Fermi surface evolution of Na-doped PbTe studied through density functional theory calculations and Shubnikov–de Haas measurements

Abstract

We present a combined experimental and theoretical study of the evolution of the low-temperature Fermi surface of lead telluride, PbTe, when holes are introduced through sodium substitution on the lead site. Our Shubnikov-de-Haas measurements for samples with carrier concentrations up to 9.4 × 10 19 cm −3 (0.62 Na atomic %) show the qualitative features of the Fermi surface evolution (topology and effective mass) predicted by our density functional (DFT) calculations within the generalized gradient approximation (GGA): we obtain perfect ellipsoidal L-pockets at low and intermediate carrier concentrations, evolution away from ideal ellipsoidicity for the highest doping studied, and cyclotron effective masses increasing monotonically with doping level, implying deviations from perfect parabolicity throughout the whole band. Our measurements show, however, that standard DFT calculations underestimate the energy difference between the L-point and Σ-line valence band maxima, since our data are consistent with occupation of a single Fermi surface pocket over the entire doping range studied, whereas the calculations predict an occupation of the Σ-pockets at higher doping. Our results for low and intermediate compositions are consistent with a non-parabolic Kane-model dispersion, in which the L-pockets are ellipsoids of fixed anisotropy throughout the band, but the effective masses depend strongly on Fermi energy.

Sharp increase in the density of states in PbTe upon approaching a saddle point in the band structure

“…In this form the orbits from the four L-pockets collapse onto a single curve and the ellipsoidal model reduces to unity (orange dashed line). The validity of this analysis is confirmed in Figure 5a for a sample at lower doping (p = 4.1 × 10 19 cm −3 ) using data reanalysed from Giraldo-Gallo et al 14 the present data follows the model up to θ ≈30 • but f (θ) deviates upwards at higher angles. It is important to note that simply increasing K does not reconcile the model and the data for realistic values.…”

“…The lower temperature range was necessitated by the small values of the Sommerfeld coefficient and higher-order phonon terms observed to low temperature in PbTe which make extrapolations from He 4 temperatures unreliable. EMPA was performed on the very same two crystals used in the quantum oscillation measurements (as well as two well characterised, physical vapour transport grown samples from our previous study 14 ) and found the crystals to be homogenous and single phase (see Appendix E), ruling out the presence of precipitates that have been found previously in quenched solid solutions 26 . EMPA was performed with a JEOL JXA-8230 SuperProbe Electron Probe Microanalyzer at the Stanford Microchemical Analysis Facility.…”

“…Recently, a detailed quantum oscillation study by Giraldo-Gallo et al 14 extended previous works 23 to characterise the Fermi surface in Na-doped PbTe (Pb 1−x Na x Te) down to 160 meV below the valence band edge. It was shown that the Fermi surface is fully described by ellipsoid of fixed ellipsoidicity located at the Lpoint with its major semi-axis along the Λ line, as shown in Figure 1a and illustrated in green in Figure 1d.…”

“…Figure 2a shows raw, symmetrised and anti-symmetrised V y data in magnetic fields up to 30 T for a heavily doped sample of Pb 1−x Na x Te, with the component that is antisymmetric in B representing the Hall effect. In Figure 2b this data is replotted as the Hall number (p H = 1/R H e) and compared to samples of lower doping that were characterised in the previous study by Giraldo-Gallo et al 14…”

“…Both experimental investigations and calculations of the electronic structure in PbTe show some significant disagreement on the relative energies at which the valence band should deviate from a single-band Kane-type dispersion, whose key features are a constant anisotropy (ellipsoidicity) and a linear dependence of the electronic effective mass as a function of energy close to the band extrema. It also remains unclear whether the band structure generally deviates from Kane-type via the appearance of a second valence-band maximum, significant nonellipsoidicity, or resonant impurity states at the Fermi level in PbTe [13][14][15][16][17][18][19][20] . As each of these scenarios should increase the density of states, the Mott relation dictates that the distinction between these scenarios becomes important in engineering a high zT .…”