Abstract
It was recently found that the anharmonic phonon–phonon scattering in tungsten is extremely weak at high frequencies, leading to a predominance of electron–phonon scattering and consequently anomalous phonon transport behaviors. In this work, we calculate the phonon linewidths of W along highsymmetry directions from first principles. We find that the weak phonon–phonon scattering can be traced back to two factors. The first is the triple degeneracy of the phonon branches at the P and H points, a universal property of elemental bodycenteredcubic (bcc) structures. The second is a relatively isotropic character of the phonon dispersions. When both are met, phonon–phonon scattering rates must vanish at the P and H points. The weak phonon–phonon scattering feature is also applicable to Mo and Cr. However, in other elemental bcc substances like Na, the isotropy condition is violated due to the unusually soft character of the lower transverse acoustic phonon branch along the ΓN direction, opening emission channels and leading to much stronger phonon–phonon scattering. We also look into the distributions of electron meanfree paths (MFPs) at room temperature in tungsten, which can help engineer the resistivity of nanostructured W for applications such as interconnects.
Introduction
It is widely accepted that, around and above room temperature, electron–phonon coupling in metals has a much weaker effect on phonon scattering than anharmonic phonon–phonon interactions.^{1} This has been verified for some common metals (Al, Ag, Au, Cu, Pt, and Ni) using firstprinciples techniques.^{2,3} However, NbC and W have been identified as exceptions by very recent calculations.^{4,5} In those materials, electron–phonon scattering is comparable to or stronger than phonon–phonon scattering, leading to an anomalously weak temperature dependence of the lattice thermal conductivity (κ_{ph}). Likewise, the unusual temperature dependence of κ_{ph} in NbSe_{3} nanowires below the chargedensitywave transition temperature was found to be related to electron–phonon coupling.^{6} A common feature of these systems is a significant κ_{ph}; specifically, κ_{ph} reaches as much as 46 W/mK in W.^{5} The situation is reminiscent of heavily doped Si, where electron–phonon scattering can also be comparable to phonon–phonon scattering. When the carrier density reaches 1 × 10^{21} cm^{−3}, electron–phonon scattering can lead to a reduction of κ_{ph} by 45% in Si.^{7}
It is difficult, without assuming a value for the Lorenz number, to decouple the lattice and electronic contributions to the thermal conductivity in experimental measurements. The phonon linewidths are the most critical intermediate physical quantities of determining κ_{ph}. These linewidths are accessible to measurement techniques, such as inelastic neutron scattering or Xray scattering, and can therefore provide direct verification of theoretical calculations. Other related quantities include the electron–phonon enhancement of electron mass, the electron–phonon spectral function α^{2}F, the electrical conductivity (σ), the electronic thermal conductivity, and the superconducting transition temperature.
In this paper, we quantify the contributions to the phonon linewidths of W due to electron–phonon and phonon–phonon interactions from first principles. We attribute the weak phonon–phonon scattering at high frequencies, crucial for its anomalous phonon transport properties,^{5} to the elemental bodycenteredcubic (bcc) structure, in which phonon frequencies are triply degenerated at the highsymmetry P and H points. We further use the example of Na, of which the phonon dispersions display unusually strong anisotropy, to illustrate that the bcc structure is a necessary but not sufficient condition to guarantee the weak phonon–phonon scattering. We also study α^{2}F and the meanfree path (MFP) distributions of electrons and phonons for W, relevant to size effects in applications such as nanointerconnects.^{8,9,10,11} Furthermore, we assess the accuracy of Allen’s approximation to calculate the resistivity.
Results and discussion
W and Mo
The calculated phonon dispersions and the contributions to the phonon fullwidth at half maximum (FWHM) for W from anharmonic phonon–phonon scattering at T = 0 K and T = 300 K, from electron–phonon interactions and from isotope scattering, along the same highsymmetry path, are plotted in Fig. 1. Note that the longitudinal acoustic (LA) branch comprises the lowestfrequency modes along the PH segment. The two transverse acoustic (TA) branches are degenerate along ΓPHΓ. At the P and H points, all three branches are degenerate as a result of the spacegroup symmetry of elemental bcc structures, a point that we will explain in more detail in Section “Symmetry analysis for the triple degeneracy at the P and H points”. The linewidths are also the same for these degenerate branches. Isotope scattering is negligible compared to phonon–phonon and electron–phonon scattering.
As expressed in Eq. (2), the contribution to the FWHM from anharmonic phononphonon scattering Γ^{pp} is determined by the equilibrium phonon occupancies, the phase space available for scattering, and the scattering matrix elements V. For the threephonon scattering processes \(\lambda + \lambda ^\prime \rightleftharpoons \lambda ^{\prime \prime}\), where a phonon mode is denoted by a composite index λ comprising both a wavevector q and a branch index p), conservation of energy and quasimomentum requires that the phase velocity of λ″ be greater than at least of one of those for λ and λ′.^{1} Therefore, for typical elemental materials in which the phonon dispersions are nearly isotropic and tend to a horizontal tangent at the zone boundary, λ″ must lie in a branch higher than λ or λ′.^{1} The lowestlying branch phonons are prevented from emission, while the highestlying branch phonons are prevented from absorption.
At 0 K, all lattice modes are unexcited. Since the contributions to Γ^{pp} from absorption processes are proportional to \(n_{\lambda ^\prime }^0  n_{\lambda ^{\prime\prime} }^0\), they are all zero in this 0 K limit, and the only nonzero contributions come from emission processes. Since the phonon dispersions of W can be described as typical with respect to the features mentioned in the previous paragraph, phonons lying on the lowestlying branch throughout the whole Brillouin zone, including the ΓPHΓN path, have vanishing Γ^{pp} at 0 K. The frequency differences between LA and TA modes are small along the PH segment, and only a small number of emission processes are allowed for TA modes. The corresponding Γ^{pp} for the TA branches at 0 K are very small, but not strictly zero. The degeneracy of the TA modes is slightly lifted along the ΓN segment. For similar reasons, the higherlying TA branch along the ΓN path has small but nonzero linewidths. There are many more emission channels for the LA modes, including LA → TA + TA and LA → TA + LA processes. The LA linewidth can therefore reach as high as 0.0045 THz at the N point [Fig. 1b].
An important feature of Γ^{pp} is that it always vanishes at the highsymmetry P and H points regardless of the temperature [Fig. 1b, c]. Although the phonon frequencies do not reach maxima at the P and H boundary points, the group velocities at those points are smaller than those of any branch in the neighborhood of the Γ point. As a result, absorption channels are still prevented by the restriction of energy and quasimomentum conservation. At any other point different from P or H, a phonon lying on a lower branch can be always scattered to a higher branch by absorbing a phonon. For instance, at the N point, TA + TA/LA → LA channels are always allowed.
Electron–phonon scattering is much weaker than anharmonic phonon–phonon scattering at the zone center. However, it dominates over the latter close to the zone boundary, mainly due to the weakness of phonon–phonon scattering, which vanishes at the P and H points. The contribution to the FWHM from electronphonon scattering Γ^{el} reaches a maximum value of over 0.08 THz around the H point. This unusual predominance of Γ^{el} at high frequencies leads to anomalous phonon transport behaviors.^{4,5} Γ^{el} is temperature independent, whereas above room temperature Γ^{ph} is approximately proportional to T. This is easy to understand since the Debye temperature of W (383 K) is only slightly higher than room temperature.^{12} Note that anharmonic phonon–phonon scattering, as calculated here, is limited to threephonon processes. Considering higher orders would increase the FWHM slightly.^{13} Additionally, the total FWHMs at P and H points will display weak temperature dependence.
The feature of weak phonon–phonon scattering is not unique to the individual bcc substance W. We have also studied another elemental bcc system Mo. The phonon dispersions of Mo are plotted in Fig. 2a. The phonon dispersion shape of Mo looks rather similar to that of W. Γ^{pp} is also zero at P and H points [Fig. 2b, c]. Furthermore, the frequency differences between the LA and TA modes along the PH path are even smaller than in W. As a result, the phonon scattering rates between P and H are smaller and almost vanishing even at T = 300 K. Considering that chromium also has similar phonon dispersions,^{14} it can be expected to also share this feature.
Just like the vanishing Γ^{pp} at P and H, the exponent of the powerlaw dependence of the acoustic phonon lifetimes on q at low temperatures and close to the zone center depends on the crystal structure, as pointed out by Herring.^{15} However, as demonstrated recently for the cases of GaAs^{16} and Si,^{17} this asymptotic powerlaw dependence can and does break down further away from the zone center, due to the interplay between Herring and nonHerring phonon–phonon processes. Note that the present work focuses on a phenomenon at the P and H points, which are located at the zone boundary, and is thus not directly connected to that universal behavior at the zone center.
Symmetry analysis for the triple degeneracy at the P and H points
As discussed above, the triple degeneracy of phonon frequencies at the P and H points is a crucial ingredient for the vanishing Γ^{pp}. This degeneracy is inherent in elemental bcc structures, which belong to the spacegroup Im\(\bar 3\)m (No. 229), so it is worth explaining its origin in more detail.
As shown in Fig. 3, there are six symmetry equivalent H points in the Brillouin zone, located along the three Cartesian axes in the figure. All those points are identical up to a translation by a certain reciprocal lattice vector. For any phonon wavevector q along a given ΓH direction, a rotation of 90° around ΓH leaves both the structure and q invariant; therefore, the two TA branches must be degenerate along the ΓH direction. To show that LA is also degenerate with those two TA modes at H, we take the example of H_{1}, a point at which the LA mode vibrates along the xaxis. The symmetry operation of rotating around the zaxis by 90° maps this LA mode to a vibration along the yaxis at H_{3}. Since H_{1} and H_{3} are identical, the y direction vibration at H_{3} is actually one TA mode at H_{1}. This proves that the LA and TA branches must be degenerate at H_{1}, and thus at each H point.
The case of P points is similar. There are eight symmetry equivalent P points in the Brillouin zone. If we restrict ourselves to translations by a reciprocal lattice vector, there are two equivalence classes, namely {P_{1}, P_{4}, P_{6}, P_{7}} and {P_{2}, P_{3}, P_{5}, P_{8}}, following the notation of of Fig. 3. For any q along a particular ΓP direction, the symmetry operation of rotating around this axis by 120° can mix up the two TA modes. Those are thus required to be degenerate along the ΓP direction. At P_{1} the LA mode vibrates along the \({\hat{\mathbf x}} + {\hat{\mathbf y}} + {\hat{\mathbf z}}\) direction. The symmetry operation of rotating around the zaxis by 180° will transform this mode into a vibration along the \( {\hat{\mathbf x}}  {\hat{\mathbf y}} + {\hat{\mathbf z}}\) direction at P_{4}. The latter actually corresponds to a certain superposition of the TA and LA modes at P_{1}, since P_{1} and P_{4} are identical. Therefore, the LA and TA modes should be degenerate at P_{1} and any P point.
Systems with soft phonons: the case of Na
In spite of its importance as a necessary condition, the triple degeneracy of phonon frequencies at the P and H points shared by all elemental bcc systems is not sufficient to guarantee that Γ^{pp} vanishes there. We illustrate this by studying the case of Na. The phonon dispersions of Na are plotted in Fig. 4a, and display strong anisotropy, particularly for the TA modes. The lower TA branches are unusually soft along the ΓN direction as compared to other directions, consistent with a bcchcp structural phase transition.^{18} This makes a huge difference in terms of phase space for threephonon scattering: in contrast to the cases of W and Mo, where the emission channels for the lowestlying branches are completely forbidden, the lowestlying phonons with wavevectors from other directions of the Brillouin zone can easily emit a phonon lying in the lower TA branch along the ΓN direction. As a result [see Fig. 4b], Γ^{pp} at 0 K, which only involves contributions from emission processes, has large nonzero values for the lowestlying branches everywhere except along the ΓN path. At 300 K, Γ^{pp} is very large for the lowestlying branch along the ΓN path [Fig. 4c], due to numerous allowed absorption processes, each one the inverse of an emission process. Given the fact that the group velocities at P are larger than those for the lowerlying TA branch along the ΓN direction, the absorption channels are also opened at P. However, absorption processes are forbidden at H. We note that many other bcc elemental substances, including Li, K, Rb, Cs,^{19} Ba^{20}, and Ta^{21} also possess soft phonons along the ΓN direction, and therefore Γ^{pp} is excepted to be similarly strong in those systems.
Electrical transport and MFPs of electrons and phonons in W
Moving on to electrical transport, our calculated Eliashberg electron–phonon spectral function α^{2}F and its transport variant α^{2}F_{tr} are plotted in Fig. 5 and lead to values of 0.29 and 0.28 for total electronphonon coupling constant λ and its transport counterpart λ_{tr}, respectively. Our value of λ is slightly larger than the literature value of 0.28, which was obtained based on estimates^{22,23} or firstprinciples calculations.^{24} The resistivity afforded by Eq. (7) is plotted in Fig. 6. It slightly overestimates the exact solution, which was obtained by solving the BTE iteratively,^{5,25} only by up to 5% at 500 K. Both Allen’s approximation and the exact solution to the BTE underestimate the measured resistivity, especially at higher temperatures.^{5}
When the temperature is higher than the Debye temperature, the average scattering rate of electrons can be obtained from Allen’s approximation^{26}
Its value at room temperature for W is 72 ps^{−1}, which falls in the middle of the actual distribution of scattering rates^{5} [Fig. 7] and in consistent with the lifetime (16 fs) estimated from the measured ρ and the calculated band structure in ref. ^{8}. The scattering rates span a factor of 2, which renders some support to the constant lifetime assumed in ref. ^{8}.
Because of the boundary contribution to scattering, the ρ of metal wires is higher than the bulk value once entering the nanoscale. Moreover, the relative increase is materialdependent, so it is possible to find a metal that is more resistive than copper in the bulk, but whose wires have lower ρ than copper wires of the same size. In fact, finding such metal nanowires to replace copper nanowires as interconnect material is a major issue for the semiconductor industry.^{8} Likewise, grain boundary scattering is another source of size effects in polycrystals.^{27,28} A qualitative way to understand those size effects for a particular material is to look at the distribution of electron MFPs and their contributions to the conductivity.^{5} As shown in Fig. 7, the largest MFP is 24 nm at the Fermi level. According to the cumulative σ presented in Fig. 8a, the MFPs are distributed in the range from 5 nm to 24 nm at room temperature, and half of σ is contributed by electrons with MFPs shorter than 18 nm. The reported average MFP in the literature ranges from 254 nm.^{8,10,29,30,31,32,33} Specifically, our calculations are in consistent with ref. ^{8}, which estimated an average of 15.5 nm from the measured ρ and calculated band structure.
In contrast, phonons have longer MFPs than electrons in W. Ninetypercent of the κ_{ph}^{5} is contributed by phonons with MFPs between 5 and 130 nm [Fig. 8b]. W has smaller electron MFPs and larger phonon MFPs than Al, Ag, and Au, which possess larger σ but much smaller κ_{ph} than W.^{2} In those three metals, phonons with MFPs between 1 and 10 nm are the predominant contributors to κ_{ph}.^{2} However, it should be noted that longer MFPs do not always imply higher conductivity: for instance, Al has almost the same σ as Au, but its average electron MFP is only half of that for Au.^{8} When the size is comparable to the characteristic MFPs, the transport properties are affected by the system size. In that regard, and since in W the MFPs of phonons are several times longer than those of electrons, size effects will result in a reduction of κ_{ph} for significantly larger sizes than needed to cause a reduction in σ. Therefore W nanostructures can be expected to show reduced values of the Lorenz number.^{5}
In summary, we report the phonon linewidths of tungsten contributed from electron–phonon and phonon–phonon interactions along highsymmetry paths, calculated through firstprinciples techniques. The electron–phonon scattering dominates except in a neighborhood of the zone center. The unusually weak phonon–phonon scattering, and in particular its vanishing strength at the triply degenerate P and H points, can be traced back to the elemental bcc structure. Although this feature is also applicable to Mo and Cr, it is not a universal phenomenon common to all elemental bcc substances. We find that in other systems like Na, the phonon–phonon scattering is strong due to the unusually soft transverse acoustic phonon along the ΓN direction.
The electrical resistivity of W obtained with Allen’s approximation agrees well with the accurate solution to the linearized Boltzmann transport equation. The room temperature meanfree paths of electrons contributing to the conductivity range from 5 to 24 nm, much shorter than those of phonons, suggesting reduced Lorenz numbers in W nanostructures.
Methods
The phonon FWHM corresponds to the scattering rate (inverse of the lifetime) divided by 2π. For a given mode denoted by λ (a composite index comprising both a wavevector q and a branch index p) the FWHM due to anharmonic phonon–phonon scattering can be calculated as^{34,35}
where N_{q} is the number of uniformly sampled q points in the Brillouin zone, and \(n_\lambda ^0\) is the Bose–Einstein occupancy for phonon frequency ω_{λ}. There are two types of threephonon processes: absorption (+) and emission (−) processes. The phonon of interest λ is scattered into phonon λ″ by absorbing/emitting phonon λ′ in the absorption/emission process, also termed as coalescence/decay process in the literature.^{17} The conservation of quasimomentum requires that q″ = q ± q′ up to a certain reciprocal lattice vector for the absorption and emission processes. The scattering matrix elements \(V_{\lambda \lambda ^\prime \lambda ^{\prime\prime} }^ \pm\) are determined by the thirdorder interatomic force constants (IFCs).^{34,35}
The phonon FWHM due to isotopic mass disorder is given by^{36}
where D(ω) is phonon density of states per unit cell, and g_{2} is the Pearson deviation coefficient of the atomic masses of isotopes. The natural isotopic distribution of W yields a value of g_{2} of 6.9668 × 10^{−5}.
The contribution to the phonon FWHM from electron–phonon interactions is almost temperature independent, and can be well estimated as^{1,25}
where \(g_{n{\mathbf{k}},{\mathbf{q}}p}^{m{\mathbf{k}} + {\mathbf{q}}}\) is the electron–phonon coupling matrix element for the electron state with band index n and wavevector k, the phonon mode (q, p), and the electron state with band index m and wavevector k + q. E_{nk} and E_{f} are the corresponding electronic and Fermi energy, respectively. N_{k} is the number of uniformly sampled k points in the Brillouin zone. The factor of 2 in 2/N_{k} accounts for the spin degeneracy in nonspinpolarized calculations.
Γ^{el} is closely related to σ. Allen obtained an approximated solution to the Boltzmann transport equation (BTE) in metals, and related the electrical resistivity ρ to the transport spectral function α^{2}F_{tr’}^{26} which is a variant of the Eliashberg electron–phonon spectral function α^{2}F. The latter can be written as^{37}
where N_{F} is the electronic density of states per unit cell and per spin at the Fermi level E_{F}. The total coupling constant can be obtained as
For transport properties, the contributions to scattering also depend on the effective change in velocity. Defining the efficiency factor α as
and multiplying the term in the sum of Eq. (4) by α, one can obtain the transport analog of the spectral function α^{2}F_{tr} and consequently the total transport coupling constant λ_{tr}. The ρ of metals can be approximately obtained as:^{26}
where V is the volume of the unit cell, x = ℏω/(2k_{B}T), and \(\left\langle {v_z^2} \right\rangle\) is the average square of the Fermi velocity along the transport direction, denoted here as z.
The electronic band structure, phonon dispersions and electron–phonon interactions were calculated using the QUANTUM ESPRESSO package,^{38} combining density functional theory (DFT), and density functional perturbation theory (DFPT).^{39} The Perdew–Zunger parametrization^{40} of the local density approximation (LDA) and Bachelet–Hamann–Schlueter type normconserving pseudopotentials^{41} were used for W. The Perdew–Burke–Ernzerhof parametrization^{42} of the generalized gradient approximation (GGA) and TrouillerMartins type normconserving Pseudopotientials were used for Mo and Na. The thirdorder.py script from the ShengBTE package^{35} was used to generate the thirdorder IFCs^{43} using 5 × 5 × 5 supercells. The \({\mathrm{\Gamma }}_\lambda ^{{\mathrm{pp}}}\) and \({\mathrm{\Gamma }}_\lambda ^{{\mathrm{iso}}}\). were obtained on a 48 × 48 × 48 q grid by using the ShengBTE package.^{35} Furthermore, the EPW package^{44} was employed to perform Wannier function interpolation from initial 8 × 8 × 8 k and q grids for the electron–phonon coupling matrix elements of W. The \({\mathrm{\Gamma }}_\lambda ^{{\mathrm{el}}}\) were then obtained on 36 × 36 × 36 k and q grids. The phonon and electron MFP analysis for W were carried out by solving the corresponding BTEs^{5} accurately with 36 × 36 × 36 q and 108 × 108 × 108 k grids, respectively. The δfunctions involved in Eqs. (2)–(5) are represented by Gaussian functions with physically motivated adaptive broadening parameters,^{25,45} eliminating the need for any adjustable parameters.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
We acknowledge support from the Natural Science Foundation of China (NSFC) under Grants No. 11704258 and No. 11574198 and the Shenzhen Science, Technology and Innovation Commission under Grant No. JCYJ20170412105922384. Y.C. also acknowledges the support from the China Postdoctoral Science Foundation under Grant No.2017M622745. J.M. also acknowledges support from NSFC under Grant No. 11804229. We thank Dr. Jesús Carrete for his critical reading of the manuscript.
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Y.C. performed calculations for W and Na. S.W. performed calculations for Mo. J.M. wrote the code. W.L. conceived the project. All authors contributed to the data analysis and wrote the paper.
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Chen, Y., Ma, J., Wen, S. et al. Bodycenteredcubic structure and weak anharmonic phonon scattering in tungsten. npj Comput Mater 5, 98 (2019). https://doi.org/10.1038/s4152401902357
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