sphere-shape tip. The cathode current is obtained by integration of the current density over the emission surface. The gate current is derived by the same integration, but over part of the emission area. A procedure to extract the values for the parameters of the model is also given. The model and the procedure has been applied to experimental devices to demonstrate its accuracy.
Index Terms—Field emission triode, Fowler–Nordheim, simu-lation model.
I. INTRODUCTION
F
IELD emission triode (FET) is a solid-state device which features a miniature vacuum tube triode formed by appli-cation of modern intregrated circuit (IC) processing technol-ogy. It can be used in the applications such as flat panel display [1]–[4], microwave tubes [5]–[7]. A typical Spindt type FET consists of a cathode, a control gate, and an anode similar to that of a conventional vacuum tube triode. When a high voltage is applied between the gate and the cathode, a very large electric field is formed on the tip of the cathode due to the sharp geometry of the tip and the short gate-to-cathode distance. Following the Fowler–Nordheim (F–N) tunneling mechanism electrons will be emitted from the surface of the cathode to the anode which is biased at a proper high voltage [8]–[10]. However, since the gate is always biased more positively than the cathode, a large gate current can flow at the zero or moderate positive anode voltage.The applications of FET in circuits have triggered a demand for its device model for circuit simulation. A simple, efficient and accurate FET model is pursued. Some works had been done on modeling the field emission diode [12]–[14] based on the F–N J-E relationship. In their approach, so-called electric field enhancement a and area b factors were used to transform the local J-E relationship to the global – characteristics. The approach is somehow questionable since the electric field is not constant across the tip and the factor b is a voltage dependent parameter [15]–[17]. A different approach, in which – equation is obtained through integration of the current density over the tip surface, is proposed by Nicolaescu and Manuscript received Jaunuary 5, 1998; revised April 13, 1998. The review of this paper was arranged by Editor C.-Y. Lu.
The authors are with the Department of Electronics Engineering, National Chiao Tung University, Hsinchu, Taiwan, R.O.C.
Publisher Item Identifier S 0018-9383(98)07451-6.
a model for the FET amplifier. Most recently, Lu et al. [21] developed an FET triode model with a parameter extraction procedure for SPICE simulation.
In this work, a new FET model which is based on the F–N tunneling mechanism, (but its J-E relationship is related with the device geometry parameters), is proposed. An extraction procedure is also derived to extract parameter values based on the device – characteristics. The model is developed on the diode as well as on the triode. Although the model is related with the device physical geometry parameters, the model is simple and easy to be incorporated into the circuit simulation program such as SPICE.
II. DEVICE MODEL
The device model is first derived for a diode and then extended for a triode.
A. The Diode Model
Field emission diode modeling is generally based on the F–N J-E relationship which relates the field emission current density, to the electric field on the surface of the tip, and the work function, This relationship is given below as [9], [10]
(1) where
where is expressed in terms of A/cm in volts/cm, and in electron volts. In this approach, instead of lumping the total device current as a global parameter by treating to be a constant over an effective area of the emission tip of the diode, we calculate the device current by integrating the current density by taking the variation of the electric field 0018–9383/98$10.00 1998 IEEE
Fig. 1. The “Saturn” model, given by Jensen et al., in which the tip is a floating sphere suspended near a ring of charge along the symmetry axis.
across the surface of the tip into account. Hence, the electric field of the emission is first considered.
Previously, some work had been done on investigating the electric field of the emission tip of different shapes [22]–[24]. For example, Veen et al. investigated the electric field distribution of a tip of a shape of a paraboloid [23]. Jensen et al. [24] gave an expression for the electric field of a Saturn model which is shown in Fig. 1, where the tip is a floating sphere suspended at the symmetry axis of a gate of a ring shape. The field at the position which is at an angle of with respect to the symmetry axis of the surface is given by
(2) where is the maximum electric field at the position of of the sphere and is a fitting parameter. In this work, the Jenson’s result is used for our approach where is taken 0.76. In fact, the electric field obtained by assuming
is almost the same as that obtained from the Veen’s model. The maximum difference between two field distributions is less than 1.5% for So, we consider that (2) is a good approximated formula for describing the electric field distribution on the surface near the apex of the tip. Hence, in this work, we assume that the shape of the tip near the apex is a sphere and its field distribution is that of (2).
In practice, the field emission current is measured as a function of the applied voltage. The conventional geometrical factor is used to correlate the electric field at the apex of the tip to the voltage [25], i.e.,
(3) With (1)–(3), the angular current density distribution can then be calculated. Fig. 2 shows a calculated angular current density distribution for several gate voltages for eV and cm which is a value calculated by using the parameter extraction procedure which is to be presented later in this paper for the first demonstrated device. It can be seen that the current density is an exponential-like function of The cathode current can then be obtained by integrating the
Fig. 2. The angular current density distribution calculated from (1)–(3) for several gate voltages. e is the critical emission angle beyond which the emission becomes negligible. The larger theVg; the larger the e:
Fig. 3. The computed value ofe in terms ofVgfrom (1)–(3). Thee is defined asJ( = e) = 0:1%J( = 0): The value of eis almost linearly proportional toVg:
current density over the emission surface of the tip, i.e., (4) where is the radius of the sphere-shape tip and is the critical emission angle beyond which the current emission becomes negligible, i.e.,
for (5)
From Fig. 2, the value of is seen to be a function of The larger the gate voltage, the larger value of This indicates that a larger results in a larger emitting area. in terms of can be computed from (1)–(3). Fig. 3 shows such a plot for the above example where is defined as It can be seen that the value of is almost linearly proportional to the gate voltage. Here, we also assume that the radius of the tip is large enough to allow the planar F–N J-E relationship to be applied
Fig. 4. The value of the term, Vg=B; computed for several work functions. The value of this term is smaller than one even the electrical field is up to
5 2 109 V/cm.
[26] and the value of in (1) is independent of electric field. Using (1)–(4), we can obtain the effective cathode current for an array of emission tips to be
(6)
where and is the number of the
emission tips.
In (6), since from (5), the obtained cathode current is:
(7) In (7), the term, , is a function of . This term can be calculated by using (1) and (3). Fig. 4 plots the value of this term for several work functions. From this plot, it is seen that it is smaller than one even for the electric field up to 5 10 V/cm. The maximum electric field value in practice must be determined by the dielectric strength of the insulating material between the gate electrode and the substrate. Since the insulator is most likely to be silicon oxide or silicon nitride, the dielectric strength is lower than 5 10 V/cm. Hence, (7) converges. Also, in (7), it is noted that is expressed in terms of the radius “ ” of the emission tip and it is proportional to instead of as in the conventional field emission diode current-voltage expressions [9], [25].
B. The Triode Model
For a triode, the current emitted from the cathode is shared, mostly by the anode and by the gate. In this case, as in Fig. 5 which is the plot of one of curves of Fig. 2, the current emitted from within the surface of the tip from to the angle is collected by the anode to be the anode current and the current
Fig. 5. The anode current component and the gate current component obtained from the current density distribution along the cathode surface respectively.a is the angle beyond which the cathode emitted charges are collected by the gate to be the gate current. The anode collects emitted currents emitted from the cathode for the angles from = 0 to a:
emitted from the surface for the angle is collected by the gate to be the gate current. The gate current can be obtained by integrating the emitted current density over the range of to from (1)–(3), i.e.,
(8) where
In (8), also from (5), hence is a function of
is a parameter related with the collecting ability of the anode. For a large anode voltage it gives a large value of For the saturation region of the FET device, equals to and all the cathode emission current becomes the anode current. For the triode region, is between 0 and Fig. 6 is the plots of the gate current versus computed from (8) for a set of different gate voltages. These plots are very similar to those measured curves of the gate currents versus the anode voltage Hence, it is assumed that, for the triode region of the FET, is linearly proportional to i.e.,
(9) where is the threshold voltage of the FET device, above which the anode starts to collect the emitted currents. The value of is dependent on the anode configuration. A smaller distance from anode to gate gives a lower
is also dependent on the gate voltage. A larger gate voltage causes a larger value of
To verify the above assumption, the values of versus are computed from the data of Fig. 6 and the measured
Fig. 6. The computed gate current as a function of the emission anglea
from (8) for different gate voltages.
Fig. 7. The plots of the computed values of a versus Va which are computed from the data of Fig. 6 and the measured gate currents based on (8) for several gate voltages. Four plotted curves are roughly linear and their extrapolated points intersect at the horizontal axis atVa= 0; 4, and 6 V for
Vg 68 V, = 70 V and = 72 V, respectively.
gate currents respectively for several gate voltages by using (8), and the results are plotted in Fig. 7. The plotted curves are approximately linear and their extrapolated values at the horizontal axis are approximately 0, 4, and 6 V for
V, V and V, respectively.
In the above equation, is a factor which relates the anode voltage to the angle It is expected to be a function of the gate voltage. Fig. 8 is the plots of versus also computed by using (8) from the same Fig. 6 and the measured gate currents but with as the varying parameters. It is seen that increases first, due to the increasing value of (see Fig. 3), with respect to then decreases due to the increasing emitting area collected by the gate (see Fig. 4). Hence it is then assumed that is a function of of second order polynomials, i.e.,
(10)
Fig. 8. The plots ofaversusVgcomputed by using (8) from Fig. 6 and the measured gate currents but withVa as the varying parameters.aincreases first then decreases with respect toVgfor allVa:
where and are parameters to be extracted in the later extraction procedure.
Since the emission current always equals the anode current plus the gate current, the anode current can be obtained by subtracting the gate current of (8) from the cathode current
of (7), i.e.,
(11) III. PARAMETER EXTRACTION PROCEDURE
In this section the extraction procedure for values of the necessary parameters of the above (7)–(10) are described. To describe the procedure more clearly, device data which were published in this area are used as demonstration examples.
The first device example used is that published in the Betsui’s paper [11]. The device was an silicon field emitter arrays of 6400 (80 80) tips. The radius of the bullet-shaped tip was reported being less than 20 nm. The spacing between tips was 4 m, so the tip density was 6.25 10 cm . The diameter of the gate aperture was 2 m, which was larger than that of its silicon dioxide mask. The distance between the anode plate and gate was 1 mm. In Betsui’s measuring apparatus, the gate was grounded and negative voltage was applied to the cathode to extract electrons. In this work, the cathode voltage is assumed to be the reference grounded voltage.
In (7)–(10), and are unknown
param-eters. In the extraction procedure, these parameters are not extracted individually. They are extracted through two param-eters: and is a parameter related with the emission area and is a parameter related with the geometrical factor :
(12a) and
Fig. 9. The computed summation of the first four terms of the summation part,63n=0(01)n(n + 1)!( Vg=B)n; of the parameter A0as a function of
Vgfor = 9:252105cm01and = 4:5 eV. It is a slowly varying function ofVgfor the whole 0–300 V gate voltage region.
With these two parameters, the emission current can be ex-pressed as:
(13) Fig. 9 plots the value of the summation of the first four terms of the summation part of the parameter versus
for taking cm and eV (It is of
enough accuracy by just taking the first four terms since the term converges fast). It can be seen that for the whole 0–300 V gate voltage region it is a slowly varying function of Hence, it can be considered to be constant during extraction. Also, it was reported that the parameter is a slowly varying function of [27], hence the parameter
is also a slowly varying function of It can also be considered to be constant during extraction.
Since we can rearrange (13) as
(14)
for the diode characteristics, we can plot versus to obtain a straight line. From the slope and the inter-section of the straight line with the coordinate axis, the values of and can be obtained. Fig. 10 shows such a plot of the device, which were operated at V. The plotted curve is seen to be a straight line. The values obtained for and are 6.27 10 A/V and 708 V, respectively. The term of which is a slowly varying function of in the gate current equation, can be obtained from the values of and from (12a) and (12b), respectively. For this example, its value is 6.22 10 1.94 10 The
values of and of this example are
compiled in Table I. From these values, if the work function and the number of tips are known, the radius of the tip can be estimated. For this example, the number of tips is 6400. A value of 19 nm is obtained for the radius of the tip if a
Fig. 10. Theln(Ic=Vg3) versus 1=Vg plot of the cathode current of the experimental device operating atVa= 260 V. It is a closely approximated
straight line asln(Ic=Vg3) = ln(6:27 2 1005) 0 (708=Vg):
TABLE I
EXTRACTEDPARAMETERS OF THEFIRSTEXAMPLE
work function of 4.5 eV and is assumed. This value is consistent with that reported in [11].
To this point, the only two parameters of unknown values are and Their values can be obtained from Fig. 7. In the figure, the slope and the intersection of straight lines with the coordinate axis for each give the values of and For this example, is determined to be approximately 0, 4, and 6 V for V, V and V, respectively, and the values of determined as a function of are plotted in Fig. 11, where the square dots are the extracted data and the solid line is the fitted curve for (10). The fitted values for parameters and are also shown in Table I.
With all the values obtained in Table I, the gate current can be calculated and reconstructed from (8) and (9). Fig. 12 shows the reconstructed gate current (the solid curves) versus with the measured data (the dotted curves). With the cathode and gate currents obtained, the anode current is calculated from (11) and is shown in Fig. 13. In the above two figures, the calculated reconstructed currents and the measured currents match very well.
To verify the accuracy of this model, a second example is also given. Fig. 14 shows the I–V characteristics of an FET triode with 100 emitter tips [28], where the dotted curves are
Fig. 11. The extracted as a function of Vg for the experimental device of [11]. The square dots are the extracted data and the solid line is the fitted curve for (10).
Fig. 12. The reconstructed gate current (the solid curves) and the measured data (the dotted curves) versusVa:
Fig. 13. The reconstructed current data (the solid curves) computed from the extracted values of the model parameters and the original measured data (the dotted curves). Two sets of characteristics match very well.
the data points obtained from [28] and the solid lines are reconstructed curves from the model. The values derived for
Fig. 14. The I–V characteristics of an FET triode with 100 emitter tips. The square curves are the measured data which were obtained from the Holland’s paper [28]. The solid lines are the reconstructed curves from the model.
TABLE II EXTRACTEDPARAMETER
¯S OF THESECONDEXAMPLE
the parameters are summarized in the Table II. This example also verify the accuracy of the model.
IV. CONCLUSION
In this work, a simple but accurate physical model for FET device has been developed. The model can be used in the circuit simulation programs such as SPICE. For the model, the cathode current based on the F–N relationship is obtained by integration of current density on the surface of the emission tip. The emission angle is introduced to describe the emission area. The gate current is derived by same integration but on a different emitting area. A procedure to extract the values for the model parameters has also been given. The model has been applied to experimental FET devices to demonstrate its accuracy.
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Chi-Wen Lu was born in Tainan, Taiwan, R.O.C., on October 11, 1965. He received the B.S. de-gree in electronic engineering from National Taiwan Institute of Technology, Taipei, in 1991, and the M.S. degree in electro-optics from National Chiao Tung University, Hsinchu, Taiwan, in 1994, where he is currently pursuing the Ph.D. degree in the Department of Electronics Engineering. His Ph.D research focuses on the modeling of field emitter transistor, the driver circuit design for field emission display, the built-in current sensor design for IDDQ testing and the methodology of IDDQ testing for deep-submicron circuit.
Chung Len Lee (S’70–M’75–SM’92), photograph and biography not avail-able at the time of publication.
