Charge Transport in Disordered Organic Materials an introduction Ronald Österbacka Deptartment of Physics Åbo Akademi Based on lectures by Prof. V.I. Arkhipov in Turku June 2003 Recommended litterature: Borsenberger & Weiss, Organic Photoreceptors for imaging systems (M. Dekker)

Outline Introduction and Motivation Definitions Electronic structure in disordered solids Positional disorder Deep traps Trap controlled transport Multiple trapping Equilibrium transport TOF Field dependence Gaussian disorder formalism Predicitions Energy relaxation photo-CELIV Summary

Introduction and Motivation Limiting factor in devices is often transport Dispersive transport means ”the longer you measure, the slower the transport” > Properties becomes dependent on measurement conditions!

Definitions Disordered organic materials include: molecularly doped polymers, p-conjugated polymers, spin- or solution cast molecular materials Mobility, m [cm2/Vs], is the velocity of the moving charge divided with electric field (F) m=v/F Conductivity: s=enm=epm Only discussing ”insulating” materials, i.e. s < 10-6S/cm Current: j=sF=epmF

Ordered and disordered materials: defects and impurities Periodic potential distribution implies the occurrence of extended (non-localized) states for any electron (or hole) that does not belong to an atomic orbital A defect or an impurity atom, embedded into a crystalline matrix, creates a point-like localized state but do not destroy the band of extended states

Disordered materials: positional disorder and potential fluctuations Positional disorder inevitably gives rise to energy disorder that can be described as random potential fluctuations. Random distribution of potential wells yields an energy distribution of localized states for charge carriers Potential landscape for electrons Potential landscape for holes

Disordered materials: deep traps Shallow localized states, that are often referred to as band-tail states, are caused by potential fluctuations. Deep states or traps can occur due to topological or chemical defects and impurities. Because of potential fluctuations the latter is also distributed over energy. Shallow (band-tail) states Deep traps

Trap-controlled transport Extended states: jc = ec pcF Mobility edge (E = 0) (Activation) energy Density-of-states distribution Localized states Important parameters: c - carrier mobility in extended states c - lifetime of carriers in extended states 0 - attempt-to-escape frequency pc - the total density of carriers in extended states (free carriers)  (E) - the energy distribution of localized (immobile) carriers

Multiple trapping equations (1) Since carrier trapping does not change the total density of carriers, p, the continuity equation can be written as Change of the total carrier density Drift and diffusion of carriers in extended states Simplifications: (i) no carrier recombination; (ii) constant electric field (no space charge) A.I. Rudenko, J. Non-Cryst. Solids 22, 215 (1976); J. Noolandi PRB 16, 4466 (1977); J. Marshall, Philos. Mag. B, 36, 959 (1977); V.I. Arkhipov and A.I. Rudenko, Sov. Phys. Semicond. 13, 792 (1979)

Multiple trapping equations (2) Trapping rate: Total trapping rate Share of carriers trapped by localized states of energy E Release rate: Attempt-to-escape frequency Boltzmann factor Density of trapped carriers

Equilibrium transport (*) Since the equilibrium energy distribution of localized carriers is established the function (E) does not depend upon time. Solving (*) yields the equilibrium energy distribution of carriers (**) and bearing in mind that Integrating (**) relates p and pc as

Equilibrium carrier mobility and diffusivity The relation between p and pc can be written as where Substituting this relation into the continuity equation yields With the equilibrium trap-controlled mobility, , and diffusivity, D, defined as

Equilibrium carrier mobility: examples DOS Energy 1) Monoenergetic localized states E = Et (E) E = 0 DOS Energy E = Et 2) Rectangular (box) DOS distribution kT

Time-of-flight (TOF) measurements Field Transient current Light Current Equilibrium transport: Time Current Equilibrium transit time ttr

Trap controlled transport: field dependent mobility E-field lowers the barrier Poole-Frenkel coefficient J. Frenkel, Phys. Rev. 54, 647-648 (1938) Problem: b does not fit!

Gaussian Disorder formalism The Gaussian Disorder formalism is based on fluctuations of both site energies and intersite distances (see review in: H. Bässler, Phys. Status Solidi (b) 175, 15 (1993) ) Long range order is neglected > Transport manifold is split into a Gaussian DOS! Distribution arises from dipole-dipole and charge-dipole interactions Field dependent mobility arises from that carriers can reach more states in the presence of the field. It has been argued that long range order do exist, due to the charge-dipole interactions. (see Dunlap, Parris, Kenkre, Phys. Rev. Letters 77, 542 (1996) ) -> Correlated disorder model

Equilibrium carrier distribution: Gaussian DOS Energy  The width of the (E) distribution is the same as that of the Gaussian DOS !

Equilibrium mobility: Gaussian DOS Energy E = 0 Ea  >  >   >  >  Activation energy of the equilibrium mobility Ea is two times smaller than the energy Em around which most carriers are localized !

Bässler model in RRa-PHT s/kT m0=2.5 ´ 10-3 cm2/Vs s=100 meV S=3.71 C=8.1´10-4 (cm/V)1/2

Disorder formalism, predictions A cross-over from a dispersive to non-dispersive transport regime is observed. The Bässler model predicts a negative field dependent mobility! Borsenberger et. al, PRB 46, 12145 (1992) A.J. Mozer et. al., Chem. Phys. Lett. 389, 438 (2004). A.J. Mozer et. al, PRB in press

Carrier equilibration: a broad DOS distribution After first trapping events the energy distribution of localized carriers will resemble the DOS distribution. The latter is very different from the equilibrium distribution. E = 0 1(E) eq(E) Energy Those carriers, that were initially trapped by shallow localized states, will be sooner released and trapped again. For every trapping event, the probability to be trapped by a state of energy E is proportional to the density of such states. Therefore, (i) carrier thermalization requires release of trapped carriers and (ii) carriers will be gradually accumulated in deeper states. Concomitantly, (i) equilibration is a long process and (ii) during equilibration, energy distribution of carriers is far from the equilibrium one.

Photo-CELIV G. Juška, et al., Phys. Rev. Lett. 84, 4946 (2000) G. Juška, et al., Phys. Rev. B62, R16 235 (2000) G. Juška, et al., J. of Non-Cryst. Sol., 299, 375 (2002) R. Österbacka et. al., Current Appl. Phys., 4, 534-538 (2004)

Photo-CELIV G. Juška, et al., Phys. Rev. Lett. 84, 4946 (2000) G. Juška, et al., Phys. Rev. B62, R16 235 (2000) G. Juška, et al., J. of Non-Cryst. Sol., 299, 375 (2002) R. Österbacka et. al., Current Appl. Phys., 4, 534-538 (2004)

Mobility Relaxation measurements The tmax shifts to longer times as a function of tdel The tmax is constant as a function of intensity Photo-CELIV is the only possible method to measure the equilibration process of photogenerated carriers.

Mobility relaxation R. Österbacka et al., Current Appl. Phys. 4, 534-538 (2004)

Summary An introduction to carrier transport in disordered organic materials is given Disorder gives rise to potential fluctuations > Energy distribution of localized states By knowing the DOS: equilibrium transport can be calculated In the disorder formalism (Bässler) carrier equilibration is a long process > Decrease of mobility as a function of time! We have shown a possible method (CELIV) to measure the equilibration process

Acknowledgements Planar International Ltd for patterned ITO A. Pivrikas, M. Berg, M. Westerling, H. Aarnio, H. Majumdar, S. Bhattacharjya, T. Bäcklund, and H. Stubb, ÅA G. Juska, K. Genevicius and K. Arlauskas, Vilnius University, Lithuania Planar International Ltd for patterned ITO Financial support from Academy of Finland and TEKES Graduate student positions open: See www.abo.fi/~rosterba