Electronic materials generally exhibit a single isotropic majority carrier type, electrons or holes. Some superlattice1,2 and hexagonal3,4,5 materials exhibit opposite conduction polarities along in-plane and cross-plane directions due to multiple electron and hole bands. Here, we uncover a material genus with this behaviour that originates from the Fermi surface geometry of a single band. NaSn2As2, a layered metal, has such a Fermi surface. It displays in-plane electron and cross-plane hole conduction in thermopower and exactly the opposite polarity in the Hall effect. The small Nernst coefficient and magnetoresistance preclude multi-band transport. We label this direction-dependent carrier polarity in single-band systems ‘goniopolarity’. We expect to find goniopolarity and the Fermi surface geometry that produces it in many metals and semiconductors whose electronic structure is at the boundary between two and three dimensions. Goniopolarity may enable future explorations of complex transport phenomena that lead to unprecedented device concepts.

Access optionsAccess options

Rent or Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Data availability

All relevant data are available from the authors.

Additional information

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.


  1. 1.

    Zhou, C., Birner, S., Tang, Y., Heinselman, K. & Grayson, M. Driving perpendicular heat flow: (p × n)-type transverse thermoelectrics for microscale and cryogenic peltier cooling. Phys. Rev. Lett. 110, 227701 (2013).

  2. 2.

    Tang, Y., Cui, B., Zhou, C. & Grayson, M. p x n-type transverse thermoelectrics: a novel type of thermal management material. J. Electron. Mater. 44, 2095–2104 (2015).

  3. 3.

    Rowe, V. A. & Schroeder, P. A. Thermopower of Mg, Cd, and Zn between 1.2 degrees K and 300 degrees K. J. Phys. Chem. Solids 31, 1–8 (1970).

  4. 4.

    Burkov, A. T., Vedernikov, M. V., Elenskii, V. A. & Kovtun, G. P. Anisotropy of thermo-EMF and electroconductivity of high-purity rhenium. Fiz. Tverd. Tela 28, 785–788 (1986).

  5. 5.

    Burkov, A. T. & Verdernikov, M. V. Anomalous aniosotropy of high temperature thermo-EMF in beryllium. Sov. Phys. Solid State 28, 3737–3739 (1986).

  6. 6.

    Gu, J. J., Oh, H. W., Inui, H. & Zhang, D. Anisotropy of mobility ratio between electron and hole along different orientations in ReGexSi1.75-x thermoelectric single crystals. Phys. Rev. B 71, 113201 (2005).

  7. 7.

    Putley, E. H. The Hall Effect and Related Phenomena 85–90 (Butterworths, 1960).

  8. 8.

    Caswell, A. E. The Hall, Ettingshausen, Nernst, and Leduc effects in cadmium, nickel, and zinc. Phys. Rev. 20, 280–282 (1922).

  9. 9.

    Mangez, J. H., Issi, J.-P. & Heremans, J. P. Transport properties of bismuth in quantizing fields. Phys. Rev. B 14, 4381–4385 (1976).

  10. 10.

    Ong, K. P., Singh, D. J. & Wu, P. Unusual transport and strongly anisotropic thermopower in PtCoO2 and PdCoO2. Phys. Rev. Lett. 104, 176601 (2010).

  11. 11.

    Tanaka, M., Hasegawa, M. & Takei, H. Growth and anisotropic physical properties of PdCoO2 single crystals. J. Phys. Soc. Jpn 65, 3973–3977 (1996).

  12. 12.

    Cutler, M. & Mott, N. F. Observation of Anderson localization in an electron gas. Phys. Rev. 181, 1336–1340 (1969).

  13. 13.

    Harrison, W. Electronic Structure and the Properties of Solids: The Physics of the Chemical Bond 34 (Dover, 1989).

  14. 14.

    Jan, J.-P. Effective masses and curvature of the Fermi surface or energy bands. Helv. Phys. Acta 41, 957–959 (1968).

  15. 15.

    MacDonald, D. K. C. & Pearson, W. B. Thermo-electricity at low temperatures I. The ‘ideal’ metals: sodium, potassium, copper. Proc. R. Soc. Lond. A 219, 373–387 (1953).

  16. 16.

    Anderson, H. G. Jr. Investigation of Thermoelectric Properties of Liquid Metals MACHLAB-254 (David W. Taylor Naval Ship Research and Development Center, 1969).

  17. 17.

    Madsen, G. K. H. & Singh, D. J. BoltzTraP. A code for calculating band-structure dependent quantities. Comput. Phys. Commun. 175, 67–71 (2006).

  18. 18.

    Arguilla, M. Q. et al. NaSn2As2: an exfoliatable layered van der Waals Zintl phase. ACS Nano 10, 9500–9508 (2016).

  19. 19.

    Kresse, G. & Hafner, J. Ab initio molecular dynamics for liquid metals. Phys. Rev. B 47, 558–561 (1993).

  20. 20.

    Kresse, G. & Hafner, J. Ab initio molecular-dynamics simulation of the liquid-metal amorphous–semiconductor transition in germanium. Phys. Rev. B 49, 14251–14269 (1994).

  21. 21.

    Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865–3868 (1996); erratum 78, 1396 (1997).

  22. 22.

    Blöchl, P. E. Projector augmented-wave method. Phys. Rev. B 50, 17953–17979 (1994).

  23. 23.

    Ong, N. P. Geometric interpretation of the weak-field Hall conductivity in two-dimensional metals with arbitrary Fermi surface. Phys. Rev. B 43, 194–201 (1991).

  24. 24.

    Førsvoll, K. & Holwech Sondheimer, I. Oscillations in the Hall effect of aluminium. Philos. Mag. 10, 921–930 (1964).

  25. 25.

    Ravich, Y. I, Efimova, B. A. & Smirnov, I. A. Semiconducting Lead Chalcogenides 111–113 (Plenum, 1970).

  26. 26.

    Fauqué, B. et al. Magnetoresistance of semi-metals: the case of antimony. Phys. Rev. Mater. 2, 114201 (2018).

  27. 27.

    Lin, Z. et al. Thermal conductivities in NaSnAs, NaSnP, and NaSn2As2: effect of double lone-pair electrons. Phys. Rev. B 95, 165201 (2017).

  28. 28.

    Heremans, J. P., Thrush, C. M. & Morelli, D. T. Thermopower enhancement in lead telluride nanostructures. Phys. Rev. B 70, 115334 (2004).

Download references


Primary funding for the synthesis (M.Q.A., N.D.C., M.R.S. and J.E.G.) and transport measurements (B.H. and J.P.H.) was provided by NSF EFRI-1433467. Primary funding for the DFT simulations was provided by the Center for Emergent Materials: an NSF MRSEC under Award DMR-1420451 (Y.W.) and AFOSR FA9550-18-1-0335 (W.W.). J.E.G. acknowledges the Camille and Henry Dreyfus Foundation for partial support. R. Ripley is acknowledged for editing text and figures.

Author information

Author notes

  1. These authors contributed equally: Bin He, Yaxian Wang.


  1. Department of Mechanical and Aerospace Engineering, The Ohio State University, Columbus, OH, USA

    • Bin He
    •  & Joseph P. Heremans
  2. Department of Materials Science and Engineering, The Ohio State University, Columbus, OH, USA

    • Yaxian Wang
    • , Wolfgang Windl
    •  & Joseph P. Heremans
  3. Department of Chemistry and Biochemistry, The Ohio State University, Columbus, OH, USA

    • Maxx Q. Arguilla
    • , Nicholas D. Cultrara
    • , Michael R. Scudder
    •  & Joshua E. Goldberger
  4. Department of Physics, The Ohio State University, Columbus, OH, USA

    • Wolfgang Windl
    •  & Joseph P. Heremans


  1. Search for Bin He in:

  2. Search for Yaxian Wang in:

  3. Search for Maxx Q. Arguilla in:

  4. Search for Nicholas D. Cultrara in:

  5. Search for Michael R. Scudder in:

  6. Search for Joshua E. Goldberger in:

  7. Search for Wolfgang Windl in:

  8. Search for Joseph P. Heremans in:


Measurements were carried out by B.H. and J.P.H.; the theory was developed by Y.W., J.P.H. and W.W.; the sample synthesis was carried out by M.Q.A., N.D.C., M.R.S. and J.E.G.; and all DFT computations were carried out by Y.W. and W.W. The paper was primarily written by B.H., Y.W., J.E.G., W.W. and J.P.H.

Competing interests

The authors declare no competing interests.

Corresponding author

Correspondence to Joseph P. Heremans.

Supplementary information

  1. Supplementary Information

    Supplementary Figures 1–6, Supplementary Notes 1–5, Supplementary References 1–10

About this article

Publication history