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Dielectric effect on electric fields in the vicinity of the metal–vacuum–dielectric junction
Ultramicroscopy 132 (2013) 41–47
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Dielectric effect on electric fields in the vicinity of the metal–vacuum–dielectric junction M.S. Chung a,*, A. Mayer b, N.M. Miskovsky c, B.L. Weiss c, P.H. Cutler c a
Department of Physics, University of Ulsan, San 29, Muga, Ulsan 680-749, Korea FUNDP, University of Namur, Rue de Bruxelles 61, B-5000 Namur, Belgium c Department of Physics, The Pennsylvania State University, University Park, PA 16802, USA b
a r t i c l e i n f o
abstract
Available online 20 December 2012
The dielectric effect was theoretically investigated in order to describe the electric field in the vicinity of a junction of a metal, dielectric, and vacuum. The assumption of two-dimensional symmetry of the junction leads to a simple analytic form and to a systematic numerical calculation for the field. The electric field obtained for the triple junction was found to be enhanced or reduced according to a certain criterion determined by the contact angles and dielectric constant. Further numerical calculations of the dielectric effect show that an electric field can experience a larger enhancement or reduction for a quadruple junction than that achieved for the triple junction. It was also found that even though it changes slowly in comparison with the shape effect, the dielectric effect was noticeably large over the entire range of the shape change. & 2012 Elsevier B.V. All rights reserved.
Keywords: Triple junction Field enhancement Dielectric effect Metal–dielectric contact
1. Introduction Dielectrics are usually considered to reduce electric field intensities. For a long time, however, strongly enhanced electric fields have often been observed as phenomena of an electron emission avalanche or a breakdown in the vicinity of the metal– dielectric contact [1–5]. Such an unexpected field enhancement, called the triple junction effect, must be due to a dielectric in contact with metal (see Fig. 1). Two significant experiments that revealed the dielectric effect were made by Geis’s group [4,6]. In 1977, they reported that the field emission was strangely enhanced when the diamond portion of a triple junction was extended into the vacuum. Several years later, they observed that the field emission continued even after the bias was removed. Ma and Sudarshan [7,8] found that voids at the contact were crucial in vacuum insulation breakdown. The contact angle of the junction was considered to play a role in the enhancement ¨ mechanism [9–14]. Schachter [15] first used a two-dimensional model to obtain the theoretical result that the field emission current density was proportional to the dielectric constant for a specific triple junction. The field enhancement was also observed at the interface of the silicon and dielectric layer [16]. In addition, Hellman et al. [17] argued that an artifact of the field evaporation process results from a change in the local electric field due to the presence of dielectric in the atom-by-atom chemistry.
*
Corresponding author. Tel.: þ82 52 259 2329; fax: þ 82 52 259 1693. E-mail address: [email protected] (M.S. Chung).
0304-3991/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ultramic.2012.12.014
Field enhancement at the triple junction is different from that produced by a protrusion or a sharpening of the metallic edge [18,19]. The electric field is enhanced or reduced according to the angular configuration of the triple junction [12–14]. The value of the dielectric constant is also a factor in determining the magnitude of enhancement. This implies that both the magnitude and direction of polarization are involved in determining the enhancement of electric field. Thus, it is interesting to find which configuration yields the largest dielectric enhancement with consideration to the triple junction as a basis. One of the most probable candidates is a quadruple junction, even though it may not appear as often as a triple junction (see Fig. 1). According to previous work [20], a certain type of quadruple junction yields a larger dielectric effect than the triple junction. Even though it is not experimentally confirmed yet, the use of the triple or quadruple junction may lead to a new type of cold electron source. In addition, a deeper understanding of the dielectric effect may be used to avoid an avalanche of electron emission or breakdown [3,21]. Thus we need to make a systematic approach to describe the field behavior in regions near the contact between metal and dielectric. Besides, the current works deals with the reduction of electric field as well as enhancement.
2. Metal–vacuum–dielectric triple junctions As seen in Fig. 1, the metal-layer contact has several configurations of junction at which the metal, vacuum, and dielectric meet [6]. Since it is a layer-by-layer contact, the junction
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where the subscripts 1 and 2 refer to the vacuum and dielectric regions, respectively and the subscript t refers to the total angle for the field-varying region. Here, we put n ¼ n1 for convenience. From ! Eqs. (2) and (3), the electric field magnitude F ¼ r F is given by F 1 ¼ A1 nr n1 , F 2 ¼ A2 n r n1 ,
Fig. 1. Triple and quadruple junctions formed on an interface. There may exist triple junctions of metal–vacuum–dielectric (small blue circle) and quadruple junctions of metal–vacuum–dielectric–vacuum (red large circle) and metal– dielectric–vacuum–dielectric (red dotted circle). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
0 o y o y1 ðvacuumÞ,
y1 o y o yt ðdielectricÞ:
ð4Þ ð5Þ
The two magnitudes F1 and F2 are independent of the angle y and their behavior is essentially characterized by n only. Now, we use the boundary conditions which are the continuities of both F and the normal component of the electric displacement, Dy ¼(e/r)(qF/qy), at y ¼ y1. The two relations obtained from these boundary conditions are
e tan ny1 ¼ tan ny2
ð6Þ
A2 =A1 ¼ sin ny1 = sin ny2 Z
ð7Þ
Eq. (6) is the equation used to determine n. Z also represents the field ratio F2/F1 for r1 ¼ r2 by Eqs. (4), (5) and (7). The analytic solutions for n are easily found in the two limiting cases of e ¼1 and e-N. For e ¼1 (in the absence of dielectric), Eq. (6) is satisfied at
n ¼ n0 ¼ p=y1 ¼ p=ð2paÞ:
Fig. 2. Triple junctions made by Geis et al. [4] (a) for experiment and suggested by Chung et al.[14] (b) for calculation. Two are reduced to the same two-dimensional junction (c), where the coordinate origin is taken at the triple junction (d). The contact angles of metal, vacuum, and dielectric are given by a, y1, and y2, respectively. The cylinder is considered to have a radius RE1 mm and a length ‘ 1 mm.
structure has a relatively large longitudinal dimension in comparison with the facial size. Thus, we assume that junctions have two-dimensional symmetry. For a triple junction, we can consider the axis of a cylinder to be composed of three portions of metal, vacuum, and dielectric, as seen in Fig. 2. As far as the electric field is concerned in the vicinity of the junction, the cylinder plays the same role as the real triple junction configuration. It is supposed that the cylinder has a length of ‘ 1 mm and a radius of RE1 mm (R{‘). Now we use the polar coordinates (r, y). The two-dimensional portions of the metal, vacuum and dielectric are given by the angles a, y1 and y2, respectively. The electric potential F is given by the solution of the two-dimensional Laplace equation @ @F @2 F r þ 2 ¼0 r ð1Þ @r @r @y
ð8Þ
This is also obtained just for the contact between two composites, metal and vacuum [22]. Next we rewrite Eq. (6) in the form 1=e ¼ tan ðpny1 Þ= tan ny2 . In the limit e-N, the right-hand side is zero when the numerator is zero or when the denominator becomes infinity. Then we have two solutions, of which the lower one is ( 1= y1 =yt ¼ 1= 1y2 =yt , 0 o y2 =yt o1=3 n1 =n0 ¼ :ð9Þ 1=3 o y2 =yt o 1 0:5= y2 =yt , The curves of nN/n0 versus y2/yt are shown in Fig. 3. Each coordinate is normalized with n0 and yt. The reason is because those graphs are independent of a. We do not need to calculate n for all values of a but only for one value, say a ¼ p. It is seen that nN/n0 has a maximum of 1.5 at y2/yt ¼1/3 and a minimum of 0.5 at y2/yt ¼1. Here, we recall that the value of y2 ¼ yt does not mean the triple junction. Thus, the minimum of nN/n0 is reached
In general, the solution is written in the form F ¼ 1 P r nn ðAn sin nn y þ Bn cos nn yÞ, where r is the distance from the n¼1
junction, An and Bn are arbitrary constants, and nn is positive [22]. Since r is limited to the vicinity of the junction, i.e., r{R, only the lowest-power term is dominant. Then F is given by [22]
F1 ¼ A1 rn sin ny, 0 o y o y1 ðvacuumÞ,
ð2Þ
F2 ¼ A2 r n sin nðyt yÞ, y1 o y o y1 þ y2 ¼ yt ¼ 2pa ðdielectricÞ, ð3Þ
Fig. 3. Enhancement parameter nN for the triple junction. The inset shows the plot of n0versus a for the metal–vacuum junction (y2 ¼ 0). The dotted line represents n when the dielectric portion is metallic.
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not at the exact point of y2 ¼ yt but just before it. Anyway, nN has a minimum of 0.25 since n0 has a minimum of 0.5. Here, the inset represents n0 ( ¼ p/(2p a)) as a function of a for a configuration of two composites, metal and vacuum. The green line also represents n0 ( ¼ p/(2p a)) as a function of y2 for a configuration of metal, vacuum, and dielectric. Next, it is interesting to see the difference between a metal and a dielectric of e ¼N. The triple junction of metal, vacuum, and metal (in fact, the double junction of metal and vacuum) gives one curve (dotted line) whereas the triple junction of metal, vacuum, and dielectric of e-N gives two curves (solid line) including the curve of n o n0 for y2/yt 40.5. This implies that the limit e-N indicates the complete polarization of dielectric which does not mean free charges. The surface polarization charge of dielectric enhances or reduces the accumulation of free charges on metal near the junction according to the value of dielectric angle. In contrast, a set of two metals in contact plays like one metal and then has only surface free charges so that the electric field reduces only with increasing the metallic angle. For real dielectrics of 1 o e oN, we numerically solve Eq. (6) in order to obtain n(e), n in short, as a function of y2, a and e. We take e ¼ 5.7 (diamond), 10.4 (GaN), 100, and 1000. The obtained values of n are plotted in Fig. 4. The effects of y2 and e are clearly seen. In the first half of 0 o y2 o yt/2, we have n0 o n o nN. In the second half of yt/2o y2 o yt, we have nN o n o n0. On the other hand, the value of e is a factor to determine how large the field enhances or reduces. As e increases, n increases in the region of n 4 n0 and decreases in the region of n o n0. That is, the dielectric effect becomes more profound for e larger. The dotted medium line of e ¼ 1 represents n(¼ n0) when dielectric is replaced with vacuum. According to Fig. 4, we can discuss the e-dependence of the maximum and the minimum of n in more detail. As e increases from 1 to N, the maximum position y2 =yt Þ max moves from 1/4 to 1/3 whereas the minimum y2 =yt Þ min moves from 3/4 to 1.0. On the other hand, the maximum value n=n0 max moves from 1.0 to 1.5 whereas the minimum n=n0 min moves from 1.0 to 0.5. As mentioned above, there is an ambiguity in the value of n/n0 at y2 ¼ yt in the limit e-N. It is 0.5, whereas should be 1 in Fig. 4. We can resolve this double-value to go back to Eq. (6). The solution of n ¼ n0 is true at y2 ¼0, yt =2ð ¼ y1 Þ, and yt , regardless of e. We should have n ¼ n0 at y2 ¼ yt for all e.This means no change in
43
F when the entire field region is occupied with vacuum only, dielectric only, or is divided equally. It is worthwhile to note that the shape of the curve is asymmetric under even though the entire field region is equally divided for enhancement or reduction. At this point, we have to stress the two properties one more since they are important for the field behavior. Firstly, it depends on the ratio y2/yt (or the ratio y2/y1) whether F is enhanced or reduced due to dielectric. Simply, n is larger than n0 for y2 o yt/2 (i.e., y2 o y1) and is smaller for y2 4 yt/2 (i.e., y2 4 y1). This implies that F is enhanced (or reduced) for y2 larger (or smaller) than y1. At y2 ¼ y1, n is equal to n0, implying no enhancement of F at all. Secondly, the value of e is only the factor to determine the shape of n/n0 versus y2/yt. As e increases, it has a higher peak and a deeper valley, meaning a larger enhancement or reduction in the field [12–14].
3. Quadruple junctions 3.1. Metal–vacuum–dielectric–vacuum quadruple junctions For more features of the dielectric effect, we consider the metal–vacuum–dielectric–vacuum (MVDV) quadruple junction formed by introducing another vacuum portion between metal and dielectric in the triple junction (see Figs. 1, 2, and 5). The symbols a, y1, y2 and y3 indicate the angles subtended by the metal, vacuum, dielectric, and vacuum, respectively. In the same way as done for the triple junction, we make a two-dimensional treatment to find the solution, F, of the Laplace equation. In the region near the junction, F can be given in the forms [20,22]
F1 ¼ A1 rn sin ny, 0 r y r y1 ðvacuumÞ,
ð10Þ
F2 ¼ rn ðA2 sin ny þ B2 cos nyÞ, y1 r y r y1 þ y2 ðdielectricÞ,
ð11Þ
F3 ¼ A3 rn sin nð2payÞ, y1 þ y2 r y r y1 þ y2 þ y3 ¼ yt ðvacuumÞ, ð12Þ where the subscript refers to the region. In each region, the electric field is given in the form F 1 ¼ A1 nr n1 , 0 r y r y1 ðvacuumÞ,
ð13Þ
1=2 F 2 ¼ A2 2 þB2 2 nrn1 , y1 r y r y1 þ y2 ðdielectricÞ,
ð14Þ
F 3 ¼ A3 nr n1 , y1 þ y2 r y r yt ðvacuumÞ:
ð15Þ
Using the boundary conditions of both F and Dy at y ¼ y1 and y1 þ y2, we have four relations for five unknowns, n and four coefficients Ai(i¼1, 2, and 3) and B2. There are three expressions for three coefficients in terms of the rest, say A2. The fourth is the
Fig. 4. Enhancement parameter n for 1 o e oN. We take e ¼ 5.7 (diamond) and 10.4 (GaN), 100, 1000. As e increases, n has higher peaks and lower valleys which move toward the larger y2.
Fig. 5. Cylindrical quadruple junctions. An additional portion, vacuum (a) or dielectric (b), is added to the triple junction. The symbol y3 always indicates the angle subtended by the additional portion with respect to the triple junction seen in Fig. 2.
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transcendental equation for n [20]: tan n y1 þ y2 þ e tan ny3 ð1eÞ tan ny1 ¼ 1e tan nðy1 þ y2 Þ tan ny3 1 þ e tan 2 ny1
ð16Þ
It seems that y1 and y3 are not exchangeable in Eq. (16). However, the exchange of y1 and y3 does not give rise to any difference in the obtained n. As expected, Eq. (16) for a quadruple junction reduces to Eq. (6) for a triple junction when y3 ¼0. We numerically solved Eq. (16) to obtain n as a function of y2 for several values of y3. The obtained results are shown in Fig. 6 for y3 ¼0 (a triple junction), 301, 601, 901, 1201, 1501 (asymmetric), and y1(symmetric). The two curves of y3 ¼301 and 1201, or 601 and 1201 are complementary but not symmetric. The reason is that y3 is fixed whereas y1 varies with y2. As seen in Fig. 6, we have two interesting properties for the MVDV junction. Firstly, all values of n are lower than those for the triple junction. This implies that the MVDV quadruple junction can yield higher enhancement of the electric field than the triple junction. Secondly, n for the symmetric quadruple junction is the entire loci of valleys for asymmetric junctions. This means that the symmetric MVDV junction always yields the lowest values of n accessible at any given y2 and the largest enhancement of F. It is also interesting that n for the symmetric MVDV junction has the minimum nmin at y2 ¼ yt/2, irrespective of values of e and a. Even though its location is fixed, the minimum value varies with a and e. At a ¼1801, we have nmin ¼0.5049 and.3826 for e ¼5.7 and 10.4, respectively. At e ¼5.7 and a ¼901, we have nmin ¼0.3366. For a large e and small a, nmin becomes small. It is recalled that as e increases for the triple junction, the minimum location moves to the larger y2 while its value decreases. 3.2. Metal–dielectric–vacuum–dielectric quadruple junctions Alternatively, we consider the metal–dielectric–vacuum–dielectric (MDVD) quadruple junction by inserting an additional dielectric portion between metal and vacuum in the triple junction (see Figs. 1, 2, and 5). This structure may appear if a void exists on the side of the dielectric at the triple junction. Symbols a, y3, y1, and y2 represent the angles subtended by metal, dielectric, vacuum, and dielectric, respectively. Numbering is made so that the quadruple junction of y3 ¼0 always reduces to the same triple junction as
Fig. 6. Enhancement parameter n for the metal–vacuum–dielectric–vacuum quadruple junction. We take y3 ¼ 0(triple junction), 301, 601, 901, 1201, 1501 (asymmetric), and y1 (symmetric) for e ¼ 5.7. The lowest curve is for y3 ¼ y1 and e ¼ 10.4.
treated before. The electric potential Fis obtained by making similar arguments as done above. Then we write F3 ¼ A3 r n sin ny for 0r y r y3, F1 ¼ r n ðA1 sin ny þB1 cos nyÞ for y3 r y r y3 þ y1, and F2 ¼ A2 rn sin nð2payÞ for y3 þ y1 r y r y3 þ y1 þ y2, where the subscripts 3, 1, and 2 refer to the regions. The use of the boundary conditions at y ¼ y3 and y3 þ y1 yields the associated transcendental equation ð1eÞ tan ny3 e tan nðy3 þ y1 Þ þ tan ny2 ¼ e þ tan nðy3 þ y1 Þ tan ny2 e þ tan 2 ny3
ð17Þ
As expected, Eqs. (16) and (17) are similar in form. The difference comes from composites and numbering. Numerical calculations of Eq. (17) were carried out to obtain n as a function of y1 for several values of y3. As seen in Fig. 7, we have two features for the MDVD junction. First, all values of n are higher than those for the triple junction. This implies that the field can be more largely reduced for the MDVD junction than for the triple junction. Second, n for the quadruple junction of y3 ¼ y2 is the entire loci of peaks for asymmetric junctions of given y3. This means that the symmetric MDVD junction always yields the highest n at given y2. It is also seen that n for the symmetric MDVD junction has the maximum at y2 ¼ yt/2, irrespective of e. 4. Electric fields and enhancements 4.1. Electric fields for the triple junction It is supposed that a free negative charge Q is distributed on the surface of the metal (see Fig. 2). The surface charge density s is given by the normal component of the electric displacement: R s ¼Dy at y ¼0 and y1 þ y2. Using Q¼ sda along with Eqs. (4) n and (5), we derive Q ¼ R ‘ 1 þ Ze A1 . The field energy W R stored in vacuum and dielectric is calculated: W¼ dt F2/8p ¼ 2 nR2n ‘ y1 þ Z2 ey2 A1 =16p. The potential is then given by [12,15] V ¼ 2W=Q ¼
n n y1 þ Z2 ey2 R A1 8p 1 þ Ze
ð18Þ
V is regarded as the electric potential imposed on the metal with free charge Q. Combining Eqs. (4), (5), (7), and (18), we have 1n 1 þ Ze R V , ð19Þ F 1 ðrÞ ¼ 8p 2 r R y1 þ Z ey2
Fig. 7. Enhancement parameter n for the metal–dielectric–vacuum–dielectric quadruple junction. We take y3 ¼0(triple junction), 301, 601, 901, 1201, 1501, and y1 (symmetric).
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F 2 ðrÞ ¼ ZF 1 ðrÞ
45
ð20Þ
Now we define the reference field F0 as F1 when dielectric is replaced with vacuum. At e ¼1, where y2 ¼ 0, y1 ¼ yt, Z ¼1 and n ¼ n0, Eq. (19) becomes F 0 ðrÞ ¼
16p R 1n0 V R yt r
ð21Þ
The field enhancement b is defined as the ratio between the two fields, F1 and F0: ! 1 þ Ze R n0 n bðrÞ ¼ F 1 ðrÞ=F 0 ðrÞ ¼ 0:5 ð22Þ 2 r 1 þ Z e1 y2 =yt It is clear that Eq. (22) yields b ¼1 at e ¼1. For e 41, b is smaller (or larger) than unity for y2 o y1 (or y2 4 y1). The prefactor in Eq. (22) varies approximately from unity to 5.5 over the entire range of y2 but does not have much difference at b ¼ bmax and bmin for different e. To make numerical calculations of b, we choose R¼1 mm and r¼0.1 nm (see Fig. 8). For e ¼5.7 (dotted line), we have the minimal enhancement bmin 0.17 at y2/yt E0.29 and the maximal enhancement bmax 19.8 at y2/yt E0.91. For e ¼10.4 (solid line), we have bmin 0.12 at y2/yt E0.30 and the maximum bmax 47.1 at y2/yt E0.93. This is the enhancement b of the vacuum field F1 (blue line). The dielectric field F2 has the enhancement Zb (red line), which is smaller than b. As seen in Figs. 4 and 8, nmin and bmax do not both take place at the same y2/ yt because of the prefactor of Eq. (22). The field enhancement factor b has its maximum at a slightly larger y2 than the power index n. It is noted that all the associated extrema nmin , bmax , F 1,max and F 2,max lie in the region 0.75o y2/yt o1.0, where y1 þ y2 ¼ yt. Fig. 9 shows the e-dependence of bmax . It appears that bmax increases almost linearly with e. The slope depends on a and r. We take the values of r¼ 0.11 and 1.0 nm for R¼1 mm. As e increases from 1 to 100 (not shown up to this value), bmax increases from 1.0 to 122.5 for a ¼01 and r¼1.0 nm (red dashed line), from 1.0 to 153.2 for a ¼01 and r ¼0.1 nm (red solid line), and from 1.0 to 669.4 for a ¼ 1801 and r ¼0.1 nm (blue solid line). The plots of a ¼901 and r ¼1.0 nm (green dashed line) coincides with that of
Fig. 9. Plot for bmax versus e. The maximal enhancement bmax is shown as a function of e for r ¼0.1 (solid line) and 1.0 nm (dashed line) and a ¼ 01, 901, and 1801. The plot for a ¼901 and r¼ 1.0 nm (green dashed line) is overlapped with that for a ¼ 01 and r¼ 0.1 nm (red solid line). It is seen that bmax increases almost linearly with increasing e. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 10. Shape dependence of the field and enhancement. As a varies from 01 to 1801, F0 decreases rapidly while b increases slowly. Thus both F1 (blue line) and F2 (red line) are shown to follow the trend of F0. We take e ¼ 5.7 (dashed line) and 10.4 (solid line). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
a ¼01 and r ¼0.1 nm (red solid line), and is not seen. The field enhancement bmax becomes very large for both a and e large. Fig. 10 shows the a-dependence of F0 and bmax . The a-dependence reflects the curvature of the metallic shape. From Eq. (21), F0 has the exponent 1 n0 ¼(p a)/(2p a), which decreases with increasing a. Then F0 decreases with increasing a. By Eq. (22), b has the exponent n0 n ¼ n0(1 n/n0), where n0 ¼ p/(2p a) and n/n0 has no a-dependence. Then b increases with increasing a. The two have opposite a-dependence. This Fig. 8. Field enhancement b of vacuum and dielectric fields for the triple junction. We take e ¼5.7 (dashed line) and 10.4 (solid line). The enhancement, b, for the vacuum field F1 (blue line) is larger than that for the dielectric field F2 (red line). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
leads to a competition between b and F0 in determining F1 (and F2) when a varies. To figure out the curvature-dependence of both enhancements, we did numerical calculations of F0,F 1,max and F 2,max as functions of a. The numerical values of F are calculated
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by setting 16pV/R¼4.0 and are given in an arbitrary unit in Fig. 10. Here, we are just interested in relative magnitudes. As a increases from 01 to 1801, F0(r) decreases from 63.7 to 1.27 (50 times smaller), irrespective of e. Instead, bmax increases from 7.77 to 19.8 (2.5 times larger) for e ¼5.7 and from 15.0 to 47.1 (3.1 times larger) for e ¼10.4. This implies that F0 changes more rapidly with a than b or Zb. As a result, both F 1,max and F 2,max are shown to decrease with increasing a. However, the dielectric enhancement is still significantly large over all a. Even though it is considerable, the dielectric effect is not just apparently visible in Fig. 10 because it changes slowly in comparison with the shape effect.
4.2. Electric fields for the quadruple junction As discussed in Section 3, the MVDV junction is effective in enhancing the field whereas the MDVD is effective in reducing the field. In the current work, we consider the MVDV junction. We apply the same calculation scheme as used for the triple junction to obtain ! Z1 Z1 þ Z2 yt R n0 n bðrÞ ¼ F 1 ðrÞ=F 0 ðrÞ ¼ 0:5 , 2 2 2 r Z1 y1 þ e 1 þ Z2 yt þ Z3 y3 ð23Þ where Z1 ¼A1/A2, Z2 ¼ B2 =A2 , and Z3 ¼A3/A2. All Zi(i¼ 1, 2, and 3) values are obtained using the boundary condition [20]. The two b(r) values given by Eqs. (22) and (23) are the same in form but are different in configuration dependence. For the symmetric quadruple junction of y3 ¼ y1, we use Eq. (23) to obtain b(r) as a function of y2/yt for e ¼5.7. The results are shown in Fig. 11. We consider r ¼0.1 (solid line) and 1.0 nm (dashed line). We also plot b(r) for the triple junction for comparison. We have the maximal enhancement bmax ¼21.8 and 66.1 for r ¼1.0 and 0.1 nm, respectively. The corresponding values are 12.0 and 19.8 for the triple junction. The bmax for the symmetric quadruple junction are 2 or 3 times larger than those for the triple junction. The dielectric effect is more effective for the quadruple junction.
5. Discussions The difference between the enhancements for the triple and quadruple junctions supports the conclusion that the electric field is enhanced or reduced by the polarization of dielectric. In the absence of dielectric, the surface charge is distributed so that the potential is constant over the metallic surface, resulting in the a-dependence of F0. If a dielectric is placed in contact with the metal, then the charge on the metal causes the dielectric to polarize. In turn, the free surface charge on the metal is also redistributed under the influence of polarization charges. The magnitude and direction of polarization depends on the contact angles of the constituents. If polarization charges attract free surface charges to the junction, then more free charges are accumulated near the junction. The field becomes stronger. This stronger field induces a larger polarization. This process goes on until a perfect polarization of the dielectric is achieved. As a result, a very strong field is formed. Such strong fields lead to enhanced field emission or dielectric breakdown [1–3,21]. The current results surely give rise to a newly supposed mechanism for field electron emission even though it is not confirmed yet experimentally. It is likely that the enhanced F is strong enough to produce space charges in dielectric near the junction. The large density of space charges is responsible for lowering and thinning the contact barrier between the metal and dielectric. In general, the dielectric has a very low vacuum barrier. Thus, the field enhancement at metal–dielectric contact may create a detour path of metal–dielectric–vacuum. This possibly leads to enhanced field emission near the metal–dielectric contact. It seems that such a detour path is more preferable in potential energy than a direct path from metal to vacuum. This implies that the triple or quadruple junction can be another type of field electron source.
6. Conclusions We described the electric field in the vicinity of a junction where metal, vacuum, and dielectric meet. The assumption of a two-dimensional symmetry of the junction leaded to formulation of the electric field. For the triple junction, the electric field was found to be enhanced or reduced according to the contact angle ratio as well as the dielectric constant. It was found that a certain type of quadruple junction yielded a larger field enhancement or reduction than the triple junction. It was noted that the enhancement of the electric field at the contact between the metal and the dielectric is the product of the two enhancements due to dielectric and shape of the metallic portion. This field enhancement may be large enough to explain the triple junction phenomenon such as the vacuum or dielectric breakdown. Understanding of the reduction mechanism is applicable to avoid undesirable breakdown.
Acknowledgment This research was supported by the Basic Science Research Program through National Research Foundation of Korea funded by the MEST (Grant no. 2011-0009500). This research was also supported by Mid-carrier Research Program through the NRF grant funded by the MEST (Grant no. R01-2008-000–20025-0). Fig. 11. Field enhancement for the symmetric quadruple junction. The symmetric metal–vacuum–dielectric–vacuum quadruple junction (blue line) yields a larger enhancement than the triple junction (red line). We take r¼ 0.1 (solid line) and 1.0 nm (dashed line). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
References [1] A. Watson, Journal of Applied Physics 38 (1967) 2019. [2] G.N. Fursei, P.N. Vorontsov-Vel’yaminov, Soviet Physics: Technical Physics 12 (1968) 1370.
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[3] R.V. Latham, Vacuum 32 (1982) 137. [4] M.W. Geis, N.N. Efremow Jr., K.E. Krohn, J.C. Twichell, T.M. Lyszczarz, R. Kalish, J.A. Greer, M.D. Taber, Lincoln Laboratory Journal 10 (1997) 3. [5] (14N) M. Jordan, Y.Y. Lau, D.M. French, R.M. Gilgenbach, P. Pengvanich, Journal of Applied Physics 102 (2007) 03301. [6] M.W. Geis, S. Deneault, K.E. Krohn, M. Marchant, T.M. Lyszczarz, D.L. Crooke, Applied Physics Letters 87 (2005) 192115. [7] X. Ma, T.S. Sudarshan, Journal of Vacuum Science & Technology B 19 (2001) 683. [8] X. Ma, T.S. Sudarshan, Journal of Vacuum Science & Technology B 16 (1998) 745. [9] T. Takuma, IEEE Transaction on Electric Insulation 26 (1991) 500. [10] O. Yamamoto, T. Takuma, Electrical Engineering Japan 131 (2000) 1. [11] B. Techaumnat, S. Hamada, T. Takuma, Journal of Electrostatics 56 (2002) 67. [12] M.S. Chung, B.-G. Yoon, P.H. Cutler, N.M. Miskovsky, Journal of Vacuum Science & Technology B 22 (2004) 1240. [13] M.S. Chung, T.S. Choi, B.-G. Yoon, Applied Surface Science 251 (2005) 177.
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[14] M.S. Chung, S.C. Hong, P.H. Cutler, N.M. Miskovsky, B.L. Weiss, A. Mayer, Journal of Vacuum Science & Technology B 24 (2006) 909. ¨ [15] L. Schachter, Applied Physics Letters 72 (1998) 421. [16] Z. Wang, B. Zhang, Q. Fu, G. Xie, Z. Li, IEEE Electron Device Letters 33 (2012) 703. ¨ [17] O.C. Hellman, J. Rusing, J.T. Sebastian, D.N. Seidman, Materials Science and Engineering: C 15 (2001). [18] W. Czarczynski, Z. Znamirowski, Surface & Coatings Technology 201 (2008) 4422. [19] P.H. Cutler, N.M. Miskovsky, P.B. Lerner, M.S. Chung, Applied Surface Science 146 (1999) 126. [20] M.S. Chung, B.-G. Yoon, P.H. Cutler, N.M. Miskovsky, B.L. Weiss, A. Mayer, Journal of Vacuum Science & Technology B 24 (2010) C2A94. [21] A.A. Emel’yanov, Technical Physics 48 (2003) 1192. [22] J.D. Jackson, J.D. Jackson, Classical Electrodynamics, 3rd ed., John Wiley & Sons, New York, 1999.
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Ultramicroscopy journal homepage: www.elsevier.com/locate/ultramic
Dielectric effect on electric fields in the vicinity of the metal–vacuum–dielectric junction M.S. Chung a,*, A. Mayer b, N.M. Miskovsky c, B.L. Weiss c, P.H. Cutler c a
Department of Physics, University of Ulsan, San 29, Muga, Ulsan 680-749, Korea FUNDP, University of Namur, Rue de Bruxelles 61, B-5000 Namur, Belgium c Department of Physics, The Pennsylvania State University, University Park, PA 16802, USA b
a r t i c l e i n f o
abstract
Available online 20 December 2012
The dielectric effect was theoretically investigated in order to describe the electric field in the vicinity of a junction of a metal, dielectric, and vacuum. The assumption of two-dimensional symmetry of the junction leads to a simple analytic form and to a systematic numerical calculation for the field. The electric field obtained for the triple junction was found to be enhanced or reduced according to a certain criterion determined by the contact angles and dielectric constant. Further numerical calculations of the dielectric effect show that an electric field can experience a larger enhancement or reduction for a quadruple junction than that achieved for the triple junction. It was also found that even though it changes slowly in comparison with the shape effect, the dielectric effect was noticeably large over the entire range of the shape change. & 2012 Elsevier B.V. All rights reserved.
Keywords: Triple junction Field enhancement Dielectric effect Metal–dielectric contact
1. Introduction Dielectrics are usually considered to reduce electric field intensities. For a long time, however, strongly enhanced electric fields have often been observed as phenomena of an electron emission avalanche or a breakdown in the vicinity of the metal– dielectric contact [1–5]. Such an unexpected field enhancement, called the triple junction effect, must be due to a dielectric in contact with metal (see Fig. 1). Two significant experiments that revealed the dielectric effect were made by Geis’s group [4,6]. In 1977, they reported that the field emission was strangely enhanced when the diamond portion of a triple junction was extended into the vacuum. Several years later, they observed that the field emission continued even after the bias was removed. Ma and Sudarshan [7,8] found that voids at the contact were crucial in vacuum insulation breakdown. The contact angle of the junction was considered to play a role in the enhancement ¨ mechanism [9–14]. Schachter [15] first used a two-dimensional model to obtain the theoretical result that the field emission current density was proportional to the dielectric constant for a specific triple junction. The field enhancement was also observed at the interface of the silicon and dielectric layer [16]. In addition, Hellman et al. [17] argued that an artifact of the field evaporation process results from a change in the local electric field due to the presence of dielectric in the atom-by-atom chemistry.
*
Corresponding author. Tel.: þ82 52 259 2329; fax: þ 82 52 259 1693. E-mail address: [email protected] (M.S. Chung).
0304-3991/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ultramic.2012.12.014
Field enhancement at the triple junction is different from that produced by a protrusion or a sharpening of the metallic edge [18,19]. The electric field is enhanced or reduced according to the angular configuration of the triple junction [12–14]. The value of the dielectric constant is also a factor in determining the magnitude of enhancement. This implies that both the magnitude and direction of polarization are involved in determining the enhancement of electric field. Thus, it is interesting to find which configuration yields the largest dielectric enhancement with consideration to the triple junction as a basis. One of the most probable candidates is a quadruple junction, even though it may not appear as often as a triple junction (see Fig. 1). According to previous work [20], a certain type of quadruple junction yields a larger dielectric effect than the triple junction. Even though it is not experimentally confirmed yet, the use of the triple or quadruple junction may lead to a new type of cold electron source. In addition, a deeper understanding of the dielectric effect may be used to avoid an avalanche of electron emission or breakdown [3,21]. Thus we need to make a systematic approach to describe the field behavior in regions near the contact between metal and dielectric. Besides, the current works deals with the reduction of electric field as well as enhancement.
2. Metal–vacuum–dielectric triple junctions As seen in Fig. 1, the metal-layer contact has several configurations of junction at which the metal, vacuum, and dielectric meet [6]. Since it is a layer-by-layer contact, the junction
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where the subscripts 1 and 2 refer to the vacuum and dielectric regions, respectively and the subscript t refers to the total angle for the field-varying region. Here, we put n ¼ n1 for convenience. From ! Eqs. (2) and (3), the electric field magnitude F ¼ r F is given by F 1 ¼ A1 nr n1 , F 2 ¼ A2 n r n1 ,
Fig. 1. Triple and quadruple junctions formed on an interface. There may exist triple junctions of metal–vacuum–dielectric (small blue circle) and quadruple junctions of metal–vacuum–dielectric–vacuum (red large circle) and metal– dielectric–vacuum–dielectric (red dotted circle). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
0 o y o y1 ðvacuumÞ,
y1 o y o yt ðdielectricÞ:
ð4Þ ð5Þ
The two magnitudes F1 and F2 are independent of the angle y and their behavior is essentially characterized by n only. Now, we use the boundary conditions which are the continuities of both F and the normal component of the electric displacement, Dy ¼(e/r)(qF/qy), at y ¼ y1. The two relations obtained from these boundary conditions are
e tan ny1 ¼ tan ny2
ð6Þ
A2 =A1 ¼ sin ny1 = sin ny2 Z
ð7Þ
Eq. (6) is the equation used to determine n. Z also represents the field ratio F2/F1 for r1 ¼ r2 by Eqs. (4), (5) and (7). The analytic solutions for n are easily found in the two limiting cases of e ¼1 and e-N. For e ¼1 (in the absence of dielectric), Eq. (6) is satisfied at
n ¼ n0 ¼ p=y1 ¼ p=ð2paÞ:
Fig. 2. Triple junctions made by Geis et al. [4] (a) for experiment and suggested by Chung et al.[14] (b) for calculation. Two are reduced to the same two-dimensional junction (c), where the coordinate origin is taken at the triple junction (d). The contact angles of metal, vacuum, and dielectric are given by a, y1, and y2, respectively. The cylinder is considered to have a radius RE1 mm and a length ‘ 1 mm.
structure has a relatively large longitudinal dimension in comparison with the facial size. Thus, we assume that junctions have two-dimensional symmetry. For a triple junction, we can consider the axis of a cylinder to be composed of three portions of metal, vacuum, and dielectric, as seen in Fig. 2. As far as the electric field is concerned in the vicinity of the junction, the cylinder plays the same role as the real triple junction configuration. It is supposed that the cylinder has a length of ‘ 1 mm and a radius of RE1 mm (R{‘). Now we use the polar coordinates (r, y). The two-dimensional portions of the metal, vacuum and dielectric are given by the angles a, y1 and y2, respectively. The electric potential F is given by the solution of the two-dimensional Laplace equation @ @F @2 F r þ 2 ¼0 r ð1Þ @r @r @y
ð8Þ
This is also obtained just for the contact between two composites, metal and vacuum [22]. Next we rewrite Eq. (6) in the form 1=e ¼ tan ðpny1 Þ= tan ny2 . In the limit e-N, the right-hand side is zero when the numerator is zero or when the denominator becomes infinity. Then we have two solutions, of which the lower one is ( 1= y1 =yt ¼ 1= 1y2 =yt , 0 o y2 =yt o1=3 n1 =n0 ¼ :ð9Þ 1=3 o y2 =yt o 1 0:5= y2 =yt , The curves of nN/n0 versus y2/yt are shown in Fig. 3. Each coordinate is normalized with n0 and yt. The reason is because those graphs are independent of a. We do not need to calculate n for all values of a but only for one value, say a ¼ p. It is seen that nN/n0 has a maximum of 1.5 at y2/yt ¼1/3 and a minimum of 0.5 at y2/yt ¼1. Here, we recall that the value of y2 ¼ yt does not mean the triple junction. Thus, the minimum of nN/n0 is reached
In general, the solution is written in the form F ¼ 1 P r nn ðAn sin nn y þ Bn cos nn yÞ, where r is the distance from the n¼1
junction, An and Bn are arbitrary constants, and nn is positive [22]. Since r is limited to the vicinity of the junction, i.e., r{R, only the lowest-power term is dominant. Then F is given by [22]
F1 ¼ A1 rn sin ny, 0 o y o y1 ðvacuumÞ,
ð2Þ
F2 ¼ A2 r n sin nðyt yÞ, y1 o y o y1 þ y2 ¼ yt ¼ 2pa ðdielectricÞ, ð3Þ
Fig. 3. Enhancement parameter nN for the triple junction. The inset shows the plot of n0versus a for the metal–vacuum junction (y2 ¼ 0). The dotted line represents n when the dielectric portion is metallic.
M.S. Chung et al. / Ultramicroscopy 132 (2013) 41–47
not at the exact point of y2 ¼ yt but just before it. Anyway, nN has a minimum of 0.25 since n0 has a minimum of 0.5. Here, the inset represents n0 ( ¼ p/(2p a)) as a function of a for a configuration of two composites, metal and vacuum. The green line also represents n0 ( ¼ p/(2p a)) as a function of y2 for a configuration of metal, vacuum, and dielectric. Next, it is interesting to see the difference between a metal and a dielectric of e ¼N. The triple junction of metal, vacuum, and metal (in fact, the double junction of metal and vacuum) gives one curve (dotted line) whereas the triple junction of metal, vacuum, and dielectric of e-N gives two curves (solid line) including the curve of n o n0 for y2/yt 40.5. This implies that the limit e-N indicates the complete polarization of dielectric which does not mean free charges. The surface polarization charge of dielectric enhances or reduces the accumulation of free charges on metal near the junction according to the value of dielectric angle. In contrast, a set of two metals in contact plays like one metal and then has only surface free charges so that the electric field reduces only with increasing the metallic angle. For real dielectrics of 1 o e oN, we numerically solve Eq. (6) in order to obtain n(e), n in short, as a function of y2, a and e. We take e ¼ 5.7 (diamond), 10.4 (GaN), 100, and 1000. The obtained values of n are plotted in Fig. 4. The effects of y2 and e are clearly seen. In the first half of 0 o y2 o yt/2, we have n0 o n o nN. In the second half of yt/2o y2 o yt, we have nN o n o n0. On the other hand, the value of e is a factor to determine how large the field enhances or reduces. As e increases, n increases in the region of n 4 n0 and decreases in the region of n o n0. That is, the dielectric effect becomes more profound for e larger. The dotted medium line of e ¼ 1 represents n(¼ n0) when dielectric is replaced with vacuum. According to Fig. 4, we can discuss the e-dependence of the maximum and the minimum of n in more detail. As e increases from 1 to N, the maximum position y2 =yt Þ max moves from 1/4 to 1/3 whereas the minimum y2 =yt Þ min moves from 3/4 to 1.0. On the other hand, the maximum value n=n0 max moves from 1.0 to 1.5 whereas the minimum n=n0 min moves from 1.0 to 0.5. As mentioned above, there is an ambiguity in the value of n/n0 at y2 ¼ yt in the limit e-N. It is 0.5, whereas should be 1 in Fig. 4. We can resolve this double-value to go back to Eq. (6). The solution of n ¼ n0 is true at y2 ¼0, yt =2ð ¼ y1 Þ, and yt , regardless of e. We should have n ¼ n0 at y2 ¼ yt for all e.This means no change in
43
F when the entire field region is occupied with vacuum only, dielectric only, or is divided equally. It is worthwhile to note that the shape of the curve is asymmetric under even though the entire field region is equally divided for enhancement or reduction. At this point, we have to stress the two properties one more since they are important for the field behavior. Firstly, it depends on the ratio y2/yt (or the ratio y2/y1) whether F is enhanced or reduced due to dielectric. Simply, n is larger than n0 for y2 o yt/2 (i.e., y2 o y1) and is smaller for y2 4 yt/2 (i.e., y2 4 y1). This implies that F is enhanced (or reduced) for y2 larger (or smaller) than y1. At y2 ¼ y1, n is equal to n0, implying no enhancement of F at all. Secondly, the value of e is only the factor to determine the shape of n/n0 versus y2/yt. As e increases, it has a higher peak and a deeper valley, meaning a larger enhancement or reduction in the field [12–14].
3. Quadruple junctions 3.1. Metal–vacuum–dielectric–vacuum quadruple junctions For more features of the dielectric effect, we consider the metal–vacuum–dielectric–vacuum (MVDV) quadruple junction formed by introducing another vacuum portion between metal and dielectric in the triple junction (see Figs. 1, 2, and 5). The symbols a, y1, y2 and y3 indicate the angles subtended by the metal, vacuum, dielectric, and vacuum, respectively. In the same way as done for the triple junction, we make a two-dimensional treatment to find the solution, F, of the Laplace equation. In the region near the junction, F can be given in the forms [20,22]
F1 ¼ A1 rn sin ny, 0 r y r y1 ðvacuumÞ,
ð10Þ
F2 ¼ rn ðA2 sin ny þ B2 cos nyÞ, y1 r y r y1 þ y2 ðdielectricÞ,
ð11Þ
F3 ¼ A3 rn sin nð2payÞ, y1 þ y2 r y r y1 þ y2 þ y3 ¼ yt ðvacuumÞ, ð12Þ where the subscript refers to the region. In each region, the electric field is given in the form F 1 ¼ A1 nr n1 , 0 r y r y1 ðvacuumÞ,
ð13Þ
1=2 F 2 ¼ A2 2 þB2 2 nrn1 , y1 r y r y1 þ y2 ðdielectricÞ,
ð14Þ
F 3 ¼ A3 nr n1 , y1 þ y2 r y r yt ðvacuumÞ:
ð15Þ
Using the boundary conditions of both F and Dy at y ¼ y1 and y1 þ y2, we have four relations for five unknowns, n and four coefficients Ai(i¼1, 2, and 3) and B2. There are three expressions for three coefficients in terms of the rest, say A2. The fourth is the
Fig. 4. Enhancement parameter n for 1 o e oN. We take e ¼ 5.7 (diamond) and 10.4 (GaN), 100, 1000. As e increases, n has higher peaks and lower valleys which move toward the larger y2.
Fig. 5. Cylindrical quadruple junctions. An additional portion, vacuum (a) or dielectric (b), is added to the triple junction. The symbol y3 always indicates the angle subtended by the additional portion with respect to the triple junction seen in Fig. 2.
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transcendental equation for n [20]: tan n y1 þ y2 þ e tan ny3 ð1eÞ tan ny1 ¼ 1e tan nðy1 þ y2 Þ tan ny3 1 þ e tan 2 ny1
ð16Þ
It seems that y1 and y3 are not exchangeable in Eq. (16). However, the exchange of y1 and y3 does not give rise to any difference in the obtained n. As expected, Eq. (16) for a quadruple junction reduces to Eq. (6) for a triple junction when y3 ¼0. We numerically solved Eq. (16) to obtain n as a function of y2 for several values of y3. The obtained results are shown in Fig. 6 for y3 ¼0 (a triple junction), 301, 601, 901, 1201, 1501 (asymmetric), and y1(symmetric). The two curves of y3 ¼301 and 1201, or 601 and 1201 are complementary but not symmetric. The reason is that y3 is fixed whereas y1 varies with y2. As seen in Fig. 6, we have two interesting properties for the MVDV junction. Firstly, all values of n are lower than those for the triple junction. This implies that the MVDV quadruple junction can yield higher enhancement of the electric field than the triple junction. Secondly, n for the symmetric quadruple junction is the entire loci of valleys for asymmetric junctions. This means that the symmetric MVDV junction always yields the lowest values of n accessible at any given y2 and the largest enhancement of F. It is also interesting that n for the symmetric MVDV junction has the minimum nmin at y2 ¼ yt/2, irrespective of values of e and a. Even though its location is fixed, the minimum value varies with a and e. At a ¼1801, we have nmin ¼0.5049 and.3826 for e ¼5.7 and 10.4, respectively. At e ¼5.7 and a ¼901, we have nmin ¼0.3366. For a large e and small a, nmin becomes small. It is recalled that as e increases for the triple junction, the minimum location moves to the larger y2 while its value decreases. 3.2. Metal–dielectric–vacuum–dielectric quadruple junctions Alternatively, we consider the metal–dielectric–vacuum–dielectric (MDVD) quadruple junction by inserting an additional dielectric portion between metal and vacuum in the triple junction (see Figs. 1, 2, and 5). This structure may appear if a void exists on the side of the dielectric at the triple junction. Symbols a, y3, y1, and y2 represent the angles subtended by metal, dielectric, vacuum, and dielectric, respectively. Numbering is made so that the quadruple junction of y3 ¼0 always reduces to the same triple junction as
Fig. 6. Enhancement parameter n for the metal–vacuum–dielectric–vacuum quadruple junction. We take y3 ¼ 0(triple junction), 301, 601, 901, 1201, 1501 (asymmetric), and y1 (symmetric) for e ¼ 5.7. The lowest curve is for y3 ¼ y1 and e ¼ 10.4.
treated before. The electric potential Fis obtained by making similar arguments as done above. Then we write F3 ¼ A3 r n sin ny for 0r y r y3, F1 ¼ r n ðA1 sin ny þB1 cos nyÞ for y3 r y r y3 þ y1, and F2 ¼ A2 rn sin nð2payÞ for y3 þ y1 r y r y3 þ y1 þ y2, where the subscripts 3, 1, and 2 refer to the regions. The use of the boundary conditions at y ¼ y3 and y3 þ y1 yields the associated transcendental equation ð1eÞ tan ny3 e tan nðy3 þ y1 Þ þ tan ny2 ¼ e þ tan nðy3 þ y1 Þ tan ny2 e þ tan 2 ny3
ð17Þ
As expected, Eqs. (16) and (17) are similar in form. The difference comes from composites and numbering. Numerical calculations of Eq. (17) were carried out to obtain n as a function of y1 for several values of y3. As seen in Fig. 7, we have two features for the MDVD junction. First, all values of n are higher than those for the triple junction. This implies that the field can be more largely reduced for the MDVD junction than for the triple junction. Second, n for the quadruple junction of y3 ¼ y2 is the entire loci of peaks for asymmetric junctions of given y3. This means that the symmetric MDVD junction always yields the highest n at given y2. It is also seen that n for the symmetric MDVD junction has the maximum at y2 ¼ yt/2, irrespective of e. 4. Electric fields and enhancements 4.1. Electric fields for the triple junction It is supposed that a free negative charge Q is distributed on the surface of the metal (see Fig. 2). The surface charge density s is given by the normal component of the electric displacement: R s ¼Dy at y ¼0 and y1 þ y2. Using Q¼ sda along with Eqs. (4) n and (5), we derive Q ¼ R ‘ 1 þ Ze A1 . The field energy W R stored in vacuum and dielectric is calculated: W¼ dt F2/8p ¼ 2 nR2n ‘ y1 þ Z2 ey2 A1 =16p. The potential is then given by [12,15] V ¼ 2W=Q ¼
n n y1 þ Z2 ey2 R A1 8p 1 þ Ze
ð18Þ
V is regarded as the electric potential imposed on the metal with free charge Q. Combining Eqs. (4), (5), (7), and (18), we have 1n 1 þ Ze R V , ð19Þ F 1 ðrÞ ¼ 8p 2 r R y1 þ Z ey2
Fig. 7. Enhancement parameter n for the metal–dielectric–vacuum–dielectric quadruple junction. We take y3 ¼0(triple junction), 301, 601, 901, 1201, 1501, and y1 (symmetric).
M.S. Chung et al. / Ultramicroscopy 132 (2013) 41–47
F 2 ðrÞ ¼ ZF 1 ðrÞ
45
ð20Þ
Now we define the reference field F0 as F1 when dielectric is replaced with vacuum. At e ¼1, where y2 ¼ 0, y1 ¼ yt, Z ¼1 and n ¼ n0, Eq. (19) becomes F 0 ðrÞ ¼
16p R 1n0 V R yt r
ð21Þ
The field enhancement b is defined as the ratio between the two fields, F1 and F0: ! 1 þ Ze R n0 n bðrÞ ¼ F 1 ðrÞ=F 0 ðrÞ ¼ 0:5 ð22Þ 2 r 1 þ Z e1 y2 =yt It is clear that Eq. (22) yields b ¼1 at e ¼1. For e 41, b is smaller (or larger) than unity for y2 o y1 (or y2 4 y1). The prefactor in Eq. (22) varies approximately from unity to 5.5 over the entire range of y2 but does not have much difference at b ¼ bmax and bmin for different e. To make numerical calculations of b, we choose R¼1 mm and r¼0.1 nm (see Fig. 8). For e ¼5.7 (dotted line), we have the minimal enhancement bmin 0.17 at y2/yt E0.29 and the maximal enhancement bmax 19.8 at y2/yt E0.91. For e ¼10.4 (solid line), we have bmin 0.12 at y2/yt E0.30 and the maximum bmax 47.1 at y2/yt E0.93. This is the enhancement b of the vacuum field F1 (blue line). The dielectric field F2 has the enhancement Zb (red line), which is smaller than b. As seen in Figs. 4 and 8, nmin and bmax do not both take place at the same y2/ yt because of the prefactor of Eq. (22). The field enhancement factor b has its maximum at a slightly larger y2 than the power index n. It is noted that all the associated extrema nmin , bmax , F 1,max and F 2,max lie in the region 0.75o y2/yt o1.0, where y1 þ y2 ¼ yt. Fig. 9 shows the e-dependence of bmax . It appears that bmax increases almost linearly with e. The slope depends on a and r. We take the values of r¼ 0.11 and 1.0 nm for R¼1 mm. As e increases from 1 to 100 (not shown up to this value), bmax increases from 1.0 to 122.5 for a ¼01 and r¼1.0 nm (red dashed line), from 1.0 to 153.2 for a ¼01 and r ¼0.1 nm (red solid line), and from 1.0 to 669.4 for a ¼ 1801 and r ¼0.1 nm (blue solid line). The plots of a ¼901 and r ¼1.0 nm (green dashed line) coincides with that of
Fig. 9. Plot for bmax versus e. The maximal enhancement bmax is shown as a function of e for r ¼0.1 (solid line) and 1.0 nm (dashed line) and a ¼ 01, 901, and 1801. The plot for a ¼901 and r¼ 1.0 nm (green dashed line) is overlapped with that for a ¼ 01 and r¼ 0.1 nm (red solid line). It is seen that bmax increases almost linearly with increasing e. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 10. Shape dependence of the field and enhancement. As a varies from 01 to 1801, F0 decreases rapidly while b increases slowly. Thus both F1 (blue line) and F2 (red line) are shown to follow the trend of F0. We take e ¼ 5.7 (dashed line) and 10.4 (solid line). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
a ¼01 and r ¼0.1 nm (red solid line), and is not seen. The field enhancement bmax becomes very large for both a and e large. Fig. 10 shows the a-dependence of F0 and bmax . The a-dependence reflects the curvature of the metallic shape. From Eq. (21), F0 has the exponent 1 n0 ¼(p a)/(2p a), which decreases with increasing a. Then F0 decreases with increasing a. By Eq. (22), b has the exponent n0 n ¼ n0(1 n/n0), where n0 ¼ p/(2p a) and n/n0 has no a-dependence. Then b increases with increasing a. The two have opposite a-dependence. This Fig. 8. Field enhancement b of vacuum and dielectric fields for the triple junction. We take e ¼5.7 (dashed line) and 10.4 (solid line). The enhancement, b, for the vacuum field F1 (blue line) is larger than that for the dielectric field F2 (red line). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
leads to a competition between b and F0 in determining F1 (and F2) when a varies. To figure out the curvature-dependence of both enhancements, we did numerical calculations of F0,F 1,max and F 2,max as functions of a. The numerical values of F are calculated
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by setting 16pV/R¼4.0 and are given in an arbitrary unit in Fig. 10. Here, we are just interested in relative magnitudes. As a increases from 01 to 1801, F0(r) decreases from 63.7 to 1.27 (50 times smaller), irrespective of e. Instead, bmax increases from 7.77 to 19.8 (2.5 times larger) for e ¼5.7 and from 15.0 to 47.1 (3.1 times larger) for e ¼10.4. This implies that F0 changes more rapidly with a than b or Zb. As a result, both F 1,max and F 2,max are shown to decrease with increasing a. However, the dielectric enhancement is still significantly large over all a. Even though it is considerable, the dielectric effect is not just apparently visible in Fig. 10 because it changes slowly in comparison with the shape effect.
4.2. Electric fields for the quadruple junction As discussed in Section 3, the MVDV junction is effective in enhancing the field whereas the MDVD is effective in reducing the field. In the current work, we consider the MVDV junction. We apply the same calculation scheme as used for the triple junction to obtain ! Z1 Z1 þ Z2 yt R n0 n bðrÞ ¼ F 1 ðrÞ=F 0 ðrÞ ¼ 0:5 , 2 2 2 r Z1 y1 þ e 1 þ Z2 yt þ Z3 y3 ð23Þ where Z1 ¼A1/A2, Z2 ¼ B2 =A2 , and Z3 ¼A3/A2. All Zi(i¼ 1, 2, and 3) values are obtained using the boundary condition [20]. The two b(r) values given by Eqs. (22) and (23) are the same in form but are different in configuration dependence. For the symmetric quadruple junction of y3 ¼ y1, we use Eq. (23) to obtain b(r) as a function of y2/yt for e ¼5.7. The results are shown in Fig. 11. We consider r ¼0.1 (solid line) and 1.0 nm (dashed line). We also plot b(r) for the triple junction for comparison. We have the maximal enhancement bmax ¼21.8 and 66.1 for r ¼1.0 and 0.1 nm, respectively. The corresponding values are 12.0 and 19.8 for the triple junction. The bmax for the symmetric quadruple junction are 2 or 3 times larger than those for the triple junction. The dielectric effect is more effective for the quadruple junction.
5. Discussions The difference between the enhancements for the triple and quadruple junctions supports the conclusion that the electric field is enhanced or reduced by the polarization of dielectric. In the absence of dielectric, the surface charge is distributed so that the potential is constant over the metallic surface, resulting in the a-dependence of F0. If a dielectric is placed in contact with the metal, then the charge on the metal causes the dielectric to polarize. In turn, the free surface charge on the metal is also redistributed under the influence of polarization charges. The magnitude and direction of polarization depends on the contact angles of the constituents. If polarization charges attract free surface charges to the junction, then more free charges are accumulated near the junction. The field becomes stronger. This stronger field induces a larger polarization. This process goes on until a perfect polarization of the dielectric is achieved. As a result, a very strong field is formed. Such strong fields lead to enhanced field emission or dielectric breakdown [1–3,21]. The current results surely give rise to a newly supposed mechanism for field electron emission even though it is not confirmed yet experimentally. It is likely that the enhanced F is strong enough to produce space charges in dielectric near the junction. The large density of space charges is responsible for lowering and thinning the contact barrier between the metal and dielectric. In general, the dielectric has a very low vacuum barrier. Thus, the field enhancement at metal–dielectric contact may create a detour path of metal–dielectric–vacuum. This possibly leads to enhanced field emission near the metal–dielectric contact. It seems that such a detour path is more preferable in potential energy than a direct path from metal to vacuum. This implies that the triple or quadruple junction can be another type of field electron source.
6. Conclusions We described the electric field in the vicinity of a junction where metal, vacuum, and dielectric meet. The assumption of a two-dimensional symmetry of the junction leaded to formulation of the electric field. For the triple junction, the electric field was found to be enhanced or reduced according to the contact angle ratio as well as the dielectric constant. It was found that a certain type of quadruple junction yielded a larger field enhancement or reduction than the triple junction. It was noted that the enhancement of the electric field at the contact between the metal and the dielectric is the product of the two enhancements due to dielectric and shape of the metallic portion. This field enhancement may be large enough to explain the triple junction phenomenon such as the vacuum or dielectric breakdown. Understanding of the reduction mechanism is applicable to avoid undesirable breakdown.
Acknowledgment This research was supported by the Basic Science Research Program through National Research Foundation of Korea funded by the MEST (Grant no. 2011-0009500). This research was also supported by Mid-carrier Research Program through the NRF grant funded by the MEST (Grant no. R01-2008-000–20025-0). Fig. 11. Field enhancement for the symmetric quadruple junction. The symmetric metal–vacuum–dielectric–vacuum quadruple junction (blue line) yields a larger enhancement than the triple junction (red line). We take r¼ 0.1 (solid line) and 1.0 nm (dashed line). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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