Thickness dependence of spacechargelimited current in spatially disordered organic semiconductors
Abstract
Charge transport properties in organic semiconductors are determined by two kinds of microscopic disorders, namely energetic disorder related to the distribution of localized states and the spatial disorder related to the morphological features of the material. From a semiclassical picture, the charge transport properties are crucially determined by both the carrier mobility and the electrostatic field distribution in the material. Although the effect of disorders on carrier mobility has been widely studied, how electrostatic field distribution is distorted by the presence of disorders and its effect on charge transport remain unanswered. In this paper, we present a modified spacechargelimited current (SCLC) model for spatially disordered organic semiconductors based on the fractionaldimensional electrostatic framework. We show that the thickness dependence of SCLC is related to the spatial disorder in organic semiconductors. For trapfree transport, the SCLC exhibits a modified thickness scaling of , where the fractionaldimension parameter accounts for the spatial disorder in organic semiconductors. The traplimited and fielddependent mobility are also shown to obey an dependent thickness scaling. The modified SCLC model shows a good agreement with several experiments on spatially disordered organic semiconductors. By applying this model to the experimental data, the standard charge transport parameters can be deduced with better accuracy than by using existing models.
IEEEexample:BSTcontrol
1 Introduction
The mobility of charge carrier is a key parameter for the performance of optoelectronic devices [2], especially for devices using organic semiconductors and polymers. The mobility in organic semiconductors strongly depends on the nature, structure, purity of the materials and device operating conditions. The charge transport in organic compounds occurs across various levels, ranging from within molecules, between molecules as well as between crystalline grains and amorphous and crystalline regions. The transport properties are determined by two kinds of microscopic disorders, namely the energetic disorder characterized by a broad distribution of localized states and the spatial disorder related to the morphological features of the material [3]. The spacechargelimited current (SCLC) is an important classical transport phenomenon in organic semiconductors where the quantum effects can be ignored at microscale and above [2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 11, 12, 13, 14, 15, 16, 17].
The mobility of a given organic material sandwiched between two planar electrodes with an applied voltage (), is commonly measured indirectly by fitting the measured current densityvoltage (JV) characteristics at high voltages to some SCLC models [4]. It is assumed that there is no barrier (ohmic contact) at the interface when the charges are injected from the electrode into the solid. The simplest SCLC model for a trapfree solid is known as the onedimensional (1D) classical MottGurney (MG) law [5], given by
(1) 
where is permittivity, and is the thickness of the solid embedded between the metal electrodes. Once the values of , , and are determined, the mobility can be calculated by fitting the JV characteristics at high region to the 1D MG law. Note the assumptions in using the MG law include constant mobility (independent of the applied electric field and charge density) and the solid is trapfree.
For a trapfilled solid with exponentially energydistributed traps, the corresponding SCLC model is known as the traplimited (TL) SCLC model [6]:
(2) 
where is the effective density of states corresponding to the energy at the bottom of the conduction band, is the total trapped electron density, and with being a parameter controlling the trap distribution.
For shallow traps or energetic disorder, the mobility varies with the electric field that Murgatroyd’s [7] model may be used to describe a fielddependent mobility in the form of
(3) 
where is representing the mobility at zero field, and is a materialspecific parameter that describes the strength of the fielddependence. The fielddependent mobility can also include other effects, such as carrierdensity dependence (Gaussian disorder model (GDM) [8]) and deep traps, which can only be resolved by having a more comprehensive model to fit with experiments over a wide range of parameters [9]. By using Eq. (3), we get [7]
(4) 
where is used as a fitting parameter.
Other than fielddependent mobility, chargecarrierdensity dependent mobility models have also been studied including a powerlaw dependence for energetically disordered semiconductors by Tanase et. al. [10], Blom et. al. [1], and others for disordered polymers [11, 12, 13].
The mobility is often extracted from JV measurements by fitting the experimental data with the theoretical models of SCLC with different mobility terms. The mobility of a given sample is determined solely based on the goodness of the fit. Such empirical methodology may not always produce accurate physical picture. For example, an inconsistency of fielddependent mobility model with the experimental data has been raised in Ref. [1]. The conventional model of Eq. (4) could not fit the experimental data due to weaker thickness dependence of measured SCLC, and hence the carrierdensity dependent mobility model was chosen to extract the mobility of PPV based diode device. However, assuming Gaussian density of states (DOS), it is known that in low carrierdensity regime the mobility should be carrierdensity independent [14]. Another recent experiment [3] has also reported that the charge transport in amorphous semiconductors is not charge density dependent but instead follows a fielddependent mobility model. In such scenarios, a new model of spacechargelimited transport is required to capture the correct thickness scaling of measured SCLC.
The transport sites in organic semiconductors are distributed both in space and energy. The combined effect of spatial and energetic disorder on charge transport has already been studied in the previous works (see [18] for a comprehensive review). However, the SCLC is a transport phenomenon closely related with both the material properties (i.e., mobility) and the electrostatic field distribution inside these materials, and thus far, the nonuniform distribution of electrostatic field due to spatial complexity of the material and its implications on the macroscopic SCLC have not been fully addressed. The complex, spatially disordered and often selforganized microstructure, in which ordered microcrystalline domains are embedded in an amorphous domain, can be considered as fractal features having important consequences for electrical properties of these materials (refer to Fig. 2 in [19] and Fig. 1 in [20]).
In this work, we present a modified SCLC model to account for the spatial disorder effect of a solid such as amorphous semiconducting polymer by treating the material as a fractal object. The key novelty of our proposed model is to utilize the fractionaldimension of space to effectively model the nonuniform distribution of electrostatic field inside these spatially disordered materials. Fig. (1) provides a schematic description of the concept that the spatially disordered organic semiconductor in real integerdimensional space can be considered effectively as spatially ordered organic semiconductor embedded in the corresponding fractionaldimensional space using mathematical framework briefly introduced in the Appendix (also see [21, 22] and references therein). Such methods have been applied in other areas including quantum field theory [23, 24], general relativity [25], thermodynamics [26], mechanics [27], hydrodynamics [28], electrodynamics [29, 30, 31, 32, 33, 34, 35, 36, 37, 38], and fractional charge transport [22, 21] to name a few.
The proposed approach has been generalized to cover three types of SCLC models: trapfree model (or MG law), traplimited (TL) model and the field dependent mobility models. We first analyze various experimental results to study the thickness (or ) dependence to show that the traditional scalings from the traditional models are not valid for spatially disordered semiconducting materials. By using our proposed models, we are able to reproduce the experimental currentvoltage measurements in [1] without using the carrierdensity dependent model, and thus solving the issues raised by recent paper [3].
2 Derivation of SCLC Scaling Laws for spatially disordered organic semiconductors
2.1 Trapfree model
Here, we derive the modified MG law for spatially disordered organic semiconductors with the assumption that the effect of nonuniform electrostatic field distribution inside spatially disordered material can be studied effectively by replacing the governing equations of classical SCLC model with the fractionaldimensional counterparts, by using the formulation described in Appendix (for more details see [21, 22] and references therein), where the fractionaldimension is related to the amount of spatial disorder. Provided that the thermal carriers are negligible in comparison to injected carrier, and assuming that the size of the electrode is much greater than the spacing , thus the derivation is conducted only in direction perpendicular to surface of the electrode. The following equations are solved in the dimensional space [39] with :
(5)  
(6)  
(7) 
where [21], is the carrier charge density, is the drift velocity, is the electric field, and is the dielectric permittivity of the material, and is the electric potential. Using , Eq. (5) gives
(8) 
It should be emphasized that is an averaged quantity independent of space variables.
Now, inserting Eq. (6) into Eq. (8), we get
(9) 
which can be rewritten in the form of Bernoulli differential equation:
(10) 
Solving Eq. (10) with zero electric field condition (at SCL regime) at the injecting electrode, , we have
(11) 
By using Eq. (11) in solving Eq. (7), we obtain
(12) 
which gives the modified MG law as a function of :
(13) 
2.2 Traplimited (TL) model
For a spatially disordered material with exponentially distributed traps in energy, the traplimited TLSCLC injection is derived here. We assume that the mobility is fieldindependent, and that the density of the trapping states per unit energy range above the valence band is described by the distribution
(14) 
where is the energy measured upward from the top of the valence band, is the total trap density, and is a characteristic constant of the distribution. Following the Mark and Helfrich (MH) approach [6], we obtain
(15) 
where and is effective density of states. In this case the relation between free and trapped carrier density is given by
(16) 
where, is total density of transport sites. By solving Eq. (6), the governing equation is
(17)  
(18) 
Integrating Eq. (17) on both sides gives
(19) 
and
(20) 
It should be noted that for a Gaussian distribution of traps, a similar equation for the traplimited current is derived, except that is then related to the depth and width of the trap distribution [40, 41]. In the case of Gaussian trap DOS centered at a distance below the conductionband edge ,the nondegenerate approximation gives [42]
(22) 
where, , is the variance of Gaussian DOS. Finally, following the MH formalism, a currentvoltage characteristic is obtained for Gaussian trap DOS, which is of the form
(23)  
which reduces to Eq. (7) in [42] at .
2.3 Fielddependent mobility model
If we simply combine the fielddependent mobility Eq. (3), and Eq. (13), the modified SCL model of fielddependent mobility for spatially disordered semiconductors is
(24) 
where is just a fitting parameter. In general, to include the field dependence of the mobility in SCLC model, coupled equations such as Eqs. (57) must be solved consistently [43]. It is however possible to derive an analytic solution if the field dependence of the drift mobility can be expressed in power law [44] given by
(25) 
with at . By using this power law of mobility, we solve Eqs. (57) to obtain an analytical solution of
(26)  
which reduces to Eq. (13) at .
3 Results and Discussions
By analyzing the thickness () dependence of the classical SCLC models, we see the dependence of (at fixed ), (at fixed ) and (at fixed ), respectively, from Eq. (1), Eq. (2), and Eq. (4). However, as predicted by corresponding modified SCLC models in Sec. II, the thickness dependence will be reduced by the fractionaldimension parameter , which accounts to the spatial disorder in the underlying solids. In other words, the thickness dependence of the modified SCLC models will provide a tool to characterize the spatial disorder in the porous organic semiconductor.
Polymer Type 






NRSPPV 
2.895 (Fig. (3a))  3 (Eq. (19))  0.965  [1]  
PFO  2.9 (Fig. (3b))  3  0.967  [45]  
conjugatedSexithienyl  2.6 (Fig. (3c))  3  0.86  [46]  
polyfluorenebased  2.5 (Fig. (3d))  3  0.83  [47]  
11 (Fig. (4a))  2+ (Eq. (27))  0.918  [48]  
NPB  0.52 (Fig. (4b))  32 (Eq. (30))  0.84  [49] 
3.1 Implications of modified SCLC model on mobility extraction
Before proceeding with the analysis of thickness dependence in some reported experimental data, it would be of interest to see the effect of variation in thickness dependence due to spatial disorder in the semiconductors on the mobility values extracted form experimental JV curves taken from [1]. We denote the extracted values of mobility by , to distinguish them from actual mobility for this device. In Fig. (2), the extracted mobility () is plotted as a function of thickness dependence parameter . The fractionaldimension parameter corresponds to the measure of spatial disorder in the semiconductor, with corresponding to zero spatial disorder. It can be seen that is sensitively influenced by . Thus it is important to check the dependence rather than assuming which may no longer be valid for complicated materials such as porous and amorphous organic materials.
3.2 Consistency of extracted from experimental data
Most organic semiconductors have spatial as well as energetic disorder. The existing mobility models incorporate combined effect of energetic and spatial disorder using a range of mobility models including field and densitydependent mobility. However, our model predicts that the spatial disorder can affect the thickness scaling of SCLC. Here, we explore the available experimental data of SCLC versus device thickness for a range of disordered organic semiconductors. It is observed that the thickness scaling of SCLC varies as predicted by our model, to which not much attention was paid previously and it was assumed trivially that thickness scaling follows standard MG law which turns out to be not true for organic semiconductor in several example cases reported in the following. Fig. (3) shows the corresponding thickness dependence for various devices using different organic materials. The results shown in Fig. ((a)a(d)d) are at fixed voltage regime (constant mobility) with varying thickness . Based on the classical MG law, we will expect a scaling of . However, due to spatial disorder, the results show a weaker thickness dependence in the range to , which corresponds to about to .
For the results shown in Fig. ((a)a) for a trapfilled organic material we have . Based on the classical TLSCLC model (without any spatial disorder), the thickness dependence should be . However, the experimental fitting shows again a weaker dependence, which corresponds to based on Eq. (21), instead of = 1. For results shown in Fig. ((b)b), the scaling is calculated from measurements at fixed electric field. As mentioned earlier for fielddependent mobility, the thickness dependence should be at a fixed field for negligible spatial disorder at = 1. However, we observe a weaker dependence, which corresponds to based on Eq.(24). Table I summarizes the results of Fig. (34) for various disordered organic semiconductor based devices and its relation to the parameter used in our models to fit with the experimental results. Our analysis suggests that the traditional scaling formulated in the classical SCLC models may not be suitable for organic semiconductors, and it will provide an inaccurate estimation of the mobility if such models are used. Note that the uncertainty in the measurement of , which is about 5 nm from normal experimental setup, is not able to explain the variation from the expected = 1 assuming the classical models are correct.
In Fig. (3), we have extracted the thickness scaling of SCLC at low voltages to avoid the fielddependent SCLC regime. However, one must be careful while extracting the slopes at higher applied voltages. As the SCLC is fielddependent at high applied voltage, the extraction of should be performed at fixed electric field strength, , rather than at fixed voltage, . To demonstrate this, we analyze the SCLC versus voltage data of polymer NPB based devices reported in Ref. [49], and plot the current density against device thickness at different voltages in Fig. ((a)a) . The varying slope of versus shows that the thickness dependence of SCLC varies at different applied voltages due to factor in fielddependent mobility model of Eq. (24). Fig. ((b)b) shows the extracted thickness dependence at different voltages which immediately reveals that the value of the extracted is inconsistent at different voltage. At highvoltage regime where fielddependence becomes nonnegligible, the extracted thickness dependence even becomes stronger than which leads to an unphysical value of . This clearly reveals the fallacy of extracting from the JL curve at fixed voltage. Instead, the should be extracted at fixed electric field strength as indicated by Eq. (24), i.e., at fixed . Fig. ((c)c) and ((d)d) shows the JL characteristics and the extracted at different , respectively. In this case, a singular value of is extracted for all applied electric field strengths. More importantly, this value of is consistent with that extracted from the lowvoltage regime of Fig. ((b)b).
In Fig. ((a)a), the roomtemperature current density versus voltage characteristics data from Ref. [45] is shown for PFO diodes of varying thickness together with various numerical models calculations. It should be noted that the classical model of Eq. (4) requires different values of to be used in order to fit with the experimental data despite the fact that the devices are composed of the same type of polymer. To address this inconsistency, we fitted the experimental data using our modified SCLC model with [extracted from Fig. ((b)b)]. Remarkably, our modified model is able to fit the experimental JV curves of all devices using a singular consistent value of . Similarly, in Fig. ((b)b) the roomtemperature current density versus voltage data from Ref. [48] for diodes with varying thicknesses is shown. The classical model in Eq. (2) fails to reproduce the experimental results with fixed for all L. In contrast, by using our modified traplimited SLC model with extracted from Fig. ((a)a), a much better agreement with experimental results is obtained at a fixed . These results show that the modified MG law can sufficiently describe the thickness dependence of SCLC for given range of applied voltages.
3.3 Fitting experimental currentvoltage characteristics and mobility extraction using modified SCLC model
In Fig. ((a)a) the experimental JV characteristics [1] (circles) of the NRSPPV based devices are shown together with various numerical models calculations: (i) (dotted lines) classical MG law based fielddependent mobility model of Eq. (4), (ii) (dashed lines) modified MG law based fielddependent mobility model of Eq. (24), and (iii) (solid lines) modified MG law based fielddependent mobility model of Eq. (26). From the figure, it is clear that the classical model (dotted lines) does not have a good agreement with the experimental results. As shown in Fig. ((a)a), NRSPPV based devices show a thickness dependence of which corresponds to . Using this = 0.965, the two modified SCL models including field dependent mobility (dashed and solid lines) are able to provide better agreements without the needs to use carrierdependent mobility assumption that have been debated in recent years [18]. It is important to note that one of the direct consequence of modified MG law in Eq. (24) is that the mobility can be considered to have a thickness dependence along with fielddependence given by . In Fig. ((b)b) we show the field and thickness dependent mobility values for the same NRSPPV based devices [1] using this model with the parameters shown in figure caption.
Finally, we analyzed the thickness dependence of experimentally measured SCLC in holeonly devices based on diketopyrrolopyrolebased polymer (PDPPDTSE) [50]. The thickness dependence of current density at fixed voltage for PDPPDTSE based devices taken from experimental currentvoltage data is shown in Fig. ((a)a). The thickness scaling of SCLC from standard to . The observed thickness dependence corresponds to spatial disorder parameter . In order to validate our model we also compared the reported mobility values for varying thickness of devices with the mobility scaling predicted by our model. It is shown in Fig. ((b)b)that the thickness scaling of measured mobility is in good agreement with the one predicted by our model ().
4 Summary
Description  Modified SCLC Model  Eqs.  


(13)  

(21)  

(23)  

(24)  

(26)  

In summary, we have presented a modified thickness scaling in SCLC model to account for the spatial disorder in organic semiconductors by introducing a parameter to imagine the solid as a fractal object sandwiched between two electrodes. The model has included different effects such as trapfree, traplimited and fielddependent mobility. To provide an easy access to the main results of this work, we have summarized the modified SCLC equations in Table II. An analysis of multiple experimental results from literature reveals that the classical SCLC models might lead to incorrect extraction of mobilities due to weak thickness dependence arising from spatial disorder. For such materials, our proposed model here would be a better choice to extract the mobility for spatially disordered organic materials as we have shown that the traditional thickness scaling is not valid anymore. By applying our model with fielddependent mobility, we are able to reproduce the experimental results of SCLC transport in PPV derivative based device without using the carrierdensity dependent mobility [1], agreeable with a recent report for amorphous polymers [3].
Note that the thickness dependence had been reported in others works. For example, Brutting et. al (see Fig. 2a in Ref. [51]) reported a weaker thickness dependence for Alq lightemitting devices than the expected at fixed electric field. John et. al (see Fig. (56) in Ref. [52]) reported a varying thickness dependence ( to ) for plasma polymerized pyrrole thin films. Boni et. al. (see Fig. 12 in Ref. [53]) also reported a possible weaker thickness dependence for PZT ferroelectric based devices. Macdonald et. al (see Fig. 1b in Ref. [54]) also reported a weaker thickness dependence due to nonplanar electrodes in conducting the experiment using tip atomic force microscopy (cAFM). This is a geometrical effect producing weaker thickness dependence of organic semiconductor devices [55] and is different from the physics studied here. It should be emphasized that our proposed models are based on a planardiode geometry, thus such nonplanar geometrical effects are not included. The extension of our models into nonplanar geometries will be pursued in future works.
Moreover, in this work we obtain the parameter from the length scaling of SCLC in the experimental results, however a complete microscopic model can be created in further extensions to determine directly from the knowledge of disorder either spatial or energetic or both.
FractionalDimensional Space Framework as Description of Complexity
There is an increasing interest in the fractional modeling of complexity in physical systems [56, 57]. In recent years, the concept of fractionaldimensional space has been used as an effective physical description of restraint conditions in complex physical systems [24, 39, 29]. The approaches to describe the fractional dimensions include fractal geometry [58], fractional calculus [59, 60], and the integration over fractionaldimensional space [23, 61]. The axiomatic basis of spaces with fractional dimension with Euclidean metric were introduced by Stillinger [23]. The fractionaldimensional generalization of first order Laplace operators was then reported by Zubair et. al. [29] as approximations of the square of the secondorder Laplace operator introduced in [23, 24]. Recently, a fractal metric based approach is considered by Tarasov [39] which provides a complete generalization of first and second order Laplace operators. In this work, we have utilized Tarasov’s approach to vector calaculus in fractionaldimensional spaces, which is summarized in the following.
In fractionaldimensional space () framework [39], it is convenient to work with physically dimensionless space variables , , , , where is a characteristic size of considered model. This provides a dimensionless integration and differentiation in dimensional space which leads to correct physical dimensions of quantities.
We define a differential operator in the form of
(27) 
where corresponds to the noninteger dimensionality along the axis and it is defined by [39]
(28) 
For the case of spatially disordered semiconductor or porous solid, the system can be effectively modeled by replacing the anisotropy with an isotropic continuum in an dimensional space, with a parameter 0 1 to measure the anisotropy or disorder of the material.
Using the operators in Eq. (27), we can generalize vector differential operators in an dimensional space. The gradient of a scalar function in fractionaldimensional space is
(29) 
where are unit base vectors of the Cartesian coordinate system. The divergence of the vector field is
(30) 
The curl for the vector field is
(31) 
where is the LeviCivita symbol. Using Eqs. (29) and (30), the scalar Laplacian in the fractionaldimensionalspace is written as [39]
[]Muhammad Zubair (S’13M’15) received his Ph.D. degree in electronic engineering from the Politecnico di Torino, Italy, in 2015. From 2015 to 2017, he was with the SUTDMIT International Design Center, Singapore. Since 2017, he has been with Information Technology University, Lahore, Pakistan.
His current research interests include charge transport, electron device modeling, computational electromagnetics, fractal electrodynamics, and microwave imaging. {IEEEbiography}[]Yee Sin Ang his bachelorâs degree in medical and radiation physics in 2010, and his PhD degree in theoretical condensed matter physics in 2014 from the University of Wollongong (UOW), Australia. He is currently a Research Fellow with the Singapore University of Technology and Design, Singapore.
His research interests include the theory and mathematical modelling of electron emission phenomena in 2D and topological materials, electron transport physics across 2D/3D, 2D material valleytronics, nanoelectronics and superconducting devices.
[]Lay Kee Ang (S’95M’00SM’08) received the B.S. degree from the Department of Nuclear Engineering, National Tsing Hua University, Hsinchu, Taiwan, in 1994, and the M.S. and Ph.D. degrees from the Department of Nuclear Engineering and Radiological Sciences, University of Michigan, Ann Arbor, MI, USA, in 1996 and 1999, respectively. Since 2011, he has been with the Singapore University of Technology and Design, Singapore.
He is currently the Interim Head and Professor of the Engineering Product Development pillar and also the Ng Teng Fong Chair Professor of SUTDZJU IDEA.
References
 P. Blom, C. Tanase, D. De Leeuw, and R. Coehoorn, “Thickness scaling of the spacechargelimited current in poly (pphenylene vinylene),” Applied physics letters, vol. 86, no. 9, p. 092105, 2005.
 M. Kuik, G.J. A. Wetzelaer, H. T. Nicolai, N. I. Craciun, D. M. De Leeuw, and P. W. Blom, “25th anniversary article: Charge transport and recombination in polymer lightemitting diodes,” Advanced Materials, vol. 26, no. 4, pp. 512–531, 2014.
 A. J. Campbell, R. Rawcliffe, A. Guite, J. C. D. Faria, A. Mukherjee, M. A. McLachlan, M. Shkunov, and D. D. Bradley, “Chargecarrier density independent mobility in amorphous fluorenetriarylamine copolymers,” Advanced Functional Materials, vol. 26, no. 21, pp. 3720–3729, 2016.
 M. A. Lampert and P. Mark, “Current injection in solids,” 1970.
 N. F. Mott and R. W. Gurney, Electronic processes in ionic crystals. Clarendon Press, 1948.
 P. Mark and W. Helfrich, “Spacechargelimited currents in organic crystals,” Journal of Applied Physics, vol. 33, no. 1, pp. 205–215, 1962.
 P. Murgatroyd, “Theory of spacechargelimited current enhanced by frenkel effect,” Journal of Physics D: Applied Physics, vol. 3, no. 2, p. 151, 1970.
 H. Bässler, “Charge transport in disordered organic photoconductors a monte carlo simulation study,” physica status solidi (b), vol. 175, no. 1, pp. 15–56, 1993.
 J. C. Blakesley, F. A. Castro, W. Kylberg, G. F. Dibb, C. Arantes, R. Valaski, M. Cremona, J. S. Kim, and J.S. Kim, “Towards reliable chargemobility benchmark measurements for organic semiconductors,” Organic Electronics, vol. 15, no. 6, pp. 1263–1272, 2014.
 C. Tanase, E. Meijer, P. Blom, and D. De Leeuw, “Unification of the hole transport in polymeric fieldeffect transistors and lightemitting diodes,” Physical Review Letters, vol. 91, no. 21, p. 216601, 2003.
 I. Fishchuk, V. Arkhipov, A. Kadashchuk, P. Heremans, and H. Bässler, “Analytic model of hopping mobility at large charge carrier concentrations in disordered organic semiconductors: Polarons versus bare charge carriers,” Physical Review B, vol. 76, no. 4, p. 045210, 2007.
 W. Pasveer, J. Cottaar, C. Tanase, R. Coehoorn, P. Bobbert, P. Blom, D. De Leeuw, and M. Michels, “Unified description of chargecarrier mobilities in disordered semiconducting polymers,” Physical review letters, vol. 94, no. 20, p. 206601, 2005.
 J. Cottaar, L. Koster, R. Coehoorn, and P. Bobbert, “Scaling theory for percolative charge transport in disordered molecular semiconductors,” Physical review letters, vol. 107, no. 13, p. 136601, 2011.
 C. Tanase, P. Blom, D. De Leeuw, and E. Meijer, “Charge carrier density dependence of the hole mobility in poly (pphenylene vinylene),” physica status solidi (a), vol. 201, no. 6, pp. 1236–1245, 2004.
 I. Fishchuk, A. Kadashchuk, J. Genoe, M. Ullah, H. Sitter, T. B. Singh, N. Sariciftci, and H. Bässler, “Temperature dependence of the charge carrier mobility in disordered organic semiconductors at large carrier concentrations,” Physical Review B, vol. 81, no. 4, p. 045202, 2010.
 I. Katsouras, A. Najafi, K. Asadi, A. Kronemeijer, A. Oostra, L. Koster, D. M. de Leeuw, and P. W. Blom, “Charge transport in poly (pphenylene vinylene) at low temperature and high electric field,” Organic Electronics, vol. 14, no. 6, pp. 1591–1596, 2013.
 T. Leijtens, J. Lim, J. Teuscher, T. Park, and H. J. Snaith, “Charge density dependent mobility of organic holetransporters and mesoporous tio2 determined by transient mobility spectroscopy: Implications to dyesensitized and organic solar cells,” Advanced Materials, vol. 25, no. 23, pp. 3227–3233, 2013.
 A. Nenashev, J. Oelerich, and S. Baranovskii, “Theoretical tools for the description of charge transport in disordered organic semiconductors,” Journal of Physics: Condensed Matter, vol. 27, no. 9, p. 093201, 2015.
 N. Tessler, Y. Preezant, N. Rappaport, and Y. Roichman, “Charge transport in disordered organic materials and its relevance to thinfilm devices: A tutorial review,” Advanced Materials, vol. 21, no. 27, pp. 2741–2761, 2009.
 R. Noriega, J. Rivnay, K. Vandewal, F. P. Koch, N. Stingelin, P. Smith, M. F. Toney, and A. Salleo, “A general relationship between disorder, aggregation and charge transport in conjugated polymers,” Nature materials, vol. 12, no. 11, pp. 1038–1044, 2013.
 M. Zubair, Y. S. Ang, and L. K. Ang, “Fractional fowlernordheim law for field emission from rough surface with nonparabolic energy dispersion,” IEEE Transactions on Electron Devices, vol. 65, no. 6, pp. 2089–2095, 2018.
 M. Zubair and L. K. Ang, “Fractionaldimensional childlangmuir law for a rough cathode,” Physics of Plasmas (1994present), vol. 23, no. 7, p. 072118, 2016.
 F. H. Stillinger, “Axiomatic basis for spaces with noninteger dimension,” Journal of Mathematical Physics, vol. 18, no. 6, pp. 1224–1234, 1977.
 C. Palmer and P. N. Stavrinou, “Equations of motion in a nonintegerdimensional space,” Journal of Physics A: Mathematical and General, vol. 37, no. 27, p. 6987, 2004.
 M. Sadallah and S. I. Muslih, “Solution of the equations of motion for einsteins field in fractional d dimensional spacetime,” International Journal of Theoretical Physics, vol. 48, no. 12, pp. 3312–3318, 2009.
 V. E. Tarasov, “Heat transfer in fractal materials,” International Journal of Heat and Mass Transfer, vol. 93, pp. 427–430, 2016.
 M. OstojaStarzewski, J. Li, H. Joumaa, and P. N. Demmie, “From fractal media to continuum mechanics,” ZAMMJournal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, vol. 94, no. 5, pp. 373–401, 2014.
 A. S. Balankin and B. E. Elizarraraz, “Map of fluid flow in fractal porous medium into fractal continuum flow,” Physical Review E, vol. 85, no. 5, p. 056314, 2012.
 M. Zubair, M. J. Mughal, and Q. A. Naqvi, Electromagnetic fields and waves in fractional dimensional space. Springer Science & Business Media, 2012.
 M. J. Mughal and M. Zubair, “Fractional space solutions of antenna radiation problems: An application to hertzian dipole,” in Signal Processing and Communications Applications (SIU), 2011 IEEE 19th Conference on. IEEE, 2011, pp. 62–65.
 Q. A. Naqvi and M. Zubair, “On cylindrical model of electrostatic potential in fractional dimensional space,” OptikInternational Journal for Light and Electron Optics, vol. 127, no. 6, pp. 3243–3247, 2016.
 M. Zubair, M. J. Mughal, and Q. A. Naqvi, “An exact solution of the cylindrical wave equation for electromagnetic field in fractional dimensional space,” Progress In Electromagnetics Research, vol. 114, pp. 443–455, 2011.
 H. Asad, M. J. Mughal, M. Zubair, and Q. A. Naqvi, “Electromagnetic greens function for fractional space,” Journal of Electromagnetic Waves and Applications, vol. 26, no. 1415, pp. 1903–1910, 2012.
 H. Asad, M. Zubair, and M. J. Mughal, “Reflection and transmission at dielectricfractal interface,” Progress In Electromagnetics Research, vol. 125, pp. 543–558, 2012.
 M. Zubair, M. J. Mughal, and Q. A. Naqvi, “An exact solution of the spherical wave equation in ddimensional fractional space,” Journal of Electromagnetic Waves and Applications, vol. 25, no. 10, pp. 1481–1491, 2011.
 M. Zubair, M. J. Mughal, Q. A. Naqvi, and A. A. Rizvi, “Differential electromagnetic equations in fractional space,” Progress In Electromagnetics Research, vol. 114, pp. 255–269, 2011.
 M. Zubair, M. J. Mughal, and Q. A. Naqvi, “On electromagnetic wave propagation in fractional space,” Nonlinear Analysis: Real World Applications, vol. 12, no. 5, pp. 2844–2850, 2011.
 M. Zubair, M. J. Mughal, and Q. A. Naqvi, “The wave equation and general plane wave solutions in fractional space,” Progress In Electromagnetics Research Letters, vol. 19, pp. 137–146, 2010.
 V. E. Tarasov, “Anisotropic fractal media by vector calculus in noninteger dimensional space,” Journal of Mathematical Physics, vol. 55, no. 8, p. 083510, 2014.
 W. Hwang and K. Kao, “Studies of the theory of single and double injections in solids with a gaussian trap distribution,” SolidState Electronics, vol. 19, no. 12, pp. 1045–1047, 1976.
 D. Abbaszadeh, A. Kunz, G. Wetzelaer, J. Michels, N. Cra?ciun, K. Koynov, I. Lieberwirth, and P. Blom, “Elimination of charge carrier trapping in diluted semiconductors,” Nature materials, vol. 15, no. 6, pp. 628–633, 2016.
 M. Mandoc, B. de Boer, G. Paasch, and P. Blom, “Traplimited electron transport in disordered semiconducting polymers,” Physical Review B, vol. 75, no. 19, p. 193202, 2007.
 I. Chen, “A model of charge injection at metalinsulator contacts,” Solid State Communications, vol. 26, no. 6, pp. 359–363, 1978.
 M. Abkowitz, J. Facci, and M. Stolka, “Timeresolved space chargelimited injection in a trapfree glassy polymer,” Chemical physics, vol. 177, no. 3, pp. 783–792, 1993.
 H. Nicolai, G. Wetzelaer, M. Kuik, A. Kronemeijer, B. De Boer, and P. Blom, “Spacechargelimited hole current in poly (9, 9dioctylfluorene) diodes,” Applied Physics Letters, vol. 96, no. 17, p. 172107, 2010.
 G. Horowitz, D. Fichou, X. Peng, and P. Delannoy, “Evidence for a linear lowvoltage spacechargelimited current in organic thin films. film thickness and temperature dependence in alphaconjugated sexithienyl,” Journal de Physique, vol. 51, no. 13, pp. 1489–1499, 1990.
 R. Coehoorn, S. Vulto, S. Van Mensfoort, J. Billen, M. Bartyzel, H. Greiner, and R. Assent, “Measurement and modeling of carrier transport and exciton formation in blue polymer light emitting diodes,” in Photonics Europe. International Society for Optics and Photonics, 2006, pp. 61 920O–61 920O.
 M. Mandoc, B. De Boer, and P. Blom, “Electrononly diodes of poly (dialkoxypphenylene vinylene) using holeblocking bottom electrodes,” Physical Review B, vol. 73, no. 15, p. 155205, 2006.
 T.Y. Chu and O.K. Song, “Hole mobility of n, n’bis (naphthalen1yl)n, n’bis (phenyl) benzidine investigated by using spacechargelimited currents,” Applied physics letters, vol. 90, no. 20, pp. 203 512–203 512, 2007.
 K. H. Cheon, J. Cho, B. T. Lim, H.J. Yun, S.K. Kwon, Y.H. Kim, and D. S. Chung, “Analysis of charge transport in highmobility diketopyrrolopyrole polymers by space charge limited current and time of flight methods,” RSC Advances, vol. 4, no. 67, pp. 35 344–35 347, 2014.
 W. Brütting, S. Berleb, and A. Mückl, “Spacecharge limited conduction with a field and temperature dependent mobility in alq lightemitting devices,” Synthetic Metals, vol. 122, no. 1, pp. 99–104, 2001.
 J. John, S. Sivaraman, S. Jayalekshmy, and M. Anantharaman, “Investigations on the mechanism of carrier transport in plasma polymerized pyrrole thin films,” Journal of Physics and Chemistry of Solids, vol. 71, no. 7, pp. 935–939, 2010.
 A. Boni, I. Pintilie, L. Pintilie, D. Preziosi, H. Deniz, and M. Alexe, “Electronic transport in (la, sr) mno3ferroelectric(la, sr) mno3 epitaxial structures,” Journal of Applied Physics, vol. 113, no. 22, p. 224103, 2013.
 G. A. MacDonald, P. A. Veneman, D. Placencia, and N. R. Armstrong, “Electrical property heterogeneity at transparent conductive oxide/organic semiconductor interfaces: mapping contact ohmicity using conductingtip atomic force microscopy,” ACS nano, vol. 6, no. 11, pp. 9623–9636, 2012.
 O. G. Reid, K. Munechika, and D. S. Ginger, “Space charge limited current measurements on conjugated polymer films using conductive atomic force microscopy,” Nano letters, vol. 8, no. 6, pp. 1602–1609, 2008.
 B. J. West, Fractional calculus view of complexity: Tomorrow’s science. CRC Press, 2015.
 V. E. Tarasov, Fractional dynamics: applications of fractional calculus to dynamics of particles, fields and media. Springer Science & Business Media, 2011.
 K. Falconer, Fractal geometry: mathematical foundations and applications. John Wiley & Sons, 2004.
 K. B. Oldham and J. Spanier, The Fractional Calculus. Academic Press, New York, 1974.
 G. Calcagni, “Geometry and field theory in multifractional spacetime,” Journal of High Energy Physics, vol. 2012, no. 1, pp. 1–77, 2012.
 A. S. Balankin, “Effective degrees of freedom of a random walk on a fractal,” Physical Review E, vol. 92, no. 6, p. 062146, 2015.