Costanzi, Barry2016-10-252016-10-252016-08https://hdl.handle.net/11299/182837University of Minnesota Ph.D. dissertation. August 2016. Major: Physics. Advisor: Dan Dahlberg. 1 computer file (PDF); x, 63 pages.The observation of noise signals with a $\frac{1}{f}$ power spectral density dependence on frequency \emph{f} is both ubiquitous in quantitative measurements across fields, and not entirely well understood. So-called ``$\frac{1}{f}$" spectra have been observed in systems spanning the realm of physics, and in other disciplines as well. Van der Ziel's model of $\frac{1}{f}$ noise as a composite of Lorentizian noise signals is the most widely accepted explanation for $\frac{1}{f}$, but experiments have for the most part only implicitly confirmed the result thus far. In this thesis, an explicit bottom-up approach to the Van der Ziel model is presented by combining random telegraph noise signals in square magnetic dots. Square dots made of the iron-nickel alloy Permalloy were fabricated to be 250 nm on a side and $\sim$ 10 nm thick. The configurational anisotropy of the dots is small enough to reduce energy barriers between adjacent magnetic states to approximately thermal energies through the application of an external field, causing two-state thermal hopping of the magnetization. This magnetization was measured through the anisotropic magnetoresistance of the dots. The random telegraph signals generate Lorentizan spectra when transformed to the frequency domain, and are shown to combine to form $\frac{1}{f}$ spectra when multiple dots are measured in series. The energy landscape of the dots is determined through easy-axis coercivity measurements, and the distribution of energy barriers predicts a range of applied fields where individual noise signals should combine to produce $\frac{1}{f}$ noise by the Van der Ziel model. Experiment shows good agreement with the predicted range of these ``noise fields" for two different series of samples with different coercivity distributions. Measurements of both individual dots and aggregate multi-dot signals show that the number of individual oscillating dots necessary to produce an aggregate $\frac{1}{f}$ signal is lower than might be expected, with $\frac{1}{f}$ observed in collections of fewer than ten oscillating dots, and in some cases as few as two. Additionally, while the statistics over multiple samples agree with the Van der Ziel model, individual collections of dots exhibiting $\frac{1}{f}$ noise can either vary signifcantly from the ideal Van der Ziel distribution, or defy the distribution description altogether when the number of dots becomes too few. This suggests that the Van der Ziel model is a sufficient but not necessary condition for observing $\frac{1}{f}$ noise in a collection of Lorentizan oscillators, and that the actual requirements to generate $\frac{1}{f}$ noise are much looser than Van der Ziel's. In systems with any type of distribution of Lorentizan signals, $\frac{1}{f}$ noise is likely due to combination of those signals. This result is relevant other systems exhibiting magnetic noise, as well as non-magnetic systems displaying both RTN and $\frac{1}{f}$ noise.en1/fDynamicsFlickerMagnetismNoiseStochasticThesis or Dissertation