EIPAM, 19 Apr 2005

Coupled electron-ion dynamics: Introduction to CEID

David Bowler [1,2], Andrew Fisher [1], Andrew Horsfield [1], Tchavdar Todorov [3], Christian Sanchez [3]

[1] University College London[2] International Center for Young Scientists, NIMS, Tsukuba

[3] Queen’s University Belfast

EIPAM, 19 Apr 2005

Those who did the work...

Hervé Ness (now CEA Saclay)

David Bowler (UCL and ICYS/NIMS)

Thanks to EPSRC, IRC in Nanotechnology, Royal Society for funding

Andrew Horsfield (UCL)

Tchavdar Todorov (Belfast)

Christian Sanchez (Belfast)

EIPAM, 19 Apr 2005

Electron-ion dynamics: context• Interactions between electronic and atomic

degrees of freedom important in many places in physics, chemistry and nanoscience:

– Local heating in nanostructures;

– Local current-voltage spectroscopy and STM-induced surface chemistry;

– Decoherence of electronic processes used for quantum information processing;

– Molecular electronics.C. Durkan, M. A. Schneider, and M. E. Welland, J. App. Phys. 86, 1280 (1999)

EIPAM, 19 Apr 2005

Models for atomic-scale electronics

Rigid-molecule (elastic) transport

Need to worry about:•Fluctuations (e.g. ring torsions)•Feedback of electrons on geometrical structure (breakdown of Born-Oppenheimer approximation)

•Local heating (diffusion, electromigration)

XMolecular and electronic motions strongly coupled

Exceptions: nanotubes, small-molecule STM (mostly)

Bloch-like states

EIPAM, 19 Apr 2005

Overview• An example of a “conventional” approach: solution of the time-

independent coupled electron-lattice Schrödinger equation• The CEID approach:

– Aim: a Car-Parrinello-like revolution for coupled electron-ion dynamics

– Analysis of the local heating problem– The Ehrenfest approximation– Going beyond Ehrenfest– First results from the DINAMO code

• Survey of future plans

EIPAM, 19 Apr 2005

Conducting polymers: the simplest model

† 2 20 1 1 1

ˆ ˆ ˆ[ ( )]( h.c.) ( )2 2i i i i i i

i i i

K MH t u u c c u u u

+

-

+

-+

-

+

-+

-

+

-

Π-electron tight-binding model linearly coupled to atomic displacements (Su, Schrieffer and Heeger, 1980)

Displacement iu

EIPAM, 19 Apr 2005

The method

Many ‘copies’ of electronic system with different states of vibrational excitation

0qn

1qn

2qn

3qn

‘Transitions’ mediated by annihilation/creation operators.

el vibProduct Hilbert space H H

(Bonca and Trugman, 1995)

EIPAM, 19 Apr 2005

The basis set

Reference system:

‘Neutral’ chain (N atoms, N electrons)

Add single carrier (electron or hole) in one of N/2 states

Include lowest Nmax states of M chosen oscillators

el maxHilbert space of overall dimension ( 1)MN N

EIPAM, 19 Apr 2005

Polarons affect conductance• Increases tunnel conductance,

because carrier has to ‘borrow’ less energy to tunnel through the molecule

Polaron-assisted

Elastic (neutral chain)

Elastic (neutral chain)

β-factor (attenuation) depends strongly on inelastic terms

Elastic (charged chain)

EIPAM, 19 Apr 2005

Heating• These large effects on current

also involve a small probability of energy loss (corresponding to excitations remaining within the molecule).

• Dominant processes are “virtual” ones where lattice vibrations are produced and then re-emitted.

• Nevertheless corresponds to substantial heating rate:

One phonon emittedinelasticphonon

1 nA 15pW

IPW

e

Polaron-dominated conductance (even chain):

EIPAM, 19 Apr 2005

Towards CEID: time-dependent conduction model

Model current as the discharge of a capacitor through a resistor.

Enables incorporation of other time-dependent effects due to ions/atoms.

Want a method that works for a general (possibly large) R having many almost classical degrees of freedom.

EIPAM, 19 Apr 2005

The Ehrenfest Approximation

Simplest approach to coupled quantum-classical dynamics: Ehrenfest approximation

Approximation: represent distributions of ionic positions and momenta by a single average value:

True distribution of ionic positions at time t:

ˆTr ( )eR t R

ˆ( ) Tr ( )R t R t

R

EIPAM, 19 Apr 2005

First results (Ehrenfest approximation)

Implemented in tight-binding (non-self-consistent so far):

Vbias=0.1V (cooling)

Vbias=1.0V (heating)

Dynamic atoms (Tinitial=300K)

Vgate

Vbias=0 Vbias=1.0V

Static atoms:

Landauer value

EIPAM, 19 Apr 2005

Is Ehrenfest good enough?

0 0 0 0 0 0 0 0

0

ˆ ˆ ˆ ( )

ˆ ˆ ˆ( ) ( ) ( ) ( ) ( ) ( ) ( )

ˆ ˆ( )

I eI

I e e eI

I e eI

H T H R

T E R E R H R H R E R H R E R

H H R H

In an exact calculation, would decompose general electron-ion Hamiltonian as

Lowest eigenvalue of He, gives Born-Oppenheimer potential surface

Expand HI and HeI about reference ionic positions R0:

,

1ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ( ) ( ) where ( )2I eI I I IJ J I el I I I

I I J I

H H T X K R X X F R X R R t

Full ionic heating rate is thenˆ1 ˆ ˆ ˆ,

iI

I eI II I

Pw H H F

M

EIPAM, 19 Apr 2005

Is Ehrenfest good enough? (2)

ˆ ˆˆ ˆ( ) Tr ( ) ; ( ) Tr ( )

d d ˆˆ; Tr ( )d d

ˆ ˆ ˆ ˆi ( ), ( ),

ˆˆ ˆ ˆˆwith Tr ( ) and Tr

I I Ie e el

I

ee e el I

I

e I I I I

R t R t P t P t

R P PF R

t M t

H R F Rt

t X

In Ehrenfest approximation: expand around instantaneous average values R(t) and P(t) of ionic position and momentum:

Ionic heating rate is now

ˆˆ I

II I

PF

M

Lose correlations between electrons and ions; heating may contain large errors (or even be wrong sign)

Average force from electrons

This term usually neglected

Ionic motion

Electronic evolution

EIPAM, 19 Apr 2005

Is Ehrenfest good enough? (3)

Calculate heating/cooling of a single Einstein oscillator, forming a 1eV potential barrier between two reservoirs and heated by electrons of different biases.

Shows ionic cooling (and heating of electrons) even for biases (~1eV) much larger than initial ionic K.E.

Ehrenfest approximation does not give correct

physics

EIPAM, 19 Apr 2005

Going Beyond Ehrenfest

ˆ ˆˆ ˆTr ( ) Tr

ˆ ˆ ˆ ˆi ( ), ( ),

ˆˆd ˆ ˆi , id

ˆd i iˆˆ ˆ ˆˆ ˆi , , ,d 2 2

e e el e IJ JJ

ee e el I

I

I Ie I

I

Ie I e IJ J

J

F F R K

H R F Rt

H Rt M

H R F Kt

Must keep the terms we formerly neglected. Do this by making a systematic moments expansion about the average ionic trajectory, keeping correlations between electrons and ions.

2

ˆ ˆwith Tr ( ) and

ˆ ˆˆ Tr

ˆ ˆ ˆTr

ˆ ˆ

ˆ

e I

I I I

I I I

I I

IJI J R

t

X

P

F F F

HK

R R

First moment approximation.

First moments of X, P

ˆTr ( )eR t R

R

ρe varies

X

EIPAM, 19 Apr 2005

Going Beyond Ehrenfest (2)

(2, ) (1) (1) (1) (1)

(2, ) (1) (1) (1) (1)

(1) (1) (1) (1)

(2, ) (1) (

(1, 2;1', 2 ') (1,1') (2, 2 ') (1, 2 ') (2,1')

(1, 2;1', 2 ') (1,1') (2, 2 ') (1,1') (2, 2 ')

(1, 2 ') (2,1') (1, 2 ') (2,1')

(1, 2;1', 2 ') (1,1')

HF

HFI I I

I I

HFI I

1) (1) (1)

(1) (1) (1) (1)

(2, 2 ') (1,1') (2, 2 ')

(1, 2 ') (2,1') (1, 2 ') (2,1')

I

I I

As a starting point, neglect electronic correlations, use Hartree-Fock approximation.

(1)2,3...

(2)3...

ˆTr [ ]

ˆ( 1)Tr [ ]

etc.

N e

N e

N

N N

Define

Then work entirely in terms of one-particle quantities by using the extended Hartree-Fock ansatz

EIPAM, 19 Apr 2005

Beyond Ehrenfest – results(1) Local Heating

Ionic energy change now contains original (classical) part plus new quantum part:

Classical (cools ions)

Quantum (heats ions)

DINAMO code (Sanchez et al)

Increasing bias

1 ˆˆ( )I I e I II I

w F P Tr FM

EIPAM, 19 Apr 2005

Beyond Ehrenfest – results(2) Inelastic Spectroscopy

CEID (at first moment level) already contains enough information to describe IETS

Sanchez, Todorov, Horsfield

Expected position of inelastic peak: 0.26 V

EIPAM, 19 Apr 2005

Our plans• We plan a three-pronged development programme for CEID over the

next four years, focussing on– Implementing the second moment approximation– Local heating and vibrational spectroscopy in nanostructures– Electron-lattice coupling and degradation in conducting polymer

films– Electron-ion energy transfer during radiation damage in solids

• We will also be working on– STM-IETS (with Geoff Thornton, Werner Hofer)– Charge transport and oxidative damage in biomolecules (with

Sarah Harris, William Barford)– Decoherence induced by electron-lattice coupling in other

quantum systems (e.g. dopant spins in semiconductors, quantum dots)

EIPAM, 19 Apr 2005

To read more:• Open-boundary Ehrenfest molecular dynamics: towards a model of

current induced heating in nanowires.  A.P. Horsfield, D.R. Bowler and A.J. Fisher.  J. Phys.: Conden. Matt. 16 L65 (2004). 

• Power dissipation in nanoscale conductors: classical, semi-classical and quantum dynamics.  A.P. Horsfield, D.R. Bowler, A.J. Fisher, T.N. Todorov and M.J. Montgomery. J. Phys.: Conden. Matt. 16 3609-3622 (2004).

• Beyond Ehrenfest: correlated non-adiabatic Molecular Dynamics.  A.P. Horsfield, D.R. Bowler, A.J. Fisher, T.N. Todorov, and C. Sanchez. J. Phys.: Conden. Matt. 16 8251-8266 (2004). 

Thank you for your attention!