The Monte Carlo simulation of the hole transport in thin films of PFO:MEH-PPV

Abstract

Hole transport is numerically studied by means of the Monte Carlo method in a single blended layer of poly(9,9$^\prime$-dihexyl fluorenyl-2,7-diyl) (PFO) and poly(2-methoxy-5-(2-ethylhexyloxy)-l,4-phenylenevinylene) (MEH-PPV), which is sandwiched between two electrodes. A bimodal Gaussian density of states is used for randomly distributed localized states in the blended organic layer and an exponential distribution function for trap density of states. In this study, a new approximation has been used for the Fermi level instead of the Boltzmann approximation due to the high charge carrier density. The current density and the mobility have been calculated for different concentrations of MEH-PPV versus voltage and 1000/T at temperatures 150–290 K. The results of calculations show that the current density and the mobility are maximized at the blending ratio of 2 wt%, and there is a linear relationship between the current density and 1000/T at different voltages. The comparison of the numerical results with the experimental data shows a very good consistency between them, particularly at low and medium voltages of the working range of organic semiconductor devices.

Appendix: An approximation for the Fermi level

To derive Eq. (16), first consider the following relation between the Fermi level and the charge carrier density [38].

$$\begin{aligned} \frac{\hbox {d}E_{\mathrm{F}} }{\hbox {d}n}=k_{\mathrm{B}} T\left[ {\frac{1}{n}+\frac{1}{\hbox {g}\left( n \right) }\frac{\hbox {dg}\left( n \right) }{\hbox {d}n}} \right] , \end{aligned}$$

(21)

where $\hbox {g}\left( n \right)$ is the mobility enhancement function.

Moreover, Pasveer et al. [36] used the following expression for $\hbox {g}\left( n \right)$ which had been obtained by them from the fitted numerical data for their purpose:

$$\begin{aligned} \hbox {g}\left( n \right)= & {} \exp \left( {\frac{1}{2}\left( \hat{ \sigma }^{{2}}-\hat{\sigma } \right) \left( {2\frac{n}{N_0 }} \right) ^{\delta }} \right) , \end{aligned}$$

(22)

$$\begin{aligned} \delta= & {} 2\frac{\ln \left( {\hat{\sigma }^{{2}}-\hat{\sigma }} \right) -\hbox {ln}\left( {\ln \left( 4 \right) } \right) }{\hat{\sigma }^{{2}}}, \end{aligned}$$

(23)

here $\hat{\sigma } =\frac{\sigma }{k_{\mathrm{B}} T}$ .

By substituting Eqs. (22) into (21), we will have:

$$\begin{aligned} \frac{\mathrm {d}E_{\mathrm {F}}}{\mathrm {d}n}= & {} k_{\mathrm {B}}T\left[ \frac{1}{n}+\frac{1}{\exp \left( \frac{1}{2}\left( \hat{\sigma }^{2}-\hat{\sigma } \right) \left( 2\frac{n}{N_{0}} \right) ^{\delta } \right) }\right. \nonumber \\&\left. \times \, \frac{\delta }{2}\left( \hat{\sigma }^{2}-\hat{\sigma } \right) \left( \frac{2}{N_{0}} \right) ^{\delta }\times \left( n \right) ^{\delta -1}\right. \nonumber \\&\left. \times \,\exp \left( \frac{1}{2}\left( \hat{\sigma }^{2}-\hat{\sigma } \right) \left( 2\frac{n}{N_{0}} \right) ^{\delta } \right) \right] \nonumber \\= & {} k_{\mathrm {B}}T\left[ \frac{1}{n}+\frac{\delta }{2}\left( \hat{\sigma }^{2}-\hat{\sigma } \right) \left( \frac{2}{N_{0}} \right) ^{\delta }\times \left( n \right) ^{\delta -1} \right] \end{aligned}$$

(24)

If we compute the integral of Eq. $\left( {\hbox {A}.\hbox {}4} \right)$, we will get:

$$\begin{aligned} E_{\mathrm{F}} =k_{\mathrm{B}} T\left[ {\hbox {ln}\left( {\frac{n}{N_0 }} \right) +{\left( n \right) ^{\delta }}/2\left( {\hat{\sigma }^{2}-\hat{\sigma } } \right) \left( {\frac{n}{N_0 }} \right) ^{\delta -1}} \right] +E_0\nonumber \\ \end{aligned}$$

(25)