Subsections

• 7.2.1 Variable Range Hopping Transport in Organic Semiconductors
• 7.2.2 Sheet Conductance of the OTFT
• 7.2.3 Drain Current
• 7.2.4 Results and Discussion

7.2 Analytical Model for Organic Thin Film Transistors

In this section we derive a basic expression for the sheet conductance based on the variable range hopping (VRH) theory. This theory describes thermally activated tunneling of carriers between localized states around the Fermi level in the tail of a Gaussian distribution. It has been used to calculate the mobility of OTFTs successfully. After some simplification for the surface potential, simple and efficient analytical expressions for the transfer characteristics and output characteristics are obtained. The model does not require as input parameters the explicit definition of the threshold and saturation voltage, which are rather difficult to evaluate for this kind of device. The obtained results are in good agreement with experimental data.

7.2.1 Variable Range Hopping Transport in Organic Semiconductors

Because most organic films have an amorphous structure and disorder is dominating the charge transport, variable-range-hopping in positionally and energetically disordered systems of localized states is widely accepted as the conductivity mechanism in organic semiconductors. Different from hopping, where the charge transport is governed by the thermally activated tunneling of carriers between localized states rather than by the activation of carriers to the extended-state transport level, the concept of variable range hopping means that a carrier may either hop over a small distance with high activation energy or hop over a long distance with a low activation energy. In an organic thin film transistor with a typical structure shown in Fig 7.1, an applied gate voltage gives rise to an accumulation of carriers in the region of the organic semiconductors

Figure 7.1: Schematic structure of an organic thin film transistor.

close to the insulator. As the carriers in the accumulation layer fill the low-energy states of the organic semiconductor, any additional carrier in the accumulation layer will require less activation energy to hop to a neighboring site. This results in a higher mobility with increasing gate voltage. In combination with percolation theory, Vissenberg studied the influence of temperature and the influence of the filled states on the conductivity based on the variable range hopping theory. The expression for the conductivity as a function of the temperature and carrier concentration is given by [43] where $\sigma_0$ is a prefactor, $\alpha$ is an effective overlap parameter, which governs the tunneling process between two localized states, and $B_c\cong$ 2.8 is the critical number of bonds per site in the percolating network [131], $T_0$ is the effective temperature, $N_t$ is the number of states per unit volume and $\delta$ is the fraction of the localized states occupied by a carrier. The carrier concentration $\delta N_t$ can be expressed in equilibrium as

$$\rho(V)=N_t\delta(V)=N_t\delta_0\exp\left(\frac{q\Phi}{k_BT_0}\right),$$ (7.1)

where $\Phi$ is the electrostatic potential, and the $\delta_0$ is the carrier occupation far from the organic-insulator interface.

7.2.2 Sheet Conductance of the OTFT

For an amorphous TFT the drain current $I_D$ can be expressed as

$$I_D=\frac{W}{L}\int_{V_G-V_{FB}-V_D}^{V_G-V_{FB}}G_S(V)dV,$$ (7.2)

where $W$ is the channel width, $L$ is the channel length, $V_{FB}$ is the flat-band voltage, and $G_S$ is the sheet conductance of the channel at $V_D=0$V. The potential $V$ is defined as $V=V_G-V_{FB}-V_0(y)$, where $V_0(y)$ is the potential at the edge of the space-charge layer where there is no band bending.

Figure 7.2: Geometric definition.

The basic definition of the channel configuration and the variables for the OTFT investigated are illustrated in Fig 7.2.

$$\sigma(\delta,T) = \sigma_0 \left[\frac{\pi N_t\delta(T_0/T)^3}{(2\alpha)^3B_c\Gamma(1-T/T_0)\Gamma(1+T/T_0)}\right]^{T_0/T},$$ (7.3)

The electrostatic potential in the space charge layer at the point ($x,y$) in the channel is expressed as $V(x,y)=V_0(y)+\Phi(x,y)$, where the $\Phi(x,y)$ is the amount of the band bending in the channel. The conductance for an element of channel length $\Delta y$ and the width $W$ can be written as

$$G_s=\frac{W}{\Delta y}\int_0^{t}\sigma dx=\frac{W}{\Delta y}\frac{\sigma(\delta_0,T)}{\sigma_0}\int_0^{t} \exp\left(\frac{q\Phi}{k_BT}\right)dx$$ (7.4)

where $t$ is the thickness of the organic layer. Changing the variable of integration yields

$$G_s=A\int_{\Phi (t(y))}^{\Phi_s(y)}\frac{\exp(q\Phi/{k_BT})}{\partial\Phi/\partial x}d\Phi,$$ (7.5)

where $\Phi_s(y)$ is the surface band bending and $A=\sigma(\delta_0,T).$ With the identity

$$\Gamma(1+x)\Gamma(1-x)=\frac{\pi x}{\sin(\pi x)}$$

we obtain

$$A=\sigma_0\left[\frac{N_t\delta_0(T_0/T)^4\sin(\pi T/T_0)}{B_c(2\alpha)^3}\right]^{T_0/T}$$

In order to solve (7.5) we need to get an expression for $\Phi(x)$. By solving Poisson's equation in the gradual channel approximation

$$\frac{\partial^2\Phi(x)}{\partial x}=-\frac{\rho(x)}{\epsilon_0\epsilon_s},$$ (7.6)

we obtain the electric field.

$$-F_x=\frac{\partial\Phi}{\partial x}\approx\sqrt{\frac{2k_BT_0N_t\delta_0}{\epsilon_0\epsilon_s}}.\exp\left[\frac{q\Phi(x)}{2k_bT_0}\right]$$ (7.7)

From (7.5) and (7.7) we obtain

$$G_s=A\int_{\Phi(t)}^{\Phi_s}\exp\left[\frac{q\Phi}{k_B}\left(\frac{1}{T}-\frac{1}{2T_0}\right)\right]d\Phi.$$ (7.8)

An expression for $\Phi_s$ is required. The surface charge density $Q_s$ is related to $\Phi_s$ by

$$Q_s=-\epsilon_0\epsilon_sF_s=\sqrt{2k_BT_0N_t\epsilon_0\epsilon_s\delta_0}\exp\left(\frac{q\Phi_s}{2k_BT_0}\right).$$ (7.9)

The surface band bending is related to the applied gate voltage by

$$V_G=V_{FB}+V_i+\Phi_s$$ (7.10)

where $V_i$ is the voltage drop across the insulator,

$$V_i=\frac{Q_s}{C_i}$$ (7.11)

where $C_i=\epsilon_i/d_i$ is the insulator capacitance per unit area. From the equations above, an expression for $\Phi_s$ is obtained

$$V_{G}-V_{FB}-\Phi_s=\gamma\exp\left(\frac{q\Phi_s}{2k_BT_0}\right)$$ (7.12)

For an accumulation mode OTFT, the surface potential is negative, $\Phi_s\leq 0$, corresponding to $V_G\leq 0$.

$$V_G=V_{FB}+\Phi_s-\frac{\sqrt{2k_BT_0\delta_0N_t\epsilon_0\epsilon_s}}{C_i}\exp\left(-\frac{q\Phi_s}{2k_BT_0}\right).$$ (7.13)

In order to reduce computation time, an explicit yet accurate relation between surface potential and gate voltage is preferable. In (7.13) we can get $\Phi_s$ using a numerical approach. However, in the accumulation mode, it holds $\exp(-\Phi_s)\gg\Phi_s$, so that an approximate expression of surface potential can be obtained as

$$\Phi_s=-\frac{2k_BT_0}{q}\ln\left[\frac{(V_{FB}-V_G)C_i}{\sqrt{2K_bT_0\delta_0N_t\epsilon_0\epsilon_s}}\right].$$ (7.14)

A comparison between numerical calculation and approximate calculation is shown in Fig 7.3 and Fig 7.4. As can be seen, the agreement is very satisfactory. Parameters are from [132,133].

With the simplified surface potential and (7.8) we can get the simplified sheet conductance as

$$G_s=\beta\left[\left(\frac{V_G-V_{FB}}{\varrho}\right)^{2T_0/T-1}-1\right]$$ (7.15)

For a thick organic semiconductor layer, $\Phi(t)=0$ and the coefficient $\beta$ is

$$\beta=\sigma_0\sqrt{\frac{2\epsilon_0\epsilon_sK_BT_0}{\delta_0N_t}}\frac{k_BT}{q(T-2T_0)}$$

$$\varrho=\frac{(2\alpha)^3B_c2k_BT_0\epsilon_0\epsilon_s}{C_i^2(T_0/T)^3\sin(\pi T/T_0)}$$

Figure 7.3: The electrostatic surface potential as a function of gate voltage obtained by the implicit relation (7.13) and the approximation (7.14) (solid line).

Figure 7.4: Sheet conductance from numerical calculation (symbols) and the approximation.

Figure 7.5: Measured (symbols) and calculated transfer characteristics of a pentacence OTFT at room temperature.

Figure 7.6: Measured (symbols) and calculated transfer characteristics of a pentacence OTFT at different temperatures at $V_{D}=-2V$.

7.2.3 Drain Current

The drain current can be calculated by substituting the expression for $G_s$ into (7.2). We obtain

$$I_D=\beta\frac{W}{L}\left[{\left(\frac{V_G-V_{FB}}{\varrho}\right)^{2T_0/T}-\left(\frac{V_G-V_{FB}-V_D}{\varrho}\right)^{2T_0/T}}\right]$$ (7.16)

in the triode region ( $V_{GS}-V_{FB}\ge V_{DS}$) and

$$I_D=\beta\frac{W}{L}\left(\frac{V_{GS}-V_{FB}}{\varrho}\right)^{2T_0/T}$$ (7.17)

in saturation ( $V_{GS}-V_{FB}\leq V_{DS}$).

7.2.4 Results and Discussion

Figure 7.7: Measured (symbols) and calculated transfer characteristics of a PTV OTFT at room temperature at $V_{D}=-2V$.

Figure 7.8: Modeled output characteristics of a pentacene OTFT.

This model has been confirmed by comparisons between experimental data and simulation results. Input parameters are taken from [132]: $W=20,000 \mu m$, $L=10\mu m$, $\epsilon_s =3$, $C_i=17F/(\mu m)^2$, $\sigma_0=3.5 S/m$, $\alpha^{-1}=3.1\AA$, $T_0=385K$.

In Fig 7.5 and Fig 7.6 the transfer characteristics of a pentacene OTFT are given for $V_{FB}=1V$ at different drain voltage and different temperature. Both figures show a good agreement between the analytical model and experimental data. Here we also model the transfer characteristics of a PTV OTFT, where some parameters are different from those for pentacence: $T_0=382$K, $\sigma_0=5.6$S/m, $\alpha^{-1}=1.5\AA$, as shown in Fig 7.7. The modeled output characteristics of the pentacene OTFT is shown in Fig 7.8.