Charge Transport in Organic Crystals

Transcription

1 Charge Transport in Organic Crystals Karsten Hannewald European Theoretical Spectroscopy Facility (ETSF) & Friedrich-Schiller-Universität Jena Karsten Hannewald (Uni Jena) July 15, 2010, TU Dresden 1

2 Coworkers & Sponsors : TU Eindhoven Peter Bobbert (Netherlands) City of since 2005: U Jena Frank Ortmann & (Germany) Friedhelm Bechstedt City of L. Matthes & F. Tandetzky (quasi-1d stacks) M. Hieckel & M. Krause (rubrene & durene) R. Maul & B. Höffling (amino acids) B. Oetzel & M. Preuss (quantum transport) R. Hambach & U. Treske (nanotubes & ribbons) Karsten Hannewald (Uni Jena) July 15, 2010, TU Dresden 2

3 Organic Electronics semiconducting polymers via doping (Nobel Prize 2000) many molecules available easy tailoring of properties (e.g., via sidegroups) easy solubility & processability flexible substrates cheap production (spin coating, printing, roll-to-roll) π-conjugated molecules Coropceanu et al., Chem. Rev Applications: OFETs, OLEDs, OPVs organic ligthing solar cells electronics & displays & modules Example: rollable OLED display (Sony, SID-2010) Karsten Hannewald (Uni Jena) July 15, 2010, TU Dresden 3

4 Organic Molecular Crystals as Benchmark Systems = crystals of organic molecules well-ordered systems ideal to study intrinsic transport properties Niemax, Tripathi & Pflaum [APL 86, (2005)] Herringbone Stacking Naphthalene Anthracene Hannewald et al., Phys. Rev. B 69, & (2004) Appl. Phys. Lett. 85, 1535 (2004) Tetracene Durene Ortmann, Hannewald & Bechstedt Phys. Rev. B 75, (2007) Appl. Phys. Lett. 93, (2008) Layered Stacking Guanine Ortmann, Hannewald & Bechstedt J. Phys. Chem. B 112, 1540 (2008) J. Phys. Chem. B 113, 7367 (2009) Karsten Hannewald (Uni Jena) July 15, 2010, TU Dresden 4

5 Search for High-Mobility Organic Crystals [from Gershenson et al., Rev. Mod. Phys. 78, 973 (2006)] Naphthalene (T=10K): 300 cm 2 /Vs (Warta & Karl, 1985) Pentacene (T=300K): 35 cm 2 /Vs (Jurchescu et al., 2004) Rubrene (T=300K) : 20 cm 2 /Vs (Podzorov et al., 2004) Durene (T=300K) : 20 cm 2 /Vs (Pflaum et al., 2008, private comm.) [ 5 cm 2 /Vs (Burshtein & Williams, 1977)] in technologically relevant regime! Karsten Hannewald (Uni Jena) July 15, 2010, TU Dresden 5

6 Measurements of Charge Transport in Organic Crystals Field-Effect Transistor Exp. Hulea et al., Nature Mat. 5, 982 (2006) Time-of-Flight Exp. Group of A.F. Morpurgo (Delft & Genf) surface mobilities allows stamp technique pioneered by groups of Gershenson & Rogers (USA) Sundar et al., Science 303, 1644 (2004) Podzorov et al., PRL 93, (2004) Group of J. Pflaum (Stuttgart & Würzburg) bulk mobilities photogenerated & injection free electrons & holes separately Review Papers - N. Karl, Landolt-Börnstein, Group III, Vol 17i, (1985) - N. Karl, Synth. Metals 133/134, 649 (2003) - R.W.I. de Boer et al., phys. stat. sol. (a) 201, 1302 (2004) - M.R. Gershenson et al., Rev. Mod. Phys. 78, 973 (2006) - Materials Today, March 2007 issue Karsten Hannewald (Uni Jena) July 15, 2010, TU Dresden 6

7 Open Questions for Charge Transport Time-of-Flight Experiment: Warta & Karl [PRB 32, 1172 (1985)] 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)] electrons holes Karsten Hannewald (Uni Jena) July 15, 2010, TU Dresden 7

8 Peculiarities of Organic Molecular Crystals Quite different than inorganic semiconductors! weak intermolecular van der Waals bonds small electronic bandwidths < 1eV strong electron-lattice interaction polarons = electrons + phonon cloud Moreover: Many important material parameters difficult to measure! bandwidths, effective masses, electron-phonon couplings? Theory & Ab-initio modelling play essential role! Karsten Hannewald (Uni Jena) July 15, 2010, TU Dresden 8

9 Goal: First-Principles Theory of Mobilities Drude formula for mobility looks simple: = e 0 τ / m* v =µ E m* effective mass τ scattering time Step 1: Polaron band narrowing Temperature dependences? Step 2: Mobility theory incl. electron-phonon scattering Needed: Microscopic models incl. electron-phonon coupling supplemented by ab-initio material parameters Karsten Hannewald (Uni Jena) July 15, 2010, TU Dresden 9

10 Polaron Band Narrowing: Theory & Modelling electron/hole bandwidths polaron bandwidths for local el-ph coupling polaron bandwidths for local + nonlocal el-ph coupl. DFT calculations Holstein model Holstein-Peierls model Holstein, Ann. Phys. (NY) 8, 325 (1959) Mahan Many-Particle Physics (1990) Hannewald et al., Phys. Rev. B 69, (2004) Karsten Hannewald (Uni Jena) July 15, 2010, TU Dresden 10

11 λ Hamiltonian for Holstein-Peierls Model electrons phonons electron-phonon coupling H = mn a + m a n + Q (b+ Q b Q +½) + Q g Qmn (b -Q +b+ Q )a+ m a n 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! polarons phonons H = ~ mn A + ma n + Q (B + Q B Q +½) mn Q polaron energies & transfer integrals Karsten Hannewald (Uni Jena) July 15, 2010, TU Dresden 11

12 λ λ λ λ ħ λ λ λ ω ε λ λ Polaron Transfer Integrals & Bandwidths Holstein model (known) ~ mn = mn e (½+Nλλ)(g2 mm + g 2 nn ) ~ mn = ( mn mn )e Temperature enters via phonon occupation number Holstein-Peierls model (new) (½+N N λ = [e Full solution qualitatively similar to a Holstein model with effective coupling constants G mm = g2 mm + ½ g2 mk m k λλ)(g mm + G /k BT 1] 1 ω λ nn ) 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, (2004)] Karsten Hannewald (Uni Jena) July 15, 2010, TU Dresden 12

13 λ λ β How to obtain material parameters? Step 1: Crystal geometry (R m ) & 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 Step 3: Electron-phonon coupling constants (g mn mn ) repeat step 2, but for the displaced (phonon-excited) geometry Naphthalene monoclinic crystal two molecules per unit cell herringbone stacking c c a b a Karsten Hannewald (Uni Jena) July 15, 2010, TU Dresden 13

14 Strong Band Narrowing in Naphthalene Crystals Bandwidth (mev) bare LUMO bare HOMO LUMO Holstein-Peierls LUMO Holstein HOMO Holstein HOMO Holstein-Peierls Temperature (K) Napthalene, Anthracene & Tetracene: Hannewald et al., Phys. Rev. B 69, (2004) Durene Crystals: Ortmann, Hannewald & Bechstedt, Appl. Phys. Lett. 93, (2008) Guanine Crystals: Ortmann, Hannewald & Bechstedt, J. Phys. Chem. B 113, 7367 (2009) Experiments? Very difficult to measure but recent progress using ARPES: Pentacene: 240meV (at 120K) 190meV (at 300K) [N. Koch et al., PRL 96, (2006)] (HOMO) 250meV (at 75K) 200meV (at 300K) [R. Hatch et al., PRL 104, (2010)] Go beyond bandwidth calculations & develop mobility theory! Karsten Hannewald (Uni Jena) July 15, 2010, TU Dresden 14

15 Mobility for Narrow Bands ( Small Polarons ): Theory & Modelling DFT calculations Holstein model Holstein-Peierls model bare electron & hole bands polaron bands with local el-ph polaron bands with local + nonlocal el-ph mobilities with local el-ph Holstein, Ann. Phys. (NY) 8, 325 (1959) Mahan Many-Particle Physics (1990) (known) mobilities with local + nonlocal el-ph Hannewald & Bobbert, Phys. Rev. B 69, (2004) Appl. Phys. Lett. 85, 1535 (2004) (new) Karsten Hannewald (Uni Jena) July 15, 2010, TU Dresden 15

16 λ ω ε λ Kubo Formulism for Electrical Conductivity Linear response theory for mobility: T 1 dt j αα~ (t)j α(0) H Current j = dp/dt = 1/i [P,H] with polarization P = e 0 m R m a + ma m current = electronic current + phonon-assisted current (new) j = j (I) + j (II) = e 0 /i (R m -R n ) mn a + ma n + (R m -R n ) Qg Qmn (b -Q +b + Q)a + ma n mn Qmn hopping term for nonlocal coupling! Idea: Evaluate Kubo formula by means of canonical transformation! Key result: Once R m, are known, quantitative predictions for anisotropy & T-dependence of mobilities = (I) + (II) can be made! mn, g mn, and [Hannewald & Bobbert, Phys. Rev. B 69, (2004)] Karsten Hannewald (Uni Jena) July 15, 2010, TU Dresden 16

17 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 = γ T electrons electrons µa µb µc' electrons 0.01 (I) µc' Temperature (Kelvin) Karsten Hannewald (Uni Jena) July 15, 2010, TU Dresden 17

18 Mobility Predictions for Other Materials c* Durene ( hole ) Theory c b Experiment b a* 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, (2008) Burshtein & Williams Phys. Rev. B 15, 5769 (1977) Karsten Hannewald (Uni Jena) July 15, 2010, TU Dresden 18

19 Anisotropic Hole Transport in Durene: Visualization Mobility Tensor HOMO Wave Function Overlap Movie: Karsten Hannewald (Uni Jena) July 15, 2010, TU Dresden 19

20 So Far: Our Approach Works Very Well novel combination of analytical theory & numerical analysis - formulas for polaron bandwidths & 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 (Uni Jena) July 15, 2010, TU Dresden 20

21 General Mobility Theory: Beyond Narrow Bands & Small Polarons Level of Theory bandwidths Ortmann, Bechstedt, Hannewald (2009) full narrow Holstein (1959) Hannewald & Bobbert (2004) local local + nonlocal el-ph coupling Temperature Karsten Hannewald (Uni Jena) July 15, 2010, TU Dresden 21

22 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? solve analytically as before Karsten Hannewald (Uni Jena) July 15, 2010, TU Dresden 22

23 Step 2 : Diagonalization of Before: Narrow-Band Theory = approximate diagonaliz. in real space Now: Generalized Theory = exact diagonalization in k-space a + L a M a + N a P (1 n ~ ' L M pol H n ) n L = n M = constant n k = Fermi distribution of polarons Major improvement: correct statistics + correct T dependence Karsten Hannewald (Uni Jena) July 15, 2010, TU Dresden 23

24 ħ Key Result: Generalized Mobility Formula Ortmann, Bechstedt & Hannewald, Phys. Rev. B 79, (2009) anisotropy momentum occupation energy conservation (incl. phonon absorption & emission) momentum conservation Σ initial state k 1 final state k 2 k 1 k 2 occupied n k1 empty (1-n k2 ) Pauli blocking Σ M : momentum k 1 momentum k 2 +Σ i q i dt : energy energy ± Σ i ω qi coherent band transport & incoherent hopping high T = narrow-band theory Karsten Hannewald (Uni Jena) July 15, 2010, TU Dresden 24

25 Improved Low-Temperature Mobilities (Example: Naphthalene Holes) Ortmann, Bechstedt & Hannewald New J. Phys. 12, (2010) N. Karl, in Landolt-Börnstein Group III, Vol.17, p.106 Hannewald & Bobbert APL 85, 1535 (2004) Karsten Hannewald (Uni Jena) July 15, 2010, TU Dresden 25

26 Summary: Charge Transport in Organic Crystals Novel theories of polaron bandwidths & mobilities in organic crystals & Ab-initio calculations of all material parameters Naphthalene temperature effects & anisotropy 3D visualization deeper understanding of charge transport Karsten Hannewald (Uni Jena) July 15, 2010, TU Dresden 26