= Electronic contrast in scanning tunneling microscopy
{title2=$z=h+\ln\rho/(2\kappa)+\mathrm{constant}$}
Separate geometry from electronic response in an original one-dimensional toy <tunnel junction>. Write the geometric surface height as $h(x)$, tip height as $z(x)$, and positive gap as $s(x)=z(x)-h(x)$. Assume low fixed tip-sample <electric potential difference>: the sampled <energy> window is narrow enough that the tip <density of states> is featureless and the sample <local density of states> varies little across that window. Neglect thermal broadening, changes of tip shape and lateral averaging. Let $\rho(x)>0$ be the effective sample <local density of states> divided by a positive reference value, so it is dimensionless. Choose a spatially constant effective barrier <energy> $\Phi>0$, a positive coefficient $C$ with units of <electric current>, and
$$
I(x,z)=C\rho(x)\exp\bigl[-2\kappa(z-h(x))\bigr],\qquad
\kappa=\frac{\sqrt{2m_e\Phi}}{\hbar}>0.
$$
Here $m_e$ is the <Electron> <mass>, $\hbar$ the reduced <Planck constant>, and $\kappa$ the <tunnelling decay constant>. This is an illustrative low-bias model, not a universal calibration formula. At fixed $I=I_0>0$, taking a <logarithm> gives
$$
\ln(I_0/C)=\ln\rho-2\kappa(z-h),\qquad
z(x)=h(x)+\frac{\ln\rho(x)}{2\kappa}+s_0,\qquad
s_0=\frac{\ln(C/I_0)}{2\kappa}.
$$
The <logarithms> have dimensionless arguments. Only constant $\rho$ makes the recorded height differ from geometric height by a constant. Multiplying $\rho$ by $r>0$ raises the apparent height by $\ln r/(2\kappa)$ without moving the geometric surface.
For a concrete ambiguity, choose $\kappa=10\,\mathrm{nm}^{-1}$, $a=0.020\,\mathrm{nm}$, lateral period $L=1\,\mathrm{nm}$, $s_0=0.50\,\mathrm{nm}$, and $q(x)=\cos(2\pi x/L)$. Compare
$$
\begin{aligned}
h_A(x)&=a q(x),&\rho_A(x)&=1,\\
h_B(x)&=0,&\rho_B(x)&=\exp\bigl(2\kappa a q(x)\bigr).
\end{aligned}
$$
Both produce exactly $z(x)=s_0+a q(x)$ at $I_0=C e^{-2\kappa s_0}=C e^{-10}$. Substitution checks this without identifying $\rho$ with geometric height: case A has constant gap $s_0$, while case B has gap $s_0+a q(x)$ and a compensating electronic factor. The chosen $\rho_B$ ranges from $e^{-0.4}$ to $e^{0.4}$, approximately $0.6703$ to $1.4918$; every gap remains positive. The chosen $\kappa$ corresponds to $\Phi\simeq3.81$ <electronvolts>, a positive effective barrier.
At fixed tip height $z_f=s_0$, both cases instead give
$$
\frac{I(x,z_f)}{I_0}=\exp\bigl[0.4q(x)\bigr].
$$
Their maximum-to-minimum <electric current> ratio is $e^{0.8}\simeq2.226$. Thus even adding this fixed-height scan at the same <electric potential difference> cannot distinguish these deliberately chosen surfaces within the toy model. Independent structural information or additional electronic measurements are required. Changing <electric potential difference> can supply such information, but also changes the assumptions used to define $C$ and $\rho$.
\Image[/scanning-tunneling-height-electronic-contrast.png]
{title=Different geometric and electronic profiles can produce the same STM image}
{description=Original toy calculations, not measured data. The corrugated surface with constant <local density of states> and the flat surface with varying <local density of states> give identical recorded heights under <constant-current scanning tunneling microscopy>. They also give identical fixed-height <electric current> contrast at this chosen bias.}
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