Nano-gap planar metal electrodes: fabrication and I-V characteristics

The nanowires/bars and nano-gap electrodes are vital components for emerging electronics and have wide ranging applications in flat-panel displays, sensors, sub-100 nm transistor circuits, and miniaturized computers/devices. Focused ion beam (FIB) has emerged as a powerful and unique tool for nanofabrication. The research work described here is concerned with (a) the FIB fabrication of planar metallic (copper and gold) nanostructures, (b) their current-voltage (I–V) measurements in situ, and (c) a viable method for extracting the realistic values of emission parameters. The planar electrodes with gap of 80-100 nm are realized by FIB milling of thin metal films. The difficulties faced in objective interpretation of their I-V data (based on known mechanisms) are highlighted. For determining the parameters (namely, effective emission area α eff, apparent work function ф, and the field enhancement factor β), Fowler-Nordheim [ln(I/V2) versus 1/V] plots showing a minimum with straight line of negative slope can be used. The striking findings demonstrated are (i) occurrence of emission from a tiny region (<1 nm2) vis-à-vis physical area (400 μm × 200 nm), (ii) significant lowering of barrier height, and (iii) enhancement of local field due to protrusions present. Typical values of α eff, ϕ, and β deduced are 52.3 Å2, 1.62 eV, and 39.3, respectively for copper planar electrodes (gap ∼100 nm); the corresponding data for the case of gold (gap ∼80 nm) are 29.1 Å2, 1.97 eV, and 12.1, respectively. Moreover, β lowering observed with bias is accompanied by increase in the emission area due to progressive smoothening of protrusions at the cathode surface. The electrodes are found rough/rocky at the nanoscale with protrusions and varying separations at places. These features make the electron emissive region small and pointed with an enhanced local electric field and effectively of a lower barrier height. The current discrepancy in the Child-Langmuir’s space charge regime is attributed to the emission occurring from a restricted area only. These findings are important for futuristic nano-devices like thermo-tunnelling refrigerator, energy harvester, etc.


Introduction
Quantum mechanical tunnelling plays a crucial role in many nanodevices (such as tunnel diodes, Josephson junctions, etc) and forms the basis for the futuristic thermo-tunnelling refrigerator and ambient energy harvesting [1][2][3][4]. Tunnelling of electrons across a nanoscale metallic gap at a small bias is used in scanning tunnelling microscopes and flash memory devices [5,6]. The phenomenon involves passage of electrons (having less energy than the work function of the cathode material) through the potential barrier and gives rise to a current via cold field emission [7]. The barrier shape is near rectangular at low field and the current evolves is termed as direct tunnelling [8]. With increase of electric field, the barrier assumes a triangular (via a trapezoidal) shape, reduces the barrier width, and delivers higher current as Fowler-Nordheim (F-N) emission [9,10]. Thus, the current shows transition from direct tunnelling to FN emission due to field induced shape changes in the barrier [8,11]. Since the nature of barrier (height, width, and shape) plays a vital role in determining the overall current at a given field, its knowledge becomes crucial in applications like thermo-tunnelling cooling and energy harvesting [12].
The nano wires/bars and nano-gap electrodes are also important for emerging electronics and have enormous potential applications in high brightness as well as contrast/fast response/low power flat-panel displays, sensors, high power microwave amplifiers, sub-100 nm transistor circuits, field emission electron sources, and miniaturized computers/devices with high speed and large storage capacity [13][14][15][16]. Further, the reliable electrical contacts are vital at the nanoscale for future electronics, i.e., Nanoelectronics, optoelectronics and photonics, particularly in light emitting source, tunable Esaki effect, tunnel field-effect transistors, etc [17][18][19][20][21]. They can bridge individual nanostructure with the macroscopic system and form devices for detecting nanoparticles [22,23]. The nano-bar structures can be obtained by electron beam lithography [24], shadow evaporation [25], mechanically controllable break junction [23], electrochemical plating [26], and electromigration [13,27]. Another quick and flexible technique based on focused ion beam (FIB)-direct lithography enables mask-less fabrication of not only nanogap electrodes but also a variety of three-dimensional prototype nanostructures (namely, pillars, cantilevers, and springs), useful for electromechanical systems, sensors, and actuators [28][29][30].
The interpretation of I-V data of nanoscale electrodes is quite challenging. In fact, numerous emission parameters deduced and reported in the literature are physically unrealistic. These resulted due to widespread use of faulty elementary FE equations for local emission current density (LECD) and discussed in detail by Forbes recently [31]. It is imperative therefore to have standard equations that describe the FE current density reliably and devise a scheme for consistent analysis of the measured data. All formulations in use lead to uncertainty in the quantitative prediction of emission current density (ECD). Nevertheless, their verification can only be made by making careful comparison with the experiments. With the inherent complications just beginning to realize/unfold in FE, the exercise has proven to be difficult in making a precise comparison of theory and experiment. Consequently, a technological subject of field electron emission is turning out to be more scientific (i.e., a FE-science problem). This manuscript focuses on its limited but a crucial part and aim to provide a comprehensive scientific description of FE by concentrating on different methodologies used for the interpretation of current-voltage data. For this, planar electrodes (cross section 400 μm × 200 nm with a gap of ∼80-100 nm) are fabricated over a substrate by FIB milling of thin metal films and their I-V characteristics measured in situ to examine suitability of the well-known bulk emission processes/mechanisms in Figure 1. (a)-(d) Schematic diagram of (a) an I-shaped metal thin film deposited on a glass substrate by thermal evaporation, crosssectional views (b) before and (c) after forming a trench (width∼100 nm) by milling with a focused gallium ion beam, (d) electrodes of cross section 400 μm × 200 nm with a gap of∼100 nm, and (e) A typical scanning electron micrograph of the electrodes having a gap of 100 nm.
understanding the structures at the nanoscale. The difficulties encountered in the interpretation of I-V data are highlighted and a scheme suggested for extracting the parameters (namely, effective emission area α eff , apparent work function f, and the field enhancement factor β) from the ln(I/V 2 ) versus 1/V plots, taking copper and gold as examples. The striking features of nanoscale electrodes are critically analyzed to find explanation for discrepancies, viz., (i) exceeding small emission area, (ii) non-linearity in F-N plot, and (iii) apparent lowering of work function.

Fabrication of planar electrodes
The steps used for realizing planar electrodes (with a gap of ∼100 nm) are given schematically in figure 1. First, a thin film (I-shape, thickness ∼200 nm) of copper is deposited on a clean glass substrate by thermal evaporation in vacuum (∼10 −6 mbar) using a mask ( figure 1(a)). To ensure its purity, initial vapor condensation on the substrate is prevented with a shutter assembly. A trench of length 400 μm, width 100 nm, and depth 200 nm is then created by milling of the copper film with a focused gallium ion beam (30 keV, size 10-20 nm) at a current of ∼100 pA in a FEI Nova NanoLab 600 (figures 1(b), (c)). The planar electrode structure of cross-section 400 μm × 200 nm with a gap of 100 nm thus created (figure 1(d)) is placed on a sample holder and inserted in the FIB chamber for measuring the I-V characteristics in vacuum (∼10 −6 mbar) by employing a Keithley Source Meter model 6430.

Results and discussion
3.1. Current-voltage characteristics Figure 1(e) shows a scanning electron micrograph of copper electrodes having a cross-section of 400 μm × 200 nm and gap of ∼100 nm. Its current-voltage (I-V) characteristics measured in vacuum∼10 −6 mbar are depicted in figure 2(a). Note that the current increases slowly in the beginning but quite rapidly beyond a certain voltage. The corresponding ln I versus ln V plot up to ∼8 V presented in figure 2(a) (inset) demonstrates V 1/2 , V, and V 3/2 dependence in the voltage range 0.2-1.0 V, 1.3-2.4 V, and 3.2-8.0 V following the modified Child-Langmuir relation, direct tunnelling within the Simmon's approximation, and classical Child-Langmuir (CL) law, respectively [32][33][34]. The sharp increase in current above∼8 V results possibly due to initiation of Fowler-Nordheim tunnelling caused by sufficiently high electric field developed across the electrodes. The ln (I/V 2 ) versus 1/V plot given in figure 2(b) contains a minimum and a straight line with negative slope above 8.3 V. The inflection point suggests transition from V 3/2 dependence to field emission. Also, ln (I/V 2 ) vs ln (1/V) plot in the inset (b2, figure 2(b)) shows a straight line with positive slope (as unity) in the bias range of 1.3-2.4V and represents direct tunnelling. All these features are discussed below in detail.

Planar electrode model
The maximum current density J CL across two parallel planar electrodes separated by a gap (d) at a voltage (V) in one dimension is given by the classical Child-Langmuir law as [34]  where e 0 is the permittivity of free space, e is the electronic charge and m is the mass of the electron. For finite electrodes (say, strip of thickness 'w' and infinite length), electron emission follows the Child-Langmuir's relation in 2-dimension, such that the current density ⁎ J CL is given by It means that the current density ⁎ J CL improves progressively with increase in (d/w) ratio and can become greater than J CL even. For example, ⁎ J J CL CL =1.1572 and 1.3145 with (d/w)=1/2 and 1, respectively. But ⁎ J CL is essentially equal to J CL for (d/w)0.03, i.e., when electrodes separation (d) is much smaller than the thickness (w). When d is in nanometer range, quantum effects such as electron tunnelling, space-charge, and exchange correlation assume importance. The limiting current density (J QM ) then takes the form [32,36] where h is the Planck's constant. Obviously, the current density varies very differently with applied voltage (V) and electrodes gap (d) in the above two cases. The ratio of J CL and J QM reduces to This J CL /J QM ratio increases linearly with voltage but quadratically with the gap (d). Consequently, J CL is expected to be orders of magnitude higher than J QM at (d)10 nm. In other words, quantum effects cause significant reduction in the current density. On the other hand, J QM is greater than J CL when d<0.35/ V (nm). Obviously, quantum effect is prevalent at the sub-nanometer separation only. Figure 3 display J CL and J QM obtained from equations (1) and (3), respectively for planar electrodes of cross-section 400 μm × 200 nm and having gap of 0.1, 0.5, 1.0, 10, 50 and 100 nm. Accordingly, J QM is insignificant in comparison to J CL for the separation1 nm. Moreover, the current no doubt displays V 3/2 dependence in the voltage range 3.2-8.0 V but is several orders lower than that deduced from equation (1), taking electrode cross section as 400 μm × 200 nm (figures 2 and 4(a)). This discrepancy can however be understood if emission is believed to occur from a restricted portion [i.e., exceedingly small area∼80 (nm) 2 or (1:10 6 )] of the cathode (figure 4(a)).

Simmon's formulation
A structure comprising of two metal electrodes with a little separation and defined by a generalized trapezoidal barrier of height (say, f) gives rise to a tunnelling current (I) described by [33] f cr 0 with A cr , e, h, and m being the electrode cross-sectional area, electronic charge, Planck's constant, and electron mass, respectively. Also, f (x) stands for the potential energy of electron in between the two electrode surfaces, d 1 and d 2 are the distances from the first electrode to the surface 1 and 2, respectively (where the potential energy corresponds to the Fermi level), Δd=(d 2 -d 1 ), and V is the voltage applied across the electrodes. A trapezoidal barrier turns rectangular at low voltages (i.e., V∼0) with Δd=(d 2 -d 1 )=d, separation between the two electrodes, and f f = , 0 the metal work function. For V<f/e, f becomes (f 0 −eV/2). In both these cases (figures 4(b1), (b2)), equation (5) yields Equation (9) can be rewritten as Thus, tunnelling current should vary linearly with voltage (equation (9)). Also, ln (I) vs ln (V) and ln (I/V 2 ) versus ln (1/V) plots ought to exhibit straight lines with positive unit slope. Incidentally, (I) values presented in figures 2(a) and (b) (insets) match with the above description in the range 1.3-2.4 V. Similarly, at high voltages V>f 0 /e, the barrier assumes a triangular shape with f=f 0 /2 and effective separation Δd=df 0 /eV (figure 4(b3)). By making substitutions, equation (5) takes the form Hence, the ln (I/V 2 ) versus 1/V plot yields a straight line with negative slope. Notice that both the slope and the intercept can determine the barrier height. Equation (11) describes the field emission and is like Fowler-Nordheim tunnelling through a triangular barrier (discussed later).
The above formulation indicates that while the ln (I/V 2 ) versus ln (1/V) plot depicts direct tunnelling and linearity with unit slope at low bias, ln (I/V 2 ) versus (1/V) plot signifies field emission and contains a straight line with negative slope at high voltages. No current should practically be observed in electrodes of significant barrier height (f o ) and/or large width (d) at low voltages, prior to initiation of field emission. Nevertheless, the (I-V) data presented in figure 2(b) demonstrate a transition from direct tunnelling to field emission despite of the electrode work function being high (copper, f 0 =4.8 eV) and the separation (d) large (100 nm). This discrepancy is resolved by invoking an apparent work function as shown later.

Fowler-Nordheim mechanism
The generalized Fowler-Nordheim field emission current equation is written as [37] f where J is the current density in Am -2 , a=e 3 /8пh and b=(8п/3) (2m e ) 1/2 /eh are constants equal to 1.54 × 10 -6 A(eV)V −2 and 6.83 × 10 9 (eV) -3/2 Vm -1 , respectively, f is the work-function of electrode, E is the field strength, v(y) and t 2 (y) are slowly varying functions of Schottky barrier lowering parameter y=3.79 × 10 -5 E 1/2 /f. If α is the emission area, the current becomes equal to I=Jα. Also, the local field strength is represented as E=β (V/d) where d is the physical separation of electrodes and β is a field enhancement factor determined by the geometry of the emitting electrode [38]. By making these substitutions, equation (12) takes the form Accordingly, the ln (I/V 2 ) versus 1/V plot corresponds to a straight line with a negative slope, whose magnitude (say, B) is given by Charbonnier and Martin [39] suggested a simple way for extracting information about the field emission source in the low current density range (10 5 -10 8 Am -2 ) from equation (13) assuming v(y)=s(y)−1.062y 2 , t 2 (y)=1.044, and s(y)=0.956. With these substitutions and putting y=3.79 × 10 -   On differentiating equation (17) with respect to V and rearranging, one gets Thus, by measuring I-V characteristics and determining (dI/dV) for each set (I, V), the value of B/V (or B) can be deduced from equation (20). The term A can then be obtained from equation (17). Three unknown quantities namely, A, β, and f are to be found with the existing two equations (18) and (19). So, they combined equations (18) and ( Equations (17) and (22) give a =´- 10 exp 23 18 2 Once (B/V) is found from equation (20) using the I-V data, emission area (α) can be obtained from equation (23). The above procedure enabled Spindt et al [40] to estimate the emission area of molybdenum cone cathodes (1.5 μm tall, tip radius∼50 nm), fabricated with e-beam evaporated thin films and electron beam microlithography, within±10%. The value of 'α' turned out to be exceedingly small (i.e., 1.3 × 10 -19 m 2 or 13Å 2 ) compared to cathode tip radius of 50 nm-suggesting emission contribution essentially from a few atoms. For the case under study of copper planar electrodes, I=1.86 × 10 -6 A at 15 V and dI/dV=4.11 × 10 -7 (figure 3), equation (23) gives α=8.82 × 10 -23 m 2 for B/V=1.31. This figure is extremely small and unrealistic with no physical significance/meaning. Obviously, the above method fails to yield realistic values of the emission area. Wong and Ingram [41] fabricated Au/Ti lateral tunnel diodes with inter-electrode separation below 50 nm on Si 3 N 4 substrates using high resolution electron beam lithography and lift-off metallization process. The I-V characteristics measured under vacuum (10 -6 Torr) and in air (760 Torr) followed Fowler-Nordheim tunnelling above a threshold voltage of 45 V. They used ln (I/V 2 ) versus 1/V plot to estimate the field enhancement factor (β) and apparent emission area from the negative gradient and intercept at the ordinate, respectively with equations like (17)(18)(19); the values arrived at were β=2.63 and α=2 × 10 -20 m 2 (or 2 Å 2 only) for a diode having inter-electrode gap of 25 nm.
Forbes [42,43] examined the issue of extracting emission area from the intercept (ln A) and negative slope (-B) of a straight line drawn by linear fitting in the Fowler-Nordheim ln(I/V 2 ) versus 1/V plot and proposed a formulation with new symbols and correction factors but finally obtained relations analogous to equation (16) of Charbonnier and Martin [39] and equation (21) of Spindt et al [40] given above. Further, the emission area (α eff ) was given by a = Г ( ) AB C 24 2 2 where C 2 is a universal constant (=7.192 × 10 13 Am -2 eV -2 ) and Г is termed as extraction parameter related to work function (f) of the emitter and the current density (J). C 2 Г is analogous to the term 6.29 × 10 13 G(f) of equation (16). Forbes [26] expressed the logarithmic current density in terms of Nordheim parameter y (= 3.79 × 10 -5 E 1/2 /f) and its functions namely, v(y) and s(y). He carried out numerical calculations using a spreadsheet by varying 'y'. The values of y and, in turn, Г were then tabulated for various log J and f. Since f and E are not known at the emission site, the value of Г can be estimated approximately. The range of Г is chosen by experimental conditions considering plausible values of f and J to estimate the emission area from equation (24).
Applying this approach to the present case of copper electrodes (cross section 400 μm × 200 nm, gap∼100 nm) and taking f=4 eV, lower bound of Г turns out to be 564 for current density∼3.2 × 10 13 A m −2 . With values Table 1. The effective emission area (α eff ), apparent work function f ( ), field enhancement factor (β), local field strength (E), Schottky barrier lowering (Δf) and work function (f+Δf) for copper electrodes corresponding to the straight lines 1 and 2 of figure 2  Ln (I/V 2 ) versus 1/V plot of copper planar electrode structure presented in figure 2(b) corresponds to a straight line with negative slope above 8.3V. This straight line is split into two covering voltage ranges 12.0-17.5 V and 18.0-20.9 V with slopes (magnitude B=16.1 and 33.9) and intercepts on the y-axis (ln A)=−17.5 and -16.5, respectively (inset of figures 2(b), (b1)). Equation (26) gives the values of α eff f 2 product as 11.3 and 136.5 Å 2 (eV) 2 , respectively. With each product, the possible combination of α eff and f can be determined (table 1). Now taking d=100 nm, the value of (β) can be deduced from equation (19).    ), apparent work function f ( ), field enhancement factor (β), local field strength (E), Schottky barrier lowering (Δf) and work function (f+Δf) for gold electrodes corresponding to the straight lines 1, 2 and 3 of figure 5(b2). These results yield the emission parameters as listed in table 2. For determining the local field E=βV/d, d is taken as 80 nm and the voltage as 30 V, 37 V and 45 V for the straight lines 1, 2 and 3, respectively (figure 5(b2)). For gold, the bulk work function is 5.1 eV but reduces substantially due to Schottky barrier-lowering (induced by high fields∼10 9 V m −1 ) and protrusions present (figure 5(a2)). A progressive rise in the emission area results due to smoothening of protrusions by local heating with current at increasing bias. These findings on copper and gold planar diodes fabricated by FIB milling of thin films amply demonstrate (i) significant decrease in the work function and (ii) an exceedingly small area of the electrode contribution to the field emission. The main reason is the local enhancement of the electric field due to protrusions and decrease in the work function of the emitter caused by pronounced lowering of the Schottky barrier at high local electric fields.
To ascertain the elemental composition, x-ray analysis of planar electrodes assembly was undertaken using an energy dispersive spectrometer. Besides, elemental distribution maps were observed in each case. Accordingly, metal was found distributed outside the trenched (or gap) region only. x-ray line scans of a typical platinum planar electrode assembly (gap∼200 nm) fabricated with FIB using the same procedure over the SiO 2 substrate for various elements (C, O, Ga, Si, and Pt) along the milled trench region are shown in figure 6(a) while the elemental distribution map of platinum is displayed in figure 6(b), conforming to the above description.
The fabricated copper and gold electrodes discussed here are of the same thickness (i.e., 200 nm). The difficulty arises when the current density (J) is regarded as emission occurring from the entire cathode geometrical cross-sectional area. The approach gives erroneous values of (J) and causes problem in accounting them vis-à-vis current equations developed for bulk, even though the latter described the voltage dependence reliably for the nanoscale electrodes too. This is what precisely demonstrated here. The actual current density is determined by the preferential emission sites (i.e., protrusions present in the fabricated electrodes due to lack of smoothness at the nanoscale) rather than the geometrical area. Also, the number of protrusions and their geometry depends on the fabrication process itself. The local field becomes high at such sites and lead to preferential emission from an infinitesimal small area. This feature is clearly revealed in figure 4(a) using the I-V data. But the emission area increases with rise in applied voltage caused by local burn out (or removal) of protrusions -leading to flattening of electrode tip at high fields (tables 1 and 2). The details of analysis procedure are given in section 3.1.1.
The I-V characteristics of yet another copper planar electrode assembly of cross-section 400 μm × 1000 nm with a gap of 200 nm (i.e., greater thickness and wide gap) is shown in figure 7(a). They also reveal a sharp increase in current (or initiation of F-N tunnelling) across the electrodes above 175 V with a minimum and a straight line having negative slope in the ln (I/V 2 ) versus 1/V plot ( figure 7(b)). The analysis exercise as before gives the slope and intercept of the straight line in the voltage range 175-210 V as B=−2196.3 and ln A=−17.6, respectively. With the α eff f 2 product 3.11 × 10 5 Å 2 (eV) 2 , the possible combinations of α eff and f  Table 3. The effective emission area (α eff ), apparent work function f ( ), field enhancement factor (β), local field strength (E), Schottky barrier lowering (Δf) and work function (f+Δf) for copper electrodes corresponding to a straight line with negative slope in inset of figure 7 are summarized in table 3 along with the values of (β) and local electric field (E) at d=200 nm and onset bias (V)=175 V. Accordingly, the apparent work function, the effective emission area the enhancement factor (β) are 3.05 eV, 8.67 × 10 5 Å 2 and 2.4, respectively corresponding to bulk copper (work function∼4.8 eV). Notice that the emission area is much larger than listed in table 1 but for higher applied voltages (above 175 V). This happened presumably due to electrodes having larger gap (200 nm) and higher thickness as the smoothness is attained in wide area above 175 V (a relatively higher bias).

Conclusions
• A 30 keV focused gallium ion beam (size 7-10 nm) combined with a scanning electron microscope in a FEI Nova Nanolab 600 can be successfully employed for controlled milling of thin metal films (thickness say, 200 nm) to fabricate planar electrode with gap of∼100 nm.
• A simple method developed using the negative slope and intercept of ln (I/V 2 ) versus. 1/V plot enables extraction of realistic emission area (α eff ) and the apparent work function f ( ) of the emitter. Typical values of α eff and f being (i) 52.3 Å 2 and 1.62 eV, respectively for copper and (ii) 29.1 Å 2 and 1.97 eV, respectively for gold planar electrode.