Time dependent QM: Nanotube in 2D
To analyse in more detail the consequences of the second tunnelling gap we have studied the effects of the geometry on the tunnelling of an electron from the STM (Scanning Tunneling Microscope) tip through the nanotube to the sample by a dynamical one particle wave function calculation.

In the first approximation a simple model was used, which takes in account only the geometry of the objects involved in the tunnelling process. An infinitely small bias was assumed. The time development of the (quantum mechanical) probability density Rho(x,z,t) is calculated from the Psi(x,z,t) solution of the time dependent Schrödinger equation which was solved numerically [1]. (Click here to read about the theoretical background!)

Selected results from this simulation was presented as part of an invited talk at E-MRS Spring Meeting, June 16-20, 1997, Strasbourg, France.

Initital conditions

The initial wave function is a two dimensional Gauusian wave packet.
initial position { x0, y0 } : { 0, 32 } Å
initial impulse { px0, py0 } : { 0, -0.756 } Å-1
DeltaX = { 10, 10 } / Sqrt[2] Å

Calculated properties of the initial wave packet:
DeltaP = { 0.133, 0.133 } Å-1
EKin = 2.31 eV
T2 = { 4.17, 4.17 } fs

Where:
DeltaX = Sqrt[ < x2 > - < x >2 ]
DeltaP = Sqrt[ < p2 > - < p >2 ]
EKin = < p2 > / ( 2 m )
T2 = the time while DeltaX doubles

The potential

The simple model potential is derived from the assumption that the electron moves freely inside the tip, and the HOPG, and inside the wall of the nanotube. The work function is taken[2] to be 2.49 eV and the Fermi energy 2.31 eV . The jellium surface is assumed to be at 0.71 Å distance from the geometrical surface of the electrodes.

Nanotube diameter: 10 Å
Tip radius: 5 Å
Tip - nanotube distance: 4 Å
Nanotube - substrate distance: 3.35 Å

Description of phenomena

The detailed simulation shows that due to the presence of the second tunnelling gap a significant fraction of the wave-packet is reflected back to the tip, i.e., the reduction of the tunnelling current occurs even when the differences in the LDOS values of the support and of the nanotube are not taken in account.

Probability of finding the electron at the tube region

Total probability of finding the electron at the presentation window

Maximum probability at the presentation window


Details of the calculation

Size of the calculation box: 153.6 x 153.6 Å (512x512 points)
Size of the presentation box: 38.4 x 38.4 Å (128x128 points)
Calculation time step: 0.0048 fs
Number of calculation time steps: 2000
Presentation time step: 0.048 fs
Total calculation time: 9.65 fs

Images

The Abs2[ Psi[ x, t ] ] phase images are available here.

Animation

The Abs2[ Psi[ x, t ] ] animation is displayed in two modes and in various file formats.

References

[1]Feit, M.D., Fleck, J.A., Steiger, A.: J. Comput. Phys. 47 412 (1982)

[2] Géza I. Márk, Dynamical calculation of tunnel current for STM, NANO IV (Fourth International Conference on Nanometer-Scale Science & Technology, 8-12 September 1996, Beijing, China) p. 186

Links

Last updated: Feb 13, 98 Géza I. Márk mark@sunserv.kfki.hu

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