  It has recently been suggested [S. B. Giddings, Phys. Lett. B {\bf 754}, 39
(2016)] that the Hawking black-hole radiation spectrum originates from an
effective quantum "atmosphere" which extends well outside the black-hole
horizon. In particular, comparing the Hawking radiation power of a
$(3+1)$-dimensional Schwarzschild black hole of horizon radius $r_{\text{H}}$
with the familiar Stefan-Boltzmann radiation power of a $(3+1)$-dimensional
flat space perfect blackbody emitter, Giddings concluded that the source of the
Hawking semi-classical black-hole radiation is a quantum region outside the
Schwarzschild black-hole horizon whose effective radius $r_{\text{A}}$ is
characterized by the relation $\Delta r\equiv r_{\text{A}}-r_{\text{H}}\sim
r_{\text{H}}$. It is of considerable physical interest to test the general
validity of Giddings's intriguing conclusion. To this end, we study the Hawking
radiation of $(D+1)$-dimensional Schwarzschild black holes. We find that the
dimensionless radii $r_{\text{A}}/r_{\text{H}}$ which characterize the
black-hole quantum atmospheres, as determined from the Hawking black-hole
radiation power and the $(D+1)$-dimensional Stefan-Boltzmann radiation law, are
a decreasing function of the number $D+1$ of spacetime dimensions. In
particular, it is shown that radiating $(D+1)$-dimensional Schwarzschild black
holes are characterized by the relation
$(r_{\text{A}}-r_{\text{H}})/r_{\text{H}}\ll1$ in the large $D\gg1$ regime. Our
results therefore suggest that, at least in some physical cases, the Hawking
emission spectrum originates from quantum excitations very near the black-hole
horizon.
