5. Transport in Doped Conjugated Materials. Nobel Prize in Chemistry 2000 “For the Discovery and...
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Transcript of 5. Transport in Doped Conjugated Materials. Nobel Prize in Chemistry 2000 “For the Discovery and...
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