In This Lecture - Bilkent Universitybulutay/573/notes/ders_2.pdf · C. Bulutay Topics on Semiconductor Physics Lecture 2 Electronic Bandstructure all-electron (FPLAPW) valence electron

C. Bulutay Lecture 2Topics on Semiconductor Physics

� Electronic Bandstructure: General Info

In This Lecture:


Electronic Bandstructure

all-electron

(FPLAPW)

valence electronPAW

Acronyms

FPLAPW: Full-potential linearized

augmented plane wave

PAW: Projector augmented wave

LMTO: Linearized muffin tin orbital

EPM: Empirical pseudopotential

method

ETB: Empirical tigh-binding

ab-initio semi-empirical

planewave pseudopotential

(FP) LMTO EPM ETB k·p


Perfect Crystal Hamiltonian (cgs units)

2 2 2 22' ''

, ' , ' ,''

1 1

2 2 2 2

j i i i i

j j j i i i i jj i i ij j j i

p P e Z Z e ZeH

m M R Rr r r R= + + + −

−− −∑ ∑ ∑ ∑ ∑� � ��

over all e’s e-nucleiover all nuclei

� 1st Approximation: core vs. valence e’s

Still Eq. Above Applies with:

core e’s+nucleus

e’s

ion core

valence e’s

semicore


� ions are much heavier (> 1000 times) than e’s

So,

for e’s: ions are essentially stationary (at eql. lattice sites {Rj0})

� 2nd Approximation: Born-Oppenheimer or adiabatic approx.

for ions: only a time-averaged adiabatic electronic potential is seen

In other words,

using the adiabatic approximation, we separate the (in principle

non-separable) perfect crystal Hamiltonian


Under adiabatic approximation…

( ) ( ) ( )0, ,ion i e j i e ion j i

H H R H r R H r Rδ−= + +� � ��

e-phonon interaction (resistance, superconductivity…)phonon spectrum electronic

band structure

Ref: Yu-Cardona


Electronic Hamiltonian

2 22'

, ' ,' 0

1

2 2

j ie

j j j i jj j j j i

p e ZeH

m r r r R= + −

− −∑ ∑ ∑ ��

over all valence e’s >1023 cm-3

� 3rd Approximation: Mean-field Approximation

2

1 ( ); ( ) ( )2

e

pH V r V r R V r

m= + + =

��

Density Functional Theory

VH+Vx+Vc

A direct lattice vector


Translational Symmetry & Brillouin Zones

( ) ( ); : Translational Operator

Introduce a function, ( ) ( ), where ( ) ( ) with

( ) ( ) ( )

R R

ikx

k k k k

ikR

R k k k

T f x f x R T

x e u x u x nR u x n

T x x R e x

ψ

ψ ψ ψ

≡ +

≡ + = ∈

= + =

ℤ

eigenvalues of T

[ ]1

1

Since , 0,

Eigenfunctions of can be expressed also as

e R

e

H T

H

=

eigenfunctions of

2 2: is only defined modulo , i.e., and are equivalent

RT

nNB k k k

R R

π π+

eigenvalues of TR


ax

Real space

Direct Lattice

Momentum space

Reciprocal lattice

k

One-dimensional Lattice

k=-π/a k=π/a

1st BZ

[Real Space] Primitive lattice vector: a

[Mom. Space] Primitive lattice vector: 2π/a


Three-dimensional Lattice

1 2 3

1 1 2 2 3 3

1 2 3

[Real space] , ,

A general direct lattice vec

Prim

tor (DLV): ;

[Mom. space] , ,

itive lattice vectors:

Primitive lattice vectors:

i

j k

a a a

R n a n a n a n

b b b

a a

= + + ∈

×

� � �

� � � �ℤ

� � �

� �� Volume of the real

1

where, 2(

j k

i

a ab

aπ

×≡

×

� ��

� �2 3

1 1 2 2 3 3

)

A general reciprocal lattice vector (RLV): ;

: 2

i

i j ij

a a

G n b n b n b n

NB b a πδ

⋅

= + + ∈

⋅ =

�

� � ��ℤ

� �

Volume of the real space primitive cell

1st Brillouin zone is the Wigner-Seitz cell of the reciprocal lattice


Direct vs. Reciprocal Lattices

1ˆ

ˆ

a a x

a a y

=

=

�

�

1

2ˆ

2

b xa

π

π

=�

�

� SC

Direct Lattice Reciprocal Lattice

also a SC lattice!2

3

ˆ

ˆ

a a y

a a z

=

=

�

� 2

3

2ˆ

2ˆ

b ya

b za

π

π

=

=

�

�

also a SC lattice!Primitive translation

vectors

1st BZ of SC lattice is again a cube


� BCCDirect Lattice Reciprocal Lattice

forms an fcc lattice!

1

2

3

ˆ ˆˆ( )2

ˆ ˆˆ ( )2

ˆ ˆ ˆ ( )2

aa y z x

aa z x y

aa x y z

= + −

= + −

= + −

�

�

�

1

2

3

2ˆ ˆ( )

2ˆˆ ( )

2ˆ ˆ ( )

b y za

b z xa

b x ya

π

π

π

= +

= +

= +

�

�

�

a is the side of the conventional cube

1st BZ of bcc lattice


Ref: Kittel


� FCCDirect Lattice Reciprocal Lattice

forms an bcc lattice!


1

2

3

ˆ ˆ( )2

ˆˆ ( )2

ˆ ˆ ( )2

aa y z

aa z x

aa x y

= +

= +

= +

�

�

�

1

2

3

2ˆ ˆˆ( )

2ˆ ˆˆ ( )

2ˆ ˆ ˆ ( )

b y z xa

b z x ya

b x y za

π

π

π

= + −

= + −

= + −

�

�

�

1st BZ of fcc lattice


Ref: Kittel


fcc 1st BZ Cardboard Model

Truncated Octahedron

Ref: Yu-Cardona

Ref: Wikipedia


More on Symmetry

� The star of a k-point

All have the same energy eigenvalues

Symmetry of

the direct latticeSymmetry of the

reciprocal lattice

� Wavefunctions can be expressed in a form such that they have definite

transformation properties under symmetry operations of the crystal

Selection rules: certain matrix elements of certain operators vanish

identically…

� Formal analysis is remedied by the use of Group theory


Symmetry Points & Plotting the Band Structure

Diamond BZ

0

5

EPM Bandstructure of Si

-15

-10

-5

KLWX ΓΓ

En

erg

y (

eV

)


Bloch Functions vs Wannier Functions

�Felix Bloch provided the important

theorem that the solution of the

Schroedinger equation for a periodic

potential must be of the special form:

Cell-periodic functions Orthonormality:* ( ) ( )

nnnk n k kkr r drψ ψ δ δ′′ ′ ′

=∫ � � ��

( ) ( ), with ( ) ( ),ik r

nk nk nk nkr e u r u r u r Rψ ⋅= = +

� �

� � � �

��

Transformation relations

Wannier functions

all space

( ) ( ) nnnk n k kk

r r drψ ψ δ δ′′ ′ ′=∫

is the crystal momentum (more on this later)

1( ; ) ( ),

1( ) ( ; )

i

i

i

nk nk nk nk

ik R

n i nkk

ik R

n inkR

k

a r R e rN

r e a r RN

ψ

ψ

− ⋅

⋅

=

=

∑

∑

� �

�

�

� �

�

�

�ℏ

��

��


Bloch vs. Wannier Functions

xBloch functions are extended

x

Wannier fn’s are localized around lattice sites Ri

Wannier form is useful in describing impurities, excitons…

But note that the Wannier functions are not unique!


Crystal Momentum

( ) ( )ik r

nk nkr T e rψ ψ⋅+ =

� �

� �

��

determines the phase factor by which a BF is multiplied under a translation in real space

k�

k�

labels different eigenstates together with the band index n

�k�

is determined up to a reciprocal lattice vector; this arbitrariness can be removed by restricting it to 1st BZ

A typical conservation law in a xtal: k q k G′+ = +� � ��

Any arbitrariness in labelling the BFs can be absorbed in these additive RLVs w/o changing the physics of the process

Physically, the lattice

supplies necessary

recoil momentum so

that linear momentum is exactly conserved


Be ware of the Complex Bandstructure

( )ik r

nke u r

⋅� �

��

“Real” bandstructure of Si

What if we allow k to become complex?

Ref: Chang-Schulman PRB 1982


Complex Bandstructure (cont’d)

Evanescent modes play an important role in low-dimensional structuresThey are required in mode matching at the boundaries etc…

Ref: Brand et al SST 1987

In This Lecture - Bilkent Universitybulutay/573/notes/ders_2.pdf · C. Bulutay Topics on Semiconductor Physics Lecture 2 Electronic Bandstructure all-electron (FPLAPW) valence electron

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