Presentazione di PowerPoint

1
Charge transport in molecular devices
Aldo Di Carlo, A. Pecchia, L. Latessa, M.Ghorghe
Dept. Electronic Eng. University of Rome Tor
Vergata, (ITALY)
P. Lugli
TU-Munich (GERMANY)
Collaborations
T. Niehaus, T. Frauenheim
University of Paderborn (GERMANY)
G. Seifert
TU-Dresden (GERMANY)
R. Gutierrez, G. Cuniberti
University of Regensburg (GERMANY)
European Commission Project
2
What about realistic nanostructured devices ?
Traditionally, nanostructures are studied via k
p approaches in the context of the envelope
function approximation (EFA). In this case, only
the envelope of the nanostructure wavefunction is
considered, regardless of atomic details.
Modern technology, however, pushes
nanostructures to dimensions, geometries and
systems where the EFA does not hold any
more. Atomistic approaches are required for the
modeling structural, electronic and optical
properties of modern nanostructured devices.
3
Transport in nanostructures
active region where symmetry is lost contact
regions (semi-infinite bulk)
The transport problem is
activeregion
contact
contact
contact
Open-boundary conditions can be treated within
several schemes

• Transfer matrix
• LS scattering theory
• Green Functions .

These schemes are well suited for localized
orbital approach like TB
4
Atomistic approaches The Tight-Binding method
We attempt to solve the one electron Hamiltonian
in terms of a Linear Combination of Atomic
Orbitals (LCAO)
The approach can be implemented ab-initio where
the orbitals are the basis functions and Hia ,jb
is evaluated numerically
5
Scalability of TB approaches
Empirical Tight-Binding
Hamiltonian matrix elements are obtained by
comparison of calculated quantities with
experiments or ab-initio results. Very efficient,
poor transferability.
Semi-Empirical Tight-Binding
Density Functional Tight-Binding
Density-functional based methods permit an
accurate and theoretically well founded
description of electronic properties for a wide
range of materials.
6
Si/SiO2 tunneling. empirical TB sp3d5s
Staedele, et al. J. Appl. Phys. 89 348 (2001
) Sacconi et al. Solid State Elect. 48 575
(04) IEEE TED in press
b-critobalite
b-quartz
tridymite
Empirical parameterizations are necessary due to
the band gap problem of ab-initio approaches
7
Tunneling Current Comparison with experimental
data
non-par. EMA
par. EMA
8
Toward ab-initio approaches. Density
Functional Tight-Binding

• Many DFT tight-binding SIESTA (Soler etc.),
  FIREBALL (Sankey),
  DMOL (Delley), DFTB (Seifert,
  Frauenheim etc.) ..
• The DFTB approach Elstner, et al. Phys. Rev. B
  58 (1998) 7260
• provides transferable and accurate interaction
  potentials. The numerical efficiency of the
  method allows for molecular dynamics simulations
  in large super cells, containing several thousand
  of atoms.
• DFTB is fully scalable (from empirical to DFT)
• DFTB allows also for TD-DFT simulations
• We have extended the DFTB to account for
  transport in organic/inorganic nanostructures by
  using Non Equilibrium Green Function approach
  self-consistently coupled with Poisson equation

9
DFTB
Tight-binding expansion of the wave
functions Porezag, et al Phys. Rev. B 51 (1995)
12947
DFT calculation of the matrix elements,
two-centers approx.
Self-Consistency in the charge density (SCC-DFTB)
10
Non equilibrium systems
The contact leads are two reservoirs in
equilibrium at two different elettro-chemical
potentials.
f2
f1

How do we fill up the states ?
How to compute current ?
11
How do we fill up states ? (Density matrix)
The crucial point is to calculate the
non-equilibrium density matrix when an external
bias is applied to the molecular device
Three possible solutions

• Ignore the variation of the density matrix (we
  keep H0) Suitable for situations very close to
  equilibrium (Most of the people do this !!!!)
• The new-density matrix is calculated in the usual
  way by diagonalizing the Hamiltonian for the
  finite system Problem with boundary conditions,
  larger systems
• The new-density is obtained from the
  Non-Equilibrium Greens Function theory
  Keldysh 60 Caroli et al. 70 Datta 90

12
DFTB Green Functions
Systems close to the equilibrium

• Molecular vibrations and current

(details Poster 16)
13
The role of molecular vibrations
T 300 K
An organic molecule is a rather floppy entity

• We compare
• Time-average of the current computed at every
  step of a MD simulation (Classical vibrations)
• Ensemble average of over the lattice fluctuations
  (quantum vibrations phonons).

A. Pecchia et al. Phys. Rev. B. 68, 235321
(2003).
14
Molecular Dynamics current
Di Carlo, Physica B, 314, 211 (2002)
The dynamics of the a-th atom is given by
The evolution of the system is performed on a
time scale of 0.01 fs
15
Molecular dynamics limitations
The effect of vibrations on the current flowing
in the molecuar device,via molecular dynamics
calculations, has been obtained
withoutconsidering the quantization effects of
the vibrational field.
The quantum nature of the vibrations (phonons) is
not considered !
However, vibration quantization can be considered
by performing ensamble averages of the current
over phonon displacements
H. Ness et al, PRB 63, 125422
How does it compare with MD calculations ?
16
The lowest modes of vibration
17
Phonons
H. Ness et al, PRB 63, 125422
The hamiltonian is a superposition of the
vibrational eigenmodes, k
The eigenmodes are one-dimesional harmonic
oscillators with a gaussian distribution
probability for qk coordinates

18
The current calculation

• The tunneling probability is computed as an
  ensemble average over the atomic positions (DFTB
  code Green Fn.)

• We average the log(T) because T is a
  statistically ill-defined quantity (is dominated
  by few events). MC integration

• The current is computed as usual

19
Transmission functions
MD Simulations
Quantum average
20
Comparison MD, Quantum PH, Classical PH
QPH phonon treatement CPH phonons treatement
without zero point energy
21
Frequency analysis of MD results
Mol. Dynamics
Fourier Transf.
A. Pecchia et al. Phys. Rev. B. 68, 235321
(2003).
22
I-V characteristics
Molecular dynamics
Quantum phonons
Harmonic approximation failure produces incorrect
results of the quantum phonontreatement of
current flowing in the molecule
23
DFTB Non-Equilibrium Green Functions

• Full Self-Consistent results
• Electron-Phonon scattering

(details Posters 34 and 37)
24
Self-consistent quantum transport
Self-consistent loop
Density Matrix
Mullikan charges Correction
SC-loop
Di Carlo et. al. Physica B, 314, 86 (2002)
25
Charge and Potential in two CNT tips
Potential Profile
Equilibrium charge density
Charge density with 1V bias
Charge neutrality of the systemis only achived
in large systems
Net charge density
Negative charge
Positive charge
26
Self-consistent charge in a molecular wire
0.5 V
1.0 V
27
CNT-MOS Coaxially gated CNT
VG
VD
VS0
5 nm
1.5 nm
Semiconducting (10,0) CNT
CNT contact
Insulator (er3.9)
28
CNT-MOS
2.10-5
0
-4.10-5
-8.10-5
Potential
Charge
Isosurfaces of Hartree potential and contour plot
of charge density transfer computed for an
applied gate bias of 0.2 V and a source-drain
bias 0f 0.0 V
29
Output characteristics
Gate coupling (capacitance) is too low. A precise
design is necessary (well tempered CNT-MOS)
30
Electron-phonon self-energy
The el-ph interaction is included to first order
(Born approximation) in the self-energy
expansion.
A. Pecchia, A. Di Carlo Report Prog. in Physics
(2004)
Born approximation
31
Simple linear chain system
?q17 meV, E0 0.06 eV
emission
resonance
absorption
incoherent
coherent
32
Inelastic scattering Current phonons
I(E)
33
IV Current phonons
No phonons
34
Conclusions
The method

• Density Functional Tight-Binding approach has
  been extended to account for current transport in
  molecular devicesby using Self-consistent
  non-equilibrium Green function (gDFTB ).
• DFTB is a good compromise between simplicity and
  reliability.
• The use of a Multigrid Poisson solver allows for
  study very complicated device geometries
• Force field and molecular dynamics can be easily
  accounted in the current calculations.
• Electron-phonon coupling can be directly
  calculated via DFTB
• Electron-phonon interaction has been included in
  the current calculations.

For the gDFTB code visit http//icode.eln.unirom
a2.it
35
Conclusions
Results

• Anharmonicity of molecular vibrations can limit
  the use of phonon concepts
• Concerning ballistic transport, temperature
  dependence of current is better described whit
  molecular dynamics than ensamble averages of
  phonon displacements
• Screening length in CNT could be long.
• Coaxially gated CNT presents saturation effects
  but gate control is critical.
• Electron-phonon scattering is not negligible
  close to resonance conditions of molecular
  devices
• All the details in A. Pecchia, A. Di Carlo Report
  Prog. in Physics (2004)

For the gDFTB code visit http//icode.eln.unirom
a2.it