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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...
David Bowler
(UCL and
ICYS/NIMS)
Hervé Ness (now
CEA Saclay)
Tchavdar Todorov
(Belfast)
Andrew
Horsfield (UCL)
Christian Sanchez
(Belfast)
Thanks to EPSRC, IRC in Nanotechnology,
Royal Society for funding
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;
C. Durkan, M. A. Schneider, and M. E. Welland, J. App.
Phys. 86, 1280 (1999)
– Molecular electronics.
EIPAM, 19 Apr 2005
Models for atomic-scale electronics
Bloch-like
states
X
Rigid-molecule
(elastic) transport
Molecular and electronic
motions strongly coupled
Need to worry about:
•Fluctuations (e.g. ring torsions)
•Feedback of electrons on
geometrical structure (breakdown
of Born-Oppenheimer
approximation)
•Local heating (diffusion,
electromigration)
Exceptions: nanotubes,
small-molecule STM
(mostly)
EIPAM, 19 Apr 2005
Overview
• An example of a “conventional” approach: solution of the timeindependent 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
+
+
+
+
+
-
-
+
-
Displacement ui
Π-electron tight-binding model linearly coupled to atomic
displacements (Su, Schrieffer and Heeger, 1980)
K
M
Hˆ  [t0   (ui 1  ui )](cˆi†cˆi 1  h.c.)   (ui 1  ui ) 2   u 2
2 i
2 i
i
EIPAM, 19 Apr 2005
The method
Product Hilbert space H el  H vib
Many ‘copies’ of electronic system with different states of
vibrational excitation
nq  3
nq  2
nq  1
nq  0
‘Transitions’ mediated by annihilation/creation operators.
(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
Hilbert space of overall dimension N el ( N max  1) M
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
(charged
chain)
Elastic (neutral chain)
β-factor (attenuation) depends
strongly on inelastic terms
Elastic (neutral 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:
Polaron-dominated
conductance (even chain):
IPinelastic
phonon
e
1 nA  15pW
One phonon emitted
W
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
True distribution of ionic
positions at time t:
R Tre  ˆ (t ) R
Approximation: represent
distributions of ionic positions and
momenta by a single average
value:
R
R (t )  Tr  Rˆ  (t ) 
EIPAM, 19 Apr 2005
First results (Ehrenfest
approximation)
Implemented in tight-binding (non-self-consistent
so far):
Static atoms:
Landauer
value
Vgate
Dynamic atoms (Tinitial=300K)
Vbias=0.1V
(cooling)
Vbias=1.0V
(heating)
Vbias=0
Vbias=1.0V
EIPAM, 19 Apr 2005
Is Ehrenfest good enough?
In an exact calculation, would decompose general electronion Hamiltonian as
Lowest eigenvalue of He, gives BornHˆ   TˆI  Hˆ e ( R)
Oppenheimer potential surface
I


  TI  E0 ( R)  E0 ( R0 )   Hˆ e ( R0 )   Hˆ e ( R)  E0 ( R)  Hˆ e ( R0 )  E0 ( R0 ) 


 I

 Hˆ  H ( R )  Hˆ

I
e
0
 

eI
Expand HI and HeI about reference ionic positions R0:

1
Hˆ I  Hˆ eI   TˆI   Xˆ I  K IJ ( R )  Xˆ J   Xˆ I  Fel ( R ) where Xˆ I  Rˆ I  RI (t )
2 I ,J
I
I
Full ionic heating rate is then
1
 Hˆ I , Hˆ eI  
w

i 
PˆI
ˆ
I FI  M
I
EIPAM, 19 Apr 2005

Is Ehrenfest good enough? (2)
In Ehrenfest approximation: expand around instantaneous average values R(t) and
P(t) of ionic position and momentum:
R (t )  Tr  Rˆ ˆ (t )  ; P (t )  Tr  Pˆ ˆ (t ) 
Ionic motion
Electronic evolution
dRI
P dP
 I ; I  Tre  ˆ e Fˆel ( R ) 
dt
M I dt
ˆ e
i
  Hˆ e ( R ), ˆ e     Fel ( R ), ˆ I 
t
I
with ˆ e  TrI  ˆ (t )  and ˆ I  TrI  Xˆ I ˆ 
Ionic heating rate is now
Average
force from
electrons
PˆI
ˆ
FI 

MI
I
This term usually
neglected
Lose correlations between electrons
and ions; heating may contain large
errors (or even be wrong sign)
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
R Tre  ˆ (t ) R
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.
X
F  Tre  ˆ e Fˆel ( R )    Tre  Kˆ IJ ˆ J 
with ˆ e  TrI  ˆ (t )  and
J
ˆ I  TrI  Xˆ I ˆ 
ˆ e
i
  Hˆ e ( R ), ˆ e     Fel ( R ), ˆ I 
t
I
ˆ
dˆ I

ˆ
i
  H e  R  , ˆ I   i I
dt
M
ˆI  TrI


 PˆI ˆ 


FˆI  FˆI  F
I
dˆI  ˆ
i
i
i
  H e  R  , ˆI  
Fˆ , ˆ e 
dt
2
2
ρe varies
 Kˆ
J
IJ
, ˆ J

First moment approximation.
EIPAM, 19 Apr 2005
 2 Hˆ
K IJ 
RI RJ
R
R
First
moments
of X, P
Going Beyond Ehrenfest (2)
As a starting point, neglect electronic correlations,
use Hartree-Fock approximation.
 (1)  NTr2,3...N [ ˆ e ]
Define
 (2)  N ( N  1)Tr3...N [ ˆ e ]
etc.
Then work entirely in terms of one-particle quantities by
using the extended Hartree-Fock ansatz
 (2, HF ) (1, 2;1', 2 ')   (1) (1,1')  (1) (2, 2 ')   (1) (1, 2 ')  (1) (2,1')
 I(2, HF ) (1, 2;1', 2 ')   I (1) (1,1')  (1) (2, 2 ')   (1) (1,1')  I (1) (2, 2 ')
  I (1) (1, 2 ')  (1) (2,1')   (1) (1, 2 ')  I (1) (2,1')
I(2, HF ) (1, 2;1', 2 ')  I (1) (1,1')  (1) (2, 2 ')   (1) (1,1')I (1) (2, 2 ')
I (1) (1, 2 ')  (1) (2,1')   (1) (1, 2 ')I (1) (2,1')
EIPAM, 19 Apr 2005
Beyond Ehrenfest – results
(1) Local Heating
Ionic energy change now
contains original (classical) part
plus new quantum part:
w
I
1
( FI  PI  Tre  FˆI  ˆI  )
MI
Quantum
(heats ions)
DINAMO code
(Sanchez et al)
Classical
(cools ions)
EIPAM, 19 Apr 2005
Increasing bias
Beyond Ehrenfest – results
(2) Inelastic Spectroscopy
CEID (at first
moment level)
already contains
enough
information to
describe IETS
Expected position of
inelastic peak: 0.26 V
Sanchez, Todorov, Horsfield
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!
EIPAM, 19 Apr 2005