C. Bulutay Lecture 2Topics on Semiconductor Physics

� Electronic Bandstructure: General Info

In This Lecture:

C. Bulutay Lecture 2Topics on Semiconductor Physics

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

C. Bulutay Lecture 2Topics on Semiconductor Physics

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

C. Bulutay Lecture 2Topics on Semiconductor Physics

� 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

C. Bulutay Lecture 2Topics on Semiconductor Physics

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

C. Bulutay Lecture 2Topics on Semiconductor Physics

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

C. Bulutay Lecture 2Topics on Semiconductor Physics

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

C. Bulutay Lecture 2Topics on Semiconductor Physics

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

C. Bulutay Lecture 2Topics on Semiconductor Physics

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

C. Bulutay Lecture 2Topics on Semiconductor Physics

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

C. Bulutay Lecture 2Topics on Semiconductor Physics

� 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

a is the side of the conventional cube

Ref: Kittel

C. Bulutay Lecture 2Topics on Semiconductor Physics

� FCCDirect Lattice Reciprocal Lattice

forms an bcc lattice!

a is the side of the conventional cube

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

a is the side of the conventional cube

Ref: Kittel

C. Bulutay Lecture 2Topics on Semiconductor Physics

fcc 1st BZ Cardboard Model

Truncated Octahedron

Ref: Yu-Cardona

Ref: Wikipedia

C. Bulutay Lecture 2Topics on Semiconductor Physics

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

C. Bulutay Lecture 2Topics on Semiconductor Physics

Symmetry Points & Plotting the Band Structure

Diamond BZ

0

5

EPM Bandstructure of Si

-15

-10

-5

KLWX ΓΓ

En

erg

y (

eV

)

C. Bulutay Lecture 2Topics on Semiconductor Physics

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

�� �

�� �

C. Bulutay Lecture 2Topics on Semiconductor Physics

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!

C. Bulutay Lecture 2Topics on Semiconductor Physics

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

C. Bulutay Lecture 2Topics on Semiconductor Physics

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

C. Bulutay Lecture 2Topics on Semiconductor Physics

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