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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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

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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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

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

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

12Measurement of Effective Mass

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.

Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations

Band structure of Si and GaAs

Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations

Fermi surfaces of conduction electrons in Si and GaAs

Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations

16Bandstructure of Ge

Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations

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Four Valleys Inside BZ for Germanium

Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations

18Measurement of Effective Mass

Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations

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Conduction Band Effective Mass

Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations

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E-k Diagram, Velocity and Effective Mass

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.

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?

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

Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations

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Sign of Effective Mass

Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations

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Bandstructure of GaAs

Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations

26Parabolic Approximations of Bands

Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations

27Valence Band Effective Mass

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

Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations

29Spin-Orbit Coupling in Crystals

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

Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations

31C-ZrO2 – Bandstructure and Total DOS

(Relativistic Effects)

Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations

32C-ZrO2 – Carrier Effective Masses

Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations

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Effective Masses of ZrO2

Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations

342D-Dispersion of Heavy and Light Hole Bands

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

Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations

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Effective Mass Values for Some Materials

Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations

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

Calculation

Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations

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Valence Band Effective Mass in Si

Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations

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

Material)

Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations

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Valence band Effective Mass in CdTe

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)

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)

Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations

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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)

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

47Effect of Strain on the Band Structure

Prof.P.Ravindran, Computational Condensed Matter Physics, Spring 2015 Carrier effective mass calculations

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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.