Microwave noise detection of a quantum dot with stub impedance matching

Noise is defined as random fluctuations of a signal in time. The fundamental requirement for noise is some sort of randomness. Noise is well-known and infamous to every experimentalist - whether he is working in the field of electronics, optics, acoustics or anywhere else - since such fluctuations are inherent and unavoidable in many systems.
For most of us, the word noise has a negative connotation. It is considered to be an unwanted disturbance superposed on a useful signal, which tends to obscure the signal's information. The natural reaction to this nuisance is trying to reduce it as much as possible, be it with a longer averaging time or an improved setup. In this respect, the signal-to-noise ratio, which compares the level of the desired signal with the level of the superposed noise, is the relevant quantity. A signal-to-noise level larger than one has to be achieved in order to observe the requested signal. In fact, noise is often a limiting factor in experiments and there are many examples where a reduced noise level led to the revelation of unexpected features.
In this sense, noise seems to be a tedious, annoying matter and it is a fair question to ask why one would make it the topic of an entire thesis. While noise is primarily an experimental affair, theoretical studies on the statistics of these fluctuations have been carried out for a long time, too. These studies draw an interesting picture. Measuring the average current through a system delivers partial information on the mechanisms responsible for conduction. But a more complete description and further information on the conduction mechanisms are given by the probability distribution of the current, containing both the average current and its fluctuations.
Even though the fluctuations appear randomly, they are caused by well-defined processes like the thermal motion of charges, the discreteness of charge carriers and the probabilistic character of scattering. Each noise source exhibits distinct characteristics. Measuring the noise properties of a system and knowing the underlying process, one might be able to infer complementary insight beyond what is possible with the mean current. Hence, a profound knowledge of the noise processes does not only help to find a way for reducing the noise level, but can also be used as a diagnostic tool.
It was Einstein who realised in 1909 that electromagnetic fluctuations differ if the energy is carried by waves or particles. He derived a linear relation between the mean energy and the corresponding fluctuations for waves, whereas the fluctuations scale with the square root of the mean energy for particles. Another example where fluctuations can provide information about the charge carriers was proposed by Schottky in 1918 in the context of vacuum tubes. Shot noise (which is not a dangerous effect at all despite its name) arises from the granularity of charge and therefore scales with the unit of charge. Indeed, the doubled charge of Cooper pairs and the fractional charge of Laughlin quasiparticles appearing in the fractional quantum Hall state was confirmed in this way. In 1928, the dependence of fluctuations due to thermal agitation was studied experimentally by Johnson and theoretically by Nyquist. In the following, the extrapolation of thermal noise to zero amplitude was used to determine the absolute zero of temperature and a value for the Boltzmann constant was deduced from the temperature dependence of thermal noise.
Typical currents that occur in nanoelectronics are tiny. Current fluctuations coming from these samples are even smaller and more challenging to detect and one has to come up with a clever measurement scheme. We are mainly interested in shot noise, whose spectral density is frequency-independent up to a few gigahertz. In contrast, electronic components add an undesired noise contribution, which is inversely proportional to the frequency f. At gigahertz frequencies, the amplitude of this 1/f-noise is considerably reduced. Moreover, measuring at high frequencies has a second advantage. Higher frequencies enable us to measure with a larger bandwidth and consequently to acquire more signal. For these reasons, we started the noise project by building up a microwave measurement scheme.
Our main interest lies in noise studies of high-resistance mesoscopic devices, such as quantum dots. However, the combination of high-frequency measurements with impedances on the order of R = 100 kOhm suffers from the large impedance mismatch to the standard characteristic impedance of the measurement line, Z0 = 50 Ohm. According to voltage division, the suppression of detectable signal power on the 50 Ohm side is on the order of (Z_0/R)^2. Hence, there is a solution needed to enhance the transmission from the device to the instrument. This is achieved with impedance matching, for which we use a so-called stub impedance-matching circuit. It is a resonant circuit based on transmission lines.
This thesis about noise detection with a stub impedance-matching circuit is structured as follows: It starts in chapter 2 with an introduction to the characteristics of microwave transmission lines, which are the building blocks of the later used microwave circuit. The development of carbon nanotube samples with an integrated stub impedance-matching circuit for noise detection as well as building up the high-frequency measurement setup were important experimental parts of this PhD project. For this reason, it is documented in detail in the thesis. A description of the stub impedance-matching circuit's properties is found in chapter 3. It also mentions impedance matching with an LC circuit and ends with a comparison of the two approaches. All fabrication considerations and recipes are collected in chapter 4. Chapter 5 gives an overview of the measurement setup, which is partially inside a dilution refrigerator. The remaining two chapters are devoted to results from a quantum dot formed in a carbon nanotube. Chapter 6 discusses RF reflectometry in the presence of a stub impedance-matching circuit. It is shown how to extract the circuit parameters and the device impedances from the reflection spectrum. Finally, noise measurements and their analysis are presented in chapter 7. The good agreement of our noise data in the single quantum dot regime with previous studies is a confirmation that the developed methods for noise detection with stub impedance matching and for calibration are well suited and allow for accurate noise results.