Calculation of axial charge spreading in carbon nanotubes and nanotube Y junctions during STM measurement

Géza I. Márk, László P. Biró, and Philippe Lambin

Phys. Rev. B 70, 115423 – Published 29 September 2004

Abstract

Distribution of the probability current and the probability density of wave packets was calculated for nanotubes and nanotube Y junctions by solving the three dimensional time-dependent Schrödinger equation for a jellium potential model of the scanning tunneling microscope (STM) tip-nanotube-support system. Four systems were investigated: an infinite single wall nanotube (SWNT) as reference case, a capped SWNT protruding a step of the support surface, a quantum dot (finite tube without support), and a SWNT Y junction. It is found that the spatial distribution of the probability current flowing into the sample is decided by the electron probability density of the tube and by the oscillation in time of the probability current, which in turn is governed by the quasibound states on the tube. For the infinite tube the width of the axial spreading of the wave packet during tunneling is about $5 nm$ . When the STM tip is above that part of the tube which protrudes from the atomic scale step of the support surface it is found that the current flows ballistically along the tube and the total transmission is the same as for the infinite tube. In the case of quantum dot, however, the finite tube is first charged in a short time then it is discharged very slowly through the tip-nanotube tunnel junction. In the Y junction both the above the junction and off the junction tip positions were investigated. For a $1.2 nm$ displacement of the tip from the junction the wave packet still “samples” the junction point which means that in STM and scanning tunneling spectroscopy experiments the signature of the junction should be still present for such tip displacement. For all tunneling situations analyzed the tunnel current is mainly determined by the tip-nanotube junction owing to its large resistance. The tunneling event through the STM model is characterized by two time scales, the nanotube is quickly “charged” with the wave packet coming from the tip then this “charge” flows into the support 50 times slower.

Received 5 February 2004

DOI:https://doi.org/10.1103/PhysRevB.70.115423

Authors & Affiliations

Géza I. Márk^* and László P. Biró

Research Institute for Technical Physics and Materials Science, H-1525 Budapest, P.O. Box 49, Hungary

Philippe Lambin

Facultés Universitaires Notre Dame de la Paix, 61, Rue de Bruxelles, B-5000 Namur, Belgium

^*Electronic address: mark@sunserv.kfki.hu; http://www.mfa.kfki.hu/int/nano/

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Vol. 70, Iss. 11 — 15 September 2004

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Images

Figure 1
Probability currents analyzed in this paper are shown by arrows. Cross section of the geometric (effective) surface of the STM tip, nanotube, and support are shown by full (broken) line.Reuse & Permissions
Figure 2
Time evolution of the probability density of the wave packet approaching the STM junction from the tip bulk and tunneling through the nanotube into the support. The left column is for the infinite tube on an atomically flat support and the right column is for the capped tube hanging above a step of the support surface. Geometries of the two systems are shown on the upper subimages. The cuboid shows the presentation box boundaries. All dimensions are in nanometers. The subsequent subimages show snapshots of an isodensity surface with density value of $ϱ (r; t) = ϱ_{0} = 2.0245 \times 10^{- 6} {nm}^{- 3}$ . The isosurface is clipped at the presentation box boundaries.Reuse & Permissions
Figure 3
Analysis of the tunneling process as the function of time and the $y$ position along the tube. Upper part (a)–(d) is for the infinite tube above an atomically flat support and lower part (e)–(h) for the capped tube protruding a $1 nm$ high step. The YZ cross sections of the potential are shown in the left subfigures. The tip is fixed at $y = 0$ . (a) and (e) “Quantum carpet” plot of the linear probability density on the tube as the function of time and the axial coordinate. White corresponds to zero density and black to $2.10 \times 10^{- 3} {nm}^{- 1}$ . (b) and (f) Linear probability density along the tube at time instants $t_{1} = 2.54 fs$ , $t_{2} = 3.75 fs$ , and $t_{3} = 4.96 fs$ . (c) and (g) Probability current density flowing into the support surface as the function of time and axial coordinate. White corresponds to zero current and black to $8.03 \times 10^{- 6} {nm}^{- 1} {fs}^{- 1}$ for (c) and $6.34 \times 10^{- 6} {nm}^{- 1} {fs}^{- 1}$ for (g). (d) and (h) Transmitted probability into the support as the function of the axial coordinate. (See the text for details.) Contour shades are drawn on a square root scale on all grayscale figures.Reuse & Permissions
Figure 4
Time cumulated transmissions of the wave packet launched from the tip bulk into the support surface (dotted line) and through the tube cross section at the presentation box boundary (dashed line). The total probability of the wave packet at the tube is also shown as the function of time by a dash-dotted line. Net transmission from the tube plus the probability on the tube is shown by the continuous line. See the text for details. (a) Infinite tube above an atomically flat support. (b) Capped tube hanging above a step of the support.Reuse & Permissions
Figure 5
Time cumulated transmissions measured below the tip apex. Dotted, dashed, and full lines are for the infinite tube above atomically flat support, for the capped tube hanging above the step, and for the quantum dot tube, respectively. The $Y Z$ cross sections of the potential for the three model situations are shown by grayscale plots near the curves. The inset shows the initial, large intensity peak (same for all the three models within the line thickness). See the text for details.Reuse & Permissions
Figure 6
Snapshot of the tunneling process through a nanotube Y junction at $t = 6.71 fs$ . (a) and (b) $Z$ integrated tube probability densities for the $d = 0 nm$ and $d = 1.2 nm$ tip displacements. (c) and (d) Probability current densities flowing into the support surface for the $d = 0 nm$ and $d = 1.2 nm$ tip displacements. Axial position of the tip is shown by small black circle on each subfigure. Contour shades are drawn on a square root scale. White corresponds to zero and black to maximum density (current), for (a) and (b) [(c) and (d)].Reuse & Permissions

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