Conversion Method of Thermionic Emission Current to Voltage for High-Voltage Sources of Electrons

Abstract

The stability of the electron thermionic emission current is one of the most important requirements for electron sources used, inter alia, in evaporators, production of rare gas excimers, and electron beam objects for high energy physics. In emission current control systems, a negative feedback signal, directly proportional to the emission current is transferred from the high-voltage anode circuit to the low-voltage cathode circuit. This technique, especially for high-voltage sources of electrons, requires the use of galvanic isolation. Alternatively, a method of converting the emission current to voltage in the cathode power supply circuit was proposed. It uses a linear cathode current intensity distribution and multiplicative-additive processing of two voltage signals, directly proportional to the values of cathode current intensity. The simulation results show that a relatively high conversion accuracy can be obtained for low values of the electron work function of the cathode material. The results of experimental tests of the dynamic parameters of the electron source and the steady-state I_e-V characteristic of the converter are presented. The implementation of the proposed I_e-V conversion method facilitates the design of the emission current controller, especially for high-voltage sources of electrons, because a negative feedback loop between the anode and cathode circuits is not required, all controller sub-components are at a common electrostatic potential.

1. Introduction

Thermionic electron sources operated in temperature limited or space charge limited mode are a key component of vacuum devices, which use the interaction of the thermionic electron beam with matter, including evaporators [1], X-ray photoelectron spectroscopy [2], electron beam facilities for high energy physics [3], production of rare gas excimers [4], and others, e.g., electron beam inspection for integrated circuit manufacturing process monitoring [5].

The stability of the emission current has a significant impact on the quality of electron beam devices. The problem of stabilization includes the conversion of the thermionic emission current to voltage, comparing its value with the reference voltage and controlling the electric power supplied to the cathode, e.g., in evaporators, or controlling the bias voltage on the control grid electrode of electron source, for a constant cathode temperature, e.g., on the Wenhelt electrode in electron microscopes. With regard to evaporators, there are also designs in which the electric power supplied to the cathode is controlled by the voltage directly proportional to the ion current [6]. In commonly used emission current control systems, the I_e-V conversion is performed in the anode power supply circuit and its output signal is transferred as a negative feedback signal to the low-voltage cathode power control circuit [7,8,9,10]. For a relatively small value of the cathode-anode potential difference and, consequently, a relatively low electron energy, e.g., in mass spectrometers, ionization vacuum gauges, a galvanic connection can be used to implement the negative feedback loop. The works [7,9] present systems in which the negative feedback signal is transferred from the anode power supply circuit to the low-voltage cathode power supply circuit by means of a current mirror, a high-voltage differential amplifier used for this purpose is described in [8,10]. These controllers ensure high-quality stabilization of the emission current, however, the value of the electron accelerating voltage is limited by the breakdown voltage of semiconductor components. For higher anode power supply voltages, the problem of stabilizing the emission current is more complex. The paper [4] describes the original system of the emission current controller for the infrared emission spectroscopy of rare gas excimers, in which the electron accelerating voltage of 100 kV is used. In the system, the measurement of the emission current is performed in the high-voltage anode power supply circuit and its result is transferred as a negative feedback signal to the cathode control circuit by means of an optical fiber. The optoelectronic connection was also used in the digital controller of the electron beam evaporator, described in [1].

The necessity to transfer the negative feedback signal from the high-voltage anode circuit to the low-voltage cathode circuit in the controllers of high-voltage sources of electrons justifies an attempt to convert the emission current into voltage in the cathode power supply circuit. The article presents the results of simulation tests of the I_e-V conversion method in the cathode power supply circuit for the electron source in a diode configuration, operating in temperature limited mode and the results of experimental tests of the dynamic parameters of the electron source and the static characteristic of the I_e-V converter.

2. Modeling of the Emission Current-to-Voltage Conversion in the Cathode Power Supply Circuit

In terms of the thermionic electron source working under temperature limited conditions, the temperature dependence of the intensity of the emission current is described by the Richardson-Dushmann equation [11]:

$$I_e=AS{T}^2\exp \left(-\frac{\phi }{k_{B}T}\right),$$

(1)

where A = 1.2 × 10⁶ Am⁻² K⁻² is the Richardson constant, S is the surface from which the electrons are emitted, T is the temperature, φ is the electron work function for the cathode material, k_B is the Boltzmann constant.

Due to the cathode’s heat capacity, the thermionic electron source is a higher-order inertial system. In order to simplify considerations, its small signal transconductance G(s) can be approximated by the following expression [12,13]:

$$G(s)=\frac{G_0}{T_{C}s+1}{e}^{-s{T}_0},$$

(2)

where G₀ is the DC transconductance, T_c is the time constant, T₀ is the delay time, s is the Laplace operator.

In order to justify the method of converting the emission current to the voltage in the cathode power supply circuit, a simplified equivalent circuit of the diode-type electron source has been presented. Figure 1a shows a scheme of the biasing system of the thermionic electron sources, which simplified equivalent circuit as presented in Figure 1b.

The cathode is divided into n elementary parts with the resistance values r₁, r₂, …, r_n, which at a fixed value of the heating voltage V_h give an emission current with the intensity values I_e1, I_e2, …, I_en. The resistances r_i of individual elements and the values of the intensity I_ei of the emission current depend on the temperature of the individual cathode elements, which can be determined using the cathode temperature distribution [14]. The current intensity I_ei is described by the Richardson-Dushmann equation:

$$I_{ei}=A{S}_i{T}_i^2\exp \left(-\frac{\phi }{k_B{T}_i}\right),$$

(3)

where: S_i is the side area of the i-th element, T_i is the temperature of the i-th cathode element. It was assumed that the electron work function is equal to φ for all cathode elements.

According to the diagram shown in Figure 1b, the intensity of the cathode current I_ci flowing through the i-th element can be expressed as follows:

$$I_{ci}=I_1+{\displaystyle \sum }_{j=i}^n{I}_{ej}$$

(4)

The cathode current intensity depends on the location along the cathode, described by the generalized variable i. Consequently, the above formula describes a linear cathode current intensity distribution. For the j = 1 limit value, the intensity I₂ of the current flowing from the cathode can be written as follows:

$$I_2=I_1+{\displaystyle \sum }_{j=1}^n{I}_{ej}.$$

(5)

As

$$I_e={\displaystyle \sum }_{j=1}^n{I}_{ej}$$

(6)

Then

$$I_2=I_1+I_e$$

(7)

The above relationship was used to convert the thermionic emission current to voltage in the cathode power supply circuit.

Based on the diagrams shown in Figure 1, the electron accelerating voltage V_e was derived. The voltage accelerating the electrons emitted from the i-th element is given by the formula:

$$V_{ei}=V_a-V_{ci}$$

(8)

In order to determine the voltage V_ci, it is helpful to determine, first the voltage V_c1 at the upper terminal of the resistor r₁ with respect to ground:

$$V_{c1}=\left({\displaystyle \sum }_{j=1}^n{I}_{ej}+I_1\right)r_1$$

(9)

Similarly, the voltage at the upper terminal of the resistor r₂ with respect to ground is described by the formula:

$$V_{c2}=V_{c1}+\left({\displaystyle \sum }_{j=2}^n{I}_{ej}+I_1\right)r_2$$

(10)

With the above in mind, a generalized equation can be expressed to describe the voltage at the upper terminal of the resistor r_i relative to ground:

$$V_{ci}={\displaystyle \sum }_{k=1}^i\left({\displaystyle \sum }_{j=k}^n{I}_{ej}{r}_k\right)+I_1{\displaystyle \sum }_{k=1}^i{r}_k.$$

(11)

Equation (8), after substituting Formula (11), takes the following form:

$$V_{ei}=V_a-\left[{\displaystyle \sum }_{k=1}^i\left({\displaystyle \sum }_{j=k}^n{I}_{ej}{r}_k\right)+I_1{\displaystyle \sum }_{k=1}^i{r}_k\right]$$

(12)

The above equation describes the electron accelerating voltage V_ei at the i-th cathode element. For a relatively long cathode, it can be assumed that the temperature distribution is uniform, then the resistances r_k and the values of the current intensity I_ei are respectively equal:

$$r_k=\frac{1}{n}{R}_C,$$

(13)

$$I_{ej}=\frac{1}{n}{I}_e,$$

(14)

where R_c is the cathode resistance for the given heating voltage V_h.

Combining the dependencies (12) and (13) and performing the summation with respect to j and k, has the following form:

$$V_{ei}=V_a-\frac{i}{n}{R}_c\left[\frac{2n-i+1}{2n}{I}_e+I_1\right]$$

(15)

Given the emission current and the electron accelerating voltage for the i-th cathode element, the electron beam power can be determined:

$$P_{en}={\displaystyle \sum }_{i=1}^n{V}_{ei}{I}_{ei}$$

(16)

Taking into account the expression (15), the formula (16) can be presented as follows:

$$P_{en}=V_a{I}_e-\frac{R_c}{n}\left(I_e{\displaystyle \sum }_{i=1}^{n}i{I}_{ei}-\frac{I_e}{2n}{\displaystyle \sum }_{i=1}^n{i}^2{I}_{ei}++\frac{I_e}{2n}{\displaystyle \sum }_{i=1}^{n}i{I}_{ei}+I_1{\displaystyle \sum }_{i=1}^{n}i{I}_{ei}\right)$$

(17)

Using the equation given in the work [15]:

$${\displaystyle \sum }_{i=1}^n{i}^2=\frac{1}{6}n\left(n+1\right)\left(2n+1\right)$$

(18)

and summing in Formula (17) with respect to i, we obtain:

$$P_{en}=V_a{I}_e-R_c{I}_e\left[\frac{2n^2+3n+1}{6n^2}{I}_e+\frac{1}{2n}{I}_1+\frac{1}{2}{I}_1\right]$$

(19)

Assuming that the cathode is divided into infinitely many elementary parts, the power of the electron beam can be determined using the relationship:

$$P_e=\underset{n\to \infty }{\lim }{P}_{en}$$

(20)

Substituting Equation (19) to the above formula, we obtain:

$$P_e=\left(V_a-\frac{1}{3}{R}_c{I}_e-\frac{1}{2}{R}_c{I}_1\right)I_e$$

(21)

Hence the electron accelerating voltage V_e can be written as follows:

$$V_e=V_a-\frac{1}{3}{R}_c{I}_e-\frac{1}{2}{R}_c{I}_1$$

(22)

The above relation shows the effect of the cathode heating current and the thermionic emission current on the value of V_e. For high-voltage sources, the electron accelerating voltage can be approximated by the anode power supply voltage V_a.

The simplified scheme shown in Figure 2 is used to explain the conversion of the emission current to voltage in the cathode power supply circuit.

The cathode power supply circuit consists of a voltage source V_h, sensing resistors R₁, R₂, and a cathode. The difference in the voltage drops across the resistors R₁, R₂ is equal to:

$$V=R_2{I}_2-R_1{I}_1$$

(23)

After taking into account the Formula (7), the following is obtained:

$$V=R_2{I}_e+\left(R_2-R_1\right)I_1$$

(24)

In order to make the voltage V independent of the current I₁, the following condition should be assumed:

$$\frac{\partial V}{\partial I_1}=0$$

(25)

For the ideal case described in this way, the voltage V = V_ideal is given by the formula:

$$V_{\mathrm{ideal}}=R_2{I}_e$$

(26)

To determine the electron accelerating voltage, the voltage drop across the resistor R₂ should be taken into account in Formula (24), hence:

$$V_e=V_a-I_1\left(\frac{1}{2}{R}_c+R_2\right)-I_e\left(\frac{1}{3}{R}_c+R_2\right).$$

(27)

In practice, given the finite values of the tolerances of the resistors δR₁ = δR₂, the Equation (26) does not have to be met. The discrepancy between the voltages V and V_ideal can be described by the relative voltage difference (V − V_ideal)/V_ideal). Using Formulas (24) and (26) we obtain:

$$\frac{V-V_{\mathrm{ideal}}}{V_{\mathrm{ideal}}}=\frac{I_1}{I_e}\frac{R_2-R_1}{R_2}.$$

(28)

Assuming that the resistances R₁ = R₂ and the tolerances of the resistors δR₁ = δR₂, for the most unfavorable distribution of the values of R₁ and R₂, the Formula (28) takes the form:

$$\frac{V-V_{\mathrm{ideal}}}{V_{\mathrm{ideal}}}=\frac{I_1}{I_e}2\delta R_1$$

(29)

In order to quantify the relative voltage difference, first the numerical relationship of the I₁/I_e ratio as a function of the electron work function was simulated, using the cathode model described by the Formula (A2) and the graphical relationship shown in Figure A3 in Appendix A. For the simulation, model tungsten wire cathodes with a diameter of d = 1.2 × 10⁻⁴ m and a length of l = 4.5 × 10⁻² m, surface-coated with materials with the electron work function φ in the range from 0.9 eV to 4.5 eV were used. The calculations were made for the set values of the intensity of the emission current: I_e = 1 mA, I_e = 10 mA, I_e = 50 mA, and I_e = 100 mA. The results are shown in

Table 1. The simulation results of the I₁/I_e quotient as a function of the electron work function φ, at the set values of the emission current: 1 mA, 10 mA, 50 mA, and 100 mA.

φ, eV	Ie = 1 mA		Ie = 10 mA		Ie = 50 mA		Ie = 100 mA
	I1, A	I1/Ie	I1, A	I1/Ie	I1, A	I1/Ie	I1, A	I1/Ie
4.5	1.64	1637.99	1.93	195.51	2.26	45.13	2.42	24.20
4.2	1.44	1442.46	1.73	171.63	1.98	39.55	2.11	20.99
3.9	1.26	1256.39	1.51	150.69	1.73	34.51	1.84	18.40
3.6	1.09	1086.97	1.30	129.77	1.49	29.81	1.59	15.85
3.3	0.93	925.03	1.11	109.76	1.27	25.46	1.36	13.60
3.0	0.78	777.17	0.93	92.59	1.07	21.42	1.14	11.40
2.7	0.64	639.48	0.77	76.63	0.89	17.71	0.94	9.34
2.4	0.52	514.53	0.62	61.78	0.72	14.32	0.76	7.61
2.1	0.40	403.68	0.49	48.44	0.56	11.25	060	5.98
1.8	0.31	304.07	0.37	36.83	0.43	8.52	0.46	4.56
1.5	0.22	217.75	0.27	26.31	0.31	6.13	0.33	3.26
1.2	0.14	143.32	0.18	17.36	0.20	4.09	0.22	2.18
0.9	0.08	80.82	0.10	10.05	0.12	2.40	0.13	1.30

:

Using the above data, the relative voltage difference ((V − V_ideal)/V_ideal) was determined, based on the Formula (29), for the tolerance of the sensing resistors R₁ and R₂ equal to δR₁ = δR₂ = 0.01%. The results are shown in Figure 3.

The analysis of the results shows that the described I_e-V conversion method in the cathode power supply circuit is predestined for electron sources with low electron work function of the cathode material. For a fixed electron work function, the relative voltage difference decreases with the increase of the emission current. An important design issue is the selection of sensing resistors with high temperature stability, because a relatively high cathode current flows through them, which causes their heating.

3. Experimental Results

3.1. Dynamic Parameters of the Electron Source with Sensing Resistors Connected in Series in the Cathode Supply Circuit

In order to assess the influence of the sensing resistors on the dynamic properties of the thermionic electron source, the time constant T_c and the delay time T₀ were determined using step response method [13]. The scheme of the measuring system is presented in Figure 4. The square wave control signal is fed from the Ao1 output of the NI 6251 data acquisition card to the input of the power amplifier A1 operating in a voltage follower configuration, the output of which is connected to the cathode power supply circuit and the input Ai0. The voltage, directly proportional to the emission current intensity, is supplied to the Ai2 input from the I_e-V anode converter built on the basis of the current mirror [7].

For the square wave period, the value of 10 s was assumed, which is sufficient for the cathode temperature and the emission current to reach the set values. Measurements were made for a Bayart-Alpert gauge electron source at a pressure of 0.1 Pa, with a thoriated tungsten cathode (d = 0.00012 m and l = 0.045 m). Sensing resistors R₁ = R₂ with nominal values of 0.100 Ω, 0.383 Ω, 0.562 Ω, 1.000 Ω were used in the measurements. The waveforms of the control voltage and the emission current step response were acquired for the set values of the constant component V_h0 and the step voltage ΔV_h. An example of the waveforms of the control voltage V_h(t), emission current I_e(t) and the step response of the first-order inertial model with a delay are shown in Figure 5.

Successive step responses I_e were registered for the set values of the constant component V_h0 and the step voltage ΔV_h = 0.05·V_h0. The values of the time constant T_c and the delay time T₀ determined on the basis of these waveforms are presented as a function of the intensity of the emission current in Figure 6.

As can be seen, with increasing emission current, the time constant decreases while the delay time remains approximately constant. According to the results of work [13], for the electron source without sensing resistors, the delay time decreases with increasing emission current. In the present work, the delay time remains approximately constant due to the use of linear, sensing resistors connected in series with the cathode, which have a linearizing effect on the temperature dependence of the resistance of the new control object (cathode + sensing resistors). The decrease in the time constant results from faster cathode-ambient thermal energy exchange at higher and higher temperatures. Analogous measurements were made for the electron source itself and for the electron source with sensing resistors 0.100 Ω, 0.562 Ω, 1.000 Ω. The summary of all the results is shown in Figure 7.

Increase in series resistance in the cathode circuit causes an increase in the time constant of the electron source circuit. For example, for I_e = 0.45 mA and resistance R₁ = R₂ = 1.000 Ω the increase of time constant is close to 75% in relation to time constant of the electron source itself. For this reason, the resistances of R₁, R₂ should be as low as possible. On the other hand, the resistance values have an influence on the measured voltage drops (Formula (23)) and the signal/noise ratio, hence the need for a compromise when choosing their values.

3.2. Static Characteristic of the I_e-V Converter

In order to verify the proposed conversion method, a prototype of the I_e-V converter was designed. A simplified scheme of the converter, hereinafter referred to as the cathode converter, is presented in Figure 8.

The converter consists of sensing resistors R₁, R₂ placed on a common heat sink and three instrumentation amplifiers IA₁, IA₂, IA₃ with differential-mode gains k_d1, k_d2, k_d3, respectively. Resistors R_G1, R_G2, R_G3 are used to adjust these gains. The common-mode gains of the amplifiers were assumed to be negligible. The output voltages V₁, V₂ are directly proportional to the voltage drops across the sensing resistors R₁, R₂, respectively:

$$V_1=k_{d1}{R}_1{I}_1,$$

(30)

$$V_2=k_{d2}{R}_2{I}_2.$$

(31)

The output voltage V of the amplifier IA₃ is equal to:

$$V=k_{d3}\left(V_2-V_1\right).$$

(32)

Combining the relations (30)–(32) and (7), we obtain:

$$V=k_{d3}\left[I_e{k}_{d2}{R}_2+I_1\left(k_{d2}{R}_2-k_{d1}{R}_1\right)\right].$$

(33)

Assuming the condition:

$$\frac{\partial V}{\partial I_1}=0,$$

(34)

the ideal case is obtained, in which the voltage V = V_ideal is expressed by the relationship:

$$V_{\mathrm{ideal}}=k_{d3}{k}_{d2}{R}_2{I}_e.$$

(35)

The discrepancy between the output voltage of the ideal converter (Formula (35)) and the output voltage of the analyzed converter (Formula (33)) is expressed by the relative voltage difference:

$$\frac{V-V_{\mathrm{ideal}}}{V_{\mathrm{ideal}}}=\frac{I_1}{I_e}\frac{\left(k_{d2}{R}_2-k_{d1}{R}_1\right)}{k_{d2}{R}_2}.$$

(36)

Appropriate selection of the k_d2 differential-mode gain value with the R_G2 resistor allows to compensate the influence of the sensing resistors tolerance on the conversion accuracy. The estimated maximum relative contributions of components in the output signals of the IA₁ and IA₂ amplifiers due to common-mode gains are 0.43% and 3.00%, respectively.

In order to determine the static dependence of the output voltage V as a function of the emission current I_e of the cathode converter, measurements were carried out in the circuit, the diagram of which is shown in Figure 9.

The system uses an electron source with a tungsten cathode installed in a Bayard-Alpert gauge. The sensing resistors R₁ = R₂ = 0.383 Ω were used. The intensity of the emission current was adjusted by the voltage V_h. The output voltage V of the cathode converter was measured with an Agilent 34461A multimeter, while the emission current was measured with a Brymen 859 multimeter in the anode power supply circuit. Figure 10 shows the static characteristic of the converter output voltage as a function of the emission current intensity.

The characteristic of the converter was approximated by the equation:

$$V=0.37 I_e+0.01.$$

(37)

The relative non-linearity of the characteristic is lower than 0.77%.

The obtained results constitute an experimental verification of the proposed method of converting the emission current into voltage. Based on the dependencies (30)–(32), it can be concluded that the multiplicative-additive processing of voltage signals directly proportional to the values of the cathode current intensity enables the conversion of the emission current into voltage in the cathode power supply circuit. Although the anode voltage V_a = 100 V was used in the measurements, it is clear that the upper value of the voltage V_a is only limited by the parameters of the electron source.

The use of a cathode converter facilitates the design of the emission current controller. The simplified scheme of an exemplary controller using a cathode converter is shown in Figure 11.

For the amplifier A1, the voltage V, directly proportional to the emission current I_e, is a negative feedback signal that is compared with the reference voltage V_ref and the amplified error signal (V_ref–V)k_v controls the cathode heating so as to maintain the pre-set value of the emission current. In the controller diagram shown, the anode is at a high potential, but it is easy to polarize the cathode circuit with a high negative voltage, and the anode is grounded, which is used in many electron beam devices.

As can be seen in Figure 11, the I_e-V cathode converter is the negative feedback loop of the A1 amplifier. Comparison of presented and reported feedback loop applications used in emission current controllers is shown in

Table 2. Comparison of feedback loop applications used in emission current controllers.

	Feedback Loop Based on the Cathode Converter (Present Work)	Feedback Loop Based on the Differential Amplifier [8,10]	Feedback Loop Based on the Current Mirror [7,9]	Feedback Loop Based on the Optical Link [4]
Ie-V conversion implementation	Cathode circuit	Anode circuit	Anode circuit	Anode circuit
Feedback signal transferring from the anode to the cathode circuit	Not required	Voltage	Current	Voltage
Electron accelerating voltage	High	Low	Low	High
Electron work function of the cathode	Low	Wide range	Wide range	Wide range
Galvanic isolation in the feedback loop	Not required	Not required	Not required	Applied
Complexity	Low	Low	Low	High

.

4. Conclusions

The method of converting the emission current to the voltage in the cathode power supply circuit was proposed. Based on the original static model of the thermionic electron source in a diode configuration, a linear distribution of the cathode current intensity was determined on the basis of which the conversion of the emission current to voltage with the use of sensing resistors in the cathode power supply circuit was justified. The results of simulation studies show that a relatively high conversion accuracy can be obtained for low values of the electron work function of the cathode material. The influence of the sensing resistors on the time constant of the electron source system was determined experimentally. A prototype of a cathode converter with a tungsten cathode was developed and its static characteristic was determined, for which the relative nonlinearity is lower than 0.77%. The use of the I_e-V cathode converter in the automatic control system of the emitted electron current eliminates the need to transfer the negative feedback signal from the high-voltage anode circuit to the low-voltage cathode circuit, which may be the key application advantage of the presented converter, especially in relation to high-voltage sources of electrons.

Currently, a prototype of a digital cathode converter applied in a digital emission current controller is being designed and tested. The investigations for electron sources with an yttrium oxide coated iridium cathode and a thoriated tungsten cathode will be performed.