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Charge distribution on and near Schottky nanocontacts
ARTICLE IN PRESS
Physica E 33 (2006) 296–302 www.elsevier.com/locate/physe
Charge distribution on and near Schottky nanocontacts Carl Ha¨gglunda, Vladimir P. Zhdanova,b, a
Department of Applied Physics, Chalmers University of Technology, S-412 96 Go¨teborg, Sweden b Boreskov Institute of Catalysis, Russian Academy of Sciences, Novosibirsk 630090, Russia Received 28 November 2005; accepted 30 March 2006 Available online 12 May 2006
Abstract Schottky nanocontacts are formed when nm-sized metal particles are located on the planar surface of a doped semiconductor. The charge distribution on and near such nanocontacts is analyzed in the case of disc-shaped particles. The results of calculations are presented as a function of particle size, semiconductor permittivity, dopant concentration, and Fermi level difference. In contrast to macroscopic junctions, the charging of the metal particle is demonstrated to be proportional to the Fermi level difference and accordingly to the potential difference between the metal and semiconductor, so that the junction exhibits a constant capacitance. The charging of the metal-vacuum metal surfaces may be appreciable, especially for relatively low values of the semiconductor permittivity. The tunneling barrier width at half height is shown to be close to, or less than, 38 of the disc diameter. r 2006 Elsevier B.V. All rights reserved. PACS: 61.46.+w; 73.63.b; 89.20.a Keywords: Semiconducting surfaces; Metal-semiconductor interfaces; Schottky junctions; Nanocontacts; Surface electronic phenomena; Catalysis; Ag; Au; Si; TiO2
1. Introduction In the case of the metal-semiconductor contact, the Fermi levels in the two materials must be coincident. This condition is fulfilled by charge redistribution near the interface. For macroscopic flat junctions with participation of a doped semiconductor, the redistribution is well known to be described by the 1D Schottky model [1]. If for example the semiconductor is of n-type, this model predicts that the surface density of charge passed to the metal and the width of the charge depletion zone in the semiconductor are given by [1] r ¼ ðN d DE F =2pÞ1=2 ,
(1)
W ¼ ½DE F =ð2pe2 N d Þ1=2 ,
(2)
Corresponding author. Boreskov Institute of Catalysis, Russian Academy of Sciences, Novosibirsk 630090, Russia. Fax: +7 46 31 7723134. E-mail address: [email protected] (V.P. Zhdanov).
1386-9477/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2006.03.152
where DE F is the difference of the Fermi energies in the semiconductor and metal before the charge redistribution, N d the dopant concentration, the semiconductor dielectric constant, and e the absolute value of the electron charge. Eqs. (1) and (2) hold provided that the size of the metalsemiconductor contact is appreciably larger than the depletion width. According to Eq. (2), the depletion width typically exceeds 100 nm. Thus, the conventional Schottky model is not applicable for metal particles on the nanoscale below 100 nm. Meanwhile, the properties of the metalsemiconductor junctions of the latter size are of high interest from the viewpoints of fabrication of semiconductor devices [2], catalysis [3], photocatalysis [4] and photovoltaics [5]. For this and other reasons, it is instructive to discuss how one should modify the Schottky model in this limit. To address this question, it is appropriate first to notice that in the case of an embedded spherical metal particle, due to symmetry, the charge should be uniformly distributed on the particle surface and accordingly the charge distribution in the semiconductor
ARTICLE IN PRESS C. Ha¨gglund, V.P. Zhdanov / Physica E 33 (2006) 296–302
can elementary be calculated in analogy with the conventional Schottky model. The corresponding equations were used by Smit et al. [6] and Vostokov and Shashkin [7] in order to analyze electron tunneling from nm-sized embedded metal particles. Ioannides and Verykios [8] suggested that the equations obtained for an embedded particle could be used for a supported particle as well. In particular, they proposed to simply truncate the embeddedparticle-semiconductor system and to ignore the redistribution of the charge, related with the deviations from the spherical symmetry. Smit et al. [6] and Donolato [9] analyzed the case of supported particles more accurately assuming that the metal particle can be represented by an infinitely thin circular plate. In Refs. [6,8,9], the charge transferred to the supported metal particle from the semiconductor is assumed to be located on the metal-semiconductor interface. The validity of this assumption is however not obvious. In particular, the charge transferred to a metal particle shaped as a sphere or truncated octahedron (for the shape of nm-sized metal particles, see e.g. Ref. [10]) and located on the semiconductor surface, can be distributed more or less uniformly on the particle surface and the corresponding potential seems to be more reasonable than that obtained assuming the transferred charge to be located on the interface [11] (see also discussion [12,13]). In general, the charge distribution on and near nm-sized metal particles located on the surface of a doped semiconductor depends on the particle shape and the semiconductor dielectric constant. In this work, we analyze the case when a disc-shaped metal nanoparticle is situated on a planar semiconductor support (such structures are now routinely fabricated using electron-beam lithography or other patterning methods [14,15]). Compared to the earlier studies [6,8,9], we allow distribution of the transferred charge not only on the metal-surface interface but also on the metal-vacuum interface. The model parameters (e.g., DE F and ) are varied in our calculations in a wide range in order to illustrate various situations. Some of the results are of particular relevance for Ag and Au on Si and TiO2 .
2. Model and general equations In our calculations, the metal particles are assumed to be of cylindrical shape with radius a and height h (Fig. 1). The bottom part of the particles is considered to contact the surface of an n-doped semiconductor. At equilibrium, charge is transferred from dopants to the metal particles, resulting in the formation of depletion regions in the semiconductor near the particles. We assume that the distance between metal particles is much larger than the size of the depletion regions. In this case, the mutual perturbation of the charge distributions near different particles is negligible, and accordingly we focus on a single metal particle here.
297
Vacuum
z
Q1 Metal particle
a h
Q2 Q3 R
Semiconductor
Fig. 1. Geometry employed in the model.
For nm-sized metal particles and typical concentrations of dopants, the size of the depletion region exceeds or, more often, appreciably exceeds the particle size. In this case, the charge located on the metal particle acts almost like a point charge on the semiconductor, and accordingly the shape of the depletion region is expected to be close to hemispherical with radius R (Fig. 1). We employ this approximation and consider that the charge of the dopants in the depletion region is distributed uniformly. The use of the depletion-region concept makes sense provided that the size of this region is larger than the Debye screening length in the semiconductor (lD ¼ ½kB T=ð4pe2 N d Þ1=2 ). At room temperature, this is the case in the examples below. For validation of the hemispherical approximation, we refer to calculations [9] indicating that the deviations of the depletion-region shape from hemispherical are relatively minor. Taking into account that the size of the metal particles under consideration is appreciably larger than the Debye screening length in a metal ðlD t1 nmÞ, we consider that the charge transferred to a metal particle is distributed on its surface. The value and distribution of the charge should be determined so that (i) the electrostatic potential inside a metal particle is constant and (ii) the shift of the potential compared to the bulk results in the coincidence of the Fermi levels in the metal and semiconductor. Mathematically, this is a variational problem. In our calculations, we use three free variational parameters. Specifically, we assume that the charges Q1 , Q2 and Q3 located on the top, side and bottom faces of the metal disc (Fig. 1), are distributed uniformly. Except for the positively charged dopants, the semiconductor is treated like a dielectric. For a given size of the depletion region and fixed charges Q1 , Q2 and Q3 , the potential inside and outside the semiconductor can be calculated accurately by using the conventional method of
ARTICLE IN PRESS C. Ha¨gglund, V.P. Zhdanov / Physica E 33 (2006) 296–302
298
image charges [16]. In particular, taking the zero point of the potential to be at z ¼ 1 (z is the coordinate perpendicular to the metal-semiconductor interface and defined so that z ¼ 0 at the interface), the potential along the axis coinciding with the central axis of the disc is given by jðzÞ ¼ j1 ðzÞ þ j2 ðzÞ þ j3 ðzÞ þ j4 ðzÞ þ j5 ðzÞ þ j6 ðzÞ,
(3)
where j1 ðzÞ ¼
2Q1 ½ða2 þ ðh zÞ2 Þ1=2 h þ z , a2 2 1=2
2
j2 ðzÞ ¼
2ð1 ÞQ1 ½ða þ ðh þ zÞ Þ ð1 þ Þa2
(4)
h z
,
(5)
z Q2 zh arcsh j3 ðzÞ ¼ arcsh , a a h
(6)
z ð1 ÞQ2 zþh j4 ðzÞ ¼ arcsh arcsh , a a ð1 þ Þh
(7)
j5 ðzÞ ¼
j6 ðzÞ ¼
3
3
2
(8) #
2 3=2
2Qs R þ z ðR þ z Þ ð1 þ Þ R3 z
þ
jðzÞ ¼ j1 ðzÞ þ j2 ðzÞ þ j3 ðzÞ þ j4 ðzÞ þ j5 ðzÞ,
3 . 2R
4Q1 ½ða2 þ ðh zÞ2 Þ1=2 h þ z , a2 ð1 þ Þ z 2Q2 zh j2 ðzÞ ¼ arcsh arcsh , a a hð1 þ Þ j1 ðzÞ ¼
4Q3 ½ða2 þ z2 Þ1=2 þ z , ð1 þ Þa2 " # Qs R3 2z3 ðR2 þ z2 Þ3=2 3 j4 ðzÞ ¼ , þ 2R R3 z j3 ðzÞ ¼
(9)
In these equations, j1 ðzÞ and j2 ðzÞ are the potentials generated by the charge Q1 and its image, j3 ðzÞ and j2 ðzÞ are the potentials related to the charge Q2 and its image, j5 ðzÞ is the potential generated by the charge Q3 and its image, j6 ðzÞ is the potential corresponding to the charge Qs located in the semiconductor, and is the semiconductor permittivity. Using the condition of charge conservation, we have Qs ¼ Q1 Q2 Q3 .
2Q1 ½ða2 þ ðh zÞ2 Þ1=2 z þ h . (15) a2 The potential inside the semiconductor at Rpzp0 can be represented as
j1 ðzÞ ¼
(16)
where
4Q3 ½ða2 þ z2 Þ1=2 z , ð1 þ Þa2 "
where DE F is the difference of the Fermi energies of the semiconductor and the metal [in terms of the work functions, DE F ¼ eðfm fs Þ]. Eqs. (3)–(14) constitute a non-linear system which must be solved by numerical iterations for specific parameter values. Once all the charges are determined by Eqs. (3)–(14), the potentials outside the metal particle and inside the semiconductor are obtained from expressions similar to Eqs. (3)–(9). In particular, for the potential outside the metal particle at z4h, Eq. (3) holds as well, with the only difference that expression (4) should be rewritten as
(10)
The radius R of the depletion zone may further be expressed in terms of Qs as 3Qs 1=3 , (11) R¼ 2peN d where it is assumed that the semiconductor depletion region charge Qs is positive. To determine the three unknown charges, we take into account that the potential inside the metal should be constant and, for our model, require that this condition is fulfilled for jðzÞ on the bottom, medium and top points of the disc. Thus, we should have ejð0Þ ¼ DE F ,
(12)
ejðh=2Þ ¼ DE F ,
(13)
ejðhÞ ¼ DE F ,
(14)
" # ð 1ÞQs 3 R3 z3 ðR2 þ z2 Þ3=2 . j5 ðzÞ ¼ ð þ 1Þ 2R R3 z
(17)
(18)
(19)
(20)
(21)
Here, j1 ðzÞ, j2 ðzÞ and j3 ðzÞ are the potentials generated by the charges Q1 , Q2 and Q3 , while j4 ðzÞ and j5 ð4Þ are the potentials related to the charge Qs and its image, respectively. The potential inside the semiconductor at zp R is also given by Eq. (16), with the only difference that expression (20) should be rewritten as " # Qs R3 þ z3 þ ðR2 þ z2 Þ3=2 3 j4 ðzÞ ¼ þ . (22) 2R R3 z The equations derived above form a basis for our numerical calculations presented in the next section. Here, it is appropriate to notice that at in the macroscopic limit when abW , the depletion region should obviously be cylindrical. With this shape of the depletion region, as expected, our model yields the same results as the conventional Schottky model. This indicates that this method for calculation of the potential can be used in other cases as well. For example, the potential for metal particles, shaped as a truncated pyramid or octahedron and contacting the semiconductor via the (1 0 0) face, can be
ARTICLE IN PRESS C. Ha¨gglund, V.P. Zhdanov / Physica E 33 (2006) 296–302
-100 ×Q1 -10 ×Q2 - Q3 Qs
5 × z1/2 W1/2 R
0.1
0.2
Electronic electrostatic potential / eV
0.4
0.5 0.6 0.7 ∆ EF / eV
0.8
0.9
1
a = 25 nm, ε = 110, Nd = 10-15 cm-3
a = 25 nm
1.2 EF = 1.1 eV 1.1 1 ε = 11.8, Nd = 1017 cm-3 0.9 ε = 110, Nd = 1015 cm-3 0.8 0.7 ∆EF = 0.6 eV 0.6 0.5 0.4 ∆EF = 0.3 eV 0.3 0.2 0.1 0 -0.1 -0.2 -200 -150 -100 -50 0
0.3
(a)
-100×Q1 -10×Q2 - Q3 Qs
5 × z1/2 W1/2 R
50
100
150
200
Distance from surface / nm Fig. 2. Electronic electrostatic potential, ejðzÞ, for parameter combinations yielding a5R and atR (dashed and solid lines, respectively).
0.1 (b)
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
400 350 300 250 200 150 100 50 0 80 70 60 50 40 30 20 10 0 1.1
600 525 450 375 300 225 150 75 0 1080 945 810 675 540 405 270 135 0 1.1
Charge Q / e
a = 25 nm, ε = 11.8, Nd = 1017 cm-3
Distance / nm
Our calculations are performed for low and moderate dopant concentrations in the range from 1015 to 1017 cm3 . The relative permittivity is either varied in a wide range or fixed at 11:8 (applicable for Si) or 110 (for TiO2 ). Typical dependence of the potential on z for these parameters and DE F ¼ 0:3, 0.6 and 1.1 eV are shown in Fig. 2 (note that since the values of the workfunctions scatters from about 4.3 to 4.6 eV for Ag and from 5.1 to 5.4 eV for Au in the literature, and the electron affinities of TiO2 and Si are both about 4.0 eV [17–19], a DE F of 0.3 eV corresponds roughly to Ag on n-Si or n-TiO2 , and a DE F of 1.1 eV to Au on n-Si or n-TiO2 , respectively). The variation of the potential inside the metal is seen to be nearly negligible (smaller that kB T at room temperature and much smaller than DE F ), which indicates that the approximations, used for calculation of the potential inside the metal particle, are fairly accurate. In the semiconductor region, the model predicts rapid drop of the potential with increasing jzj. Close to z ¼ R, one can observe a shallow potential well. This well we believe is related with the hemispherical approximation of the shape of the depletion region. For an accurate solution (beyond this approximation), the well is expected to disappear. The depth of the well is low (of the order of kB T and much smaller than DE F ) and accordingly it can safely be neglected. In general, the hemispherical approximation appears to be quite accurate provided that the depletion zone radius R exceeds 2a. This condition holds for sufficiently small particles, high semiconductor permit-
Charge Q / e
3. Results of calculations
tivity, low dopant concentration, and/or high Fermi level difference, as in Fig. 2. Due to the combination of the electric fields generated by the charges located in the semiconductor and on the surface of the metal particle, the electric field outside the semiconductor (at z4h) does not vanish (in the macroscopic case, this field is zero). With increasing z, it rapidly drops as well. Fig. 3 shows that irrespective of the dopant concentration and permittivity of the semiconductor, the amount of charge transferred to the three faces of the metal disc varies approximately linearly with DE F . This is in contrast to the conventional Schottky model where the surface density of 1=2 charge is proportional to DE F [see Eq. (1)], and means that the junction capacitance does not vary with bias for
Distance / nm
calculated by using, respectively, three or four variable charges corresponding to differently oriented faces.
299
∆EF / eV
Fig. 3. Charge distribution and barrier characteristics as a function of the Fermi level difference. The macroscopic barrier width W 1=2 is shown for comparison.
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100 × Q1 a = 25 nm, ∆ E = 0.6 eV, N = 1015 cm-3 F d 10 × Q2 - Q3 Qs
0 (a)
20
40 60 Relative permittivity ε
80
100
-100 × Q1 a = 25 nm, ∆ E = 0.3 eV, N = 1016 cm-3 F d -10 × Q2 - Q3 Qs
-100 × Q1 a = 25 nm, ∆ EF = 1.1 eV, Nd = 1016 cm-3 -10 × Q2 - Q3 Qs
0 (b)
20
40
60
80
100
400 350 300 250 200 150 100 50 0 400 350 300 250 200 150 100 50 0
200 175 150 125 100 75 50 25 0 600 525 450 375 300 225 150 75 0
Charge Q / e
100 × Q1 a = 25 nm, ∆ E = 0.6 eV, N = 1017 cm-3 F d 10 × Q2 - Q3 Qs
Charge Q / e
small particles. The average fractions of charge on the disc are q1 ¼ 0:025, q2 ¼ 0:242 and q3 ¼ 0:733 for ¼ 11:8 [Fig. 3(a)] and q1 ¼ 0:005, q2 ¼ 0:041 and q3 ¼ 0:954 in the case for ¼ 110 [Fig. 3(b)]. The increased fraction of charge on the metal-vacuum interface with decreasing is reasonable as one can expect a transition from the situation close to the conventional Schottky model with location of the charge on the metal-semiconductor interface (i.e., with q3 ’ 1) to the situation in vacuum (with ¼ 1 and q1 ¼ q3 ). The corresponding dependence of the radius of the depletion zone is exhibited in the lower panels of Fig. 3. Since the total charge grows linearly with DE F , R is 1=3 proportional to DE F . As already noted, the potential rapidly drops inside the semiconductor with increasing jzj. For this reason, a better measure with respect to the junction transport properties is the barrier full-width at half-maximum on the z-axis [9], denoted by z1=2 and also shown in Fig. 3. For ¼ 11:8 [Fig. 3(a)], z1=2 grows as R at low DE F but then somewhat slower than R at high DE F . For ¼ 110 [Fig. 3(b)], this slowdown occurs at much lower DE F due to limitations imposed by the metal-particle size, as further articulated below. It is of interest that z1=2 is generally much lower than thepcorresponding macroscopic ffiffiffi barrier width, W 1=2 ¼ ð1 1= 2ÞW , also shown in Fig. 3. This results in an enhanced tunneling current as found both experimentally and theoretically [6]. For fixed value of DE F , the interfacial charge Q3 increases linearly with the permittivity, and accordingly the increase of the permittivity does not result in the decrease of the electric field strength inside the semiconductor. Charging of the outer metal disc surfaces, however, is only weakly dependent on the permittivity and dopant concentration, at least at \10 (Fig. 4). Q2 is an order of magnitude larger than Q1 , and they are both mainly determined by DE F . Our calculations (not shown) indicate that the dependence of various properties of the system on the disc height is weak. The dependence on the disc radius is however more appreciable. Here, there are two asymptotic cases. The first one is close to the macroscopic limit, where the total charge transferred is proportional to the junction area. This situation takes place when R (or ) is relatively small, as e.g. shown in Fig. 5(a) at pa2 4400 nm2 . In this case, R / a2=3 / A1=3 , where A is the junction area, and the barrier width is proportional to R [see the lower panel of Fig. 5(a)] and approaches the macroscopic depletion width. The second case occurs when R (or ) is large or equivalently, a5ð4=3ÞW ’ W [this condition can be derived from Eqs. (8) and (9) assuming Q1 ¼ Q2 ¼ 0], as in Fig. 5(b). Since in this case the influence of Q3 dominates, the potential at z ¼ 0 is given by Eq. (19), i.e., Q3 ’ aDE F ð1 þ Þ=4e [the line corresponding to this expression is indicated by ’+’ in the upper panel of Fig 5(b)]. In addition, Qs ’ Q3 , and hence R ’ ½3að1 þ ÞDE F =ð8pe2 N d Þ1=3 / A1=6 . The barrier fullwidth at half-height is, however, in this case not proportional to R, but given by z1=2 ¼ 3a=4 [see Eq. (19)]. This
Charge Q / e
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Charge Q / e
300
Relative permittivity ε
Fig. 4. Charge distributions for different dopant concentrations as a function of semiconductor permittivity.
constitutes an upper limit of the width for this geometry and holds regardless of assumed depletion region shape. This limit is indicated by ’+’-marks in the lower panel of Fig. 5(b). Thus, for particles of atW =10, the barrier width is pinned by the disc size, independent of the metal work function, potential bias, semiconductor dopant concentration, and permittivity. As already noted, the charges on the external surfaces, Q1 and Q2 , are relatively insensitive to the semiconductor permittivity and dopant concentration, and this holds for any disc size on the nanoscale as can be seen from comparison of Fig. 5 (a) and (b). Finally, we show [Fig. 5 (c) and (d)] the dependence of the results on the disc size for intermediate dopant concentration N d ¼ 1016 cm3 and DE F ¼ 0:3 (c) and 1.1 eV (d). These examples are of relevance for Ag and Au discs on TiO2 , respectively. As already noted, the charging of the metal disc is directly proportional to the Fermi level difference, and by comparison of the upper
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3000 4000 5000 6000 7000
-100 ×Q1 -10 × Q2 - Q3 Qs
5×z1/2 W1/2 R
1000 2000
(c)
3000
4000
5000
6000 7000
400 350 300 250 200 150 100 50 0 280 245 210 175 140 105 70 35 0
1000 2000 3000 4000 5000
Disc area π a 2 / nm2
-100 ×Q1 -10 ×Q2 - Q3 Qs
5×z1/2 W1/2 R
0 (d)
Charge Q / e Distance / nm
6000 7000
Disc area π a 2 / nm 2 ε = 110, ∆EF = 1.1 eV, Nd = 1016 cm-3
Charge Q / e
ε = 110, ∆ EF = 0.3 eV, Nd = 1016 cm-3
0
0 (b)
Disc area π a 2 / nm2
800 700 600 500 400 300 200 100 0 800 700 600 500 400 300 200 100 0
1000 2000 3000 4000 5000 6000 7000
1400 1225 1050 875 700 525 350 175 0 400 350 300 250 200 150 100 50 0
Charge Q / e
1000 2000
5×z1/2 W1/2 R
Distance / nm
0 (a)
-100 × Q1 -10 × Q2 - Q3 Qs
Distance / nm
5 × z1/2 W1/2 R
ε = 110, ∆EF = 0.6 eV, Nd = 1015 cm-3 Charge Q / e
-100 ×Q1 -10 × Q2 - Q3 Qs
600 525 450 375 300 225 150 75 0 120 105 90 75 60 45 30 15 0
Distance / nm
ε = 11.8, ∆EF = 0.6 eV, Nd = 1017 cm-3
301
Disc area π a 2 / nm 2
Fig. 5. Size dependence of the charge distribution and barrier characteristics at (a) ¼ 11:8, (b) ¼ 110, and (c) and (d) at different DE F .
panels in Fig. 5(c) and (d), we conclude that this generally holds on the nanoscale, as long as a5W .
tional photoactive nanoparticles’’). The authors thank B. Kasemo for his interest and discussions.
4. Conclusion We have performed a detailed analysis of the specifics of the Schottky nanocontacts formed when nm-sized metal particles are located on the planar surface of a doped semiconductor. The properties of such contacts were shown to change compared to those of the macroscopic Schottky junctions. In our calculations, metal particles were assumed to be of disc-like shape. However, the approach is readily applicable to particles of other shapes as well. Acknowledgments This work was supported by the Swedish Foundation for Strategic Research (Material science project ‘‘Multifunc-
References [1] R. Ludeke, in: K. Horn et al. (Eds.), Electronic Structure; Handbook of Surface Science, vol. 2, Elsevier, Amsterdam, 2000, p. 749. [2] K.L. Wang, J. Nanosci. Nanotech. 2 (2002) 235. [3] H.-J. Freund, M. Ba¨umer, H. Kuhlenbeck, Adv. Catal. 45 (2000) 131. [4] Y. Tian, T. Tatsuma, J. Am. Chem. Soc. 127 (2005) 20. [5] D.M. Schaadt, B. Feng, E.T. Yu, Appl. Phys. Lett. 86 (2005) 063106. [6] G.D.J. Smit, S. Rogge, T.M. Klapwijk, Appl. Phys. Lett. 81 (2002) 3852. [7] N.V. Vostokov, V.I. Shashkin, J. Exp. Theor. Phys. 99 (2004) 211. [8] T. Ioannides, X.E. Verykios, J. Catal. 161 (1996) 560. [9] C. Donolato, J. Appl. Phys. 95 (2004) 2184. [10] F. Baletto, R. Ferrando, Rev. Mod. Phys. 77 (2005) 371. [11] V.P. Zhdanov, Surf. Sci. 512 (2002) L331. [12] T. Ioannides, Surf. Sci. 530 (2003) 216. [13] V.P. Zhdanov, Surf. Sci. 530 (2003) 219.
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[14] P. Hanarp, D.S. Sutherland, J. Gold, B. Kasemo, Coll. Surf. A: Physicochem. Eng. Asp. 214 (2003) 23. [15] A.A. Tseng, A. Notargiacomo, J. Nanosci. Nanotech. 5 (2005) 683. [16] L.D. Landau, E.M. Lifshitz, Electrodynamics of Continuous Media, Pergamon, Oxford, 1984.
[17] D.A. Neamen, Semiconductor Physics and Devices, Irwin, Boston, 1992, p. 325. [18] R. Jerisian, J.P. Loup, J. Gautron, Thin Solid Films 115 (1984) 229. [19] D.R. Lide, et al. (Eds.) Handbook of Chemistry and Physics, CRC Press, Boca Raton, FL, 2004, pp. 12–124.
Physica E 33 (2006) 296–302 www.elsevier.com/locate/physe
Charge distribution on and near Schottky nanocontacts Carl Ha¨gglunda, Vladimir P. Zhdanova,b, a
Department of Applied Physics, Chalmers University of Technology, S-412 96 Go¨teborg, Sweden b Boreskov Institute of Catalysis, Russian Academy of Sciences, Novosibirsk 630090, Russia Received 28 November 2005; accepted 30 March 2006 Available online 12 May 2006
Abstract Schottky nanocontacts are formed when nm-sized metal particles are located on the planar surface of a doped semiconductor. The charge distribution on and near such nanocontacts is analyzed in the case of disc-shaped particles. The results of calculations are presented as a function of particle size, semiconductor permittivity, dopant concentration, and Fermi level difference. In contrast to macroscopic junctions, the charging of the metal particle is demonstrated to be proportional to the Fermi level difference and accordingly to the potential difference between the metal and semiconductor, so that the junction exhibits a constant capacitance. The charging of the metal-vacuum metal surfaces may be appreciable, especially for relatively low values of the semiconductor permittivity. The tunneling barrier width at half height is shown to be close to, or less than, 38 of the disc diameter. r 2006 Elsevier B.V. All rights reserved. PACS: 61.46.+w; 73.63.b; 89.20.a Keywords: Semiconducting surfaces; Metal-semiconductor interfaces; Schottky junctions; Nanocontacts; Surface electronic phenomena; Catalysis; Ag; Au; Si; TiO2
1. Introduction In the case of the metal-semiconductor contact, the Fermi levels in the two materials must be coincident. This condition is fulfilled by charge redistribution near the interface. For macroscopic flat junctions with participation of a doped semiconductor, the redistribution is well known to be described by the 1D Schottky model [1]. If for example the semiconductor is of n-type, this model predicts that the surface density of charge passed to the metal and the width of the charge depletion zone in the semiconductor are given by [1] r ¼ ðN d DE F =2pÞ1=2 ,
(1)
W ¼ ½DE F =ð2pe2 N d Þ1=2 ,
(2)
Corresponding author. Boreskov Institute of Catalysis, Russian Academy of Sciences, Novosibirsk 630090, Russia. Fax: +7 46 31 7723134. E-mail address: [email protected] (V.P. Zhdanov).
1386-9477/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2006.03.152
where DE F is the difference of the Fermi energies in the semiconductor and metal before the charge redistribution, N d the dopant concentration, the semiconductor dielectric constant, and e the absolute value of the electron charge. Eqs. (1) and (2) hold provided that the size of the metalsemiconductor contact is appreciably larger than the depletion width. According to Eq. (2), the depletion width typically exceeds 100 nm. Thus, the conventional Schottky model is not applicable for metal particles on the nanoscale below 100 nm. Meanwhile, the properties of the metalsemiconductor junctions of the latter size are of high interest from the viewpoints of fabrication of semiconductor devices [2], catalysis [3], photocatalysis [4] and photovoltaics [5]. For this and other reasons, it is instructive to discuss how one should modify the Schottky model in this limit. To address this question, it is appropriate first to notice that in the case of an embedded spherical metal particle, due to symmetry, the charge should be uniformly distributed on the particle surface and accordingly the charge distribution in the semiconductor
ARTICLE IN PRESS C. Ha¨gglund, V.P. Zhdanov / Physica E 33 (2006) 296–302
can elementary be calculated in analogy with the conventional Schottky model. The corresponding equations were used by Smit et al. [6] and Vostokov and Shashkin [7] in order to analyze electron tunneling from nm-sized embedded metal particles. Ioannides and Verykios [8] suggested that the equations obtained for an embedded particle could be used for a supported particle as well. In particular, they proposed to simply truncate the embeddedparticle-semiconductor system and to ignore the redistribution of the charge, related with the deviations from the spherical symmetry. Smit et al. [6] and Donolato [9] analyzed the case of supported particles more accurately assuming that the metal particle can be represented by an infinitely thin circular plate. In Refs. [6,8,9], the charge transferred to the supported metal particle from the semiconductor is assumed to be located on the metal-semiconductor interface. The validity of this assumption is however not obvious. In particular, the charge transferred to a metal particle shaped as a sphere or truncated octahedron (for the shape of nm-sized metal particles, see e.g. Ref. [10]) and located on the semiconductor surface, can be distributed more or less uniformly on the particle surface and the corresponding potential seems to be more reasonable than that obtained assuming the transferred charge to be located on the interface [11] (see also discussion [12,13]). In general, the charge distribution on and near nm-sized metal particles located on the surface of a doped semiconductor depends on the particle shape and the semiconductor dielectric constant. In this work, we analyze the case when a disc-shaped metal nanoparticle is situated on a planar semiconductor support (such structures are now routinely fabricated using electron-beam lithography or other patterning methods [14,15]). Compared to the earlier studies [6,8,9], we allow distribution of the transferred charge not only on the metal-surface interface but also on the metal-vacuum interface. The model parameters (e.g., DE F and ) are varied in our calculations in a wide range in order to illustrate various situations. Some of the results are of particular relevance for Ag and Au on Si and TiO2 .
2. Model and general equations In our calculations, the metal particles are assumed to be of cylindrical shape with radius a and height h (Fig. 1). The bottom part of the particles is considered to contact the surface of an n-doped semiconductor. At equilibrium, charge is transferred from dopants to the metal particles, resulting in the formation of depletion regions in the semiconductor near the particles. We assume that the distance between metal particles is much larger than the size of the depletion regions. In this case, the mutual perturbation of the charge distributions near different particles is negligible, and accordingly we focus on a single metal particle here.
297
Vacuum
z
Q1 Metal particle
a h
Q2 Q3 R
Semiconductor
Fig. 1. Geometry employed in the model.
For nm-sized metal particles and typical concentrations of dopants, the size of the depletion region exceeds or, more often, appreciably exceeds the particle size. In this case, the charge located on the metal particle acts almost like a point charge on the semiconductor, and accordingly the shape of the depletion region is expected to be close to hemispherical with radius R (Fig. 1). We employ this approximation and consider that the charge of the dopants in the depletion region is distributed uniformly. The use of the depletion-region concept makes sense provided that the size of this region is larger than the Debye screening length in the semiconductor (lD ¼ ½kB T=ð4pe2 N d Þ1=2 ). At room temperature, this is the case in the examples below. For validation of the hemispherical approximation, we refer to calculations [9] indicating that the deviations of the depletion-region shape from hemispherical are relatively minor. Taking into account that the size of the metal particles under consideration is appreciably larger than the Debye screening length in a metal ðlD t1 nmÞ, we consider that the charge transferred to a metal particle is distributed on its surface. The value and distribution of the charge should be determined so that (i) the electrostatic potential inside a metal particle is constant and (ii) the shift of the potential compared to the bulk results in the coincidence of the Fermi levels in the metal and semiconductor. Mathematically, this is a variational problem. In our calculations, we use three free variational parameters. Specifically, we assume that the charges Q1 , Q2 and Q3 located on the top, side and bottom faces of the metal disc (Fig. 1), are distributed uniformly. Except for the positively charged dopants, the semiconductor is treated like a dielectric. For a given size of the depletion region and fixed charges Q1 , Q2 and Q3 , the potential inside and outside the semiconductor can be calculated accurately by using the conventional method of
ARTICLE IN PRESS C. Ha¨gglund, V.P. Zhdanov / Physica E 33 (2006) 296–302
298
image charges [16]. In particular, taking the zero point of the potential to be at z ¼ 1 (z is the coordinate perpendicular to the metal-semiconductor interface and defined so that z ¼ 0 at the interface), the potential along the axis coinciding with the central axis of the disc is given by jðzÞ ¼ j1 ðzÞ þ j2 ðzÞ þ j3 ðzÞ þ j4 ðzÞ þ j5 ðzÞ þ j6 ðzÞ,
(3)
where j1 ðzÞ ¼
2Q1 ½ða2 þ ðh zÞ2 Þ1=2 h þ z , a2 2 1=2
2
j2 ðzÞ ¼
2ð1 ÞQ1 ½ða þ ðh þ zÞ Þ ð1 þ Þa2
(4)
h z
,
(5)
z Q2 zh arcsh j3 ðzÞ ¼ arcsh , a a h
(6)
z ð1 ÞQ2 zþh j4 ðzÞ ¼ arcsh arcsh , a a ð1 þ Þh
(7)
j5 ðzÞ ¼
j6 ðzÞ ¼
3
3
2
(8) #
2 3=2
2Qs R þ z ðR þ z Þ ð1 þ Þ R3 z
þ
jðzÞ ¼ j1 ðzÞ þ j2 ðzÞ þ j3 ðzÞ þ j4 ðzÞ þ j5 ðzÞ,
3 . 2R
4Q1 ½ða2 þ ðh zÞ2 Þ1=2 h þ z , a2 ð1 þ Þ z 2Q2 zh j2 ðzÞ ¼ arcsh arcsh , a a hð1 þ Þ j1 ðzÞ ¼
4Q3 ½ða2 þ z2 Þ1=2 þ z , ð1 þ Þa2 " # Qs R3 2z3 ðR2 þ z2 Þ3=2 3 j4 ðzÞ ¼ , þ 2R R3 z j3 ðzÞ ¼
(9)
In these equations, j1 ðzÞ and j2 ðzÞ are the potentials generated by the charge Q1 and its image, j3 ðzÞ and j2 ðzÞ are the potentials related to the charge Q2 and its image, j5 ðzÞ is the potential generated by the charge Q3 and its image, j6 ðzÞ is the potential corresponding to the charge Qs located in the semiconductor, and is the semiconductor permittivity. Using the condition of charge conservation, we have Qs ¼ Q1 Q2 Q3 .
2Q1 ½ða2 þ ðh zÞ2 Þ1=2 z þ h . (15) a2 The potential inside the semiconductor at Rpzp0 can be represented as
j1 ðzÞ ¼
(16)
where
4Q3 ½ða2 þ z2 Þ1=2 z , ð1 þ Þa2 "
where DE F is the difference of the Fermi energies of the semiconductor and the metal [in terms of the work functions, DE F ¼ eðfm fs Þ]. Eqs. (3)–(14) constitute a non-linear system which must be solved by numerical iterations for specific parameter values. Once all the charges are determined by Eqs. (3)–(14), the potentials outside the metal particle and inside the semiconductor are obtained from expressions similar to Eqs. (3)–(9). In particular, for the potential outside the metal particle at z4h, Eq. (3) holds as well, with the only difference that expression (4) should be rewritten as
(10)
The radius R of the depletion zone may further be expressed in terms of Qs as 3Qs 1=3 , (11) R¼ 2peN d where it is assumed that the semiconductor depletion region charge Qs is positive. To determine the three unknown charges, we take into account that the potential inside the metal should be constant and, for our model, require that this condition is fulfilled for jðzÞ on the bottom, medium and top points of the disc. Thus, we should have ejð0Þ ¼ DE F ,
(12)
ejðh=2Þ ¼ DE F ,
(13)
ejðhÞ ¼ DE F ,
(14)
" # ð 1ÞQs 3 R3 z3 ðR2 þ z2 Þ3=2 . j5 ðzÞ ¼ ð þ 1Þ 2R R3 z
(17)
(18)
(19)
(20)
(21)
Here, j1 ðzÞ, j2 ðzÞ and j3 ðzÞ are the potentials generated by the charges Q1 , Q2 and Q3 , while j4 ðzÞ and j5 ð4Þ are the potentials related to the charge Qs and its image, respectively. The potential inside the semiconductor at zp R is also given by Eq. (16), with the only difference that expression (20) should be rewritten as " # Qs R3 þ z3 þ ðR2 þ z2 Þ3=2 3 j4 ðzÞ ¼ þ . (22) 2R R3 z The equations derived above form a basis for our numerical calculations presented in the next section. Here, it is appropriate to notice that at in the macroscopic limit when abW , the depletion region should obviously be cylindrical. With this shape of the depletion region, as expected, our model yields the same results as the conventional Schottky model. This indicates that this method for calculation of the potential can be used in other cases as well. For example, the potential for metal particles, shaped as a truncated pyramid or octahedron and contacting the semiconductor via the (1 0 0) face, can be
ARTICLE IN PRESS C. Ha¨gglund, V.P. Zhdanov / Physica E 33 (2006) 296–302
-100 ×Q1 -10 ×Q2 - Q3 Qs
5 × z1/2 W1/2 R
0.1
0.2
Electronic electrostatic potential / eV
0.4
0.5 0.6 0.7 ∆ EF / eV
0.8
0.9
1
a = 25 nm, ε = 110, Nd = 10-15 cm-3
a = 25 nm
1.2 EF = 1.1 eV 1.1 1 ε = 11.8, Nd = 1017 cm-3 0.9 ε = 110, Nd = 1015 cm-3 0.8 0.7 ∆EF = 0.6 eV 0.6 0.5 0.4 ∆EF = 0.3 eV 0.3 0.2 0.1 0 -0.1 -0.2 -200 -150 -100 -50 0
0.3
(a)
-100×Q1 -10×Q2 - Q3 Qs
5 × z1/2 W1/2 R
50
100
150
200
Distance from surface / nm Fig. 2. Electronic electrostatic potential, ejðzÞ, for parameter combinations yielding a5R and atR (dashed and solid lines, respectively).
0.1 (b)
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
400 350 300 250 200 150 100 50 0 80 70 60 50 40 30 20 10 0 1.1
600 525 450 375 300 225 150 75 0 1080 945 810 675 540 405 270 135 0 1.1
Charge Q / e
a = 25 nm, ε = 11.8, Nd = 1017 cm-3
Distance / nm
Our calculations are performed for low and moderate dopant concentrations in the range from 1015 to 1017 cm3 . The relative permittivity is either varied in a wide range or fixed at 11:8 (applicable for Si) or 110 (for TiO2 ). Typical dependence of the potential on z for these parameters and DE F ¼ 0:3, 0.6 and 1.1 eV are shown in Fig. 2 (note that since the values of the workfunctions scatters from about 4.3 to 4.6 eV for Ag and from 5.1 to 5.4 eV for Au in the literature, and the electron affinities of TiO2 and Si are both about 4.0 eV [17–19], a DE F of 0.3 eV corresponds roughly to Ag on n-Si or n-TiO2 , and a DE F of 1.1 eV to Au on n-Si or n-TiO2 , respectively). The variation of the potential inside the metal is seen to be nearly negligible (smaller that kB T at room temperature and much smaller than DE F ), which indicates that the approximations, used for calculation of the potential inside the metal particle, are fairly accurate. In the semiconductor region, the model predicts rapid drop of the potential with increasing jzj. Close to z ¼ R, one can observe a shallow potential well. This well we believe is related with the hemispherical approximation of the shape of the depletion region. For an accurate solution (beyond this approximation), the well is expected to disappear. The depth of the well is low (of the order of kB T and much smaller than DE F ) and accordingly it can safely be neglected. In general, the hemispherical approximation appears to be quite accurate provided that the depletion zone radius R exceeds 2a. This condition holds for sufficiently small particles, high semiconductor permit-
Charge Q / e
3. Results of calculations
tivity, low dopant concentration, and/or high Fermi level difference, as in Fig. 2. Due to the combination of the electric fields generated by the charges located in the semiconductor and on the surface of the metal particle, the electric field outside the semiconductor (at z4h) does not vanish (in the macroscopic case, this field is zero). With increasing z, it rapidly drops as well. Fig. 3 shows that irrespective of the dopant concentration and permittivity of the semiconductor, the amount of charge transferred to the three faces of the metal disc varies approximately linearly with DE F . This is in contrast to the conventional Schottky model where the surface density of 1=2 charge is proportional to DE F [see Eq. (1)], and means that the junction capacitance does not vary with bias for
Distance / nm
calculated by using, respectively, three or four variable charges corresponding to differently oriented faces.
299
∆EF / eV
Fig. 3. Charge distribution and barrier characteristics as a function of the Fermi level difference. The macroscopic barrier width W 1=2 is shown for comparison.
ARTICLE IN PRESS
100 × Q1 a = 25 nm, ∆ E = 0.6 eV, N = 1015 cm-3 F d 10 × Q2 - Q3 Qs
0 (a)
20
40 60 Relative permittivity ε
80
100
-100 × Q1 a = 25 nm, ∆ E = 0.3 eV, N = 1016 cm-3 F d -10 × Q2 - Q3 Qs
-100 × Q1 a = 25 nm, ∆ EF = 1.1 eV, Nd = 1016 cm-3 -10 × Q2 - Q3 Qs
0 (b)
20
40
60
80
100
400 350 300 250 200 150 100 50 0 400 350 300 250 200 150 100 50 0
200 175 150 125 100 75 50 25 0 600 525 450 375 300 225 150 75 0
Charge Q / e
100 × Q1 a = 25 nm, ∆ E = 0.6 eV, N = 1017 cm-3 F d 10 × Q2 - Q3 Qs
Charge Q / e
small particles. The average fractions of charge on the disc are q1 ¼ 0:025, q2 ¼ 0:242 and q3 ¼ 0:733 for ¼ 11:8 [Fig. 3(a)] and q1 ¼ 0:005, q2 ¼ 0:041 and q3 ¼ 0:954 in the case for ¼ 110 [Fig. 3(b)]. The increased fraction of charge on the metal-vacuum interface with decreasing is reasonable as one can expect a transition from the situation close to the conventional Schottky model with location of the charge on the metal-semiconductor interface (i.e., with q3 ’ 1) to the situation in vacuum (with ¼ 1 and q1 ¼ q3 ). The corresponding dependence of the radius of the depletion zone is exhibited in the lower panels of Fig. 3. Since the total charge grows linearly with DE F , R is 1=3 proportional to DE F . As already noted, the potential rapidly drops inside the semiconductor with increasing jzj. For this reason, a better measure with respect to the junction transport properties is the barrier full-width at half-maximum on the z-axis [9], denoted by z1=2 and also shown in Fig. 3. For ¼ 11:8 [Fig. 3(a)], z1=2 grows as R at low DE F but then somewhat slower than R at high DE F . For ¼ 110 [Fig. 3(b)], this slowdown occurs at much lower DE F due to limitations imposed by the metal-particle size, as further articulated below. It is of interest that z1=2 is generally much lower than thepcorresponding macroscopic ffiffiffi barrier width, W 1=2 ¼ ð1 1= 2ÞW , also shown in Fig. 3. This results in an enhanced tunneling current as found both experimentally and theoretically [6]. For fixed value of DE F , the interfacial charge Q3 increases linearly with the permittivity, and accordingly the increase of the permittivity does not result in the decrease of the electric field strength inside the semiconductor. Charging of the outer metal disc surfaces, however, is only weakly dependent on the permittivity and dopant concentration, at least at \10 (Fig. 4). Q2 is an order of magnitude larger than Q1 , and they are both mainly determined by DE F . Our calculations (not shown) indicate that the dependence of various properties of the system on the disc height is weak. The dependence on the disc radius is however more appreciable. Here, there are two asymptotic cases. The first one is close to the macroscopic limit, where the total charge transferred is proportional to the junction area. This situation takes place when R (or ) is relatively small, as e.g. shown in Fig. 5(a) at pa2 4400 nm2 . In this case, R / a2=3 / A1=3 , where A is the junction area, and the barrier width is proportional to R [see the lower panel of Fig. 5(a)] and approaches the macroscopic depletion width. The second case occurs when R (or ) is large or equivalently, a5ð4=3ÞW ’ W [this condition can be derived from Eqs. (8) and (9) assuming Q1 ¼ Q2 ¼ 0], as in Fig. 5(b). Since in this case the influence of Q3 dominates, the potential at z ¼ 0 is given by Eq. (19), i.e., Q3 ’ aDE F ð1 þ Þ=4e [the line corresponding to this expression is indicated by ’+’ in the upper panel of Fig 5(b)]. In addition, Qs ’ Q3 , and hence R ’ ½3að1 þ ÞDE F =ð8pe2 N d Þ1=3 / A1=6 . The barrier fullwidth at half-height is, however, in this case not proportional to R, but given by z1=2 ¼ 3a=4 [see Eq. (19)]. This
Charge Q / e
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Charge Q / e
300
Relative permittivity ε
Fig. 4. Charge distributions for different dopant concentrations as a function of semiconductor permittivity.
constitutes an upper limit of the width for this geometry and holds regardless of assumed depletion region shape. This limit is indicated by ’+’-marks in the lower panel of Fig. 5(b). Thus, for particles of atW =10, the barrier width is pinned by the disc size, independent of the metal work function, potential bias, semiconductor dopant concentration, and permittivity. As already noted, the charges on the external surfaces, Q1 and Q2 , are relatively insensitive to the semiconductor permittivity and dopant concentration, and this holds for any disc size on the nanoscale as can be seen from comparison of Fig. 5 (a) and (b). Finally, we show [Fig. 5 (c) and (d)] the dependence of the results on the disc size for intermediate dopant concentration N d ¼ 1016 cm3 and DE F ¼ 0:3 (c) and 1.1 eV (d). These examples are of relevance for Ag and Au discs on TiO2 , respectively. As already noted, the charging of the metal disc is directly proportional to the Fermi level difference, and by comparison of the upper
ARTICLE IN PRESS C. Ha¨gglund, V.P. Zhdanov / Physica E 33 (2006) 296–302
3000 4000 5000 6000 7000
-100 ×Q1 -10 × Q2 - Q3 Qs
5×z1/2 W1/2 R
1000 2000
(c)
3000
4000
5000
6000 7000
400 350 300 250 200 150 100 50 0 280 245 210 175 140 105 70 35 0
1000 2000 3000 4000 5000
Disc area π a 2 / nm2
-100 ×Q1 -10 ×Q2 - Q3 Qs
5×z1/2 W1/2 R
0 (d)
Charge Q / e Distance / nm
6000 7000
Disc area π a 2 / nm 2 ε = 110, ∆EF = 1.1 eV, Nd = 1016 cm-3
Charge Q / e
ε = 110, ∆ EF = 0.3 eV, Nd = 1016 cm-3
0
0 (b)
Disc area π a 2 / nm2
800 700 600 500 400 300 200 100 0 800 700 600 500 400 300 200 100 0
1000 2000 3000 4000 5000 6000 7000
1400 1225 1050 875 700 525 350 175 0 400 350 300 250 200 150 100 50 0
Charge Q / e
1000 2000
5×z1/2 W1/2 R
Distance / nm
0 (a)
-100 × Q1 -10 × Q2 - Q3 Qs
Distance / nm
5 × z1/2 W1/2 R
ε = 110, ∆EF = 0.6 eV, Nd = 1015 cm-3 Charge Q / e
-100 ×Q1 -10 × Q2 - Q3 Qs
600 525 450 375 300 225 150 75 0 120 105 90 75 60 45 30 15 0
Distance / nm
ε = 11.8, ∆EF = 0.6 eV, Nd = 1017 cm-3
301
Disc area π a 2 / nm 2
Fig. 5. Size dependence of the charge distribution and barrier characteristics at (a) ¼ 11:8, (b) ¼ 110, and (c) and (d) at different DE F .
panels in Fig. 5(c) and (d), we conclude that this generally holds on the nanoscale, as long as a5W .
tional photoactive nanoparticles’’). The authors thank B. Kasemo for his interest and discussions.
4. Conclusion We have performed a detailed analysis of the specifics of the Schottky nanocontacts formed when nm-sized metal particles are located on the planar surface of a doped semiconductor. The properties of such contacts were shown to change compared to those of the macroscopic Schottky junctions. In our calculations, metal particles were assumed to be of disc-like shape. However, the approach is readily applicable to particles of other shapes as well. Acknowledgments This work was supported by the Swedish Foundation for Strategic Research (Material science project ‘‘Multifunc-
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[14] P. Hanarp, D.S. Sutherland, J. Gold, B. Kasemo, Coll. Surf. A: Physicochem. Eng. Asp. 214 (2003) 23. [15] A.A. Tseng, A. Notargiacomo, J. Nanosci. Nanotech. 5 (2005) 683. [16] L.D. Landau, E.M. Lifshitz, Electrodynamics of Continuous Media, Pergamon, Oxford, 1984.
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