hole mobility in semiconductor nanowires : temperature and strain effects
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HOLE MOBILITY IN SEMICONDUCTOR NANOWIRES:TEMPERATURE AND STRAIN EFFECTS
Mariama Rebello de Sousa Dias
Advisor: Prof. Dr. Victor Lopez Richard
SUMARY Electron phonon interaction via deformation
potential; Interaction potential; Matrix element calculation; Deformation Potential;
Transition Rate; Lifetime; Mobility; Strain; Results; Molecular Dynamics Simulation.
ELECTRON PHONON INTERACTION VIA DEFORMATION POTENTIAL;
Wave function of an infinite nanowire: jqii mJn ,
ziyx
ziyx
iyx
iyx
mJ j
2)(6
121,
23
2)(6
121,
23
)(2
123,
23
)(2
123,
23
:,
yx
Zikyx
npn
ziknpn
npn
zikinnpn
i LLeynsenxnsen
Jr
ensenr
J
Jr
eer
J
SquareerSemicylindCylinder
z
zz
22
22
zz qiq
qiq
qIe
eaeaV
qqUzV
ˆˆ
2)()()(
2/1~
Interaction potential
iqi
qIeiiqi
qIeiqq
qpezz eaqUeaqU
VqH
ˆ)(ˆ)(
2)( ''
2/1
zz
zz
kpnqiz
kpnqqqjIej
kpnqiz
kpnqqqjIej
qqpe
enanmJqUmJ
enanmJqUmJMH
,,,','
,,,','
'
'
ˆ1,)(','
ˆ1,)(',')(
Interaction element calculation
11ˆ
1ˆ
qqqq
qqqq
nnna
nnna
qkknnnnknnqi
knn
qkkppnnkpnqi
kpn
qkkppnnkpnqi
kpn
zzyyxxzyx
z
zyx
zzz
z
z
zzz
z
z
e
e
e
,',,,,,,
,'',',,,,','
,'',',,,,','
,,',,
'
'
)()(2
3)(0
zDzDa
qU heIe
Deformation Potential
where and are proportional to)(zDe )(zDh r
0
03
100
310
23,
23
03
10000
31
21,
23
00
3100
03
121,
23
000
31
03
1023,
23
23,
23
21,
23
21,
23
23,
23,','
y
x
z
y
x
z
z
y
x
z
y
x
jj
DiD
iD
DiD
iD
iDD
iDiD
DiD
mJrmJ
For the valence band
0srs For the conduction band
In the direction [001]LHHH couples with
qzze
qzza
zz kEkEHkEkEHkkSpepe
)()'()()'(2)',(22
qzzqkkqqzzqkkqq q
zz kEkEnkEkEnaVqdkkS
zzzz
)()'(1)()'(4
)',( ,','20
220
qkk
zqqkk
zq
qq zzzzpe a
iDna
iDnMH
,'0
,'0 2
12
)(
So,
22 )2/()()'(2/1)()'(
qzz
qzz kEkEkEkE
Replace the Dirac delta by a Lorentzian,
TRANSITION RATE CALCULATION
And using the momentum conservation,
22,'
22,'
20
220
)2/()()'(2/11
)2/()()'(2/1
4)',(
qzzqkkq
qzzqkkq
q qzz
kEkEn
kEkEn
aVqdkkS
zz
zz
qzzqz kEqkEkEkEz
)()()()( ,
2... 1,nwith 11112
2... 1,nwith 11112
2... 1, 0,nwith 11112
22
22
2
222
0
2
22
22
0
2
2
122
22
0
2
qzizf
zify
y
x
x
zf
z
zf
qzzizf
np
ifzf
z
zf
qzzz
np
ifzf
z
zf
mmk
mmLn
Ln
mqk
mq
mSquare
kmmrmmm
qkmq
merSemicylind
kmmrmmm
qkmq
mCylinder
a b c
dAdqqkSve
VqkSkkSk z
qz
kzz
z z
),(.
),()',()(
1 3/1
'
aaacb
bacia
aacb
baciaqdcbqaq
q2)2/(44
2)2/(2
)2/(44
2)2/(2)2/(
2/2
2
2
2
222
2
C
LIFETIME CALCULATION
ea2
0
20
3/1
C1C4..)(
1
qz
nnaV
dAdve
Vk
2
23/1 rV yxLLV 3/1
4
23/1 rV)(
* zkme
MOBILITY
STRAIN
Subband HH Displacment Subband HL
where
qzz kEqkE )()(
QPEhh
qizfz EkEEqkE )()(
solh
QQPE
22
||11
12112
cccaaP cv ||
11
1211 2
cccbQ
Elastic moduliDeformation Potential
Spin-orbit split-off energy
RESULTS
MOLECULAR DYNAMICS SIMULATION
•Will provide values of strain field distribution in the wires
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