Charge Transport in Disordered Charge Transport in Disordered Organic MaterialsOrganic Materials

an introductionan introduction

Ronald ÖsterbackaDeptartment 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)

2

OutlineOutline• 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

4

DefinitionsDefinitions

•Disordered organic materials include: molecularly doped polymers, -conjugated polymers, spin- or solution cast molecular materials

•Mobility, [cm2/Vs], is the velocity of the moving charge divided with electric field (F) =v/F

•Conductivity: =en=ep

•Only discussing ”insulating” materials, i.e. < 10-6S/cm

•Current: j=F=epF

5

Ordered and disordered materials: Ordered and disordered materials: defects and impuritiesdefects 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

Coordinate

En

erg

y

Coordinate

En

erg

yA 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

6

Disordered materials: positional disorder Disordered materials: positional disorder and potential fluctuationsand potential fluctuations

Coordinate

En

erg

y

Potential landscape for electrons

Potential landscape for holes

En

erg

y

Density of states

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

7

Disordered materials: deep trapsDisordered materials: deep traps

Coordinate

En

erg

y

Shallow (band-tail) states

Deep trapsE

ne

rgy

Density of states

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.

8

Trap-controlled transportTrap-controlled transport

Mobility edge (E = 0)

Localized states

Important parameters:c - carrier mobility in extended states

c - lifetime of carriers in extended states

0 - attempt-to-escape frequency

(Act

ivat

ion )

en e

rgy

Density-of-states distribution

Extended states: jc = ec pcF

r(E)E = 0

En

erg

y

DOS, g(E)

pc - the total density of carriers in extended states (free carriers)

r (E) - the energy distribution of localized (immobile) carriers

EdEpp c r

9

Multiple trapping equations (1)Multiple trapping equations (1)

Since carrier trapping does not change the total density of carriers, p, the continuity equation can be written as

t

p

2

2

x

pD

x

pF c

cc

c

0

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)

10

Multiple trapping equations (2)Multiple trapping equations (2)

r(E)E = 0

En

erg

yDOS, g(E)

Trapping rate:

0cp

Total trapping rate

Share of carriers trapped by localized states of energy E

Release rate:

EkT

E r

exp0

Attempt-to-escape frequency

Boltzmann factor

Density of trapped carriers

EkT

EpEg

Nt

Ec

t

r

r

exp1

00

tN

EEg r tN

Eg

11

Equilibrium transportEquilibrium transport

EkT

EpEg

Nt

Ec

t

r

r

exp1

00

Since the equilibrium energy distribution of localized carriers is established the function r(E) does not depend upon time.

0

Solving (*) yields the equilibrium energy distribution of carriers

kT

EEg

N

pE

t

c exp00

r

Integrating (**)

(*)

(**)

relates p and pc as

p

kT

EEgdE

N

pp

t

cc exp

00

kT

EEgdE

N

p

t

c exp00

EdEpp c rand bearing in mind that

12

Equilibrium carrier mobility and diffusivityEquilibrium carrier mobility and diffusivity

pTpc

The relation between p and pc can be written as

where 1

00 exp

kT

EEgdENT t

t

p

2

2

x

pD

x

pF c

cc

c

0

Substituting this relation into the continuity equationyields

02

2

x

pDT

x

pFT

t

pcc 0

2

2

x

pTD

x

pFT

t

p

With the equilibrium trap-controlled mobility, , and diffusivity, D, defined as

cTT cDTTD

13

Equilibrium carrier mobility: examplesEquilibrium carrier mobility: examples

1) Monoenergetic localized states E = 0DOS

En

ergy

E = Et

tt EENEg

kT

ET t

c exp00

2) Rectangular (box) DOS distribution E = 0DOS

En

ergy

E = Et

ttt

t EEEgEEE

NEg ,0;,

kT

r(E)

1exp

00

kTE

kT

ET

t

tc

14

Time-of-flight (TOF) measurementsTime-of-flight (TOF) measurements Field

Light

Cu

rren

t

L

cc

L

c txpdxL

Fetxjdx

Lj

00

,,1

Transient current

Equilibrium transport:

ceqc TtxpTtxp ,,,

L

eq txpdxL

Fej

0

,

Time

Cu

rren

t

ttr

Equilibrium transit time

F

Lt

eqtr

treq Ft

L

15

Trap controlled transport: Trap controlled transport: field dependent mobilityfield dependent mobility

J. Frenkel, Phys. Rev. 54, 647-648 (1938)

•E-field lowers the barrier

•Poole-Frenkel coefficient

2/1

03 ePF

100 150 200 250 300 35010-6

10-5

10-4

p [c

m2 /V

s]

F1/2 [V/cm]1/2

Problem: does not fit!

16

Gaussian Disorder formalismGaussian 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 DOSEquilibrium carrier distribution: Gaussian DOS

DOSr(E)E

ner

gy

E = 0

2

2

2exp

2

EN

Eg t

2

2

2exp

2

1

r mEE

E

kTEm

2 The width of the r(E)

distribution is the same as that of the Gaussian DOS !

18

Equilibrium mobility: Gaussian DOSEquilibrium mobility: Gaussian DOS

2

200

2exp

2 kTT c

kT

Eac exp

200

22

2m

a

E

kTE

DOSr(E)E

ner

gyE = 0

kTEm

2

Ea

Activation energy of the equilibrium mobility Ea is two times smaller than the energy Em around which most carriers are localized !

2 3 4 5 6 7 8 9 1010-15

10-13

10-11

10-9

10-7

10-5

10-3

10-1

Mo

bili

ty,

a.

u.

1000/T, K-120 40 60 80

10-15

10-13

10-11

10-9

10-7

10-5

10-3

10-1

Mo

bili

ty,

a.

u.

(1000/T)2, K-2

> > > >

19

Bässler model in RRa-PHTBässler model in RRa-PHT

200 300 4001x10-8

1x10-7

1x10-6

1x10-5

305 K 290 K 285 K 260 K 240 K 215 K

[cm

2 /Vs]

F0.5 [(V/cm)0.5]

2/1222/12

0 F Cexp3

2exp),,(

F

0=2.5 10-3 cm2/Vs=100 meV=3.71 C=8.110-4 (cm/V)1/2

10-1 100 101 102 103 10410-10

10-9

10-8

10-7

10-6

10-5

10-4

RC

RRa-PHTd=2.5m

ttr(50V)

50V 46V 42V 38V 34V 30V 26V 22V 18V 14V 10V 6V 2V

j [A

]

t [s]

/kT

20

Disorder formalism, predictionsDisorder formalism, predictions

A.J. Mozer et. al., Chem. Phys. Lett. 389, 438 (2004). A.J. Mozer et. al, PRB in press

A cross-over from a dispersive to non-dispersive transport regime is observed.

Borsenberger et. al, PRB 46, 12145 (1992)

The Bässler model predicts a negative field dependent mobility!

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Carrier equilibration: a broad DOS Carrier equilibration: a broad DOS distributiondistribution

DOS

req(E)

En

ergy

E = 0 After first trapping events the energy distribution of localized carriers will resemble the DOS distribution. The latter is very different from the equilibrium distribution.

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.

r1(E)

22

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)

0 1000 2000 3000 40000

5

10

15

j(0)=A/d

j [A

/cm

2 ]

t [s]

0 1000 2000 3000 4000

AU=At

t [s]

Photo-CELIVPhoto-CELIV

23

0 1000 2000 3000 40000

5

10

15

j0

tmax

j

j [A

/cm

2 ]

t [s]

0 1000 2000 3000 4000

tdel

AU=At

t [s]

Photo-CELIVPhoto-CELIV

)0(36.013

2

2max

2

jj

At

d

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)

24

Mobility Relaxation measurementsMobility Relaxation measurements

0 1 2

0

5

10

15

20

50 s 200 s 500 s 1000 s 2000 s 10 ms dark

j [A

/cm

2 ]

t [ms]

The tmax shifts to longer times as a function of tdel

0 1 2

0

5

10

15

20

t [ms]

8 J 4.5 J 2.8 J 0.94 J 1.59 J 0.33 J dark

j [A

/cm

2 ]The tmax is constant as a function of intensity

Photo-CELIV is the only possible method to measure theequilibration process of photogenerated carriers.

25

Mobility relaxationMobility relaxation

10-4 10-3 10-2 10-110-7

10-6

10-5

t-0.58

[cm

2 /Vs]

tdel

+ tmax

[s]

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

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SummarySummary

• 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

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AcknowledgementsAcknowledgements

•Planar International Ltd for patterned ITO

•Financial support from Academy of Finland and TEKES

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

Graduate student positions open: See www.abo.fi/~rosterba