Jul 15, 2001 - Université de Bourgogne, IUT du Creusot, Le2i, 12 rue de la Fonderie, 71200 Le Creusot, France ... The force density on the hyperboloidal probe medium is calculated as a ..... column gives the corresponding gap sizes.
JOURNAL OF APPLIED PHYSICS
VOLUME 90, NUMBER 2
15 JULY 2001
Electrostatic force density for a scanned probe above a charged surface A. Passian and A. Wig Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996-1200
F. Meriaudeau Universite´ de Bourgogne, IUT du Creusot, Le2i, 12 rue de la Fonderie, 71200 Le Creusot, France
M. Buncick Department of Physics, University of Memphis, Memphis, Tennessee 38152
T. Thundat and T. L. Ferrell Photometrics Group, Life Sciences Division, Oak Ridge National Laboratory, Bethel Valley Road, Oak Ridge, Tennessee 37830 and Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996-1200
共Received 30 March 2000; accepted for publication 24 April 2001兲 The Coulomb interaction of a dielectric probe tip with a uniform field existing above a semi-infinite, homogeneous dielectric substrate is studied. The induced polarization surface charge density and the field distribution at the bounding surface of the dielectric medium with the geometry of half of a two sheeted hyperboloid of revolution located above the dielectric half space interfaced with a uniform surface charge density is calculated. The force density on the hyperboloidal probe medium is calculated as a function of the probe tip shape. The calculation is based on solving Laplace’s equation and employing a newly derived integral expansion for the vanishing dielectric limit of the potential. The involved numerical simulations comprise the evaluation of infinite double integrals involving conical functions. © 2001 American Institute of Physics. 关DOI: 10.1063/1.1380224兴
I. INTRODUCTION
charge density at the boundary of the tip. Numerical simulations and the discussion of the results are summarized in Sec. IV with the conclusions given in Sec. V.
The development of various scanning probe microscopes 共SPMs兲 to reach higher lateral and vertical resolutions in extracting data from material surfaces has revealed numerous features ranging down to topographic and atomic scale features. A common prerequisite for the operation of SPMs is the presence of a probe tip in close proximity of the bounding surfaces of the sample, interacting with the attributes of the surface under investigation. The complicated small magnitude forces experienced by the probe tip of an atomic force microscope 共AFM兲1 is an example of such an interaction. Several experimental and theoretical approaches have been undertaken to characterize the forces prevailing in the respective ranges in the tip–sample region. These include the long-range electrostatic forces and contact forces in the angstrom domain,2 and if the substrate surface is electrically charged there is a further complication. Electrostatic considerations via application of a small potential difference between the tip and the sample have been utilized in AFM experiments to obtain topographic images.3 Here we invoke a direct analytical approach to calculate the force density on a hyperboloidal dielectric probe tip. The results presented in this work can be utilized to study the importance of tip shapes. The dielectric material considered is linear, isotropic, and homogeneous and is characterized by a dielectric function. We confine our simulations to the nanometer sized region of the tip extremity where shape variation is more pronounced with respect to the tip–sample separation. Section II introduces a short background to the formulation of the problem. In Sec. III we derive the expressions for the magnitude of the induced field and polarization 0021-8979/2001/90(2)/1011/6/$18.00
II. BACKGROUND
The calculation is carried out in spheroidal coordinate system where the tip has been modeled as half of a towsheeted hyperboloid of revolution, as shown in Fig. 1. These coordinates can automatically include both the probe and sample surfaces. Prolate spheroidal coordinates of a point with cylindrical coordinates ( , ,z) are related via x⫽ 共 , 兲 cos , y⫽ 共 , 兲 sin , 共1兲
z⫽z, z⫹i ⫽z 0 cosh共 ⫹i 兲 , with the following domains for the spheroidal variables: 0⭐ ⬍⬁,
0⭐ ⭐ ,
0⭐ ⭐2 ,
共2兲
and z 0 being a scale factor that defines the focal distance of the confocal hyperboloids spanned by Eq. 共1兲. For simplicity we make the substitutions ⫽cosh and ⫽cos . Thus with fixed values ⫽ t and ⫽0, Eq. 共1兲 maps the tip hyperboloid, and the xy plane, respectively. As is evident from Eq. 共1兲, the effect of a variation in z 0 , corresponds to zooming in or out over an arbitrary region of space. Due to the periodicity required azimuthally, the general solution of Laplace’s equation in the spheroidal coordinate system takes the form of a Fourier series in as 1011
© 2001 American Institute of Physics
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where, for a common amplitude A 0 (q), the partial ansatz ⌽ i , i⫽1,2,3 for various regions defined by the step functions ⍜共兲 are written, taking into account the symmetry of the problem as ⌽ i 共¯r 兲 ⫽
冕
⬁
0
0 A 0 共 q 兲 i 共 q, ; t 兲 P ⫺ 共 1/2兲 ⫹iq 共 兲 dq,
共6兲
where, in order for the potential to be continuous, we write the scalar functions i for i⫽1,2,3 as FIG. 1. A probe tip above a charged substrate surface. An uncharged hyperboloidal dielectric medium with ⫽ˆ above a uniformly charged dielectric half space with ⫽ ¯ . The surface of the hyperboloid is represented by t ⫽cos t . The charge density is given by , and z 0 is the focal distance of the hyperboloid.
兺
m⫽0
m 共 , 兲共 2⫺ ␦ m0 兲 cos m ,
共3兲
where m ( , ) can be expressed in terms of an infinite m integral involving conical functions P ⫺(1/2)⫹iq (u) with u 4,5 ⫽ or u⫽ as arguments. Here q苸 关 0,⬁ 关 is a real and continuous variable which allows Laplace’s equation to possess a continuous infinity of particular solutions. The magnitude of conical functions with argument cos ⫽苸关⫺1,1兴 increase exponentially with q , while for arguments beyond this range, these functions are oscillating and falling off with q and . Van Nostrand6 derives the following orthogonality relation for these functions with argument 苸 关 1,⬁ 关 , specialized for m⫽0:
冕
⬁
0
0 0 2 共 q, ; t 兲 ⫽ P ⫺ 共 1/2兲 ⫹iq 共 兲 ⫺ ␣ q 共 t 兲 P ⫺ 共 1/2兲 ⫹iq 共 ⫺ 兲 , 共7兲 0 3 共 q, ; t 兲 ⫽ 共 1⫺ ␣ q 共 t 兲兲 P ⫺ 共 1/2兲 ⫹iq 共 ⫺ 兲 ,
and where we have introduced the following notations for convenience:
⬁
⌽ 共¯r 兲 ⫽
0 1 共 q, ; t 兲 ⫽  q 共 t 兲 P ⫺ 共 1/2兲 ⫹iq 共 兲 ,
0 0 q tanh q P ⫺ 共 1/2兲 ⫹iq 共 兲 P ⫺ 共 1/2兲 ⫹iq 共 ⬘ 兲 dq
⫽␦共 ⫺⬘兲,
共4兲
which will be useful in the derivations presented in Sec. III.
III. FORCE DENSITY
The modeling geometry for the dielectric probe tip, with dielectric constant ⫽ˆ , located above a uniform surface charge density residing on a semiinfinite dielectric substrate, with dielectric constant ⫽ ¯ , is depicted in Fig. 1. As shown, the surface of the hyperboloidal tip is represented by ⫽cos t⫽t , and the substrate surface by ⫽0, with the ¯)⫽ ␦ ( ). Selecting these valcharge density given by (r ues for the boundaries of the problem, puts the probe tip’s apex at a distance z⫽z 0 t above the planar substrate and confocal with it. The dielectric constant for the space between the tip and substrate has been set to ⫽1 here, but can take on other appropriate values. The scalar electric potential is assumed to be given by ⌽ 共¯r 兲 ⫽⌽ 1 共¯r 兲 ⍜ 共 t ⫺ 兲 ⫹⌽ 2 共¯r 兲 ⍜ 共 ⫺ t 兲 ⍜ ⫹⌽ 3 共¯r 兲 ⍜ 共 ⫺z 兲 ,
冉 冊 ⫺ 2
␣ q共 t 兲 ⫽
冋
册
 q共 t 兲 ⫽
1⫺ q 共 t 兲 , ˆ ⫺ q 共 t 兲
ˆ ⫺1 K共 t兲, ˆ ⫺ q 共 t 兲 共8兲
with q ( t )⫽K( t )/K ⬘ ( t ) and K( t ) and K ⬘ ( t ) being the ratios of the conical functions and their respective partial derivatives, i.e., K共 t兲⫽
0 P⫺ 共 1/2兲 ⫹iq 共 t 兲
,
0 P⫺ 共 1/2兲 ⫹iq 共 ⫺ t 兲
冏
0 P⫺ 共 1/2兲 ⫹iq 共 兲 K ⬘共 t 兲 ⫽ 0 P ⫺ 共 1/2兲 ⫹iq 共 ⫺ 兲
共9兲
. t
In Eq. 共5兲, incorporating the continuity of the normal component of the displacement field everywhere except when ⫽0, and using relation 共4兲, and noting that K ⬘ (0) ⫽⫺1, yields the following expression for the potential in the region between the tip and the substrate ⌿ 2 共¯r 兲 ⫽ 共 4 z 0 兲
冕 ⬘ ⬘ 冕 ⬁
⬁
d
1
0
0 U共 q 兲P⫺ 共 1/2兲 ⫹iq 共 兲
0 0 0 ⫻ P⫺ 共 1/2兲 ⫹iq 共 ⬘ 兲关 P ⫺ 共 1/2兲 ⫹iq 共 兲 ⫺ P ⫺ 共 1/2兲 ⫹iq 共 0 兲 0 0 ⫹ ␣ q 共 t 兲共 P ⫺ 共 1/2兲 ⫹iq 共 0 兲 ⫺ P ⫺ 共 1/2兲 ⫹iq 共 ⫺ 兲兴 dq,
共10兲 where ⌿ 2 has been with the case where potential difference potential,7 and U(q) U共 q 兲⬅ f 共 q 兲
冉
taken as ⌽(r ¯)⫺⌽(r ¯) 兩 ⫽0 in analogy no dielectric media are present and a of this type produces the correct has been defined as
冊
q tanh q 0 P ⫺ 共 1/2兲 ⫹iq 共 0 兲 , cosh q
共11兲
with f (q) given by 共5兲
f 共 q 兲⫽
1 . ¯ 兲 ⫹ 共 1⫺ ¯ 兲 ␣ q 共 t 兲兴 关共 1⫹
共12兲
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J. Appl. Phys., Vol. 90, No. 2, 15 July 2001
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We now observe that from the definitions Eqs. 共8兲, 共9兲, and based on the asymptotic behavior of the conical functions8 for qⰇ1 and fixed argument, q ( t )→⫺1, and since 2 t ⫺ ⬍0 is always satisfied for any realistic tip shape, then as seen from Eq. 共A1兲 in the Appendix, ␣ q ( t ) →0 and U(q)→0 and thus f (q) can be written as f 共 q 兲⫽
1 ⫹g 共 q 兲 , 1⫹ ¯
共13兲
where g(q)→0 for qⰇ1, allowing the static field corresponding to the potential in Eq. 共10兲 to be written as ¯E 2 共¯r 兲 ⫽
冉 冊 1
1⫹ ¯
¯E 0 共¯r 兲 ⫹
4
冑 2 ⫺ 2
共 I ,I ,0兲 ,
共14兲
FIG. 2. The variation of the function Eq. 共8兲 with t , displaying the effect of polarization. These curves correspond, from bottom to top, to the range of tip shapes: t ⫽0.12,0.37,0.80,1.26,1.42,1.49,1.53. As can be seen, the sharper the tip, the smaller the contribution from ␣ q ( t ), whereas for higher values of t this function decreases very slowly with q.
¯)⫽ lim ¯E 2 (r ¯) and is given by where ¯E 0 (r ¯ ,ˆ →1
E 共¯r 兲 ⫽
¯0
2
冑 2 ⫺ 2
共 冑 ⫺1, 冑1⫺ ,0兲 , 2
2
共15兲
¯ →1 limit of the which is the corresponding field to the ˆ , potential in Eq. 共10兲. In this limit, an integral expansion has been utilized7 to allow for ⌿ 0 to be extracted from the potential given in Eq. 共10兲. The second term in Eq. 共14兲 is given by the following integrals: 共 I ,I ,0兲 ⫽
冕 ⬘ ⬘ 冕 ⬁
⬁
d
1
0
1 共 ⫺D 共 q 兲 P ⫺ 共 1/2兲 ⫹iq 共 兲 ,H 共 q 兲
0 0 ⫻ P⫺ 共 1/2兲 ⫹iq 共 兲 ,0 兲 P ⫺ 共 1/2兲 ⫹iq 共 ⬘ 兲 dq,
共16兲
where D (q) and H (q) have been defined as D 共 q 兲 ⫽h 1 共 q 兲 d 1 共 q 兲 ⫹h 2 共 q 兲 d 2 共 q 兲 ,
共17兲
1 1 H 共 q 兲 ⫽h 1 共 q 兲 P ⫺ 共 1/2兲 ⫹iq 共 兲 ⫹h 2 共 q 兲 P ⫺ 共 1/2兲 ⫹iq 共 ⫺ 兲 ,
with
冉 冉
冊 冊
q tanh q 0 f 共 q 兲 ␣ q共 t 兲 P ⫺ 共 1/2兲 ⫹iq 共 0 兲 , cosh2 q
共18兲
and 0 0 d 1共 q 兲 ⫽ P ⫺ 共 1/2兲 ⫹iq 共 兲 ⫺ P ⫺ 共 1/2兲 ⫹iq 共 0 兲 , 0 0 d 2共 q 兲 ⫽ P ⫺ 共 1/2兲 ⫹iq 共 0 兲 ⫺ P ⫺ 共 1/2兲 ⫹iq 共 ⫺ 兲 .
2 共 1,0,0 兲 . 1⫹ ¯
冉 冊
ˆ ⫺1 E 1共 , t 兲 . 4
共21兲
In a similar manner, as in the derivation of Eq. 共14兲, we get the following expression for p :
冋
w共 ⬁ 兲 0 E 共,t兲 2
冕 ⬘ ⬘ 冕
p 共 , t 兲 ⫽ 共 ˆ ⫺1 兲 共19兲
As implied by Eqs. 共12兲 and 共13兲, ␣ q ( t ) appears here to display the strength of the polarization whose shape dependence is shown in Fig. 2. Two interesting limiting scenarios can be discussed here. First when t ⬇1, that is the lower region of the probe hyperboloid becomes a very sharp needle with its apex located almost at z 0 , in which case ␣ q ( t ) Ⰶ1 for all q and thus g(q)Ⰶ1. Consequently, h 1 (q)Ⰶ1, and h 2 (q)Ⰶ1 which in turn results in a minimum contribution from the second term in Eq. 共14兲, i.e., the field along the hyperboloid and away from the curved part tends to a constant. In fact, in the limit t →1 we have I , I →0 and the field is given by
共20兲
The second case is when t →0, that is a very wide probe residing extremely close to the surface. At such values of tip shape as is evident from Fig. 2, ␣ q ( t ) does not fall off with q fast enough for the inner integral in Eq. 共16兲 to converge in which case the surviving field component proportional to I diverges (I ⬇0). This divergent integral, Eq. 共A4兲 is given in the Appendix. As will be seen from the expressions for the induced polarization charge density and the force density, these quantities will also diverge due to the same divergent integral, Eq. 共28兲. We also note here that as the dielectric media vanish, we have h 1 (q)→0 and h 2 (q)→0, so that D (q)→0 and H (q)→0 and thus the field, as given by Eq. 共14兲, reduces properly. The polarization surface charge density p can now be obtained from
p共 , t 兲 ⫽
q tanh q 0 h 1共 q 兲 ⫽ g共 q 兲P⫺ 共 1/2兲 ⫹iq 共 0 兲 , cosh2 q h 2共 q 兲 ⫽
¯E 2 共¯r 兲 →
⫹
冑
2
⫺ 2t
⬁
⬁
d
1
0
q tanh q 1 P⫺ 共 1/2兲 ⫹iq 共 0 兲
册
0 0 ⫻w 1 共 q 兲 P ⫺ 共 1/2兲 ⫹iq 共 兲 P ⫺ 共 1/2兲 ⫹iq 共 ⬘ 兲 dq ,
共22兲 where w(q)⫽ f (q)  q ( t ) has been written as w(q)⫽w(⬁) ⫹w 1 (q) with w 共 ⬁ 兲 ⫽ lim w 共 q 兲 ⫽ q→⬁
2 ¯ 兲共 1⫹ˆ 兲 共 1⫹
.
共23兲
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FIG. 3. The translation of the previously confocal hyperboloids to display the shape variation. A scale factor compensation of z ⬘0 ⫽z 0 / ⬘ has been applied to each hyperboloid represented by ⬘.
Now due to the symmetry of the problem, the force on the tip will be parallel to the symmetry axis of the hyperboloid and thus the force density is given by dF z 共 , t 兲 2 z 20
⫽ p 共 , t 兲 冑1⫺ 2t 共 t 冑 2 ⫺1E 1 共 , t 兲 ⫹ 冑1⫺ 2t E 1 共 , t 兲兲 d .
共24兲
IV. NUMERICAL RESULTS AND DISCUSSIONS
The numerical simulations of the results attained in the last section are based on the evaluation of the infinite double integrals with the integration in the ranges q苸 关 0,⬁ 关 and 苸 关 0,⬁ 关 . The precision of the results depends critically on the precision with which the conical functions are evaluated for high values of their argument and lower index, and on the 0 (cosh ) oscilintegration method. The function P ⫺(1/2)⫹iq lates with a frequency proportional to q 8 making the evaluation of the q integration less effective for fixed distant observation points and a large value for the source point. The evaluation of the conical functions have been based on the integral representation for appropriate range of values for q, , and , and on the proper asymptotic expansion for higher or especial values of these attributes.9,10 Asymptotic expansions have also been considered in the evaluation of the involved double integrals. All numerical integrations were based on a five-point Newton–Cotes algorithm.
FIG. 4. The relative polarization charge density for various tip shapes t in the range 0.17⭐ t ⭐0.65. Each curve represents Eq. 共22兲 for a t in this range. Here the induced polarization charge density has been normalized by , the substrate charge density.
FIG. 5. The component of the total field evaluated at the tip boundary t , assuming a charge density of ⫽1.0 C/cm2. We observe that this tangential component of the field vanishes at the apex of the tip, that is for →1 for all tip shapes, displaying the symmetry of the problem.
The results presented here have been adopted for tip shapes, charge density, and a tip dielectric material used in contemporary AFM experiments.11 The charge density has been taken as ⫽1.0 C/cm2 and the examined tip angles range from t ⫽0.17 for a very sharp tip to t ⫽0.65 for a relatively dull tip. Figure 3 shows a series of tip shapes considered in the simulations here, including the corresponding modeling hyperboloids for the range mentioned above. Here, in order to display the tip shape variation only, a scale factor compensation of z 0 / ⬘ has been applied to each hyperboloid represented by ⬘. To get an estimate for the tip apex radius corresponding to the range t 苸 关 0.17,0.65兴 , a sphere can be fitted to the tip’s extremity which generates the range of radii R苸 关 1.0,10.0兴 nm, for z 0 ⫽10.0 nm, which is within the range of microfabricated tips. These values will obviously be different if one was to employ another modeling surface such as that of a parabola with a different apex curvature. The dielectric constant of the tip and substrate media, has been set to ˆ ⫽ ¯ ⫽4.0, which approximately equals the lower end of the range 4.8 –9.4 pertaining to the dielectric constant of silicon nitride (Si3N4) at 100 kHz.12 As confirmed by Eqs. 共8兲 and 共12兲, the higher the dielectric constant of the tip, the stronger the interaction with the field. Figure 4 shows the induced charge density, Eq. 共22兲, as a function of tip shape and the distance along the hyperboloid away from the apex 共increasing 兲. The magnitude of the charge density decays faster for the sharper tip, as one may expect. The tangential component of the field at the boundary of the tip is shown in Fig. 5. This figure also displays the
FIG. 6. The total field evaluated at the tip boundary t , assuming a charge density of ⫽1.0 C/cm2.
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J. Appl. Phys., Vol. 90, No. 2, 15 July 2001
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TABLE I. Evaluation of the force as a function of the tip shape t and the scale factor z 0 for four typical values of the shape parameter.a The second column displays the diameter of a sphere with a curvature equal to that of the extremity of the hyperboloidal tip. It should be noted that this is a direct function of z 0 , and can be adjusted to get other appropriate values. The third column gives the corresponding gap sizes.
FIG. 7. The force density on the dielectric probe tip, calculated from the induced polarization charge density and the field at the boundary.
symmetry of the model as this component vanishes in the vicinity of the apex, and reaches a constant when the curvature of the hyperboloid settles towards cot t , the limit of the curvature. Also seen in this figure is the effect of the finite source mesh for the wider tips whose lateral dimensions stretch further out from the axis of the hyperboloid. We note here that since in prolate spheroidal coordinates, the differential surface elements is smaller for smaller t , the corresponding area spanned along the hyperboloid, for fixed , will consequently be smaller. The magnitude of the total field on the tip is shown in Fig. 6. Again the field falls off to a constant value away from the apex. The force density, Eq. 共7兲, plotted in Fig. 7 exhibits the combined features of the induced charge density and the field. Finally, the behavior of the total force as a function of the shape is given by integrating the density equation 共7兲 over the finite apex region of the tip extending far enough for the curvature of the tip to have little or no variation. Figure 8 displays this result where the projection of the considered surfaces onto the z axis has been taken as z⫽z 0 t (cosh⫺1) with ⫽5.0. Table I summarizes the force and the corresponding geometrical parameters of the system for four experimentally relevant values of the tip shape parameter. V. CONCLUSION
In summary, it has been demonstrated that the direct analytical approach presented can produce results that are useful in characterizing the overall influence of the tip shape. All the simulated quantities indicate that the tip shape dependency is confined to the region where tip curvature variation is non-negligible as a function of . For the tip angles t in
a
t
D( t ,z 0 ) 共nm兲
d( t ,z 0 ) 共nm兲
F 共nN兲
12.0° 17.0° 18.0° 35.0°
1.0 1.8 2.0 8.1
9.78 9.56 9.51 8.19
0.37 0.95 1.03 4.84
See Ref. 11.
the range t 苸 关 0.17,0.65兴 considered, the variation in the curvature of the tip is less than 3.6% for as low as 2.0. In the employed coordinate system, there is no explicit way to compensate for the inherent translation of the tip towards the substrate with vanishing t . However, the effect is accounted for implicitly by the variation in the surface element along the hyperboloid. This effect is consistent with the self similarity associated with this coordinate system as indicated by Eq. 共14兲, and 共22兲 which lack an explicit dependence on z 0 . Thus, as suggested by Fig. 4, as the induced polarization charge density decays faster for the sharper tip away from the apex, the average force experienced will be smaller relative to the broader tip. ACKNOWLEDGMENT
This work was supported by the Oak Ridge National Laboratory managed by Lockheed Martin Energy Research Corp. for the U.S. Department of Energy under Contract No. DE-AC05-96OR22464. APPENDIX
The asymptotic behavior of ␣ q ( t ) and  q ( t ) can be derived by the use of the asymptotic forms of the involved functions8
␣ q共 t 兲 ⬇ ˆ
冉 冊冉 冊 共 ˆ ⫺1 兲 e q 共 2 t ⫺ 兲 1 3 8 tanh t ⫺ 8 tanh t ⫹ q q ⫹ 1 3 8 tanh t ⫹ 8 tanh t ⫺ q q
2 , 1⫹ ˆ q→⬁
 q共 t 兲 →
→ 0, q→⬁
共A1兲 共A2兲
and similarly for U(q) U共 q 兲⬇
冑2 qqe ⫺ 共 /2兲 q ¯ 兲 ⫹ 共 1⫺ ¯ 兲 ␣ q 共 t 兲兴 关共 1⫹
→ 0.
→
冑2 qqe ⫺ 共 /2兲 q ¯兲 共 1⫹ 共A3兲
q→⬁
FIG. 8. Total force on a segment of the hyperboloid.
When the probe angle becomes very large, the filed component E is approximately given by the following divergent integral:
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I ⬇
冕 ⬘ ⬘ 冕 ⬁
⬁
d
1
0
0 0 ⫻ P⫺ 共 1/2兲 ⫹iq 共 兲 P ⫺ 共 1/2兲 ⫹iq 共 ⬘ 兲 dq.
共A4兲
G. Binnig, C. F. Quate, and Ch. Gerber, Phys. Rev. Lett. 56, 930 共1986兲. B. Cappella and G. Dietler, Surf. Sci. Rep. 34, 1 共1999兲. 3 R. Erlandsson, G. M. McClelland, C. M. Mate, and S. Chiang, J. Vac. Sci. Technol. A 6, 266 共1988兲. 4 T. L. Ferrell, Nucl. Instrum. Methods Phys. Res. B 96, 483 共1995兲. 5 T. L. Ferrell, Phys. Rev. B 50, 14738 共1994-I兲. 1 2
R. G. Van Nostrand, J. Math. Phys. 10, 276 共1954兲. A. Passian, H. Simpson, and T. L. Ferrell 共unpublished, 2000兲. 8 M. I. Zhurina and L. N. Karmazina, Tables and Formulae for The Spherical Functions P ⫺(1/2)⫹i (x) 共Pergamon, New York, 1966兲. 9 N. K. Chukhrukidze, USSR Comput. Math. Math. Phys. 6, 86 共1966兲. 10 K. S. Ko¨lbig, Comput. Phys. Commun. 23, 51 共1981兲. 11 The microfabricated commercial AFM tips are in the range: Ultra-Levers 共Park Scientific Instruments兲: t ⫽12°, Olympus: t ⫽17°, Micro-Levers: t ⫽18°, Micro-Levers without a tip: t ⫽35°. 12 J. T. Milek, Handbook of Electronic Materials 共IFI/Plenum, New York, 1971兲 Vol. 3. 6
0 q tanh共 q 兲 sech2 共 q 兲 P ⫺ 共 1/2兲 ⫹iq 共 0 兲
7
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