= Solution
Inside the barrier, the stationary <Schrödinger equation> gives an exponentially decaying <wavefunction> with inverse penetration length
$$
\kappa=\frac{\sqrt{2m_e(V_0-E)}}\hbar=7.245\times10^9\,\mathrm{m}^{-1}.
$$
The <amplitude> falls approximately as $e^{-\kappa a}$, so the leading exponential estimate for the <transmission probability> is
$$
\boxed{T\sim e^{-2\kappa a}=e^{-14.49}=5.09\times10^{-7}.}
$$
For comparison, <wavefunction matching at a finite potential step> at both edges gives the <rectangular-barrier tunnelling probability>
$$
T=\left[1+\frac{V_0^2\sinh^2(\kappa a)}{4E(V_0-E)}\right]^{-1}=1.30\times10^{-6}.
$$
The difference is the interface prefactor: for an opaque barrier it is $16E(V_0-E)/V_0^2=2.56$. Both give a transmission of order $10^{-6}$; the first is the requested exponential estimate.
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