4 Discussion

The surface potential of a spherical probe or dust grain is determined by the electron and ion currents to its surface. In a low-temperature plasma, where $T_i \ll T_e$, the surface potential is mainly determined by $T_e$. Since $v_{\textrm {the}}\gg v_{\textrm {thi}}$, the surface potential is always negative with respect to the plasma potential. Here, $v_{\textrm {the}}$ and $v_{\textrm {thi}}$ are the electron and ion thermal velocities, respectively. In an unmagnetized rf discharge plasma (at $B = 0\ \textrm {T}$), the surface potential of a spherical object of the radius $r > \lambda _{De}$ is estimated using the theoretical value of $\alpha$ in the transition region between OML and TS regions (Willis et al. Reference Willis, Coppins, Bacharis and Allen2010, Reference Willis, Coppins, Bacharis and Allen2012). A slight variation in $T_e$ (see figure 4) with increasing the power and pressure demonstrates a negligible change in $V_s$ in the unmagnetized plasma (see figure 2).

With the application of a magnetic field, the gyroradius of electrons ($r_{\textrm {ge}} = m_e v_{\textrm {the}}/e B$) and of ions ($r_{\textrm {gi}} = m_i v_{\textrm {thi}}/e B$) decreases with increasing $B$-field. Due to the mass differences, electrons are magnetized at lower magnetic field than ions, i.e. $r_{\textrm {ge}} \ll r_{\textrm {gi}}$. The electrons and ions are considered to be magnetized when the gyration frequency of the respective species ($\omega _{\textrm {ce}/\textrm {ci}}$) is higher than the collisional frequency ($\nu _{e-n/i-n}$), i.e. $r_{\textrm {ge}/i}<\lambda _{e-n/i-n}$. Here $\lambda _{e/i}$ is the collisional mean free path for the respective species. In our experiments ($p = 15$ to 50 Pa and $P = 3.5$ to 12 W), $n$ and $T_e$ are observed to varybetween ${\sim }6\times 10^{14}\ \textrm {m}^{-3}$ and $3\times 10^{15}\ \textrm {m}^{-3}$ and 3–5 eV, respectively. The mean free path of electrons, $\lambda _{e-n} = 1/n_g \sigma _{e-n} \sim 14\text {--}3\ \textrm {mm}$ and ions, $\lambda _{i-n} = 1/n_g \sigma _{i-n} \sim 0.2\text {--}0.08\ \textrm {mm}$. Here $\sigma _{e-n} \sim 2\times 10^{-20}\ \textrm {m}^{2}$ and $\sigma _{i-n} \sim 1\times 10^{-18}\ \textrm {m}^{2}$ are the collision cross-sections of electrons and ions with argon atoms, respectively

(Benyoucef et al. Reference Benyoucef, Yousfi, Belmadani and Settaouti2010), and $n_g$ is the neutral gas density. The electron gyroradius varies between $r_{\textrm {ge}} \sim 0.5\text {--}0.7\ \textrm {mm}$ for $B = 0.01\ \textrm {T}$ at given discharge conditions. Therefore, the condition $r_{\textrm {ge}} < \lambda _{e-n}$ meets even below $B = 0.01\ \textrm {T}$. With an increasing of the strength of the magnetic field ($B > 0.01\ \textrm {T}$), $r_{\textrm {ge}}$ continuously decreases and electrons are fully magnetized. Ions are assumed to be at room temperature, i.e. $T_i \sim 0.03\ \textrm {eV}$. The ion gyroradius $r_{\textrm {gi}}$ at $B = 0.2\ \textrm {T}$ is estimated as $\sim$0.5 mm, which indicates that ions get magnetized at high magnetic field, $B > 0.2\ \textrm {T}$. It essentially means that in the range of the magnetic field ($B< 0.2\ \textrm {T}$), only electrons are magnetized but ions are assumed to be unmagnetized. In a magnetized plasma, the currents $I_e$ and $I_i$ to the surface of a spherical probe are altered when the condition, $r_{\textrm {ge}/i} < \lambda _{De}$, is satisfied. Here $\lambda _{De}= \sqrt {\epsilon _0 k_B T_e/e^2 n_e}$ is the electron Debye length. In the present work, $\lambda _{De}$ varies between $\sim$0.2 and 0.5 mm for the given range of plasma parameters. It shows that the electron current gets changed as a $B$-field is introduced. The ions do not fulfil this criteria at $B< 0.2\ \textrm {T}$, therefore, the ion current to the surface of the spherical object is considered to be unaffected.

In an unmagnetized plasma, a constant flux of energetic electrons is lost to the chamber wall. The magnetic field confines the electrons, which definitely reduces the electrons loss to the chamber wall. Therefore, it is expected that the density of the energetic electrons would be increased as the magnetic field is introduced. To see the effect of the $B$-field on the electron population, the EEDF is measured using a tungsten cylindrical probe of length $l_p = 8\ \textrm {mm}$ and radius $r_p = 0.15\ \textrm {mm}$. The probe is positioned perpendicular to the discharge axis or magnetic field lines. At discharge condition ($P = 12\ \textrm {W}$ and $p = 30\ \textrm {Pa}$), $r_p <\lambda _{De}$, therefore, the conventional probe theory of a cylindrical probe is used to get the EEDF. It should be noted that the plasma anisotropy in the presence of the magnetic field depends on the parameter $B/p$ and this value should be higher than $3\times 10^{-2}$ T/Pa (Behnke et al. Reference Behnke, Passoth, Csambal, Tichý, Kudrna, Trunec and Brablec1999). Since in our set of experiments, the ratio of $B/p$ varies from ${\sim }1\times 10^{-4}$ to $2\times 10^{-3}$ T/Pa, it does not exceed $3\times 10^{-2}$ T/Pa. Therefore, we do not expect any substantial anisotropy of the plasma or EEDF in our measurements. It should also be noted that the second derivative probe method gives a reliable EEDF in the range of the diffusion parameter (Demidov et al. Reference Demidov, Ratynskaia, Armstrong and Rypdal1999) $\varPsi = r_p ({\ln (\pi l_p/4 r_p)}/{\gamma r_{\textrm {ge}}})< 30$ (Popov et al. Reference Popov, Ivanova, Dimitrova, Kovačič, Gyergyek and Čerček2012). Here, $\gamma$ is constant and we assumed $\gamma \sim 4/3$ for our pressure regime. In our case, the diffusion parameter has the value $\varPsi < 15$ for $B < 0.1\ \textrm {T}$; therefore, this method is used to get the EEDF to show the increase in the energetic electron population as the $B$-field is turned on. The EEDF is estimated from the second derivative of the probe I-V characteristics with respect to the probe voltage (Godyak Reference Godyak1990; Kudrna & Passoth Reference Kudrna and Passoth2007; Popov et al. Reference Popov, Ivanova, Dimitrova, Kovačič, Gyergyek and Čerček2012),

(4.1)\begin{equation} F(E)= \frac{2 \sqrt{2 m_e}}{A_p e^3} \sqrt{E} \dfrac{\textrm{d}^2I_e}{\textrm{d}V^2}, \end{equation}

where $E = \textrm {eV} = e(V_p - V_b)$, $I_e$ is the electron current to probe, $V_b$ is the probe bias, $V_p$ is the plasma potential and $A_p$ is the area of probe. The EEDF with magnetic field at $p = 30\ \textrm {Pa}$ and $P = 12\ \textrm {W}$ is shown in figure 7(a). The population of the cold (or lower energy) electrons, which are reaching the probe, decreases with increasing $B$-field whereas the population of the energetic electrons increases at low $B < 0.05\ \textrm {T}$ (see inset image). It means that the energetic electrons can easily reach this probe surface at low $B$-field.

Figure 7.

(a) Electron energy distribution function (EEDF) with external magnetic field at gas pressure, $p = 30\ \textrm {Pa}$ and rf power, $P = 12\ \textrm {W}$. The inset image represents the enhancement of the energetic electron population with the application of a magnetic field. (b) The plots of ln(EEPF) for given EEDF at various strengths of the magnetic field.

It is obvious that the average electron energy will increase due to the increase of the density of energetic electrons. It means $T_e$ is expected to increase while the magnetic field is introduced. Since the variation in $T_e$ affects the surface potential of a spherical object (see (3.3)), it is measured using a double or single probe at various strengths of the $B$-field. The inverse slope of $\ln (\textrm {EEPF})=$ $\ln ({F(E)}/{\sqrt {E}})$ with respect to $E$ (see figure 7b) gives $T_e$. It should be noted that the single probe used to obtain the EEDF is not compensated and overestimates $T_e$, therefore, errors concerning $T_e$ are expected at low $B$-field. At higher $B$ ($B > 0.09\ \textrm {T}$), the secondary plasma around the probe tip during the positive bias does not give true I–V characteristics, which is also a cause of error in the $T_e$ measurement even though it is rf compensated. In view of this, a double probe is used to obtain the approximate value of $T_e$ up to $B \sim 0.15\ \textrm {T}$. The double probe theory (Johnson & Malter Reference Johnson and Malter1950) estimates reliable plasma parameters ($n$ and $T_e$) in rf discharges if electrons obey the Maxwellian distribution, i.e. the EEDF should be Maxwellian in the presence of the $B$-field. In figure 7(b), $\ln (\textrm {EEPF})$ is plotted against the electron energy $E$ for different values of $B$. The $\ln (\textrm {EEPF})$ against $E$ shows the characteristics of a Maxwellian plasma (Godyak, Piejak & Alexandrovich Reference Godyak, Piejak and Alexandrovich1993) in the experiments. Figure 8 represents the variation of the electron temperature ($T_e$) with magnetic field at $p = 30\ \textrm {Pa}$ and $P = 6.5$ and $12\ \textrm {W}$.

Figure 8.

Electron temperature ($T_e$) variation with magnetic field at pressure, $p = 30 \ \textrm {Pa}$ and rf powers, $P = 6.5$ and 12 W.

In the magnetized plasma, the net electron current $I_e$ to the spherical probe is a sum of the electron currents to different positions on the probe (Patacchini et al. Reference Patacchini, Hutchinson and Lapenta2007). In the present experiments, it is difficult to estimate the electron current as a function of position (with respect to the magnetic field direction) on the spherical probe surface (Patacchini et al. Reference Patacchini, Hutchinson and Lapenta2007). Therefore, a simple model which considers the net electron current ($I_e$) in two possible directions along the $B$-field ($I_{e\parallel }$) and transverse to the $B$-field ($I_{e\perp }$) is used to explain the observed results qualitatively. Here $I_{e\parallel }$ and $I_{e\perp }$ are assumed to be the electron current components at a given position on the spherical probe with respect to the magnetic field direction. The total electron current to the surface of a spherical probe is $I_e = I_{e\parallel } + I_{e\perp }$, which determines the floating surface potential of a probe in a magnetized plasma. It should be noted that the electron motion transverse to the $B$-field is much more hindered than that along the $B$-field in the moderately collisional plasma. In other words, $I_{e\perp }$ is reduced much more than $I_{e\parallel }$ in a magnetized plasma (Sanmartin Reference Sanmartin1970; Bohm, Burhop & Messey Reference Bohm, Burhop, Messey, Guthrie and Wakerling1994), i.e. $D_{e\perp } < D_{e\parallel }$, where $D_{e\perp }$ and $D_{e\parallel }$ are the transverse and longitudinal diffusion coefficients, respectively.

There are two possible diffusion processes: the first one is the drain diffusion and the second one is the collisional diffusion. In drain diffusion, electrons may change their direction and cross the $B$-field during the motion in rf oscillating sheath field of a spherical object. In collisional diffusion, the gyrating electrons collide with background neutrals and diffuse across the $B$-field with a higher rate (Bohm et al. Reference Bohm, Burhop, Messey, Guthrie and Wakerling1994). The drain diffusion mainly dominates over the collision diffusion in a low pressure magnetized plasma. Since the present work is performed in a moderately collisional plasma, the collisional diffusion process is considered to be more effective. For moderately collisional low-temperature magnetized plasmas, the collisional transverse diffusion coefficient is $D_{e\perp } = D_{e0}/(1 + \omega ^2_{\textrm {ce}} \tau ^2_e)$, where $D_{e0} = \lambda _{e-n} v_{\textrm {the}}/3$ is the diffusion coefficient in the absence of a $B$-field, $\omega _{\textrm {ce}} = e B/m_e$ is the electron cyclotron frequency and $\tau _e = \lambda _{e-n}/v_{\textrm {the}}$ is the electron–neutral collision time (Chen Reference Chen1984; Bohm et al. Reference Bohm, Burhop, Messey, Guthrie and Wakerling1994; Curreli & Chen Reference Curreli and Chen2014).

In a collisionless magnetized plasma, the electron motion is restricted by the magnetic field in the transverse direction. Therefore, the $D_{e\parallel }$ (or $I_{e\parallel }$) play a dominant role in the determination of the surface potential (or charge) of the dust grain in the presence of magnetic field (Patacchini et al. Reference Patacchini, Hutchinson and Lapenta2007). The logarithmic plots of transverse diffusion coefficient, $\ln ({D_{e\perp })}$, are shown in figure 9 which show a reduction in $D_{e\perp }$ after a magnetic field strength of 0.01 T. The value of $D_{e\perp }$ increases with the pressure while the magnetic field strength ($B > 0.02\ \textrm {T}$) is kept constant, which is also illustrated in figure 9.

Figure 9.

Logarithm plots of transverse diffusion coefficient ($D_{e\perp }$) for different argon pressures at various strengths of magnetic field.

In the low magnetic field regime ($B < 0.05\ \textrm {T}$), the increase in $T_e$ (see figure 8) enhances the surface potential (more negative) of a spherical probe according to (3.3). Now, the probe collects a higher net current ($I_e$) than $I_{e0}$. Here, $I_{e0}$ represents an equilibrium electron current to the spherical probe in unmagnetized plasma. This can also be understood on the basis of energetic electron population. Experimentally, the dominating role of energetic electrons in the charging process of a spherical object or dust grain in the plasma have been confirmed (Arnas, Mikikian & Doveil Reference Arnas, Mikikian and Doveil1999). Since the Larmor radius of the energetic electrons ($T_e > 20\ \textrm {eV}$) lies between 1.5 mm and 0.4 mm for $B < 0.05\ \textrm {T}$, energetic electrons are considered to be weakly magnetized. Since $D_{e\perp }$ decreases in this $B$-field regime ($B < 0.05\ \textrm {T}$), the net electron current to spherical surface should be lowered. However, the opposite behaviour (large $I_e$ or $V_s$) is observed at low $B$-field because of the confinement of energetic electrons (or higher $T_e$). The higher charges on the dust grains or more negative surface potential in a weakly magnetized plasma is also observed numerically by Yukihiro et al. (Reference Yukihiro, Gakushi, Takatoshi and Osamu2009). They claimed a larger absorption cross-section for electron capture on the dust surface in the presence of a magnetic field. In the present study, higher charges on a spherical probe are due to the increase of $T_e$ for the higher magnetic field (see figure 8).

At higher $B$-field ($B > 0.05\ \textrm {T}$), a lower value of the mean free path ($r_{\textrm {ge}} < \lambda _{e-n}$) increases the electron–neutral collision frequency, resulting in a reduction of $T_e$. A slight reduction in $T_e$ at higher $B$ is seen in figure 8. At fixed gas pressure, transverse diffusion coefficient ($D_{e\perp }$) decreases with the increasing magnetic field (see figure 9) which causes a reduction in $I_{e\perp }$. It is also observed in numerical simulations that $I_{e\parallel }$ starts to decrease with an increase of the $B$-field. However, this reduction in $I_{e\parallel }$ is small compared with the reduction in $I_{e\perp }$ in the magnetic field regime ($B < 0.15\ \textrm {T}$). Therefore, it is assumed that the reduction in $I_{e\perp }$ to the spherical probe could be a possible cause of lower or less negative surface potential at high $B$-field ($B > 0.05\ \textrm {T}$), which is shown in figure 5.

Since $D_{e\perp }$ (or $I_{e\perp }$) varies with $1/B^2$ at a given pressure, $V_s$ should have a $1/B^2$ dependence. In figure 10(a), $V_s$ is plotted against $1/B^2$ for different spherical probes between $B = 0.06\ \textrm {T}$ and 0.2 T. Figure 10(a) clearly indicates that $V_s$ decreases linearly with $1/B^2$ between $B \sim 0.06\ \textrm {T}$ and 0.15 T for an rf power of 12 W. However, the upper $B$-field value shifts to a slightly lower value ($B > 0.10\ \textrm {T}$) for the low power discharge ($P = 6.5$). Since plasma density is different in both cases ($P = 12\ \textrm {W}$ and 6.5 W) at $p = 30\ \textrm {Pa}$, different rates of reduction of $V_s$ with $1/B^2$ are expected. The plots of $V_s$ against $1/B^2$ for different pressures at given power ($P = 12\ \textrm {W}$) are shown in figure 10(b). We also see a linear variation of $V_s$ with $1/B^2$ below $B < 0.12\ \textrm {T}$ and a nonlinear reduction in $V_s$ above $B > 0.12\ \textrm {T}$ (above $p \geq 30\ \textrm {Pa}$). For the low pressure case ($p = 15\ \textrm {Pa}$), the nonlinear behaviour of $V_s$ is observed at a magnetic field of $>$0.10 T. The different linear rates of $V_s$ at different pressures (figure 10b) are expected because of different plasma backgrounds (plasma density and $T_e$) at a fixed input rf power and different gas pressures. In a similar plasma background, we could expect a constant linear rate of $V_s$ with $1/B^2$ at different pressures according to the theoretical estimation as shown in figure 9. It shows that the reduction in $I_e$ (or $V_s$) is mainly due to the lower value of $I_{e\perp }$ in this magnetic field regime.

Figure 10.

(a) Surface potential variation with $1/B^2$ for different spherical probes at pressure $p = 30\ \textrm {Pa}$ and rf powers, $P = 6.5\ \textrm {W}$ and 12 W. The values of $1/B^2$ are correspond to $B = 0.06\ \textrm {T}$ to 0.2 T (b) $V_s$ variation against $1/B^2$ for bronze probe ($r = 1.7\ \textrm {mm}$) at rf power 12 W and different pressures. The values of $1/B^2$ are correspond to $B = 0.04\ \textrm {T}$ to 0.2 T.

At strong magnetic field strength, the contribution of $I_{e\perp }$ to $I_e$ starts to reduce and a larger contribution comes from $I_{e\parallel }$ (Sanmartin Reference Sanmartin1970). Hence the role of $D_{e\parallel }$ (or $I_{e\parallel }$) becomes more important in the determination of the surface potential. In the present experiments, $V_s$ does not decrease linearly with $1/B^2$ above $B > 0.12\ \textrm {T}$ (figure 10) except for the low density plasmas ($p = 30\ \textrm {Pa}$, $P = 6.5\ \textrm {W}$ and $p = 15\ \textrm {Pa}$, $P = 12\ \textrm {W}$). In moderately collisional plasma, a reduction in $I_{e\parallel }$ is expected while the magnetic field increases but it could be comparable or larger than that of $I_{e\perp }$ at strong $B$-field (Sanmartin Reference Sanmartin1970). The nonlinear characteristics of $V_s$ against $1/B^2$ above $B > 0.12\ \textrm {T}$ (see figure 10) is expected due to a dominant role of $I_{e\parallel }$ along with $I_{e\perp }$ for determining the surface potential. In some other experiments and simulations a reduction in $V_s$ of a spherical object with increasing magnetic field has also been reported (Dote, Amemiya & Ichimiya Reference Dote, Amemiya and Ichimiya1964; Lange Reference Lange2016; Tadsen et al. Reference Tadsen, Greiner and Piel2018).

It is known that the current $I_{e\perp }$ to the spherical probe decreases with increasing $\omega _{\textrm {ce}} \tau _e$ or decreasing $D_{e\perp }$. At given finite $B$-field, $D_{e\perp }$ increases with increasing electron–neutral collisions or gas pressure, which leads to an increase of $I_{e\perp }$. At finite $B$-field ($B > 0.01\ \textrm {T}$), $D_{e\perp }$ has a slightly larger value at higher pressure ($p = 50\ \textrm {Pa}$) than at lower pressure ($p = 15\ \textrm {Pa}$), as shown in figure 9. The difference in $D_{e\perp }$ for different pressures at finite $B$-field is one of the possible causes for the different values of $V_s$ in the presence of a magnetic field (figure 5b). The spherical surface collects more electron current at higher pressure ($\,p = 50\ \textrm {Pa}$) because of the large value of $D_{e\perp }$. The lower value of $D_{e\perp }$ causes a smaller electron current to the spherical probe. Therefore, the spherical probe has a higher value (more negative) of $V_s$ at $p = 50\ \textrm {Pa}$ than at $p = 15\ \textrm {Pa}$ (figure 5) in the magnetized plasma ($B > 0.05\ \textrm {T}$). These results confirm the dominating role of the collisional diffusion over the drain diffusion in a moderately collisional magnetized plasma.

It should be noted that in an unmagnetized plasma ($B = 0\ \textrm {T}$), the dust charge or surface potential decreases with an increase of the pressure due to the higher ion–neutral collision frequency (Khrapak et al. Reference Khrapak, Ratynskaia, Zobnin, Usachev, Yaroshenko, Thoma, Kretschmer, Höfner, Morfill and Petrov2005). It is also known that collisions of plasma species with neutrals retard the motion along the magnetic field, hence reduction in $I_{e/i \parallel }$ is expected with increasing the gas pressure. Since ions are unmagnetized below the magnetic field of $<$0.2 T, the role of ions is negligible compared with the electrons in the determination of the surface potential in this given magnetic field regime ($B < 0.2\ \textrm {T}$). Due to the smaller gyroradius of electrons (higher gyrofrequency) above $B > 0.05 \ \textrm {T}$, the electron–neutral collision frequency is found to be larger. It means that collisions lower the $I_{e\parallel }$ more effectively than $I_{i \parallel }$ in the magnetized discharge. Hence the role of ion current for determining the surface potential at different pressures ($p = 15$ to 50 Pa) is assumed to be negligible above $B > 0.05\ \textrm {T}$. The higher negative value of $V_s$ at higher pressure also confirms a dominant role of $I_{e\perp }$ in the reduction of the net electron current ($I_e$) or the surface potential in the magnetized plasma.

The difference in $V_s$ for magnetic (stainless steel) and non-magnetic (bronze) spherical objects (figure 6) is understood on the basis of field line distribution around a spherical body in a magnetized plasma. Since $I_{e\perp }$ decreases faster than $I_{e\parallel }$ with the magnetic field, $I_{e\perp }$ mainly responsible for the reduction in $I_e$ (or $V_s$). The magnetic flux density on either side of the magnetic sphere is less than that inside of it in the presence of a magnetic field (see figure 3.7 of Fagan (Reference Fagan2013)), which enhances $I_{e\perp }$ to the object surface due to the large value of $D_{e\perp }$ that varies with $1/B^2$. Thus, the electron current, $I_e$, to the magnetic sphere increases, making the surface of the spherical object more negative (higher $V_s$) in the presence of $B$-field. In the case of the non-magnetic object (copper bronze), the magnetic flux density on either side of the sphere is slightly larger than that inside of it due to the diamagnetic characteristics of copper. The $B$-field line density on either side of the non-magnetic sphere is expected to be higher than that around the magnetic sphere, which reduces $I_{e\perp }$ to the object surface. The lower value of the electron current ($I_e$) to the non-magnetic sphere makes the surface less negative. Hence a magnetic sphere has a higher value (more negative) of $V_s$ than that of a non-magnetic sphere (see figure 6a) above a finite value of $B$-field ($B > 0.03\ \textrm {T}$).

The qualitative description presented here provides a better understanding of the observed surface potential variation for magnetic and non-magnetic spherical probes (or large dust grains) in a magnetized rf discharge. We have provided a possible charging mechanism of magnetic and non-magnetic particles based on the reduction of electron current to spherical surface. To the best of our knowledge, at present there is no analytical or simulation work for a moderately collisional magnetized plasma to support our claim. Therefore it may be possible that another charging mechanism of magnetic and non-magnetic spherical particles plays a role in a magnetized plasma.