Charge transport in organic semiconductors is commonly described within the framework of the Gaussian Disorder Model (GDM).208 Energetic and positional disorder result in localization of charge carriers on their transport sites, be it a single molecule or a conjugated segment of a polymer chain. Due to the associated random fluctuations of intermolecular interactions charge transport takes place via hopping among (uncorrelated)1 neighbouring sites in an inhomoge-nously broadened density of states (DOS) of Gaussian shapeρ(E):
ρ(E) = √ 1 2πσ2exp
"
−(E−E0)2 2σ2
#
(4.1) with E0 being the energy at the centre of the DOS and σ being its standard deviation. The hopping rateνijamong two sitesiandjwith energiesEiandEj, respectively is usually expressed in terms of a Miller-Abrahams-rate if relaxation effects are small compared to disorder effects:
νij = ν0exp (−2γr)
exp−Ekj−Ei
BT
Ej > Ei
1 Ej ≤ Ei (4.2)
where ν0 denotes a frequency prefactor, γ the inverse localization length and r the intersite distance. The first exponential accounts for the electronic wavefunction overlap while the second one is simply a Boltzmann factor accounting for thermally activated jumps upward in energy.
Downward jumps are not hindered. If relaxation effects are relevant, i.e. when electron-phonon coupling is significant, Marcus-type rates are often used instead of Miller-Abrahams hopping rates.211
According to this model injected or photogenerated charge carriers will execute a random walk inside DOS and subsequently relax energetically until they reach a quasi-equilibrium energy located athE∞i = −kσ2
BT = −σσb below the centre of the DOS. This results in the formation of a so called occupied DOS centred at hE∞i with the same variance σ2 as the DOS. Charge transport then takes place via thermal activation to a so called transport energy Etr (figure 4.1(a)).212 Choosing the energy scale such that the transport energy is located at 0 eV and taking both energetic (σb) and positional (Σ) disorder as well as the presence of an external electric fieldF into account, the charge carrier mobilityµ(T, F) as a function of temperatureT and fieldF then takes the form:208,213
1 This assumption is justified in case of amorphous (glassy) organic solids with small structural correlation lengths in the order of only a few nm.208 If energetic correlations are present, e.g. due to the presence of permanent or induced dipoles, more complex, extended models need to be considered.209,210
µ(T, F) = µ0exp
"
− 2σb
3 2#
exph σb2−Σ2√
Fi Σ = 1.5 exph σb2−2.25√
Fi Σ < 1.5 (4.3)
r 𝐸𝑡𝑡
𝐸 𝜌(𝐸) 𝐸∞
𝜌𝑂𝑂(𝐸)
∆𝑅𝑗𝑗 DOS
𝑖 →𝑗 ∶ ~𝑒−2𝛾∆𝑅𝑖𝑖𝑒− 𝐸𝑗𝑖𝐵−𝐸𝑇𝑖 𝑗→𝑘 ∶ ~𝑒−2𝛾∆𝑅𝑖𝑗
i j k
r
𝐸 𝜌(𝐸)
DOS 𝜌𝑡(𝐸) (1) (2)
a)
b)
Figure 4.1.: (a) Illustration of charge transport via hopping in an inhomogenously broadened DOS ρ(E) according to the GDM. After initial relaxation to a quasi-equilibrium energy hE∞i establishing an occupied DOS ρOc(E) subsequent transport takes place via thermally activated hops to the transport energy Etr (i → j). Downward jumps in energy are not impeded (j → k). ∆Rjk is the distance between two sites j and k. E and r denote energetic and spatial coordinates, respectively. (b) Hopping transport in a Gaussian DOS ρ(E) in the presence of additional trapping sites. The associated trap DOS is assumed to be Gaussian, too. Charge transport is impeded due to repeated trapping (1) and release (2) of charge carriers.
Common methods to experimentally assess the mobility in organic semiconductors range from the early Time of Flight experiments (TOF)214–218 and the widely used Space-Charge-Limited-Current (SCLC) measurements,218–223 where relatively thick layers of several 100 nm up to 1µm are needed, over the Charge Extraction by Lineraly Increasing Voltage (CELIV) technique, which allows to measure in thickness and carrier density regimes relevant to actual devices,85,224–227to highly sophisticated methods like Time Resolved Microwave Conductivity (TRMC)228and Time Resolved Electric Field Induced Second Harmonic Generation (TREFISH),229 which enables systematic probing of mobility on different length and time scales, making it possible for example to directly study intra- and interchain transport in conjugated polymers. In the course of this
28
thesis, two selected methods are used to characterize the charge carrier mobility at low to medium as well as high charge carrier densities, respectively: MIS-CELIV and OFET measurements. The first technique is a modification of the original CELIV method where an additional insulating layer (I) is inserted into an OPV device or a single layer heterojunction between the organic semiconductor (S) and the metal electrode (M). By applying a prebias voltage charge carriers are selectively injected into the sample and accumulated at the semiconductor/insulator interface forming a charge reservoir that may subsequently extracted by applying a voltage ramp.85,230 Mobilityµmay then be determined from the current response of the sample according to85,230,231
µ = 2d2s current densities with previously applied prebias and for pure capacitive response, respectively.
The indicessandi refer to the organic semiconductor and the insulator, respectively,d denotes the layer thickness and the relative permittivity. A = dVdt is the slope of the applied voltage ramp, ttr is the transit time of a charge carrier from the insulator/semiconductor interface to the opposite electrode and t2j0 is the time at which the current density reaches two times of the capacitive plateau value j0. This technique allows to selectively study electron and hole mobilities depending on the position of the insulating layer. The measurement principle is illustrated in figure 4.2(a). Detailed descriptions of the method and the related data analysis are given in the works by Armin et al., Juška et al. and Sandberg et al.85,230,231
In contrast to CELIV, OFET measurements yield mobility values characteristic for high charge carrier densities.9,232–234 The general principle for a bottom gate/top contact p-type OFET is shown in figure 4.2(b). An OFET features a three electrode architecture, with a gate electrode that is separated from the organic semiconductor via an insulating layer (gate dielectric) and two further electrodes, source and drain, that are in direct contact with the semiconductor. The current between source and drain is controlled via the gate voltage Vg applied between source and gate electrode, as above a certain threshold voltage VT the associated field draws positive charges (for Vg < 0V) from the source into the semiconductor which then accumulate at the interface to the gate dielectric forming a conducting channel. This in turn allows a current flow between source and drain when a (drain) voltageVdis applied between these two electrodes. For voltages Vd Vg, the current Id between source and drain (drain current) increases linearly withVd. In this regime, the mobility may be determined according to9,233
µlin = L W CiVd
∂Id
∂Vg
|Vd (4.5)
withLand W being the channel length and with, respectively andCibeing the capacitance per unit area. For voltagesVdhigher thanVg−VT, the accumulation layer will start to get depleted and the drain current will start to saturate. In this regime, the mobility may be determined according to9,233 Detailed information about the working principle and various aspects related to the operation of OFETs may especially be found in the works of Sirringhaus and coworkers.232–235
Source
Figure 4.2.: (a) Illustration of a typical sample structure in a MIS-CELIV experiment (left) and the according measurement principle (right). Charge carriers, in this case holes, are injected from one electrode and accumulated at the semiconductor(S)/insulator(I) interface by applying a prebias voltageVof f. These charge carriers are then extracted by a subsequent voltage ramp of durationtp. The mobility may finally be determined from the resulting current (density) response j(t). The associated response is illustrated for low (dashed), medium (dotted) and high (solid) prebias voltages. j0 denotes the current density due to pure capacitive response, t2j0 the time at which j(t) = 2j0 and ttr the transit time of the charge carriers needed to travel through the device. Further details are given in the text. (b) Schematic of a typical bottom gate/top contact p-type OFET architecture. Vg and Vd denote gate and drain voltage, respectively.
An important issue of practical relevance in terms of charge transport is the presence of addi-tional energy sites below the DOS that may impede the motion of charges via trapping.213,221,223 232,236,237Thesetrap statesmay be due to structural or chemical defects or impurities.43,221,238,239
Especially in the context of crosslinking polymers to form insoluble networks and stabilize mor-phologies the latter two aspects could play a role, as chemical crosslinks are formed and side products or remnants of initiators may remain in the organic film. This issue is addressed in the course of my publication on the influence of crosslinking on the hole mobility in a series PF2/6 derivatives (see chapter 8.2.3 and 11).
In the simplest way, one can think of traps effectively increasing the disorder parameterσthereby broadening the DOS, as long as the concentration of traps is small compared to the number of states in the original DOS (figure 4.1(b)).236,240. The mobility is reduced in this case since charge carriers have to be released from trap states by thermal activation before they can
con-30
tinue their movement (”trapping-and-release transport”). In case of high trap concentrations, charge transport will rather take place within the trap DOS (”trap transport”).240 This may be the case in sufficiently doped organic semiconductors. The effect of traps on charge transport has been extensively studied by several groups, especially by Blom and coworkers.210,213,221,236,240–244
By investigating a wide range of semiconducting polymers Blom and coworkers found that the frequently observed inferiority of electron transport in organic semiconductors is related to the existence of a universal trap level located at about 3.6 eV below the vacuum level that is present in all the investigated compounds. As origin of these traps they suggest hydrated-oxygen com-plexes to be likely candidates.221 As a consequence, trapping should not occur for acceptor materials with electron affinities larger than 3.6 eV, which is usually the case for the commonly used fullerene derivatives,32,34,179meaning that their mobility should not be altered in a device.
In general, charge carrier mobility plays an important role concerning the overall performance of an organic solar cell as it governs the transport of charge carriers through the device and away from D/A interfaces thereby greatly affecting the probability of recombination of charges both geminately and non-geminately. Low mobilities in general increase the probability of geminate recombination (GR) as well as (NGR) because in this case charges are more likely to reside in the vicinity of a D/A interface for a longer time and it also takes longer for them to be extracted at a given electric field, thereby increasing the probability of an encounter between charges of different sign in the bulk. A straightforward indication of significant recombination in a device is the presence of a so called S-Shape (usually in the forth quadrant) of the J-V-characteristic (Figure 4.3), which is accompanied by a reduction of the Fill Factor (FF).83,245–247
The effect of NGR due to low charge carrier mobility may be greatly reduced by using a PHJ architecture. In this case percolation paths to the electrodes for separated charge carriers of opposite sign are well defined and separated from each other. This makes it possible to decouple interface related effects from morphology related effects. Bimolecular recombination during extraction will not occur as long as the illumination intensity is low enough that recombination between charges stemming from different e-h-pairs is unlikely.43,44,75 Nevertheless, GR will still be enhanced for low mobilities as separated charges will on average stay in closer vicinity of the D/A interface during their lifetime τ. This situation is characterized by a small µτ-product.
Furthermore, according to Tress et al. a space charge will built up in the complete device due to the slow extraction as compared to the fast generation giving rise to an increased carrier concentration at the D/A interface and therefore higher recombination probability.83,86
In addition to that, Hahn et al. point out that also back diffusion to the D/A interface is connected to the mobility of the charge carriers via the Stokes-Einstein relation meaning that a lower charge carrier mobility also results in a lower diffusion coefficient.75 Therefore, the probability of a charge carrier to recombine with a counter charge rather than being extracted at the electrode is increased for low mobilities. In this respect, charge carrier mobility also affects the photogenerated current near VOC, i.e. at low internal electric fields, where charge carrier motion is dominated by diffusion instead of drift, and with this also the fill factor. This conclusion even holds for low illumination intensities and low carrier densities. In this respect, Hahn et al. found that even secondary geminate recombination may occur when charges diffuse back into the Coulomb capture radius of their sibling charge after they had already left it and that this process is mainly determined by charge carrier mobility.75The latter effect will also be more pronounced for larger layer thicknesses as in this case back diffusion of charge carriers to
the D/A interface becomes more probable as compared to extraction, especially at low electric fields close toVOC. This leads to an enhanced S-Shape for thicker layers with low charge carrier mobility (see figure 4.3(a)).75,248,249 Smaller layer thicknesses will in this case be of advantage in terms of recombination although this will be at the cost of absorption ability in the device.
a) b)
J
V
under illumination in the dark
J
V
layer thickness 𝜇ℎ ≪ 𝜇𝑒
Figure 4.3.:(a) Formation of an S-Shape in the J-V-characteristic as function of layer thickness in PHJ for imbalanced mobilities, e.g. µh µe. (b) Schematic showing a kink in forward direction as a result of the presence of an injection barrier. In this case, extraction is still more efficient than injection, leading to a crossing of J-V-characteristics under illumination and in the dark.
Imbalanced mobilities, which means a difference by a factor of more than 10-100,83,84,86 in addition result in the build up of a space charge at the side where charges with lower mobility are extracted. This in turn reduces the electric potential drop on the opposite side of the device in order to maintain overall charge neutrality and with this increases the probability of recombination.86 In this respect also the presence of charge carrier traps may be interpreted in this way as they effectively lower charge carrier mobility towards the respective electrode.245The same reasoning as given above concerning the impact of mobility on backdiffusion of charges to the D/A interface also applies to the layer with lower mobility in the case of imbalanced mobilities.75
Also in this case, the effect of GR can be reduced in PHJ devices by decreasing the thickness of the layer with lower mobility, because this enhances the probability of extraction in relation to backdiffusion to the D/A interface75 and reduces the electric potential drop over the layer with low mobility.86Hahn et al. for example showed that the FF may eventually become independent of illumination intensity up to 1 sun (at AM1.5 conditions) even for a mobility imbalance of 103 −104 when using a thin donor layer of only 14nm and a thickness ratio of 1:2 for donor as compared to acceptor. This also means that bimolecular processes are no longer important and recombination effects are purely geminate. Similar results were also obtained by Tress.86 Therefore, using a proper design of the sample, limitations due to extraction and device geome-try can be avoided. This renders PHJ solar cells useful model systems to study processes at the D/A interface like the dissociation of charge transfer states.84 For this reason bilayer samples with small donor layer thickness were also used in my publication about the effect of electron de-localization on CT dissociation at D/A interfaces in organic solar cells (cf. chapters 8.2.1 and 9).
32
In line with the work by Tress et al.83, Athanasopoulos et al.84 also find that a mobility imbal-ance of up to two orders of magnitude is still bearable and only has a minor effect on charge separation efficiency. Yet, in addition, based on entropic arguments, they point out that trans-port dimensionality has a strong effect, so that filamentary 1D transtrans-port would deteriorate the charge separation yield. This is in line with several works on the role of entropy in the CT dissociation process as discussed in chapter 3.3.
Besides low or imbalanced mobilities also injection or extraction barriers can account for an S-shaped J-V-characteristic.245–247,250 In the case of hindered extraction charge carriers pile up at the barrier thereby generating a space charge that partially screens the internal electric field in the device. This in turn gives rise to enhanced recombination at a D/A interface and a larger reverse bias is needed to extract the charges as well.247,250 On the other hand, also an injection barrier may cause an S-shape. Here, the barrier decreases the built-in potential, so that at a certain voltage belowVOC the field becomes positive - e.g. in the donor layer in a PHJ in case of a hole injection barrier - and current is extracted against the internal electric field.
Consequently, carrier motion is driven only by diffusion making extraction inefficient.250 An injection barrier may also cause a kink in forward direction (in the first quadrant), if extraction is still more efficient than injection (figure 4.3(b)). In this case the photogenerated current is larger than the injected current until the barrier is overcome and significant injection sets in.87,250 The presence of injection barriers may also be desired, if OPV materials are intended to be used as photosensors, where a low dark current is essential for the resulting on/off ratio of the device and concomitantly the sensitivity of the detector.251–254 This concept is used in my fifth publication to realize organic bidirectional phototransistor-like devices capable of both electrical and optical switching (see chapters 8.2.5 and 13).