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Problems in standard DFT calculations of transport across single molecules

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Generalize TDCDFT to include dissipation. Produces a TD KS master equation ... Master equation for dissipation. H=Hel Hph Kel-ph ... – PowerPoint PPT presentation

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Title: Problems in standard DFT calculations of transport across single molecules


1
Problems 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
2
Outline
  • Rigorous density functional theory
  • Errors in transport calculations
  • Within standard formulation
  • Test standard model for weak bias
  • Beyond standard model for finite bias

3
He atom in Kohn-Sham DFT
Dashed-line EXACT KS potential
4
Exact ground-state DFT and now time-dependent DFT
In time-dependent external field
For a given interaction and statistics HS KS
RG KS
5
Benzene 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).
6
Lesson 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.

7
Transport 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

9
Recent example
10
Three 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.

11
I. 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?

12
Derivative discontinuity Perdew,Parr,Levy and
Balduz, PRL 82
m
ehomo(N) -I
m
ehomo(N1)-A
ehomo(N) -I
13
KS potential of H atom
14
Derivative 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

15
Effect 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
16
Missing derivative discontinuity
  • Local functionals miss derivative discontinuity
  • Resonances smeared in LDA, yielding overestimated
    current

17
Sanvito calculations
Tohar, Filipetti, Sanvito, and KB (PRL, to
appear).
  • For weak coupling, see much lower conductance
    when SIC turned on.

V(eV)
18
Aside 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)
19
II. 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

21
Static 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
23
Conclusions 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.

24
III. 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).

26
Finite 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!

27
Results 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

28
Model 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

29
Intrinsic 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
30
Master 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

31
Master 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

32
1-1 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.

33
Kohn-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)

34
Return 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

35
Funding
  • Rutgers
  • Research Corp.
  • Petroleum Research Fund
  • NJ Commision on Sci and Tech
  • NSF
  • DOE
  • ONR
  • AFOSR
  • Science Foundation Ireland

36
Summary
  • 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.
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