simulation of charge t ransport in oxides and organic semiconducting materials

Simulation of Charge Transport in Oxides and Organic Simulation of Charge Transport in Oxides and Organic Semiconducting MaterialsSemiconducting Materials

Jochen Blumberger

University College London

Trieste, 23.05.2014

Workshop on Materials Challenges in Devices for Fuel Solar Production and Employment

OverviewI. Theoretical background II. CT in oxide materials

III. CT in organic solar cell materials

IV. CT in bacterial nanowiresElectrode

e-

Food

Food

Microbe

Photoelectrochemical cell

e-

oxide good charge transport

essential for high efficiency

e-

Clarke and Durrant, Chem Rev. (2010)

Organic photovoltaic cell

good charge transport essential for high efficiency

Ishii, S. et al. Appl. Environ. Microbiol. 71, 7838 (2006).

Electrode

Food

e-Microbe

Mediatorless microbial fuel cell

Summers, Z. M. et al. Science 330, 1413 (2010).

good charge transport essential for high efficiency

Why computation ?

Nature of charge carrier (localised/delocalised)

Mechanism of charge transport (band/hopping/wavepacket)

Atomistic interpretation of measured charge mobilities, I-V curves

NanoStructure-property relationships

Overview

• I. Theoretical Background

• II. Electron tunneling between O-vacancies in MgO

• III. Electron transport in fullerenes

• IV. Electron transport in bacterial nanowires

Which theory is adequate?

Cha

rge

mob

ilit

y

electron-phonon couplingor reorganisation energy or trapping energy


Cha

rge

mob

ilit

y


metals

band theory

redox in solutionredox proteins

Small polaron hoppingET theories (Marcus)


Cha

rge

mob

ilit

y


metals

band theory

redox in solution redox proteins

Small polaron hoppingET theories (Marcus)

?

holes/e- in oxidesholes/e- in organicsemiconductors

Electron transfer theory (Thermally activated polaron hopping)

initial diabatic state final diabatic state

e-e-

Diabatic and adiabatic electronic states

Q reaction coordinate

λ reorganization energy

Hab electronic coupling matrix element

initial diabatic state

final diabatic statee-e-

adiabatic ground & 1st excited ET state

Diabatic and adiabatic electronic states

e-e-

Transition state theory

Nuclear tunneling

Landau-Zener theory

Diabatic states from constrained DFT (CDFT)

Idea: - Construct Hamiltonian in diabatic = charge localized basis

-Create charge localized states by adding an external

potential to Hamiltonian

=

Martin Karplus (1963) (?), Arieh Warshel (1993), Troy Van Voorhis (2005), John Tully

(2008),…

Charge constrained DFT (CDFT)

1. Define donor, acceptor and external potential w(r)

Q. Wu and T. van Voorhis, Phys. Rev. A 72, 024502 (2005)



2. Add Vw(r) to KS-equation

and vary V so that charge constraint

is fulfilled.




2. Add Vw(r) to KS-equation

and vary V so that charge constraint

is fulfilled.


diabatic state B ψB, EB = FB-VBN

1 AD qq

e-

e-

1 AD qq

diabatic state AψA, EA = FA-VAN



Q Wu and T Van Voorhis, Phys. Rev. A 72, 024502 (2005)

diabatic state B ψB, EB = FB-VBN

1 AD qq

e-

e-

1 AD qq

diabatic state AψA, EA = FA-VAN

CDFT Implementation in the CPMD code

• CDFT weight function for charge constraint:

where ρi are promolecular atomic densities (pseudo AO, Slater, Gaussians)

• CDFT wavefunction optimisation, geometry optimisation and BOMD

• GGA, hybrid and range separated hybrid functionals (with new HFX parallelisation)

• Troullier-Martins or Goedecker-Hutter pseudo potentials for core electrons

• Available in CPMD Version 3.15.1 (thanks to M. Boero and T. Laino)

H. Oberhofer, JB, J. Chem. Phys. 131, 064101 (2009)H. Oberhofer, JB, J. Chem. Phys. 133, 244105 (2010)

Benchmarking CDFT electronic couplings (Hab):The HAB11 database

A. Kubas, F. Hoffmann, A. Heck, H. Oberhofer, M. Elstner JB, J. Chem. Phys. 140, 104105 (2014).

Hab CDFT error wrt MRCI+Q/NEVPT2

Relatively small dependence on %HFX25-50% HFX gives excellent accuracy

A. Kubas, F. Hoffmann, A. Heck, H. Oberhofer, M. Elstner JB, J. Chem. Phys. 140, 104105 (2014).

Overview





Electron tunneling between F+-F0 centres in MgO

positively charged oxygen vacancy

neutral oxygenvacancy

F0F+

Initial diabatic state from CDFT

Isosurface of spin density, PBE0 (CPMD code):

K. P. McKenna, JB Phys. Rev B. 86, 245110 (2012).

Final diabatic state from CDFT

Isosurface of spin density, PBE0 (CPMD code):


Electronic coupling

• CDFT calculation with PBE0 for

• F-centres along 100, 110, 111, 211 and 310 crystallographic directions

• distance between F-centres ranging from 3 to 16 Angstroms


Hab CDFT for hole transfer in MgO

5 10 15defect separation (Angstroms)

PBE with different %ages of Hartree-Fock exchange (HFX):

JB, K. P. McKenna, Phys. Chem. Chem. Phys. 15, 2184 (2013).

Strong dependence on % HFX

Coupling decay constant versus band gap Eg

Eg: band gap of MgO

0%

25%

50%75%

100%

PBE + x % HFX


Square barrier tunneling model

Exact solution of Schroedinger equation:

Use functional that gets band gap right (PBE0 i.e. 25%HFX)

Parallelisation of HFX in CPMD

PBE

PBE0 PBE0hfx parallelized

V. Weber, T. Laino, A. Curioni (IBM Zurich)http://cpmd.org/the-code/performance-and-scale-out

Mg80O78+

Accounting for finite size effects on electronic couplingJB, K. P. McKenna, Phys. Chem. Chem. Phys. 15, 2184 (2013).

Electronic coupling versus defect separation

Overall approximately exponential and isotropic decay

CDFT, PBE0, for MgO 100, 110, 111, 211 and 310 crystallographic directions


Reorganization energy versus defect separation

CDFT, PBE0, for MgO 100, 110, 111, 211 and 310 crystallographic directions

long range: Marcus like = const – 1/d short range: non-Marcus due to large distortions of e--Mg2+ distances


Hole transfer rates between F-center defects in MgO K. P. McKenna, JB Phys. Rev B. 86,45110 (2012).

5

6

7

8

9

10

11

12

13

442 6 8 10 12 14 16 18 20

log

(kE

T/s

-1)

distance between F centers (Angstroms) 0

hello hellohello

Three ET regimes

Hab << λ Hab > 3/8 λ Hab < 3/8 λ

“non-adiabatic ET” “adiabatic ET” delocalized carrier

no ET rate defined

5

6

7

8

9

10

11

12

13

442 6 8 10 12 14 16 18 20

log

(kE

T/s

-1)

distance between F centers (Angstroms) 0

crossover incoherent coherenttransport

Rates & Transport regimesK. P. McKenna, JB Phys. Rev B. 86, 245110 (2012).

no polaron no rate

hole hopping

hole hopping

Overview





Electron transport in fullerenes

e-

exciton dissociation efficiency

electron mobility nanoscale/mesoscale structure

Clarke and Durrant, Chem Rev. (2010)

Dabirian et al. PCCP 12, 4473 (2010).

Kim et al. ACS Nano 3, 2557 (2009).

distorted bcc (model)

hexagonal (no X-ray)

monoclinic w. solvent

Rispens et al. Chem Commun 2116, (2003).

triclinic w. solvent

Phenyl-C61-Butyric acid Methyl ester (PCBM)

First solvent-free PCBM single crystalG. Paterno, J. Spencer, JB, F. Cacialli and co-workers, J. Mater. Chem. C (2013)

monoclinic4 PCBM/unit cell

obtained by slow drying from chlorobenzene See also Casalegno et al.

Chem. Commun. 49, 4525 (2013)


MD exp

density (g/cm3) 1.653 1.631

RDF 1. peak (A) 10.05 10.2

coordination number 7 7

very good agreement with experimental X-ray structure

Electronic couplings vs distance in PCBM crystalsF. Gajdos, H. Oberhofer, M. Dupuis, JB J. Phys. Chem. Lett. 4, 1012 (2013).

I

(Gajdos)

Electronic couplings vs distance in PCBM crystalsF. Gajdos, H. Oberhofer, M. Dupuis, JB J. Phys. Chem. Lett. 4, 1012 (2013).

constructive orbital overlap

destructiveorbital overlap

Polaron hopping

calculate Hab , for all possible hops

hopping rate kET for all possible hops

time

field

e-

is currently the state-of-the-art, used by many groups: Bredas, Nelson, Andrienko,…

Polaron hopping

calculate Hab , for all possible hops

hopping rate kET for all possible hops

experiment

time

field

e-

mobility factor of 6 too large not bad….

is currently the state-of-the-art, used by many groups: Bredas, Nelson, Andrienko,…

Kinetic Monte Carlo

Polaron does not exist

|HAB| = up to 150 meV

λ = 150 meV

|HAB| > 3/8 λ

no barrier, ET rate not defined

(though one can still insert into the rate equation and get some number)

Non-adiabatic MD for ET in organic materials

field

time

Don’t integrate electron dynamics out (as done in rate theory)

Solve coupled electron-nuclear dynamics

Non-adiabatic MD for ET in organic materials

field

time

Don’t integrate electron dynamics out (as done in rate theory)

Solve coupled electron-nuclear dynamics

Fast Tully surface hopping molecular dynamics:

Wavefunction expansion:

work in progress

“Message passing” parametrisation of non-adiabatic MD

off-diagonal Hamiltonian

elements:

can be calculated

ultrafast

Speed-up of 6 orders of magnitude, little loss of accuracy

Overview





Very long-ranged electron transport (ET) in biology

pili

e-extracellular ET via conductive pili

Summers et al. Science 330, 1413 (2010).

1 m

extracellular ET via conductive pili

Very long-ranged electron transport (ET) in biology

I-V measurementson pili

nA over 0.6 m

1 Siemens/cm

El-Naggar et al., PNAS 107, 18127 (2010).

AFM tip

e-

Summers et al. Science 330, 1413 (2010).

Au

m

What is mediating the ET?

10-100 nm

5-10 nm

1 nm

Protein thermal fluctuations

Electronic coupling

Protein-protein interactions

Cellular/environmental interactions

MtrF: a deca-heme “nanowire” protein

• binds 10 hemes

• all hemes bis-his (low spin)

• staggered octaheme bisected by planar tetra-heme chain

4.5 nm

6.5 nm

Why a tri-furcated electron transfer path?

Band structure of deca-heme protein

CB water

CB protein

VB water

VB protein

‘VB heme’

‘CB heme’

~ 0.3 eV

Computation of heme reduction potentials

redox

oxred

AMBER03/TIP3P

exp: from proteinfilm voltammetry

assignment of heme reduction potentials

M Breuer, PP Zarzycki, JB, KM Rosso, J. Am. Chem. Soc. 134, 9868 (2012).

Free energy landscape for electron transport

1

6

4

3

2

5

7

8

9

10

largest barrier for forward flow: 0.35 eV

largest barrier for reverse flow: 0.34 eV

M Breuer, PP Zarzycki, JB, KM Rosso, J. Am. Chem. Soc. 134, 9868 (2012).

How can protein cope with these barriers?

Electronic coupling versus heme-heme distance

2.1±1.4 0.4±0.3 0.2±0.2

stacked T-shaped coplanar

in meV

FODFT(PBE), 100 ns MD for each heme-pair

M Breuer, KM Rosso, JB PNAS 111, 611 (2014)

1

stacked

T-shaped

coplanar

Free energy (solid) vs electronic coupling Hab (circles)

(Breuer)

stacked

T-shapedco-planar

ET is a balancing act between electronic coupling and free energy

Strongest electronic couplings where free energy barriers are largest

Stacked hemes facilitate transport into protein interior


Heme-to-heme ET rates in MtrF

(Breuer)

rates in the range (100 s)-1 to (100 ps) -1 range

rates ~ the same in and directions

full : e- shaded: e-


MV+e-

Solution kineticsGF White et al., PNAS 110, 6346 (2013).

How can we make contact to experiment?

MV+e-

Solution kinetics I-V measurements

a

GF White et al., PNAS 110, 6346 (2013). MY El-Naggar et al., PNAS 107, 18127 (2010).

How can we make contact to experiment?

Electron flux via Master Equation

5

4

7

6

8

3

2

1

• Electron hopping: i = D, heme 1-8, A

(i)red + (i+1)ox (i)ox + (i+1)red

• Electron flux between two neighbour hemes:

P(i) = electron population on heme i (≤1)

• Steady state flux:

solve iterativelyD

A

k1D kD1

kA8 k8A

k9,10

k10,9J


Electron flux J through MtrF

simulation

experimental lower limit

protein limited

5

4

7

6

8

3

2

1

MV+

Fe2O3

kin

kout

k9,10

k10,9J

kin >>kout

Electron flux J through MtrF

simulation


protein limited

5

4

7

6

8

3

2

1

MV+

Fe2O3

kin

kout

k9,10

k10,9J

kin >>kout

experiment: 8x103 electrons/sec (lower limit) simulation: 104-105 electrons/sec “a first indication” that computed rates are OK

simulation


acceptor limited protein limited

5

4

7

6

8

3

2

1

I-V characteristic of MtrF

kin, EETe-

e- kout, EET

J = -I

V

anode

cathode

5

4

7

6

8

3

2

1

J

I-V characteristic of MtrF

kin, EETe-

e-

simulation: ~ pAexperiment: ~ nA !

kout, EET

J = -I

V

anode

cathode

Conclusions: Three ET regimes

Hab << λ Hab > 3/8 λ Hab < 3/8 λ

“non-adiabatic ET” “adiabatic ET” delocalized carrier

Acknowledgment

CDFT & FODFT implementationDr Harald Oberhofer (U Cambridge, TU Munich)

MgODr Keith McKenna (York)Prof Alex Shluger (UCL)

fullerenesFruzsina Gajdos (UCL)Jacob Spencer (UCL)

Dr. Michel Dupuis (PNNL)Prof Franco Cacialli (UCL)

£££:

Biological nanowireMarian Breuer (UCL)

Dr Kevin Rosso (PNNL)Prof Julea Butt (UEA)

Sensitivity on definition of charge constraint

Charge constraint 1: 6 Mg coord. O vacancies

Charge constraint 2: all atoms in left/right half


CPMD INPUT file: donor (D) atoms, acceptor (A) atoms

weight function w (r) = wA,D (pseudo AO, Slater, Gaussian)

charge difference (NC = +1 or -1 for transfer of 1 electron)

initial guess for Lagrange multiplier V=V0

wavefunction optimization

is charge constraint fulfilled ? i.e.

Yes, ψA, EA = FA-VANC

new Lagrange multiplier

No

(Oberhofer)

simulation of charge t ransport in oxides and organic semiconducting materials

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