- State Fourier’s law for heat transport in solids and define thermal diffusivity.
- Derive the kinetic-theory expression for lattice thermal conductivity from a phonon picture.
- Distinguish boundary, defect, isotope/alloy, normal, and umklapp scattering mechanisms.
- Use Matthiessen’s rule to discuss regimes and crossover scales in \(\kappa(T)\).
- Explain why crystalline insulators typically show a low-temperature peak in \(\kappa(T)\), whereas glasses often show a plateau.
10.1 Roadmap
- Start from the macroscopic description: heat current, Fourier’s law, and the diffusion equation.
- Build the microscopic picture: phonons as heat carriers and the relaxation-time approximation.
- Identify the scattering mechanisms that limit the phonon mean free path.
- Use those mechanisms to understand the characteristic temperature dependence of \(\kappa(T)\) in crystals.
- Contrast crystalline and amorphous solids and prepare the transition to total thermal transport in real materials.
| Symbol | Meaning |
|---|---|
| \(\mathbf{J}_h\) | heat current density |
| \(\kappa\) | lattice thermal conductivity |
| \(D_{th}\) | thermal diffusivity |
| \(c\) | volumetric heat capacity entering the heat equation |
| \(c_V\) | constant-volume heat capacity per unit volume |
| \(\omega_\lambda(q)\) | phonon frequency of branch \(\lambda\) and wave vector \(q\) |
| \(n_\lambda(q)\) | phonon occupation number of mode \((q,\lambda)\) |
| \(\tau_\lambda(q)\) | phonon relaxation time |
| \(\ell_\lambda(q)\) | phonon mean free path, \(\ell_\lambda(q)=v_\lambda(q)\tau_\lambda(q)\) |
| \(v_{\lambda,x}(q)\) | \(x\)-component of the phonon group velocity |
| \(G\) | reciprocal-lattice vector in three-phonon momentum conservation |
| \(\Theta_D\) | Debye temperature |
10.2 From Fourier’s Law to Thermal Diffusivity
Lecture 9 ended with the statement that anharmonicity gives phonons finite lifetimes and therefore finite thermal resistance. We now turn that statement into a transport theory.
The macroscopic constitutive law is Fourier’s law: \[ \mathbf{J}_h=-\boldsymbol{\kappa} \nabla T. \]
Here \(\mathbf{J}_h\) is the heat current density, \(\nabla T\) is the temperature gradient, and \(\boldsymbol{\kappa}\) is the thermal-conductivity tensor. In isotropic solids, \(\boldsymbol{\kappa}\) reduces to the scalar \(\kappa\).
The minus sign enforces the physically correct direction: heat flows from hot to cold.
A local energy-balance equation for a solid with no internal heat source reads \[ c \frac{\partial T}{\partial t}+\nabla\cdot \mathbf{J}_h=0. \]
Here \(c\) is the volumetric heat capacity relevant for the slow temperature field. In a solid, the difference between \(c_P\) and \(c_V\) is usually small, so in the present context one often writes \(c\simeq c_V\).
Combining energy conservation with Fourier’s law gives \[ c \frac{\partial T}{\partial t} = \nabla\cdot\left(\boldsymbol{\kappa} \nabla T\right). \]
If \(\kappa\) is spatially uniform and isotropic, this becomes the heat equation \[ \frac{\partial T}{\partial t} = D_{th} \nabla^2 T, \qquad D_{th}=\frac{\kappa}{c}. \]
Here \(D_{th}\) is the thermal diffusivity. It measures how fast a temperature disturbance spreads, whereas \(\kappa\) measures how much heat current is driven by a given gradient.
Interpretation:
- large \(\kappa\) does not automatically mean fast equilibration if the heat capacity is also large;
- diffusivity is the transport quantity most directly tied to a relaxation time for the temperature field;
- once we obtain \(\kappa\) microscopically, we immediately also obtain \(D_{th}\).
10.3 Microscopic Picture: Phonons Carry Heat
10.3.1 Heat Current in Terms of Phonon Modes
In the phonon picture, each mode \((q,\lambda)\) carries energy \(\hbar\omega_\lambda(q)\) and moves with group velocity \[ v_{\lambda,x}(q)=\frac{\partial \omega_\lambda(q)}{\partial q_x}. \]
Here \(v_{\lambda,x}(q)\) is the component of the group velocity along the transport direction \(x\).
The microscopic heat current density in the \(x\)-direction can therefore be written as \[ J_{h,x} = \frac{1}{V} \sum_{q,\lambda} \hbar\omega_\lambda(q) \left[ \frac{1}{2}+n_\lambda(q) \right] v_{\lambda,x}(q). \]
Here \(V\) is the crystal volume and \(n_\lambda(q)\) is the occupation number of mode \((q,\lambda)\).
In thermal equilibrium, \(J_{h,x}=0\). The reason is simple: \(n_\lambda(q)=n_\lambda(-q)\), while \(v_{\lambda,x}(q)=-v_{\lambda,x}(-q)\), so the contributions from opposite wave vectors cancel pairwise. The zero-point term \(\frac{1}{2}\hbar\omega\) also cancels in transport.
Hence only the deviation from local equilibrium contributes: \[ J_{h,x} = \frac{1}{V} \sum_{q,\lambda} \hbar\omega_\lambda(q) \delta n_\lambda(q) v_{\lambda,x}(q) \]
with \(\delta n_\lambda(q)=n_\lambda(q)-n_\lambda^{0}(q)\) and \(n_\lambda^{0}(q)\) is the Bose occupation number in local thermal equilibrium.
10.3.2 Relaxation-Time Approximation
To determine \(\delta n_\lambda(q)\), we use a stationary Boltzmann equation in the relaxation-time approximation.
This means we first define occupation number in terms of two contributions: \[ \begin{align} & \frac{\partial n_\lambda}{\partial t} = \left(\frac{\partial n_\lambda}{\partial t}\right)_{\mathrm{diff}} + \left(\frac{\partial n_\lambda}{\partial t}\right)_{\mathrm{scatt}},\\ & \left(\frac{\partial n_\lambda}{\partial t}\right)_{\mathrm{scatt}} = -\frac{\delta n_\lambda(q)}{\tau_\lambda(q)}, \end{align} \]
where \(\tau_\lambda(q)\) is the relaxation time of mode \((q,\lambda)\).
Furthermore, we consider only stationary states, i.e., \(\frac{\partial n_\lambda}{\partial t}=0\).
For a weak temperature gradient along \(x\), \[ \left(\frac{\partial n_\lambda}{\partial t}\right)_{\mathrm{diff}} = -v_{\lambda,x}(q) \frac{\partial n_\lambda^{0}(q)}{\partial T} \frac{\partial T}{\partial x}. \]
This means that phonons arriving at \(x\) carry the occupation of a neighboring region with slightly different temperature.
Combining the two terms gives \[ \delta n_\lambda(q) = -\tau_\lambda(q) v_{\lambda,x}(q) \frac{\partial n_\lambda^{0}(q)}{\partial T} \frac{\partial T}{\partial x}. \]
Substituting into the heat-current formula yields \[ J_{h,x} = -\frac{1}{V} \sum_{q,\lambda} \hbar\omega_\lambda(q) \tau_\lambda(q) v_{\lambda,x}^2(q) \frac{\partial n_\lambda^{0}(q)}{\partial T} \frac{\partial T}{\partial x}. \]
Comparison with Fourier’s law gives the mode-sum expression \[ \kappa = \frac{1}{V} \sum_{q,\lambda} \hbar\omega_\lambda(q) \tau_\lambda(q) v_{\lambda,x}^2(q) \frac{\partial n_\lambda^{0}(q)}{\partial T}. \]
For a cubic or isotropic solid, \[ \langle v_{\lambda,x}^2(q)\rangle=\frac{1}{3}v_\lambda^2(q), \] so \[ \kappa = \frac{1}{3V} \sum_{q,\lambda} \hbar\omega_\lambda(q) v_\lambda^2(q) \tau_\lambda(q) \frac{\partial n_\lambda^{0}(q)}{\partial T}. \]
Now define the mode contribution to the volumetric heat capacity as \[ c_{V,\lambda}(q) = \frac{1}{V} \hbar\omega_\lambda(q) \frac{\partial n_\lambda^{0}(q)}{\partial T}. \]
Then \[ \kappa = \frac{1}{3} \sum_{q,\lambda} c_{V,\lambda}(q) v_\lambda^2(q) \tau_\lambda(q). \]
If we introduce the mean free path \[ \ell_\lambda(q)=v_\lambda(q)\tau_\lambda(q), \] the same result can be written as \[ \kappa = \frac{1}{3} \sum_{q,\lambda} c_{V,\lambda}(q) v_\lambda(q) \ell_\lambda(q). \]
In a one-velocity kinetic picture, this simplifies to the following expression for the thermal conductivity:
Here \(c_V\) is the total volumetric heat capacity of the heat-carrying modes, \(v\) is a characteristic group velocity, and \(\ell\) is a characteristic phonon mean free path.
This immediately gives a microscopic interpretation of thermal diffusivity: \[ D_{th} = \frac{\kappa}{c} \simeq \frac{1}{3}v\ell \] if \(c\simeq c_V\) for the relevant temperature range.
Sanity checks:
- phonons with small group velocity contribute little, even if they are strongly occupied;
- optical phonons and zone-edge modes therefore tend to carry heat poorly;
- the central transport problem is to determine which modes are occupied and how far they travel before scattering.
10.4 Scattering Mechanisms and Regimes
The formula \[ \kappa \simeq \frac{1}{3}c_V v \ell \] shows that the temperature dependence of \(\kappa\) is controlled mainly by two ingredients:
- the heat capacity \(c_V(T)\);
- the mean free path \(\ell(T)\).
Lecture 8 already established the behavior of \(c_V(T)\). The new physics in this lecture is due to \(\ell(T)\).
10.4.1 Matthiessen’s Rule
If several scattering mechanisms act independently, the total scattering rate is approximately additive: \[ \frac{1}{\tau} = \frac{1}{\tau_B} + \frac{1}{\tau_D} + \frac{1}{\tau_I} + \frac{1}{\tau_U} +\cdots \] or, equivalently, \[ \frac{1}{\ell} = \frac{1}{\ell_B} + \frac{1}{\ell_D} + \frac{1}{\ell_I} + \frac{1}{\ell_U} +\cdots. \]
Here the subscripts denote boundary, defect, isotope/alloy, and umklapp scattering.
Interpretation: the shortest relevant mean free path dominates.
10.4.2 Boundary Scattering: The Casimir Regime
At sufficiently low temperature, intrinsic phonon-phonon scattering becomes weak and the mean free path is cut off by the sample size or by surface roughness: \[ \ell_B \sim d. \]
Here \(d\) is a characteristic sample dimension. This is the boundary-limited or Casimir regime.
Consequences:
- \(\ell_B\) is approximately temperature independent;
- sample geometry and surface quality matter directly;
- with \(c_V\propto T^3\) at low \(T\), one obtains \(\kappa\propto T^3\) in a clean crystal.
10.4.3 Point Defects, Isotopes, and Alloys
Point defects distort the local mass or force-constant pattern and scatter phonons. For small defects, the scattering is Rayleigh-like, which gives approximately \[ \ell_D^{-1}\propto n_D \omega^4. \]
Here \(n_D\) is the defect density and \(\omega\) is the phonon frequency.
Thus high-frequency phonons are scattered especially strongly.
The same logic applies to isotopes: random mass disorder breaks perfect periodicity and limits \(\ell\). Alloy disorder is an even stronger version of this effect, because it introduces both mass and local bonding disorder. In practice:
- isotope purification raises the peak value of \(\kappa\);
- alloying strongly suppresses \(\kappa\);
- disorder mainly removes long mean-free-path phonons.
10.4.4 Normal and Umklapp Processes
Three-phonon scattering is enabled by anharmonicity. The momentum-conservation condition has two forms: \[ q_1+q_2=q_3 \qquad \text{(normal process)}, \] \[ q_1+q_2=q_3+G \qquad \text{(umklapp process)}. \]
Here \(G\) is a reciprocal-lattice vector.
The key distinction is not merely geometric but transport-theoretic.
For normal processes, the total crystal momentum of the phonon gas is conserved: \[ Q=\sum_{q,\lambda} n_\lambda(q),\hbar q=\mathrm{const}. \]
Within the simple kinetic picture used here, normal processes redistribute phonons but do not directly relax the net heat-carrying drift.
Umklapp processes are different: they transfer crystal momentum to the lattice through \(G\), so they do relax the drifting phonon distribution and generate thermal resistance.
This is the central microscopic answer to the question: why is lattice thermal conductivity finite in a perfect crystal at all?
10.4.5 Thermal Activation of Umklapp Scattering
Umklapp processes require large enough wave vectors that the sum of incoming phonon wave vectors can leave the first Brillouin zone. For that reason, they are thermally activated at low temperature.
We can express this as \[ \tau_U^{-1}(T)\propto e^{-\Theta_D/(bT)}, \] with \(b\) of order unity, often close to \(2\) in simple estimates.
Hence:
- at high \(T\), umklapp scattering is abundant and strong;
- on cooling, umklapp scattering collapses rapidly;
- once it becomes weaker than defect or boundary scattering, those extrinsic mechanisms set the scale of \(\ell\).
10.5 Why Crystals Peak: Typical \(\kappa(T)\) in Insulating Crystals
We now combine the scattering mechanisms with the known phonon heat capacity.
10.5.1 High Temperatures
For \(T\gtrsim \Theta_D\), the phonon heat capacity is approximately constant, \(c_V\approx \mathrm{const}\), while umklapp scattering dominates and typically gives \[ \ell_U\propto \frac{1}{T}. \]
Therefore \[ \kappa(T)\propto \frac{1}{T} \] at sufficiently high temperature.
10.5.2 Intermediate Temperatures
Upon cooling from room temperature, the decrease in umklapp scattering is initially stronger than the decrease in \(c_V\). As a result, \(\ell\) grows quickly and \(\kappa\) rises.
This is the regime where high-purity crystals can become excellent heat conductors.
10.5.3 Low Temperatures
Once intrinsic phonon-phonon scattering has become weak, defect and boundary scattering take over. In a clean specimen with boundary-limited transport, \(\ell\simeq d\) is nearly constant and the low-temperature Debye law gives \[ c_V\propto T^3 \qquad\Longrightarrow\qquad \kappa\propto T^3. \]
Thus the crystal shows a characteristic maximum in \(\kappa(T)\) between the low-temperature \(T^3\) rise and the high-temperature \(1/T\) fall.
This is why crystalline insulators typically peak:
- cooling suppresses umklapp resistance and initially increases \(\kappa\);
- further cooling eventually suppresses the heat capacity itself;
- the competition forces a maximum.
10.5.4 Purity Dependence
Because isotope and defect scattering truncate \(\ell\), cleaner samples show:
- a larger peak value of \(\kappa\);
- a peak shifted to lower temperature or made sharper;
- stronger sensitivity to sample size in the boundary-limited regime.
That is why isotope-pure Si and Ge show such dramatic improvements over natural-abundance material.
10.6 Why Glasses Plateau
The behavior of amorphous solids is qualitatively different because long-range translational order is absent. As a result, the plane-wave phonon picture becomes progressively less reliable once the mean free path approaches the wavelength.
Experimentally, many glasses show three regimes.
10.6.1 High-Temperature Regime
At high temperature, the dominant vibrational wavelength is already very short, and estimates of \(\ell\) can become comparable to or even smaller than the wavelength itself. In that regime, the usual weakly scattered-phonon picture is no longer fully satisfactory.
10.6.2 Plateau Regime
At intermediate temperatures, many glasses show an extended plateau in \(\kappa(T)\).
This is the essential point: in the same temperature window, the heat capacity still rises appreciably. Therefore a nearly constant \(\kappa\) implies that the effective mean free path must decrease strongly with temperature: \[ \kappa \sim \frac{1}{3}c_V v \ell \qquad\Rightarrow\qquad \ell(T)\ \text{must fall as}\ c_V(T)\ \text{rises}. \]
The plateau is evidence for strong disorder-induced scattering, with proposed mechanisms including scattering from disorder, localization tendencies, and coupling to soft modes.
So the intuitive picture between crystalline and amorphous solids is:
- crystals peak because their mean free path grows enormously on cooling before \(c_V\) collapses;
- glasses plateau because disorder keeps \(\ell\) short and strongly temperature dependent even where \(c_V\) is still rising.
10.6.3 Low-Temperature Regime
At very low temperature, amorphous solids often show approximately \[ \kappa(T)\propto T^2. \]
This differs from the crystalline \(T^3\) law and is commonly associated with resonant scattering of long-wavelength vibrations from tunneling two-level systems.
Optional / Further Reading: Why Many Glasses Show a \(T^2\) Law at Very Low Temperature
A standard phenomenology introduces a broad distribution of tunneling two-level systems in the disordered structure. Long-wavelength phonons can resonantly exchange energy with those defects. The common argument is that the resulting scattering rate scales roughly linearly with \(T\), while the phonon heat capacity still scales as \(T^3\), which leads to \[ \kappa(T)\propto T^2. \]
This mechanism is not needed for the main kinetic-theory derivation of \(\kappa\), but it explains why amorphous solids do not revert to the crystalline \(T^3\) law at the lowest temperatures.
10.7 Outlook on the Next Lecture: From Lattice to Total Thermal Transport
In this lecture we treated phonons as the only heat carriers, which is the correct starting point for insulating crystals and glasses.
In metals and doped semiconductors, however, the total thermal conductivity is the sum of multiple channels: \[ \kappa_{\mathrm{tot}}=\kappa_{ph}+\kappa_e+\cdots. \]
Here \(\kappa_{ph}\) is the lattice contribution and \(\kappa_e\) is the electronic contribution:
- the phonon part is controlled by heat capacity, group velocity, and scattering;
- the electronic part will be controlled by band structure, carrier statistics, and momentum-relaxing collisions;
- in good metals, \(\kappa_e\) often dominates, whereas in insulators the present lecture gives the whole story to leading order.
The next lecture will therefore extend thermal transport from insulating lattices to conducting solids, where heat and charge transport become linked.
- Fourier’s law describes heat flow macroscopically, while thermal diffusivity describes the time evolution of temperature disturbances.
- In a phonon gas, the heat current vanishes in equilibrium and appears only because a temperature gradient skews the phonon occupations.
- The central kinetic result is that lattice thermal conductivity is set by heat capacity, phonon speed, and mean free path.
- Boundary scattering dominates in very clean samples at low temperature, defect and isotope/alloy scattering suppress long mean free paths, and umklapp scattering controls the high-temperature resistance of crystals.
- In the simple transport picture used here, normal processes redistribute phonons but do not directly relax the net heat current, whereas umklapp processes do.
- Crystalline insulators peak in \(\kappa(T)\) because umklapp scattering weakens strongly on cooling before the phonon heat capacity collapses.
- Glasses plateau because structural disorder keeps the effective mean free path short and strongly temperature dependent over a broad temperature range.
10.8 Problem Set
Heat Equation and Thermal Diffusivity. Starting from Fourier’s law and local energy conservation, derive the heat equation for a homogeneous isotropic solid and identify the thermal diffusivity.
Kinetic-Theory Form of Thermal Conductivity. Starting from \[ J_{h,x} = \frac{1}{V} \sum_{q,\lambda} \hbar\omega_\lambda(q) \delta n_\lambda(q) v_{\lambda,x}(q) \] together with \[ \delta n_\lambda(q) = -\tau_\lambda(q) v_{\lambda,x}(q) \frac{\partial n_\lambda^{0}(q)}{\partial T} \frac{\partial T}{\partial x} \] derive the isotropic result \[ \kappa = \frac{1}{3} \sum_{q,\lambda} c_{V,\lambda}(q) v_\lambda^2(q) \tau_\lambda(q). \]
Matthiessen’s Rule. Suppose \[ \frac{1}{\ell(T)} = \frac{1}{d} + A\omega^4 + \frac{B}{T} \] is used as a schematic model for boundary, defect, and high-temperature umklapp scattering. Explain which term dominates at low and high temperature, and what qualitative shape you expect for \(\kappa(T)\).
Normal Versus Umklapp. Explain why normal three-phonon processes do not directly produce thermal resistance in the simple kinetic picture, whereas umklapp processes do.
Crystals Versus Glasses. Using the kinetic relation \(\kappa\sim \frac{1}{3}c_Vv\ell\), explain in words why crystals can show a pronounced maximum in \(\kappa(T)\) while glasses often show a plateau instead.