Parametrization of the Gaussian Disorder Model to Account for the High Carrier Mobility in Disordered Organic Transistors

Correct parameterization of the Gaussian disorder model (GDM) on spatially random sites is necessary for a complete description of charge transport in disordered materials and concomitant device characteristics. Because the GDM on spatially random sites considers both energetic and spatial disorder, it is superior to the GDM on a cubic lattice. However, analytical arguments and experimental evidence are still lacking for correct parameterization of the model over a wide range of model parameters, energetic and spatial disorder, and electric fields. We show that the model requires a set of parameters to correctly account for high mobility and its charge density dependence, and we develop such a model. The model is implemented in a numerical simulation tool for comparison with the measured device characteristics. Accurate agreement with experimental data, particularly with the high mobility values in organic field-effect transistors, is achieved throughout a wide range of temperature by adjusting both the localization length and the attempt-to-escape frequency.

Figure 1

Parametrization process of the SRGDM.

Figure 2

Reduced mobility $\ln [{{\mu }^{\mathrm{\prime }}}_{\mathrm{red}}(0)]$ fitting between numerical solution of the VM model (black dots) and analytical expression from Eq. (8), $c_1\exp (-c_2{\hat{\sigma }}^2)$ (red lines) at $N_0{a}^3={10}^{-1}$, 10⁻², 10⁻³, and 10⁻⁴.

Figure 3

(a) ${\mu }_{\max }$ versus μ modulation with $V_{GS}$ for various OFETs at room temperature. Donor-acceptor (DA) copolymer for IDTBT [1] and poly[2,5-(2-octyldodecyl)-3,6-diketopyrrolopyrrole-alt-5,5-(2,5-di(thien-2-yl)thieno[3,2-b]thiophene)] (DPP-DTT) [2, 3]; semicrystalline (SemiC.) polymer for poly(3-hexylthiophene-2,5-diyl) (P3HT) [4] and PBTTT [1, 5]; disordered polymer for chemically modified PPV [6, 7, 8]; polycrystalline (PC) molecule for T6 [9], pentacene [10, 11, 12, 13, 14, 15], 5,11-bis(triethylgermylethynyl)anthradithiophene (diF-TEG-ADT) [16], and dihexylsexithiophene (DH6T) [17]; single-crystal (SC) molecule for rubrene [52, 53, 54]. Hatched area in red illustrates SC molecules employing bandlike transport, and other shaded areas illustrate materials employing hopping transport. Also, filled symbols represent contact-resistance-corrected μ and open symbols represent contact-resistance-uncorrected μ. (b) Gaussian widths extracted from the literature for rubrene [22, 55, 56], IDTBT [1, 57], PBTTT [1, 58, 59], pentacene [60, 61, 62, 63], and PPVs [6, 19, 64].

Figure 4

Mobility calculated by EGDM (solid line) and SRGDM (dotted line) with various $N_0{a}^3$ and calculated $r({\epsilon }_t)$ in the cubic lattice and spatial disorder for (a),(c) $\sigma =0.05\mathrm{eV}$ and (b),(d) $\sigma =0.15\mathrm{eV}$. In the calculation, ${\nu }_0={10}^{12}{\mathrm{s}}^{-1}$, $B_c=2.735$, and $N_0=3\times {10}^{21}\mathrm{c}{\mathrm{m}}^{-3}$ are used.

Figure 5

(a) Experimentally measured and numerically calculated Arrhenius plots for IDTBT, PBTTT, and pentacene. Calculation considers $\sigma =0.059\mathrm{eV}$, $N_0=7.4\times {10}^{20}\mathrm{c}{\mathrm{m}}^{-3}$, and $p/N_0=0.0126$ for IDTBT; $\sigma =0.11\mathrm{eV}$, $N_0=8.9\times {10}^{20}\mathrm{c}{\mathrm{m}}^{-3}$, and $p/N_0=0.0243$ for PBTTT; and $\sigma =0.15\mathrm{eV}$, $N_0=3\times {10}^{21}\mathrm{c}{\mathrm{m}}^{-3}$, and $p/N_0=3\times {10}^{-4}$ for pentacene. (b) Slopes of Arrhenius plot, $\mathrm{\Delta }$, with respect to a from (a). Horizontal and vertical dotted lines indicate ${\mathrm{\Delta }}_{\mathrm{exptl}\text{.}}$ and extracted a, respectively. (c) Direct calculation of ${\epsilon }_a={\epsilon }_t-{\epsilon }_F$ using Eq. (3).

Figure 6

Molecules and molecular orientation in thin films of (a) IDTBT, (b) PBTTT, and (c) pentacene. Black arrows indicate the direction of charge transport and red double-headed arrows indicate ID of π-π stacking.

Figure 7

Calculated results of (a) ${\epsilon }_t$ and (b) $r({\epsilon }_t)$ with respect to localization length a. Schematic illustration of the hopping range in (c) energetic and (d) spatial diagrams for different localization lengths a: small a in violet color; large a in red color.

Figure 8

(a) Extracted a and ν₀ from Fig. 5 for three different materials: IDTBT, PBTTT, and pentacene. (b) Direct calculation of MA upward, ${\nu }_{\uparrow }$, and downward, ${\nu }_{\downarrow }$, transition rates by Eq. (2) for each ${\nu }_0$ from (a).

Figure 9

Transfer curves on (a)–(c) semilogarithmic and (d)–(f) linear scales for IDTBT- (V_DS = −60 V), PBTTT- (V_DS = −70 V), and pentacene-based (V_DS = −2 V) OFETs. Symbols illustrate measured data and solid lines show simulated results without interface trap states. For pentacene OFET, dotted lines for low temperatures, T = 260 and 230 K, represent simulated results with interface trap states. Channel length L and width W are indicated in the figure.