I was concerned about the modes of heat transfer during laser irradiation, specifically the lead-lag characteristics between electronic heat transfer and phonon mode heat transfer.. While searching, I came across Thermo-physical Properties of Materials, by G. Grimvall.  It lists photon and phonon modes of heat transfer. In page 279 there is a mention about  electronic contribution to thermal conductivity. May be this can lead us somewhere, to explain the increased cavity depth in case grounding is removed and conductivity is reduced as a result.

Dear Sushanta, thanks for voting.

In this case, the grounding connection must hard enough to change the conductivity of copper (the depth of the crater varies more than 2 times). This effect would be difficult not to notice. It seems to be another mechanism. Probably this mechanism is manifested by increasing the depth of the weld under electrical fields (I read such publication) and in vacuum conditions. For some reason this effect is most pronounced on copper.

This may be due to the fact that laser plasma is created due to a micro explosion in a thin layer under the surface of the target material. the energy of the incident laser beam is   

I think that you need to consider the charge separation in the expanding plasma. On the role of charges in the laser processing of materials written in the works of S. Klimentov from General Physics Institute. Similarly, in what it was I do not remember. It are avalable at Research Gate.

 I also believe that the charge separation plays an important role in that. But I see this separation on the contact “plasma - liquid metal”. Perhaps, the key role belongs to a Fontan-like system of currents around beam impact zone (Gurgen Askaryan, 1968)

I may have an explaination.  When laser light processing metal specimen, photon "hit"  electron at the surface first. Electron got hited get kinetic energy which drive it away from surface to inner part of the metal specimenor just emit. During which, the running elctron will hit other electron or lattice. The hitting will accrelrate both electron and lattice vibration. that is how energy of laser turn into heat. There are free electron inside metal. so the hitting between electrons play the major role in enery transpot.

The ground (earth) could be considered as a very large negative electrode which will provide infinite amont of (cold, slow moving) electron.

PS： if my theory fit, the smaller specimen is ,the bigger difference  maybe detected.

I hope this wil help.