Normal Modes of Vibration OneDimensional Model 1 The

Normal Modes of Vibration One-Dimensional Model # 1: The Monatomic Chain

1 - Dimensional Model # 1: The Monatomic Chain • Consider a Monatomic Chain of Identical Atoms with nearest-neighbor, “Hooke’s Law” type forces (F = - Kx) between the atoms. • This is equivalent to a force-spring model, with masses m & spring constants K.

• Consider a typical interatomic potential as is illustrated in the figure: V(r) Repulsive 0 a = lattice constant r 1, r 2 = “atomic radii” a Attractive r 2 r 1

• Assume V(r) is known & expand it in a Taylor’s series in displacements about the equilibrium separation a, keeping only up through quadratic terms in the displacements (r-a). V(r) Repulsive 0 a Attractive r 2 r 1 This potential is the same as that associated with a spring with spring constant K:

One Dimensional Model # 1: The Monatomic Chain • This is the simplest possible solid. Assume that the chain contains a large number (N ) of atoms with identical masses m. Let the atomic separation be a distance a. • Assume that the atoms move only in a direction parallel to the chain. Assume that only nearest-neighbors interact with each other (the forces are short-ranged). a a Un-2 a Un-1 a Un+1 a Un+2

The Monatomic Chain • Consider the simple case of a monatomic linear chain with only nearest-neighbor interactions. • Expand the energy near the equilibrium point for the nth atom. Then, the Newton’s 2 nd Law equation of motion is: . . a a 0 Un-1 Un l Un+1

• This can be seen as follows. The total force on the nth atom is the sum of 2 forces. • The force to the right is: a a • The force to the left is: Un-1 Un Un+1 Total Force = Force to right – Force to left. . The Equation of Motion of each atom is of this form. Only the value of ‘n’ changes.

• Assume that all atoms oscillate with the same amplitude A & the same frequency ω. Assume harmonic solutions for the displacements un of the form: Undisplaced Position: Displaced Position: • Put all of this into the equation of motion: . . • Now, carry out some simple math manipulation:

Equation of Motion for the nth Atom Cancel Common Terms & Get:

• Mathematical Manipulation finally gives: • After more manipulation, this simplifies to This is the solution to the normal mode eigenvalue problem for the monatomic chain. • The maximum allowed frequency is:

This is the solution to the normal mode eigenvalue problem for the monatomic chain. • The physical significance of these results is: For the monatomic chain, the only allowed vibrational frequencies ω must be related to the wavenumber k = (2π/λ) or the wavelength λ in this way. • This result is often called the “Phonon Dispersion Relation” for the chain, even though these are classical lattice vibrations & there are no (quantum mechanical) phonons in the classical theory.

“Phonon Dispersion Relations” or Normal Mode Frequencies or ω versus k relation for the monatomic chain. w C –л/a A B 0 л/a 2 л/a k • Because of BZ periodicity with a period of 2π/a, only the first BZ is needed. Points A, B & C correspond to the same frequency. They all have the same instantaneous atomic displacements.

Monatomic Chain Dispersion Relation Some Physics Discussion • We started from the Newton’s 2 nd Law equations of motion for N COUPLED simple harmonic oscillators. If one atom starts vibrating, it does not continue with constant amplitude, but transfers energy to the others in a complicated way. That is, the vibrations of individual atoms are not simple harmonic because of this exchange of energy among them. • On the other hand, our solutions represent the oscillations of N UNCOUPLED simple harmonic oscillators.

Monatomic Chain Dispersion Relation • Our solutions represent the oscillations of N UNCOUPLED simple harmonic oscillators. • As we already said, these are called the Normal Modes of the system. They are a collective property of the system as a whole & not a property of any of the individual atoms. Each mode represented by ω(k) oscillates independently of the other modes. Also, it can be shown that the number of modes is the same as the original number of equations N. Proof of this follows.

• To establish which wavenumbers are possible for the one-dimensional chain, we can reason as follows: Not all values are allowed because of periodicity. In particular, the nth atom is equivalent to the (N+n)th atom. This means that the assumed solution for the displacements: must satisfy the periodic boundary condition: • This, in turn requires that there an integer number of wavelengths in the chain. So, in the first BZ, there are only N allowed values of k.

• Physical significance of wave numbers k outside the First Brillouin Zone [-(π/a) k (π/a)]? • At the Brillouin Zone edge: • This k value corresponds to the maximum frequency. • A detailed analysis of the displacements shows that, in that mode, every atom is oscillating π radians out of phase with it’s 2 nearest neighbors. That is, a wave at this value of k is A STANDING WAVE.

k • At the Brillouin Zone edge: l • The maximum frequency occurs at this k. In that mode, every atom is oscillating π radians out of phase with it’s 2 nearest neighbors. That is, a wave at this value of k is A STANDING WAVE. Black: k = π/a or = 2 a Note! This is also the for Bragg reflection! Green: k = (0. 85)π/a or = 2. 35 a x

Dispersion Relation: w C -π/a B A π/a 2π/a kk • Points A & C are symmetrically equivalent. They have the same frequency & the same atomic displacements. • The group velocity vg = (dω/dk) at both of them is negative, so that a wave at that ω & that k moves to the left. Point B in the ω(k) diagram has the same frequency & displacement as points A and C, but vg = (dω/dk) there is positive, so that a wave at that ω & that k moves to the right. • Adding a multiple of 2π/a to k does not change either ω or vg, so point A contains no physical information that is different point B.

The Monatomic Chain k = ( /a) = (2 / ); = 2 a k 0;

For visualization purposes, it is sometimes useful to visualize a plane of atoms, made up of a large number of parallel chains like the one we just analyzed. See the next few slides:

• Lets briefly look in more detail at the group velocity, vg. The dispersion relation is: So, the group velocity is: vg (dω/dk) = a(K/m)½cos(½ka) vg = 0 at the BZ edge [k = (π/a)] • This tells us that a wave with λ corresponding to a zone edge wavenumber k = (π/a) will not propagate. • That is, it must be a standing wave! • At BZ edge, the displacements have the form (for site n): Un= Uoeinka = Uo e i(nπ/a) = Uo(-1)n

Group Velocity, vg in the 1 st BZ vg (dω/dk) = a(K/m)½cos(½ka) At the 1 st BZ Edge: vg = 0 • This means that a wave with λ corresponding to a zone edge wavenumber k = (π/a) Will Not Propagate! • That is, it must be a Standing Wave!

One-Dimensional Model # 2: The Diatomic Chain

• Consider a Diatomic Chain of Two Different Atom Types with nearest-neighbor, Hooke’s Law type forces (F = - kx) between the atoms. This is equivalent to a force-spring model with two different types of atoms of masses, M & m connected by identical springs of spring constant K. (n-2) (n-1) K (n) K M m (n+1) K M (n+2) K m M a) a b) Un-2 Un-1 Un Un+1 Un+2 • This is the simplest possible model of a diatomic crystal. a is the repeat distance, so, the nearest-neighbor separation is (½)a

• This model is complicated due to the presence of 2 different atom types, which, in general, move in opposite directions. M m Un-2 Un-1 m M Un Un+1 M Un+2 • The GOAL is to find the dispersion relation ω(k) for this model. There are 2 atom types, with masses M & m, so there will be 2 equations of motion, one for M & one for m. Equation of Motion for M -1 -1 Equation of Motion for m

• As before, assume harmonic (plane wave) solutions for the atomic displacements Un: M Un-2 m Un-1 m M Un Un+1 M Un+2 Displacement for M Displacement for m α = complex number which determines the relative amplitude & phase of the vibrational wave. Put all of this into the 2 equations of motion. Carry out some simple math manipulation as follows:

Equation of Motion for the nth Atom (M) Cancel Common Terms

Equation of Motion for the (n-1)th Atom (m) Cancel Common Terms

• The Equation for M becomes: (1) • The Equation for m becomes: (2) • (1) & (2) are two coupled, homogeneous, linear algebraic equations in the 2 unknowns α & ω as functions of k. • More algebra gives: • Combining (1) & (2) & manipulating:

• Cross multiplying & manipulating with (1) & (2): The 2 roots are:

• So, the resulting quadratic equation for ω2 is: • The two solutions for ω2 are: • Since the chain contains N unit cells, there will be 2 N normal modes of vibration, because there are 2 N atoms and 2 N equations of motion for masses M & m.

Solutions to the Normal Mode Eigenvalue Problem ω(k) for the Diatomic Chain w A ω+ = “Optic” Modes B C ω- = “Acoustic” Modes –л/a 0 л/a 2 л/a k • There are two solutions for ω2 for each wavenumber k. That is there are 2 branches to the “Phonon Dispersion Relation” for each k. From an analysis of the displacements, it can be shown that, at point A, the two atoms are oscillating 180º out of phase, with their center of mass at rest. Also, at point B, the lighter mass m is oscillating & M is at rest, while at point C, M is oscillating & m is at rest.

Diatomic Chain Model: Kittel’s Notation! The solution is: K K Optic Modes (Optic Branch) Near BZ Center (qa << 1) • The Optic Mode becomes: (ω+)2 2 K(M 1 + M 2)/(M 1 M 2) Gap or ω+ constant • The Acoustic Mode becomes: (ω-)2 (½) Kq 2/(M 1 + M 2) qa Acoustic Modes (Acoustic Branch) or ω- vsq vs sound velocity in the crystal. Just like an acoustic wave in air!

The Diatomic Chain Solution: K K ( = 2 a) Optic Modes (Optic Branch) Near BZ edge [q = (π/a)] (Assuming M 1 > M 2) The Optic Mode becomes: (ω+)2 2 K/M 2 Gap The Acoustic Mode becomes: (ω-)2 2 K/M 1 Acoustic Modes (Acoustic Branch) So, at the BZ edge, the vibrations of wavelength = 2 a for the 2 modes behave as if there were 2 uncoupled masses M 1 & M 2, vibrating independently with identical springs of constant K.

• Again briefly examine limiting solutions at points 0, A, B & C. In the long wavelength region near k = 0 (ka « 1), sin(ka/2) ≈ ½ka. w A –л/a 0 B C л/a 2 л/a A Taylor’s series expansion, using for, small x: k

w A –л/a 0 The root with the minus sign gives the minimum value of the B C л/a acoustic branch: 2 л/a k The root with the positive sign gives the maximum value of the optic branch: Substituting these values of ω into the expression for the relative amplitude α & using cos(ka/2) ≈1 for ka << 1 gives the corresponding value of α: OR

Substituting into the expression for the relative amplitude α: This solution corresponds to long-wavelength vibrations near the center of the BZ at k = 0. In that region, M & m oscillate with same amplitude & phase. Also in that region ω = vsk, where vs is the velocity of sound & has the form: w A –л/a 0 B C л/a 2 л/a k

Similarly, substituting into the relative amplitude gives: This solution corresponds to point A in the figure. This value of α shows that, in that mode, the two atoms are oscillating 180º out of phase with their center of mass at rest. w A B C –л/a 0 л/a 2 л/a k

• The other limiting solutions for ω2 are for ka = π. In this case sin(ka/2) =1, so w A –л/a 0 B C л/a 2 л/a OR (C) (B) • At point C in the plot, which is the maximum acoustic branch point, M oscillates & m is at rest. • By contrast, at point B, which is the minimum optic branch point, m oscillates & M is at rest. k

Eigenmodes of the chain at q = 0 Center of the unit cell is not moving! Optical Mode: These atoms, if oppositely charged, would form an oscillating dipole which would couple to optical fields with < a

Acoustic & Optic Branches • Despite the fact that diatomic chain model is one-dimensional, it’s results for the vibrational normal modes ω contain considerable qualitative physics that carries over to the observed vibrational frequencies for many real materials. • So, much of the physics contained in the diatomic chain results can teach us something about the physics contained in the normal modes of many real materials. • In particular, ALL MATERIALS with 2 atoms per unit cell are observed to have two very different kinds of vibrational normal modes. These are called The Acoustic Branch & The Optic Branch

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 Branch • This branch is always at much higher vibrational 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.

A Longitudinal Optic Mode The vibrational amplitude is highly exaggerated!

A Transverse Optic Mode for the Diatomic Chain The vibrational amplitude is highly exaggerated! For the case in which the lattice has some ionic character, with + & - charges alternating:

A Long Wavelength Longitudinal Acoustic Mode The vibrational amplitude is highly exaggerated!

A Short Wavelength Longitudinal Acoustic Mode The vibrational amplitude is highly exaggerated!

A Transverse Acoustic Mode for the Diatomic Chain The vibrational amplitude is highly exaggerated! For the case in which the lattice has some ionic character, with + & - charges alternating:

Acoustic vs. Optic Phonons Which has lower energy? Why? Acoustic Mode Lower Energy, Less Compression of Springs Optic Mode