It is a fairly common in for cryogenic researchers to wish to measure a capacitance inside a low temperature environment. Unlike many electrical properties of materials capacitance is unaffected by the temperature of the environment or the magnetic field, meaning that it becomes a useful property to measure.
This practical guide is focussed on explaining how to correctly set up a coaxial feed-through into a cryogenic environment. Coax cables present several challenges to those who are not familiar with their use. If used incorrectly they can conduct unmanageable thermal loads into your cryostat and coax designed to avoid this is typically highly resistive. Most coax is pretty inflexible, especially if flexed at cryogenic temperatures and the additional grounded braid means that particular care must be taken to avoid ground loops.
After running a coax into a cryostat, one of the first issues encountered is that the cooling power of the system is reduced. The fridge will not get to the same base temperature and will take longer to cool in general. This is usually the predictable of result of the coax core being both electrically isolated and pretty well thermally isolated from the outer and consequently being close to ambient temperature even deep inside the cryostat. This effect can be calculated from the cross-section and conductivity integrals as follows;
From Fourier’s law of conduction
Where qcond is the steady state heat conduction, U is the conductance and is the temperature difference in the direction of heat conduction. This is analogous to the current flowing being equal to the conductivity multiplied by the potential difference. U is dependent on the thermal conductivity of the material, λ, the area through which and the length along that the heat is conducted (A and L respectively). Inconveniently, in many cases λ is itself temperature dependent (especially so at low temperature). In those scenarios the following equation is easily derived by considering the material to consist of infinitesimally thin layers each with their own temperature and value of λ;
Because the function that defines λ must be empirically measured, for most practical purposes it is much more convenient to define the above integral as the difference of two integrals from 4K to T which can be looked up from an empirically derived table of values.
Before fitting a cable into a cryostat it is necessary to calculate the heat load that you will introduce. Let us consider the fitting of 1 metre of Ultra Miniature Cryogenic Coax C supplied by Lake Shore Cryotronics. The core wire is 0.203 mm diameter copper and the outer consists of aluminised polyester layer incorporating a second 0.203 mm diameter copper drain wire. Because the majority of the heat will be conducted through the two copper wires, a good approximation for the heat conducted into the cryostat can be considered by assuming that the cabling is solid copper with a cross-section equal to the combined cross section of the core and drain wires. Plugging in the numbers gives 10.5 mW thermal load, for 1 meter of Lake Shore C miniature cryogenic coax table one end of which is held at 300K and the other at 4K.
10.5 mW is tolerable for most cryostats operating from 1K upwards. Some experimental set-ups will not have the spare capacity to soak up this thermal load, so steps will have to be taken to reduce it. If only small currents need be carried by the coax, then a higher resistance stainless steel coax may be used, which will also reduce the conducted heat by up to a factor of 50 but will have approximately a factor of 8 times higher resistance than the copper wire.
A second useful strategy is to “thermally anchor” the wire so that it makes good thermal contact to a cold plate within the cryostat before it extends the final stretch into the coldest part of the cryostat. This won’t decrease the thermal load that the coax puts on the cryostat but will mean the most of the thermal load is taken where there is more cooling power and the sample will be least affected. This works particularly well with cryogen free systems, when the cable can be anchored to the cryocooler’s intermediate stage, and sub-kelvin systems where the cable can be anchored to the 1K pot, helium bath, or cryocooler. In order to make sure the coax is well thermally anchored, it usually necessary to have a length of cable typically 50-100 times the cable diameter in contact with cold metal. One way to do this is to wind a length of cable onto a copper bobbin, then coat it with potting epoxy or varnish. By winding half one way and half the other, one can minimise inductance and noise pick-up.
In a metal, there is a fundamental relationship between the heat conduction at cryogenic temperature and the electrical conduction – the Wiedemann Franz law. Crudely this is because the phonons are frozen out at low temperature and the only heat carriers left are electrons, which are also the charge carriers. Consequently, there is often a trade-off between allowing too much heat flow into the cryostat and using cabling with prohibitively high impedance. Usually for a particular experiment and cryostat, you can have a ‘budget’ for heat flow permitted and electrical resistance tolerated and can select cabling accordingly. Where only extremely low heat loads are permitted, but the cryostat always stays below 8-9K, it may be possible to use superconducting coax which has an extremely low thermal conductivity and negligible resistance.
Lack of flexibility
Because of its cross-sectional structure, coaxial cable is typically damaged if forced into bending around too tight a corner. All cable is different and you should check the manufacturer’s recommendation, but it’s usually possible to bend the cable into a curve with a radius a few times the cable diameter if it’s only done once, or somewhat larger if flexed repeatedly.
If the bending occurs at cryogenic temperature the problem is particularly acute. Like most other cabling the insulation on cryogenic coax is more brittle at lower temperature and can crack or break off if flexed while cold.
A common problem in complex measurement systems is that that of ground loops, especially so with coax as each cable will have a grounded outer which makes wiring to ground without creating a ground loop all the more complicated. Ground loops occur when something in the circuit is connected to ground via more than one route. Because these routes will have non-zero resistances, circulating currents caused by electromagnetic interference can introduce variations in voltage in the ‘ground’. From there it can couple into the measurement through the parasitic capacitance between the shield and the thing it is shielding. Usually most of this noise will be at the AC mains supply frequency (50 Hz in the much of the world but 60 Hz in the Americas and Asia), but it’s not unusual to see local TV/radio frequencies or mobile phone transmissions. In the low noise environment of a research cryostat you might pick up something from your other measurements or thermometry which will make grounds loops harder to identify. The knack to avoiding ground loops with coax cable is to attach the braid to ground at only one end of the cable. For capacitance measurements, it is usually best to have the braid connected to the capacitance bridge or LCR meter, and nothing else. This means that the body of connectors should be connected to the braid of the cable, but not to the cryostat. Both the cryostat and the capacitance bridge should be connected to a safety ground – especially if the cryostat contains a superconducting magnet.