Anomalous photo-thermoelectric transport due to anisotropic energy

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Anomalous photo-thermoelectric transport due to anisotropic energy dispersion in WTe2 Qisheng Wang, Can Yesilyurt, Fucai Liu, Zhuo Bin Siu, Kaiming Cai, Dushyant Kumar, Zheng Liu, Mansoor Bin Abdul Jalil, and Hyunsoo Yang Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.9b00513 • Publication Date (Web): 12 Mar 2019 Downloaded from http://pubs.acs.org on March 12, 2019

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Anomalous photo-thermoelectric transport due to anisotropic energy dispersion in WTe2 Qisheng Wang1, §, Can Yesilyurt1, §, Fucai Liu2, §, Zhuo Bin Siu1, Kaiming Cai1, Dushyant Kumar1, Zheng Liu2, Mansoor B. A. Jalil1, Hyunsoo Yang1,* 1Department

of Electrical and Computer Engineering, National University of Singapore, 117576 Singapore

2Center

for Programmable Materials, School of Electrical and Electronic Engineering, Nanyang Technology University, Singapore 639798

Correspondence and requests for materials should be addressed to H.Y. (email: [email protected], Tel: (+65)-65167217).

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ACS Paragon Plus Environment

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ABSTRACT: Band structures are vital in determining the electronic properties of materials. Recently, the two-dimensional (2D) semimetallic transition metal tellurides (WTe2 and MoTe2) have sparked broad research interest because of their elliptical or open Fermi surface, making it distinct from the conventional 2D materials. In this study, we demonstrate a centrosymmetric photo-thermoelectric voltage distribution in WTe2 nanoflakes, which has not been observed in common 2D materials such as graphene and MoS2. Our theoretical model shows the anomalous photo-thermoelectric effect arises from an anisotropic energy dispersion and micrometre-scale hot carriers diffusion length of WTe2. Further, our results are more consistent with the anisotropic tilt direction of energy dispersion being aligned to the b axis rather than a axis of the WTe2 crystal, which is consistent with the previous firstprinciple calculations as well as magneto-transport experiments. Our work shows the photothermoelectric current is strongly confined to the anisotropic direction of the energy dispersion in WTe2, which opens an avenue for interesting electro-optic applications such as electron beam collimation and electron lenses.

KEYWORDS: 2D materials, photo-thermoelectric effect, anisotropic energy dispersion, WTe2 crystal, electro-optic applications

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The band structure plays a central role to the electronic properties of materials. The emerging 2D materials provide a versatile family of band structures which contribute to the intriguing condensed matter phenomena and electronic applications1-5 such as linear energy dispersion in graphene6, spin-valley coupling7,

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and edge topological states in transition

metal dichalcogenides (TMDCs)9. However, the energy dispersion is normally isotropic in 2D materials such as graphene and MoS2 where the energy valleys are located at high symmetry K points. Lowering the crystal symmetry could induce anisotropic energy spectra which enable new avenues in electronic and optoelectronic properties. For example, the asymmetric band structure of black phosphorus leads to the anisotropic excitons3 and Hall mobility10, and polarization-sensitive optical conductivity11. Further, the recent work demonstrated an integrated digital inverter utilizing the in-plane anisotropic electronic features of ReS212. Recently, the semimetallic transition metal tellurides (WTe2 and MoTe2) with twofold lattice symmetry have sparked broad research interest13-16. The unusual electronic structure13, 17, 18 of WTe2 and MoTe2 generates novel physics such as the superconductivity19, Weyl phase13, and extremely large magnetoresistance16. Interestingly, the WTe2 and MoTe2 presents the anisotropic energy dispersion20,

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which leads to an elliptical Fermi surface

configuration22, making it distinct from the conventional 2D materials6,

23, 24.

The next

question is naturally that how the anisotropic energy dispersion affects the electronic and optoelectronic properties. Herein, we investigate the effect of anisotropic energy dispersion on the photothermoelectric transport via the photovoltage (PV) mapping in WTe2 nanoflake devices. WTe2 possesses an orthorhombic Bravais lattice (Fig. S1) with a layered structure which facilitates the mechanical exfoliation of WTe2 nanoflakes. The crystal structure is layered along the z-direction (c axis), where the band structure shows two-fold symmetry on the a-b 3


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plane25. We observe a strong photo-thermoelectric voltage in WTe2 nanoflakes with centrosymmetric distribution, which has not been observed in common 2D materials26-29 such as graphene27, 28, 30 and MoS229. The scanning PV mapping method is utilized to probe the anisotropic energy dispersion-induced anomalous photo-thermoelectric transport. The scanning PV microscopy (Fig. 1a) is equipped with a laser source (photon energy of 1.9 eV). The WTe2 nanoflake device (Fig. 1b) is mounted to a piezostage. The six electrodes (5 nm Cr/ 50 nm Au) pairs surround the nanoflake. All electrode pairs present Ohmic contacts with the WTe2 nanoflake (Fig. S2). The magneto-transport measurement confirms the semimetallic nature of our WTe2 nanoflakes with electron doping at room temperature (Note S2). The z-coordinate is parallel to the c axis of WTe2 crystal structure. The electrodes pair orientation (θ) with respect to the a axis of WTe2 crystal is characterized by Raman spectra (Fig. S3). The PV is measured using the lock-in technique with grounding one electrode (Fig. 1b). The reflection image (Fig. S4) of the device allows the identification of electrode locations (dashed yellow line in Fig. 1c) in the PV images. We have investigated five devices which show similar results. One typical device is shown in the main text. Another representative device is shown in Fig. S13. More details of devices fabrication, characterization and measurements are shown in Methods of Supporting Information. The PV image (Fig. 1c and Fig. 4c) of WTe2 at zero source bias is in stark contrast to that in common 2D materials27-29 (see control sample of MoS2 in Note S3). The PV signals (Fig. S6) in MoS2 mainly emerge at the contact region of MoS2 and electrodes, and its distribution is nearly axisymmetric with respect to electrodes pair. However, the PV signals (Fig. 1c) of the WTe2 nanoflake are distributed almost across the entire scanning area even when the laser spot is 15 μm away from electrodes edges. This distance is up to 10 times larger than the diameter (1.5 µm) of laser spot. In addition, the PV map (Fig. 1c) of WTe2 4


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nanoflake displays a nearly centrosymmetric profile. Here the centrosymmetric and axisymmetric PV profile satisfies │PV (x, y)│≈│PV (-x,-y)│ and │PV (x, y)│≈│PV (-x, y)│, respectively. We first exclude the photoconductive effect. In the photoconductive effect, the external bias separates the photo-generated electrons and holes which drift towards electrodes, giving rise to the generation of photocurrent or photovoltage. However, in our case, no external bias is applied. We can further rule out the bolometric effect. The bolometric effect arises from the change of conductivity due to laser heating. The change of conductivity induces the change of voltage, which can be detected with an external bias. However, as mentioned above, we have not applied any bias to the devices. In order to further understand the physical origin of strong PV signals in the entire WTe2 nanoflake, we extract the data of PV and reflectance (R) along the black dashed line in Fig. 1c and Fig. S4, respectively. Figure 2a shows the derivative of reflectance (blue line) and PV data (blank squares). The derivative of reflectance data allows us to clearly identify the boundaries of electrodes and WTe2. The black solid line in Fig. 2a is the Gaussian fitting to the PV data with a diffraction limited laser spot size of 1.5 µm. The red solid line is the results after subtracting the Gaussian component from the PV data. The photovoltaic effect cannot be the origination of photovoltage in WTe2 either. The photovoltaic effect is the result of built-in potential in the junction area such as the p-n junction or the Schottky junction. In our device, the contact area of gold electrodes-WTe2 nanoflakes possibly forms a Schottky barrier. The photo-excited electrons and holes are separated by a built-in potential. If the PV is due to the internal field of the Schottky barrier, the photovoltage data should follow the Gaussian profile, and the PV peak should be centered at the electrodes-WTe2 interface. However, the PV distribution extends into electrodes with a long tail, indicating a strong photo-thermoelectric effect which dominates the PV in WTe2. 5


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Therefore, the PV is determined by the local light excitation-induced temperature gradient rather than a Schottky barrier height26. We then fit the diffusion length of the thermalized carriers by laser excitation. As shown in Fig. 2b, we extract the PV and reflectance (R) data across one electrode as indicated by the red lines in Fig. 1c and Fig. S4, respectively. We choose the PV data along the b axis of WTe2 crystal due to the large electron transmission probability and small distortion of electron transport along this direction as demonstrated below. The fitting curve (black line) based on the diffusion theory (Note S4) indicates a room temperature photo-excited carrier diffusion length as long as 3.2 µm. In order to elucidate the physical origin of the long hot carriers diffusion length, we have directly measured the carrier relaxation time via performing the ultrafast transient reflectivity in single crystal WTe2. As shown in Fig. S11, two distinct decay processes are observed. The first process of fast decay (τ1 = 2.9 ps) can be ascribed to electron-phonon thermalization. The second process of slow decay (τ2 = 0.22 ns) is the result of phonon-assisted electron-hole recombination. A flat offset (blue dashed line) at longer time scales is dominated by the lattice heat diffusion31. The carrier relaxation time of WTe2 is two orders of magnitude larger than graphene32. The carrier diffusion length (L) can be estimated according to L=vτ, where v is the electron velocity. We assume v is the Fermi velocity. Take the Fermi energy EF of 20 meV33, we get v=8.4×104 m s-1 through 𝑣 = 2𝐸𝐹 𝑚∗

, where m* is the static electron mass. Then L is estimated to be 18 µm which is the

same scale as the fitting result (3.2 µm) in Fig. 2b. The long photo-excited carrier diffusion length indicates weak phonon scattering in WTe2, which is consistent with the low thermoconductivity in WTe2 suggested by the first principle calculation34 and thermal transport35. Furthermore, we have estimated the magnitude of the Seebeck coefficient of WTe2 (Notes S6). The Seebeck coefficient S is estimated to be ~312 µV/K from our model at room temperature. The estimated S is the same order of magnitude to that (tens of µV/K) of 6


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bulk WTe236-38, and two orders larger than that (a few µV/K) of few-layer WTe239. The temperature-dependent S of thin-flake WTe2 shows that electrons dominate the transport at high temperatures39. However, with decreasing temperature, additional holes compete with electrons, leading to a sign change of S39. This result is consistent with our Hall transport data in Fig. S5. The responsivity (R) reaches 19 V/W at a laser power (P) of 1.6 µW (Fig. 2c). The strong photo-thermoelectric effect enables to observe the PV signals in the entire WTe2 nanoflake. In addition, a long photo-excited carrier diffusion length in WTe2 indicates its great potential in far-infrared photodetectors40. We now discuss the physical model for the effect of anisotropic energy dispersion on the electron transport by focused laser excitation. We model the system by assuming the generated PV to be induced by the photo-thermoelectric effect due to the temperature gradient between the laser-focused region and electrodes. Despite the fact that WTe2 exhibits an over-tilted characteristic (type-II), this feature is not robust against temperature since the tilt of the band structure highly depends on the lattice constants25. The band structure of WTe2 is expected to lose the type-II Weyl characteristic but still show anisotropic dispersion at room temperature at which our experiment is performed25. Therefore, we focus on a system with the band structure shown in Fig. S9. We use a generic Hamiltonian with an anisotropic band structure along the z-direction to investigate transport properties of the system. We consider the contributions from two different valleys which carry opposite chiralities and are tilted along opposite directions along the z-direction. The detailed model can be found in Note S1. To investigate the transport anisotropy of the system, we calculate the thermoelectric current along all directions of the two-dimensional plane (i.e., the WTe2 surface scanned by the laser). The transport direction is denoted by m as illustrated in Fig. 3c. In systems with isotropic energy dispersion, the transport properties are expected to be same along all 7


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directions. However, an anisotropic dispersion breaks the rotational symmetry of momentum, and an anisotropic Fermi surface occurs in such systems. Consequently, the electron transport becomes highly dependent on the elliptical shape of the Fermi surface, and hence the direction of transport, which is defined by the angle thermoelectric conductance as a function of

as shown in Fig. 3c. The total

is given by the Landauer-Buttiker formula41, (1)

where

is the angle between Fermi wave vector k F and k y . The current flux is calculated by , which is used to sum up all the incident k-vectors that has non-zero component

along the transport direction.

is the current at an incidence angle

normalized by the density of states, and nh functions nh(c) (E, E F ) 

1 e

( E  h ( c ) ) k BTh ( c )

1

and energy

E

and nc are the Fermi-Dirac distribution

for the hot and cold regions, i.e., under and outside

the laser spot, respectively (see Note S1 for details). The temperature of the laser-focused spot and electrodes are denoted by Th and Tc respectively. In the regular expression of Landauer-Buttiker formula, the difference between electrochemical potentials (

) at

source and drain generates the current. However, the difference in temperature also gives rise to non-zero

, although the electrochemical potential throughout the

device is constant and equal to the Fermi energy (

). Fig. 3a shows nh and nc ,

assuming laser increases the temperature locally by 10 K. This temperature difference gives rise to non-zero

as shown in Fig. 3b, which is the factor that generates

the thermoelectric current in the proposed model. Using Eq. (1), the normalized conductance of the system is calculated as a function of transport direction symmetry about the transmission direction (Fig. 3d). 8


, which exhibits a two-fold

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The number of electrons that reach the electrodes is proportional to SG  D , where ρ is the resistivity, D is the distance between the laser spot and electrodes, and S is the crosssection area of transport channel. Therefore, the PV measured by the electrodes A and B can be described as PV 

S  G , A G , B     (Fig. 4a). Both the experimental PV maps obtained   DA DB 

by scanning the surface of WTe2 devices (Fig. 1c and 4c) and the calculated PV images (Fig. 1d and 4d) exhibit a nearly centrosymmetric profile. Our theoretical model suggests the anomalous PV distribution is very sensitive to the direction and strength of the anisotropy in the band structure. However, small changes in other parameters such as the laser power would only change the strength of PV but not its distribution characteristics. In order to verify this, we carry out PV imaging experiments with varying the laser power in another device (Note S5). The PV intensity increases with increasing the laser power (Fig. S8a). However, the characteristic of the centrosymmetric PV distribution remains the same regardless of the laser power, which is in close agreement with the simulations in Fig. S8b. In addition, we have performed the measurement of photovoltage mapping using a diode laser with a wavelength of 850 nm. As shown in Fig. S14, the photovoltage image displays a nearly centrosymmetric profile as that (Fig. 1) under the laser irradiation of 650 nm. Although the gold absorption is more at 650 nm than that at 850 nm42, the photovoltage distribution is almost the same. The amount of absorption in gold electrodes possibly affects the magnitude of photovoltages. However, the centrosymmetric profile of photovoltage is intrinsically determinated by the anisotropic band structure of WTe2. We note that the twofold symmetric Fermi surface is crucial to the centrosymmetric PV profiles as described in the model. The strong photo-thermoelectric effect with a micro-meter thermalized carrier diffusion length is another factor that makes the observation possible.

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The characteristic of PV distribution is directly related to the direction of the band anisotropy, because the elliptical shape of the Fermi surface originates from the anisotropic band structure. Therefore, we are able to identify the anisotropic tilt direction of energy dispersion by comparing the experimental results with the theoretical model. The PV images of four different angle configurations in Fig. 1c and 4c exhibit a centrosymmetric distribution, but with different profiles. The determinant factor for the resultant anisotropic PV distribution is the relative angle between the anisotropic direction of the energy dispersion and the electrodes pair orientation. We achieve the best fitting of the calculated PV images (Fig. 1d and 4d) to the experimental results (Fig. 1c and 4c) when we set the direction of the anisotropy to wt as depicted in Fig 4b. Therefore, the band anisotropic direction is inferred to be closer to b axis than a axis, which is consistent with the previous first-principle calculations13 as well as transport experiments25. Our results open up an experimental means for controlling electron transports by optical manipulation. The common 2D materials such as graphene28 and MoS229, 43 typically exhibit an axisymmetric photocurrent distribution as shown in Fig. S6. The hexagonal crystal lattice with conserved inversion symmetry gives rise to an isotropic energy dispersion along two directions in k-space, which ensures that the Fermi surface is always a perfect circle. Therefore, the transmission does not depend on the current direction. However, the inversion symmetry breaking in WTe2 results in a much larger conductance along the direction that is parallel to the anisotropic vector wt (Fig. 4b). That indicates that the optically excitedelectrons flow is strongly confined to the anisotropic direction of the energy dispersion in WTe2. However, we observe an axisymmetric rather than centrosymmetric photovoltage distribution (Fig. S12) in the bulk WTe2 flake (125 nm). This can be explained by the fact that the thick sample provides an additional diffusion path for hot carriers along the out-ofplane direction. As a result, the hot carriers transport is no longer confined into two 10


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dimensional plane, which weakens the effect of anomalous photothermoelectric transport aroused by anisotropic energy bands. Apart from electrical and magnetic fields, our work provides a novel optical route to control carrier transports based on the anisotropic energy dispersion of WTe2, which opens the door to interesting electro-optic applications such as electron beam collimation44 and electron lenses45. ASSOCIATED CONTENT Supporting Information Methods, electrical transport data, raman characterization, magneto-transport of a WTe2 nanoflake, PV mapping in a MoS2 device, laser power dependence of PV distribution, The band structure of the Weyl semimetal thin film at k y = 0, Ultrafast reflectivity. This material is available free of charge via the Internet at http://pubs.acs.org. AUTHOR INFORMATION Corresponding Author *Email: [email protected], Tel: (+65)-65167217 Author Contributions §These

authors contributed equally.

Notes The authors declare no completing financial interest. ACKNOWLEDGMENT The work is partially supported by RIE2020 AME Programmatic Grant: Spin-Orbit Technologies for Intelligence at the Edge. Z. L. acknowledges the supported by the National Research Foundation Singapore under NRF RF Award No. NRF-RF2013-08, MOE Tier 2 grant MOE2016-T2-2-153 and MOE2015-T2-2-007.

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(25) Lv, Y.-Y.; Li, X.; Zhang, B.-B.; Deng, W.; Yao, S.-H.; Chen, Y.; Zhou, J.; Zhang, S.-T.; Lu, M.-H.; Zhang, L. Phys. Rev. Lett. 2017, 118, 096603. (26) Lee, M.-J.; Ahn, J.-H.; Sung, J. H.; Heo, H.; Jeon, S. G.; Lee, W.; Song, J. Y.; Hong, K.H.; Choi, B.; Lee, S.-H.; Jo, M.-H. Nature Commun. 2016, 7, 12011. (27) Xia, F.; Mueller, T.; Golizadeh-Mojarad, R.; Freitag, M.; Lin, Y.-m.; Tsang, J.; Perebeinos, V.; Avouris, P. Nano Lett. 2009, 9, 1039-1044. (28) Mueller, T.; Xia, F.; Freitag, M.; Tsang, J.; Avouris, P. Phys. Rev. B 2009, 79, 245430. (29) Buscema, M.; Barkelid, M.; Zwiller, V.; van der Zant, H. S.; Steele, G. A.; CastellanosGomez, A. Nano Lett. 2013, 13, 358-363. (30) Park, J.; Ahn, Y.; Ruiz-Vargas, C. Nano Lett. 2009, 9, 1742-1746. (31) Dai, Y.; Bowlan, J.; Li, H.; Miao, H.; Wu, S.; Kong, W.; Shi, Y.; Trugman, S.; Zhu, J.X.; Ding, H. Phys. Rev. B 2015, 92, 161104. (32) Jadidi, M. M.; Suess, R. J.; Cheng, T.; Cai, X.; Watanabe, K.; Taniguchi, T.; Sushkov, A. B.; Mittendorff, M.; Hone, J.; Drew, H. D. Phys. Rev. Lett 2016, 117, 257401. (33) Zhu, Z.; Lin, X.; Liu, J.; Fauqué, B.; Tao, Q.; Yang, C.; Shi, Y.; Behnia, K. Phys. Rev. Lett. 2015, 114, 176601. (34) Liu, G.; Sun, H. Y.; Zhou, J.; Li, Q. F.; Wan, X.-G. New J. Phys. 2016, 18, (3), 033017. (35) Mleczko, M. J.; Xu, R. L.; Okabe, K.; Kuo, H.-H.; Fisher, I. R.; Wong, H. S. P.; Nishi, Y.; Pop, E. ACS Nano 2016, 10, 7507-7514. (36) Kabashima, S. J. Phys. Soc. Jap. 1966, 21, 945-948. (37) Champion, J. A. Brit. J. Appl. Phys. 1965, 16, 1035-1037. (38) Wu, Y.; Jo, N. H.; Ochi, M.; Huang, L.; Mou, D.; Bud'Ko, S. L.; Canfield, P. C.; Trivedi, N.; Arita, R.; Kaminski, A. Phys. Rev. Lett. 2015, 115, 166602. (39) Rana, K. G.; Dejene, F. K.; Kumar, N.; Rajamathi, C. R.; Sklarek, K.; Felser, C.; Parkin, S. S. Nano Lett. 2018, 18, 6591-6596. (40) Cai, X.; Sushkov, A. B.; Suess, R. J.; Jadidi, M. M.; Jenkins, G. S.; Nyakiti, L. O.; Myers-Ward, R. L.; Li, S.; Yan, J.; Gaskill, D. K.; Murphy, T. E.; Drew, H. D.; Fuhrer, M. S. Nature Nanotechnol. 2014, 9, 814-819. (41) Datta, S. World Scientific Publishing Company: 2012, 1. (42) Wu, D.; Peng, J.; Cai, Z.; Weng, J.; Luo, Z.; Chen, N.; Xu, H. Optics express 2015, 23, (18), 24071-24076. (43). Xiao, D.; Liu, G.-B.; Feng, W.; Xu, X.; Yao, W. Phys. Rev. Lett. 2012, 108, 196802. (44) Park, C.-H.; Son, Y.-W.; Yang, L.; Cohen, M. L.; Louie, S. G. Nano Lett. 2008, 8, 29202924. (45) Cheianov, V. V.; Fal'ko, V.; Altshuler, B. L. Science 2007, 315, 1252-1255. (46) Belopolski, I.; Sanchez, D. S.; Ishida, Y.; Pan, X.; Yu, P.; Xu, S.-Y.; Chang, G.; Chang, T.-R.; Zheng, H.; Alidoust, N.; Bian, G.; Neupane, M.; Huang, S. –M.; Lee, C.-C.; Song, Y.; Bu, H.; Wang, G.; Li, S.; Eda, G.; Jeng, H.-T.;Kondo, T.; Lin, H.; Liu, Z.; Song, F.; Shin, S.; Hasan, M. Z. Nature Commun. 2016, 7, 13643.

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Figure 1. The centrosymmetric PV distribution. a, Scanning optical microscope. The zdirection is parallel to the c axis of WTe2 crystal. b, Optical image of a WTe2 nanoflake device with six pairs of electrodes. a, b and c are the crystals axes of WTe2. θ is the angle of electrodes pair with respect to the a axis. c, PV mapping with electrodes along the a axis (θ=0). Yellow dashed lines show the positions of electrodes. d, Calculated PV map in case of θ=0. The electrodes are represented by the yellow bars.

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Nano Letters

Figure 2. Photo-thermoelectric properties. a, PV (squares) and derivative reflectance (blue solid line) data extracted along the black dashed line in Fig. 1c and Fig. S4, respectively. Black solid line is the Gaussian fitting to the PV profile centered at the interface between electrodes and WTe2 nanoflake. Red solid line is the result of PV data after subtracting Gaussian component. b, PV (squares) and derivative reflectance (blue solid line) profiles extracted along red solid lines in Fig. 1c and Fig. S4, respectively. Black solid line is the fitting curve to the PV data, which reveals a carrier diffusion length of 3.2 μm in WTe2. Light gray regions in a and b guide the electrodes. c. Laser power dependence of PV and responsivity.

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Figure 3. The two-fold symmetric conductance profile. a, The Fermi distribution functions at laser-focused regions ( nh ) and electrodes ( nc ). The temperature difference at these points induce

K and

, as shown in b, where

K . c, The anisotropic

Fermi surface of the system at 225 meV. The details about the band structure are provided in Note S1. Unlike the non-tilted isotropic systems where the Fermi surfaces are perfect circles, the conductance of the anisotropic system depends on the transmission direction (i.e., m ).  is the angle between transmission direction and k z . d, The two-fold symmetry conductance of the WTe2 system.

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Nano Letters

Figure 4. The anisotropic direction of energy dispersion. a, Schematic illustration of the theoretical model, where the blue and red squares represent the electrodes, while the purple dot indicates the laser spot. b, The characteristic of two-fold symmetric conductance profile. c, Experiment PV data. 30, 60, and 90 indicates the angle of electrode pairs with respect to the a axis of WTe2 crystal. d, Calculated normalized PV results. The yellow bars represent the electrodes. The Fermi energy E F  235 meV. The temperature at laser-focused region (electrodes) are taken as

K(

K ). The best fitting of the calculated PV

images to the experimental results is achieved as we set the direction of the anisotropy to wt , indicating the band anisotropic direction is closer to b than a axis.

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For TOC only

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Anomalous photo-thermoelectric transport due to anisotropic energy

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