Vibrational Normal Modes

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16 Νοε 2013 (πριν από 3 χρόνια και 8 μήνες)

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Vibrational Normal Modes

or

“Phonon” Dispersion Relations

in Crystalline Materials

“Phonon” Dispersion Relations

in Crystalline Materials


So far, we’ve discussed results for the

Phonon


Dispersion
Relations

ω(k)

(or
ω(q)
) only in model, 1
-
dimensional lattices.


Now, we’ll have a
Brief Overview

of the
Phonon
Dispersion Relations

ω(k)

in real materials.


Both experimental results & some of the past theoretical
approaches to obtaining predictions of
ω(k)

will be discussed.


As we’ll see, some past “theories” were quite complicated in
the sense that they contained
N (N >> 1)

parameters which were
adjusted to fit experimental data. So, (my opinion)

They were really models &
NOT

true theories
.


As already mentioned, the modern approach is to solve the
electronic problem first, then calculate the force constants for the
lattice vibrational predictions by taking 2
nd

derivatives of the total
electronic ground state energy with respect to the atomic positions.


Part I



This will be a
general discussion

of
ω(k)

in crystalline
solids, followed by the presentation of some representative
experimental results for
ω(k)

(obtained mainly in neutron
scattering experiments) for several materials.

Part II



This will be a
brief survey of various

Lattice Dynamics
models
, which were used in the past to try to understand the
experimental results.


As we’ll see, some of these models were
quite complicated

in
the sense that they contained
LARGE NUMBERS

of adjustable
parameters which were fit to experimental data.


The modern method is to
first solve the electronic problem
.
Then, the force constants which for the vibrational problem are
calculated by taking various 2
nd

derivatives of the electronic
ground state energy with respect to various atomic displacements.

Two Part Discussion

The Classical Vibrational Normal Mode Problem

(in the Harmonic Approximation)

ALWAYS

reduces to solving:

Here,
D(q)

The Dynamical Matrix

D(q) ≡
The spatial Fourier Transform of the

“Force Constant” Matrix
Φ

q


wave vector,
I ≡

identity matrix

ω
2


ω
2
(q)

vibrational mode eigenvalue

NOTE
!



There are, in general, 2 distinct types of vibrational
waves
(2 possible wave
polarizations
)
in solids:

Longitudinal


Compressional
:

The vibrational amplitude is
parallel

to the wave propagation direction.

and

Transverse


Shear
:

The vibrational amplitude is
perpendicular

to the wave propagation direction.


For each wave vector

k
,
these 2 vibrational
polarizations will give

2 different solutions

for

ω(k)
.


We also know that
there are
,
at least
,
2 distinct
branches of

ω(k)

(2 different functions
ω(k)

for each
k
)

The Acoustic Branch


This branch received it’s name because it
contains long wavelength vibrations of the
form
ω = v
s
k
, where
v
s

is the velocity of
sound. Thus, at long wavelengths, it’s
ω
vs.
k

relationship is identical to that for
ordinary
acoustic

(sound)
waves

in a medium like air.

The Optic Branch

Discussed on the next page:


The Optic Branch


This branch is always at much higher frequencies than
the acoustic branch. So, in real materials, a probe at
optical frequencies is needed to excite these modes.


Historically, the term

Optic


came from how these
modes were discovered. Consider an ionic crystal in
which atom 1 has a positive charge & atom 2 has a
negative charge. As we’ve seen, in those modes, these
atoms are moving in opposite directions.
(So, each unit cell
contains an oscillating dipole.)

These modes can be excited
with optical frequency range electromagnetic radiation.


We’ve already seen that the
2 branches

have
very different vibrational frequencies
ω(k)
.

So, when discussing the vibrational frequencies
ω(k)
,

it is necessary to distinguish between

Longitudinal & Transverse Modes

(
Polarizations
)

&

At the same time

to distinguish between

Acoustic & Optic Modes.


So,
there are four distinct kinds of modes

for
ω(k)
.


The terminologies used, with their abbreviations are:

Longitudinal Acoustic Modes



䱁⁍
潤敳

Transverse Acoustic Modes



呁⁍
潤敳

Longitudinal Optic Modes



䱏⁍
潤敳

Transverse Optic Modes



呏⁍
潤敳

The vibrational amplitude is highly exaggerated!

A Transverse Acoustic Mode for the Diatomic Chain


The type of relative motion illustrated here carries over
qualitatively

to real three
-
dimensional crystals.

This figure illustrates the case in which
the lattice has

some ionic character
, with + &
-

charges alternating:

A Transverse Optic Mode for the Diatomic Chain


The type of relative motion illustrated here carries over
qualitatively

to real three
-
dimensional crystals.

The vibrational amplitude is highly exaggerated!

This figure illustrates the case in which
the lattice has

some ionic character
, with + &
-

charges alternating:

Polarization & Group Velocity

Frequency,


Wave vector, K

0

(p
/a)

Vibrational Group


Velocity
:

dK
d
v
g


Speed of Sound
:

dK
d
v
K
s

0
lim


A crystal with 2 atoms or more per unit cell

will
ALWAYS

have
BOTH

Acoustic & Optic Modes
.

If there are
n atoms per unit cell

in 3 dimensions,

there will
ALWAYS

be
3

Acoustic Modes &
3n
-
3

Optic Modes
.


Acoustic
Modes

Lattice Constant,

a

x
n

y
n

y
n
-
1

x
n+1

Polarization

Frequency,


W慶攠癥捴crⰠ
K

0

p




TO

Optic

Modes

LA & LO

TA & TO

For 2 atoms per unit

cell in 3 d, there are a

total of
6 polarizations

The transverse modes

(
TA & TO
) are often

doubly degenerate, as

has been assumed in

this illustration.


LA

TA

Acoustic

Modes

Direct: FCC
Reciprocal: BCC

1
st

Brillouin Zones
:

For the FCC, BCC, & HCP Lattices

Direct: HCP

Reciprocal: HCP

(rotated)

Direct: BCC

Reciprocal: FCC

1
st

Brillouin Zone

of
FCC Lattice


Direct Lattice

Reciprocal Lattice


Measured

Phonon Dispersion Relations in
Si

(Inelastic, “Cold” Neutron Scattering)

1
st

BZ for the

Si

Lattice

(diamond; FCC,
2 atoms/unit cell)

Normal Mode Frequencies


⡫(


Plotted for
k

along high symmetry
directions in the
1
st

BZ
.

k

ω

Normal Modes of Silicon

L = Longitudinal, T =

Transverse

O =

Optic, A =

Acoustic

1
st

BZ for the

GaAs

Lattice

(zincblende; FCC, 2
atoms/unit cell)

ω

k

Theoretical

(?)

Phonon Dispersion

Relations in
GaAs

Normal Mode Frequencies


⡫(


Plotted for
k

along high symmetry
directions in the
1
st

BZ
.


For
Diamond Structure

materials, such as
Si
, &
Zincblende
Structure

materials, such as
GaAs
,
for each wavevector
q
, there are

6 branches

(modes) to the

“Phonon Dispersion Relations”

ω
(q)


For
Diamond Structure

materials, such as
Si
, &
Zincblende
Structure

materials, such as
GaAs
,
for each wavevector
q
, there are

6 branches

(modes) to the

“Phonon Dispersion Relations”

ω
(q)


These are:

3
Acoustic Branches

1
Longitudinal

mode:
LA branch

or
LA mode

+ 2
Transverse

modes:

TA branches

or
TA modes

In the acoustic modes,
the atoms vibrate

in phase with their neighbors
.




For
Diamond Structure

materials, such as
Si
, &
Zincblende
Structure

materials, such as
GaAs
,
for each wavevector
q
, there are

6 branches

(modes) to the

“Phonon Dispersion Relations”

ω
(q)


These are:

3
Acoustic Branches

1
Longitudinal

mode:
LA branch

or
LA mode

+ 2
Transverse

modes:

TA branches

or
TA modes

In the acoustic modes,
the atoms vibrate

in phase with their neighbors
.


and

3
Optic Branches

1
Longitudinal

mode:
LO branch

or
LO mode

+ 2
Transverse

modes:
TO branches

or
TO modes

In the optic modes,
the atoms vibrate

out of phase with their neighbors
.

Pb

Cu

1
st

BZ for the

FCC

Lattice

Measured

Phonon Dispersion Relations in

FCC Metals

(Inelastic, “Cold” Neutron Scattering)

1
st

BZ for the

FCC

Lattice

Unit Cell for

the FCC

Lattice

Al

Measured

Phonon Dispersion Relations in

FCC Metals

(Inelastic X
-
Ray Scattering)

Measured

Phonon Dispersion Relations for C in the

Diamond Structure
(Inelastic X
-
Ray Scattering)

1
st

BZ for the

Diamond

Lattice



L

1
st

BZ for the

Diamond

Lattice

Measured

Phonon Dispersion Relations for Ge in the

Diamond Structure
(Inelastic “Cold” Neutron Scattering)

Measured

Phonon Dispersion Relations for KBr

in the NaCl Structure
(FCC, 1 Na & 1 Cl in each unit cell)

(Inelastic, “Cold” Neutron Scattering)



L

1
st

BZ for the

Diamond

Lattice

Measured

&
Calculated

Phonon Dispersion Relations

for
Zr in the BCC Structure

(Inelastic, “Cold” Neutron Scattering)

Data Points,
2 Different Theories: Solid & Dashed Curves)

1
st

BZ for the

BCC

Lattice

Models for Normal Modes
ω(k)

in 3 Dimensions

Outline

of Calculations with

Newton’s 2
nd

Law Equations of Motion



2
,
( ) ( )
1
( ) ( )
2
n i n i n i n i
n i n i n i n i n i n i m j
n i n i m j
n i n i m j
r s r s
r s r s s s s
r r r
   
      
  
  
 
 
   
    
  
 
Assuming the
Harmonic Approximation

N unit cells
, each with

n atoms
means that there are



3Nn Coupled Newton’s 2
nd

Law Equations of Motion

0
0

(r)



Interatomic Potential


s


Displacements from Equilibrium


In the
harmonic approximation
, expand



楮⁡

Taylor’s series of displacements
s
about

the

equilibrium positions.

Cut off the series

at the

term that is quadratic in the displacements.

The following illustrates this procedure:


n
th

unit cell




Lattice Dynamics in 3 Dimensions
-

Outline Calculations
of
ω(k)

in the
Harmonic Approximation





(r)



Interatomic Potential

s


Displacements from Equilibrium

Expand


in a Taylor’s series in displacements
s

about
equilibrium. Keep only up to quadratic terms:


n
th

unit cell










Force Constant

Matrix




2
2
,
( )
( )
1 1
2 2
m j
n i n i
n i
n i m j
m j
n i n i
n i n i n i n i m j
n i n i m j
n i
m j
n i n i m j
m j
r s
r r
r s
F H M s s s
s
M s s

 

 

 
     
  


   



 
 
 
 
     

  
 


Analogous to 1 d

F =
-
(d

⽤砩




Hamiltonian

in the
Harmonic Approx.

Resulting
Newton’s 2
nd

Law

Equation of Motion

2
,
( ) ( )
1
( ) ( )
2
n i n i n i n i
n i n i n i n i n i n i m j
n i n i m j
n i n i m j
r s r s
r s r s s s s
r r r
   
      
  
  
 
 
   
    
  
 
N unit cells
, each with

n atoms
means that there are



3Nn Coupled Newton’s 2
nd

Law Equations of Motion

Force Constant Matrix Properties

are analogous to elastic coefficients
m j
n i n i m j
m j
m j
n i
M s s mx kx
k

   



     


( )
0
fromtranslational invariance
0
m j n j
n i m i
m j m n j
n i i
m j
n i
m
 
 
 
 




  
  
 

Analogous to the 1d


Harmonic Oscillator

Analogous to the 1 d

Spring Constant

Various symmetries of the


Force Constant Matrix

Schematic view

of
the lattice.



Formally Solve the Equations of Motion



Use a Spatial Fourier Series Approach

( )
1
( ) ( ) ( )
n
i t
i
n i i n i n i
s u e T s e s
M

   



  
qr
qa
a
q q q
( )
2
( )
( )
0
1
( ) ( )
1 1
dynamical matrix (does not depend on )
n m
p
n m
i
m j
i n i j
m j
i
i
j m j p j
i n i i
m p
j
i n
u e u
M M
D e e
M M M M
D

  

 
  
  
   





 
   


 
q r r
q r
q r r
q q
r
(

2 2
( ) ( ) ( ) 0
j j j
i i j i i j
j j
u D u D u
  
     
 
 

   
 
q q q
(

2 2
det 0 for each:eigenvalues ( )
s
d r
 
   
D(q) I q q
After some work, the equations of motion become:

So, the mathematics of

All of

FORMAL Lattice Dynamics

can

be summarized

as finding solutions to





The remainder is the use of various models
& theories for the
“force constants”

which
enter the force constant matrix
Ф

& thus
the dynamical matrix
D
.


There are many different models & theories

which were designed to determine the force
constants which enter the dynamical matrix
D
.
These can broadly be divided into 4 groups:




1.

Force Constant Models




2.

Shell Models




3.

Bond Models




4.

Bond Charge Models


Within each group, there are
MANY

variations
on these models
!


Going down the list:

The models get more complex &
(in my opinion) harder to understand in terms of the
physics behind them.


Common Features of All Models

(or Theories):


1.

All
model the ion
-
ion interactions

with some

parameters
in the force constant matrix

.


2.

All find these parameters by
fitting to various

experimental quantities
.

A few of the
many quantities

used to do the fitting are:


Bulk Modulus; Shear Modulus; BZ center LO, TO, LA,
& TA frequencies; BZ edge LO, TO, LA, TA frequencie

+ Many Others


Since the goal was to explain neutron scattering data, people
tried to
use non
-
neutron scattering data

to fit the parameters.


3.

All
used the

fitted parameters

in the matrix


瑯t

compare to neutron scattering data & to predict



results of neutron scattering experiments.

Force Constant Models


These models are the
crudest approach

taken &
the closest
in spirit & actual calculations

to the 1d models we discussed.


They model the force constant matrix


with as few
parameters as possible & fit to data mentioned
.


Assumption
:

The atoms (the ion core + valence electrons) are
HARD SPHERES
,

coupled by “springs”, characterized
by spring constants (~ like the 1d models)


They include
short range forces only
. But have
no
Coulomb forces
!


There are various types of “springs”:


1
st
, 2
nd
, 3
rd
, 4
th
, 5
th
, …
neighbor coupling!!


The spring forces have directional dependences, with

different spring constants for

coupling in different directions
.


The
“best” force constant models

require
12 to
20

DIFFERENT

force constants

per material!

A Rhetorical Question:

Is this physically reasonable & satisfying
?


Such models give good

(q)

for the

Group IV
covalent solids
:

C (diamond), Si, Ge,
α
-
Sn


But, they
FAIL

for many covalent & ionic compounds, such as


The III
-
V & II
-
VI materials, GaAs, CdTe, etc.


This happens because

Coulomb (ionic) forces are ignored
!


Also,
the bonds in these compounds are
partially ionic

(there is a charge separation).

A Rhetorical Question
!!

Is a 15 to 20 adjustable parameter “theory”

REALLY A THEORY
?



A quote in several references:

“The parameters are not easily understood
from a physical point of view.”


(In my opinion, this is putting it mildly!)


Often, these models need up to

5
th

& 6
th

neighbor (or higher)

force constants!



A
physically realistic qualitative expectation

for relative size of the force constants
connecting neighbors at various distances is:

The force constant size should decrease
as the distance increases
.


However, it’s been found that, in order to get a good
fit to data, some of these models require instead that
the size of some force constants must increase with
increasing distance
!!

For example:

Φ
4nn
>
Φ
1nn


& other,
absurd, completely unphysical results
!


In addition, no matter how many force constants are
assumed,
these models cannot explain a lot of data
!


For example, the flattening of

TA

near the BZ edges.


Often, these models were found to
work ok

for
purely covalent solids
like

C (diamond), Si, Ge,…


but to do a
poor job on
ionic compounds

in which
Coulomb Effects

are important!


To deal with these problems, “better” theories or
models were introduced. One such group of
models is called

The Shell Models

Shell Models


The
force constant models all
assume “hard sphere” atoms

(ion core + valence electrons).
From our discussion of bonding &
from electronic properties studies, we know that this is a

Very BAD

assumption for covalently bonded
solids as well as for many other solid types!


Our knowledge of bonding & electronic properties tells that:

The valence electrons are
NOT

rigidly attached to the ions!

The Main Idea of the Shell Model
:

Each atom is modeled as a rigid ion core plus an
“independent” valence electron shell.


Also, the valence electron shell AND the ion core can move
.

That is, the
Atoms are
Deformable
!

So, in the extensions of the force constant
models to the Shell Models,

the atoms are deformable
!

That is,

The ions & valence electron shells
are all moving
.


Also,
Coulomb Interactions are included by putting
charges on the shells & the ion cores
.


In these models, the atomic displacements induce dipole
moments on the atoms.

So, there are dipole
-
dipole interactions between
unit cells as well as force constants to couple
the cells.

Best Shell Model results

for

(q)


Ge
-

A good fit to neutron data is found with only
5 parameters
!


GaAs & other Compounds
-


A good fit to neutron data is found with

~
10
-

12 parameters
.


That is, the combined force constant shell model
doesn’t
do much better than pure force constant models
!

Physics Criticisms


1.

The valence electrons in covalent materials
are
NOT


in the shells around the ion cores!


2.

The valence electrons in these materials
ARE

in the

covalent bonds between the cores!


3.

The fitting parameters are ~unphysical & have limited

use for modeling properties other than

(q).

Other Physics Criticisms


These models make an
artificial division

of valence electrons between atoms which
are covalently bonded together.


Actually, these valence electrons are
shared

in the covalent bonds!


So, people introduced “better” theories
or models, such as the
Bond Models
.



In covalent materials, the valence electrons are in
the covalent bonds
between

the atoms & along
the directions from an atom’s near neighbors.


Bond models:

Extend the
“Valence Force
Field” Method

to covalent solids.


Valence Force Field Method

(VFFM):


Used in theoretical molecular chemistry to
explain vibrational properties of covalent
molecules.


In this model, the vibrations are analyzed in terms
of “valence forces” for bond stretching &
bending.

Bond Models

VFFM Advantages
:


The force constants for bond stretching &
bending are ~ characteristic of particular
bonds & are transferable from one molecule
to another, which contains same bond (e.g.
the force constants for a
C
-
C

bond are ~ the
same no matter what solid it is in!

Bond Models
:


The force constants are ~ the same for an
A
-
B

bond in a solid as they are in molecules
.





Extension of the VFFM
to covalent solids,
2 atoms / unit cell.


The bond potential energy
V

is expanded
about equilibrium positions for all possible
degrees of freedom of bending, stretching,
etc. of bond.


Expansion stopped at 2nd order in
deviations from equilibrium.

Simple harmonic oscillators in all
degrees of vibrational freedom!


Disappointment
!



Despite the greater physical appeal &
(hopefully) the better physical realism of
such models, to get good fits to
ω
(q)

(neutron scattering data), the bond models
need ~ a similar number of parameters as
the shell models!

So, after all the work on the Bond
Models,
it turns out that there is no real
advantage of them over the shell models!

The Keating Model
≡ The
VFFM with 2 or
3 parameters + a charge parameter.


Good for elastic properties at long wavelengths (later).


BAD for frequencies
!

Other models &
extensions of the VFFM:

5 or 6 parameters


Often do well for trends in frequencies &
BAD

for other vibrational properties
(like elastic properties)!


So, people introduced “better” theories
(models) like the

Bond Charge Models


The most difficult part of modeling the force
constant matrix is accurately including the long
-
range (Coulomb) electron
-
ion interaction.


The Shell Models + charges:
Attempts
to simulate this. However, fails to account for
dielectric
screening
. Also, for covalent bonds,
charge is not on atoms, but between them!


The Bond Models:
Account for covalent
bonding, but neglects Coulomb screening.

Bond Charge Models

Review of Screening:


Look at a specific ion at origin.


Let the Coulomb interaction with one electron



V
o
(r).


But, the presence of other electrons reduces this:


The presence of all other charges (ions & electrons) near
ion of interest causes effective interaction to be reduced
.


It is shown in EM courses that the true potential is of form

V(r)


V
o
(r)exp(
-
r/r
o
)


Usually, this is simulated in simpler way:

V(r)


V
o
(r)/
ε

Here,
ε

=

dielectric constant


Screening in classical E&M:

V(r)


V
o
(r)/
ε

ε

=
dielectric constant:

Really,
ε

=
ε
(q,
ω
)

but neglect this.


This is too simple to work well for vibrational


spectra! Reason: the implicit assumption is that


valence electrons are “free” , except for Coulomb


interactions with ion of interest.


We could treat Coulomb effects (ion with charge
Ze
) by

V(r)


-
⡚(
2
)/(
ε
r),


but this is too crude.


Instead,
localize some of valence electrons on the
bonds
,

so that some screen in this way & others don’t.

Bond Charge Model


A portion of valence electron charge is
localized on bonds (between atoms).


Another portion is “free” & contributes to
screening.


The portion which contributes to screening
is an empirical, adjustable parameter.

NOTE
!


This is a model! Don’t it take too seriously or
literally. It is designed to simulate actual
effects.
It’s physical significance is questionable.


In this model, the valence electrons are
divided into two parts:


1. “Free” charge
which contributes to screening


2. Localized charge

in the bonds between atoms


The fraction of the valence electron charge which
is localized on bonds
is an adjustable parameter
,


The
bond charge fraction Z
b

is defined

by

Z
b
e


-
㉥2
ε


This is the theory definition.

In practice,
Z
b

is an
adjustable parameter


The
bond charge fraction
:

Z
b
e


-
㉥2
ε


When
Z
b

is determined, often it is found that

Z
b
e < 1.0e

!


Don’t take this too seriously!! Remember that

It is a crude MODEL
!



In addition, there must be spring
-
like force
constants for coupling between the ions.


Bond Charge Model Results for Si

Z
b
e


〮0㔠攠††
㰠eㄮ1攩


This actually compares favorably to the
Si

charge density calculated by state of
the art electronic structure
(pseudopotential) codes.


It also compares favorably with X
-
Ray
experiments on the
Si

charge density.


Both show a build
-
up of


0.4 e
per
bond between

Si
atoms!

Bond Charge Model Results for Si

Z
b
e


〮㌵⁥†††⠼†ㄮづ


The 4 valence electrons from each atom are divided into


1.
Bond Charges

localized on each bond

Z
b
e


〮㌵⁥
.



Point charges are assumed. In reality the bond

charge is spread out over a volume.


2. “
Free

Charges

which screen.


Each
Si

contributes
Z
b
e/2
to the bond charge
and
(4
-
2Z
b
)e
to the free charge.


In addition, this has to be combined with force
constants to couple ions.



The Bond Charge Model

combines the
“best” of force constant, shell, & bond models!


A further refinement:

Adiabatic Bond Charge Model

(ABCM).


Allows bond charges to follow motion
of ions so that they are not located
exactly in middle of bond.


Gets good
ω
(q)
for
Si

& other materials
with
only 4 parameters
!