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
1
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
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Part 1: Organic Molecular Crystals
= Crystals of Organic Molecules
Niemax, Tripathi & Pflaum
[Appl. Phys. Lett. 86, 122105 (2005)]
• Example → Semiconducting Oligo-Acene Crystals:
Naphthalene Anthracene Tetracene
Pentacene
Gap ≈ 5 eV
≈ 4.1eV
≈ 3.1 eV
≈ 2.2 eV
• Crystals = well-ordered systems
→ ideal to study intrinsic transport properties
• Applications → OFET, OLED
Karsten Hannewald: "Polaron Transport in Organic Semiconductors"
QuantSim09, Warwick/UK, 27 Aug 2009
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Charge Transport in Organic Crystals
• weak intermolecular van der Waals bonds
• electronic bandwidths < 1eV
• strong electron-lattice interaction → polarons = electrons + phonon cloud
Open Questions:
Expt.: Warta & Karl [PRB 32, 1172 (1985)]
• band transport or phonon-assisted hopping?
• temperature dependence?
• electrons vs. holes?
• mobility anisotropy ↔ stacking motif?
• visualization of transport?
holes
“There are still great challenges for theoreticians.”
Norbert Karl [Synth. Met. 133 & 134, 649 (2003)]
electrons
→ Goal: First-Principles Theory of Mobilities
Karsten Hannewald: "Polaron Transport in Organic Semiconductors"
QuantSim09, Warwick/UK, 27 Aug 2009
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Part 2:
Polaron Band Narrowing
electron/hole
bandwidths
polaron bandwidths for
local el-ph coupling
DFT calculations
Holstein model
Holstein, Ann. Phys. (NY) 8, 325 (1959)
Mahan “Many-Particle Physics” (1990)
Karsten Hannewald: "Polaron Transport in Organic Semiconductors"
polaron bandwidths for
local + nonlocal el-ph coupl.
Holstein-Peierls model
Hannewald et al.,
Phys. Rev. B 69, 075211 (2004)
QuantSim09, Warwick/UK, 27 Aug 2009
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Hamiltonian for Holstein-Peierls Model
electrons
phonons
electron-phonon coupling
H = ∑ mna+man + ∑ Q (b+Q bQ +½) + ∑ QgQmn(b-Q+b+Q)a+man
λ
mn
Q=(q, )
Qmn
m=
=n: on-site energies mm
m≠
≠n: transfer integrals mn
m=
=n: local coupling (Holstein)
m≠
≠n: nonlocal coupling (Peierls)
Idea: Nonlocal canonical transformation into polaron picture!
f → F = eS f eS+ , S = ∑ Cmna+man , Cmn = ∑ gQmn(b-Q–b+Q)
mn
polarons
H = ∑ ~mnA+mAn + ∑
mn
Q
Q
phonons
+
Q (B Q BQ +½)
polaron energies & transfer integrals
Karsten Hannewald: "Polaron Transport in Organic Semiconductors"
QuantSim09, Warwick/UK, 27 Aug 2009
6
Polaron Transfer Integrals & Bandwidths
λ
– ∑ (½+N )(G mm+ G nn)
)e
mn
λ
λ
λ
λ
λ
λ
• Temperature enters via phonon occupation number N = [e
• Full solution qualitatively similar to a Holstein model with
ω
ħ
mn–
λ
~ =(
mn
–∑ (½+N )(g2 mm+ g2 nn)
e
mn
λ
~ =
mn
Holstein-Peierls model (new)
λ
Holstein model (known)
/kBT–1]–1
λ
λ
λ
effective coupling constants G mm = g2 mm + ½ ∑ g2 mk
m≠k
ω
λ
λ
ε
Key result: Once mn, g mn, and
are known, these formulas allow
quantitative studies of polaron bandwidth narrowing
due to local and/or nonlocal electron-phonon coupling!
[Hannewald et al., Phys. Rev. B 69, 075211 (2004)]
Karsten Hannewald: "Polaron Transport in Organic Semiconductors"
QuantSim09, Warwick/UK, 27 Aug 2009
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How to obtain material parameters?
)
λ
• Step 1: Crystal geometry (Rm) & phonons (
→ ab-initio DFT-LDA calculations, VASP code
• Step 2: Transfer integrals (
mn)
for HOMO & LUMO
→ ab-initio band-structure calculations & fit to tight-binding model
mn∼
λ
• Step 3: Electron-phonon coupling constants (g
mn)
→ repeat step 2, but for the displaced (phonon-excited) geometry
Naphthalene
•
c’
c
β
monoclinic crystal
• two molecules per unit cell
• herringbone stacking
a
Karsten Hannewald: "Polaron Transport in Organic Semiconductors"
b
a
QuantSim09, Warwick/UK, 27 Aug 2009
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Strong Band Narrowing in Naphthalene Crystals
700
Bandwidth (meV)
Napthalene, Anthracene & Tetracene:
Hannewald et al., Phys. Rev. B 69, 075211 (2004)
bare LUMO
600
bare HOMO
LUMO Holstein
500
Durene Crystals:
Ortmann, Hannewald & Bechstedt,
Appl. Phys. Lett. 93, 222105 (2008)
HOMO Holstein
400
300
HOMO
Holstein-Peierls
200
100
0
Guanine Crystals:
Ortmann, Hannewald & Bechstedt,
J. Phys. Chem. B 113, 7367 (2009)
LUMO
Holstein-Peierls
0
50
100
150
200
Temperature (K)
250
300
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)]
⇒ Go beyond bandwidth calculations & develop mobility theory!
Karsten Hannewald: "Polaron Transport in Organic Semiconductors"
QuantSim09, Warwick/UK, 27 Aug 2009
9
Part 3:
Narrow-Band Mobility Theory
DFT calculations
bare electron
& hole bands
Holstein model
Holstein-Peierls model
polaron bands with
local el-ph
polaron bands with
local + nonlocal el-ph
mobilities with
local el-ph
mobilities with
local + nonlocal el-ph
Holstein, Ann. Phys. (NY) 8, 325 (1959)
Mahan “Many-Particle Physics” (1990)
Hannewald & Bobbert,
Phys. Rev. B 69, 075212 (2004)
Appl. Phys. Lett. 85, 1535 (2004)
(known)
(new)
Karsten Hannewald: "Polaron Transport in Organic Semiconductors"
QuantSim09, Warwick/UK, 27 Aug 2009
10
Kubo Formula for Electrical Conductivity
α
~ T–1 ∫ dt ⟨j (t)j (0)⟩H
α
• Mobility:
α
∞
–∞
• Current j = dP/dt = 1/i [P,H] with polarization P = e0
+
mRma mam
current = electronic current + phonon-assisted current (new)
j
=
j(I)
+
j(II)
= e0/i ∑ (Rm-Rn) mna+man + ∑ (Rm-Rn) QgQmn(b-Q+b+Q)a+man
mn
Qmn
only nonzero for nonlocal coupling!
• Idea: Evaluate Kubo formula by means of canonical transformation!
[→ details later in Part 4 of this talk]
Karsten Hannewald: "Polaron Transport in Organic Semiconductors"
QuantSim09, Warwick/UK, 27 Aug 2009
11
α
λ
due to j(I) ≈ Holstein model with effective G = g2 jj + ½ ∑ g2 jk
λ
(I)
λ
• Mobility
Charge-Carrier Mobilities
k≠j
α
-∞
n≠m
λ
(t)] e– 2t2
Γ
λ
G [1+2N –
λ
2 –2 ∑
mn) e
λ
~ T–1 ∑ (R m–R n)2 ∫dt (
α
α
(I)
∞
temperature
Φ
anisotropy
incoherent hopping
[N = phonon occupation]
[ (t) = (1+N ) e–i t + N e+i t]
→ phonon emission & absorption
λ
λ
ω
ω
λ
λ
λ
incl. phonon-assisted current j(II) gives new hopping term:
→ (
mn – mn)
2
+ ½ ∑ (h g
mn)
2
(t)
λ
2
λ
mn)
λ
(
λ
α
• Total mobility
λ
Φ
band narrowing
ω
λ
λ
ε
Key result: Once Rm, mn, g mn, and
are known, quantitative predictions
for anisotropy & T-dependence of mobilities can be made!
[Hannewald & Bobbert, Phys. Rev. B 69, 075212 (2004)]
Karsten Hannewald: "Polaron Transport in Organic Semiconductors"
QuantSim09, Warwick/UK, 27 Aug 2009
12
Naphthalene: Theory Describes Experiments Well
Experiment
Ab-Initio Theory
Hannewald & Bobbert [APL 85, 1535 (2004)]
Warta & Karl [Phys. Rev. B 32, 1172 (1985)]
100
holes
µa
µb
10
µc'
holes
holes
1
Mobility
γ = 2.5
2.0
-γ
T
1.5
0.1
1.0
electrons
electrons
µa
µb
µc'
electrons
0.01
(I)
µc'
0.001
10
30
100
300
Temperature (Kelvin)
Karsten Hannewald: "Polaron Transport in Organic Semiconductors"
QuantSim09, Warwick/UK, 27 Aug 2009
13
Mobility Predictions for Other Materials
Durene
( hole)
Theory
c
c*
Experiment
b
a*
b
a*
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)
Ortmann, Hannewald &
Bechstedt, Appl. Phys.
Lett. 93, 222105 (2008)
Karsten Hannewald: "Polaron Transport in Organic Semiconductors"
Burshtein & Williams
Phys. Rev. B (1977)
QuantSim09, Warwick/UK, 27 Aug 2009
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Anisotropic Hole Transport in Durene: Visualization
Mobility Tensor
HOMO Wave Function Overlap
Movie: 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
15
So Far: Our Approach Works Very Well …
→ novel combination of analytical theory & numerical analysis
- easy-to-interpret formulas 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!
… But There is a Problem at Very Low Temperatures
• Experiment: Plateau
• Theory: 1/T-Singularity
Ad-hoc modification of
narrow-band theory proposed
[Kenkre: Phys. Lett. A 305,, 443 (2002)]
→ Goal: Extend polaron concept
to arbitrary bandwidths !
Karsten Hannewald: "Polaron Transport in Organic Semiconductors"
QuantSim09, Warwick/UK, 27 Aug 2009
16
Ortmann,
Bechstedt,
Hannewald
(2009)
Hannewald
& Bobbert
(2004)
el-ph
coupling
lo
no cal
nl +
oc
al
bandwidths
ll
fu
Level of Theory
Part 4:
General Mobility Theory → Beyond Narrow Bands
Holstein
(1959)
w
lo
rro
ca
l
na
Temperature
Karsten Hannewald: "Polaron Transport in Organic Semiconductors"
QuantSim09, Warwick/UK, 27 Aug 2009
17
Mobility from Kubo Formula
H = H el + H el − ph + H ph
Problem: Diagonalization of H necessary, but not exactly possible
Step 1 : Polaron Transformation
polaron correlator → ?
Karsten Hannewald: "Polaron Transport in Organic Semiconductors"
solve analytically as before
QuantSim09, Warwick/UK, 27 Aug 2009
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Step 2 : Diagonalization of
Before: Narrow-Band Theory
Now: Generalized Theory
= approximate diagonaliz. in real space
= exact diagonalization in k-space
aL+ aM a N+ aP
~
H 'pol
→ nL (1 − nM )
nL = nM = constant
nk = Fermi distribution of polarons
Major improvement: correct statistics + correct T dependence
Karsten Hannewald: "Polaron Transport in Organic Semiconductors"
QuantSim09, Warwick/UK, 27 Aug 2009
19
Key Result: Generalized Mobility Formula
Ortmann, Bechstedt & Hannewald, Phys. Rev. B 79, 235206 (2009)
anisotropy
momentum
energy conservation
(incl. phonon absorption & emission)
k1k2
ΣM :
∫
∞
−∞
dt :
initial state k1
→
final state k2
occupied nk1
→
empty (1-nk2)
momentum k1
→
momentum k2 +Σi qi
energy
→
energy
Karsten Hannewald: "Polaron Transport in Organic Semiconductors"
momentum
conservation
Pauli
blocking
± Σi ωqi
ħ
Σ
occupation
QuantSim09, Warwick/UK, 27 Aug 2009
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Polaron Band Transport & Hopping Included
0th order in g2
vs higher orders:
coherent (band) transport:
→ like Boltzmann equation but with polaron velocities:
→ for low T: µ → const because
incoherent (phonon-assisted) hopping:
→ high-T limit covers narrow-band approximation & Marcus theory
Ortmann, Bechstedt & Hannewald, Phys. Rev. B 79, 235206 (2009)
Karsten Hannewald: "Polaron Transport in Organic Semiconductors"
QuantSim09, Warwick/UK, 27 Aug 2009
21
Improved Low-Temperature Mobilities
(Example: Naphthalene Holes)
Ortmann, Bechstedt & Hannewald
(Org. Electr., 2009, submitted)
N. Karl, in Landolt-Börnstein
Group III, Vol.17, p.106
Karsten Hannewald: "Polaron Transport in Organic Semiconductors"
Hannewald & Bobbert
APL 85, 1535 (2004)
QuantSim09, Warwick/UK, 27 Aug 2009
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Summary & Outlook
• Novel theories of polaron bandwidths & mobilities in organic crystals
• explicit formulas for T dependence & anisotropy
• ab-initio calculations of all material parameters
• deeper understanding & 3D visualization of charge transport
(1) Bandwidths incl. el-ph coupling: local → local + nonlocal (Holstein-Peierls model)
(2) Mobilities:
Next Goal:
Unification!
Hannewald
& Bobbert
(2004)
ll
fu
Holstein
(1959)
w
rro
na
Karsten Hannewald: "Polaron Transport in Organic Semiconductors"
el-ph
coupling
lo
ca
l
bandwidths
lo
n o cal
nl +
oc
al
Ortmann,
Bechstedt,
Hannewald
(2009)
QuantSim09, Warwick/UK, 27 Aug 2009
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