DOI 10.1007/s10825-008-0200-z
Three-dimensional simulation of field emission triode structure
using carbon-nanotube emitters
Yiming Li· Ta-Ching Yeh
Published online: 28 January 2008
© Springer Science+Business Media LLC 2008
Abstract Field emission (FE) triode arrays with the anodic aluminum oxide (AAO) template carbon nanotubes (CNTs) as the field emitters are successfully fabricated and ana-lyzed. Both the experimental measurement and numerical calculation are conducted to examine the electron conduc-tion properties of AAO-CNTs. Using a three-dimensional finite-difference time-domain particle-in-cell method, a set of Maxwell equations and Lorentz equation is solved self-consistently, where the FE current is computed with Fowler– Nordheim equation. We explore the FE characteristics of AAO-CNTs in the triode structure. After calibration with the measured data, we study the evolution of current density and the convergence of the electron beams on anode plate with different gate voltages, anode heights, and SiO2thickness. The current density and focus performance maintain a trade-off relationship, where a larger current density accompanies more divergent electron beam.
Keywords Carbon nanotubes· Anodic aluminum oxide · Field emission· Three-dimensional simulation ·
Finite-difference time-domain particle-in-cell method· Current density· Electron beam
1 Introduction
Carbon nanotubes (CNTs) are promising for advanced cold-cathode flat panel displays due to their chemical stability, mechanical strength, and electron emission properties [1–3].
Y. Li (
)· T.-C. YehDepartment of Communication Engineering, National Chiao Tung University, Hsinchu, Taiwan
e-mail:[email protected]
A triode type field emission (FE) display possesses more stable and effective emission properties, and higher quality screen, compared with a diode one. To obtain uniform and high FE current, it is necessary to grow vertically aligned CNT arrays on a large area with suitable tube density and di-ameter. A template method, anodic aluminum oxide (AAO) nano-template, is thus developed [4,5]. AAO has vertical pore channel, highly ordered pore arrangement, and uni-form pore size. FE triode arrays with AAO-CNTs as emitters were fabricated by standard integrated circuit (IC) fabrica-tion process in our recent work [6].
In this article, we employ a three-dimensional (3D) particle-in-cell (PIC) method coupled with the Maxwell equations to simulate the electron emission behavior in the CNT FE triode structure with the AAO template. Effect of the gate voltage, the height of SiO2(distance between the CNT and the gate), and the anode height in the triode struc-ture are further discussed for the best emission properties.
2 Experiment and simulation
The structure of AAO-CNTs and its cross section on the
x–z plane are shown in Fig. 1. To fabricate AAO-CNTs’ FE triodes, standard IC processes were used [6]. After the preparation of AAO-CNTs emitters, tetraethoxysilane oxide was deposited on AAO-CNTs’ layer as the dielectric layer of the triode by plasma enhanced CVD, and followed by the evaporation-deposition of an Al layer as the gate electrode. The pattern of triodes with a diameter of 6.5 µm was pho-tolithographically defined by a mask aligner. The top layer of Al was etched by a high density plasma reactive ion etch (RIE) system using the gas mixture of BCl3 and Cl2 as the etchant source. The oxide dielectric was then RIE etched using a gas mixture of CHF3 and Ar as the plasma source.
Fig. 1 (a) A schematic plot of the structure. (b) and (c) show a single circle of AAO-CNT and its cross section view
Fig. 2 The entire simulation box includes the whole structure
After the process of RIE, buffered oxide etchant was used to remove the remnant SiO2. The inset of Fig.4a shows a SEM image of the fabricated sample. More than 8000 CNTs were grown within a single AAO-CNTs’ structure.
To explore the electron-emission properties in AAO-CNTs, the electromagnetic PIC scheme [7,8] is used. Fig-ure2 shows the entire simulation box including the whole structure, as shown in Fig.1a. Starting from a specified ini-tial state, we evaluate the electrostatic field along the emitter surface which is determined for a given geometry and ap-plied voltage. In the FE process, the electron emission is modeled by the Fowler–Nordheim (F–N) equation [9]:
J=AE 2 ϕt2 exp −Bv(y)ϕ3/2 E , (1)
where A= 1.541 × 10−6A eV/V2 and B is a fitting para-meter which depends on the applied voltage; E is the nor-mal component of the electric field at the emitter surface,
ϕ = 5 eV is the work function of the emission material, t2= 1.1, and v(y) = 0.95−y2with y= 3.79×10−5E1/2/ϕ
is in SI unit. Determined by (1), the emission current den-sity is function of the local electric field, the work function of emitter material and the fitting parameters. The weighted
charge density and current density at the grids are subse-quently calculated. The obtained charge density and cur-rent density are successively used as sources in the Maxwell equations for advancing the electromagnetic fields. For solv-ing Maxwell equations, we use Yee-cells on a staggered time-grid. We then perform a time integration of the Fara-day’s law, the Ampere’s law, and the non-relativistic Lorentz equation, ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ∂B ∂t = −∇ × E, ∂E ∂t = − J ε + 1 με∇ × B, F= q (E + v × B) , and ∂x ∂t = v, (2)
where the velocity is derived from the deviation of the po-sition of charge particle, subject to constraints provided by Gauss’s law and the divergence of B,
∇ · E =ρ
ε and ∇ · B = 0. (3)
We notice that E and B are the electric and magnetic fields, x is the position of charge particle, and J and ρ are the current density and charge density resulting from charge particles. The time-dependent Maxwell’s equations are solved to obtain electromagnetic fields, where the Lorentz equation is solved for non-relativistic particle trajectories. We use a center-difference solution scheme to solve the fully time-dependent Maxwell’s equations. It is one of the simplest time-dependent field algorithms. However, center-difference is very susceptible to the high-frequency noise typically produced by relativistic particles. Thus we use the centered-difference algorithm only for non-relativistic parti-cles.
The charge particles are moved according to the Lorentz force F using the fields advanced in each time step. The pro-cedure is repeated for each time step until the specified num-ber of time steps is reached. The 3D FDTD-PIC, as shown in Fig.3, thus approaches to self-consistent simulation, where the space-charge effects [10] are automatically included.
3 Results and discussion
Simulated results are calibrated with the measured data be-fore analyzing different structures. Figure4a shows the mea-sured (solid line) and simulated (dotted line) current-voltage (I–V) curves with the SiO2thickness of 0.7 µm, the anode height of 100 µm and the gate voltage (VG) of 0 V when the anode voltage (VA) varying from 10 V to 1000 V.
Figure 4b shows the 3D electron trajectories and the cross-sectional plot along the x–z plane. The anode plate collects the current emitted from CNTs. The F–N plot is shown in the inset of Fig.4a. It indicates the true nature of
Fig. 3 (a) The computational scheme. (b) The flowchart of PIC procedure and the corresponding equations
Fig. 4 (a) The I–V curve of the CNT device. The SEM image of fabricated AAO-CNT sample and the F–N plot are shown in the inset. (b) 3D electron trajectories of the AAO-CNTs’ structure and the x–z plane with VG= 0 V, SiO2= 0.7 µm and anode height= 100 µm
FE. The solid line is the experiment data and the dash line is the simulation result using the calibrated model. We are fur-ther going to discuss the current density and the convergence of the electron beams on the anode plate with different gate voltages, anode heights, and SiO2thickness.
Figure5a shows the evolution of collected current with different VG, and Fig.5b presents the current density versus
VG. The electron beams of the three different gate voltages are shown in the inset. When VG= 20 V, it will have the maximum current and take the least time to get the satura-tion all over the three. It is because of the higher electric
field and larger injection velocity at the anode. But its elec-tron beam divergence seriously and will cause the loss of power. A best focus performance is when VG= 9 V. At this moment, the bundle of the electron beam is close to a cylin-der. If we decrease the gate voltage to 2 V, the electron beam will become over-focus and result in divergence on the an-ode plate.
Figure 6a shows the collected time-dependent current with different anode height. The maximum current is with anode height of 50 µm. It also takes the least time to reach the saturation. The current density as a function of anode
Fig. 5 (a) Plot of the collected current with different VG, where the anode height= 100 µm and the thickness of SiO2= 0.7 µm. (b) Plot of the saturated current density versus VG. The electron trajectories of the x–z plane are shown in the inset
Fig. 6 (a) Plot of the collected current with different anode height: 50 µm, 70 µm, 100 µm, where VG= 10 V and SiO2 thickness= 0.7 µm. (b) Plot of the saturated current density versus the anode height. The electron trajectories of the x–z plane are shown in the inset
Fig. 7 (a) Plot of the collected current with different SiO2 thickness: 0.5 µm, 1 µm, 2 µm, where anode height= 100 µm and VG= 10 V. (b) Plot of the saturated current density versus SiO2thickness. The electron trajectories of the x–z plane are shown in the inset
height is shown in Fig. 6b. When the anode height is get-ting larger, the current density becomes smaller because of the weaker electric field intensity. And it takes longer time to saturate with longer anode height. The focus performance with respect to anode heights, shown in the inset of Fig.6b, is all under-focus. Nevertheless, when the anode height is 50 µm, it is slightly over-focus, compared with the straight
electron beam with the anode height of 70 µm. Figure 7a shows the collected current with different SiO2 thickness. We find that when the gate is closer to CNT, it will eject more current and faster ejection speed because of the larger electric field intensity. However, Fig.7b shows that if the gate is very close to CNT, the electron beam will diverge seriously. When the SiO2 thickness increases to 2 µm, the
Table 1 The current densities and angles of electron beams with dif-ferent gate voltages, anode heights, and SiO2thickness
Current density Angle (mA/cm2) (degree) Gate voltage (V) 2 12.21 −20 9 71.49 0 20 272.14 15 Anode height (µm) 50 172.95 −5 70 120.2 −1 100 83.1 3 SiO2thickness (µm) 0.5 124.17 5 1 51.6 0 2 17.58 −10
Table 2 Affections of current densities from different shapes Dimensions Current density
(µm2) (mA/cm2)
Circle 38.4835 19.97
Square 38.44 18
Rectangle 38.5 19.58
Trapezoid 38.5 19.64
electron beam will over-focus. An optimal setting on the thickness of SiO2 is 1 µm. The results are summarized in Table1.
The shape of a single circle of AAO-CNTs, as shown in the inset of Fig.4a, may not be a circular form because of the process variation. We have examined the FE property of the structure with different shapes of the cross section of
x–y plane, such as circle, square, rectangle, and trapezoid. The environmental settings are the anode height is 100 µm, the gate voltage is 5 V, the anode voltage is 1000 V, and the thickness of SiO2is 0.7 µm. As listed in Table2, we can see when the four shapes have about the same volumes; their current densities are very close. As shown in Fig.8, we can see that the four shapes are about to have the same collected current and saturation time. So in a certain range the process variation of the shape will not affect the characteristic of the AAO-CNTs much.
4 Conclusions
We have studied the FE property for AAO-CNTs in the triode structure using a FDTD-PIC method. After calibra-tion, we found that the evolution of collected current on the anode as well as the current density depends upon the applied voltage, the thickness of SiO2, and the distance be-tween the anode and structure. The time for stable collected current is crucial for advanced FE technology. For more
spe-Fig. 8 Collected current and ejection speed versus different shapes of a single circle of AAO-CNTs
cific calculation, we are examining the effect of finite num-ber of CNTs, random height and position of CNTs on FE property.
Acknowledgements This work was supported in part by Taiwan National Science Council (NSC) under NSC-96-2221-E-009-210 and Contract NSC-96-2752-E-009-003-PAE, and by the ChungHwa Pic-ture Tubes, Ltd., under a 2006–2008 grant. Authors thank Prof. F.-M. Pan for supporting sample characterization.
References
1. Saito, Y., Uemura, S.: Field emission from carbon nanotubes and its application to electron sources. Carbon 38, 169 (2000) 2. Huh, Y., Lee, J.Y., Lee, J.H., Lee, T.J., Lyu, S.C., Lee, C.J.:
Selec-tive growth and field emission of vertically well-aligned carbon nanotubes on hole-patterned silicon substrates. Chem. Phys. Lett. 375, 388 (2003)
3. Ito, F., Tomihari, Y., Okada, Y., Konuma, K., Okamoto, A.: Carbon-nanotube-based triode-field-emission displays using gated emitter structure. IEEE Electron Device Lett. 22, 426 (2001)
4. Mei, X., Blumin, M., Kim, D., Wu, Z., Ruda, H.E.: Molecular beam epitaxial growth studies of ordered GaAs nanodot arrays us-ing anodic alumina masks. J. Cryst. Growth 251, 253 (2003) 5. Chen, P.-L., Kuo, C.-T.: Self-organized titanium oxide nanodot
ar-rays by electrochemical anodization. Appl. Phys. Lett. 82, 2796 (2003)
6. Lin, C.-C., Chang, K.-C., Pan, F.-M., Kuo, C.-T., Liu, M., Mo, C.-N.: Growth of carbon nanotube field emitters in the triode struc-ture using anodic aluminum oxide as the template. Diam. Relat. Mater. 16, 1388 (2007)
7. Birdsall, C.K., Langdon, A.B.: Plasma Physics via Computer Sim-ulation. McGraw-Hill, New York (1985)
8. Verboncoeur, J.P., Langdon, A.B., Gladd, N.T.: An object-oriented electromagnetic PIC code. Comput. Phys. Commun. 87, 199–211 (1995)
9. Fowler, R.H., Nordheim, L.W.: Containing papers of a mathemat-ical and physmathemat-ical character. R. Soc. Proc. A 119, 173–181 (1928) 10. Stern, T.E., Gossling, B.S., Fowler, R.H.: Further studies in the
emission of electrons from cold metals. R. Soc. Proc. A 124, 699– 723 (1929)
