Skip to main content
SpringerLink
Log in
Book cover

Theory of Thermodynamic Measurements of Quantum Systems Far from Equilibrium pp 13–42Cite as

Temperature and Voltage

  • Chapter
  • First Online:

  • 284 Accesses

Part of the book series: Springer Theses ((Springer Theses))

Abstract

Outside equilibrium, the definition of basic thermodynamic observables such as temperature and voltage are muddled by a competing panoply of “operation definitions” which are often contradictory. Here we define temperature and voltage on an equal footing by means of an equilibrium probe reservoir (such as an STM) coupled locally to the nonequilibrium system of interest; The temperature and voltage measurement are defined by requiring vanishing charge and heat dissipation into the probe. We show that temperature and voltage measurements are unique when they exist. We further derive a necessary and sufficient condition for the existence of a positive temperature solution. We then show that, when a positive temperature solution doesn’t exist, there must exist a negative temperature solution. The latter condition corresponds to a net population inversion. Therefore, a solution always exists. Our results suggest that a local temperature measurement without a simultaneous local voltage measurement, or vice-versa, is a misleading characterization of the state of the nonequilibrium system of interest. These results show an intimate connection to statements of the second law of thermodynamics. We see that the uniqueness of the (simultaneous) measurement of temperature and voltage is related to the Onsager’s statement of the second law of thermodynamics. Therefore, as an intermediate step, we provide the first proof for Onsager’s phenomenological statement (1931) for the case of quantum thermoelectric transport.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    Voltage here refers to the electrochemical potential (μ) and not just the electrostatic potential (V). A voltmeter in fact measures the electrochemical potential difference which of course includes the electrostatic contribution μ = μ 0 + eV.

  2. 2.

    This is further made clear in Sect. 2.5.2 [see Eq. (2.56)]. The local spectrum introduced in Sect. 2.5.2 is proportional to the probe-system transmission function in the broadband limit.

  3. 3.

    \(\mathcal {L}^{(\nu )}_{ps}(\mu _{p},T_{p})\) are finite even if \(\mathcal {T}_{ps}(\omega )\) and \(\omega \mathcal {T}_{ps}(\omega )\) do not obey the finite measure conditions of Postulate 2.1 due to the exponentially decaying tails of the Fermi-derivative. We merely need \(\mathcal {T}_{ps}(\omega )\) to grow slower than exponentially for ω →±∞.

  4. 4.

    Furthermore, the tangent vector [see Eq. (2.34)] along \(I^{(1)}_{p}=c\) cannot be of magnitude zero since \(\mathcal {L}^{(2)}_{ps}\) is strictly positive for T p ∈ (0, ∞). Therefore, the contour \(I^{(1)}_{p}=c\) does not terminate for finite values of T p and μ p. This implies the existence of a function τ c : (−∞, ∞) → (0, ∞) which defines

    $$\displaystyle \begin{aligned} T_{p}=\tau_{c}(\mu_{p}) \end{aligned}$$

    for each point on \(I^{(1)}_{p}(\mu _{p},T_{p})=c\).

  5. 5.

    Transport in a vast majority of mesoscopic and nanoscale conductors are dominated by elastic processes at room temperature. Theorem 2.1 proves the Onsager’s phenomenological statement of the second law (1931) for quantum thermoelectric transport where elastic processes dominate the transport.

References

  1. A.S. Eddington, The Nature of the Physical World (The University Press, Cambridge, 1929)

    MATH  Google Scholar 

  2. D. Ruelle, J. Stat. Phys. 98(1), 57 (2000). https://doi.org/10.1023/A:1018618704438. http://dx.doi.org/10.1023/A:1018618704438

    MathSciNet  Google Scholar 

  3. J. Casas-Vázquez, D. Jou, Rep. Prog. Phys. 66(11), 1937 (2003). http://stacks.iop.org/0034-4885/66/i=11/a=R03

    ADS  Google Scholar 

  4. G. Lebon, D. Jou, Understanding Non-equilibrium Thermodynamics (Springer, Berlin, 2008). https://doi.org/10.1007/978-3-540-74252-4

    MATH  Google Scholar 

  5. L.F. Cugliandolo, J. Phys. A Math. Theor. 44(48), 483001 (2011). http://stacks.iop.org/1751-8121/44/i=48/a=483001

    MathSciNet  Google Scholar 

  6. P.A. Jacquet, C.A. Pillet, Phys. Rev. B 85, 125120 (2012). https://doi.org/10.1103/PhysRevB.85.125120. http://link.aps.org/doi/10.1103/PhysRevB.85.125120

    ADS  Google Scholar 

  7. C.A. Stafford, Phys. Rev. B 93, 245403 (2016). https://doi.org/10.1103/PhysRevB.93.245403. http://link.aps.org/doi/10.1103/PhysRevB.93.245403

    ADS  Google Scholar 

  8. M. Esposito, M.A. Ochoa, M. Galperin, Phys. Rev. Lett. 114, 080602 (2015). https://doi.org/10.1103/PhysRevLett.114.080602. http://link.aps.org/doi/10.1103/PhysRevLett.114.080602

    ADS  Google Scholar 

  9. A. Shastry, C.A. Stafford, Phys. Rev. B 92, 245417 (2015). https://doi.org/10.1103/PhysRevB.92.245417. http://link.aps.org/doi/10.1103/PhysRevB.92.245417

    ADS  Google Scholar 

  10. P. Muralt, D.W. Pohl, Appl. Phys. Lett. 48(8), 514 (1986). http://dx.doi.org/10.1063/1.96491. http://scitation.aip.org/content/aip/journal/apl/48/8/10.1063/1.96491

    ADS  Google Scholar 

  11. T. Kanagawa, R. Hobara, I. Matsuda, T. Tanikawa, A. Natori, S. Hasegawa, Phys. Rev. Lett. 91, 036805 (2003). https://doi.org/10.1103/PhysRevLett.91.036805. http://link.aps.org/doi/10.1103/PhysRevLett.91.036805

    ADS  Google Scholar 

  12. A. Bannani, C.A. Bobisch, R. Moeller, Rev. Sci. Instrum. 79(8), 083704 (2008). https://doi.org/10.1063/1.2968111

    ADS  Google Scholar 

  13. F. Luepke, S. Korte, V. Cherepanov, B. Voigtlaender, Rev. Sci. Instrum. 86(12), 123701 (2015). https://doi.org/10.1063/1.4936079

    ADS  Google Scholar 

  14. C.C. Williams, H.K. Wickramasinghe, Appl. Phys. Lett. 49(23), 1587 (1986). http://dx.doi.org/10.1063/1.97288. http://scitation.aip.org/content/aip/journal/apl/49/23/10.1063/1.97288

    ADS  Google Scholar 

  15. K. Kim, J. Chung, G. Hwang, O. Kwon, J.S. Lee, ACS Nano 5(11), 8700 (2011). https://doi.org/10.1021/nn2026325. http://dx.doi.org/10.1021/nn2026325. PMID: 21999681

    Google Scholar 

  16. Y.J. Yu, M.Y. Han, S. Berciaud, A.B. Georgescu, T.F. Heinz, L.E. Brus, K.S. Kim, P. Kim, Appl. Phys. Lett. 99(18), 183105 (2011). http://dx.doi.org/10.1063/1.3657515. http://scitation.aip.org/content/aip/journal/apl/99/18/10.1063/1.3657515

    ADS  Google Scholar 

  17. K. Kim, W. Jeong, W. Lee, P. Reddy, ACS Nano 6(5), 4248 (2012). https://doi.org/10.1021/nn300774n. http://dx.doi.org/10.1021/nn300774n. PMID: 22530657

    Google Scholar 

  18. F. Menges, H. Riel, A. Stemmer, B. Gotsmann, Nano Lett. 12(2), 596 (2012). https://doi.org/10.1021/nl203169t. http://dx.doi.org/10.1021/nl203169t. PMID: 22214277

    ADS  Google Scholar 

  19. H.L. Engquist, P.W. Anderson, Phys. Rev. B 24, 1151 (1981). https://doi.org/10.1103/PhysRevB.24.1151. http://link.aps.org/doi/10.1103/PhysRevB.24.1151

    ADS  Google Scholar 

  20. Y. Dubi, M. Di Ventra, Nano Lett. 9, 97 (2009)

    ADS  Google Scholar 

  21. P.A. Jacquet, J. Stat. Phys. 134(4), 709 (2009). https://doi.org/10.1007/s10955-009-9697-1. http://dx.doi.org/10.1007/s10955-009-9697-1

    ADS  MathSciNet  MATH  Google Scholar 

  22. Y. Dubi, M. Di Ventra, Phys. Rev. E 79, 042101 (2009). https://doi.org/10.1103/PhysRevE.79.042101. http://link.aps.org/doi/10.1103/PhysRevE.79.042101

    ADS  Google Scholar 

  23. A. Caso, L. Arrachea, G.S. Lozano, Phys. Rev. B 83, 165419 (2011). https://doi.org/10.1103/PhysRevB.83.165419. http://link.aps.org/doi/10.1103/PhysRevB.83.165419

    ADS  Google Scholar 

  24. D. Sánchez, L. Serra, Phys. Rev. B 84, 201307 (2011). https://doi.org/10.1103/PhysRevB.84.201307. http://link.aps.org/doi/10.1103/PhysRevB.84.201307

    ADS  Google Scholar 

  25. A. Caso, L. Arrachea, G.S. Lozano, Phys. Rev. B 81(4), 041301 (2010). https://doi.org/10.1103/PhysRevB.81.041301

    ADS  Google Scholar 

  26. J.P. Bergfield, S.M. Story, R.C. Stafford, C.A. Stafford, ACS Nano 7(5), 4429 (2013). https://doi.org/10.1021/nn401027u

    Google Scholar 

  27. J.P. Bergfield, M.A. Ratner, C.A. Stafford, M. Di Ventra, Phys. Rev. B 91, 125407 (2015). https://doi.org/10.1103/PhysRevB.91.125407. http://link.aps.org/doi/10.1103/PhysRevB.91.125407

    ADS  Google Scholar 

  28. L. Ye, D. Hou, X. Zheng, Y. Yan, M. Di Ventra, Phys. Rev. B 91, 205106 (2015). https://doi.org/10.1103/PhysRevB.91.205106. http://link.aps.org/doi/10.1103/PhysRevB.91.205106

    ADS  Google Scholar 

  29. Y. Chen, M. Zwolak, M. Di Ventra, Nano Lett. 3, 1691 (2003)

    ADS  Google Scholar 

  30. Y. Ming, Z.X. Wang, Z.J. Ding, H.M. Li, New J. Phys. 12, 103041 (2010)

    ADS  Google Scholar 

  31. M. Galperin, A. Nitzan, M.A. Ratner, Phys. Rev. B 75, 155312 (2007). https://doi.org/10.1103/PhysRevB.75.155312. http://link.aps.org/doi/10.1103/PhysRevB.75.155312

    ADS  Google Scholar 

  32. Y. de Wilde, F. Formanek, R. Carminati, B. Gralak, P.A. Lemoine, K. Joulain, J.P. Mulet, Y. Chen, J.J. Greffet, Nature 444, 740 (2006). https://doi.org/10.1038/nature05265

    ADS  Google Scholar 

  33. Y. Yue, J. Zhang, X. Wang, Small 7(23), 3324 (2011)

    Google Scholar 

  34. J.J. Greffet, C. Henkel, Contemp. Phys. 48(4), 183 (2007). https://doi.org/10.1080/00107510701690380

    ADS  Google Scholar 

  35. J. Meair, J.P. Bergfield, C.A. Stafford, P. Jacquod, Phys. Rev. B 90, 035407 (2014). https://doi.org/10.1103/PhysRevB.90.035407. http://link.aps.org/doi/10.1103/PhysRevB.90.035407

    ADS  Google Scholar 

  36. Y. Meir, N.S. Wingreen, Phys. Rev. Lett. 68, 2512 (1992)

    ADS  Google Scholar 

  37. J.P. Bergfield, C.A. Stafford, Nano Lett. 9, 3072 (2009)

    ADS  Google Scholar 

  38. G. Stefanucci, R. van Leeuwen, Nonequilibrium Many-Body Theory of Quantum Systems: A Modern Introduction (Cambridge University Press, Cambridge, 2013)

    MATH  Google Scholar 

  39. R.A. Horn, C.R. Johnson (eds.), Matrix Analysis (Cambridge University Press, New York, 1986)

    Google Scholar 

  40. J.P. Bergfield, C.A. Stafford, Phys. Rev. B 90, 235438 (2014). https://doi.org/10.1103/PhysRevB.90.235438. http://link.aps.org/doi/10.1103/PhysRevB.90.235438

    ADS  Google Scholar 

  41. C.J. Chen, Introduction to Scanning Tunneling Microscopy, 2nd edn. (Oxford University Press, New York, 1993)

    Google Scholar 

  42. S.V. Kalinin, A. Gruverman, Scanning Probe Microscopy: Electrical and Electromechanical Phenomena at the Nanoscale (Springer, Berlin, 2007). https://doi.org/10.1007/978-0-387-28668-6

    Google Scholar 

  43. H. Pothier, S. Guéron, N.O. Birge, D. Esteve, M.H. Devoret, Phys. Rev. Lett. 79, 3490 (1997). https://doi.org/10.1103/PhysRevLett.79.3490. https://link.aps.org/doi/10.1103/PhysRevLett.79.3490

    ADS  Google Scholar 

  44. L. Onsager, Phys. Rev. 37, 405 (1931). https://doi.org/10.1103/PhysRev.37.405. http://link.aps.org/doi/10.1103/PhysRev.37.405

    ADS  Google Scholar 

  45. U. Sivan, Y. Imry, Phys. Rev. B 33, 551 (1986). https://doi.org/10.1103/PhysRevB.33.551. http://link.aps.org/doi/10.1103/PhysRevB.33.551

    ADS  Google Scholar 

  46. M. Büttiker, IBM J. Res. Dev. 32, 63 (1988)

    Google Scholar 

  47. M. Büttiker, Phys. Rev. B 40(5), 3409 (1989). https://doi.org/10.1103/PhysRevB.40.3409

    ADS  Google Scholar 

  48. A.D. Benoit, S. Washburn, C.P. Umbach, R.B. Laibowitz, R.A. Webb, Phys. Rev. Lett. 57, 1765 (1986). https://doi.org/10.1103/PhysRevLett.57.1765. http://link.aps.org/doi/10.1103/PhysRevLett.57.1765

    ADS  Google Scholar 

  49. K.L. Shepard, M.L. Roukes, B.P. van der Gaag, Phys. Rev. B 46, 9648 (1992). https://doi.org/10.1103/PhysRevB.46.9648. http://link.aps.org/doi/10.1103/PhysRevB.46.9648

    ADS  Google Scholar 

  50. R. de Picciotto, H.L. Stormer, L.N. Pfeiffer, K.W. Baldwin, K.W. West, Nature 411(6833), 51 (2001). http://dx.doi.org/10.1038/35075009

    ADS  Google Scholar 

  51. B. Gao, Y.F. Chen, M.S. Fuhrer, D.C. Glattli, A. Bachtold, Phys. Rev. Lett. 95, 196802 (2005). https://doi.org/10.1103/PhysRevLett.95.196802. http://link.aps.org/doi/10.1103/PhysRevLett.95.196802

    ADS  Google Scholar 

  52. M. Büttiker, Phys. Rev. B 32, 1846 (1985). https://doi.org/10.1103/PhysRevB.32.1846. http://link.aps.org/doi/10.1103/PhysRevB.32.1846

    ADS  Google Scholar 

  53. M. Büttiker, Phys. Rev. B 33, 3020 (1986). https://doi.org/10.1103/PhysRevB.33.3020. http://link.aps.org/doi/10.1103/PhysRevB.33.3020

    ADS  Google Scholar 

  54. J.L. D’Amato, H.M. Pastawski, Phys. Rev. B 41, 7411 (1990). https://doi.org/10.1103/PhysRevB.41.7411. http://link.aps.org/doi/10.1103/PhysRevB.41.7411

    ADS  Google Scholar 

  55. T. Ando, Surf. Sci. 361362, 270 (1996). http://dx.doi.org/10.1016/0039-6028(96)00400-1. http://www.sciencedirect.com/science/article/pii/0039602896004001; Proceedings of the Eleventh International Conference on the Electronic Properties of Two-Dimensional Systems

  56. D. Roy, A. Dhar, Phys. Rev. B 75, 195110 (2007). https://doi.org/10.1103/PhysRevB.75.195110. http://link.aps.org/doi/10.1103/PhysRevB.75.195110

    ADS  Google Scholar 

  57. M. de Jong, C. Beenakker, Physica A 230(1–2), 219 (1996). http://www.sciencedirect.com/science/article/pii/0378437196000684

    ADS  Google Scholar 

  58. S.A. van Langen, M. Büttiker, Phys. Rev. B 56, R1680 (1997). https://doi.org/10.1103/PhysRevB.56.R1680. http://link.aps.org/doi/10.1103/PhysRevB.56.R1680

    ADS  Google Scholar 

  59. H. Förster, P. Samuelsson, S. Pilgram, M. Büttiker, Phys. Rev. B 75, 035340 (2007). https://doi.org/10.1103/PhysRevB.75.035340. http://link.aps.org/doi/10.1103/PhysRevB.75.035340

    ADS  Google Scholar 

  60. K. Saito, G. Benenti, G. Casati, T.c.v. Prosen, Phys. Rev. B 84, 201306 (2011). https://doi.org/10.1103/PhysRevB.84.201306. http://link.aps.org/doi/10.1103/PhysRevB.84.201306

  61. V. Balachandran, G. Benenti, G. Casati, Phys. Rev. B 87, 165419 (2013). https://doi.org/10.1103/PhysRevB.87.165419. http://link.aps.org/doi/10.1103/PhysRevB.87.165419

    ADS  Google Scholar 

  62. K. Brandner, K. Saito, U. Seifert, Phys. Rev. Lett. 110, 070603 (2013). https://doi.org/10.1103/PhysRevLett.110.070603. http://link.aps.org/doi/10.1103/PhysRevLett.110.070603

    ADS  Google Scholar 

  63. M. Bandyopadhyay, D. Segal, Phys. Rev. E 84, 011151 (2011). https://doi.org/10.1103/PhysRevE.84.011151. http://link.aps.org/doi/10.1103/PhysRevE.84.011151

    ADS  Google Scholar 

  64. S. Bedkihal, M. Bandyopadhyay, D. Segal, The Eur. Phys. J. B 86(12), 506 (2013). https://doi.org/10.1140/epjb/e2013-40971-7

    ADS  Google Scholar 

  65. S. Bedkihal, M. Bandyopadhyay, D. Segal, Phys. Rev. B 88, 155407 (2013). https://doi.org/10.1103/PhysRevB.88.155407. http://link.aps.org/doi/10.1103/PhysRevB.88.155407

    ADS  Google Scholar 

  66. R. Clausius, Ann. Phys. 169(12), 481 (1854). https://doi.org/10.1002/andp.18541691202.

    ADS  Google Scholar 

  67. J.P. Bergfield, M.A. Solis, C.A. Stafford, ACS Nano 4(9), 5314 (2010)

    Google Scholar 

  68. L. Onsager, Phys. Rev. 38, 2265 (1931). https://doi.org/10.1103/PhysRev.38.2265. https://link.aps.org/doi/10.1103/PhysRev.38.2265

    ADS  Google Scholar 

  69. R. Pathria, P. Beale, Statistical Mechanics, 3rd edn. (Elsevier, Butterworth-Heinemann, Oxford, 2011)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Chemistry, University of Toronto, Toronto, ON, Canada

    Abhay Shastry

Authors
  1. Abhay Shastry
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Shastry, A. (2019). Temperature and Voltage. In: Theory of Thermodynamic Measurements of Quantum Systems Far from Equilibrium. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-030-33574-8_2

Download citation

Publish with us

Policies and ethics

Search

Navigation