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Boundary Conditions at Heterojunctions
zkykxkizkykxki zyxzyx erer
111
zkykxki zyxetr
22
(2) Probability current must be continuous at the interface:
1
22
1
22
1
21
2
1 222 z
z
y
y
x
xc m
km
k
mk
EE
2
22
2
22
2
22
2
2 222 z
z
y
y
x
xc m
km
k
mk
EE
tmk
rmk
etmik
eremik
xmxm
x
x
x
x
zkyki
x
xzkykizkyki
x
x
xxxx
zyzyzy
2
2
1
1
2
2
1
1
0
2
20
1
1
1
11
Conservation of probability current at the interface
1cE2cE
cE
x0
r t r
1 r
2
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Transmission and Reflection at Heterojunctions
tr 1 tmk
rmk
x
x
x
x
2
2
1
1 1
We have two equations in two unknowns:
The solution is:
1221
1221
1221 11
12
xxxx
xxxx
xxxx kmkmkmkm
rkmkm
t
zyeffx
x
x
x kkVmk
mk
,22 1
21
2
2
22
2
Where:
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:
and the electron is completely reflected from the hetero-interface
1r
1cE2cE
cE
x0
r t
26
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Semiconductor Quantum Wells
1cE2cE
cE2cE
1vE
2vE2vE
AlGaAs AlGaAsGaAsA thin (~1-10 nm) narrow bandgap material sandwiched between two wide bandgap materials
GaAs
GaAsInGaAs quantum well (1-10 nm)
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
InG
aA
s
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Semiconductor Quantum Well: Conduction Band Solution
1cE
2cE
cE2cE
x
0Assumptions and solutions:
e
cc mk
EkE2
22
11
e
cc mk
EkE2
22
22
rErEm
rEriE
ce
c
111
22
111
2
ˆ
rErEm
rEriE
ce
c
222
22
222
2
ˆ
zkykix
zkykix
zy
zy
exk
exkAr
sin
cos1
2
2
2
2 Lxee
eeBr zkykiLx
zkykiLx
zy
zy
2
2
2
2 Lxee
eeBr zkykiLx
zkykiLx
zy
zy
L
Symmetric
Anti-symmetric
27
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
22
2||
22
2
2||
22
1
2
22
xce
ec
e
xc
kEm
m
kE
m
kkEE
1cE2cE
cE2cE
x
0
L
Energy conservation condition:
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:
x
xce
x
x
x
xce
x
x
k
kEm
kLk
k
kEm
kLk
22
22
2
2cot
2
2tan
Wavevector kx cannot be arbitrary!Its value must satisfy these transcendental equations
222|| zy kkk
Semiconductor Quantum Well: Conduction Band Solution
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
1cE
2cE
cE2cE
x
0
L
x
xce
x
x
x
xce
x
x
k
kEm
kLk
k
kEm
kLk
22
22
2
2cot
2
2tan
2Lkx2
23 2
250
Different red curves for Increasing Ec values
Graphical solution:
• Values of kx are quantized• Only a finite number of solutions are possible – depending on the value of Ec
In the limit Ec ∞ the values of kxare:
Lpkx ( p = 1,2,3……..
Semiconductor Quantum Well: Conduction Band Solution r
28
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Electrons in Quantum Wells: A 2D Fermi Gas
1cE
2cE
cE2cE
x
0
L
epc
ee
xc
m
kEE
m
k
mk
EE
2
222||
2
1
2||
222
1
Since values of kx are quantized, the energy dispersion can be written as:
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!
1E
2E
In the limit Ec ∞ the values of Ep are:22
2
Lp
mE
ep
p = 1,2,3……..
222|| zy kkk
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
1cE
2cE
cE2cE
x
0
L1E
2E
Energy Subbands in Quantum Wells
e
pcc m
kEEkpE
2,
2||
2
1||
p =1,2,3……..
The energy dispersion for electrons in the quantum wells can be plotted as shown
It consists of energy subbands (i.e. subbands of the conduction band)
Electrons in each subband constitute a 2D Fermi gas||k
1cE11 EEc
21 EEc
E
kz
ky
31 EEc
E
222|| zy kkk
29
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
1cE
2cE
cE2cE
x
0
L1E
2E
Density of States in Quantum Wells
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:
pfc EkpEf
kdn ||2
||2
,2
2
fEIf we want to write the above as:
fQWE
EEfEgdEnc
1
Then the question is what is the density of states gQW(E ) ?||k
1cE
11 EEc
21 EEc 31 EEc
||k
1cE
11 EEc
21 EEc 31 EEc
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Density of States in Quantum Wells
fE
||k
1cE
11 EEc
21 EEc 31 EEc
||k
1cE
11 EEc
21 EEc 31 EEc
pfc EkpEf
kdn ||2
||2
,2
2
Start from:
e
pcc m
kEEkpE
2,
2||
2
1||
And convert the k-space integral to energy space:
fp
pce
E
pf
e
EE
EEfEEEm
dE
EEfm
dEn
c
pc
12
2
1
1
This implies:
ppc
eQW EEE
mEg 12
EgQW
1cE 11 EEc 21 EEc 31 EEc
2em
22
em
23
em
30
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Density of States: From Bulk (3D) to QW (2D)
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 applicationsii) Control and design carrier scattering rates, recombination rates, mobilities, for electronic applications iii) Achieve ultra low-power electronic and optoelectronic devices
Eg D3k
1cE
E E E
Eg D2
2em
22
em
23
em
||k
1cE
11 EEc
21 EEc
31 EEc
EE
Eg D2
2em
22
em
23
em
||k
1cE
11 EEc
21 EEc
31 EEc
E
EgQW
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Semiconductor Quantum Well: Valence Band Solution1vE
2vEvE2vE
x
0Assumptions and solutions:
h
vv mk
EkE2
22
11
v
vv mk
EkE2
22
22
rErEm
rErEm
rEriE
vh
vh
v
111
22
111
22
111
2
2
ˆ
rErEm
rErEm
rEriE
vh
vh
v
222
22
222
22
222
2
2
ˆ
zkykix
zkykix
zy
zy
exk
exkAr
sin
cos1
2
2
2
2 Lxee
eeBr zkykiLx
zkykiLx
zy
zy
2
2
2
2 Lxee
eeBr zkykiLx
zkykiLx
zy
zy
L
Symmetric
Anti-symmetric
31
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
22
2||
22
2
2||
22
1
2
22
xvh
ev
xv
kEm
m
kE
mh
kkEE
Energy conservation condition:
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:
x
xvh
x
x
x
xvh
x
x
k
kEm
kLk
k
kEm
kLk
22
22
2
2cot
2
2tan
Wavevector kx cannot be arbitrary!
1vE
2vEvE2vE
x
0
L
Semiconductor Quantum Well: Valence Band Solution
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
1vE
2vE
vE
2vE
x
0
L
x
xvh
x
x
x
xvh
x
x
k
kEm
kLk
k
kEm
kLk
22
22
2
2cot
2
2tan
2Lkx2
23 2
250
Different red curves for Increasing Ev values
Graphical solution:
• Values of kx are quantized• Only a finite number of solutions are possible – depending on the value of Ev
In the limit Ev ∞ the values of kxare:
Lpkx ( p = 1,2,3……..
Semiconductor Quantum Well: Valence Band Solution
32
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
1vE
2vE
vE
2vEx
0
L
hpv
hh
xv
m
kEE
m
k
mk
EE
2
222||
2
1
2||
222
1
Since values of kx are quantized, the energy dispersion can be written as:
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!
1E
2E
In the limit Ev ∞ the values of Ep are:22
2
Lp
mE
hp
p = 1,2,3……..
Semiconductor Quantum Wells: A 2D Fermi Gas
Light-hole/heavy-hole degeneracy breaks!
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Density of States in Quantum Wells: Valence Band
fE
pfv EkpEf
kdp ||2
||2
,12
2
Start from:
h
pvv m
kEEkpE
2,
2||
2
1||
And convert the k-space integral to energy space:
fp
pvh
E
pf
hEE
EEfEEEm
dE
EEfm
dEp
v
pv
1
1
12
2
1
1
This implies:
ppv
hQW EEE
mEg 12
EgQW
1vE11 EEv 21 EEv 31 EEv
2hm
22
hm
23hm
||k
1vE
11 EEv
21 EEv
31 EEv
E
33
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Growth of Semiconductor Heterostructures: MBE
Low pressure (10-11 Torr), near-equilibrium, chemical reaction free, layer-by-layer growth
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Growth of Semiconductor Heterostructures: MOCVD or MOVPE
Adsorption
TM-In
PH3
CH4
Growth of InP by MOCVD
Atm pressure (760 Torr) growth, involves gas flow and chemical reactions
34
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Epitaxial Growth and Lattice Mismatch
A lattice mismatch between the epitaxial layer and the substrate means that the layer grown will be strained (biaxial strain):
sub
sub
a
aa
0
0
Tensile strain
Compressive strain
if the thickness h of the coherently strained layer exceeds a certain critical thickness hc the coherent strain relaxes and this process generates crystal dislocations (crystal defects). Critical thickness is given by:
b
hbh c
c lncos1
cos14
2
2ab
a
asub
Poisson ratio
and are both equal to 60-degrees for diamond and zinc-blende lattices
for diamond and zinc-blende lattices
Matthews-Blakeslee Formula
ECE 407 – Spring 2009 – Farhan Rana – Cornell University
Strain Compensation
b
hbh c
c lncos1
cos14
2
How does one calculate the critical thickness for a multiple layer stack?
321
332211
hhh
hhhavg
Strain compensation can be used to grow much thicker dislocation-free layers!
Substrate
h1
h2
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