prof.p. ravindran, - folk.uio.nofolk.uio.no/ravi/cutn/ccmp/9-effectivemass1.pdf11 1 00 00 00 l ct t...
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![Page 1: Prof.P. Ravindran, - folk.uio.nofolk.uio.no/ravi/cutn/ccmp/9-EffectiveMass1.pdf11 1 00 00 00 L cT T m mm m m m m m LT 0.98 0.19 00 Altogether 6 valleys. Prof.P.Ravindran, Computational](https://reader034.vdocument.in/reader034/viewer/2022042301/5ecc281e0aafd800fe193fc8/html5/thumbnails/1.jpg)
Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations
http://folk.uio.no/ravi/CMT15
Prof.P. Ravindran, Department of Physics, Central University of Tamil
Nadu, India
&Center for Materials Science and Nanotechnology,
University of Oslo, Norway
Carrier effective mass calculations
1
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Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations
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Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations
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Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations
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This shows a wave with the group velocity and phase velocity going in
different directions.[1] The group velocity is positive (i.e.
the envelope of the wave moves rightward), while the phase velocity is
negative (i.e. the peaks and troughs move leftward).
Solid line: A wave packet. Dashed line: The envelope of the wave packet. The
envelope moves at the group velocity.
Phase Velocity and Group Velocity
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Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations
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Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations
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Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations
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Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations
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![Page 9: Prof.P. Ravindran, - folk.uio.nofolk.uio.no/ravi/cutn/ccmp/9-EffectiveMass1.pdf11 1 00 00 00 L cT T m mm m m m m m LT 0.98 0.19 00 Altogether 6 valleys. Prof.P.Ravindran, Computational](https://reader034.vdocument.in/reader034/viewer/2022042301/5ecc281e0aafd800fe193fc8/html5/thumbnails/9.jpg)
Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations
9Effective mass
21
, 2
1c ij
i j
Em
k k
Effective mass tensorSilicon
E
ky
kx
22 2 2 2 22 2 2
2 2 2 2 2 2
, , ,
1 1 1( )
2 2 2 2 2 2
yx zg x y z
x y z c x c y c z
pE E E p pE E p p p
k k k m m m
p
1
1 1
1
0 0
0 0
0 0
x
c y
z
m
m m
m
Only diagonal termsIn Si
1
1 1
1
0 0
0 0
0 0
L
c T
T
m
m m
m
0 00.98 0.19L Tm m m m
Altogether 6 valleys
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Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations
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Effective mass in GaAs
kx
Ec(k)
ky
00.067cm m
In GaAs
1
1 1
1
0 0
0 0
0 0
c
c c
c
m
m m
m
Effective mass is a scalar2E / 2 cp m
InAs00.023cm m
,
1 xx
x c x
pEv
k m
Velocity 1 E
x cm
pv
k
![Page 11: Prof.P. Ravindran, - folk.uio.nofolk.uio.no/ravi/cutn/ccmp/9-EffectiveMass1.pdf11 1 00 00 00 L cT T m mm m m m m m LT 0.98 0.19 00 Altogether 6 valleys. Prof.P.Ravindran, Computational](https://reader034.vdocument.in/reader034/viewer/2022042301/5ecc281e0aafd800fe193fc8/html5/thumbnails/11.jpg)
Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations
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![Page 12: Prof.P. Ravindran, - folk.uio.nofolk.uio.no/ravi/cutn/ccmp/9-EffectiveMass1.pdf11 1 00 00 00 L cT T m mm m m m m m LT 0.98 0.19 00 Altogether 6 valleys. Prof.P.Ravindran, Computational](https://reader034.vdocument.in/reader034/viewer/2022042301/5ecc281e0aafd800fe193fc8/html5/thumbnails/12.jpg)
Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations
12Measurement of Effective Mass
![Page 13: Prof.P. Ravindran, - folk.uio.nofolk.uio.no/ravi/cutn/ccmp/9-EffectiveMass1.pdf11 1 00 00 00 L cT T m mm m m m m m LT 0.98 0.19 00 Altogether 6 valleys. Prof.P.Ravindran, Computational](https://reader034.vdocument.in/reader034/viewer/2022042301/5ecc281e0aafd800fe193fc8/html5/thumbnails/13.jpg)
Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations
Reciprocal lattice of a face-centred cubic lattice
Brillouin boundaries for a face-centred cubic lattice.
![Page 14: Prof.P. Ravindran, - folk.uio.nofolk.uio.no/ravi/cutn/ccmp/9-EffectiveMass1.pdf11 1 00 00 00 L cT T m mm m m m m m LT 0.98 0.19 00 Altogether 6 valleys. Prof.P.Ravindran, Computational](https://reader034.vdocument.in/reader034/viewer/2022042301/5ecc281e0aafd800fe193fc8/html5/thumbnails/14.jpg)
Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations
Band structure of Si and GaAs
![Page 15: Prof.P. Ravindran, - folk.uio.nofolk.uio.no/ravi/cutn/ccmp/9-EffectiveMass1.pdf11 1 00 00 00 L cT T m mm m m m m m LT 0.98 0.19 00 Altogether 6 valleys. Prof.P.Ravindran, Computational](https://reader034.vdocument.in/reader034/viewer/2022042301/5ecc281e0aafd800fe193fc8/html5/thumbnails/15.jpg)
Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations
Fermi surfaces of conduction electrons in Si and GaAs
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Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations
16Bandstructure of Ge
![Page 17: Prof.P. Ravindran, - folk.uio.nofolk.uio.no/ravi/cutn/ccmp/9-EffectiveMass1.pdf11 1 00 00 00 L cT T m mm m m m m m LT 0.98 0.19 00 Altogether 6 valleys. Prof.P.Ravindran, Computational](https://reader034.vdocument.in/reader034/viewer/2022042301/5ecc281e0aafd800fe193fc8/html5/thumbnails/17.jpg)
Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations
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Four Valleys Inside BZ for Germanium
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Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations
18Measurement of Effective Mass
![Page 19: Prof.P. Ravindran, - folk.uio.nofolk.uio.no/ravi/cutn/ccmp/9-EffectiveMass1.pdf11 1 00 00 00 L cT T m mm m m m m m LT 0.98 0.19 00 Altogether 6 valleys. Prof.P.Ravindran, Computational](https://reader034.vdocument.in/reader034/viewer/2022042301/5ecc281e0aafd800fe193fc8/html5/thumbnails/19.jpg)
Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations
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Conduction Band Effective Mass
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Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations
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E-k Diagram, Velocity and Effective Mass
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Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations
21Physical Meaning of the Band Effective Mass
Near the bottom of a nearly-free electron band m* is approximately constant, but
it increases dramatically near the inflection point and even becomes negative (!)
near the zone edge.
The effective mass is
inversely proportional to the
curvature of the energy
band.
![Page 22: Prof.P. Ravindran, - folk.uio.nofolk.uio.no/ravi/cutn/ccmp/9-EffectiveMass1.pdf11 1 00 00 00 L cT T m mm m m m m m LT 0.98 0.19 00 Altogether 6 valleys. Prof.P.Ravindran, Computational](https://reader034.vdocument.in/reader034/viewer/2022042301/5ecc281e0aafd800fe193fc8/html5/thumbnails/22.jpg)
Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations
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Physical Meaning of the Band Effective Mass
Of course, for a free
electron,
In a 3-D solid we would find that m* is
a second-order tensor with 9
components:
and
The effective mass concept if useful because it allows us to retain the notion of a
free-electron even when we have a periodic potential, as long as we use m* to
account for the effect of the lattice on the acceleration of the electron.
m
kE x
2
22
zyxjikk
E
m ji
,,,1
*
1 2
2
m
m
m
2
2
*
But what does it mean to have a varying “effective mass” for different
materials?
![Page 23: Prof.P. Ravindran, - folk.uio.nofolk.uio.no/ravi/cutn/ccmp/9-EffectiveMass1.pdf11 1 00 00 00 L cT T m mm m m m m m LT 0.98 0.19 00 Altogether 6 valleys. Prof.P.Ravindran, Computational](https://reader034.vdocument.in/reader034/viewer/2022042301/5ecc281e0aafd800fe193fc8/html5/thumbnails/23.jpg)
Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations
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Concept of a Hole in an Otherwise Filled Band
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Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations
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Sign of Effective Mass
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Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations
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Bandstructure of GaAs
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Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations
26Parabolic Approximations of Bands
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Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations
27Valence Band Effective Mass
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Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations
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Spin-Orbit Coupling
This is known as spin-orbit interaction
![Page 29: Prof.P. Ravindran, - folk.uio.nofolk.uio.no/ravi/cutn/ccmp/9-EffectiveMass1.pdf11 1 00 00 00 L cT T m mm m m m m m LT 0.98 0.19 00 Altogether 6 valleys. Prof.P.Ravindran, Computational](https://reader034.vdocument.in/reader034/viewer/2022042301/5ecc281e0aafd800fe193fc8/html5/thumbnails/29.jpg)
Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations
29Spin-Orbit Coupling in Crystals
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Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations
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Note: Δo increases with Z of the elements in compounds
Value of Valence Band Spin-Orbit Splitting Δo in
semiconductors
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Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations
31C-ZrO2 – Bandstructure and Total DOS
(Relativistic Effects)
![Page 32: Prof.P. Ravindran, - folk.uio.nofolk.uio.no/ravi/cutn/ccmp/9-EffectiveMass1.pdf11 1 00 00 00 L cT T m mm m m m m m LT 0.98 0.19 00 Altogether 6 valleys. Prof.P.Ravindran, Computational](https://reader034.vdocument.in/reader034/viewer/2022042301/5ecc281e0aafd800fe193fc8/html5/thumbnails/32.jpg)
Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations
32C-ZrO2 – Carrier Effective Masses
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Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations
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Effective Masses of ZrO2
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Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations
342D-Dispersion of Heavy and Light Hole Bands
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Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations
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Values of A, B, and C in semiconductors
![Page 36: Prof.P. Ravindran, - folk.uio.nofolk.uio.no/ravi/cutn/ccmp/9-EffectiveMass1.pdf11 1 00 00 00 L cT T m mm m m m m m LT 0.98 0.19 00 Altogether 6 valleys. Prof.P.Ravindran, Computational](https://reader034.vdocument.in/reader034/viewer/2022042301/5ecc281e0aafd800fe193fc8/html5/thumbnails/36.jpg)
Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations
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Effective Mass Values for Some Materials
![Page 37: Prof.P. Ravindran, - folk.uio.nofolk.uio.no/ravi/cutn/ccmp/9-EffectiveMass1.pdf11 1 00 00 00 L cT T m mm m m m m m LT 0.98 0.19 00 Altogether 6 valleys. Prof.P.Ravindran, Computational](https://reader034.vdocument.in/reader034/viewer/2022042301/5ecc281e0aafd800fe193fc8/html5/thumbnails/37.jpg)
Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations
37Spin-Orbit Coupling in Si from First Principle band structure
Calculation
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Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations
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Valence Band Effective Mass in Si
![Page 39: Prof.P. Ravindran, - folk.uio.nofolk.uio.no/ravi/cutn/ccmp/9-EffectiveMass1.pdf11 1 00 00 00 L cT T m mm m m m m m LT 0.98 0.19 00 Altogether 6 valleys. Prof.P.Ravindran, Computational](https://reader034.vdocument.in/reader034/viewer/2022042301/5ecc281e0aafd800fe193fc8/html5/thumbnails/39.jpg)
Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations
39First-Principle Band Structure of CdTe (Thinfilm Solar Cell
Material)
![Page 40: Prof.P. Ravindran, - folk.uio.nofolk.uio.no/ravi/cutn/ccmp/9-EffectiveMass1.pdf11 1 00 00 00 L cT T m mm m m m m m LT 0.98 0.19 00 Altogether 6 valleys. Prof.P.Ravindran, Computational](https://reader034.vdocument.in/reader034/viewer/2022042301/5ecc281e0aafd800fe193fc8/html5/thumbnails/40.jpg)
Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations
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Valence band Effective Mass in CdTe
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Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations
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S.Karazhanov, P.Ravindran et al PRB (2006)
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Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations
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P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 15 March 2010 Carrier effective mass calculations
Band structure and density of states for orthorhombic AlH3
AlH3 ZnO
0.27m0(F,Z) 2.74m0, ( || A)
2.27m0, ( || A)
0.32m0 (L) 2.74m0, ( A)
0.35m0 ( A)
Topmost valence band and bottommost conduction band are well
dispersive
MgAl forms a shallow acceptor band gap.
S.Karazhanov, P.Ravindran et al. (2008)
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Kohn-Sham (solid lines) and
quasiparticle (dashed lines)
band structures calculated with SIC
pseudopotentials for zinc blende InN.
The In
4d electrons are frozen into the core. The
lattice constant a0=4.9670 Å.
FURTHMÜLLER et al. PHYSICAL
REVIEW B 72, 205106 2005
Eg = 0.59 (0.67) eV
Electron effective mass: m*=0.048m0
Effective Mass in InN (From pseudopotential calculation)
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Summary:
The carrier effective mass in semiconductors can be derived from the band
structure obtained from abinitio density functional Calculations.
The effects from spin-orbit coupling is important to understand the carrier
mobility in solar cell materials with high atomic number.
Parameters important to improve solar cell efficiencies such as Spin-Orbit
Coupling and Effective Masses can be predicted accurately by theory.