Thermal and flicker noise

Noise as a density

Noise is the random motion of charge that every resistor and transistor adds to a signal. It has no single value at an instant, so we describe its strength by frequency. The power spectral density is the noise power in each 1 Hz slice of frequency, in V²/Hz or A²/Hz. SPICE prints its square root, in V/√Hz or A/√Hz.

Thermal noise comes from the heat motion of carriers. A resistor R at temperature T makes a white current noise of 4kT/R A²/Hz, or a voltage noise of 4kTR V²/Hz across it. White means the same at every frequency. k is Boltzmann's constant, so the noise grows with temperature and with resistance.

Bench A drives a resistor with a 1 A AC current source. The source only names the input of the analysis: noise v(n) I1 ... reports the noise at node n. Three resistors give exactly the hand values.

Two kinds of noise in a transistor

Bench B biases one NMOS at 10 µA with its drain held at 0.9 V. The drain source VD holds the drain still, so all of the channel's noise current flows through VD. H₁, a current-controlled voltage source, copies that current to a node at 1 V per ampere and adds no noise of its own.

The flat floor is channel thermal noise: the thermal noise of the channel's carriers. In saturation it is usually written 4kTγg_m A²/Hz, with γ = 2/3 for a long-channel device. That gives 1.378 pA/√Hz here, but the model gives 2.389 pA/√Hz, so its γ is 2.00. The 40/4 device gives 2.06. That γ is a model result, not a measurement of silicon: the SKY130 model cards select BSIM4's holistic thermal noise model (tnoimod = 1). Use the run's γ when you compare a hand estimate with this simulator.

Flicker noise, or 1/f noise, comes from carriers trapped and released by defects at the gate oxide. Its power density falls with frequency and scales as 1/(WLC_ox): a larger gate averages over more traps. In the SKY130 NMOS model it falls as 1/f^0.84, the model card's EF parameter. The flicker corner is the frequency where the flicker and thermal parts are equal.

A PMOS of the same size at the same current is quieter at low frequency on this process. Its input-referred noise at 10 Hz is 1.313 µV/√Hz against the NMOS's 1.949, and its corner is 111.9 kHz. Its γ from the run is 0.99. That is one reason low-frequency designs often pick a PMOS input pair; check it on each process.

How a noise analysis adds it up

A noise analysis linearizes the circuit at its operating point, as ac does. Then it turns on each noise source alone, carries it to the output, and adds the results as powers. Independent noise sources are uncorrelated, so their powers add and their voltages do not.

At 10 MHz the lab's device has 1.010 pA/√Hz of flicker noise and 2.389 pA/√Hz of thermal noise. The total is √(1.010² + 2.389²) = 2.594 pA/√Hz, exactly what the run prints, and well below their sum of 3.399.

The input source matters only for inoise_spectrum, the output noise divided by the gain from that source. Here the gain from VG is g_m = 171.8 µS, so the gate noise is the drain noise over g_m. Totals integrate over the sweep, so the sweep's limits are part of any total.

noise v(out) VG dec 10 100k 1g 1

noise v(out) VG dec 10 1 1g 1

Try it

Run it and read the lines: the bias vgs and gm, the drain-current density at 10 Hz, 1 kHz and 10 MHz, the flicker corner and the thermal floor, and the γ implied by that floor. Open Plots for the spectra.

Now make the device 40/4 and run again: the flicker density at 10 Hz should fall about four times and the corner should drop below 100 kHz. Then try .temp 125 and watch the thermal floor rise (change the 27 in the gamma line too).

Check yourself

* the thermal noise density of 10 k and of 40 k, and their ratio
I1 0 a DC 0 AC 1
R1 a 0 10k
I2 0 b DC 0 AC 1
R2 b 0 40k
.control
noise v(a) I1 dec 10 1 1meg
noise v(b) I2 dec 10 1 1meg
ac dec 10 1 1meg
let ratio = noise3.onoise_spectrum/noise1.onoise_spectrum
meas ac ratio FIND ratio AT=1k
.endc
.end