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Edge Effects on the Intrinsic Loss Mechanisms in Graphene Nanoresonators

Harold S. Park's picture

Graphene has recently become one of the most studied materials in the world, mainly due to its unique 2D crystal structure and its exceptional electrical and mechanical properties.  One of the most exciting applications for graphene is as a basic building block for NEMS.  In particular, due to its extremely small mass and exceptional stiffness, it is being investigated extensively as an ultra-sensitive nanoscale element for sensing incredibly small amounts of mass, force, pressure, etc.  Recently, Bunch et al. (2007) (http://www.sciencemag.org/cgi/content/abstract/315/5811/490) suspended graphene and measured its quality (Q)-factor, which was found to be extremely low; low Q-factors are critical as they place fundamental limits as to the mass, force and pressure sensitivity of the graphene NEMS.

More recently, we have studied, using classical molecular dynamics, possible intrinsic (extrinsic effects such as gas damping and clamping losses were neglected) loss mechanisms that may be responsible for the experimentally observed low Q-factors.  In doing so, we have found interesting results, recently published in Nano Letters (http://pubs.acs.org/doi/abs/10.1021/nl802853e), which suggest that free edge effects (analogous to surface effects in nanomaterials, and thus surface losses in nanowire-based NEMS) dominate the intrinsic losses, and thus the Q-factors of graphene nanoresonators.  

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Rui Huang's picture

Dear  Harold,

I very much enjoyed reading your paper, but do have many questions in mind. First and foremost, I am trying to understand what you called the intrinsic loss mechanism or intrinsic energy dissipation.

As I learned from elementary vibtation theory, there should be no intrinsic energy loss in an elastic system (be it linear or nonlinear). In other words, under the condition of free vibration, the total energy (internal strain enegry + kinetic energy) in an elastic system would remain constant, thus having an infinitely large quality factor Q. Of course, in a real resonator, there is always energy loss, due to various inelastic mechanisms, some of which are intrinsic (e.g., internal lattice friction, viscoelastic) and some others are extrinsic (e.g., boundary clamping, air damping).

With such a background understanding, I find it difficult to accept the edge effect as an intrinsic loss mechanism. If I understand it correctly, during your MD simulations, the graphene remains an elastic system. Then, the question is, if there is energy loss under the free vibration condition, where did the energy go? As you shown in Fig. 2, the total energy remains constant, but both the kinetic energy and external energy decrease as you stated in the text (not obvious in the figure though). Also I have to admit that I may have misunderstood what you mean by "external energy". In any case, how can both the energy components decrease while the sum of them remains constant?

I hope to discuss with you in more depth, but better to keep it simple to begin with. Thank you.

RH

Harold S. Park's picture

Dear Rui:

Thanks for reading the paper over, and I appreciate your question, which I would like to answer as follows.  For the purposes of discussion, I'd like to refer you to the following papers by Chu et al (http://arxiv.org/abs/0705.0015), and Jiang et al (http://prola.aps.org/abstract/PRL/v93/i18/e185501).  In particular, let's consider the Chu paper first.  There, they consider intrinsic loss mechanisms in silicon nanowires, with a particular emphasis on how surface effects, and the breaks in geometric symmetry they induce (see Figure 2 of their paper), result in intrinsic energy loss.  In their article, energy is "lost" (note that they studied the free oscillation of their nanowires in an NVE ensemble, just as in our work) in the sense that due to the loss of oscillation symmetry caused by the surfaces, energy in the main vibrational modes is transferred to other vibrational modes.  In their paper, this is shown through the center of mass plot in Figure 1.  There, initially all of the energy is in the induced mode of vibration, or in the x-direction.  However, the surface effects cause energy to transfer to the other modes of oscillation, where the y and z directions are studied in the Chu work.  You can see the energy transfer that occurs over time, which is where the "loss" comes from. 

The way I like to think about it is that if the oscillation were perfect, and there were no intrinsic losses, the x-direction center of mass in Figure 1 in the Chu work would remain constant over time; however, it does not.  Instead, the energy in the main initial mode (the plucking in the x-direction) gets transferred to all the other vibrational modes in the nanowire that get activated by the surface-induced break in symmetry. Because these other (minor) vibrational modes "eat up" energy, i.e. take energy away from the main mode of vibration, what the center of mass plot (Figure 1) in the Chu work, and also the external energy plots in my paper represent is most simply the coherence (or loss of coherence) of the oscillation of the main/induced mode of vibration.

You can see this by looking at Figure 1 of my paper - there, with periodic boundary conditions, the graphene has no edge effects, and thus there is nothing that breaks the natural oscillation of the monolayer, and thus the external energy remains constant over time.  However, in Figure 2, there are no PBCs, and so the edge modes quickly interact with and pollute the induced (main) mode of oscillation.  Because then the energy in the induced (main) mode is "taken away", i.e. continually transferred into other, non-coherent modes of vibration, the main mode "loses energy", and thus the external energy decreases as a function of time.

I've enjoyed reading your works on graphene elastic properties as well, and look forward to continued discussion.

Regards,

 

Harold

Rui Huang's picture

Dear Harold,

Thank you for answering my question. Although I have not read the other two papers you mentioned, I can understand coupling of a "main" vibration mode with other modes possibly induced by surfaces or edges as a result of symmetry breaking. In fact, mode coupling occurs essentially in all real resonators. For example, a somewhat similar edge mode was noted by Mindlin (1960) in vibrations of a circular elastic disk. A more relevant question regarding your paper may be: What is the external energy? Also, what is the "main" vibration mode for the graphene?

Thanks again for discussing with me.

Best regards, 

RH

Harold S. Park's picture

Hi Rui:

In the simulation, before the free oscillation occurs, the graphene monolayer is equilibrated at a specified temperature within an NVT ensemble.  Then, the oscillations are induced by applying a specified velocity field to the monolayer.  At this point, before the graphene begins to oscillate, the potential energy of the system has a certain value, which we scale to an external energy value of 0, as in Figure 1(c) of my paper.  Once the graphene begins oscillating, then the external energy increases and decreases about 0, as in Figure 1(c) of my paper.

The main vibrational mode is specified by whatever velocity profile is applied.  In our case, we applied a sinusoidal velocity profile between the two long edges of our graphene monolayer to approximate the first harmonic mode of oscillation of the graphene sheet.

Regards,

 

Harold

 

 

Rui Huang's picture

Dear Harold,

According to your response, the "external energy" is calculated from the "internal" potential energy, with some scaling to make it zero before oscillation. Am I right? It sounds a little strange to me calling this "external energy". In static atomistic modeling (e.g., molecular mechanics), it would be called "strain energy".

Now back to the discussion of "intrinsic loss". When calculating the energy (both potential and kinetic energy), did you separate the edge mode from the main mode? If so, how did you do it? If not, there would be no energy loss, I think. Your arguement that the edge modes "eat up" the energy from the main mode would be justified if the energies (both kinetic and "external") you plotted are for the main mode only. I am curious how you separated the modes in the energy calculation.

Thanks.

RH

Harold S. Park's picture

Hi Rui:

The term external energy came from the paper of Jiang et al (see my first response); I decided not to change the terminology, though I admit that another term might be more precise.  And yes, you are right - it's computed from the potential energy.

About your second question - Figures 2 and 3 in my paper are the total external energy of the monolayer - the modes are not separated out.  The energy is "lost" because the other modes do not allow the monolayer to oscillate coherently and with the same amplitude as the main mode;because the main mode is the major contributor to the potential energy, and because the energy is removed from the main mode, the energy is "lost".

Regards,

 

Harold

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