# Critical Evaluation of Organic Thin-Film Transistor Models

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Equivalent Circuit Model

**S’**, drain

**D’**, and gate

**G’**terminals. At the gate terminal, the threshold voltage ${V}_{T}$ is implemented as an external bias, and the source and drain terminals are connected to Ohmic contact resistances ${R}_{S,0}$ and ${R}_{D,0}$. The experimentally-accessible terminals are labeled source

**S**, drain

**D**, and gate

**G**. The assignment of the elements in the equivalent circuit model to the location in a real device is indicated in gray. Figure 1b shows a schematic drawing of a bottom-gate, bottom-contact TFT; Figure 1c shows optical microscopy images of bottom-gate, bottom-contact DNTT TFTs with channel lengths of 2, 8, 40, and 80 $\mathsf{\mu}$m from top to bottom; and Figure 1d shows a photograph of a set of pentacene TFTs on a flexible plastic substrate.

**G’**, source

**S’**, and drain

**D’**terminals divided by ${V}_{0}$. The Heaviside function $\Theta \left(x\right)$ is equal to 1 for $x\ge 0$ and equal to 0 for $x<0$. ${C}_{I}$ is the gate capacitance per unit area, and ${r}_{S,0}={R}_{S,0}W$ and ${r}_{D,0}={R}_{D,0}W$ are the channel-width-normalized source and drain resistances, respectively. The drain current ${I}_{D}$ as the output parameter is thus implicitly determined by two input parameters ${V}_{GS}$ and ${V}_{DS}$, six fitting parameters ${V}_{T}$, ${\mu}_{0}$, ${r}_{S,0}$, ${r}_{D,0}$, $\beta $, and $\gamma $, two constants ${L}_{0}$ and ${V}_{0}$, and three geometry parameters L, W, and ${C}_{I}$. The gate capacitance per unit area ${C}_{I}$ is considered here as a geometry parameter since it is determined by the thickness and the permittivity of the gate dielectric.

#### 2.2. Fitting Procedure

#### 2.3. Device Fabrication

## 3. Results

#### 3.1. Conventional Transmission Line Method

- The measured output characteristics (gray symbols in Figure 3a–d) must be linear for very small drain-source voltages (${V}_{DS}\to 0$ V), and the slope of the curves must decrease monotonically as the absolute value of the drain-source voltage ${V}_{DS}$ increases. An S-shape of the output curves in this regime is an indicator of a non-Ohmic contact resistance.

#### 3.2. TSFA with Constant-Mobility Model

#### 3.3. TSFA with Field- and Charge-Carrier-Density-Dependent Mobility

#### 3.4. Testing Other Organic-TFT Technologies

## 4. Summary and Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Panel (

**a**) shows the equivalent circuit model based on an ideal field-effect transistor in the gradual channel approximation with a field- and charge-carrier-density-dependent mobility connected to the Ohmic source and drain resistances ${R}_{S,0}$ and ${R}_{D,0}$. The threshold voltage ${V}_{T}$ is implemented in the form of an external bias. The terminals of the ideal transistor are labeled source

**S’**, drain

**D’**, and gate

**G’**, and the experimentally-accessible terminals are labeled source

**S**, drain

**D**, and gate

**G**. The corresponding location of the elements in a real device is indicated in gray. In (

**b**), a schematic drawing of a bottom-gate, bottom-contact thin-film transistor (TFT) is illustrated. In (

**c**), optical microscopy images of bottom-gate, bottom-contact dinaphtho[2,3-b:2’,3’-f]thieno[3,2-b]thiophene (DNTT) TFTs with channel lengths of 2, 8, 40, and 80 $\mathsf{\mu}$m from top to bottom can be seen, and (

**d**) shows a photograph of a set of pentacene TFTs on a flexible plastic substrate.

**Figure 2.**Parameter extraction in the framework of the conventional transmission line method (TLM), performed here on a set of bottom-gate, bottom-contact p-channel TFTs based on the small-molecule semiconductor DNTT. In (

**a**), the on-state resistance ${r}_{on}=W\partial {V}_{DS}/\partial {I}_{D}$ for ${V}_{DS}\to 0$ V, extracted from the measured output characteristics, is plotted as a function of the channel length for different gate-source voltages ${V}_{GS}$. From a linear fit to the data, the inverse slope $\Delta L/\Delta {r}_{on}$ and the y-axis intersect ${r}_{on}(L=0)$ are extracted. The inset shows a magnification of the region in which the linear fits intersect and the extracted ${r}_{on}$ values for the smallest channel length of $L=2$ $\mathsf{\mu}$m (symbols). In (

**b**), $\Delta L/\Delta {r}_{on}$ is plotted as a function of the gate-source voltage ${V}_{GS}$, and from the linear fit to the data, the threshold voltage ${V}_{T}=1.25$ V and the intrinsic channel mobility ${\mu}_{TLM}=3.2$ cm${}^{2}$/Vs are obtained. In (

**c**), ${r}_{on}(L=0)={r}_{Sh}{L}_{T}+{r}_{C,0}$ is plotted as a function of the sheet resistance ${r}_{Sh}=\left|{V}_{0}\right|{\left[{V}_{0}{C}_{I}{\mu}_{TLM}({V}_{GS}-{V}_{T})\right]}^{-1}$, and from the linear fit to the data, the transfer length ${L}_{T}=3.4$ $\mathsf{\mu}$m and the total Ohmic contact resistance ${r}_{C,0}=0.14$ k$\Omega $cm are obtained.

**Figure 3.**Measured output and transfer characteristics (gray symbols) and calculated output and transfer characteristics (black lines) of bottom-gate, bottom-contact DNTT TFTs with channel lengths L of 2, 8, 40, and 80 $\mathsf{\mu}$m. The TFT with a channel length of 2 $\mathsf{\mu}$m has a channel width of 20 $\mathsf{\mu}$m, while the TFTs with channel lengths of 8, 40, and 80 $\mathsf{\mu}$m have a channel width of 200 $\mathsf{\mu}$m. Note that the gray symbols appear as an apparent thick line due to the close spacing of the data points. In (

**a**–

**h**), the output and transfer curves were calculated using the transistor parameters extracted using the conventional TLM analysis. In (

**i**–

**p**), the output and transfer curves were calculated using our two-step fitting approach (TSFA) with the constant-mobility model underlying the conventional TLM.

**Figure 4.**Dependence of the transistor parameters extracted using the TSFA for the theoretical transistor model underlying the conventional TLM on the channel length L. In (

**a**), it can be seen that the threshold voltage ${V}_{T}$ shows only a very small dependence on the channel length. Panel (

**b**) indicates that the dependence of the charge-carrier mobility ${\mu}_{TSFA}$ on the channel length L (symbols) is not properly described by the equation ${\mu}_{TSFA}={\mu}_{TLM}L/(L+{L}_{T})$ (solid line), with ${L}_{T}$ being the transfer length. In (

**c**), the distinct linear increase of the contact resistance ${r}_{C,0}={r}_{S,0}+{r}_{D,0}$ with increasing channel length L cannot be explained at all. As a consequence, the model does not pass the second step.

**Figure 5.**Results of the TSFA for the model with a field- and charge-carrier-density-dependent mobility. In (

**a**–

**h**), the measured output and transfer characteristics (gray symbols) and the calculated output and transfer characteristics (black lines) of bottom-gate, bottom-contact DNTT TFTs with channel lengths L of 2, 8, 40, and 80 $\mathsf{\mu}$m and channel widths W of 20 or 200 $\mathsf{\mu}$m are shown, indicating good agreement. Note that the gray symbols appear as an apparent thick line due to the close spacing of the data points. In (

**i**,

**j**), the channel-length dependence of the mobility prefactor ${\mu}_{0}$ and the combined contact resistance ${r}_{C,0}={r}_{S,0}+{r}_{D,0}$ indicates a failure of the model.

**Figure 6.**Selected examples of output characteristics for each set of TFTs (

**a**–

**d**) and Ohmic component of the combined contact resistance ${r}_{C,0}$ plotted as a function of the channel length for each set of organic TFTs (

**e**–

**h**). In (a,e), bottom-gate, top-contact DNTT TFTs are investigated. In (b–d) and (f–h), bottom-gate, bottom-contact pentacene and C${}_{60}$ TFTs with Au contacts functionalized with 2-phenylpyrimidine-5-thiol (BP0-down), 4-(2-mercaptophenyl)pyrimidine (BP0-up), or biphenyl-4-thiol (BP0) are analyzed. For all TFTs, despite different TFT architectures, different organic semiconductors, and different contact materials, a clear channel-length dependence of the Ohmic component of the contact resistance ${r}_{C,0}$ is observed. This leads to a failure of the transistor model in all cases. The substantial fluctuations of ${r}_{C,0}$ in the short-channel-length C${}_{60}$ TFTs in (g,h) (including transistors with ${r}_{C,0}=0$) reflect the fact that the uncertainty of the Ohmic contact resistances for these small channel lengths is on the order of the actual value. This large uncertainty does not obscure the clear increase of the contact resistance ${r}_{C,0}$ with the channel length L. The failure appears to occur for the same reason as seen for the bottom-gate, bottom-contact DNTT TFTs, because the symptom of an incorrect spacing in the saturation regime of the output characteristics is present here as well.

**Table 1.**Device architecture, materials employed for the organic semiconductor and the source and drain contacts, gate-dielectric thickness ${d}_{{\mathrm{Al}}_{2}{\mathrm{O}}_{3}}$, and the range of channel lengths L and channel widths W of the TFTs analyzed in this work. Device architectures are the bottom-gate, bottom-contact (BGBC) and the bottom-gate, top-contact (BGTC) structure. The Au contacts of the BGBC TFTs were functionalized with either pentafluorobenzenethiol (PFBT), 2-phenylpyrimidine-5-thiol (BP0-down), 4-(2-mercaptophenyl)pyrimidine (BP0-up), or biphenyl-4-thiol (BP0). DNTT TFTs with channel lengths $L\le 4$ $\mathsf{\mu}$m have a channel width of 20 $\mathsf{\mu}$m, and DNTT TFTs with channel lengths $L>4$ $\mathsf{\mu}$m have a channel width of 200 $\mathsf{\mu}$m.

Name/Reference | Architecture | Semiconductor | Contact | ${\mathit{d}}_{{\mathrm{Al}}_{2}{O}_{3}}$ (nm) | L ($\mathsf{\mu}$m) | W ($\mathsf{\mu}$m) |
---|---|---|---|---|---|---|

DNTT-BC [29] | BGBC | DNTT | Au/PFBT | 5.3 | 2–80 | 20–200 |

DNTT-TC [21] | BGTC | DNTT | Au | 5.3 | 4–100 | 20–200 |

Pentacene [22] | BGBC | Pentacene | Au/BP0-down | 18 | 4.85–52.90 | 1000 |

C${}_{60}$-BP0-up [22] | BGBC | C${}_{60}$ | Au/BP0-up | 18 | 3.0–100.5 | 1000 |

C${}_{60}$-BP0 [22] | BGBC | C${}_{60}$ | Au/BP0 | 18 | 3.6–51.0 | 1000 |

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## Share and Cite

**MDPI and ACS Style**

Krammer, M.; Borchert, J.W.; Petritz, A.; Karner-Petritz, E.; Schider, G.; Stadlober, B.; Klauk, H.; Zojer, K. Critical Evaluation of Organic Thin-Film Transistor Models. *Crystals* **2019**, *9*, 85.
https://doi.org/10.3390/cryst9020085

**AMA Style**

Krammer M, Borchert JW, Petritz A, Karner-Petritz E, Schider G, Stadlober B, Klauk H, Zojer K. Critical Evaluation of Organic Thin-Film Transistor Models. *Crystals*. 2019; 9(2):85.
https://doi.org/10.3390/cryst9020085

**Chicago/Turabian Style**

Krammer, Markus, James W. Borchert, Andreas Petritz, Esther Karner-Petritz, Gerburg Schider, Barbara Stadlober, Hagen Klauk, and Karin Zojer. 2019. "Critical Evaluation of Organic Thin-Film Transistor Models" *Crystals* 9, no. 2: 85.
https://doi.org/10.3390/cryst9020085