  Nano-constriction based spin Hall nano-oscillators (SHNOs) are at the
forefront of spintronics research for emerging technological applications such
as oscillator-based neuromorphic computing and Ising Machines. However, their
miniaturization to the sub-50 nm width regime results in poor scaling of the
threshold current. Here, we show that current shunting through the Si substrate
is the origin of this problem and study how different seed layers can mitigate
it. We find that an ultra-thin Al$_{2}$O$_{3}$ seed layer and SiN (200 nm)
coated p-Si substrates provide the best improvement, enabling us to scale down
the SHNO width to a truly nanoscopic dimension of 10 nm, operating at threshold
currents below 30 $\mu$A. In addition, the combination of electrical insulation
and high thermal conductivity of the Al$_{2}$O$_{3}$ seed will offer the best
conditions for large SHNO arrays, avoiding any significant temperature
gradients within the array. Our state-of-the-art ultra-low operational current
SHNOs hence pave an energy-efficient route to scale oscillator-based computing
to large dynamical neural networks of linear chains or two-dimensional arrays.
