Introduction

Tremendous efforts in development of novel nanomechanical fabrication techniques and devices have witnessed a variety of emerging nanoelectronics, including field emission diodes1,2,3, field effect transistors4,5,6,7,8,9,10,11,12, ultrafast switches13,14,15,16,17, biosensors18,19,20, and molecular transistors21,22. The typical device structure has a vacuum nanogap as an electron transport channel, which is expected to show extraordinary performances in terms of the high electron traveling velocity, fast switching rates, high operating temperatures, and exceptional radiation tolerance. Specially designed with asymmetric electrode structures and operating at low voltages, these devices rely heavily on field electron emission (FE)3,6,9,17,23,24,25. The state-of-the-art ultrafast nanoelectronic switches reported in Nature has successfully achieved picosecond-level switching speeds with lower power consumption by utilizing electron field emission induced nanoplasma13,14. However, 1950s type planar FE theory, based on the 1956 Murphy-Good (MG) FE equation (see Supplementary Note 1)26,27,28, does not work well for nanodevices. This is because the traditional planar potential barrier, the so-called Schottky-Nordheim barrier (Supplementary Eq. (S1)), does not adequately describe the potential-energy variation near a significantly curved surface29,30. Further, when the gap distance is at the nanometer scale, quantum effects may occur, particularly if there is significant space-charge in the gap31,32,33. These quantum effects are, firstly that it may become necessary to describe the electron motion in the gap by using the Schrödinger equation (rather than by using classical space-charge arguments), secondly that it may become necessary to consider exchange-and-correlation interactions between electrons in the space-charge. Both effects are difficult to clearly distinguish when the size of the emitter is comparable to the nanogap spacing.

Generally, for typical sharp emitters, it is common practice to assume that the average electrostatic field F0 is enhanced by a field enhancement factor (FEF) of γ in Supplementary Eq. (S2): F = γF0 = γV/d, where F is the absolute magnitude of the electrostatic field at emitter surface, and γ depends on the definition of the distance d, either as the distance between planar plates (plate FEF), particularly for a post standing on one of a pair of parallel plates, or as the distance between the emitter apex and the anode plate (gap FEF)34,35. However, this approach by itself may fail to accurately predict the field emission behavior of emitter tips with apex radii R less than 50 nm, as the width of tunneling barrier is getting very sensitive to the physical dimension36,37,38,39,40,41,42,43. Previous studies have reported that during field emission from nanotips, the electron total-energy distributions sometimes display multiple well-separated peaks rather than one strong peak at Femi level, indicating a considerable variation of the structure of electron states inside the nano-emitter44,45. Currently, most FE models proposed in the literatures for nano-emitters either are focused on the corrections of the supply function and transmission coefficient due to probable quantum confinement effects of emitter with a non-planar geometry or are mainly aimed to calculate the image charge potential of space charges with cathode alone more accurately by including the geometric effects in the related mathematical formulation42,46,47,48,49. It is important to note that the supply function and transmission coefficient can be strongly geometrically dependent. This complexity suggests that it is nearly impossible to have an analytical multi-dimensional field emission model for a given emitter. This has prompted this experimental attempt to obtain an empirical scaling.

Moreover, when developing a FE model for nanogaps, a sophisticated approach has been to obtain space charge limited field emitted current density by solving the coupled Schrödinger and Poisson equations self-consistently50,51,52,53,54. Therefore, the question of whether quantum effects of this kind need to be considered turns out to be a very fundamental issue for further elucidating the intrinsic and experimental field emission characteristics at the nanoscale.

Here, we studied the experimental field emission characteristics of the pure tungsten nanoelectrodes across vacuum nanogaps by utilizing a home-built in situ electrical experiment system. We explored how the field emission characteristics vary as a function of the ratio of apex radius to gap length (R/d in the range of 0.04 to 48) and revealed a consistent scaling relationship between them. This scaling is shown to recover the accepted limits at R/d « 1 (ultrasharp tip) and R/d » 1 (planar gap). Furthermore, after excluding the space charge quantum effects, we successfully proposed a modified FE equation that accounts for geometric asymmetry effects and related R/d dependence. Finally, the I-F curves and corresponding FN plots based on this FE equation are compared with previous experimental results.

Results and discussion

In situ field emission measurement at nanoscale

Experimental field emission characteristics from various nanoscopic emitters in nanogaps are tested using an in situ electrical experiment system, which comprises a transmission electron microscopy (JEOL-2100F TEM) and an in situ electrically biased TEM holder (ZepTools Technology)55,56. The electrode structure is tungsten nanotip (W, cathode)—gold plate (Au, anode), and the pure tungsten nanotips with different apex radii of curvature are prepared by a two-step process: the ex-situ double‑electrolyte electrochemical etching method57,58 and the in situ Joule melting method59. Details of the in situ electrical measurement system and the preparation of tungsten nanotips are described in the “Methods” section. In this work, we investigated the field emission characteristics of tungsten nanotips with apex radii of curvature Rm (measured from TEM images) ranging from 2 nm to 190 nm across various nanogaps (5 nm ≤ dm ≤ 100 nm), which covers a wide range of Rm/dm = 0.04 to 48. Notably, in our experiments, the gap distance dm refers to the distance between the emitter apex and the anode plate. The TEM images of these tungsten nanotips are shown in Fig. 1.

Fig. 1: The transmission electron microscopy (TEM) images of tungsten nanotips with apex radii (Rm) ranging from 2 nm to 190 nm.
figure 1

a Rm = 2 nm. b Rm = 3 nm. c Rm = 4 nm. d Rm = 5 nm. e Rm = 6 nm. f Rm = 17 nm. g Rm = 20 nm. h Rm = 43 nm. i Rm = 55 nm. j Rm = 60 nm. k Rm = 70 nm. l Rm = 100 nm. m Rm = 170 nm. n Rm = 180 nm. o Rm = 190 nm.

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Field emission characteristics

Figure 2a, b presents the experimental field emission Im-Vm curves (solid lines) and the corresponding Fowler-Nordheim (FN) plots (solid lines) of the tungsten nanotip with an apex radius of 4 nm, across nanogaps dm ranging from 5 nm to 80 nm. As depicted, the current increases rapidly as the voltage reaches the turn-on threshold, and with increasing nanogap distance, the Im-Vm curves shift towards higher voltages. Notably, all FN plots exhibit a nearly linear relationship, indicating the typical field emission behavior. The dotted lines in Fig. 2a, b represent theoretical IMG-VMG-eff data generated as described in the following Section. The experimental field emission Im-Vm curves of all tungsten nanotips offer valuable insights into how both the radius of curvature Rm of nanotips and the gap spacing dm affect their emission characteristics. Figure 2c–e illustrate the applied voltages Vm (measured at Im = 2 nA), the FEFs γa, and the γa·Vm as functions of R/d in logarithmic coordinates (scatterplots). Here, γa is obtained from finite element method (FEM) simulations (see “Methods” section and Supplementary Fig. 3), where the nanotips are modeled using the sphere-on-cone approach, based on the morphology of all the tungsten nanotips in Fig. 1. Notably, although the nanotip models are constructed using parameters (Rm, dm) measured from TEM images, they may not perfectly match the experimental nanotips. Thus, the radius of curvature of nanotip models and the gap distance in FEM simulations are denoted by Rt and dt, to distinguish them from the experimental nanotips, even though Rt = Rm and dt = dm. From the FEM electrostatics simulations, γa can be obtained as γa = Fa/FG, where Fa represents the electrostatic surface-field magnitude at the emitter apex, and FG is the average field amplitude in the gap between the emitter apex and the anode plate, given by FG = Vm/dt. It’s important to note that our experiments are conducted in a near anode configuration, where the total emitter height Hm » Rm and Hm » dm, meaning that γ is not sensitive to H. To minimize the effect of the simulated height on the electrostatic fields, we set the height of the simulation models to be sufficiently large (at Ht/Rt = 400) as shown in the “Methods” section and Supplementary Fig. 2. The relationships \({V}_{{{{\rm{m}}}}}\propto {({R}_{{{{\rm{m}}}}}/{d}_{{{{\rm{m}}}}})}^{-{n}_{1}}\), \({\gamma }_{{{{\rm{a}}}}}\propto {({R}_{{{{\rm{t}}}}}/{d}_{{{{\rm{t}}}}})}^{-{n}_{2}}\), and \({\gamma }_{{{{\rm{a}}}}}\cdot {V}_{{{{\rm{m}}}}}\propto {(R/d)}^{-{n}_{3}}\) are used to obtain the fitted values of n1, n2, and n3 based on R/d, respectively, as shown in Fig. 2c–e. As expected, the required applied voltages Vm (at a fixed current) and the γa decrease with higher values of R/d. Notably, the relationship between γa and Rt/dt is nearly identical for all tungsten nanotip sphere-on-cone models with different Rt at different gap spacing dt (Fig. 2d). For Rm/dm values less than 0.1, n1 increases slowly to about 0.2, rises rapidly to 0.9 as Rm/dm approaches 10, and then increases more slowly towards 1 for Rm/dm greater than 10, reflecting a transition to planar geometry with the relationship approximated as Vm ∝ dm/Rm (Fig. 2c). Similarly, for Rt/dt < 0.1, the electrostatic field distribution between the emitter apex and the anode plane is non-uniform with n2 decreasing slowly to about 0.8, becoming more uniform as Rt/dt increases to 10 (where n2 drops rapidly to about 0.1), and approaching a planar case for Rt/dt > 10 with γa nearing 1 or n2 = 0 (Fig. 2d). More importantly, for all nanotips in our experiments, n3 is approximately 1 (Fig. 2e), with acceptable fluctuations due to the differences between the experimental nanotips and the sphere-on-cone models. Therefore, we disregard the subscript differences “m” and “t” for R and d, obtaining the following relationship: γa·Vm ∝ d/R, which further leads to Fa = γa·FG ∝ 1/R. This finding implies that the local electrostatic field strength at the emitter apex required to obtain the same field emission current remains largely unchanged across the range of R/d = 0.04 to 48, with values for different nanotips depending on their apex radii of curvature. Moreover, within this range of R/d, even with a minimum R of 2 nm and d of 5 nm, no significant deviations from this relationship γa·Vm ∝ d/R (Fa = γa·FG ∝ 1/R) are observed in Fig. 2e. This suggests that the effects due to the non-planar shape of the emitter may play a critical role in nanoscale field emission process. Notably, these relationships hold across various field emission currents in the nanoampere (nA) range. Additional details, including the logarithmic relationship between applied voltages Vm and Rm/dm, as well as between γa·Vm and R/d at different field emission currents of (10, 50, and 100 nA), are provided in Supplementary Fig. 4.

Fig. 2: Field emission (FE) characteristics at nanoscale.
figure 2

a The experimental field emission current-voltage Im-Vm curves (solid lines) of the tungsten nanotip with an apex radius of 4 nm across different nanogaps. The dotted lines represent the theoretical IMG-VMG-eff data based on the Murphy-Good (MG) FE equation (VMG-eff = VMG/CMG ≈ Vm, where CMG is correction factor). b The corresponding experimental Fowler-Nordheim (FN) plots (solid lines) and the corresponding FN plots (dotted lines) based on the MG FE equation (VMG-eff = VMG/CMG ≈ Vm) of the tungsten nanotip with an apex radius of 4 nm. c The applied voltages Vm at a field emission current (Im) of 2 nA as a function of the ratio of experimental apex radius to gap length (Rm/dm) in logarithmic coordinates (scatterplots), along with the corresponding negative exponent n1 (solid line). d The field enhancement factors γa obtained from simulation as a function of simulated apex radius to gap length (Rt/dt) in logarithmic coordinates (scatterplots), along with the corresponding negative exponent n2 (solid line). e The product γa·Vm of the field enhancement factors γa and the applied voltages Vm at Im = 2 nA as a function of R/d in logarithmic coordinates (scatterplots), along with the corresponding negative exponent n3 (solid line).

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Notably, the aforementioned relationships are based on experimental data or simulated data, and the specific analytical scaling (i.e., the specific analytical relationship between γa·Vm and R/d has yet to be derived (which may not be possible). Generally, the electrostatic field at the emitter apex can be expressed as Fa = βa·Vm = Vm/(kaR), where βa is the conversion factor connecting the electrostatic surface-field magnitude at the emitter apex and the applied voltages, and ka is a shape factor to be determined either experimentally or numerically60. The previous theoretical analytical scaling60 at the apex of the microtip with the prolate-spheroidal shape (which also used the assumption in Supplementary Note 1) shows ka = 0.5 × ln(4 d/R) and ka = d/R, respectively, for R/d « 1 (sharp tip) and R/d » 1 (planar limit). By writing Fa = βa·d·Vm/d = γa·FG, we have γa = βa·d = d/(kaR), which gives γa = 2/(R/d) and γa = 1 (see the derivation in Supplementary Note 2), respectively, for R/d « 1 (sharp tip) and R/d » 1 (planar limit), which is consistent with results shown in Fig. 2d. It is important to note that the geometrical shape of our experimental tip (in Fig. 1) is not exact similar to the assumed in ref. 60, but the discussion above shows qualitative agreement.

Geometric effects at nanoscale

To further determine whether the space charge quantum effects are significant in our experiment, we used a self-consistent quantum model, which has been reported in refs. 53,54, to obtain the field emission characteristics JQ for the tungsten (cathode)-gold (anode) parallel-plate electrode structure50,51,52,53,54. In this model, the one-dimensional modified Poisson-Schrödinger equation combined with the Jeffreys-Wentzel-Kramers-Brillouin tunneling model is solved numerically for calculating electron field emission current density54. The local work function (ϕ) and Fermi Energy (EF) used for tungsten are given as 4.5 eV and 5.782 eV, respectively, and for gold, they are 5.1 eV and 5.535 eV, respectively61. Figure 3 shows the normalized electron current density μQC = JQ/JCL (black lines) compared to μMC = JMG/JCL (red lines) at different distances between well-separated parallel planar plates (dsep = 1 nm, 5 nm, and 10 nm) based on theoretical calculations, where JCL is the classical Child-Langmuir (CL) law62,63, and the JMG is the MG FE equation without the space charge effects (see Supplementary Note 1 and Supplementary Eq. (S2)). When the gap distance dsep is greater than 5 nm and the electrostatic field FP (FP = VP/dsep, VP is the applied voltage between the plates) is less than 10 V/nm, the field emission characteristics considering space charge quantum effects are consistent with those obtained by MG FE equation. This indicates that the space charge quantum effects can be ignored so that we can conclude the effects due to the non-planar shape of the emitter are more important for R/d = 0.04 to 48 as studied in our experiment. Moreover, it should be noted that when dsep is 1 nm and FP is less than 4 V/nm, the emission current density μQC is higher than μMC, due to direct tunneling (see Supplementary Note 3 and Supplementary Fig. 5)54.

Fig. 3: The theoretical normalized electron emission current density in nanogaps.
figure 3

a The normalized electron emission current density μQC = JQ/JCL (black lines) and μMC = JMG/JCL (red lines) as a function of electrostatic fields FP = 1 to 12 V/nm (FP = VP/dsep, VP is the applied voltage between the plates, dsep is distance between well-separated parallel planar plates) at dsep = 1 nm for the tungsten (cathode)-gold (anode) parallel-plate electrode structure, based on theoretical calculations. Here, JCL is the classical Child-Langmuir (CL) law, JMG is the MG FE equation, and JQ is obtained by a self-consistent quantum model53,54. b μQC = JQ/JCL (black lines) and μMC = JMG/JCL (red lines) at dsep = 5 nm. c μQC = JQ/JCL (black lines) and μMC = JMG/JCL (red lines) at dsep = 10 nm.

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Since the space charge quantum effects can be ignored in our experiment, we further compare the experimental current-voltage Im-Vm characteristics with theoretically predicted characteristics. The representative geometry models used are shown in Supplementary Fig. 3. For a given voltage between the emitter and the anode plate, the electrostatic field distribution over the emitter-shape-model surface is first calculated by finite-element methods (FEM), where the FEFs γa can also be obtained. Then, the total predicted emission current is obtained by integration of local emission current density (LECD) over the shape-model surface, using the planar emission approximation and the MG FE equation (see Supplementary Note 1 and Supplementary Eq. (S2)). The planar emission approximation assumes that the LECD JL at any particular location “L” on the emitter surface is given by some specified planar emission equation, using the values at “L” of the local work function and the surface electrostatic field magnitude34. For a specific model and emission current, we find that theoretically predicted voltage values VMG are less than those Vm found experimentally for the same gap spacing d. This suggests that (other things being equal) MG FE theory may be over-predicting LECDs. To quantify the discrepancy, we introduced a correction factor CMG < 1. Specifically, due to the consistent scaling relationship at different currents (Im = 2 nA, 10 nA, 50 nA, and 100 nA) as shown in Fig. 1 and Supplementary Fig. 4, we selected the voltage at I = 2 nA to calculate CMG, (CMG = [VMG(IMG = 2 nA)]/[Vm(Im = 2 nA)]), then the effective voltage for all VMG values can be expressed as: VMG-eff = VMG/CMG ≈ Vm. The theoretical IMG-VMG-eff curves and corresponding theoretical FN plots obtained are shown in Fig. 2a, b (dotted lines), which are closely aligned with the experimental data. Figure 4a illustrates the numerical values of correction factors CMG as a function of R/d in logarithmic coordinates. As R/d increases, CMG initially increases from approximately 0.4 and gradually approaches 0.7. This confirms experimentally the discrepancies between the actual field emission and theoretical predictions64.

Fig. 4: The geometric effects in nanoscale field emission.
figure 4

a The correction factors CMG as a function of the ratio of apex radius to gap length R/d in logarithmic coordinates (scatterplots). b The effective field-voltage conversion factors βeff-MG as a function of R/d in logarithmic coordinates (scatterplots), along with the corresponding exponent n4 (solid line). c The shape factors kMG as a function of R/d in logarithmic coordinates (scatterplots), along with the corresponding negative exponent n5 (solid line). The black line shows the fitting curve by kMG = 1.680 × (R/d + 0.468)−1.066. d The field emission Im-Feff-MG curves of the tungsten nanotip with an apex radius of 4 nm across different nanogaps. e The corresponding Fowler-Nordheim (FN) plots of Im-Feff-MG curves. f The effective emission area Aeff-MG obtained by finite element method (FEM) simulation.

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Figure 4b, c show the effective field-voltage conversion factors βeff-MG (βeff-MG = CMGβa = CMGγa/d) and shape factors kMG (βeff-MG = 1/(kMGR)) as functions of R/d in logarithmic coordinates. The relationships \({\beta }_{{{{\rm{eff}}}}-{{{\rm{MG}}}}}\propto {(R/d)}^{{n}_{4}}\) and \({k}_{{{{\rm{MG}}}}}\propto {(R/d)}^{-{n}_{5}}\) are used to fit n4 and n5. For R/d < 0.1, n4 increases slowly to about 0.2, rises rapidly to 0.9 as R/d approaches 10, and then increases more slowly towards 1 for R/d > 10, following βeff-MG ∝ R/d (Fig. 4b). This is consistent with the n1 trend reported in Fig. 2c. Surprisingly, the scaling of kMG is valid for all tungsten nanotips with different R, which converges onto a single line with R/d (Fig. 4c). Analogously, for R/d values less than 0.1, n5 rises slowly to about 0.2, increases rapidly to 0.9 as R/d approaches 10, and then rises more slowly towards 1 for R/d > 10, following kMG ∝ d/R (for planar geometry) and indicating a nearly uniform electrostatic field distribution between the emitter apex and the anode plane. As we can see, the kMG values ranging from 0.02 to 4 provide a clear and detailed characterization of field emission of a nanotip in a nanogap. We believe that these scaling laws should be valid for any metallic nanotip of apex radius R in a nanogap of spacing d, in the range of R/d = 0.04 to 48 reported experimentally in this work. For the ease of applications, the values of kMG can be fitted into a simple analytical expression depending of R/d, which is kMG = 1.680 × (R/d + 0.468)−1.066 (based on Fig. 4c, with a R-squared fitting of 0.986). Thus, the electrostatic field F in the MG FE equation (Supplementary Eq. (S2)) should be substituted by the effective electrostatic field Feff-MG

$${F}_{{{\rm{eff}}}-{{{\rm{MG}}}}}={\beta }_{{{\rm{eff}}}-{{{\rm{MG}}}}}{V}_{{{\rm{m}}}}={C}_{{{\rm{MG}}}}{\beta }_{{{\rm{a}}}}{V}_{{{\rm{m}}}}={C}_{{{\rm{MG}}}}{{{\gamma }}}_{{{\rm{a}}}}{V}_{{{\rm{m}}}}/d={C}_{{{\rm{MG}}}}{{{\gamma }}}_{{{\rm{a}}}}{F}_{{{\rm{G}}}}={V}_{{{\rm{m}}}}/\left({k}_{{{\rm{MG}}}}R\right)$$
(1)

where Feff-MG is the effective apex-field. Then, the total field emission current can be estimated by

$${I}_{{{{\rm{m}}}}}={A}_{{{{\rm{eff}}}}-{{{\rm{MG}}}}}\, {J}_{{{{\rm{eff}}}}-{{{\rm{MG}}}}}={A}_{{{{\rm{eff}}}}-{{{\rm{MG}}}}}a{\phi }^{-1}{F}_{{{{\rm{eff}}}}-{{{\rm{MG}}}}}^{2}{t}^{-2}(y)\exp \left[-\frac{v(y)b{\phi }^{3/2}}{{F}_{{{{\rm{eff}}}}-{{{\rm{MG}}}}}}\right]$$
(2)

where Jeff-MG is the effective apex emission current density, Aeff-MG is effective emission area, a (= 1.54 × 10−6 [A eV V−2]) and b (= 6.83 [eV−3/2 V nm−1]) are constants, y is the Nordheim parameter (where y = (1.20 eV V−1/2 nm1/2) F1/2/ϕ, and F is in V/nm), and t(y) and v(y) are well-known mathematical functions26,27,28.

Based on the proposed equation, the applied voltage Vm in field emission characteristics of the tungsten nanotip with a radius of curvature of 4 nm shown in Fig. 2a, b can be replaced by the effective apex-field, Feff-MG. Figure 4d, e show the field emission Im-Feff-MG curves and their corresponding FN plots. Notably, as d increases, the Im-Feff-MG curves gradually converge for R/d ≤ 0.27. The observed differences in the curves at d = 5 nm (R/d = 0.8) and d = 10 nm (R/d = 0.4) can be explained by the effective emission area Aeff-MG in Fig. 4f. As the gap increases with R/d ≤ 0.27, the effective emission area increases and converges gradually. However, for d = 5 nm (R/d = 0.8), especially or d = 10 nm (R/d = 0.4), the effective emission area remains below this convergent value, indicating that the nanotip requires a higher electrostatic field to maintain the same field emission current Im. This causes the deviations of the Im-Feff-MG curves and the corresponding FN plots when R/d > 0.27, differing from the reported collapsing log10(I/F2) ~ 1/F curves in ref. 65, where the change of emission area is not considered. Despite this difference at R/d > 0.27, the collapsing I-V curves and the FN plots reported in ref. 65 are consistent with our findings for R/d < 0.27. To be specific, the I-V curves can collapse onto one single I-VR(d) curve, with R(d) approximating a power law R(d) ∝ d−λ (where λ ≈ 0.22 for d = 3–300 nm and R ≈ 5 nm). Furthermore, they found that when d > 10 nm, V ∝ dλ with λ ≈ 0.2–0.35, while λ towards to a larger value for d < 10 nm65. This well matches our universal scaling \({V}_{{{{\rm{m}}}}}\propto {d}^{{n}_{1}}\) for a nanotip with R as the radius of curvature, where n1 increases slowly at approximately 0.2 for R/d < 0.1, which proves the validity of our proposed models for field emission of nanotip in nanogap as studied in this work.

Based on the above analysis, the effects of the emitter’s non-planar shape are more important for R/d = 0.04 to 48 as studied in our experiment. Therefore, we utilize the Kyritsakis-Xanthakis (KX) FE equation42, including surface curvature corrections for local electrostatic potential and the image charge potential which is valid only along the emitter’s axis (see Supplementary Note 1 and Supplementary Eq. (S4)), to derive the correction factor CKX, effective field-voltage conversion factors βeff-KX and kKX as functions of R/d, along with the corresponding power exponents, as provided in Supplementary Note 4 and Supplementary Fig. 6. This approach, based on the planar emission approximation, is similar to the process used for deriving CMG, βeff-MG and kMG. As shown in Supplementary Fig. 6, for R/d < 1, CKX is greater than CMG, increasing from 0.5 (whereas CMG increases from 0.4 in Fig. 4a), due to the inclusion of the curvature correction effects. For R/d > 1, CKX is consistent with CMG, and approaches 0.7. Therefore, despite these corrections, the KX FE equation (ref. 42), which uses an Earthed-sphere model to assess surface curvature effects, still does not accurately predict the field emission current of nanotips.

Interestingly, the investigation of electron tunneling process between the planar substrate and the non-planar tips (spherical or hyperboloidal geometry) decades ago has already concluded that electron field emission current density in a nanogap is significantly regulated by the two factors including the asymmetry in the geometries of cathode and anode, and the overall image charge potential of space charge with both electrodes. Those effects on the overall image charge potential have been carefully examined in refs. 29,30 for sphere-plane electrode configuration with nanogaps using the rigorous electrodynamic models under spherical and prolate spheroidal coordinates. It was found that the non-planar geometry of cathode could directly result in the anisotropic distribution of image charge potential spatially surrounding the cathode. Most importantly, the overall image charge potential of sphere-plane system in nanogap was predicted to be less negative near the cathode, compared to that of plane-plane configuration, indicating an increasing of electron transmission barrier height in the former case due to the anticipated geometric asymmetry in electrode pair. Furthermore, the increasing of geometric asymmetry further diminishes the lowering of electron emission barrier height, because the complete multiple-image potential goes to less negative, relative to that of plane-plane geometry. For example, using the exact image charge potential model given in ref. 30 for hyperboloidal tip model (tungsten electrode with R = 10 nm and d = 1, 2, and 3 nm, respectively), for R/d = 10, 5.0, and 3.3, the absolute values are reduced by 1.0%, 1.9%, and 2.9%, compared to those of plane-plane system, respectively. The raising of electron transmission energy barrier height with the increasing of the geometric asymmetry in nanogap for anode-cathode system can reduce the emission current density in comparison with that of plane-plane configuration under the same electrostatic field. This may be the main reason for the presence of the correction factor VMG = CMGVm < Vm between experimental field emission curves and the MG FE equation, as shown in Fig. 4a. These insight inspires further theoretical exploration to refine predictions for nanoscale field emission phenomena, where the three-dimensional image charge potential between space charges and the bulk charges in the both electrodes (anode and cathode) should be especially considered30.

In this study, we delved into the experimental relationship between field emission characteristics of nanotips in nanogaps and the ratio of apex radius to gap length (R/d). We demonstrated that space charge quantum effects are not considerable in the range of R/d = 0.04 to 48, while the effects of quantum tunnelling due to the emitter shape are predominant. Based on our experimental results, we proposed a modified field emission (FE) equation, where the effective electrostatic field at the emitter apex is given by Feff-MG = Vm/(kMGR), with a detailed analytical expression for the shape factor, kMG = 1.680 × (R/d + 0.468)−1.066. These findings are believed to be valid for any metallic nanotip in the range of R/d = 0.04 to 48. It is important to note that despite the success and limitation of 1D field emission models (many different versions), all models require some approximation to derive some scaling laws or equations to compare with experimental results. Physics-consistent 3D model for arbitrary shape of emitter is likely to be impossible. In this paper, we performed a clear measurement of field emission from an emitter of radius R = 2 to 190 nm in a nanogap of d = 5 nm to 100 nm and provide a first experimental-proved scaling that is valid over a wide range of R/d (4 order of magnitude). Discussion of this scaling to other models can be referred to Supplementary Note 4. Our work underscores the importance of emitter shape effects in dictating the emission properties, and highlights the need for precise theoretical models to accurately predict and harness these effects in practical applications, which is crucial for understanding and improving the performance of nanoelectronic devices.

Methods

In situ electrical experiment system

The in situ electrical experiment system consists of the transmission electron microscopy (JEOL-2100F TEM) and an in situ electrically biased TEM holder (ZepTools Technology). This setup achieves a spatial resolution of 0.1 nm and allows for nanogap adjustment with an accuracy of 0.04 nm. During field emission measurements, the system can apply a direct current voltage of up to 150 V across nanogaps, with a current measurement resolution of 0.1 nA. The in situ electrically biased TEM holder and the schematic diagram of the field emission measurement are shown in Supplementary Fig. 1a, b. All measurements are performed at room temperature and a vacuum level of ~10−5 Pa.

Fabrication of the pure tungsten nanotips

The preparation of the tungsten nanotips involves two steps. The first step is an ex-situ double‑electrolyte electrochemical etching method57,58, and the schematic is shown in Supplementary Fig. 1c. Specifically, a stepping motor (TSTA-1050, 7-star, Beijing) precisely controls the motion of a 99.97% pure tungsten wire with a diameter of 0.3 mm. The tungsten wire passes through a 5 M NaOH electrolyte lamella (flake, purity 97%, Aladdin) and is immersed perpendicularly in a saturated NaCl solution (AR, Aladdin). A DC power source (B2091A, Keysight, USA) supplies a voltage of 5 V to a stainless-steel anode immersed in the saturated NaCl solution and a ring inert nickel-chromium cathode (diameter 10 mm). The tungsten wire is etched only in the NaOH lamella, with the NaCl solution acting as a conducting element between the tungsten wire and the stainless-steel anode. Once the tungsten wire breaks, the electric circuit is automatically cut off as the lower fragment of the tungsten wire drops. However, although the apex radius of the prepared tungsten nanotip by first step is less than 50 nm, there is an unavoidable oxide layer with a few nanometers thick on the tungsten nanoelectrode surface due to the ex-situ preparation, which significantly affects the field emission characteristics.

The second step is an in situ Joule melting method. Initially, the tungsten nanoelectrodes prepared by the first step and the gold plate (prepared by pressing and cutting) are mounted on the TEM electrical sample holder, as shown in Supplementary Fig. 1a. Next, the interelectrode gap is adjusted in situ to less than 20 nm under TEM. A ramped DC voltage is then applied across the nanogap to induce electrical breakdown, with the tungsten nanotip acting as the cathode and the gold plate as the anode. Following the breakdown, the tungsten nanotips with radii of curvature from 2 nm to 200 nm can be prepared in situ. The surfaces of the tungsten nanotips prepared by this method are very clean and free of contaminants such as oxide layer. This is demonstrated by the representative high-resolution TEM images and the selected area electron diffraction results of the pure tungsten nanotip with an apex radius of 5 nm, as shown in Supplementary Fig. 1d.

Finite element method (FEM) numerical simulations

The FEFs are obtained using a commercial FEM tool (COMSOL Multiphysics). First, the sphere-on-cone nanotip models are established based on the morphology of all the tungsten nanotips shown in Fig. 1. Material parameters are based on the default values provided by the COMSOL internal library. It’s worth noting that the heights of practical tungsten nanotips are approximately 1 cm, i.e., Hm » Rm and Hm » dm (in a near anode configuration), and in order to reduce the effect of the simulated heights on the simulated electrostatic fields, we set the height of the simulated models at Ht/Rt = 400. The reason is that according to the Supplementary Fig. 2, when Ht/Rt is greater than 400, the selected heights have little effect on the FEF, given that the minimum Rt/dt ratio in our experiments is 0.04. Next, the electrostatic fields are simulated using the electrostatics module. The FEF γa can be obtained as γa = Fa/FG, where Fa represents the electrostatic surface-field magnitude at the emitter apex, and FG is the average field amplitude in the gap between the emitter apex and the anode plate, given by FG = V/d. Then, the total predicted emission current is obtained by integration of LECD over the shape-model surface, using the planar emission approximation and the MG FE equation26,27,28 or KX FE equation42 (see Supplementary Note 1 and Supplementary Eqs. (S2) and (S4)). The planar emission approximation assumes that the LECD JL at any particular location “L” on the emitter surface is given by some specified planar emission equation, using the values at “L” of the local work function and the surface electrostatic field magnitude34. Supplementary Fig. 3 shows the electrostatic field distribution, as well as field emission current density JMG and JKX (two-dimensional linear current density) for the nanotip with an apex radius of 4 nm at d = 5 nm and V = 20 V.