Vibrational Normal Modesor “Phonon” Dispersion Relations

in Crystalline Materials

“Phonon” Dispersion Relationsin 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 2nd 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 2nd 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 MatrixD(q) ≡ The spatial Fourier Transform of the

“Force Constant” Matrix Φq ≡ wave vector, I ≡ identity matrix

ω2 ≡ ω2(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 give2 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 = vsk, where vs 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 BranchDiscussed 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 betweenLongitudinal & Transverse Modes (Polarizations)

&

At the same time to distinguish betweenAcoustic & Optic Modes.

• So, there are four distinct kinds of modes for ω(k).• The terminologies used, with their abbreviations are:

Longitudinal Acoustic Modes LA ModesTransverse Acoustic Modes TA ModesLongitudinal Optic Modes LO ModesTransverse Optic Modes TO Modes

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 hassome 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 hassome ionic character, with + & - charges alternating:

Polarization & Group VelocityF

req

uen

cy,

Wave vector, K0 /a)

LA Modes

TA Modes

Vibrational Group Velocity:

dK

dvg

Speed of Sound:

dK

dv

Ks

0

lim

A crystal with 2 atoms or more per unit cellwill 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

xn ynyn-1 xn+1

Polarization

Fre

qu

ency

,

Wave vector, K0 /a

LO

TO

Optic Modes

LA & LO

TA & TO

For 2 atoms per unit cell in 3 d, there are a total of 6 polarizationsThe transverse modes (TA & TO) are oftendoubly degenerate, ashas been assumed in this illustration.

LA

TA

Acoustic Modes

Direct: FCC Reciprocal: BCC

1st Brillouin Zones: For the FCC, BCC, & HCP Lattices

Direct: HCPReciprocal: HCP

(rotated)

Direct: BCC Reciprocal: FCC

1st Brillouin Zone of FCC Lattice

Direct Lattice Reciprocal Lattice

Measured Phonon Dispersion Relations in Si(Inelastic, “Cold” Neutron Scattering)

1st BZ for the Si Lattice

(diamond; FCC, 2 atoms/unit cell)

Normal Mode Frequencies (k) Plotted for k along high symmetry

directions in the 1st BZ.

k

ω

Normal Modes of Silicon

L = Longitudinal, T = Transverse O = Optic, A = Acoustic

1st BZ for the GaAs Lattice

(zincblende; FCC, 2 atoms/unit cell)

ω

k

Theoretical (?) Phonon DispersionRelations in GaAs

Normal Mode Frequencies (k) Plotted for k along high symmetry

directions in the 1st 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 Branches1 Longitudinal mode: LA branch or LA mode

+ 2 Transverse modes: TA branches or TA modesIn 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 Branches1 Longitudinal mode: LA branch or LA mode

+ 2 Transverse modes: TA branches or TA modesIn the acoustic modes, the atoms vibrate

in phase with their neighbors.and

3 Optic Branches1 Longitudinal mode: LO branch or LO mode

+ 2 Transverse modes: TO branches or TO modesIn the optic modes, the atoms vibrate

out of phase with their neighbors.

Pb

Cu

1st BZ for the FCC Lattice

Measured Phonon Dispersion Relations in FCC Metals(Inelastic, “Cold” Neutron Scattering)

1st BZ for the FCC Lattice

Unit Cell for the FCCLattice

Al

Measured Phonon Dispersion Relations in FCC Metals(Inelastic X-Ray Scattering)

Measured Phonon Dispersion Relations for C in theDiamond Structure (Inelastic X-Ray Scattering)

1st BZ for the Diamond Lattice

L

1st BZ for the Diamond Lattice

Measured Phonon Dispersion Relations for Ge in theDiamond Structure (Inelastic “Cold” Neutron Scattering)

Measured Phonon Dispersion Relations for KBrin the NaCl Structure (FCC, 1 Na & 1 Cl in each unit cell)

(Inelastic, “Cold” Neutron Scattering)

L1st BZ for the

Diamond Lattice

Measured & Calculated Phonon Dispersion Relationsfor Zr in the BCC Structure (Inelastic, “Cold” Neutron Scattering)

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

1st BZ for the BCC Lattice

Models for Normal Modes ω(k) in 3 Dimensions Outline of Calculations with

Newton’s 2nd Law Equations of Motion

2

,

( ) ( )1( ) ( )

2n i n i n i n i

n i n i n i n i n i n i m jn i n i m jn i n i m j

r s r sr 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 2nd Law Equations of Motion

0

0

(r) Interatomic Potential s Displacements from Equilibrium

In the harmonic approximation, expand in aTaylor’s series of displacements s about theequilibrium positions. Cut off the series at theterm that is quadratic in the displacements.

The following illustrates this procedure:

nth unit cell

Lattice Dynamics in 3 Dimensions - Outline Calculations of ω(k) in the Harmonic Approximation

(r) Interatomic Potentials Displacements from Equilibrium

Expand in a Taylor’s series in displacements s about equilibrium. Keep only up to quadratic terms:

nth unit cell

“Force Constant” Matrix 2

2

,

( )

( ) 1 1

2 2

m j n i n in i

n i m j

m jn i n in i n i n i n i m j

n i n i m jn i

m jn i n i m j

m j

r s

r r

r sF H M s s s

s

M s s

Analogous to 1 dF = -(d/dx)

Hamiltonian in the Harmonic Approx.

Resulting Newton’s 2nd Law Equation of Motion

2

,

( ) ( )1( ) ( )

2n i n i n i n i

n i n i n i n i n i n i m jn i n i m jn i n i m j

r s r sr s r s s s s

r r r

N unit cells, each with n atoms means that there are 3Nn Coupled Newton’s 2nd Law Equations of Motion

Force Constant Matrix Properties

are analogous to elastic coefficients

m jn i n i m j

m j

m jn i

M s s mx kx

k

( )0 from translational invariance

0

m j n jn i m i

m j m n jn i i

m jn 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( ) ( ) ( )ni t i

n i i n i n is u e T s e sM

qr qaaq q q

( )2

( )( )0

1( ) ( )

1 1

dynamical matrix (does not depend on )

n m

pn m

im ji n i j

m j

iij m j p ji n i i

m p

ji n

u e uM M

D e eM M M M

D

q r r

q rq r r

q q

r

2 2( ) ( ) ( ) 0j j ji i j i i j

j j

u D u D u

q q q

2 2det 0 for each : eigenvalues ( )sd r D(q) I q q

After some work, the equations of motion become:

So, the mathematics ofAll of FORMAL Lattice Dynamics canbe 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 Models2. Shell Models3. Bond Models4. 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 OthersSince 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 to

compare to neutron scattering data & to predictresults 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”: 1st, 2nd, 3rd, 4th, 5th, … neighbor coupling!!

• The spring forces have directional dependences, withdifferent 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

5th & 6th 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:

Φ4nn > Φ1nn

& 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 Criticisms1. 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

Vo(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) Vo(r)exp(-r/ro)• Usually, this is simulated in simpler way:

V(r) Vo(r)/εHere, ε = dielectric constant

• Screening in classical E&M: V(r) Vo(r)/ε

ε = dielectric constant:Really, ε = ε(q,ω) but neglect this.

• This is too simple to work well for vibrationalspectra! Reason: the implicit assumption is thatvalence electrons are “free” , except for Coulombinteractions with ion of interest.

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

V(r) -(Ze2)/(εr),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 Zb is defined by

Zbe -2e/ε• This is the theory definition.

In practice, Zb is an adjustable parameter

• The bond charge fraction:

Zbe -2e/ε• When Zb is determined, often it is found that

Zbe < 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 SiZbe 0.35 e (< 1.0e)

• 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 SiZbe 0.35 e (< 1.0e)

• The 4 valence electrons from each atom are divided into

1. Bond Charges localized on each bond Zbe 0.35 e.

Point charges are assumed. In reality the bond charge is spread out over a volume.2. “Free” Charges which screen.Each Si contributes Zbe/2 to the bond charge and (4-2Zb)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!