[Pw_forum] Acoustic sum rule in 2D surfaces

[Lorenzo Paulatto]

On Saturday, 20 May 2017 02:34:03 CEST Charlie Ruffman wrote:
> We typically calculate at least one imaginary mode of vibration (at ~ 34
> cm-1), and were wondering if it was valid to apply a "crystal" acoustic sum
> rule, which removes the imaginary frequency. Also, does anyone have insight
> into whether we should be applying a different type of acoustic sum rule
> than crystal (eg. one-dim or zero-dim)?

Acoustic sum rule sets three translational modes at Gamma to zero energy, this 
is done by ensuring that the total force acting on each atom, when they all 
move, is zero. If you have any other negative frequency ASR won't help.

For one-dim and zero-dim rules are fit for nano-wires and molecules 
respectively, they would give wrong results for 2D systems. In principle there 
is a special sum rule that apply to 2D systems, which guarantees then the 
acoustic phonon branch with an out-of-plane displacement pattern 
(polarization) will go to zero quadratically with the q-vector when you move 
toward Gamma.

The out-of-plane acoustic mode means that the sheet is vibrating out-of-plane 
like a drum skin, in the long wavelength limit  this corresponds to the entire 
plane rotating (it takes a while to visualize this).

In practice this sum rule is not imposed by matdyn, we all use the normal 3D 
sum rule. Since there is always a bit of interaction between periodic images, 
despite the vacuum, the mode will eventually become linear in q very close to 
Gamma. But if you don't sample the BZ with a very fine mesh you won't even 
notice. Also if the sheet is under tension, i.e. if there is a non-zero stress 
component in plane, than the mode won't be quadratic close to Gamma any more, 
you may want to keep this in mind when comparing to experiments.

I have no idea how this specific sum rule should be imposed, I think that maybe 
Nicola Bonini did it for Graphene anharmonic effects (which are especially 
tricky to treat). I do not think it is really necessary in general.

I hope I did not say anything wrong, it's been a while since I worked on this 
stuff.

regards

-- 
Dr. Lorenzo Paulatto
IdR @ IMPMC -- CNRS & Université Paris 6
+33 (0)1 44 275 084 / skype: paulatz
http://www.impmc.upmc.fr/~paulatto/
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