Copyright � 2006 American Scientific Publishers All rights reserved Printed in the United States of America Journal of Computational and Theoretical Nanoscience Vol. 3, 28�41, 2006 Theoretical Analysis of Diamond Mechanosynthesis. Part III. Positional C2 Deposition on Diamond C(110) Surface Using Si/Ge/Sn-Based Dimer Placement Tools
Jingping Peng,1 Robert A. Freitas, Jr.,2 Ralph C. Merkle,3 James R. Von Ehr,1 John N. Randall,1 and George D. Skidmore1
2 1 Zyvex Corporation, 1321 North Plano Road, Richardson, TX 75081, USA Institute for Molecular Manufacturing, Suite 354, Palo Alto, CA 94301, USA 3 Georgia Institute of Technology, Atlanta, GA 30332, USA This paper extends an ongoing computational and theoretical investigation of the vacuum mechanosynthesis of diamond on a clean C(110) diamond surface from carbon dimer (C2 ) precursors, using Si-, Ge-, and Sn-substituted triadamantane-based positionally-controlled DCB6 dimer placement tools. Interactions between the dimer placement tools and the C(110) surface are investigated by means of stepwise ab initio molecular dynamics (AIMD) simulations, using Density Functional Theory (DFT) with generalized gradient approximation (GGA), implemented in the VASP software package. The Ge-based tool tip provides better functionality over a wider range of temperatures and circumstances (as compared with the Si or Sn tool tips). The transfer of a single carbon dimer from the Si-based tool tip onto C(110) is not controllable at 300 K but is workable at 80 K; the Ge-based tool remains workable up to 300 K. Geometry optimization suggests the Sn-based tool deposits reliably but the discharged tool is distorted after use; stepwise AIMD retraction simulations (at 300 K for the Sn tip) showed tip distortion with terminating Sn atoms prone to being attracted towards the surface carbon atoms. Stepwise AIMD shows successful placement of a second dimer in a 1-dimer gapped position, and successful intercalation of a third dimer into the 1-dimer gap between two previously deposited dimers, on clean C(110) at 300 K using the Ge tool. Maximum tolerable dimer misplacement error, investigated by stepwise AIMD quantification, is 0.5 � in x (across trough) and 1.0 � in y (along trough) for a positionally-correct isolated C2 deposition, and 1.0 � in x and 0.3 � in y for C2 intercalation between two gapped ad-dimers. Rotational misplacement tolerances for dimer placement are �30 for the isolated dimer and -10 /+22 5 for the intercalated dimer in the xy plane, with a maximum tolerable "in plane" tip rolling angle of 32.5 and "out-of-plane" tip rocking angle of 15 for isolated dimer. Classical molecular dynamics (MD) analysis of a new Ge tooltip + handle system at 80 K and 300 K found that dimer positional uncertainty is halved by adding a crossbar in the most compliant direction. We conclude that the Si-based and Ge-based tools can operate successfully at appropriate temperatures, including up to room temperature for the Ge-based tool. RESEARCH ARTICLE Keywords: Adamantane, AIMD, Carbon, Density Functional Theory, Diamond, Dimer Placement, Germanium, Mechanosynthesis, Nanotechnology, Positional Control, Silicon, Tin, Tooltip, VASP. 1. INTRODUCTION
Merkle and Freitas1 have proposed the use of Si-, Ge-, Sn-, and Pb-substituted derivatives of the hydrocarbon cage molecule triadamantane as end effectors (placement tools) in an AFM-based nanopositioning apparatus for the vacuum mechanosynthesis of diamond nanostructures, via the pick-and-place mechanochemistry of carbon dimers
Author to whom correspondence should be addressed. onto an existing diamond seed cleaved along the C(110) surface plane. With a carbon dimer covalently attached to two terminal Si, Ge or Sn atoms and the substituted triadamantane tooltip (DCB6) either attached to a scanning probe or integrated into an extended diamond lattice, the carbon dimer can be positioned and deposited onto a growing diamond substrate. The success of this process is based on the premise that a typical C�Si, C�Ge or C�Sn bond is weaker than a typical C�C bond2 3 and will dissociate
doi:10.1166/jctn.2006.003 J. Comput. Theor. Nanosci. 2006, Vol. 3, No. 1 1546-198X/2006/3/001/014 28 Theoretical Analysis of Diamond Mechanosynthesis Peng et al. first, leaving the carbon dimer covalently attached to the diamond surface. This paper (Part III) extends an ongoing computational and theoretical investigation of the vacuum mechanosynthesis of diamond on the clean C(110) surface using positionally-constrained carbon dimer (C2 ) precursors. Part I provided a detailed atomic picture of the dimermediated surface chemistry during the gas-phase growth of dehydrogenated diamond C(110) from C2 plasmas, deducing some of the many possible stabilized defects that can be formed early in the dimer-mediated diamond growth process.4 Part II analyzed the chemical stability and recharging of dimer placement tools from the view of organo-synthesis, presented reaction path potential energy profiles and analysis of a small number of ab initio molecular dynamics (AIMD) simulations of tool retraction events, and established preliminary constraints on the required positional precision needed to avoid the formation of stable defects during positional dimer placement to achieve diamond growth.5 The present work reports new results of electronic structure geometry optimization and stepwise AIMD simulations of the placement of isolated carbon dimers onto the clean diamond C(110) surface using Si-, Ge-, and Sn-based C2 dimer placement tools under conditions of constant number of particles N , volume V , and temperature T (the canonical or constant NVT ensemble), including: (1) studies of deposition and tooltip retraction event sequences; (2) stepwise AIMD simulations of the placement of a second dimer in a gapped position and the subsequent intercalation of a third dimer into the 1-dimer gap between these two previously deposited dimers, on the clean C(110) surface; and (3) stepwise AIMD analysis of the maximum tolerable dimer misplacement errors, both rotational and translational, that will yield a positionallycorrect C2 deposition onto the diamond C(110) surface. The classical molecular dynamics (MD) analysis of temperature effects on the positional uncertainty and control of the terminal carbon dimer for representative tooltips and extended handle structures are also performed in order to aid future specification of experimental protocols to achieve practical diamond mechanosynthesis. 2. COMPUTATIONAL CONSIDERATIONS
Paper II reported our studies, conducted using ab initio electronic structure calculations and constant number of atoms, volume, and energy ab initio molecular dynamics (AIMD) simulations, of the positionally-controlled placement of C2 carbon dimers on the clean (dehydrogenated) diamond C(110) surface.5 From the reaction path potential energy plots for deposition and retraction of the Si/Ge triadamantane tool, considering that there are two pathways for retraction and assuming the lowest energy reaction pathway would be followed, it was initially concluded that the Si/Ge dimer placement tools would not leave the
29 terminal carbon dimer bonded to the diamond substrate surface during retraction. A small number of AIMD stepwise simulations were performed to confirm this conclusion and to investigate the effects of internal energy on the tool retraction event. These AIMD simulations were run under conditions of constant number of atoms N , volume V , and energy E (the microcanonical or constant NVE ensemble) and were initiated at 300 K. The results of the AIMD simulations generally supported the preliminary interpretation from the reaction path potential energy plots, but one of the five AIMD simulations using the Ge tool provided a successful dimer deposition. This left two questions unanswered for future work: (1) whether two pathways must always exist and be accessible during dimer placement tool retraction, and (2) whether an assumption of constant NVE conditions is applicable for AIMD simulations of the real process of deposition/retraction events using a dimer placement tool. In the present work, we address these two outstanding issues and extend previous lines of investigation. Regarding the first unanswered question, both pathways of the branched mechanosynthetic reaction (i.e., following one pathway the dimer remains on the surface, while following the other pathway the dimer remains on the tool tip) are accessible only if the probabilities of breaking the two C(dimer)�C(surface) bonds and breaking the two C(dimer)�Si/Ge/Sn(tip) bonds are approximately equal during the process, such that the retraction is able to proceed in either direction, leaving the pathway with the lower reaction barrier to dominate. Obviously, the picture of two pathways is oversimplified. The real situation is more complex--as will be seen in the results, the two simplistic pathways do not capture the simulated phenomena well enough. The typical failure event is not an undeposited dimer remaining on the tool tip, but rather is a dimer that has rotated such that one of its carbon atoms remains bonded to the tip with its other carbon atom still bonded to the surface, leaving an unrecoverable situation. During dimer placement tool retraction, the carbon dimer interacts directly with the two corresponding carbon atoms of the surface and the two terminal Si/Ge/Sn atoms of the tool but also with neighboring atoms of both surface and tool. The dimer placement tool retraction is found to be a dynamic and complex process with all atoms librating randomly in all directions and constantly changing relative positions, especially in the region of interest (ROI)--in our discussion, the C (surface)�C (dimer)� Si/Ge/Sn (tool) atoms--with net forces that stretch ROI bonds changing during the retraction, and with these forces changing more drastically at higher system temperatures. The analysis provided from a static view of the potential energy curves and from simple theoretical models is therefore not sufficient to predict tool retraction behavior, which requires a dynamic approach. A properly designed ab initio molecular dynamics (AIMD) simulation approach
J. Comput. Theor. Nanosci. 3, 28-41, 2006 RESEARCH ARTICLE Peng et al. Theoretical Analysis of Diamond Mechanosynthesis can be used to mimic the real behavior of a system at a specific temperature. To fully simulate a process the speed of real events must be taken into account, but a full AIMD simulation would become an extremely time consuming task, making it impractical to perform complete AIMD calculations in our study. For this reason, the stepwise AIMD simulation was adopted. The stepwise AIMD simulation is not designed to mimic the continuous "real" progress of an event, but rather to check whether or not some phenomenon of interest--say, the breakage of specific bonds--will occur during the progress of an event. Using this method for modeling tooltip retraction, we artificially raise the tool with a reasonable step-size, then fix the positions of some atoms on the top of the tool and perform AIMD simulation under proper conditions for a reasonable period of time with a suitable time-step. The tool displacement step-size should be small enough not to affect the phenomenon of interest, and the simulation time should be long enough to capture the phenomenon of interest. It is of first importance to choose the proper conditions under which such stepwise AIMD simulations are performed. Regarding the second unanswered question, the choice of using a constant NVT ensemble or a constant NVE ensemble to sample the dimer placement tool retraction process is determined by whether energy can diffuse away from the ROI during the time allowed in a practicable operation. If the practicable retraction speed is quite slow such that any changes in internal energy can be redistributed, letting the system reach equilibrium with the environment, then a constant NVT ensemble is appropriate. If the practicable retraction speed can be extremely fast so that internal energy changes are confined within the ROI, then a constant NVE ensemble should be considered. Comparing the energy transfer speed in the system to the motional speed of a typical scanning tunneling microscope (STM) tip (the closest existing laboratory device to that which might be required to conduct this experiment) provides reasonable justification for using a constant NVT approach. The C�C stretching frequency is about 1200 cm-1 , (Ref. [6]) corresponding to 3 6 � 1013 Hz or a period of vibration of 28 fs. The stretching frequency is 905 cm-1 for C�Si,7 a 37 fs period of vibration, and 708 cm-1 or a 47 fs period for isolated C�Ge dimers.7 The energy transfer rate in diamond may be estimated by the acoustic speed in diamond,3 which is 1.75 � 104 m/s or 5.7 fs/�. Since in an experimental apparatus the vertical movement of an STM tip is typically 1�2 microns per second, equivalent to 1011 fs/� for lifting a mechanosynthetic tip up or down near the substrate surface, and in no case faster than 1 mm/sec (108 fs/�), then the practical process of carbon dimer placement tool deposition/retraction can be considered slow enough to keep the system equilibrated with the environment. Therefore, constant NVT conditions should be adopted, not constant NVE conditions, to simulate the real process.
J. Comput. Theor. Nanosci. 3, 28-41 , 2006 In order to investigate the behaviors of our dimer placement tools based on the above considerations, we performed constant NVT stepwise AIMD simulations on the following systems: Si tooltip retraction at 300 K and 80 K, Si tooltip deposition/retraction at 300 K, Ge tooltip retraction at 300 K, Ge tooltip deposition/retraction at 300 K, and Sn tooltip deposition/retraction at 300 K, all on the clean diamond C(110) surface slab. In the study of this initial series, tooltips were moved only along the z-coordinate with no tooltip tilting, rotation, or forced lateral movements of any kind. As in previous work,5 the clean diamond C(110) surface slab was constructed of 4 layers of carbon with the bottommost carbon layer terminated by hydrogen atoms. The model consisted of 160 carbon atoms and 40 hydrogen atoms and was confined to a periodic box with supercell dimensions of 14.245 � and 12.604 � along the edges surrounding the surface plane. The bottommost carbon layer and terminating hydrogen atoms of the surface slab were fixed in bulk-like positions during the simulations, with the z-coordinate defined as perpendicular to the surface. A series of total energy minimization calculations were carried out to model the C2 deposition process using the Si/Ge/Sn-based dimer placement tool, a 46-atom molecule consisting of 42 C and H atoms arranged in a fused triadamantane cage, plus 2 additional C atoms in the bound C2 dimer and 2 atoms of Si, Ge, or Sn as the dimersupporting atoms. Initially the toolbound carbon dimer was aligned with the ideal position of the global minimum (GM) of one ad-dimer on the clean C(110) surface. Subsequent tooltip steps were controlled in 0.1�0.2 � increments by fixing the positions of six hydrogen atoms on the topmost 6 carbon atoms of the tool, a constraint used in all later simulations. The lowest total energy configuration of the deposition process was then taken as the starting point for stepwise tool retraction AIMD simulations. The combined stepwise deposition/retraction AIMD simulations were performed for most tasks in this study. All the calculations were based on Density Functional Theory (DFT) with the generalized gradient approximation (GGA) and performed using the Vienna Ab initio Simulation Package (VASP).8 The present stepwise AIMD simulations employed a time step of 1 fs and a simulation time at each step of 200 fs, which is longer by a factor of 4�6 than the stretching time per vibration of the bonds in the ROI (vs only a 25 fs simulation time at each step in the prior work5 ). The present work also used a 0.1�0.2 � tool movement step size (0.5 eV to reach the global minimum and thus constitute stabilized defects relative to the desired lattice structure. That is, placing C2 dimers, one immediately next to the other, is prone to defect formation (Section 3.2). To avoid this difficulty, we examined possible defect states available to a C2 dimer intercalated between two previously-placed isolated C2 dimers having a one-dimer gap between them, again within a trough of a clean diamond C(110) surface. After optimizing the global minimum structure of the two initial ad-dimers, which had the same configurations, a third C2 ad-dimer was positioned parallel to the C(110) surface (consistent with the rigidity of the dimer placement tool) and was intercalated into the gap from various heights at 5 representative rotational angles (0 , �45 , �90 ) around the center of the blank spot without in-plane rotational constraints. The energy minimization calculations showed that the third ad-dimer relaxed from all five initial orientations to just one local minimum, a metastable state, at the position about 1.2 � above the two previously-placed isolated C2 dimers (Fig. 6). The metastable 3-dimer cluster then converted to the desired global minimum (Fig. 7) by passing through a transition state with a barrier of only +0.22 eV (about 5 kcal/mole). The barrier height was determined using a series of partial optimization calculations in which the ad-dimer z-coordinate was restrained to progressively decrease in 0.01 � step size from LM to GM. No local minima leading to defect configurations were found. This suggests a possible procedure for
J. Comput. Theor. Nanosci. 3, 28�41 , 2006 Metastable State Fig. 6. Third C2 dimer, intercalated into the gap between two C2 dimers previously deposited on C(110) at various rotational angles, converges to a single metastable state. positional dimer deposition that minimizes accessibility to defect states, wherein C2 dimers are initially placed in every other position, rather than immediately adjacent, during the first pass, after which the blank spots between each existing pair of ad-dimers in a row are filled with ad-dimers during the second pass. A stepwise AIMD simulation was conducted to test this procedure. In the starting configuration the carbon dimer was bound on the tooltip with the same orientation as in the metastable state and positioned 4.0 � above the diamond C(110) surface. The tool was then lowered stepwise toward the surface. When the C2 dimer reached a position 3.4 � above the surface, after 200 fs constant NVT simulation at 300 K the C2 dimer formed loose connections with the corresponding proximal carbon atoms of the previously deposited two ad-dimers, simultaneously weakening the original bonds between the tool-bound carbon dimer and the Ge atoms on the tooltip. After lifting the tool upward in the usual stepwise retraction from this point, the final separation of the carbon dimer from the tooltip and the subsequent adsorption of the C2 ad-dimer onto the diamond surface occurred at a height of 4.2 � above the RESEARCH ARTICLE + 0.22 eV Metastable Transition � 4.10 eV Global Minimum Fig. 7. Low barrier from local minimum (metastable state) to transition state, relaxing to global minimum, for third C2 dimer intercalated into the gap between two C2 dimers previously deposited on C(110). 34 Theoretical Analysis of Diamond Mechanosynthesis Peng et al. surface after 200 fs constant NVT simulation at 300 K, whereupon the carbon dimer relaxed to its global minimum arrangement between the previously deposited 2 carbon ad-dimers. Thus a dimer deposition procedure directed first to producing gapped sites, then to filling the gaps, can successfully deposit fully populated dimer rows on C(110) surface. 3.5. Dimer Misalignment Tolerance During Placement on Clean C(110) Surface To specify an experimental protocol to achieve practical diamond mechanosynthesis it is necessary to determine the maximum tolerable dimer misplacement error that will still result in a positionally correct C2 deposition onto the diamond C(110) surface, either as an isolated ad-dimer or as an intercalation between two gapped dimers. For these studies, the xyz coordinate directions are defined the same as for the C(110) surface slab and do not refer to a tool-centered coordinate frame: The x-coordinate is perpendicular to the C(110) surface troughs, the y-coordinate is parallel to the troughs, and the z-coordinate is normal to the surface plane. This set of directions is adopted because the alignment of tools in practical operations will most probably refer to the coordinate frame of the substrate. Two principal classes of placement misalignment error are identified--rotational and translational. RESEARCH ARTICLE 3.5.1. Rotational Misalignment During Isolated Dimer Placement In rotational placement error, the C2 dimer may approach the target placement site rotated within the horizontal plane parallel to the diamond surface at some angle relative to the ideal lattice orientation for deposition. Rotational placement error may consist of an uncertainty component and a displacement component. Regarding the uncertainty component, a diamond AFM tip is torsionally stiff, so the primary source of horizontal rotational uncertainty will be thermal vibrations of the tooltip-bound dimer. A 1-ps 300 K molecular mechanics simulation of the extended tool5 found the maximum inplane rotation of a tool-attached dimer to be less than �8 with an average of �2 . This represents a 0.04� 0.16 � displacement in the horizontal plane which is well within even the most conservative defect-avoidance placement accuracy of 0.2�0.5 �,4 and also within the roomtemperature tool-handle thermal uncertainties for all tools (Section 3.6). As for the displacement component of the horizontal rotational placement error, the rotational allowance of placing an isolated carbon dimer on the clean C(110) surface was examined for the simplest case in which the rotational deviation is around the center of and with respect to the LM1 (single-dimer local minimum)4 position. The clockwise (CW) rotation viewed from the top was defined
35 as "minus" and the counterclockwise (CCW) rotation as "plus." As in previous AIMD simulations, the 6 terminating hydrogen atoms which are bound to the topmost 2 carbon atoms and their neighboring 4 carbon atoms in the tooltip were fixed. During the stepwise AIMD simulations of deposition, trial CW rotations of -30 , -32.5 , and -35 and a trial CCW rotation of +30 yielded controllable behavior, whereas CCW rotations of +32.5 , +35 , +45 , and +60 resulted in defect formation. During the stepwise AIMD simulations of retraction, a trial CW rotation of -30 and a trial CCW rotation of +30 left the carbon dimer bonded to the surface, relaxed to its global minimum arrangement, whereas CW rotations of -32.5 and -35 and CCW rotations of +32.5 and +35 resulted in uncontrollable behavior. Thus stepwise AIMD predicts the maximum horizontal rotational allowance of the Ge tool is -30 to +30 for adding a single isolated carbon dimer to C(110). A related source of possible placement error might occur when the tooltip is rolled to some intermediate angle relative to the vertical (such that the vertical tool axis is no longer perpendicular to the deposition plane) while maintaining the toolbound C2 dimer parallel to LM1 in the horizontal plane, as previously described for single-dimer depositions on C(110). Tip rolling is likely to be required in practical situations where a dimer must be delivered to a side position (e.g., a vertical face) on a workpiece rather than to a top position, or where two tooltips must operate in close proximity near a workpiece and it becomes necessary to tilt both tools away from vertical to reduce steric congestion. The simplest case is where the tool is first oriented with the C2 dimer parallel to LM1 as if in preparation for a single-dimer deposition on C(110), but the tool is then rotated through some angle roll around an axis defined by the line connecting the two carbon dimer atoms on the tip. After such tilting, the tool is subsequently moved only in the z direction, as before. During deposition, the dimer will still approach the C(110) surface with the dimer axis parallel to the xy surface plane, but the tool will now be tilted to the C(110) surface, not normal to it. Stepwise AIMD simulations of the Ge tool placed in this tilted orientation at 300 K found that the tool still deposits the dimer successfully on the C(110) surface at roll = 30 and 32.5 , but shows uncontrollable behavior at roll = 35 , giving a maximum tolerable "in-plane" tip rolling angle of roll max = 32 5 . Another rotational placement error is the case where the C2 dimer which is attached to the tooltip arrives at the diamond surface no longer parallel to the horizontal plane. Here again, the vertical rotational placement error may consist of an uncertainty component and a displacement component. Regarding the uncertainty component, a 1-ps 300 K molecular mechanics simulation of the extended tool5 found the maximum out-of-plane rocking angle of a toolattached dimer due to thermal motions to be less than �5
J . Comput. Theor. Nanosci. 3, 28-41, 2006 Peng et al. Theoretical Analysis of Diamond Mechanosynthesis with an average of �1 . This represents a 0.02�0.10 � displacement in the vertical direction which is within the room-temperature tool-handle z-axis thermal uncertainties for all tools (Section 3.6). As for the displacement component, state-of-the-art nanopositioners such as the PicoCubeTM from Physik Instrumente typically introduce minimal off-axis displacement-from-vertical tilts of only