The yellow dot (an extra charge) can be trapped at a site close to the conducting region of a nanostructure (yellow dot), or at another site slightly further away (grey dot). These two different configurations result in a slightly different electrostatic potential profile in the conducting region and could thus lead to a small difference in the current measured. In this exercise we will show that an ensemble of such fluctuators in the vicinity of a conducting region can indeed produce current noise with a 1/f power spectraldensity. We first focus on the simple situation sketched in Fig. 2, where a single charge hops back and forth randomly between two traps. We assume that the rate at which the charge hops 1/t is the same for both directions, i.e., in a tiny time interval dt the chance that the charge hops to the opposite site is dt/t . Let us consider the source-drain current with a constant bias applied, and distinguish between a measured source-drain current 1 (when the charge is in trap 1) and a current I2 (when the charge is in trap 2). (a) Sketch a typical time-dependent plot of the source-drain current (neglecting other sources of noise for simplicity). What is the average current (I)? (b) We treat the successive jumps as rare uncorrelated events. Calculate (SI(t)S(0)), (2) in terms of h, h, and t. Make sure that your result also works for t< 0. Hint. Depending on the method you choose to evaluate this correlator, you might need the relations 2n 2n+1 Σ ΣΧ = sinhx = coshx and (2n)! n=0 (2n + 1)! n=0 (c) Calculate the noise power spectral density in the current. (d) Now we assume that there is a large ensemble of such two-site fluctuators present. Let us set up the simplest model possible: The two sites constituting each fluctuator are

Introductory Circuit Analysis (13th Edition)

13th Edition

ISBN:9780133923605

Author:Robert L. Boylestad

Publisher:Robert L. Boylestad

Chapter1: Introduction

Section: Chapter Questions

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Transcribed Image Text:1. 1/f noise At low temperatures, a very common type of noise in 2DEG systems hosted in semi- conductor heterostructures manifests itself as random fluctuations of the electric poten- tial, resulting in noise with a power spectral density that behaves over a large range as S(@) 1/w ; for that reason this noise is commonly called 1/fnoise. The dominating source of such 1/f naise can differ from system to system, but one contribution is believed to come from charges that get suddenly trapped at a certain site (e.g., an impurity), spend some time in the trap, and then escape from the trap or jump to another trap. One example of how such a jumping charge could lead to noise, for instance when we try to measure a source-drain current through some constriction, is sketched in Fig.2: D Figure 2: A charge (yellow dot) can be trapped at two different sites, one close to the conducting region and one slightly further away (the "empty" grey dot). The yellow dot (an extra charge) can be trapped at a site close to the conducting region of a nanostructure (yellow dot), or at another site slightly further away (grey dot). These two different configurations result in a slightly different electrostatic potential profile in the conducting region and could thus lead to a small difference in the current measured. In this exercise we will show that an ensemble of such fluctuators in the vicinity of a conducting region can indeed produce current noise with a 1/f power spectraldensity. We first focus on the simple situation sketched in Fig. 2, where a single charge hops back and forth randomly between two traps. We assume that the rate at which the charge hops 1/t is the same for both directions, i.e., in a tiny time interval dt the chance that the charge hops to the opposite site is dt/t . Let us consider the source-drain current with a constant bias applied, and distinguish between a measured source-drain current 1 (when the charge is in trap 1) and a current I2 (when the charge is in trap 2). (a) Sketch a typical time-dependent plot of the source-drain current (neglecting other sources of noise for simplicity). What is the average current (I)? (b) We treat the successive jumps as rare uncorrelated events. Calculate (81(1)S(0)), (2) in terms of Iı, 2, and t. Make sure that your result also works for t< 0. Hint. Depending on the method you choose to evaluate this correlator, you might need the relations 2n 2n+1 Σ Σ = sinhx = coshx and (2n)! n=0 (2n + 1)! n=0 (c) Calculate the noise power spectral density in the current. (d) Now we assume that there is a large ensemble of such two-site fluctuators present. Let us set up the simplest model possible: The two sites constituting each fluctuator are 1

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