Analysing organic transistors based on interface approximation

For the basic understanding of operation in organic transistors, temperature (T) dependence of transistor characteristics has been investigated on the basis of the multiple trapping and release model.^1–9 Historically, it dates back to Horowitz's work, in which the trap density of states (DOS) was obtained by analysing T-dependent transfer characteristics of dihexylsexithiophene transistors.^1–3 When we investigate organic transistors following the conventional amorphous silicon (a-Si) transistors (Fig. 1),^10–17 depth of the accumulation layer (x) is smaller than the molecular length (1–3 nm), and it is appropriate to keep x constant; this is called interface approximation. Charge on the second organic layer is estimated to be as small as one tenth of the first-layer charge at the gate voltage V_G of 100 V, though it amounts to half at V_G = 10 V.³ Accordingly, the interface approximation is a good approximation at large V_G. Photoelectron spectroscopic investigation has suggested that high-purity organic semiconductors do not exhibit such band bending as conventional inorganic semiconductors, but an abrupt energy shift occurs at the interface.¹⁸ In addition, it has been suggested that in thin-film transistors, where the bulk potential is not fixed in contrast to the conventional metal-oxide-semiconductor field-effect transistors (MOS-FET), the band bending is considerably reduced due to the global energy-level shift.¹⁹ If the potential is flat, charge exists only at the interface, and the interface model applies perfectly.

To estimate trap DOS in organic transistors, Lang's method has been widely used,^20,21 where the trap DOS is extracted from the Arrhenius plot of the drain current I_D.^7,22,23 This method is based on the interface approximation, but there are many other methods developed for a-Si thin-film transistors, which assume variable x.^10–17 Kalb et al. have analysed organic transistors based on these methods and proved that various methods afford approximately consistent exponential trap distribution.^24–28 Although Lang's analysis has been extensively used due to its simpleness for organic transistors, the corresponding formulas of transistor operation have not been investigated very well.²⁹ 

In the present work, we investigate a method to analyse T dependence of transistor characteristics thoroughly under the interface approximation. Some of similar formulas have been given in a previous paper,²⁹ but not investigated extensively. Although the interface approximation is a good approximation only at large V_G, we have found that this model affords a clear physical meaning which is valid in the overall V_G region. In order to examine the validity of the present model, we investigate T dependence of hexamethylenetetrathiafulvalene (HMTTF) and dibenzotetrathiafulvalene (DBTTF) transistors. These organic semiconductors are chosen because the T-dependent characteristics have not been studied yet, though these materials exhibit excellent transistor performance. The reported mobility is μ = 11 cm²/Vs for HMTTF crystals and μ = 6.9 cm²/Vs for thin films,^30,31 as well as μ = 0.55 cm²/Vs for DBTTF films.^32–35 The transistor characteristics are, however, comparatively susceptible to surface treatments owing to the small ionization potentials. Ordinary organic semiconductors like pentacene have been investigated extensively,^24–27 and even band-like transport has been observed in the single crystals, where the mobility increases with lowering the temperature.^36–41 We investigate three different surface treatments, on which particularly the DBTTF transistors have shown large dependence. According to Lang's method, the trap DOS is estimated, and based on the trap DOS, the T-dependent transistor characteristics as well as the V_G dependence of the activation enregy E_a are theoretically reproduced. Recently, we have carried out similar analysis for n-channel transistors based on naphthalenediimine (NDI) and dicyanoquinonediimine (DCNQI),⁴² and extracted a critical V_G, above which band transport is expected. In the course of the present study, we have found that it is important to exclude T dependence of the on-set voltage V_on, which is a factor somewhat beyond the conventional framework. We have also investigated how the presence of a discrete trap level modifies the transistor characteristics. Finally, we investigate a method to evaluate contact resistance only from the transfer characteristics.

We consider a thin-film transistor with bottom-gate and top-contact geometry. In a-Si transistors, the T-dependent transistor properties have been explained assuming the existence of mid-gap localized states. The density of the localized states are conventionally represented by an exponential trap distribution model.¹⁴ 

As shown in Fig. 1, E is measured from the conduction band edge, E_c, which corresponds to the mobility edge in a-Si. Figure 1 is for a n-type transistor, but the formalism is entirely the same for p-type transistors. T_G represents the distribution width of the exponential function, and N_G means the trap density N(E) at E_c.

When the gate voltage V_G is applied, charges Q = CV_G are accumulated at the interface (Fig. 1), where C is the capacitance of the gate dielectric per unit area. The free charges, Q_f, excited up to E_c (Fig. 1) by an activation energy of E_a, are given by

where N_c is the intrinsic DOS at E_c, and q is the elemental charge.

In the standard MOS transistor, the depth of the accumulation layer x changes depending on the surface potential φ = E_c − E_F − E_a (Fig. 1),⁴³ 

where ɛ is the dielectric constant of the semiconductor. When the typical volume of an organic molecule, 0.3 nm³, is used,^30,45,46 the three-dimensional number density of organic molecule N_a is 3.3 × 10²¹ cm⁻³, and x is obtained to be in the order of 0.1 nm. Even when the carrier number is one hundredth of this value, namely corresponding to the actual carrier density at V_G = 50 V, x is estimated to be 1 nm. Since this x is still smaller than the length of a molecule (several nm), x should be considered to be constant (interface approximation). This approximation is grounded on the relatively largel size of an organic molecule, and holds more appropriately at large carrier density realized at large V_G.

In the conventional model of a field-effect transistor,^10–17 x changes depending on V_G,⁴³ and the interface charge is proportional to Q^1/2. However, in the interface approximation, x does not change, and the interface charge is directly proportional to Q. In the thin-film model with floating bulk potential, the band bending is considerably reduced due to the global energy-level shift, particularly near the drain electrode and in the saturated region.¹⁹ To investigate the potential distribution precisely, we have to solve Poisson's equation,⁴⁴ but from Eq. (3), the resulting comparatively small φ leads to small x, and the interface model becomes more appropriate.

By integrating Eq. (1) up to E_a, the total trapped charge is obtained.

Here a step-like distribution as shown in Fig. 1, or the T = 0 K distribution, is assumed. The results obtained by integrating the Fermi distribution have been given previously for a-Si transistors.¹¹ We have carried out the following analysis by using the Fermi distribution as well, which slightly modifies the shape of the transfer characteristics particularly in the large V_G region, but does not change the essential results. Since the step approximation is sufficient for our purpose, we only report the results of the step approximation in the present paper.

From Eqs. (2) and (4), the whole field-induced carriers are represented by a sum of the trapped and free charges.

In order to generally estimate transistor characteristics, E_a is reduced starting from E_c − E_F, and Q_t and Q_f are calculated. When the accumulated carriers reach to CV_G, this E_a value is recognized as the potential at V_G. Then E_a is obtained as a function of V_G (Fig. 2(a)), and Q_f is estimated as a function of V_G from Eq. (2) (Fig. 2(a)). Usually, only Q_f is considered to contribute to I_D. This corresponds to the multiple trapping and release model; instead of considering each hopping process, we consider that Q_f represents the statistical sum of the transporting carrier number, and Q_t is the trapped carrier number. Since I_D is basically proportional to Q_f, we can obtain the transistor characteristics.

In the actual situation, Q_t is much larger than Q_f, and we may assume Q_t = CV_G instead of Eq. (5). Then E_a is obtained from Eq. (4).

The calculated E_a decreases rapidly starting from E_c − E_F when the transistor turns on, and approaches to zero at large V_G (Fig. 2(a)). Later, we will show that experimentally obtained E_a also follows this logarithmic dependence (see Fig. 6). Putting this equation in Eq. (2), we obtain,

In order to calculate I_D precisely, the gradual channel approximation is applied.⁴⁷ The channel potential V(y) changes depending on the position, y, from the source y = 0 to the drain y = L, so the accumulated charge at y is given by C(V_G − V(y)). The drain current is estimated from σ = neμ, where

Since ne = Q_f,

Integration of this formula from y = 0 to L gives the expression of the drain current.

Here, the left-hand side affords simply I_DL, and the integration in the right-hand side is transformed to the integration from V = V_D to 0. By integrating Eq. (7) in Eq. (10), we obtain,

This is a formula for T-dependent transistor characteristics under the interface approximation. When I_D is plotted against V_D, this equation represents the standard output characteristics. This equation corresponds to the usual equation of trap-free transistor characteristics.⁴⁷ 

An equation similar to Eq. (11) has been obtained for the variable x case for a-Si transistors,¹¹ where the index is 2T_G/T instead of T_G/T. This is the reason that in Kalb's analysis of trap DOS using various methods,²⁶ only Lang's method has afforded apparently twice larger T_G.

In the saturated region, V_D increases up to V_G, and the second term in the final factor in Eq. (11) vanishes to afford,

This equation gives transfer characteristics in the saturated region.²⁹ If T is not much different from T_G, T_G/T is approximately one, and I_D is proportional to V_G². Then a straight line is obtained in the usual I_D^1/2 vs. V_G plot. The factor T/(T_G + T) affords 1/2, and I_D is proportional to (1/2)V_G² similarly to the conventional formula of trap-free transfer characteristics in the saturated region derived from Eq. (12).

At low temperatures, V_G in Eq. (13) follows a larger power law, and a downward curvature is observed. This is the reason that a downward curvature is obtained for T_G = 600 K in Fig. 2(b). T_G changes the curvature as shown in Fig. 2(b), but N_G changes only the slope as expected from Eq. (13). The deviation from the straight line in the I_D^1/2 vs. V_G plot has been usually attributed to the traps, but it should be ascribed not to the amount of the traps N_G, but to the presence of deep states. Similarly to Eq. (4), the total number of the trap states below E_c is obtained by integrating Eq. (1).

Without the presence of traps, I_D is proportional to (1/2)V_G², but traps diminish this by N_c/N_GkT_G times around T ∼ T_G, where N_c/N_GkT_G is the ratio of the free and trapped carriers. Roughly speaking, Eqs. (11) and (13) mean that not only the transfer characteristics but also the output characteristics are reduced by a factor of N_c/N_GkT_G. The results based on Eq. (13) are plotted in Fig. 2(b), which are in good agreement with the numerical calculation based on Eq. (3). Accordingly, we do not need to distinguish the results from Eq. (13) from the numerical calculation from Eq. (3). However, when N_GkT_G and N_c are comparable, Eq. (13) overestimates I_D about twice. In order to investigate the validity of the above model, next we will analyse characteristics of HMTTF and DBTTF transistors.

HMTTF and DBTTF were obtained according to the published procedure.^48,49 The materials were purified by sublimation at 230 °C for HMTTF and 200 °C for DBTTF under the vacuum of 10⁻⁴ Pa. Highly doped n-type silicon wafers with thermally grown silicon dioxide layers of 300 nm thickness were treated with polystyrene (PS) and hexamethyldisilazane (HMDS).^50,51 In addition, bare SiO₂ surface was investigated; in order to investigate particularly OH enhanced substrate, the substrate was used after the ultraviolet irradiation (Technovision model 208). The capacitance of the gate dielectric was C = 13.7 nF/cm². The active layers of HMTTF and DBTTF with the approximate thickness of 100 nm were fabricated by thermal evaporation under the vacuum of 10⁻³ Pa. In order to form the top-contact transistors, Au was evaporated through a metal mask (channel length L = 100 μm and channel width W = 1000 μm). These six kinds of transistors were cooled from 290 K to 200 K with a low-temperature micro prober system (Riko International), and the transfer characteristics were measured with an interval of 10 K.

Left-column graphs in Figs. 3 and 4 show T-dependent transfer characteristics of HMTTF and DBTTF transistors with various surface treatments. For the PS-treated HMTTF transistor, the mobility estimated from the gradient of the I_D^1/2 vs. V_G plot is μ = 0.73 cm²/Vs and the on-off ratio is 3 × 10⁵ at 290 K. The performance gradually decreases in the order of PS > HMDS > bare, but the difference is not large for HMTTF. For DBTTF transistors (Fig. 4), comparatively large change is observed, where the performance decreases in the same order. In particular, the DBTTF-bare transistor suffers from very large influence (Fig. 4(c)), though the HMTTF-bare transistor is not largely different from the other two.

With decreasing the temperature, the drain current I_D decreases, and the threshold voltage V_T shifts to the negative direction. When I_D is plotted with respect to T inverse for respective V_G, approximately linear dependence is obtained as shown in Fig. 5. Although the lines are not strictly straight over a wide T range, we estimate E_a from the average slope between 290 K and 200 K, which is plotted in Fig. 6 as a function of V_G. The E_a value is largest at the turn-on voltage, and gradually decreases with increasing V_G, which agrees with the logarithmic function represented by Eq. (6). If we may assume that E_a obtained from the T dependence is equal to the gap E_a = E_c − E_F − φ between E_F and E_c at the interface (Fig. 1), the decreasing E_a in Fig. 6 represents the manner how E_c at the interface approaches to E_F as a function of V_G (Fig. 1), where the decreasing E_a is due to the increasing band bending. At the left end, E_a is large, suggesting large E_a = E_c − E_F in the flat-band (off) state. In general, PS-treated transistors show small E_a, indicating small T-dependence over all V_G range, where the T-dependence is particularly small in the large V_G region.

All Arrhenius plots seem to pass a single point, and the Meyer-Neldel rule is satisfied.⁴ However, since the Arrhenius plots are not strictly straight over a wide T range, the crossing point, called the Meyer-Neldel temperature, T_MN, depends on the used T region. We have previously shown that the theoretically calculated transistor characteristics do not afford a straight line as well.⁴² The Meyer-Neldel rule has been attributed to an exponential DOS,^52,53 and T_MN is related to the DOS distribution.⁵² It has been, however, argued that the Meyer-Neldel rule is derived more generally from multiexcitation process.^54,55 Although the Meyer-Neldel rule has been universally observed in organic transistors,^4–8 it seems to have only a phenomenological meaning, and T_MN does not have a clear physical meaning.⁴² 

V_G is related to the total accumulated charge as Q = CV_G, but in the subthreshold region, most charge enters the trap state Q_t. When E_a changes, the corresponding V_G variation is equal to the carrier number trapped within the energy interval. According to Lang,^20,21 the trap DOS is estimated from,

When dE_a/dV_G is calculated differentiating the E_a plot in Fig. 6, and its inverse is plotted with respect to E_a, we obtain E_a dependence of N(E) as shown in Fig. 7. E_a is the energy measured downward from E_c (Fig. 1), and E_a = 0 corresponds to E = E_c. Since N(E) is plotted logarithmically, the approximate straight dependence implies that N(E) follows an exponential relation as represented by Eq. (1). Similar exponential DOS has been obtained not only from transistor characteristics but also from space-charge-limited-current spectroscopy and Kelvin probe microscopy.^56–58 Note that this method is based on the interface approximation because it simply assumes Q = CV_G.

In Fig. 7, the low energy part shows a considerable increase, suggesting the presence of tail states. In a-Si transistors, the existence of tail states is important and double exponential distribution has been assumed.¹¹ However, the steep rise in the N(E) plot (Fig. 7) in small E_a comes from the flat E_a dependence in the E_a plot at large V_G (Fig. 6).

In Fig. 6, the HMTTF-PS transistor shows nearly zero E_a at large V_G, but E_a does not drop to zero for other transistors. If E_a in the E_a plot approaches to a finite E_a value and becomes flat, the N(E) plot does not contain any data below this E_a, and N(E) diverges toward an finite E_a. For example, the N(E) plot for the DBTTF-PS transistor (Fig. 7(a)) does not contain any point below 0.08 eV. This is an evidence of a relatively large number of shallow traps, but it is practically difficult to estimate the exact amount of the tail states from the N(E) plot (Fig. 7).

The obtained N(E) does not afford a very straight line particularly when the trap number is small as in the case of PS-treated transistors. Nonetheless, approximate straight lines are assumed as depicted in Fig. 7, and the corresponding N_G and T_G values are extracted as shown in Table I. The present two-dimensional density N(E) ∼ 10¹⁴ /eV cm² approximately corresponds to the three-dimensional density 10²⁰ /eV cm³ when the accumulation layer thickness is assumed to be 7.5 nm.^24–26 The present values of N_G ∼ 10¹⁴–10¹⁵ /eV cm² are about in the same order as the previous results for pentacene.^23–26 This is also consistent with the previously reported two-dimensional densities.⁶ The trap density of the present TTF transistors is not particularly larger than that of the usual pentacene transistors.

The PS-treated transistors show smaller N(E) than the HMDS-treated transistors over all E_a region (Fig. 7), indicating the existence of smaller numbers of trap states. This is related to small E_a at large V_G in the E_a plot (Fig. 6). Since there is no trap states to be filled, E_c at the interface drops close to E_F. As a result, most field-induced carriers directly enter Q_f. In the Arrhenius plot in Fig. 5, this leads to flat T-dependence at large V_G, and the carrier transport approaches to the band transport.⁴² The small T-dependence at large V_G is also obvious from the raw transfer characteristics particularly for HMTTF-PS in Fig. 2(a).

In Lang's original paper,²¹ they have concluded the presence of a polaron state from the small hump of the N(E) plot. Our plot in Fig. 7 shows small structures that are not obvious in the original E_a plot (Fig. 6). Later we show a method to prove the existence of a polaron state directly comparing the transfer characteristics.

The total number of trap states below E_c is given by N_GkT_G as shown in Eq. (15). When q/C = 1.17 × 10⁻¹¹ V/cm² is multiplied, trap numbers are transformed to the unit of V. The qN_GkT_G/C value corresponds to V_G necessary to induce the same number of carriers. These voltages are listed in Table I. The bare-SiO₂ transistor shows large qN_GkT_G/C, and the PS-treated transistors show small qN_GkT_G/C. The order of qN_GkT_G/C (PS < HMDS < ozone) is well correlated with the order of the transistor performance. In general, the HMTTF transistors show smaller qN_GkT_G/C than the DBTTF transistors.

From the lattice volume of these crystals,^30,45,46 the area of a TTF molecule (0.23 nm² for HMTTF and 0.29 nm² for DBTTF) afford the two-dimensional number density to be 4.3 × 10¹⁴ /cm² for HMTTF and 3.5 × 10¹⁴ /cm² for DBTTF. By multiplying q/C = 1.17 × 10⁻¹¹ V/cm², these numbers give V_G = 4000 V and 5000 V, respectively. These are V_G that are necessary to accumulate one carrier per molecule. These voltages are obviously much above the dielectric breakdown voltages of SiO₂, and to induce carriers in this order, electrolyte-gated transistors with large C are necessary.⁵⁹ The total trap numbers, qN_GkT_G/C ∼ several hundred volts (Table I), are in the order of one tenth of the molecular numbers.

It has been conventionally used that the subthreshold swing (S) is associated with the ratio of the trap number N_t to the free carrier number N_f.^60–63 

Since N_t ≫ N_f, the S value is approximately proportional to N_t/N_f. The S values at room temperature are listed in Table I. When S at room temperature is divided by 2.3kT = 0.060 eV, we can obtain a rough estimate of the trap number N_t/N_f as listed in Table I. N_t/N_f is a good indicator of the total trap number.

We can obtain transistor characteristics inversely from the trap distribution using the theoretical formulas. This is much easier than calculation using simulators,^19,64,65 In the DBTTF transistors (Fig. 4), the onset voltage V_on is shifted by 15, 35, and 80 V for PS, HMDS, and bare transistors, respectively. In contrast, V_on of the HMTTF transistors is close to 0 V (Fig. 3). Such an onset shift is treated by adding the surface charge Q_s to Eq. (5).

The onset voltage V_on = Q_s/C corresponds to V_G at which I_D starts to increase. This is usually smaller than the threshold voltage obtained by extrapolating the I_D^1/2-plot, so we designate this as V_on.

Equation (13) predicts a straight line when log I_D is plotted with respect to log V_G. Figure 8(a) shows such a plot for the HMTTF-PS transistor. Since the curvature at small V_G depends on V_on sensitively, V_on is adjusted so as to give a straight line; this is a usual way to estimate the flat-band potential in a-Si transistors.¹² When V_on is chosen appropriately, the observed transfer characteristics afford a straight line in good agreement with Eq. (13). The slope corresponds to T_G/T +1, so we can estimate T_G from this plot.

In the actual simulation, T_G and N_G are determined so as to reproduce the observed transfer characteristics as shown in right-column graphs in Figs. 3 and 4. The curvature of the transfer characteristics is determined by T_G (Fig. 2(b)), whereas E_a at large V_G mainly depends on the total trap number, qN_GkT_G/C (Eq. (6)). From the Arrhenius plot of the simulated characteristics, E_a is obtained, and the parameters are adjusted so as to reproduce the resulting E_a vs. V_G plot. In Fig. 6, the calculated E_a is plotted as solid curves, which agree well with the observations. The used parameters are listed in Table II.

In the DBTTF-HMDS transistors, V_on is obviously T-dependent (Fig. 4(b)), and V_on changes at least 10 V within these temperatures. However, V_on shows a very small shift when Eq. (13) is used; an example is depicted in Fig. 8(b). Such a V_on shift has been universally observed in organic transistors.^5–7 In addition, we cannot entirely exclude the possibility of bias stress effect,²⁷ because the measurement has been done from high temperature to low temperature. In order to reproduce the observed V_on shift, we have to use an unrealistically large amount of traps. Sometimes, even qN_GkT_G/C larger than the molecular number (5000 V) is insufficient to reproduce the observed V_on shift. Although the theoretical background of the V_on shift is not certain, it has been included in the analysis of a-Si transistors.^66,67 We have to suppose that the flat-band potential is T-dependent, or the surface charge Q_s is T-dependent. We have investigated activated and power-law T-dependence of V_on = Q_s/C, but all are reduced to an approximately linear dependence, because the T range is not very large. For reproducing Figs. 3 and 4, we have assumed linearly T-dependent V_on, where the starting and the ending V_on values are listed in Table II.

Finally, μ₀N_c is adjusted so as to reproduce the absolute value of I_D. When Q_f/C is given in the unit of V as shown in Fig. 2(a), we obtain the I_D value from Eq. (13) by multiplying μ₀ and (W/L)C = 1.37 × 10⁻⁷ C/Vcm². As Eq. (13) shows, we cannot determine μ₀ and N_c independently. Since the intrinsic mobility μ₀ should be larger than the observed mobility, the N_c values in Table II are estimated by assuming μ₀ = 10 cm²/Vs. However, if N_GkT_G/N_c is assumed to be equal to N_t/N_f obtained from the S values (Table I), this μ₀ is mostly appropriate for DBTTF, but should be somewhat reduced for HMTTF.

In principle, E_c − E_F should be larger than the maximum E_a in the experimental E_a plot. From Eq. (6), however, the experimental E_a tends to diverge when V_G approaches to V_on. When the transistor once enters the on state at large V_G, the transistor characteristics does not depend on E_c − E_F. Experiments using the Kelvin probe microscopy have indicated E_c − E_F to be 0.2–0.4 eV.⁵⁸ It has been pointed out that in many organic semiconductors the Fermi level is pinned to such a value probably due to the existence of a polaron level located by such an amount inside the energy gap.⁶⁸ In the on state, the E_a plot does not depend on the starting E_c − E_F. If the off state shows obvious temperature dependence as the DBTTF-PS transistor in Fig. 4(a), the off current follows,

Accordingly, E_c − E_F is the activation energy of the bulk (off-state) conductivity. In the present simulation, E_c − E_F is assumed to be 0.22 eV for DBTTF-bare and 0.30 eV for others. However, when the off current does not clearly depend on T, and when the off current is determined by the experimental measurement limit as shown in Fig. 4(b), we cannot determine E_c − E_F by this method.

It is noteworthy that even the rather degraded characteristics observed in the DBTTF-bare transistor (on/off = 10 in Fig. 4(c)) is satisfactorily reproduced as shown in Fig. 4(f) by using large N_G. The present model is applicable even when the bulk transport and the field-effect transport are comparable.

As a whole, the T_G and qN_GkT_G/C values used in the simulation (Table II) are roughly in good agreement with the values obtained from Lang's method (Table I). Both the amounts of T_G and qN_GkT_G/C decrease in the order of bare > HMDS > PS for HMTTF and DBTTF (Table II). We can quantitatively estimate the trap DOS by using the present method.

Recently, the DOS distribution has been extracted from the analysis of ESR lineshape,^69,70 where relatively complicated DOS has been obtained. Since only the shallow states influence the transfer characteristics, the present single exponential model provides a good approximation.

N_c in Table II are in the order of several volts, suggesting that free carriers participating in conduction are about 1/1000 of the number of molecules (4000–5000 V). If we use 4000–5000 V for N_c, unreasonably large off current is obtained, where the resulting characteristics are close to Fig. 4(f). qN_GkT_G/C in Table II is typically several 100 V, and nearly one hundred times larger than N_c. N_c is not largely different for a given organic semiconductor, but the trap number, qN_GkT_G/C, changes largely depending on the surface treatments. Previous DOS analyses for organic thin-film transistors afford similar qN_GkT_G/C values between 500 V and 2000 V,²⁶ where the three-dimensional N_G = 2 × 10²¹ cm³/eV with T_G = 500 K corresponds to approximately 1000 V. However, those for single-crystal transistors are smaller by one to three orders, namely 1–100 V.^20,27 Carefully prepared thin-film transistors afford as small trap number as qN_GkT_G/C ∼ 80 V like the present HMTTF-PS transistors. As we have pointed out previously,⁴² we can expect band transport when V_G exceeds the trap number, qN_GkT_G/C.

Deviation of the actually observed transistor characteristics from the above ideal transistor characteristics is explained mainly from the non exponential trap distribution. In order to show this, a small hump in the HMTTF-PS transistor characteristics (Fig. 3(a)) at the subthreshold region is investigated. Such characteristics are reproduced by adding an energetically narrow DOS to the continuous exponential DOS (inset in Fig. 9(a)). The presence of such a discrete state, probably due to the influence of oxygen, has been indicated in other organic semiconductors from the transistor characteristics,²⁷ space-charge-limited-current spectroscopy,⁵⁶ Kelvin probe microscopy,⁵⁷ and ESR.^69,70 The simulated characteristics depicted in Fig. 9(a) reproduces the hump very well (Fig. 3(a)).

As shown in Fig. 9(b), when an unoccupied trap level is added, the on current drops because free carriers are not produced until the discrete level is filled.⁷¹ We have also carried out a simulation assuming a finite width of the localized level, but we have obtained essentially the same transfer characteristics, so here we show the results based on a discrete energy level. With deepening the discrete level E_p, the drop becomes significant, and when E_p is close to or deeper than E_c − E_F, the drop is complete. This practically results in a V_on shift by an amount of the traps, qN_p/C = 30 V. Accordingly, the introduction of the deep discrete level leads to a threshold shift. The shift directly gives the trap number. If the discrete level is originally occupied, the on state starts from the negative V_G region (Fig. 9(c)). Such a case is expected to occur when oxygen produces D⁺ species. When the discrete level is shallow, a pure V_on shift occurs. The interface charge Q_s, introduced in the previous section, is recognized to come from such a localized state. The hump structure in the HMTTF-PS characteristics (Fig. 3(a)) is obvious because the number of trap states is small, but in other characteristics, the relatively large number of trap states leads to the V_on shift.

Recently, controlled V_on shift by intentional chemical doping has been reported in C₆₀ transistors.⁷² Since the molecular density of C₆₀ corresponds to about 1000 V,⁷³ 10⁻³ doping of two-electron donating ruthenium dimer is expected to lead to 2 V of threshold shift. This is in perfect agreement with the observation.

If we assume a more complicated DOS, we can obtain a variety of transfer characteristics. For example, the characteristics in the DBTTF-bare transistor (Fig. 4(c)) is reproducible by assuming the trap levels. If the DOS is given, the transfer characteristics are uniquely obtained from the present method, though the estimation of the trap DOS from the transfer characteristics is not unique. Lang has concluded the presence of a polaron state from the N(E) plot,²¹ but if a structure is evident even in a transfer characteristics, the present method is a more direct evidence of the discrete state.

Using the present framework, we can investigate the contact effect. The observed transfer characteristics starts to deviate from the ideal theoretical equation in the large V_G region (Fig. 8(a)); I_D becomes slightly flatter than the expectation from Eq. (13). This is considered to be because the pinch off is lost at V_G = V_D + V_on, and the transistor goes into the linear region from the saturated region. When we measure the transfer characteristics above V_G > V_D, this is more evident as shown in Fig. 10(a). Differenciating the transfer characteristics, we observe a clear inflection point as shown in Fig. 10(b).

The theoretical equation of trap-free transistor characteristics affords the relation |$I_D \propto V_G^2$|$ID∝VG2$ in the saturated region (Eq. (14)), whereas I_D∝V_G in the linear region (Eq. (12)). The equation with traps (Eq. (11)) affords different curvatures as shown in Fig. 10(c), but the curvature exhibits some change at V_G = V_D. The trap-free case is equivalent to T_G/T = 1 in Fig. 10(c). When the transfer characteristics are differentiated, we observe a clear inflection point as shown in Fig. 10(d). This resembles the observed behavior in Fig. 10(b). We can explain the remarkable drop of the derivative, though it is not certain why T_G always seems to be smaller than T.

The inflection point, however, appears at a V_G lower than V_D. This is regarded as due to the contact voltage drop. As shown in Figs. 10(e) and 10(f), only the drain voltage drop but not the source voltage drop influences the shift of the inflection point, because only the former changes the voltage at which the pinch off is lost. In order to show this strictly, when there is a voltage drop at the source I_DR_S due to the contact resistance R_S, Eq. (12) should be integrated from I_DR_S to V_D, to afford

We differentiate this equation by V_D and putting ∂I_D/∂V_D = 0 to give V_G = V_D. Accordingly, the V_G value at which the pinch off is lost, is not affected by the source voltage drop. In contrast, the drain voltage drop I_DR_D is included in Eq. (12) by integrating it from 0 to V_D - I_DR_D.

We differentiate this equation by V_D and putting ∂I_D/∂V_D = 0 to give V_G = V_D - I_DR_D. Figure 10(f) obviously shows that the pinch off is lost at a lower voltage than V_D by the drain voltage drop I_DR_D.

Figure 10(g) shows the drain voltage drop evaluated from the peaks of the differentiated transfer characteristics. The voltage drop is in the range of 10–20% of the total voltage, and approximately temperature independent. This is in agreement with the previous estimation from the four-probe method.⁶ The PS-treated transistors tend to show comparatively large voltage drops. However, since this is the ratio to the channel resistance, the magnitude of the drain resistance seems to be unchanged. In this method, we can estimate the drain resistance independently from the source resistance. It is also characteristic that we can estimate it without using a special device alignment as the four-probe method, and only using a single device in contrast to the transfer-line method, only by measureing the transfer characteristics up to the linear region.

T-dependent transistor characteristics of HMTTF and DBTTF are analysed thoroughly using the interface approximation (Eqs. (11) and (13)). The present analysis should be consistent with Lang's trap DOS analysis. The observed transistor characteristics do not afford a very straight exponential trap distribution (Fig. 7), and the N(E) plot has some structures that are not obvious either in the E_a plot nor in the original transfer characteristics. Nonetheless, assuming a simple exponential trap distribution, we can sufficiently reproduce the overall feature of the transfer characteristics (Figs. 3 and 4) as well as the V_G dependence of E_a (Fig. 6). In the analysis, it is important to exclude the influence of the T-dependent V_on shift, because, although the origin is not certain, the observed V_on shift is obviously beyond the expectation from the trap model. The downward curvature of the transfer characteristics implies large T_G and the presence of relatively deep traps. The small T dependence at large V_G means small total trap number N_GkT_G.

Under the interface approximation, two-dimensional carrier density is represented in a unit of V by multipling q/C. From the present analysis, the total trap number, qN_GkT_G/C, is estimated to be several 100 V (Table II), which should be compared with the molecular number, 4000–5000 V. Owing to the traps, the output and transfer characteristics are reduced by the ratio of the free and trapped carriers, N_c/N_GkT_G, and this number is usually in the order of one hundredth. In the present model, E_a becomes negative when V_G exceeds qN_GkT_G/C. In a finely fabricated thin-film transistor, qN_GkT_G/C becomes as small as 80 V, above which we have a possibility to observe band transport.

We have investigated that the presence of a descrete level gives rise to a hump in the transfer characteristics. The theoretical framework given here is useful to discuss various aspects of organic transistors. For example, this model explains the behavior that the transistor goes from the saturated region into the linear region above V_G > V_D, and we can estimate the drain contact resistance by simply observing the transfer characteristics at large V_G > V_D.