Experimental techniques: bridge measurements

When we teach undergraduates about materials and measuring electrical resistance, we tend to gloss over the fact that there are specialized techniques for this - it's more than just hooking up a battery and an ammeter.  If you want to get high precision results, such as measuring the magnetoresistance \(\Delta R(B)\), where \(B\) is a magnetic field, to a part in \(10^{5}\) or better, more sophisticated tools are needed.  Bridge techniques compose a class of these, where instead of, say, measuring the voltage drop across a sample with a known current, instead you measure the difference between that voltage drop and the voltage drop across a known reference resistor.   

Why is this good?  Well, imagine that your sample resistance is something like 1 kOhm, and you want to look for changes in that resistance on the order of 10 milliOhms.  Often we need to use relatively low currents because in condensed matter physics we are doing low temperature measurements and don't want to heat up the sample.  If you used 1 microAmp of current, then the voltage drop across the sample would be about 1 mV and the changes you're looking for would be 10 nV, which is very tough to measure on top of a 1 mV background.  If you had a circuit where you were able to subtract off that 1 mV and only look at the changes, this is much more do-able.

Wheatstone bridge, from wikipedia

Sometimes in undergrad circuits, we teach the Wheatstone bridge, shown at right.  The idea is, you dial around the variable resistor \(R_{2}\) until the voltage \(V_{G} = 0\).  When the bridge is balanced like this, that means that \(R_{2}/R_{1} = R_{x}/R_{3}\), where \(R_{x}\) is the sample you care about and \(R_{1}\) and \(R_{3}\) are reference resistors that you know.  Now you can turn up the sensitivity of your voltage measurement to be very high, since you're looking at deviations away from \(V_{G} = 0\).   

You can do better in sensitivity by using an AC voltage source instead of the battery shown, and then use a lock-in amplifier for the voltage detection across the bridge.  That helps avoid some slow, drift-like confounding effects or thermoelectric voltages. 

Less well-known:  Often in condensed matter and nanoscale physics, the contact resistances where the measurement leads are attached aren't negligible.  If we are fortunate we can set up a four-terminal measurement that mitigates this concern, so that our the voltage measured on the sample is ideally not influenced by the contacts where current is injected or collected.  

A Kelvin bridge, from wikipedia

Is there a way to do a four-terminal bridge measurement?  Yes, it's called a Kelvin bridge, shown at right in its DC version.  When done properly, you can use variable resistors to null out the contact resistances.  This was originally developed back in the late 19th/early 20th century to measure resistances smaller than an Ohm or so (and so even small contact resistances can be relevant).  In many solid state systems, e.g., 2D materials, contact resistances can be considerably larger, so this comes in handy even for larger sample resistances.  

There are also capacitance bridges and inductance bridges - see here for something of an overview.  A big chunk of my PhD involved capacitance bridge measurements to look at changes in the dielectric response with \(10^{-7}\) levels of sensitivity.

One funny story to leave you:  When I was trying to understand all about the Kelvin bridge while I was a postdoc, I grabbed a book out of the Bell Labs library about AC bridge techniques that went back to the 1920s.  The author kept mentioning something cautionary about looking out for "the head effect".  I had no idea what this was; the author was English, and I wondered whether this was some British/American language issue, like how we talk about electrical "ground" in the US, but in the UK they say "earth".  Eventually I realized what this was really about.  Back before lock-ins and other high sensitivity AC voltmeters were readily available, it was common to run an AC bridge at a frequency of something like 1 kHz, and to use a pair of headphones as the detector.  The human ear is very sensitive, so you could listen to the headphones and balance the bridge until you couldn't hear the 1 kHz tone anymore (meaning the AC \(V_{G}\) signal on the bridge was very small).  The "head effect" is when you haven't designed your bridge correctly, so that the impedance of your body screws up the balance of the bridge when you put the headphones on.  The "head effect" = bridge imbalance because of the capacitance or inductance of your head.  See here.