Nanostructures Laboratory

KFKI MFA Budapest

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3D calculation of wave packet tunneling through a supported carbon nanotube

Introduction Model potential Results Animation Acknowledgments Links References


Investigating the distribution of the scanning tunneling microscope (STM) current through a nanostructured material is a subject of great current interest. In this work, the transmission of an electron wave packet was calculated through a jellium potential model of a carbon nanotube under the tip of a STM. The theory shows that the wave packet spreads along the nanotube while it tunnels through it and the nanotube-support tunneling channel is also extended along the direction of the tube as compared to the tip-nanotube channel which remains narrow.

These results were presented in the IWEPNM2001 conference.
Click here for details!

Model potential

The tunneling problem was regarded as a problem in potential scattering theory. In the present approach we used a simple jellium potential which does not take into account the atomic structure. At this level of approximation, all CNTs are metallic. The model system geometry is shown below.
Model system used in the calculation. The carbon nanotube is modeled by a cylinder of 0.5 nm radius floating above the support plane at a distance of 0.335 nm. The STM tip is taken as a hyperboloid of 0.5 nm apex radius. The tip/nanotube tunnel gap is 0.4093 nm.


The time development of a Gaussian wave packet approaching the tunnel junction from inside of the tip bulk was calculated by numerically solving the time dependent 3D Schrödinger equation with the split time FFT method.
The Rho(x,y,z,t) = Abs( Psi(x,y,z,t) )^2 time dependent probability density function is visualized by the time development of a constant density surface.
Initial Gaussian wave packet, spherical symmetric density clipped at the upper boundary of the presentation box. Wave packet begins to enter into the tip apex region. Wave packet enters into the tip-nanotube interface. The part reflected back into the tip bulk forms interference patterns with the incoming wave (disc like structures at tip bulk).
Wave packet flows around the tube and simultaneously tunnels through it. The incoming and outgoing waves form interferences in the tip apex region. Wave packet tunnels through the nanotube-support junction and begins to enter into the support. The two wave packet parts (one moving on the left and another on the right side of the tube) meet at the lowest point, standing wave patterns begin to form along the tube circumference. The probability density is gradually spreading along the tube axis. Nanotube-support tunnel channel begins to open.

Snapshots of the probability density of the wave packet approaching the STM junction from the tip bulk and tunneling through the nanotube into the support. (Click on each image to see a larger version!) Constant density surface is clipped at the presentation box boundaries. The color scaled insets are X-Z (perpendicular to the tube) and Y-Z (along the tube) plane cuts of the density.


This GIF animation (size 1.5 M) shows the time development of a constant probability density surface clipped at the presentation box boundaries.
Click here to see larger version in MPEG format (size 1.4 M).
Click here to see larger version  in DivX AVI (MPEG-4) format (size 4.2 M). The driver is available here.


This work was partly supported by the EC, contract NANOCOMP, HPRN-CT-2000-00037, by OTKA grants T 30435 and T 25928 in Hungary, and the Belgian PAI P4/10 project. The help of Cs. S. Daróczi in preparing the animations is gratefully acknowledged.


Research Laboratoires involved in this Work

Time Dependent Schrödinger Equation

Scanning Tunneling Microscopy (STM)

Carbon Nanotubes


  1. Márk,Géza,I.;Biró,László,P.;Gyulai,József:
    Simulation of STM images of 3D surfaces and comparison with experimental data: carbon nanotubes;
    Phys.Rev.B 58, 12645(1998)

  2. Márk,Géza,I.;Biró,László,P.;Gyulai,József:;Thiry,Paul,A.;Lucas,Amand,A.;Lambin,Philippe:
    Simulation of scanning tunneling spectroscopy of supported carbon nanotubes;
    Phys.Rev.B 62, 2797(2000)

Last updated: Dec 15, 2003 by Géza I. Márk ,
This page was accessed  times since April 26, 2001.