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UTor Vergata
Charge transport in molecular devices
Aldo Di Carlo, A. Pecchia, L. Latessa, M.Ghorghe*Dept. Electronic Eng. University of Rome “Tor Vergata”, (ITALY)
T. Niehaus, T. FrauenheimUniversity of Paderborn (GERMANY)
IH P R esea rch Tra in ing N e tw orkEuropean Commission Project
P. LugliTU-Munich (GERMANY)
Collaborations
G. Seifert TU-Dresden (GERMANY)
R. Gutierrez, G. Cuniberti*University of Regensburg (GERMANY)
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What about realistic nanostructured devices ?
1D (quantum wells): 100-1000 atoms in the unit cell
2D (quantum wires): 1000-10’000 atoms in the unit cell
3D (quantum dots): 100’000-1’000’000 atoms in the unit cell
OrganicsMolecules, Nanotubes, DNA: 100-1000 atoms (or more)
Inorganics
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.
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Transport in nanostructures
The transport problem is:
activeregioncontact
con
tact
contact
active region where symmetry is lost +
contact regions (semi-infinite bulk)
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
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Atomistic approaches: The Tight-Binding method
The approach can be implemented “ab-initio” where the orbitals are the basis functions and Hi,jis evaluated numerically
0
jsite,atomic orbitals,
,,
jjiji CESH
jiji HH , jijiS ,
i
iiin C
site,atomic orbitals,
Rrr
We attempt to solve the one electron Hamiltonian in terms of a Linear Combination of Atomic Orbitals (LCAO)
Ci
orbital
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Scalability of TB approaches
Density-functional based methods permit an accurate and theoretically well
founded description of electronic properties for a wide range of materials.
Density Functional Tight-Binding
Empirical Tight-Binding
Semi-Empirical Tight-Binding
Hamiltonian matrix elements are obtained by comparison of calculated quantities with experiments or ab-initio results. Very efficient, poor transferability.
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SiO2
p-SiPoly-Si-gate
Si/SiO2 tunneling:. empirical TB sp3d5s*
Empirical parameterizations are necessary due to the band gap problem of ab-initio approaches
-critobalite -quartz
tridymite
Staedele, et al. J. Appl. Phys. 89 348 (2001 )Sacconi et al. Solid State Elect. 48 575 (‘04) IEEE TED in press
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Tunneling Current: Comparison with experimental data
-cristobalite
-quartz
tridimite-cristobalite
Good agreement between experimental and TB results for the -cristobalite polimorph
Slope of the current density is related to the microscopic structure of SiO2
non-par. EMA
par. EMA
g
CBCB E
EEmEm 1
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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
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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.
(2)0
,
12
repi i i
i
E n H q q E
Self-Consistency in the charge density (SCC-DFTB)
II order-expansion of Kohn-Sham energy functional[Elstner, et al. Phys. Rev. B 58 (1998) 7260]
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Non equilibrium systems
The contact leads are two reservoirs in equilibrium at two different elettro-chemical potentials.
How do we fill up the states ?
f2f1
How to compute current ?
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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:
1. Ignore the variation of the density matrix (we keep H0)Suitable for situations very close to equilibrium(Most of the people do this !!!!)
2. The new-density matrix is calculated in the usual way by diagonalizing the Hamiltonian for the finite system
Problem with boundary conditions, larger systems
3. The new-density is obtained from the Non-Equilibrium Green’s Function theory [Keldysh ‘60] [Caroli et al. ’70] [Datta ’90]
. . .
. . .
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DFTB+
Green Functions
Systems close to the equilibrium
• Molecular vibrations and current
(details: Poster 16)
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The role of molecular vibrations
An organic molecule is a rather floppy entity
T= 300 K
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).
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Molecular Dynamics + current
The dynamics of the -th atom is given by
The evolution of the system is performed on a time scale of ~ 0.01 fs
FR
2
2
dt
dM
RR d
dE
d
dHn rep
kkkk =
Hamiltonian matrix
Calculation of the forces
Atomic position update
t=t+t
Molecular dynamics
Hamiltonian matrix
Calculation of the forces
Atomic position update
Current calculation [ J(t) ]t=t+t
Molecular dynamics + current
Di Carlo, Physica B, 314, 211 (2002)
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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 ?
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The lowest modes of vibration
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The hamiltonian is a superposition of the vibrational eigenmodes, k:
232 2
21
1 1
2
N
k k kk k k
H m qm q
2 2 2
, exp( )2 ( , ) 2 ( , )
BE k k kk k B
th k B th k B
m m qP q K T
E K T E K T
Phonons
The eigenmodes are one-dimesional harmonic oscillators with a gaussian distribution probability for qk coordinates:
1,
2th thE n kT
H. Ness et al, PRB 63, 125422
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The current calculation
• The tunneling probability is computed as an ensemble average over the atomic positions (DFTB code + Green Fn.)
2
221 3 1 3
1log ( ) ... log ({ ... })
2
k
k
q
N Nk k
T E dq dq T q q e
1 2
2( ) exp( log ( ) )[ ( ) ( )]
ei E T E f E f E dE
h
• The current is computed as usual:
• We average the log(T) because T is a statistically ill-defined quantity (is dominated by few events). MC integration
aDR
rDL GGtrT
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Transmission functions
MD Simulations Quantum average
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Comparison: MD, Quantum PH, Classical PH
QPH = phonon treatementCPH = phonons treatement without zero point energy
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Frequency analysis of MD results
Mol. Dynamics
Fourier Transf.
S-Au stretchCC stretch
Ph-twist
A. Pecchia et al. Phys. Rev. B. 68, 235321 (2003).
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I-V characteristics
Molecular dynamics Quantum phonons
Harmonic approximation failure produces incorrect results of the quantum phonontreatement of current flowing in the molecule
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DFTB+
Non-Equilibrium Green Functions
• Full Self-Consistent results
• Electron-Phonon scattering
(details: Posters 34 and 37)
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BULK
Surface
BULK
Surface
Devic
eSelf-consistent quantum transport
SELF-CONS. DFTB WITH POISSON
3D MULTI-GRID
Self-consistent loop:
1
2dE G
i
0q q S
Density Matrix
Mullikan charges Correction
2 4
'Hn V n
H G nSC-loop
2[()()()()]
eITrEGEEGEdE
h
Di Carlo et. al. Physica B, 314, 86 (2002)
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Equilibrium charge density
Charge density with 1V bias
Net charge density
Negative charge Positive charge
Charge and Potential in two CNT tips
Potential Profile
Charge neutrality of the systemis only achived in large systems
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Self-consistent charge in a molecular wire
1.0 V
0.5 V
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CNT-MOS: Coaxially gated CNT
VD VS=0
VG
Semiconducting (10,0) CNT
Insulator (εr=3.9)
5 nm 1.5 nm
x
yz
CNT contact
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CNT-MOS
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
Potential Charge
2.10-5
-8.10-5
0
-4.10-5
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Output characteristics
Gate coupling (capacitance) is too low. A precise designis necessary (well tempered CNT-MOS)
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Electron-phonon self-energy
Born –approximation
20( ) ' ( ') ( ')
2 q qq
iE dE G E E D E
The el-ph interaction is included to first order (Born approximation) in the self-energy expansion.
()()()()ra
GEGEEGE
2[()()()()]
eITrEGEEGEdE
h
Directly from DFTB hamiltonian
[A. Pecchia, A. Di Carlo Report Prog. in Physics (2004)]
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Simple linear chain system
q=17 meV, E0 = 0.06 eV
ab
sorp
tion
em
issio
n
reson
an
ce
incoherent
coherent
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Inelastic scattering: Current + phonons
I(E
)
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0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.002468
1012141618202224
Cu
rre
nt
(A)
Applied Voltage [V]
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.002468
1012141618202224
Cu
rre
nt
(A)
Applied Voltage [V]
T=0 K
Coherent
Incoherent
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.002468
1012141618202224
Cu
rre
nt
(A)
Applied Voltage [V]
T=150 K
Coherent
Incoherent
No phonons
IV Current + phonons
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Conclusions
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.uniroma2.it
The method
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Conclusions
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)
Results
For the gDFTB code visit: http://icode.eln.uniroma2.it