Lateral field emission devices have been characterized and degradation tested for >1000 h to study stability and reliability. Two types of planar device structures, diode and bowtie, were studied. These nanoscale devices have 10–20 nm tip to tip or tip to collector dimensions with the tips fabricated from Au/Ti. Typical currents of 2–6 nA at 6 V were measured. The devices were placed on lifetime tests in a vacuum of <10−8 Torr and biased at 6 V DC for >1000 h. Seven total devices were tested with one failing at 300 h. and three of the devices showed <5% degradation in current until 1400 h when testing was stopped, and three other devices showed a sudden drop of ≈20% ranging from 700 to 900 h. Optical microscope images of one of the devices that failed catastrophically at 350 h show physical arc damage where the bond pad narrows to the emitter trace. Scanning electron microscope images of a bowtie part that completed 1400 h of operation showed no obvious erosion or damage to the tips.

Planar, Au/Ti diode, and bowtie type field emitter devices are being developed as stable, long-lived electron sources for an increasing range of applications, such as ultrafast optical spectroscopy,1 femtosecond surface plasmonic,2 and vacuum nanochannel transistors.3,4 Nanovacuum channel transistors (NVCTs), relying on ballistic electron transport in vacuum, promise high operating speed, low energy loss, high temperature operation (>400 °C),5,6 and radiation immunity performance.4,5 Vacuum microelectronic devices are favorable for a variety of applications ranging from sensors7 and field emission displays8 to high-performance integrated circuits (ICs).9 In the past, only a few practical implementations of NVCT devices at the circuit level have been reported even though the concepts and modeling of vacuum ICs have been described.10,11 Improvement in NVCT device stability, particularly cathode reliability, is required for the practical implementation of vacuum microelectronic devices in ICs. Past work9 has shown successful fabrication and implementation of nanodiamond-vacuum field emission-based differential amplifiers as an example of vacuum ICs, which suggests the possibility of future, real world applications of planar, vacuum nanoscale field emitter devices. Lateral field emission devices12 utilizing Au/Ti as the emitters with low threshold voltage, stable emission current, and high voltage gain have been observed, allowing their further implementation into high speed, nanoscale vacuum electron devices.

Despite the interest in such field emission devices, a primary criticism is in the reliability and lifetime performance. Some variability is inevitable in field emitter devices, even with carefully characterized manufacturing processes and high-specification voltage sources, because of the nonlinear emission current response to both the tip apex size distribution and small gate voltage fluctuations. While significant improvements in reliability and lifetime of field emission displays were achieved in the 1990s, little of this development was published. Hence, data on gated field emission reliability and degradation are limited, particularly for nanoscale structures. The aim of this article is to assess the long-term stability performance of Au/Ti, two terminal bowtie and diode field emission structures.12 A detailed device fabrication discussion is provided in Sec. II. Experimental setup and test methods are provided in Sec. III including the lifetime test system, followed by device characterization and lifetime results in Sec. IV, and failure analysis in Sec. V.

The field emitter die was fabricated using a novel method, and the detailed fabrication process is described elsewhere.13 The nanodevices were fabricated on 30 nm thermal SiO2 on silicon substrates. The patterning was done with PMMA-based electron beam lithography (EBL). Then, a 5 nm adhesion layer of Ti and a 20 nm layer of Au were deposited via electron-beam evaporation followed by a metal liftoff process. The device nanogap size was tuned by changing the lithographic dose, with the smallest average gap size being ≈10 nm. Then, SiO2 was etched with a two-step process of CF4 reactive-ion etching (RIE) and buffered oxide (BOE) wet etch to create an undercut, which prevents charging during operation. The contact pads were fabricated via a subsequent photolithography step. Two device types, diode and bowtie, were studied in this effort. Figure 1 shows the field emission scanning electron microscope (SEM) image of (a) diode and (b) bowtie structures with emitter to electrode gaps of 10–20 nm. These main structures of the devices are ≈100 nm wide reducing down to narrow tips with radii <10 nm. The fabricated devices had a distribution of gap sizes that differ from the nominal size due to process variations. Also, current limiters such as a ballast resistor14 were not present, which normally increases the probability of tip failure by arcs.15–17 After fabrication, a section of wafer was diced and attached to a printed circuit board where the pads were wire bonded out for connection through a multipin ribbon cable. The entire circuit board was placed in vacuum for characterization and testing. A photograph of the board with the test die is shown in Fig. 1(c).

FIG. 1.

SEM images of the lateral devices (a) diode structure, (b) bowtie structure, and (c) image of a test die wire-bonded to a printed circuit board with multiple diode and bowtie structures. This board is placed in the vacuum test chamber for characterization and lifetime testing.

FIG. 1.

SEM images of the lateral devices (a) diode structure, (b) bowtie structure, and (c) image of a test die wire-bonded to a printed circuit board with multiple diode and bowtie structures. This board is placed in the vacuum test chamber for characterization and lifetime testing.

Close modal

Two separate test chambers were used for characterization and long-term stability studies. I–V characterization was performed in a stainless test chamber. It was equipped with electrical feedthroughs, thermocouple feedthroughs, and a multipin high vacuum feedthrough connected to a multiwire ribbon cable. A turbomolecular pump backed by a roughing pump is used to maintain high vacuum (<5 × 10−8 Torr) inside the chamber. The schematic test jig is shown in Fig. 2(a) and consists of an aluminum platform, two low temperature cofired ceramic (LTCC) isolators, a molybdenum heating block (for future high temperature characterization studies), and the test die and ribbon cables connected to the test die. Initial I–V characterization experiments were carried out using a Keysight B2902A source measurement unit (SMU). A DC sweep up to 8 V was applied to the collector structure while the emitter was kept at 0 V for the diode device, or one emitter was held at 0 V while the other was biased up to 6 V for the bowtie structures. Reverse bias testing was performed for each device type.

FIG. 2.

(a) Schematic of test schematic jig in the characterization chamber consisting of an aluminum holder with the test circuit board sitting on top connected by a ribbon cable and (b) schematic of the lifetime chamber system test jig with the circuit board placed on an aluminum backplate and connections with a ribbon cable. The heating block shown in the schematic will be used in future high temperature experiments. (c) Test circuit schematic consists of an in-house developed transimpedance amplifier, a data acquisition (DAQ) system, and an arc detection circuit which triggers if an electrical arc takes place between the tip and the collector.

FIG. 2.

(a) Schematic of test schematic jig in the characterization chamber consisting of an aluminum holder with the test circuit board sitting on top connected by a ribbon cable and (b) schematic of the lifetime chamber system test jig with the circuit board placed on an aluminum backplate and connections with a ribbon cable. The heating block shown in the schematic will be used in future high temperature experiments. (c) Test circuit schematic consists of an in-house developed transimpedance amplifier, a data acquisition (DAQ) system, and an arc detection circuit which triggers if an electrical arc takes place between the tip and the collector.

Close modal

A separate test chamber equipped with electrical and thermocouple feedthroughs was used for the lifetime test. Along with a turbomolecular pump and roughing pump, an ion pump is also used to maintain the high vacuum (≈6 × 10−9 Torr). The separate lifetime chamber was used so that without any interruption, long-term stability study can be carried out.

The test jig schematic for the lifetime chamber can be seen in Fig. 2(b) and consists of an aluminum platform, a molybdenum heating block, an LTCC isolator, an aluminum back plate, and the test die circuit board. However, the characterization carried out in this work does not include any high temperature results as the die chip is mounted on a PCB board. Future work will include long-term stability experiments in a high temperature environment, where the molybdenum heating block will be used to heat up the devices. The emitters are connected to the electrical feedthroughs through a Kapton coated ribbon cable which is attached to the board. The system can carry out tests on three devices simultaneously at the same bias voltage. For these experiments, a constant DC bias of 6 V was applied to the collector of the diode or to one of the emitters for the bowtie structure, and the emission current data were recorded using a labview data acquisition system with a sample interval of 1 min.

Figure 2(c) shows the schematic of circuit which was used to carry out the long-term stability test. The in-house developed circuit includes a transimpedance amplifier along with an in-house developed arc detection circuit.

The transimpedance amplifier circuit is capable of measuring sub-nA output current and was calibrated against the Keysight SMU by measuring I–V curves before and after lifetime testing in the chamber. The arc detection circuit was introduced to detect electrical arcs between the emitter and collector electrodes. An electrical transient from an arc triggers the circuit15–17 so that the time of the arc can be recorded. Two lifetime tests were carried out for three die at a time; however, the arc detection circuit was only available during the second set of long-term stability tests. The output from the transimpedance amplifier was connected to a PC using a labview program for data acquisition through a National Instruments NI PXI 6133 chassis.

Several I–V measurements were carried out to ensure that the characteristics were repeatable. The I–V characteristics of a device labeled diode 2 are shown in Fig. 3(a). From the graph, it can be seen that the collector current is repeatable with a <2% difference between two consecutive sweeps. In the plot, the observed collector current is ≈9.6 nA for an applied collector voltage of 8 V, and above 3 V, the collector current begins to increase exponentially and can be described using the Fowler–Nordheim18 (F–N) model as indicated by the linear response in the F–N plot shown in Fig. 3(b). However, this current could also be a combination of surface leakage19 and Schottky20 emission. A separate, detailed study is going on to explain the nature of this current.

FIG. 3.

(a) I–V characteristics of diode 2 and (b) the corresponding F–N plot.

FIG. 3.

(a) I–V characteristics of diode 2 and (b) the corresponding F–N plot.

Close modal

I–V characteristic results for bowtie structure 1 are shown in Fig. 4(a), The maximum observed collector current for an applied voltage of 6 V is ≈5 nA; however, a collector current as low as 0.90 nA was also observed for other bowtie structures. Similar to the diode devices, above the ≈3 V turn-on field, the linear nature of the current is observed in the F–N plot which is shown in Fig. 4(b). However, similar to the diode device, a more detailed study on the characteristics and nature of the emission is ongoing.

FIG. 4.

(a) I–V characteristics of bowtie 1 and the corresponding (b) F–N plot.

FIG. 4.

(a) I–V characteristics of bowtie 1 and the corresponding (b) F–N plot.

Close modal

One of the primary issues of a gated field emission device is dielectric or surface leakage21–23 due to adsorbates. To explore the dielectric leakage or surface leakage from adsorbates on the insulator between the emitter and the collector, a reverse to forward DC bias sweep was applied to the collector on diode devices. As an example, results for diode device 1 are plotted in Fig. 5(a). From Fig. 5(a), it is observed that the current (3.5 nA) during reverse bias at −8 V is approximately 30% of the collector current (11 nA) at a forward bias of 8 V.

FIG. 5.

(a) I–V sweep on diode 1 from an applied DC voltage of −8 to +8 V. For an applied DC voltage of −8 V, the collector current was ≈3.5 nA whereas for +8 V, the current was ≈11 nA, which is ≈3 times. (b) I–V sweep on bowtie 1 for an applied DC voltage of −6 V shows that the collector current was ≈4.2 nA whereas for +6 V, the current was ≈5 nA.

FIG. 5.

(a) I–V sweep on diode 1 from an applied DC voltage of −8 to +8 V. For an applied DC voltage of −8 V, the collector current was ≈3.5 nA whereas for +8 V, the current was ≈11 nA, which is ≈3 times. (b) I–V sweep on bowtie 1 for an applied DC voltage of −6 V shows that the collector current was ≈4.2 nA whereas for +6 V, the current was ≈5 nA.

Close modal

This reverse bias current could be surface leakage current which could be reduced by a high temperature bakeout process6 or UV exposure24 which will be part of future device studies. However, high temperature could not be applied in this experiment as these devices were tested on a printed circuit board. Figure 5(b) shows the reverse to forward DC bias I–V characteristics data for bowtie 1. From Fig. 5(b), it is observed that the emission current during reverse bias at −6 V is approximately ≈4.2 nA; whereas at a forward bias of +6 V, the collector current is ≈5 nA. This current is almost symmetric as, for the bowtie devices, the emitter and collector terminal can be alternated. The small difference in current can be attributed to geometrical differences between the two tips.

One of the most important aspects of a field emission cathode is stability over time. To check the stability of the collector current from both the structure types over time, the devices were operated for over 1000 h at an applied DC potential of 6 V. These lifetime data are plotted in Fig. 6. During the test, data were recorded every 1 min using a labview program. The first lifetime test runs included four devices: diode 1, diode 2, diode 3, and bowtie 1. Diode 1 failed after a continuous operation of ≈350 h. This device was removed, and another device replaced it. However, all the other devices (diodes 2, 3, and bowtie 1) operated for over 750 h. There are transients seen in the data, e.g., around 250 and 500–600 h in Figs. 6(a) and 6(b). These transients are believed to be noise on the data acquisition system creating false data. After the sudden transient, the measured current went back to the pretransient value. These phenomena suggest that an error in measurement took place in which the data acquisition system stopped taking data and only noise was measured. The system was reset and further issues were not observed.

FIG. 6.

Long-term stability curve for fixed DC voltage of 6 V. (a) Diode structure 2, (b) diode structure 3, (c) bowtie structure 1, (d) diode structure 4, (e) bowtie structure 2, and (f) bowtie structure 3. The second set of lifetime tests [(d)–(f)] were carried out for 1400 h.

FIG. 6.

Long-term stability curve for fixed DC voltage of 6 V. (a) Diode structure 2, (b) diode structure 3, (c) bowtie structure 1, (d) diode structure 4, (e) bowtie structure 2, and (f) bowtie structure 3. The second set of lifetime tests [(d)–(f)] were carried out for 1400 h.

Close modal

In the overall lifetime data, there are two stages of cathode degradation. The first one (for diode 2) is a sharp decrease in the current around 1100 h. Prior to this decrease, the current was very stable. This is possibly the result of damage to the emission site induced by an electrical arc. However, the arc detection circuit was not available for this test. Hence, a definitive conclusion cannot be drawn on the occurrence of an arc. For bowtie structure 1 [Fig. 6(c)], a large current fluctuation was observed at ≈925 h, followed by a rapid current decrease indicating possible degradation caused by a series of small arcs at ≈950 h. Prior to these changes, the current was again very stable. The other type of current decrease (diode 3) can be observed in Fig. 6(b) which shows an increase in current at 450 h followed by a gradual degradation between 700 and 750 h. The current then stabilizes and remains approximately constant until the experiment was terminated. These results imply very little degradation until the sharp transitions, although diode 3 has more complicated results.

Lifetime experiments were carried out on a second set of devices as presented in Figs. 6(d)6(f). Three different devices, diode 4, bowtie 2, and bowtie 3, were tested simultaneously with a constant DC voltage (6 V) configuration similar to the first set of experiments. To clearly understand the deviation over time, a 25-point averaging filter was applied. From the filtered data, it was obtained that diode 4 and bowtie 2 were able to produce relatively constant current over a period of >1400 h with the current deviation of <5%.

For the case of bowtie 3 [Fig. 6(f)], although it can clearly be seen that the device worked for more than 1400 h with a deviation of <5%, there were arcs occurring during the initial hours of the test. These arcs were measured by the arc detection circuit and seen in the lifetime data during the first 155 h of operation. Figure 7(a) shows the triggers in the arc detection circuit, and Fig. 7(b) shows the corresponding current fluctuations for bowtie 3 during the first 180 h. This graph confirms the occurrence of electrical transients in the first 155 h of operation. These are likely very small cathodic arcs17 or surface discharges, which do not appear to damage the structure. After that time, the arc detection circuit measured no new arcs, and the current fluctuations significantly reduced and disappeared. Overall, the devices showed very stable collector current. For all the lifetime data, a different noise level is observed which could be caused by the differences in devices structures, as the measurement circuit remains the same for all the tests.

FIG. 7.

(a) Arc circuit trigger and (b) the corresponding collector current fluctuations over time confirm the occurrence of arcs in bowtie 3 device.

FIG. 7.

(a) Arc circuit trigger and (b) the corresponding collector current fluctuations over time confirm the occurrence of arcs in bowtie 3 device.

Close modal

After the lifetime tests, the devices were again I–V characterized to check the I–V curves and the measured values from the lifetime measurement hardware. Results of these measurements are shown in Fig. 8 comparing the curves before and after lifetime testing. From the I–V sweeps on the first set of the devices [Figs. 8(a)8(c)], it can be seen that although the peak current output is much lower for the after lifetime cases compared to the before lifetime test, the devices are still operational where diode 2 shows a degradation of ≈20% at 6 V. Similarly, for diode 3 and bowtie 1, the degradation was 30% and 14%, respectively, at 6 V. At lower voltage, both before and after data points overlapped with each other. This could be due to the measurement error. However, with higher applied voltage, the difference becomes distinct as can be seen in Fig. 8. The data confirm the lifetime current measurements as well. The I–V characterization data on the second set of the devices are shown in Figs. 8(d)8(f). From the I–V sweeps, it can be seen that the peak current outputs for the after lifetime cases are comparable with the before lifetime I–V sweeps, with a deviation ranging from ≈2% to 8%. Again, these I–V data match the lifetime current data and establish the relative current stability for >1400 h of operation.

FIG. 8.

Before and after lifetime test I–V curve for (a) diode 2, (b) diode 3, (c) bowtie 1, (d) diode 4, (e) bowtie 2, and (f) bowtie 3 at a fixed DC voltage of 6 V.

FIG. 8.

Before and after lifetime test I–V curve for (a) diode 2, (b) diode 3, (c) bowtie 1, (d) diode 4, (e) bowtie 2, and (f) bowtie 3 at a fixed DC voltage of 6 V.

Close modal

Figure 9 shows the quantitative comparison of before and after collector current for diodes 2 (a) and 4 (b) in percentage, as an example. For the first set of lifetime test, the observed degradation for diode 2 [Fig. 9(a)] at 6 V was ≈20%, similarly for diode 3 and bowtie 1, the degradation was 30% and 14%, respectively, after the lifetime test. For the second set of tests, the observed degradation for diode 4 [Fig. 9(b)] at 6 V was ≈5%, similarly for bowtie 2 and 3, the degradation was 8% and 6%, respectively, after the lifetime test. For both cases, data are compared for the after turn-on voltage. From the graph, it can be inferred that at lower voltage, error in measurement caused the overlap between before and after cases. However, at higher voltage, the difference in before and after is clear.

FIG. 9.

Before and after lifetime test collector current difference in percentage for (a) diode 2 and (b) diode 4, at a fixed DC voltage of 6 V.

FIG. 9.

Before and after lifetime test collector current difference in percentage for (a) diode 2 and (b) diode 4, at a fixed DC voltage of 6 V.

Close modal

After the lifetime cycle, devices were removed and examined by optical microscopy and SEM. It was observed that diode 1 failed after 350 h. It was examined and the damaged structure is shown in an optical image in Fig. 10. The arc damage is located where the bond pad transitions to the metal trace that extends to the emitter tip. Note that the arc detection circuit was not available, so only the device current drop was observed.

FIG. 10.

Diode structure 1 optical microscope image after device failure. Damage caused by an arc is clearly visible (red dotted circle). This damage occurred at the beginning of the emitter structure transition to the bond pad.

FIG. 10.

Diode structure 1 optical microscope image after device failure. Damage caused by an arc is clearly visible (red dotted circle). This damage occurred at the beginning of the emitter structure transition to the bond pad.

Close modal

Unfortunately, a number of devices were damaged during shipping, so a thorough post lifetime examination was not possible for all devices. Bowtie 2, which was undamaged, was examined under SEM to observe any physical damage after lifetime testing. Figures 11(a) and 11(b) show the before and after lifetime test image of the bowtie 2 device which was not damaged during the test and operated for 1400 h. This device shows no obvious damage or erosion of the emitter tips during the lifetime testing.

FIG. 11.

SEM image of bowtie 2 (a) before and (b) after the lifetime test. No damage was observed after the test for this device.

FIG. 11.

SEM image of bowtie 2 (a) before and (b) after the lifetime test. No damage was observed after the test for this device.

Close modal

Two types of planar field emitters were characterized for field emission performance and long-term current output stability. From the I–V characterization study, for an applied potential of 8 V, the observed collector current was ≈1–11 nA among four diode devices. Among four bowtie devices, the observed collector current was ≈0.9–5 nA for a maximum applied potential of 6 V. This current disparity can be attributed to the nonuniformity in the vacuum gap in between electrodes tips and the tip sharpness distribution. A lifetime study was carried out on seven different devices for periods of over 1000 h with a data acquisition interval of 1 min. It is noted that the measured current could be field emission, surface leakage, or Schottky emission. Hence, the lifetime study is of the total collected current and not just field emission current. Although the first diode structure failed after a 350 h of continuous operation, the other six devices showed stable FE performance over a period of >1000 h with three parts tested for >1400 h.

During the first set of tests, current degradation varies from ≈14% to 30%, for three devices (diode 2, diode 3, and bowtie1) measured after 1000 h. In these cases, sharp transitions in current were observed which may have been due to small arcs or surface discharges. In the second set of tests, which were carried out on three devices (diode 4, bowtie 1, and bowtie 2), the measurements showed only ≈5%–8% of current degradation. It is important to note that these devices achieved >1000 h lifetimes without ballast resistors to control emission. Devices were examined by optical microscopy and SEM, and it was clearly observed that the device pad was damaged by arc or discharge on the device that failed at 350 h. However, it was also observed that at least one of the devices was undamaged or showed no obvious erosion after lifetime testing.

These devices show great promise as possible field emission electron sources for femtosecond surface plasmon devices and nanoscale vacuum channel-based transistor devices, where a few nano amps of emission current with superior stability and long life is required.

Material support for this work is provided by the Air Force Office of Scientific Research under Grant No. FA9550-18-1-0436. The authors would also like to thank support from undergraduate students Patrick Ward, John McClarin, Robby Davis, Liz Gaffney, Jessica Carlson, David Vogel, and Gerardo Herrera. R.B. and M.T. contributed equally to this work.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

1.
T.
Rybka
,
M.
Ludwig
,
M. F.
Schmalz
,
V.
Knittel
,
D.
Brida
, and
A.
Leitenstorfer
,
Nat. Photonics
10
,
667
(
2016
).
2.
A.
Kubo
,
K.
Onda
,
H.
Petek
,
Z.
Sun
,
Y. S.
Jung
, and
H. K.
Kim
,
Nano Lett.
5
,
1123
(
2005
).
3.
H.
DuyáNguyen
and
J.
SangáKang
,
Nanoscale
11
,
3129
(
2019
).
4.
J.-W.
Han
,
D.-I.
Moon
, and
M.
Meyyappan
,
Nano Lett.
17
,
2146
(
2017
).
5.
W.
Kang
,
J.
Davidson
,
K.
Subramanian
,
B.
Choi
, and
K.
Galloway
,
IEEE Trans. Nucl. Sci.
54
,
1061
(
2007
).
6.
R.
Bhattacharya
,
N.
Karaulac
,
W.
Chern
,
A.
Akinwande
, and
J.
Browning
,
J. Vac. Sci. Technol. B
39
,
023201
(
2021
).
7.
H.
Busta
,
J.
Pogemiller
, and
B.
Zimmerman
,
J. Micromech. Microeng.
3
,
49
(
1993
).
8.
H. S.
Uh
,
S. J.
Kwon
, and
J. D.
Lee
,
J. Vac. Sci. Technol. B
15
,
472
(
1997
).
9.
S.-H.
Hsu
, “
Development of Vertical Nanodiamond Vacuum Field Emission Microelectronic Integrated Devices
,”
Ph.D. thesis
(
Vanderbilt University
,
2014
).
10.
R.
Greene
,
H.
Gray
, and
G.
Campisi
, in
IEEE International Electron Devices Meeting
,
Washington, DC
(IEEE, Washington DC,
1985
), pp.
172
175
.
11.
L.
Zhang
,
A. Q.
Gui
, and
W.
Carr
,
J. Micromech. Microeng.
1
,
126
(
1991
).
12.
Y.
Yang
,
M.
Turchetti
,
P.
Vasireddy
,
W. P.
Putnam
,
O.
Karnbach
,
A.
Nardi
,
F. X.
Kärtner
,
K. K.
Berggren
, and
P. D.
Keathley
,
Nat. Commun.
11
,
3407
(
2020
).
13.
S.
Guerrera
and
A. I.
Akinwande
,
Nanotechnology
27
,
295302
(
2016
).
14.
S. A.
Guerrera
,
L. F.
Velasquez-Garcia
, and
A. I.
Akinwande
,
IEEE Trans. Electron Devices
59
,
2524
(
2012
).
15.
M.
Gilmore
,
N.
McGruer
,
J.
Browning
, and
W.
Bintz
,
Rev. Sci. Instrum.
64
,
581
(
1993
).
16.
J.
Browning
,
N. E.
McGruer
,
W.
Bintz
, and
M.
Gilmore
,
IEEE Electron Device Lett.
13
,
167
(
1992
).
17.
S.
Meassick
,
Z.
Xia
,
C.
Chan
, and
J.
Browning
,
J. Vac. Sci. Technol. B
12
,
710
(
1994
).
18.
R. H.
Fowler
and
L.
Nordheim
,
Proc. R. Soc. London, Ser. A
119
,
173
(
1928
).
19.
F.
Allen
,
J.
Eisinger
,
H.
Hagstrum
, and
J.
Law
,
J. Appl. Phys.
30
,
1563
(
1959
).
20.
G.
Schwind
,
G.
Magera
, and
L.
Swanson
,
J. Vac. Sci. Technol. B
24
,
2897
(
2006
).
21.
A.
Di Bartolomeo
,
F.
Giubileo
,
L.
Iemmo
,
F.
Romeo
,
S.
Russo
,
S.
Unal
,
M.
Passacantando
,
V.
Grossi
, and
A. M.
Cucolo
,
Appl. Phys. Lett.
109
,
023510
(
2016
).
22.
P.
Schwoebel
and
C.
Spindt
,
J. Vac. Sci. Technol. B
12
,
2414
(
1994
).
23.
N. E.
McGruer
,
K.
Warner
,
P.
Singhal
,
J.
Gu
, and
C.
Chan
,
IEEE Trans. Electron Devices
38
,
2389
(
1991
).
24.
R.
Bhattacharya
,
N.
Karaulac
,
G.
Rughoobur
,
W.
Chern
,
A. I.
Akinwande
, and
J.
Browning
,
J. Vac. Sci. Technol. B
39
,
033201
(
2021
).