An Analytical Surface Potential and Effective Charge Density Approach Based Drain Current Model for Amorphous InGaZnO Thin-Film Transistors

Abstract

An analytical surface-potential-based drain current model for amorphous indium–gallium–zinc–oxide (a-InGaZnO) thin film transistors (TFTs) is proposed by introducing an effective charge density approach in this paper. This approach gives two initial approximate values of the effective state density and the effective thermal voltage by using the dominant state of the free charge density in total charge density, and then obtains a high-precision one-exponent equivalent transformation for three-exponent total charge density. Based on this approach, we have solved the problem that the physical meaning of the transition area in the regional method is not clear and a one-piece analytical surface potential solution to Poisson’s equation is successfully derived. Furthermore, the drain current is also explicitly derived from the charge sheet model and I-V characteristics of a-InGaZnO TFTs are reproduced from the above obtained model. Finally, accurate and effective surface-potential model and drain current model are obtained and verified by experimental data, respectively. Good verification results prove that the proposed model could become an accurate and suitable tool for being embedded into a circuit simulation.

1. Introduction

It is well known that amorphous indium–gallium–zinc–oxide (a-InGaZnO) thin film transistors (TFTs) have received much attention from researchers and industries because of its good electrical properties [1]. At present, a-InGaZnO TFTs are mainly used in sensors [2], memory [3], wearable devices [4] and liquid crystal display technology [5] due to its advantages of high mobility [6], good uniformity [7], low temperature fabrication [8] and so on. Especially in the display field, a-InGaZnO TFTs have the advantages of low cost [9] and compatibility [10] compared with a-Si TFT or a-Si: H TFT, which make them easier and more suitable for manufacturing liquid crystal displays and other devices. These devices are often used as the main tools for information exchange, which also enables them to be produced and applied on a large scale in society. However, as people’s demand for these products begins to increase, this requires researchers to establish more correct models to improve the utilization of a-InGaZnO TFTs. Hence, in order to improve the performance and reduce the cost of these products, it is necessary to have an analytical and accurate model to understand the electrostatic characteristics of a-InGaZnO TFTs.

Recently, there are many physics-based drain current models [11,12,13] proposed to predict the I-V characteristics of a-InGaZnO TFTs. In fact, surface-potential-based drain current model has become the industrial standard for a physics-based compact model. However, for a-InGaZnO TFTs, the surface-potential-based drain current models [14,15,16,17,18] have the difficulty of a trade-off between accuracy and efficiency for deriving an analytical surface potential solution to Poisson’s equation including three-exponent total charge density, i.e., the free charge density, the tail and deep trap state densities. Up to today, there are two types of approaches for deriving an analytical surface potential solution: regional approach [19,20,21,22,23,24] and effective charge density approach [25,26,27,28,29,30,31]. In fact, the regional approach has been widely used in the modeling of Polysilicon TFTs [19,20,21], MoS2 FETs [22], organic TFTs [23], a-InGaZnO TFTs [24], etc. It has even developed into an industrial standard method for modelling surface-potential-based compact model. However, the critical disadvantages of the regional approach are that the unclear physical meaning in the transition region decreases simulation accuracy, and multiple times computations of surface potential in the different regions reduce simulation efficiency. In other words, the regional approach acquires a trade-off between accuracy and efficiency in compact modeling, but the further improvement rooms are left for us. As an improvement of the regional approach, the effective charge density approach is more accurate and efficient. However, there are still some problems needed to be overcome for modelling a-InGaZnO TFTs compactly by using the effective charge density approach. Surface-potential-based Polysilicon TFTs’ compact models [25,26] are inappropriate for modelling a-InGaZnO TFTs directly. M. Bae’s model [27] is based on the effective charge density approach, which is still essentially the regional approach. J. Park et al. [28], T. Qin et al. [29] and Y. Hernández-Barrios et al. [30] ignored the effects from deep trap density on electrostatic characteristics of a-InGaZnO TFTs, which lead to the inaccuracy of the model directly and would have a great impact on circuit simulation. W. Deng et al. [31] proposed an analytical drain current model of a-InGaZnO TFTs including the deep trap density. Surface potential is solved from Poisson’s equation including free charge, tail trap and deep trap densities by utilizing the Lambert W function [32]. It is noted that the initial estimations of surface potential still use the regional approach and smooth function, which would lead to heavier computation requirements. Briefly, an accurate and effective surface-potential-based drain current model of a-InGaZnO TFTs is still urgently needed for circuit simulators.

In this paper, we present an analytical surface-potential-based drain current model for a-InGaZnO TFTs using the effective charge density approach. By giving two initial approximations of the effective state density and the effective thermal voltage, we have successfully transformed three exponent Poisson’s equation into one exponent transcend equation with using effective charge density approach. Then, we overcome the problem of the unclear physical meaning of the traditional region method in the transition region and derive an analytical one-piece surface potential solution to the transcend equation. In addition, the regional method, our surface potential model and numerical results are used to compare and successfully verify the high accuracy of our model. Based on surface potential, the drain current is derived according to the charge sheet model. Finally, we use numerical iteration results and experimental results to verify the a-InGaZnO TFTs’ model. Symbols and lines keep excellent consistency in all valid ranges, which means that a better trade-off between accuracy and efficiency in the proposed model is successfully obtained through using the effective charge density approach, and such a model is more suitable for being applied into circuit simulators.

2. Analytical One-Piece Surface Potential Solution

A cross-section schematic of bottom-gate a-InGaZnO TFTs is shown in Figure 1. Glass is used for substrate, a reverse-staggered metal gate is located in the bottom of the silicon dioxide layer [33], the a-InGaZnO thin film is prepared above silicon dioxide layer, and two symmetrical electrodes (source and drain) are situated on the top of a-InGaZnO thin film. Here, the thickness of the oxide layer is t_ox, and the thickness of a-InGaZnO thin film is t_igzo.

Generally, the free charge density and the complete density of trap states including exponential tail states and deep states in the a-InGaZnO thin film directly affect the electrostatic characteristics of a-InGaZnO TFTs. In the case of non-degenerate, considering the gradual channel approximation and ignoring percolation conduction mechanism [34], a one-dimensional Poisson equation is listed as

$$\frac{d^2\phi }{d{x}^2}=\frac{q}{{\epsilon }_{igzo}}{N}_{toi}=\frac{q}{{\epsilon }_{igzo}}\left[n_0\exp \left(\frac{\phi -V_{ch}}{V_T}\right)+N_T\exp \left(\frac{\phi -V_{ch}}{E_T/q}\right)+N_D\exp \left(\frac{\phi -V_{ch}}{E_D/q}\right)\right]$$

(1)

where q is the charge of the electron, ε_igzo is the a-InGaZnO permittivity, N_toi is the sum of charge density, φ is the electrostatic potential in the channel, V_ch is the channel potential. In the right part of Equation (1), the free charge density is given by n₀·exp[(φ − V_ch)/V_T]. Here, V_T is the thermal voltage, which can be calculated by V_T = kT/q, where k is the Boltzmann constant, T is the temperature. n₀ = N_C·exp[(V_fp − E_C/q)/V_T], where N_C is the effective density of states in the conduction band edge E_C, and V_fp is the quasi-Fermi potential. Furthermore, the tail state density of trap states is expressed as N_T·exp[(φ − V_ch)/(E_T/q)] and the deep state density of trap states is given by N_D·exp[(φ − V_ch)/(E_D/q)], respectively. Here, N_T = g_c1·[πkT/sin(πkT/E_T)]·exp(−E_C/E_T) and N_D = g_c2·[πkT/sin(πkT/E_D)]·exp(−E_C/E_D), where g_c1 is the density of tail state at E_C and E_T is the inverse slope of tail state, and g_c2 is the density of deep state at E_C and E_D is the inverse slope of deep state. The boundary conditions for Equation (1) are: (dφ/dx)_x=tigzo = 0, φ_s = φ (0).

As we can see, deriving the electrostatic potential in Equation (1) is a big challenge because of the three exponential terms contained in it. However, we can give an equivalent transform of the three charge densities into one kind of effective charge density by introducing the effective charge density approach. In this way, it will be possible to solve the electrostatic potential in Equation (1). Hence, we can rewrite Equation (1) as follows.

$$\frac{d^2\phi }{d{x}^2}=\frac{q}{{\epsilon }_{igzo}}{N}_{eff}=\frac{q}{{\epsilon }_{igzo}}\left[n_{eff}\exp \left(\frac{\phi -V_{ch}}{V_{eff}}\right)\right]$$

(2)

where N_eff represents the effective charge density, n_eff is the effective state density, V_eff is the effective thermal voltage. Obviously, N_eff should be equal to N_toi for the same device model.

In fact, a-InGaZnO TFTs work usually in the above-threshold region. On this common condition, the free charge density of TFTs always occupies a dominant position in the total charge density, i.e., the free charge density is always larger than trap density in this dominant range. Therefore, the effective state density should be closer to the n₀ to make the model more accurate. In addition, since n₀ is much larger than N_T and N_D, we give n_eff an approximate value that meets the above requirement, i.e.,

$$n_{eff}=n_0+N_T+N_D$$

(3)

Then, we can obtain the expression between n_eff and V_eff based on potential according to Equations (1) and (2):

$$n_{eff}\exp \left(\frac{\phi -V_{ch}}{V_{eff}}\right)=\left[n_0\exp \left(\frac{\phi -V_{ch}}{V_T}\right)+N_T\exp \left(\frac{\phi -V_{ch}}{E_T/q}\right)+N_D\exp \left(\frac{\phi -V_{ch}}{E_D/q}\right)\right]$$

(4)

Using the Gauss law at the front interface for Equation (1), the relationship between the gate voltage and the surface potential is also obtained as

$${\left(V_{gs}-V_{fb}-{\phi }_s\right)}^2=\frac{2q{\epsilon }_{igzo}}{{C_{ox}}^2}\left\{n_0{V}_T\left[\exp \left(\frac{{\phi }_s-V_{ch}}{V_T}\right)-1\right]+N_T\frac{E_T}{q}\left[\exp \left(\frac{{\phi }_s-V_{ch}}{E_T/q}\right)-1\right]+N_D\frac{E_D}{q}\left[\exp \left(\frac{{\phi }_s-V_{ch}}{E_D/q}\right)-1\right]\right\}$$

(5)

where C_ox is the gate oxide capacitance per unit area, V_gs is the gate voltage, V_fb is the flat band voltage, φ_s is the surface potential. It is noted that Equation (5) also contains three complex exponential parts. In order to solve the surface potential φ_s, we assume that only the exponential part of the first term is considered on the right side of Equation (5). Therefore, Equation (5) is rewritten as

$${\left(V_{gs}-V_{fb}-{\phi }_s\right)}^2=\frac{2q{\epsilon }_{igzo}}{{C_{ox}}^2}\left\{n_0{V}_T\left[\exp \left(\frac{{\phi }_s-V_{ch}}{V_T}\right)-1\right]\right\}$$

(6)

Considering V_T is much less than φ_s and the term exp[(φ_s − V_ch)/V_T] >> 1, 1 can be ignored. Then, using Lambert W function to solve Equation (6), we obtain

$${\phi }_{s0}=V_{gs}-V_{fb}-2V_T{W}_0\left[\frac{\sqrt{2q{\epsilon }_{igzo}{n}_0{V}_T}}{2V_T{C}_{ox}}\exp \left(\frac{V_{gs}-V_{fb}-V_{ch}}{2V_T}\right)\right]$$

(7)

In Equation (7), it shows the relationship between gate voltage and surface potential when free charge is dominant in the above-threshold region. This is completely consistent with the idea that n_eff is close to n₀. Hence, φ_s0 can be taken as initial value to obtain the expression of V_eff in Equation (4), yielding:

$$V_{eff}=\frac{{\phi }_{s0}-V_{ch}}{\ln \left[\frac{N_{toi}({\phi }_{s0})}{n_{eff}}\right]}$$

(8)

Now, using the Gauss law at the front interface again for Equation (2), we can get

$${\left(V_{gs}-V_{fb}-{\phi }_s\right)}^2=\frac{2q{\epsilon }_{igzo}}{{C_{ox}}^2}\left\{n_{eff}{V}_{eff}\left[\exp \left(\frac{{\phi }_s-V_{ch}}{V_{eff}}\right)-1\right]\right\}$$

(9)

Finally, the Lambert W function is used again to solve Equation (9) according to the obtained expressions of n_eff and V_eff, the truly surface potential φ_s can be expressed as

$${\phi }_s=V_{gs}-V_{fb}-2V_{eff}{W}_0[\frac{\sqrt{2q{\epsilon }_{igzo}{n}_{eff}{V}_{eff}}}{2V_{eff}{C}_{ox}}\exp \left(\frac{V_{gs}-V_{fb}-V_{ch}}{2V_{eff}}\right)]+\omega$$

(10)

Here, we use the Schroder series ω [35] to improve the accuracy of the surface potential solution, i.e.,

$$\omega =-\frac{y}{y^{\prime }}\left(1+\frac{y{y}^{″}}{2{y^{\prime }}^2}+\frac{3y^2{y^{″}}^2-y^2{y}^{\prime }{y}^{\left(3\right)}}{6{y^{\prime }}^4}\right)$$

$$y={\left(V_{gs}-V_{fb}-{\phi }_s\right)}^2-\frac{2q{\epsilon }_{igzo}}{{C_{ox}}^2}\left\{n_0{V}_T\left[\exp \left(\frac{{\phi }_s-V_{ch}}{V_T}\right)-1\right]+N_T\frac{E_T}{q}\left[\exp \left(\frac{{\phi }_s-V_{ch}}{E_T/q}\right)-1\right]+N_D\frac{E_D}{q}\left[\exp \left(\frac{{\phi }_s-V_{ch}}{E_D/q}\right)-1\right]\right\}$$

where y′, y″ and y⁽³⁾ are the first, the second and the third derivatives of y versus φ_s, respectively.

3. Drain Current Model

According to the charge sheet model (CSM), the drain current can be expressed as the sum of drift current (I_ds1) and diffusion current (I_ds2). Therefore, we can get the expression of drain current based on surface potential as

$$I_{ds}=I_{ds1}+I_{ds2}={\displaystyle \underset{{\phi }_{ss}}{\overset{{\phi }_{sd}}{\int }}-{\mu }_{eff}\frac{W}{L}{Q}_i\left({\phi }_s\right)}d{\phi }_s+{\displaystyle \underset{{\phi }_{ss}}{\overset{{\phi }_{sd}}{\int }}{\mu }_{eff}\frac{W}{L}{V}_{T}d{Q}_i\left({\phi }_s\right)}$$

(11)

where W is the channel width, L is the channel length, φ_ss represents the solutions of φ_s at the source, corresponding to the case of V_ch = 0, φ_sd represents the solutions of φ_s at the drain, corresponding to the case of V_ch = V_ds, respectively. Here, μ_eff is the effective mobility, which is generalized using a universal power law with appropriate values of μ₀ and γ, Q_i (φ_s) is the mobile charge density based on surface potential of the unit area in the channel. μ_eff using the empirical model and Q_i (φ_s) using the Gauss law can be expressed as

$${\mu }_{eff}={\mu }_0{\left(V_{gs}-V_{fb}\right)}^{\gamma }$$

(12)

$$Q_i\left({\phi }_s\right)=-C_{ox}\left(V_{gs}-V_{fb}-{\phi }_s\right)-Q_T-Q_D$$

(13)

where μ₀ and γ are the fitting parameters of effective mobility, Q_T is the tail state trapped charge density per unit area, Q_D is the deep trapped charge density per unit area. Q_T and Q_D can be expressed by integral as follows.

$$Q_T={\displaystyle \underset{0}{\overset{t_{igzo}}{\int }}-q{N}_T\exp \left(\frac{\phi -V_{ch}}{E_T/q}\right)dx}$$

(14)

$$Q_D={\displaystyle \underset{0}{\overset{t_{igzo}}{\int }}-q{N}_D\exp \left(\frac{\phi -V_{ch}}{E_D/q}\right)dx}$$

(15)

Finally, we can get the analytical expression of drain current based on surface potential as

$$I_{ds}={\mu }_{eff}{C}_{ox}\frac{W}{L}{\left\{\left[\left(V_{gs}-V_{fb}\right)+Q_T+Q_D\right]{\phi }_s-\frac{1}{2}{{\phi }_s}^2+V_T{Q}_i\left({\phi }_s\right)\right\}}_{{\phi }_{ss}}^{{\phi }_{sd}}$$

(16)

4. Model Verification and Discussion

In this section, we will gradually verify the analytical solution derived from the previous equation and the effective charge density approach. In addition, the necessary physical analysis and discussion of future experiments will also be given, as follows:

First of all, in order to verify the effective charge density approach, we compared the relationship between the surface potential and various charge densities, as shown in Figure 2 and Figure 3. According to Equation (3), we give an effective density of states (n_eff) to replace the sum of the three densities of states (the free charge density n_free, the tail state density n_tail the deep state density n_deep) contained in Equation (2) so as to explore the relationship between the total charge density and the effective charge density through the set parameters. All the parameter values have been listed in the figures. In fact, in order to better discuss the physical meaning of the three densities of states (n_free, n_tail, n_deep) and their respective effects on the total charge density (N_toi), we also compared the relationship between the three density of states, the density of states in the subthreshold region (n_deep + n_tail) and the surface potential. In Figure 2a, we can observe that the circular symbol corresponds to the red solid line one by one, which also means that the total charge density and the effective charge density (N_eff) always maintain excellent consistency within the range of surface potential. In addition, the free charge density and the total charge density also maintain a good consistency in most of the range of the surface potential (about 0.15 V–0.6 V), which also conforms to the physical meaning of the effective state density proposed in Equation (3). In Figure 2b, the subthreshold region is amplified to further discuss the fitting effect and physical significance of different charge densities. We can observe that even in such a small range as the subthreshold region, the effective charge density remains in excellent agreement with the numerical results. Compared with the free charge density, the tail state density and the deep state density are closer to the numerical results. It is noteworthy that when φ_s is relatively small (subthreshold region), the density of deep states is dominant, which also has a great impact on the accuracy of modeling, as shown in Figure 3b. However, with the increase of E_T, n_deep + n_tail is more dominant, as shown in Figure 2b. In fact, in the actual device work, the sum of the tail state density and the deep state density is usually used as the trap state density and to dominate the electrical characteristics of the subthreshold region. This also shows the physical significance of different intervals of the model.

Subsequently, in order to verify the accuracy of the surface potential model, we use two groups of different experimental data (data1 and data2) to compare the proposed model, the regional approach with the numerical results of the surface potential φ_s, and give the corresponding absolute errors, as shown from Figure 4, Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9. The parameter values set by these two sets of data are listed in Figure 4 and Figure 6. From these two figures, we can see that the state density and size of the verified model are different. In Figure 4, we can observe that the numerical results represented by the symbols and the results of our proposed model represented by the red solid line maintain excellent consistency in all ranges, while the results of the regional method represented by the blue solid line are significantly different after the gate voltage exceeds 1 V. In fact, the results obtained by the regional method are greater than the numerical results in most ranges. In addition, the absolute error corresponding to data1 is also calculated, as shown in Figure 5. It is obvious that the absolute error obtained by our surface potential model is kept at 10⁻⁷ in a small part of the range, and is far less than 10⁻⁷ in most of the range, while the absolute error obtained by the regional method is almost kept at 10⁻². It is worth noting that we have enlarged the pairing error curve calculated in the subthreshold area in Figure 5. According to the enlarged error diagram, it is obvious that when the gate voltage is 0 V–0.2 V, the absolute error of our model remains at 10⁻⁷, and when the gate voltage exceeds 0.2 V, our absolute error drops sharply. However, the absolute error of the regional method in the subthreshold region is basically kept at 10⁻⁵. Even at the point with the smallest error (x = 0.35 V), the ordinate of our model calculated by the proposed approach is still smaller than that obtained by the regional method. In fact, when the gate voltage of our model exceeds 0.2 V, the absolute error is basically stable below 10⁻¹⁰, while the error of the regional method in the subthreshold region is almost kept at 10⁻⁵, and the absolute error in above the threshold region is kept at 10⁻². This directly reflects the extremely high accuracy of our surface potential model. In fact, the error of our model results in all ranges is far lower than that of the regional method. In addition, the regional method often lacks sufficient physical meaning in the transition region, while the effective charge density approach is continuous in all regions, which also shows the advantages of the latter. Similarly, the process of data2 validation is shown in Figure 6 and Figure 7. The accuracy of our model results is much higher than that of the regional method, and the absolute error is at least four orders of magnitude lower than that of the regional method. In particular, the error of our model is also lower than that of the minimum error point (x = 0.27 V) calculated by the regional method. It is worth noting that Figure 5 and Figure 7 have the same trend. We can observe that when the gate voltage is relatively small, the absolute error of the proposed model has a sharp downward trend. This is because when the gate voltage increases, the free charge density begins to dominate, which is also consistent with the calculation method in the model. When the gate voltage increases, the error decreases rapidly, which makes our model closer to the actual analysis model. Figure 8 and Figure 9, respectively, show the surface potential φ_s of the model under different channel voltages V_ch, which further illustrates the high accuracy of the proposed model.

Finally, we compared the experimental results in [36] with our drain current model, as shown in Figure 10 and Figure 11. In Figure 10, the extracted parameter values have been listed. Here, we verify the transfer characteristics of our model and experimental results when the drain source voltage is 1 V and 20 V, respectively. It can be observed that the symbols represented by the experimental results are consistent with the solid lines represented by our results. Similarly, in Figure 11, we also verified the output characteristics of the model under different gate voltages (0 V, 10 V, 20 V). In fact, the transfer and output characteristics of the model results are in good agreement with the experimental results, which better indicates that the model successfully predicts the I–V characteristics of a-InGaZnO TFT. The good prediction ability reflected by the model not only helps to understand the electrical characteristics, but also can further improve the performance and quality of the product. Therefore, the a-InGaZnO TFT model proposed in this paper is accurate and effective, and can be used for process optimization and circuit simulation of thin film transistors.

Future work is mainly divided into the following two points: The first one is the modeling mechanism in non-degenerate state. In fact, in the actual doping process, it is easy for the doping concentration to be too high which causes the model to be in a degenerate state. This means that the quasi-Fermi level will appear above the conduction band, which refers that the position of the quasi-Fermi level needs to be considered in the simulation process. In addition, the Boltzmann approximation can no longer be used in the calculation process. The equation derivation in the degenerate state must use the Fermi Dirac approximation, which will make the model more accurate and more complex. Therefore, it is a difficult problem to find a highly accurate and efficient analytical solution in the degenerate state. The second one is that the Lambert W function is still used as the analytical solution of the surface potential in this paper. In recent years, Lambert W function has received a lot of attention in numerical iteration due to its ability to transform exponential equations into analytical solutions. Even in modeling engineering, analytical solutions of surface potential and drain current are often expressed by Lambert W function. In fact, Lambert W function is not an elementary function, which may reduce efficiency and affect the accuracy of modeling. Therefore, reducing or no longer using Lambert W function is also an urgent problem to be solved in the process of modeling. The above two points will be overcome and realized in future work.

5. Conclusions

An analytical surface-potential-based drain current model of a-InGaZnO TFTs has been proposed in this paper. Firstly, the model uses two approximation values to derive a one-piece exponential equation and obtains an exact analytical solution of the surface potential by utilizing the effective charge density approach. Then, the problem of the unclear physical meaning of the regional method is solved by the proposed surface potential model, which is verified by comparing with the regional method. And the obtained absolute error curves prove the proposed model has high accuracy. In addition, the charge sheet model are also used to derive the drain current model according to the surface potential obtained above. Finally, the proposed drain current model is compared with the experimental data of a-InGaZnO TFTs. Excellent consistency between symbols and lines efficiently predict the electrical characteristics of the drain current model. As a result, such an accurate and effective model could become a good tool to produce the I-V characteristics of a-InGaZnO TFTs.