Quantum mechanics

To simplify, a sensor (tip or probe) follows the surface of the object.

The sensors sweeps or scans the studied surface. A computer adjusts (via a subservience service) in real-time the height of the sensor to maintain a constant current (tunneling current) and registers this height to recreate the surface.

In order to do so, a conductive tip is placed in front of the surface to study with a positioning system of great precision (realized with the help of piezoelectricity).

The current resulting to the passage of electrons between the surface and the sensors, by quantum tunneling can be measured : the free electrons of the metal hardly come out to the surface so if the sensor approaches them really closely but without touching them, an electric current can be registered.

In most cases, the current decreases very rapidly -exponentially- in function of the distance separating the surface from the sensor, with a characteristic distance of a few dozen of nanometers.

When the sensor above the sample is set in motion with a movement of sweeping, the height of the latter is adjusted in order to conserve a constant intensity of tunneling current, thanks to a retroaction buckle.

The profile of the surface can thus be determined with a precision inferior to the inter-atomic distances.

The expression of the wave function is :

\(\varphi (x) = \left\{ \begin{array}{l}A{e^{ikx}} + B{e^{ - ikx}}\;\;\;\;\;\;\;(x < 0) \\C{e^{\alpha x}} + D{e^{ - \alpha x}}\;\;\;\;\;\;\;(0 < x < L) \\F{e^{ikx}}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(x > L) \\\end{array} \right.\)

With :

\(k = \sqrt {\frac{{2mE}}{{{\hbar ^2}}}} \;\;\;\;\;\;\;\;and\;\;\;\;\;\;\;\;\;\alpha = \sqrt {\frac{{2m(-E + {V_0})}}{{{\hbar ^2}}}}\)

The two border conditions, in \(x=0\) and \(x=L\) (continuity of the wave-function and its derivative), give :

\(\left\{ \begin{array}{l}A + B = C + D\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;and\;\;\;\;\;\;\;\;\;\;\;ik(A - B) = \alpha (C - D) \\C{e^{\alpha L}} + D{e^{ - \alpha L}} = F{e^{ikL}}\;\;\;\;and\;\;\;\;\;\;\;\;\;\;\;\alpha (C{e^{\alpha L}} - D{e^{ - \alpha L}}) = ikF{e^{ikL}} \\\end{array} \right.\)

Numeric value :

\(\alpha \approx {10^{10}}{m^{ - 1}}\;;\;\alpha L \approx 10\;so\;{e^{\alpha L}} > > {e^{ - \alpha L}}\)

Then :

\(t = \frac{F}{A} \approx \frac{4}{{\left( {1 + \frac{{i\alpha }}{k}} \right)\left( {1 - \frac{{ik}}{\alpha }} \right){e^{\alpha L}}{e^{ikL}}}} = \frac{{4{e^{ - \alpha L}}}}{{2 + \frac{{i\alpha }}{k} - \frac{{ik}}{\alpha }}}{e^{ - ikL}}\)

The transmission factor \(T\) of the potential barrier is :

\(T = \frac{{\left| {{J_t}} \right|}}{{\left| {{J_i}} \right|}} = \frac{{\left( {p_t^2/2m} \right){{\left| F \right|}^2}}}{{\left( {p_i^2/2m} \right){{\left| A \right|}^2}}} = {\left| {\frac{{4{e^{ - \alpha L}}}}{{2 + \frac{{i\alpha }}{k} - \frac{{ik}}{\alpha }}}} \right|^2}\)

Hence :

\(T = \frac{{16{k^2}{\alpha ^2}}}{{{{({\alpha ^2} + {k^2})}^2}}}{e^{ - 2\alpha L}} = \frac{{16E({U_0} - E)}}{{U_0^2}}\exp \left( { - 2\alpha L} \right)\)

The ratio of the two respective transmission coefficients \(T_1\) and \(T_2\) for the distances \(L_1=1\) \(nm\) and \(L_2=1,1\) \(nm\) is :

\(\frac{{{T_1}}}{{{T_2}}} = \exp (2\alpha ({L_2} - {L_1})) \approx {e^2} \approx 7\)

The electric current is divided by almost \(10\).

The atoms which had the dimension of a few nanometers can now be revealed by the variations of electric current during the sweeping of the surface of the sample by the sensor.

One simple idea is to chose \(U_0=5\) \(eV\), the average value for \(U_0\).

Then \(\alpha\) diminishes of approximately \(0,9\) and \(T_1/T_2\) is approximately equal to \(6\).

The transmission of the barrier can also be calculated, by noticing that :

\(\frac{{dT}}{T} = - 2\alpha dL\)

Then :

\(\ln T = - 2\int_0^L {\sqrt {\frac{{2m(U(x) - E)}}{{{\hbar ^2}}}} dL = } - 2\int_0^L {\frac{{\sqrt {2m({U_0} - \Delta {U_0}(x/L) - E)} }}{\hbar }dL}\)