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Investigation of layers’ properties using brush discharge: Mobility of charge carriers in the layer is zero
Journal of Electrostatics 68 (2010) 429e438
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Investigation of layers’ properties using brush discharge: Mobility of charge carriers in the layer is zero R. Rinkunas*, S. Kuskevicius Faculty of Physics, Vilnius University, Sauletekio 9, LT-10222 Vilnius, Lithuania
a r t i c l e i n f o
a b s t r a c t
Article history: Received 28 April 2010 Accepted 7 June 2010 Available online 20 June 2010
This work is mainly based on the paper “R. Rinkunas, S. Kuskevicius, A contactless method of resistance measurement, Tech. Phys., 59 (2009) 133e137”. This paper contains a proposed contact less method for measuring resistivity of various materials, as well as various ambient parameters related to resistivity, e.g., humidity, intensity of illumination, sample thickness, etc. The mentioned paper describes experimental applications of the proposed method for measuring resistances in the range from 107 U to 1013 U. In this work, a model of the method proposed previously is presented. On the basis of that model, it has been determined that during charging of an insulating layer of a material (on whose surface the deposited ions are immobile), the charge flux becomes wider as it approaches the surface of the insulator. For example, the diameter of the charge flow region may increase from 0.2 mm (near the needle tip) up to 2 cm near the surface of the insulator. [Those numbers correspond to the distance h ¼ 1 mm between the needle and the substrate, insulating layer thickness 40 mm and needleesubstrate voltage of 4000 V. A change of those parameters would cause a change of the size of the spot on the layer surface]. It has been determined experimentally that resistance of the air gap between the needle and the substrate is linearly dependent only on h, whereas the electromotive force, which is responsible for the electric current from the needle to the substrate, also depends only on h. The radial coordinate of the points where the gradient of the electric charge density is largest is equal to h/2 (a zero radial coordinate corresponds to the point that is directly below the needle). During transfer of charge carriers from the needle onto the surface of the insulating layer, the largest potential is obtained at the point corresponding to radial coordinate r ¼ 0, but this potential is still smaller than the electromotive force that causes electric current in the circuit (i.e., the difference between the power supply voltage and the voltage on the capacitor formed by the needle and the substrate, when no charge has been deposited yet). The time dependence of charging current and of the potential difference between the needle and the substrate is not monotonic: at first the current increases, then it begins to decrease, and the potential difference at first decreases, then it begins to increase. The initial parts of those dependences can be explained by the “breakdown” of the capacitor formed by the needle and the substrate, and the subsequent time dependence is determined by the increase of the insulating layer potential due to accumulation of charge on it. Ó 2010 Elsevier B.V. All rights reserved.
Keywords: Insulator Discharge Resistance
1. Introduction The work [1] contains a proposed method of measuring resistance of various layers at the time when the layer is exposed to an ion beam. This method can be also used for measuring various properties of the environment or of the layer that influence resistance of the investigated layer: humidity, illumination intensity, ionizing radiation intensity, layer thickness, resistance of island
* Corresponding author. Tel.: þ370 5 68674515; fax: þ370 5 236 6003. E-mail address: [email protected] (R. Rinkunas). 0304-3886/$ e see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.elstat.2010.06.003
layers. This method of measuring layer resistance has attractive properties that distinguish it among other known methods: for example, measurements based on vacuum evaporation of metal electrodes on the layer surface and measuring resistance between those electrodes [2,3] cause damage to the investigated sample. The electrodes can be pressed to the sample surface (as in a twoelectrode or four-electrode method [4]). However, the contact between those electrodes and the sample may be insufficiently good (e.g., if the sample has a rough surface), or the pressure may break the sample. Contact-free methods of resistance measurements are known, too. Using the dosed charging method [5], a weak corona discharge current causes a constant and uniform charging
R. Rinkunas, S. Kuskevicius / Journal of Electrostatics 68 (2010) 429e438
of the entire sample (so that it is impossible to measure values of the sample resistance at separate points of the sample), therefore the entire sample resistance is measured during the charging. The duration of a single measurement is several minutes, and this time is sufficient for a significant change of the sample resistance. Using the scanning method [6], a distribution of the sample resistance can be investigated, but this method uses a voltage of only several volts, which may be a problem when the sample resistance depends on voltage and it is necessary to know the resistance at a high substrate potential. The method proposed in [1] does not damage the samples; the ion contact ensures a reliable contact with the surface of the investigated sample; it is possible to measure spatial distribution of the sample resistance; resistance is measured at high potentials that ionize the surrounding air, therefore the measured value of the sample resistance is exactly the same that would be observed under operating conditions of those high voltages; measurement duration is a fraction of a second. It was determined empirically in [1] that the proposed method can be used for measuring resistances from 107 U to 1013 U. When investigating sample with resistances larger than 1013 U or smaller than 107 U, the shape of the measured time dependence of the sample charging current does not depend on the sample resistance. Therefore, it is not possible to separate the influence of mobile charges from the influence of immobile charges on the time dependence of charging current. Also, it is unclear how much the dimensions of the area containing the deposited charge change during the resistance measurements; what is the charge distribution in that spot, what is the precision of measuring resistance distribution in various points of the sample. In order to answer those questions, we have developed a model describing charging of the investigated insulating layer placed on the conducting substrate, with a needle used for charging. The calculation results have been compared with experimental data. In this work, we will discuss a case when the charge carriers deposited onto the investigated layer surface are immobile. 2. Material and methods Fig. 1 shows a circuit diagram of the equipment used for charging the insulating layer 2 with ion beam emitted from the needle 1. The insulating layer 2 is placed with its metalized surface on the metal substrate 3. This layer is charged using the voltage source 4 (its potential US is chosen such that electric field between the needle and the substrate exceeds the electric field needed for ionization of this air gap). The time dependence of the current (or charge) is measured using the device 7. The charge carriers are deposited from the needle 1 onto the insulating layer 2. Its potential increases until the potential difference between the needle 1 and the substrate 3 becomes equal to US. Then the current from the needle 1 to the substrate 3 stops. The resistor R limits the charging current and thus protects the insulating layer from damage. The material of the insulating layer 2 should not accumulate electric charge: after discharging a charged insulating layer, no charge should remain on the layer surface or in its bulk. It is known that dielectrics used in capacitors have a property called dielectric absorption (soakage) e when voltage recurs between the capacitor terminals after its charging and temporary short-circuiting (that is why high voltage capacitors are always kept with short-circuited terminals). In order to find out the extent of dielectric absorption of polyethylene terephthalate, which is the material of the insulating layer 2, we created a polyethylene terephthalate capacitor and evaporated Al electrodes on both sides of this layer. This capacitor was charged to 4000 V and then it was short-circuited for a period of 0.1 s. Then we measured the time dependence of the capacitor voltage. The maximum voltage is 0.32 V. This value is reached at
5 1 R
2
Us
3
4 6
7 Fig. 1. Circuit diagram of the measurement equipment. 1 e needle for charging the insulating layer 2, which is placed with its metalized side onto the metal substrate 3. 4 e high voltage source with potential US; 5 e resistor R for limiting the charging current; 6 e a small resistance (compared to R) used for measuring charging current or charge; 7 e a measuring device for measuring charging current or charge of the insulating layer 2 (it may be an oscilloscope with memory, or a computer circuit board).
1 min after the short circuit. Afterwards, the capacitor voltage continuously decreases (see Fig. 2). The just-mentioned voltage is very small in comparison with the operating potential of the insulating layer (4000 V). Therefore, we can conclude that polyethylene terephthalate is a suitable material for an insulating layer 2 (see Fig. 1). Other suitable materials are polystyrene, polypropylene, NPO ceramic and Teflon. 2.1. Notation used in this work C1 e the capacitance between the needle 1 (Fig. 1) and the substrate 3; Csam e capacitance of the investigated sample, determined by geometric dimensions of the substrate 3 (the insulating layer 2 is above the substrate 3); Clr e capacitance of 1 m2 of the insulating layer 2;
0,35 potential difference, V
430
0,30 0,25 0,20 0,15 0,10 0,05 0 100 200 300 400 500 600 700 time since short-circuiting the terminals, s
Fig. 2. Time dependence of a 40 mm-thick polyethylene terephthalate capacitor potential at different moments of time since the short circuit. Prior to measuring the time dependence, the capacitor was charged to 4000 V, and then its terminals were short-circuited for a period of 0.1 s.
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d e thickness of the insulating layer 2; ELn(z) e electric field strength at a height z from the layer surface, created by charge density sL ðr; hÞ; e0 e electric constant; e e dielectric permittivity of the insulating layer 2; h e distance between the needle 1 and the substrate 3; I e electric current flowing out from the needle 1; r e radial coordinate on the substrate 2, measured from the point directly below the needle 1 to any point on the substrate 3 (or on the insulating layer 2, which is on the substrate 3); ri e the radius of the smallest circular spot containing all the charge deposited on the insulating layer 2 at the moment of time ti (if r > ri, there is no charge on the surface of the insulating layer 2: sL(r > ri,h) ¼ 0); rmax e the largest radial coordinate of the charge deposited on the surface of the insulating layer 2. In this case, UT ¼ US, therefore electric current between the needle 1 and the substrate 3 stops flowing; rsam e the radius of the sample (the insulating layer 2 is circular in shape and it is placed on the substrate 3 of the same radius); rqmax e the radial coordinate of the points corresponding to the maximum gradient of deposited charge density on the surface of the insulating layer 2; qp(r,h) e a fraction of charge (relative to the total charge of the needle Qn) deposited at a given moment of time on the surface of the insulating layer 2; Qn e the charge of the needle 1; QL(ri,h) e the charge deposited on the surface of the insulating layer 2 at a given moment of time; QL(rmax,h) e the maximum charge deposited on the surface of the insulating layer 2 (in this case UT ¼ US and the current between the needle 1 and the substrate 3 stops flowing); R e the resistance 5 limiting the charging current; RL e resistance of the air gap between the needle 1 and the insulating layer 2; s(r,h) e density of charge induced on the substrate 3 by the charge of the needle Qn; sL(r,h) e density of charge deposited on the surface of the insulating layer 2 at a given moment of time; t e time; Un e potential difference between the needle 1 and substrate 3 (see Fig. 1), created by the needle charge Qn; UEMF e electromotive force e is the voltage causing the current to flow between the needle 1 and the substrate 3; ULn e potential difference between the needle 1 and the surface of the insulating layer 2, created by the charge density sL(r,h); ULp e potential difference between the substrate 3 and the surface of the insulating layer 2, caused by the charge density sL(r,h); UP e potential difference between the needle 1 and substrate 3, caused by charge density s(r,h) existing in the substrate; US e potential difference of the source 4; UT e the total potential difference between the needle 1 and substrate 3, caused by all charges (existing in the substrate e s(r,h), the needle e Qn and on the surface of the insulating layer e sL(r,h)); z e the coordinate indicating the height above the substrate.
431
the final radius of the charged region on the layer surface would be rmax. In this case, UT ¼ US. An increase of the material resistivity would not change rmax. Therefore, the time dependence of charging current, which is used in this work for determination of the layer resistance, would not change, too. It has been determined empirically [1] that the maximum value of the layer resistance measurable with this method is 1013 U. The model of charging of an insulating layer proposed in this work can be checked by comparing the calculated and measured time dependences of the current. In order to calculate the currents, it is necessary to know the potential differences created by charges in the needle, metal substrate and on the insulating layer 2. Those potential differences can be calculated only when the distribution of charges in the substrate and on the insulating layer 2 is known (the charge in the needle is assumed to be point-like). If a potential difference exists between the needle 1 and the substrate 3, then a charge Qn is accumulated in the needle. This charge induces the charge of the opposite sign in the conductive substrate 3. The induced charge will be denoted s(r,h), where h is the distance between the needle and the substrate. The entire charge induced in the substrate is equal to the charge in the needle. However, the charge in the needle is point-like, whereas the charge in the substrate is distributed on the substrate surface symmetrically relative to the needle. According to the image charge theory [7e9], the interaction between the charges s(r,h) and Qn is the same as interaction between the point charge Qn and its mirror-image charge Qn (see Fig. 3). Using the method of image charges [7e9], charge density distribution on the substrate can be calculated (see Fig. 4):
sðr; hÞ ¼
Qn h 3 2p 2 2 h þ r2
(1)
The needle charge Qn can be calculated, if the potential difference UT between the needle and the substrate is known, and if the capacitance C1 between the needle and the substrate is known:
Qn ¼ UT C1
(2)
Capacitance C1 is determined by the needle spike shape, size and distance of the needle to the substrate. Thus theoretically calculate this capacity is a complex task. That’s why the capacity is measured experimentally: for example, when h ¼ 1 mm, C1 ¼ 75 fF. If the entire charge Qn of the needle 1 was removed from the needle and distributed on the insulating layer 2 (where the distributed charge is immobile) according to the same law as the distribution of charge on the substrate 3 (Fig. 4), then the charge
3. Results and discussion If resistivity of a material is very large, then the deposited charge will not have enough time to spread on the surface during its charging. The sample will be charged by an ion beam which spreads after leaving the needle tip and while moving towards the insulating layer. In the mentioned case of an extremely large resistivity,
Fig. 3. The point charge þQn above a metal plate and its image Qn in that metal plate.
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diameter of the developed area, m
charge density
(r,h), C/m
4x10 Q (r ,h) A
A
3x10
1 2x10 2
r
10 3 0 -0,004 -0,002 0,000 0,002 coordinate x, m
0,004
Fig. 4. Dependence of substrate charge density induced by the point charge Qn of the needle on distance x to the point that is directly below the needle. Potential difference between the needle and the substrate is UT ¼ 3500 V; h ¼ 0.001 m (curve 1), h ¼ 0.002 m (curve 2); h ¼ 0.003 m (curve 3). The other notations will be explained below.
distribution on the substrate would not change, but the electric field between the needle and substrate would disappear, so that potential difference between the needle and the substrate would become zero. However, it is impossible to remove the entire charge Qn onto the surface of the insulating layer 2, because the electric currents that flow out of the needle are small. The charge is deposited on the place where electric field is strongest (initially the largest electric field corresponds to the point where charge density is highest e immediately below the needle). On the basis of those considerations, we assume that the charge is initially deposited on the point that is immediately below the needle, and later on the deposited charge will spread in a circular area with radius r ¼ ri (see Fig. 4). The charge deposited in this area is equal to QL(ri,h). If this assumption is correct, then the calculation results should coincide with measurement data. Electric field created by charge QL(ri,h) deposited on the insulating layer is compensated by the field created by the opposite charge on the substrate, therefore the total electric field created by both those charges between the needle and the substrate would be practically equal to zero. However, the remaining charge on the substrate (i.e., the charge that is not compensated by the charge QL(ri,h)) will create electric field between the needle and the substrate. Therefore, in this case the needle would only “see” the charge that corresponds to the curve below the line A‑A in Fig. 4. Hence, the “focusing” effect of the electric field would be weakened and the charge would be deposited approximately uniformly in the entire area pri2 , which corresponds to radial coordinate values 0 < r < ri. An increase of the total deposited charge QL(ri,h) causes an increase of the radius ri (i.e., the line AeA in Fig. 4 shifts down). This increase of the radius can be verified experimentally. This is done by placing a screen (mask) over the insulating layer 2 (the screen does not touch the surface of the layer 2). That screen has a round hole centered on the needle. The hole’s diameter is much larger than the diameter of the streamer. If the charge transfer from the needle 1 onto the insulating layer 2 was only by means of a narrow streamer, followed by spreading of the charge over the surface of the layer 2, then, after placing the screen (mask) on the layer, the screen would have no effect on the size of the charged area of the insulating layer 2. However, the curves presented in Fig. 5 indicate that the screen limits the charged area of the layer 2. This limiting effect becomes more pronounced when the distance between the screen and the layer is decreased or when the hole diameter is decreased (see curves 2, 3, 4). Therefore, it follows that during charging of the insulating layer 2, the diameter of the streamer increases from 0.2
0,024 0,022 1
0,020
2
0,018 0,016
3
0,014 0,012
4
0,010 0,008 0,001 0,002 0,003 0,004 0,005 0,006 0,007 h, m
Fig. 5. Dependence of the developed area diameter on distance h between the needle and the substrate. Curve 1 corresponds to absence of the screen. Other curves were measured after placing a screen with a circular hole on the insulating layer. The hole diameter is 7 mm (curve 4) or 12 mm (curves 2, 3). That screen is raised over the layer 2 (Fig. 1) using an insulating rubber spacer with a 35 mm hole. Thickness of the spacer is 0.2 mm (curves 3, 4) or 0.8 mm (curve 2). US ¼ 4000 V, d ¼ 40 mm.
mm in the beginning of charging up to the diameter of the charged area (24 mm). The charge QL(ri,h) increases until the potential difference between the needle and the substrate reaches the value US. At that time, the radius of the charged area of the insulating layer is ri ¼ rmax. This spreading of the charge can be observed by developing the insulating layer with an electrographic developer after charging of the layer with a needle electrode (see Fig. 6). However, the electrographic developer does not develop the regions with a small charge density (charge density s(r,h) decreases with increasing radial coordinate r, as evident from Fig. 4), therefore, in order to see the value of rmax, a potential relief has to be formed on the insulating layer surface (e.g., by pressing a grounded metal foil to the charged insulating layer for a short time). In this case, the charge
Fig. 6. A potential relief was formed on the surface of a 40 mm-thick polyethylene terephthalate layer (e.g., by pressing a grounded metal foil to the surface of the charged layer for a short time). Such layer was used as the insulating layer 2 (Fig. 1) and charged by the needle 1 with US ¼ 4000 V. The value rmax w 20 mm was obtained.
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deposited on the layer during its charging with a needle electrode would “wipe” the previous charge; hence the value of rmax would be seen in the developed image even if the small charge density of the needle-charged area is not developed. If the charge density distribution on the insulating layer is known (see Eq. (1)), then the charge QL(ri,h) deposited onto the surface of the insulating layer 2 can be calculated as follows:
ZSi QL ðri ; hÞ ¼
sðr; hÞ dS sðri ; hÞ p ri2
0
Zri ¼
2 p r sðr; hÞ dr sðri ; hÞ p ri2
(3)
0
where Si is the area of the circle with radius ri. Now, we define a quantity qp e the fraction of the charge (relative to the total charge in the substrate) deposited on the surface of the insulating layer 2 (Fig. 1). From Equations (1) and (3), the following expression of qp is obtained:
qp ðri ; hÞ ¼
QL ðri ; hÞ ¼ 1 Qn
h ri2
þ h2
1 2
h ri2 3 2 2 ri2 þ h2
(4)
opposite charge in the substrate. This deposited charge (and the corresponding charge in the substrate) could redistribute if conductivity of the insulating layer were non-zero. This motion of charge on the investigated layer can be applied for measuring resistance of conducting layers. In order to calculate time dependence of charging current of an insulating layer (or the time dependence of charge deposited on that layer or the time dependence of the layer potential), the distribution of the deposited charge QL(r,h) must be known. In Fig. 4, the total charge deposited on the insulating layer is denoted QL(ri,h) (it corresponds to the hatched area). From this diagram we can see that distribution of the charge density on the insulating layer is similar to distribution of charge density in the substrate (Equation (1)), but there are some differences, too: when r ri, charge density in the substrate is non-zero, whereas the charge on the insulating layer is already zero. The charge on the insulating layer is only present inside the circular area corresponding to r < ri. Based on those considerations and on Equation (1), we can express the distribution of charge deposited on the insulating layer 2 as follows:
" Qn sL ðr; hÞ ¼ 2p
(5)
In this case, Qn is the charge that was in the needle at the beginning of the experiment e while there was no charge deposited on the insulating layer (because during the experiment the needle charge decreases due to charge transfer onto the surface of the insulating layer). Qn is also equal to the total charge in the substrate (it does not change during this experiment). While there is no charge deposited on the insulating layer surface, the only sources of electric field are the needle charge and the substrate charge induced by the needle charge (Fig. 4). The substrate charge is tied to the needle charge via the electric field lines; therefore the substrate charge distribution is fixed. However, if a charge is transferred from the needle onto the surface of the insulating layer, then this charge (QL(ri,h)) becomes “tied” to the
h2 þ r 2
1 0,8 2 3 0,6
0,4
0,2
0,0 0,000 0,005 0,010 0,015 0,020 0,025 0,030 r i, m Fig. 7. Dependence of the relative charge qp(ri,h) (defined as the ratio of the deposited charge and the total needle charge Qn) on the radius ri of the area where the deposited charge is distributed. Curve 1 e h ¼ 0.001 m; curve 2 e h ¼ 0.002 m; curve 3 e h ¼ 0.003 m.
3 2
h h2 þ ri2
5 3 /r < ri 2
(6)
According to this equation, sL(r < rmax,h) > 0 and sL(r > rmax,h) ¼ 0, i.e., density of charge deposited on the insulating layer abruptly drops to zero when r > rmax. This is confirmed by the experiment: the charged area has a sharp edge (see Fig. 6). If the charge distribution on the insulating layer is known (see Equation (6)), it is possible to determine the radial coordinate corresponding to the largest gradient of charge density (if the charge is assumed to consist of separate particles, then the points with the largest gradient of particle density would correspond to the best conditions of particle spreading on the layer surface). At first, let us discuss the case when the total charge of the needle has been deposited on the surface of the insulating layer (then ri is infinite, so that Equations (1) and (6) become identical to each other). At the point of the maximum gradient of charge density, the first derivative relative to r has a maximum value, whereas the second derivative is zero (ds2(r,h)/dr2 ¼ 0). Solution of the latter equation gives the following expression of the radial coordinate rqmax corresponding to the maximum gradient of charge density on the surface of the insulating layer 2:
rqmax ¼
relative charge q p(r i,h)
3 h
sL ðr; hÞ ¼ 0/r ri
This function is shown in Fig. 7, where we see that qp increases with increasing ri and approaches 1. The total charge deposited on the surface of the insulating layer 2 is equal to
QL ðri ; hÞ ¼ qp ðri ; hÞ Qn
433
h 2
(7)
From this equation we see that the place of the maximum charge density gradient depends only on h. The corresponding value of that gradient is
d s rqmax ; h 4:8 Qn ¼ dr p h3
(8)
If ri < h/2, then the radial coordinate corresponding to the maximum charge density gradient on the insulating layer surface is rqmax ¼ ri. If ri > h/2, then the maximum charge density gradient corresponds to rqmax ¼ h/2. Using the known values of charge density distribution on the substrate (Equation (1)), charge density distribution on the surface of the insulating layer (Equation (6)), needle charge Qn and the insulating layer thickness d, we can calculate the electric field between the needle and the substrate. Then it is possible to calculate the total potential difference between the needle and the substrate UT caused by the charges in the needle, the substrate and on the insulating layer:
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R. Rinkunas, S. Kuskevicius / Journal of Electrostatics 68 (2010) 429e438
UT ¼ Un þ ULp þ UP ULn
(9)
where Un is the potential difference between the needle and the substrate caused by the point charge Qn in the needle:
Un ¼ Qn =C1
(10)
ULp is the potential difference between the substrate and the surface of the insulating layer caused by the charge density sL(r,h) on the surface of the insulating layer; ULn is the potential difference between the needle and the surface of the insulating layer caused by the charge density sL(r,h) on the surface of the insulating layer; UP is the potential difference between the needle and substrate caused by the charge in the substrate:
UP ¼ Qn =ð8 p e0 hÞ
(11)
The potential differences ULn and ULp are created by the charge deposited on the surface of the insulating layer. They are calculated in the same way as the potential difference UP: we calculate electric field (created by the charge density sL ðr; hÞ (Equation (6)), which is bounded by the circle with radius ri) perpendicular to the substrate at height z. The total electric field at height z is calculated by summing the fields created by individual charge elements (similarly to the technique used in [9]):
Zri ELn ðzÞ ¼ 0
sL ðr; hÞ z dS 4 p e0 l3
(12)
where dS ¼ 2 p r dr; l ¼ ðr 2 þ z2 Þ1=2 ; When ELn(z) is known, ULn can be calculated:
Zh Zri
Zh ULn ¼
ELn ðzÞ dz ¼ d
d
0
sL ðr; hÞ z dS dz 4 p e0 l3
(13)
ULn is found by combining Equations (6) and (13). ULp is calculated similarly. Then all potential differences are:
Qn ULn ¼ 4 p e0
Zh Zh d 0
dr dz Qn ULp ¼ 4 p e0 e
0 1 hzr @ 1 1 A 3 32 2 32 2 2 2 r 2 þ z2 2 r þh r þh
Zd Zri 0 0
i
0 1 hzr @ 1 1 A 3 3 2 3 r 2 þ z2 2 r 2 þ h2 2 r þ h2 2 ð14Þ
dr dz Qn UP ¼ 8 p h e0 Qn Un ¼ C1 UT ¼ Un þ ULp þ UP ULn
i
Here, e is dielectric permittivity of the insulating layer (Fig. 1). In the case of polyethylene terephthalate layer that is used in this work, e ¼ 3.4 [10,11]. The signs of the various potentials in the expression of the total potential UT (Equation (9)) can be explained as follows: in all equations given above, only absolute values of potentials are used. Therefore, in order to determine the sign of a particular potential, one has to take into account the direction of electric field created by the corresponding charge. Let us assume that the needle charge Qn is positive. Then that charge will create electric field directed from the needle to the substrate. The substrate charge is negative; hence the direction of electric field created by the substrate charge
between the substrate and the needle is the same as direction of the field created by the needle charge. Therefore, the potentials of those two fields add up. The charge deposited on the surface of the insulating layer has the same sign as the needle charge. Hence, the direction of the field created by the deposited charge inside the insulating layer is the same as the direction of the field created by the needle charge, therefore the potential ULp has the same sign as the potentials of the fields created by the needle charge (Un) and the substrate charge (UP). However, direction of the field created by the deposited charge between the insulating layer and the needle is opposite to the direction of the other mentioned fields. Therefore, the potential difference ULn must be subtracted from the other mentioned potential differences when calculating the total potential difference between the needle and the substrate (UT). In this way, Equation (9) is obtained. Direct measurements of the potential UT are difficult, because the needle charge is small and connection of a measuring device to the needle would distort the measurement data. Besides, the capacitor formed by the needle and the substrate would have to be charged prior to measuring the potentials, and a part of the needle charge would be deposited on the insulating layer already during the charging. A simpler method of experimental testing of the discussed model is by measuring the time dependence of current flowing through the circuit formed by the mentioned capacitor (needle e air gap e insulating layer e substrate), a constant voltage source US and a limiting resistance R (5) connected in series (see Fig. 1). For comparing the calculation results with measurement data, one has to know the time dependence of the needle charge Qn. In order to determine that dependence, let us first discuss the case when a uniform electric field is present between the electrodes and the field strength is sufficient for the corona discharge, and the number of neutral molecules in the discharge region is very large. In this case, a slight increase of electric field strength causes a significant increase of the number of charge carriers. This causes an increase of electric current and an increase of the voltage drop on the resistances that are connected in series. Therefore, the potential difference between the electrodes decreases and the electric field strength decreases, too. This causes a decrease of charge carrier generation and a corresponding decrease of electric current. But then the voltage drop on the resistances decreases, so that the electrode potential difference increases and the process repeat itself. As a result, constant field strength is established between the electrodes. This field strength corresponds to a particular potential difference between the electrodes. This is the mechanism of formation of the Townsend discharge [12e14] (its characteristic property is formation of a certain potential difference between the electrodes). In our case, one electrode is in the shape of a needle and the other one is in the shape of a plate (the substrate). Therefore, electric field between the electrodes is not uniform; the strongest electric field is created near the needle. However, in our case the number of neutral air molecules is very large, too (an increase of electric field strength causes an increase of the number of charge carriers). In our case, the charge carriers are created near the needle. Therefore, a constant value of the needle electric field strength is maintained. This means that the needle charge is constant (that charge determines the magnitude of electric field created by the needle). Thus, in our case the Townsend discharge condition is
Qn ¼ const
(15)
However, the potential difference between the needle and the substrate is not constant. It depends on electric current, because resistance RL of the air gap between the needle and the substrate is non-zero. If potential difference between the needle and the substrate was constant and independent of R (i.e., of electric
R. Rinkunas, S. Kuskevicius / Journal of Electrostatics 68 (2010) 429e438
current), then RL would have to be zero, but it is not the case, because from Equation (20) below it follows that RL is non-zero. In this case, under conditions of a closed circuit one has to calculate time dependences of current, potential and charge. Equation (14) were calculated for the case when the charge accumulated in the needle was gradually transferred onto the surface of the insulating layer, and the needle charge was not replenished. However, in the case of a closed circuit shown in Fig. 1, Equation (14) would change. The voltage source US would replenish the needle charge. The substrate charge would gradually increase. Its value would be equal to the sum of the needle charge and the charge deposited on the insulating layer surface. However, in this case both the distribution of the deposited charge and the distribution of the substrate charge are still determined by the same laws as in the previously-discussed case. Therefore, all the earlier equations derived for an open circuit are applicable to the case of a closed circuit, but the charge Qn must be replaced by the charge (Qn þ QL) except the constant charge in the needle (Equation (15)) [This is essentially the same assumption that was made when deriving Equation (6). If this assumption is correct, then values of the currents, charges and potentials and their time dependences, which follow from that assumption, must coincide with measurement data]:
Qn þ QL ðr;hÞ ULn ¼ 4 p e0 dr dz Qn þ QL ðr;hÞ ULp ¼ 4 p e0 e
Zh Zri d 0
Zd Zri 0 0
0
1
hzr @ 1 1 A 3 32 2 32 2 2 2 2 2 r þz r þh r þ h2 i
0 1 hzr @ 1 1 A 3 32 2 32 r 2 þ z2 2 r 2 þ h2 r þ h2 i
dr dz
ð16Þ
Qn þ QL ðr;hÞ 8 p h e0 Qn Un ¼ C1 UT ¼ Un þ ULp þ UP ULn
UP ¼
435
Resistance RL can be measured. This is done by removing the insulating layer 2 in the circuit shown in Fig. 1 and by measuring charging currents at different values of R. In this way, the curves shown in Fig. 8 are obtained. From Fig. 8 we see that inverse current is linearly dependent on R, but the intercept of this straight line is not zero. This means that resistance of the air gap between the needle and the substrate is not equal to zero, but it is equal to some constant. There is a wellknown technique for determining resistance RL of the air gap between the needle and the substrate from the presented graph (based on the point where the straight line shown in Fig. 8 crosses the y axis). The same graph can be used for calculating the electromotive force UEMF of this process (the electromotive force is given by the slope of the straight line in Fig. 8). The linear dependence of inverse current on Rð1=IwRÞ indicates that a change in the limiting resistance R (and a corresponding change in the current) does not cause any change in RL and UEMF. Those quantities depend only on the distance h between the substrate and the needle. The presented experimental results, as well as results of additional experiments, indicate that the following empirical dependences of RL and UEMF on h can be used (all units in SI):
RL ¼ 7:35 109 h UEMF ¼ 500 ð1 logð1000 hÞÞ
(20)
Those expressions are valid when 0.5 mm < h < 8 mm and US ¼ 4000 V. When h ¼ 1 mm, then from Equation (20) it follows that UEMF ¼ 500 V. As mentioned, UEMF does not depend on R. Therefore, potential difference US UEMF ¼ 3500 V stays the same at all values of current (which depends on R). When the current is zero (i.e., when there is no voltage drop on resistances R and RL), this potential difference is equal to the voltage drop between the needle and the substrate. This is the case of discharge ignition and discharge quenching. When h ¼ 1 mm, the measured needlee substrate potential difference corresponding to discharge ignition is approximately 3502 V, and the measured needleesubstrate potential difference corresponding to discharge quenching is approximately 3497 V. Thus, those potential differences are very close to the potential difference US UEMF ¼ 3500 V.
In the case of a closed circuit, the distribution of charge density deposited on the insulating layer surface is
3 h2 þ r 2
3
sL ðr;hÞ ¼ 0/r ri
2
h h2 þ ri2
5/r < ri 3 2
a (17)
Using the known charge distribution on the surface of the insulating layer (Equation (17)), we can find ri ¼ rmax corresponding to UT ¼ US. In this case, there is no current between the needle and the substrate. When ri ¼ rmax, the maximum charge QLmax is deposited on the surface of the insulating layer (from Equation (5)):
QLmax ðrmax ; hÞ ¼ qp ðrmax ; hÞ Qn
(18)
Using the known potential difference between the needle and the substrate UT (caused by the charges in the needle, the substrate and on the insulating layer) and the known source potential US, we can calculate the current in the circuit formed by the source US, the limiting resistance R, the needle and the substrate, connected in series:
U UT I ¼ S R þ RL
(19)
where RL is the resistance of the air gap between the needle and the insulating layer 2 (Fig. 1).
1/I, relative units
h
0,10 0,08 0,06 0,04 0,02 0,00
b 1/I, relative units
" Qn þ QL ðr;hÞ sL ðr;hÞ ¼ 2p
0 10 20 30 40 50 R, MΩ
2,2 2,0 1,8 1,6 1,4
0 10 20 30 40 50 R, MΩ
Fig. 8. Dependence of inverse charging current of a metal plate on the limiting resistance R (No. 5 in Fig. 1). The distance h between the needle and the metal plate is 1 mm (a) or 8 mm (b). The source potential is US ¼ 4000 V.
R. Rinkunas, S. Kuskevicius / Journal of Electrostatics 68 (2010) 429e438
Q ðr ; hÞ QL ðri1 ; hÞ ti ¼ ti1 þ L i Ii
(21)
0,020
3x10 -9 2
0,015
2x10 -9 0,010
radius ri, m
1 charge Q L(r i,h), C
10-9 0,005
0
0
3x10 -5
6x10 -5
9x10 -5
0,000
time t, s Fig. 9. Calculated time dependences of the charge QL deposited on the insulating layer surface (curve 1) and of the radius of the charged area ri (curve 2). The time dependences have been calculated using Equations (16)e(21). The insulating layer thickness is d ¼ 40 mm, the distance between the needle and the substrate is h ¼ 1 mm, the source potential is US ¼ 4000 V.
Using this method, the time dependences of I, UT, ULn, ULp, UP, QLi, ri have been calculated. Some of them are shown in Figs. 9e11. As we see in the figure, the charge deposited on the insulating layer increases monotonically, approaching the maximum value QLmax. The radius of the charged area also increases monotonically, approaching rmax (the calculated value is rmax ¼ 21 mm, when US ¼ 4000 V and h ¼ 1 mm. This value is in accord with the measured value rmax ¼ 20 mm e see Fig. 6). As shown earlier, the needleesubstrate potential difference corresponding to breakdown of the air gap is 3500 V. Therefore, this potential corresponds to the start of the time dependence of potential UT (curve 3 in Fig. 10) (electric current begins to flow between the needle and the substrate only when the capacitor C1 formed by the needle and the substrate is charged to this potential). At the start of the time dependence, the calculated potential difference UT does not increase (as it could have been expected), but decreases instead (this can be explained by “breakdown” of the capacitor needleesubstrate, because it is known that breakdown of a capacitor causes a decrease of its potential). However, eventually another capacitor e the surface of the insulating layer (Csam) e begins to charge. Therefore, UT begins increasing and it increases until reaches the value US (in this case 4000 V). Since the current is calculated, according to equation (21), this time dependence of UT causes a corresponding time dependence of the current (curve 1). 4000
1,6x10 -4 3 1,2x10 -4
3500 8,0x10 -5 2
3000
potential U T, V
Current density on the surface of the insulating layer must be limited, because large current density can damage the layer. From [15] we know that current magnitude determines the type of the discharge (corona discharge, brush discharge, spark or arc). From Equation (20) we obtain that when the distance between the needle and the insulating layer is h ¼ 0.5 mm, then the resistance is RL ¼ 3.6 MU, and when the distance is h ¼ 8 mm, this resistance is RL ¼ 60 MU. Therefore, at the initial moment of time, electric current flowing into the surface of the layer may be between 0.15 mA and 8 mA. The corresponding current density is 300e5000 A/m2 (the diameter of the streamer formed between the needle and the substrate is approximately 0.2 mm e this has been determined by photographing the streamer). Those currents are slightly larger than corona discharge current (1e30 mA). Therefore, it is necessary to choose such values of current density that do not damage the investigated materials (during arc welding, samples are certainly “damaged”, but in this case current density reaches about 107 A/ m2). The current is largest at the initial moment of time after the breakdown of the air gap between the needle and the surface of the insulating layer. Therefore, resistance R must be chosen so that even the maximum current does not damage the layer and the results are reproducible. However, if the charging current decreases to values corresponding to corona discharge, then it becomes more difficult to measure sample properties (its resistance, humidity, etc.), because in the case of corona discharge the process of charge deposition on the layer surface is mainly determined by the layer potential and is less dependent on the layer structure (e. g., surface irregularities), and the charge is deposited on a large area at once. In the case discussed here, the currents used are larger than those during a corona discharge, and the current is concentrated into a small area of the surface (this ensures a better spatial resolution on the surface of the investigated layer). This area is rapidly charged to high potentials. Therefore, strong tangential electric fields are formed when charges are deposited on the layer. Those fields can cause secondary generation of charge carriers on the layer surface. For example, if the layer surface is uneven or if islands of another material have been evaporated on the layer, then those surface irregularities would cause ionization of air under conditions of strong tangential electric fields. As a result, the charging area would spread and it would cause an increase of the maximum charge that can be deposited on the layer (even when the layer surface is smooth). Hence, a larger charging current would be measured when the layer surface is uneven than when it is even. Thus, when a layer is charged with a larger charging current, then properties of the layer surface has a stronger influence on the value and time dependence of charging current. The time dependence of charging current is experimentally measured using an oscilloscope (No. 7 in Fig. 1). The first step of calculating the time dependence of charging current is determination of the maximum charge QLmax, which can be deposited on the surface of the insulating layer 2 (when that charge is deposited, then UT ¼ US and the current between the needle and the substrate drops to zero). The charge QLmax is calculated with the program “Mathcad” using Equations (16) and (17) and the condition UT ¼ US. When QLmax is known, then it can be divided into, e.g., 100 parts and then the entire deposition of QLmax on the surface of the insulating layer can be modeled in i ¼ 100 steps. Each value of QL(ri,h) corresponds to a particular value of the charged area radius ri (this follows from Equations (4) and (5)). When ri and QL(ri,h) are known, UTi can be calculated from Equation (16). The current Ii is calculated from Equation (19), and the time is calculated using this equation:
current I, A
436
4,0x10 -5 1 0
0
3x10 -5 6x10 -5 time t, s
9x10-5
2500
Fig. 10. Time dependences of the charging current I of the insulating layer (curves 1 and 2) and of the potential difference between the needle and the substrate (curve 3). Conditions are the same as in Fig. 9. Curves 1 and 3 have been calculated. The scatter graph (curve 2) presents measurement results.
dielectric layer potential U Lp , V
R. Rinkunas, S. Kuskevicius / Journal of Electrostatics 68 (2010) 429e438
where Csam is capacitance of the capacitor formed by the substrate and the surface of the insulating layer. In this case, charge density does not depend on r:
350 300
sL ðr; h; tÞ ¼ sL ðh; tÞ ¼
250
150 100 50
ULn ¼ 2x10 -5 4x10 -5 6x10 -5 8x10 -5 time t, s
0
10 -4
ULp ¼
Fig. 11. The calculated time dependence of the insulating layer potential below the needle (ULp) during charge deposition on the layer. Conditions are the same as in Fig. 9.
However, the experimental points (curve 2) coincide with the measured curve only at the beginning of the time dependence. Later the calculated current decreases (curve 1), whereas the experimental current stays constant (curve 2). This is because the calculations have been done under the assumption of infinite surface resistivity of the insulating layer, whereas in reality the surface resistivity is finite (this is evident from the experimental fact that the charged area grows during charging of the insulating layer). Fig. 11 indicates that surface potential of the insulating layer increases during charge deposition, approaching a constant value (375 V), but does not reach the value UEMF ¼ 500 V. The value of the maximum potential of the charged area, measured with equipment designed for measuring point potentials (e.g., the device “Relief”), is about 350e370 V, which is in good agreement with calculation results. The calculation method used in this work can be easily checked by applying it to the case when the charge deposited on the insulating layer surface spreads over entire surface immediately (in this case, the surface resistivity of the insulating layer is zero). This case corresponds to charging of a capacitor, therefore a well-known expression of the time dependence of the capacitor charging current must be applicable:
UEMF ðRþR ÞCsam L e R þ RL t
(22)
6x10 -5 5x10 -5 current I, A
QL ðh; tÞ
(23)
2 p rsam
where rsam is the radius of the insulating layer and the substrate (a circular insulating sample is placed on the metal substrate of the same shape). The potential ULn is calculated by combining Equations (13) and (23). The other potentials are obtained similarly:
200
0
I ¼
437
4x10 -5 3x10 -5 1
2x10 -5 10-5 0 0,000
2
0,005
0,010
0,015
0,020
0,025
time t, s Fig. 12. Time dependence of the current flowing out of the needle. Curve 1 e calculated using Equation (22); curve 2 e calculated using Equations (19) and (24). rsam ¼ 2 cm; h ¼ 1 mm; d ¼ 40 mm.
QL ðh;tÞ 2 e 4p rsam 0
Zh Zrsam d
QL ðh;tÞ 2 e e 4 p rsam 0
0
rz 3 drdz 2 r þz2 2
Zd Zrsam 0
0
rz r 2 þz2
Qn QL ðh;tÞ þ UP ¼ 2 e 8 p e0 h 4 p rsam 0 Un ¼
3 drdz 2
Zh Zrsam 0
0
rz
(24)
3 drdz r 2 þz2 2
Qn C1
UT ¼Un þULp þUP ULn The potential UT is calculated from Equation (24), and then the current is calculated from Equation (19). The same current is also calculated from Equation (22). Values of the current calculated using both those equations is almost equal to each other (see Fig. 12). This proves that our calculation method can be applied for calculation of potentials and time dependences of currents, charges and potentials. 4. Conclusion This work presents a model describing the process of charging of an insulating layer placed on a conducting substrate, when charges deposited on the surface of the insulating layer are immobile. Using this model, the following conclusions have been made: 1. During charging of the insulating layer with a needle electrode, the diameter of the charged area of the layer surface grows from the streamer diameter (0.2 mm) up to the maximum diameter of the charged area that can be achieved under given charging conditions (for example, this diameter is 2 cm, when the distance h between the needle and the substrate is 1 mm, insulating layer thickness is 40 mm, and when a 4000 V voltage source is connected between the needle and the substrate. A change of those parameters causes a change of the charged area size on the surface of the insulating layer). 2. The radial coordinate corresponding to the largest gradient of charge density deposited on the surface of the insulating layer is r ¼ h/2 (the radial coordinate r is measured from the point that is directly below the needle). 3. The charges deposited on the insulating layer surface create the maximum potential ULp at the point r ¼ 0 (because charge density is largest at that point), but this potential is still smaller than the electromotive force UEMF, which causes electric current in the circuit (i.e., than the difference between the external source voltage and the voltage of the capacitor formed by the needle and the substrate when no charge has been deposited yet: ULp < UEMF ¼ US UT). 4. The time dependence of the charging current of the insulating layer, as well as the time dependence of the potential difference
438
R. Rinkunas, S. Kuskevicius / Journal of Electrostatics 68 (2010) 429e438
between the needle and the substrate are not monotonous: at first, the current grows, but later on it begins to decrease, whereas the potential difference between the needle and the substrate at first decreases, but later on it begins to increase. The initial parts of those time dependences can be explained as a result of “breakdown” of the capacitor formed by the needle and the substrate, and the subsequent time dependence is governed by the increase of the insulating layer potential due to charge accumulation on it. 5. It has been determined experimentally that resistance of the air gap between the needle and the substrate increases linearly with increasing distance h between the needle and the substrate. The electromotive force (which causes the current from the needle to the substrate) also depends only on the mentioned distance h between the needle and the substrate. Acknowledgement The authors are grateful to Prof. E. Montrimas and doc. A. Poskus for their interest and help in this work. References [1] R. Rinkunas, S. Kuskevicius, A contactless method of resistance measurement. Tech. Phys. 59 (2009) 133e137.
[2] M. Kasu, M. Kubovic, A. Aleksov, N. Teofilov, Y. Taniyasu, R. Sauer, Influence of epitaxy on the surface conduction of diamond film. Diamond Relat. Mater. 13 (2004) 226e232. [3] I.K. Kikoin, Tablici Fizicheskich Velichin. Atomizdat, Moskow, 1976. [4] N. Bowler, Theory of four-point alternating current potential drop measurements on conductive plates. J. Phys. D: Appl. Phys. 39 (2006) 584e589. [5] P.J. Zhilinskas, E. Montrimas, T. Lozovski, Measuring the parameters of dielectric layers by periodically charging the surface. Tech. Phys. 51 (2006) 1372e1378. [6] M. Tachiki, T. Fukuda, K. Sugata, H. Seo, H. Umezawa, H. Kawarada, Control of adsorbates and conduction on CVD-grown diamond surface, using scanning probe microscope. Appl. Surf. Sci. 159e160 (2000) 578e582. [7] M.H. Nayfeh, M.K. Brussel, Electricity and Magnetism. J. Wiley & Sons, New York Chichester Brisbane, 1985. [8] E.J. Konopinski, Electromagnetic Fields and Relativistic Particles. McGraw-Hill Book Company, New York, 1981. [9] D.J. Griffiths, Introduction to Electrodynamics, second ed. Prentice Hall, Englewood cliffs (New Jersey), 1989. [10] S. Osaki, Molecular orientation and dielectric anisotropy in polyimide films as determined by the microwave method. Macromolecules 31 (1998) 1661e1664. [11] D.U. Erbulut, S.H. Masood, V.N. Tran, I. Sbarski, A novel approach of measuring the dielectric properties of pet preforms for stretch blow moulding. J. Appl. Polym. Sci. 109 (2008) 3196e3203. [12] S. Flugge, Handbuch der Physik. Springer, Berlin, 1955. [13] J.W. Gewartowski, H.A. Watson, P. Bell Telephone Laboratories. Communications Development Training, Principles of Electron Tubes Including Gridcontrolled Tubes, Microwave Tubes, and Gas Tubes. Van Nostrand, Princeton, N.J.; London, 1965. [14] E. Kuffel, W.S. Zaengl, J. Kuffel, in: E. Kuffel, W.S. Zaengl, J. Kuffel (Eds.), High Voltage Engineering: Fundamentals, second ed. Newnes, Oxford, 2000. [15] M. Yamaguma, T. Kodama, Observation of propagating brush discharge on insulating film with grounded antistatic materials. IEEE Trans Ind Appl 40 (2004) 451e456.
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Journal of Electrostatics journal homepage: www.elsevier.com/locate/elstat
Investigation of layers’ properties using brush discharge: Mobility of charge carriers in the layer is zero R. Rinkunas*, S. Kuskevicius Faculty of Physics, Vilnius University, Sauletekio 9, LT-10222 Vilnius, Lithuania
a r t i c l e i n f o
a b s t r a c t
Article history: Received 28 April 2010 Accepted 7 June 2010 Available online 20 June 2010
This work is mainly based on the paper “R. Rinkunas, S. Kuskevicius, A contactless method of resistance measurement, Tech. Phys., 59 (2009) 133e137”. This paper contains a proposed contact less method for measuring resistivity of various materials, as well as various ambient parameters related to resistivity, e.g., humidity, intensity of illumination, sample thickness, etc. The mentioned paper describes experimental applications of the proposed method for measuring resistances in the range from 107 U to 1013 U. In this work, a model of the method proposed previously is presented. On the basis of that model, it has been determined that during charging of an insulating layer of a material (on whose surface the deposited ions are immobile), the charge flux becomes wider as it approaches the surface of the insulator. For example, the diameter of the charge flow region may increase from 0.2 mm (near the needle tip) up to 2 cm near the surface of the insulator. [Those numbers correspond to the distance h ¼ 1 mm between the needle and the substrate, insulating layer thickness 40 mm and needleesubstrate voltage of 4000 V. A change of those parameters would cause a change of the size of the spot on the layer surface]. It has been determined experimentally that resistance of the air gap between the needle and the substrate is linearly dependent only on h, whereas the electromotive force, which is responsible for the electric current from the needle to the substrate, also depends only on h. The radial coordinate of the points where the gradient of the electric charge density is largest is equal to h/2 (a zero radial coordinate corresponds to the point that is directly below the needle). During transfer of charge carriers from the needle onto the surface of the insulating layer, the largest potential is obtained at the point corresponding to radial coordinate r ¼ 0, but this potential is still smaller than the electromotive force that causes electric current in the circuit (i.e., the difference between the power supply voltage and the voltage on the capacitor formed by the needle and the substrate, when no charge has been deposited yet). The time dependence of charging current and of the potential difference between the needle and the substrate is not monotonic: at first the current increases, then it begins to decrease, and the potential difference at first decreases, then it begins to increase. The initial parts of those dependences can be explained by the “breakdown” of the capacitor formed by the needle and the substrate, and the subsequent time dependence is determined by the increase of the insulating layer potential due to accumulation of charge on it. Ó 2010 Elsevier B.V. All rights reserved.
Keywords: Insulator Discharge Resistance
1. Introduction The work [1] contains a proposed method of measuring resistance of various layers at the time when the layer is exposed to an ion beam. This method can be also used for measuring various properties of the environment or of the layer that influence resistance of the investigated layer: humidity, illumination intensity, ionizing radiation intensity, layer thickness, resistance of island
* Corresponding author. Tel.: þ370 5 68674515; fax: þ370 5 236 6003. E-mail address: [email protected] (R. Rinkunas). 0304-3886/$ e see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.elstat.2010.06.003
layers. This method of measuring layer resistance has attractive properties that distinguish it among other known methods: for example, measurements based on vacuum evaporation of metal electrodes on the layer surface and measuring resistance between those electrodes [2,3] cause damage to the investigated sample. The electrodes can be pressed to the sample surface (as in a twoelectrode or four-electrode method [4]). However, the contact between those electrodes and the sample may be insufficiently good (e.g., if the sample has a rough surface), or the pressure may break the sample. Contact-free methods of resistance measurements are known, too. Using the dosed charging method [5], a weak corona discharge current causes a constant and uniform charging
R. Rinkunas, S. Kuskevicius / Journal of Electrostatics 68 (2010) 429e438
of the entire sample (so that it is impossible to measure values of the sample resistance at separate points of the sample), therefore the entire sample resistance is measured during the charging. The duration of a single measurement is several minutes, and this time is sufficient for a significant change of the sample resistance. Using the scanning method [6], a distribution of the sample resistance can be investigated, but this method uses a voltage of only several volts, which may be a problem when the sample resistance depends on voltage and it is necessary to know the resistance at a high substrate potential. The method proposed in [1] does not damage the samples; the ion contact ensures a reliable contact with the surface of the investigated sample; it is possible to measure spatial distribution of the sample resistance; resistance is measured at high potentials that ionize the surrounding air, therefore the measured value of the sample resistance is exactly the same that would be observed under operating conditions of those high voltages; measurement duration is a fraction of a second. It was determined empirically in [1] that the proposed method can be used for measuring resistances from 107 U to 1013 U. When investigating sample with resistances larger than 1013 U or smaller than 107 U, the shape of the measured time dependence of the sample charging current does not depend on the sample resistance. Therefore, it is not possible to separate the influence of mobile charges from the influence of immobile charges on the time dependence of charging current. Also, it is unclear how much the dimensions of the area containing the deposited charge change during the resistance measurements; what is the charge distribution in that spot, what is the precision of measuring resistance distribution in various points of the sample. In order to answer those questions, we have developed a model describing charging of the investigated insulating layer placed on the conducting substrate, with a needle used for charging. The calculation results have been compared with experimental data. In this work, we will discuss a case when the charge carriers deposited onto the investigated layer surface are immobile. 2. Material and methods Fig. 1 shows a circuit diagram of the equipment used for charging the insulating layer 2 with ion beam emitted from the needle 1. The insulating layer 2 is placed with its metalized surface on the metal substrate 3. This layer is charged using the voltage source 4 (its potential US is chosen such that electric field between the needle and the substrate exceeds the electric field needed for ionization of this air gap). The time dependence of the current (or charge) is measured using the device 7. The charge carriers are deposited from the needle 1 onto the insulating layer 2. Its potential increases until the potential difference between the needle 1 and the substrate 3 becomes equal to US. Then the current from the needle 1 to the substrate 3 stops. The resistor R limits the charging current and thus protects the insulating layer from damage. The material of the insulating layer 2 should not accumulate electric charge: after discharging a charged insulating layer, no charge should remain on the layer surface or in its bulk. It is known that dielectrics used in capacitors have a property called dielectric absorption (soakage) e when voltage recurs between the capacitor terminals after its charging and temporary short-circuiting (that is why high voltage capacitors are always kept with short-circuited terminals). In order to find out the extent of dielectric absorption of polyethylene terephthalate, which is the material of the insulating layer 2, we created a polyethylene terephthalate capacitor and evaporated Al electrodes on both sides of this layer. This capacitor was charged to 4000 V and then it was short-circuited for a period of 0.1 s. Then we measured the time dependence of the capacitor voltage. The maximum voltage is 0.32 V. This value is reached at
5 1 R
2
Us
3
4 6
7 Fig. 1. Circuit diagram of the measurement equipment. 1 e needle for charging the insulating layer 2, which is placed with its metalized side onto the metal substrate 3. 4 e high voltage source with potential US; 5 e resistor R for limiting the charging current; 6 e a small resistance (compared to R) used for measuring charging current or charge; 7 e a measuring device for measuring charging current or charge of the insulating layer 2 (it may be an oscilloscope with memory, or a computer circuit board).
1 min after the short circuit. Afterwards, the capacitor voltage continuously decreases (see Fig. 2). The just-mentioned voltage is very small in comparison with the operating potential of the insulating layer (4000 V). Therefore, we can conclude that polyethylene terephthalate is a suitable material for an insulating layer 2 (see Fig. 1). Other suitable materials are polystyrene, polypropylene, NPO ceramic and Teflon. 2.1. Notation used in this work C1 e the capacitance between the needle 1 (Fig. 1) and the substrate 3; Csam e capacitance of the investigated sample, determined by geometric dimensions of the substrate 3 (the insulating layer 2 is above the substrate 3); Clr e capacitance of 1 m2 of the insulating layer 2;
0,35 potential difference, V
430
0,30 0,25 0,20 0,15 0,10 0,05 0 100 200 300 400 500 600 700 time since short-circuiting the terminals, s
Fig. 2. Time dependence of a 40 mm-thick polyethylene terephthalate capacitor potential at different moments of time since the short circuit. Prior to measuring the time dependence, the capacitor was charged to 4000 V, and then its terminals were short-circuited for a period of 0.1 s.
R. Rinkunas, S. Kuskevicius / Journal of Electrostatics 68 (2010) 429e438
d e thickness of the insulating layer 2; ELn(z) e electric field strength at a height z from the layer surface, created by charge density sL ðr; hÞ; e0 e electric constant; e e dielectric permittivity of the insulating layer 2; h e distance between the needle 1 and the substrate 3; I e electric current flowing out from the needle 1; r e radial coordinate on the substrate 2, measured from the point directly below the needle 1 to any point on the substrate 3 (or on the insulating layer 2, which is on the substrate 3); ri e the radius of the smallest circular spot containing all the charge deposited on the insulating layer 2 at the moment of time ti (if r > ri, there is no charge on the surface of the insulating layer 2: sL(r > ri,h) ¼ 0); rmax e the largest radial coordinate of the charge deposited on the surface of the insulating layer 2. In this case, UT ¼ US, therefore electric current between the needle 1 and the substrate 3 stops flowing; rsam e the radius of the sample (the insulating layer 2 is circular in shape and it is placed on the substrate 3 of the same radius); rqmax e the radial coordinate of the points corresponding to the maximum gradient of deposited charge density on the surface of the insulating layer 2; qp(r,h) e a fraction of charge (relative to the total charge of the needle Qn) deposited at a given moment of time on the surface of the insulating layer 2; Qn e the charge of the needle 1; QL(ri,h) e the charge deposited on the surface of the insulating layer 2 at a given moment of time; QL(rmax,h) e the maximum charge deposited on the surface of the insulating layer 2 (in this case UT ¼ US and the current between the needle 1 and the substrate 3 stops flowing); R e the resistance 5 limiting the charging current; RL e resistance of the air gap between the needle 1 and the insulating layer 2; s(r,h) e density of charge induced on the substrate 3 by the charge of the needle Qn; sL(r,h) e density of charge deposited on the surface of the insulating layer 2 at a given moment of time; t e time; Un e potential difference between the needle 1 and substrate 3 (see Fig. 1), created by the needle charge Qn; UEMF e electromotive force e is the voltage causing the current to flow between the needle 1 and the substrate 3; ULn e potential difference between the needle 1 and the surface of the insulating layer 2, created by the charge density sL(r,h); ULp e potential difference between the substrate 3 and the surface of the insulating layer 2, caused by the charge density sL(r,h); UP e potential difference between the needle 1 and substrate 3, caused by charge density s(r,h) existing in the substrate; US e potential difference of the source 4; UT e the total potential difference between the needle 1 and substrate 3, caused by all charges (existing in the substrate e s(r,h), the needle e Qn and on the surface of the insulating layer e sL(r,h)); z e the coordinate indicating the height above the substrate.
431
the final radius of the charged region on the layer surface would be rmax. In this case, UT ¼ US. An increase of the material resistivity would not change rmax. Therefore, the time dependence of charging current, which is used in this work for determination of the layer resistance, would not change, too. It has been determined empirically [1] that the maximum value of the layer resistance measurable with this method is 1013 U. The model of charging of an insulating layer proposed in this work can be checked by comparing the calculated and measured time dependences of the current. In order to calculate the currents, it is necessary to know the potential differences created by charges in the needle, metal substrate and on the insulating layer 2. Those potential differences can be calculated only when the distribution of charges in the substrate and on the insulating layer 2 is known (the charge in the needle is assumed to be point-like). If a potential difference exists between the needle 1 and the substrate 3, then a charge Qn is accumulated in the needle. This charge induces the charge of the opposite sign in the conductive substrate 3. The induced charge will be denoted s(r,h), where h is the distance between the needle and the substrate. The entire charge induced in the substrate is equal to the charge in the needle. However, the charge in the needle is point-like, whereas the charge in the substrate is distributed on the substrate surface symmetrically relative to the needle. According to the image charge theory [7e9], the interaction between the charges s(r,h) and Qn is the same as interaction between the point charge Qn and its mirror-image charge Qn (see Fig. 3). Using the method of image charges [7e9], charge density distribution on the substrate can be calculated (see Fig. 4):
sðr; hÞ ¼
Qn h 3 2p 2 2 h þ r2
(1)
The needle charge Qn can be calculated, if the potential difference UT between the needle and the substrate is known, and if the capacitance C1 between the needle and the substrate is known:
Qn ¼ UT C1
(2)
Capacitance C1 is determined by the needle spike shape, size and distance of the needle to the substrate. Thus theoretically calculate this capacity is a complex task. That’s why the capacity is measured experimentally: for example, when h ¼ 1 mm, C1 ¼ 75 fF. If the entire charge Qn of the needle 1 was removed from the needle and distributed on the insulating layer 2 (where the distributed charge is immobile) according to the same law as the distribution of charge on the substrate 3 (Fig. 4), then the charge
3. Results and discussion If resistivity of a material is very large, then the deposited charge will not have enough time to spread on the surface during its charging. The sample will be charged by an ion beam which spreads after leaving the needle tip and while moving towards the insulating layer. In the mentioned case of an extremely large resistivity,
Fig. 3. The point charge þQn above a metal plate and its image Qn in that metal plate.
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diameter of the developed area, m
charge density
(r,h), C/m
4x10 Q (r ,h) A
A
3x10
1 2x10 2
r
10 3 0 -0,004 -0,002 0,000 0,002 coordinate x, m
0,004
Fig. 4. Dependence of substrate charge density induced by the point charge Qn of the needle on distance x to the point that is directly below the needle. Potential difference between the needle and the substrate is UT ¼ 3500 V; h ¼ 0.001 m (curve 1), h ¼ 0.002 m (curve 2); h ¼ 0.003 m (curve 3). The other notations will be explained below.
distribution on the substrate would not change, but the electric field between the needle and substrate would disappear, so that potential difference between the needle and the substrate would become zero. However, it is impossible to remove the entire charge Qn onto the surface of the insulating layer 2, because the electric currents that flow out of the needle are small. The charge is deposited on the place where electric field is strongest (initially the largest electric field corresponds to the point where charge density is highest e immediately below the needle). On the basis of those considerations, we assume that the charge is initially deposited on the point that is immediately below the needle, and later on the deposited charge will spread in a circular area with radius r ¼ ri (see Fig. 4). The charge deposited in this area is equal to QL(ri,h). If this assumption is correct, then the calculation results should coincide with measurement data. Electric field created by charge QL(ri,h) deposited on the insulating layer is compensated by the field created by the opposite charge on the substrate, therefore the total electric field created by both those charges between the needle and the substrate would be practically equal to zero. However, the remaining charge on the substrate (i.e., the charge that is not compensated by the charge QL(ri,h)) will create electric field between the needle and the substrate. Therefore, in this case the needle would only “see” the charge that corresponds to the curve below the line A‑A in Fig. 4. Hence, the “focusing” effect of the electric field would be weakened and the charge would be deposited approximately uniformly in the entire area pri2 , which corresponds to radial coordinate values 0 < r < ri. An increase of the total deposited charge QL(ri,h) causes an increase of the radius ri (i.e., the line AeA in Fig. 4 shifts down). This increase of the radius can be verified experimentally. This is done by placing a screen (mask) over the insulating layer 2 (the screen does not touch the surface of the layer 2). That screen has a round hole centered on the needle. The hole’s diameter is much larger than the diameter of the streamer. If the charge transfer from the needle 1 onto the insulating layer 2 was only by means of a narrow streamer, followed by spreading of the charge over the surface of the layer 2, then, after placing the screen (mask) on the layer, the screen would have no effect on the size of the charged area of the insulating layer 2. However, the curves presented in Fig. 5 indicate that the screen limits the charged area of the layer 2. This limiting effect becomes more pronounced when the distance between the screen and the layer is decreased or when the hole diameter is decreased (see curves 2, 3, 4). Therefore, it follows that during charging of the insulating layer 2, the diameter of the streamer increases from 0.2
0,024 0,022 1
0,020
2
0,018 0,016
3
0,014 0,012
4
0,010 0,008 0,001 0,002 0,003 0,004 0,005 0,006 0,007 h, m
Fig. 5. Dependence of the developed area diameter on distance h between the needle and the substrate. Curve 1 corresponds to absence of the screen. Other curves were measured after placing a screen with a circular hole on the insulating layer. The hole diameter is 7 mm (curve 4) or 12 mm (curves 2, 3). That screen is raised over the layer 2 (Fig. 1) using an insulating rubber spacer with a 35 mm hole. Thickness of the spacer is 0.2 mm (curves 3, 4) or 0.8 mm (curve 2). US ¼ 4000 V, d ¼ 40 mm.
mm in the beginning of charging up to the diameter of the charged area (24 mm). The charge QL(ri,h) increases until the potential difference between the needle and the substrate reaches the value US. At that time, the radius of the charged area of the insulating layer is ri ¼ rmax. This spreading of the charge can be observed by developing the insulating layer with an electrographic developer after charging of the layer with a needle electrode (see Fig. 6). However, the electrographic developer does not develop the regions with a small charge density (charge density s(r,h) decreases with increasing radial coordinate r, as evident from Fig. 4), therefore, in order to see the value of rmax, a potential relief has to be formed on the insulating layer surface (e.g., by pressing a grounded metal foil to the charged insulating layer for a short time). In this case, the charge
Fig. 6. A potential relief was formed on the surface of a 40 mm-thick polyethylene terephthalate layer (e.g., by pressing a grounded metal foil to the surface of the charged layer for a short time). Such layer was used as the insulating layer 2 (Fig. 1) and charged by the needle 1 with US ¼ 4000 V. The value rmax w 20 mm was obtained.
R. Rinkunas, S. Kuskevicius / Journal of Electrostatics 68 (2010) 429e438
deposited on the layer during its charging with a needle electrode would “wipe” the previous charge; hence the value of rmax would be seen in the developed image even if the small charge density of the needle-charged area is not developed. If the charge density distribution on the insulating layer is known (see Eq. (1)), then the charge QL(ri,h) deposited onto the surface of the insulating layer 2 can be calculated as follows:
ZSi QL ðri ; hÞ ¼
sðr; hÞ dS sðri ; hÞ p ri2
0
Zri ¼
2 p r sðr; hÞ dr sðri ; hÞ p ri2
(3)
0
where Si is the area of the circle with radius ri. Now, we define a quantity qp e the fraction of the charge (relative to the total charge in the substrate) deposited on the surface of the insulating layer 2 (Fig. 1). From Equations (1) and (3), the following expression of qp is obtained:
qp ðri ; hÞ ¼
QL ðri ; hÞ ¼ 1 Qn
h ri2
þ h2
1 2
h ri2 3 2 2 ri2 þ h2
(4)
opposite charge in the substrate. This deposited charge (and the corresponding charge in the substrate) could redistribute if conductivity of the insulating layer were non-zero. This motion of charge on the investigated layer can be applied for measuring resistance of conducting layers. In order to calculate time dependence of charging current of an insulating layer (or the time dependence of charge deposited on that layer or the time dependence of the layer potential), the distribution of the deposited charge QL(r,h) must be known. In Fig. 4, the total charge deposited on the insulating layer is denoted QL(ri,h) (it corresponds to the hatched area). From this diagram we can see that distribution of the charge density on the insulating layer is similar to distribution of charge density in the substrate (Equation (1)), but there are some differences, too: when r ri, charge density in the substrate is non-zero, whereas the charge on the insulating layer is already zero. The charge on the insulating layer is only present inside the circular area corresponding to r < ri. Based on those considerations and on Equation (1), we can express the distribution of charge deposited on the insulating layer 2 as follows:
" Qn sL ðr; hÞ ¼ 2p
(5)
In this case, Qn is the charge that was in the needle at the beginning of the experiment e while there was no charge deposited on the insulating layer (because during the experiment the needle charge decreases due to charge transfer onto the surface of the insulating layer). Qn is also equal to the total charge in the substrate (it does not change during this experiment). While there is no charge deposited on the insulating layer surface, the only sources of electric field are the needle charge and the substrate charge induced by the needle charge (Fig. 4). The substrate charge is tied to the needle charge via the electric field lines; therefore the substrate charge distribution is fixed. However, if a charge is transferred from the needle onto the surface of the insulating layer, then this charge (QL(ri,h)) becomes “tied” to the
h2 þ r 2
1 0,8 2 3 0,6
0,4
0,2
0,0 0,000 0,005 0,010 0,015 0,020 0,025 0,030 r i, m Fig. 7. Dependence of the relative charge qp(ri,h) (defined as the ratio of the deposited charge and the total needle charge Qn) on the radius ri of the area where the deposited charge is distributed. Curve 1 e h ¼ 0.001 m; curve 2 e h ¼ 0.002 m; curve 3 e h ¼ 0.003 m.
3 2
h h2 þ ri2
5 3 /r < ri 2
(6)
According to this equation, sL(r < rmax,h) > 0 and sL(r > rmax,h) ¼ 0, i.e., density of charge deposited on the insulating layer abruptly drops to zero when r > rmax. This is confirmed by the experiment: the charged area has a sharp edge (see Fig. 6). If the charge distribution on the insulating layer is known (see Equation (6)), it is possible to determine the radial coordinate corresponding to the largest gradient of charge density (if the charge is assumed to consist of separate particles, then the points with the largest gradient of particle density would correspond to the best conditions of particle spreading on the layer surface). At first, let us discuss the case when the total charge of the needle has been deposited on the surface of the insulating layer (then ri is infinite, so that Equations (1) and (6) become identical to each other). At the point of the maximum gradient of charge density, the first derivative relative to r has a maximum value, whereas the second derivative is zero (ds2(r,h)/dr2 ¼ 0). Solution of the latter equation gives the following expression of the radial coordinate rqmax corresponding to the maximum gradient of charge density on the surface of the insulating layer 2:
rqmax ¼
relative charge q p(r i,h)
3 h
sL ðr; hÞ ¼ 0/r ri
This function is shown in Fig. 7, where we see that qp increases with increasing ri and approaches 1. The total charge deposited on the surface of the insulating layer 2 is equal to
QL ðri ; hÞ ¼ qp ðri ; hÞ Qn
433
h 2
(7)
From this equation we see that the place of the maximum charge density gradient depends only on h. The corresponding value of that gradient is
d s rqmax ; h 4:8 Qn ¼ dr p h3
(8)
If ri < h/2, then the radial coordinate corresponding to the maximum charge density gradient on the insulating layer surface is rqmax ¼ ri. If ri > h/2, then the maximum charge density gradient corresponds to rqmax ¼ h/2. Using the known values of charge density distribution on the substrate (Equation (1)), charge density distribution on the surface of the insulating layer (Equation (6)), needle charge Qn and the insulating layer thickness d, we can calculate the electric field between the needle and the substrate. Then it is possible to calculate the total potential difference between the needle and the substrate UT caused by the charges in the needle, the substrate and on the insulating layer:
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R. Rinkunas, S. Kuskevicius / Journal of Electrostatics 68 (2010) 429e438
UT ¼ Un þ ULp þ UP ULn
(9)
where Un is the potential difference between the needle and the substrate caused by the point charge Qn in the needle:
Un ¼ Qn =C1
(10)
ULp is the potential difference between the substrate and the surface of the insulating layer caused by the charge density sL(r,h) on the surface of the insulating layer; ULn is the potential difference between the needle and the surface of the insulating layer caused by the charge density sL(r,h) on the surface of the insulating layer; UP is the potential difference between the needle and substrate caused by the charge in the substrate:
UP ¼ Qn =ð8 p e0 hÞ
(11)
The potential differences ULn and ULp are created by the charge deposited on the surface of the insulating layer. They are calculated in the same way as the potential difference UP: we calculate electric field (created by the charge density sL ðr; hÞ (Equation (6)), which is bounded by the circle with radius ri) perpendicular to the substrate at height z. The total electric field at height z is calculated by summing the fields created by individual charge elements (similarly to the technique used in [9]):
Zri ELn ðzÞ ¼ 0
sL ðr; hÞ z dS 4 p e0 l3
(12)
where dS ¼ 2 p r dr; l ¼ ðr 2 þ z2 Þ1=2 ; When ELn(z) is known, ULn can be calculated:
Zh Zri
Zh ULn ¼
ELn ðzÞ dz ¼ d
d
0
sL ðr; hÞ z dS dz 4 p e0 l3
(13)
ULn is found by combining Equations (6) and (13). ULp is calculated similarly. Then all potential differences are:
Qn ULn ¼ 4 p e0
Zh Zh d 0
dr dz Qn ULp ¼ 4 p e0 e
0 1 hzr @ 1 1 A 3 32 2 32 2 2 2 r 2 þ z2 2 r þh r þh
Zd Zri 0 0
i
0 1 hzr @ 1 1 A 3 3 2 3 r 2 þ z2 2 r 2 þ h2 2 r þ h2 2 ð14Þ
dr dz Qn UP ¼ 8 p h e0 Qn Un ¼ C1 UT ¼ Un þ ULp þ UP ULn
i
Here, e is dielectric permittivity of the insulating layer (Fig. 1). In the case of polyethylene terephthalate layer that is used in this work, e ¼ 3.4 [10,11]. The signs of the various potentials in the expression of the total potential UT (Equation (9)) can be explained as follows: in all equations given above, only absolute values of potentials are used. Therefore, in order to determine the sign of a particular potential, one has to take into account the direction of electric field created by the corresponding charge. Let us assume that the needle charge Qn is positive. Then that charge will create electric field directed from the needle to the substrate. The substrate charge is negative; hence the direction of electric field created by the substrate charge
between the substrate and the needle is the same as direction of the field created by the needle charge. Therefore, the potentials of those two fields add up. The charge deposited on the surface of the insulating layer has the same sign as the needle charge. Hence, the direction of the field created by the deposited charge inside the insulating layer is the same as the direction of the field created by the needle charge, therefore the potential ULp has the same sign as the potentials of the fields created by the needle charge (Un) and the substrate charge (UP). However, direction of the field created by the deposited charge between the insulating layer and the needle is opposite to the direction of the other mentioned fields. Therefore, the potential difference ULn must be subtracted from the other mentioned potential differences when calculating the total potential difference between the needle and the substrate (UT). In this way, Equation (9) is obtained. Direct measurements of the potential UT are difficult, because the needle charge is small and connection of a measuring device to the needle would distort the measurement data. Besides, the capacitor formed by the needle and the substrate would have to be charged prior to measuring the potentials, and a part of the needle charge would be deposited on the insulating layer already during the charging. A simpler method of experimental testing of the discussed model is by measuring the time dependence of current flowing through the circuit formed by the mentioned capacitor (needle e air gap e insulating layer e substrate), a constant voltage source US and a limiting resistance R (5) connected in series (see Fig. 1). For comparing the calculation results with measurement data, one has to know the time dependence of the needle charge Qn. In order to determine that dependence, let us first discuss the case when a uniform electric field is present between the electrodes and the field strength is sufficient for the corona discharge, and the number of neutral molecules in the discharge region is very large. In this case, a slight increase of electric field strength causes a significant increase of the number of charge carriers. This causes an increase of electric current and an increase of the voltage drop on the resistances that are connected in series. Therefore, the potential difference between the electrodes decreases and the electric field strength decreases, too. This causes a decrease of charge carrier generation and a corresponding decrease of electric current. But then the voltage drop on the resistances decreases, so that the electrode potential difference increases and the process repeat itself. As a result, constant field strength is established between the electrodes. This field strength corresponds to a particular potential difference between the electrodes. This is the mechanism of formation of the Townsend discharge [12e14] (its characteristic property is formation of a certain potential difference between the electrodes). In our case, one electrode is in the shape of a needle and the other one is in the shape of a plate (the substrate). Therefore, electric field between the electrodes is not uniform; the strongest electric field is created near the needle. However, in our case the number of neutral air molecules is very large, too (an increase of electric field strength causes an increase of the number of charge carriers). In our case, the charge carriers are created near the needle. Therefore, a constant value of the needle electric field strength is maintained. This means that the needle charge is constant (that charge determines the magnitude of electric field created by the needle). Thus, in our case the Townsend discharge condition is
Qn ¼ const
(15)
However, the potential difference between the needle and the substrate is not constant. It depends on electric current, because resistance RL of the air gap between the needle and the substrate is non-zero. If potential difference between the needle and the substrate was constant and independent of R (i.e., of electric
R. Rinkunas, S. Kuskevicius / Journal of Electrostatics 68 (2010) 429e438
current), then RL would have to be zero, but it is not the case, because from Equation (20) below it follows that RL is non-zero. In this case, under conditions of a closed circuit one has to calculate time dependences of current, potential and charge. Equation (14) were calculated for the case when the charge accumulated in the needle was gradually transferred onto the surface of the insulating layer, and the needle charge was not replenished. However, in the case of a closed circuit shown in Fig. 1, Equation (14) would change. The voltage source US would replenish the needle charge. The substrate charge would gradually increase. Its value would be equal to the sum of the needle charge and the charge deposited on the insulating layer surface. However, in this case both the distribution of the deposited charge and the distribution of the substrate charge are still determined by the same laws as in the previously-discussed case. Therefore, all the earlier equations derived for an open circuit are applicable to the case of a closed circuit, but the charge Qn must be replaced by the charge (Qn þ QL) except the constant charge in the needle (Equation (15)) [This is essentially the same assumption that was made when deriving Equation (6). If this assumption is correct, then values of the currents, charges and potentials and their time dependences, which follow from that assumption, must coincide with measurement data]:
Qn þ QL ðr;hÞ ULn ¼ 4 p e0 dr dz Qn þ QL ðr;hÞ ULp ¼ 4 p e0 e
Zh Zri d 0
Zd Zri 0 0
0
1
hzr @ 1 1 A 3 32 2 32 2 2 2 2 2 r þz r þh r þ h2 i
0 1 hzr @ 1 1 A 3 32 2 32 r 2 þ z2 2 r 2 þ h2 r þ h2 i
dr dz
ð16Þ
Qn þ QL ðr;hÞ 8 p h e0 Qn Un ¼ C1 UT ¼ Un þ ULp þ UP ULn
UP ¼
435
Resistance RL can be measured. This is done by removing the insulating layer 2 in the circuit shown in Fig. 1 and by measuring charging currents at different values of R. In this way, the curves shown in Fig. 8 are obtained. From Fig. 8 we see that inverse current is linearly dependent on R, but the intercept of this straight line is not zero. This means that resistance of the air gap between the needle and the substrate is not equal to zero, but it is equal to some constant. There is a wellknown technique for determining resistance RL of the air gap between the needle and the substrate from the presented graph (based on the point where the straight line shown in Fig. 8 crosses the y axis). The same graph can be used for calculating the electromotive force UEMF of this process (the electromotive force is given by the slope of the straight line in Fig. 8). The linear dependence of inverse current on Rð1=IwRÞ indicates that a change in the limiting resistance R (and a corresponding change in the current) does not cause any change in RL and UEMF. Those quantities depend only on the distance h between the substrate and the needle. The presented experimental results, as well as results of additional experiments, indicate that the following empirical dependences of RL and UEMF on h can be used (all units in SI):
RL ¼ 7:35 109 h UEMF ¼ 500 ð1 logð1000 hÞÞ
(20)
Those expressions are valid when 0.5 mm < h < 8 mm and US ¼ 4000 V. When h ¼ 1 mm, then from Equation (20) it follows that UEMF ¼ 500 V. As mentioned, UEMF does not depend on R. Therefore, potential difference US UEMF ¼ 3500 V stays the same at all values of current (which depends on R). When the current is zero (i.e., when there is no voltage drop on resistances R and RL), this potential difference is equal to the voltage drop between the needle and the substrate. This is the case of discharge ignition and discharge quenching. When h ¼ 1 mm, the measured needlee substrate potential difference corresponding to discharge ignition is approximately 3502 V, and the measured needleesubstrate potential difference corresponding to discharge quenching is approximately 3497 V. Thus, those potential differences are very close to the potential difference US UEMF ¼ 3500 V.
In the case of a closed circuit, the distribution of charge density deposited on the insulating layer surface is
3 h2 þ r 2
3
sL ðr;hÞ ¼ 0/r ri
2
h h2 þ ri2
5/r < ri 3 2
a (17)
Using the known charge distribution on the surface of the insulating layer (Equation (17)), we can find ri ¼ rmax corresponding to UT ¼ US. In this case, there is no current between the needle and the substrate. When ri ¼ rmax, the maximum charge QLmax is deposited on the surface of the insulating layer (from Equation (5)):
QLmax ðrmax ; hÞ ¼ qp ðrmax ; hÞ Qn
(18)
Using the known potential difference between the needle and the substrate UT (caused by the charges in the needle, the substrate and on the insulating layer) and the known source potential US, we can calculate the current in the circuit formed by the source US, the limiting resistance R, the needle and the substrate, connected in series:
U UT I ¼ S R þ RL
(19)
where RL is the resistance of the air gap between the needle and the insulating layer 2 (Fig. 1).
1/I, relative units
h
0,10 0,08 0,06 0,04 0,02 0,00
b 1/I, relative units
" Qn þ QL ðr;hÞ sL ðr;hÞ ¼ 2p
0 10 20 30 40 50 R, MΩ
2,2 2,0 1,8 1,6 1,4
0 10 20 30 40 50 R, MΩ
Fig. 8. Dependence of inverse charging current of a metal plate on the limiting resistance R (No. 5 in Fig. 1). The distance h between the needle and the metal plate is 1 mm (a) or 8 mm (b). The source potential is US ¼ 4000 V.
R. Rinkunas, S. Kuskevicius / Journal of Electrostatics 68 (2010) 429e438
Q ðr ; hÞ QL ðri1 ; hÞ ti ¼ ti1 þ L i Ii
(21)
0,020
3x10 -9 2
0,015
2x10 -9 0,010
radius ri, m
1 charge Q L(r i,h), C
10-9 0,005
0
0
3x10 -5
6x10 -5
9x10 -5
0,000
time t, s Fig. 9. Calculated time dependences of the charge QL deposited on the insulating layer surface (curve 1) and of the radius of the charged area ri (curve 2). The time dependences have been calculated using Equations (16)e(21). The insulating layer thickness is d ¼ 40 mm, the distance between the needle and the substrate is h ¼ 1 mm, the source potential is US ¼ 4000 V.
Using this method, the time dependences of I, UT, ULn, ULp, UP, QLi, ri have been calculated. Some of them are shown in Figs. 9e11. As we see in the figure, the charge deposited on the insulating layer increases monotonically, approaching the maximum value QLmax. The radius of the charged area also increases monotonically, approaching rmax (the calculated value is rmax ¼ 21 mm, when US ¼ 4000 V and h ¼ 1 mm. This value is in accord with the measured value rmax ¼ 20 mm e see Fig. 6). As shown earlier, the needleesubstrate potential difference corresponding to breakdown of the air gap is 3500 V. Therefore, this potential corresponds to the start of the time dependence of potential UT (curve 3 in Fig. 10) (electric current begins to flow between the needle and the substrate only when the capacitor C1 formed by the needle and the substrate is charged to this potential). At the start of the time dependence, the calculated potential difference UT does not increase (as it could have been expected), but decreases instead (this can be explained by “breakdown” of the capacitor needleesubstrate, because it is known that breakdown of a capacitor causes a decrease of its potential). However, eventually another capacitor e the surface of the insulating layer (Csam) e begins to charge. Therefore, UT begins increasing and it increases until reaches the value US (in this case 4000 V). Since the current is calculated, according to equation (21), this time dependence of UT causes a corresponding time dependence of the current (curve 1). 4000
1,6x10 -4 3 1,2x10 -4
3500 8,0x10 -5 2
3000
potential U T, V
Current density on the surface of the insulating layer must be limited, because large current density can damage the layer. From [15] we know that current magnitude determines the type of the discharge (corona discharge, brush discharge, spark or arc). From Equation (20) we obtain that when the distance between the needle and the insulating layer is h ¼ 0.5 mm, then the resistance is RL ¼ 3.6 MU, and when the distance is h ¼ 8 mm, this resistance is RL ¼ 60 MU. Therefore, at the initial moment of time, electric current flowing into the surface of the layer may be between 0.15 mA and 8 mA. The corresponding current density is 300e5000 A/m2 (the diameter of the streamer formed between the needle and the substrate is approximately 0.2 mm e this has been determined by photographing the streamer). Those currents are slightly larger than corona discharge current (1e30 mA). Therefore, it is necessary to choose such values of current density that do not damage the investigated materials (during arc welding, samples are certainly “damaged”, but in this case current density reaches about 107 A/ m2). The current is largest at the initial moment of time after the breakdown of the air gap between the needle and the surface of the insulating layer. Therefore, resistance R must be chosen so that even the maximum current does not damage the layer and the results are reproducible. However, if the charging current decreases to values corresponding to corona discharge, then it becomes more difficult to measure sample properties (its resistance, humidity, etc.), because in the case of corona discharge the process of charge deposition on the layer surface is mainly determined by the layer potential and is less dependent on the layer structure (e. g., surface irregularities), and the charge is deposited on a large area at once. In the case discussed here, the currents used are larger than those during a corona discharge, and the current is concentrated into a small area of the surface (this ensures a better spatial resolution on the surface of the investigated layer). This area is rapidly charged to high potentials. Therefore, strong tangential electric fields are formed when charges are deposited on the layer. Those fields can cause secondary generation of charge carriers on the layer surface. For example, if the layer surface is uneven or if islands of another material have been evaporated on the layer, then those surface irregularities would cause ionization of air under conditions of strong tangential electric fields. As a result, the charging area would spread and it would cause an increase of the maximum charge that can be deposited on the layer (even when the layer surface is smooth). Hence, a larger charging current would be measured when the layer surface is uneven than when it is even. Thus, when a layer is charged with a larger charging current, then properties of the layer surface has a stronger influence on the value and time dependence of charging current. The time dependence of charging current is experimentally measured using an oscilloscope (No. 7 in Fig. 1). The first step of calculating the time dependence of charging current is determination of the maximum charge QLmax, which can be deposited on the surface of the insulating layer 2 (when that charge is deposited, then UT ¼ US and the current between the needle and the substrate drops to zero). The charge QLmax is calculated with the program “Mathcad” using Equations (16) and (17) and the condition UT ¼ US. When QLmax is known, then it can be divided into, e.g., 100 parts and then the entire deposition of QLmax on the surface of the insulating layer can be modeled in i ¼ 100 steps. Each value of QL(ri,h) corresponds to a particular value of the charged area radius ri (this follows from Equations (4) and (5)). When ri and QL(ri,h) are known, UTi can be calculated from Equation (16). The current Ii is calculated from Equation (19), and the time is calculated using this equation:
current I, A
436
4,0x10 -5 1 0
0
3x10 -5 6x10 -5 time t, s
9x10-5
2500
Fig. 10. Time dependences of the charging current I of the insulating layer (curves 1 and 2) and of the potential difference between the needle and the substrate (curve 3). Conditions are the same as in Fig. 9. Curves 1 and 3 have been calculated. The scatter graph (curve 2) presents measurement results.
dielectric layer potential U Lp , V
R. Rinkunas, S. Kuskevicius / Journal of Electrostatics 68 (2010) 429e438
where Csam is capacitance of the capacitor formed by the substrate and the surface of the insulating layer. In this case, charge density does not depend on r:
350 300
sL ðr; h; tÞ ¼ sL ðh; tÞ ¼
250
150 100 50
ULn ¼ 2x10 -5 4x10 -5 6x10 -5 8x10 -5 time t, s
0
10 -4
ULp ¼
Fig. 11. The calculated time dependence of the insulating layer potential below the needle (ULp) during charge deposition on the layer. Conditions are the same as in Fig. 9.
However, the experimental points (curve 2) coincide with the measured curve only at the beginning of the time dependence. Later the calculated current decreases (curve 1), whereas the experimental current stays constant (curve 2). This is because the calculations have been done under the assumption of infinite surface resistivity of the insulating layer, whereas in reality the surface resistivity is finite (this is evident from the experimental fact that the charged area grows during charging of the insulating layer). Fig. 11 indicates that surface potential of the insulating layer increases during charge deposition, approaching a constant value (375 V), but does not reach the value UEMF ¼ 500 V. The value of the maximum potential of the charged area, measured with equipment designed for measuring point potentials (e.g., the device “Relief”), is about 350e370 V, which is in good agreement with calculation results. The calculation method used in this work can be easily checked by applying it to the case when the charge deposited on the insulating layer surface spreads over entire surface immediately (in this case, the surface resistivity of the insulating layer is zero). This case corresponds to charging of a capacitor, therefore a well-known expression of the time dependence of the capacitor charging current must be applicable:
UEMF ðRþR ÞCsam L e R þ RL t
(22)
6x10 -5 5x10 -5 current I, A
QL ðh; tÞ
(23)
2 p rsam
where rsam is the radius of the insulating layer and the substrate (a circular insulating sample is placed on the metal substrate of the same shape). The potential ULn is calculated by combining Equations (13) and (23). The other potentials are obtained similarly:
200
0
I ¼
437
4x10 -5 3x10 -5 1
2x10 -5 10-5 0 0,000
2
0,005
0,010
0,015
0,020
0,025
time t, s Fig. 12. Time dependence of the current flowing out of the needle. Curve 1 e calculated using Equation (22); curve 2 e calculated using Equations (19) and (24). rsam ¼ 2 cm; h ¼ 1 mm; d ¼ 40 mm.
QL ðh;tÞ 2 e 4p rsam 0
Zh Zrsam d
QL ðh;tÞ 2 e e 4 p rsam 0
0
rz 3 drdz 2 r þz2 2
Zd Zrsam 0
0
rz r 2 þz2
Qn QL ðh;tÞ þ UP ¼ 2 e 8 p e0 h 4 p rsam 0 Un ¼
3 drdz 2
Zh Zrsam 0
0
rz
(24)
3 drdz r 2 þz2 2
Qn C1
UT ¼Un þULp þUP ULn The potential UT is calculated from Equation (24), and then the current is calculated from Equation (19). The same current is also calculated from Equation (22). Values of the current calculated using both those equations is almost equal to each other (see Fig. 12). This proves that our calculation method can be applied for calculation of potentials and time dependences of currents, charges and potentials. 4. Conclusion This work presents a model describing the process of charging of an insulating layer placed on a conducting substrate, when charges deposited on the surface of the insulating layer are immobile. Using this model, the following conclusions have been made: 1. During charging of the insulating layer with a needle electrode, the diameter of the charged area of the layer surface grows from the streamer diameter (0.2 mm) up to the maximum diameter of the charged area that can be achieved under given charging conditions (for example, this diameter is 2 cm, when the distance h between the needle and the substrate is 1 mm, insulating layer thickness is 40 mm, and when a 4000 V voltage source is connected between the needle and the substrate. A change of those parameters causes a change of the charged area size on the surface of the insulating layer). 2. The radial coordinate corresponding to the largest gradient of charge density deposited on the surface of the insulating layer is r ¼ h/2 (the radial coordinate r is measured from the point that is directly below the needle). 3. The charges deposited on the insulating layer surface create the maximum potential ULp at the point r ¼ 0 (because charge density is largest at that point), but this potential is still smaller than the electromotive force UEMF, which causes electric current in the circuit (i.e., than the difference between the external source voltage and the voltage of the capacitor formed by the needle and the substrate when no charge has been deposited yet: ULp < UEMF ¼ US UT). 4. The time dependence of the charging current of the insulating layer, as well as the time dependence of the potential difference
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between the needle and the substrate are not monotonous: at first, the current grows, but later on it begins to decrease, whereas the potential difference between the needle and the substrate at first decreases, but later on it begins to increase. The initial parts of those time dependences can be explained as a result of “breakdown” of the capacitor formed by the needle and the substrate, and the subsequent time dependence is governed by the increase of the insulating layer potential due to charge accumulation on it. 5. It has been determined experimentally that resistance of the air gap between the needle and the substrate increases linearly with increasing distance h between the needle and the substrate. The electromotive force (which causes the current from the needle to the substrate) also depends only on the mentioned distance h between the needle and the substrate. Acknowledgement The authors are grateful to Prof. E. Montrimas and doc. A. Poskus for their interest and help in this work. References [1] R. Rinkunas, S. Kuskevicius, A contactless method of resistance measurement. Tech. Phys. 59 (2009) 133e137.
[2] M. Kasu, M. Kubovic, A. Aleksov, N. Teofilov, Y. Taniyasu, R. Sauer, Influence of epitaxy on the surface conduction of diamond film. Diamond Relat. Mater. 13 (2004) 226e232. [3] I.K. Kikoin, Tablici Fizicheskich Velichin. Atomizdat, Moskow, 1976. [4] N. Bowler, Theory of four-point alternating current potential drop measurements on conductive plates. J. Phys. D: Appl. Phys. 39 (2006) 584e589. [5] P.J. Zhilinskas, E. Montrimas, T. Lozovski, Measuring the parameters of dielectric layers by periodically charging the surface. Tech. Phys. 51 (2006) 1372e1378. [6] M. Tachiki, T. Fukuda, K. Sugata, H. Seo, H. Umezawa, H. Kawarada, Control of adsorbates and conduction on CVD-grown diamond surface, using scanning probe microscope. Appl. Surf. Sci. 159e160 (2000) 578e582. [7] M.H. Nayfeh, M.K. Brussel, Electricity and Magnetism. J. Wiley & Sons, New York Chichester Brisbane, 1985. [8] E.J. Konopinski, Electromagnetic Fields and Relativistic Particles. McGraw-Hill Book Company, New York, 1981. [9] D.J. Griffiths, Introduction to Electrodynamics, second ed. Prentice Hall, Englewood cliffs (New Jersey), 1989. [10] S. Osaki, Molecular orientation and dielectric anisotropy in polyimide films as determined by the microwave method. Macromolecules 31 (1998) 1661e1664. [11] D.U. Erbulut, S.H. Masood, V.N. Tran, I. Sbarski, A novel approach of measuring the dielectric properties of pet preforms for stretch blow moulding. J. Appl. Polym. Sci. 109 (2008) 3196e3203. [12] S. Flugge, Handbuch der Physik. Springer, Berlin, 1955. [13] J.W. Gewartowski, H.A. Watson, P. Bell Telephone Laboratories. Communications Development Training, Principles of Electron Tubes Including Gridcontrolled Tubes, Microwave Tubes, and Gas Tubes. Van Nostrand, Princeton, N.J.; London, 1965. [14] E. Kuffel, W.S. Zaengl, J. Kuffel, in: E. Kuffel, W.S. Zaengl, J. Kuffel (Eds.), High Voltage Engineering: Fundamentals, second ed. Newnes, Oxford, 2000. [15] M. Yamaguma, T. Kodama, Observation of propagating brush discharge on insulating film with grounded antistatic materials. IEEE Trans Ind Appl 40 (2004) 451e456.