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Although both Eqs. (8)–(10) are governed by the same underlying e–ph Hamiltonian, and share the same matrix element $g_{mn}^{\nu}(k,q)$, the relevant lifetimes and transport mechanisms depend strongly on the dominant carrier type, the density of states, and the screening properties of the material. In the next sections we will study different materials, their properties and how their thermal conductivity is affected by e–ph interaction.
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3 Lattice Thermal Conductivity in Metals: The Dominant Role of Electron–Phonon Interaction
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In crystalline metals the steady-state heat flux obeys Fourier’s law
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$$J = -(k_e + k_{\text{ph}}) \nabla T, \quad \kappa_{\text{tot}} = k_e + k_{\text{ph}},$$
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(12)
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where $k_e$ and $k_{\text{ph}}$ (W m$^{-1}$ K$^{-1}$) are the electronic and lattice contributions, respectively. The additive decomposition $\kappa_{\text{tot}} = k_e + k_{\text{ph}}$ holds when e–ph drag is negligible. With drag, cross-coefficients appear in the coupled BTE and the simple sum is an approximation. First-principles calculations for 18 elemental and intermetallic metals show that once e–ph scattering is included, $k_{\text{ph}}$ may supply 1–40% of $\kappa_{\text{tot}}$ at 300 K, contradicting the common assumption that phonons are negligible in metals[9]. This is shown in Fig. 1(a).
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Working in the lattice framework, within the single-mode relaxation-time approximation (RTA) the lattice conductivity tensor is
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$$\kappa_{\text{ph}}^{\alpha\beta} = \sum_{\lambda} c_{v,\lambda} v_{\lambda}^{\alpha} v_{\lambda}^{\beta} \tau_{\lambda}^{\text{tot}},$$
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(13)
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with mode label $\lambda = (q,\nu)$, $c_{v,\lambda} = k_B x_\lambda^2 e^{x_\lambda} / (e^{x_\lambda} - 1)^2$, $x_\lambda = \hbar\omega_\lambda/k_BT$, and $v_{\lambda}^{\alpha} = \partial\omega_\lambda/\partial q_\alpha$. Matthiessen’s rule gives
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$$\left(\tau_{\lambda}\right)^{-1} = \left(\tau_{\lambda}^{\text{ph-ph}}\right)^{-1} + \left(\tau_{\lambda}^{\text{e-ph}}\right)^{-1}.$$
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(14)
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At very low temperatures ($T \lesssim \Theta_D/10$), beyond-RTA treatments with normal-process vertex corrections (e.g., Callaway-/hydrodynamic-formalisms) are required for quantitative accuracy.
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3.1 Phonon–phonon scattering
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For a given phonon mode $\lambda = (q,\nu)$ the inverse lifetime due to three–phonon interactions is obtained from Fermi’s Golden Rule applied to the cubic-anharmonic lattice Hamiltonian. Writing the anharmonic term in second-quantised form yields a matrix element $V_{\lambda\lambda_1\lambda_2}$ describing the quantum amplitude for decay ($\lambda \rightarrow \lambda_1 + \lambda_2$) and combination ($\lambda + \lambda_1 \rightarrow \lambda_2$) processes. Summing the corresponding transition probabilities over all final modes and enforcing both energy and crystal–momentum conservation leads to
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$$\frac{1}{\tau_{\lambda}^{\text{ph-ph}}} = \frac{\pi}{16\hbar N} \sum_{\lambda_1\lambda_2} |V_{\lambda\lambda_1\lambda_2}|^2 D_{\lambda\lambda_1\lambda_2},$$
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(15)
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where $N$ is the number of $q$ points and $D_{\lambda \lambda_1 \lambda_2}$ is a compact phase–space factor containing the Bose–Einstein occupations and the $\delta$-functions that impose energy and (Normal/Umklapp) momentum conservation. The term $|V_{\lambda \lambda_1 \lambda_2}|^2$ encapsulates the strength of the lattice anharmonicity via third-order interatomic force constants, while $D_{\lambda \lambda_1 \lambda_2}$ determines which scattering channels are thermally active.
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### 3.2 Electron–phonon scattering
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For a given phonon mode $\lambda = (q, \nu)$, Fermi’s Golden Rule gives the probability per unit time for an electron to scatter by emitting or absorbing that phonon. The inverse phonon lifetime reads
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$$ (\tau_{e-ph}^{-1}) = \frac{2\pi}{\hbar} \sum_{ijk} |g_{j,k+q,i,k}|^2 (f_{ik} - f_{jk+q}) \delta(\epsilon_{ik} - \epsilon_{jk+q} + \hbar \omega_\lambda), $$
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(16)
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where the summation spans all initial and final electronic bands and wave vectors. The difference of Fermi–Dirac factors enforces Pauli blocking, and the Dirac delta conserves energy while allowing for phonon absorption or emission. This expression shows that a large electronic density of states around the Fermi level, sizeable matrix elements, or high phonon frequencies all shorten $\tau_{e-ph}$ and thus diminish lattice heat transport.
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The coupling amplitude that mediates these transitions is
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$$ g_{j,k+q,i,k} = \sqrt{\frac{\hbar}{2\omega_\lambda}} \langle \psi_{j,k+q} | \partial_\lambda U | \psi_{i,k} \rangle, $$
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(17)
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where $\psi_{i,k}$ is the Bloch wavefunction of band $i$ and $\partial_\lambda U$ is the derivative of the self-consistent crystal potential with respect to the normalized lattice displacement of mode $\lambda$. The prefactor $\sqrt{\hbar/2\omega_\lambda}$ fixes the single-quantum normalisation. Equation (16) therefore quantifies how efficiently electrons damp individual phonon modes and set the electron-limited contribution to the phonon thermal conductivity.
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### 3.3 Klemens–Williams analytic limit
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Treating the conduction band as a free electron gas that interacts with long-wavelength acoustic phonons through a deformation potential, Klemens and Williams obtained a closed-form e–ph lifetime
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$$ (\tau_{e-ph}^{-1}) = \frac{\pi}{3} \frac{v_g}{v_F} \frac{n_e D_{e-ph}^2}{\mu E_F} \omega, $$
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(18)
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where $v_g$ is the acoustic phonon group velocity, $v_F$ the electronic Fermi velocity, $n_e$ the conduction-electron density, $D_{e-ph}$ a deformation-potential-like constant, $\mu$ the shear modulus (Pa), and $E_F$ the Fermi energy. This form is the Klemens–Williams long-wavelength limit for electron damping of acoustic modes and is linear in $\omega$. While the prefactor captures how stronger e–ph coupling, higher carrier density, or a smaller Fermi velocity accelerate phonon damping.
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Insertion of the e–ph lifetime into a Debye model for the lattice conductivity yields
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$$ \kappa_{ph}^{(ph-ph+e-ph)} = \kappa_{ph}^{(ph-ph)} \left[ 1 - \frac{\omega_i}{\omega_D} \ln \left( \frac{\omega_D}{\omega_i} + 1 \right) \right], \quad \omega_i = \omega_D \frac{v_g}{v_F} \frac{\pi n_e}{3B \gamma_G} \frac{D_{e-ph}^2}{k_B T E_F}, $$
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(19)
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where $\omega_D$ is the Debye cut-off, $B$ the bulk modulus, and $\gamma_G$ the Grüneisen parameter. The ratio $\omega_i/\omega_D$ acts as a compact figure of merit: stronger deformation potentials, higher carrier densities, or lower Fermi velocities $v_F$ increase $\omega_i$, amplifying the logarithmic term and more strongly suppressing $\kappa_{ph}$. In weakly coupled (noble) metals, by contrast, $\omega_i$ stays small, so the lattice contribution is only marginally reduced.
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### 3.4 Electron transport under e–ph control
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While the preceding subsections quantify how electrons limit phonon heat flow, the same matrix elements govern electronic transport. Starting from the linearised Boltzmann equation and Onsager reciprocity, the electrical conductivity tensor is
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$$ \sigma^{\alpha \beta} = -\frac{e^2 n_s}{V} \sum_{ik} \frac{\partial f_{ik}}{\partial \epsilon} v_{ik}^\alpha v_{ik}^\beta \tau_{ik}, $$
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(20)
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where $\sigma^{\alpha \beta}$ are the Cartesian components of the electrical conductivity tensor, $\tau_{ik}$ is the lifetime of the electronic states, $n_s$ is the spin degeneracy factor, and $V$ is the normalisation volume of the crystal used in the Brillouin-zone sum.
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Unless otherwise stated, the Brillouin-zone sums are over all bands/valleys and the explicit spin factor \( n_s \) accounts for spin degeneracy only (no double counting). The derivative \( \partial f_{\mathbf{k}} / \partial \epsilon \) strongly peaks at the Fermi energy, so only states near \( E_F \) contribute. The electronic part of the thermal conductivity follows from Onsager reciprocity
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\[
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\kappa_e = K - T \sigma S^2,
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\]
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(21)
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with \( K \) the open-circuit heat–current coefficient and \( S \) the Seebeck coefficient. Both transport coefficients are therefore governed by the same microscopic lifetime \( \tau_{ik} \), which can be written as
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\[
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\frac{1}{\tau_{ik}} = \frac{1}{\tau_{ik}^{e-e}} + \frac{1}{\tau_{ik}^{e-ph}} + \cdots
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\]
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(22)
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In metals where the electron lifetime is dominated by e–ph interactions,
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\[
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(\tau_{ik}^{e-ph})^{-1} = \frac{2\pi}{\hbar} \sum_{j\lambda} |g_{j,k+q,i,k}^\lambda|^2 \left[ (n_\lambda + f_{j,k+q}) \delta(\epsilon_{ik} + \hbar \omega_\lambda - \epsilon_{j,k+q}) \right. \\
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\left. + (n_\lambda + 1 - f_{j,k+q}) \delta(\epsilon_{ik} - \hbar \omega_\lambda - \epsilon_{j,k+q}) \right],
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\]
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(23)
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with \( n_\lambda \) the Bose occupation of phonon mode \( \lambda \). Larger \( |\mathbf{q}|^2 \), a denser electronic DOS near \( E_F \), or a higher thermal phonon population all shorten \( \tau_{ik} \), thereby reducing both \( \sigma \) and \( \kappa_e \) [44, 45, 46]. Ziman’s Bloch–Grüneisen analysis [44] and Grimvall’s self-energy formalism [45] remain standard references and analytical benchmarks for low-temperature power laws and the separation of momentum- and energy-relaxation times. Allen’s treatment of the Eliashberg spectral function established quantitative rules for resistivity slopes and Lorenz-number renormalisation in strongly coupled metals [46, 47].
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Density-functional perturbation theory combined with Wannier interpolation marked the first-principles advance and delivers state-resolved \( g_{j,k+q,i,k}^\lambda \) on ultradense grids, enabling parameter-free predictions of \( \sigma(T) \) and \( \kappa_e(T) \) for transition metals [48, 9]. These calculations indicate that e–ph scattering alone can reproduce the measured resistivity of Cu, Ag and Au in good agreement with experiment once vertex corrections are included [49].
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Furthermore, numerical solvers and codes such as BOLTZTRAP2 [50] and EPW [51] implement the linearised Boltzmann equation with onboard e–ph matrix elements, while SHENGBTE extensions have begun to treat the coupled electron and phonon problems on equal footing [52]. Together these analytical and computational tools provide a seamless bridge from the long-established Bloch–Grüneisen picture to modern ab initio, mode-resolved transport in metals.
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### 3.5 Key damping parameters
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For a given phonon mode \( \lambda = (\mathbf{q}, \nu) \), Eq. (16) shows that the electron–phonon damping rate is controlled by three microscopic ingredients.
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(i) **Electronic phase space.** The joint availability of initial–final electronic states near \( E_F \) scales with the Fermi-level density of states \( N(E_F) \) (and the detailed band velocities). A larger \( N(E_F) \) increases the joint DOS, shortens phonon lifetimes, and lowers \( \kappa_{ph} \); \( d \)-band transition metals typify this high-DOS regime.
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(ii) **Energy window set by the phonon frequency.** The \( \delta \)-function in Eq. (16) restricts scattering to an energy shell of width \( \hbar \omega_\lambda \). In simple three-dimensional (nearly free-electron) metals the joint DOS grows roughly linearly with \( \omega_\lambda \), so high-frequency modes experience stronger damping than long-wavelength acoustics. (In monatomic fcc metals such as Cu/Ag/Au there are no optical branches; the “high-\( \omega \)” remark applies to multi-atom metals/intermetallics.)
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(iii) **Matrix-element strength.** The coupling \( |g_{j,k+q,i,k}^\lambda|^2 \) measures the sensitivity of the self-consistent potential to atomic displacements and reflects bonding character and deformation potentials; long-range polar (Fröhlich-like) contributions are strongly screened in good metals but can matter in polar intermetallics or at low carrier density.
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In combination, large \( N(E_F) \), higher \( \omega_\lambda \), and stronger \( |g_\lambda|^2 \) conspire to shorten phonon lifetimes and suppress the lattice contribution to heat flow. This explains why \( d \)-band transition metals and many ordered intermetallics can lose up to \( \sim 40\% \) of \( \kappa_{ph} \) once e–ph scattering is included, whereas noble metals—with strong screening and low \( N(E_F) \)—are only weakly affected.
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### 3.6 Dimensionality, screening, and electron–electron corrections
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**Confinement effects.** Femtosecond pump–probe measurements on Au and Ag nanoparticles show that electron cooling times decrease markedly with size—from \( \sim 850 \) fs at diameters \( \sim 25–30 \) nm to \( \sim 500 \) fs at \( \sim 3 \) nm—reflecting enhanced surface scattering and modified e–ph coupling in confined geometries [53, 54, 55]. Because the dominant
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heat-carrying phonons in metals already have short mean-free paths, additional boundary scattering in thin films, nanowires, and grain-refined materials further suppresses the lattice contribution $\kappa_{\text{ph}}$ [56] [57]. This can be seen in Fig. 1(b).
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**Electrostatic screening.** Both e–ph and e–e rates depend on the screened Coulomb interaction, well approximated in metals by a Yukawa (Thomas–Fermi/Lindhard) potential,
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$$V(r) = \frac{e^2}{4\pi\epsilon_0} \frac{e^{-k_s r}}{r},$$
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(24)
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where the screening wave vector $k_s$ may be taken as the Thomas–Fermi value $k_{\text{TF}}$ or, more generally, as the static/dynamic Lindhard function $k_s(q,\omega)$ [14] [27]. Stronger (“hard”) screening (large $k_s$) reduces the range of long-range fields and weakens Fröhlich-like couplings; weaker (“soft”) screening has the opposite effect [44] [48].
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**Electron–electron corrections.** For transport coefficients, the e–e contribution is sensitive to how screening is treated. Partial-wave (phase-shift) solutions of the e–e scattering problem with a consistently screened potential show that first-Born estimates overstate the e–e thermal resistivity by roughly a factor of two; using the same $k_s$ in both treatments restores agreement and avoids the larger ($\sim 5\times$) discrepancies quoted in older literature [58] [59]. Accurate accounting of e–e scattering is therefore essential when deconvolving $\kappa_e$ and $\kappa_{\text{ph}}$ in nanoscale metals and in ultrafast pump–probe analyses [55] [56].
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| Metal class | $\kappa_{\text{ph}}^{(\text{ph-ph})}$ | $\kappa_{\text{ph}}^{(\text{ph-ph+e-ph})}$ | $\kappa_{\text{ph}}/\kappa_{\text{tot}}$ | Main limiting factor(s) |
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|-------------|-----------------|-----------------|-----------------|--------------------------|
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| Noble (Cu, Ag, Au) | 3–10 | 2–6 | < 10% | Low $N(E_F)$, weak EPC, strong screening |
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| Transition metals (Ni, Co, Pt) | 5–30 | 3–18 | 10–40% | Large $N(E_F)$, high $\omega_\lambda$, strong deformation-potential coupling |
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| Ordered intermetallics | similar to parent | significantly lower | variable | Optical-branch EPC dominates (multi-atom bases) |
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*Ranges are indicative values in W m$^{-1}$ K$^{-1}$ at $T \approx 300$ K. $\kappa_{\text{ph}}^{(\text{ph-ph})}$: only phonon–phonon scattering; $\kappa_{\text{ph}}^{(\text{ph-ph+e-ph})}$: phonon–phonon plus electron–phonon scattering. EPC $\equiv$ electron–phonon coupling; $N(E_F)$ $\equiv$ electronic density of states at the Fermi level; $\omega_\lambda$ $\equiv$ phonon frequency.
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The table compares typical room-temperature lattice thermal conductivities for three metal families, first assuming ph–ph scattering alone, $\kappa_{\text{ph}}^{(\text{ph-ph})}$, and then after e–ph damping is included, $\kappa_{\text{ph}}^{(\text{ph-ph+e-ph})}$. In noble metals such as Cu, Ag, and Au a low electronic density of states together with strong screening reduce the lattice channel only modestly, from $\sim 3$–10 to $\sim 2$–6 W m$^{-1}$ K$^{-1}$, so phonons carry $< 10\%$ of the total heat flow. Transition metals (Ni, Co, Pt) possess a large $N(E_F)$, stiffer phonons, and strong deformation-potential couplings; e–ph scattering therefore cuts the lattice conductivity from $\sim 5$–30 to $\sim 3$–18 W m$^{-1}$ K$^{-1}$, leaving phonons responsible for 10–40 % of overall conduction. Ordered intermetallics such as CuAu or Cu$_3$Au behave similarly to their parent elements but can experience additional damping because optical branches couple strongly to electrons in their multi-atom bases. Hence the combined magnitude of $N(E_F)$, phonon frequency, and the e–ph matrix element dictates how severely the lattice channel is suppressed, ranging from almost negligible in noble metals to quantitatively significant in d-band and intermetallic systems.
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### 3.7 Summary and open challenges
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Electron–phonon scattering, not ph–ph interactions, is the principal bottleneck for lattice heat transport in most bulk metals. Modern first-principles e–ph frameworks now deliver mode-by-mode values for the phonon conductivity $\kappa_{\text{ph}}$ and the electronic conductivity $\kappa_e$, replacing semi-empirical Debye–Klemens models that can miss the mark by more than a factor of three. Building on these advances, three avenues stand out for future work:
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**Extend e–ph coupling data to new regimes,** Current databases focus on non-magnetic, near-equilibrium metals at 300 K. Including finite-temperature magnetism, strong spin–orbit coupling, and ultrafast laser-excited nonequilibrium states will make predictions relevant to magnetic memory devices, spintronics, and pump–probe experiments.
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Figure 1: (a) Phonon vs. electron share of $\kappa_{\text{total}}$ at 300 K: Phonons contribute < 10% of $\kappa_{\text{tot}}$ in noble/alkali/NIC metals but 10–40% in transition/TIC metals (even though $\kappa_{\text{ph}} \approx 3–15 \text{ W m}^{-1} \text{ K}^{-1}$). Higher $\sigma$ in nobles and stronger e-ph coupling in transitions explain the contrast. (b) Mean free paths at 300 K: Average mean free paths (MFPs) at 50% accumulation of thermal conductivity show phonons $\lesssim 10 \text{ nm}$ for all 18 metals, while electrons are $\sim 5–25 \text{ nm}$; electron MFPs generally exceed phonon MFPs, implying a stronger size effect in $\kappa_e$ for metal nanostructures. Figures taken from Tong et al.\cite{9}
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**Benchmark against experiment.** Systematic comparisons with Lorenz-number measurements, time-domain thermoreflectance, and femtosecond electron diffraction will validate (or refine) first-principles lifetimes, ensuring that theoretical gains translate into quantitative accuracy.
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**Exploit nanoscale size effects.** Phonon mean-free-path spectra show that, once limited by e–ph coupling, the dominant heat-carrying phonons are scattered after travelling only a few to a few tens of nanometres. Tailoring layer thicknesses, grain sizes, or nanowire diameters offers a practical route to engineer heat flow in nano-interconnects, metallic superlattices, and thermoelectric barriers.
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In short, a concerted push to widen e–ph datasets, anchor them to precise measurements, and integrate size-dependent design rules will turn today’s qualitative understanding into predictive control of thermal transport across the metallic landscape.
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### 4 Electron–phonon–limited lattice heat transport in semiconductors
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#### 4.1 Ab initio foundation
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We use the density functional theory (DFT) to obtain the electronic eigenenergies $\epsilon_{n,k}$ and wave-functions $\psi_{n,k}$, followed by density-functional perturbation theory (DFPT) for phonon frequencies $\omega_{\nu,q}$ and eigendisplacements $e_{\nu,q}$. Within the DFT/DFPT formalism\cite{28}, the linear, self-consistent change in the Kohn–Sham potential associated with a phonon mode $(\nu, q)$ can be written as a mass–weighted sum of Cartesian derivatives:
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$$
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\Delta_{\nu,q} V(r) = \sum_{\kappa,\alpha} \frac{e^{\nu,q}_{\kappa,\alpha}}{\sqrt{M_{\kappa}}} \partial_{q,\kappa,\alpha} V(r), \quad \partial_{q,\kappa,\alpha} V(r) = \sum_{R_p} e^{iq,R_p} \frac{\partial V(r)}{\partial R_{p,\kappa,\alpha}},
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$$
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(25)
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where $e^{\nu,q}_{\kappa,\alpha}$ are the phonon eigenvectors and $M_{\kappa}$ the ionic masses. The corresponding electron–phonon (e–ph) matrix element is\cite{48}
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$$
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g^{\nu}_{mn}(k, q) = \sqrt{\frac{\hbar}{2\omega_{\nu,q}}} \langle \psi_{m,k+q} | \Delta_{\nu,q} V | \psi_{n,k} \rangle,
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$$
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(26)
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