The Breakdown of the Static Lattice Model. • The free electron model was refined
by introducing a crystalline external potential. • This allows much progress, but ...
Lattice Vibrations Chris J. Pickard 500
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The Breakdown of the Static Lattice Model • The free electron model was refined • The classical static lattice can only be by introducing a crystalline external valid for T=0K potential • It is even wrong for T=0K: ∆x∆p ≥ ¯h ⇒ Zero point motion • This allows much progress, but is not the full story • This is a particular problem for insulators: unless kB T > Eg there • Ions are not infinitely massive, nor are no degrees of freedom to account held in place by infinitely strong forces for their many properties
Equilibrium Properties cv saturated
– the electronic degrees of freedom alone cannot explain experiment • Density and cohesive energy – zero point motion is important for solid neon and argon, and dominant for solid helium (a quantum solid )
cubic (lattice)
linear (electronic) T
• Specific heat of a metal:
• Thermal expansion of insulators – electronic contribution negligible for kbT < Eg
Transport Properties • Conductivity
– no vibrations, no superconductors
– in a perfect metallic crystal there are no collisions and perfect • Thermal conductivity of insulators conduction – the electronic degrees of freedom – lattice vibrations provide the are not sufficient scattering mechanisms • Superconductivity – interaction between two electrons via lattice vibrations
• Transmission of sound – sound waves are carried vibrations of the lattice
by
Interaction with Radiation • Reflectivity of ionic crystals – Sharp maximum in infrared, far below ¯hω = Eg – E-field applies opposite forces on ± ions • Inelastic scattering of light – small frequency shifts (Brillouin or Raman scattering) – understood via lattice vibrations
• X-ray scattering – thermal vibrations and zero point motion diminish the intensity of the peaks – there is a background in directions not satisfying the Bragg condition • Neutron scattering – momentum transfer with the lattice is discrete, and provides a probe of the lattice vibrations
A Classical Theory of the Harmonic Crystal • A general treatment of the deviation of ions from their equilibrium positions is intractable, so proceed in stages:
Bravais lattice remains as an average of the instantaneous configurations
1. Treat small deviations classically • Denote the position of an atom whose 2. Proceed to a quantum theory mean position is R by r(R): 3. Examine implications of larger r(R) = R + u(R) movements • To treat the small deviations, we • The dynamics of the lattice is governed by the classical Hamiltonian: assume each ion stays in the vicinity P P(R)2 of its equilibrium position R, and the R 2M + U
The Harmonic Approximation • 3D Taylor expand the potential energy around the equilibrium configuration: f (r + a) = f (r) + a · ∇f (r)+ 1 2 3 (a · ∇) f (r) + O(a ) 2
U
• At equilibrium the net force is zero, and the potential energy is given by: U = U eq + U harm r
U
harm
=
1 2
P
RR0 ,µν
• The general form for U harm is: uµ(R)Dµν (R − R0)uν (R0)
The Adiabatic Approximation • The quantities D in the harmonic expansion are in general very difficult to calculate
configuration ⇒ the wavefunctions change as the ions move
• Make the adiabatic approximation by • In ionic crystals the difficulties are the separating the typical timescales of long ranged coulomb interactions the motion of the electrons and ions • In covalent/metallic crystals the – the electrons are in their groundstate for any configuration difficulty comes from the fact that the contribution to the total energy of the valence electrons depends on the ionic • D is still difficult to calculate
Normal Modes of a 1D Bravais Lattice K M
K M
K M
K M
K M
K M
K M
K M
Masses M and springs K
M
• Consider ions of mass M separated by distance a • For simplicity, assume neighbour interactions only
u([N+1]a) = u(a) ; u(0) = u(Na)
Born-von Karman BCs
nearest
• In the Harmonic approximation, this is equivalent to masses connected by springs of strength K: harm Mu ¨(na) = − ∂U , U harm = ∂u(na) P 1 2 K [u(na) − u((n + 1)a)] n 2
Normal Modes of a 1D Bravais Lattice ω( k)
• Seek solutions of the form: u(na, t) ∝ ei(kna−ωt)
2Κ Μ
• The PBCs ⇒ eikN a = 1 ⇒ k = with n integer, N solutions and − πa ≤ k < πa
2π n a N
• Substitution into qthe dynamical eqn. K |sin(ka/2)| gives: ω(k) = 2 M −π/a
0
π/a
k
• The group and phase velocities differ substantially at the zone boundaries
Normal Modes of a 1D Bravais Lattice with a Basis ω( k) O 2(K+G) Μ
2K Μ
2G Μ
A
−π/a
0
π/a
k
• The analysis can be repeated
• The are 2 solutions for each k ⇒ 2N solutions in total: √ K+G 1 2 ω = M ± M K 2 + G2 + 2KGcos ka • There are acoustic and optical modes O A
Normal Modes of a 3D Bravais Lattice ω( k)
• The matrix is D(k) = P dynamical −ik·R , where D(R − R0) R D(R)e is the second derivative of U with respect to the displacement of ions at R and R0 at eqbm.. L
• The solution of the dynamical equation is given by the eigenequation M ω 2e = D(k)e, where e is the polarization vector
T2 T1
0
k
• 3N solutions for each ion in the basis
Normal Modes of a Real Crystal • The dynamical matrix can be built up from first principles calculations
500
400
• Can use a supercell approach to study certain high symmetry k-vectors
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ω (cm )
300
• For arbitary k use linear response theory – a perturbation theory
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Dispersion curves for Silicon
K
Connections with the theory of Elasticity
• The classical theory of elasticity slowly over the atomic length scale ignores the microscopic atomic structure • Using the symmetries of D: P Uharm = 14 RR0 • The continuum theory of elasticity can [u(R0) − u(R)]D(R − R0)[u(R0) − u(R be derived from the theory of lattice vibrations • Slowly varying displacements ⇒ u(R0) = • Consider displacements that vary u(R) + (R0 − R) · ∇u(r)|r=R ³ ´³ ´ P P ∂u ( R ) ∂u ( R ) µ 1 1 ν harm Eσµτ ν Eσµτ ν = − 2 R Rσ Dµν (R)Rτ U = 2 R,µνστ ∂xσ ∂xτ
A Quantum Theory of the Harmonic Crystal • In a Quantum theory the system can be in a set of discrete stationary states
independent oscillators ⇒ the 3N classical normal modes
• These stationary states are the • eigenstates of the harmonic P P (R)2 harm = R 2M + Hamiltonian: H P 1 0 0 RR0 u(R)D(R − R )u(R ) 2 • • The result is: an N -ion harmonic crystal is equivalent to 3N
The energy in each mode is discrete, and the is: P total energy E = ks(nks + 12 )¯ hωs(k) The integer nks is the excitation number of the normal mode in branch s at wave vector k
Normal Modes or Phonons • So far we have described the state in • Instead of saying that the mode k,s is in the nks excited state we say there terms of the excitation number nks are nks phonons of type s with wave • This is clumsy if describing processes vector k involving the exchange of energy – between normal modes, or other • Photons ⇒ of the correct frequency systems (electrons, neutrons or X- are visible light Phonons ⇒ of the correct frequency rays) are sound • As for the QM theory of the EM field we use an equivalent corpuscular • Don’t forget phonons/normal modes are equivalent description
Classical Specific Heat: Dulong-Petit • The thermal energy density is given • by averaging over all configurations weighted by e−βE with β = k 1T B R −βH H dΓe • u = V1 R dΓe−βH R 1 ∂ = − V ∂β ln dΓe−βH
The thermal energy density is: u = ueq + 3nkB T, (n = N/V ) The specific heat is independent of T : cv = 3nkB
• This is not observed experimentally – only approximately at high • By a change of variables: temperature where the harmonic R making eq dΓe−βH = e−βU β −3N × constant approximation is bad anyway
The Quantum Mechanical Lattice Specific Heat • The QM thermal energy density is: P P u = V1 i Eie−βEi / i e−βEi P −βE 1 ∂ i = − V ∂β ln i e
• The energy density is : P ¯ ωs(k)[ns(k) + 12 ] u = V1 ks h
• The mean excitation number ns(k) = • The sum is over the stationary states eβ¯hωs1(k)−1 is the Bose-Einstein with energy: distribution function P 1 i hωs(k), Ei = ks(nks + 2 )¯ niks = 0, 1, 2, . . . • The specific heat is given by: P P −βE −β¯ hωs (k)/2 h ¯ ωs k e ∂ 1 i = Π e = c k s v −β¯ h ω ( k ) β¯ h i ks ∂T e ωs (k) −1 s V 1−e
The High-Temperature Lattice Specific Heat • When kB T À ¯hωs(k) all the normal modes are highly excited
cv =
1 V
P
∂ ks ∂T kB T
=
3N V kB
• Writing β¯hωs(k) = x, then x is small • The next term is constant in T and we can expand: 1 1 x x2 3 • We might try to correct the Dulong= [1 − + + O(x )] x e −1 x 2 12 Petit law, but anharmonic terms • Keeping just the leading term we are likely to dominate where the regain the Dulong-Petit law: expansion holds – or the crystal melts!
ω( k)
L
T2 T1
0
k
The Low-Temperature Lattice Specific Heat • In the limit of a large crystal integrate over the P 1st RBrillouin zone: h ¯ ω s (k) ∂ dk cv = ∂T 3 β¯ i (2π) e hωs (k) −1
1. Ignore the optical modes 2. Use the dispersion relationship: ˆ ωs(k) = cs(k)k 3. Integrate over all k
• Modes with ¯hωs(k) À kB T will not contribute – but the acoustic branches ˆ = hcs(k)k will at long enough wavelengths for • Making the substitution β¯ x we obtain, and c as the average any T speed of sound: • Make some approximations: cv =
∂ ∂T
const ×
(kB T )4 (¯ hc)3
∝ T3
Intermediate Temperature: The Debye and Einstein Models • The T 3 relation only remains valid while the thermal energy is small compared to the energy of phonons with a non-linear dispersion (much • lower than room temperature)
acoustic modes, all with ω = ck, and integrate up to kD Einstein: optical modes represented by modes of ωE
• The Debye and Einstein models • The Debye temperature divides the approximate the dispersion relations quantum and classical statistical regimes: √ 3 • Debye: all branches modelled by 3 kB ΘD = ¯hωD = ¯ hckD = ¯ hc 6π 2n
Measuring Phonon Dispersion Relations • Normal mode dispersion relations phonon ωs(k) can be extracted from experiments in which lattice vibrations • The same applies or X-rays or visible exchange energy with an external light probe • Energy lost (or gained) by a neutron • Neutrons carry more momentum than ⇒ emission (or absorption) of a photons in the energy range of interest
Neutron Scattering by a Crystal • Neutrons only interact strongly with ∆nks = n0ks − nks the atomic nuclei, and so will pass through a crystal, possibly with a • The conservation of crystal 2 changed E = p /2Mn and p momentum: P hk∆nks + K p 0 − p = ks ¯ • Conservation laws allow the extraction of information from the scattering • This is the same crystal momentum as for the Bloch states – important for theories of electron-phonon scattering
• The conservation of energy: P E 0 = E − ks ¯hωs(k)∆nks,
• Different numbers of phonons can be involved in a scattering event
Zero Phonon Scattering • The final state is identical to the initial state
changes by ¯ hK: q0 = q + K
• These are just the von Laue conditions • Energy conservation implies that the energy of the neutron is unchanged • We can extract the same (elastically scattered): q 0 = q crystallographic information of the • Crystal momentum conservation static lattice as from X-ray diffraction implies that the neutron’s momentum experiments
One Phonon Scattering • The situation where one phonon is • For the absorption case: p02 p2 p 0 −p absorbed or emitted conveys the most hωs( h¯ ) 2Mn = 2Mn + ¯ information • In an experiment we control p and E • The conservation laws imply: E 0 = E ± ¯hωs(k) p0 = p ± ¯hk + ¯hK
• We can choose a direction in which to measure, and record the energy E 0 to map out the dispersion curves ωs(k)
• The additive K can be ignored • Multi-phonon scattering because ωs(k + K) = ωs(k) produce a background
events
