= Solution
The dimensionally meaningful small-potential condition is \b[$|e\Phi|\ll k_BT$], not a comparison of volts directly with <energy>. Expanding $\sinh u=u+O(u^3)$ gives the physical linearized equation
$$
\Phi''=\kappa_D^2\Phi,\qquad
\boxed{\lambda_D=\kappa_D^{-1}=
\sqrt{\frac{\epsilon_0\epsilon_r k_BT}{2e^2n_0}}}.
$$
For a single plane with bulk <boundary condition> $\Phi(\infty)=0$, $\Phi(x)=\Phi_0e^{-x/\lambda_D}$. At fixed <temperature> and <permittivity>, \b[$\lambda_D\propto n_0^{-1/2}$]: the curve decreases with increasing density, diverging in the ideal zero-salt limit.
Literal expansion of the printed minus-sign equation instead gives $\Phi''=-\kappa_D^2\Phi$, with sine/cosine solutions. A nonzero such solution cannot decay to zero at infinity. That literal equation does not describe a <Debye length>. The figure and following ion profiles use the physically derived plus-sign equation, with this correction explicit.
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