  The test masses in next-generation gravitational-wave interferometers may
have a semiconductor substrate, most likely silicon. The stochastic motion of
charge carriers within the semiconductor will cause random fluctuations in the
material's index of refraction, introducing a noise source called Thermal
Charge Carrier Refractive (TCCR) noise. TCCR noise was previously studied in
2020 by Bruns et al., using a Langevin force approach. Here we compute the
power spectral density of TCCR noise by both using the Fluctuation-Dissipation
theorem (FDT) and accounting for previously neglected effects of the standing
wave of laser light which is produced inside the input test mass by its
high-reflecting coatings. We quantify our results with parameters from Einstein
Telescope, and show that at temperatures of 10 K the amplitude of TCCR noise is
up to a factor of $\sqrt{2}$ times greater than what was previously claimed,
and from 77 K to 300 K the amplitude is around 5 to 7 orders of magnitude lower
than previously claimed when we choose to neglect the standing wave, and is up
to a factor of 6 times lower if the standing wave is included. Despite these
differences, we still conclude like Bruns et al. that TCCR noise should not be
a limiting noise source for next-generation gravitational-wave interferometers.

General Relativity and Quantum Cosmology;Instrumentation and Methods for Astrophysics;Materials Science;Instrumentation and Detectors