Constant Temperature Interface Characterization of Silicon Bicrystals




Ben Cho, Alfonso Reina, Daniel Go
MATSE 385 Final Project Report

PowerPoint | Movie1 | Movie2 | Movie3

 

1. Bicrystals and interface between flat surfaces

The theoretical investigation of bicrystal behavior is extremely relevant to a wide range of matrerials phenomena, such as grain boundary interfaces in metals, and atomic friction recently observed by AFM techniques. There has been much effort implemented to understand the relations between a material’s microstructure and properties. Study of the simplified situation of a single grain boundary interface can give insight into the much more complicated phenomena underlying microstructural dynamics. Therefore there has been a simplification of systems of interest as depicted in fig.1 [1]. The discussions and simulations contained in this project refer to the case illustrated in fig 1a. This configuration represents a preliminary investigation of the more complex case referring to polycrystals.


Figure 1: Different kinds of polycrystals

At the micron scale level inside a material, there are multiple grains that comprise a typical crystalline solid as illustrated in figure 1d. These different regions are bounded by interfaces known as grain boundaries. For a bicrystal, this interface is the region uniting two differently oriented lattices of the same material. Therefore, the study of these characteristic regions of bicrystal systems provides insight into the nature of grain boundaries.

In addition, friction is responsible for a wide range of important phenomena in scientific research despite a paucity of knowledge about the atomistic causes underlying its behavior. Atomic friction is currently a topic of great scientific interest due to its impact in AFM microscopy and nanoscience. The work undertaken by Gyalog et al. is a good example of the issues being addressed [2]. For instance, the friction between extended atomically smooth surfaces is relevant to investigate phenomena occurring in currently performed nanosled experiments, where defined islands of C60 are dragged over a NaCl substrate. Varying results for the interfacial friction behavior observed between two atomically flat surfaces have been obtained for different models. Figure 2 illustrates the interfacial friction dependence on the sliding direction between two surfaces of materials. The authors of this research paper simulated particles classically by assigning specific spring constants to them. In addition, fig.3 shows data relating friction force per particle as function of the misfit angle between the two slabs of material [2]. It can be seen that there are well-defined angle-dependent maxima and minima of the frictional force value.


Figure 2: Friction force for different sliding directions



Figure 3: Friction force for different misfit angles.

The project here presented has been limited to the case of silicon bicrystals generated by rotating two parallel slabs with respect to each other and with along the (001) zone axis as depicted in fig. 4. Nose-Hoover thermostat control has been implemented for constant temperature simulation. Additionally, lateral translations were added to the rotated configuration of the two slabs in order to analyze the position-dependent energetic behavior of the resulting system.

2. Objectives

Generate a silicon bicrystal by rotating two silicon slabs with respect to one another.
Implement a Nose-Hoover thermostat control to employ constant temperature simulations of the aforementioned silicon slabs.
Study the energetics and structure of the bulk system and interface for a specific rotational angle as a function of lateral translation and temperature.

3. Computer Simulations

3.1 OHMMS

OHMMS (Object-Oriented High-Performance Multi-Scale Materials Simulator) is a C++ based program created by Dr. Jeongnim Kim. Under her guidance, we implemented a Nose-Hoover thermostat control to the OHMMS code. The items that were added to her program are:

A modification to an already existing Leapfrog integrator in OHMMS to include the Nose-Hoover thermostat.
An algorithm to generate the atomic positions of two silicon slabs rotated with respect to one another, given angle of rotation, lateral translation, slab separation, and system size.

3.2 Preparation of bicrystal sample

To simulate twist bonded interfaces one starts with two perfect supercells and rotate one with respect to the other by a certain angle. To ensure periodic boundary conditions for both crystal parts, we need an enlarged supercell. For certain angular rotations lattice points of the bottom lattice will coincide with lattice points of the upper lattice. This is known as a coincident site lattice [1 ].



Figure 4: Coincident lattice for rotated simple cubic 2D slabs


Therefore, for certain angles, there is a periodicity cell that defines the particular points at which both lattices coincide. We expect important differences in the behavior between cases with different periodicity cell of their coincidence sites, e.g. different “magic” angles. Angles that give this coincidence condition are of the form [2]:

cos = a / N cos = b / N a, b, and N are integers

The integers a, b and N comprise a Pythagorean triplet. Every Pythagorean triplet can be generated from two integers p and q (p > q) by a=p2 – q2, b=2pq, N= p2 + q2, and the permitted angles are given by:

tan /2 = q/p


Fig. 5 illustrates the geometrical significance of a, b and N inside the pattern formed by the two rotated slabs with lattice constant equal to one for simplicity [2]. The number of special angles that can work for a specific system strongly depends in the size of the lattices. Therefore, if a specific angle which corresponds to a specific triplet, there is a minimum size the slabs should have. Pythagorean triplets are presented with their respective angle of rotation in table 1 shown below.


Figure 5: Schematic illustration of a, b and N and the corresponding vectors
t1 and t2 of the smallest common periodicity cell in a coincidence site lattice.

a

b

N

1

0

1

90

3

4

5

36.8698

5

12

13

22.6198

7

24

25

16.2602

9

40

41

12.6803

11

60

61

10.3888

13

84

85

8.7974

15

112

113

7.6281

17

144

145

6.7329

19

180

181

6.0255

Table 1: Examples of pythagorean triplets with their respective rotation angles

Table 1 is only a partial listing of the many Pythagorean triplets possible. When implementing the coincidence site lattice theory to a silicon lattice composed from a unit cell like the one shown in fig 6, the coincidence angles are the same as the ones calculated in the previous table due to the square geometry of the silicon unit cell (see fig. 5). The only difference is that the lattice constant is not 1 but 5.43 Angstroms for the three directions, which permit us to use the same angles calculated for the Pythagorean triplets.


Figure 6: Silicon unit cell

When rotating two parallel silicon slabs as described previously, it is possible to obtain the vectors that span the coincidence site lattice (CSL) or smallest periodicity cell represented in fig. 5 by the green points. These are shown below:

1 = N( e1 cos j/2 + e2 sin /2 )

2 = N( -e1 sin j/2 + e2 cos /2 )

In order to include all the atoms that are contained in this coincidence site lattice and to simplify the algorithm to reproduce the twisted bicrystal structure for the simulation, the grayed-out section of fig. 4 was chosen to be the base supercell. This was then expanded periodically to generate a sample of the bicrystal with specific dimensions depending on the desired sample geometry. Due to the complexity and time required to run Molecular Dynamics and Nose-Hoover algorithm, this project has been limited to the specific rotation angle of 36.8698, which corresponds to the (3,4,5) Pythagorean triplet. To generate the coordinates of our rotated supercell, we first created an expanded version of the base supercell and applied rotation to the coordinates using the relationships [2]:

Xnew = X cos - Ysin
Y new = X sin + Ycos

The expanded lattice was used to account for the area left unoccupied over the original unit cell as a result of the rotation, as illustrated in figure 7a. Using the expanded supercell, and applying rotation, and discarding points that fall outside of our original cell boundaries, we obtain a slab structure as shown in figure 7b.


(a)                                                                  (b)

Figure 7: (a) Illustration showing effect of rotation only.
(b) Illustration of final slab structure


3.3 Nose-Hoover Thermostat, Leap Frog integrator and code implementation

Molecular Dynamics simulations have typically been done with a constant energy ensemble or microcanonical distribution. This is somewhat inappropriate when considering that the majority of experiments in the laboratory are made at constant energy conditions rather than constant overall energy of the studied system. Nose proposed a way to accomplish constant temperature simulations by the introduction of an extended Hamiltonian of the system, which had an additional term Q corresponding to an artificial mass interacting with the system. Later, Hoover modified this Hamiltonian and its respective equations of motion and so the Nose-Hoover dynamics were obtained. The Nose-Hoover extended energy is described by :

    

And this results in the Nose-Hoover equations of motion:

             Where = s pi / Q and = ln s (pi is the associated momenta of s). The variable s represents an extended position variable related to the artificial mass Q with its canonical momenta, p. Several explicit time reversal methods have been proposed for the Nose-Hoover algorithm. The method we consider in this project is based in the Verlet Algorithm in its leapfrog form [4]. The integration is made implementing of the following iterative equations of motion:

 







3.4 Potentials

3.4.1 Stillinger-Weber Potential

The Stillinger-Weber Potential is one of the first attempts to simulate silicon with a classical model. The potential is based on a two-body and three-body term. It is built to favor four-fold coordination bonding between atoms just as seen in the silicon structure.



This potential gives a very good description of crystalline silicon but doesn’t work well for conditions different from those it was designed for. Therefore, it does not give accurate information of surface structures and energies, due to the fact that surface atoms have coordination different for that of bulk atoms.

3.4.2 Tight Binding Potential

In Tight-Binding schemes for silicon each atom is associated with a finite set of orbitals, or atomic basis states, each of which can be occupied by two electrons. To describe bonding in silicon requires a minimal (s,p) basis consisting of one s orbital and a set of three rotationally related p orbitals for each atom. Four bonding electrons are distributed among these orbitals. Occupation of the s and p orbitals by an electron requires on-site energies Es and Ep, respectively. The coupling of the orbitals of nearby atoms promotes bonding.

The disadvantage of Tight Binding is that it is a more computationally intensive due to the fact that it considers multi-atom interactions over a relatively long length scale, which requires a significant increase in computing power.

Note: Due to the interfacial surfaces to be studied through this project, Tight Binding potential was chosen as a first option for the simulations presented. Later on, Stillinger-Weber was used due to the unavailability of computational resources necessary to compute a 400 atoms Tight-Binding simulation necessary for our generated rotated slabs. It is expected that the results obtained with Stillinger-Weber are not technically correct. However, it is possible to investigate energetic behavior of the silicon slabs using Stillinger-Weber to find characteristic behavior of the interface for different configurations. Due to the short range nature of the Stillinger-Weber potential, the distance between slabs was made relatively small in order to ensure interaction between surface atoms of the two slabs.

3.5 Temperature controlled simulations for 36.8698 degree rotation and different translations.

The first step following generation of the lattice positions of the rotated lattice was to determine the lateral translation of this rotated lattice that would result in a lowest energy configuration.
Ten possible lattice translations were investigated, 5 along each of 2 possible translation angles, 0 and 45, in increments of L/10 and (L*20.5)/10 where L=5.43 angstrom (bulk Si lattice constant as shown in fig.8.

 


Figure 8. Vectors for initial lattice translations

Next, to find which of these lattice configurations corresponded to the lowest energy state, the following procedure was performed:

At 0K, steepest descent algorithm from OHMMS was implemented to promote the relaxation of the system at 0 K. This way, it was possible to relax the system and allow it to rearrange into a low-energy configuration characteristic of the lattice translation applied.
The system was heated up to a temperature of 1000 K using the leapfrog-Nose-Hoover algorithm in OHMMS . Entire system and interface energies were recorded at the end of the simulation.
Finally, the sample was cooled down to 0K using steepest descent algorithm from OHMMS. Once again, entire system and interface energies were recorded.
The final configuration energies (interfacial and entire system) were compared to find which one of the ten translations gave the lowest (most stable) arrangement of the interface atoms.
The configuration found at the previous point was used to simulate different constant energy behavior of the silicon bicrystal at: 200, 400, 800, 1000, 1200, 1400, 1600, 2000, and 3000 K.

4. Results

To implement correctly the developed Nose-Hoover thermostat, it was necessary to find an appropriate value for the artificial mass of the algorithm. The criteria used to determine it was that the correct value should give a minimum standard deviation or minimum in temperature fluctuations, the nearest value of the temperature mean to the temperature pursued and of course a Gaussian distribution of the temperature values for the particular MD run. In a Nose-Hoover simulation, the energy that is conserved is not the system under study but the entire system including the Nose artificial mass. To analyze the reliability of the thermostat code programmed, energy conservation was recorded for some MD runs of the twisted silicon slabs. Fig. 9a shows the temperature trace obtained for one of the attempts to implement the Nose thermostat code presented in Bond’s paper in one of the twisted slab systems. It is evident that the overall energy keeps increasing with a constant slope. The problem was that the equation of motion of the friction factor had the term gkT, which made the equation size dependent due to the multiplication between g and T. Thus, a minor modification of the Nose-Hoover implementation presented in Bond’s paper was required, consisting of dividing by g, thus normalizing the quantity as seen in the equations of motions shown previously. After fixing the problem and repeating the simulation, energy conservation was accomplished. This is illustrated in Fig. 9b.

 


Figure 9a: System energy calculated using incorrect Nose-Hoover thermostat


Figure 9b: System energy calculated using correct Nose-Hoover thermostat



Figure 10: Thermostat-controlled system temperature (1000 K)


A schematic of the atomic positions for the translation resulting in the lowest energy configuration is shown in figure 11a. It may be seen that in this initial state there is relatively little bonding in the interface, thus this is a quite unrealistic configuration. Performing a steepest descent calculation on this arrangement results in the configuration shown in figure 11b. The formation of a significant number of tetrahedral bonds is evident.

 


Figure 11a: Initial system configuration


Figure 11a: System configuration after 1st steepest descent calculation


After determining the most suitable value for the artificial mass in the Nose-Hoover thermostat for the twisted silicon slab system, the MD simulations were performed as described in the previous section. Figures 12a and 12b show the results obtained for the (a) total and (b) interface energies of the system as a function of lateral translation of the upper rotated slab. Well defined minima can clearly be identified, which correspond to stable configurations of the twisted slab system.



Figure 12a: Total system energy results for different lateral translations



Fig. 12b: Surface energy results for different translations


Atomic images from the constant temperature MD runs for the lateral translation of the rotated slab leading to minimum system energy are shown in figures 13a, b, c, and d for temperatures of 200, 600, 1200 and 2000 K.



Fig. 13a Constant temperature configuration at 200 K



Fig. 13b Constant temperature configuration at 600 K



Fig. 13c Constant temperature configuration at 1200 K


Fig. 13d Constant temperature configuration at 2000 K


It can clearly be seen that the lattice becomes more disordered as the temperature of the system is increased, and that past the melting point all long-range order in the system is lost. Unfortunately, from simple visual examination, the manner in which the character of bonding at the interface changes with temperature is not immediately apparent.
The interface energy as a function of temperature is plotted in figure 14. Additionally on this graph, interface energy for low and high energy translations is shown for T = 1000 K. The interface energy exhibits a rising trend with temperature. The difference in surface energy for high and low energy translations was 0.000497. No clear difference in interfacial structure is visually apparent for these two structures, as shown in figure 15 a (low) and b (high).

 


Fig. 14: Interface energy as a function of temperature

 

Figure 15a: Interface configuration obtained for a low energy lateral translation




Figure 15b: Interface configuration obtained for a high energy lateral translation

 

Finally, the total system energy as a function of temperature is shown in Figure 16. The onset of melting is apparent as a large jump in the system energy near the theoretical melting temperature of Si (1687 K).




Fig. 16: Constant temperature configuration at 2000 K

 

5. Conclusions

To restate our experimental objectives, we set out to:

Generate a silicon bicrystal by rotating two silicon slabs with respect to each other.
Apply a Nose-Hoover thermostat control to implement constant temperature simulations.
Study the energetics and structure of a specific low energy rotational case as a function of temperature and lateral translation.

With respect to the first point, it was possible to implement Coincidence Lattice Site theory to find a smallest common periodicity cell when rotating two silicon slabs with respect of each other. Therefore, periodic boundary conditions were developed correctly by the usage of this method to fabricate the twisted slab system. Although we focused the results to only one case of rotation, there are many other combinations of rotation angle and supercell dimension that can be implemented.

Referring to the second point, Nose-Hoover thermostat was successfully implemented, correctly maintaining a constant temperature ensemble along with a verlet-based leapfrog propagator. The criteria used to find a suitable Q mass value was that for that specific value, it was imperative to have a minimum in the standard deviation of the temperature trace, a maximum proximity to the temperature desired and a Gaussian distribution of these temperature values through the MD run. Also, another indicator of the correct functionality of this thermostat is the conservation of the total energy (slabs system plus Nose Q mass system or heat bath) as shown in figure 9b.

We found a clear dependence of the system and interfacial energy on lattice translation, with well defined minima corresponding to low-energy atomic configurations. The interfacial energy of the slabs increased with increasing temperature, while the difference in interfacial energy between high and low energy translations seemed to be minimal at best, with a value of
EHIGH-ELOW = 0.000497. Below the melting temperature we found the system energy to be constant with increasing temperature. Once the temperature reached >1600 K we observed a large jump in the system energy corresponding to melting of the slabs.

6. Future Investigations

There are a number of areas where this study can be continued. First is the quantitative statistical analysis of interfacial bonding and structure as a function of temperature, lateral translation, and slab separation. Also, this study can be extended to other rotation angles. Ideally a generic lattice expansion algorithm could be implemented to allow automatic calculation of coincidence site geometry and supercell size. Finally simulation employing alternate potentials such as MEAM or tight-binding can provide more a accurate simulation of atomistic behavior and serve as a valuable complement to the information obtained in this study.

7. Appendix

Nose-Leapfrog algorithm code snippet

//////////////////////////////////////////////////////////////////
// (c) Copyright 2003- by Jeongnim Kim, Benjamin Cho, Daniel Go, Alfonso Reina
//////////////////////////////////////////////////////////////////
//////////////////////////////////////////////////////////////////
// Jeongnim Kim
// National Center for Supercomputing Applications &
// Materials Computation Center
// University of Illinois, Urbana-Champaign
// Urbana, IL 61801
// e-mail: jnkim@ncsa.uiuc.edu
// Tel: 217-244-6319 (NCSA) 217-333-3324 (MCC)
//
// Supported by
// National Center for Supercomputing Applications, UIUC
// Materials Computation Center, UIUC
// Department of Physics, Ohio State University
// Ohio Supercomputer Center
//
//////////////////////////////////////////////////////////////////
// -*- C++ -*-

#include "Propagators/NoseLeapFrog.h"
#include "Potentials/PotentialBase.h"
#include "ParticleBase/ParticleUtility.h"
using namespace OHMMS;

NoseLeapFrog::NoseLeapFrog():
ToApplyBConds(false), T_ref(0.0), Dt(1.0), Chi(0.0)
{
nose_ke_ind = E.add("Nose-KE");
nose_pe_ind = E.add("Nose-PE");

addParameter(T_ref,"temperature","K");
addParameter(Dt,"timestep","fsec");
addParameter(Q,"Q","fsec-1");
addParameter(Chi,"Chi","nose-var1");
addParameter(Theta,"Theta","nose-var2");
}


bool
NoseLeapFrog::propagate(PotentialBase* pot) {

ParticlePos_t& F = *myPtcl->getVectorAttrib(FORCE_TAG);
ParticlePos_t& V = *myPtcl->getVectorAttrib(VELOCITY_TAG);
ParticleScalar_t& Eloc = *myPtcl->getScalarAttrib(ENERGY_TAG);
ParticlePos_t& R0 = *myPtcl->getVectorAttrib(TRAJECTORY_TAG);
Tensor_t stress;
// ************* degrees of freedom *************
Scalar_t g = static_cast<Scalar_t>(OHMMS_DIM*myPtcl->getTotalNum()-1);

if(!Initialized) {
reset(false,false);
pot->set(myPtcl); // assign a particle set to PotentialBase::myPtcl
Initialized = true;

F = 0.0e0;
Eloc = 0.0e0;
E[PotEnergy] = pot->evalForce(F, Eloc, stress, 0);
evalAccel(F);
}

// ************ Eval velocity at t+dt/2 *************
Scalar_t v0 = 1.0/(1.0+Chi*Dtover2);
Scalar_t v1 = Dtover2/(1.0+Chi*Dtover2);

V = v0*V+v1*F;

// ************* Eval position at t+dt *************
R0 += Dt*V;

// ************* Eval Theta at t+dt/2 *************
Theta += Dtover2*Chi;

// ************* Eval Chi at t+dt *************
evalKineticEnergy();
Chi += DtoverQ*(2*E[KineticEnergy]-g*KE_ref);

// ************* Eval Theta at t+dt *************
Theta += Dtover2*Chi;

// ************* Update the force with R0 at t+dt *************
myPtcl->updateR(R0);
F = 0;
Eloc = 0;
stress = 0;
E[PotEnergy] = pot->evalForce(F, Eloc, stress, 0);
evalAccel(F);

// ************* Eval velocity at t+dt *************
v0 = 1.0/(1.0+Chi*Dtover2);
v1 = Dtover2/(1.0+Chi*Dtover2);
V = v0*V+v1*F;

// ************* Eval Kinetic Energy at t+dt *************
evalKineticEnergy();

// ************* Eval Nose-KE and Nose-PE *************
E[nose_ke_ind] = Chi*Chi*Q*0.5;
E[nose_pe_ind] = g*KE_ref*Theta;

// ************* Eval total energy *************
E[FreeEnergy] = E[PotEnergy] + E[KineticEnergy] + E[nose_ke_ind] + E[nose_pe_ind];

Counter++; // increment the internal counter
bool moving = (E[Temperature] > zeroKelvin);

E[Now] += Dt;
return moving;
}

void NoseLeapFrog::reset(bool trescale, bool wrapping)
{

resetBase();

// reset, time coefficients with new or old MD parameters
Counter = -1;
Dtover2 = Dt/2.0;
DtoverQ = Dt/Q;

if(!Initialized) {

KE_ref = Kelvin_to_eV*T_ref;
//Q *= amu_to_mass_unit;

ParticlePos_t& V = *myPtcl->getVectorAttrib(VELOCITY_TAG);
ParticlePos_t& R0 = *myPtcl->getVectorAttrib(TRAJECTORY_TAG);

///////////////////////////////////
// these values are always in cartesian coordinates
///////////////////////////////////
R0.InUnit = false;

///only R0 has to be added to the position data to which boundary conditions apply
myPtcl->addPositionAttrib(R0);

//if(fabs(E[Now]) < zeroKelvin && T_ref > zeroKelvin) {
if(fabs(E[Now]) < zeroKelvin) {
///needs to assign Velocity, if not initialized
if(T_ref>zeroKelvin && E[Temperature]<zeroKelvin) {
setVelocity(T_ref);
}
}
convert(myPtcl->Lattice, myPtcl->R, R0);
}

// scale temperature with the new parameters
if(trescale) setTemperature(trescale,T_ref);

if(Initialized && wrapping) myPtcl->applyBConds();
}

...

8. Acknowledgement

We would like to give our thanks to many people whose guidance and suggestions are critical in making our simulation project possible. Many thanks go to Dr. Jeongnim Kim of Materials Computation Center for her generosity in time and effort on helping us debugging the Nose-Hoover thermostat in her OHMMS program, and suggesting the topic as well as the practical issues on the simulation. Many thanks go to Professor Stephen Bond of the Department of Computer Science at UIUC, for his published work on simplified implementation of Nose-Hoover algorithm. His private discussion with one member of the group is appreciated. Many thanks go to Professor Kurt Scheerschmidt at Max Planck Institute of Microstructure Physics of Halle, Germany, for his invaluable suggestions on the use of Pythagorean triplets and Coincidence Site Lattice theory for determining the correct misfit angles. Finally, many thanks go to Professor Johnson for his general discussion, excitement, patience and encouragement on his students' projects.



9 . References


1) Phillips, Rob. Crystals, Defects and Microstructures. Cambridge University Press
April 2000.
United Kingdom

2) Gyalog, Tibor et al. Atomic Friction. Zeitschrift Fur Physik B.

3) Foll, Helmul. Defects in Crystals. University of Kiel.
http://www.tf.uni-kiel.de/matwis/amat/def_en/index.html

4) Bond, Stephen et al. The Nose-Poincare Method for Constant Temperature
Molecular Dynamics. Journal of computational Physics 151(1999)114-134