Hostname: page-component-686fd747b7-mrck6 Total loading time: 0 Render date: 2025-02-13T14:42:46.726Z Has data issue: false hasContentIssue false
  • English
  • Français

Thermodynamical Properties and Stability of Crystalline Membranes in the Quantum Regime

Published online by Cambridge University Press:  11 February 2015

B. Amorim ,
R. Roldán ,
E. Cappelluti ,
A. Fasolino ,
F. Guinea  and
M. I. Katsnelson
B. Amorim
Affiliation:
Instituto de Ciencia de Materiales de Madrid, CSIC, Cantoblanco, E28049 Madrid, Spain
R. Roldán
Affiliation:
Instituto de Ciencia de Materiales de Madrid, CSIC, Cantoblanco, E28049 Madrid, Spain
E. Cappelluti
Affiliation:
Istituto dei Sistemi Complessi, CNR, U.O.S. Sapienza, v. dei Taurini 19, 00185 Roma, Italy
A. Fasolino
Affiliation:
Radboud University Nijmegen,Institute for Molecules and Materials, NL-6525AJ Nijmegen, The Netherlands
F. Guinea
Affiliation:
Instituto de Ciencia de Materiales de Madrid, CSIC, Cantoblanco, E28049 Madrid, Spain
M. I. Katsnelson
Affiliation:
Radboud University Nijmegen,Institute for Molecules and Materials, NL-6525AJ Nijmegen, The Netherlands
Get access

Abstract

We study the thermodynamical properties and lattice dynamics of two-dimensional crystalline membranes, such as graphene and related compounds, in zero temperature limit, where quantum effects are dominant. We find out that, just as in the high temperature classical limit, a fundamental role is played by the anharmonic coupling between in-plane and out-of plane lattice modes, which leads to a strong reconstruction of the dispersion relation of the out-of-plane mode. We identify a crossover temperature, T*, bellow which quantum effects are dominant. We estimate that for graphene T* ∼ 70 - 90 K. Inclusion of anharmonic effects makes the thermal expansion finite in the thermodynamic limit, and below T* it tends to zero as a power law as T→0 as required by the third law of thermodynamics. The specific heat also goes to zero as a power law as T→0, but with a exponent that differs from the one predicted by the harmonic theory.

Keywords

crystallinespecific heatthermal stresses
Type
Articles
Information
MRS Online Proceedings Library (OPL) , Volume 1727: Symposium K – Graphene and Graphene Nanocomposites , 2015 , mrsf14-1727-k03-03
Copyright
Copyright © Materials Research Society 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Novoselov, K. S., Jiang, D., Schedin, F., Booth, T. J., Khotkevich, V. V., Morozov, S. V. and Geim, A. K., Proc. Natl. Acad. Sci. (USA) 102, 10451–10453, (2005).CrossRefGoogle Scholar
Katsnelson, M., Graphene: Carbon in Two Dimensions, (Cambridge University Press, Cambridge, 2012), chap. 9, pp. 205–242.CrossRefGoogle Scholar
Katsnelson, M. I. and Fasolino, A., Acc. Chem. Res. 46, 97–105, (2013).CrossRefGoogle Scholar
Statistical Mechanics of Membranes and Surfaces, edited by Nelson, D., Piran, T. and Weinberg, S. (World Scientific, Singapore, 2004)..CrossRefGoogle Scholar
Nelson, D. and Peliti, L., J. Phys. Paris 48, 1085, (1987).CrossRefGoogle Scholar
Aronovitz, J. A. and Lubensky, T. C., Phys. Rev. Lett. 60, 2634–2637, (1988).CrossRefGoogle Scholar
Aronovitz, J., Golubovic, L. and Lubensky, T., J. Phys. France 50, 609–631, (1989).CrossRefGoogle Scholar
Los, J. H., Katsnelson, M. I., Yazyev, O. V., Zakharchenko, K. V. and Fasolino, A., Phys. Rev. B 80, 121405, (2009).CrossRefGoogle Scholar
Kownacki, J.-P. and Mouhanna, D., Phys. Rev. E 79, 040101, (2009).CrossRefGoogle Scholar
Braghin, F. L. and Hasselmann, N., Phys. Rev. B 82, 035407, (2010).CrossRefGoogle Scholar
Mermin, N. D., Phys. Rev. 176, 250–254, (1968).CrossRefGoogle Scholar
Abraham, F. F. and Nelson, D. R., Science 249, 393–397, (1990).CrossRefGoogle Scholar
Abraham, F. F. and Nelson, D. R., J. Phys. France 51, 2653–2672, (1990).CrossRefGoogle Scholar
Gourier, C., Daillant, J., Braslau, A., Alba, M., Quinn, K., Luzet, D., Blot, C., Chatenay, D., Grübel, G., Legrand, J.-F. and Vignaud, G., Phys. Rev. Lett. 78, 3157–3160, (1997).CrossRefGoogle Scholar
Schmidt, C., Svoboda, K., Lei, N., Petsche, I., Berman, L., Safinya, C. and Grest, G., Science 259, 952–955, (1993).CrossRefGoogle Scholar
Fasolino, A., Los, J. and Katsnelson, M. I., Nature materials 6, 858–861, (2007).CrossRefGoogle Scholar
Katsnelson, M. and Geim, A., Philos. Trans. R. Soc. A 366, 195–204, (2008).CrossRefGoogle Scholar
Kim, E.-A. and Neto, A. H. C., EPL (Europhysics Letters) 84, 57007, (2008).CrossRefGoogle Scholar
Mariani, E. and von Oppen, F., Phys. Rev. Lett. 100, 076801, (2008).CrossRefGoogle Scholar
Gazit, D., Phys. Rev. B 80, 161406, (2009).CrossRefGoogle Scholar
Efetov, D. K. and Kim, P., Phys. Rev. Lett. 105, 256805, (2010).CrossRefGoogle Scholar
Mariani, E. and von Oppen, F., Phys. Rev. B 82, 195403, (2010).CrossRefGoogle Scholar
Castro, E. V., Ochoa, H., Katsnelson, M. I., Gorbachev, R. V., Elias, D. C., Novoselov, K. S., Geim, A. K. and Guinea, F., Phys. Rev. Lett. 105, 266601, (2010).CrossRefGoogle Scholar
San-Jose, P., González, J. and Guinea, F., Phys. Rev. Lett. 106, 045502, (2011).CrossRefGoogle Scholar
Gornyi, I. V., Kachorovskii, V. Y. and Mirlin, A. D., Phys. Rev. B 86, 165413, (2012).CrossRefGoogle Scholar
Gibertini, M., Tomadin, A., Guinea, F., Katsnelson, M. I. and Polini, M., Phys. Rev. B 85, 201405, (2012).CrossRefGoogle Scholar
Amorim, B. and Guinea, F., Phys. Rev. B 88, 115418, (2013).CrossRefGoogle Scholar
Meyer, J., Geim, A., Katsnelson, M., Novoselov, K., Booth, T. and Roth, S., Nature 446, 60, (2007).CrossRefGoogle Scholar
Brivio, J., Alexander, D. T. L. and Kis, A., Nano Lett. 11, 5148–5153, (2011).CrossRefGoogle Scholar
López-Polín, G., Gómez-Navarro, C., Parente, V., Guinea, F., Katsnelson, M. I., Pérez-Murano, F. and Gómez-Herrero, J., ArXiv e-prints 1406.2131 (2014).Google Scholar
Roldán, R., Fasolino, A., Zakharchenko, K. V. and Katsnelson, M. I., Phys. Rev. B 83, 174104, (2011).CrossRefGoogle Scholar
Paczuski, M., Kardar, M. and Nelson, D. R., Phys. Rev. Lett. 60, 2638–2640, (1988).CrossRefGoogle Scholar
Guitter, E., David, F., Leibler, S. and Peliti, L., Phys. Rev. Lett. 61, 2949–2952, (1988).CrossRefGoogle Scholar
Guitter, E., David, F., Leibler, S. and Peliti, L., J. Phys. France 50, 1787–1819, (1989).CrossRefGoogle Scholar
Paczuski, M. and Kardar, M., Phys. Rev. A 39, 6086–6089, (1989).CrossRefGoogle Scholar
Le Doussal, P. and Radzihovsky, L., Phys. Rev. Lett. 69, 1209–1212, (1992).CrossRefGoogle Scholar
Bowick, Mark J., Catterall, Simon M., Falcioni, Marco, Thorleifsson, Gudmar and Anagnostopoulos, Konstantinos N., J. Phys. I France 6, 1321–1345, (1996).CrossRefGoogle Scholar
Gazit, D., Phys. Rev. E 80, 041117, (2009).CrossRefGoogle Scholar
Zakharchenko, K. V., Katsnelson, M. I. and Fasolino, A., Phys. Rev. Lett. 102, 046808, (2009).CrossRefGoogle Scholar
Zakharchenko, K. V., Roldán, R., Fasolino, A. and Katsnelson, M. I., Phys. Rev. B 82, 125435, (2010).CrossRefGoogle Scholar
Costamagna, S. and Dobry, A., Phys. Rev. B 83, 233401, (2011).CrossRefGoogle Scholar
Hasselmann, N. and Braghin, F. L., Phys. Rev. E 83, 031137, (2011).CrossRefGoogle Scholar
Lebedev, V. V. and Kats, E. I., Phys. Rev. B 85, 045416, (2012).CrossRefGoogle Scholar
Košmrlj, A. and Nelson, D. R., Phys. Rev. E 88, 012136, (2013).CrossRefGoogle Scholar
Essafi, K., Kownacki, J.-P. and Mouhanna, D., Phys. Rev. E 89, 042101, (2014).CrossRefGoogle Scholar
Košmrlj, A. and Nelson, D. R., Phys. Rev. E 89, 022126, (2014).CrossRefGoogle Scholar
de Andres, P. L., Guinea, F. and Katsnelson, M. I., Phys. Rev. B 86, 144103, (2012).CrossRefGoogle Scholar
Popov, V. N., Phys. Rev. B 66, 153408, (2002).CrossRefGoogle Scholar
Mounet, N. and Marzari, N., Phys. Rev. B 71, 205214, (2005).CrossRefGoogle Scholar
Zimmermann, J., Pavone, P. and Cuniberti, G., Phys. Rev. B 78, 045410, (2008).CrossRefGoogle Scholar
Amorim, B., Roldán, R., Cappelluti, E., Fasolino, A., Guinea, F. and Katsnelson, M. I., Phys. Rev. B 89, 224307, (2014).CrossRefGoogle Scholar
Nelson, D., in Statistical Mechanics of Membranes and Surfaces, edited by Nelson, D., Piran, T. and Weinberg, S. (World Scientific, Singapore, 2004), chap. 6, pp. 131–148.CrossRefGoogle Scholar
Landau, L. and Lifshitz, E., edited by, Course of Theoretical Physics vol. 7: "Theory of Elasticity" , (Pergamon Press, Oxford, 1959), chap. 11, pp. 44–62.Google Scholar
Chaikin, P. M. and Lubetsky, T. C., Principles of condenced matter physics, (Cambridge University Press, Cambridge, 2003), chap. 10, pp. 630–635.Google Scholar
We use grapheme as an example of a crystalline membrane. Typical parameters for the single-layer grapheme at T=0 are taken (see Refs. 17 and 39) μ=9.44 eV Å-2, λ=3.25 eV Å-2 and κ=0.82 eV. At T=300 K, we used the values: μ=9.95 eV Å-2, λ=2.57 eV Å-2 and κ=1.1 eV. Graphene has a density ρ/ħ2 =1104 eV-1Å-2 and its lattice constant is given by a=2.64 Å, from which we obtain a Debye momentum $q_D = \sqrt {8\pi /\left( {3^{1/2} a^2 } \right)} = 1.55$ Å-1 .Google Scholar
Ochoa, H., Castro, E. V., Katsnelson, M. I. and Guinea, F., Phys. Rev. B 83, 235416, (2011).CrossRefGoogle Scholar
Galitskii, V. and Migdal, A., Zh. Eksp. Teor. Fiz. 34, 139, (1958).Google Scholar
Koltun, D. S., Phys. Rev. C 9, 484–497, (1974).CrossRefGoogle Scholar