Quantized thermoelectric Hall effect induces giant power factor in a topological semimetal

Thermoelectrics are promising by directly generating electricity from waste heat. However, (sub-)room-temperature thermoelectrics have been a long-standing challenge due to vanishing electronic entropy at low temperatures. Topological materials offer a new avenue for energy harvesting applications. Recent theories predicted that topological semimetals at the quantum limit can lead to a large, non-saturating thermopower and a quantized thermoelectric Hall conductivity approaching a universal value. Here, we experimentally demonstrate the non-saturating thermopower and quantized thermoelectric Hall effect in the topological Weyl semimetal (WSM) tantalum phosphide (TaP). An ultrahigh longitudinal thermopower \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{xx} \sim 1.1 \times 10^3 \, \mu \, {\mathrm{V}} \, {\mathrm{K}}^{ - 1}$$\end{document}Sxx~1.1×103μVK−1 and giant power factor \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sim 525 \, \mu \, {\mathrm{W}} \, {\mathrm{cm}}^{ - 1} \, {\mathrm{K}}^{ - 2}$$\end{document}~525μWcm−1K−2 are observed at ~40 K, which is largely attributed to the quantized thermoelectric Hall effect. Our work highlights the unique quantized thermoelectric Hall effect realized in a WSM toward low-temperature energy harvesting applications.

gradually increased the temperature difference from zero until the I2 transport agent started to flow. This process seems to be furnace-and distance-specific. In our case, the optimal temperatures for the two zones are 900 o C and 950 o C, respectively, and the distance between the two heating zones is constantly optimized. With the help of the transport agent I2, the TaP source materials transferred from the cold end of the tube to the hot end and condensed at the hot end in a single-crystalline form in 14 days. The resulting products of TaP single crystals are centimeter-sized and have a metallic luster.
A typical crystal is shown in Figure S1.

Sample Preparation for Measurements
To conduct high-precision electrical and thermal transport measurements on TaP, we performed a thinning-down process on the crystals. Due to the very high electrical and thermal conductivities of TaP, it is difficult to do high-precision electrical and thermal transport measurements on the as-grown crystals. To magnify the electrical resistance and the temperature gradient in the electrical and thermal transport measurements, one piece of crystal was polished down to a thin slab along the c-axis. Figures S2a and b display top and side views of the thinned-down crystal we used for the thermal transport Page 4 of 35 measurement (namely thermoelectric measurement) whose thickness is only 0.17 mm. Figure

Carrier Concentration and Mobility
The electrical and thermal transport measurements were carried out with the electrical transport option (ETO) and the thermal transport option (TTO) of physical property measurement system (PPMS), respectively. The data about the quantum oscillations were measured with the ETO whereas the data about the thermoelectric (including resistivity) with the TTO. When we performed the ETO measurements we adopted a standard six-probe configuration and connected the longitudinal and transverse probes to two independent measurement channels. The details about the ETO measurement can be found in Figure S3a. To experimentally validate the prediction of a quantized thermoelectric Hall effect requires information about the carrier concentration and mobility. To extract this information, we carried out a delicate electrical transport measurement with the ETO of the PPMS. The measurement was done using a standard six-probe geometry, schematically shown in Figure S3a. With the symmetric probe configuration, the measured longitudinal resistivity ρxx is symmetric with respect to the applied magnetic field, while the transverse resistivity ρyx is antisymmetric, as shown in Figure S3b and c.
In both ρxx and ρyx, strong Shubnikov-de Haas (SdH) oscillations can be observed at low temperatures. The oscillation is preserved up to 25K, indicating high-quality crystallization in our sample, as the temperature damping effect would otherwise eliminate the quantum oscillation at this relatively high temperature.
Because the contacts on the sample were made manually with silver epoxy, the measured data exhibit slight asymmetry due to slight misalignment of the contacts. To eliminate the effect of the contact misalignment, we averaged the ρxx and ρyx using the equations listed below: Then we calculated the longitudinal and transverse conductivities σxx and σxy using the following equations: The field dependence of σxx and σxy at various temperatures is shown in Figures S3d

Analysis of Quantum Oscillation
Since the carrier pockets analysis based on quantum oscillation can be influenced by the choice of background of magnetoresistance (MR), we adopted three independent methods using 1) background-free curvature approach ( Figure S4), 2) a T=25K data without quantum oscillation as background ( Figure S5), and 3) a fitted background to a linear-quadratic function ( Figure S6), all of which lead to a consistent conclusion of the existence of a low frequency carrier pocket F a  2.3T~4T . This enables the possibility that the carrier pockets of W2 Weyl point can indeed reach the desired n=0 LL. Figure S4. MR analysis using curvature-based background subtraction. In this approach, a second-order derivative against magnetic field B is taken to the MR data, where all linear, constant, and quadratic terms will be automatically wiped out without need to manually choosing background. Although this method is seldom used, this may offer an alternative but strong approach for MR analysis. a MR data up to B=14T, at T=2K, 4K and 6K. ¶

Landau Level and Quantum Limit
The quantized thermoelectric Hall effect considered in this work is theoretically predicted to exist in the quantum limit of Dirac/Weyl semimetals 1, 2 . Therefore, to examine the validity of the theoretical prediction, we first verify that the quantum limit condition is satisfied by Weyl fermions in our TaP sample. To do this, we performed a thorough analysis of the quantum oscillations observed in the electrical transport measurement, as shown in Figure 1 in the main text and discussed in detail in the previous section. The quantum oscillation data ΔMR shown in Figure 1d of the main text was obtained by subtracting a smooth background from the magnetoresistance (MR) data, Figure 1c, where MR is defined according to: From the fast Fourier transform (FFT) analysis depicted in Figure 1e, we observe four noticeable oscillation frequencies: After performing a standard signal filtering process by performing inverse FFT to the two relatively low frequencies of 4T and 18T individually, we isolate the two oscillation components from the pristine data and determine the corresponding Landau levels (LLs) by assigning an integer (half-integer) value to the oscillation maxima (minima), as shown in Figure 1f. From the LL index fan, we conclude that in our TaP sample, the α shown in Figure S7. The thermal conductivity is directly calculated by the PPMS using the applied heater power, the resulting temperature difference ΔT detected between the two thermometers, and the sample dimension. The voltage drop DV between the two thermometers is monitored simultaneously, which yields the Seebeck signals by calculation of  DV DT . A magnetic field was applied along the c axis for detecting the proposed quantized thermoelectric Hall effect. Figure S7b shows where L and W represent the length-wise and the width-wise separation between the two thermometers. Figure S7c displays the obtained longitudinal thermal conductivity κxx as a function of temperature at different magnetic fields. From the inset of Figure S7c, we see that the applied magnetic field gradually suppresses the longitudinal thermal conductivity. This phenomenon is consistent with the giant magnetoresistance, as both originate from the greatly elevated electron scattering induced by the magnetic field.
The magnitude of the thermal conductivity of TaP is very large compared to most materials, which explains the importance of thinning the sample prior to measurement.
The Seebeck signals at 0T, 9T and -9T are plotted in Figure S7d, from which giant magnetic field-induced Seebeck signals can be observed at 9T and -9T.
The  After performing the thermal transport measurement at a certain temperature, a subsequent electrical transport measurement at the same temperature is made with the TTO. The inset of Figure S8a shows the schematic diagram for the electrical transport measurement in the diagonal offset geometry. In the presence of a magnetic field, the system applies an electrical current along the a or b axis, and the voltmeter between the diagonal offset probes detects the voltage drop which contains both longitudinal and transverse components. The longitudinal resistivity ρxx and the transverse resistivity (also called Hall resistivity) ρyx are separated using the following equations: From the plot of zT in Figure S8d, we note that, although the power factor (shown in Figure 2f in the main text) is record-breaking in magnitude, the zT does not attain a very high value due to the significant thermal conductivity.
It should be noted that the giant magnetic field-induced Seebeck coefficients cannot be observed in the case of B∥a∥jQ, which is evidenced by comparison of two geometries in

Thermoelectric Hall Conductivity up to 9T
To validate the quantized thermoelectric Hall effect, particularly the quantized plateau of the thermoelectric Hall coefficient αxy in the high magnetic field limit, we calculated αxy using the following equation: (S10) To obtain αxy as a function of magnetic field for different temperatures, we replotted Sxx . (S11) where the notation . (S12) and v F is treated as v F eff . The function s(x) is the entropy per carrier, given by is the Fermi-Dirac distribution. The data to be fitted using Eq.
(S11) is shown in Figure S14c, and we extrapolate the fitted function to even larger magnetic fields, revealing we are near the onset of the quantized limit. The value of a xy / T approached in this limit is ~0.4AK -2 m -1 . The corresponding fitted parameters are given in Figures S14d and e.
To verify this fit, we additionally fit our low-temperature data up to T=50K using the expression for a xy / T which also includes a finite scattering time and is thus a more expressive form for data with weak scattering present 2 : . (S14) where the cyclotron frequency w c is given by and once more, v F is treated as v F eff . This fit is shown in Figure S14f  Similarly, we fit our high-temperature data, T>50K, in the limit of weak scattering using . (S16) which is shown in Figure S14i with corresponding fitted parameters plotted in Figures S14j and k.

X-Ray and Neutron Scattering Measurement Details
Inelastic neutron scattering measurements were performed on the HB1 triple-axis spectrometer at the High-Flux Isotope Reactor at the Oak Ridge National Laboratory.
We The basic principles of such instrumentations are discussed elsewhere 7,8 .
Measurements of the phonon modes along high-symmetry lines in the Brillouin zone of TaP were performed using both inelastic x-ray scattering and inelastic neutron scattering. Selected raw intensity spectra along high symmetry direction Γ to Σ are shown in Figure S19

Separation of Phonon and Electron Contributions to Thermal Conductivity
To check the compliance or violation of the Wiedemann-Franz law, the phononic and electronic contributions to thermal conductivity need to be separated.
To separate the phononic and electronic contributions, we fit κxx versus B curves with the following empirical equation: (S17) where βe(T) is proportional to the zero-field electronic mean free path of electrons, and m is related to the nature of the electron scattering 9,10 .

Computational Details
All the ab initio calculations are performed by Vienna Ab Initio Package (VASP) 11,12 with projector-augmented-wave (PAW) pseudopotentials and Perdew-Burke-Ernzerhof (PBE) for exchange-correlation energy functional 13 . The geometry optimization of the conventional cell was performed with a 6 × 6 × 2 Monkhorst-Pack grid of k-point sampling. The second-order and third-order force constants was calculated using a real space supercell approach with a 3 × 3 × 1supercell, same as Ref 14 . The Phonopy package 15 was used to obtain the second-order force constants. The thirdorder.py and ShengBTE packages 16 were used to obtain the third-order force constants and relaxing time approximation was used to calculate the thermal conductivity. A cutoff radius of about 0.42 nm was used, which corresponds to including the fifth nearest neighbor when determining the third-order force constants. To get the equilibrium distribution function and scattering rates using the third-order force constants, the first Brillouin zone was sampled with 30×30×10 mesh.