#### Next topic

RF and Microwave Measurements

# Noise¶

Any physical system has two types of noise, thermal noise (important at high temperature, relatively low frequency), and quantum noise (uncertainty principle, important at low temperature, very high frequency).

This means even if we make perfect resistors and perfect transistors in fabrication, we will still have to deal with these noises.

For the frequencies of interest to electrical engineering, we are mostly interested in thermal noise. The so called transistor shot noise is actually a result of thermal noise as well at a microscopic level.

## Resistor Thermal Noise¶

Johnson first measured open circuit noise voltage of resistor in 1926, and found it to be independent of material, frequency, and only dependent on R and T. Its physical origin is random walk of electrons, or velocity fluctuation first studied by Einstein in 1906 and then formulated by Langevin.

Shortly after, Nyquist explained the 4kTR noise observed by Johnson using a transmission line cavity and second law of thermodynamics, without involving its microscopic origin. However, it relies on the assumption of thermal dynamic equilibrium, that is, no applied voltage and current.

Experimentally the same 4kTR noise was observed at non-equilibrium, or when DC or AC currents are flowing. It was puzzling for a long time until relatively recently.

It took several decades for us to quantitatively connect the microscopic voltage noise 4kTR with the microscopic velocity fluctuation studied by Einstein.

## Transistor Noise¶

The major noise sources in a bipolar transistor are the base resistance thermal noise, or Johnson noise, the base current shot noise and the collector current shot noise. The base resistance thermal noise is typically described by a noise voltage with a spectral density of , and the shot noise is described by a spectral density of , with being the DC base current or collector current. These descriptions are based on macroscopic views. The standard derivation of the magic shot noise assumes a Poisson stream of an elementary charge . These charges need to overcome a potential barrier, and thus flow in a completely uncorrelated manner. In a bipolar transistor, the base current shot noise results from the flow of base majority holes across the EB junction potential barrier. The reason that appears in the base shot noise is that the amount of hole current overcoming the EB barrier is determined by the minority hole current in the emitter, . Similarly, the collector current shot noise results from the flow of emitter majority electrons over the EB junction potential barrier, and has a spectral density of .

Surprisingly, however, both the resistor noise and the shot noise can be attributed to the same physical origin at the microscopic level, the Brownian motion of electrons and holes, also referred to as diffusion noise as the same mechanism is responsible for diffusion. The thermal motion of carriers gives rise to fluctuations of carrier velocities, and hence fluctuations of current densities. The density of such current density fluctuation is according to microscopic treatment of carrier motion .

The current density fluctuation at each location propagates towards device terminals, giving rise to device terminal voltage or current noise. The problem of noise analysis is now equivalent to solving the transfer functions of noise propagation at each location and summing over the whole device space. These transfer functions can be solved analytically for ideal transistor operation with simplified boundary conditions, or numerically for arbitrary device structures, and the later process is referred to as microscopic noise simulation.

Various mathematical methods have been developed, all based on the impedance field method developed by Schockley and his colleagues , and its various generalizations. A very satisfying early application is the successful derivation of the Johnson noise, a macroscopic model result, from the microscopic noise source density. The impedance field approach is equivalent to the Green function based approaches, which can be rigorously derived from the general master equation. Efficient numerical algorithms have been developed, which enabled the recent implementation of noise analysis in commercial device simulators, e.g. DESSIS from ISE and Taurus from TMA.

## Methodology¶

We research measurement, understanding, microscopic level simulation and macroscopic level compact modeling of noise in transistors and circuits, from low-frequency noise to high frequency noise as well as phase noise.

For high frequency noise that is of particular interest for low-noise RF and microwave amplifier design, we take a unified approach of modeling the macroscopic level terminal current and voltage noise as results of microscopic level Brownian motion and related carrier transport or propagation.

For these new noise models to be useful to IC designers, we develop techniques of implementing our new compact models in non-linear circuit simulators and propotype such implementation techniques with low-noise amplifier designs.