Scattering of phonons by high-concentration isotopic impurities in ultrathin graphite

As the isotopic concentration of ultrathin graphite is varied from natural abundance to nearly pure ${}^{13}\mathrm{C}$, the thermal conductivity displays a slight dependence on the isotope concentration at temperatures near its maximum $\sim 150\mathrm{K}$. The strength of phonon-isotope scattering in the high-isotope impurity-concentration regime is found to be well below that given by a commonly used incoherent and independent isotope impurity scattering model. This finding is in agreement with some recent theoretical predictions that coherent multiple scattering of phonons is important in the measured thermal conductivity of low-dimensional materials in the high-isotope impurity-concentration regime.

Figure 1

SEM images of (a) UGF, (b) the surface of as-grown UGF, and (c) the surface of UGF after postsynthesis annealing at 3000 °C. (d) XRD of the annealed UGF shown in comparison with the reflection positions of highly oriented pyrolytic graphite (HOPG) [56]. The intensities have been normalized to the {0002} peak. [(d), inset] High-resolution XRD of the {0004} peak.

Figure 2

Temperature-dependent electrical properties of UGF after postsynthesis annealing. The isotopic concentration of ${}^{13}\mathrm{C}$ in ${}^{12}\mathrm{C}$ for the UGF samples is 1.1% (green down triangles), 50.2% (red diamonds), and 99.2% (blue circles). (a) Measured low-bias electrical resistance $\left(R_{\mathrm{UGF}}\right)$ and (b) normalized low-bias resistance change with temperature at each temperature $(d{R}_{\mathrm{UGF}}/dT)/R_{\mathrm{UGF}}$. (c) Electrical resistivity $\left({\rho }_{\mathrm{UGF}}\right)$ of the UGF. (d) Solid electrical resistivity of the ultrathin graphite $\left({\rho }_{\mathrm{UG}}\right)$ within the UGF, calculated using the foam theory of Lemlich [27], ${\rho }_{\mathrm{UG}}=(\varphi /3){\rho }_{\mathrm{UGF}}$, where φ is the volume fraction. Shown in comparison are values for highly oriented pyrolytic graphite deposited at 2250 °C (HOPG, gray crosses) [57], HOPG deposited at 2200 °C, subsequently heat-treated HOPG (HT-HOPG) at 3200 °C (gray asterisks) [10], and single-crystal graphite (SC-G, gray stars) [58]. [(d), inset] ${\rho }_{\mathrm{UG}}$ of the 50.2- and 99.2-at. $\%{}^{13}\mathrm{C}$ samples normalized by that of the 1.1-at. $\%{}^{13}\mathrm{C}$ sample. The legend shown in panel (a) applies to panels (a)–(d).

Figure 3

(a) Measured thermal conductance of the UGF samples $\left(G_{\mathrm{UGF}}\right)$ determined with (filled symbols) and without (open symbols) the inclusion of radiation in the heat transfer model, i.e., Eqs. (3) and (5) in Ref. [25], respectively. The isotopic concentration of ${}^{13}\mathrm{C}$ in ${}^{12}\mathrm{C}$ for the UGF samples is 1.1% (green down triangles), 50.2% (red diamonds), and 99.2% (blue circles). (b) Measured effective thermal conductivity of the $\mathrm{UGF}\left({\kappa }_{\mathrm{UGF}}\right)$ versus temperature. The legend shown in panel (a) applies to panels (a) and (b).

Figure 4

(a) Solid thermal conductivity of the UG $\left({\kappa }_{\mathrm{UG}}\right)$ versus temperature for ${}^{13}\mathrm{C}$ concentration of 1.1% (green down triangles), 50.2% (red diamonds), and 99.2% (blue circles). The lines are the calculated thermal conductivity based on the fitting parameters discussed in the text for different ${}^{13}\mathrm{C}$ concentrations. (b) Calculated solid thermal conductivity normalized by the theoretical value for isotopically pure ${}^{12}\mathrm{C}$ graphite $\left({\kappa }_{\mathrm{UG}}/{\kappa }_{{}^{12}\mathrm{C}}\right)$ as a function of ${}^{13}\mathrm{C}$ isotopic concentration at temperatures of 152 K (purple down triangles) and 303 K (orange circles). In one calculation, the isotope-scattering coefficient $B$ is taken to be $B_i=\pi /4$ for the in-plane mode and $B_o=\pi /2$ for the out-of-plane modes. In another calculation, the two $B$ coefficients are reduced to $B_i=\pi /10.8$ for the in-plane mode and $B_o=\pi /5.4$.

Figure 5

(a) Raman spectra normalized to the $G$ peak intensity using 2.41-eV laser energy. The isotopic concentration of ${}^{13}\mathrm{C}$ in ${}^{12}\mathrm{C}$ for the UGF samples is 1.1% (green line), 50.2% (red line), and 99.2% (blue line). [(a), inset] The Raman peak positions (ω) normalized by those for 1.1-at. $\%{}^{13}\mathrm{C}\mathrm{UGF}$ exhibit the predicted dependence on the average atomic mass for Raman-active phonon modes, i.e., $\omega /{\omega }_{1.1\%{}^{13}\mathrm{C}}={\left(\overline{M}/{\overline{M}}_{1.1\%{}^{13}\mathrm{C}}\right)}^{-1/2}$. The Raman peaks are labeled on the 1.1-at. $\%{}^{13}\mathrm{C}$ spectra in the main panel where circles and down triangles correspond to near-zone-center LO and 2LO processes, respectively, and asterisks, diamonds, and stars denote near-zone-boundary iTO + LA, 2iTO along $\Gamma \text{–}\mathrm{K}$, and 2iTO along Κ–Μ processes, respectively. (b)–(d) Raman spectra of graphite foams with (b) 1.1%, (c) 50.2%, and $99.2\%{}^{13}\mathrm{C}$ excited by 2.54-eV (blue), 2.4-eV (green), and 1.96-eV (red) laser energies. Legend in (b) applies to all panels (b)–(d).

Figure 6

(a) Calculated phonon dispersion for bulk graphite with 1.1-at. $\%{}^{13}\mathrm{C}$ (green solid lines), 50.2-at. $\%{}^{13}\mathrm{C}$ (red dashed-dotted lines), and 99.2-at. $\%{}^{13}\mathrm{C}$ (blue dashed lines) isotopic concentrations using the virtual crystal approximation. Experimental dispersion data obtained from first-order (filled symbols) and derived from second-order (open symbols) Raman scattering using an ab initio electron band-structure calculation for graphite is shown for UGF with ${}^{13}\mathrm{C}$ concentrations of 1.1% (green down triangles), 50.2% (red diamonds), and 99.2% (blue circles). (b) Near-zone-center phonon group velocity (sound velocity, $v_{\mathrm{s}}\equiv {\lim }_{\mathbf{q}\to 0}{\nabla }_{\mathbf{q}}\omega )$ taken along Γ-Μ for the LA and in-plane transverse acoustic (iTA) polarizations as a function of ${}^{13}\mathrm{C}$ isotopic concentration. [(b), inset] The sound velocities normalized by that for ${}^{12}\mathrm{C}$ graphite exhibit the expected dependence on atomic mass, i.e., $v_{\mathrm{s}}/v_{\mathrm{s},{}^{12}\mathrm{C}}={\left(\overline{M}/M_{{}^{12}\mathrm{C}}\right)}^{-1/2}$. (c) Calculated volumetric specific heat $c$ versus temperature for ${}^{13}\mathrm{C}$ concentrations of 1.1 at. % (green solid line), 50.2 at. % (red dash-dot line), and 99.2 at. % (blue dashed line). [(c), inset] Due to the reduced group velocity of isotopically enriched graphite, $c$ for 50.2- and 99.2-at. $\%{}^{13}\mathrm{C}$ is higher than that for pure ${}^{12}\mathrm{C}$ at the temperature range of calculation, i.e., below 500 K.

Figure 7

First-order (left panel) and second-order (right panel) Raman modes measured in the UGF samples with (a) 1.1%, (b) 50.2%, and (c) $90.2\%{}^{13}\mathrm{C}$ in comparison with ab initio phonon dispersions. Red, green, and blue symbols indicate experimental data obtained using laser exitations at 1.96 eV (632.8 nm), 2.41 eV (514.5 nm), and 2.54 eV (488.0 nm), respectively.

Figure 8

Top-view photograph of the experimental setup for measuring the electrical and thermal properties of UGF. Units of integers printed on ruler are in centimeters.

Figure 9

ab initio electron band structure of graphite. The red lines indicate the π and ${\pi }^{*}$ bands, i.e., valence and conduction bands, respectively, along the $\Gamma \text{–}\mathrm{K}\text{–}\mathrm{M}$ direction used in the electron-scattering analysis. Dashed lines indicate σ and ${\sigma }^{*}$ bands.

Figure 10

Double-resonant Raman-scattering processes. (a) Brillouin zone of graphite showing electron energy contours (black dashed lines). Intervalley scattering processes $q_1$ (orange solid arrow) involve two phonons belonging to the iTO branch along the $\Gamma \text{–}\mathrm{K}$ direction. Intervalley scattering processes $q_2$ (teal solid arrow) involve two phonons belonging to the iTO branch along the Κ-Μ direction as $q_2$ can be translated with a reciprocal lattice translational vector to lie along K–M. Intravalley scattering processes $q_3$ (magenta solid arrow) involve two phonons belonging to the LO branch. The corresponding dashed arrows have the same directions and lengths as the solid arrows and are used to denote the phonon wave vectors originating from the Brillouin-zone center (Γ point). (b)–(d) Energy wave-vector diagrams indicating corresponding scattering processes in (a). Photon energy resulting in the creation of an electron-hole pair $E_1$, corresponding electron wave vector $k_1$, energy loss by phonon generation $\hslash {\omega }_{\mathrm{ph}}$, and corresponding electron wave vector $k_2$, are indicated on each graph and are exaggerated for clarity.

Figure 11

(a) Solid thermal conductivity of the ultrathin graphite $\left({\kappa }_{\mathrm{UG}}\right)$ versus temperature for ${}^{13}\mathrm{C}$ concentration of 1.1% (green down triangles), 50.2% (red diamonds), and 99.2% (blue circles). The lines are the calculated thermal conductivity based on the fitting parameters discussed in Appendix pp4 for different ${}^{13}\mathrm{C}$ concentrations. (b) Calculated solid thermal conductivity normalized by the theoretical value for isotopically pure ${}^{12}\mathrm{C}$ graphite $\left({\kappa }_{\mathrm{UG}}/{\kappa }_{{}^{12}\mathrm{C}}\right)$ as a function of ${}^{13}\mathrm{C}$ isotopic concentration at temperatures of 152 K (purple down triangles) and 303 K (orange circles). In one calculation, the isotope-scattering coefficient $B$ is taken to be $B_i=\pi /4$ for the in-plane mode and $B_o=\pi /2$ for the out-of-plane modes. In another calculation, the two $B$ coefficients are reduced to $B_i=\pi /50$ for the in-plane mode and $B_o=\pi /25$.