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 = H e + H n + H e-n H e = Electron Kinetic Energy + Interactions with 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 = H e + H n + H e-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 = H e + H n + H e-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: 1s 2 2s 2 2p 6 3s 2 3p 2 Core Electrons = 1s 2 2s 2 2p 6 (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]  H e-n  H e-i H n  H i

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! Example The Si free atom electron configuration is: 1s 2 2s 2 2p 6 3s 2 3p 2 Valence Electrons = 3s 2 3p 2 (unfilled shell!) In the solid, these hybridize with electrons on neighboring atoms. This results in very 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 = H e + H i + H e-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. H e = Electron Kinetic Energy + Interactions with other Electrons Ion Kinetic Energy + Interactions with other Ions Electron-Ion Interaction Energy

The Hamiltonian (Mechanical Energy) for a Perfect, Periodic Crystal: N e electrons, N i ions; N e, N i ~ (huge!) Notation: i = electron; j = ion classical The classical, many-body Hamiltonian is: (Gaussian units!) H = H e + H i + H e-i H e = Pure electronic energy = KE(e - ) + PE(e - -e - ) H e = ∑ i (p i ) 2 /(2m i ) + (½)∑ i ∑ i´ [e 2 /|r i - r i´ |] (i  i´) 0 H i = Pure ion energy = KE(i) + PE(i-i) H i = ∑ j (P j ) 2 /(2M j ) + (½)∑ j ∑ j´ [Z j Z j´ e 2 /|R j - R j´ |] (j  j´) 0 H e-i = Electron-ion interaction energy = PE(e - -i) H e-i = - ∑ i ∑ j [Z j e 2 /|r i - R j |] 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!! (m e /M i ) ~ (<< 1) (or smaller)  Classically, the massive ions move much slower than the very small mass electrons! Approximation # 2: The Born-Oppenheimer (Adiabatic) Approximation

Typical ionic vibrational frequencies: υ i ~ s -1  The time scale of the ion motion is : t i ~ s Electronic motion occurs at energies of about a bandgap: E g = hυ e = ħω ~ 1 eV  υ e ~ s -1  t e ~ 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, H e 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 an Average Electron Potential.  In the ion Hamiltonian, H i, the electrons can be treated in an average way!

Implementation: Born-Oppenheimer (Adiabatic) Approximation Write the coordinates for the vibrating ions as R j = R jo + δR j, R jo = equilibrium ion position δR j = (small) deviation from equilibrium position The many body electron-ion Hamiltonian is (schematic!): H e-i ~ = H e-i (r i,R jo ) + H e-i (r i,δR j ) The New many body Hamiltonian in this approximation is: H = H e (r i ) + H e-i (r i,R jo ) + H i (R j ) + H e-i (r i,δR j ) (1)

The Many body Hamiltonian in this approximation: H = H e (r i ) + H e-i (r i,R jo ) + H i (R j ) + H e-i (r i,δR j ) (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  H E [1 st 2 terms of (1)] + H I [2 nd 2 terms of (1)] H E  Electronic Part (Gives energy bands. Ch. 7 of Kittel) H I  Ionic Part (Gives the lattice vibrations). We will now focus on H I only.

Before making the Born-Oppenheimer Approximation, the Ion Hamiltonian was: H I = H i + H e-i =  j [(P j ) 2 /(2M j )] + (½)  j  j´ [Z j Z j´ e 2 /|R j -R j´ |] -  i  j [Z j e 2 /|r i -R j |] Because the ion ions are moving (vibrations), the ion positions R j are obviously time dependent. After a tedious implementation of the Born Oppenheimer Approximation, the Ion Hamiltonian becomes: H I   j [(P j ) 2 /(2M j )] + E e (R 1,R 2,R 3,…R N ) E e  Average electronic total energy for all ions at positions R j. Equivalently, E e  Average of the ion-ion interaction + electron-ion interaction.

So, we will use the Ion Hamiltonian in the form: H I   j [(P j ) 2 /(2M j )] + E e (R 1,R 2,R 3,…R N ) E e  Total (average) electronic ground state total energy of the many electron problem as a function of all ion positions R j

So, we will use the Ion Hamiltonian in the form: H I   j [(P j ) 2 /(2M j )] + E e (R 1,R 2,R 3,…R N ) E e  Total (average) electronic ground state total energy of the many electron problem as a function of all ion positions R j This results in the fact that E e acts as an effective Potential for the ion motion

So, the total average electronic ground state energy E e acts as an effective potential energy for the ion motion. Note that E e 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 E e to a high degree of accuracy (as a function of the R j ). 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 about the physics of lattice vibrations. Lattice Dynamics

Consider the coordinates of each vibrating ion: R j  R jo + δR j R jo  equilibrium ion position δR j  vibrational displacement amplitude As long as the solid is far from it’s melting point, it is always true that |δR j | << a, where a  lattice constant. If this were not true, the solid would melt or “fall apart”! Lets use this fact to expand E e in a Taylor’s series about the equilibrium ion positions R jo. In this approximation, the ion Hamiltonian becomes: H I  ∑ j [(P j ) 2 /(2M j )] + E o (R jo ) + E'(δR j ) (Note that this is schematic; the last 2 terms are functions of all j) E o (R jo ) = a constant & irrelevant to the motion E'(δR j ) = an effective potential for the ion motion

Now, expand E'(δR j ) in a Taylor’s series for small δR j The expansion is about equilibrium, so the first-order terms in δR j = R j - R jo are ZERO. That is, (  E e /  R j ) 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 u jk = (δR j - δR k ) (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 H I with the Vibrational Hamiltonian H v. H v = ∑ j [(P j ) 2 /(2M j )] + E'(δR j ) E'(δR j ) is a function of all δR j & is quadratic in the displacements δR j.

For the rest of the discussion, we will only be discussing The Vibrational Hamiltonian H v in the Harmonic Approximation: H v = ∑ j [(P j ) 2 /(2M j )] + E'(δR j ) E'(δR j ) is a function of all δR j & is quadratic in the displacements δR j. 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 E e must be used. Anharmonic terms are necessary to explain the observed Linear Expansion Coefficient α. In the harmonic approximation, α  0!! Thermal Conductivity Κ In the harmonic approximation, Κ   !!

Now, hopefully, a notation simplification: U kℓ  Displacement of Ion k in Cell ℓ P kℓ  Momentum of Ion k in Cell ℓ & of course P kℓ = M k (dU kℓ /dt) (p = mv) With this change, the Vibrational Hamiltonian in the Harmonic Approximation is: H v = (½)∑ kℓ M k (dU kℓ /dt) 2 + (½)∑ kℓ ∑ kℓ U kℓ  Φ(kℓ,kℓ)  U kℓ Φ(kℓ,kℓ)  “Force Constant Matrix” (or tensor) H v = 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 2 nd derivatives of the total electronic energy function E′ Φ(kℓ,kℓ)  (∂ 2 E′/∂U kℓ ∂U kℓ ) 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)  (∂ 2 E′/∂U kℓ ∂U kℓ ) 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 SOME of 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: r2r2 r1r1 V(r) 0 a Repulsive Attractive

The Vibrational Hamiltonian in the Harmonic Approximation has the form: H v = (½)∑ kℓ M k (dU kℓ /dt) 2 + (½)∑ kℓ ∑ kℓ U kℓ  Φ(kℓ,kℓ)  U kℓ 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 2 nd Law!

The Classical Newton’s 2 nd 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): F kl = M k (d 2 U kℓ /dt 2 ) = - ∑ kℓ Φ(kℓ,kℓ)  U kℓ (These are “Hooke’s Law” type forces!) The Force Constant matrix Φ(kℓ,kℓ) has two physical contributions: 1. A direct, ion-ion, Coulomb repulsion 2. An Indirect interaction The 2 nd 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 2 nd 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 Discussion:

So, for various models of the vibrating solid, we will be 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). 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 H v As already mentioned, H v  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 H v written in terms of N coupled SHO’s to H v 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.