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)
Karsten Hannewald: "Polaron Transport in Organic Semiconductors" QuantSim09, Warwick/UK, 27 Aug 2009 2
Outline
(1) Organic Molecular Crystals
(2) Polaron Band Narrowing
(3) Narrow-Band Mobility Theory
(4) General Mobility Theory →→→→ Beyond Narrow Bands
(5) Summary & Outlook
Karsten Hannewald: "Polaron Transport in Organic Semiconductors" QuantSim09, Warwick/UK, 27 Aug 2009 3
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
Karsten Hannewald: "Polaron Transport in Organic Semiconductors" QuantSim09, Warwick/UK, 27 Aug 2009 4
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:
Karsten Hannewald: "Polaron Transport in Organic Semiconductors" QuantSim09, Warwick/UK, 27 Aug 2009 5
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)
Karsten Hannewald: "Polaron Transport in Organic Semiconductors" QuantSim09, Warwick/UK, 27 Aug 2009 6
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
~
Karsten Hannewald: "Polaron Transport in Organic Semiconductors" QuantSim09, Warwick/UK, 27 Aug 2009 7
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)]
~ ~
Karsten Hannewald: "Polaron Transport in Organic Semiconductors" QuantSim09, Warwick/UK, 27 Aug 2009 8
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
Karsten Hannewald: "Polaron Transport in Organic Semiconductors" QuantSim09, Warwick/UK, 27 Aug 2009 9
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)
Karsten Hannewald: "Polaron Transport in Organic Semiconductors" QuantSim09, Warwick/UK, 27 Aug 2009 10
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
Karsten Hannewald: "Polaron Transport in Organic Semiconductors" QuantSim09, Warwick/UK, 27 Aug 2009 11
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:
Karsten Hannewald: "Polaron Transport in Organic Semiconductors" QuantSim09, Warwick/UK, 27 Aug 2009 12
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
-∞
∞
Karsten Hannewald: "Polaron Transport in Organic Semiconductors" QuantSim09, Warwick/UK, 27 Aug 2009 13
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
Karsten Hannewald: "Polaron Transport in Organic Semiconductors" QuantSim09, Warwick/UK, 27 Aug 2009 14
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
Karsten Hannewald: "Polaron Transport in Organic Semiconductors" QuantSim09, Warwick/UK, 27 Aug 2009 15
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
Karsten Hannewald: "Polaron Transport in Organic Semiconductors" QuantSim09, Warwick/UK, 27 Aug 2009 16
→ 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
Karsten Hannewald: "Polaron Transport in Organic Semiconductors" QuantSim09, Warwick/UK, 27 Aug 2009 17
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
Karsten Hannewald: "Polaron Transport in Organic Semiconductors" QuantSim09, Warwick/UK, 27 Aug 2009 18
Problem: Diagonalization of H necessary, but not exactly possible
solve analytically as beforepolaron correlator → ?
Mobility from Kubo Formula
phphelel HHHH ++= −
Step 1 : Polaron Transformation
Karsten Hannewald: "Polaron Transport in Organic Semiconductors" QuantSim09, Warwick/UK, 27 Aug 2009 19
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
−→++
Karsten Hannewald: "Polaron Transport in Organic Semiconductors" QuantSim09, Warwick/UK, 27 Aug 2009 20
: 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
Karsten Hannewald: "Polaron Transport in Organic Semiconductors" QuantSim09, Warwick/UK, 27 Aug 2009 21
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)
Karsten Hannewald: "Polaron Transport in Organic Semiconductors" QuantSim09, Warwick/UK, 27 Aug 2009 22
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)
Karsten Hannewald: "Polaron Transport in Organic Semiconductors" QuantSim09, Warwick/UK, 27 Aug 2009 23
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!