Prof.P. Ravindran, - folk.uio.nofolk.uio.no/ravi/cutn/ccmp/9-EffectiveMass1.pdf11 1 00 00 00 L cT T...

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

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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

https://en.wikipedia.org/wiki/Group_velocity#cite_note-nemirovsky2012negative-1

https://en.wikipedia.org/wiki/Envelope_(waves)

https://en.wikipedia.org/wiki/Wave_packet


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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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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


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12Measurement of Effective Mass


Reciprocal lattice of a face-centred cubic lattice

Brillouin boundaries for a face-centred cubic lattice.


Band structure of Si and GaAs


Fermi surfaces of conduction electrons in Si and GaAs


16Bandstructure of Ge


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Four Valleys Inside BZ for Germanium


18Measurement of Effective Mass


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Conduction Band Effective Mass


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E-k Diagram, Velocity and Effective Mass


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.


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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?


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Concept of a Hole in an Otherwise Filled Band


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Sign of Effective Mass


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Bandstructure of GaAs


26Parabolic Approximations of Bands


27Valence Band Effective Mass


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Spin-Orbit Coupling

This is known as spin-orbit interaction


29Spin-Orbit Coupling in Crystals


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Note: Δo increases with Z of the elements in compounds

Value of Valence Band Spin-Orbit Splitting Δo in

semiconductors


31C-ZrO2 – Bandstructure and Total DOS

(Relativistic Effects)


32C-ZrO2 – Carrier Effective Masses


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Effective Masses of ZrO2


342D-Dispersion of Heavy and Light Hole Bands


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Values of A, B, and C in semiconductors


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Effective Mass Values for Some Materials


37Spin-Orbit Coupling in Si from First Principle band structure

Calculation


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Valence Band Effective Mass in Si


39First-Principle Band Structure of CdTe (Thinfilm Solar Cell

Material)


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Valence band Effective Mass in CdTe


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S.Karazhanov, P.Ravindran et al PRB (2006)


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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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47Effect of Strain on the Band Structure


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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.

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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