Fermi surface studies of the low-temperature structure of sodium

Sodium is the most abundant alkali-metal element and has one of the simplest electronic structures of any metal. At ambient conditions, sodium forms a body-centered-cubic lattice. However, during cooling, it undergoes a partial martensitic phase transition to a complex mixture of rhombohedral polytypes commencing from 36 K. Although the Fermi surface (FS) of bcc sodium has been extensively studied, not much attention has been given to the FS of the martensite structure. Here we report results for the Fermi surface and quantum oscillation (QO) frequencies of several energetically favorable crystal structures of Na at low temperature from first-principles calculations. Interestingly we find that despite drastic differences in the crystal structures of the candidate low-temperature phases of sodium, for all these phases the strongest quantum oscillation peak is centered at 28 kT. Our theoretical results are accompanied by experimental data of QO on a multigrain sodium sample at $T=0.3$ K and $B_{\max }=18$ T exhibiting a sharp peak at 28 kT, independent of the sample orientation. The persistence of this peak even in the presence of the structural transitions has an implication for using the quantum oscillations of polycrystalline sodium for high magnetic field calibration.

Figure 1

The bcc, fcc, and $9R$ structures of sodium at 1 atmosphere pressure.

Figure 2

Brillioun zones, Fermi surfaces, extremal orbits, and dHvA/SdH frequencies for (a) bcc and (b) fcc Na computed for 0 K at 1 atmosphere pressure.

Figure 3

Fermi surface calculations for bcc and fcc Na computed at 5 and 75 GPa. (a) Brillouin zone and Fermi surface and (b) the dHvA/SdH frequencies for the magnetic field orientations ($\theta =0$ and $\varphi =0$).

Figure 4

Brillouin zones, Fermi surfaces, selected extremal orbits, and dHvA/SdH frequencies for hcp Na at 0 K and 1 atmosphere computed for crystal orientations relative to the magnetic field specified by ($\varphi =0^{\circ }$ and $\theta =0^{\circ }$). The definitions of these angles are shown in Fig. S5. Here (a) and (b) show the Fermi surfaces of bands 9 and 10, respectively. The hcp band structure is shown in Fig. S3. The combined FSs of all bands are shown in (c).

Figure 5

Brillouin zones, Fermi surfaces, selected extremal orbits, and dHvA/SdH frequencies for $9R$ Na at 0 K and 1 atmosphere computed for crystal orientations relative to the magnetic field specified by ($\varphi =0^{\circ }$ and $\theta =0^{\circ }$). Here (a), (b), and (c) show the surfaces originating from bands 13, 14, and 15, respectively. The $9R$ band structure is shown in Fig. S4. The combined FSs of all bands are shown in (d).

Figure 6

Spectrum of the dHvA/SdH frequencies for all magnetic field orientations for the Na structures: bcc, fcc, hcp, and $9R$. The dHvA/SdH frequencies are collected in uniform bin sizes equal to 0.05 kT. All magnetic orientations are considered by varying the angles $\theta$ and $\varphi$ from $0^{\circ }$ to ${360}^{\circ }$ and ${90}^{\circ }$, respectively, in increments of $1^{\circ }$. The experimental data were collected at temperature 350 mK using magnetic fields between 16 and 18 T. The dotted vertical line in the figure is a guide for the eye. The Na samples were cut from a high-purity reagent grade dry Na (Sigma-Aldrich) and were embedded.

Figure 7

Fourier spectrum of measured oscillations at four different sample orientations. The short vertical dashed lines indicate the peak positions of the distributions. The long vertical lines (solid and dashed) are the computed frequencies for bcc and fcc at 1 atmosphere pressure, as indicated in the legend. The blue shaded region indicates the variation of the fcc frequency when the pressure is varied by $\pm 0.1$ GPa.

Figure 8

A histogram analysis of the peaks frequencies observed in fermiology measurement for all the measured orientations (see Supplemental Material for Fourier spectra of the sample at all orientations [29]).