The Anomalous Photo‐Nernst Effect of Massive Dirac Fermions In HfTe5

The quantum geometric Berry curvature results in an anomalous correction to the band velocity of crystal electrons with a corresponding transverse (thermo)electric conductivity. However, time‐reversal symmetry typically constrains the direct observation and exploitation of anomalous transport to magnetic compounds. Here, it is demonstrated the anomalous Hall and Nernst conductivities are essential for describing the optoelectronic transport in thin films of the non‐magnetic, weakly gapped semimetal HfTe5 subject to an external magnetic field. A focused photoexcitation adresses the symmetries of the local Nernst conductivity, which unveils a hitherto hidden, anomalous photo‐Nernst effect of three‐dimensional (3D) massive Dirac fermions. The experimental temperature and density dependencies are compared with a semiclassical Boltzmann transport model. For HfTe5 thin films with the Fermi level close to the gap, the model suggests that the anomalous photo‐Nernst currents originate from an intrinsic Berry curvature mechanism, where the Zeeman interaction effectively breaks time‐reversal symmetry of the massive Dirac fermions already at moderate external magnetic fields.

remaining degeneracies and enables a net Hall or Nernst voltage in the linear response regime.The impact of Berry curvature is maximized in the vicinity of (avoided) band crossings, since the magnitude of (k) scales with inverse power laws of the gap and momentum measured relative to the node. [10]In Dirac or Weyl semimetals, (k) has even divergent behavior forming the analog of a fictitious magnetic monopole in k-space.While in metals the regular Nernst conductivity is typically small due to Sondheimer cancellation, theory predicts a strongly enhanced anomalous Nernst conductivity arising from the Berry curvature for 3D Dirac semimetals. [11]e transition-metal pentatellurides form a particularly interesting platform for transverse transport phenomena.[14][15][16][17] Being van der Waals materials, they can be exfoliated into two-dimensional (2D) systems, [18] with predictions of a quantum spin Hall phase in monolayers. [12]Yet, the exact bulk electronic structure remains under debate due to fact that small changes of the lattice parameters, e.g., due to strain, strongly modify the fundamental gap or induce secondary carrier pockets. [19,20][23] High quality crystals achieve low charge carrier densities ≈1 × 10 16 cm −3 with high low-temperature mobilities ≈1 × 10 6 cm 2 V −1 s −1 [24,25] making it possible to reach the quantum limit at fields as low as 0.3 T. [25] Experimentally, indications of anomalous electric and thermoelectric responses [26][27][28][29][30][31][32] have been reported, which are, however, contrasted by multiple studies reporting quasi-quantized, 3D Hall transport under nominally similar conditions. [24,33,34]ere, we combine local magneto-photocurrent measurements and conventional magnetotransport to elucidate the anomalous Hall and Nernst effects of HfTe 5 thin films.We demonstrate that the photocurrents allow us to selectively address the anomalous Nernst effect, while we track the evolution of the chemical potential and Fermi surface as function of temperature via magnetotransport.For describing the Hall conductivity, we introduce a phenomenological two-pocket model with an additional anomalous contribution, whose qualitative behavior is reproduced by Boltzmann transport calculations, when taking into account the Berry curvature of the effective Dirac Hamiltonian.Interestingly, we find that nominally identical, bulk reference crystals show exclusively a quasi-quantized Hall effect.We speculate that for transferred thin films, external factors, such as strain evidenced in our samples by subtle Raman shifts, play an important role in determining the electronic structure and low-temperature transport properties of HfTe 5 .

Results
Figure 1d sketches the photocurrent measurement.A focused optical excitation induces a local temperature gradient ∇T driving a transversal photo-Nernst current j N , which in turn is detected as a global photocurrent via source-drain contacts.Figures 2a and 2b spatial maps of the photocurrent I photo with an applied outof-plane magnetic field, where each pixel represents the current generated by the local laser excitation at the corresponding position.For B = 9 T (Figure 2a), we find a strong photoresponse of approximately I photo = 1 μA localized to the sample boundary (dashed lines).The photocurrent pattern extends between source and drain along the sample edges, with opposite signs at opposite edges.As the polarity of the magnetic field is reversed, the polarity of the edge photocurrent reverses as well (Figure 2b).We rule out the edge photocurrent to originate from a 1D edge state, which could hypothetically appear from the 2D surface state in the strong topological insulator phase of HfTe 5 . [35]In such a scenario, the photocurrent amplitude would be expected to oscillate as the external magnetic field drives transitions through the 2D Landau levels. [36]Rather, photocurrent line scans (along the dotted line in Figure 2b) reveal a step-like, quickly saturating behaviour up to the highest magnetic fields of our experiments (Figure 2c).
Since HfTe 5 is a weakly gapped semimetal, we expect the carrier recombination after optical excitation to occur on ultrafast (fs to ps) timescales, while on longer timescales a heated carrier bath prevails. [37]Therefore, the dominant photoresponses are thermal Nernst and Seebeck effects. [38]In literature, the photo-Nernst effect was discussed in graphene at relatively low fields, [39] where it arises from the conventional, linear Hall conductivity.In the present case, however, the non-linear anomalous term dominates, such that we coin the observed photoresponse the anomalous photo-Nernst effect.The global Nernst current is nonzero only for an edge excitation, because, for a center excitation, the local photothermal currents circulate around the isotropic temperature profile. [40]Along the crystal edge, the excitation results in a net, perpendicular temperature gradient, which in turn drives currents parallel to the edge.The Nernst effect is consistent with the detected symmetries of the photocurrent, i.e., a sign reversal of the current for excitation at the upper versus lower boundary and for positive versus negative magnetic fields.[46] Note that the above symmetries also exclude further Nernst currents arising from a potential out-of-plane temperature gradient at the laser position.The latter photocurrents have been widely studied in ferromagnetic thin films with a typical in-plane magnetization. [5]e further fabricated devices with on-chip heaters that allow us to directly match the conventional Nernst voltage with the photo-Nernst currents on the same sample (Figure S2, Supporting Information).Figure 2d shows the edge photocurrent as function of magnetic field for temperatures from 1.6 to 240 K. Up to 100 K, we observe a sharp, step-like onset of the photocurrent.At even higher temperatures, the high-field photocurrent becomes linear, while the step-like feature at low magnetic fields diminishes and finally reverses polarity.Based on these observations, we define the anomalous contribution to the photocurrent I A photo as the amplitude of the step-like photocurrent after removing the linear high-field background.
The anomalous photocurrent (Figure 3a) shows a clear sign change around 100 K, which we attribute to a transition from n-type at low temperatures to p-type behavior at high temperatures. [47]Consistent with previous literature reports, the Hall conductivity (Figure 3b) exhibits an overall sign change around 80 K and the longitudinal resistivity exhibits a resistance anomaly at a temperature close-by (T anomaly = 45 K). [22,24,48,49] Note that in a multi-band transport, as we introduce below, the sign changes of Hall and Nernst coefficient do not necessarily need to occur at the exact same temperature. [50]Further, due to the similar magnitudes of Hall and longitudinal resistivity, as well as due to the rather large variation of the longitudinal magnetoresistance, we found that the transformation from sheet resistivity to conductivity is crucial to correctly model and interpret the data (for raw resistivity data, see Figure S3, Supporting Information).For simplicity, we assume  yy =  xx in the transformation.We focus first on the p-type regime at 190 K (Figure 3c).Here, the Hall conductivity is well described by a simple Drude term with density n = 1.3 × 10 18 cm −2 and mobility μ = 1.4 × 10 4 cm 2 V −1 s −1 .In the n-type regime, i.e., for temperatures below 80 K, fit models with only one electron pocket fail to prop-erly describe the shape of the conductivity.The best agreement is found by assuming two electron pockets, which we assign to the weakly gapped Dirac cone and a secondary electron pocket, and an additional step-like contribution, which we interpret as an anomalous Hall effect (for comparison of different fitting models, see Figure S4, Supporting Information).For 1.6 K, our fit model yields a first pocket with n 1 = − 2.4 × 10 16 cm −2 and μ 1 = 5.7 × 10 4 cm 2 V −1 s −1 and a second pocket with n 2 = − 2.2 × 10 17 cm −2 and μ 2 = 4 × 10 3 cm 2 V −1 s −1 .Our assumption of a secondary electron pocket is in line with photoemission studies on ZrTe 5 films [22] and HfTe 5 films, [51] where the emergence of a secondary pocket away from Γ, but close in energy to the Dirac gap, was directly observed.Density functional theory predicts that the exact energetic position of this secondary pocket depends very sensitively on the choice of lattice parameters. [19]The observed temperature dependence of carrier type and density is hence likely caused by a combination of the thermal activation of unintentional dopants as well as the temperature dependence of the lattice constant with the concomitant change in the underlying electronic structure.Figure 3d depicts exemplarily the magnetotransport data and the fit model for the lowest temperature of 1.6 K.The extracted anomalous Hall effect is only weakly temperature dependent (inset of Figure 3c).The anomalous Hall and the Drude conductivity enter the model with opposite signs.Therefore, the fit at 1.6 K, which was applied only from -4 T to 4 T (Figure 3d), predicts a change of sign of the overall Hall voltage, once the magnitude of the ordinary Hall conductivity becomes smaller than that of the anomalous Hall conductivity.Indeed, we observe such a sign change at B = 5.5 T in measurements across the full magnetic field range of 9 T (Figure 3e).We note that a qualitatively similar sign change of both Hall, [52] and Nernst coefficient [26] has been observed for bulk ZrTe 5 samples that exhibit a quasiquantized Hall effect, albeit at much larger magnetic fields of B ≈ 15 − 25 T. The sign change was attributed to a band closing due to crossing of the Landau levels in the magnetic fields beyond the quantum limit.Within our semi-classical picture, the sign change observed here arises from the competition between anomalous and ordinary Hall conductivity in the gapped phase of HfTe 5 .Above 60 K, the Hall conductivities can also be fitted satisfactorily by simple two-pocket models and, at even higher temperatures, even one-pocket models.To exclude erroneous interpretations due to overfitting, we hence restrict the modeling of the anomalous Hall effect to the low-temperature regime, where the chosen model clearly outperforms simpler models.
We find that the qualitative behavior of the conductivities observed in experiment, including the identification of a roughly step-like anomalous contribution, are in agreement with a semi-classical transport calculation.Our calculation is based on a low-energy Hamiltonian describing HfTe 5 in the regime where only a single Dirac pocket close to the Γ-point is occupied. [53]his model reflects the fact that HfTe 5 is close to the transition between weak and strong topological insulator phases. [12,19,35]ia the Zeeman effect, a magnetic field breaks time reversal symmetry and induces a finite Berry curvature (Figure 4a). [54]sing model parameters consistent with previous studies, [21,24,52] we follow Refs.[10, 55-57] in setting up a Boltzmann theory for charge transport (for details, see Supporting Information).Importantly, as the field varies, we self-consistently adjust the chemical potential in order to keep the particle number fixed.

The results of this calculation qualitatively reproduce the central features of the experimental data. The Fermi-surface contribution to the Hall response 𝜎 FS
xy shows qualitatively the expected Drude-like behavior (Figure 4b).The anomalous Hall response  A xy , which purely originates from Berry curvature effects by definition, saturates at relatively small fields of ca. 3 − 4 T, and persists for all temperatures relevant to the experiments (Figure 4c).The saturation-field coincides with the field strength at which the bottom of one of the conduction bands crosses the chemical potential, which naturally explains the fielddependence of  A xy .For both contributions to the total Hall effect, our realistically chosen model parameters yield a response that is of the same order of magnitude as observed in experiments.The amplitudes of both contributions depend sensitively on the electronic density, and the Fermi surface contribution is furthermore proportional to the scattering time (see Figure S5, Supporting Information).At elevated densities (Figure 4d), which experimentally corresponds to elevated temperatures as well, the anomalous Hall conductivity acquires a close to linear behavior, because the saturation occurs only at very large fields.In turn, the differentiation between the anomalous and Fermi surface contribution becomes challenging within the experimentally available field strengths.While qualitatively reproducing the experimental data, our theoretical results still exhibits certain quantitative deviations from it.The amplitudes of the experimentally observed anomalous Hall effect is for example about three times larger than what is found in theory.Interestingly, a similar behavior has recently been observed in a study based on ab initio theory. [28]Furthermore, based on the experimentally motivated fit model, we determine a positive sign of the anomalous Hall conductivity and a negative sign for the Drude Hall conductivity for B > 0 T. Within the simplified theoretical model, the relative sign of anomalous and Fermi surface responses is controlled by the sign of the Dirac mass parameter, i.e., by whether the sample is in a strong (different sign) or weak (same sign) topological insulator phase.In principle, this suggests that the thin films are in a strong topological insulator phase at low temperatures.

Discussion
Lastly, we would like to discuss and comment on some of the seemingly contradicting reports regarding the magnetotransport properties of HfTe 5 , and similarly ZrTe 5 .On the one hand, several studies reported a quasi-quantized 3D Hall effect, both for HfTe 5 [24] and ZrTe 5 , [21,34,52] with clear Shubnikov-de-Haas oscillations, a single Fermi surface, low carrier densities, and a linear low-field Hall resistance.On the other hand, other studies found non-linear low-field Hall conductivities both for HfTe 5 [28]   and ZrTe 5 [27,28,58] which were interpreted either by an anomalous contribution [28] or multiband models. [58,59]Interestingly, we could reproduce both behaviors in our transport study.Our bulk reference samples showed quasi-quantized Hall plateaus (Figure S6, Supporting Information).Consistently, these reference samples did not show any anomalous photocurrents, but rather clear quantum oscillations (Figure S7, Supporting Information).Importantly, the anomalous transport discussed above was observed in thin films samples fabricated from the very same reference bulk crystals.We hypothesize that these differences between bulk and thin may be partially related to strain for the following reasons.For the case of ZrTe 5 , experiment [20,27,60,61] and theory [19,35] suggest that the electronic structure depends sensitively on the lattice parameters.Furthermore, angle resolved photoemission spectroscopy of HfTe 5 revealed direct evidence for strain tuning between a (gapped) Dirac phase, under tensile strain, and a strong topological insulator phase, under com-pressive strain. [62]Lastly, on the commonly used Si/SiO 2 substrates, thin pentatelluride films may experience an effective tensile strain upon cooling, due to their larger thermal expansion coefficient.In this scenario, other sources of strain, such as from the transfer or exfoliation process, need to be considered as well.To experimentally support this strain hypothesis, we carried out Raman measurements on our thin films and reference crystals, where we found small, but systematic differences (below 1 cm −1 ) between the Raman modes of the bulk reference and the exfoliated films (Figure S8, Supporting Information).Finally, we note that although several studies already reported on the impact of strain on transport and band structure in ZrTe 5 and HfTe 5 , a systematic understanding of thin films versus bulk is still lacking for this class of materials, which will be important for a prospective integration into more complex, thin-film based device structures.

Conclusion
Overall, we demonstrate that the magneto-optoelectronic response of HfTe 5 thin films is governed by the intrinsic anomalous Nernst conductivities, which arise from the time-reversal symmetry breaking due to the external magnetic field and which become dominant already at moderate magnetic fields below 1 T. Due to the expected Sondheimer cancellation of the regular Nernst effect, thermoelectric responses may be more sensitive to unconventional transverse transport in ultra-low density Dirac semimetals, such as HfTe 5 , when compared to purely electric responses.The contrasting behavior observed in bulk crystals and exfoliated thin films suggests that external material parameters, such as strain, play a crucial role in determining the electronic structure and Berry-curvature induced anomalous Hall and Nernst responses.

Experimental Section
Crystal Growth: The HfTe 5 single crystals were grown by chemical vapor transport according to ref. [63].
Sample Preparation: Crystals of HfTe 5 with thickness varying from approximately 100 nm to 2 μm were exfoliated onto Si/SiO 2 substrates (oxide thickness 290 nm) using the scotch tape technique from bulk single crystals of HfTe 5 .The process of exfoliation and optical inspection was carried out inside a controlled atmosphere, i.e., a glove box with oxygen and water concentrations less than 0.5 ppm.For Hall bar geometries, we chose uniform, rectangular shaped crystals based on their optical contrast and size.The contact geometries were patterned using a maskless lithography tool and a positive tone resist.To remove the natural surface oxide, which forms during exposure of the HfTe 5 crystals to ambient conditions, Ar-sputtering was performed prior to the deposition of 25 nm titanium (Ti) and 350 nm gold (Au).The Ar-sputtering and the subsequent deposition were done in the same vacuum chamber without air exposure between the process steps.In this way, low resistance Ohmic contacts were reliably formed.The thickness of the investigated crystals was later determined by atomic force microscopy.
Optoelectronic Transport Measurements: All electrical transport measurements up to ±9 T were performed in a closed-cycle cryostat with a sample insert suitable for optoelectronic measurements.For the magnetotransport measurements, the longitudinal  xx and Hall resistivity  xy were measured with standard lock-in techniques by applying a source drain bias with fixed amplitude V SD = 5 mV at frequency f = 333 Hz.The electrical bias was always applied along the a-axis of the crystals and the magnetic field was directed out-of-plane along the b-axis.All measurements were done at a base temperature of 1.6 K unless specified otherwise.For photocurrent measurements a 800 nm cw-laser (P laser = 500 μW) was used.The laser was focused onto the sample by an aspheric lens (NA = 0.55) resulting in a spatial resolution of approximately 800 nm.The photocurrent generated by the local excitation I photo was measured using a lock-in amplifier synchronized to an optical chopper (f = 333 Hz).A transimpedance preamplifier was connected to the drain contact to provide the virtual ground.The source contact was maintained at zero bias via a voltage source.All other contacts were floating or connected to voltage amplifiers with a high input impedance of 1 TΩ.

Figure 1 .
Figure 1.a) Nernst effect in metals.Transverse electron (blue) and hole (red) currents are driven by the combined action of a temperature gradient ∇T and a time-reversal symmetry breaking magnetic field B resulting in a transverse Nernst voltage ΔV. b) In ferromagnetic metals, the magnetization M results in an anomalous Nernst effect.c) In Weyl-or Dirac-semimetals with broken time-reversal symmetry, a non-zero Berry curvature  (k) gives rise to an anomalous Nernst effect.d) Optoelectronic scheme to locally address transverse Nernst currents j N .A local temperature gradient arises from a focused laser.The global current is collected by metal electrodes (yellow) in a short-circuit configuration.

Figure 2 .
Figure 2. Photocurrent maps at a) B = 9 T, and b) B = − 9 T. The edges of the HfTe 5 crystal are marked with dashed lines.The electric contacts are shaded in light yellow.The photocurrent I photo is measured between the unbiased source and drain contacts.All other contacts are floating.Scale bar, 5 μm.c) Scan across the dotted line in (b) as a function of B. d) Temperature dependence of edge photocurrent.At low temperatures, the photocurrent shows a step-like magnetic field dependence.

Figure 3 .
Figure 3. a) Magnitude of the anomalous photocurrent, I A photo as a function of temperature.b) Hall conductivity  xy as a function of temperature.Inset shows the temperature dependence of longitudinal resistivity  xx at B = 0 T. The resistivity anomaly peaks at around T = 45 K. Hall conductivity at c) 190 K and d) 1.6 K. Grey solid line is the experimental data, navy blue dashed line is the fitting result based on our model that includes one pocket (two pockets) and an anomalous term.e) Hall conductivity at 1.6 K for magnetic fields from 0 T to 9 T. The inset (c) shows the temperature dependence of the anomalous Hall effect.

Figure 4 .
Figure 4. a) Band structure and calculated Berry curvature of the gapped Dirac model with g = 37, m = 10 meV, at different applied magnetic field strengths.At B = 0, the bands are doubly degenerate and the Berry curvature cancels.For B ≠ 0, the bands are Zeeman split and a net Berry curvature emerges.The chemical potential (dashed line) at zero-field is 20 meV, corresponding to a carrier density 5.2 × 10 17 cm −3 .As the field varies, the chemical potential is adjusted such that the carried density is kept fixed.Hall conductivity from the b) Fermi surface and the c) anomalous Hall contribution at different temperatures.d) Calculated anomalous Hall conductivity  A xy for varying electron densities.