Faculty of Mathematics, Physics
and Informatics
Comenius University Bratislava

Quantum graphene: flat but rough

An article on the structure of quantum graphene was published on 4 April 2018 in Physical Review B within the Rapid Communications section devoted to the publication of short papers presenting highly original and significant material. The paper has been selected by the editors to be an Editors' Suggestion. The authors are Juraj Hašík and Roman Martoňák from the Department of Experimental Physics of the Faculty of Mathematics, Physics and Informatics of the Comenius University in Bratislava and Erio Tosatti from the International School for Advanced Studies, Trieste, Italy.

04. 04. 2018 14.29 hod.
By: Roman Martoňák

Graphene represents the thinnest layer of carbon which is just one atom thick. This fascinating system is in the last decade of substantial interest of theoretical and experimental physicists. The discovery of graphene introduced the new paradigm of 2D materials and was awarded Nobel Prize (A. Geim, K. Novoselov 2010). Due to its interesting electronic properties graphene is considered as possible future material for electronics. It might sound surprising, however, that even the very existence of graphene is interesting as it apparently contradicts the laws of physics. Freely suspended graphene represents a 2D membrane in 3D space which is very different from an ordinary 3D crystal. Thermal motion of atoms at any temperature destroys the flatness of graphene and induces ripples whose size grows with the size of the sample. A simple calculation shows that these ripples are so large that the membrane becomes crumpled and loses its planar character. Fortunately, in this case the simple calculation does not work. More accurate calculations show that the membrane still remains flat. The ripples predicted by the simple calculation are so large that they start interfering with each other and become smaller. Classical physics based on Newton and Boltzmann thereore predicts that the membrane creates ripples but survives the thermal motion. It is never perfectly flat because the ripples would completely disappear only at absolute zero and this temperature cannot be reached. 

Since about hundred years we know, however, that the atoms do not obey the laws of classical physics. Even if we could cool matter down to absolute zero and get rid of the thermal motion, the atoms would not stop moving. Atoms obey the laws of quantum physics and Bohr, Heisenberg and Schrödinger taught us that they always move because of so-called zero-point motion. In a classical world an atom without thermal motion would be localized at precise point and atoms in graphene would form perfect regular hexagons. In the quantum world the atoms shiver and induce ripples on graphene even in complete lack of thermal motion. What do these quantum ripples in graphene look like? Are they smaller or larger than the classical ones and can they be seen or measured? Can they crumple graphene? Surprisingly not much is known about this fundamental property of graphene. A simple calculation in this case suggests that quantum membrane does not crumple and more accurate calculations confirm this conclusion. The true behaviour can be found either from experiment or from numerical quantum simulations. 

Juraj Hašík, student of Theoretical physics at the Faculty of Mathematics, Physics and Informatics studied the problem in his Diploma Thesis (defended in 2015) under the guidance of Prof. Roman Martoňák from the Department of Experimental Physics. In the thesis he implemented, coded and tested the Path Integral Quantum Monte Carlo method for graphene. He further continued this research after moving for PhD Studies to International School of Advanced Studies, Trieste, where he also collaborated with Prof. Erio Tosatti. In extensive numerical simulations performed on the supercomputer Aurel in the Computing Center of the Slovak Academy of Sciences he studied the sample of graphene of roughly square shape and linear size about 110 Angström. He managed to cool it down to the temperature of 0.6 K, about order of magnitude lower compared to that reached in similar simulations in literature. At this temperature the sample fully obeys the laws of quantum mechanics. The quantum simulations confirmed that the simple calculation in this case indeed agrees with the more accurate one and quantum ripples do not crumple graphene. However, quantum ripples are quite different from their classical counterpart. They are weaker on long distances and that's why they don't crumple the membrane. On short distances, however, they are stronger and make the membrane locally more rough. Simply stated, quantum graphene at very low temperature is flatter and at the same time rougher than the classical one. Interestingly, at low temperatures below 50 K the fluctuations of normal angles to graphene representing local deviations from the average plane of the membrane reach about 2 degrees. This is a surprisingly large value and represents a spectacular manifestation of quantum zero-point motion in a macroscopic object. This value could be experimentally measured, e.g. by electron diffraction and we expect that the corresponding experiment might be performed soon. Graphene thus offers us a unique "window" to the microscopic world where one can see and separate thermal and quantum zero-point motion of carbon atoms. 

Juraj Hašík, Erio Tosatti and Roman Martoňák: Quantum and classical ripples in graphene, Physical Review B 97, 140301(R) – Published 4 April 2018