Does the stacking of peltiers device shows promising results in achieving the cryogenic temperature? I have tried doing the experiment, the overall temperature of the system increased around 80ºC and then the dissipation of that heat was the prime issue? Heatsinks usually fail to dissipate this high amount of heat. Is there any alternative technology to achieve cryogenic temperature electrothermally without producing much amount of heat? Can any modifications in stacking help?
There is no way to get around the physics in this case. The fundamental issue here is that peltier effect refrigerators are individually extremely inefficient, and stacking them in essence "stacks" their inefficiencies.
Why are they inefficient? Because the material they are made from ideally should have zero electrical resistance and infinite thermal resistance at the same time. No known substance can exhibit both of these characteristics, and so we are left with "compromise" material sets which do allow for the device to function, but most of the electrical energy that is put into the system gets wasted as heat.
Does the stacking of peltiers device shows promising results in achieving the cryogenic temperature?
If you consider actually working in devices that are working in satellites to be "promising", then yes.
I have tried doing the experiment, the overall temperature of the system increased around 80ºC and then the dissipation of that heat was the prime issue? Heatsinks usually fail to dissipate this high amount of heat.
Then you aren't thinking your design through properly. You need a multi-stage cooler, and going from the cold end to the hot end, each stage needs to be larger than the last. The ones I saw looked like pagoda roofs, with a small cooler at the cold end, and a cooler with over 100 times the area at the hot end.
It's been way to long for me to remember the specifics, but you need to start with what you want to keep cold, what temperature and heat flux you need to deal with. Then design a stage that takes that heat out most efficiently, and call it your new "thing I want to cool", and use that as input for your next stage. Keep going until you're done.
When I was shown these devices we looked at all the complications, looked at the fact that we were doing something that was earthbound and therefor close to maintenance facilities, and decided "eh, we'll stick with moving parts".