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)

UTor Vergata

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.

UTor Vergata

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

UTor Vergata

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

UTor Vergata

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.

UTor Vergata

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

UTor Vergata

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

UTor Vergata

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

UTor Vergata

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]

UTor Vergata

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 ?

UTor Vergata

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]

. . .

. . .

UTor Vergata

DFTB+

Green Functions

Systems close to the equilibrium

• Molecular vibrations and current

(details: Poster 16)

UTor Vergata

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).

UTor Vergata

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)

UTor Vergata

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 ?

UTor Vergata

The lowest modes of vibration

UTor Vergata

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

UTor Vergata

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

UTor Vergata

Transmission functions

MD Simulations Quantum average

UTor Vergata

Comparison: MD, Quantum PH, Classical PH

QPH = phonon treatementCPH = phonons treatement without zero point energy

UTor Vergata

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).

UTor Vergata

I-V characteristics

Molecular dynamics Quantum phonons

Harmonic approximation failure produces incorrect results of the quantum phonontreatement of current flowing in the molecule

UTor Vergata

DFTB+

Non-Equilibrium Green Functions

• Full Self-Consistent results

• Electron-Phonon scattering

(details: Posters 34 and 37)

UTor Vergata

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)

UTor Vergata

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

UTor Vergata

Self-consistent charge in a molecular wire

1.0 V

0.5 V

UTor Vergata

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

UTor Vergata

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

UTor Vergata

Output characteristics

Gate coupling (capacitance) is too low. A precise designis necessary (well tempered CNT-MOS)

UTor Vergata

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)]

UTor Vergata

Simple linear chain system

q=17 meV, E0 = 0.06 eV

ab

sorp

tion

em

issio

n

reson

an

ce

incoherent

coherent

UTor Vergata

Inelastic scattering: Current + phonons

I(E

)

UTor Vergata

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

UTor Vergata

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

UTor Vergata

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