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

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 $9.4\times {10}^{19}{\mathrm{cm}}^{-3}$ (0.62 Na at. $\%$) 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 $\mathrm{\Sigma }$-line valence band maxima, since our data are consistent with a single-band Fermi surface over the entire doping range studied, whereas the calculations predict an occupation of the $\mathrm{\Sigma }$ pockets at higher doping. Our results for low and intermediate compositions are consistent with a nonparabolic 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.

Figure 1

Energy dispersion for stoichiometric PbTe along the high symmetry directions of the fcc Brillouin zone, calculated in this work using density functional theory (DFT) (for details see text). A direct gap, underestimated compared with experiment as is usual in DFT calculations, is observed at the $L$ point, and a second valence band maximum occurs along the $\mathrm{\Sigma }$ high-symmetry line. A representative Fermi surface, which emerges as the Fermi energy is shifted into the valence band by Pb vacancies or hole-dopant impurities, is shown in the inset. For the choice of Fermi level shown (green-dashed line), the Fermi surface contains eight half-ellipsoids (shaded in red) centered at the $L$ point and oriented along the [111] directions ($L$ pockets), and $12 \mathrm{\Sigma }$ pockets (shaded in blue) centered close to the midpoint of the [110] $\mathrm{\Sigma }$ line and oriented along the [100] directions.

Figure 2

Upper panel: Fermi surface of hole-doped PbTe calculated in this work using the rigid band approximation. Lower panel plots: The corresponding (110)-plane angle evolution of the cross-sectional areas (in frequency units) of the calculated Fermi surface pockets. The four columns correspond to monovalent impurity concentrations of (a) $x=0.02\%$ ($p_L=0.27\times {10}^{19}{\mathrm{cm}}^{-3}$ and $p_{\mathrm{\Sigma }}=0$); (b) $x=0.81\%$ ($p_L=3.5\times {10}^{19}{\mathrm{cm}}^{-3}$ and $p_{\mathrm{\Sigma }}=8.6\times {10}^{19}{\mathrm{cm}}^{-3}$); (c) $x=1.56\%$ ($p_L=6.1\times {10}^{19}{\mathrm{cm}}^{-3}$ and $p_{\mathrm{\Sigma }}=17.4\times {10}^{19}{\mathrm{cm}}^{-3}$); and (d) $x=2.61\% \left(p_{\text{total}}=36.2\times {10}^{19}{\mathrm{cm}}^{-3}\right)$. The frequencies of the $L$ pockets are shown in red, and compared with those expected in a perfect ellipsoidal model shown as black lines. The evolution of the $\mathrm{\Sigma }$ pockets is shown in blue. These pockets appear at a dopant concentration of $x=0.11\% \left(p_L\approx {10}^{19}{\mathrm{cm}}^{-3}\right)$. In column (d), the $\mathrm{\Sigma }$ and $L$ pockets have merged, forming a cube-shape Fermi surface; cross sections that cannot be identified separately with $\mathrm{\Sigma }$ or $L$ are shown in purple. We plot frequencies up to 600 T, noting, however, that frequencies up to 8 kT occur, corresponding to the large-square Fermi surface orbits. As described in the main text, our quantum oscillation studies reveal that for carrier densities up to at least $9.4\times {10}^{-19}{\mathrm{cm}}^{-3}$ value, the maximum studied in this report, the Fermi surface is found to only comprise $L$ pockets (shown in red), implying a larger band offset between the $L$ and $\mathrm{\Sigma }$ pockets than predicted by these and other DFT calculations.

Figure 3

Sodium contribution to the calculated band structure around the Fermi energy for the 128-atom supercell. (a) Sodium projected density of states (pDOS). (b) Difference in the total DOS with and without the impurity $\mathrm{\Delta }\mathrm{DOS}={\mathrm{DOS}}_{\mathrm{with}\mathrm{Na}}-{\mathrm{DOS}}_{\mathrm{undoped}}$. Note the drop in DOS just below the top of the valence band (set to 0 eV), consistent with a lifting in degeneracy of the highest valence bands (see also Fig. 4).

Figure 4

Calculated band structure with and without sodium impurity for the 128-atom supercell $\left(x\approx 1.6\%\right)$. (a) The zero of energy was set at the top of the valence band for both cases. Note the lifting of the degeneracy of the top valence bands (marked by arrows); apart from this, the bands coincide almost perfectly. (b) The carrier density for both cases was fixed to a concentration corresponding to $x=1.6\%$. The Fermi energy is moved more into the valence band than expected from the rigid band approximation because of the lifting of degeneracy.

Figure 5

Evolution of three calculated cross-sectional areas (in frequency units) with density of holes in the $L$ pockets $\left(p_L\right)$. The dashed curve in all the plots shows the functional dependence of $p_L^{2/3}$ expected for a perfect ellipsoidal model. The dotted vertical line indicates the $L$-pocket hole density above which the $\mathrm{\Sigma }$ pockets start to be populated. (a) Frequency associated with the $L$-pockets minimum cross-sectional area $f_{\text{min}}$. (b) Frequency associated with the $L$-pockets' cross-sectional area in the [100] direction $f_{\left[100\right]}$. (c) Frequencies associated with the $L$-pockets' maximum cross-sectional area. The green circles correspond to the orbits in the longitudinal direction of the $L$ pocket $\left(f_{\parallel }\right)$—for perfect ellipsoidal $L$ pockets they would correspond to the largest possible frequencies; the blue triangles correspond to the orbits associated with the largest cross-sectional area $f_{\max }$, which for large concentrations do not correspond anymore to longitudinal orbits on the $L$ pockets. The inset shows two representative orbits ($f_{\parallel }$ in green and $f_{\max }$ in blue) on the distorted $L$ pocket (shown in red) for a concentration $x=1.56\% \left(p_L=6\times {10}^{19}{\mathrm{cm}}^{-3}\right)$.

Figure 6

Evolution of calculated cyclotron effective masses [Eq. (A5)] as a function of density of holes in the $L$ pocket $\left(p_L\right)$ at three high symmetry directions: $\parallel$ or in the longitudinal direction of the $L$ pocket, in the [100] direction, and $\perp$ or in the transverse direction of the $L$ pocket (corresponding to a magnetic field oriented along the [111] direction). The variation with $p_L$ provides striking evidence for the nonparabolicity of the bands.

Figure 7

Magnetoresistance measurements for ${\mathrm{Pb}}_{1-x}{\mathrm{Na}}_x\mathrm{Te}$ samples of different Na concentrations [row (a) $x=0$, row (b) $x=0.13\%$, row (c) $x=0.26\%$, row (d) $x=0.4\%$, and row (e) $x=0.62\%]$ as a function of magnetic field rotated along the (110) plane. The first column shows the measured resistivity as a function of applied magnetic field. The second column shows the background-free resistivity, obtained as explained in the main text, as a function of inverse field. The third column shows the amplitude of the normalized FFT, represented by the color scale, as a function of the angle of the magnetic field from the [100] direction (horizontal axis), and the frequency (vertical axis). The last column replots column three, with a comparison to a perfect ellipsoidal model calculation superimposed (solid lines for fundamental frequencies, and dashed lines for higher harmonics). The parameters used for the perfect ellipsoidal model calculation for each set of data are summarized in Table 1. For samples with $x=0.13\%,0.4\%$, and $0.62\%$, small deviations from the (110) plane of rotation are evidenced in the splitting of the angle evolution of the intermediate branch, and they were considered in the perfect ellipsoidal model comparison. For the two highest concentrations, combination frequency terms due to magnetic interaction effects are observed. These are identified in the fourth column plots by the light-blue-dotted lines (sum of fundamental branches) and gray-dotted lines (difference of fundamental branches).

Figure 8

(a) Longitudinal magnetoresistance for a Na-doped PbTe sample with $x=0.4\%$ and Hall number $p_H=6.3\times {10}^{19}{\mathrm{cm}}^{-3}$, for different directions of the applied magnetic field, with respect to the [100] crystalline axis, as the field is rotated in the (100) plane. (b) As in (a), as a function of inverse magnetic field, after eliminating the background, therefore only preserving the oscillatory part. (c) The color scale in both plots represents the amplitude of the Fourier transform of the data shown in (b), as a function of the angle from the [100] direction (horizontal axis), and the frequency (vertical axis). For these plots, the field is rotated in the (100) plane. The right-hand side figure replots the figure in the left, but with a perfect ellipsoidal model calculation superimposed on the data, up to the third harmonic (black lines). For the model, the plane of rotation is offset by $5.5^{\circ }$ (about the [100] axis). The parameters used for the calculations are the same as those used for the (110) plane of rotation data in Fig. 7: $f_{\text{min}}=81.4$ T and $f_{\text{max}}=307$ T.

Figure 9

FFT of the background-free resistivity data of Fig. 7, as a function of the angle from the [100] direction and the frequency. A perfect ellipsoidal model calculation has been superimposed on the data, up to the third harmonic (black lines). In order to better guide the comparison with the perfect ellipsoidal model, the exact frequencies of the local maxima of the FFT for each angle (labeling only FFT peaks with amplitude $1\%$ or more of the largest peak for each angle) are indicated by black dots. The parameters used in the perfect ellipsoidal model for each plot are: (a) $f_{\text{min}}=97$ T, $f_{\text{max}}=370$ T; and (b) $f_{\text{min}}=97$ T, $f_{\text{max}}=460$ T. For both plots, an offset of $4^{\circ }$ from the (110) plane of rotation (about the [110] axis) is considered, to account for the splitting seen in the middle branch. Additionally, the combination frequency terms are shown in light-blue-dotted lines (sum of fundamental branches) and gray-dotted lines (difference of fundamental branches). None of the fits presented here give a satisfactory description of the data, suggesting deviations from perfect ellipsoidicity.

Figure 10

Temperature dependence of the amplitude of the oscillating component of magnetoresistance for ${\mathrm{Pb}}_{1-x}{\mathrm{Na}}_x\mathrm{Te}$ samples, with magnetic field along the [111] direction (${55}^{\circ }$ from the [100] direction, in the (110) plane). The left-column plots show the background-free data at different temperatures. The right-column plots show the fits of the data to the LK formula in Eq. (1), using the four most dominant frequencies observed in the FFT of the lowest temperature curve. From this fit, the values of cyclotron effective mass and Dingle temperature, for each frequency term, are obtained. The values obtained for the transverse cyclotron mass and Dingle temperature are summarized in Table 2.

Figure 11

Temperature dependence of the amplitude of the oscillating component of magnetoresistance for a ${\mathrm{Pb}}_{1-x}{\mathrm{Na}}_x\mathrm{Te}$ sample with $x=0.24\%$, and magnetic field oriented close to ${35}^{\circ }$ from the [100] direction, along the (110) plane. For this orientation, the cross-sectional area of two of eight $L$ pockets corresponds to the maximum cross-sectional area of the ellipsoids. The left-column plot shows the background-free data at different temperatures. The right-column plot shows the fit of the data to the LK formula in Eq. (1), using the five most dominant frequencies observed in the FFT of the lowest temperature curve. From this fit, the values of cyclotron effective mass and Dingle temperature, for each frequency term, are obtained.

Figure 12

Carrier concentration calculated from Luttinger's theorem and the volume of the $L$ pockets extracted from the comparison between the data and a perfect ellipsoidal model, as a function of the Hall number, for Na-doped PbTe (black squares), and obtained using the ellipsoid parameters from previous studies in Refs. [14, 15] (blue stars). The dashed line shows the expected behavior for a single-parabolic band, for which the carrier density enclosed by the Fermi surface, as determined through Luttinger's theorem, matches the carrier density measured using the Hall effect. All the measured samples lie on this line, and the deviations seen for the highest Na doping (gray square) are attributed to deviations from perfect ellipsoidicity.

Figure 13

Evolution of the characteristic frequencies of the $L$ pockets with Hall number, for Na doping, as determined from this study, and for self-doped samples from the works in Refs. [14, 15]: (a) Frequency associated with the $L$ pockets' minimum cross-sectional area $f_{\text{min}}$. (b) Frequency associated with the $L$ pockets' cross-sectional area in the [100] direction $f_{\left[100\right]}$. (c) Frequency associated with the $L$ pockets' maximum cross-sectional area $f_{\text{max}}$. The blue-star symbols are data points obtained by previous quantum oscillation studies from other authors [14, 15], in self-doped PbTe with different levels of Pb vacancies (the last star in $f_{\text{min}}$, in green, was obtained by Na doping). The $f_{\text{max}}$ data point for the highest Na concentration is represented by a gray square, in order to emphasize that deviations from perfect ellipsoidicity seen in this sample result in a less accurate determination of $f_{\text{max}}$. The dashed line in all the plots is the functional dependence of $p_H^{2/3}$ expected for a perfect ellipsoidal model with fixed anisotropy.

Figure 14

Anisotropy parameter of the $L$ pockets, $K={(f_{\text{max}}/f_{\text{min}})}^2$, extracted from the data, as a function of the Hall number for Na-doped samples, as determined from this study (black squares and gray square for the highest Na composition), and for self-doped samples from the works in Refs. [14, 15] (blue stars). The horizontal gray line shows the average value of $K=14.3\pm 0.4$ for this range of concentrations.

Figure 15

Cyclotron effective mass $m^{\text{cyc}}$ along different directions with respect to the ($L$-pocket) ellipsoid semimajor axis, for a ${\mathrm{Pb}}_{1-x}{\mathrm{Na}}_x\mathrm{Te}$ sample with $x=0.24\%$. The data points were obtained through fits to the LK formula of the oscillating components of magnetoresistance along three different crystallographic directions: [111] [square symbols, Fig. 10], [100] [filled circles, Fig. 20], and ${35}^{\circ }$ from [100] in the (110) plane (diamonds, Fig. 11). The dashed lines represent the angle dependence of the cyclotron mass (fundamental and higher harmonics) for a perfect parabolic dispersion and perfect ellipsoidal model, as presented in Eq. (D7), and using an anisotropy parameter $K=14.3\pm 0.4$ (which implies $m_{\parallel }^{\text{cyc}}/m_{\perp }^{\text{cyc}}=3.78\pm 0.05$). The shadowed region around the dashed lines represents the error bar in $m^{\text{cyc}}\left(\theta \right)$ estimated from propagation of errors in $K,\theta$, and $m_{\perp }^{\text{cyc}}$.

Figure 16

Effective cyclotron mass $m^{\text{cyc}}$ along three high symmetry directions for ${\mathrm{Pb}}_{1-x}{\mathrm{Na}}_x\mathrm{Te}$ samples, as a function of the Hall number. Cyclotron effective masses were determined through fitting the curves in Figs. 10, 11, and 20 to the LK formula in Eq. (1). $m_{\perp }^{\text{cyc}}$ is the cyclotron mass in the transverse direction of the $L$-pocket ellipsoid, or [111] direction (red-solid squares); $m_{\left[100\right]}^{\text{cyc}}$ is the cyclotron mass in the [100] direction of the crystal lattice (gray-solid circles); and $m_{\parallel }^{\text{cyc}}$ is the cyclotron mass in the longitudinal direction of the $L$-pocket ellipsoid (blue-solid diamond). The red-dashed line represents a guide to the eye for the trend observed in the longitudinal cyclotron mass. The blue-dashed line is the trend expected for the longitudinal cyclotron mass given the anisotropy parameter of $m_{\parallel }^{\text{cyc}}/m_{\perp }^{\text{cyc}}=\sqrt{K}=3.78$. The small open triangles show the $L$-pocket effective cyclotron masses determined through our DFT calculations, as shown in Fig. 6. The agreement between the calculated masses and the experimental values for the transverse and [111] direction is outstanding.

Figure 17

Dingle temperature in the transverse direction ${\mathrm{\Theta }}_{D,\perp }$ obtained through fitting of the curves in Fig. 10 to the LK formula in Eq. (1), as a function of carrier concentration $p_H$. We find that the Dingle temperature is independent of carrier concentration, with a value of ${\mathrm{\Theta }}_{D,\perp }=(9.7\pm 0.4)$ K, indicated by the dashed-gray line. This value of ${\mathrm{\Theta }}_{D,\perp }$ results in a value of $\tau =(0.125\pm 0.005)$ ps for the carrier relaxation time along the transverse direction.

Figure 18

Lattice parameter of the cubic unit cell of ${\mathrm{Pb}}_{1-x}{\mathrm{Na}}_x\mathrm{Te}$ samples with different Na content, as a function of temperature. Filled symbols represent the experimentally determined values, and the solid lines represent a fit to a low order polynomial, as explained in the text, from which the zero-temperature lattice parameter can be extrapolated.

Figure 19

Evolution of SdH frequencies (a) and cyclotron masses (b) as a function of density of holes in the $L$-pocket $p_L$ for two different volumes: 6.50 Å (open symbols and superscript $l$) and 6.44 Å (filled symbols and no superscript). The larger volume corresponds also to a larger band offset 120 vs 74 meV. The dashed vertical lines indicate the $L$-pocket hole density above which the $\mathrm{\Sigma }$ pockets start to be populated.

Figure 20

Temperature dependence of the amplitude of the oscillating component of magnetoresistance for ${\mathrm{Pb}}_{1-x}{\mathrm{Na}}_x\mathrm{Te}$ samples, with magnetic field oriented in or close to the [100] direction. The left-column plots of each composition show the background-free data at different temperatures. The right-column plots show the fits of the data to the LK formula in Eq. (1), using the two most dominant frequencies observed in the FFT of the lowest temperature curve (three most dominant for the $x=0.62\%$ sample). From these fits, the values of cyclotron effective mass and Dingle temperature, for each frequency term, are obtained.