We have investigated trap density of states (trap DOS) in n-channel organic field-effect transistors based on *N*,*N* ’-bis(cyclohexyl)naphthalene diimide (Cy-NDI) and dimethyldicyanoquinonediimine (DMDCNQI). A new method is proposed to extract trap DOS from the Arrhenius plot of the temperature-dependent transconductance. Double exponential trap DOS are observed, in which Cy-NDI has considerable deep states, by contrast, DMDCNQI has substantial tail states. In addition, numerical simulation of the transistor characteristics has been conducted by assuming an exponential trap distribution and the interface approximation. Temperature dependence of transfer characteristics are well reproduced only using several parameters, and the trap DOS obtained from the simulated characteristics are in good agreement with the assumed trap DOS, indicating that our analysis is self-consistent. Although the experimentally obtained Meyer-Neldel temperature is related to the trap distribution width, the simulation satisfies the Meyer-Neldel rule only very phenomenologically. The simulation also reveals that the subthreshold swing is not always a good indicator of the total trap amount, because it also largely depends on the trap distribution width. Finally, band transport is explored from the simulation having a small number of traps. A crossing point of the transfer curves and negative activation energy above a certain gate voltage are observed in the simulated characteristics, where the critical *V*_{G} above which band transport is realized is determined by the sum of the trapped and free charge states below the conduction band edge.

## I. INTRODUCTION

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 cm^{2}/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 cm^{2}/Vs, on/off ratio = 2 × 10^{6}, *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.

## II. EXPERIMENTAL

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/cm^{2}) 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)).

## III. VARIABLE TEMPERATURE CHARACTERISTICS

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 cm^{2}/Vs and on/off = 10^{5}) than DMDCNQI (*μ* = 0.0044 cm^{2}/Vs and on/off = 10^{4}) 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} = d*I*_{D}/d*V*_{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}

## IV. TRAP DOS

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 × 10^{21} cm^{−3} for Cy-NDI and 2.12 × 10^{21} 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 by^{33}

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*_{G}*kT*_{G} is converted from the dimension of cm^{−2} to V by multiplying *q*/*C* (1.17 × 10^{−11} V/cm^{2}) 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*_{G}*kT*_{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.

. | . | Lang . | log V_{G} plot
. | ||
---|---|---|---|---|---|

. | . | T_{G} (K)
. | qN_{G}kT_{G}/C (V)
. | T_{G} (K)
. | qN_{G}kT_{G}/C (V)
. |

Cy-NDI | deep | 1000 | 40 | 1000 | 32 |

tail | 310 | 76 | |||

DMDCNQI | deep | 880 | 100 | ||

tail | 250 | 4.5 × 10^{4} | 225 | 1.2 × 10^{5} |

. | . | Lang . | log V_{G} plot
. | ||
---|---|---|---|---|---|

. | . | T_{G} (K)
. | qN_{G}kT_{G}/C (V)
. | T_{G} (K)
. | qN_{G}kT_{G}/C (V)
. |

Cy-NDI | deep | 1000 | 40 | 1000 | 32 |

tail | 310 | 76 | |||

DMDCNQI | deep | 880 | 100 | ||

tail | 250 | 4.5 × 10^{4} | 225 | 1.2 × 10^{5} |

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*_{G}*kT*_{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.

## V. SIMULATION

### A. Model

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.

Compound . | T_{MN} (K)
. | N_{G} (/cm^{2} eV)
. | T_{G} (K)
. | N_{G}kT_{G} (/cm^{2})
. | E_{C} – E_{F} (eV)
. | μ (cm^{2}/Vs)
. | S (V/decade)
. | N_{T}/N_{F}^{b}
. |
---|---|---|---|---|---|---|---|---|

Cy-NDI (OTS) | 2000 | 3.8 × 10^{13} | 1000 | 3.3 × 10^{12} (40 V)^{c} | 0.32 | 0.5 | 2.2 | 0.8 |

DMDCNQI (OTMS) | 500 | 1.7 × 10^{17} | 250 | 3.7 × 10^{15} (45000 V)^{c} | 0.37 | 4.7 | 1.3 | 900 |

Compound . | T_{MN} (K)
. | N_{G} (/cm^{2} eV)
. | T_{G} (K)
. | N_{G}kT_{G} (/cm^{2})
. | E_{C} – E_{F} (eV)
. | μ (cm^{2}/Vs)
. | S (V/decade)
. | N_{T}/N_{F}^{b}
. |
---|---|---|---|---|---|---|---|---|

Cy-NDI (OTS) | 2000 | 3.8 × 10^{13} | 1000 | 3.3 × 10^{12} (40 V)^{c} | 0.32 | 0.5 | 2.2 | 0.8 |

DMDCNQI (OTMS) | 500 | 1.7 × 10^{17} | 250 | 3.7 × 10^{15} (45000 V)^{c} | 0.37 | 4.7 | 1.3 | 900 |

^{a}

*N*_{C} is 4.1 × 10^{12} /cm^{2} (50 V)^{c} both for Cy-NDI and DMDCNQI.

^{b}

*N*_{T}/*N*_{F} = *N*_{G}*kT*_{G}/*N*_{C}

^{c}

Using *q*/*C* = 1.17 × 10^{−11} V/cm^{2}

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 *μ*_{0} is the intrinsic mobility. By substituting the field *F*_{y} = –d*V*/d*y*, *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):

### B. Investigation of trap-limited charge transport

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*_{G}*kT*_{G} are converted from the dimension of cm^{−2} to V by multiplying *q*/*C* (1.17 × 10^{−11} V/cm^{2}) 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*_{G}*kT*_{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*_{G}*kT*_{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*_{G}*kT*_{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 *μ*_{0}*N*_{C}, so we can determine *μ*_{0}*N*_{C} from the magnitude of *I*_{D}, but *μ*_{0} and *N*_{C} are not independent. In Cy-NDI, however, *Q*_{T} and *Q*_{F} are comparable because *N*_{G}*kT*_{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 (*μ*_{0}) of 0.5 cm^{2}/Vs. In DMDCNQI, we assume the same *qN*_{C}/*C* = 50 V, but only *μ*_{0}*N*_{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*_{G}*kT*_{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*_{G}*kT*_{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*_{G}*kT*_{G}/*N*_{C} with a constant *T*_{G} = 500 K. The resulting parameters are listed in Table III. With increasing *N*_{G}*kT*_{G}, the *S* value increases approximately proportional to *N*_{G}*kT*_{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*_{G}*kT*_{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*_{G}*kT*_{G}, but changes very little by changing *T*_{G} when *N*_{G}*kT*_{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*_{G}*kT*_{G}, and Cy-NDI with large *T*_{G} and a small number of traps exhibits a large *S* value.

Parameters . | N_{G} × 1
. | N_{G} × 5
. | N_{G} × 10
. | T_{G} × 0.5
. | T_{G} × 1
. | T_{G} × 2
. |
---|---|---|---|---|---|---|

N_{G}kT_{G}/N_{C} | 1 | 5 | 10 | 1 | 1 | 1 |

S | 1.9 | 5 | 7.2 | 1 | 1.9 | 4.8 |

μ (cm^{2}/Vs) | 0.04 | 0.006 | 0.002 | 0.05 | 0.04 | 0.03 |

Parameters . | N_{G} × 1
. | N_{G} × 5
. | N_{G} × 10
. | T_{G} × 0.5
. | T_{G} × 1
. | T_{G} × 2
. |
---|---|---|---|---|---|---|

N_{G}kT_{G}/N_{C} | 1 | 5 | 10 | 1 | 1 | 1 |

S | 1.9 | 5 | 7.2 | 1 | 1.9 | 4.8 |

μ (cm^{2}/Vs) | 0.04 | 0.006 | 0.002 | 0.05 | 0.04 | 0.03 |

^{a}

The default values are *N*_{C} = 4.1 × 10^{12} /cm^{2} (50 V), *N*_{G}*kT*_{G} = 4.1 × 10^{12} /cm^{2} (50 V), and *T*_{G} = 500 K.

### C. Band transport

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*_{G}*kT*_{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*_{G}*kT*_{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*_{G}*kT*_{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}

## VI. CONCLUSION

*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*_{G}*kT*_{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*.