Stopping potential and ion beamlet control for micro-resistive patterning through sub-Debye length plasma apertures

Ion beams are becoming increasingly important for research in basic science^1,2 technology³ as well as material studies.⁴ With the rapid development of nanotechnology there is a major requirement of easy, economic as well as time efficient ion beam processing techniques, which cannot be fulfilled by conventional ion sources. For nanometer scale modifications, use of focused ion beam (FIB) systems is one viable option but conventional FIB sources can only produce single beam of gallium ions, and is not suitable for processing large area samples. In order to be able to process large samples, instead of single ion beam, multiple ion beamlet system is a promising candidate. This can be used for large-scale surface patterning, creation of functional surfaces on soft matter such as polymers,⁵ complex mask-making for micro-electronics, bio patterning and mask-less ion beam lithography.^6,7 Ions being much heavier than electrons (1837 times for the lightest element hydrogen), therefore, the de Broglie wavelength of ions is two orders of magnitude shorter than electrons at the same energy, and higher resolution than electron beam are achievable. Multiple beamlets can be utilized to generate pre-programmed patterns over a large sample area which makes it economic as well as time efficient.

There are a few efforts world-wide^6–8 to develop such multiple ion beamlet systems (MIBS). Notable among them are: (a) Maskless Micro-ion-beam Reduction Lithography (MMRL) by Leung et al. at Lawrence Berkley National Laboratory (LBNL) which employed filament generated plasmas.⁶ In these plasmas, it is found that the radial plasma uniformity often has to be compromised and can lead to beam current non-uniformity on the substrate.⁵ More recently, there is another report on multi-ion beam lithography from Appleton et al. at University of Florida,⁹ that is capable of generating 16 × 16 beamlets with beam energy ∼15 – 40 keV. In another work in Europe, an ion projection lithography (IPL)⁷ system is developed where a broad beam and a stencil mask are used to generate patterns and the image of the mask is projected on the sample.

A MIBS^10,11 has been developed to investigate interaction of low energy (1 - 5 keV) ion beams with different substrate materials at micrometer resolutions. A compact microwave driven plasma (ion) source developed in the laboratory is employed to obtain the beamlets. Currently efforts are in place to reduce the focused beamlet diameter to the submicron regime by controlling the PE aperture and beamlet focusing.

In this article we demonstrate the patterning capability of the MIBS using switching technique. This is a useful technique with various applications. Firstly, one can generate computer controlled patterns from a switchable plasma electrode (PE). Secondly, these patterns can be directly transferred to the substrate (which is quite useful for maskless lithography) and finally, localized changes can be made on substrates at micrometer length scales, thereby increasing the possibility of industrial applications.

To realize the above, a switching technique is developed to control individual beamlets separately by applying electrostatic potential on a specially designed PE. Switchable plasma electrodes (PE) with a 3×3 array of through apertures having aperture diameters 800 µm, 400 µm and 200 µm are first investigated for generating focused ion beamlets. The stopping potential V_s (i.e. the voltage at which beam current goes to zero and the beamlet from the individual aperture is shutoff) is determined for each aperture diameter. To reduce the beamlet diameter further to a few microns or even in the sub-micron regime, the PE aperture diameters are reduced to 50 µm and 30 µm, which are below the Debye length λ_D of the plasma in the system. The effect of this reduction on the overall beamlet characteristics is investigated. The beamlets are extracted in a controlled manner and current-voltage (I-V) profile is measured along with individual beamlet current to confirm the uniformity. The focused ion beamlets are employed to create localized resistive regions in a patterned manner on different substrate materials by ion beam irradiation. Specific pattern designs are created. The change in electrical properties of thin films upon irradiation with low energy broad ion beams have been investigated in details in our recent work,¹² where variation of sheet resistance in metallic nano films with beam energy, fluence and ionic species are demonstrated. The prime advantage of the micro beamlets is that one can change the properties of the sample material locally at desired locations, and this is demonstrated in this work.

The article is arranged as follows. In section II, the experimental set-up for beamlet extraction and switching in described. The experimental results are reported in section III. Finally, in section IV, some of the important results are discussed and a summary is presented.

The schematic of the experimental system with all components is shown in Fig. 1(a). Microwaves (MW) of 2.45 GHz are guided through a quartz window (W) into a magnetic multicusp (MC) constructed with permanent magnets, for plasma generation. For details on fabrication of the multicusp please see reference 13–16. The multicusp helps in near boundary resonances and in confinement of the plasma. The microwaves are produced by a magnetron oscillator capable of producing continuous mode microwaves of peak power 1.8 kW and pulsed power of 8 kW. The wave guide (WG) circuitry consists of an isolator for protecting the magnetron from the reflected power, a directional coupler for measuring the forward and reflected powers and a triple stub tuner for impedance matching. The vacuum chamber (VC) is evacuated by a turbo molecular pump (TMP) backed by a rotary pump to 2 × 10^-7 Torr. The gas is fed through a valve (Gas Inlet (GI)). The gas flow is controlled by a Mass Flow Controller and the pressure is maintained in the range of 8.5 × 10^-5 – 5.5 × 10^-4 Torr during plasma generation. The multiple ion beamlet system is attached to a flange which is then attached to the MC. For further details please refer to references 13–16.

To control the individual beamlets separately a new plasma electrode having switching capability is designed. To stop a beamlet current it is necessary to apply a repulsive potential (positive voltage) to the individual apertures of the PE. For this, design of a special kind of PE with voltage controlling system is made. The PE has 3 layers (cf. Fig. 1(c)). The first layer is a continuous metallic layer (except the nine apertures) which faces the plasma, the second layer is made of an insulator and the third layer is a patterned electrode with voltage controllability, where 0 - 120 volts can be applied to stop the beam as shown in Fig. 1(c). The schematic diagram and digital photograph of the PE are shown in Fig. 1(b) and 1(d) respectively. A thin aluminium foil is used as the sample and is kept on the collector (C) plate at a distance 5 mm away from PE (cf. Fig. 1(c)). The collector is maintained at a high negative potential (1 - 3 kV) to extract the ion beamlets. The layered design of the PE is made on PCB (printed circuit board) using a software named Protel. PEs with different aperture sizes viz. 800 µm, 400 µm and 200 µm with the beam controlling system are made on PCB (for a design refer to Fig. 1(d)).

Next a switching circuit is constructed which can control the voltages on the individual pads of the PE for controlling the individual beamlets. The schematic diagram of the switching circuit is shown in Fig. 2(a). The circuit consists of a 40 pin ATMEGA-16L Microcontroller unit (MCU), Relays (HJR-3FF-S-Z) and MOSFETs (FB A14 IRF830B) and resistances. A digital image of the fabricated circuit is shown in Fig. 2(b).

To test the PE, at first the total extracted current is measured by varying the extraction voltage in the range 0.01 to 3.0 kV. The PE has two sides: the plasma facing side has a continuous metal layer kept at ground potential and the other side contains the connection pads for the switching circuit. The variation of the total beam current with the extraction voltage is shown in Fig. 3(a) for the 800 µm, 400 µm and 200 µm plasma electrode apertures. After an initial rapid growth (until ∼100 volts) the beam current increases uniformly with extraction voltage. In this region the rate of increase of current i.e. the slope of the I_b – V_E characteristics increases with aperture diameter, where I_b is the total beam current and V_E is the extraction voltage. The slopes vary from 8.4 - 0.66 μA/kV as shown in the Fig. 3(a). Until -3 kV there is no sign of any beam current saturation. The maximum ion current was found to be 32 μA at an extraction voltage of -3 kV for 800 µm plasma electrode aperture.

The experiments are continued by fabricating PE with much smaller aperture diameters of 50 µm and 30 µm and the total beam current is measured first. In the experiments with larger size apertures, the PE was kept in contact with the multicusp and thus it was at ground potential. In the grounded situation, it is interesting to note that we did not obtain any extracted beamlet current for the smaller apertures of 50 µm and 30 µm. Therefore, the PE was floated and kept ∼ 4 mm away from the multicusp. At this condition, the PE acquires a potential equal to the floating potential of the plasma. The floating potential V_f is given by,¹⁷ $V f =− κ B T e e ln m i 4 π m e$⁠, where T_e is electron temperature and m_i and m_e are the masses of ion and electron respectively. In our experimental conditions with argon plasma, T_e at the extraction region is ∼ 5 eV and V_f is obtained to be ∼ - 21.7 volt. Therefore, the positive ions are accelerated through this extra potential and could pass through the smaller PE apertures. The extracted current obtained in this condition is plotted in Fig. 4. Similar trend of current is observed, although the magnitude of the current is reduced quite a bit for these smaller apertures as shown in Fig. 4. The slope of I_b – V_E plot is much smaller as expected for these smaller apertures and increases with aperture diameter in agreement with the larger aperture size PEs.

Except for the reduced aperture diameter, another possible reason for the reduction in the current is that the plasma electrode is kept 4 mm away from the edge of the multicusp and the ion density as well as ion saturation current falls sharply outside the multicusp.^10,18

All the results presented here are corrected for the leakage current of the system. To measure the leakage current we apply a high voltage (0 – 3 kV) to the collector in the absence of plasma and note the current. In all the cases the leakage current is much smaller ∼ 6 – 10 times less than the actual beam current.

Next the switching circuit is tested to determine the stopping potential V_s for each aperture diameter. For this the total beam current (at a particular extraction potential) is measured by varying the voltage on the beamlet control pads (please see Fig 1(c)). The result is shown in Fig. 5(a) for the three larger apertures (800 µm, 400 µm and 200 μm).

The beam current decreases gradually with increase in applied voltage and goes to zero at the stopping potential. Figure 5(b) show the results for smaller apertures (50 µm and 30 µm) and a similar trend is observed. These data can be fitted reasonably well with a Hill equation having variable slope given by,

where y_max and y_min are the values at the top and bottom asymptotes, log x₀ and p are respectively the centre and Hill slope of the fit. It is observed that the magnitude of the Hill slope, p increases from 0.026 to 0.112 with decreasing aperture diameter i.e. beam current falls faster for smaller apertures. This indicates that for larger size apertures the field penetration from the extraction side is more resulting in larger beam energy and beam current at the exit side of PE.

In case of mono-energetic beam the nature of the graphs of total beam current vs. switching voltage (as shown in Fig. 5) would have been a “step-function”. Rather they follow Hill equation (Eq. (1)) and the extracted ions show a range of energies with a mean. Therefore, by taking the derivative of total beam current (I_b) with respect to V_s, we can estimate the energy spread of the beam. This is similar as determining the ion energy spread in a retarding field energy analyzer.

The data can be well fitted with a Gaussian. Figure 6(a) and 6(b) show the Gaussian fit for 200 µm and 30 µm apertures respectively. The extracted axial ion beam energy spread at the exit of the plasma electrode is obtained from the FWHM of the Gaussian curve. The FWHM increases with aperture diameter as shown in Fig. 6 and goes from 15.5 volt to 21.5 volt as the aperture is increased from 50 to 200 µm.

The variation of V_s with PE aperture size is shown in Fig. 7, which shows almost a linear behaviour. We tried a linear fit for the data points with an adjacent R-square value of 0.982 which suggests a good linear fitting. The numbers at the data points shows the standard deviations in drilling 9 apertures for each of the cases (800 µm, 400 µm, 200 µm, 50 µm and 30 µm apertures). We wanted to compare this result with a standard beam optics code SIMION,¹⁹ the simulations are performed for each aperture and the corresponding stopping potential is determined. The result is shown by the black squares in Fig. 7. Although the simulation values are little lower than the experimental data, they follow same linear nature of the experimental data. The difference is about ∼ 9 % (for 800 µm) to ∼ 30 % (for 400 µm).

After the determination of V_s, we measured the individual beamlet current from all the nine apertures separately using the controller circuit. In the nine-aperture (3 × 3) system, the apertures can now be selectively switched on and off. The results are shown in Fig. 8. Figure 8(a) shows the individual currents for the 200 µm aperture whereas in Fig. 8(b) the results for the 50 µm aperture are shown. In both the cases the extraction potential is kept fixed at -1 kV. There is a small variation in currents for the 200 µm apertures, but it is almost uniform for the 50 µm apertures. The reason for this slight non-uniform distribution of beam currents in 3 × 3 array is that the apertures are not identical to each other. There are small variations in geometrical parameters (i.e. diameter and roundness) of the apertures. Apertures of 200 µm and larger were drilled mechanically and thus more variation in shape and size are observed. Whereas the smaller apertures are laser drilled and provide more accurate and almost identical apertures and therefore more uniform beamlet current through all the apertures. The numbers adjacent to the data points in Fig. 7 indicate the errors (standard deviation) in diameter for different apertures.

Employing the switching circuit we can generate different patterns on substrates. To demonstrate the patterns, a computer code was written to run the microcontroller and subsequently to control the beamlets. The patterns are obtained on aluminium foils. In Fig. 9 the digital photographs of the some of the patterns are shown. In Fig. 9(a) all the apertures were open. Figures 9(b), 9(c), 9(d) and 9(e) show a square shape, a ‘U’ shape, a ‘+’ sign and a right angled triangle respectively. The details of the patterns are given in Table I.

Figure 9(d) and 9(e) show the beam spot sizes at the substrate for 50 μm PE aperture with extraction voltage of 2.5 kV and 2 kV. To verify some of the experimental results of beam irradiation we performed SIMION simulation to obtain the spot size at focal point. For that the gap between PE and collector (where the substrate material is placed) is kept fixed and the extraction potential is varied and the beam diameter is measured. A change in extraction potential changes the electric field between the PE and the collector. In an earlier report, Chowdhury et al.¹¹ has explained how this electric field determines the focal point of the beam. Here, for different PE apertures, we tried to achieve lowest possible beam spot on the substrate kept at a fixed distance away from PE. The result is shown in Fig. 10(a) for the three bigger apertures (800 µm, 400 µm and 200 µm) and Fig. 10(b) for the two smaller apertures (50 µm and 30 µm). For 50 µm aperture, the focal point is obtained for 0.3 keV and corresponding spot size is ∼ 6 µm. But experimentally we obtained much larger spot (37.3 µm as shown in Fig. 9(d)). The reason is that at 0.3 keV we did not get any prominent spot at the substrate. We had to increase the extraction voltage which shifted the focal point and a defocused larger spot is obtained at the substrate.

To measure the change in resistance at micron scale, first we designed a mask for thin film deposition. The design of the mask is shown in Fig. 11. It is basically a 200 µm thick strip with four pads (A, B, C and D) for contacts. The pads are of square shaped with dimension 500 μm × 500 μm. Pads A and B are connected to a constant current source and pads C and D measure the corresponding voltage drop in the strip. The strip (area =200 μm × 800 µm) is then irradiated with ion beam as shown by the shaded region in Fig. 10 and we can calculate the difference in resistance due to irradiation as compared to the pristine sample at micrometer length scales.

Using the fabricated mask, Cu thin film samples are prepared on glass substrate in a thermal evaporation system. The sheet resistances (R_S) of the pristine (as-deposited) samples are measured and found to be almost same (∼ 1.25 Ω). The samples are then irradiated with argon ion beam with varying fluences, keeping the energy the same at 0.5 kV and the change in R_S is measured. For better sensitivity the samples are irradiated with higher fluence. The results are summarized in Table II. Therefore, with MIBS one can create an array of local resistive regions of diameter ∼ 30 µm where the sheet resistance of the irradiated regions are 2 – 6 % higher, than the unirradiated region. The resistance can be further controlled by changing the fluence and changing the mass of the ionic species.

For generating focused ion beamlets the aperture diameters are varied from values that are higher than the Debye length λ_D of the plasma to lower than λ_D. The Debye length λ_D is given by $λ D = ε 0 κ B T e n e e 2$⁠, where ε₀ and κ_B are free space permittivity and Boltzmann’s constant and T_e, n_e and e are the electron temperature, electron density and electronic charge respectively. For our experimental conditions at the ion beam extraction point (i.e. T_e ∼ 5 eV and n_e ∼3 × 10¹⁰ per cc) the value of the Debye length λ_D is ∼100 μm. The results of total beam current variation with extraction potential is shown in Fig. 3 (aperture diameter >λ_D) and Fig. 4 (aperture diameter >λ_D). For aperture larger than λ_D, the plasma electrode was kept at ground potential and the total beam current is measured in collector (as shown in Fig. 3), but this condition was not sufficient for apertures smaller than λ_D. The main reason is, for larger size apertures the field penetration from the extraction side is more, compared to apertures smaller than λ_D, which makes extraction easier. To extract through the smaller apertures, PE was kept at floating condition so that ions can gain more energy equivalent to the plasma floating potential while passing through the smaller apertures. When the apertures are bigger than λ_D, the sheath does not cover the aperture fully. On the contrary, for apertures that are smaller than λ_D, the sheath fully covers the aperture and quasi-neutral nature of plasma is not present there. In this case, the extra floating potential helps in extracting the beamlets.

The extracted beam current profile can be divided into two parts. As seen from Fig. 4 for the case of 50 μm diameter PE aperture (also the same is true for apertures of other dimensions indicated in Fig. 3), the total beam current first shows a rapid increase with the extraction voltage up to ∼50 V and then reaches a value about 0.48 nA. This region is called space-charge-limited flow region. The current then increases slowly with the extraction voltage, but saturation is not reached even for a value of -3 kV where the maximum ion current is ∼1.98 nA. The second region is called extraction-voltage-limited flow region. The slope of the current-voltage characteristics in this region increases with aperture diameter as the increment of extraction voltage has more prominent effect for bigger apertures due to higher field penetration. The complete I-V profile is similar to that of vacuum diode characteristics. The transition from the space-charge-limited region to the extraction-voltage-limited region is gentler for apertures below λ_D. We’ll now further analyse and discuss both the regimes.

For a small applied potential on the collector, the beam current at the collector is smaller than the total current available for extraction. However, the extraction of the ion beam continues even at small voltages, although the currents are quite small. As a result, the ions form a cloud near the PE and the space charge controls the amount of current in this region. This limitation is set by the repelling effect that the space charge exerts on the ions which try to emerge from the apertures. Normally, in this region, the current-voltage relationship generally obeys the Child-Langmuir law, which demands $I∝ V 3 2$⁠. However this relationship assumes that the current source and the collectors are infinite, planar, parallel, equipotential surfaces and the ions are emitted with zero initial velocity. The experimental system satisfies none of the above assumptions. It has been observed that when the aperture size becomes comparable to λ_D, the nature of the extracted beam current deviates from the conventional Child-Langmuir’s law. The actual I-V relationship that is found for our experiments is:

This equation has been plotted with the experimental data in Fig. 12 and shows a reasonable agreement. The details of the calculation are given in Appendix  A.

As seen in Fig. 4, after the space-charge-limited flow regime, the ion current tends to saturate. This corresponds to a situation when all the ions emitted through the PE apertures, reaches the collector. Ideally, beyond the space-charge-limited region, the total ion current should be independent of the collector potential. But this does not happen in reality. From Fig. 4, we note that the ion current actually increases slowly with increasing collector potential – a phenomenon similar to the famous Schottky effect in vacuum diode valves. In that case, the collector potential actually lowers the work function of the emitting surface and thus enhances the emission of electrons. For our experimental situation, it will be interesting to investigate how the collector potential modifies the potential at the centre of the aperture due to the formation of plasma sheath near the PE boundary wall. This is investigated next. Fig. 13 shows the ion beam trajectory and the equipotential lines that have been simulated by using the AXCEL-INP²⁰ code. The equipotential surfaces due to the collector potential bend through the PE aperture and penetrate into the plasma sheath region. These curved potential lines act as lens for the beam and focus the beam as shown in Fig. 13.

The part that has been simulated is therefore the plasma region and the region between the PE and C. The figure shows the emergence of the beam from the plasma side through the PE aperture and getting focused onto the collector C.

For this kind of circular aperture of diameter a, the current density is calculated to be,

where, φ₀ is the collector potential, z₀ = 2 mm, the distance between PE and collector, $E 1 = φ 1 z 1$⁠, φ₁ is the potential at the centre of the aperture with respect to the bulk plasma potential and z₁ may be taken as equal to the sheath thickness (= n×Debye length) (we have taken n = 2 for the calculations). Eq. (3) has been plotted with the experimental data and shown in Fig. 14 and agrees with the experimental I-V relationship within the extraction-voltage-limited flow regime. Thus it gives a satisfactory explanation of the observed Schottky-like effect for our system. The steps to reach Eq. (3) are outlined in Appendix  B.

We have determined the switch voltage for each aperture from the variation of total beam current with switch voltage (as shown in Fig. 5). The results show that switch voltage increases with increasing aperture diameter and almost follow a linear behaviour (as shown in Fig. 7). The reason for this can be obtained from the potential structure of the electrode system. We know the potential at the centre of the circular aperture of radius a for this kind of electrode system is given by $φ ( 1 ) = E 0 − E 1 a π$ where E₀ and E₁ are the electric fields on either side of the aperture (see Appendix A2 for details). This potential is responsible for extraction of ions through the aperture and is proportional to the aperture radius a. Therefore, the extracted ions will have higher energy for larger aperture and thus higher switch voltage is required to stop the ions. This is a consequence of field penetration through the apertures.

As beam energy is not mono energetic (as evident from Fig. 5), we have determined the extracted ion energy spread from the derivative of the total beam current (I_b) with respect to V_s as shown in Fig. 6. The energy spread (FWHM of the Gaussian fit of these derivatives) is higher for larger aperture and confirms earlier results.

To test the uniformity in beam currents from all 9 apertures of the 3 × 3 arrays are measured individually. Slight variations are observed in individual beamlet currents (as shown in Fig. 8) arising from the fact that there are small variations in geometrical parameters (i.e. diameter and roundness) of the apertures. The standard deviation in diameter while drilling the 9 apertures are given in Fig. 7. Some patterns are generated with these 3 × 3 aperture arrays (as shown in Fig. 9) and irradiation parameters are summarized in Table I. The variation of beam spot size with the extraction voltage is shown in Fig. 10. As the voltage between the PE and the collector determines the focal point of the beam,¹¹ we determined the spot size in a view to locate the focal point on the substrate by adjusting the extraction voltage. The irradiated regions of the samples show higher value of sheet resistance than the pristine one and has been confirmed from the micron-scale resistivity measurement with the help of the circuit fabricated as shown in Fig. 11.

Switchable plasma electrodes with a 3×3 array of through apertures having aperture diameters 800 µm, 400 µm, 200 µm, 50 µm and 30 µm are investigated for generating focused multiple ion beamlets. We have demonstrated the patterning capability of the MIBS using a beam switching technique which has many useful applications. For this, individual control over the beamlets is required which is achieved by using small repulsive potential (0 - 120 volt) to the individual apertures in the plasma electrode.

The beam current profile consists of (a) space-charge-limited flow region and (b) extraction-voltage-limited flow region and is very similar to the vacuum diode characteristics. In the space-charge-limited flow region, the relationship deviates from the Child-Langmuir law $( I α V 3 2 )$ and takes the form: $I α V 1 2$⁠. In the extraction-voltage-limited flow region, the extracted beam current keeps on increasing slowly instead of getting saturated. This is similar to the Schottky effect in vacuum diodes. This effect has been found to be due to the penetration of the equipotential surfaces due to the collector potential. When the aperture size reduces to below the Debye length, then the effect of perturbing collector potential decreases thereby making the slope of the I-V characteristic in the extraction-voltage-limited flow region lesser and the extraction becomes more and more governed by the sheath potential at the centre of the aperture. With the inclusion of switching electrode, the system behaves like a vacuum triode valve and a characteristic similar to the mutual or transfer characteristic of a vacuum triode valve has been obtained.

The stopping potential V_s is determined for each of the aperture diameters and is found to be linearly decreasing with decreasing aperture diameter which is also confirmed from SIMION simulations. The beamlets are extracted in a controlled manner and individual beamlet current is measured to confirm the uniformity. We have created some patterns with 3×3 array of beamlets. These patterned irradiated spots (i.e. regions with high surface defects) can act as active centres for further treatment of the substrate in terms of chemical etching, electrochemical treatment,²² selective deposition and formation of self-organized microstructures.²¹ Change is sheet resistance due to micron beam irradiation is measured and 2 – 6% change is observed with varying beam fluence.

In order to find out the current-voltage relationship for our system in the space-charge-limited flow region, let us assume that the I-V relationship is given by:

Since, at V = 0, I = 0, we are left with b = 0.

Taking natural log on both sides we obtain,

Therefore, a plot between ln(I) and ln(V) will be a straight line whose slope is γ and intercept on the y axis is ln(a).

From Fig. 15, it is found that the slope is 0.6 and intercept is ln (a) = 0.6. Therefore the actual I-V relationship that is found for our experiments is:

Now, to calculate the I-V relationship in the extraction-voltage-limited region we take the following approach. The potential at the centre of the circular aperture of radius a for this kind of arrangement is given by:²³ 

where E₀ and E₁ are the electric fields on both sides of the aperture - in this case they are electric fields due to the collector and the electric field due to the formation of sheath respectively as marked in Fig. 13.

The ion current density at any point inside the sheath is²⁴ given by

where T_e is the electron temperature, M_i is the ionic mass, and, n_e0 is the plasma electron density. $η=− e φ k T e$⁠, where φ is the potential at any point inside the sheath with respect to the potential of the bulk plasma and I(η) is a term which is a function of η and it needs to be determined.

Assuming the electrons obey the Boltzman distribution, the electron current at the PE wall is given by

At equilibrium, the ion and the electron current will be equal at the PE wall. This gives,

which implies

Therefore substituting for $I η w$ in Eq. (B2) we obtain,

where φ_w is the value of potential φ at the PE wall. In the presence of the collector potential, φ_w should be replaced by $( φ w + φ 1 )$⁠, where $φ 1$ is given by Eq. (B1).

Defining the constant pre-exponential factor of Eq. (B6) by A, we can write the current in the presence of the collector potential as

By virtue of Eq. (B1),