1. 5. Transport in Doped Conjugated Materials

2. Nobel Prize in Chemistry 2000
“For the Discovery and Development of Conductive Polymers”
Hideki Shirakawa
University of Tsukuba
Alan Heeger
University of California
at Santa Barbara
Alan MacDiarmid
University of
Pennsylvania

3. Conducting Polymers

4. 5.1. Electron-Phonon Coupling
Excitations
Charges E E
Lowest excitation state +1
Relaxation
effects
Relaxation
effects
Absorption
Ionization
Emission
Ground state GS Q Q
Charge Transport
Optical processes

5. 5.1.1. Geometry Relaxation a b c d e f g a b c d e f g AM1(CI) 14 1315 a b c d e f g 12 16 a b c d e f g
AM1(CI)
 Polaron / Radical-ion
 Polaron-exciton

6. 5.1.2. Geometrical structure vs. Doping level
Radical-cation / Polaron
+ 
Dication / Bipolaron ++

7. Within Koopmans approximation
Geometrical Structure vs. Electronic structure E
Bond length alternation r A B
Within Koopmans approximation

8. Allowed optical transition
Electronic structure upon Doping E
Neutral
Polaron
Bipolaron L
Allowed optical transition H
Spin =0
Charge = 0
Spin =1/2
Charge = +1
Spin =0
Charge = +2

9. Conductivity: σ=p|e|μ
Electronic structure in the solid phase
Anions: A-
with polarons E
lumo
+  A- A- Y
homo A- A-
pol
neutral
X, Y or Z
X, Z
Or bipolarons E
lumo A- A- ++ A- A- A- ++
homo A- A-
bipol
neutral
X, Y or Z A-
Conductivity: σ=p|e|μ
Increase of doping level= higher charge carrier density ”p” larger conductivity

10.  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.

11. Polaron or bipolaron states
5.2. Variable range hopping conduction
The energy difference between filled and empty states is related to the activation energy necessary for an electron hop between two sites
Accessible states
Polaron or bipolaron states
Band edge
Valence Band
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.

12. 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
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).
 The hopping probability from site i to site in a narrow band formed by doped molecules is:
(1)

13. Narrow “polaronic band” made of localized states (obtained upon doping of conjugated molecules)
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)]:
(4)

14. 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 kijET increases
2) Second term-activation energy
<E>=ε0
If ε0 large, the activation barrier is large and the charge transfer is difficult, kijET drops
The maximum for the average hopping probability is obtained for an optimal band width:

15. Mott’s law Optimal band
(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)
Mott’s law
The numerical coefficient η is not determined in this course

16. 5.2.1. Average hopping length <r>
<r> = average distance rij between states in the optimal band
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:
B is a numerical factor related to N(εf)
 In Mott’s theory, the hopping length changes with the temperature.
That’s why this model is also called ”variable range hopping”.

17. Limits of Mott’s law
 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.

18. Where x is determined by details of the phonon-assisted hopping
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:
Efros-
Shklovskii
Where x is determined by details of the phonon-assisted hopping

19. 5.3.1. Chemical doping: protonation
5.3. Example: polyaniline (PAni) ES EB
Chemical doping: protonation
Emeraldine base
Emeraldine salt
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

20. 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

21. 5.4. Metal-Insulator transition
C.O. Yom et al., Synthetic Metals 75 (1995) 229
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.
Conductivity increases
ρ=1/σ
Temperature increases
Metallic regime for ρr < 3: ρ(T) approaches a finite value as T0
Critical regime for ρr= 3: ρ(T) follows power-law dependence
Insulating regime ρr > 3: ρ(T) follows Mott’s law
ρ(T)=aT-β (0.3<β<l)
Ln ρ(T)=(T0/T)1/4

22. The reduced activation energy: W= -T [dlnρ(T)/dT] = -d(lnρ)/d(lnT)
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.
Zabrodskii plot
The reduced activation energy:
W= -T [dlnρ(T)/dT] = -d(lnρ)/d(lnT)
Metallic regime: W>0
Critical regime: W(T)= constant
Insulating regime: W<0
Disorder increases
C.O. Yom et al. /Synthetic Metals 75 (1995)

23. e- PEDOT c = 7.8 Å a = 14 Å 3.4 Å b = 6.8 Å PEDOT-Tos
K.E. Aasmundtveit et al. Synth. Met. 101, (1999)
3.4 Å
a = 14 Å
c = 7.8 Å e-
b = 6.8 Å
PEDOT-Tos
The arrow indicates the critical regime
At low T, metal regime occurs and charge carriers are delocalized
Kiebooms et al. J. Phys. Chem. B 1997, 101, 11037