This blog post accompanies the webinar "Interfaces: How to make or break a nanodevice". In this webinar, we discussed the role of interfaces for nanodevices: why they are important, how they can destroy the properties of a nanodevice, and how you can characterize interfaces using different measurement techniques. The recording is available here on our YouTube channel.
Thank you to everybody who joined the live event and participated actively by asking many relevant questions. The answers to all questions are provided in the summary of the webinar or at the end of this blog post.
Interfaces and how to characterize them using lock-in amplifiers
In the first part of the webinar, we looked into several types of interfaces related to metallic, semiconducting, and superconducting materials. Interfaces are needed to make contacts to a device, to achieve control over the device properties, e.g. by electrostatic gating, and to introduce functionality, e.g. by induced superconductivity, or spin injection. As the dimensions of a device decrease, interfaces become more and more important. Interface roughness or individual charge fluctuators can strongly influence the device properties. An interesting example is the interface of semiconductor-superconductor hybrid devices, where it has been shown that the hardness of the induced superconducting gap can be significantly increased if the superconductor is grown in-situ on a pristine semiconductor surface, as opposed to evaporation of the superconductor after the semiconductor has been exposed to air .
When characterizing nanodevices and interfaces, there are several important considerations. First, you need to choose the right measurement environment. For this, the energy scale of the device properties needs to be compared to the temperature and the measurement voltage and frequency. In the table below you can see approximate values for different energy scales for room temperature, 1 K and 10 mK.
|290 K||5.8 THz||25 mV|
|1 K||20 GHz||100 uV|
|10 mK||200 Mhz||1 uV|
The second consideration is the right measurement equipment. In the webinar, we introduced the basic working principle of a lock-in amplifier, which allows to achieve a high signal-to-noise (SNR) by modulating the signal with a precise frequency and measuring the signal at precisely the same frequency. By choosing a modulation frequency that is outside of the 1/f noise and far away from external noise sources, the noise of the measurement can be reduced significantly. For more information, please refer to our whitepaper Principles of Lock-in Detection and the related video.
The Zurich Instrument MFLI is optimized for low-frequency measurements up to 5 MHz, with a very low input noise of 2.5 nV/√Hz. It offers a current input channel and a differential voltage input channel, as well as a differential voltage output channel, and several trigger and auxiliary channels. The two input channels can be used simultaneously, for example for 4-point measurements when contact resistances would otherwise be a limiting factor. During the webinar, a question was asked about the maximum number of channels which is available. Using the MF-MD Multi-demodulator option, it is possible to do up to 4 lock-in measurements simultaneously. This can be used to measure current and voltage simultaneously, but also to characterize the same input signal at up to 4 different frequencies. This can be useful to measure at several harmonics, e.g. for 3-omega measurements, or to get the first derivative dI/dV and the second derivative d2I/dV2 simultaneously. Additionally, the MFLI even allows performing measurements at DC.
In the webinar. we gave detailed examples of how to characterize different material interfaces and the expected measurement results in terms of the current or the conductance. It is interesting to note, that for a superconductor-normal metal interface, the conductance as a function of applied bias voltage strongly depends on the transparency of the interface. This has been predicted theoretically  and was also confirmed experimentally .
Characterization of QD devices
In the second part of the webinar, we focused on quantum dots (QDs) and their characterization. Quantum dots are small islands where charge carriers are confined. This leads to Coulomb blockade, resulting in the characteristic Coulomb diamond structure when measuring transport through the device as a function of an external gate voltage and the source-drain bias voltage.
Fig 1: Schematic illustration of a QD device and the characteristic Coulomb diamond stability diagram when measuring the conductance as a function of gate voltage and source-drain bias voltage.
Interfaces play a crucial role in the fabrication and the properties of a quantum dot device. To make a QD device, one typically starts with a low-dimensional material, such as a 2-dimensional electron or hole gas, or a 1-dimensional nanowire. The charge carriers can then be confined to a 0-dimensional island by additional potential barriers. Often this is done by electrostatic gates, where the interface between the gate and the device material is important to achieve a well-defined confinement potential. Alternatively, it is possible to create QDs by controlled changes in the material composition or the crystal structure in a nanowire. However, it is important that the changes are well-defined, whereas an uncontrolled occurrence of interfaces in a nanowire can lead to trapped charges which cannot be controlled individually, and thereby result in noise in the measurement.
Low-frequency transport measurements
Different measurement techniques can be used to characterize QD devices. These include DC measurements, low-frequency lock-in measurements, and RF reflectometry. The MFLI Lock-in Amplifier which was introduced in the first part of the webinar is an excellent tool for low-frequency transport measurements. In the recording of the webinar, you can see a demo of how to set up the measurement using the LabOne User Interface. The demo was recorded during a real measurement using the MFLI and a MOSFET as a device under test. The connections to the MFLI are shown in Figure 2.
Setup for a low-frequency lock-in measurement using the MFLI lock-in amplifier.
In the demo, we also highlighted the importance of optimizing the settings of the low-pass filter. One of the questions during the Q&A was whether there is a rule-of-thumb for the low-pass filter bandwidth. As a starting point, you can use a low-pass filter bandwidth which is 1/10 of the modulation frequency. However, the best settings depend on the requirements in terms of SNR and measurement speed, and the noise floor of your experiment. The filter order is the second parameter that can be adjusted to achieve the rejection of noise. An ideal way to optimize the low-pass filter setting is to record time traces of the measurements, for example using the built-in Plotter tool of the LabOne User Interface shown below. Here, a modulation frequency of 314 Hz was used. With a low-pass filter bandwidth of 100 Hz (right side), you can see the 2-omega component of the lock-in measurement which is not filtered out correctly. In this case, reducing the filter bandwidth to 10 Hz (left side) leads to a very precise measurement result.
Fig 2: Screenshot of the Plotter tool in the LabOne User Interface, showing the measured amplitude R over a certain time interval. A signal with a frequency of 314 Hz is measured with a low-passfilter bandwidth of 10 Hz (left side) and 100Hz (right side). The fluctuation of the measurement signal when using a high low-pass filter bandwidth is related to the 2-omega component of the lock-in measurement.
RF reflectometry measurements
The optimization of the low-pass filter settings brings up the topic of measurement speed. For transport measurements, frequencies below 1 kHz are used in order to avoid the reflection of the signal. This requires a rather low low-pass filter bandwidth, which translates to a large time constant. The same consideration applies to DC measurements, where averaging is required to achieve a high SNR. Typically, every data point takes around 100 ms. A map with 100 x 100 pixels, such as a Coulomb diamond stability diagram, therefore, takes around 15 min.
The properties of QD devices typically depend on many parameters, such as several gate voltages or magnetic fields in different directions. Hence, characterizing a QD device from all angles requires many different measurements. To achieve this in a reasonable amount of time, RF reflectometry measurements are an attractive alternative. For this, an RF signal is applied to the device and the reflected amplitude is measured. The principle relies on the fact that an RF signal is partly reflected at an interface where the impedance changes. The reflected amplitude and phase are directly related to the impedance change at the interface. For a QD, the impedance of the device changes when the state of the quantum dot changes, i.e. when tunneling of a charge carrier onto the device is possible or not. Hence, measuring the reflected amplitude and phase of an RF signal enables mapping out the properties of a quantum dot device as a function of gate voltage and source-drain bias.
Typical frequencies for RF reflectometry measurements are around 100 MHz. This shows that there are two main advantages of the technique: 1) The signal is far away from any low-frequency noise sources, such as the 1/f noise), and 2) a much larger low-pass filter bandwidth can be used, enabling to record a full Coulomb diamond stability diagram in just a few seconds.
To achieve a measurement that is sensitive to impedance changes of the QD, it is important that the impedance of the device is matched to the impedance of the cables, which is typically 50 Ohm. This is achieved by connecting a tank circuit to the QD device . The simplest version of a tank circuit consists of an inductance and a capacitance as shown in the figure below (see also this blog for more details). However, the impedance of the resonant circuit, its resonance frequency, and its quality factor are important parameters. Therefore, a more sophisticated resonant circuit is usually designed for a real experiment (for a conclusive review of the topic, please see Ref. ).
Schematic of a setup for RF reflectometry measurements. The signal is generated using the Zurich Instruments UHFLI lock-in amplifier and connected to the resonant circuit using a directional coupler. The amplitude and phase of the reflected signal is precisely measured using the signal input of the instrument.
The Zurich Instruments UHFLI Lock-in Amplifier is an excellent instrument for RF reflectometry measurements: It allows to generate signals with a frequency of up to 600 MHz and precisely measure the amplitude and phase of the input signal. The UHFLI also offers additional tools and functionality which is required to characterize a QD device, such as the Arbitrary Waveform Generator Option (UHF-AWG) to provide the source-drain and the gate voltages synchronized with the data acquisition. To start a measurement, one first needs to characterize the properties of the resonant circuit and find its resonance frequency. The built-in Sweeper tool allows to quickly characterize the resonant circuit, by sweeping the frequency and recording the amplitude and phase. One can then choose a fixed frequency and start mapping out the QD device. In the webinar, we showed a demo of how the resonance frequency of 224 MHz of a quartz resonator can be determined with the UHFLI in a reflection measurement in less than one minute. A recording from the Sweeper tool is shown in the figure below.
Demo how to characterize a resonator by a reflection measurement using the Zurich Instruments UHFLI and the LabOne Sweeper tool.
Finally, it is interesting to highlight, that the UHFLI Lock-in Amplifier enables to perform measurements on up to 8 QDs in parallel, using a single signal input and RF line into the fridge. This is achieved using the Multi-Frequency Option (UHF-MF), which allows to generate and demodulate a signal at 8 frequencies simultaneously. By designing a chip with 8 devices, where the resonance frequencies of the individual resonant circuits is lightly different, these devices can be measured in parallel. Such a parallel measurement is of high interest for example for the characterization of QDs for qubits, where many different devices need to be characterized to find the ideal material system, device design and measurement conditions.
Here is a summary of the questions asked during the webinar. Please note that some of the questions are also answered in the summary of the webinar above.
Low-frequency lock-in measurements
1. Is it possible to perform 3-omega measurements with the MFLI? Can the second derivative d2I/dV2 be measured, for example for inelastic co-tunneling spectroscopy?
Yes, these are common applications of the MFLI. Higher harmonics can easily be measured by setting the demodulation frequency to a harmonic of the modulation frequency. It is even possible to measure up to 4 harmonics simultaneously by using the Multi-Demodulator Option (MF-MD), which increases the number of demodulators to four. The frequency of the four demodulators can be set individually, or they can be assigned to a harmonic of a common numerical oscillator. Similarly, measuring d2I/dV2 is possible using the second harmonic of the modulation frequency. Also here it is possible to measure dI/dV and d2I/dV2 simultaneously using the Multi-Demodulator option.
2. How many channels can you measure with the MFLI? How is the common-mode rejection performance of the lock-in amplifiers?
There are two aspects regarding the number of measurement channels: 1) the number of physical input channels, and 2) the number of demodulators or lock-in units. In terms of physical input channels, the MFLI has a current input channel, a differential voltage input channel and two auxiliary input channels. The current and voltage input channels are designed for highly sensitive measurements, providing low input noise and variable input ranges. The auxiliary channels are typically used to measure additional signals, where a slightly lower sensitivity is sufficient. Regarding the number of demodulators, the basic version of the MFLI provides a single demodulator that can be assigned to any of the input channels. Simultaneous measurement of all four channels is possible using the Multi-Demodulator Option (MF-MD). This option offers four demodulators that can be assigned to different input channels or to measure at different frequencies of the same input channel. The common-mode rejection ratio (CMRR) is a specification of a differential amplifier indicating the ability to obtain the difference between two inputs while rejecting the components that do not differ from the signal. A high CMRR is important if the signal of interest is represented by a small fluctuation superimposed on a large offset. The CMRR is defined as CMRR = 20*log(differential gain / common-mode gain) and is frequency dependent. For the MFLI the CMRR is > 60 dB.
3. Is an external voltage or current pre-amplifier required for improving SNR before measurement by the lock-in?
For all Zurich Instruments lock-in amplifiers, the input range can be adjusted to the signal in order to achieve the highest resolution and the highest signal to noise ratio. The minimum range is 1 nA and 1 mV for the current and the voltage input of the MFLI, and 10 mV for the voltage inputs of the UHFLI. If the measurement signal is in this range, it can be precisely measured using the instruments without an external pre-amplifier. For lower signals, an external pre-amplifier is beneficial to achieve the best SNR. In some applications, cryogenic pre-amplifiers which are very close to the device in the fridge can also be of interest.
4. Is it possible to measure the second-order correlation functions using the instruments?
Second-order correlation functions require post-processing of the data which is acquired from the instruments. For this, the measurement data is first acquired with a high resolution using either the LabOne User Interface or the Application Progamming Interfaces (APIs), which are offered for various programming languages including Python, MATLAB and LabView. To capture large datasets of the digitized input signal at the full sampling speed, the Digitizer Options (MF-DIG and UHF-DIG) enable to increase the data storage on the device. An example of how to acquire and post-process the data can be found in this blog about cross-correlation measurements.
RF reflectometry measurements
1. Why is the tank circuit needed? Do you have recommendations for the tank circuit elements? Are there RF measurement techniques that utilize transmission through the device?
The tank circuit is needed to achieve a measurement that is highly sensitive to impedance changes of the device. Typically the device has a high impedance on the order of hundreds of kOhms, which needs to be matched to the impedance of the cables (typically 50 Ohms) to get the highest sensitivity. There are many considerations to be taken into account when designing the tank circuit, including the ideal modulation frequency, quality factor and impedance. Often it is desirable to tune the properties of the tank circuit during the measurement, in which case some of the elements need to be adjustable inside the fridge, as shown in Ref. 6. Depending on the application it is also possible to do RF measurements in transmission. However, for QD devices, which typically have a high impedance, reflectometry measurements are usually the method of choice. A conclusive overview discussing the different measurement geometries (reflection and transmission) as well as the design of the tank circuit can be found in this recent review article .
2. What is the difference in RF reflectometry achieved by a conventional vector network analyzer (VNA) and a lock-in amplifier?
A basic reflectometry measurement can be done both with a lock-in amplifier and with a VNA. However, a lock-in amplifier such as the Zurich Instruments UHFLI typically has a lower input noise, it offers more flexibility in terms of setting the filter settings and input range, resulting in a higher SNR, and it provides additional functionality which is required for more sophisticated experiments. For example, an arbitrary waveform generator (AWG) can be added to the UHFLI with an upgrade option (UHF-AWG). This enables to create additional voltage signals which are precisely synchronized with the RF measurement, which is required for changing the source-drain and gate voltages in a QD measurement. The LabOne software also provides a Data Acquisition Module with enables to record the measurement data synchronized with the voltage sweeps. Additionally, the AWG can be used to create customized waveforms for qubit measurements, and it allows to react on the measurement results, e.g. to play a pulse to re-initialize the qubit. Finally, the UHFLI with the Multi-Frequency Option (UHF-MF) enables parallel measurements of up to 8 devices with a single instrument, which is not possible with a conventional VNA.
3. Can you measure samples with very high impedance such as oxide-based materials?
RF reflectometry is an ideal measurement technique for samples where the impedance changes as a function of an external parameter. While the relative impedance change can be precisely measured, an accurate value of the impedance would require modeling of the device and sophisticated calibration and compensation routines. For applications where an accurate measurement of the impedance is the primary interest, direct measurements of current and voltage are more suitable, which can for example be achieved with the Zurich Instruments MFIA Impedance Analyzer. A more detailed description of dielectric spectroscopy can be found on our application page.
4. What is the bandwidth of the lock-in? What is the fastest measurement speed?
The input bandwidth is 500 kHz / 5 MHz with the MFLI Lock-in Amplifier, and 600 MHz with the UHFLI Lock-in Amplifier. The highest low-pass filter bandwidth is 200 kHz and 5 MHz, respectively (please refer to the product webpages of the MFLI and the UHFLI for the complete specifications). For low-frequency transport measurements with the MFLI, the fastest measurement speed depends on the choice of modulation frequency, the noise floor of the setup, and the experimental requirements in terms of SNR. Typically, a low-pass filter bandwidth below 10 Hz is used to filter out the 2-omega component and get a high SNR, which limited the measurement speed to approximately 100 ms per data point. For RF reflectometry measurements, where frequencies above 100 MHz are typically used, it is in principle possible to use the highest low-pass filter bandwidth of 5 MHz, which would enable to get a data point in less than 100 ns. However, in practice, other components in the setup add additional limitations, such that approximately 1 us per data point is achieved in a real measurement.
 W. Chang et al. Nature Nanotechnology 10, 232 (2015)
 G. E. Blonder et al. Transition from metallic to tunneling regimes in superconducting microconstrictions. Physical Review B, 25, 4516 (1982)
 M. Kjaergaard et al. Nature Communications 7, 12841 (2016)
 N. Ares et al. Sensitive Radio-Frequency Measurements of a Quantum Dot by Tuning to Perfect Impedance Matching. Physical Review Applied 5, 034011 (2016)
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