Evidence for massive bulk Dirac fermions in Pb_1−xSn_xSe from Nernst and thermopower experiments

Abstract

Topological surface states protected by mirror symmetry are of interest for spintronic applications. Such states were predicted to exist in the rocksalt IV–VI semiconductors, and several groups have observed the surface states in (Pb,Sn)Te, (Pb,Sn)Se and SnTe using photoemission. An underlying assumption in the theory is that the surface states arise from bulk states describable as massive Dirac states, but this assumption is untested. Here we show that the thermoelectric response of the bulk states displays features specific to the Dirac spectrum. By relating the carrier density to the peaks in the quantum oscillations, we show that the first (N=0) Landau level is non-degenerate. This finding provides robust evidence that the bulk states are indeed massive Dirac states. In the lowest Landau level, S_xx displays a striking linear increase versus magnetic field characteristic of massive Dirac fermions. In addition, the Nernst signal displays a sign anomaly in the gap-inverted phase at low temperatures.

Introduction

The rocksalt IV–VI semiconductors have been identified by Fu et al.^1,2 as a novel class of insulators—the topological crystalline insulators—which display surface states that are protected by crystalline symmetry. The topological surface states in topological crystalline insulators are to be contrasted with those in the widely investigated Z2 invariant topological insulators, which are protected by time-reversal invariance^3,4. Angle-resolved photoemission spectroscopy (ARPES) experiments have obtained evidence for the surface states in Pb_1−xSn_xSe (ref. 5), SnTe (ref. 6) and Pb_1−xSn_xTe (ref. 7).

In the alloys Pb_1−xSn_xTe and Pb_1−xSn_xSe, the bulk electrons occupy four small Fermi surface (FS) pockets located at the L points in k space (inset, Fig. 1). The conduction band is predominantly derived from the cation Pb 6p orbitals, whereas the uppermost valence band is predominantly anion 4p (or 5p) orbitals (ordering similar to the atomic limit)⁸. As the Sn content x increases, the system undergoes gap inversion when x exceeds a critical value x_c (refs 9, 10, 11, 12). In samples with x≥x_c, gap inversion occurs when the temperature T is lowered below the gap-inversion temperature T_inv. The ARPES experiments^5,6,7 confirm that the predicted topological surface states appear in the gap-inverted phase.

Figure 1: Thermopower and Nernst effect in a topological crystalline insulator.

The panels provide an overview of how the thermopower S_xx and Nernst signal S_xy vary with field B in a high-mobility crystal (sample 1) of Pb_1−xSn_xSe with x=0.23. (a, b) Curves of S_xx versus B at selected T (sample 1). At each T, the V-profile bracketing B=0 reflects the rapid crossover from small μB to large μB regime. (c) The Nernst signal S_xy/T from 60 to 300 K. The sharp peaks reflect the semiclassical response. An anomalous sign change occurs at T_inv=180 K. (b,d) Fits equations 1 and 2 (thin curves) to S_xx and S_xy at low B. For S_xy (d), we have had to invert the sign. At 30.3 K, the best-fit values of μ, and _H are 51,404 cm² V⁻¹ s⁻¹, 61.5 eV⁻¹ and 104.6 eV⁻¹, respectively. At 4.71 K, the corresponding values are 113,250cm² V⁻¹ s⁻¹, 52.3 eV⁻¹ and 81.3 eV⁻¹. The inset shows the L (111) points on the hexagonal faces of the Brillouin Zone.

The new topological ideas invite a fresh look at the bulk states of the IV–VI semiconductors. To date, the gap inversion appears to have no discernible effect on transport properties (the resistivity, Hall coefficient and thermopower vary smoothly through T_inv). This is surprising given that transport probes the states at the Fermi level. Moreover, a long-standing prediction^8,13 is that the bulk electrons occupy states described by the massive Dirac Hamiltonian. This assumption underlies the starting Hamiltonian of Hsieh et al.² However, no experimental test distinguishing the massive Dirac from the Schrödinger Hamiltonian has appeared to our knowledge.

We have grown crystals of Pb_1−xSn_xSe (x=0.23) in which the n-type carriers have high mobilities (μ=114,000 cm² V⁻¹ s⁻¹ at 4 K). The low electron density (3.46 × 10¹⁷ cm⁻³) enables the quantum limit to be reached at 7.7 T (measurements reveal that holes are absent). In addition to resistivity, we have used both thermopower and the Nernst effect to probe the states in fields up to 34 T. Surprisingly, the Nernst signal is observed to change its sign at T_inv. To date, this appears to be the only transport or thermodynamic quantity that is strongly affected by gap inversion.

In a thermal gradient and an applied magnetic field , the diffusion of carriers produces an electric field E, which is expressed as the thermopower signal S_xx=−E_x/|∇T| and the Nernst signal S_xy=E_y/|∇T|. In the semiclassical regime, the Mott relation¹⁴ simplifies S_xx and S_xy to the form (see Methods section)

where . The dependence on B appears only in the conductivity matrix elements σ_ij(B) (for brevity, we write σ≡σ_xx). The parameters =∂lnσ/∂ζ and _H=∂lnσ_xy/∂ζ are independent of the mobility μ (ζ is the chemical potential). Equation 1 describes the crossover in S_xx from (at B=0) to _H when μB>>1. Correspondingly, S_xy increases linearly from 0 to peak at the value 1/2(_H−) at B=1/μ before falling as 1/B when μB>>1. For n-type carriers, both and σ_xy are negative, and S_xx<0. From equation 2, the Nernst signal S_xy is positive if _H> (we discuss the sign convention in the Methods section). In terms of the exponents β and β_H defined by σ(E)~E^β and σ_xy~$E^{{\beta }_{\text{H}}}$, we have =β/E_F and _H=β_H/E_F.

Even for one-band systems, equations 1 and 2 have not received much experimental attention, possibly because real materials having only a single band of carriers with a low density (and high mobility) are rare. The analysis of S_ij is complicated by the extreme anisotropy of the FS pockets in many semi-metals. For recent Nernst measurements on Bi and Bi₂Se₃, see refs 15, 16. For results on S_ij in graphene, see refs 17, 18, 19. The angular variation of the SdH period in Bi is investigated in refs 20, 21. In the annealed crystals of Pb_1−xSn_xSe (with x~x_c) investigated here, our measurements reveal only n-type carriers (holes are absent). The FS pockets are very small (the quantum limit is reached at 7.7 T) and nearly isotropic (as shown by field-tilt measurements). We find that equations 1 and 2 provide a very good fit to S_xx and S_xy in the semiclassical regime.

Here we provide evidence that the bulk electronic states at the L points are in fact massive Dirac states. A characteristic feature of the massive Dirac spectrum is that in a magnetic field, the lowest Landau level (LL) is non-degenerate with respect to the spin degrees, whereas all higher LLs are doubly degenerate. Knowing the carrier density, we show that the field at which the chemical potential jumps to the lowest LL accurately determines its spin degeneracy to be 1. This confirms the starting assumption of ref. 2. The unusual thermoelectric response is also investigated deep in the quantum limit. In addition, we show that the sign of the Nernst signal is anomalous (relative to standard Boltzmann theory) within the gap-inverted phase.

Results

Semiclassical regime

Figure 1a-1b plots curves of the thermopower S_xx versus B for selected T. From 250 to 160 K, the dominant feature is the rapid increase in weak B followed by saturation to a B-independent plateau at large B. As noted, the Nernst signal (shown as S_xy/T in Fig. 1c) changes from positive to negative as T is decreased below 180 K (identified with T_inv).

As shown in Fig. 1b, the curves of S_xx versus B fit very well to equation 1 in the semiclassical regime (|B|<1T). Likewise, below 100 K the curves of S_xy also fit well to equation 2 up to an overall sign (Fig. 1d). Although the fit parameters (μ, , _H) for S_xx are independent of those for S_xy, we find that they agree with each other (at the level of ±2%) below 60 K (see Methods section). At each T, the two curves, S_xx(B) and S_xy(B), are described by just three parameters. This provides a potent self-consistency check of equations 1 and 2. As a further test, we have also fitted the measured conductivity tensor σ_ij(B) and obtained similar values for μ below 100 K (Methods). By and large, the close fits to both tensors S_ij and σ_ij demonstrate that we have one band of carriers below 100 K.

The semiclassical expressions are no longer valid when quantum oscillations appear at higher B. In particular, the giant step at 7.7 T in the curves at 30 and 40 K (Fig. 1a) is a relic of the quantum regime that remains resolvable up to 100 K. The step has a key role in the discussion later. In the opposite extreme above 100 K, the two sets of fit parameters begin to deviate. The disagreement is especially acute near 180 K, where S_xy changes sign. We reason that the one-band assumption breaks down because of strong thermal activation of holes as the gap closes and re-opens across T_inv. The evidence comes from the T dependence of the Hall density n_H=B/eρ_yx (solid circles in Fig. 2a). Whereas n_H is nearly T-independent below 80 K, in agreement with the one-band model, it deviates upwards above 180 K. Thermal activation of a large population of holes leads to partial cancellation of the Hall E-field and a reduction in |ρ_yx|.

Figure 2: Temperature dependence of Hall density, thermopower and Nernst slope.

(a) The T dependence of the Hall density n_H=B/ρ_yxe inferred from the Hall resistivity ρ_yx and the zero-B thermopower S (T)≡S_xx(T, B=0) in Pb_1−xSn_xSe (x=0.23). The Hall signal is n-type at all T. Below 20 K, n_H equals 3.46 × 10¹⁷ cm⁻³ (sample 1). n_H increases significantly above 200 K signalling thermal activation of holes across the band gap. (b) The T dependence of the initial slope of the Nernst signal dS_xy/dB (B→0) to show the sign change at T_inv. The error bars correspond to 1 s.d. in determining dS_xy/dB in the limit B→0.

In Fig. 2a, we have also plotted the zero-H thermopower S≡S_xx(0) to bring out its nominally T-linear variation below 100 K (bold curve). The large value of the slope S(T)/T=1.41 μV K⁻² implies an unusually small E_F.

As discussed, the Nernst signal changes sign at T_inv=180 K. The T dependence of its initial slope dS_xy/dB (B→0) is displayed in Fig. 2b. From the fits to equations 1 and 2, we may address the interesting question whether the sign anomaly occurs in the gap-inverted phase (T<T_inv) or in the uninverted phase. On both sides of T_inv, the fits of S_xx imply _H> (that is, |S_xx| always increases as μB goes from 0 to values >>1). As S_xy~(_H−), we should observe a positive S_xy. Hence, the sign anomaly occurs in the gap-inverted phase (in Fig. 1d, we multiplied the curves by an overall minus sign). The sign of the Nernst signal below T_inv disagrees with that inferred from equations 1 and 2, despite the close fit. Further discussion of the sign anomaly is given below (see Discussion section). However, we note that the sign of S_xy is independent of the carrier sign. As seen in Fig. 2a, both S and n_H vary smoothly through T_inv without a sign change.

Quantum oscillations

As shown in Fig. 3, oscillations in S_xx and S_xy grow rapidly below 60 K to dominate the weak-B semiclassical profile. The most prominent feature in S_xx is the large step decrease at the field B₁=7.7 T (at which the chemical potential ζ jumps from the N=1 LL to the N=0 LL). In the Nernst curves, plotted as S_xy/T in Fig. 3b, the quantum oscillations are more sharply resolved. As S_xy is the off-diagonal term of the tensor S_ij, its maxima (or minima) are shifted by 1/4 period relative to the extrema of the diagonal S_xx (analogous to the shift of σ_xy relative to σ). This shift is confirmed in Fig. 4a, which plots the traces of S_xx and S_xy versus 1/B. For the analysis below, we ignore the weak spin splitting, which is resolved in the N=1 LL (and barely in N=2).

Figure 3: Quantum oscillations in the thermopower and Nernst signal of PbSnSe.

(a) Curves of S_xx/T versus B at T=4.71 to 100 K of Pb_1−xSn_xSe (x=0.23, sample 1). |S_xx/T| displays a maximum when ζ is at the DOS maximum in each LL. At 4.71 K, the N=1 LL (5–7 T) displays a weak spin splitting. The giant step at 7.7 T occurs when ζ enters the N=0 LL. (b) Curves of S_xy/T for the same T. The sharp resonance-like peaks at low fields are the semiclassical response of large μ electrons. Below 30 K, they are eclipsed by strong quantum oscillations.

Figure 4: Analysis of the quantum oscillations.

(a) Comparison of curves of S_xx and S_xy versus 1/B at 4.71 K in Pb_1−xSn_xSe (x=0.23). The maxima in S_xy are shifted by a period relative to the maxima in S_xx. The N=1 LL shows a weak spin splitting. The sketch (inset) shows the peaks in the DOS of each LL for 3D massive Dirac fermions. (b) Index plot of B_n corresponding to the maxima in |S_xx| (solid circles) and S_xy (triangles) versus the integers n. The straight line is the relation _F=2π (n+γ)/ where _F is the FS section and γ the Onsager phase. The maxima in S_xy are shifted by in n. From the slope, we infer the Fermi wavevector k_F=0.0134 Å⁻¹ and n_e=8.20 × 10¹⁶ cm⁻³ (per valley). The inset shows the LL energy E versus B in the massive Dirac spectrum for k_z=0 (ref. 22).

Figure 4b shows the index plot of 1/B_n (inferred from the maxima in |S_xx| and S_xy) plotted versus the integers n. From the slope of the line, we derive the FS section S_F=5.95 T=5.67 × 10¹⁶ m⁻². Assuming a circular cross-section, we have k_F=0.0134 Å⁻¹. The electron concentration per FS pocket is then =8.2 × 10¹⁶ cm⁻³. As there are four pockets, the total carrier density is 4n_e=3.28 × 10¹⁷ cm⁻³, in good agreement with the Hall density n_H at 4 K (3.46 × 10¹⁷ cm⁻³).

Using sample 2, we have tracked the variation of the SdH period versus the tilt angle θ of B. Figure 5a plots the fields B₁ and B₂ versus θ (B is rotated in the y–z plane). Here, B₁ and B₂ are the fields at which ζ jumps from N=1→0 and from N=2→1, respectively. To our resolution, the SdH period is nearly isotropic. The fields B₁ and B₂ are also independent of tilt angle when B is rotated in the x–y plane. This justifies treating the FS pockets as nominally spherical.

Figure 5: Dependence of SdH period on field-tilt angle and high-field thermopower.

(a) The dependence of the transition fields B₁ and B₂ versus tilt angle θ of B in sample 2 (x=0.23) inferred from magnetoresistance (B₁ and B₂ are the fields at which ζ jumps from LL with N=1→0 and N=2→1, respectively). Within our resolution, no angular dependence of B₁ and B₂ is observed. B is rotated in the y–z plane (sketch in inset). (b) High-field measurements of S_xx/T to 34 T at several T in sample 1. The B-linear dependence smoothly extends through the region 22–28 T where the transition (0, −)→(0, +) should have appeared. The interval 22–28 T is indicated by the horizontal green bar.

The N=0 Landau level

We next address the question whether the bulk states in the inverted phase are Dirac fermions or Schrödinger electrons. The two cases differ by a distinctive feature in their LL spectrum that is robust against small perturbations. In the quantum limit, the massive Dirac Hamiltonian exhibits an interesting twofold difference in degeneracy between the N=0 and N=1 levels. Wolff²² considered a three-dimensional (3D) massive Dirac Hamiltonian with spin–orbit interaction but no Zeeman energy term. More recently, Serajedh et al.²³ included the Zeeman energy term as well as a Rashba term in the massive two-dimensional (2D) Dirac Hamiltonian. Other 3D massive Dirac cases are discussed by Bernevig and Hughes²⁴. All these authors find that N=0 LL is non-degenerate with respect to spin degrees, whereas the LLs with N≠0 are doubly spin degenerate (we discuss in Methods section a pedagogical example, which shows that this anomaly is related to the conservation of states). By contrast, for the Schrödinger case, all LLs are doubly degenerate.

In Pb_1−xSn_xSe, the ability to measure accurately both n_e and the ‘jump’ field B₁ provides a crisp confirmation of this prediction.

The energy of the N^th LL is , where m_D is the Dirac mass and the magnetic length²⁴. At B₁, E_F lies just below the bottom of the N=1 LL so that all the electrons are accomodated in the N=0 LL. Integrating the density of states (DOS) for one spin polarization in the N=0 LL from m_Dv² to E_F, we find (see Methods section)

Ignoring the small spin splitting, we equate B₁ with 7.7 T. Equation 3 then gives n_e↑=9.0 × 10¹⁶ cm⁻³, which agrees within 10% with the measured n_e (the agreement is improved if we correct for spin splitting). All the electrons are accomodated by an N=0 LL that is non-degenerate, in agreement with the prediction for massive Dirac states^22,23,24, but disagreeing with the Schrödinger case by a factor of 2. As the singular spin degeneracy of the N=0 LL cannot be converted to a double degeneracy, the experiment uncovers a topological feature of the bulk states that is robust. As predicted in refs 22, 23, 24, the N=0 LL has only one spin state (0,+); the spin-down partner (0,−) is absent.

To check this further, we extended measurements of S_xx to 34 T to search for the transition from the sublevel (0,−) to (0,+) (which should occur if N=0 LL were doubly degenerate). From extrapolation of the spin split N=1 and N=2 LLs, we estimate that the transition (0,−)→(0,+) should appear in the interval 22–28 T. As shown in Fig. 5b, the measured curves show no evidence for this transition to fields up to 34 T.

Finally, we note an interesting thermopower feature in the quantum limit. At fields above B₁, S_xx displays a B-linear profile that extends to 34 T (Fig. 5). The B-linear behaviour is most evident in the curve at 44 K. As T is decreased to 18.6 K, we resolve a slight downwards deviation from the linear profile in the field interval 10–20 T. The B-linear profile appears to be a characteristic property of massive Dirac fermions in the quantum limit. We discuss below a heuristic, semiclassical approach that reproduces the observed profile.

Discussion

We summarize the electronic parameters inferred from our experiment and relate them to ARPES measurements.

As noted, the FS section derived from the index plot (Fig. 4b) corresponds to a total electron density 4n_e=3.28 × 10¹⁷ cm⁻³, in good agreement with the Hall density n_H at 4 K (3.46 × 10¹⁷ cm⁻³).

We may estimate E_F from the slope of the thermopower S(T)/T=1.41 μV K⁻². Using the Mott expression S(T)=(π²/3)(k_B/e)(k_BT/E_F)β, we find for the Fermi energy E_F=17.0 β meV. For the massive Dirac dispersion, we have , which implies that β has the minimum value 3 (if the mobility increases with E, β is larger). Using the lower bound, β=3, S/T gives E_F=51 meV.

These numbers may be compared with ARPES measurements. We estimate the Fermi velocity from the expression v~E_F/ħk_F (valid when E_F≫m_Dv² with m_D the Dirac mass). Using our values of E_F and k_F, we find v=5.74 × 10⁵ m s⁻¹ as the lower bound. Although ARPES experiment cannot resolve v in the conduction band, the ARPES estimate⁵ for the hole band velocity is 5.6 × 10⁵ m s⁻¹, in good agreement with our lower bound. It is likely to be that the conduction band has a higher velocity (which would then require β>3).

One of our findings is that gap inversion changes the sign of the Nernst signal. As the energies of states involved in gap inversion are very small, the resulting dispersion can be hard to resolve by ARPES measurement. Transport quantities would appear to be more sensitive to these changes. As noted, however, most transport quantities are either unaffected or only mildly perturbed. The Hall effect and thermopower are unchanged in sign across T_inv (Fig. 2a). Although n_H shows a gradual increase, this is largely attributed to thermal activation of holes across a reduced gap for T>T_inv. Hence, the dramatic sign change observed in S_xy stands out prominently; its qualitative nature may provide a vital clue.

It has long been known¹⁰ that in the lead rocksalt IV–VI semiconductors, the energy gap E_g undergoes inversion as the Sn content x increases from 0. Moreover, within a narrow range of x, gap inversion is also driven by cooling a sample (the critical temperature is x dependent within this interval). Strauss¹⁰ performed early optical transmission measurements of E_g in a series of single-crystal films of Pb_1−xSn_xSe, with x ranging from 0 to 0.35. For x=0.25, he reported that E_g closes at 195 K. A slight interpolation of his data shows that at our doping x=0.23, E_g should vanish at 179 K, remarkably close to our T_inv=180 K. The recent ARPES measurements of Dziawa et al.⁵ is consistent with E_g closing between 100 and 200 K. Given the ARPES resolution, these results are all consistent with our inference that our T_inv corresponds to the gap inversion temperature. Hence, we reason that the Nernst signal changes sign either at, or very close to, the gap inversion temperature. The inverted sign of S_xy below 180 K in Fig. 1c,d occurs in the gap-inverted phase. We refrain from making the larger claim that this is also the topological transition because we are unable to resolve the surface states in our experiments.

The fits of S_xy to equation 2 (Fig. 1d) shows that the curves below 100 K are well described by the Boltzmann–Mott expression assuming a single band of carriers, but there is an overall sign disagreement. Despite the sign problem, the analysis singles out the physical factors that fix the sign and delineates the scope of the problem. For example, reversing the sign of both β and β_H inverts the sign of S_xy, but also that of S_xx. Alternately, one might try reversing the signs of β and β_H, and e simultaneously. This will invert the sign of S_xy but leave S_xx unchanged. However, ρ_yx is forced to change sign.

The analysis assumes that in the gap-inverted phase, the FS is simply connected. This may not be valid. Gap inverion may lead to the existence of a small pocket surrounded by a larger FS sheet (topologically similar to the FS of the ‘giant Rashba’ material BiTeI²⁵). As the small pocket dominates the thermoelectric response, the Nernst effect may be detecting this novel situation. These issues will be left for future experiments.

We may attempt to understand the striking B-linear profile of S_xx/T in Fig. 5 using a semiclassical approach. In N=0 LL, the long-lived quasiparticles complete a large number of cyclotron orbits between scattering events (for example, from μB~220, we estimate this number is ~35 at 20 T). The scattering results in the drift of the orbit centres X in a direction transverse to the applied −▿T. Ignoring the fast cyclotron motion, we may apply the Boltzmann equation to X. The thermopower is then given by the high-B limit of equation 1, S_xx(T,H)→β_H′/E_F, where E_F is now measured from the bottom of N=0 LL and β_H′ differs from the weak-field β_H. In this picture, the B dependence of S_xx arises solely from how E_F changes with B.

For B>B₁, only N=0 LL is occupied. From equation 14 (Methods), we have the relation between E_F, n_e and B, viz.

In the limit E_F≫m_Dv², we obtain the relation E_F~1/B. This immediately implies that S_xx/T increases linearly with B as observed. Setting g_s=1, we derive from equation 4 the rate of increase

Repeating this calculation for the Schrödinger case, we get instead S_xx/T~B².

From Fig. 5, the thermopower slope ∂(S_xx/T)/∂(B)=8.71 × 10⁻⁸ V K⁻² T⁻¹. Using the above values of v and n_e in equation 5, we find (∂S_xx/T)/∂(B)=6.1 β′_H × 10⁻⁸ V K⁻² T⁻¹. The value of β′_H is not known. Comparison of the calculated slope with experiment suggests β′_H~1.5. Hence, this back-of-the-envelop estimate can account for the rate at which S_xx/T increases with B.

Methods

Semiclassical fits to S_xx and S_xy

In the presence of a magnetic field B, an electric field E and a temperature gradient −∇T (in an infinite medium), the total current density is given by¹⁴ J=σ·E+α·(−∇T). Here, σ_ij is the conductivity tensor and α_ij is the thermoelectric tensor. Setting J=0 (for a finite sample) and solving for E, we have E=−ρ·α·(−∇T), with ρ=σ⁻¹ the resistivity tensor.

In the geometry with and , the components of the E-field (for an isotropic system) are

The thermoelectric tensor S_ij is given by E_i=S_ij∂_jT (S_xx>0 for hole carriers and S_xy>0 if E_y>0 when H_z>0).

The Mott relation¹⁴,

where k_B is Boltzmann’s constant, e is the elemental charge and ζ the chemical potential, has been shown to hold under general conditions, for example, in the quantum Hall Effect (QHE)^26,27. Using equation 8, equations 6 and 7 reduce to equations 1 and 2, respectively.

The fits of S_ij to these equations displayed in Fig. 1d were carried out using the one-band, Boltzmann–Drude expressions for the conductivity tensor, viz.

where the total carrier density N_e is 4n_e (n_e is the density in each of the FS pocket at the L points).

In the geometry with and , we define the sign of the Nernst signal to be that of the y component of the E-field E_y. More generally, if E_N is the E-field produced by the Nernst effect, the sign of the Nersnt signal is that of the triple product E_N·B × (−∇T). This agrees with the old convention based on ‘Amperean current’²⁸ and with the one adopted for vortex flow in superconductors²⁹.

At each T, we have fitted the measured curves of S_xx and S_xy versus B to equations 1 and 2 using equations 9 and 10 for the conductivity tensor. The separate fits of S_xx and S_xy yield two sets of the parameters μ, and _H, which are displayed in Fig. 6 (solid triangles and open circles, respectively). The three-parameter fit places strong constraints on the curves of S_xx and S_xy. Disagreement between the two sets signals that the one-band model is inadequate.

Figure 6: Variation of fit parameters versus temperature.

The T dependence of the parameters μ (a), (b) and _H (c) obtained from best fits to the curves of S_xx (solid triangles) and S_xy (open circles). The fits are most reliable below 100 K where the values obtained from fitting S_xx and S_xy are in agreement. Above 100 K, disagreement between the two sets is significant, especially close to T_inv=180 K. Above 200 K, the one-band model is no longer valid.

Below 100 K, the two sets agree well, whereas closer to T_inv they begin to deviate. The reason is that equation 2 cannot account for the change of sign in the Nernst signal given the relative magnitudes of and _H fixed by the curves of S_xx. Above 200 K, the two sets are inconsistent because thermal excitations of holes across the small band gap is important at elevated T and the one-band assumption becomes inadequate. This is evident in the onset above 200 K of significant T dependence in the Hall density n_H (see Fig. 2a).

We remark that S_xx=V_x/δT is directly obtained from the observed voltage difference V_x and the temperature difference δT between longitudinal electrical contacts (their spatial separation L_x is immaterial). However, for the Nernst signal, we have S_xy=(V_y/δT)(L_x/L_y), where L_y is the spatial separation between the transverse contacts. Hence, the aspect ratio L_y/L_x is needed to convert the observed Nernst voltage V_y to S_xy. The ratio L_y/L_x is measured to be 4±0.4. The fits are improved significantly if this value is refined to 4.20, which we adopt for the curves at all T.

Fits to equation 9 and 10 of the conductivity tensor measured in the same sample are shown in Fig. 7 for weak B at selected T from 5 to 150 K. The fits yield values of the mobility μ similar to those shown in Fig. 6a. The inferred carrier density N_e is also similar to the measured Hall density n_H.

Figure 7: Fits of the conductivity tensor.

The measured weak-B conductivity σ_xx (a) and Hall conductivity σ_xy (b) of Pb_1−xSn_xSe (x=0.23) versus B at selected T from 5 to 150 K, together with the fits to equation 9 and 10 (thin solid curves).

Indexing the quantum oscillations

For the 3D systems, one identifies the index field B_n as the field at which the DOS displays a sharp maximum (diverging as in the absence of disorder). From the quantization rule for areas in k-space, B_n is related to the FS cross-section _F as

where and γ (the Onsager phase) is for Schödinger electrons. The plot in Fig. 3b follows equation 11. From its slope, we obtain _F. The intercept γ is close to zero in Fig. 3b. We will discuss γ elsewhere.

We note that, in the 2D systems in the QHE regime, the index field is the field at which the chemical potential ζ falls between adjacent LLs, where the DOS vanishes, and the Hall conductance displays a plateau. The difference between 2D and 3D systems arises because the integer n counts the number of edge states in the QHE case, whereas n indexes the DOS peaks in the 3D case. One needs to keep this in mind in interpreting γ.

We have verified that the slope in Fig. 4b is insensitive to the tilt angle θ of B relative to the crystalline axes. As shown in Fig. 5a, the SdH period is virtually independent of θ within the experimental uncertainties, consistent with negligible anisotropy in the small FS pockets. The good agreement between _F and n_H (Hall density) at 5 K is also evidence for a negligible anisotropy.

Spin degeneracy in N=0 LL

Knowledge of the field B₁ (the transition from N=1 to the N=0 LL) and the electron density per valley n_e suffices to determine the spin degeneracy of the N=0 LL.

For the 3D Dirac case²⁴, the energy in the Nth LL is

with m_D the Dirac mass, k_z the component of k along B and the magnetic length.

For N=0 LL, we solve for k_z(E)

where E₀₀=m_Dv².

Let us assume that only N=0 LL is occupied. To obtain the relation linking E_F, B and n_e, we integrate the 3D DOS (E)dE=(g_Lg_s/π)dk_z, with g_s the spin degeneracy and the 2D LL degeneracy per spin. Using equation 13, we have

This equation is valid until B is reduced to the jump field B₁, whereafter electrons enter the N=1 LL. At the jump field, E_F lies just below the bottom of N=1 LL, that is, . Using this in equation 14, we have

In relation to equation 3, we showed that equation 15 gives a value equal (within 10%) to the total electron density per valley if g_s=1, that is, when B>B₁, all the electrons can be accomodated by the N=0 LL with only one spin polarization. This is direct evidence for the non-degeneracy of the N=0 LL.

Interestingly, equation 15 is identical for the isotropic Schrödinger case, for which

where ω_c=eB/m and m is the mass. However, for N=0 LL of the Schrödinger spectrum, we must have g_s=2, so it can be excluded.

A simple example of massive Dirac spectrum

An example illustrating the non-degeneracy of N=0 LL is the spinless fermion on the 2D hexagonal lattice (valley degeneracy replaces spin degeneracy in this example). The sublattices A and B have distinct on-site energies ε_A and ε_B as in BN. The Dirac cones remain centred at the inequivalent ‘valleys’ K and K′ in k-space (inset, Fig. 8). Both valleys acquire a mass gap.

Figure 8: The massive Dirac spectrum in the 2D hexagonal lattice.

We assume distinct on-site energies on sublattices A and B (as in boron nitride). The Dirac cones sit at the high-symmetry points K and K′ on the edge of the Brillouin Zone (upper inset). In a magnetic field B, the N=0 LL at K moves to a value above E=0 (curve 1), whereas the N=0 LL at K′ moves below E=0 (curve 2). The sum of the two spectra (K+K′) is symmetric about E=0 (curves 3). However, the N=0 LLs are non-degenerate, whereas all other LLs (N≠0) have a valley degeneracy of 2. As B increases, all LLs fan out except for the N=0 levels (curves 3).

For states close to the valley at K, the 2D massive Dirac Hamiltonian is

in the basis (1,0)^T (pseudospin up) and (0,1)^T (pseudospin down), where k is measured from K and m>0 represents the gap parameter proportional to ε_A–ε_B (we set the velocity v to 1). In a field B, we replace k by π=k–e A with the vector gauge A=(0,Bx,0). Introducing the operators

with π_±=π_x±iπ_y, and eigenstates |N› satisfying

we diagonalize the Hamiltonian to get eigenenergies E_N given by

(for brevity, we will write E for E_N).

For positive E, the (unrenormalized) two-spinor eigenstates are (for N=0,1,⋯)

For the negative energy states, the corresponding eigenvectors are (N=1,2,⋯)

Setting N=0 in equation 21, we find that the positive-energy state |Ψ_0,+›=(|0›,0)^T (pseudospin up). For E<0, however, the lower entry in equation 22 is non-determinate (0/0). This implies that the state N=0 does not exist for E<0. Thus, for the valley at K, there is only one LL with N=0. It has positive energy E₀=|m|; the corresponding LL at −|m| is absent (the spectrum of K is sketched as curve 1 in Fig. 8).

Repeating the calculation for K′, we find the opposite situation (the Hamiltonian is the conjugate of equation 17). Now N=0 LL has energy E₀=−|m|, but N=0 LL is absent in the positive spectrum (the spectrum of K′ is the curve 2 in Fig. 8).

A transport experiment detects the sum of the two spectra (curves of K+K′ at different B are collectively labelled as 3 in Fig. 8). In the total spectrum, the two N=0 LLs are non-degenerate, whereas all LLs with N≠0 have a valley degeneracy of 2. The difference simply reflects the conservation of states. In the limit m→0, we recover the spectrum of graphene. If, at finite m, each of the N=0 LLs had a valley degeneracy of 2, we would end up with an N=0 LL in graphene with fourfold valley degeneracy.

The authors in refs 22, 23, 24 and others have shown that the non-degeneracy of the N=0 LL also holds in massive Dirac systems even when a Rashba term and a Zeeman energy term are included.