A Cartoon About Solid State Chemistry!

Chapter 4: Phonons I – Crystal Vibrations

A Cartoon About Solid State Physics!

Change“Good-Chemist/Bad-Chemist”

to “Good-Physicist/Bad-Physicist”

This then becomes:

Chapter 4: Phonons I – Crystal Vibrations

Lattice Dynamics is a VERY LARGE

subfield of solid state physics!

Lattice Dynamics or “Crystal Dynamics”

Lattice Dynamics is a VERY LARGE

subfield of solid state physics!

It is also a VERY OLD subfield!

Lattice Dynamics or “Crystal Dynamics”

Lattice Dynamics is a VERY LARGE

subfield of solid state physics!

It is also a VERY OLD subfield!

It is also a “Dead” subfield!!• That is, it is no longer an area of active research!

Lattice Dynamics or “Crystal Dynamics”

• Most of our discussion will be very general & will apply to any crystalline solid.

• We start the discussion more generally than Ch. 4 does:

• For all of this discussion, we will use a large amount of material from many sources outside of Ch. 4!

• At the beginning of this discussion, the material may seem abstract.

But, don’t worry!• Before this discussion is finished, it should

hopefully be less abstract & it also should be a discussion that any upper level undergraduate in science or engineering should be able to understand.

• Lets start the discussion more generally than Ch. 4 does.• Some of what we discuss in the following will be

useful to us later when we discuss electronic band structures. Specifically, we know that,

From the theory viewpoint,a solid is a system with a

VERY LARGE number of coupled atoms.

• The form of the coupling between the atoms depends on the type of bonding that holds the solid together. Many possible bonding mechanisms were discussed in Kittel’s Ch. 3.

From the theory viewpoint, a solid is a system with a VERY LARGE number of coupled atoms.

• The form of the coupling between the atoms depends on the type of bonding that holds the solid together. Many possible bonding mechanisms were discussed in Kittel’s Ch. 3.

• A solid can be considered as a system of

Many Coupled Electrons & Nuclei. So, lets start the discussion by looking at the many electron, many nuclei Hamiltonian H (total mechanical energy) for a solid.

H = He + Hn + He-n

He = Electron Kinetic Energy + Interactionswith other Electrons

Electron-Nuclear Interaction Energy

Nuclear Kinetic Energy + Interactions

with other Nuclei

The Classical, Many-Body Hamiltonian (Mechanical Energy) for the solid has the form:

H = He + Hn + He-n

• Exactly solving the equations of motion resulting from this Hamiltonian is impractical & intractable, even with the most powerful computers of 2013!

Some Approximations obviously must be made!• By making some approximations, which are

rigorously justified in many advanced texts, after a lot of work, the complexity of the problem is significantly reduced.

The Classical, Many-Body Hamiltonian (Mechanical Energy) for the solid has the form:

H = He + Hn + He-n

Some Approximations obviously must be made!• By making some approximations, which are rigorously justified in

many advanced texts, after a lot of work, the complexity of the problem is significantly reduced.

• The usual starting point (without even acknowledgement that approximations are being made!) for MOST Solid State texts (including Kittel) is to discussions after such approximations have already been made.

• Later (Ch. 7) we’ll discuss that, for electronic properties calculations (electronic bands, etc.) these approximations reduce H a to a

One Electron Hamiltonian!• For the lattice vibrational problem of interest here, the two most

important approximations will now be briefly discussed.

Approximation #1: Separate the electrons into 2 types: Core Electrons & Valence Electrons

• Core Electrons ≡ Those in the filled, inner shells of the atoms. They play NO role in determining electronic properties of the solid! Example:

The Si free atom electron configuration is: 1s22s22p63s23p2

Core Electrons = 1s22s22p6 (filled shells!)• The core electrons are localized near the nuclei.

We lump the core shells & nuclei together.• So, in the Hamiltonian, we make the replacements:

Nuclei Ions [Core Electron Shells + Nucleus Ion Core]

He-n He-i Hn Hi

The Valence Electrons• These are the electrons in the unfilled, outer shells

of the free atoms. These determine the electronic properties of the solid & take part in the bonding!

ExampleThe Si free atom electron configuration is:

1s22s22p63s23p2

Valence Electrons = 3s23p2 (unfilled shell!)

• In the solid, these hybridize with electrons on neighboring atoms. This results invery strong covalent bonds with the 4 Si nearest-neighbors in the Si lattice

So, the Classical, Many-Body Hamiltonian(Mechanical Energy) for the solid is now:

H = He + Hi + He-i

• Later, when we focus on electronic properties calculations (bandstructures, etc.), we will make some approximations, to reduce this many electron Hamiltonian to a

One Electron Hamiltonian!• Now, however, we will focus our attention on

The Ion Motion Part of H.

He = Electron Kinetic Energy + Interactionswith other Electrons

Ion Kinetic Energy + Interactions

with other Ions

Electron-Ion Interaction

Energy

The Hamiltonian (Mechanical Energy) for a Perfect, Periodic Crystal:

Ne electrons, Ni ions; Ne, Ni ~ 1023 (huge!) Notation: i = electron; j = ion

The classicalclassical, many-body Hamiltonian is: (Gaussian units!)

H = He + Hi + He-i

He = Pure electronic energy = KE(e-) + PE(e- -e-)

He= ∑i(pi)2/(2mi) + (½)∑i∑i´[e2/|ri - ri´|] (i i´) 0

Hi = Pure ion energy = KE(i) + PE(i-i)

Hi = ∑j(Pj)2/(2Mj) + (½)∑j∑j´[ZjZj´ e2/|Rj - Rj´|] (j j´)0

He-i = Electron-ion interaction energy = PE(e--i)

He-i= - ∑i∑j[Zje2/|ri - Rj|]Lower case r, p, m: Electron position, momentum, mass

Upper case R, P, M: Ion position, momentum, mass

• This approximation allows the separation of the electron & ion motions. A rigorous proof of it requires detailed, many body Quantum Mechanics.

Qualitative (semiquantitative) justification:The very small ratio of the electron & ion masses!!

(me/Mi) ~ 10-3 (<< 1) (or smaller)

Classically, the massive ions move

much slowerthan the very small mass electrons!

Approximation # 2: The Born-Oppenheimer (Adiabatic) Approximation

• Typical ionic vibrational frequencies: υi ~ 1013 s-1

The time scale of the ion motion is: ti ~ 10-13 s • Electronic motion occurs at energies of about a bandgap:

Eg= hυe = ħω ~ 1 eV υe ~ 1015 s-1 te ~ 10-15 s

• So, classically, the

Electrons Respond to the Ion Motion~ Instantaneously!

As far as the electrons are concerned, the ions are ~ stationary!

In the electron Hamiltonian, He the ions can be treated as stationary!

Born-Oppenheimer (Adiabatic) Approximation

Now, lets look at the Ions:• The massive ions cannot follow the rapid,

detailed electron motion.

The Ions ~ see anAverage Electron Potential.

In the ion Hamiltonian, Hi,the electrons can be treated

in an average way!

Implementation: Born-Oppenheimer (Adiabatic) Approximation

• Write the coordinates for the vibrating ions as

Rj = Rjo + δRj, Rjo = equilibrium ion position

δRj = (small) deviation from equilibrium position

• The many body electron-ion Hamiltonian is (schematic!):

He-i ~ = He-i(ri,Rjo) + He-i(ri,δRj)

• The New many body Hamiltonian in this approximation is:

H = He(ri) + He-i(ri,Rjo) + Hi(Rj)

+ He-i (ri,δRj) (1)

• The Many body Hamiltonian in this approximation:

H = He(ri) + He-i(ri,Rjo) + Hi(Rj) + He-i (ri,δRj) (1)• For electronic energy band structure calculations (Ch. 7 of

Kittel), we will neglect the last 2 terms.• However, we will

Keep ONLY THEM for the vibrational properties calculations.

• Now, rewrite the Hamiltonian H (Eq. (1)) in the form:

H HE [1st 2 terms of (1)] + HI [2nd 2 terms of (1)]

HE Electronic Part(Gives energy bands. Ch. 7 of Kittel)

HI Ionic Part(Gives the lattice vibrations).

• We will now focus on HI only.

• Before making the Born-Oppenheimer Approximation, the Ion Hamiltonian was:

HI = Hi + He-i = j[(Pj)2/(2Mj)] + (½)jj´[ZjZj

´e2/|Rj-Rj´|] - ij[Zje2/|ri-Rj|]• Because the ion ions are moving (vibrations), the ion

positions Rj are obviously time dependent.• After a tedious implementation of the Born Oppenheimer

Approximation, the Ion Hamiltonian becomes:

HI j[(Pj)2/(2Mj)] + Ee(R1,R2,R3,…RN) Ee Average electronic total energy

for all ions at positions Rj.Equivalently,

Ee Average of the ion-ion interaction+ electron-ion interaction.

• So, we will use the Ion Hamiltonian in the form:

HI j[(Pj)2/(2Mj)] + Ee(R1,R2,R3,…RN) Ee Total (average) electronic ground state

total energy of the many electron problem

as a function of all ion positions Rj

• So, we will use the Ion Hamiltonian in the form:

HI j[(Pj)2/(2Mj)] + Ee(R1,R2,R3,…RN)

Ee Total (average) electronic ground state

total energy of the many electron problem

as a function of all ion positions Rj

• This results in the fact that

Ee acts as an effective Potential

for the ion motion

• So, the total average electronic ground state energy

Ee acts as an effective potential

energy for the ion motion. • Note that Ee depends on the electronic

states of all e- AND the positions of all ions! To calculate it from first principles,

the many electron problem

must first be solved!

• With modern computational techniques, it is possible to:

1. Calculate Ee to a high degree of accuracy (as afunction of the Rj).

2. Then,Use the calculated electronic structure of

the solid to compute & predict it’s vibrational properties.

• This is a HUGE computational problem! With modern computers, this can be done & often is done.

• But, historically, this was very difficult or even impossible to do.

• Therefore, people used many different empirical models instead.

• Most work in this area was done long before the existence of modern computers!

• This is an OLD field. It is also essentially DEAD in the sense that little, if any, new research is being done.

• The work that was done in this field relied on phenomenological (empirical), non-first principles, methods.

However, it is still useful

to briefly look at some of these empirical models

because doing so will (hopefully)

teach us something aboutthe physics of lattice vibrations.

Lattice Dynamics

• Consider the coordinates of each vibrating ion:Rj Rjo + δRj

Rjo equilibrium ion position δRj vibrational displacement amplitude

• As long as the solid is far from it’s melting point, it is always true that |δRj| << a, where a lattice constant. If this were not true, the solid would melt or “fall apart”!

• Lets use this fact to expand Ee in a Taylor’s series about the equilibrium ion positions Rjo. In this approximation, the ion Hamiltonian becomes:

HI ∑j[(Pj)2/(2Mj)] + Eo(Rjo) + E'(δRj)(Note that this is schematic; the last 2 terms are functions of all j)

Eo(Rjo) = a constant & irrelevant to the motionE'(δRj) = an effective potential for the ion motion

• Now, expand E'(δRj) in a Taylor’s series for small δRj

• The expansion is about equilibrium, so the first-order terms in δRj = Rj - Rjo are ZERO.

That is, (Ee/Rj)o = 0.• Stated another way, at equilibrium, the total force on

each ion j is zero by the definition of equilibrium!• The lowest order terms are quadratic in the quantities

ujk = (δRj - δRk) (j,k, neighbors)

• If the expansion is stopped at the quadratic terms, the Hamiltonian can be rewritten as the energy

for a set of coupled 3-dimensional simple harmonic oscillators.

“The Harmonic Approximation”

The Harmonic Approximation

• A change of notation!

Replace the Ion Hamiltonian HI with the Vibrational

Hamiltonian Hv.

Hv = ∑j[(Pj)2/(2Mj)] + E'(δRj)E'(δRj) is a function of all δRj & is

quadratic in the displacements δRj.

• For the rest of the discussion, we will only be discussingThe Vibrational Hamiltonian Hv in the Harmonic Approximation:

Hv = ∑j[(Pj)2/(2Mj)] + E'(δRj)E'(δRj) is a function of all δRj & is quadratic in the displacements δRj.

Caution!!• There are limitations to the harmonic approximation! Some

phenomena are not explained by it. For these, higher order (Anharmonic) terms in the expansion of Ee must be used. Anharmonic terms are necessary to explain the observed

Linear Expansion Coefficient α. In the harmonic approximation, α 0!!

Thermal Conductivity ΚIn the harmonic approximation, Κ !!

• Now, hopefully, a notation simplification:Ukℓ Displacement of Ion k in Cell ℓPkℓ Momentum of Ion k in Cell ℓ

& of course Pkℓ = Mk(dUkℓ/dt) (p = mv)

• With this change, the

Vibrational Hamiltonianin the Harmonic Approximation is:

Hv = (½)∑kℓMk(dUkℓ/dt)2 + (½)∑kℓ∑kℓUkℓΦ(kℓ,kℓ)Ukℓ

Φ(kℓ,kℓ) “Force Constant Matrix” (or tensor)• Hv = The standard classical Hamiltonian for a

system of coupled simple harmonic oscillators!

• Look at the details & find that the matrix elements of the force constant matrix Φ are proportional to 2nd derivatives of the total electronic energy function E′

Φ(kℓ,kℓ) (∂2E′/∂Ukℓ∂Ukℓ)E′ = Ion displacement-dependent portion of

the electronic total energy.• So, in principle, Φ(kℓ,kℓ) could be

calculated using results from the electronic structure calculation.

• This was impossible before the existence of modern computers. Even with computers it can be computationally intense!

Φ(kℓ,kℓ) (∂2E′/∂Ukℓ∂Ukℓ)

E′ = Ion displacement-dependent portion of the electronic total energy.

• Before computers, Φ(kℓ,kℓ) was usually determined empirically within various models. That is, it’s matrix elements were expressed in terms of parameters which were fit to experimental data. Even though we now can, in principle, calculate them exactly, it is

still useful to look at SOMEof these empirical models

because doing so will (hopefully) TEACH US something about the physics of

lattice vibrations in crystalline solids.

• To illustrate the procedure for treating the interatomic potential in the harmonic approximation, consider just two neighboring atoms.

• Assume that they interact with a known potential V(r). See Figure. Expand V(r) in a Taylor’s series in displacements about the equilibrium separation, keeping only up through quadratic terms in the displacements:

This potential energy is the same as that associated with a spring with spring constant K:

ardr

VdK

2

2

)( arKForce

r2 r1

V(r)

0 a

Repulsive

Attractive

The Vibrational Hamiltonianin the Harmonic Approximation

has the form:

Hv = (½)∑kℓMk(dUkℓ/dt)2 + (½)∑kℓ∑kℓUkℓΦ(kℓ,kℓ)Ukℓ

• This is a Classical Hamiltonian!• So, when we use it, we are obviously treating the

motion classically. So we can describe lattice motion using Hamilton’s Equations of Motion or, equivalently,

Newton’s 2nd Law!

The Classical Newton’s 2nd Law Equations of Motion for a system of coupled harmonic

oscillators are all of the form:(Analogous to F = ma = -kx for a single mass & spring):

Fkl = Mk(d2Ukℓ/dt2) = - ∑kℓΦ(kℓ,kℓ)Ukℓ(These are “Hooke’s Law” type forces!)

• The Force Constant matrix Φ(kℓ,kℓ) has two physical contributions:1. A direct, ion-ion, Coulomb repulsion2. An Indirect interaction

• The 2nd one is mediated by the valence electrons. The motion of one ion causes a change in its electronic charge distribution & this causes a force on it’s ion neighbors.

We will use the Newton’s 2nd Law equations of motion to find the allowed vibrational frequencies in various materials. In classical mechanics (see Goldstein’s graduate text or any undergraduate mechanics text) this means

Finding the normal mode vibrational frequencies of the system.

• Here, only a brief outline or summary of the procedure will be given.

• So, this will be an outline of how

“Phonon Dispersion Curves” ω(q)are calculated (q is a wavevector).

GOAL of the Following DiscussionGOAL of the Following Discussion::

• So, for various models of the vibrating solid, we will beFinding the normal mode vibrational

frequencies of the system.• Here, only a brief outline or summary of the procedure will be given.

• So, this will be an outline of how

“Phonon Dispersion Curves” ω(q)are calculated (q is a wavevector).

• I again emphasize that this is a Classical Treatment! That is, this treatment makes no direct reference to PHONONS. This is because

Phonons are Quantum Mechanical Quasiparticles.Here, first we’ll outline the method to find the classical normal modes. Once those are found, then we can quantize & start talking about Phonons. Shortly (Ch. 5) we’ll briefly summarize phonons also.

The Classical Treatment of the Vibrational Hamiltonian Hv

• As already mentioned, Hv Energy of a collection of

N coupled simple harmonic oscillators (SHO)• The classical mechanics procedure to solve such a problem is:

1. Find a coordinate transformation to re-express Hv written in terms of N coupled SHO’s to Hv written in

terms of N uncoupled (1d) (independent) SHO’s. 2. The frequencies of the new, uncoupled (1d) SHO’s are

The NORMAL MODE FREQUENCIES The allowed vibrational frequencies

for the solid.3. The amplitudes of the uncoupled SHO’s are

The NORMAL MODE Coordinates The amplitudes of the allowed vibrations for the solid.