The past semester, I have been teaching in the physics third year lab. One of the experiments consists in emasuring the electrical resistivity of a graphene layer with a lock-in amplifier.

Before this lab, to measure a resistence, students are used to use a d.c. potentiometer to measure an electrical resistance, so I could notice some qualm about the new technique. The d.c. potentiometer is an easy system for such applications, but prone to measurement errors because of thermoelectric and drift effects that can be hard to eliminate, as well as issues when the measured signal falls below the error voltage of the instrument. For this reason, methods involvig the use of an a.c. current allow for more precise electrical measurements. In fact, if one applies an a.c. current and measure the differential output voltage, between successive maxima and minima (or, more conveniently, the root mean square of this value, $V_{rms}$) and interval this over a number of modulation cicles, one can get an improved estimate of the voltage difference, that eliminates the thermal drifts and low-frequency noise.

I think this method is of particular importance in modern mesoscopic physics, and justifies a few words on this blog.

## The lock-in method.

The basic experimental arrangement in lock-in detection in a four-probe configuration consists in the following: a sine-wave current is generated at a frequency $f_0$ and constant amplitude and passed through a pair of contacts (A and B for example) of the measured device, and a voltage drop across another pair of contacts (C and D) is measured with the lock-in amplifier. This acts as a highly-tuned demodulator, and gives a d.c. output that is proportional to the amplitude of the a.c. voltage $V$ between C and D. The essential feature of the lock-in amplifier is that it works at the reference signal of frequency $f_0$, that has the same frequency as the sample. This methods allows to measure a current and a voltage with a precision of about 1 part in 104.

### Importance of phase-sensitive detection

As I just wrote, a lock-in is a phase-sensitive detector, measuring the difference voltage of interest, using a synchronous reference voltage. Detection with respect to a synchronous reference allows to use long averaging time to be able to improve the detection below background noise level. This is important in comparison to the use of amplitude demodulation with non-linear devices (like enveloppe detectors), that would make no difference between the signal and noise components. The phase-sensitive detector would measure only the frequency of interest $f_0$. Lock-in amplifiers are usually able to measure both the amplitude of the voltage, and the phase difference.

## Noise reduction.

In a typical measurement, the resistance of metallic samples can vary from 10-3Ω at room temperature, to 10-6Ω at low temperatures. In the Kelvin-range, a few millivolts can heat a device substantially and counterbalance all the efforts made to reach low-temperatures. Under such conditions, the maximum range that can decently be used is about 10µV, so the measured voltage drop would be in the order of the nanovolt. In a typical experiment, leads from room temperature to liquid-helium temperatures would carry a thermoelectric voltage (noise) in the order of the µV to the mV. It is thus necessary to be able to measure below the noise level. Synchronous detection allows to single out the component of the signal at a specific reference frequency and noise, therefore reducing the measured noise.

There are three main types of noise: Johnson (or thermal) noise, shot noise and 1/f noise.

### Johnson noise

Every resistive system would generate a noise voltage at its therminal, that is due to the thermal fluctuations of the electron density. We can characterise the noise as: $V_{JN} = \sqrt{4k_B T R \Delta f}$ where $k_B$ is Boltzmann’s constant, $T$ the temperature of the sample of interest, $R$ its resistance and $\Delta f$ the bandwith of the measurement. This means that the amount of noise that is captured by the lock-in is determined directly by the bandwidth of the input signal. State of the art lock-ins would have a bandwidth frequency around 500kHz, that means an effective noise at 300K, at the amplifier input that is $V_{JN} \approx 40\sqrt{R}$nV. Here, high input bandwidth allow to reduce Johnson noise.

### Shot noise

Shot noise comes from the discrete nature of the charge carriers. As there is always non-uniformities in the electron flow generating current noise, the shot noise would be given by: $I_{SN} = \sqrt{2qI_{a.c.}\Delta f}$ where $q$ is the electron charge, $I_{a.c.}$ the current and $\Delta f$ the bandwidth. With a lock-in, this is minimal (that is $I_{SN} \approx 400\sqrt{I_{a.c.}}$10-9).

### 1/f noise (or Flicker noise)

Flicker noise results fluctuations in resistance due to a current flowing through the resistor. It can be between 10nV and 1µV per volt applied across a resistor, with a 1/f spectrum. As a result, measurements at low frequencies are noisier.

• Betts, J.A, Signal processing, modulation and noise, London 1970
• Meade, ML, Lock-in amplifiers: principles and applications, London 2013
• Principles of lock-in detection and the state of the art, Zurich Instruments white paper, 2016.