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Quantum line cut

We have performed quantum mechanical probability current calculations through a simple two dimensional (2D) jellium potential modeling the STM tip - nanotube - support system. STM constant current feedback loop was simulated by moving the tip along a line (''quantum line cut'' ) where the transferred probability was constant.

The 2D model potential has the value zero outside the effective surfaces of the electrodes and the value $-9.81 \, eV$ inside. ($E_F = 5 \, eV$ , $W = 4.81 \, eV$).

Tunneling probability is calculated from time dependent scattering of a Gaussian wave packet approaching the model potential from the bulk of the tip. The $\vec j (x,z,t)$probability current density was calculated along a horizontal line inside the support bulk (at $z_j = -0.5 \, nm$). Line integration of $\vec j (x,z,t)$ along this line gives the I(t) probability current and the tunneling probability is obtained as $P_{tunnel} = \int_0^{t_{max}} I(t) dt$ .

Probability density of the scattered wave packet is shown on Fig. 6..

Figure 6: Probability density of scattered wave packet for selected time instants and for selected Xapex lateral tip displacements. Effective surfaces of electrodes are shown by red lines. Size of presentation window is $3.84 \, nm$. Contour lines are drawn on sqare root scale. Each frame is normalized to its maximum density. Initial momentum of Gaussian wave packet is $k_z = - \sqrt{{ 2 E_F} / m_e } = -11.46 \, {nm} ^ {-1}$ and its initial with is $\Delta k_x = \Delta k_z = 1.336 \, {nm} ^ {-1}$ .

Table I: Average time tspent (in fs) the quantum particle spends in different regions. ''Forbidden'' region is the region of zero potential. tspent for ''total interface'' region is sum of the values for ''forbidden'' and ''tube'' regions.
Xapex (nm) Forbidden Tube Total Interface = Forbidden + Tube
0.0 0.071 0.084 0.155
0.8 0.079 0.123 0.202
1.6 0.060 0.020 0.080
2.0 0.057 0.0001 0.057

STM constant current loop was simulated by finding for each fixed Xapex lateral tip displacement that appropriate Zapex vertical tip displacement which yielded a setpoint tunneling probability value of $P_{setpoint} = 3 \cdot 10^{-3}$. ( Xapex , Zapex ) tip displacement values resulting from this procedure are shown on Fig. 7. by crosses.

Figure 7: Comparison of geometric and quantum line cut. Thick red line is the geometric line cut. Crosses show calculated points of quantum line cut.

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Next: Discussion Up: Simulation of STM images Previous: Geometric line cut