Multiple Quantum Wells - Georgia Institute of...
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1/3/2006 ECE-6451 1 Multiple Quantum Wells Reading: Brennan - 2.5. Lecture by Shamas M. Ummer School of Electrical and Computer Engineering Georgia Institute of Technology
Transcript of Multiple Quantum Wells - Georgia Institute of...
1/3/2006 ECE-6451 1
Multiple Quantum Wells
Reading: Brennan - 2.5.
Lecture by Shamas M. Ummer
School of Electrical and Computer EngineeringGeorgia Institute of Technology
1/3/2006 ECE-6451 2
Multiple Quantum Wells
• This technology is used extensively in semiconductor laser technology
• Multiple Quantum Wells are grown using MBE, CBE and MOCVD
• Typical materials used:– GaAs/AlGaAs– InGaAs/InP– GaN/AlGaN
1/3/2006 ECE-6451 3
I II III IV V VI VII
0 b a+b a+2b 2(a+b) 2a+3b
I II III IV V VI VII
0 b a+b a+2b 2(a+b) 2a+3b
V=V0
V=0
V=V0
V=0
Multiple Quantum Wells – General Structure (E<V0)
We already have the necessary tools to solve this problem. We take the same approach as we did for the potential well and potential barrier problem, i.e., we solve for the wave function in each region and then we apply the boundary conditions between the regions and finally, solve for the reflection and transmission probabilities.
The general structure of a Multiple Quantum well with potential V0, barrier width “b” and well width “a” is shown below:
Let us consider the case when an electron with energy less than V0 is incident on the structure:
V=V0 V=V0
1/3/2006 ECE-6451 4
Multiple Quantum Wells – Assumptions (for both cases)
Two important Assumptions are made for this problem:
– Electron effective mass is assumed to be the same in all regionsThis allows us to use the simple boundary conditions we have been using for
solving our barrier and well problems
– No field is applied to the structure Therefore, the device is translationally symmetric, i.e., the electron energy
doesn’t change according to position.
1/3/2006 ECE-6451 5
Multiple Quantum Wells – Wave Functions Review (E<V0)
Now, we can begin solving the problem. For all the regions, the wave function can be found by solving the Schrödinger Equation. For region I:
xikxik eRe 111
−+=Ψ⇒
efficientcoreflectionR
mEk
−=
= 212h
11021
22
2Ψ=Ψ+
Ψ− EVdx
dmh
where,
0 b a+b a+2b 2(a+b)2a+3b
Incidentwave
Reflectedwave
V0 V0 V0
0 0
1/3/2006 ECE-6451 6
Multiple Quantum Wells – Wave Functions Review (E<V0)
For region II, we have a potential of V0:
22022
22
2Ψ=Ψ+
Ψ− EVdx
dmh
xkxk eCeC 22222
−−+ +=Ψ⇒ 20
2)(2
h
EVmk −=
For region III,
where,
where,
xkCxkC 22222 sinhcosh −+ +=ΨOr, in terms of hyperbolic functions,
33023
22
2Ψ=Ψ+
Ψ− EVdx
dmh
212h
mEk =
xkCxkC 13133 sincos −+ +=Ψ
xikxik eCeC 11333
−−+ +=Ψ
0 b a+b a+2b 2(a+b) 2a+3b
V0 V0 V0
0
+x directionwave
-x directionwave
0
0 b a+b a+2b 2(a+b) 2a+3b
V0 V0 V0
0
Or,
Similarly, we can solve for the wave functions for the other regions with each region with a potential having similar solutions as Ψ2 and the regions without a potential having solutions similar to Ψ3
+x directionwave
-x directionwave
1/3/2006 ECE-6451 7
Multiple Quantum Wells – Wave Functions Review (E<V0)
For Region IV:
,11555
xikxik eCeC −−+ +=Ψ
,22666
xkxk eCeC −−+ +=Ψ
,17
xikTe=Ψ
,22444
xkxk eCeC −−+ +=Ψ
212h
mEk =
20
2)(2
h
EVmk −=
20
2)(2
h
EVmk −=
212h
mEk =
xkCxkC 15155 sincos −+ +=Ψ
xkCxkC 24244 sinhcosh −+ +=Ψ
xkCxkC 26266 sinhcosh −+ +=Ψ
0 b a+b a+2b 2(a+b) 2a+3b
V0 V0 V0
0
0 b a+b a+2b 2(a+b) 2a+3b
V0 V0 V0
0
0 b a+b a+2b 2(a+b) 2a+3b
V0 V0 V0
0
0 b a+b a+2b 2(a+b) 2a+3b
V0 V0 V0
0
For Region V:
For Region VI:
For Region VII:
Note: For region VII, only a +x component exists as the wave extends out to infinity and there is no barrier to reflect the wave back
1/3/2006 ECE-6451 8
Multiple Quantum Wells – Boundary Conditions (I-II interface) (E<V0)
Now, we can apply the boundary conditions between each region. For regions I and II, we can use two boundary conditions:
( ) ( )
( )
( )
)2.(
coshsinh,
coshsinh2
2
)1.(1
sinhcosh
)0()0(
2211
022
222
2
0
11
222
222*
222*22
11*111
*11
0201
2
022220
21
11
11
11
EqCmkR
mik
mik
xkCmkxkC
mkeR
mike
mik
xkCmkxkC
mk
mij
eRmike
mik
mij
jj
EqCR
xkCxkCeRe
xx
xx
xikxik
x
xikxikx
xxxx
xx
xikxik
−
=
−+
=
−
−+
−
==
+
=
−+
=
−
=−
+=
−∴
+=Ψ∇Ψ−Ψ∇Ψ=
−=Ψ∇Ψ−Ψ∇Ψ=
=
=+
+=+
=Ψ==Ψ
vvh
vvh
0 b a+b a+2b 2(a+b) 2a+3b
V0 V0 V0
• Boundary Condition 1 (Continuity of the wave function):
• Boundary Condition 2 (Probability Current Density):
NOTE: For every interface, the boundary conditions are similar. The only parameter that changes is the x value at the interface.
1/3/2006 ECE-6451 9
Multiple Quantum Wells – Boundary Conditions (I-II interface) (E<V0)
)2.(
)1.(1
2211
2
EqCmkR
mik
mik
EqCR
−
+
=−
=+
−
−−
−=
−=
=
−
−
+
−
+−
−
+
2
22
1
1
1
2
22
1
11
2
2211
001
1
1
21
001111
,1
001111
CC
mk
mikmik
ikm
R
CC
mk
mik
mik
R
RforSolving
CC
mk
Rmik
mik
But, equation 1 and 2 can also be expressed in matrix form
−
−−
=
=
==
−
−
acbd
bcadAthen
dcba
Aif
CABthenCABifHEREUSEDIDENTITIESMATRIX
1,.2
,.1:
1
1
(CONTD……)
1/3/2006 ECE-6451 10
Multiple Quantum Wells – Boundary Conditions (I-II interface) (E<V0)
−=
−
−−
−=
−
+
−
+
2
22
1
1
2
22
1
1
1
001
221
221
1
001
1
1
21
CC
mk
ikm
ikm
R
CC
mk
mikmik
ikm
R
[ ] )3.(1
2
2 EqCC
YR III
=
−
+
−
(CONTD.)
where, YI-II is a 2x2 matrix that is obtained by multiplying [a] and [b]. YI-II is called the transfer matrix for the region I-II interface (A transfer matrix describes the boundary conditions at each interface).
Note: The subscripts of Y#1-#2 stands for the regions #1-#2 interface it represents
[a] [b]x
1/3/2006 ECE-6451 11
Multiple Quantum Wells – Boundary Conditions (II-III interface) (E<V0)
We can follow the same procedure we did for the regions I-II interface for obtaining the transfer matrix for the region II-III. Again, we start by using the boundary conditions @ x = b for regions II and III:
( )
( )
bkCmkbkC
mkbkC
mkbkC
mk
xkCmkxkC
mk
mij
xkCmkxkC
mk
mij
jjBC
bkCbkCbkCbkC
bxbxBC
x
x
bxxbxx
131
131
222
222
131
131*
333*33
222
222*
222*22
32
13132222
32
cossincoshsinh,
cossin2
coshsinh2
:2
sincossinhcosh,
)()(:1
−+−+
−+
−+
==
−+−+
+−=+∴
+−=Ψ∇Ψ−Ψ∇Ψ=
+=Ψ∇Ψ−Ψ∇Ψ=
=
+=+∴
=Ψ==Ψ
vvh
vvh
0 b a+b a+2b 2(a+b) 2a+3b
V0 V0 V0
−=
−
+
−
+
3
3
11
11
11
2
2
22
22
22
cossin
sincos
coshsinh
sinhcosh
CC
bkmkbk
mk
bkbk
CC
bkmkbk
mk
bkbk
Again, we can represent these two equations in in the matrix form:
(CONTD……)
1/3/2006 ECE-6451 12
Multiple Quantum Wells – Boundary Conditions (II-III interface) (E<V0)
[ ]
[ ] [ ][ ] )4.(1
,3.
cossin
sincos
coshsinh
sinhcosh
3
3
2
2
3
3
2
2
3
3
11
11
11
222
222
22
2
EqCC
YYCC
YR
haveweEqinabovethengSubstitutiCC
YCC
CC
bkmk
bkmk
bkbk
bkbkmk
bkbkmk
km
CC
IIIIIIIIIII
IIIII
=
=
=
−
−
−=
−
+
−−−
+
−
−
+
−−
+
−
+
−
+
,
cossin
sincos
coshsinh
sinhcosh
2
2
3
3
11
11
11
2
2
22
22
22
−=
−
+
−
+
−
+
CC
forSolving
CC
bkmkbk
mk
bkbk
CC
bkmkbk
mk
bkbk
(CONTD.)
x
Note: Now, it should be obvious to see why we use the Y#1-#2 notation - we avoid rewriting the rather nasty matrix multiplication with this notation.
1/3/2006 ECE-6451 13
Multiple Quantum Wells – Boundary Conditions (III-IV interface) (E<V0)
We continue finding the transfer matrices. Using the boundary conditions @ x = a + b for regions III and IV:
( )
( )
)(cosh)(sinh)(cos)(sin,
coshsinh2
cossin2
:2
)(sinh)(cosh)(sin)(cos,
)()(:1
242
242
131
131
242
242*
444*44
131
131*
333*33
43
24241313
43
bakCmkbakC
mkbakC
mkbakC
mk
xkCmkxkC
mk
mij
xkCmkxkC
mk
mij
jjBC
bakCbakCbakCbakC
baxbaxBC
x
x
baxxbaxx
+++=+++−
∴
+=Ψ∇Ψ−Ψ∇Ψ=
+−
=Ψ∇Ψ−Ψ∇Ψ=
=
+++=+++∴
+=Ψ=+=Ψ
−+−+
−+
−+
+=+=
−+−+
vvh
vvh
++
++=
++−
++
−
+
−
+
4
4
22
22
22
3
3
11
11
11
)(cosh)(sinh
)(sinh)(cosh
)(cos)(sin
)(sin)(cos
CC
bakmkbak
mk
bakbak
CC
bakmkbak
mk
bakbak
Again, these equations can be represented in matrix form,
0 b a+b a+2b
2(a+b)
2a+3b
V0 V0 V0
(CONTD………)
1/3/2006 ECE-6451 14
Multiple Quantum Wells – Boundary Conditions (III-IV interface) (E<V0)
[ ]
[ ][ ] [ ][ ][ ] )5.(1
4.
)(cosh)(sinh
)(sinh)(cosh
)(cos)(sin
)(sin)(cos
,
)(cosh)(sinh
)(sinh)(cosh
)(cos)(sin
)(sin)(cos
4
4
3
3
4
4
3
3
4
4
22
22
22
111
111
13
3
3
3
4
4
22
22
22
3
3
11
11
11
EqCC
YYYCC
YYR
EqinthisngSubstituti
CC
YCC
CC
bakmkbak
mk
bakbak
bakbakmk
bakbakmk
km
CC
CC
forsolving
CC
bakmkbak
mk
bakbak
CC
bakmkbak
mk
bakbak
IVIIIIIIIIIIIIIIIIIII
IVIII
=
=
=
++
++
++
+−+=
++
++=
++−
++
−
+
−−−−
+
−−
−
+
−−
+
−
+
−
+
−
+
−
+
−
+
(CONTD.)
x
This calculation continues iteratively until the carrier crosses through all the interfaces.
1/3/2006 ECE-6451 15
Multiple Quantum Wells – Boundary Conditions (IV-V interface) (E<V0)
( )
( )
)2(cos)2(sin)2(cosh)2(sinh,
cossin2
coshsinh2
:2
)2(sin)2(cos)2(sinh)2(cosh,
)2()2(:1
151
551
242
242
151
551*
555*55
242
242*
444*44
2524
15152424
34
bakCmkbakC
mkbakC
mkbakC
mk
xkCmkxkC
mk
mij
xkCmkxkC
mk
mij
jjBC
bakCbakCbakCbakC
baxbaxBC
x
x
baxxbaxx
+++−=+++∴
+−=Ψ∇Ψ−Ψ∇Ψ=
+=Ψ∇Ψ−Ψ∇Ψ=
=
+++=+++∴
+=Ψ=+=Ψ
−+−+
−+
−+
+=+=
−+−+
vvh
vvh
Using the boundary conditions @ x = a + 2b for regions IV and V we have:
0 b a+b a+2b 2(a+b) 2a+3b
V0 V0 V0
(CONTD……..)
++−
++=
++
++
−
+
−
+
5
5
11
11
11
4
4
22
22
22
)2(cos)2(sin
)2(sin)2(cos
)2(cosh)2(sinh
)2(sinh)2(cosh
CC
bakmkbak
mk
bakbak
CC
bakmkbak
mk
bakbak
Again, these equations can be represented in matrix form,
1/3/2006 ECE-6451 16
Multiple Quantum Wells – Boundary Conditions (IV-V interface) (E<V0)
[ ]
[ ][ ][ ] [ ][ ][ ][ ] )6.(1
,5.
)2(cos)2(sin
)2(sin)2(cos
)2(cosh)2(sinh
)2(sinh)2(cosh
,
5
5
4
4
5
5
4
4
5
5
11
11
11
222
222
24
4
4
4
EqCC
YYYYCC
YYYR
EqinabovethengSubstitutiCC
YCC
CC
bakmkbak
mk
bakbak
bakbakmk
bakbakmk
km
CC
CC
forSolving
VIVIVIIIIIIIIIIIIVIIIIIIIIIII
VIV
=
=
=
++−
++
++−
+−+=
−
+
−−−−−
+
−−−
−
+
−−
+
−
+
−
+
−
+
(CONTD.)
++−
++=
++
++
−
+
−
+
5
5
11
11
11
4
4
22
22
22
)2(cos)2(sin
)2(sin)2(cos
)2(cosh)2(sinh
)2(sinh)2(cosh
CC
bakmkbak
mk
bakbak
CC
bakmkbak
mk
bakbak
x
1/3/2006 ECE-6451 17
Multiple Quantum Wells – Boundary Conditions (V-VI interface) (E<V0)
( )
( )
)22(cosh)22(sinh)22(cos)22(sin,
coshsinh2
cossin2
:2
)22(sinh)22(cosh)22(sin)22(cos,
))(2())(2(:1
262
262
151
151
262
262*
666*66
151
151*
555*55
)(26)(25
26261515
65
bakCmkbakC
mkbakC
mkbakC
mk
xkCmk
xkCmk
mij
xkCmkxkC
mk
mij
jjBC
bakCbakCbakCbakC
baxbaxBC
x
x
baxxbaxx
+++=+++−
∴
+=Ψ∇Ψ−Ψ∇Ψ=
+−
=Ψ∇Ψ−Ψ∇Ψ=
=
+++=+++∴
+=Ψ=+=Ψ
−+−+
−+
−+
+=+=
−+−+
vvh
vvh
Again, these equations can be expressed in matrix form,
Using the boundary conditions @ x = 2(a + b) for regions V and VI:
++
++=
++−
++
−
+
−
+
6
6
22
22
22
5
5
11
11
11
)22(cosh)22(sinh
)22(sinh)22(cosh
)22(cos)22(sin
)22(sin)22(cos
CC
bakmkbak
mk
bakbak
CC
bakmkbak
mk
bakbak
0 b a+b a+2b 2(a+b) 2a+3b
V0 V0 V0
(CONTD……..)
1/3/2006 ECE-6451 18
Multiple Quantum Wells – Boundary Conditions (V-VI interface) (E<V0)
[ ]
[ ][ ][ ][ ] [ ][ ][ ][ ][ ] )7.(1
,6.
)22(cosh)22(sinh
)22(sinh)22(cosh
)22(cos)22(sin
)22(sin)22(cos
,
6
6
5
5
6
6
5
5
6
6
22
22
22
111
111
15
5
5
5
EqCC
YYYYYCC
YYYYR
EqinabovethengSubstitutiCC
YCC
CC
bakmkbak
mk
bakbak
bakbakmk
bakbakmk
km
CC
CC
forSolving
VIVVIVIVIIIIIIIIIIIVIVIVIIIIIIIIIII
VIV
=
=
=
++
++
++
+−+=
−
+
−−−−−−
+
−−−−
−
+
−−
+
−
+
−
+
−
+
(CONTD.)
++
++=
++−
++
−
+
−
+
6
6
22
22
22
5
5
11
11
11
)22(cosh)22(sinh
)22(sinh)22(cosh
)22(cos)22(sin
)22(sin)22(cos
CC
bakmkbak
mk
bakbak
CC
bakmkbak
mk
bakbak
x
1/3/2006 ECE-6451 19
Multiple Quantum Wells – Boundary Conditions (VI-VII interface) (E<V0)
( )
( )bik
xikx
x
baxxbaxx
bik
TeikbakCmkbakC
mk
Teikmi
j
xkCmkxkC
mk
mij
jjBC
TebakCbakC
baxbaxBC
1
1
1
1262
262
1*777
*77
262
262*
666*66
)32(7)32(6
2626
76
)32(cosh)32(sinh,
2
coshsinh2
:2
)32(sinh)32(cosh,
)32()32(:1
=+++∴
=Ψ∇Ψ−Ψ∇Ψ=
+=Ψ∇Ψ−Ψ∇Ψ=
=
=+++∴
+=Ψ=+=Ψ
−+
−+
+=+=
−+
vvh
vvh
Lastly, using the boundary conditions @ x = 2a + 3b for regions VI and VII:
=
++
++
−
+
000
)32(cosh)32(sinh
)32(sinh)32(cosh
1
1
16
6
22
22
22 TTeik
TeCC
bakmkbak
mk
bakbakbik
bik
0 b a+b a+2b 2(a+b) 2a+3b
V0 V0 V0
Again, these equations can be represented in matrix form,
(CONTD……)
1/3/2006 ECE-6451 20
Multiple Quantum Wells – Boundary Conditions (VI-VII interface) (E<V0)
[ ]
[ ][ ][ ][ ][ ] [ ][ ][ ][ ][ ][ ] )8.(0
1
,7.
0
000
)32(cosh)32(sinh
)32(sinh)32(cosh
,
000
)32(cosh)32(sinh
)32(sinh)32(cosh
6
6
6
6
122
2
222
26
6
6
6
16
6
22
22
22
1
1
1
1
EqT
YYYYYYCC
YYYYYR
EqinabovethegSubsitutin
TY
CC
Teik
e
bakbakmk
bakbakmk
km
CC
CC
forSolving
TTeik
TeCC
bakmk
bakmk
bakbak
VIIVIVIVVIVIVIIIIIIIIIIIVIVVIVIVIIIIIIIIIII
VIIVI
bik
bik
bik
bik
=
=
=
++−
+−+=
=
++
++
−−−−−−−
+
−−−−−
−−
+
−
+
−
+
−
+
(CONTD.)
(CONTD…….)
x
We have finally got a useful relationship between the reflection co-efficient and transmissionco-efficient. However, multiplying the transfer matrices is very complex as will be seen from thenext slide.
1/3/2006 ECE-6451 21
Multiple Quantum Wells – Final Matrix (E<V0)
[ ][ ][ ][ ][ ][ ] )8.(0
1EqYYYYYY
R VIIVIVIVVIVIVIIIIIIIIIII
=
−−−−−−
τ
The complexity in multiplying these transfer matrices can be seen from the final matrix below:
++−
+−+×
++
++
++
+−+×
++−
++
++−
+−+×
++
++
++
+−+×
−
−
−×
−=
000
)32(cosh)32(sinh
)32(sinh)32(cosh
)22(cosh)22(sinh
)22(sinh)22(cosh
)22(cos)22(sin
)22(sin)22(cos
)2(cos)2(sin
)2(sin)2(cos
)2(cosh)2(sinh
)2(sinh)2(cosh
)(cosh)(sinh
)(sinh)(cosh
)(cos)(sin
)(sin)(cos
cossin
sincos
coshsinh
sinhcosh
001
221
221
1
1
1
122
2
222
2
22
22
22
111
111
1
11
11
11
222
222
2
22
22
22
11
1
11
1
1
11
11
11
22
2
22
2
2
2
1
1
Teik
e
bakbakmk
bakbakmk
km
bakmk
bakmk
bakbak
bakbakmk
bakbakmk
km
bakmk
bakmk
bakbak
bakbakmk
bakbakmk
km
bakmk
bakmk
bakbak
bakkmbak
bakkmbak
km
bkmk
bkmk
bkbk
bkkmbk
bkkmbk
km
mk
ikm
ikm
R
bik
bik
YI-II
YII-III
YIII-IV
YIV-V
YV-VI
YVI-VII
1/3/2006 ECE-6451 22
Multiple Quantum Wells – Reflection and Transmission Probabilities (E<V0)
[ ][ ][ ][ ][ ][ ] )8.(0
1EqYYYYYY
R VIIVIVIVVIVIVIIIIIIIIIII
=
−−−−−−
τ
ATAT 11 =⇒=
ACCTR ==
** 11
=
AATT
The Transmission Co-efficient, “T”, can then be found by using:
Therefore, the Transmission Probability and Reflection Probability is given by:*
**
** 1111
=
−=−=
AC
ACRROR
AATTRR
The Reflection Co-efficient, “R”, would then be given by:
(CONTD.)
)9.(0
1Eq
DCBA
R
=
τ
Equation 8 can now be used to calculate the transmission and reflection co-efficients. We would need a powerful math tool in order to multiply the complicated transfer matrices. After this multiplication is performed, we would obtain a 2x2 matrix as shown below:
1/3/2006 ECE-6451 23
I II III IV V VI VII
0 b a+b a+2b 2(a+b) 2a+3b
V=V0
V=0
Multiple Quantum Wells – General Structure (E>V0)
Again, we take the same approach as we did earlier: we solve for the wave function in each region and then we apply the boundary conditions between the regions and finally, solve for the reflection and transmission probabilities.
Now, let us consider the case when an electron with energy E greater than V0 is incident on the structure:
1/3/2006 ECE-6451 24
Multiple Quantum Wells – Wave Functions Review (E>V0)
For region I, the wavefunction is the same as the E<V0 case as there is no potential involved here:
xikxik eRe 111
−+=Ψ⇒
efficientcoreflectionR
mEk
−=
= 212h
11021
22
2Ψ=Ψ+
Ψ− EVdx
dmh
where,
0 b a+b a+2b 2(a+b) 2a+3b
Incidentwave
Reflectedwave
V0
0
0
1/3/2006 ECE-6451 25
Multiple Quantum Wells – Wave Functions Review (E>V0)
For region II, we have a potential of V0:
22022
22
2Ψ=Ψ+
Ψ− EVdx
dmh
xikxik eCeC 22222
−−+ +=Ψ⇒ 20
2)(2
h
VEmk
−=
For region III,
where,
where,
xkCxkC 22222 sincos −+ +=ΨOr, in terms of cosine and sine functions,
33023
22
2Ψ=Ψ+
Ψ− EVdx
dmh
212h
mEk =
xkCxkC 13133 sincos −+ +=Ψ
xikxik eCeC 11333
−−+ +=Ψ
+x directionwave
-x directionwave
0
Or,
Similarly, we can solve for the wave functions for the other regions with each region with a potential having similar solutions as Ψ2 and the regions without a potential having solutions similar to Ψ3
+x directionwave
-x directionwave
0 b a+b a+2b 2(a+b) 2a+3b
V0
0
0 b a+b a+2b 2(a+b) 2a+3b
V0
0
1/3/2006 ECE-6451 26
Multiple Quantum Wells – Wave Functions Review (E>V0)
For Region IV:
,11555
xikxik eCeC −−+ +=Ψ
,22666
xikxik eCeC −−+ +=Ψ
,17
xikTe=Ψ
,22444
xikxik eCeC −−+ +=Ψ
212h
mEk =
20
2)(2
h
VEmk
−=
20
2)(2
h
VEmk
−=
212h
mEk =
xkCxkC 15155 sincos −+ +=Ψ
xkCxkC 24244 sincos −+ +=Ψ
xkCxkC 26266 sincos −+ +=Ψ
0 b a+b a+2b 2(a+b) 2a+3b
V0
0
0 b a+b a+2b 2(a+b) 2a+3b
V0
0
0 b a+b a+2b 2(a+b) 2a+3b
V0
0
0 b a+b a+2b 2(a+b) 2a+3b
V0
0
For Region V:
For Region VI:
For Region VII:
Note: For region VII, only a +x component exists as the wave extends out to infinity and there is no barrier to reflect the wave back
1/3/2006 ECE-6451 27
Multiple Quantum Wells – Boundary Conditions (I-II interface) (E>V0)
Now, we can apply the boundary conditions between each region. For regions I and II, we can use two boundary conditions:
( ) ( )
( )
( )
)11.(
cossin,
cossin2
2
)10.(1
sincos
)0()0(
2211
022
222
2
0
11
222
222*
222*22
11*111
*11
0201
2
022220
21
11
11
11
EqCmk
Rmik
mik
xkCmkxkC
mkeR
mike
mik
xkCmk
xkCmk
mij
eRmike
mik
mij
jj
EqCR
xkCxkCeRe
xx
xx
xikxik
x
xikxikx
xxxx
xx
xikxik
−
=
−+
=
−
−+
−
==
+
=
−+
=
−
=−
+−=
−∴
+−=Ψ∇Ψ−Ψ∇Ψ=
−=Ψ∇Ψ−Ψ∇Ψ=
=
=+
+=+
=Ψ==Ψ
vvh
vvh
0 b a+b a+2b 2(a+b) 2a+3b
• Boundary Condition 1 (Continuity of the wave function):
• Boundary Condition 2 (Probability Current Density):
1/3/2006 ECE-6451 28
Multiple Quantum Wells – Boundary Conditions (I-II interface) (E>V0)
)11.(
)10.(1
2211
2
EqCmk
Rmik
mik
EqCR
−
+
=−
=+
−
−−
−=
−=
=
−
−
+
−
+−
−
+
2
22
1
1
1
2
22
1
11
2
2211
001
1
1
21
001111
,1
001111
CC
mk
mikmik
ikm
R
CC
mk
mik
mik
R
RforSolving
CC
mk
Rmik
mik
But, equation 10 and 11 can also be expressed in matrix form
−
−−
=
=
==
−
−
acbd
bcadAthen
dcba
Aif
CABthenCABifHEREUSEDIDENTITIESMATRIX
1,.2
,.1:
1
1
(CONTD……)
1/3/2006 ECE-6451 29
Multiple Quantum Wells – Boundary Conditions (I-II interface) (E>V0)
−=
−
−−
−=
−
+
−
+
2
22
1
1
2
22
1
1
1
001
221
221
1
001
1
1
21
CC
mk
ikm
ikm
R
CC
mk
mikmik
ikm
R
[ ] )12.(1
2
2 EqCC
YR III
=
−
+
−
(CONTD.)
where, YI-II is the transfer matrix for the region I-II interface
[a] [b]x
1/3/2006 ECE-6451 30
Multiple Quantum Wells – Boundary Conditions (II-III interface) (E>V0)
We can follow the same procedure we did for the regions I-II interface for obtaining the transfer matrix for the region II-III. Again, we start by using the boundary conditions @ x = b for regions II and III:
( )
( )
bkCmkbkC
mkbkC
mkbkC
mk
xkCmk
xkCmk
mij
xkCmkxkC
mk
mij
jjBC
bkCbkCbkCbkC
bxbxBC
x
x
bxxbxx
131
131
222
222
131
131*
333*33
222
222*
222*22
32
13132222
32
cossincossin,
cossin2
cossin2
:2
sincossincos,
)()(:1
−+−+
−+
−+
==
−+−+
+−=+−∴
+−=Ψ∇Ψ−Ψ∇Ψ=
+−=Ψ∇Ψ−Ψ∇Ψ=
=
+=+∴
=Ψ==Ψ
vvh
vvh
0 b a+b a+2b 2(a+b) 2a+3b
−=
− −
+
−
+
3
3
11
11
11
2
2
22
22
22
cossin
sincos
cossin
sincos
CC
bkmk
bkmk
bkbk
CC
bkmk
bkmk
bkbk
Again, we can represent these two equations in in the matrix form:
(CONTD……)
1/3/2006 ECE-6451 31
Multiple Quantum Wells – Boundary Conditions (II-III interface) (E>V0)
[ ]
[ ] [ ][ ] )13.(1
,12.
cossin
sincos
cossin
sincos
3
3
2
2
3
3
2
2
3
3
11
11
11
222
222
22
2
EqCC
YYCC
YR
haveweEqinabovethengSubstitutiCC
YCC
CC
bkmk
bkmk
bkbk
bkbkmk
bkbkmk
km
CC
IIIIIIIIIII
IIIII
=
=
=
−
−=
−
+
−−−
+
−
−
+
−−
+
−
+
−
+
,
cossin
sincos
cossin
sincos
2
2
3
3
11
11
11
2
2
22
22
22
−=
−
−
+
−
+
−
+
CC
forSolving
CC
bkmk
bkmk
bkbk
CC
bkmk
bkmk
bkbk
(CONTD.)
x
Note: Now, it should be obvious to see why we use the Y#1-#2 notation - we avoid rewriting the rather nasty matrix multiplication with this notation.
1/3/2006 ECE-6451 32
Multiple Quantum Wells – Boundary Conditions (III-IV interface) (E>V0)
We continue finding the transfer matrices. Using the boundary conditions @ x = a + b for regions III and IV:
( )
( )
)(cos)(sin)(cos)(sin,
cossin2
cossin2
:2
)(sin)(cos)(sin)(cos,
)()(:1
242
242
131
131
242
242*
444*44
131
131*
333*33
43
24241313
43
bakCmk
bakCmk
bakCmk
bakCmk
xkCmk
xkCmk
mij
xkCmk
xkCmk
mij
jjBC
bakCbakCbakCbakC
baxbaxBC
x
x
baxxbaxx
+++−=+++−∴
+−=Ψ∇Ψ−Ψ∇Ψ=
+−=Ψ∇Ψ−Ψ∇Ψ=
=
+++=+++∴
+=Ψ=+=Ψ
−+−+
−+
−+
+=+=
−+−+
vvh
vvh
++−
++=
++−
++
−
+
−
+
4
4
22
22
22
3
3
11
11
11
)(cos)(sin
)(sin)(cos
)(cos)(sin
)(sin)(cos
CC
bakmkbak
mk
bakbak
CC
bakmkbak
mk
bakbak
Again, these equations can be represented in matrix form,
0 b a+b a+2b 2(a+b) 2a+3b
V0
(CONTD………)
1/3/2006 ECE-6451 33
Multiple Quantum Wells – Boundary Conditions (III-IV interface) (E>V0)
(CONTD.)
This calculation continues iteratively until the carrier crosses through all the interfaces.
[ ]
[ ][ ] [ ][ ][ ] )14.(1
,13.
)(cos)(sin
)(sin)(cos
)(cos)(sin
)(sin)(cos
,
4
4
3
3
4
4
3
3
4
4
22
22
22
111
111
13
3
4
4
EqCC
YYYCC
YYR
EqinabovethengSubstitutiCC
YCC
CC
bakmk
bakmk
bakbak
bakbakmk
bakbakmk
km
CC
CC
forSolving
IVIIIIIIIIIIIIIIIIIII
IVIII
=
=
=
++−
++
++
+−+=
−
+
−−−−
+
−−
−
+
−−
+
−
+
−
+
−
+
++−
++=
++−
++
−
+
−
+
4
4
22
22
22
3
3
11
11
11
)(cos)(sin
)(sin)(cos
)(cos)(sin
)(sin)(cos
CC
bakmk
bakmk
bakbak
CC
bakmk
bakmk
bakbak
x
1/3/2006 ECE-6451 34
Multiple Quantum Wells – Boundary Conditions (IV-V interface) (E>V0)
( )
( )
)2(cos)2(sin)2(cos)2(sin,
cossin2
cossin2
:2
)2(sin)2(cos)2(sin)2(cos,
)2()2(:1
151
551
242
242
151
551*
555*55
242
242*
444*44
2524
15152424
34
bakCmk
bakCmk
bakCmk
bakCmk
xkCmk
xkCmk
mij
xkCmk
xkCmk
mij
jjBC
bakCbakCbakCbakC
baxbaxBC
x
x
baxxbaxx
+++−=+++−∴
+−=Ψ∇Ψ−Ψ∇Ψ=
+−=Ψ∇Ψ−Ψ∇Ψ=
=
+++=+++∴
+=Ψ=+=Ψ
−+−+
−+
−+
+=+=
−+−+
vvh
vvh
Using the boundary conditions @ x = a + 2b for regions IV and V we have:
0 b a+b a+2b 2(a+b) 2a+3b
V0
(CONTD……..)
++−
++=
++−
++
−
+
−
+
5
5
11
11
11
4
4
22
22
22
)2(cos)2(sin
)2(sin)2(cos
)2(cos)2(sinh
)2(sin)2(cos
CC
bakmk
bakmk
bakbak
CC
bakmk
bakmk
bakbak
Again, these equations can be represented in matrix form,
1/3/2006 ECE-6451 35
Multiple Quantum Wells – Boundary Conditions (IV-V interface) (E>V0)
[ ]
[ ][ ][ ] [ ][ ][ ][ ] )15.(1
,14.
)2(cos)2(sin
)2(sin)2(cos
)2(cos)2(sinh
)2(sin)2(cos
,
5
5
4
4
5
5
4
4
5
5
11
11
11
222
222
24
4
4
4
EqCC
YYYYCC
YYYR
EqinabovethengSubstitutiCC
YCC
CC
bakmkbak
mk
bakbak
bakbakmk
bakbakmk
km
CC
CC
forSolving
VIVIVIIIIIIIIIIIIVIIIIIIIIIII
VIV
=
=
=
++−
++
++
+−+=
−
+
−−−−−
+
−−−
−
+
−−
+
−
+
−
+
−
+
(CONTD.)
++−
++=
++−
++
−
+
−
+
5
5
11
11
11
4
4
22
22
22
)2(cos)2(sin
)2(sin)2(cos
)2(cos)2(sinh
)2(sin)2(cos
CC
bakmkbak
mk
bakbak
CC
bakmkbak
mk
bakbak
x
1/3/2006 ECE-6451 36
Multiple Quantum Wells – Boundary Conditions (V-VI interface) (E>V0)
( )
( )
)22(cos)22(sin)22(cos)22(sin,
coshsin2
cossin2
:2
)22(sin)22(cos)22(sin)22(cos,
))(2())(2(:1
262
262
151
151
262
262*
666*66
151
151*
555*55
)(26)(25
26261515
65
bakCmkbakC
mkbakC
mkbakC
mk
xkCmk
xkCmk
mij
xkCmkxkC
mk
mij
jjBC
bakCbakCbakCbakC
baxbaxBC
x
x
baxxbaxx
+++−=+++−∴
+−=Ψ∇Ψ−Ψ∇Ψ=
+−=Ψ∇Ψ−Ψ∇Ψ=
=
+++=+++∴
+=Ψ=+=Ψ
−+−+
−+
−+
+=+=
−+−+
vvh
vvh
Again, these equations can be expressed in matrix form,
Using the boundary conditions @ x = 2(a + b) for regions V and VI:
++−
++=
++−
++
−
+
−
+
6
6
22
22
22
5
5
11
11
11
)22(cos)22(sin
)22(sin)22(cos
)22(cos)22(sin
)22(sin)22(cos
CC
bakmk
bakmk
bakbak
CC
bakmk
bakmk
bakbak
0 b a+b a+2b 2(a+b) 2a+3b
V0
(CONTD……..)
1/3/2006 ECE-6451 37
Multiple Quantum Wells – Boundary Conditions (V-VI interface) (E>V0)
[ ]
[ ][ ][ ][ ] [ ][ ][ ][ ][ ] )16.(1
,15.
)22(cos)22(sin
)22(sin)22(cos
)22(cos)22(sin
)22(sin)22(cos
,
6
6
5
5
6
6
5
5
6
6
22
22
22
111
111
15
5
5
5
EqCC
YYYYYCC
YYYYR
EqinabovethengSubstitutiCC
YCC
CC
bakmk
bakmk
bakbak
bakbakmk
bakbakmk
km
CC
CC
forSolving
VIVVIVIVIIIIIIIIIIIVIVIVIIIIIIIIIII
VIV
=
=
=
++−
++
++
+−+=
−
+
−−−−−−
+
−−−−
−
+
−−
+
−
+
−
+
−
+
(CONTD.)
++−
++=
++−
++
−
+
−
+
6
6
22
22
22
5
5
11
11
11
)22(cos)22(sin
)22(sin)22(cos
)22(cos)22(sin
)22(sin)22(cos
CC
bakmk
bakmk
bakbak
CC
bakmk
bakmk
bakbak
x
1/3/2006 ECE-6451 38
Multiple Quantum Wells – Boundary Conditions (VI-VII interface) (E>V0)
( )
( )bik
xikx
x
baxxbaxx
bik
TeikbakCmkbakC
mk
Teikmi
j
xkCmk
xkCmk
mij
jjBC
TebakCbakC
baxbaxBC
1
1
1
1262
262
1*777
*77
262
262*
666*66
)32(7)32(6
2626
76
)32(cos)32(sin,
2
cossin2
:2
)32(sin)32(cos,
)32()32(:1
=+++∴
=Ψ∇Ψ−Ψ∇Ψ=
+−=Ψ∇Ψ−Ψ∇Ψ=
=
=+++∴
+=Ψ=+=Ψ
−+
−+
+=+=
−+
vvh
vvh
Lastly, using the boundary conditions @ x = 2a + 3b for regions VI and VII:
=
++−
++
−
+
000
)32(cosh)32(sinh
)32(sinh)32(cosh
1
1
16
6
22
22
22 TTeik
TeCC
bakmkbak
mk
bakbakbik
bik
0 b a+b a+2b 2(a+b) 2a+3b
V0
Again, these equations can be represented in matrix form,
(CONTD……)
1/3/2006 ECE-6451 39
Multiple Quantum Wells – Boundary Conditions (VI-VII interface) (E>V0)
[ ]
[ ][ ][ ][ ][ ] [ ][ ][ ][ ][ ][ ] )17.(0
1
,16.
0
000
)32(cos)32(sin
)32(sin)32(cos
,
000
)32(cos)32(sin
)32(sin)32(cosh
6
6
6
6
122
2
222
26
6
6
6
16
6
22
22
22
1
1
1
1
EqT
YYYYYYCC
YYYYYR
EqinabovethegSubsitutin
TY
CC
Teik
e
bakbakmk
bakbakmk
km
CC
CC
forSolving
TTeik
TeCC
bakmkbak
mk
bakbak
VIIVIVIVVIVIVIIIIIIIIIIIVIVVIVIVIIIIIIIIIII
VIIVI
bik
bik
bik
bik
=
=
=
++
+−+=
=
++−
++
−−−−−−−
+
−−−−−
−−
+
−
+
−
+
−
+
(CONTD.)
(CONTD…….)
x
Multiplying the transfer matrices is again very complex as will be seen from the next slide.
1/3/2006 ECE-6451 40
Multiple Quantum Wells –Final Matrix (E>V0)
[ ][ ][ ][ ][ ][ ] )17.(0
1Eq
TYYYYYY
R VIIVIVIVVIVIVIIIIIIIIIII
=
−−−−−−
The complexity in multiplying these transfer matrices can be seen from the final matrix below:
++
+−+×
++−
++
++
+−+×
++−
++
++
+−+×
++−
++
++
+−+×
−
−×
−=
000
)32(cos)32(sin
)32(sin)32(cos
)22(cos)22(sin
)22(sin)22(cos
)22(cos)22(sin
)22(sin)22(cos
)2(cos)2(sin
)2(sin)2(cos
)2(cos)2(sinh
)2(sin)2(cos
)(cos)(sin
)(sin)(cos
)(cos)(sinh
)(sin)(cos
cossin
sincos
cossin
sincos
001
221
221
1
1
1
122
2
222
2
22
22
22
111
111
1
11
11
11
222
222
2
22
22
22
111
111
1
11
11
11
222
222
2
2
1
1
Teik
e
bakbakmk
bakbakmk
km
bakmk
bakmk
bakbak
bakbakmk
bakbakmk
km
bakmkbak
mk
bakbak
bakbakmk
bakbakmk
km
bakmkbak
mk
bakbak
bakbakmk
bakbakmk
km
bkmkbk
mk
bkbk
bkbkmk
bkbkmk
km
mk
ikm
ikm
R
bik
bik
YI-II
YII-III
YIII-IV
YIV-V
YV-VI
YVI-VII
1/3/2006 ECE-6451 41
Multiple Quantum Wells – Reflection and Transmission Probabilities (E>V0)
[ ][ ][ ][ ][ ][ ] )17.(0
1EqYYYYYY
R VIIVIVIVVIVIVIIIIIIIIIII
=
−−−−−−
τ
ATAT 11 =⇒=
ACCTR ==
** 11
=
AATT
The Transmission Co-efficient, “T”, can then be found by using:
Therefore, the Transmission Probability and Reflection Probability is given by:*
**
** 1111
=
−=−=
AC
ACRROR
AATTRR
The Reflection Co-efficient, “R”, would then be given by:
(CONTD.)
)18.(0
1Eq
DCBA
R
=
τ
Equation 17 can now be used to calculate the transmission and reflection coefficients. Again, we would need a powerful math tool in order to multiply the complicated transfer matrices. After the transfer matrices are multiplied, we would obtain a 2x2 matrix as shown below:
1/3/2006 ECE-6451 42
Multiple Quantum Wells –Example 1
Consider a GaAs/AlGaAs multiple quantum well structure with a = 50 Ao and b= 50 Ao:
I II III IV V VI VII
0 50 100 150 200 250
V=0.347eV
V=0
(Ao)
1/3/2006 ECE-6451 43
Multiple Quantum Wells –Example 1
Brennan Fig. 2.5.2
Important Observations from the Transmissivityplot:
• There are three peak values:The peak value of 0.08 eV is the lowest
value for which an electron can transmit through the entire structure. The other peak (transmission) values are all above the potential (0.347eV)
• Transmission is strongly dependent on incident energy – higher the incident energy, higher the chances of transmission.
• The peak values also corresponds to the eigenvalues of the quantum well. Therefore, its possible to obtain eigenvalues from a transmissivity plot.
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Multiple Quantum Wells –Example 2
I II III IV V VI VII
0 25 75 100 150 175
V=0.347eV
V=0
(Ao)
Now, consider a GaAs/AlGaAs multiple quantum well structure with a smaller barrier width of 25 A
o:
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Multiple Quantum Wells –Example 1 versus Example 2
Brennan Fig. 2.5.7Brennan Fig. 2.5.2
For example 1, For example 2,
For the transmissivity plot of example 2, there are two peaks at low energy whencompared to one peak for the previous example. This is due to the factthat a smaller barrier width was used. Therefore, transmissivity increases withsmaller barrier widths.
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Multiple Quantum Wells –Additional Comments
• Coupled Quantum Well (Super Lattices)– Multiple quantum wells with a non-zero transmission co-efficient
throughout the structure– More to be discussed in ECE-6453
