Modeling the Temporal Response of Gated ZnO Nanowire Field Emitter by Considering the Charging and Self-Heating Effect for Improving the Response Speed

Abstract

ZnO nanowire is a promising candidate for large-area gated field emitter arrays. How to improve its temporal response is one of the key problems to be solved for applications. In this work, a device model for a gated ZnO nanowire field emitter with consideration of charging and self-heating effect has been established to investigate its temporal response. It is found that while the charging effect is responsible for the delay at the beginning of the pulse, the self-heating effect which induces delay due to the thermal conduction process can shorten the charging time because of its lowering of nanowire resistance. The response time can be minimized when these two effects are balanced at an optimal field which is below the critical field for thermal runaway. We further investigate the optimal response time of a nanowire with the same resistance but a different length, radius, and electrical properties. The results imply that a lower heat capacity and higher critical temperature for thermal runaway are in favor of a shorter response time, which must be taken into account in the reduction in nanowire resistance for improving response speed. All the above should be useful for the device design of a fast-response gated ZnO nanowire field emitter array.

1. Introduction

Field emitter array has important applications in field emission display [1,2,3,4,5], parallel electron-beam lithography [6,7,8,9], and X-ray source [10,11,12,13,14,15]. Until now, only CNT [16], Si nanotip [17], and ZnO nanowire [18] have been active in studies of field emitter array applications. Among them, ZnO nanowire has advantages in terms of its simple synthesis method, low cost, high uniformity, and compatibility with microfabrication techniques, which makes it outstanding in large-area gated field emitter arrays [18]. One of the key problems in its commercialization is how to further improve response speed, which is important for realizing the addressing or fast imaging function in a flat-panel X-ray source with desirable frequency.

So far, the temporal responses of several kinds of quasi-one-dimensional field emitters have been reported. It is found that the response time of CNT can be down to ~100 ns [19,20], while that for individual SiC nanowire is ~1 ms [21]. For ZnO nanowire, the response time is tens of microseconds in a MOSFET controlled device [22]. Here, it should be noted that the plasma-induced electron emission [23,24,25] is not considered since it requires a pulsed voltage of more than 10⁴ V, which is difficult to apply in a gated structure. This large difference in response time from different materials is related to the charging process under the pulsed driving voltage. Basically, in a gated field emitter device, as shown in Figure 1a, there are two charging processes. One is due to the capacitance C₀ between gate and cathode electrodes that induces a delay for the actual voltage on the gate U_g. The other is due to the capacitance C between gate and emitter that induces a delay for the actual voltage difference U_c. The equivalent circuit can be seen schematically in Figure 1b, where R and R₀ are the resistance of the emitter and electrode, respectively. Compared to CNT, SiC and ZnO nanowire have higher resistance due to their wide-bandgap semiconductor properties (for SiC nanowire, its piezoresistive effect further increases its resistance [21]), resulting in a longer delay for U_c as well as the field emission current I_FE (note that the time for turning on and off I_FE, as denoted by t₁ and t₃ in Figure 1c, is the response time). Therefore, to improve the response speed of a gated ZnO nanowire field emitter, one needs to reduce its nanowire resistance, which can be realized by tunning the resistivity or geometrical structure of the nanowire.

On the other hand, the self-heating effect which usually exists in the field emission of ZnO nanowire [26,27] could help to further improve the response speed since the semiconductor nanowire’s resistance will be reduced under a high temperature. However, as the temperature increases, a longer delay to obtain a steady thermal field emission current will be induced, which may reduce the response speed. To obtain the optimal response time, the influence of the self-heating effect on the temporal response of the gated ZnO nanowire field emitter needs to be investigated. Moreover, considering that the self-heating process is mainly determined by the electrical properties and geometrical structure of the nanowire, knowledge of the impact of the self-heating effect is also important to guide the tunning of nanowire resistance. All of these require not only experimental but also theoretical studies on the temporal response of pulsed field emission from ZnO nanowire. In theoretical works, although the charging process and self-heating effect have been considered separately in modeling the temporal response of the nanowire field emitter, a device model that takes into account both of them is still lacking, which is necessary for the design of a fast-response gated field emitter array device.

In this work, such a model has been established to investigate the temporal response of a gated ZnO nanowire field emitter. Simulations have been performed under single and continuous pulses, showing the dependence of response time on the pulsed field, that decreases first and then increases. Calculations of the optimal response time of nanowire with the same resistance but different electrical properties or geometrical structure have also been carried out to understand the proper way to tune nanowire resistance.

2. Theoretical Model

Figure 1a illustrates the device structure, where U_A is the applied anode voltage, U_G is the applied gate voltage (the voltage source is treated as ideal with no delay), and the cathode is grounded. Considering that U_A is a constant which has no influence on the response speed of the device, the equivalent circuit for calculating the charging time can be presented as Figure 1b. According to Kirchhoff law,

$$U_c\left(t\right)+R\left(C\frac{d{U}_c\left(t\right)}{dt}+I_{FE}\right)=U_g,$$

(1)

$$U_g\left(t\right)+R_0{C}_0\frac{d{U}_g\left(t\right)}{dt}=U_G,$$

(2)

where t denotes time.

To calculate I_FE, the formula of the thermal field emission current density j_TFE of thermal oxidized synthesized ZnO nanowire can be adopted, which can be expressed as [26]

$$\begin{array}{l}{j}_{TFE}=\frac{4\pi mek{T}_e}{h^3}A{\displaystyle {\int }_{E_C}^{W_l}\ln (1+e^{\frac{E_{Fs}-W}{k{T}_e}}){\left[1+e^{\frac{8\pi {\left(2m{\left|W\right|}^3\right)}^{1/2}}{3heF}v\left(y\right)}\right]}^{-1}dW}\\ +\frac{4\pi mek{T}_e}{h^3}A{\displaystyle {\int }_{W_l}^{\infty }\ln (1+e^{\frac{E_{Fs}-W}{k{T}_e}})dW}\end{array},$$

(3)

where e denotes the electron charge, m is the electron effective mass, k is the Boltzmann constant, h is the Planck constant, v(y) is an elliptic function, $y={\left[e^{3}F/\left(4\pi {\epsilon }_r{\epsilon }_0{W}^2\right)\right]}^{1/2}$, A is the ratio factor balancing the conductance between the nanowire body and surface, T_e is the effective electron temperature, E_Fs is the surface Fermi level, F is the surface electrical field, W is the electron energy normal to the surface, $W_l=-{\left[e^{3}F/\left(8\pi {\epsilon }_r{\epsilon }_0\right)\right]}^{1/2}$, ε₀ is the vacuum dielectric constant, and ε_r is the relative dielectric constant of ZnO. Here, F = F_a + F_g, where F_a is the anode field and F_g = βU_c is the actual gate field (β is the voltage-to-field conversion factor). With consideration of both the tip and sidewall emission from the nanowire when it is self-heating, I_FE can be calculated as the integral of j_TFE over the whole surface of the nanowire.

On the other hand, the self-heating effect of ZnO nanowire can be described by the one-dimensional time-dependent thermal conduction equation, as follows:

$$\begin{array}{l}\pi r^{2}c{\rho }_m\frac{\partial T\left(x,t\right)}{\partial t}dx=\pi r^2\kappa \frac{{\partial }^{2}T\left(x,t\right)}{\partial x^2}dx-2\pi rdx\sigma \left(T{\left(x,t\right)}^4-T_0^4\right)\\ -E_N\frac{2\pi r{j}_{TFE}\left(F\left(x\right),T\left(x,t\right)\right)dx}{e}+I{\left(x\right)}^2\rho \left(T\left(x,t\right)\right)dx/\pi r^2\end{array}$$

(4)

where T₀ denotes the environment temperature, E_N is the electron energy difference caused by the Nottingham effect, r, c, ρ_m, κ, σ, and ρ are the radius, thermal capacitance, mass density, thermal conductivity, Stephan–Boltzmann constant with an emissivity coefficient, and resistivity of the nanowire, respectively. A thermal excitation model with the expression as $N_d{e}^{-E_d/kT}$, where N_d and E_d are the defect state density and the excitation energy, has been used to describe the temperature-dependent ρ of ZnO nanowire. The boundary conditions for Equation (4) are as follows:

(i)

$\frac{\partial T\left(L,t\right)}{\partial x}=-\sigma {\kappa }^{-1}\left(T{\left(L,t\right)}^4-T_0^4\right)-{\kappa }^{-1}\frac{I{E}_N}{e\pi r^2}$

(ii)

$T\left(0,t\right)=T_0$

(iii)

$T\left(x,0\right)=T_0$

It should be pointed out that since I_FE has a spatial distribution along the nanowire, $R={\displaystyle {\int }_0^L\rho \left(T\left(x,t\right)\right)dx/\pi r^2}$ and $R{I}_{FE}={\displaystyle {\int }_0^{L}I\left(x\right)\rho \left(T\left(x,t\right)\right)dx/\pi r^2}$, where L is the nanowire length. More details for the self-heating model and material property parameter values can be found in our previous work [26]. Considering that this is a one-dimensional model and that the actual nanowire may have a thinner tip and thicker base structure, the latter case can have a faster heat dissipation and higher maximum steady temperature.

3. Simulation Results

According to our previous work [26], N_d = 2 × 10²⁶ m⁻³, E_d = 0.23 eV, L = 5 μm, r = 10 nm. In addition, considering that the electric field for the field emission of ZnO nanowire is ~5.4 V/nm and the applied gate voltage of the gated device is ~150 V, we assume F_a = 1 V/nm and β = 3 × 10⁷ m⁻¹. By setting C = 3 fF and R₀C₀ = 2 μs to obtain a same order of magnitude delay for both U_g and U_c (noting that if RC is much smaller than R₀C₀, the influence of the self-heating effect cannot be investigated since the delay for U_c will be nearly equal to that of U_g), the temporal response of U_c, T_max (maximum temperature of nanowire), and I_FE under a single square-wave pulse with different U_G has been simulated, where the result under a pulse width of 40 μs is shown in Figure 2a–c. Here, U_G = U_g = U_c = 0 V, when t = 0 s and T₀ = 300 K. From the normalized U_c versus t characteristics, as shown in the inset of Figure 2a, it is seen that the delays for both the rise and fall of U_c will reduce when U_G increases. Unlike the fall region, the normalized U_c curves under different U_G have the same shape from the beginning to the rise at a certain time (~10 μs). After that, the normalized U_c under a higher U_G will increase faster, which induces a shorter rising time. The underlying mechanism is as follows: At the beginning, U_c is not large enough to drive a considerable amount of I_FE to cause heat accumulation. Only when I_FE increases to a certain value that can induce an increase in T_max will R then reduce, which reduces the rising time of U_c. This is consistent with the result shown in Figure 2b, that T_max remains around 300 K for ~10 μs from the beginning before it starts to increase. In the normalized I_FE versus t characteristics, as shown in the inset of Figure 2c, t₁ and t₃ are defined as I_FE rises to 90% and reduces to 10% of its steady value (for the case that I_FE keeps increasing during the pulse, its maximal value will be used in the definition instead of the steady value). It is worth noting that this definition is different from the usual one for voltage that is calculated from 10% to 90% of its steady value. This is because it takes nearly half of the total rising time for I_FE to rise from 0 to 10%, as seen in the inset of Figure 2c, which is due to the exponential relationship between I_FE and U_c. It is seen that t₁ is much longer than t₃ because at the beginning of the pulse, the nanowire is at room temperature, which has a larger R, and the self-heating effect will induce an additional delay for I_FE to reach its steady state. With t₁ and t₃, the total response time of the gated ZnO nanowire field emitter can be obtained, which is the sum of them. Figure 2d shows the dependence of response time on U_G and its corresponding maximum surface field F_max (defined as the field when I_FE reaches its steady or maximum value), where the dot dash line represents the critical field for the thermal runaway of the simulated ZnO nanowire, which refers to U_G of 164 V. Note that the range of U_G in the simulation is below 165 V to avoid the melting of the nanowire, of which the melting point is 1000 K (although U_G of 165 V can induce thermal runaway, its critical time is much longer than the pulse width of 40 μs, for which the corresponding T_max will not reach the melting point). It is seen that while t₃ decreases monotonously with U_G due to the reduction in R, t₁ will first decrease then increase with U_G, which is caused by the combination of charging and self-heating effects. As can be seen in Figure 2c, I_FE has a slower increasing rate at both the beginning and end during the time of t₁. While the former region is limited by the charging process, the later one is due to the self-heating effect, which is related to the thermal conduction on the nanowire. As U_G increases, the time for the former region will be reduced due to the reduction in the delay for U_c. However, a higher T_max under a larger U_G also causes a longer time for heat conduction, which increases the time for the later region. As a result, the minimum t₁ can be obtained only when the above two effects are balanced (the optimal value is under U_G of 155 V in our case). Since t₁ is much longer than t₃, the total response time t₁ + t₃ has the same relationship with U_G and F_max as that of t₁. Considering that the optimal pulsed field (corresponds to U_G of 155 V) is below the critical field for thermal runaway (corresponds to U_G of 164 V), the critical temperature T_c for thermal runaway is important for optimization because a lower T_c will reduce the upper limit of T_max under the optimal field, which induces a longer t₁ due to the larger R.

The effect of pulse width on the minimum response time t₁ + t₃ and the corresponding U_G as well as the maximum T_max and F_max during the pulse is presented in Figure 3, where Figure 3a is the result for t₁ + t₃ and T_max; Figure 3b is the result for U_G and F_max. It is seen that when the pulse width is long enough for I_FE to reach its steady state (larger than ~40 μs in our case), the minimum t₁ + t₃ as well as the optimal T_max, U_G, and F_max are independent in terms of pulse widths, which are ~25 μs, ~710 K, 155 V, and ~5.16 V/nm, respectively. However, when the pulse width is below ~40 μs, the minimum t₁ + t₃ will reduce, and the optimal T_max, U_G, and F_max will increase as the pulse width decreases. This is because when the pulse width is too short, I_FE keeps increasing during the whole pulse. As mentioned above, I_FE has a slower increasing rate at the beginning and end during its rising process. To minimize t₁, the increasing rate for I_FE at the end of the pulse needs to be as slow as possible (in other words, the operation time t₂, which is equal to the time for I_FE to increase from 90% of its maximum to the end of the pulse, needs to be as long as possible), which needs a larger U_G as well as F_max to reduce the delay for U_c. As a result, T_max will also increase. It is worth pointing out that although a shorter pulse width induces a shorter response time, the operation time t₂ will also reduce to nearly zero. Furthermore, when the pulse width is too short, U_g does not have enough time to rise to the applied U_G. Therefore, this abnormal operation situation is not under consideration in the following section.

In the actual application, such as the application of the flat-panel X-ray source, the gated ZnO nanowire field emitter may need to work under continuous pulses. To investigate the influence of the self-heating effect in this case, its temporal response under continuous pulses with different U_G is also simulated, for which the result with a pulse frequency of 25 kHz and U_G of 164 V is shown in Figure 4. To maximize the duty ratio, we set the time for the off state (U_G = 0 V) to be t₃, which has been calculated from the single pulse case. Figure 4a shows the U_G, U_g, and U_c versus t characteristics. It is clearly seen that under such a short time for the off state, both U_g and U_c cannot decrease to zero. However, for I_FE, as shown in Figure 4c, it can be reduced to 10% of its maximum, which means that the device can be turned off. The underlying mechanism is due to the exponential relationship between I_FE and U_c. From Figure 4c, the total response time t₁ + t₃ can be extracted as ~17.3 μs, which is shorter than that in the single pulse case (~27.4 μs) under the same U_G with a pulse width of 40 μs. This is because during the short time of the off state, T_max cannot cool down to room temperature, as shown in Figure 4b. As a result, R can remain at a lower value during the continuous pulses, which reduces the delay for U_c as well as I_FE. Considering that the period of the pulses is 40 μs, t₂ is ~22.7 μs, and the duty ratio for I_FE can be calculated as ~57%, which is much smaller than that for the applied U_G (~96%).

To further investigate the optimal response time under the continuous pulses, its dependence of U_G has been simulated, which is shown in Figure 5a. Similar to the single pulse case, as shown in Figure 2d, t₁ + t₃ under continuous pulses also first decreases then increases with U_G. The major difference between them is when U_G is larger than the critical value for thermal runaway (164 V), the response time will rapidly increase in order to avoid the melting of the nanowire. Considering that the nanowire will not cool down to room temperature during each pulse, if the nanowire cannot reach a steady self-heating state under a given U_G, the heat accumulation during each pulse can cause the nanowire to become thermal runaway at the end. Therefore, when U_G is larger than 164 V, it requires a longer t₃ to cool down the nanowire, which induces a rapid increase in t₁ + t₃. A comparison of the duty ratio for I_FE and U_G is also given in Figure 5b. It is seen that the maximum duty ratio for I_FE is ~66% at U_G of 155 V, which is smaller than that for U_G (~96%). To increase the duty ratio for I_FE, one needs to apply the continuous pulses with a lower frequency, which can increase the time of t₂.

Finally, we discuss the optimization of nanowire resistance for improving the response speed. Assuming that R is reduced by one half, it can be realized by shortening L to L/2 (e.g., 2.5 μm), increasing r to $\sqrt{2}r$ (e.g., ~14.14 nm), increasing N_d to 2N_d (e.g., 4 × 10²⁶ m⁻³), or decreasing E_d to E_d−kTln(2)/e (e.g., ~0.2121 eV), respectively. The optimized response time t₁ + t₃ and the corresponding T_max for each case are calculated as shown in

Table 1. Comparison on the temporal response of ZnO nanowire with different L, r, N_d, and E_d.

No.	L (mm)	r (nm)	Nd (m−3)	Ed (eV)	t1 + t3 (μs)	Tmax (K)	Tc (K)
1	5	10	2 × 1026	0.23	24.9	712.4	1045.2
2	2.5	10	2 × 1026	0.23	12.4	767.6	1125
3	5	14.14	2 × 1026	0.23	17.2	687.6	1122.9
4	5	10	4 × 1026	0.23	15.4	700.3	1203.9
5	5	10	2 × 1026	0.2121	18.2	629.9	964.7

. Here, other parameters remain unchanged. It is seen that all the cases have a shorter t₁ + t₃ than that for the original nanowire (case 1) because of their reduced R. By shortening L (case 2), one can obtain the shortest t₁ + t₃ due to its smaller volume that causes a smaller heat capacity. Considering that it takes a shorter time to reach the same temperature, the optimal T_max can be higher, which leads to a lower R and shorter t₁ + t₃. In comparison, we obtain a longer t₁ + t₃ than case 2 by increasing r (case 3) since it has a larger heat capacity, which results in a lower optimal T_max as well as a higher R. Therefore, in the modification of the geometrical structure of nanowire, one needs to take into account the nanowire volume, which determines its heat capacity. To further reduce R, the nanowire resistivity needs to be reduced, for which the result can be seen in the comparison between cases 4 and 5. As mentioned before, a higher T_c benefits for a higher optimal T_max, which leads to a shorter response time due to the lower R. For a better comparison, T_c for each case is also listed in Table 1. Considering that T_c increases with both N_d and E_d according to our previous work [26], a shorter t₁ + t₃ can be obtained by increasing N_d rather than decreasing E_d. Based on these results, the guideline that one needs to keep T_c as high as possible and the nanowire volume (heat capacity) as small as possible when reducing the nanowire resistance for improving the response speed is suggested. It is worth noting that although C is related to the geometrical structure of gate and nanowire, which can be different among cases 1, 2, and 3, this conclusion is still validated. This is because C will increase with r due to the larger tip surface area, which induces an even longer delay. While L reduces, the gate can be fabricated at the same distance with respect to the nanowire tip, which has a negligible influence on C. Considering that in the actual case of nanowires field emitter arrays, they inevitably have differences in their electrical properties and geometrical structures, which may cause deviation from the above conclusions, such as the relationship between response time and gate voltage, it is recommended that the model is validated by testing the pulsed field emission of individual nanowire, which can avoid the average effect.

4. Conclusions

To summarize, a device model for a gated ZnO nanowire field emitter with consideration of charging and the self-heating effect has been established. Both simulations under single and continuous pulses show that the optimal field which minimizes the response time is below the critical field for thermal runaway, indicating that a higher critical temperature is favored for improving the response speed. Compared to the single pulse case, the continuous pulses case has a shorter response time because of the heat accumulation during each pulse, which keeps a low nanowire resistance. Possible ways to further improving its response speed are also discussed, which shows that the heat capacity as well as the critical temperature are two important factors in the optimization of nanowire resistance. All of these are important for the device design of a fast-response gated ZnO nanowire field emitter array.