Mechanism of nanostructure movement under an electron beam and its application in patterning

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Written by Kevin Tian, AP 225, Fall 2011

--Ktian 05:44, 28 October 2011 (UTC)

Title: Mechanism of nanostructure movement under an electron beam and its application in patterning

Authors: Agnese Seminara, Boaz Pokroy, Sung H. Kang, Michael P. Brenner, and Joanna Aizenberg

Journal: Physical Review B, Vol 83, Issue 23, Page 5438 (2011) [6 pages]

Keywords: self-assembly, charged interface, electrostatics, patterning, electron-beam

Note: For clarity of presentation, intermediate equations will be provided but not numbered. Important equations will be numbered.

Paper Summary

For all the fanfare surrounding the development of electron microscopy, there are still many disadvantages to using techniques involving an electron beam. Since the sample features inevitably interacts with the electron beam in a seemingly uncontrolled fashion there has been a limit to the effectiveness of using electron microscopy (as well as e-beam writing). This includes limitations to resolution, the distortion of surface features and alteration of material properties. However the precise mechanism through which this occurs is, for the most part, not well quantified. This paper gives an experiments study of e-beam-induced motion of a nanopost array and attempts to quantitatively describe the phenomenon using a combination of electrostatics and mechanical deformation.

The essential proposal is that the e-beam bombardment generates an a spatially varying distribution of electrons in the sample, which then causes electrostatic forces and torques of sufficient magnitude to deform them in a visible manner. The model is compared against controlled experiments that support the model. The potential applications of the theoretical understanding are then demonstrated by the creation of pseudo-3D structures using only e-beam induced deformation.


Figure 1.
Table 1.


The systematic study of e-beam induced motion is setup as follows:

  • An array of cylindrical nanoposts of height H and radius r were fabricated from epoxy resin via soft lithography (See Figure 1a). These structures are attached on a supporting surface of the same material.
    • Polymeric nanoposts were selected for their relatively low force requirement for motion (due to high aspect ratio)
  • The entire sample is sputter-coated with a single 5-10nm layer of either gold or carbon.
    • Gold coating done with a Cressington 208 HR Sputter coater
    • Carbon coating done with a Baltech CED 030 Carbon Arc Coater.
  • The sample is placed into an SEM and grounded.
    • Imaging was done in a Zeiss Ultra 55 Field-emission Scanning Electron Microscope at Harvard CNS.

After imaging the surface for 1-5sec in the SEM, the nanoposts were observed to bend (as seen in Figure 1b). The bending showed three characteristic features: 1) Nanoposts near the center of the scanning window showed virtually no deflection 2) Nanoposts near the edge of the scanning window showed maximum deflection 3) All deflections were directed towards the center of the scanning window. These patterns were reproducible and persisted for several seconds before relaxing to their original positions (after reducing magnification). This demonstrates that the deflection is reversible and related to the presence of the electron flow.

This experiment was repeated for varying beam energies, beam currents and materials.

  • Beam Energies (1-30keV)
    • Deflection was not a monotonic function of beam energy
    • A maximum deflection was observed at 5keV, after which deflection decreased
  • Beam Current (10-150pA)
    • Deflection monotonically increase with beam current
  • Materials (Epoxy, n-type Si, Alumina) - see Table 1
    • Only epoxy and alumina were observed to deflect. n-type Si did not deflect.
    • Since Alumina is stiffer than Si, the basis is clearly not in mechanical stiffness.
    • However we notice n-type Si is a conductor, whereas the other two are insulators. This suggests an electrostatic interaction.


Theoretical Models for Implanted Charge

Figure 2.

The essential electrostatic model the authors propose is relatively simple.

  • 1) Electrons from the e-beam are implanted as a result of backscattering.
  • 2) These electrons cause charging and induction, producing electric fields inside the nanoposts. These fields cause forces and torques that cause deflection.

The authors then propose a framework that seems a little confusing to me, where the arguments for the implanted charge model depend on Monte Carlo simulations for justifications. Yet these simulations are used to compute the values based on the model they propose.

Regardless, the idea is boiled down to the fact that based on electron trajectory simulations, there is reason to believe the charge distribution is inhomogenously implanted. The detailed distribution of charge is (apparently according to the authors, the reasons for this are not described) unable to be determined experimentally or through simulations. Thus the center of the charge distribution is left as a free parameter.

This then reduces to an electrostatic equilibrium problem coupled to the mechanical bending of the nanoposts.

The model then considers a single nanopost, of radius r, with an e-beam in 'spot mode' at a distance a from the nanopost (see Figure 2a for a sketch). The e-beam has an incoming current i. The logic then is as follows:

  • After impact the electrons are backascattered with backscattering coefficient <math>\eta\,\!</math>
  • The Probability that electrons backscatter with angle <math>\theta\,\!</math> is <math>P(\theta)=cos(\theta)\,\!</math>
  • The fraction of electrons impacting the nanopost from this backscattering is <math> r \over (\pi a)</math>
  • Of this fraction impacting the nanopost a smaller fraction are further backscattered away (with nanopost backscattering coefficient <math>\eta_n\,\!</math>)
    • This leaves <math>1-\eta_n\,\!</math> fraction of backscattered electrons implanting themselves in the nanopost.
    • <math>\eta_n\,\!</math> depends on the angle of impact, <math>\theta\,\!</math>.
  • This leads to the total current per unit height impacting the nanopost, a distance z from the substrate:
    • <math>(1) \qquad i_n = \eta(1-\eta_n) {r \over (\pi a)} P(z) i </math>,
    • where <math> P(z) = P(\theta) \left \vert {d \theta \over d z} \right \vert = {a z \over (a^2 + z^2)^{3/2}} </math>

We now defined a charge per unit height at distance z from substrate, <math>\lambda(z,t)</math> then we make the claim that:

  • <math>\dot \lambda(z,t) = i_n - \lambda / \tau \,\!</math>
    • Charge relaxation time <math>\tau \sim \epsilon / K\,\!</math>. K is conductivity and <math>\epsilon=\epsilon_0 \epsilon_r</math> is the permittivity of the material times the permittivity of free space.
  • In steady state consitions we get:
    • <math>(2) \qquad \lambda(z) = i_n(z) \tau \,\!</math>
    • This ignores longitudinal currents that are a result of secondary emissions (which have been justified by the authors as negligible in effect).

Monte Carlo Simulations

As observed in equations (1) and (2), we require the backscattering coefficients in order to evaluate <math>\lambda(z)</math>. Monte Carlo simulations were used to obtain these values by simulating the electron trajectories (this was accomplished by an electron flight simulator from Ted Pella, Inc., see Figure 2b). These simulations provide the backscattering fraction directly for each angle of impact. The results are averaged over 6 simulations.

The simulations provide an explanation for the the maximum deflection occurring at intermediate energies. When repeated for different electron energies (Figure 2c) it was noticed that electrons travel to different depths depending on their energy. Too low and the electrons don't penetrate the nanopost at all (2keV). Too high and the electrons simply travel straight through the nanopost entirely (10keV). Either case results in virtually no electron implantation, and thus minimal bending. There is essentially a window of energies that allow for implantation of electrons in the nanopost (just enough energy to penetrate the nanopost but not enough to go entirely through), thus explaining this intermediate energy being the maximal deflection case.

The authors note that due to this energy barrier of 2keV at the nanopost interface, this is indicative of a non-uniform electron distribution. Electrons with energy less than the interfacial energy are decelerated to rest before leaving the sample. Therefore the expectation is that charges will be biased to the nanopost side opposite to electron impact.

Electrostatically Driven Bending Model

Figure 3.

Due to the complex nature of the problem, Finite Element Method (FEM) simulations are then performed in order to obtain a theoretical result to compare to the experimental ones. The purpose of the FEM simulations is to determine the fores and torques generated from uneven charge density <math>\lambda</math> (see Equation 2) by using the Monte Carlo parameters <math>\eta , \eta_n</math>. However the FEM simulation was based on equations derived from classical electrostatics, which will be discussed below.

The very well known starting point for any equilibrium electrostatic problem is the Poisson equation:

<math>(3) \qquad \nabla^2 \phi = -{\rho \over \epsilon}</math>

For an equilibrium electrostatic potential <math>\phi</math> , implanted charge density <math>\rho</math> and permittivity of the material <math>\epsilon</math>. Regarding Equation 3 we note that since the implanted charge is distributed along a (square) cylinder of cross section <math>l^2</math> centered at a distance from the nanopost axis <math>x_c</math>, we can write <math>\rho = \rho (z) = \lambda (z) / l^2</math> . However for brevity we shall still refer to the quantity as <math>\rho</math>, though this statement becomes useful for interpreting the results. To illustrate the physical picture, see Figure 3b (depicting the nanopost with described quantities marked).

The domain of the problem is described in Figure 3a (essentially a 2D domain with grounded external boundaries with a singular nanopost zoomed in to be shown in Figure 3b).

We now consider the action of an electric field <math>\mathbf{E}</math> on a static charge <math>\rho</math> and induced surface charge <math>\sigma</math> that generates an electrostatic force w and torque N. This yields the following equations:

<math>\mathbf{E}=-\nabla \phi</math>

<math>\sigma = \epsilon \mathbf{E} \cdot \hat n</math> , for <math>\hat n</math> unit vector perpendicular to the coating

For surface area of the nanopost S, distance from nanopost center x and superscripts + or - representing the quantity for the nanopost side exposed to or opposite the e-beam respectively we have the following expression for forces:

<math> (4) \qquad \mathbf{N} = \int\limits_S \rho \mathbf{E} \times \mathbf{x}~d^2 x</math>

<math> (5) \qquad \mathbf{w} = \int\limits_S \rho \mathbf{E}~d^2 x + r(\sigma^{-} \mathbf{E}^{-} + \sigma^{+} \mathbf{E}^{+}) </math>

Solving for the small bending of a thin rod (see Landau and Lifshitz Theory of Elasticty (1959) for details), we obtain the following:

<math>(6) \qquad {d \mathbf{M} \over d z} = - \mathbf{N} - \hat z \times \mathbf{f}</math>

<math>(7) \qquad {d \mathbf{F} \over d z} = - \mathbf{w}</math>

where we have shear force F and bending moment M. We defined the bending moment as:

<math>\mathbf{M} = -E I \partial^{2}_z \mathbf{u}</math>

Where...<math>\mathbf{u} = u \hat x</math> is the displacement of the centerline, <math>I = {\pi r^4 \over 4}</math> is the area moment of inertia, and E is Young's Modulus.

FEM Simulation Results

Using FEM based software (COMSOL Multiphysics) one can compute the electrostatic forces and torques as described in Equations 4 and 5. Figure 3c shows the shape of the carbon-coated nanopost at maximum bending. Figure 3d shows carbon- and gold-coated nanopost's maximum tip displacements as a function of <math>x_c</math>. These calculations predict inward bending when <math>x_c < 0</math> (when charge accumulates towards side opposite to e-beam exposure), which was verified with samples bending inward only when grounded.

Figure 4.

Figure 4a illustrates the <math>\phi</math> distribution for <math>x_c=-0.08\mu m</math>. It was observed that the most significant contribution to nanopost bending was <math>E_z</math>, of which the average intensity profile over z is given in Figure 4b. Figures 4c-e describe the force per unit length <math>F_z=\lambda E_z</math>, torque per unit length <math>N = x_c F_z</math> and bending moment M.

It is noted that in order for the steady state charge condition to be satisfied (the basis for Equation 2), current must flow to ground until the charge implanted in each portion of the nanopost (<math>i_n</math>) is equal to the charge flowing to ground through the coating (<math>\lambda / \tau</math>). The force contribution from these currents are neglected on the basis that these currents contribute negligibly compared to the static components. An order of magnitude estimate yields a <math>10^{-18} nN</math> contribution to torque from the current (using values from Figure 1's caption), whereas static charge torque is on the order of <math>1nN-400nN</math> from tip to base of nanopost, as seen in Figure 4d (see page 5 of the paper for the detailed reasoning). The authors used a number of approximations I'm not completely confident apply, however the essential reasoning appears good enough to yield the approximation they desire.


Comparison to Experiment

Figure 5.

By extension of the single nanopost case with a focused e-beam point, one can easily explain the scanning mode actuation. Essentially the scanning is fast enough that charge is distributed across all nanoposts simultaneously. However nanoposts on the edges are exposed to only an e-beam from the inside of the scanning window, yielding a case similar to the single nanopost consideration, and thus maximum bending (since for all posts inside there is exposure to e-beams from varying directions).

Nanoposts in the center of the scanned region however are exposed equally to an e-beam from all diretions and thus receive equal amounts of backscattered electrons from all directions, negative any symmetry breaking electron implantation. This would explain the minimal deflection of nanoposts nearer the center of the scanning region. All regions in between thus have an effect somewhere in-between the two extremes.

The model also explains the lack of deflection in n-doped silicon, as the implanted charge is directly proportional to the charge relaxation time <math>\tau</math>. Since this quantity is significantly smaller for doped Silicon than it is for epoxy, it would take a significantly longer time to charge up doped-Si. What I would have liked for the authors to do to test this model is to see if it would be possible to still deflect doped silicon by engineering the situation to minimize implantation charging time by waiting longer and altering relevant parameters as per equations 1 and 2. However this point was skipped over. The results for Alumina also matched theoretically (roughly speaking, no uncertainties are given).

Figure 5 demonstrates some separate testing done to verify the theoretical basis. Focusing the e-beam near a nanopost causes it to bend towards the focus point (Figure 5a). Focusing the e-beam in the center of a nanopost causes no deflection of the cylinder (Figure 5b).

Furthermore by changing some properties of the sample it should be possible to alter the direction of bending. This is illustrated in Figures 5c and 5d. Although direct numerical simulations of ungrounded cases are not possible due to the coupling of length scales too far apart (nm to mm) it is still possible to experimentally alter the secondary backscattering yield (making it significant, whereas before with the grounded case it wasn't) by simply not grounding the sample. Upon removing ground gold-coated samples bent inwards (Figure 5c) whereas carbon-coated samples bent ouwards (Figure 5d).

Discussion & Conclusions

~~Potential Applications~~

Figure 6.

In order to illustrate the potential of the tool as a patterning technique, examples are presented where this technique can be utilized. It is known that fixing the e-beam at a position between several nanoposts will cause all of the cylinders to bend towards the e-beam focal point. When the nanopost tips contact each other they have the tendency to stick to each other (the exact reasoning is left outside the paper's scope unfortunately). The authors speculate this is likely to be due to Van der waal's interactions, however this is entirely unconfirmed.

Regardless, large scale patterning is possible by either writing various clusters through use of point scans (as seen in Figure 6a) or by using line scans (as seen in Figure 6b). The impressive fact is that this is entirely possible with any SEM system with a minimum writing voltage of approximately 4keV. Compared to current e-beam writing techniques that, not only require special equipment but also, require voltages as high as 90 keV. It is believed this value can be lowered further, thus enhancing this technique's potential for application in a wide variety of materials.

~~Personal Notes~~

The general concept seems to be upheld with the comparison with results, and clearly there is some support for the theory based on their experimental evidence. However some parts of the theory do not sit well with me personally. This originates primarily from several claims that are not explained in detail, nor are any references cited as additional material.

Intuitively one can certainly guess that the charge from an e-beam would likely not be implanted in a homogenous fashion. However the fact that a detailed charge distribution cannot even be simulated seems dubious to me. I see no reason preventing the simulation or an experimental determination of the distribution. The reason for the claim is never detailed in the paper. Also the longitudinal current estimation I did not quite follow the reasoning for, as it was not immediately clear to me why their approximations were valid for the system in consideration. However (for now) I will put this down to my lack of recent experience with electromagnetism.

Also the "verification" from experiment was rather rough in terms of a quantitative verification. As far as I could tell there were many approximations in the comparison between experiment and theory, such as for the amount of tip deflection, waiting times before charging of the tip, etc. Quantities were given but not to any consistent degree of accuracy. More quantitative experimental data would have been good.

There's several points where the paper leaves off several points as "not being in the scope of the paper" which is rather disappointing, but understandable.