Figure 1 shows the experimental setup of DC (cold cathode) magnetron sputtering unit to generate a glow discharge in a glass tube between two circular, parallel and movable metallic discs acting as electrodes. Two parallel electrodes made of aluminum, are enclosed in the discharge cell, one of the two electrodes is grounded represented cathode electrode and the other is movable to change the axial distance represented the anode electrode, where the axial distance (L) between tip of the probe and the cathode electrode, 5 cm and with 3 cm in radius (R) taking into account that we will deal with two conditions R > L and R < L. The discharge unit is evacuated using a rotary pump to a base pressure of 7 mtorr. A pressure gauge is connected to the discharge tube to measure the inside gas pressure. A stationary DC-glow discharge was generated between two both electrodes, for different parameters as shown in Table 1.

Figure 1 also shows the schematic diagram of the spherical single Langmuir probe circuit. The probe made of molybdenum wire (diameter 3.0-mm and length 0.5 mm) and the tip of a probe is inside the glow discharge plasma. The single probe (between the cathode fixed at the ground potential, and the anode) move axially to measure the axial potential distribution between the two electrodes. Also, it moves radially from the center to the edge of the cathode electrode, and a potential V_P is applied to the probe.

The potential distribution was measured at discharge currents of 10 mA and gas pressure of (0.2 - 1.2 torr) for He as shown in Figure 2, shows that the potential

distribution can be divided into three regions. In region I (AB) (cathode fall), the potential increases sharply within a small discharge length. Whenever, gas breakdown takes place in the tube, a rapid growth in the rate of ionization, near the cathode is detected. Meanwhile the electrons transfer in the electric field much more faster than positive ions (due to their masses), electrons are swept rapidly towards the anode leaving a dense positive space charge near the cathode. Thus, the electric field is distorted and most of the applied potential is dropped across a narrow space in front of the cathode. Region II (BC) (negative glow) indicates that the potential decreases slightly and hence the electric field will be weak, since this region contains many free electrons. In region III (CD) (positive column), the potential distribution is nearly constant and linear since the positive and negative carriers densities are closely, equal. The positive column can be extended to any length to fill the remaining space between the end of the negative glow and the anode.

Values of the electric field distribution are obtained by differentiating the measured potential distribution of Figure 2. Figure 3 and Figure 4 show the electric field distribution for He discharge. In the cathode fall region at edge and center, high electric field is observed which is decreased sharply away from the

cathode. This is related to the intense positive space charge which lies in front of the cathode. This acts as an accelerator for the electrons towards anode. Thus, the electrons emitted from the cathode are then accelerated away, until they reach the negative glow region where the electric field becomes weak (zero and sometimes it reaches a negative value). In this region, the gained kinetic energy of the electrons is dissipated in collisions with the atoms of the gas and thus secondary electrons would be produced. In the positive column region constant and linear electric field is needed to maintain discharge along the large length of the column which is required to carry the discharge current.

The region investigated in the present work is the abnormal region. The radial dependence of the electron temperature in low-density plasma at edge and center of the cathode electrode has been studied for different He pressures.

The plasma parameters like the electron temperature and electron density can be determined from the current-voltage (I-V) characteristic curve of the single Langmuir probe, based on the theory and the fundamental technique discussed in detail in many articles [10] [11] .

Figure 3 and Figure 4 show I-v characteristic curves of the single probe at different He pressures at edge and center of the cathode electrode for the abnormal glow region [12] . The probe was moved to investigate the radial distribution from the center to the edge of the cathode, in the direction perpendicular to the direction of the electric field lines. Figure 5 and Figure 6 show the radial temperatures and densities distributions at different pressures from center to the edge. T_e is decreased and n_e increased due to the general trend that values of T_e and n_e are inversely proportional [13] .

Increasing the helium working pressures from 0.2 to 1.2 torr, the temperature T_e decreases at the center from 9.7 eV to 5.5 eV and at the edge from 4.1 to 1.15 eV. Values of densities n_e increased at center n_e from 0.6 × 10⁹ cm⁻³ to 1.3 × 10⁹ cm⁻³ and at the edge from 2.5 to 3.95 × 10⁹ cm⁻³.

D S ν i = ( R 2.405 ) 2 (17)

A proof of the Schottky condition will be discussed briefly by two methods in Appendix A, where (R cm) represents the radial distance from the center to the edge through the cathode electrode measured by the Langmuir probe moved radially, and by substituting Equation (17) into (16), then

( R 2.4 ) 2 = k T e m e ν e ν i − m i ν i 2 n e n + (18)

Substituting the following equation into Equation (18)

ν e = N n 〈 Q e - n υ e - n 〉 (19a)

ν e = ( 3.55 × 10 16 P ) 〈 Q e - n 〉 ( 2 k T e m e ) 1 / 2 (19b)

ν i = ( 3.55 × 10 16 P ) 〈 Q i - n 〉 ( 2 k T i m i ) 1 / 2 (19c)

where P in torr, Q e - n and Q i - n are the cross sections of electron-neutral and ion-neutral collisions, respectively results in

( R 2.4 ) 2 = k T e [ m e ( 2 k T e m e ) 1 / 2 ( 2 k T i m i ) 1 / 2 ( 3.55 × 10 16 P ) 2 〈 Q i 〉 〈 Q e 〉 ] − [ n e n + m i [ ( 3.55 × 10 16 P ) 2 〈 Q i 〉 ] 2 ( 2 k T i m i ) ] (20)

then

( R P 2.4 ) 2 = 1 [ ( 4 m e k T i m i k T e ) 1 / 2 ( 3.55 × 10 16 ) 2 〈 Q i 〉 〈 Q e 〉 ] − [ n e n + ( 3.55 × 10 16 〈 Q i 〉 ) 2 ( 2 k T i m i k T e ) ] (21)

But from [20] [21] μ e μ i ≅ 7.64 ( m i ) 0.5

Or

m i ≅ ( 1 7.64 μ e μ i ) 2 (22a)

Then substituting (7) into (22a) gives

1 m i ≅ ( 7.64 k T i k T e ) 2 (22b)

Moreover, substituting (22b), m i m e ≅ 2000 and ν i ν e ≅ 10 − 2 into (21) gives

( R P 2.4 ) 2 = 1 [ ( 7.64 k T i k T e ) ( 4 m e k T i k T e ) 1 / 2 ( 3.55 × 10 16 ) 2 〈 Q i 〉 〈 Q e 〉 ] − [ n e n + ( 3.55 × 10 16 〈 Q i 〉 ) 2 ( 7.64 k T i k T e ) 2 ( 2 k T i k T e ) ] (23)

( R P ) 2 = 2.4 2 [ ( 7.64 k T i k T e ) ( 4 m e k T i k T e ) 1 / 2 ( 3.55 × 10 16 ) 2 〈 Q i 〉 〈 Q e 〉 ] − [ n e n + ( 3.55 × 10 16 〈 Q i 〉 ) 2 ( 7.64 k T i k T e ) 2 ( 2 k T i k T e ) ] (24)

Substituting the values of the cross sections, masses of ions and electrons into (24) gives

( R P ) 2 = 2.4 2 ( 3.55 ) 2 ( 7.64 ) ( 2 ) [ [ ( k T i k T e ) ( m e k T i k T e ) 1 / 2 ] − [ n e n + 7.64 ( k T i k T e ) 2 ( k T i k T e ) ] ] (25)

or

( R P ) 2 = 2.4 2 ( 3.55 ) 2 ( 7.64 ) ( 2 ) ( k T i k T e ) 3 / 2 [ m e 1 / 2 − 7.64 n e n + ( k T i k T e ) 3 / 2 ] (26)

Then finally

R P = 0.1729 1 ( k T i k T e ) 3 / 4 [ m e 1 / 2 − 7.64 n e n + ( k T i k T e ) 3 / 2 ] 1 / 2 (27)

Using the data of the electron temperatures and densities shown in Figure 4 and Figure 5 respectively, into the Equation (27), taking into account that T e ≫ T i ( T i = 0.1 T e ), m e m i ≅ 2000 [22] and for the spherical probe, the ion density is given by [23] :

n + = I + 0.6 ( k T e m i ) 1 / 2 A P (28)

Knowing T_e, ( A p = 4 π r p 2 ) area of spherical probe) and calculating positive ion current I₊ from the I-V characteristic curve of the spherical single probe [24] , n₊ can be determined.

Figure 7 shows the radial distribution of the electron temperature (RDFT) theoretically using Equation (27), in the abnormal cathode fall region from the center to the edge of the electrode as a function of He as an inert gas. The RDFT from the theoretical method presented above accurately shows a dramatic radial reduction of the electron temperature for any region (cathode fall or negative glow or positive column) and for any applied pressure. Furthermore, the decrements of T_e at the edge of the electrode are more pronounced at the center due to edge effect [25] .

D S ν i = ( R 2.405 ) 2 + ( L π ) 2 (29)

and by substituting Equation (29) into (16), then

( R 2.4 ) 2 + ( L π ) 2 = k T e m e ν e ν i − m i ν i 2 n e n + (30)

Substituting by Equations (19a, b and c) into Equation (30), respectively results in

( R 2.4 ) 2 + ( L π ) 2 = k T e [ m e ( 2 k T e m e ) 1 / 2 ( 2 k T i m i ) 1 / 2 ( 3.55 × 10 16 P ) 2 〈 Q i 〉 〈 Q e 〉 ] − [ n e n + m i [ ( 3.55 × 10 16 P ) 2 〈 Q i 〉 ] 2 ( 2 k T i m i ) ] (31)

then

( R 2.4 ) 2 + ( L π ) 2 = 1 [ ( 4 m e k T i m i k T e ) 1 / 2 ( 3.55 × 10 16 ) 2 〈 Q i 〉 〈 Q e 〉 ] − [ n e n + ( 3.55 × 10 16 〈 Q i 〉 ) 2 ( 2 k T i m i k T e ) ] (32)

Substitution from (22a and b) into (32) gives

( R P 2.4 ) 2 + ( L π ) 2 = 1 [ ( 7.64 k T i k T e ) ( 4 m e k T i k T e ) 1 / 2 ( 3.55 × 10 16 ) 2 〈 Q i 〉 〈 Q e 〉 ] − [ n e n + ( 3.55 × 10 16 〈 Q i 〉 ) 2 ( 7.64 k T i k T e ) 2 ( 2 k T i k T e ) ] (33)

( R P ) 2 = 2.4 2 [ ( 7.64 k T i k T e ) ( 4 m e k T i k T e ) 1 / 2 ( 3.55 × 10 16 ) 2 〈 Q i 〉 〈 Q e 〉 ] − [ n e n + ( 3.55 × 10 16 〈 Q i 〉 ) 2 ( 7.64 k T i k T e ) 2 ( 2 k T i k T e ) ] − 2.4 2 ( L π ) 2 (34)

Substituting the values of the cross sections, masses of ions and electrons into (34) gives finally

R P = ( 0.0299 ( k T i k T e ) 3 / 2 [ m e 1 / 2 − 7.64 n e n + ( k T i k T e ) 3 / 2 ] − 5.76 ( L π ) 2 ) (35)

For 1.2 torr of applied pressure, Figure 8 shows comparison between the radial distribution of the electron temperature (RDFT) for L < R using Equation (35) and for L > R using Equation (27), in the abnormal cathode fall region from the center to the edge of the electrode as a function of He as an inert gas.

The reasons for the radial distribution of the electron temperatures for both cases.

L < R and L > R are the following:

a) The temperature decreases for increasing pressure may be due to the relation between the temperature and the pressure given by [26] :

| T e | = 6.5 | P − 0.066 | (36)

b) As the pressure increase leads to a further increase in the breakdown voltage, a sharp increase in electron density and a higher electron-electron collision frequency, leads to the decrease of the electron temperatures [27] .

c) The present theoretical and experimental results are compatible and agree fairly well with the behavior of the data of von Engel [28] [29] [30] for the radial distribution of the electron temperature.

For L < R, RDFT accurately shows a dramatic decrease by 60% less than RDFT values for L > R. This may be attributed to:

The cathode fall, negative glow and positive column regions are compressed when L < R, i.e. the cathode fall region is compressed in thickness and henceforth higher potentials are expected. Thus, strong electric field is produced and therefore ions would accelerate, and more efficient sputtering processes take place. The produced plasma temperatures decrease, and densities increases; this means that the rate of plasma loss by diffusion decreased similarly as in the applied of magnetized DC plasma [23] , therefore the current and current density are increased.