Polaron Transport in Organic Semiconductors: Theory and ... · Karsten Hannewald: "Polaron Transport in Organic Semiconductors" QuantSim09, Warwick/UK, 27 Aug 2009 16 →novel combination

Karsten Hannewald: "Polaron Transport in Organic Semiconductors" QuantSim09, Warwick/UK, 27 Aug 2009 1

Polaron Transport in Organic Semiconductors: Theory and Modelling

Karsten Hannewald, Frank Ortmann & Friedhelm Bechstedt

European Theoretical Spectroscopy Facility (ETSF) & Friedrich-Schiller-Universität Jena (Germany)


Outline

(1) Organic Molecular Crystals

(2) Polaron Band Narrowing

(3) Narrow-Band Mobility Theory

(4) General Mobility Theory →→→→ Beyond Narrow Bands

(5) Summary & Outlook


Part 1: Organic Molecular Crystals

= Crystals of Organic Molecules

• Example → Semiconducting Oligo-Acene Crystals:Naphthalene Anthracene Tetracene Pentacene

•••• Crystals = well-ordered systems→ ideal to study intrinsic transportproperties

• Applications → OFET, OLED

Niemax, Tripathi & Pflaum[Appl. Phys. Lett. 86, 122105 (2005)]

Gap ≈ 5 eV ≈ 4.1eV ≈ 3.1 eV ≈ 2.2 eV


Charge Transport in Organic Crystals

• weak intermolecular van der Waals bonds • electronic bandwidths < 1eV • strong electron-lattice interaction → polarons = electrons + phonon cloud

Expt.: Warta & Karl [PRB 32, 1172 (1985)]

holes

electrons

• band transport or phonon-assisted hopping?• temperature dependence?• electrons vs. holes?• mobility anisotropy ↔ stacking motif?• visualization of transport?

“There are still great challenges for theoreticians.”Norbert Karl [Synth. Met. 133 & 134, 649 (2003)]

→→→→ Goal: First-Principles Theory of Mobilities

Open Questions:


Part 2: Polaron Band Narrowing

electron/holebandwidths

Holstein, Ann. Phys. (NY) 8, 325 (1959)Mahan “Many-Particle Physics” (1990)

polaron bandwidths forlocal el-ph coupling

polaron bandwidths forlocal + nonlocalel-ph coupl.

DFT calculations Holsteinmodel Holstein-Peierlsmodel

Hannewald et al., Phys. Rev. B 69, 075211 (2004)


Hamiltonian for Holstein-Peierls Model

H = ∑ mna+man + ∑ Q (b+

Q bQ +½) + ∑ QgQmn(b-Q+b+Q)a+

manmn QmnQ=(q,

λ

)

electrons phonons electron-phonon coupling

m====n: on-site energiesmm

m≠≠≠≠n: transfer integralsmn

m====n: local coupling (Holstein)m≠≠≠≠n: nonlocal coupling (Peierls)

Idea: Nonlocal canonical transformation into polaron picture!

H = ∑ mnA+mAn + ∑ Q (B+

Q BQ +½) mn Q

polarons phonons

polaron energies & transfer integrals

f → F = eS f eS+ , S = ∑ Cmna+man , Cmn= ∑ gQmn(b-Q–b+

Q)mn Q

~


Polaron Transfer Integrals & Bandwidths

N λ = [e

ħ ωλ /kBT–1]–1• Temperature enters via phonon occupation number

Holstein model (known)

mn= mne–∑ (½+N λ)(g2 λ

mm+ g2 λnn)λ Holstein-Peierls model (new)

• Full solution qualitatively similar to a Holstein model witheffective coupling constants

Key result: Once εmn, g λ

mn, and ω λ are known, these formulas allow quantitative studies of polaron bandwidth narrowing due to local and/or nonlocal electron-phonon coupling!

mn= ( mn– mn)e–∑ (½+N λ)(G λmm+ G λ

nn)λGλ

mm= g2λmm + ½∑ g2λ

mkm≠k

[Hannewald et al., Phys. Rev. B 69, 075211 (2004)]

~ ~


How to obtain material parameters?

• Step 1: Crystal geometry (Rm) & phonons ( λ)→→→→ ab-initio DFT-LDA calculations, VASP code

• Step 2: Transfer integrals ( mn) for HOMO & LUMO→→→→ ab-initio band-structure calculations & fit to tigh t-binding model

• Step 3: Electron-phonon coupling constants (g λmn∼∼∼∼ mn)

→→→→ repeat step 2, but for the displaced (phonon-excited) geometry

a

bc

a

βc’

•••• monoclinic crystal •••• two molecules per unit cell •••• herringbone stacking

Naphthalene


Strong Band Narrowing in Naphthalene Crystals

0 50 100 150 200 250 3000

100

200

300

400

500

600

700

LUMOHolstein-Peierls

LUMO Holstein

HOMOHolstein-Peierls

HOMO Holstein

B

andw

idth

(m

eV)

Temperature (K)

bare LUMO

bare HOMO

⇒⇒⇒⇒ Go beyond bandwidth calculations & develop mobility theory!

Experiments? → Very difficult to measure but recent progress using ARPES:Pentacene HOMO: 240 meV (at 120 K) → 190 meV (at 300 K)[N. Koch et al., Phys. Rev. Lett. 96, 156803 (2006)]

Napthalene, Anthracene & Tetracene:Hannewald et al., Phys. Rev. B 69, 075211 (2004)

Durene Crystals:Ortmann, Hannewald & Bechstedt, Appl. Phys. Lett. 93, 222105 (2008)

Guanine Crystals:Ortmann, Hannewald & Bechstedt,J. Phys. Chem. B 113, 7367 (2009)


Part 3: Narrow-Band Mobility Theory

bare electron& hole bands

Holstein, Ann. Phys. (NY) 8, 325 (1959)Mahan “Many-Particle Physics” (1990)

(known)

polaron bands withlocal el-ph

polaron bands withlocal + nonlocalel-ph

DFT calculations Holsteinmodel Holstein-Peierlsmodel

Hannewald & Bobbert, Phys. Rev. B 69, 075212 (2004)Appl. Phys. Lett. 85, 1535 (2004)

(new)

mobilities withlocal el-ph

mobilities withlocal + nonlocalel-ph


Kubo Formula for Electrical Conductivity

j = j (I) + j (II)

= e0/i ∑ (Rm-Rn) mna+man + ∑ (Rm-Rn) QgQmn(b-Q+b+

Q)a+man

mn Qmn

current = electronic current + phonon-assisted current (new)

only nonzero for nonlocalcoupling!

α ~ T–1 ∫ dt ⟨j α(t)j α(0)⟩H

• Current j = dP/dt = 1/i [P,H] with polarization P = e0 mRma+mam

• Idea: Evaluate Kubo formula by means of canonical transformation![→ details later in Part 4 of this talk]

–∞

∞

• Mobility:


Charge-Carrier Mobilities

α(I) ~ T–1∑ (R αm–R αn)2 ∫dt ( mn)2 e–2∑ λ G λ [1+2N λ– Φ λ(t)] e–

Γ2t2

n≠m

• Mobility �α (I) due to j(I) ≈ Holstein model with effective Gλ = g2λjj + ½∑ g2λ

jk

( mn)2 → ( mn– mn)2 + ½∑ λ(h λg λmn)2 λ(t)

anisotropy temperature

• Total mobility � α incl. phonon-assisted current j(II) gives new hopping term:

Key result: Once Rm, εmn, g λ

mn, and ω λ are known, quantitative predictionsfor anisotropy & T-dependence of mobilities can be made!

[Hannewald & Bobbert, Phys. Rev. B 69, 075212 (2004)]

band narrowing[N λ = phonon occupation]

incoherent hopping[

Φ λ(t) = (1+N λ) e–i ω λt + N λ e+i ω λt]→ phonon emission & absorption

k≠j

-∞

∞


Naphthalene: Theory Describes Experiments Well

ExperimentWarta & Karl [Phys. Rev. B 32, 1172 (1985)]

Ab-Initio TheoryHannewald & Bobbert [APL 85, 1535 (2004)]

10 100

100

1

0.1

0.001

0.01

T

µµµµc'(I)

-γγγγ

10

1.5

1.0

2.0

γγγγ = 2.5

30 300

electrons

holes

µµµµa µµµµb µµµµc'

Mob

ility

Temperature (Kelvin)

µµµµa µµµµb µµµµc'

holes

holes

electrons

electrons


Ortmann, Hannewald & Bechstedt, Appl. Phys. Lett. 93, 222105 (2008)

Theory Experiment

Burshtein & WilliamsPhys. Rev. B (1977)

c*

Naphthalene:Hannewald & Bobbert, Appl. Phys. Lett. 85, 1535 (2004)

Anthracene & Tetracene: Hannewald & Bobbert, AIP Conf. Proc. 772, 1101 (2005)

Guanine:Ortmann, Hannewald & Bechstedt, J. Phys. Chem. B 113, 7367 (2009)

Mobility Predictions for Other Materials

b

c

a*a*

Durene( hole)

b


Anisotropic Hole Transport in Durene: Visualization

Mobility Tensor HOMO Wave Function OverlapMovie: http://ftp.aip.org/epaps/appl_phys_lett/E-APPLAB-93-052847/254195_0_vid_0_k71b6k.mpeg


→ novel combination of analytical theory & numerical analysis- easy-to-interpretformulas for mobilities (incl. el-ph coupling in all orders)- important effects included: 3D anisotropy, temperature, band narrowing & hopping- material parameters from ab-initio calculations→ No fits to experiment!

So Far: Our Approach Works Very Well …

… But There is a Problem at Very Low Temperatures

→ Goal: Extend polaron concept to arbitrary bandwidths !

Ad-hoc modification of

narrow-band theory proposed

[Kenkre: Phys. Lett. A 305,,,, 443 (2002)]

• Experiment: Plateau • Theory: 1/T-Singularity


Part 4: General Mobility Theory →→→→ Beyond Narrow Bands

loca

l

loca

l +

nonl

ocal

Holstein(1959)

Hannewald& Bobbert

(2004)

el-phcoupling

band-widths

full

narrow

Ortmann, Bechstedt, Hannewald

(2009)

Temperature

Leve

l of T

heor

y


Problem: Diagonalization of H necessary, but not exactly possible

solve analytically as beforepolaron correlator → ?

Mobility from Kubo Formula

phphelel HHHH ++= −

Step 1 : Polaron Transformation


Now: Generalized Theory= exact diagonalization in k-space

Before: Narrow-Band Theory= approximate diagonaliz. in real space

nkkkk = Fermi distribution of polarons

Major improvement: correct statistics + correct T dependence

nLLLL = nM = constant

Step 2 : Diagonalization of

)1('~ MLHPNML nnaaaapol

−→++


: energy →→→→ energy

occupied nkkkk1 →→→→ empty (1-nkkkk2)

Σ initial state kkkk1111 →→→→ final state kkkk2222

ΣMMMM : momentum kkkk1 →→→→ momentum kkkk2 +Σi qqqqi

± Σi

ħωqqqqi

Pauli

blocking

Key Result: Generalized Mobility Formula

anisotropy occupation

momentum

conservation

momentum

energy conservation

(incl. phonon absorption & emission)

∫∞

∞−dt

Ortmann, Bechstedt & Hannewald, Phys. Rev. B 79, 235206 (2009)

k1k2


0th order in g2

vs higher orders:

coherent (band) transport:

→ high-T limit covers narrow-band approximation & Marcus theory

→ like Boltzmann equation but with polaron velocities:

→ for low T: µ → const because

incoherent (phonon-assisted) hopping:

Polaron Band Transport & Hopping Included

Ortmann, Bechstedt & Hannewald, Phys. Rev. B 79, 235206 (2009)


Hannewald & BobbertAPL 85, 1535 (2004)

N. Karl, in Landolt-BörnsteinGroup III, Vol.17, p.106

Ortmann, Bechstedt & Hannewald(Org. Electr., 2009, submitted)

Improved Low-Temperature Mobilities(Example: Naphthalene Holes)


Summary & Outlook

• Novel theories of polaron bandwidths & mobilities in organic crystals• explicit formulas for T dependence& anisotropy• ab-initiocalculations of all material parameters • deeper understanding & 3D visualization of charge transport

loca

l

loca

l +

nonl

ocal

Holstein(1959)

Hannewald& Bobbert

(2004)

el-phcoupling

band-widths

full

narrow

Ortmann, Bechstedt, Hannewald

(2009)

(1) Bandwidthsincl. el-ph coupling: local → local+ nonlocal(Holstein-Peierlsmodel)

(2) Mobilities:Next Goal:Unification!

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Polaron Transport in Organic Semiconductors: Theory and ... · Karsten Hannewald: "Polaron Transport in Organic Semiconductors" QuantSim09, Warwick/UK, 27 Aug 2009 16 →novel combination