1. How to understand transport?
Part 1: Transport in disordered organic materials
Part 2: Contacts

2. 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

3. 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

4. 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

5. Anderson localization: the problem
Crystalline system
Coordinate x
U(x)
A periodic potential energy distribution yields an extended wave function
(x)2
Disordered system
What will happen in an (energetically) disordered system? Can the wave function be still extended?
Coordinate x
U(x)
(x)2
Or it will be localized within a single potential well?
(x)2

6. Anderson localization: localized states in 1D disordered systems
Disordered organic materials and polymers:
Inorganic disordered semiconductors:
- larger density of states;
- smaller density of states;
- larger localization radius.
- smaller localization radius.
Hopping transport!

7. Master equation for carrier hopping
Et(i)
ft(1)
Carrier jump rate from a starting site of energy Est to a target site of energy Et over the distance r is often described by the Miller-Abrahams expression as:
Et(1)
ft(i) r1 ri
Est
fst
Et(2) r3 r2
downward hopping
ft(3)
ft(2)
Et(3)
tunneling
Hopping master equation:
upward hopping
The problem is averaging !
fj is the occupational probability

8. Variable-range hopping around the Fermi level
Energy
DOS
E = 0
g(E) = g0
Vacant sites
Fermi level
Occupied sites
Hopping conductivity includes both energetically upward and downward jumps. However, upward jumps are more difficult and they are the rate-limiting steps.
Prof. Arkhipov showed that for hopping transport the occurrence of the effective transport level virtually reduces hopping to multiple trapping except for the initial regime of downward hopping relaxation.
V. I. Arkhipov et al., PRB 64, (2001)

9. 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

10. 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)

11. 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

12. 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

13. 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

14. 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

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

16. 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

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

18. 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 !

19. Trap controlled transport: field dependent mobility
E-field lowers the barrier
Poole-Frenkel coefficient
Schematic picture, Poole-Frenkel
J. Frenkel, Phys. Rev. 54, (1938)
Problem: b is too large for organics!

20. Electric field dependent m
A.J. Mozer et. al, PRB 71, (2005)
Jumps against the field might be more favorable due to weak electronic coupling in large positional disorder! H. Bässler, Phys. Status Solidi (b) 175, 15 (1993)

21. Carrier equilibration: a broad DOS distribution
After first trapping events the energy distribution r(E) of localized carriers will resemble the DOS distribution. The latter is very different from the equilibrium distribution, req(E).
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.

22. Dispersive to non-dispersive transport
s/kT = 2.8
At high temperatures (s/kT small):
-> Quasi equilibrium is obtained!
Pautmeir et. al., Phil. Mag. Lett 59, 325 (1989)
At low temperatures (s/kT large):
-> Non-equilibrium
-> Transport characteristics become sample dependent!
s/kT = 4.6
s/kT = 6.8

23. Temperature dependence of m
Dispersive
A.J. Mozer et. al, PRB 71, (2005)

24. Concentration dependence on m
Big discrepancy between mobility mesaurements
FETs always give high m and lower activation energy
LEDs give low m and higher activation energy
Caused by the broad DOS
Tanase et al., PRL 91, (2001)

25. Coulomb traps (a) Weak doping (b) Heavy doping xm xm U(x) U(x) -eFx
(a)
-eFx
U(x) xm 
(b)
At low doping levels, Coulomb potential wells of ionized dopants do not overlap and their depths do not depend upon dopant concentration
At high doping levels, they overlap strongly and their depths decreases with increasing dopant concentration
Coulombic energy disorder decreases at high doping levels!

26. Concentration dependence of the mobility in chemically doped disordered organics
 = 65 meV

27. Mobility in a chemically doped organic semiconductor vs FET mobility
H. Shimotani et al., Appl. Phys. Lett. 86, (2005).
V.I.Arkhipov et al., Phys. Rev. B, B 72, (2005).

28. Summary: transport
Potential fluctuations give rise to energy distributions of localized states
Transport is by hopping; similarities with multiple trapping and retrapping
In disordered organic materials we usually observe a broad Gaussian DOS
Broad DOS implies a carrier relaxation during which the transport is dispersive (time-dependent)
Field-dependence m(F)µ exp(bF½ ), with the b coefficient being DOS dependent
Mobility dependends on carrier concentration
Effect of Coulomb traps of

29. Contacts Definitions: Work function: f
The work needed to eject one electron from the surface of a metal or dielectric
Electron affinity: c
Minimum excess energy to leave the dielectric or semiconductor
Fermi level: EF
The energy level below which all levels are filled, and above all are empty

30. Contacts
When bringing an electrode in close contact with a semiconductor

31. Influence of contacts:
The current through a device can either be:
Transport limited OR
Contact limited
If transport limited, then the contacts act as an infinite charge reservoir (”Ohmic”).
Let’s consider a charge reservoir (electrode) at x=0, and a sink at x=d (interelectrode distance)

32. Ideal trap-free solids:
Poisson equation:
Electric field
Free carriers
Trapped carriers = 0!
(B)
Current density:
Multiplying (A) with 2F(x) and substituting (B) into (A) we get
(C)

33. Space-charge limited current
By integrating (C) :
We get the potential by integrating the field over the distance:
And by choosing x=d and the applied potential V=V(d):
Child’s law!

34. SCLC currents Scales as V2 and d-3.
At low voltages the J-V is ohmic i.e. Linear current density as a function of voltage. The cross-over voltage is the voltage required for the carrier transit time being larger than the dielectric relaxation time

35. Contact limited transport
If the contact do not form an ohmic contact it will limit the amount of carriers that can be transported through the material.
Two types of limiting contacts:
Neutral contact: There is no space charge at either side of the contact. Also called ”flat-band” condition. The built-in field is defined as:
Blocking contact: A space charge is formed causing ”band-bending” at the contact
where

36. Blocking contact (Schottky)
The barrier is lowered due to the combined force of the ”image potential” and applied field. The potential barrier height from the Fermi energy of the metal is given by
Applied field
Image force

37. Injection currents

38. Field-enhanced thermionic emission
Where the Richardson constant A* depends on the material, but in vacuum
And fB= fB(F) depends on the type of contact

39. Injection into a disordered organic semiconductor
The energy disorder hinders the escape over the barrier, practically due to hops are short compared to the Schottky barrier
The effective activation energy is incerased due to energy disorder
Gartstein & Conwell, CPL 255, 93 (1996)

40. Doping increases injection
By doping the semiconductor, the injection becomes more feasible.
Eventually the injection is Ohmic.
Impossible to have a blocking contact for highly conducting organic materials.
Hosseini et al., JAP 97 (-05)

41. Summary: Contacts Transport can be bulk-limited or contact limited
If bulk limited, then the contact is said to be Ohmic -> SCLC type transport
If contacts are not ohmic, they can be either ”neutral” or ”blocking”
Disorder lowers the injection probability
Doping enhances injection in a disordered material