  Hermitian, pluriclosed metrics with vanishing Bismut-Ricci form give a
natural extension of Calabi-Yau metrics to the setting of complex, non-K\"ahler
manifolds, and arise independently in mathematical physics. We reinterpret this
condition in terms of the Hermitian-Einstein equation on an associated
holomorphic Courant algebroid, and thus refer to solutions as Bismut
Hermitian-Einstein. This implies Mumford-Takemoto slope stability obstructions,
and using these we exhibit infinitely many topologically distinct complex
manifolds in every dimension with vanishing first Chern class which do not
admit Bismut Hermitian-Einstein metrics. This reformulation also leads to a new
description of pluriclosed flow in terms of Hermitian metrics on holomorphic
Courant algebroids, implying new global existence results, in particular on all
complex non-K\"ahler surfaces of nonnegative Kodaira dimension. On complex
manifolds which admit Bismut-flat metrics we show global existence and
convergence of pluriclosed flow to a Bismut-flat metric, which in turn gives a
classification of generalized K\"ahler structures on these spaces.
