How are charges transported in conjugated polymer films? Since polymers do not have a three dimensional periodical lattice structure, charge transport in polymers cannot be described by standard semiconductor models. As these systems show energetic and spatial disorder, the concept of band conduction of free charge carriers does not apply. In this section, a summary is given of how charge carrier transport in conjugated polymers and akin materials is described theoretically and how it is characterized experimentally.
The field of molecularly doped polymers is much older than that of conjugated poly-mers and valuable insights can be gained from studying this field. As early as in the 1970s the charge transport in molecularly doped polymers was studied by performing time-of-flight (TOF) measurements. In this type of experiment, a sample is sandwiched between two non-injecting electrodes.∗A short light pulse is used to illuminate one side of the sample through an transparent electrode. Under the action of an applied field, charge carriers of the same electrical polarity as the illuminated electrode will traverse the sample. By monitoring the current flow in the external circuit, the charge carrier mo-bility can be determined as a function of the applied voltage. In these TOF experiments, the mobility µ of carriers in molecularly doped polymers, can empirically be described by[11–15]
µ=µ0exp(γ√
F), (1.1)
where µ0 is the zero-field mobility, F is the field strength, and γ is the field activation parameter.
∗Note, that no direct physical contact between the electrodes and the sample is necessary.
1.3.1 Hopping transport in disordered systems
How can the results summarized in Eq. (1.1) be rationalized? As these materials are disordered, the concept of band conduction does not apply. Instead, localized states are formed and charge carriers proceed from one such a state to another (hopping), thereby absorbing or emitting phonons to overcome the energy difference between those states.
Conwell[16]and Mott[17]proposed the concept of hopping conduction in 1956 to de-scribe impurity conduction in inorganic semiconductors. Miller and Abrahams calcu-lated that the transition rate Wij for phonon-assisted hopping from an occupied state i with an energy ǫito an unoccupied state j with energy ǫjis described by[18]
Wij=ν0exp(−2γRij)
(exp
−ǫkj−ǫBTi ǫi<ǫj
1 ǫi≥ǫj, (1.2)
where ν0is the attempt-to-jump frequency, Rij is the distance between the states i and j, γ is the inverse localization length, kBis Boltzmann’s constant, and T is temperature.
The wave function overlap of states i and j is described by the first exponential term in Eq. (1.2), while the second exponential term accounts for the temperature dependence of the phonon density.
In his pioneering work, B¨assler described the transport in disordered organic systems as a hopping process in a system with both positional and energetic disorder.[10]The hopping rates between sites were assumed to obey Eq. (1.2) and the site energies varied according to a Gaussian distribution with a standard deviation σ. Such a system cannot be solved analytically. By performing Monte Carlo simulations, the following expression for the charge carrier mobility µ was proposed[10]
µ=µ∞e− 3kBT2σ spacing, and Σ describes the positional disorder.
Although Eq. (1.3) predicts a functional dependence on field strength similar to Eq. (1.1), the agreement with experiments is limited to high fields.[13]Gartstein and Con-well found that the agreement with experiments could be improved by taking spatial correlations between site energies into account.[19]In this model, the mobility takes the form[20,21]
∗Since, Eq. (1.3) is an expression which describes the outcome of Monte Carlo simulations, this is a purely mathematical definition of µ∞and does not mean that it has the physical meaning of the mobility at infinite temperature. At best, it may be interpreted as the mobility if there would be no barriers to hopping at all.
where q is the elementary charge, a is the intersite spacing, and Γ is the positional dis-order of transport sites. This model was successfully used to describe the transport of charges in molecularly doped polymers.[20]
1.3.2 Transport in conjugated polymers
The stretched exponential dependence on field strength as described by Eq. (1.1) was also observed for conjugated polymers.[22]Subsequently, Eq. (1.4) was also applied success-ful to explain the charge transport in conjugated polymers[23,24]as well as other organic systems.[25]
In the foregoing discussion, only the dependence of the mobility on temperature and field strength was taken into account. When the applied voltage is increased in a TOF experiment, only the field across the sample changes. However, in organic solar cells, as well as organic LEDs, changing the applied voltage does not merely change the field.
Due to the nature of the contacts, it influences the charge carrier density as well. Recently, it has been shown that the mobility of charge carriers in conjugated polymers also has an important dependence on charge carrier density.[26–29]Moreover, it was shown that the increase of the mobility with increasing bias voltage (and concomitant increase in carrier density) observed in polymer diodes is, at least for some systems and temperatures, completely due to an increase in charge carrier density.[26]
Throughout this thesis, the increase of the mobility with increasing bias voltage is interpreted as an effect of the field only. It should be noted, however, that the polymers used in this thesis show only a rather small dependence of the mobility on bias, suggest-ing that the influence of either field strength or carrier density for the system described here is quite weak. Additionally, as we will see in chapter 2, the carrier density in solar cells is fairly modest.
Several alternative models exist for explaining charge transport; one of them is the so-called polaron model which was first applied to inorganic crystals[30] and later to conjugated polymers.[31]An excess charge carrier in a solid causes a displacement of the atoms in its vicinity thus lowering the total energy of the system. This displacement of atoms results in a potential well for the charge carrier, thereby localizing it. The charge carrier and its concomitant atomic deformation is called a polaron.
The transition rate for polaron hopping from site i to site j is given by[32]
Wij ∝ 1
where Er is the intramolecular reorganization energy. The resulting charge carrier mo-bility is of the form[33]
µ=µ0exp
The polaron contribution to the activation of the mobility is, as predicted by this model, rather low; it amounts to 25–75 meV,[33]which is much smaller than the activation due to disorder.
1.3.3 Measuring the charge carrier mobility
When an insulator is contacted by an electrode that can readily inject a sufficiently large number of charge carriers — a so-called Ohmic contact — and another electrode that can extract these charges, the current flow will be limited by a buildup of space charge.
These space-charge-limited (SCL) currents can be used as a simple, yet reliable, tool to determine the mobility in an experimental configuration that is relevant for solar cells.
Considering only one charge carrier (either electrons or holes), the SCL current density JSCLflowing across a layer with thickness L is given by[34]
JSCL= 9 8εµVint2
L3 , (1.7)
where ε is the dielectric constant of the material and Vint is the internal voltage drop across the active layer. When the mobility is of the form as given in Eq. (1.1), one can approximate JSCLby[35]
JSCL= 9
8εµ0e0.891γ√Vint/LVint2
L3 . (1.8)
The internal voltage in an actual device is related to the applied voltage Vaby
Vint=Va−Vbi−VRs, (1.9)
where Vbi is the built-in voltage which arises from the difference in work function of the bottom and top electrode and VRs is the voltage drop across the series resistance of the substrate (typically 30–40 Ω). The built-in voltage is determined from the current-voltage characteristics as the current-voltage at which the current-current-voltage characteristic becomes quadratic, corresponding to the SCL regime.
By judiciously choosing the electrode materials, the injection of either carrier type can be suppressed or enhanced, thereby enabling one to selectively assess either the hole or electron mobility. The way to do this, is to make sure that the work function of one of the electrodes is close to the energy level of the transport band under investigation, while there exists a large barrier for injection of the other carrier type into the material. Thus, in order to study the hole transport in conjugated polymers, high work function metals, such as gold and palladium, are used. Conversely, low work function metals can be used as Ohmic contacts for electron injection.
1.3.4 Conjugated polymers used in this thesis
Up to now the photoactive polymers used in this research have not been specified. The polymer poly(2-methoxy-5-(3’,7’-dimethyl octyloxy)-p-phenylene vinylene) (MDMO-PPV) had for a long time been the workhorse in polymer photovoltaics. Consequently, its
Figure 1.2: The chemical structures of the BEH-PPV, MDMO-PPV, and P3HT.
charge transport properties are well documented, making this polymer well suited for modeling purposes. Recently, another polymer has emerged: poly(3-hexylthiophene) (P3HT),∗ which is used in the most efficient polymer solar cells to date.[2] The final polymer considered in this thesis is poly(2,5-bis(2’-ethylhexyloxy)-p-phenylene viny-lene) (BEH-PPV). The chemical structure of these polymers is shown in Fig. 1.2.
The charge transport in MDMO-PPV has been extensively studied: Typically, the zero field mobility amounts to 5×10−11m2/V s.[36] Surprisingly, the hole mobility of MDMO-PPV is enhanced when mixed with 6,6-phenyl C61-butyric acid methyl ester (PCBM), as reported by several researchers:[37,38]When 80% (by weight) of this blend consists of PCBM, the hole mobility of the polymer phase is equal to 2×10−8m2/V s, an encrease of more than two orders of magnitude as compared to pristine MDMO-PPV.
This spectacular behavior of the hole mobility in MDMO-PPV is the main reason for its succes as a donor in BHJ solar cells with PCBM.
P3HT is unique in its own right: Padinger et al. observed that solar cells made from P3HT and PCBM showed a great increase in the efficiency upon thermal annealing.[39]
Mihailetchi et al. have shown that this enhancement is in part due to an increase in the mobility:[40]In its pristine form the hole mobility amounts to 10−8m2/V s, see Fig. 1.3.
For comparison, Fig. 1.3 also shows the electron mobility of the PCBM phase in these blends. When blended with PCBM, the hole mobility initially decreases, however, upon annealing the hole mobility in the P3HT phase of the blend with PCBM is restored to its pristine value, as depicted in Fig. 1.3.[40]