Title: Problems in standard DFT calculations of transport across single molecules
1Problems in standard DFT calculations of
transport across single molecules
Kieron Burke and friends Department of Chemistry
and Chemical Biology Department of Physics and
Astronomy Rutgers Princeton Institute for Science
and Technology of Materials PRISM
http//dft.rutgers.edu
2Outline
- Rigorous density functional theory
- Errors in transport calculations
- Within standard formulation
- Test standard model for weak bias
- Beyond standard model for finite bias
3He atom in Kohn-Sham DFT
Dashed-line EXACT KS potential
4Exact ground-state DFT and now time-dependent DFT
In time-dependent external field
For a given interaction and statistics HS KS
RG KS
5Benzene Fruitfly of TDDFT
First-principles density-functional calculations
for optical spectra of clusters and
nanocrystals I. Vasiliev, S. Ogut, and J.R.
Chelikowsky, Phys. Rev. B 65, 1 15416 (2002).
6Lesson from basics
- Rigorous DFT If we had exact XC functional,
wed get exact energy. - Yields correct energetics and densities, but eg
KS gap not the fundamental gap. - In TDDFT, exact TD XC kernel produces exact
transitions, which differ from KS transitions.
7Transport in single molecules
- Numerous transport experiments on organic
molecules - Theory standard approach Landauer-Büttiker
scattering states/Greens-functions
fromground-state DFT for extended molecule - Comparison with experimentqualitative agreement
in many cases, BUTconductance 1-3 orders of
magnitude too high
8 Present-day calculations
- Use ground-state DFT within Landauer picture
- Fix left and right chemical potentials
- Solve self-consistently for KS Greens function
- Use scattering states or Greens functions to
calculate conductance. - Often (confusingly) called NEGF (TranSIESTA)
- No empirical parameters, suggests confidence
level of other DFT calculations
9Recent example
10Three different levels
- Within standard model are calculations good
enough? (origin of overestimate of conductance). - Test standard model weak bias allows linear
response. - Beyond standard model how to do a microscopic
DFT derivation.
11I. Problems within standard model
- Assume all is well with doing ground-state KS DFT
calculation, and applying Landauer-Buttiker
formula to result. - Is this procedure likely to give accurate result?
12Derivative discontinuity Perdew,Parr,Levy and
Balduz, PRL 82
m
ehomo(N) -I
m
ehomo(N1)-A
ehomo(N) -I
13KS potential of H atom
14Derivative discontinuity
- Consider isolated system weakly coupled to
reservoir. - When infinitesimal charge added to system, KS
potential jumps discontinuously by I-A. - Missing in smooth density functionals.
- Present in SIC functionals and exact exchange
15Effect on resonant tunneling (Koentopp, Evers,
et al. PRB 04, xxx05).
- double barrier resonance shape and position
- compare smooth functional with exact result
- conductance of benzenedithiolHF instead of
DFT/GGA
Peaks too broad, wrong postion
-2
T reduced by 100
16Missing derivative discontinuity
- Local functionals miss derivative discontinuity
- Resonances smeared in LDA, yielding overestimated
current
17Sanvito calculations
Tohar, Filipetti, Sanvito, and KB (PRL, to
appear).
- For weak coupling, see much lower conductance
when SIC turned on.
V(eV)
18Aside Resonance widths in transport(Maxime
Dion, KB, in progress)
- When energy near a resonance, both transmission
and electron occupation of the scatterer
dramatically increase. - Often assumed that the width of the transmission
is proportional to that of the occupation. - We are constructing a methodology to extract
exact results in limit of narrow resonances.
Tanaguchi and Buttiker, PRB (1999)
19II. Testing the standard model
- Standard model seems reasonable, but can we
derive it? - Instead of general case, look at weak bias alone
(V-gt0), where Kubo response applies.
20 Polarizability in DFT
- Standard DFT quantum chemical calculation
Calculate static polarizability of molecule using
linear response
21Static density response eqns
- Three different ways to calculate same
- Full non-local susceptibility in response to
external field - Proper susceptibility in response to total
potential - KS susceptibility in response to KS pot
22 Standard model misses XC response for weak bias
- In linear response
- Standard calcs (Landauer-Buttiker) approximate
susc with KS value, ie equivalent of Hartee
response! - Local approxs yield zero correction
interacting system
KS-system
in 1D
23Conclusions from II.
- Standard model Hartree response (at best)
- XC corrections hard to calculate.
- Recall part I, theyll most likely show up in
exact X calculations. - See also Na Sai et al, PRL 05.
24III. Finite bias approach
- Many groups now working on microscopic derivation
- Charging big electrodesTodorov, Di Ventra,
Vignale. - Non-equilibrium Greens functions Kurth, Rubio,
Gross, Almbladh, Stefanucci - Dissipation Car and Burke
25 No isolated systems in DFT
- The HK theorem fails for systems with regions of
zero density. - Cannot have isolated subsystems, and turn on
interaction (even with TDDFT).
26Finite Electric Fields WITHOUT open boundaries
- Treats electric field in periodic potential
correctly no left and right chemical potentials - No empirical parameters
- Leads chemically accurate (CP code)
- Entire problem becomes time-dependent, with no
ground state, but only steady-state solutions - So must have dissipation!
27Results for finite bias
- Evolve QM in master eqn, not Schrödinger
equation. - Generalize TDCDFT to include dissipation
- Produces a TD KS master equation
- Steady-state solutions look like Landauer.
- Predicts optical bistability, heating, etc.
- KB, Roberto Car, Ralph Gebauer, PRL 2005
28Model results of Gebauer and Car
- One dimensional systems
- Time-dependent Hartree
- Find steady-state solution of Master equation in
presence of E-field. - Finite ring
- Finite temperature
29Intrinsic bistability at high fields
Self-consistent steady state calculation each
point is obtained by starting from scratch
A self-consistent steady state calculation which
mimics an adiabatic sweep shows intrinsic
bistability
30Master equation for dissipation
- HHelHphKel-ph
- Assume relaxation time much longer than time for
transitions or phonon periods - Coarse-grain over electronic transitions and
average over bath fluctuations - Master equation for system density matrix
31Master equation continued
- Operator C is from Fermis golden rule applied to
Kel-ph - Transition probabilities satisfy detailed balance
- Builds in irreversibility to evolution
- Allows off-diagonal density matrix elements, so
not a pure state evolution - Prototype lifetime of two-level atom coupled to
quantized photon field
321-1 correspond. for Master eqn
- Assume potential is Taylor-expandable about t0.
- Consider two potentials that differ by more than
a time-dep constant - Show that current densities must then differ
- Use (restored) continuity to prove densities
differ.
33Kohn-Sham Master equation
- Define a Kohn-Sham Master equation yielding same
r(r,t) from vs (r,t), but choose Cs to
equilibrate to the Mermin-Kohn-Sham Ss(0)
34Return to weak bias
- Usual Kubo calculation yields adiabatic
conductivity - Our approach produces true isothermal
conductivity - Can show, as Cs-gt0, it becomes ih in Kubo formula
35Funding
- Rutgers
- Research Corp.
- Petroleum Research Fund
- NJ Commision on Sci and Tech
- NSF
- DOE
- ONR
- AFOSR
- Science Foundation Ireland
36Summary
- There is a rigorous approach to DFT.
- Several significant problems in standard DFT
transport - Missing derivative discontinuity
- Missing non-local XC corrections to weak bias
- No microscopic derivation for finite bias.
- Thanks to friends and funders.