  In Loop Quantum Gravity (LQG) the quantisation of General Relativity leads to
precise predictions for the eigenvalues of geometrical observables like volume
and area, up to the value of the only free parameter of the theory, the
Barbero-Immirzi (BI) parameter. With the help of the eigenvalues equation for
the area operator, LQG successfully derives the Bekenstein-Hawking entropy of
large black holes with isolated horizons, fixing at this limit the BI parameter
as $\gamma \approx 0.274$. In the present paper we show some evidence that a
black hole with angular momentum $\hbar$ and Planck mass is an eigenstate of
the area operator provided that $\gamma = \sqrt{3}/6 \approx 1.05 \times
0.274$. As the black hole is extremal, there is no Hawking radiation and the
horizon is isolated. We also suggest that such a black hole can be formed in
the head-on scattering of two parallel Standard Model neutrinos in the mass
state $m_2$ (assuming $m_1 = 0$). Furthermore, we use the obtained BI parameter
to numerically compute the entropy of isolated horizons with areas ranging up
to $250\,l_P^2$, by counting the number of micro-states associated to a given
area. The resulting entropy has a leading term ${\cal S} \approx 0.25\, {\cal
A}$, in agreement to the Bekenstein-Hawking entropy. As the identification of
the above eigenstate rests on the matching between classical areas and quantum
area eigenvalues, we also present, on the basis of an effective quantum model
for the Schwarzschild black hole recently proposed by Ashtekar, Olmedo and
Singh, an expression for the quantum corrected area of isolated horizons, valid
for any black hole mass. Quantum corrections are shown to be negligible for a
Planck mass black hole, of order $10^{-3}$ relative to the classical area.
