Low-temperature features in the heat capacity of unary metals and intermetallics for the example of bulk aluminum and ${\mathrm{Al}}_3\mathrm{Sc}$

We explore the competition and coupling of vibrational and electronic contributions to the heat capacity of Al and ${\mathrm{Al}}_3\mathrm{Sc}$ at temperatures below 50 K, combining experimental calorimetry with highly converged finite-temperature density functional theory calculations. We find that semilocal exchange-correlation functionals accurately describe the rich feature set observed for these temperatures, including electron-phonon coupling. Using different representations of the heat capacity, we are therefore able to identify and explain deviations from the Debye behavior in the low-temperature limit and in the temperature regime 30–50 K as well as the reduction of these features due to the addition of Sc.

Figure 1

Convergence of the $k$ mesh for the electronic contribution to the heat capacity $C_P^{\text{el}}$ in $k_B$ for $T$ = 5, 10, 15, and 20 K. The $N$ on the $x$ axis denotes the $k$-mesh size ($N^3$). The gray shaded area corresponds to a width of magnitude $1\times {10}^{-3}{k}_B$ (boundary values depend on temperature) within which the converged $C_P^{\text{el}}$ fluctuates.

Figure 2

Orientation imaging microscopy using EBSD analysis of the ${\mathrm{Al}}_3\mathrm{Sc}$ sample. The recorded EDX spectrum is shown in the inset, and it reveals the presence of solely Al and Sc atoms.

Figure 3

Isobaric heat capacity $C_P$ plots for Al and ${\mathrm{Al}}_3\mathrm{Sc}$. (a) and (d) Measured ($\times$ symbols) and calculated (solid lines: GGA, dashed lines: LDA) $C_P$ for Al and ${\mathrm{Al}}_3\mathrm{Sc}$ in $k_B/\text{atom}$, respectively. The labels Qh and El correspond to the quasiharmonic and electronic contributions, respectively. The inset provides a close-up to highlight the electronic contribution. (b) and (e) Renormalized $C_P/T^3$ curves for Al and ${\mathrm{Al}}_3\mathrm{Sc}$ in J ${\mathrm{g}}^{-1} {\mathrm{K}}^{-4}$. The $⧫$ symbols are experimental data for Al taken from Ref. [78]. The inset shows $C_P/T^3$ up to 150 K on a linear $T$ axis. The magenta curve labeled “silica” is the renormalized $C_P/T^3$ for amorphous silica taken from Ref. [81] for comparison with the current work. (c) and (f) Plots of $C_P/T$ in mJ ${\mathrm{mol}}^{-1} {\mathrm{K}}^{-2}$ versus $T^2$ to estimate the coefficients $C_1$ and $C_3$ for Al and ${\mathrm{Al}}_3\mathrm{Sc}$. The upper twin $x$ axis shows the absolute temperature (in K). The solid (dashed) blue and maroon lines correspond to the GGA (LDA) calculations without (W/o) and with electron-phonon (el-ph) coupling, respectively. The $△$ and $\square$ symbols are the experimental data for Al taken from Refs. [20, 82] respectively. The gray, green, and orange dashed lines are the linear fits to the available low-temperature experiments represented by symbols of the same color. The colored numbers written next to the $y$ axes are the $y$ intercepts ($C_1$) corresponding to different calculated curves. The coefficients $C_1$ and $C_3$ obtained from the fit to the available experiments are given in the respective colors.

Figure 4

Linear fit (red dashed line) to the present experiments in the representation given by Eq. (10) up to approximately 60 K. The upper $x$ axis represents the absolute temperature $T$. Inset: Close-up of the low-temperature region up to $T\approx 32$ K ($T^2=1000$ K).

Figure 5

(c) Calculated phonon dispersion curves of Al (solid red lines) and ${\mathrm{Al}}_3\mathrm{Sc}$ (dashed blue lines) plotted along a path in the BZ for ${\mathrm{Al}}_3\mathrm{Sc}$. (b) and (d) The corresponding vibrational density of states (VDOS) $g\left(\nu \right)$ and (a) and (e) reduced VDOS $g\left(\nu \right)/{\nu }^2$ for Al and ${\mathrm{Al}}_3\mathrm{Sc}$, respectively. The brown curve represents the Debye VDOS $[g\left(\nu \right)\propto {\nu }^2]$ up to the respective Debye frequencies. The magenta curve (taken from Ref. [84]) represents the reduced VDOS for amorphous silica for comparison. The labels $P_1,P_2,P_3,P_4$ mark the observed peaks in the VDOS. The orange arrow at point $R$ in (c) marks the selected phonon mode to study the effect of electronic temperature on the phonons (see Sec. 2b).

Figure 6

Effect of excluding the 4–6-THz frequency band on the low-temperature peak (orange and green dashed curves) in the renormalized heat capacity $C_V/T^3$ plot (right $y$ axis) for Al and ${\mathrm{Al}}_3\mathrm{Sc}$. The solid orange and green curves correspond to $C_V/T^3$ considering all the frequency modes in the phonon dispersion. The fading gray curves are the Einstein $C_V/T^3$ calculated at frequency $\nu$, corresponding to the peaks $P_1,P_2,P_3$, and $P_4$ in the VDOS of ${\mathrm{Al}}_3\mathrm{Sc}$ [Fig. 5], respectively. The dashed gray curve joins the maxima (gray circles) of the Einstein $C_V/T^3$ curves. Note that theory is given as $C_V/T^3$, whereas experimental data are given as $C_P/T^3$, resulting in an insignificant difference.