Electrical and optical numerical modeling of DP-PPV based polymer light emitting diode

In this paper, an in-depth study of the electrical and optical characteristics of Polymer Light Emitting Diodes ITO/PEDOT: PSS/DP-PPV derivatives/Al(Ca) is presented. Three polymer materials are considered; poly(2,3-diphenyl-5-(4heptyloxy-4’-oxytrimethylenediphenyl)-phenylenevinylene) (P1), poly(2,3-diphenyl-5-[4-(4-pentylcyclohexyl)phenoxy]-propyl-pphenylene vinylene) (P2) and poly(2,3-diphenyl-5-(2-(1, 4, 5triphenyl-1H-2-imidazoloyl)-1-oxytrimethylene phenyl) phenylene vinylene) (P3). The J-V characteristics are investigated using a device model which includes the injection, transport, and recombination mechanisms. The electron and hole mobility of each material are fitted to experimental data. The charge balance factor CBF, the external quantum efficiency EQE, the Langevin recombination rates and the singlet exciton densities profiles are studied. The results are found to be in a good agreement with experimental data especially in the low current densities region, indicating that these PLEDs electrons mobility affects greatly the JV characteristics compared with holes mobility, and that is because electrons band offset smaller than that of holes. The best performance is obtained with P1 device with a CBF value of almost unity and an EQE still low (4.7 %) but similar to other PPVs and MEH-PPVs based devices.


I. INTRODUCTION
The rapid evolution of Organic Light Emitting Diode "OLEDs" gave rise to the exploration and development of Organic Semiconductors ''OSCs'' fields; Organic Field Effect Transistors OFETs, Sensors and solar cells [1]- [6].OLEDs flat panel display is the next generation technology for a wide variety of applications such as TV's and handheld displays like smart phones, tablet computers and cameras.Since the discovery of conjugated polymers [7] and the manufacturing of the first organic light emitting diode [8] with its unique properties (better power efficiency, fast response time, wide viewing angle, flexible and light weight substrates), extensive researches were made to understand and to improve these devices efficiency i.e. the injection, transport, recombination mechanisms .OLEDs [16], [17] or graphene [18], [19] instead of the conventional Indium Tin oxide (ITO) were used.Furthermore, many advanced materials were explored in the manufacturing and studying OLEDs on the nanoscale like the quantum dots [20], carbon nanotubes [21] and nanowires [22].
For several years, great effort has been devoted to the study of another PPVs derivative because of its interesting properties, which is the Poly (2, 3-Diphenyl-1, 4-Phenylene Vinylene) (DP-PPV) [26]- [34].Many scientists working on this material were mostly focused on the design and synthesis routes for highly efficient PLEDs, by characterizing the topographies of the DP-PPV based device surfaces [26], attaching phenyl groups to PPV to obtain a fully conjugated polymer [27] or investigating the influence of the materials structure on its properties [28].Through these chemical modifications, better results were achieved: higher electroluminescence efficiencies [26], solubility improvement, efficient and highly thermal luminescence, high photoluminescence efficiency (while preserving the energy gaps of the parent polymer) [27] and a super high brightness and an efficient luminance [29].
The tremendous progress done by these scientists [30] - [34] was our starting point for studying the device physics of the DP-PPV.This is the first study to undertake a longitudinal analysis of DP-PPV based devices properties.We propose, therefore, a new approach to this field of research by exploring the electrical and optical characteristics of these polymer based devices, through a numerical model of a bipolar single layer OLED with undoped organic layer trap free.The Ref. 30  triphenyl-1H-2-imidazoloyl)-1-oxytrimethylene phenyl) phenylene vinylene) (P 3 ), the J-V characteristics, luminance, electroluminescence and photoluminescence were presented.Nevertheless, there are still some interesting and relevant characteristics to be addressed.Electrical and optical characteristics are studied: the influence of the injection barriers and carriers transport on current density, Charge Balance Factor "CBF", recombination power efficiency "CBFp", External Quantum Efficiency "EQE" , recombination rate and singlet excitons density.
The paper is organized as follows: section 2 and 3 describe the single layer device model, Sec. 4 presents and discusses the results of the bipolar model calculations, and Sec. 5 summarizes our conclusions.

A. Charge drift-diffusion model
To fully understand the electrical and optical characteristics of the DP-PPV based devices, the mechanisms of injection, transport and recombination of electrons /holes are studied.For the modeling, the inorganic semiconductor based devices equations [9] - [15] are used i.e. the timeindependent continuity equations, with drift diffusion, coupled to Poisson's equation: Where J n (J p ) is the electron (hole) current density, q is the electronic charge, G is the carrier generation rate (very small for materials with an energy gap larger than 2 eV ), and R is the carrier recombination rate considered as a Langevin type: ε = ε 0 ε r is the static dielectric constant with ε 0 the vacuum permittivity and ε r the organic semiconductors permittivity.n(p) is the electron (hole) density, and μ n (μ p ) is the electron (hole) mobility expected to be Poole-Frenkel electric field dependent: Where E is the electric field, μ 0 is the zero field mobility and E 0 is the characteristic electric field.
The following expressions of J n and J p are obtained from Einstein relation: k B is Boltzmann constant and T is the temperature.
The total current density for the device is expressed as follows: Where J h,e and J h,e ́ are the injected holes and electrons current densities or majority's carrier density current and minority's carrier current densities respectively.
The recombination current density J r is defined by: Poisson's equation is: The electrostatic potential V is related to E by:

B. Boundary conditions
The equilibrium free carrier concentrations at the interfaces, considering the hole injection at the left (x=0) and the electron injection at the right (x=L) are expressed by [35]: Where N c and N v (assumed to be equal) are the density of negatively and positively chargeable sites in the film, E g the band gap energy, ϕ b1 (ϕ b2 ) are the electron (hole) energy barrier.

C. Numerical procedure
Using finite difference discretization, the following expressions of the main equations are obtained: Poisson's equation: Where h is the discretization step.
Coupled continuity and drift-diffusion equations: The EQE which is the ratio between the number of emitted photons and injected charge carriers is expressed as follows: Where γ is the charge carrier balance factor CBF [36], [37] or the recombination efficiency [12], it is expressed as follows: In ideal conditions, the CBF is unity (γ = 1), means a complete recombination process and/or a balanced number of injected holes and electrons.
η S/T : The Singlet/ Triplet factor that describes the probability for the formation of an exciton that is allowed to decay radiatively according to the spin selection rules, it has a value of 25% for the fluorescent emitters and 100% for phosphorescent ones.
η out : The light out-coupling efficiency.In a simple model based on ray optics, it is expressed as a rough estimation as: Where n r is the refractive index of the organic layer with typical values of 1.6-1.8.
q eff , ϕ f or η PL : are respectively the effective radiative quantum efficiency, effective fluorescence quantum efficiency or photoluminescence quantum efficiency.
The power Efficiency EQEp which is the ratio of the power output to the power input is defined by: η P = γ P .η S/T .q eff .η out (18) Where γ P is the recombination power efficiency or CBFp which is the ratio of recombination power output to electric power input and is expressed as follows: The Singlet excitons density S(x) is defined as follows 38: Where: Ds is the singlet exciton diffusion coefficient defined by : L D is the singlet exciton diffusion length and τ s the exciton lifetime.As it is reported [30], P 1 and P 2 contain liquid crystal side groups and P 3 contains a charge transport group.The anode is contact limited since the holes energy barriers are greater than 0.4 eV [12], which also means that the current flow is contact limited.
The parameters used in this calculation [Table 1], were mostly obtained from the references [30], [38]: The

DP-PPV
3 L (nm) 50 (cm -3 ) 2.5×10 19    (cm -3 ) 2.5×10 19     (nm) 1   (ns) 1 The main concern of this paper is to study the electrical and optical properties of DP-PPV based devices.To our knowledge, this is the first study to deal with this polymer based devices physics, while we refer to the earlier work [30], the focus is different.We have addressed not only the J-V characteristics for P 1 , P 2 and P 3 that were compared with experimental data [30] to confirm our results, but also, electric field, carriers densities, carriers mobility, recombination and singlet exciton densities profiles are studied.We have considered also other properties like recombination and recombination power efficiencies along with the external quantum and the quantum power efficiencies.We consider these structures "electron only" devices as they have electrons injected from an ohmic contact and holes injected from a contact limited contact.Device P 1 is characterized with holes and electrons barriers smaller compared to the other two devices.The mobility at the interface cathode/polymer (x=L), where electrons are injected, is lower by 3 orders of magnitude compared to the mobility at the interface anode/polymer (x=0), where holes are injected, as seen from Fig. 3.This device has an easy injection of the less mobile electrons and a more or less difficult injection of the more mobile holes, this leads to a more balanced combination of injection and transport carriers compared to the other two devices, and as a result, the recombination current density is equal to the device current density.P 2 has higher electrons and holes barriers, and equal mobilities (Fig 3), so it requires a higher applied voltage.This leads to an unbalanced combination of injection and transport mechanisms and that is why the recombination current density is too low.The third device P 3 has an electron barrier between the two, a hole barrier close to that of P 2 , and a higher mobility which ensures a better transport mechanism leading to a lower applied voltage.In this case, the injection and the transport of carriers are more balanced reducing the shift between J and Jr.  From Fig. 4 (a) and (b), the electrons density of the three devices is much higher than that of holes because of the electrons barriers that are too low compared with holes barriers.

A. Device characteristics
In P 1 , because of the electrons barrier being too low compared with holes one, the carriers densities are accumulated at cathode/ organic layer interface, leading to a variable and higher electric field (Fig. 4(c)) and carriers distributions.In the case of P 2 and P 3 , The electrons and holes barriers are higher compared with P 1 ; hence the carriers densities are not only decreased but also more uniform.
From Fig. 4 (c), the electric field is less variable for P 3 as there is an accumulation at the cathode/electrode interface (Fig 4(a)).As for P 2 , the electric field is higher because of the applied voltage and more uniform as there is no accumulation of the carriers densities.
Figure 5 shows the Langevin recombination rates (a) and singlet exciton density profiles (b) for P 1 (solid), P 2 (dash) and P 3 (dot).From Fig. 5 (a), the recombination rates are uniform across the device except for P 1 , but lower by 6, 9 and 10 orders of magnitude for P 1 , P 2 and P 3 respectively compared to other PPV derivatives materials [12], [38].
Similar results were obtained for a hole only device [12] (single carrier device) and because of the electron barriers lower than the hole ones, our devices are essentially electron only devices.The nonuniformity of P 1 is due to the non-uniformity of the electric field which affects the mobility as seen in

B. Device efficiency
The CBF (the recombination efficiency) and the EQE for P 1 (solid), P 2 (dash) and P 3 (dot) as a function of the applied current density are presented in Fig. 6.  whereas, because of the higher barrier, the holes accumulate at the vicinity of the anode/ organic layer interface.The low electrons barrier and the high holes mobility, reduces the accumulation of both carriers near their respective interfaces, hence improving the CBF and EQE (even if the EQE increase is less than 5 % which is the case for all PPV's based materials [39] - [41]).P 2 has electrons and holes injected from an ohmic contact and contact limited contact respectively with equal mobilities (Fig. 3), leading to the accumulation of holes near its interface.The combination of injection/ transport mechanism is deeply imbalanced lowering the CBF and consequently the EQE.
Figure 7 shows the recombination power efficiency CBFp (a) and the external quantum power efficiency EQEp (b) versus device current density for P 1 (solid), P 2 (dash) and P 3 (dot).The recombination power efficiency (ratio of output to input power) is lower for the second device P 2 because a high voltage is applied.In the case of P 3 , a low voltage is applied, as a result, the CBFp value is higher, while for P1, an intermediate applied voltage yields to a CBFp value between the two.
As a result, the EQEp which depends on the CBFp has similar trend even if its largest value is 5%.
In spite of these results, it is noticed that this model presents some limitations since it is derived from inorganic semiconductors physics and as the organic semiconductors have disordered molecular structures or even amorphous unlike the inorganic ones which have a crystalline structure.
From this physical modeling, we know now that balancing injection and transport mechanisms ensures a better PLEDs efficiency; however, that is not always possible.The injection mechanism would be more efficient by reducing the height barriers at both interfaces as much as possible; the main idea is to match the electrode material with the electronic structure of the electroluminescent polymer.Nevertheless, when the barrier height is too small, less than a tenth of electron volts, the barrier will not have a triangular form as the model assumes, so the electron will not tunnel, and the thermionic emission phenomena becomes too important.In addition, the evaporated electrodes will not have the outlined work-function values listed from literature, as they are full of defects.The transport mechanism which is ensured mainly by carriers mobility and the space charge limited effects would also be influenced as the barriers are too small [42].Another problem with modeling is that the experimental data does not always fit with the physical model described above, in some cases, the match is only for a small region and the model here would be considered inapplicable, and in some other cases, the fit is for a large region [43], and here, the model can be considered unfailing.
Another drawback, is that sometimes, organic semiconductors electronic structures can't be well matched with electrodes material since the barriers are too high, and in this case, an efficient injection via Fowler Nordheim tunneling will not be possible anymore, so in order to overcome this, hole or/and electrons blocking layer is added between electrode(s) and the emitting polymer [42].This layer, blocks the majority carriers (electrons or/ and holes), where they are increased at the interface of the blocking layer and a strong internal electric field is created by the minority carriers side, which balance electrons and holes current densities.
To ameliorate injection efficiency, impurities or defects are added, electrons or holes acceptor groups are end-capped which allow electron tunneling or these groups are incorporated into the polymeric chain thus copolymers are created.
In general, one should be aware that up until now, no perfect model exists to describe the electrical and optical characteristics for theses kind of devices [43].As a matter of fact, we just have to choose the best appropriate model.

Fig. 1 .
Fig.1, we can see that the electrons energy barriers for the three devices are lower than 0.2-0.3eV, which means that the cathode is an ohmic contact and the current flow is space charge limited.

Figure 2
Figure2presents the variation of the current density as a function of the applied voltage for P 1 , P 2 and P 3 .Fig.2(a) shows the simulated J-V characteristics, and we see that the simulated results are in good agreement with experimental data from[30].Fig.2(b) shows current density J (lines) and recombination current density Jr (dash) versus applied voltage, the curves of Jr and J fall nearly on top of one another for P 1 , P 2 and P 3 , the recombination current density is lower especially for P 2 .The carriers mobility for the three devices are shown in Fig.3.

Figure 4
Figure 4 presents carriers (electrons and holes) densities and electric field distributions for P 1 (solid), P 2 (dash) and P 3 (dot).Electrons and holes densities distributions are shown in Fig.4 (a) and (b) respectively and the electric field distributions in fig.4(c).

Fig. 4 .
Fig.4.Calculated distributions of carrier densities for the three devices ; electrons concentration (a) and holes concentrations (b), calculated distributions of the electric field intensity (c) and electron mobility (d) for P 1 (solid) , P 2 (dash) and P 3 (dots).

Fig 3 .
Fig 3. P 2 and P 3 have higher electron mobility with a harder injection of holes, unbalancing the combination of injection and transport mechanisms, lowering the recombination rate for these devices.When electrons and holes meet, they form singlet and triplet excitons, 25% of which recombine radiatively and 75% non-radiatively respectively, as a result, a very low singlet excitons density (Fig.5(b)).

Fig. 6 (
Fig.6 (a) and (b) show that the CBF and the EQE are the highest for the first device P 1 and the lowest for P 2 .These devices present a difficult injection of the more mobile holes and easy injection of the less mobile electrons, somewhat balancing the injection and transport mechanisms.The injected carriers accumulate in the vicinity of the cathode/ organic layer interface because of the low mobility,

TABLE I .
PARAMETERS USED FOR MODELING.