5. Transport in Doped Conjugated Materials

Nobel Prize in Chemistry 2000

“For the Discovery and Development of Conductive Polymers”

Alan HeegerUniversity of California at Santa Barbara

Alan MacDiarmid University of Pennsylvania

Hideki Shirakawa University of Tsukuba

Conducting Polymers

5.1. Electron-Phonon Coupling

E

Q

Absorption

Relaxation effects

Emission

Ground state

Lowest excitation state

Excitations Charges

Ionization

GS

+1

E

Q

Relaxation effects

Optical processes Charge Transport

5.1.1. Geometry Relaxation

Polaron / Radical-ion

Polaron-exciton

AM1(CI)

141513

12 16a

a b c d e f g

b c d e f g

5.1.2. Geometrical structure vs. Doping level

Radical-cation / Polaron

+

Dication / Bipolaron

++

5.1.3. Geometrical Structure vs. Electronic structure

E

Bond length alternation r

A

B

With

in K

oopm

ans

appr

oxim

atio

n

5.1.4. Electronic structure upon Doping

E

H

L

Polaron BipolaronNeutral

Spin =1/2Charge = +1

Spin =0Charge = +2

Spin =0Charge = 0

Allo

wed

opt

ical

tr

ansi

tion

+

5.1.5. Electronic structure in the solid phase

++

++

Or bipolarons

Anions: A-

with polarons

E

E

X, Y or Z

X, Y or Zbipol neutral

pol neutral

A-

A-

X, Z

Y

A-

A-

A-

A-

A-

A-

A-

A-A-

A-

lumo

homo

lumo

homo

Conductivity: σ=p|e|μIncrease of doping level= higher charge carrier density ”p” larger conductivity

According to the doping level, the charge carrier density and the nature of the charge carriers can be tuned

Energy disorder comes from (i) the position of the counter-ions, (ii) the polarization energy that is site dependent, and (iii) the crystal defects.

Spatial disorder arises from a variation in the density in charge carriers, crystal defects, position of the counter-ions.

At moderate doping level and room temperature, charge carriers in an organic crystal are localized. The energy levels involved in the transport from one site to the other (empty, filled or half filled) by hopping are spread over an energy range.

This situation is similar to disordered inorganic semiconductors that are slightly doped. In those materials, the charge transport can be described with the variable range hopping.

5.2. Variable range hopping conduction

Polaron or bipolaron states

The charge transport occurs in a narrow energy region around the Fermi level. The charge can hop from a localized filled to a localized empty state that are homogeneously distributed in space and around εf. i.e. with a constant density of states N(ε) over the range [εf – ε0, εf – ε0].

N(ε)dε= number of states per unit volume in the energy range dε.2ε0 is the width of the “band” involved in the transport. The localized character of a state is determined by the parameter r0.

Accessible states

The energy difference between filled and empty states is related to the activation energy necessary for an electron hop between two sites

Valence Band

Band edge

In the semi-classical electron transfer theory by Marcus, the rate of charge transfer between two sites i and j is:

E = activation energy

t= transfer integral

N(ε)= density of states

kT

Etk

ijETij exp2

02exp

r

rt ij

The localization radius r0 in Mott’s theory appears to be related to the rate of fall off of t with the distance rij between the two sites i and j (see previous chapter).

kT

E

r

rTrP

ijij

0exp),(

The hopping probability from site i to site in a narrow band formed by doped molecules is:

(1)

In this “band”, the average activation energy barrier necessary to be overcome to transfer an electron from a filled to an empty state is <Eij>=ε0. (2)

The concentration C(ε0) of states in the solid characterized by the band width 2ε0 is [N(εf) 2ε0]= number of states per volume in the band.

The average distance between sites involved is <rij>= [C(ε0)]-1/3= [N(εf) 2ε0]-1/3 (3)

The average hopping probability between two states [inject (2) and (3) in (1)]:

Narrow “polaronic band” made of localized states (obtained upon doping of conjugated molecules)

(4)

kTrTP 0

0

3/100

0

)2εN(εexp),(

P(0)=exp-(1/0+0)

0

0 2 4 6 8 10 12

P(

0)

0,0

0,2

0,4

0,6

0,8

1,0

P(0)=exp-(1/0): from <rij>

P(0)=exp-(0): from <E>=0

The maximum for the average hopping probability is obtained for an optimal band width:

1) First term- electronic coupling<rij>= [N(εf) 2ε0]-1/3

If wide band, i.e. ε0 large, many states are available per volume it is easy to find a neighbor site j such that Eij<ε0

<rij> decreases, t increases and kij

ET increases

2) Second term-activation energy<E>=ε0

If ε0 large, the activation barrier is large and the charge transfer is difficult, kij

ET drops

4/13

0

4/3max0

max0

)()(

rN

kTT

f

kTrTP 0

0

3/100

0

)2εN(εexp),(

(i) kET or P(ε0,T) is proportional to the mobility of the charge carrier

(ii) Conductivity σ = n|e| , with n the density of charge carrier

(iii) The conductivity of the entire system is determined in order of magnitude by the optimal band (States out of the band only slightly contribute to σ).

Conductivity σ (T) ÷ P(ε0max,T)

4/100 /exp)( TTT

30

0 )( rkNT

f

The numerical coefficient η is not determined in this course

Mott’s law

Optimal band

5.2.1. Average hopping length <r>

<r> = average distance rij between states in the optimal band

4/1

00

3/1max0 /)( TTrNr f

In Mott’s theory, the hopping length changes with the temperature.

That’s why this model is also called ”variable range hopping”.

As T decreases, the hopping length <r> grows.

Indeed, as T decreases, the hopping probability decreases, so the volume of available site must be increased in order to maximize the chance of finding a suitable transport route.

However the probability ω per unit time for such large hops is small:

4/1/exp TB B is a numerical factor related to N(εf)

Col 1 vs Col 2 <R>=(1/T)1/4

T

0 2 4 6 8 10 12

<R

> a

nd

0,0

0,2

0,4

0,6

0,8

1,0

1,2

1,4

=exp(-1/T1/4)

1/100 /exp)( dTTT

Mott’s theory was developed for hopping transport in highly disordered system with localized states characterized by a localization length r0. Not too small values of r0 (also related to the transfer integral t) are necessary to be in the VRH regime.

If r0 is too small, i.e. if the carrier wavefunction on one site is very localized, then hopping occurs only between nearest neighbors: this is the nearest-neighbor hopping regime.

The situation of high disorder, thus the homogeneous repartition of levels in space and energy, is not strictly true for polymers with their long coherence length and aggregates. However, it has some success for an intermediate doping and conductivity.

A more general expression is given with d the dimensionality of the transport.

5.2.2. Limits of Mott’s law

When the coulomb interaction between the electron which is hopping and the hole left behind is dominant, then the conductivity dependence is

In general in the semiconducting regime:

2/100 /exp)( TTT

x

x

TT

TTT/1

/10

)(ln

/exp)(

Where x is determined by details of the phonon-assisted hopping

Efros-Shklovskii

ES

EB

Emeraldine base

Emeraldine salt

5.3. Example: polyaniline (PAni)

The doping of PAni is done by protonation, while with the other conjugated polymer it is achived by electron transfer with a dopant or electrochemically

5.3.1. Chemical doping: protonation

5.3.2. Secondary doping: solvent effect

CSA-= camphor sulfonate

Cl-= Chlorine anion

Morphology of the polymer chain is modified

M. Reghu et al. PRB, 1993, 47, 1758

5.4. Metal-Insulator transition

The resistivity ratio: ρr= ρ(1.4K)/ ρ(300K)

The temperature dependence of the resistivity of PANI-CSA is sensitive to the sample preparation conditions that gives various resistivity ratios that are typically less than 50 for PANI-CSA.

Metallic regime for ρr < 3: ρ(T) approaches a finite value as T0Critical regime for ρr= 3: ρ(T) follows power-law dependenceInsulating regime ρr > 3: ρ(T) follows Mott’s law

ρ(T)=aT-β (0.3<β<l)Ln ρ(T)=(T0/T)1/4

C.O. Yom et al., Synthetic Metals 75 (1995) 229

ρ=1/σC

ondu

ctiv

ity in

crea

ses

Temperature increases

C.O. Yom et al. /Synthetic Metals 75 (1995) 229-239

Zabrodskii plot

The reduced activation energy: W= -T [dlnρ(T)/dT] = -d(lnρ)/d(lnT)

Metallic regime: W>0Critical regime: W(T)= constantInsulating regime: W<0

The systematic variation from the critical regime to the VRH regime as the value of ρr increases from 2.94 to 4.4 is shown in the W versus T plot. This is a classical demonstration of the role of disorder-induced localization in doped conducting polymers.

Dis

orde

r in

crea

ses

PEDOT

K.E. Aasmundtveit et al. Synth. Met. 101, 561-564 (1999)

3.4 Åa = 14 Å

c =

7.8

Å

e-

e-

b =

6.8

Å

PEDOT-Tos

The arrow indicates the critical regime

Kiebooms et al. J. Phys. Chem. B 1997, 101, 11037

At low T, metal regime occurs and charge carriers are delocalized