Dilution-induced current-density increase in disordered organic semiconductor devices: A kinetic Monte Carlo study

Using three-dimensional kinetic Monte Carlo simulations, we systematically investigate the effect of dilution with an inert material on the current density in unipolar sandwich-type disordered organic semiconductor devices. Such a dilution technique was studied experimentally by Abbaszadeh et al. [Nat. Mater. 15, 628 (2016)], who observed a dilution-induced increase of the current density. The authors explained the effect as a result of a reduced density of trapped charges (“trap dilution”), assuming an exponential density of trap states. Our simulations support this explanation, and show under which conditions this trap-dilution-induced increase of the current density becomes more than outweighed by the negative effect of the dilution-induced decrease of the mobility. The effect is studied for sets of systematically varied material parameters, including systems with a Gaussian shape of the host and trap DOS.

Figure 1

Electron densities of states (schematic) of the material systems that are studied in this paper. The DOS of the transport material is taken to be a $\delta$ function (a) or a Gaussian [(b) and (c)]. The systems have a superimposed exponential electron trap DOS [(a) and (b)], or a Gaussian trap DOS (c). The trap DOS can be taken to be shifted by an energy $\mathrm{\Delta }E$ with respect to that of the transport material. The electron DOS of an inert diluting material does not overlap with these distributions, and is not shown.

Figure 2

(a) Effect of dilution by inert host molecules on the current density $J$, normalized by its predilution value $J_0$, for a device with a DOS as shown in Fig. 1 and with parameters as given in Table 1, calculated using 3D KMC FD simulations. The inset shows the schematic flat-band energy-level diagram. The material blend that is sandwiched in between the two electrodes is composed of a transport material with a $\delta$-function-shaped DOS (blue colored box), trap molecules with an exponentially shaped DOS traps (red colored spheres), and an inert host (gray colored box). (b) Effect of dilution on the mobility at a field $F=5\times {10}^7$ V/m in a trap-free material (black full curve and symbols) and in a material containing traps, as specified in Table 1 (blue dashed curves). The curves are obtained using Eqs. (11) and (13), as indicated in the figure and as discussed in the text. The red full curve and symbols show a mobility curve with an $f_{\mathrm{dil}}$ dependence that is equal to the ratio $J/J_0$ that is shown in panel (a). The mobility increase with $f_{\mathrm{dil}}$ that is shown by the blue dashed curves is somewhat different from the increase of the current density that is shown by the red curve and symbols due to (i) a variation of $c_{\mathrm{tot}}$ and $F$ throughout the device, and (ii) a variation with $f_{\mathrm{dil}}$ of the contribution of direct-trap-to-trap transport.

Figure 3

Profiles of the electron concentration calculated using 3D KMC FD simulations on (a) transport sites ($c_{\mathrm{T}}$) and (b) trap sites ($c_{\mathrm{trap}}$), for the device shown in Fig. 1, at various dilution fractions $f_{\mathrm{dil}}$. The legends to the curves that are given in (a) apply also to (b).

Figure 4

Charge-carrier mobility, obtained from 3D KMC PBC simulations, for systems with an exponential trap DOS as specified in Table 1 (full curves and filled symbols). (a),(b) Dependence of the steady-state mobility on the concentration on active sites, $c_{\mathrm{act}}$ [panel (a)], and the total electron concentration, $c_{\mathrm{tot}}$ [panel (b)], at an electric field $F=5\times {10}^7\mathrm{V}/\mathrm{m}$. (c) Electric field dependence of steady-state mobility at $c_{\mathrm{act}}={10}^{-4}$. (d) Simulated time evolution of the mobility in the KMC simulations, resulting from carrier relaxation, at $F=5\times {10}^7\mathrm{V}/\mathrm{m}$ and $c_{\mathrm{act}}=3\times {10}^{-4}$. In panel (a), the dotted curves and open symbols are mobilities neglecting Coulomb correlation. The dashed lines show the trap-free limit of mobilities. In panel (b), the gray zone indicates the range of typical values of $c_{\mathrm{tot}}$ in the center of the default devices containing these materials (see Table 2).

Figure 5

Charge-carrier mobility for systems that consist of a transport material with a $\delta$-function-shaped DOS that is diluted by a mol fraction $f_{\mathrm{dil}}$ of an inert material, for various values of the wave-function decay length $\lambda$, studied at a field of $F=5\times {10}^7\mathrm{V}/\mathrm{m}$. Dashed curves show the $f_{\mathrm{dil}}$ dependence that is predicted by Eq. (11) and symbols and full curves show the results of 3D KMC PBC simulations. The results for $\lambda =0.5$ nm are used in Secs. 4c and 4d.

Figure 6

Current density as a function of the fraction of inert diluting molecules for the device specified in Table 1, obtained from 3D KMC FD simulations (symbols and full curves) and from an approximate model [Eq. (16), dashed curves], when varying (a) the fraction of traps ($f_{\mathrm{trap}})$, (b) the applied voltage ($V$), (c) the device thickness ($L$), (d) the characteristic trap temperature ($T_{\mathrm{trap}}$), and (e) the wave-function decay length ($\lambda$). The red curves show the result for the default parameter set.

Figure 7

Current density $J$ at $V=5\mathrm{V}$ obtained from 3D KMC FD simulations as a function of the fraction of inert diluting molecules, $f_{\mathrm{dil}}$, for transport materials with a Gaussian DOS with a width ${\sigma }_{\mathrm{T}}$, as indicated. The simulation parameters that are not specific to the trap sites are as given in the first part (“Common parameters”) of Table 1. (a) Transport material without traps. (b) Systems with an exponential trap DOS ($f_{\mathrm{trap}}=0.01$ and $T_{\mathrm{trap}}=$ 1500 K). (c),(d) Systems with a Gaussian trap DOS with $f_{\mathrm{trap}}=0.01$ and ${\sigma }_{\mathrm{trap}}$ as indicated, for the DOS types that are shown in panel (e). For DOS-types B and C, the trap depth is $\mathrm{\Delta }E=0.3\mathrm{eV}$.

Figure 8

Dependence of the thermal equilibrium electron concentration on transport sites, $c_{\mathrm{T}}$, on the total electron concentration, $c_{\mathrm{tot}}$, calculated from Eq. (17), for (a) systems with an exponential trap DOS, studied in Fig. 7, and (b)–(d) systems with a Gaussian trap DOS, with parameters that are specified in the figures. Panels (b),(c) refer to panels (c),(d), respectively, of Fig. 7. Curves are given for transport materials with ${\sigma }_{\mathrm{T}}=0\mathrm{eV}$ (black), 0.05 eV (blue), and 0.10 eV (red), and for dilution fractions $f_{\mathrm{dil}}$ increasing from 0 to 0.8 in steps of 0.2, in the direction of the arrows. The gray zones indicate the region of total charge concentrations around the value of $(1.4-4)\times {10}^{-4}$ that is approximately found in the device center of 100-nm devices, studied at 5 V (see Tables 2 and SI [29]). The vertical dashed lines in panel (c) mark the boundaries $c_{\mathrm{tot},1}$ and $c_{\mathrm{tot},2}$ of the intermediate concentration range in which the largest sensitivity of the current density to $f_{\mathrm{dil}}$ is expected (see the text in Sec. 5). Dashed curves with a slope of 1 on the double-log scale used show the (partially extrapolated) $c_{\mathrm{T}}(c_{\mathrm{tot}})$ functions in the Boltzmann and trap-filled regimes for ${\sigma }_{\mathrm{T}}=0\mathrm{eV}$ materials. Panel (d) shows a set of curves for parameters for which a strong trap dilution effect is expected.