Trap density of states in n-channel organic transistors: variable temperature characteristics and band transport

Considerable research efforts have been devoted to organic field-effect transistors last decades for the possibility of low-cost alternatives of silicon electronics,^1–3 and there has been enormous progress in developing organic semiconductors, where the best performance exceeds amorphous silicon transistors (a-SI).^4–8 Along with improving device performance, underlying charge transport mechanism has been extensively studied because it is closely related to device key parameters such as mobility, threshold voltage, subthreshold swing, and electrical and environmental stability. Charge transport of organic transistors using single crystals and thin-films has been studied from temperature (T)-dependent transistor characteristics. Intrinsic carrier transport, in particular, band transport, has been explored in organic single crystals,^9–12 and trap-limited charge transport has been examined in organic thin films, which have higher density of trap states due to the structural defect, grain boundaries, and chemical impurities.^13–27 The multiple trapping and release model for charge transport has been proposed and widely accepted,^13,14 and there have been many attempts to determine the trap density of states (trap DOS). However, the trap DOS has been analyzed mainly for p-channel materials,^13–20 and comparatively limited works have been done for n-channel materials.^15,21–23 In particular, the trap DOS has not been systemically studied from the viewpoint of the basic transistor operation.

In a-Si transistors, charge transport has been understood by a distribution of localized states in the band gap of semiconductors.^28–30 The mid-gap states are studied from T-dependent transfer characteristics, and the distribution of the trap density of states is typically described as an exponential function (Fig. 1),³¹ 

where N_G is the trap density at the conduction band edge E_C, and T_G is the distribution width of the trap DOS. Once such a DOS distribution is assumed, we can obtain the analytical formulas that describe the operation of a-Si transistors.^29,30 Trap DOS is sometimes considered as having two different regions, deep states and tail states.

Numerous determination methods of the distribution of the trap DOS have been reported in organic transistors. Lang et al. have extracted the trap DOS from the Arrhenius plot of the transconductance for single-crystal pentacene transistors,^32,33 and similar analysis has been applied to other thin-film organic semiconductors.^15,16 In addition, a variety of methods to calculate the trap DOS have been investigated, and an exponential distribution of the DOS has been established in pentacene thin-film transistors,^17–19 and in single crystal transistors.²⁰ The DOS distribution has been also extracted from space-charge-limited-current spectroscopy and Kelvin probe microscopy.^34–36 Recently, the DOS distribution has been estimated from the analysis of ESR lineshape,^37,38 where the existence of significant discrete states has been concluded. However, since the shallow states mainly influence the transport properties, the exponential model is a good approximation to understand the transistor characteristics. Since the results of a-Si are not directly applied to organic transistors due to different circumstances that need interface approximation,¹⁴ in-depth research on organic transistors is strongly required for further understanding of operational mechanisms in organic transistors.

In the present paper, we report on trap states in n-channel transistors using N,N ’-bis(cyclohexyl)naphthalene diimide (Cy-NDI) and dimethyldicyanoquinonediimine (DMDCNQI). The molecular structures are shown in the insets of Figs. 2(a) and 3(a). Cy-NDI shows outstanding performance among the n-channel materials, with the reported mobility of 6.2 cm²/Vs due to the good thin-film morphology and the highly symmetrical brickwork packing of the planar molecules.³⁹ A small-molecule organic electron acceptor, DMDCNQI, forms air-stable n-channel transistors,^40,41 and recently, improved performance has been attained (μ = 0.23 cm²/Vs, on/off ratio = 2 × 10⁶, V_T = 0 V) using the low vacuum evaporation method, though the threshold voltage is more susceptible to the surface treatment due to the trap states induced during the low vacuum evaporation.⁴² In order to study the trap states, we have investigated the characteristics of these transistors in the temperature range between 260 K and 200 K. From the Arrhenius plot of the transconductance measured at low temperatures, we have studied gate voltage (V_G) dependence of the activation energy (E_A), and extracted the trap DOS from Lang's method,³³ as well as another method based on the E_A vs. log V_G plot. Furthermore, the T-dependent transistor characteristics is calculated inversely from the trap DOS. We have conducted the simulation assuming a single exponential DOS based on the interface approximation, and extracted the information about the trapped and free charge states. Finally, we have examined band transport by the simulation method and investigated the influence of the trap states.

Cy-NDI and DMDCNQI were synthesized according to the published method.^43,44 Before use, Cy-NDI and DMDCNQI were purified by sublimination at 220 °C and 130 °C, respectively, under the vacuum of 10⁻⁴ Pa. A highly doped n-type silicon wafer with a thermally grown silicon oxide (300 nm, C = 13.7 nF/cm²) was used for the substrate. Silicon oxide surface was treated with self-assembled monolayers of octadecyltrichlorosilane (OTS) for Cy-NDI and octadecyl-trimethoxysilane (OTMS) for DMDCNQI in prior to the evaporation of the materials.^45,46 Cy-NDI and DMDCNQI were thermally deposited on the substrates with a thickness of 50 nm, where the evaporation was carried out under 10⁻⁴ Pa for Cy-NDI, but 10⁻³ Pa for DMDCNQI due to the high vapor pressure.⁴¹ In order to fabricate transistors with top-contact configuration, the source-drain electrode was patterned with shadow masks by thermal deposition of Au (channel length L = 100 μm and channel width W = 1000 μm) onto the organic thin films. The transfer characteristics of the transistors were measured in the T range from 260 K to 200 K with an interval of 10 K under the vacuum of 10⁻⁴ Pa with a low-temperature micro prober system (Riko International). Since the transistor characteristics above 270 K suffered from the influence of moisture, we only used the results below 260 K. Mobilities μ and threshold voltages V_T were estimated from the gradient and the extrapolation to I_D = 0 A in the plot of I_D^1/2 vs. V_G in the large V_G region (Figs. 2(b) and 3(b)).

Figures 2(a) and 3(a) show transfer characteristics of Cy-NDI and DMDCNQI measured at various temperatures. Cy-NDI exhibits better n-channel performance (μ = 0.19 cm²/Vs and on/off = 10⁵) than DMDCNQI (μ = 0.0044 cm²/Vs and on/off = 10⁴) at 260 K, though the threshold voltage (V_T = 54 V) is much larger than DMDCNQI (V_T = − 6 V). As lowering T, I_D decreases and V_T moves toward the positive (off) direction, but these two n-channel transistors show contrasting characteristics. The T-dependence is relatively small at high V_G in Cy-NDI, but in DMDCNQI, large T-dependence is observed in the entire V_G region. It is reflected in the T-dependence of μ and V_T (Fig. 4(a)). In Cy-NDI, μ changes with a very small thermal activation energy of 0.0086 eV (Fig. 4(a)), and V_T shifts as small as 3.2 V in this T range (Fig. 2(b)). In DMDCNQI, however, μ exhibits a larger activation energy of 0.14 eV (Fig. 4(a)) and V_T shows about twice larger shift of 5.9 V (Fig. 3(b)).

The Arrhenius plots of the transconductance g_M = dI_D/dV_G are shown in Figs. 2(c) and 3(c). The activation energy E_A is extracted from the least-squares fits and shown in Fig. 4(b). E_A is largest just above the onset voltage V_ON (40 V and 2 V, respectively), which are 0.23 eV for Cy-NDI and 0.17 eV for DMDCNQI. E_A decreases monotonically as increasing V_G. We can consider E_A as the energy difference between the conduction band edge and the Fermi level E_F at the interface as described in Fig. 1.

where E_C is the bulk conduction band level, q is the elemental charge, and ψ is the surface potential. Electron carriers require E_A to be activated into the conduction band edge. When V_G is applied, charge is accumulated at the interface, so the conduction band starts to bend by ψ, and E_A is getting smaller as increasing V_G. This explanation agrees well with the plot of E_A vs. V_G in Fig. 4(b). Cy-NDI and DMDCNQI have different flat band potentials V_ON; this is considered as the V_G value until which the conduction band is flat. Accordingly, we can convert the horizontal axis to V_G – V_ON, and compare the E_A vs. V_G plot in the same scale as shown in Fig. 4(b). E_A of Cy-NDI drops rapidly down to 0.018 eV, but E_A in DMDCNQI does not decrease below a certain E_A value of 0.16 eV even at the highest V_G. Consequently, in Cy-NDI the T-dependence of the transfer characteristics is very small at large V_G (Fig. 2(a)), whereas the T-dependence is large in the whole region in DMDCNQI (Fig. 3(a)). The large E_A is attributed to a comparatively large number of charge traps existing in the band gap; the traps prevent the conduction band edge from approaching to the Fermi level, because the charge induced by V_G is trapped. The E_A values of 0.0086 eV and 0.14 eV obtained from the Arrhenius plots of μ respectively in Cy-NDI and DMDCNQI (Fig. 4(a)) are about the same as these E_A values in the high V_G limit. Several humps observed in the E_A vs. V_G plot in Cy-NDI suggest the existence of discrete states.

In Figs. 2(c) and 3(c), the extensions of the least-squares fits seem to cross at a single point. This point is called the Meyer-Neldel temperature, T_MN. Cy-NDI shows higher T_MN = 2000 K than 500 K in DMDCNQI. The Meyer-Neldel rule is observed universally in variable T characteristics of organic transistors,^24,25,27,47 and it is considered as an evidence of an exponential trap DOS, where T_MN is related to the distribution width.⁴⁸ It has been, however, also argued that T_MN originates from multiexitation process in general.^49,50

Contact effect is sometimes important in two-probe organic transistors. Contact resistance of DMDCNQI transistors has been estimated to be 100 kΩ cm by the transfer line method,⁴⁰ and is much smaller than the channel resistance which amounts to more than 1000 kΩ cm. Accordingly, contact resistance of DMDCNQI transistors is relatively small. In addition, our estimation shows the ratio of contact resistance to the overall resistance is almost T independent as reported in Ref. 22. This is because the contact resistance shows similar activation behavior to the organic semiconductors. Therefore, the activation energy E_A extracted from transconductance is not considered to be seriously affected by the contact effect.²² 

As depicted in Fig. 1, when V_G is applied, charge Q = CV_G is accumulated at the interface between the organic semiconductor and the gate insulator. In the a-Si transistors, spatial charge distribution in the accumulation layer is critical to determine the distribution of the mid-gap states,^29,30 which is generally estimated by solving Poisson's equation.⁵¹ However, in the case of organic transistors, this process is simplified because the thickness of accumulation layer is typically shorter than the length of one monolayer.¹⁴ In the metal-insulator-semiconductor (MIS) structure,⁵² the thickness of the accumulation layer x is given by

where ɛ_S is the dielectric constant of the semiconductor and N_A is the three-dimensional carrier density. Assuming ψ = 0.3 V, and N_A to be the same as the molecular density, 1.95 × 10²¹ cm⁻³ for Cy-NDI and 2.12 × 10²¹ cm⁻³ for DMDCNQI from the crystallographic data,^39,53 we estimate the thickness x to be in the order of 0.1 nm. This is shorter than the length of one monolayer. Even if N_A is one hundredth of the molecular density, namely corresponding to the actual carrier density at V_G = 50 V, x is calculated to be in the order of 1 nm; this is still smaller than the length of a molecule. Therefore, it is reasonable to consider that x is practically constant in organic transistors (interface approximation).^14,54 In inorganic semiconductors, this approximation does not hold because the doped carrier number N_A is small, ɛ_S is large, and the atomic spacing is shorter than the size of an organic molecule.

As shown in Fig. 1, the gate induced charge is distributed to the trapped charge Q_T and to the free charge Q_F as CV_G = Q_T + Q_F. In the subthreshold regime of the actual transistors, most of induced charge fill the trapped states Q_T, so that Q_T is approximated to Q_T = CV_G.

When the thermal activation energy E_A obtained from the T-dependence (Fig. 4(b)) is equated to E_A = E_C − E_F − qψ in Fig. 1, the trap DOS is given by³³ 

As a result, N(E) is calculated from the inverse of the derivative of E_A to V_G (Fig. 4(b)).

Accordingly, we can determine the trap DOS of Cy-NDI and DMDCNQI as shown in Fig. 4(c). The approximately linear dependence indicates the exponential distribution as represented by Eq. (1), where the vertical intercept at E_A = 0 affords N_G and the slope determines T_G. There is, however, a steep increase coming from the tail states in the right end. The amount of the total traps is given by integrating Eq. (1).

N_GkT_G is converted from the dimension of cm⁻² to V by multiplying q/C (1.17 × 10⁻¹¹ V/cm²) in order to compare them with the corresponding V_G values. As listed in Table I, DMDCNQI has a much larger number of traps (qN_GkT_G/C = 45000 V) than Cy-NDI (40 V), though the distribution width T_G = 250 K is smaller than 1000 K in Cy-NDI.

In the small E_A region, a steep increase is found in N(E) for both compounds. The increase looks like a tail state; particularly, the tail states appear near E_A = 0 V in Cy-NDI.²⁰ However, DMDCNQI does not have any data below 0.16 eV in Fig. 4(b) because E_A does not drop lower than this E_A even at high V_G. Consequently, N(E) (open circles) tends to diverge at this E_A (Fig. 3(c)). This may be attributed to the existence of substantial shallow traps near the conduction band edge. The tail state has been established in a-Si transistors, where a double exponential distribution of the trap DOS has been assumed.²⁹ However, it is difficult to precisely determine the tail states in the present case, particularly when there are a comparatively large number of traps as in the case of DMDCNQI.

Instead, we obtain Q_T by integrating Eq. (1),

where a step function with the T = 0 K distribution is assumed. If we approximate Q_T = CV_G, we attain the relation,

When the V_G axis in Fig. 4(b) is plotted logarithmically, we obtain approximately a straight line as shown in Fig. 4(d). Here, we use V_G – V_ON for the horizontal axis. In the interface approximation, the V_G intercept directly affords the total number of the trapped states qN_GkT_G/C (V) below E_C. Additionally, T_G is obtained from the slope.

Figure 4(d) seems to be composed of two lines, implying a double exponential distribution. As listed in Table I, Cy-NDI has deep states that have about the same parameters as obtained from Lang's plot (Fig. 3(c)), but additional tail states are observed. The tail-state parameters in DMDCNQI are similar to the trap DOS extracted from Lang's method (Fig. 3(c)), but additional deep states, consisting of a few points, are obtained. The log V_G plot provides basically consistent results with Lang's method, but more precisely reveals the presence of tail and deep states. Moreover, the drawback of the conventional method partly comes from the differentiation process, which enlarges the scatter of the results, but more precise analysis is possible from the log V_G plot. It should be noted that the clear physical meaning of the log V_G plot is based on the interface approximation.

Largely different variable T characteristics of Cy-NDI and DMDCNQI lead to the contrasting trap DOS. There is a close relationship between the T dependence of transistor characteristics and the trap DOS. Previously, the trap DOS has been extracted from the variable T characteristics through the E_A dependence of Q_T at various V_G. By following the process inversely, we can obtain T-dependent transistor characteristics from the trap DOS. The total gate-induced charge is distributed to Q_T and the free charge Q_F,

where Q_S = CV_ON comes from the surface charge. N_C is the intrinsic DOS of the free carriers at E_C.

If N_T, T_G, N_C, E_C – E_F, and Q_S are taken as the fitting parameters, E_A vs. V_G is specifically determined, so Q_F at various V_G is obtained. The parameters are estimated by comparing the simulation and the experiment. Figure 5(a) shows the simulated E_A vs. V_G plot as well as Q_T and Q_F, derived from the Cy-NDI parameters listed in Table II. As shown in Fig. 5(a), E_A vs. V_G depends on T, but the T-dependence is very small. This is because Q_T is independent of T, while Q_F is T dependent. The sum of Q_T and Q_F is always equals to the total induced charge Q_TOT = CV_G, and the distribution ratio changes depending on T. As T decreases, induced carriers are less activated into Q_F, so Q_F decreases, and Q_T increases.

Next, I_D is calculated from Q_F by the gradual channel approximation assuming that only Q_F contributes to I_D.⁵⁵ The induced free charge Q_F depends on the position along the channel (y), which is given by C(V_G – V(y)). Then, I_D is

where W is the channel width, F_y is the electric field at y, and μ₀ is the intrinsic mobility. By substituting the field F_y = –dV/dy, I_D can be rewritten as

The final expression for I_D can be obtained by integrating Eq. (10) from y = 0 to L (from the source to the drain):

Finally, the transfer characteristics is calculated from Eq. (11), and also the output characteristics is reproduced as shown in Fig. 5(b).

Variable T characteristics of Cy-NDI and DMDCNQI are numerically calculated by the simulation method assuming a single exponential trap DOS, and compared with the experiments as depicted in Figs. 2 and 3. As shown in Eq. (8), Q_T is determined from N_G and T_G, while Q_F from N_C, and Q_S from V_ON. These parameters are estimated by fitting the calculated characteristics to the experiments. Each parameter has different influence on the T-dependent transfer characteristics. N_G and T_G used in the simulation (Table II) are the same as those obtained from Lang's method (Table I). The T-dependence of the off current is obtained as

but this is hindered in the present experiments because the off current is lower than the noise level of the measurement system (Figs. 2(a) and 3(a)). Because E_C – E_F affords practically no influence to the on current, it is determined so as to agree with the activation energy of the off current. All the used parameters for Cy-NDI and DMDCNQI are listed in Table II. N_C and N_GkT_G are converted from the dimension of cm⁻² to V by multiplying q/C (1.17 × 10⁻¹¹ V/cm²) in order to compare them with the corresponding V_G values.

For Cy-NDI, the simulated T-dependent transfer curves (Figs. 2(d) and 2(e)) reproduce well the experimental transfer curves (Fig. 2(a)) as well as the observed V_T shift (Fig. 2(b)). Only the T-dependence in the off region is not observed in the actual experiment because I_D is lower than the noise level. To further investigate the simulated characteristics, Fig. 2(f) shows the Arrhenius plot obtained from the simulated characteristics. The extracted E_A vs. V_G (solid curves in Fig. 4(b)) is in good agreement with the experimental E_A vs. V_G (open circles), which is also approximately the same as the initially calculated E_A vs. V_G (Fig. 5(a)). However, the Meyer-Neldel rule does not hold in the simulated Arrhenius plot (Fig. 2(e)). The simulated characteristics affords more parallel Arrhenius plot, and the extrapolated crossing point appears more close to 1/T ∼ 0, or sometimes at a negative 1/T. As a consequence, T_MN from the simulation is, if any, very large. This is rather surprising because the following E_A vs V_G plot is not much different. This suggests the phenomenological nature of the Meyer-Neldel rule. The trap DOS distribution (close symbols in Fig. 4(c)) determined by applying Lang's analysis (Eq. (4)) to the simulated characteristics (Figs. 2(d) and 2(e)) is consistent with the experiment (open symbols in Fig. 4(c)). The straight line corresponds to the input parameters, N_G and T_G, and the trap DOS obtained from the simulated transfer characteristics agrees with the used N_G and T_G values. The estimated T_G = 1000 K and the total amount of traps qN_GkT_G/C = 40 V are exactly the same as the starting values. This demonstrates that the simulation is self-consistent. It is noteworthy that the trap amount (40 V) in as small as the free carrier DOS N_C (50 V). Interestingly, in the trap DOS obtained from the simulated characteristics, a steep increase, suggesting the tail states, is observed near the conduction band, even though we have assumed only a single exponential DOS function. This tail state is attributed to the flat E_A region which appears when E_A approaches to zero. Note that Q_T = CV_G is not a good approximation in this region, so that Eq. (4) is not fully justified.

The simulated characteristics of DMDCNQI (Fig. 3(d)) shows good agreement with the experiment as well (Fig. 3(a)), though the V_T shift is slightly different. Figures 3(b) and 3(e) show the I_D^1/2 vs. V_G plots of the experiment and the simulation. In Fig. 3(b), the slope is inclined and becomes small above V_G = 10 V. A straight line comes from the simple saturation regime, and actually the simulated result in Fig. 3(e) shows a straight line. This gives rise to a different V_T shift from the experiment (Fig. 3(b)). It has been shown that DMDCNQI has the deep states together with the tail states (Fig. 4(c)), so the different V_T is attributed to the existence of the deep states and the simulation assuming a single exponential distribution leads to the different V_T shift. The Arrhenius plot of the simulated I_D (Fig. 3(e)) shows excellent similarity to the experiment (Fig. 3(b)), but the Meyer-Neldel rule does not hold. The E_A vs. V_G plot (Fig. 4(b)) is mostly reproduced but slightly deviated near V_ON. At the initial fitting of the E_A vs. V_G curve to the experiment, a substantial amount of traps N_GkT_G, as large as 45000 V, is assumed in order to reproduce the high E_A in the high V_G region. Because this value is about ten times larger than the molecular density of DMDCNQI (4000 V), the deviation should be ascribed to the overestimation of N_GkT_G. It is essential that, even when we adopt the log V_G plot (Fig. 4(d)), the results obtained from the Arrhenius plot do not contain any information on the states below the smallest E_A at the largest measured V_G, which is as large as 0.16 eV in DMDCNQI. It is also likely that the deviation stems from the existence of deep states suggested from the log V_G plot in Fig. 4(d). The trap DOS obtained from the simulation in Fig. 3(d) is indicated as close triangles in Fig. 4(c) together with the straight line corresponding to the initially assumed DOS function. The simulation is approximately consistent with the experiment, where T_G is estimated at 250 K.

As Eqs. (11) and (12) show, in the Q_T >> Q_F limit, I_D is proportional to μ₀N_C, so we can determine μ₀N_C from the magnitude of I_D, but μ₀ and N_C are not independent. In Cy-NDI, however, Q_T and Q_F are comparable because N_GkT_G (40 V) and N_C (50 V) are approximately the same. In such a case, N_C changes the temperature dependence, and we can uniquely determine qN_C/C = 50 V so as to reproduce the observed characteristics. This leads to the intrinsic mobility (μ₀) of 0.5 cm²/Vs. In DMDCNQI, we assume the same qN_C/C = 50 V, but only μ₀N_C is meaningful (Table II).

In both Cy-NDI and DMDCNQI, T_MN = 2000 K and 500 K are twice as large as T_G = 1000 K and 250 K, respectively (Table II). Although T_MN and T_G are not identical, the phenomenological parameter T_MN seems to have a close relation to T_G. Large T_G means a flat line in the N(E) vs. E_A plot, so that large T_G is connected with a relatively large amount of deep traps.

The subthreshold swing (S) is conventionally represented as follows,^23,56–58

where N_T and N_F are the numbers of the trapped charge and the free charge, respectively, which should correspond to N_GkT_G and N_C in the simulation. Since N_T is much larger than N_F in the actual transistors, the S value is proportional to the ratio of N_T and N_F. In the present experiments, however, a different tendency is observed. Even though DMDCNQI has almost 1000 times larger N_T/N_F ratio, DMDCNQI gives a smaller S value of 1.3 V than 2.3 V of Cy-NDI (Table II). In order to show this coming from the different T_G, we have conducted additional simulation by changing N_GkT_G in Fig. 6(a) and T_G in Fig. 6(b), and the S value is investigated. Figure 6(a) shows simulations by changing N_GkT_G/N_C with a constant T_G = 500 K. The resulting parameters are listed in Table III. With increasing N_GkT_G, the S value increases approximately proportional to N_GkT_G/N_C, whereas the mobility μ decreases. This agrees with the conventional prediction and Eq. (13). However, the S value also largely depends on T_G as shown in Fig. 6(b). By increasing T_G with keeping N_GkT_G constant, the S value remarkably increases, though μ shows smaller change than Fig. 6(a). This is because the deep states afford more profound effect on increasing the S value than the tail states. μ changes drastically by changing N_GkT_G, but changes very little by changing T_G when N_GkT_G is kept, where mainly the turn-on point in the subthreshold region shifts to the right. Therefore, it is important to consider not only N_T/N_F but also T_G for estimating the S values. In the present case, only four-time difference of T_G is more important than the three-order difference of N_GkT_G, and Cy-NDI with large T_G and a small number of traps exhibits a large S value.

Variable T characteristics of Cy-NDI implies, particularly from the E_A vs. V_G plot, that E_A approaches to zero when the trap number is small. In that case, it is possible that E_A becomes negative at even higher V_G, so band transport is realized. Band transport is one of the most important challenges in organic transistors, so it has been intensively investigated in single-crystal transistors.^9–12 

The simulation model is based on the multiple trapping and release theory,^13,14 which assumes band transport in the conduction band, with the additional existence of the mid-gap trapped states. Alternatively, we can regard E_C as the mobility edge. In principle, when all the trapped states are filled by the induced charge, further induced charge becomes free carriers moving like the band carriers. Therefore, in this section, we investigate the band transport from a different aspect, where E_A of the transconductance becomes negative.

When we put E_A = 0 in Eq. (8), we obtain

This equation gives the critical V_G above which E_A becomes negative. The parameters listed in Table II for Cy-NDI afford V_G = 130 V. Since this is still large, we show the simulated characteristics in Fig. 7 assuming qN_c/C = 5 V (one tenth of the actual one) and qQ_S/C = 0 V. As expected, E_A becomes negative above V_G = 45 V (Fig. 7(a)). The resulting simulated characteristics shown in Fig. 7(b) crosses at V_G = 58 V, above which I_D increases as lowering T. The critical V_G value of E_A is not identical to the crossing point of the I_D vs. V_G curve, because E_A vs. V_G is T-dependent, and the E_A zero point gradually changes when T changes.

We next investigate how the different trap DOS affects the critical V_G. When N_GkT_G is taken half (20 V) with keeping the same T_G, the E_A zero point moves from V_G = 45 V to 25 V (Fig. 7(a)), as expected, and the I_D crossing point moves from V_G = 58 V to 35 V (Fig. 7(c)). Figure 7(d) shows the simulation with half T_G and the same N_GkT_G, where the critical V_G is the same. From this, we conclude that the critical V_G of E_A corresponds to the sum of the trap number (N_GkT_G) below E_C and the free carrier DOS (N_C) at E_C. Negative E_A in I_D is observed somewhat above this V_G. In the transistors having small number of traps, band transport occurs when V_G sufficiently exceeds the total number of traps. Negative temperature dependence of mobility has been observed in the large V_G region in the actual organic transistors.^12,59–63

T-dependent transistor characteristics are investigated for n-channel organic transistors using Cy-NDI and DMDCNQI. The trap DOS is determined from the Arrhenius plot of the transconductance, where the E_A vs. log V_G plot is useful rather than the conventional differentiation method. Cy-NDI and DMDCNQI show contrasting trap DOS; Cy-NDI has a small number of deep states with large T_G, whereas DMDCNQI has a much large number of tail states with small T_G. Although T_G and T_MN are highly correlated, the obtained T_MN is twice larger than T_G. The variable T characteristics are remarkably well reproduced for Cy-NDI and DMDCNQI in the simulation, and we have investigated how the trap DOS affects the T-dependent transfer characteristics. The obtained N_GkT_G and T_G from the fitting parameters are consistent with the trap DOS determined directly from the experiment. The Arrhenius plot of the simulated transconductance shows good similarity to the experiment, so simulation is self-consistent. The simulation, however, satisfies the Meyer-Neldel rule at relatively small or negative 1/T_MN, suggesting only the phenomenological nature of the Meyer-Neldel rule. The simulation also reveals that the S value is not simply proportional to the trap number, but also largely depends on T_G. It is shown that the variable T characteristics is uniquely calculated from the trap DOS by simulation, which provides a powerful tool to investigate the trap-limited charge transport. As an example, it is demonstrated that band transport is realized within the present model. The model predicts that when V_G is larger than the sum of the trapped and free charge states located below E_C, E_A becomes negative, and I_D increases with lowering T.