Phonons and Lattice Dynamics
A crystal is the physical realization of the loaded string of Oscillations and Mechanical Waves: a periodic array of masses coupled by springs, whose normal modes are travelling waves and whose quanta — phonons — are bosons obeying the statistics of Quantum Statistics. Almost every thermal property of a solid follows from that identification. This chapter establishes the lattice itself as an observed structure, by X-ray diffraction [Friedrich:1912] [Bragg:1913a] rather than by assumption; derives its normal modes [Born:1912]; quantizes them; and then confronts the resulting heat capacity with calorimetric data, where the classical Dulong–Petit law [Dulong:1819] fails at low temperature and the quantum models of Einstein [Einstein:1907a] and Debye [Debye:1912a] succeed. The dispersion relations that the theory predicts are not inferred but measured, branch by branch, by inelastic neutron scattering [Brockhouse:1955] [Brockhouse:1958] — the cleanest available demonstration that the phonon is a real excitation with a definite energy and momentum. The chapter closes on anharmonicity, without which a crystal would neither expand when heated [Gruneisen:1912] nor conduct heat at a finite rate [Peierls:1929b].
It sits between the general theory of phase transitions (Phase Transitions and Critical Phenomena), which supplies the notion of a broken translation symmetry, and the electronic chapters Electrons in Solids: Band Theory and Semiconductors, which inherit its reciprocal lattice and Brillouin zones wholesale; its transport section uses the machinery of Nonequilibrium Thermodynamics and Transport, and the electron–phonon coupling it introduces is what pairs electrons in Superconductivity and Superfluidity. Standard monographs are [Born:1954] [Ashcroft:1976] [Kittel:2005].
Phonons and Lattice Dynamics: all derivations of this chapter are pending.
The crystal lattice
Bravais lattices and the crystallographic restriction
[Reserved: the enumeration of the fourteen distinct lattices of points invariant under a discrete translation group in three dimensions [Bravais:1850]; primitive vectors, unit cells and the Wigner–Seitz construction; the crystallographic restriction theorem — only two-, three-, four- and six-fold rotation axes are compatible with translational periodicity — proved from the trace of the rotation in the lattice basis, an application of Linear Algebra and Representation Theory.]
Point groups, space groups and the basis
[Reserved: the 32 crystallographic point groups and the 230 space groups obtained by adjoining screw axes and glide planes, enumerated independently by Fedorov and by Schoenflies [Schoenflies:1891]; lattice plus basis as the description of a real crystal; Neumann's principle relating the symmetry of a physical tensor to that of the crystal, which is what makes a cubic crystal optically isotropic but elastically anisotropic (Continuum Mechanics and Elasticity).]
The reciprocal lattice and Brillouin zones
[Reserved: the reciprocal lattice as the Fourier dual of the direct lattice, with \(\ee^{\ii\vect{G}\cdot\vect{R}}=1\), and the Ewald construction that makes diffraction geometry visible [Ewald:1921]; the first Brillouin zone as the Wigner–Seitz cell of the reciprocal lattice and as the complete set of inequivalent wave vectors [Brillouin:1930]; Bloch's theorem stated here for lattice waves and reused for electrons in Electrons in Solids: Band Theory; zone boundaries as the locus of Bragg reflection.]
Quasicrystals: the exception that fixed the rule
[Reserved: the observation of sharp diffraction spots with icosahedral symmetry in a rapidly cooled aluminium–manganese alloy, which is forbidden for any periodic lattice [Shechtman:1984]; the resolution — long-range order without periodicity — and the consequent redefinition of “crystal” by diffraction rather than by translational symmetry; what survives of the phonon description, and the additional phason modes that do not.]
A rapidly solidified aluminium–manganese alloy gives an electron diffraction pattern with spots as sharp as those of a good crystal, and with icosahedral point symmetry — including five-fold axes, which the crystallographic restriction forbids to any lattice periodic in three dimensions [Shechtman:1984]. The two observations are individually routine and jointly contradictory under the definition of a crystal then in force. The resolution is that sharp diffraction requires long-range order and not periodicity, and the definition was changed accordingly.
The crystallographic restriction, proved from the integrality of the matrix of a lattice rotation in the lattice basis and hence of its trace, which admits only two-, three-, four- and six-fold axes; and the demonstration that a non-periodic but long-range-ordered point set can nevertheless have a Fourier transform consisting of sharp discrete peaks, so that sharp diffraction does not entail a periodic lattice
Experiment: the lattice made visible
Laue diffraction of X-rays
[Reserved: the 1912 Munich experiment in which a collimated X-ray beam through a copper sulphate crystal produced a symmetric pattern of spots on a photographic plate [Friedrich:1912], settling two questions at once — that X-rays are waves, and that crystals are periodic on the \(10^{-10}\,\mathrm{m}\) scale; von Laue's diffraction conditions and their equivalence to the reciprocal-lattice condition [vonLaue:1912]; the apparatus, exposure times and the control observations.]
Bragg's law and structure determination
[Reserved: the reformulation of the Laue conditions as reflection from lattice planes, \(n\lambda=2d\sin\theta\) [Bragg:1913a]; the ionization spectrometer and the first structures solved, sodium chloride and diamond, which showed that the rock-salt crystal contains no NaCl molecule; measured lattice constants in SI with their modern uncertainties, and the silicon lattice parameter as a route to the Avogadro constant (Measurement, SI Units, and the Theory of Errors).]
A collimated X-ray beam passed through a single crystal emerges not as a diffuse halo but as a pattern of sharp, discrete spots arranged with the point symmetry of the crystal [Friedrich:1912]. The pattern is the joint signature of two facts, neither previously established: that X-rays are waves, and that a crystal is a lattice periodic on the scale of \(10^{-10}\,\mathrm{m}\). Read as reflection from families of parallel lattice planes of spacing \(d\), the observed directions obey
with \(\theta\) the glancing angle measured from the planes and not from their normal [Bragg:1913a]. Equation (125.1) turns the diffraction pattern into a measuring instrument: it converts measured angles into lattice spacings, and it is what showed that the rock-salt crystal contains no NaCl molecule at all, only alternating ions.
Derivation. Consider a family of parallel lattice planes of spacing \(d\) and a plane wave of wavelength \(\lambda\) incident at glancing angle \(\theta\). Two rays reflected specularly from adjacent planes travel paths differing by the two segments \(d\sin\theta\) dropped from the upper plane to the lower and back, that is by \(2d\sin\theta\). Specular reflection makes the phase difference between rays scattered within one plane vanish identically, so the whole family adds in phase if and only if the inter-plane path difference is a whole number of wavelengths, which is Equation (125.1). Two features of the observation follow at once. The spots are discrete because the condition selects isolated angles for each \(d\); a non-periodic scatterer has no family of equally spaced planes and returns a continuous distribution. And Equation (125.1) has no solution unless \(\lambda\leq2d\), which is why visible light shows no such pattern and why the measurement had to wait for X-rays. The plane picture is the crystallographers' form of the von Laue conditions; both say that the scattering vector must be a reciprocal-lattice vector (Section 125.1.3).
∎Structure factor and systematic absences
[Reserved: the geometric structure factor as the Fourier transform of the basis, and the atomic form factor as that of the electron density; systematic absences as the fingerprint of centring and of screw axes, and how a space group is read off a diffraction pattern; the phase problem and why intensities alone do not determine a structure; the Fourier machinery of Fourier Analysis and Integral Transforms.]
Heat capacity of solids
The Dulong–Petit law and its failure
[Reserved: the empirical constancy of the molar heat capacity of the solid elements at \(3R=24.94\,\mathrm{J}/\mathrm{mol}/\mathrm{K}\) [Dulong:1819], and its derivation from equipartition over \(3N\) classical oscillators (Statistical Mechanics); the anomalies known by 1900 — diamond, boron and silicon far below the value at room temperature — and the low-temperature calorimetry showing every solid's heat capacity falling to zero [Nernst:1911], which no classical theory can produce.]
Near room temperature the molar heat capacity of the solid elements is very nearly the same for all of them, close to \(3R=24.94\,\mathrm{J}/\mathrm{mol}/\mathrm{K}\), irrespective of the mass of the atom, the stiffness of the bonding or the crystal structure [Dulong:1819]. The regularity fails in two directions. Already at room temperature diamond, boron and silicon sit far below the value; and calorimetry at low temperature shows the heat capacity of every solid falling towards zero as the temperature does [Nernst:1911]. A universal constant that is right at high temperature and wrong at low temperature is the signature of a classical limit.
Derivation. A crystal of \(N\) atoms has \(3N\) vibrational degrees of freedom, and near equilibrium each is a harmonic normal mode (Section 125.4.1). A classical harmonic oscillator in equilibrium at temperature \(T\) has, by equipartition, mean kinetic energy \(\tfrac{1}{2}k_{B}T\) and mean potential energy \(\tfrac{1}{2}k_{B}T\), so \(\avg{E}=k_{B}T\) per mode, independent of its frequency. Hence
which for one mole is \(3R\). Every material property has cancelled: the masses and the force constants enter only through the frequencies, and the frequencies have dropped out. That is precisely why the result cannot be repaired. A classical theory containing no energy scale of its own has nothing with which to compare \(k_{B}T\), so it cannot produce a temperature below which the heat capacity falls. The observed collapse therefore requires a scale that the classical theory does not possess, and the only candidate is \(\hbar\omega\): modes with \(\hbar\omega\gg k_{B}T\) are frozen out, which is why the light, stiffly bound solids — diamond, boron, beryllium — are the ones that deviate first (Section 125.3.2).
∎The Einstein model
[Reserved: the first application of quantization to matter rather than radiation — \(3N\) independent oscillators of a single frequency, with \(x=\theta_{E}/T\) giving \(C=3Nk_{B}x^{2}\ee^{x}/(\ee^{x}-1)^{2}\) [Einstein:1907a]; the quantitative account of the diamond anomaly with one fitted frequency; the exponential low-temperature falloff, which is too fast, and why: the model has no low-frequency modes; the historical weight of this paper as evidence for quanta outside Black-Body Radiation and Planck's Hypothesis.]
The Debye model and the cubic law
[Reserved: replacement of the single frequency by an elastic continuum with a linear dispersion cut off at \(\omega_{D}\) so that the mode count is \(3N\) [Debye:1912a]; the interpolation formula and the limiting law \(C=\frac{12\pi^{4}}{5}Nk_{B}(T/\theta_{D})^{3}\); the Debye temperature as the single material parameter, from \(105\,\mathrm{K}\) for lead to \(2230\,\mathrm{K}\) for diamond; the exact parallel with the photon gas of Black-Body Radiation and Planck's Hypothesis, differing only in the cutoff and the three polarizations.]
Calorimetric tests at low temperature
[Reserved: the measurement that separates the two contributions — \(C/T\) plotted against \(T^{2}\) gives a straight line whose intercept is the electronic term \(\gamma T\) and whose slope is the lattice \(T^{3}\), verified for copper, silver and gold between \(1\,\mathrm{K}\) and \(5\,\mathrm{K}\) [Corak:1955]; the extracted Debye temperatures compared with those from elastic constants, a consistency test between two unrelated experiments; where the model fails at intermediate \(T\), and why.]
Well below its Debye temperature the lattice contribution to the heat capacity of a solid is not exponentially small but follows a power law,
with a single material parameter \(\theta_{D}\) ranging from about \(105\,\mathrm{K}\) for lead to about \(2230\,\mathrm{K}\) for diamond [Debye:1912a]. In a metal the lattice term is accompanied by the electronic term of Quantum Statistics, and the two are separated experimentally by plotting \(C/T\) against \(T^{2}\): the result is a straight line whose intercept is electronic and whose slope is the lattice term, as verified for copper, silver and gold between \(1\,\mathrm{K}\) and \(5\,\mathrm{K}\) [Corak:1955]. The \(\theta_{D}\) so obtained agrees with the value computed from independently measured elastic constants, which is a consistency test between two unrelated experiments and not a fit.
Derivation. Debye's approximation replaces the true dispersion by an isotropic elastic continuum, \(\omega=v_{s}k\) on three polarizations, and truncates the spectrum at a frequency \(\omega_{D}\) chosen so that the total number of modes is \(3N\). Counting modes in a box, the number with wavevector below \(k\) is \(3\cdot V k^{3}/6\pi^{2}\), so the density of modes in frequency is \(g(\omega)=3V\omega^{2}/2\pi^{2}v_{s}^{3}\), and imposing \(\int_{0}^{\omega_{D}}g\,\dd\omega=3N\) lets it be written without reference to \(v_{s}\) as
Each mode is a quantized oscillator whose thermal occupation is the Bose function of Quantum Statistics with zero chemical potential, since phonon number is not conserved. Dropping the temperature-independent zero-point term,
with \(x=\hbar\omega/k_{B}T\) and \(k_{B}\theta_{D}=\hbar\omega_{D}\). For \(T\ll\theta_{D}\) the upper limit may be sent to infinity with exponentially small error, and \(\int_{0}^{\infty}x^{3}(\ee^{x}-1)^{-1}\dd x=\pi^{4}/15\), so \(U=(3\pi^{4}/5)Nk_{B}T^{4}/\theta_{D}^{3}\) and differentiation gives Equation (125.3). The physical content is in the low-frequency end of Equation (125.4): because a crystal has acoustic modes of arbitrarily low frequency, there are always modes with \(\hbar\omega<k_{B}T\) available to excite, and the number of them scales as \(T^{3}\) in three dimensions. This is exactly the photon-gas calculation of Black-Body Radiation and Planck's Hypothesis with two changes, three polarizations instead of two and a finite cutoff instead of none; the cutoff is what the photon gas lacks and is why radiation has no Dulong–Petit regime. Einstein's earlier model, which gives every mode the same frequency, has no low-frequency modes at all and therefore predicts an exponential falloff, too fast to match the data [Einstein:1907a].
∎Lattice dynamics
The Born–von Kármán theory
[Reserved: the harmonic expansion of the crystal potential in atomic displacements, the dynamical matrix \(D(\vect{k})\) and the eigenvalue problem \(\omega^{2}(\vect{k})\vect{e}=D(\vect{k})\vect{e}\) [Born:1912]; periodic boundary conditions and the resulting discrete, finite set of allowed wave vectors in the Brillouin zone; the monatomic chain and its dispersion \(\omega=2\sqrt{K/M}\abs{\sin(ka/2)}\) as the worked case; the long-wavelength limit reproducing the elastic waves of Continuum Mechanics and Elasticity; the monograph [Born:1954].]
Acoustic and optical branches
[Reserved: the diatomic chain, whose \(3p\) branches for \(p\) atoms per cell split into three acoustic branches with \(\omega\to0\) as \(k\to0\) and \(3p-3\) optical branches with a finite gap [Born:1912]; the physical distinction — neighbouring atoms move in phase in an acoustic mode and against each other in an optical one; why only optical modes couple to light at \(k\approx0\); group velocity vanishing at the zone boundary, connecting to Oscillations and Mechanical Waves.]
Quantization: the phonon
[Reserved: each normal mode quantized as a harmonic oscillator, giving energies \((n_{\vect{k}s}+\tfrac{1}{2})\hbar\omega_{\vect{k}s}\) and an occupation obeying the Bose–Einstein distribution of Quantum Statistics with zero chemical potential, since phonon number is not conserved; crystal momentum \(\hbar\vect{k}\) defined only modulo a reciprocal-lattice vector, and why it is not true momentum; zero-point motion and its measured consequences, including the failure of helium to solidify under its own vapour pressure (Superconductivity and Superfluidity).]
Polar crystals and the Lyddane–Sachs–Teller relation
[Reserved: the long-range Coulomb field of a longitudinal optical mode in an ionic crystal, splitting it from the transverse mode, and the relation \(\omega_{LO}^{2}/\omega_{TO}^{2}=\epsilon(0)/\epsilon(\infty)\) [Lyddane:1941]; the reststrahlen band of total reflection between the two frequencies as its direct observation; polaritons as the mixed phonon–photon excitations, and their measured dispersion; the link to dielectric response in Electrodynamics in Matter.]
Measured dispersion relations
Inelastic neutron scattering
[Reserved: why the thermal neutron is the right probe — its de Broglie wavelength matches interatomic spacings while its energy, about \(25\,\mathrm{meV}\), matches phonon energies, so energy and momentum transfer are measured simultaneously; the triple-axis spectrometer and the first phonon measurement, in a single crystal of aluminium [Brockhouse:1955]; the complete branch-by-branch dispersion of germanium [Brockhouse:1958], compared with Born–von Kármán force-constant fits; conservation laws with crystal momentum, including umklapp processes.]
A neutron scattered by a crystal exchanges a definite energy and a definite wavevector with it, and the pairs so measured do not fill the plane: they lie on curves. Scanning energy transfer against wavevector transfer in a single crystal of aluminium [Brockhouse:1955] and, branch by branch through the Brillouin zone, in germanium [Brockhouse:1958], one measures \(\omega(\vect{k})\) directly. Three features of the measured curves are the substance of lattice dynamics. The dispersion is linear only at long wavelength; it flattens towards the zone boundary, where the group velocity vanishes, so a crystal is not an elastic continuum. It is periodic in the reciprocal lattice, so \(\vect{k}\) is defined only modulo a reciprocal-lattice vector. And a crystal with \(p\) atoms in its primitive cell shows \(3p\) branches, of which three are acoustic, vanishing at \(\vect{k}=0\), and the remaining \(3p-3\) are optical, with a finite frequency there. The existence of a sharp \(\omega\) for each \(\vect{k}\) is what entitles one to call the excitation a particle.
Derivation. The simplest case exhibits every feature. Take a chain of identical masses \(M\) at spacing \(a\), coupled to nearest neighbours by springs of constant \(K\); let \(u_{n}\) be the displacement of the \(n\)-th mass. In the harmonic approximation
The equations are invariant under \(n\to n+1\), so their normal modes may be labelled by the eigenvalue of that translation: try \(u_{n}=A\,\ee^{\ii(kna-\omega t)}\). Substituting into Equation (125.6) gives \(-M\omega^{2}=K(\ee^{\ii ka}+\ee^{-\ii ka}-2)=-4K\sin^{2}(ka/2)\), that is
Equation (125.7) is periodic under \(k\to k+2\pi/a\), so nothing outside the first Brillouin zone \(\abs{k}\leq\pi/a\) is new: a wave shorter than twice the spacing is indistinguishable from a longer one, because only the discrete \(u_{n}\) exist. For \(ka\ll1\) it reduces to \(\omega\approx a\sqrt{K/M}\,\abs{k}\), the acoustic branch, whose slope is the sound velocity of Continuum Mechanics and Elasticity. At the zone boundary \(\dd\omega/\dd k=0\): the group velocity vanishes and the mode is a standing wave, which is the lattice-dynamical counterpart of Bragg reflection at the same wavevector. Repeating the calculation with two different masses per cell splits the single branch into an acoustic one that still vanishes at \(k=0\), in which neighbours move together, and an optical one with a finite gap there, in which they move against each other [Born:1912]; the general statement of \(3p\) branches follows from diagonalizing the \(3p\times3p\) dynamical matrix. Quantizing each mode as an oscillator makes \(\hbar\omega(\vect{k})\) the energy of one phonon, which is what the neutron measures.
∎Inelastic X-ray and light scattering
[Reserved: Raman scattering as the optical route to zone-centre optical phonons, from its discovery in liquids [Raman:1928] to its routine crystallographic use; Brillouin scattering for acoustic phonons and the sound velocities it yields; high-resolution inelastic X-ray scattering at synchrotron sources, which reaches phonons in samples too small or too absorbing for neutrons [Burkel:2000]; the complementary selection rules of the three techniques.]
The Debye–Waller factor
[Reserved: the attenuation \(\ee^{-2W}\) of Bragg intensities by thermal displacement, with \(2W=\langle(\vect{Q}\cdot\vect{u})^{2}\rangle\), derived by Debye [Debye:1913] and corrected for the quantum zero-point term by Waller [Waller:1923]; the crucial fact that thermal motion weakens the spots but does not broaden them, the intensity moving into a diffuse background; recoilless nuclear resonance absorption as the same factor in the gamma-ray regime [Moessbauer:1958], and the mean square displacements it measures.]
Anharmonicity
Thermal expansion and the Grüneisen parameter
[Reserved: the theorem that a strictly harmonic crystal does not expand, so thermal expansion measures the cubic term directly; the Grüneisen parameter \(\gamma_{G}=-\dd\ln\omega/\dd\ln V\) and the relation \(\alpha=\gamma_{G}\kappa_{T}C_{V}/V\) [Gruneisen:1912]; the observed near-constancy of \(\gamma_{G}\approx2\) across many solids and the parallel temperature dependence of \(\alpha\) and \(C_{V}\); negative thermal expansion in silicon and in framework structures as the instructive exception.]
The volume thermal expansion coefficient \(\alpha\) of a solid is not an independent material function of temperature: measured over a wide range it is proportional to the heat capacity \(C_{V}\) of the same sample, falling towards zero at low temperature in the same way and saturating at the same temperature, so that the dimensionless ratio
is nearly independent of temperature and takes a value of order two across a wide range of solids [Gruneisen:1912]. A strictly harmonic crystal would not expand at all — the mean position of a harmonic oscillator is independent of its energy — so the whole of the effect measures the anharmonic part of the interatomic potential, and Equation (125.8) says that one number suffices to describe it. The exceptions are as instructive as the rule: silicon and several open framework structures contract on heating over some temperature ranges.
The Grüneisen relation: that a phonon frequency depends on volume, quantified by the logarithmic derivative of frequency with respect to volume; the resulting thermal pressure of the phonon gas; and the equality of that pressure gradient with the elastic restoring stress, which yields the expansion coefficient as the product of the compressibility, the heat capacity and the mean logarithmic derivative, together with the sign reversal produced by transverse modes that stiffen under expansion
Phonon–phonon scattering and umklapp
[Reserved: three-phonon processes as the leading anharmonic interaction; Peierls's decisive observation that normal processes conserve total crystal momentum and therefore cannot degrade a heat current, so only umklapp processes — in which the summed wave vector lands outside the Brillouin zone and is returned by a reciprocal-lattice vector — give a finite thermal resistance [Peierls:1929b]; the resulting \(\ee^{\theta_{D}/bT}\) freezing-out of umklapp at low temperature; phonon lifetimes measured as neutron linewidths.]
Thermal conductivity of insulators
[Reserved: the kinetic-theory expression \(\kappa=\tfrac{1}{3}C v\ell\) and the characteristic peak in \(\kappa(T)\) where umklapp scattering gives way to boundary scattering [Ziman:1960]; isotope scattering as an independently controllable term, demonstrated by the roughly \(50\,\mathrm{\%}\) rise in the conductivity of diamond on isotopic enrichment [Anthony:1990]; measured conductivities in \(\mathrm{W}/\mathrm{m}/\mathrm{K}\) for representative insulators; the transport formalism of Nonequilibrium Thermodynamics and Transport.]
A pure dielectric crystal conducts heat at a finite rate, and its conductivity is a strongly non-monotonic function of temperature: it rises steeply on cooling, passes through a pronounced maximum at a temperature of order a few per cent of the Debye temperature, and then falls again towards zero. Both flanks are informative. The high-side fall is exponential in \(\theta_{D}/T\); the low-side fall follows the heat capacity, and its magnitude depends on the size and surface of the specimen, which no bulk material property should. That the conductivity is limited by the crystal's own imperfections is shown decisively by isotopes: enriching diamond in carbon-12 raises its room-temperature thermal conductivity by roughly \(50\,\mathrm{\%}\), without changing its chemistry, its structure or its phonon spectrum to any comparable degree [Anthony:1990].
Why a harmonic crystal has infinite thermal conductivity, and why anharmonicity alone is not enough: that three-phonon processes conserving the total crystal momentum cannot degrade a heat current, so that a finite thermal resistance requires umklapp processes in which the summed wavevector leaves the Brillouin zone and is returned by a reciprocal-lattice vector; the exponential freezing-out of those processes at low temperature; and the kinetic-theory expression for the conductivity in terms of heat capacity, group velocity and mean free path, whose crossover to boundary-limited and isotope-limited scattering produces the observed maximum