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 nondegenerate. 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 gapinverted 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 timereversal invariance^{3,4}. Angleresolved photoemission spectroscopy (ARPES) experiments have obtained evidence for the surface states in Pb_{1−x}Sn_{x}Se (ref. 5), SnTe (ref. 6) and Pb_{1−x}Sn_{x}Te (ref. 7).
In the alloys Pb_{1−x}Sn_{x}Te and Pb_{1−x}Sn_{x}Se, 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 gapinversion temperature T_{inv}. The ARPES experiments^{5,6,7} confirm that the predicted topological surface states appear in the gapinverted phase.
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 longstanding 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}_{−x}Sn_{x}Se (x=0.23) in which the ntype carriers have high mobilities (μ=114,000 cm^{2} V^{−1} s^{−1} at 4 K). The low electron density (3.46 × 10^{17} 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 relation^{14} 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 ntype 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 oneband 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 semimetals. For recent Nernst measurements on Bi and Bi_{2}Se_{3}, 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−x}Sn_{x}Se (with x~x_{c}) investigated here, our measurements reveal only ntype 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 fieldtilt 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 nondegenerate 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 gapinverted phase.
Results
Semiclassical regime
Figure 1a1b 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 Bindependent 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 selfconsistency 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 oneband assumption breaks down because of strong thermal activation of holes as the gap closes and reopens 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 Tindependent below 80 K, in agreement with the oneband model, it deviates upwards above 180 K. Thermal activation of a large population of holes leads to partial cancellation of the Hall Efield and a reduction in ρ_{yx}.
In Fig. 2a, we have also plotted the zeroH thermopower S≡S_{xx}(0) to bring out its nominally Tlinear 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 gapinverted 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 gapinverted 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 weakB semiclassical profile. The most prominent feature in S_{xx} is the large step decrease at the field B_{1}=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 offdiagonal 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 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 × 10^{16} m^{−2}. Assuming a circular crosssection, we have k_{F}=0.0134 Å^{−1}. The electron concentration per FS pocket is then =8.2 × 10^{16} cm^{−3}. As there are four pockets, the total carrier density is 4n_{e}=3.28 × 10^{17} cm^{−3}, in good agreement with the Hall density n_{H} at 4 K (3.46 × 10^{17} 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 B_{1} and B_{2} versus θ (B is rotated in the y–z plane). Here, B_{1} and B_{2} 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 B_{1} and B_{2} are also independent of tilt angle when B is rotated in the x–y plane. This justifies treating the FS pockets as nominally spherical.
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. Wolff^{22} considered a threedimensional (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 twodimensional (2D) Dirac Hamiltonian. Other 3D massive Dirac cases are discussed by Bernevig and Hughes^{24}. All these authors find that N=0 LL is nondegenerate 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−x}Sn_{x}Se, the ability to measure accurately both n_{e} and the ‘jump’ field B_{1} 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 length^{24}. At B_{1}, 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_{D}v^{2} to E_{F}, we find (see Methods section)
Ignoring the small spin splitting, we equate B_{1} with 7.7 T. Equation 3 then gives n_{e↑}=9.0 × 10^{16} 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 nondegenerate, 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 spindown 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 B_{1}, S_{xx} displays a Blinear profile that extends to 34 T (Fig. 5). The Blinear 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 Blinear 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 × 10^{17} cm^{−3}, in good agreement with the Hall density n_{H} at 4 K (3.46 × 10^{17} 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)=(π^{2}/3)(k_{B}/e)(k_{B}T/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_{D}v^{2} with m_{D} the Dirac mass). Using our values of E_{F} and k_{F}, we find v=5.74 × 10^{5} m s^{−1} as the lower bound. Although ARPES experiment cannot resolve v in the conduction band, the ARPES estimate^{5} for the hole band velocity is 5.6 × 10^{5} 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 known^{10} 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). Strauss^{10} performed early optical transmission measurements of E_{g} in a series of singlecrystal films of Pb_{1−x}Sn_{x}Se, 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 gapinverted 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 gapinverted 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 BiTeI^{25}). 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 Blinear profile of S_{xx}/T in Fig. 5 using a semiclassical approach. In N=0 LL, the longlived 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 highB 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 weakfield β_{H}. In this picture, the B dependence of S_{xx} arises solely from how E_{F} changes with B.
For B>B_{1}, 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_{D}v^{2}, 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~B^{2}.
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 backoftheenvelop 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 by^{14} 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 Efield (for an isotropic system) are
The thermoelectric tensor S_{ij} is given by E_{i}=S_{ij}∂_{j}T (S_{xx}>0 for hole carriers and S_{xy}>0 if E_{y}>0 when H_{z}>0).
The Mott relation^{14},
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 oneband, 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 Efield E_{y}. More generally, if E_{N} is the Efield 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 superconductors^{29}.
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 threeparameter fit places strong constraints on the curves of S_{xx} and S_{xy}. Disagreement between the two sets signals that the oneband model is inadequate.
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 oneband 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}.
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 kspace, B_{n} is related to the FS crosssection _{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 B_{1} (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 case^{24}, 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 E_{00}=m_{D}v^{2}.
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_{L}g_{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 B_{1}, 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>B_{1}, all the electrons can be accomodated by the N=0 LL with only one spin polarization. This is direct evidence for the nondegeneracy 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 nondegeneracy 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 onsite energies ε_{A} and ε_{B} as in BN. The Dirac cones remain centred at the inequivalent ‘valleys’ K and K′ in kspace (inset, Fig. 8). Both valleys acquire a mass gap.
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–eA 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) twospinor 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 positiveenergy state Ψ_{0,+}›=(0›,0)^{T} (pseudospin up). For E<0, however, the lower entry in equation 22 is nondeterminate (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 E_{0}=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 E_{0}=−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 nondegenerate, 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 nondegeneracy of the N=0 LL also holds in massive Dirac systems even when a Rashba term and a Zeeman energy term are included.
Additional information
How to cite this article: Liang, T. et al. Evidence for massive bulk Dirac fermions in Pb_{1−x}Sn_{x}Se from Nernst and thermopower experiments. Nat. Commun. 4:2696 doi: 10.1038/ncomms3696 (2013).
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Acknowledgements
We acknowledge helpful discussions with B.A. Bernevig, F.D.M. Haldane and M.Z. Hasan. N.P.O., T.L. and S.P.K. acknowledge support by the Army Research Office (ARO W911NF1110379). R.J.C., Q.G. and J. X. were supported by a MURI grant on Topological Insulators (ARO W911NF1210461) and the US National Science Foundation (grant number DMR 0819860). T.L acknowledges scholarship support from the Japan Student Services Organization. Highfield measurements were performed at the National High Magnetic Field Laboratory, which is supported by NSF (Award DMR084173), by the State of Florida, and by the Department of Energy.
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Affiliations
Department of Physics, Princeton University, Jadwin Hall, PO Box 708, Princeton, New Jersey 08544, USA
 Tian Liang
 , Jun Xiong
 , Max Hirschberger
 , Sunanda P. Koduvayur
 & N.P. Ong
Department of Chemistry, Princeton University, Frick Chemistry, Princeton, New Jersey 08544, USA
 Quinn Gibson
 & R.J. Cava
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Contributions
T.L., Q.G., R.J.C. and N.P.O. planned and carried out the experiment. T.L. and N.P.O. analysed the data and wrote the manuscript. J.X., M.H. and S.P.K. assisted with the measurements and analyses. Q.G. and R.J.C. grew the crystals. All authors contributed to editing the manuscript.
Competing interests
The authors declare no competing financial interests.
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Correspondence to N.P. Ong.
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