EpiCalc Version 5.0 for Windows 95 Instructions and help
In a nutshell:
EpiCalc© will take two-dimensional lattice parameters and angles for the substrate and the overlayer as input. It will then calculate a dimensionless potential energy, V/Vo, and the transformation matrix elements as a function of overlayer rotation angle through a user-defined range at user-defined increments. The value of V/Vo indicates the degree of commensurism ; an incommensurate overlayer returns V/Vo=1, a coincident overlayer returns V/Vo=0.5, and a commensurate overlayer returns V/Vo=0 for a non-hexagonal substrate, or -0.5 for a hexagonal substrate. The user specifies what value of V/Vo should be considered a 'hit.' The on-screen output consists of a plot of V/Vo as a function of rotation angle, the minimum V/Vo for the calculated range, the angle at which this minimum is achieved, and the four elements of the transformation matrix at this angle.The plot and the calculated values at each interval can be saved or printed.
For the commensurate overlayer, each point of the overlayer lattice is directly over a point on the substrate lattice. For the coincident overlayer, each point lying on the supercell vertices is directly over a point on the substrate lattice; there exists some local incommensurism in a coincident overlayer. To allow for some distortion or reconstruction of the overlayer, EpiCalc© also allows the user to provide a range and calculation interval for lattice distortion. EpiCalc© will then calculate V/Vo as a function of both overlayer rotation angle and lattice parameter. In this case, output also includes the lattice parameters at which each calculation was performed. Since the function on which EpiCalc© is based is so simple, EpiCalc© has the advantage of short calculation times when compared to more traditional potential energy calculations, even for large overlayer sizes.
Program Instructions:
The following step-by-step instructions explain how the EpiCalcÓ program is used for determining the degree of commensurism.
I. Substrate and Overlayer Parameters
The first step is to identify the particular substrate and overlayer to be used in the computations and enter the necessary structural data.
A) Substrate parameters
For the substrate, the parameters a1 and a2 are the lattice vectors (in Angstroms) and alpha is the angle between these vectors (in degrees). The substrate symmetry must also be identified as hexagonal or non-hexagonal by checking the appropriate radio button. Non-hexagonal symmetry is the default selection. The calculation is different for hexagonal symmetry because epitaxial registry is checked along three directions rather than two.
B) Overlayer parameters
The parameters b1 and b2 are the lattice vectors (in Angstroms) for the overlayer and beta is the angle (in degrees) between these vectors. The size of the overlayer can also be changed by entering the overlayer dimensions in the b1 and b2 directions. These values are adjusted by typing in new integers in the text boxes after the words "b1 direction" and "b2 direction." The program then creates an overlayer of size M x N, where M = 2b1 + 1 and N = 2b2 + 1. The overlayer dimensions are calculated this way because the program requires M and N to be odd numbers. Large values of M and N will not increase calculation time because calculation time is independent of overlayer size. Because several unit cells are required to identify lattice registry involving several unit cells, using relatively large overlayer sizes (approximately 25 in each direction) is imperative.
C) Structure Database
1. Loading Saved Structures
Lattice parameters for some specific structures have been stored in a database. To obtain access to these structures, the database must be opened by clicking on the icon that looks like storage drawers next to the label "Click to open structure database." The default substrate is named "Substrate." To load a structure from the database, click on the downward-pointing arrow next to the word "Substrate" and select from the list that appears. The parameters for this structure are loaded into the appropriate boxes.
2. Updating / removing files from the database
Files may be added to or deleted from the database by using the "Update" and "Remove" buttons below the substrate and overlayer parameters.
3. Azimuth Rotation Range
The azimuth angle is defined to be the angle between vector a1 of the substrate and vector b1 of the overlayer. For each calculation, the program computes the dimensionless potential over a range of azimuth angles. The default values are a lower bound of 0 degrees, upper bound of 60 degrees, and spacing of 0.25 degrees. These options have the largest effect on computation time. A larger increment can be used to identify a region of interest. Then the increment can be reduced at the same time as the rotation range to reduce calculation time. This region of interest technique is especially important for more time consuming calculations involving lattice parameter iteration.
II. Type of Calculation
The EpiCalcÓ program supports two variations of the main calculation routine: fixed or iterated lattice parameters. The option to search for geometric solutions is available with either calculation.
A) No Iteration Calculation
A calculation without iteration uses constant values for the overlayer and substrate parameters. The computations are started by selecting the "No iteration" radio button and clicking the "Calculate" button on the main window. The dimensionless potential is calculated over the user-defined azimuth rotation range. The calculation without iteration should be used when the overlayer and substrate parameters are to remain fixed.
1. Geometric Solutions
If the geometric solutions option is checked, the program repeats the calculation of V/Vo versus azimuth angle, now using the integral multiples of b1 and b2 up to a user-defined limit. The default value for the maximum multiple is 4.
B) Iteration Calculation
A calculation with iteration allows the user to enter a range of values and a calculation interval to use for a1, a2, and alpha (substrate iteration) or b1, b2, and beta (overlayer iteration). The iteration calculation is used to allow the overlayer or substrate to undergo some lattice distortion. When the "Iteration" radio button is selected, an "Iteration Parameters" window is displayed for entering these values. This window can be closed and opened by clicking the "Show Parameters" check box. The user can choose to fix a particular parameter while varying the others if desired. If it is known that a1 = a2 in length, selecting the "Require a1 = a2" radio button will speed the calculations for a substrate iteration run. The "Allow a1 <> a2" radio button is selected by default.
1. Selection Criteria
Another important number the user can enter on the "Iteration Parameters" window is the selection criteria to use for the dimensionless potential. When the program finds a value of V/Vo that is less than the critical value, a "hit" has occurred and the parameters associated with this value are saved for future reference. The default critical value is 0.80.
2. Geometric Solutions
The option to search for geometric solutions is also available for the iteration calculations, but only when the overlayer is iterated. In this case, the values of b1 and b2 in the user-defined ranges are tested along with integral multiples of these numbers.
C) Interrupting a Calculation
During both the no iteration and iteration calculations, a window appears showing the progress of the computation. The run can be stopped at any time by clicking on the "Abort" button.
III. Results and Display Options
This section describes how the results are tabulated and displayed. The display options depend on the type of calculation that was performed.
A) General Results
The major purpose of the program is to identify the mode of epitaxy for a particular substrate / overlayer combination. The type of commensurism is determined from the values of V/Vo, as described earlier. For any calculation, the minimum value of V/Vo is likely to be of interest. The azimuth angle at the minimum value of V/Vo and the four elements of the transformation matrix relating the lattice vectors of the overlayer and substrate are general results that are displayed on the main window after any type of calculation is performed.
B) Results for No Iteration Calculation
1. Graph of V/Vo versus Azimuth Angle
For a calculation without iteration, the program generates a graph of V/Vo versus azimuth angle. The azimuth angle or angles at which a minimum in the dimensionless potential occurs can easily be located. The graph is automatically displayed when the calculations are completed. The graph can opened and closed by clicking on the "Show / Hide Graph" button under the "Display Options" on the main window. To print the graph, either click on the "Print" button or select the "Print" option under the File menu. The data points for the graph, the lattice parameters, and the elements of the transformation matrix are also included on the printout. This information can also be stored by clicking on the "Save" button or selecting the "Save" option under the File menu.
When the program searches for geometric solutions, the graph of V/Vo versus azimuth angle is created as follows. If the maximum multiple to use is set to 4, then the program performs the usual calculations at each azimuth angle for b1xb2, 2b1xb2, 3b1xb2, 4b1xb2, 1b1x 2b2, . . . 4b1x 4b2. At each azimuth angle, 16 values of V/Vo have been calculated. The program finds the minimum value and uses this on the plot. In this way, all the minima are included on the same plot rather than on 16 different plots. Note that if the user chooses to search for geometric solutions and then enters 1 as the maximum multiple, the graph reduces to the case without the geometric search.
C) Results for Iteration Calculation
1. Iteration Results Table
For an iteration calculation, the results are automatically displayed in tabular format. Each row of the table contains a value of V/Vo that was below the critical value and the lattice parameters associated with this value. The elements of the transformation matrix and the area of the overlayer supercell (b1*b2*sin beta) are also listed. This chart can be printed or saved. The iteration results table will be displayed even if the calculation has been interrupted by clicking on the "Abort" button, assuming that at least one "hit" has occurred.
2. Histogram
Another display option available for the iteration calculations is a histogram that shows the total number of hits at each azimuth angle. The histogram can be opened and closed by clicking the "Show / Hide Histogram" button. As with the iteration table, the histogram can be displayed even if the calculation was not entirely completed. The bins in the histogram have a size of one degree. For example, all hits at an azimuth angle between 1.5 and 2.5 degrees will be displayed at an angle of 2 degrees in the histogram. The histogram is useful for identifying the azimuth angle at which the orientation between the overlayer and substrate is favorable. A large number of hits will usually be clustered at the optimum angle.
Background
The EpiCalc© program is an epitaxial calculation program used for analyzing the geometric agreement between overlayer and substrate structures. Epitaxy generally is used to describe lattice registry, or, equivalently, the degree of "phase matching" between two opposing lattice planes. It has been observed that properties of certain films can be influenced significantly by interactions between the primary overlayer and the substrate upon which it forms. These interactions commonly are associated with overlayer-substrate epitaxy in which the overlayer and substrate lattices are "in phase," so that their interatomic potentials are reinforced. However, the actual structure of an overlayer will reflect a competition between the energy lowering achieved by epitaxy and the energetic penalty associated with any reconstruction of the overlayer lattice from its native form that may be required in order to achieve that epitaxy.
The EpiCalc© program uses an efficient analytical method for determining the type of epitaxy. By purely geometric considerations, a dimensionless potential energy, V / Vo, that is related to the degree of commensurism can be calculated. The specific categories of commensurism and values of V / Vo are as follows:
V / Vo = -0.5 corresponds to a commensurate overlayer on a hexagonal substrate
V / Vo = 0 corresponds to a commensurate overlayer on a non-hexagonal substrate
V / Vo = 0.5 corresponds to a coincident overlayer
V / Vo = 1 corresponds to an incommensurate overlayer
The two-dimensional interface consisting of the substrate and overlayer can be described by seven parameters: a1 and a2 are the lattice vectors for the substrate and alpha is the angle between these vectors; b1 and b2 are the lattice parameters of the overlayer and beta is the angle between these vectors; the azimuth angle, q , is the angle between a1 and b1. A transformation matrix, C, relates the overlayer and substrate parameters:
The values of the elements in this transformation matrix depend on the mode of epitaxy. For commensurism, every matrix element is an integer. For coincidence, either px and qx, or py and qy are integers. For incommensurism, none of the matrix elements are integers.
The EpiCalcÓ program calculates the dimensionless potential over a user-defined range of azimuth angles and searches for the global minimum. The program can be used with fixed lattice parameters or can test a range of values to allow for some lattice distortion. The program also has the ability to search for "geometric solutions," meaning that integral multiples of the lattice parameters b1 and b2 will be used in the calculations. A "geometric solution" is found when an overlayer supercell is in phase with the substrate lattice. This calculation reveals the dimensions of the supercell.
Calculations by the EpiCalcÓ program are much less computationally intensive than potential energy (PE) calculations and reach comparable results. The computational time for this program is independent of overlayer size, whereas PE calculation time increases exponentially with increasing overlayer area. The computational efficiency of this method enables a convenient examination of numerous possible reconstructed overlayer configurations in which the lattice parameters are bracketed around those of the native overlayer, thereby allowing examination of possible epitaxy-driven overlayer reconstructions. The EpiCalcÓ program accurately predicts observed overlayer orientations and these epitaxial configurations can be used in subsequent PE calculations that allow for other degrees of freedom. Recently, this program has been used to verify the existence of coincidence in specific structures and has been used in the design of molecular films.
A few words on the theory:
Determination of the mode of epitaxy and optimum overlayer orientation can be accomplished using EpiCalc, a geometric lattice misfit algorithm developed in our laboratory. EpiCalc relies on an analytical function devised in our laboratory that enables calculation of V/Vo, which we termed a "dimensionless potential" (eq. A1).
(A1)
term A: 2MN
term B:
term C:
term D: 1 / (2MN)
Note: M = 2m+1 and N = 2n+1; m and n are the values input into EpiCalc, resulting in M and N always being odd. This is required to achieve non-zero values for numerators in terms B and C in (A1) so that the discrete values of V/Vo (1, 0.5, and 0), which are necessary to distinguish the different modes of epitaxy, can be realized. The function in (A1) behaves as follows for the different modes of epitaxy.
I. Commensurism: every matrix element is an integer. In this case, terms B and C are both equal to 0/0 (sin (integer•p) = 0). Therefore, L’Hopitals rule (eq. A1) must be invoked.
(A1)
Then, the numerator of term B becomes
and the denominator becomes
Therefore,
term B =
Similarly,
term C =
For the case of two identical, square lattices, px = py = 1 and qx = qy = 0. Then,
.
2. Point-on-line Coincidence: px and qx, or py and qy must be integers. This will result in either term A or term B being indeterminate with a value of 0/0. Therefore, L’Hopitals rule must be used to solve the indeterminate term. For example, if only py and qy are integers,
term C = = MN.
However,
term B = .
A property of coincidence is that px and qx are fractions with values such that the products M•px and N•qx will be integers if M and N are integer multiples of the supercell dimensions. If M and N are not integer multiples of the supercell, M•px and N•qx are not integers and term B = ± 1. Therefore,
As M • N becomes large compared to 1, the ± 1 term becomes negligible and V/Vo . Therefore, large values of M and N will enable convergence toward an unambiguous minimum for coincidence. Conversely, if M and N are integer multiples of the supercell, M•px and N•qx will be integers and term B = 0. However, this condition requires that the lattice parameters be exactly the coincident values so that the matrix elements assume values that ensure that M•px and N•qx
are integers. It is improbable that the user will select lattice
parameters for the calculation that are the exact values required for
coincidence. Consequently, in most calculations M•px and N•qx can be slightly non-integral and term B (or term C if px and qx are
the pair of integral matrix coefficients) will be slightly non-zero.
Therefore, EpiCalc is most reliable when M and N are large values.
3. Coincident (not point-on-line): coincident matches that are not point-on-line will not be recognized by EpiCalc since this condition does not cause any of the matrix elements to approach 1. These matches can be found using the program, however, by recognizing that point-on-line coincidence is achieved for these matches when a integral multiple of the primitive lattice constants are used. In the program, just choose to search for geometric solutions, and select an integer number of multiples as a maximum. Remember that a larger multiple results in a less favorable coincident match, since there will be more lattice nodes on incommensurate sites. We usually use a value of 3 or 4. These matches will show up separately on the histogram of results.
4. Incommensurism: px, qx, py and qy are not integers. Therefore, M•px, N•qx, M•qy and N•py will not be integers and terms B and C will be equal to ± 1. Then,
As M· N becomes large, the ± 1 terms become negligible and V/Vo ® 1.
For more information, see:
A. C. Hillier and M. D. Ward, "Epitaxial Interactions Between Molecular Overlayers and Ordered Substrates," Phys. Rev. B, 1996, 54, 19, 14037.
J. A. Last, D. E. Hooks, A. C. Hillier, and M. D. Ward, "The Physicochemical Origins of Coincident Epitaxy in Molecular Overlayers: Lattice Modeling vs Potential Energy Calculations," J. Phys. Chem. B, In press (ASAP contents July 26 release).
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Last updated on 04/10/00