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Ultrahigh Electron Thermal Conductivity in T‐Graphene, Biphenylene, and Net‐Graphene

Ultrahigh Electron Thermal Conductivity in T‐Graphene, Biphenylene, and Net‐Graphene IntroductionThe discovery of buckminsterfullerene (C60) molecule[1] was followed by the synthesis of other carbon allotropes,[2,3] including carbon nanotubes[4] and graphene.[5] While their structure is dominated by the presence of six‐membered periodic carbon rings, nonhexagonal carbon rings are also encountered. For example, the C60 structure hosts 12 pentagons, which are responsible for the Gaussian curvature of the fullerene.[6] In carbon nanotubes and graphene, four‐, five‐, seven‐, and eight‐membered rings are encountered as extended[7] and mechanically‐induced[8–10] five‐five‐seven‐seven (Stone–Wales) and four‐four‐eight‐eight defects, as well as chemical vapor deposition growth‐induced grain boundaries.[11] The creation of curvature can be understood by viewing these rings as the cores of edge dislocations in the otherwise perfect hexagonal structure (cylindrical or flat).[10] Due to the extremely low bending rigidity combined with extremely high in‐plane stiffness of graphene,[12,13] the extension in the out‐of‐plane dimension becomes an effective way to relax the core strain and bring the stiff CC bonds closer to their preferred equilibrium value.Theoretical explorations predicted that 2D carbon allotropes with regular nonhexagonal carbon rings can still be flat. For example, first‐principles calculations[14–16] revealed stability for T‐graphene, which comprises only tetrarings shown in Figure 1a. With the progress in materials synthesis, 2D carbon allotropes with non‐hexagonal rings are now a reality. Notably, Fan et al.[17] and Liu et al.[18] reported successful growth of flat biphenylene and net‐graphene, both comprising periodically arranged four‐, six‐, and eight‐membered rings. As indicated in Figure 1a, these 2D carbon allotropes are significantly higher in energy ΔE than graphene. T‐graphene is 0.52 eV per atom above graphene, while biphenylene and net‐graphene have ΔE of 0.47 and 0.30 eV per atom, respectively. Nevertheless, finite‐temperature structural relaxations of these allotropes to graphene in stress‐free conditions must be considerably hindered by the large kinetic barriers for CC bond rotations.[8,19]1Figurea) Computed crystal structures for graphene, T‐graphene, biphenylene, and net‐graphene. The color code shows the bonds lengths. The unit cells are indicated with dark line. ΔE is the total energy measured with respect to graphene. b) Computed κtotal = κph + κe versus T. Additionally, experimental thermal conductivity data for suspended (Balandin et al.,[25] Faugeras et al.,[37] Chen et al.,[38] Lee et al.,[39] Chen et al.,[40] Li et al.,[41] and Xie et al.[42]) and supported (Seol et al.[51]) graphene with different experimental sample size as well as the previous theoretical predictions of graphene (Lindsay et al.[45] and Fugallo et al.[46]) are provided for comparison. c) Thermal conductivity of single‐layer 2D materials at T = 300 K. Literature data (theoretical) for graphene,[35,45] h‐BN,[52] 2D WS2 and MoS2,[53] 2D MoSe2,[53] 2D BeN4,[34] silicene,[54] phosphorene,[55] SnSe,[56] and 2D MoO3.[36] The label on top of the bars denotes 3ph only, 3+4ph, and 3+4ph+pe scattering events for computing κph.Understanding the physical properties of 2D carbon allotropes, including mechanical,[20,21] electronic,[21,22] and optical[23] is a current focus, and some recently reports reveled many intriguing properties such as high‐temperature superconductivity[24] and large lithium storage capacity.[17] In view of the ultrahigh thermal conductivity of graphene[25] of phononic origin, the thermal properties are a particular focus. Without the out‐of plane relaxations, the CC bonds of the three 2D carbon allotropes are strained to different extents (CC bonds in the 1.38–1.50 Å range) with respect to the bonding (CC bonds 1.42 Å) of graphene shown in Figure 1a. The corresponding lattice constants of those 2D carbon allotropes are provided in Figure S1, Supporting Information. In strained bonds, anharmonicity effects[26,27] can decrease significantly the phonon thermal conductivity (κph).[28–31]In this work, we conduct a comprehensive ab initio calculation to consistently compare the phonon‐ and electron‐based conduction of pristine graphene with that of the T‐graphene, biphenylene, and net‐graphene carbon allotropes. There is an established notion that thermal energy transport in 2D carbon materials is largely phonon‐based[32] and that the negligible electron‐based conduction can become important only under n‐doping.[33,34] Contrary to this notion, here we find that in 2D carbon allotropes with nonhexagonal rings, the electron thermal conductivity (κe), as calculated by considering electron–phonon (el‐ph) scatterings, is of central importance. The large κe compensates the drop in κph which is calculated here by including not only the three‐phonon (3ph) but also the four‐phonon (4ph) and phonon–electron (pe) scatterings. The latter contribution is neglected by other approaches,[28,29] but recently it was shown to cause strong suppression of κph in graphene,[35] MoO3,[36] and BeN4.[34]Results and Discussionsκtotal of 2D Carbon AllotropesFigure 1b plots the calculated κtotal = κph + κe (averaged within the in‐plane directions) of graphene and the three 2D carbon allotropes in the temperature range of 200 to 1000 K. The temperature‐dependent κph and κe along x and y directions are shown in Figure S2, Supporting Information and the values at T = 300 K are given in Table S1, Supporting Information. The available experimental data for graphene, which is also shown, exhibits a large spread which may originate in both the sample variability (in terms of various defects and grain boundaries) and the measurement technique. We note that Raman measurements[25,37–40] (orange solid dots) generally give larger thermal conductivity values than the thermal bridge methods[41,42] (navy solid dot) which are extended from the measurement of nanowire[43] and nanotubes.[44] Interestingly, the Raman data is in agreement with the graphene calculations which consider only 3ph scatterings[45,46] while the other measurements are instead in better agreement with the current calculations, which consider both 3ph and 4ph scatterings. It should be also noted that measurements of the thermal conductivity with the Raman method are subjected to rather large uncertaintinities[47] caused by the use of estimates of the light absorption coefficient and the temperature increase of the sample absorbing the light. Further, the method assumes that the temperature increase of the sample equals the temperature increase of the acoustic phonons, and uses this latter temperature increase to calculate the value of the thermal conductivity via the slope of the peak shift versus temperature shift. A theoretical work,[47] which has modeled the microscopic process of light energy absorption in graphene, uncovered severe overestimation of the thermal conductivity[25] by the Raman method, as also confirmed experimentally.[48] In this context, our result here further supports the critique of the Raman technique.[47]It is well known that phonon (lattice vibration) is the major heat carrier in graphene and the electronic contribution can be neglected.[49] According to conventional wisdom, we expect a similar behavior in the sp2 2D allotropes. Surprisingly, in spite of the strained CC bonds, κtotal of the three carbon allotropes are rather similar with the value for graphene. For a broader view, in Figure 1c we place our calculation results in the context of other important 2D materials. It can be seen that κtotal at T = 300 K for the T‐graphene, biphenylene, and net‐graphene is in the high range, comparable to h‐BN and larger than MoS2. Interestingly, the recently predicted graphene‐like borophene[50] displays a twofold higher thermal conductance than graphene. This reveals a weak ballistic thermal transport ability for 2D carbon allotropes compared to graphene‐like borophene.κph and κe ComponentsTo understand the high κtotal in 2D carbon allotropes, we analyze separately their κph and κe components, Figure 2a), as well as their phononic and electronic band structures, Figure 2b,c, respectively. As expected, we find that κtotal of graphene is dominated by κph, which is larger than in the other three 2D carbon allotropes. On the contrary, κe of the other three 2D carbon allotropes are above or close to their κph components. We also note that the κe behavior is fully consistent with the electronic band structure of Figure 2c. In graphene and the electronic contribution can be neglected due to the almost zero electron density‐of‐states (DOS) originating from the Dirac point shown in Figure 2c at the Fermi level.[49] However, metallic features are observed in T‐graphene, biphenylene, and net‐graphene according to the electronic band dispersion crossover at the Fermi level shown in Figure 2c, hence the non‐negligible electronic contribution to heat conduction.2Figurea) κph and κe versus T, b) harmonic phonon dispersion, and c) electronic band structure from DFT calculations and Wannier interpolation[57] for graphene, T‐graphene, biphenylene, and net‐graphene. In (a), the left inset is the κph attributed to different scattering events and the right inset is the scattering rates due to 3ph, 4ph, and pe interaction processes at T = 300 K. The most recently available literature data are provided for validation as marked in the left inset of (a) with: κph3+4ph$\kappa _{{\rm{ph}}}^{{\rm{3 + 4ph}}}$ for graphene,[35] κph3ph$\kappa _{{\rm{ph}}}^{{\rm{3ph}}}$ for T‐graphene,[28] κph calculated by only 3ph scatterings[29] and EMD[30] for biphenylene, and κph by NEMD for net‐graphene.[31] In (b), the TA, LA, ZA, and one transverse optical (ZO) phonon modes are labeled and the Brillouin zone are shown. In (c), the zero of energy is set to the Fermi level.We note in passing that by including 3+4ph+pe scatterings, which is the most complete scheme of phonon‐limited scattering rate predictions based on ab initio calculations, our κph calculations are surpassing the available literature data provided (red dot circles) in the left insets of Figure 2a. Specifically, we show κph3+4ph$\kappa _{{\rm{ph}}}^{{\rm{3 + 4ph}}}$ adopting the solution of Boltzmann transport equation (BTE) with the optimized Tersoff potential[58] for graphene,[35] κph3ph$\kappa _{{\rm{ph}}}^{{\rm{3ph}}}$ calculated from density functional theory (DFT) for T‐graphene,[28] κph calculated from the DFT with only 3ph scatterings[29] and the equilibrium molecular dynamics[30] (EMD) for biphenylene, and κph by nonequilibrium EMD (NEMD) for net‐graphene.[31] Overall, our calculations agree with the literature data, indicating the reliability of our implementations, and further allow for the following insights: i) The 4ph scattering effects induce a suppression (κph3ph$\kappa _{{\rm{ph}}}^{{\rm{3ph}}}$‐κph3+4ph$\kappa _{{\rm{ph}}}^{{\rm{3 + 4ph}}}$)/κph3ph$\kappa _{{\rm{ph}}}^{{\rm{3ph}}}$ at T = 300 K by 79%, 78%, 51%, and 58% for graphene, T‐graphene, biphenylene, and net‐graphene, respectively. This indicates the importance of 4ph scatterings when computing the κph. ii) The pe scatterings induce a further suppression (κph3+4ph$\kappa _{{\rm{ph}}}^{{\rm{3 + 4ph}}}$‐κph3+4ph+pe$\kappa _{{\rm{ph}}}^{{\rm{3 + 4ph + pe}}}$)/κph3+4ph$\kappa _{{\rm{ph}}}^{{\rm{3 + 4ph}}}$ at T = 300 K by 0.01%, 15%, 34%, and 3% for graphene, T‐graphene, biphenylene, and net‐graphene, respectively. This demonstrates the need for considering the pe interactions to predict κph. iii) These two points are next supported by the data in the right inset of Figure 2a showing for each structure the magnitude of the 3ph, 4ph, and pe scattering rates. In particular, we note that the weak pe coupling behavior in graphene does not extend to the other three 2D allotropes.Origin of High κeFocusing on electron‐based conduction, Figure 3a compares the calculated κe versus T for graphene, T‐graphene, biphenylene and net‐graphene. We can see that graphene has the smallest κe. At T = 300 K, see inset of Figure 3a, and the electronic contribution to heat conduction κe/κtotal in graphene is only of ≈10%. Remarkably, this contribution increases to 81%, 60%, and 66% in T‐graphene, biphenylene, and net‐graphene, respectively. The calculated electrical conductivity σ versus T plotted in Figure 3b displays a similar trend as κe versus T. For a broader view, we placed our predicted σ values in the context of typical metals[59] shown in the right inset of Figure 3b. T‐graphene, biphenylene, and net‐graphene show larger σ values than metals at T = 300 K, indicating their robust capability of electrical transport in potential electronic devices.3FigureComparison of a) κe versus T, (b) σ versus T, c) electron relaxation time, and d) Fermi velocity and electron DOS at Fermi level for graphene, T‐graphene, biphenylene, and net‐graphene. The inset of (a) is the ratio of κe/κtotal at T = 300 K. In (b), the left inset is the Lorenz ratio and the right inset is the electrical conductivity of some typical metals.[59] The experimental data[51,60–62] of σ in (b) are provided for comparison. Note: the σ is evaluated with single layer thickness of 0.334 nm.[51]The σ calculations displayed in Figure 3b are confirming the weak electronic contribution to electron transport in graphene. We have compared our predictions with the measured σ of graphene as shown in Figure 3a. Similar to the thermal conductivity measurement in graphene, a large spread in the experiment data is also observed for σ. Our results overall locate the same order of the experimental data, indicating the reliability of our calculations. Our predictions in general further agree with the calculations using the Allen's model[63] (see Section S4, Supporting Information) as shown in Figure S3, Supporting Information, which obviously validates our implementation. For all four structures, we have also calculated their room‐temperature intrinsic mobility μ = σ/ne (where e is the elementary charge and n is the charge carrier density). We obtained 0.22, 3.81, 1.22, and 1.51 (104 cm2 V−1 s−1) for graphene, T‐graphene, biphenylene, and net‐graphene, respectively. Due to the lowest electron density of states near the Fermi level (at the Dirac point), graphene has the smallest μ. The order of magnitude (104 cm2 V−1 s−1) obtained for T‐graphene, biphenylene, and net‐graphene is close to what was reported for other 2D carbon allotropes.[64]While our theoretical framework is able to clearly distinguish the κe and κph components, in practice it is very difficult to directly measure them. Some promising experimental methods[65] have been developed for direct κph measurements, such as alloying technique,[66] superconducting technique,[67] and magnetothermal technique[68,69] but they are either very complicated to conduct, or limited to extremely low T. The electrical conductivity can be accurately and easily obtained. The Lorenz ratio L = κe/σT is generally adopted to estimate the phonon thermal conductivity as κph = κtotal − LσT in which L is generally assumed to be the Sommerfeld value L0 = 2.44 × 10−8ΩWK−2.[59,70] We note that this approximation is inadequate especially at low temperatures (T ≪ TD, where TD is Debye temperature) since L has large differences compared to L0.[59] Indeed, in the inset of Figure 3b we observe the large deviation between L and L0 in the 2D carbon allotropes, that is, L < L0, indicating that using the Sommerfeld value in this method underestimates (overestimates) κph (κe). Our result supports the violation of the Wiedemann–Franz law in 2D Dirac materials[71] because the ratio of electron thermal conductance and electric conductance is a chemical‐potential‐dependent quantity.For all four structures we display their electron lifetime (τe), Figure 3c, and their velocity (vF) and DOS (NF) at the Fermi level, Figure 3d. Invoking the Drude's free electron model, these are key quantities for κe since κeDrude=(π2/3kB2T)NFvF2τe$\kappa _{\rm{e}}^{{\rm{Drude}}} = ({\pi ^2}/3k_{\rm{B}}^2T){N_{\rm{F}}}v_{\rm{F}}^2{\tau _{\rm{e}}}$.[59] We see that graphene has the largest τe compared to other allotropes, a result that is attributed the extremely weak el‐ph coupling, as shown later in Figure 5d. However, the largest τe does not lead to largest κe (or σ). With its Dirac point is located at the Fermi level, graphene has an extremely small NF, which leads to a very small κe. T‐graphene, biphenylene, and net‐graphene have similar NF values, which are much larger than graphene. Among them, we find that T‐graphene has the largest κe (or σ) as it has the largest vF and τe (E − EF within kBT ≈0.026 eV at 300 K). Our results and the insights uncovered here for T‐graphene, biphenylene, and net‐graphene could also be shared with other theoretically predicted 2D carbon allotropes displaying metallic character, such as penta‐graphene, a‐graphene, g‐graphene, Ψ‐graphene, and pop‐graphene.[3,31,64]4FigureThe comparison of a) κph3ph$\kappa _{{\rm{ph}}}^{{\rm{3ph}}}$ and κph3+4ph$\kappa _{{\rm{ph}}}^{{\rm{3 + 4ph}}}$ at T = 300 K, b) volumetric heat capacity at T = 300 K, c) phonon group velocity, d) phonon scattering rates from normal and Umklapp processes as well as the lifetime due to 3ph scatterings at T = 300 K, e) phonon DOS, f) Grüneisen parameter, g) phase space of 3ph and 4ph scattering events, h) the strength of 100 largest third‐ and fourth‐IFCs matrix elements, and i) the ELF for graphene, T‐graphene, biphenylene, and net‐graphene. The inset of (a) is the spectral κph3ph(ω)$\kappa _{{\rm{ph}}}^{{\rm{3ph}}}(\omega )$ at T = 300 K. The inset of (c) is the average group velocity. The inset of (f) is the Grüneisen parameter in the range of ≈1–2 THz. The ELF is visualized by using VESTA package.[74] Note: the representation of the color in (a–h) is gray for graphene, red for T‐graphene, blue for biphenylene, and green for net‐graphene, respectively.5Figurea) The normalized contribution of all acoustic modes and ZO modes to κph attributed to 3ph, 3+4ph and 3+4ph+pe interactions, in which the inset is the accumulated κph with respect to phonon frequency ω, b,c) 3ph, 4ph, and pe scatterings for all acoustic modes, and d) the Eliashberg spectral function αtr2$\alpha _{{\rm{tr}}}^2$F and transport el‐ph coupling strength λtr at T = 300 K for graphene. The corresponding contents are fore–h) T‐graphene, i–l) bipheylene, and (m–p) net‐graphene.Origin of Low κphTo understand the reasons behind lowering κph in the 2D allotropes, as detailed for example in Figure 4a, we have carried a detailed comparison of the intrinsic phonon properties. The BTE derived κph, see Equation (6), is conditioned by the volumetric heat capacity Cv, group velocity vg, and relaxation time τ. By analyzing Figure 4b–d, where we have separately plotted these quantities for all four structures, we can see that all Cv, vg, and τ contribute to the lowering of κph with respect to graphene. First, we see that the Cv of graphene is much larger than that of T‐graphene, biphenylene, and net‐graphene, Figure 4b. Second, we find that the graphene has the largest vg, a result which is more clear using the average ⟨vg⟩ shown in the inset of Figure 4c. Note that since the κph of all graphene allotropes is mainly attributed to the phonon modes less than 20 THz (mainly acoustic modes) shown in the inset of Figure 4a, in Figure 4c we have plotted the vg of the acoustic modes only. Third, we find a distinct enhancement of the phonon scatterings in T‐graphene, biphenylene, and net‐graphene compared to graphene. On one hand, in graphene the normal phonon scattering rates are one to two orders of magnitude larger than the Umklapp phonon scattering rates shown in Figure 4d. This indicates a hydrodynamic phonon transport regime where the phonon transport is much less dissipative, leading to extraordinary large κph.[72] On the other hand, in T‐graphene, biphenylene, and net‐graphene, we find in Figure 4d that the normal and Umklapp phonon scattering rates are at comparable levels at 300 K. This indicates the breakdown of hydrodynamic phonon transport (dominant in graphene) and a significant increase in dissipation.Focusing next on the anharmonic effects, we find that all the important indicators concur to an enhancement of anharmonicity in T‐graphene, biphenylene, and net‐graphene. In Figure 4e we first find that compared to graphene there is a significant increase of phonon DOS for T‐graphene, biphenylene, and net‐graphene in 1–10 THz. Hence, less phonon interactions can occur in graphene. Second, in Figure 4f the Grüneisen parameter,[73] which is a measure of the anharmonicity, is smallest in graphene. Third, in Figure 4g we see that the phonon phase space reflecting all available phonon scattering processes for simultaneously satisfying the energy and momentum conservation in graphene is much smaller than that in T‐graphene, biphenylene, and net‐graphene. Forth, the anharmonic force constants of Figure 4h show that graphene has the smallest strength of the third and fourth order interatomic force constants (IFCs, Φ).The strong anharmonicity displayed by T‐graphene, biphenylene, and net‐graphene in comparison with graphene originates in changes of the covalent bonding of strained bonds in reduced crystal symmetry environment. In Figure 4i, we characterize bonding with the electron localization function (ELF).[75,76] The position where the electron is localized has a high ELF value. As expected, ELF is largest at the center of all CC bonds, indicating the dominance of covalent bonding in the 2D carbon allotropes. We can further observe that while in graphene, the distribution of ELF is symmetric with respect to the CC bond axis, evident deviations are visible in T‐graphene, biphenylene, and net‐graphene. In the latter structures, ELF indicates that electrons spread toward the center of the larger eight‐membered and six‐membered rings and away from the center of the smaller four‐membered rings. We interpret these ELF changes as a shift in sp2 bonding caused by electrostatic repulsion, which spills the electronic charge from the smaller into the larger neighboring rings.The ZA modes are known to be very important in the phonon transport in 2D materials.[32,51] We investigate their contribution to κph arising from 3ph, 4ph and pe scatterings. In graphene, Figure 5a, we can see that the relative contribution of the ZA branch to κph is reduced from 80% with only 3ph scattering to 30% after including 4ph scattering. Further including pe scatterings gives minor changes. As shown in Figure 5b, the reduction of κph3+4ph$\kappa _{{\rm{ph}}}^{{\rm{3 + 4ph}}}$ is caused by the enhancement of the 4ph scattering rates of ZA modes which are 3 to 4 orders larger than the 3ph scattering rates. These results do not follow the general rule of bulk materials that 4ph scattering is important only in strongly anharmonic materials.[77,78] The behavior obtained in graphene is attributed to the reflection symmetry selection rule, which is unique to 2D materials. Previous works[51,79] showed that reflection symmetry of graphene (or any 2D crystal) requires that the nth order anharmonic IFCs vanish: Φα1…αnl1b1…lnbn=0$\Phi _{{\alpha _1}{\rm{ \ldots }}{\alpha _n}}^{{l_1}{b_1}{\rm{ \ldots }}{l_n}{b_n}} = 0$ when the number of out‐of‐plane components in the string α1...αn is odd. This leads to a selection rule that forbids any n‐phonon scattering process involving an odd number of flexural (out‐of‐plane) modes. Therefore, the three‐ZA (ZA+ZA → ZA) scattering processes are forbidden. This means that the major 3ph scattering phase space for ZA modes is eliminated and the ZA modes are the major heat carriers accounted for by κph3ph$\kappa _{{\rm{ph}}}^{{\rm{3ph}}}$. However, for 4ph scatterings, the four‐ZA processes ZA+ZA→ZA+ZA and ZA→ZA+ZA+ZA are allowed. There is abundant 4ph scatterings for ZA modes, leading to the suppression of κph3+4ph$\kappa _{{\rm{ph}}}^{{\rm{3 + 4ph}}}$. The ZA modes behave similarly in T‐graphene, biphenylene, and net‐graphene, where we can see significant suppression of κph3+4ph$\kappa _{{\rm{ph}}}^{{\rm{3 + 4ph}}}$ in Figure 5e,i,m as well as extremely large 4ph scattering rates in Figure 5f,j,n.A different trend to the contribution to κph is found for the TA and LA modes, Figure 5a,e,i,m. For these modes, a large percentage of 4ph processes than 3ph processes are forbidden, as shown in Figure 5c,g,k,o. In addition, from the inset of Figure 5a,e,i,m, we can see that the pe scattering effect on the κph is relative small in graphene, T‐graphene, and net‐graphene but it is strong in biphenylene. The transport el‐ph coupling strength λtr = 0.27 (see definition in Section S5, Supporting Information) for biphenylene shown in Figure 5l is larger than that of 0.01 for graphene in Figure 5d, 0.12 for T‐graphene in Figure 5h, and 0.06 in Figure 5p for net‐graphene, indicating the stronger el‐ph coupling in biphenylene than other 2D carbon allotropes.ConclusionIn conclusion, using the Boltzmann transport equation formalism in combination with density functional theory calculations, we uncovered 2D carbon allotropes with both electron‐based and phonon‐based thermal conduction. Although electron‐based thermal conduction is small in graphene, it is ultrahigh in 2D carbon allotropes with non‐hexagonal rings. Phonon‐based conduction remains important, although the effect is deteriorated by the anharmonicity effects caused by the changes in CC bonding resulted from strain and lattice symmetry changes. The electron‐based conduction compensates for the κph decrease and renders large κtotal for T‐graphene, biphenylene, and net‐graphene, close to those of graphene, h‐BN and MoS2, which exhibit phonon‐based thermal conduction. Other theoretically predicted 2D carbon allotropes, such as penta‐graphene, α‐graphene, g‐graphene, Ψ‐graphene, and pop‐graphene, display metallic character, and thus could share the impressive attributes uncovered here for T‐graphene, biphenylene, and net‐graphene. 2D carbon allotropes may contribute to realization of the next‐generation technologies requiring nanostructures with ability to transfer of thermal energy by a combination of diffusion of electrons and phonon vibrations.Experimental SectionElectronic Thermal Conductivity and Electrical ConductivityBy solving the BTE of electron,[80] the electrical conductivity tensor (σαβ) and electronic thermal conductivity tensor (κeαβ$\kappa _{\rm{e}}^{\alpha \beta }$) can be obtained as1{σαβ=LEEαβ,κeαβ=−(LTTαβ−LTEαβLETαβLEEαβ)\[\left\{ \begin{array}{c}{\sigma ^{\alpha \beta }} = L_{{\rm{EE}}}^{\alpha \beta },\\\kappa _{\rm{e}}^{\alpha \beta } = - \left( {L_{{\rm{TT}}}^{\alpha \beta } - \frac{{L_{{\rm{TE}}}^{\alpha \beta }L_{{\rm{ET}}}^{\alpha \beta }}}{{{L_{{\rm{E}}{{\rm{E}}^{\alpha \beta }}}}}}} \right)\end{array}\right.\]in which the coefficients are2LEEαβ=−e2nsNkV∑ik∂fik0∂εikvikαvikβτikep\[L_{{\rm{EE}}}^{\alpha \beta } = - \frac{{{e^2}{n_s}}}{{{N_k}V}}\mathop \sum \limits_{ik} \frac{{\partial f_{ik}^0}}{{\partial {\varepsilon _{ik}}}}v_{ik}^\alpha v_{ik}^\beta \tau _{ik}^{{\rm{ep}}}\]3LETαβ=−ensNkVT∑ik(εik−μ)∂fik0∂εikvikαvikβτikep\[L_{{\rm{ET}}}^{\alpha \beta } = - \frac{{e{n_s}}}{{{N_k}VT}}\mathop \sum \limits_{ik} \left( {{\varepsilon _{ik}} - \mu } \right)\frac{{\partial f_{ik}^0}}{{\partial {\varepsilon _{ik}}}}v_{ik}^\alpha v_{ik}^\beta \tau _{ik}^{{\rm{ep}}}\]4LTEαβ=−ensNkV∑ik(εik−μ)∂fik0∂εikvikαvikβτikep\[L_{{\rm{TE}}}^{\alpha \beta } = - \frac{{e{n_s}}}{{{N_k}V}}\mathop \sum \limits_{ik} \left( {{\varepsilon _{i{{\bf k}}}} - \mu } \right)\frac{{\partial f_{ik}^0}}{{\partial {\varepsilon _{ik}}}}v_{ik}^\alpha v_{ik}^\beta \tau _{ik}^{{\rm{ep}}}\]5LTTαβ=−nsNkVT∑ik(εik−μ)2∂fik0∂εikvikαvikβτikep\[L_{{\rm{TT}}}^{\alpha \beta } = - \frac{{{n_s}}}{{{N_k}VT}}\mathop \sum \limits_{ik} {\left( {{\varepsilon _{ik}} - \mu } \right)^2}\frac{{\partial f_{ik}^0}}{{\partial {\varepsilon _{ik}}}}v_{ik}^\alpha v_{ik}^\beta \tau _{ik}^{{\rm{ep}}}\]where e is the elementary charge, ns is the number of electrons per state, V is the volume of the unit cell, Nk the total number of k‐points sampled in the first Brillouin zone, T is the temperature, fik0$f_{i{{\bf k}}}^0$ is the electron distribution function, vik=1ℏ∂εik∂k${{{\bf v}}_{ik}} = \frac{1}{\hbar }\frac{{\partial {\varepsilon _{ik}}}}{{\partial k}}$ is the electron velocity, α and β denotes the directional component. The summation in these three equations was over all the electrons enumerated using electronic wave vector k\[k\] and band index i. μ is the chemical potential. τikep$\tau _{ik}^{{\rm{ep}}}$ is the electron lifetime (or relaxation time). It should be noted that only the el‐ph scattering rates (1/τikep$1/\tau _{ik}^{{\rm{ep}}}$) were considered in computing the κe. The expression of 1/τikep$1/\tau _{ik}^{{\rm{ep}}}$ can be found in Section S1, Supporting Information.Phononic Thermal ConductivitySolving the BTE of phonon combined with the Fourier's law of heat conduction, the phonon thermal conductivity tensor (κpαβ$\kappa _{\rm{p}}^{\alpha \beta }$) can be written as a summation over all phonon modes λ = (q, ν) of polarization ν, wave vector q$q$ and frequency ωλ as6κpαβ=1Nq∑λcλvλ,αvλ,βτλp\[\kappa _{\rm{p}}^{\alpha \beta } = \frac{1}{{{N_q}}}\mathop \sum \limits_\lambda {c_\lambda }{v_{\lambda ,\alpha }}{v_{\lambda ,\beta }}\tau _\lambda ^{\rm{p}}\]where α and β are indexing the Cartesian directions, Nq is the total number of q‐points sampled in the first Brillouin zone, while cλ, vλ, and τλp$\tau _\lambda ^{\rm{p}}$ denote the volumetric heat capacity, phonon group velocity, and phonon lifetime (or relaxation time), respectively. Note that cλ=(ℏωλ/V)(∂nλ0/∂T)${c_\lambda } = (\hbar {\omega _\lambda }/V)(\partial n_\lambda ^0/\partial T)$, where nλ0$n_\lambda ^0$ is the Bose–Einstein distribution function and V the volume of the primitive cell, and vλ, α = ∂ωλ/∂qα. The phonon scattering rate (1/τλp$1/\tau _\lambda ^{\rm{p}}$) is a summation of the phonon–phonon (1/τλpp$1/\tau _\lambda ^{{\rm{pp}}}$), pe (1/τλpe$1/\tau _\lambda ^{{\rm{pe}}}$), phonon–impurity (1/τλim$1/\tau _\lambda ^{{\rm{im}}}$), phonon–isotope (1/τλiso$1/\tau _\lambda ^{{\rm{iso}}}$), and phonon–grain boundary (1/τλgb$1/\tau _\lambda ^{{\rm{gb}}}$) scattering rates. Here, the κp of tr‐BeN4 based on ab initio computed 1/τλpp$1/\tau _\lambda ^{{\rm{pp}}}$ with 3ph and 4ph scatterings as well as 1/τλpe$1/\tau _\lambda ^{{\rm{pe}}}$ scattering rates are predicted, which can be summed using Matthiessen's rule[80] with 1/τλp=1/τλ3ph+1/τλ4ph+1/τλpe$1/\tau _\lambda ^{\rm{p}} = 1/\tau _\lambda ^{{\rm{3ph}}} + 1/\tau _\lambda ^{{\rm{4ph}}} + 1/\tau _\lambda ^{{\rm{pe}}}$. The detailed expressions of 1/τλ3ph$1/\tau _\lambda ^{{\rm{3ph}}}$, 1/τλ4ph$1/\tau _\lambda ^{{\rm{4ph}}}$, and 1/τλpe$1/\tau _\lambda ^{{\rm{pe}}}$ can be found in Section S2, Supporting Information.Ab Initio Computational DetailsThe DFT and density functional perturbation theory (DFPT) calculations were carried using the VASP[81] package with Perdew–Burke–Ernzerhof exchange and correlation functional.[82] The 3ph scattering rates were calculated using ShengBTE[73] and the 4ph scattering rates were computed using an in‐house code.[36,77,83] The el‐ph coupling matrix were calculated using DFPT with norm‐conserving pseudopotentials[84] with generalized gradient approximation[82] exchange and correlation functional as implemented in QUANTUM ESPRESSO.[85] They were initially obtained on coarse electron and phonon wave vector grids and then implemented to finer grids using the maximally localized Wannier functions basis as implemented in the el‐ph Wannier (EPW)[86,87] to obtain the el‐ph scatterings for computing κe and κph. Importantly, an in‐house modified EPW code[70,88,89]was used to predict the el‐ph coupling scattering rates in order to compute the huge dense k‐ and q‐points more efficiently. The simulation parameters and computational details are provided in Section S3, Supporting Information.AcknowledgementsThe authors thank Dr. Shouhang Li for valuable discussions. Simulations were performed at the Tianhe2‐JK of Beijing Computational Science Research Center. Z.T. acknowledges the support by National Natural Science Foundation (No. 52106068), China Postdoctoral Science Foundation (No. 2020M680127), Guangdong Basic and Applied Basic Research Foundation (Nos. 2020A1515110838 and 2021A1515011688), and Shenzhen Science and Technology Program (No. RCBS20200714114919142). 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Ultrahigh Electron Thermal Conductivity in T‐Graphene, Biphenylene, and Net‐Graphene

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Wiley
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© 2022 Wiley‐VCH GmbH
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1614-6832
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DOI
10.1002/aenm.202200657
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Abstract

IntroductionThe discovery of buckminsterfullerene (C60) molecule[1] was followed by the synthesis of other carbon allotropes,[2,3] including carbon nanotubes[4] and graphene.[5] While their structure is dominated by the presence of six‐membered periodic carbon rings, nonhexagonal carbon rings are also encountered. For example, the C60 structure hosts 12 pentagons, which are responsible for the Gaussian curvature of the fullerene.[6] In carbon nanotubes and graphene, four‐, five‐, seven‐, and eight‐membered rings are encountered as extended[7] and mechanically‐induced[8–10] five‐five‐seven‐seven (Stone–Wales) and four‐four‐eight‐eight defects, as well as chemical vapor deposition growth‐induced grain boundaries.[11] The creation of curvature can be understood by viewing these rings as the cores of edge dislocations in the otherwise perfect hexagonal structure (cylindrical or flat).[10] Due to the extremely low bending rigidity combined with extremely high in‐plane stiffness of graphene,[12,13] the extension in the out‐of‐plane dimension becomes an effective way to relax the core strain and bring the stiff CC bonds closer to their preferred equilibrium value.Theoretical explorations predicted that 2D carbon allotropes with regular nonhexagonal carbon rings can still be flat. For example, first‐principles calculations[14–16] revealed stability for T‐graphene, which comprises only tetrarings shown in Figure 1a. With the progress in materials synthesis, 2D carbon allotropes with non‐hexagonal rings are now a reality. Notably, Fan et al.[17] and Liu et al.[18] reported successful growth of flat biphenylene and net‐graphene, both comprising periodically arranged four‐, six‐, and eight‐membered rings. As indicated in Figure 1a, these 2D carbon allotropes are significantly higher in energy ΔE than graphene. T‐graphene is 0.52 eV per atom above graphene, while biphenylene and net‐graphene have ΔE of 0.47 and 0.30 eV per atom, respectively. Nevertheless, finite‐temperature structural relaxations of these allotropes to graphene in stress‐free conditions must be considerably hindered by the large kinetic barriers for CC bond rotations.[8,19]1Figurea) Computed crystal structures for graphene, T‐graphene, biphenylene, and net‐graphene. The color code shows the bonds lengths. The unit cells are indicated with dark line. ΔE is the total energy measured with respect to graphene. b) Computed κtotal = κph + κe versus T. Additionally, experimental thermal conductivity data for suspended (Balandin et al.,[25] Faugeras et al.,[37] Chen et al.,[38] Lee et al.,[39] Chen et al.,[40] Li et al.,[41] and Xie et al.[42]) and supported (Seol et al.[51]) graphene with different experimental sample size as well as the previous theoretical predictions of graphene (Lindsay et al.[45] and Fugallo et al.[46]) are provided for comparison. c) Thermal conductivity of single‐layer 2D materials at T = 300 K. Literature data (theoretical) for graphene,[35,45] h‐BN,[52] 2D WS2 and MoS2,[53] 2D MoSe2,[53] 2D BeN4,[34] silicene,[54] phosphorene,[55] SnSe,[56] and 2D MoO3.[36] The label on top of the bars denotes 3ph only, 3+4ph, and 3+4ph+pe scattering events for computing κph.Understanding the physical properties of 2D carbon allotropes, including mechanical,[20,21] electronic,[21,22] and optical[23] is a current focus, and some recently reports reveled many intriguing properties such as high‐temperature superconductivity[24] and large lithium storage capacity.[17] In view of the ultrahigh thermal conductivity of graphene[25] of phononic origin, the thermal properties are a particular focus. Without the out‐of plane relaxations, the CC bonds of the three 2D carbon allotropes are strained to different extents (CC bonds in the 1.38–1.50 Å range) with respect to the bonding (CC bonds 1.42 Å) of graphene shown in Figure 1a. The corresponding lattice constants of those 2D carbon allotropes are provided in Figure S1, Supporting Information. In strained bonds, anharmonicity effects[26,27] can decrease significantly the phonon thermal conductivity (κph).[28–31]In this work, we conduct a comprehensive ab initio calculation to consistently compare the phonon‐ and electron‐based conduction of pristine graphene with that of the T‐graphene, biphenylene, and net‐graphene carbon allotropes. There is an established notion that thermal energy transport in 2D carbon materials is largely phonon‐based[32] and that the negligible electron‐based conduction can become important only under n‐doping.[33,34] Contrary to this notion, here we find that in 2D carbon allotropes with nonhexagonal rings, the electron thermal conductivity (κe), as calculated by considering electron–phonon (el‐ph) scatterings, is of central importance. The large κe compensates the drop in κph which is calculated here by including not only the three‐phonon (3ph) but also the four‐phonon (4ph) and phonon–electron (pe) scatterings. The latter contribution is neglected by other approaches,[28,29] but recently it was shown to cause strong suppression of κph in graphene,[35] MoO3,[36] and BeN4.[34]Results and Discussionsκtotal of 2D Carbon AllotropesFigure 1b plots the calculated κtotal = κph + κe (averaged within the in‐plane directions) of graphene and the three 2D carbon allotropes in the temperature range of 200 to 1000 K. The temperature‐dependent κph and κe along x and y directions are shown in Figure S2, Supporting Information and the values at T = 300 K are given in Table S1, Supporting Information. The available experimental data for graphene, which is also shown, exhibits a large spread which may originate in both the sample variability (in terms of various defects and grain boundaries) and the measurement technique. We note that Raman measurements[25,37–40] (orange solid dots) generally give larger thermal conductivity values than the thermal bridge methods[41,42] (navy solid dot) which are extended from the measurement of nanowire[43] and nanotubes.[44] Interestingly, the Raman data is in agreement with the graphene calculations which consider only 3ph scatterings[45,46] while the other measurements are instead in better agreement with the current calculations, which consider both 3ph and 4ph scatterings. It should be also noted that measurements of the thermal conductivity with the Raman method are subjected to rather large uncertaintinities[47] caused by the use of estimates of the light absorption coefficient and the temperature increase of the sample absorbing the light. Further, the method assumes that the temperature increase of the sample equals the temperature increase of the acoustic phonons, and uses this latter temperature increase to calculate the value of the thermal conductivity via the slope of the peak shift versus temperature shift. A theoretical work,[47] which has modeled the microscopic process of light energy absorption in graphene, uncovered severe overestimation of the thermal conductivity[25] by the Raman method, as also confirmed experimentally.[48] In this context, our result here further supports the critique of the Raman technique.[47]It is well known that phonon (lattice vibration) is the major heat carrier in graphene and the electronic contribution can be neglected.[49] According to conventional wisdom, we expect a similar behavior in the sp2 2D allotropes. Surprisingly, in spite of the strained CC bonds, κtotal of the three carbon allotropes are rather similar with the value for graphene. For a broader view, in Figure 1c we place our calculation results in the context of other important 2D materials. It can be seen that κtotal at T = 300 K for the T‐graphene, biphenylene, and net‐graphene is in the high range, comparable to h‐BN and larger than MoS2. Interestingly, the recently predicted graphene‐like borophene[50] displays a twofold higher thermal conductance than graphene. This reveals a weak ballistic thermal transport ability for 2D carbon allotropes compared to graphene‐like borophene.κph and κe ComponentsTo understand the high κtotal in 2D carbon allotropes, we analyze separately their κph and κe components, Figure 2a), as well as their phononic and electronic band structures, Figure 2b,c, respectively. As expected, we find that κtotal of graphene is dominated by κph, which is larger than in the other three 2D carbon allotropes. On the contrary, κe of the other three 2D carbon allotropes are above or close to their κph components. We also note that the κe behavior is fully consistent with the electronic band structure of Figure 2c. In graphene and the electronic contribution can be neglected due to the almost zero electron density‐of‐states (DOS) originating from the Dirac point shown in Figure 2c at the Fermi level.[49] However, metallic features are observed in T‐graphene, biphenylene, and net‐graphene according to the electronic band dispersion crossover at the Fermi level shown in Figure 2c, hence the non‐negligible electronic contribution to heat conduction.2Figurea) κph and κe versus T, b) harmonic phonon dispersion, and c) electronic band structure from DFT calculations and Wannier interpolation[57] for graphene, T‐graphene, biphenylene, and net‐graphene. In (a), the left inset is the κph attributed to different scattering events and the right inset is the scattering rates due to 3ph, 4ph, and pe interaction processes at T = 300 K. The most recently available literature data are provided for validation as marked in the left inset of (a) with: κph3+4ph$\kappa _{{\rm{ph}}}^{{\rm{3 + 4ph}}}$ for graphene,[35] κph3ph$\kappa _{{\rm{ph}}}^{{\rm{3ph}}}$ for T‐graphene,[28] κph calculated by only 3ph scatterings[29] and EMD[30] for biphenylene, and κph by NEMD for net‐graphene.[31] In (b), the TA, LA, ZA, and one transverse optical (ZO) phonon modes are labeled and the Brillouin zone are shown. In (c), the zero of energy is set to the Fermi level.We note in passing that by including 3+4ph+pe scatterings, which is the most complete scheme of phonon‐limited scattering rate predictions based on ab initio calculations, our κph calculations are surpassing the available literature data provided (red dot circles) in the left insets of Figure 2a. Specifically, we show κph3+4ph$\kappa _{{\rm{ph}}}^{{\rm{3 + 4ph}}}$ adopting the solution of Boltzmann transport equation (BTE) with the optimized Tersoff potential[58] for graphene,[35] κph3ph$\kappa _{{\rm{ph}}}^{{\rm{3ph}}}$ calculated from density functional theory (DFT) for T‐graphene,[28] κph calculated from the DFT with only 3ph scatterings[29] and the equilibrium molecular dynamics[30] (EMD) for biphenylene, and κph by nonequilibrium EMD (NEMD) for net‐graphene.[31] Overall, our calculations agree with the literature data, indicating the reliability of our implementations, and further allow for the following insights: i) The 4ph scattering effects induce a suppression (κph3ph$\kappa _{{\rm{ph}}}^{{\rm{3ph}}}$‐κph3+4ph$\kappa _{{\rm{ph}}}^{{\rm{3 + 4ph}}}$)/κph3ph$\kappa _{{\rm{ph}}}^{{\rm{3ph}}}$ at T = 300 K by 79%, 78%, 51%, and 58% for graphene, T‐graphene, biphenylene, and net‐graphene, respectively. This indicates the importance of 4ph scatterings when computing the κph. ii) The pe scatterings induce a further suppression (κph3+4ph$\kappa _{{\rm{ph}}}^{{\rm{3 + 4ph}}}$‐κph3+4ph+pe$\kappa _{{\rm{ph}}}^{{\rm{3 + 4ph + pe}}}$)/κph3+4ph$\kappa _{{\rm{ph}}}^{{\rm{3 + 4ph}}}$ at T = 300 K by 0.01%, 15%, 34%, and 3% for graphene, T‐graphene, biphenylene, and net‐graphene, respectively. This demonstrates the need for considering the pe interactions to predict κph. iii) These two points are next supported by the data in the right inset of Figure 2a showing for each structure the magnitude of the 3ph, 4ph, and pe scattering rates. In particular, we note that the weak pe coupling behavior in graphene does not extend to the other three 2D allotropes.Origin of High κeFocusing on electron‐based conduction, Figure 3a compares the calculated κe versus T for graphene, T‐graphene, biphenylene and net‐graphene. We can see that graphene has the smallest κe. At T = 300 K, see inset of Figure 3a, and the electronic contribution to heat conduction κe/κtotal in graphene is only of ≈10%. Remarkably, this contribution increases to 81%, 60%, and 66% in T‐graphene, biphenylene, and net‐graphene, respectively. The calculated electrical conductivity σ versus T plotted in Figure 3b displays a similar trend as κe versus T. For a broader view, we placed our predicted σ values in the context of typical metals[59] shown in the right inset of Figure 3b. T‐graphene, biphenylene, and net‐graphene show larger σ values than metals at T = 300 K, indicating their robust capability of electrical transport in potential electronic devices.3FigureComparison of a) κe versus T, (b) σ versus T, c) electron relaxation time, and d) Fermi velocity and electron DOS at Fermi level for graphene, T‐graphene, biphenylene, and net‐graphene. The inset of (a) is the ratio of κe/κtotal at T = 300 K. In (b), the left inset is the Lorenz ratio and the right inset is the electrical conductivity of some typical metals.[59] The experimental data[51,60–62] of σ in (b) are provided for comparison. Note: the σ is evaluated with single layer thickness of 0.334 nm.[51]The σ calculations displayed in Figure 3b are confirming the weak electronic contribution to electron transport in graphene. We have compared our predictions with the measured σ of graphene as shown in Figure 3a. Similar to the thermal conductivity measurement in graphene, a large spread in the experiment data is also observed for σ. Our results overall locate the same order of the experimental data, indicating the reliability of our calculations. Our predictions in general further agree with the calculations using the Allen's model[63] (see Section S4, Supporting Information) as shown in Figure S3, Supporting Information, which obviously validates our implementation. For all four structures, we have also calculated their room‐temperature intrinsic mobility μ = σ/ne (where e is the elementary charge and n is the charge carrier density). We obtained 0.22, 3.81, 1.22, and 1.51 (104 cm2 V−1 s−1) for graphene, T‐graphene, biphenylene, and net‐graphene, respectively. Due to the lowest electron density of states near the Fermi level (at the Dirac point), graphene has the smallest μ. The order of magnitude (104 cm2 V−1 s−1) obtained for T‐graphene, biphenylene, and net‐graphene is close to what was reported for other 2D carbon allotropes.[64]While our theoretical framework is able to clearly distinguish the κe and κph components, in practice it is very difficult to directly measure them. Some promising experimental methods[65] have been developed for direct κph measurements, such as alloying technique,[66] superconducting technique,[67] and magnetothermal technique[68,69] but they are either very complicated to conduct, or limited to extremely low T. The electrical conductivity can be accurately and easily obtained. The Lorenz ratio L = κe/σT is generally adopted to estimate the phonon thermal conductivity as κph = κtotal − LσT in which L is generally assumed to be the Sommerfeld value L0 = 2.44 × 10−8ΩWK−2.[59,70] We note that this approximation is inadequate especially at low temperatures (T ≪ TD, where TD is Debye temperature) since L has large differences compared to L0.[59] Indeed, in the inset of Figure 3b we observe the large deviation between L and L0 in the 2D carbon allotropes, that is, L < L0, indicating that using the Sommerfeld value in this method underestimates (overestimates) κph (κe). Our result supports the violation of the Wiedemann–Franz law in 2D Dirac materials[71] because the ratio of electron thermal conductance and electric conductance is a chemical‐potential‐dependent quantity.For all four structures we display their electron lifetime (τe), Figure 3c, and their velocity (vF) and DOS (NF) at the Fermi level, Figure 3d. Invoking the Drude's free electron model, these are key quantities for κe since κeDrude=(π2/3kB2T)NFvF2τe$\kappa _{\rm{e}}^{{\rm{Drude}}} = ({\pi ^2}/3k_{\rm{B}}^2T){N_{\rm{F}}}v_{\rm{F}}^2{\tau _{\rm{e}}}$.[59] We see that graphene has the largest τe compared to other allotropes, a result that is attributed the extremely weak el‐ph coupling, as shown later in Figure 5d. However, the largest τe does not lead to largest κe (or σ). With its Dirac point is located at the Fermi level, graphene has an extremely small NF, which leads to a very small κe. T‐graphene, biphenylene, and net‐graphene have similar NF values, which are much larger than graphene. Among them, we find that T‐graphene has the largest κe (or σ) as it has the largest vF and τe (E − EF within kBT ≈0.026 eV at 300 K). Our results and the insights uncovered here for T‐graphene, biphenylene, and net‐graphene could also be shared with other theoretically predicted 2D carbon allotropes displaying metallic character, such as penta‐graphene, a‐graphene, g‐graphene, Ψ‐graphene, and pop‐graphene.[3,31,64]4FigureThe comparison of a) κph3ph$\kappa _{{\rm{ph}}}^{{\rm{3ph}}}$ and κph3+4ph$\kappa _{{\rm{ph}}}^{{\rm{3 + 4ph}}}$ at T = 300 K, b) volumetric heat capacity at T = 300 K, c) phonon group velocity, d) phonon scattering rates from normal and Umklapp processes as well as the lifetime due to 3ph scatterings at T = 300 K, e) phonon DOS, f) Grüneisen parameter, g) phase space of 3ph and 4ph scattering events, h) the strength of 100 largest third‐ and fourth‐IFCs matrix elements, and i) the ELF for graphene, T‐graphene, biphenylene, and net‐graphene. The inset of (a) is the spectral κph3ph(ω)$\kappa _{{\rm{ph}}}^{{\rm{3ph}}}(\omega )$ at T = 300 K. The inset of (c) is the average group velocity. The inset of (f) is the Grüneisen parameter in the range of ≈1–2 THz. The ELF is visualized by using VESTA package.[74] Note: the representation of the color in (a–h) is gray for graphene, red for T‐graphene, blue for biphenylene, and green for net‐graphene, respectively.5Figurea) The normalized contribution of all acoustic modes and ZO modes to κph attributed to 3ph, 3+4ph and 3+4ph+pe interactions, in which the inset is the accumulated κph with respect to phonon frequency ω, b,c) 3ph, 4ph, and pe scatterings for all acoustic modes, and d) the Eliashberg spectral function αtr2$\alpha _{{\rm{tr}}}^2$F and transport el‐ph coupling strength λtr at T = 300 K for graphene. The corresponding contents are fore–h) T‐graphene, i–l) bipheylene, and (m–p) net‐graphene.Origin of Low κphTo understand the reasons behind lowering κph in the 2D allotropes, as detailed for example in Figure 4a, we have carried a detailed comparison of the intrinsic phonon properties. The BTE derived κph, see Equation (6), is conditioned by the volumetric heat capacity Cv, group velocity vg, and relaxation time τ. By analyzing Figure 4b–d, where we have separately plotted these quantities for all four structures, we can see that all Cv, vg, and τ contribute to the lowering of κph with respect to graphene. First, we see that the Cv of graphene is much larger than that of T‐graphene, biphenylene, and net‐graphene, Figure 4b. Second, we find that the graphene has the largest vg, a result which is more clear using the average ⟨vg⟩ shown in the inset of Figure 4c. Note that since the κph of all graphene allotropes is mainly attributed to the phonon modes less than 20 THz (mainly acoustic modes) shown in the inset of Figure 4a, in Figure 4c we have plotted the vg of the acoustic modes only. Third, we find a distinct enhancement of the phonon scatterings in T‐graphene, biphenylene, and net‐graphene compared to graphene. On one hand, in graphene the normal phonon scattering rates are one to two orders of magnitude larger than the Umklapp phonon scattering rates shown in Figure 4d. This indicates a hydrodynamic phonon transport regime where the phonon transport is much less dissipative, leading to extraordinary large κph.[72] On the other hand, in T‐graphene, biphenylene, and net‐graphene, we find in Figure 4d that the normal and Umklapp phonon scattering rates are at comparable levels at 300 K. This indicates the breakdown of hydrodynamic phonon transport (dominant in graphene) and a significant increase in dissipation.Focusing next on the anharmonic effects, we find that all the important indicators concur to an enhancement of anharmonicity in T‐graphene, biphenylene, and net‐graphene. In Figure 4e we first find that compared to graphene there is a significant increase of phonon DOS for T‐graphene, biphenylene, and net‐graphene in 1–10 THz. Hence, less phonon interactions can occur in graphene. Second, in Figure 4f the Grüneisen parameter,[73] which is a measure of the anharmonicity, is smallest in graphene. Third, in Figure 4g we see that the phonon phase space reflecting all available phonon scattering processes for simultaneously satisfying the energy and momentum conservation in graphene is much smaller than that in T‐graphene, biphenylene, and net‐graphene. Forth, the anharmonic force constants of Figure 4h show that graphene has the smallest strength of the third and fourth order interatomic force constants (IFCs, Φ).The strong anharmonicity displayed by T‐graphene, biphenylene, and net‐graphene in comparison with graphene originates in changes of the covalent bonding of strained bonds in reduced crystal symmetry environment. In Figure 4i, we characterize bonding with the electron localization function (ELF).[75,76] The position where the electron is localized has a high ELF value. As expected, ELF is largest at the center of all CC bonds, indicating the dominance of covalent bonding in the 2D carbon allotropes. We can further observe that while in graphene, the distribution of ELF is symmetric with respect to the CC bond axis, evident deviations are visible in T‐graphene, biphenylene, and net‐graphene. In the latter structures, ELF indicates that electrons spread toward the center of the larger eight‐membered and six‐membered rings and away from the center of the smaller four‐membered rings. We interpret these ELF changes as a shift in sp2 bonding caused by electrostatic repulsion, which spills the electronic charge from the smaller into the larger neighboring rings.The ZA modes are known to be very important in the phonon transport in 2D materials.[32,51] We investigate their contribution to κph arising from 3ph, 4ph and pe scatterings. In graphene, Figure 5a, we can see that the relative contribution of the ZA branch to κph is reduced from 80% with only 3ph scattering to 30% after including 4ph scattering. Further including pe scatterings gives minor changes. As shown in Figure 5b, the reduction of κph3+4ph$\kappa _{{\rm{ph}}}^{{\rm{3 + 4ph}}}$ is caused by the enhancement of the 4ph scattering rates of ZA modes which are 3 to 4 orders larger than the 3ph scattering rates. These results do not follow the general rule of bulk materials that 4ph scattering is important only in strongly anharmonic materials.[77,78] The behavior obtained in graphene is attributed to the reflection symmetry selection rule, which is unique to 2D materials. Previous works[51,79] showed that reflection symmetry of graphene (or any 2D crystal) requires that the nth order anharmonic IFCs vanish: Φα1…αnl1b1…lnbn=0$\Phi _{{\alpha _1}{\rm{ \ldots }}{\alpha _n}}^{{l_1}{b_1}{\rm{ \ldots }}{l_n}{b_n}} = 0$ when the number of out‐of‐plane components in the string α1...αn is odd. This leads to a selection rule that forbids any n‐phonon scattering process involving an odd number of flexural (out‐of‐plane) modes. Therefore, the three‐ZA (ZA+ZA → ZA) scattering processes are forbidden. This means that the major 3ph scattering phase space for ZA modes is eliminated and the ZA modes are the major heat carriers accounted for by κph3ph$\kappa _{{\rm{ph}}}^{{\rm{3ph}}}$. However, for 4ph scatterings, the four‐ZA processes ZA+ZA→ZA+ZA and ZA→ZA+ZA+ZA are allowed. There is abundant 4ph scatterings for ZA modes, leading to the suppression of κph3+4ph$\kappa _{{\rm{ph}}}^{{\rm{3 + 4ph}}}$. The ZA modes behave similarly in T‐graphene, biphenylene, and net‐graphene, where we can see significant suppression of κph3+4ph$\kappa _{{\rm{ph}}}^{{\rm{3 + 4ph}}}$ in Figure 5e,i,m as well as extremely large 4ph scattering rates in Figure 5f,j,n.A different trend to the contribution to κph is found for the TA and LA modes, Figure 5a,e,i,m. For these modes, a large percentage of 4ph processes than 3ph processes are forbidden, as shown in Figure 5c,g,k,o. In addition, from the inset of Figure 5a,e,i,m, we can see that the pe scattering effect on the κph is relative small in graphene, T‐graphene, and net‐graphene but it is strong in biphenylene. The transport el‐ph coupling strength λtr = 0.27 (see definition in Section S5, Supporting Information) for biphenylene shown in Figure 5l is larger than that of 0.01 for graphene in Figure 5d, 0.12 for T‐graphene in Figure 5h, and 0.06 in Figure 5p for net‐graphene, indicating the stronger el‐ph coupling in biphenylene than other 2D carbon allotropes.ConclusionIn conclusion, using the Boltzmann transport equation formalism in combination with density functional theory calculations, we uncovered 2D carbon allotropes with both electron‐based and phonon‐based thermal conduction. Although electron‐based thermal conduction is small in graphene, it is ultrahigh in 2D carbon allotropes with non‐hexagonal rings. Phonon‐based conduction remains important, although the effect is deteriorated by the anharmonicity effects caused by the changes in CC bonding resulted from strain and lattice symmetry changes. The electron‐based conduction compensates for the κph decrease and renders large κtotal for T‐graphene, biphenylene, and net‐graphene, close to those of graphene, h‐BN and MoS2, which exhibit phonon‐based thermal conduction. Other theoretically predicted 2D carbon allotropes, such as penta‐graphene, α‐graphene, g‐graphene, Ψ‐graphene, and pop‐graphene, display metallic character, and thus could share the impressive attributes uncovered here for T‐graphene, biphenylene, and net‐graphene. 2D carbon allotropes may contribute to realization of the next‐generation technologies requiring nanostructures with ability to transfer of thermal energy by a combination of diffusion of electrons and phonon vibrations.Experimental SectionElectronic Thermal Conductivity and Electrical ConductivityBy solving the BTE of electron,[80] the electrical conductivity tensor (σαβ) and electronic thermal conductivity tensor (κeαβ$\kappa _{\rm{e}}^{\alpha \beta }$) can be obtained as1{σαβ=LEEαβ,κeαβ=−(LTTαβ−LTEαβLETαβLEEαβ)\[\left\{ \begin{array}{c}{\sigma ^{\alpha \beta }} = L_{{\rm{EE}}}^{\alpha \beta },\\\kappa _{\rm{e}}^{\alpha \beta } = - \left( {L_{{\rm{TT}}}^{\alpha \beta } - \frac{{L_{{\rm{TE}}}^{\alpha \beta }L_{{\rm{ET}}}^{\alpha \beta }}}{{{L_{{\rm{E}}{{\rm{E}}^{\alpha \beta }}}}}}} \right)\end{array}\right.\]in which the coefficients are2LEEαβ=−e2nsNkV∑ik∂fik0∂εikvikαvikβτikep\[L_{{\rm{EE}}}^{\alpha \beta } = - \frac{{{e^2}{n_s}}}{{{N_k}V}}\mathop \sum \limits_{ik} \frac{{\partial f_{ik}^0}}{{\partial {\varepsilon _{ik}}}}v_{ik}^\alpha v_{ik}^\beta \tau _{ik}^{{\rm{ep}}}\]3LETαβ=−ensNkVT∑ik(εik−μ)∂fik0∂εikvikαvikβτikep\[L_{{\rm{ET}}}^{\alpha \beta } = - \frac{{e{n_s}}}{{{N_k}VT}}\mathop \sum \limits_{ik} \left( {{\varepsilon _{ik}} - \mu } \right)\frac{{\partial f_{ik}^0}}{{\partial {\varepsilon _{ik}}}}v_{ik}^\alpha v_{ik}^\beta \tau _{ik}^{{\rm{ep}}}\]4LTEαβ=−ensNkV∑ik(εik−μ)∂fik0∂εikvikαvikβτikep\[L_{{\rm{TE}}}^{\alpha \beta } = - \frac{{e{n_s}}}{{{N_k}V}}\mathop \sum \limits_{ik} \left( {{\varepsilon _{i{{\bf k}}}} - \mu } \right)\frac{{\partial f_{ik}^0}}{{\partial {\varepsilon _{ik}}}}v_{ik}^\alpha v_{ik}^\beta \tau _{ik}^{{\rm{ep}}}\]5LTTαβ=−nsNkVT∑ik(εik−μ)2∂fik0∂εikvikαvikβτikep\[L_{{\rm{TT}}}^{\alpha \beta } = - \frac{{{n_s}}}{{{N_k}VT}}\mathop \sum \limits_{ik} {\left( {{\varepsilon _{ik}} - \mu } \right)^2}\frac{{\partial f_{ik}^0}}{{\partial {\varepsilon _{ik}}}}v_{ik}^\alpha v_{ik}^\beta \tau _{ik}^{{\rm{ep}}}\]where e is the elementary charge, ns is the number of electrons per state, V is the volume of the unit cell, Nk the total number of k‐points sampled in the first Brillouin zone, T is the temperature, fik0$f_{i{{\bf k}}}^0$ is the electron distribution function, vik=1ℏ∂εik∂k${{{\bf v}}_{ik}} = \frac{1}{\hbar }\frac{{\partial {\varepsilon _{ik}}}}{{\partial k}}$ is the electron velocity, α and β denotes the directional component. The summation in these three equations was over all the electrons enumerated using electronic wave vector k\[k\] and band index i. μ is the chemical potential. τikep$\tau _{ik}^{{\rm{ep}}}$ is the electron lifetime (or relaxation time). It should be noted that only the el‐ph scattering rates (1/τikep$1/\tau _{ik}^{{\rm{ep}}}$) were considered in computing the κe. The expression of 1/τikep$1/\tau _{ik}^{{\rm{ep}}}$ can be found in Section S1, Supporting Information.Phononic Thermal ConductivitySolving the BTE of phonon combined with the Fourier's law of heat conduction, the phonon thermal conductivity tensor (κpαβ$\kappa _{\rm{p}}^{\alpha \beta }$) can be written as a summation over all phonon modes λ = (q, ν) of polarization ν, wave vector q$q$ and frequency ωλ as6κpαβ=1Nq∑λcλvλ,αvλ,βτλp\[\kappa _{\rm{p}}^{\alpha \beta } = \frac{1}{{{N_q}}}\mathop \sum \limits_\lambda {c_\lambda }{v_{\lambda ,\alpha }}{v_{\lambda ,\beta }}\tau _\lambda ^{\rm{p}}\]where α and β are indexing the Cartesian directions, Nq is the total number of q‐points sampled in the first Brillouin zone, while cλ, vλ, and τλp$\tau _\lambda ^{\rm{p}}$ denote the volumetric heat capacity, phonon group velocity, and phonon lifetime (or relaxation time), respectively. Note that cλ=(ℏωλ/V)(∂nλ0/∂T)${c_\lambda } = (\hbar {\omega _\lambda }/V)(\partial n_\lambda ^0/\partial T)$, where nλ0$n_\lambda ^0$ is the Bose–Einstein distribution function and V the volume of the primitive cell, and vλ, α = ∂ωλ/∂qα. The phonon scattering rate (1/τλp$1/\tau _\lambda ^{\rm{p}}$) is a summation of the phonon–phonon (1/τλpp$1/\tau _\lambda ^{{\rm{pp}}}$), pe (1/τλpe$1/\tau _\lambda ^{{\rm{pe}}}$), phonon–impurity (1/τλim$1/\tau _\lambda ^{{\rm{im}}}$), phonon–isotope (1/τλiso$1/\tau _\lambda ^{{\rm{iso}}}$), and phonon–grain boundary (1/τλgb$1/\tau _\lambda ^{{\rm{gb}}}$) scattering rates. Here, the κp of tr‐BeN4 based on ab initio computed 1/τλpp$1/\tau _\lambda ^{{\rm{pp}}}$ with 3ph and 4ph scatterings as well as 1/τλpe$1/\tau _\lambda ^{{\rm{pe}}}$ scattering rates are predicted, which can be summed using Matthiessen's rule[80] with 1/τλp=1/τλ3ph+1/τλ4ph+1/τλpe$1/\tau _\lambda ^{\rm{p}} = 1/\tau _\lambda ^{{\rm{3ph}}} + 1/\tau _\lambda ^{{\rm{4ph}}} + 1/\tau _\lambda ^{{\rm{pe}}}$. The detailed expressions of 1/τλ3ph$1/\tau _\lambda ^{{\rm{3ph}}}$, 1/τλ4ph$1/\tau _\lambda ^{{\rm{4ph}}}$, and 1/τλpe$1/\tau _\lambda ^{{\rm{pe}}}$ can be found in Section S2, Supporting Information.Ab Initio Computational DetailsThe DFT and density functional perturbation theory (DFPT) calculations were carried using the VASP[81] package with Perdew–Burke–Ernzerhof exchange and correlation functional.[82] The 3ph scattering rates were calculated using ShengBTE[73] and the 4ph scattering rates were computed using an in‐house code.[36,77,83] The el‐ph coupling matrix were calculated using DFPT with norm‐conserving pseudopotentials[84] with generalized gradient approximation[82] exchange and correlation functional as implemented in QUANTUM ESPRESSO.[85] They were initially obtained on coarse electron and phonon wave vector grids and then implemented to finer grids using the maximally localized Wannier functions basis as implemented in the el‐ph Wannier (EPW)[86,87] to obtain the el‐ph scatterings for computing κe and κph. Importantly, an in‐house modified EPW code[70,88,89]was used to predict the el‐ph coupling scattering rates in order to compute the huge dense k‐ and q‐points more efficiently. The simulation parameters and computational details are provided in Section S3, Supporting Information.AcknowledgementsThe authors thank Dr. Shouhang Li for valuable discussions. Simulations were performed at the Tianhe2‐JK of Beijing Computational Science Research Center. Z.T. acknowledges the support by National Natural Science Foundation (No. 52106068), China Postdoctoral Science Foundation (No. 2020M680127), Guangdong Basic and Applied Basic Research Foundation (Nos. 2020A1515110838 and 2021A1515011688), and Shenzhen Science and Technology Program (No. RCBS20200714114919142). 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Journal

Advanced Energy Materials – Wiley

Published: Jul 1, 2022

Keywords: 2D materials; ab initio calculations; Boltzmann transport equation; carbon allotropes; electron thermal conductivity

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