In This Lecture - Bilkent Universitybulutay/573/notes/ders_7.pdf · C. Bulutay Topics on...

C. Bulutay Lecture 7Topics on Semiconductor Physics

Empirical Pseudopotential Method (EPM)

Pseudopotential-based full zone k.p technique

In This Lecture:

-15

-10

-5

0

5

KLWX ΓΓ

Ene

rgy (

eV

)

EPM Bandstructure of Si


Empirical Pseudopotential Method (EPM)

We start with the effective one-electron crystal Hamiltonian

Effective xtal potentialA direct lattice

vector

2

0

( ); ( ) ( )2

c c c

pH V r V r R V r

m= + + =

Reciprocallattice vectors

Ref: Harrison


Reciprocal and direct lattice vectors are related by

(to be used soon)

(k)

Ref: Harrison

,G Gδ ′Ωxtal volume

Project this to G ,ku ′


This results in an eigenvalue equation for each value of k:

(k)

This is an infinite set! → Truncate the matrix size within a sphere of RLVs

Introduces Energy cut off

This is an infinite set! → Truncate the matrix size within a sphere of RLVs

xtal volume

Ref: Harrison

V(q): a Fourier xform of xtal potential


Hence, this reveals a diagonal kinetic energy contribution:

Next, concentrate on the potential term for which we shall insertan atomistic form that is situated at each atomic site ra:

Ref: Harrison


Geometrical (static) structure factor, S(G-G’)

Separate into

Ref: Harrison

Direct Lattice (Bravais)

within the basis

lattice

basis

crystal structure


N : # primitive cells within the computation volume

(primitive) cell volume

lattice constant

Ref: Harrison

e.g: For FCC lattice


Atomic Basis for Diamond & ZB se/c’s

Ga-

As+

Si

Si

Diamond: same atoms Zinc blende: cation-anion

Ref: Harrison


Pseudopotential Form Factors

Atomic Psp Form Factors

Symmetric Anti-Symm

Psp Form Factors

Symmetric/Anti-Symm Structure Factors

Ref: Harrison


Due to structure factors of Diamond & ZB only few form factors are needed

Here comes the final form of EPM

Ref: Harrison

Form factors are

extracted by

fitting the b/s to

experimental

data


Sometimes as in strained lattices, form factors are needed over all wave vectors

-0.4

-0.2

0.0

0.2

Si

V (

Ryd

berg

s)

0.0 Si

Ge

V (

Ryd

berg

s)

0 2 4 6 8 10 12 14 16

-1.2

-1.0

-0.8

-0.6 C

V (

Ryd

berg

s)

G2 in units of (2π/a)

2

0 2 4 6 8 10 12 14 16

-0.4

-0.2

V (

Ryd

berg

s)

G2 in units of (2π/a)

2

Lowest non-vanishing componentsused for a fixed lattice constant, a


Now from symmetric/antisymmetric back to atomic form factors

Note the change in atomic form factors as extracted from differentcompound band structures. This brings in the issue of the transferabilityof the atomic form factors; one partial remedy is non-local EPM

Ref: Harrison


Charge Densities from EPM

( )

, ,

*

( ) ( ) i G k r

n k n kG

r a G eψ

ρ ψ ψ

+ ⋅=

=

∑

∑

Ref: Harrison

,

*

,,

( ) ( ) ( )n k n k

n k

r r rρ ψ ψ=∑


Lowest RLVs in FCC

Ref: Harrison


EPM for the wurtzite structure

From NRL crystal structure database


For WZ there are four atoms in the basis in contrast to two for ZB.Since the volume per atom is the same, a given volume of reciprocalspace for WZ contains approximately twice as many RLVs.

As in the ZB case the psp for WZ can be written as

( ) ( ) ( ) ( ) ( ),

( ) ( ) ( ), ( ) ( ) ( ),

S A

S f A f

S an cat A an cat

f f f f f f

V G S G V G S G V G

V G V G V G V G V G V G

= +

= + = −

1 2 3

( ) ( ) ( ), ( ) ( ) ( ),

( ) cos 2 cos( ),6 6 4

( ) cos 2 sin( ),6 6 4

where

f f f f f f

S

A

V G V G V G V G V G V G

L M NS G Nu

L M NS G Nu

G Lb Mb Nb

π π

π π

= + = −

= + +

= + +

= + +

see prev. page


Examples on WZ EPM: GaN & AlN

0.0

0.2

0.4

0.6AlN

GaN

GaN, AlN: EPM Form Factors

Va

0 2 4 6

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

AlN GaN

Vs

V (

Ry)

q (in units of 2π/a)

Observe that form factors

rapidly decay as demanded

from a pseudopotential


Examples on WZ EPM: GaN & AlN


Pseudopotential-based full-zone k·p

4- or 8-band k·p result in ever-concave-up or down

bands [unrealistic away from the band edge]

In principle, k·p technique becomes exact as the

number of bands included goes to infinity

The problem there lies in proliferation of Luttinger-like The problem there lies in proliferation of Luttinger-like

parameters which cannot be extracted from

experiments

Wang & Zunger have proposed a pseudopotential

based k·p technique where all desired band interaction

parameters are taken from ab initio/empirical

pseudopotential computations


Formulation

Start with the effective one-e Hamiltonian described by a local potential

we obtain the well-known k.p expression in terms of cell-periodic parts

Based on the Reference:

C. Bulutay, Turkish Journal of Physics, 30, 287 (2006).

Cell-periodic fn’s satisfy a Sturm-Liouville type eigenvalue eq. Hence, they

are complete when all (infinite) bands are considered at a fixed wv, say k0

An arbitrary fn, such as

can be expanded in terms

of this set


This leads to the following eigenvalue eq. which becomes exact as b

N →∞

where

and momentum matrix element

Since we cannot include infinite # bands at a fixed wv, we can modify the

formulation by forming an expansion basis over various bands and wv,


Since cell-periodic fn’s are not necessariy normal at different wv’s we introduce

Then the expansion coefficients satisfy

Projecting to results in a generalized eigenvalue equationProjecting to results in a generalized eigenvalue equation

where

Generalized momentum matrix element

linking indirect transitions


These can readily be either extracted or generated from ab initio DFT

or EPM bandstructure computations:

So, necessary input for multiband k.p:

eigenvectors obtained from EPM/DFTeigenvectors obtained from EPM/DFT

Hence ,in terms of these eigenvectors we can generate


Demonstration on indirect bandgap se/c’s:

Si, Ge, SiC, Diamond


Si: 4-band k·p

Black: EPM, Red: PP-based k·p

The selected expansion states are: from the Γ point we include the band indices

2 to 4 (with index 4 corresponding to valence band maximum), from the X point

band 5 making altogether 4 states.


Si: 8-band k·p

The selected expansion states are all from the Γ point from bands 1 to 8.



Si: Full-Zone with 13 and 15 band k·p

0

10

En

erg

y (

eV

)

0

10

En

erg

y (

eV

)



bands 3 to 6, and from the L point bands 3 and 4, making altogether 13 states.

In the figure on the right we append the first Γ state and the fifth K states to

further improve the overall bandstructure.

-10

Si

13x13

KW LX ΓΓ

En

erg

y (

eV

)

-10

Si

15x15

KW LX ΓΓE

ne

rgy (

eV

)

Blue: EPM, Black symbols: PP-based k·p


Ge: 15-band k·p



bands 3 to 6, and from the L point bands 3 and 4, first Γ state and the fifth K

states making altogether 15 states.



SiC (Cubic Phase: 3C) 15-band k·p

[Same expansion basis set as before]



Diamond: 15-band k·p

AgreementAgreement

over 40 eV

range

[Same expansion basis set as before]


In This Lecture - Bilkent Universitybulutay/573/notes/ders_7.pdf · C. Bulutay Topics on...

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