MOLECULAR DYNAMICS SIMULATION OF MECHANICAL,

THERMAL PROPERTIES AND GROWTH OF SELECTED NANO-

STRUCTURES

MIN TJUN KIT

8 September 2017

School of Physics USM

Introduction: Carbon Nanostructures

• Fullerene (0-D), carbon nanotubes (1-D), graphene (2-D) and graphite (3-D)

• Most of these carbon nanostructures are sp2-hybridized.

• The bond length between carbon chains is approximately 0.142 nm

Introduction: Carbon Nanostructures

Introduction: Carbon Nanostructures

Introduction: Graphene

• The first graphene sheet was synthesized through “scotch tape cleaving” method of on three-dimensional graphite.

• Discovered by two Russian-émigré scientists at the University of Manchester, Andrei Geimand Kostya Novoselov.

• They won Nobel Prize of Physics at 2010

Introduction: Graphene

• Why graphene?

– high electrical conductivity

– high carrier mobility

– superior thermal conductivity

– unusual mechanical properties such as high-in-plane stiffness

– extremely hard

Introduction: Graphene

• Hannon and Tromp [1]

Introduction: Graphene

• Borysiuk et al [2]

Introduction: Graphene

• And some computer simulations were performed by other groups.

– Tang et al. [3]

– Lampin et al. [4]

– Jakse et al. [5]

Introduction: Silicene

• A two-dimensional nanosheet made up of silicon atoms arranged in the form of honey comb lattice.

• Silicene, unlike graphene which is not flat but has a buckled configuration where the out-of-plane buckle parameter is predicted to be 0.44 Å.

• New form of 2D material

Introduction: Silicene

Introduction: Silicene

• Having a close resemblance to graphene, silicene offers many possibilities

– photovoltaic, optoelectronic devices

– thin-film solar cell

– hydrogen storage

– easier to get integrated into nanodevices which are mainly silicon-based.

Introduction: Silicene

• There are few successful synthesis of silicene on supported substrate.

• Many theoretical studies and simulations on the structural, mechanical, electronic and thermal properties of silicene.

• But one has yet to see any report of experimentally synthesized free-standing silicene.

• The properties of a free-standing silicene are mainly can be studied through computer simulations.

Introduction: Zinc Oxide

Introduction: Zinc Oxide

Introduction: Zinc Oxide

• Experimentally, if we heat a ZnO wurtzitesurface to an elevated temperature and investigate the resultant surface using photoluminescence (PL) measurement, the spectrum should reflect the amount of point vacancies created. We expect that an increase of annealing temperature will create more point vacancies.

Introduction: Molecular Dynamics Simulations

• Why we need to do simulation?– While experiments form the core of scientific research, all

experimental methods have intrinsic and practical limitations in e.g. spatial and temporal resolution.

– On the other hand, an analytical solution to given certain problem may be too difficult or impractical (many-body problems)

– A third option is to use numerical applications of various theoretical models, such as computer simulations

– Such studies can give complementary information on the problem studied, essentially forming a bridge between theory and experiment.

– Computer simulations may also have predictive power!

Introduction: Molecular Dynamics Simulations

Introduction: Molecular Dynamics Simulations

Introduction: Molecular Dynamics Simulations

Introduction: Molecular Dynamics Simulations

Introduction: Molecular Dynamics Simulations

Introduction: Molecular Dynamics Simulations

• Boundary Conditions

Introduction: Molecular Dynamics Simulations

Introduction: Molecular Dynamics Simulations

Methodology: Construction of 6H-SiC Substrate

There are various polymorph of SiC

Construction of 6H-SiC Substrate

Primitive vectors and atomic position which is necessary for construction of single unit cell of 6H-SiC

Visualization of the original unit cell’s atomic configuration as specified in dataraw.xyz. The coordinates of the atoms are also shown. There is a total of 12 atoms in the unit cell.

Sublayer of

Si and C

Sublayer of

Si and C

Sublayer of

Si and C

Sublayer of

Si and C

Sublayer of

Si and C

Construction of 6H-SiC Substrate

Detailing the coordinates of the atoms in a carbon-rich SiC substrate unit cell. Note that now only 11 atoms remain which is needed to create a carbon rich layer

Construction of 6H-SiC Substrate

11 atoms per unit cell left as one Si atom has been removed. Left with carbon rich layer on the top.

A 1584-atom supercell mimicking a carbon-rich SiC substrate. It is made up of 12 x 12 x 1 unit cells. Top view (left) and side view (right). Yellow: Carbon; Blue: Silicon.

Construction of 6H-SiC Substrate

Binding Energy

The binding energy (per atom) Eb for an infinite free graphene layer calculated with the Tersoff (solid circles or dashed line) and TEA (open circles or full line) potentials at different values of lattice constant a0. The lowest values of Eb are a0 = 2.53 and 2.56 Å for Tersoff and TEA potentials, respectively. Notice that the Eb corresponding to Tersoff and TEA potentials crosses at a0 = 2.584 Å and for a0 > 2.6 Å the Ebfor TEA potential is relatively lower in energy.

34

Simulation method of graphene growth (one layer)

2.52 Å< 1 Å = 0.63 Å

Conjugate gradient minimization

Simulated annealing

2.0 Å

Simulated Annealing

Once the data file for the Carbon-rich SiC substrate is prepared, we proceed to the next step to growth a single layer graphene via the process called simulated annealing

Simulated Annealing

• To implement the above procedure, a fixed value of target annealing temperature was first chosen, e.g. Tanneal = 900 K.

• We monitor the LAMMPS output while the system undergoes equilibration at the target annealing temperature (after the temperature has been ramped up gradually from 1 K).

• If graphene is formed at a given target annealing temperature, the following phenomena during equilibrium (at that annealing temperature) will be observed:– An abrupt formation of hexagonal rings by the carbon rich layer – an abrupt drop of biding energy

• In actual running of the LAMMPS calculation, we repeat the above procedure for a set of selected target annealing temperature one-by-one, Tanneal = 400 K, 500K, 1100K, 1200 K …, 2000 K.

• The essential parameters used in annealing the substrate for single layered graphene growth: – damping coefficient: 0.005– Timestep: 0.5 fs.– Heating rate from 300 K -> target temperatures, 5 x 1013

K/s.– Cooling rate: From target temperatures -> 1 K, 1 x 1013

K/s.– Target temperatures: 700 K, 800 K, …, 2000 K.– Steps for equilibration: (i) At 1K, 5000 steps. (ii) At 300 K,

20,000 steps, (iii) target annealing temp -> target annealing temp, 60,000 steps.

One-layer graphene overlaid on Si-terminated 6H-SiC(0001) obtained by the simulated annealing method for Tersoff (second column) and TEA (third column) potentials In the second and third columns at the bottom corner on the right, the integer is the hexagon number. The average distance of separation between the graphene buffer layer and surface is about 2.43 Å for TEA potential.

Single layer graphene

Comparison of the average bond-length (Å) versus annealing temperature T (in units Kelvin) between results calculated using TEA (open circle) and Tersoff (solid circle) potentials. Our criterion of a bond-length is ≤ 1.6 Å for the formation of graphene. This criterion yields T = 1200 and 1500 K for TEA and Tersoff potentials, respectively

Single layer graphene

• Same as previous figure except for the binding energy (per atom) Eb.

Single layer graphene

Pair correlation function g(r) of carbon atoms obtained using TEA potential at different annealing temperature T (in units of Kelvin) for the one-layer graphene which emerges for T ≥ 1200 K. At T < 1200 K, it displays typical crystalline structure.

Single layer graphene

MD simulations were done using Tersoff potential. The one-layer grapheneemerges at T ≥ 1500 K. At T < 1300 K, it displays typical crystalline structure. Only few hexogons are seen at T = 1400 K.

Two-layer graphene• Prepare a two-layered carbon-rich substrate by further knocking off two layers of Si

atom, and then shift the topmost carbon atom layers to form two carbon rich layers.• Thickness of the substrate is z=1.• With the substrate thickness of z=1, the substrate will eventually heavily distorted

during anneal. So in simulation we use the thickness z=2

Conjugate gradientminimization

Simulated annealing Conjugate gradientminimization

Two-Layer graphene

• A 6H-SiC unit cell with a thickness z = 2 substrate unit cell is shown.

Two layer graphene

Two-layer graphene

Two-layer graphene overlaid on 6H-SiC(0001) obtained by simulated annealing method with TEA potential. In the second and third columns at the bottom corner on the right, the integer is the hexagon number. The first graphene “buffer” layer refers to one closest to the top surface of substrate and has an average distance of separation about 2.37 Å, and the second layer corresponds to one next to the first graphene layer and these graphene layers are separated by an average distance about 3.13 Å.

Two-Layer graphene

(a) The average bond-length (Å) versus annealing temperature T= (in units of Kelvin) obtained by the simulated annealing using TEA potential for two-layer graphene. Notations used are: first-layer graphene, open circle; second-layer, solid circle. (b) The binding energy (per atom) Eb (in units of eV) versus annealing temperature T (in units of Kelvin) obtained by simulated annealing using TEA potential for two-layer graphene.

Three Layer graphene• Construction of

third layer graphene in 6H-SiC substrate

1.9 Å

Conjugate gradient minimization

Simulated annealing

Three Layer graphene

Three-layer graphene overlaid on 6H-SiC(0001) obtained by simulated annealing method with TEA potential. The first graphene“buffer” layer refers to one closest to the top surface of substrate and has an average distance of separation about 2.6 Å, and the third layer corresponds to one next to the second graphene layer and these graphene layers are separated by an average distance about 3.2 Å. And the seperation between the second layer and the first layer (the top most layer) is 2.8 Å.

Methodology: Silicene

• Construction of silicene– To obtain an infinitely large free-standing silicene sheet, we first construct a

surface bulk Si structure using a diamond unit cell with a lattice constant of 5.431 Å.

– This is done by replicating a total of Nx ×Ny × Nz diamond unit cells to form a simulation box, subjected to periodic boundary condition.

– The structure is then re-oriented such that the (1 1 1) surface is aligned along the z-axis.

– Si diamond lattice in the (1 1 1) direction resembles a silicene sheet with a non-zero buckling parameter.

– After re-orientating all atom-containing xy-planes perpendicular to the z-axis are removed from the bulk until only a single sheet is left.

– The removal of the silicon atoms creates a region of vacuum with 41.5 Å thick along the z-direction on both sides of the remaining silicon atom plane, which now resembles a free-standing silicene sheet with surface area of 130.563 Å ×150.761 Å, consisting of 9600 atoms and a buckling parameter of 0.44 Å

Simulation Details

1. We employ in our simulations the empirical potentials modified SW [6] to describe the interatomic interactions of Si-Si.

2. The free standing silicene is then relaxed by using conjugate gradient minimization technique.

3. The simulated annealing procedure is proceeded by MD simulation in the NVT ensemble using the Nose-Hoover thermostat with a time step ∆𝑡 = 0.5 fs. The system is then equilibrated at 1 K for time interval of 5 × 104 ∆𝑡 and then raise to 𝑇 = 300 K at the heat rate of 1 × 108 K/s, before the system is equilibrated for a time interval of 2.5 × 104 ∆𝑡.

4. We have performed a convergence test by running trial MD melting simulations at various heating rates by fixing the target temperature 𝑇 = 2000 𝐾. From our convergence test, resultant melting temperature obtained is invariant if the heating rate 2 × 1010 K/s or larger. The system is heated from 300 K for a total 1.7 × 107 ∆t (equivalent to 85 ns) at the mentioned heating rate until it reaches 2000 K.

5. Evolution of the silicene MD trajectory in the simulation is monitored visually as well as quantitatively. Radial distribution function, and caloric curve of the silicene sheet are numerically sampled and measured. In addition, we also measure a numerical descriptor, known as `global similarity index', 𝜉𝑖, to gauge the melting process. Detailed description of the global similarity index is discussed in the following slides

Global Similarity Index

• Chemical similarity is defined as compounds having underlying microscopic similarity.

• It can be used to predict a variety of important properties including biological activity, chemical reactivity and the chemical properties.

• The definition of the global similarity index is based on generic chemical similarity idea for detecting configurational changes along the trajectory during the heating process.

Global Similarity Index

• 𝑑𝑠,𝑖 and 𝑑𝑠,0 represent the sorted distance of atoms relative to the average positions (center of mass) of all the atoms in the cluster for the 𝑖th (denoted as the subscript 𝑖) and the 0th “frame” .

• 𝑛 corresponds to the number of atoms, which is an integer equals to the number of pairs of 𝑘𝑠,𝑖. The value of 𝜉𝑖 = 1 corresponds to totally identicalness and 𝜉𝑖 → 0 for vast difference.

𝜉𝑖 =1

𝑛 𝑘𝑠 ,𝑖 + 1

−1

𝑛

𝑠=1

𝑘𝑠 ,𝑖 = 𝑑𝑠 ,𝑖 − 𝑑𝑠 ,0

Global Similarity Index

• Average positions, a.k.a., mean, that enters the definition of the parameter d_scan be non-uniquely defined.

• Different definitions of mean capture different aspects of configurationalinformation contained in the system.

• Several definitions of mean are possible, namely

– Arithmetic mean

– Harmonic mean

– Quadratic mean

Global Similarity Index

• However, during a melting process, variations in the configuration of the atoms 𝑎 𝑝𝑟𝑖𝑜𝑟𝑖 are not known. In principle, variation in a certain mode of motion among the atoms could be more sensitively picked up by certain mean than the other. To cover all possibilities, the parameter 𝑑𝑠 that enters the definition of 𝜉𝑖calculated by averaging overall of the above three types of mean.

𝜉𝑖 =1

𝑁𝐶𝑂𝑅 𝜉𝑖

𝐶𝑂𝑅

𝐶𝑂𝑅

Global Similarity Index

• COR stands for center of reference, COR = {arithmetic mean, harmonic mean, quadratic mean}.

• Including all types of COR can in principle increase the sensitivity as well as accuracy of the index in capturing variations in the geometrical configuration of the system.

Results and Discussions: Melting of Silicene

• Initially the silicene has equilibrium bond length of ~2.5 Å. An abrupt change can be viewed at 1500 K in which the sheet is tearing apart and the melting process is thus begins. After the melting point, the Si particles settle down to form four smaller “islands” which has equilibrium bond length of ~2.8 Å.

• Pair correlation function of the system at various temperatures. At room temperature a sharp peak occurs at ~2.5 Å. The distribution curve begins to widen up and the peak at ~2.5 Å is lowered as target temperature 𝑇 increases. The sudden appearance of a peak at ~3.0 Å in the distribution curve for 𝑇 =1500 K indicates the occurrence of melting of the silicene sheet

• Potential energy plot (caloric curve) of the system (eV) against temperature (K) measured for target temperature, 𝑇 = 2000 K. The arrow indicates the point at which particles begin to vibrate more violently and vigorously such that vacancies begin to appear at selected areas on the silicene sheet as Si particles are ejected from their respective sites. The sharp and abrupt drop of potential energy at 1500 K indicates occurrence of melting at that temperature.

• Global similarity index plot against temperature. The silicene structure is compared with its 300 K state. The value of 1 means both comparing structure are identical. As the temperature increases, the comparing structures become more and more dissimilar. As the kinetic energy of system increases, the Si particles vibrate (there maybe rotational or translational motion based on statistical influence of choice) more violently. The melting occurs at 1500 K, and the structure failed to be identified as a silicene.

Methodology: Zinc Oxide

• Structure of ZnO

• Slab comprising 15×15×3 unit cells• The slab was sandwiched between two vacuum layers of thickness

100 Å

• ThicknThe surface area of the slab in the supercell was 1783 Å2essof the slab itself was 15 Å.

• 2700 atoms

Simulation Details

• Reaxff [7] potential of ZnO is employed in the system.• The step size throughout the simulation is Δt = 0.5 fs. The structure is first

optimized at 0.1 K using the built-in conjugate gradient minimizer.• The temperature of the system is then heated up to 𝑇𝑟 = 300 K in 5000

steps. The system is further equilibrated for 20000 steps at 300 K. At the end of the equilibration at 𝑇𝑟, the temperature is raised to a chosen target temperature, T, at a rate 5 × 1013 K/s. The evolution of the system at annealing temperature is then followed at constant T for a total of 𝑁𝑠𝑡𝑒𝑝 =250000.

• The temperature of the system is then quenched from 𝑇 to 0.1 K at a rate of 5 × 1010 K/s.

• Nose-Hoover thermostat (NVT) is used throughout the simulation to control the temperature. The damping constant for the thermostat is set to 5 fs. The total steps in each simulation are dependent upon the target temperature T. The total duration for the simulations range from 315 000 steps (0.16 ns) to 387 000 steps (0.20 ns).

Results and Discussion: Zinc Oxide

• Ratio of O atoms sublimated (normalized to original number of O atoms on the surface) for T = 300 to 1300 K (right).

• Green luminescence (512 nm) intensity of the two ZnOsamples at different temperatures (left).

• Since the parametrisation of the ZnO ReaxFF does not include sublimation of O atoms from ZnO surfaces, the value of Tt from MD falls so closely to the experimentally measured range of 873 K − 973 K is a very suggestive finding.

• If we accept an uncertainty of ∼ ± 100 K in the threshold temperature, which is reasonably acceptable for a typical MD calculation, then the following picture seems appropriate

– Oxygen point defects on the wurtzite ZnO (0 0 0 ത1) surface are created only at or above the threshold annealing temperature.

– The abundance of oxygen point defects on the surface increases as the annealing temperature T increases (where T ≥ Tt).

• The qualitative agreement between the MD results and the two experimental measurements discussed above is encouraging.

• It provides a motivation to capitalize the MD simulation as a reliable means to derive useful information of the detailed mechanism of the ZnO surface undergoing annealing at the atomistic level, which is otherwise difficult to obtain via experimental approach alone.

Partial Charge distribution

• Snapshots at various stages of the partial charge distribution in the slab when undergoing annealing at 𝑇 = 300 K. (a) At the beginning, the slab has only gone through energy minimization at 0.1 K without any thermal treatment. (b) and (c) are snapshots during which the slab is being annealed at the temperature plateau 𝑇. (d) Slab at the end of thermal history

• Snapshots at various stages of the partial charge distribution in the slab when undergoing annealing at T = 1300 K.

• Partial charge distributions at the end of an MD run for eight annealing temperatures ranging from 400 K to 1100 K.

• Charge density 𝜌(𝑧) as a function of depth from the (0 0 0 ത1) surface, 𝑧, for annealing temperature 𝑇 = 500 K, 𝑇 = 700 K, 𝑇 = 800 K and 𝑇 =1000 K. The vertical axis is in units of 𝑒/Å3. The 𝑧-axis is in units of Å. Note the qualitative change of the density profile (especially the region close to the (0 0 0 ത1) end) when crossing from 𝑇 = 700 K to 𝑇 = 800 K.

• Average partial charge per sublimated atom as a function of annealing temperature.

• When the sublimated atoms first appeared at 𝑇 = 𝑇𝑡, the average net charge per atom was ∼ − 0.5𝑒, and this value increased to ∼ − 0.7𝑒 at 𝑇 =1300 K.

• When the O atoms were sublimated from the surface, they carry away with them a net negative charge, leaving behind a substrate that has an overall net positive charge on the surface.

Conclusion

• Appealing to the Tersoff and a Tersoff-type empirical potentials in classical MD simulations, and in conjunction with the simulated annealing technique, we grew layers of graphene on the Si-terminated 6H-SiC surface.

• Through analyzing the structures of layers of graphene, in terms of the carbon-carbon average bond-length, binding energy, and pair correlation function, we give much credence to the TEA potential of Erhart and Albe.

• The relative areas of coverage of different graphene layers which shed light on the structural stability are consistent also with experiments, i.e., the first buffer layer is less stable in contrast to the second layer. The TEA potential reflects all these features nicely.

Conclusion

• We have also performed a MD experiment to measure the melting point of a finite but large free-standing silicene sheet using optimized SW potential.

• The melting scenario of the silicene sheet has been visually captured. Analysis of the statistical data sampled from the MD simulation including caloric curve, radial distribution function and global similarity index, reveals that the silicene sheet melts at 1500 K.

• This result serves the purpose of being an independent measurement of the melting temperature of free-standing silicene. Our value is consistent with that obtained by Berdiyorov et. al. who uses ReaxFF [8], but is in contrast to which predicts 1750 K using ARK [9] parameter set.

• The prediction of melting of free-standing silicene sheet will serve as reference information to be verified by experiments when they become available.

Conclusion

• We also investigated the MD annealing simulations of a wurtzite ZnO slab, with a thickness of 15 Åand a surface area of 1783 Å2. The simulations are performed using ZnO ReaxFF potential.

• There are two polar surfaces on ZnO slab, namely the (0 0 0 1) and (0 0 0 ത1). The slab is thermally annealed from 0.1 K in stages up to a target temperature 𝑇𝑡, and then further equilibrated at the constant 𝑇 plateau for sufficiently large number of steps before the temperature is gradually quenched to 0.1 K.

• The MD results show that O atoms on the (0 0 0 ത1) surface are sublimated whenever 𝑇 ≥ 𝑇𝑡, where 700 K < 𝑇𝑡 ≤ 800 K. No other atom(i.e. Zn atom) sublimates from either surface. The ratio of O atoms sublimated to the total number of O atoms on the surface increases as 𝑇 (𝑇 ≥𝑇𝑡 ) increases. The existence of a threshold temperature 𝑇𝑡 at which O atoms from the (0 0 0 ത1) surface begins to sublimate qualitatively agrees with experiment data .

• The MD simulation results also show that the sublimated O atoms do so in pairs. Due to the geometrically asymmetric condition, atoms at the surface are expected to form dimers. The O pairs seen leaving the surface in the MD sublimation originate from these dimers.

• We have also investigated the partial charge density of the slab as a function of depth from the surface, 𝑧, after each slab has gone through an annealing history.

• The oscillatory form of the charge density of the slab resembles Friedel oscillations which are expected for charge density near a surface. A qualitative change in the partial charge distribution pattern is observed when the annealing temperature crosses 𝑇 = 700 K to 𝑇 = 800 K. Thermally-driven sublimation of atoms only occurs in the O-terminated (0 0 0 ത1) surface, whereas no sublimation of either atom type would occur in the (0 0 0 1) surface which is Zn-terminated, presumably due to a strong attractive force exerted by the Zn atoms at these surfaces.

References

1. J. B. Hannon and R. M. Tromp. Pit formation during graphene synthesis on SiC(0001): In situ electron microscopy. Physical Review B, 77, 241404(R), 2008.

2. J. Borysiuk, R. Bozek, W. Strupi´nski, A. Wysmołek, K. Grodecki, R. Stepniewski, and J. M. Baranowski. Transmission electron microscopy investigations of epitaxial graphene on C-terminated 4H–SiC. Journal of Applied Physics, 108, 013518, 2010.

3. C. Tang, L. Meng, H. Xiao, and J. Zhong. Atom-by-atom simulations of graphene growth by decomposition of SiC (0001): Impact of the substrate steps. Applied Physics Letter, 103, 141602, 2013.

4. C. Lampin, C. Priester, C. Krzeminski, and L. Magaud. Graphene buffer layer on Si-terminated SiCstudied with an empirical interatomic potential. Journal of Applied Physics, 107, 103514 , 2010.

5. N. Jakse, R. Arifin, and S. K. Lai. Growth of graphene on 6H-SiC by molecular dynamics simulation. Condense Matter Physics, 14, 43802, 2011.

6. Xiaoliang Zhang, Han Xie, Ming Hu, Hua Bao, Shengying Yue, Guangzhao Qin, and Gang Su. Thermal conductivity of silicene calculated using an optimized Stillinger-Weber potential. Phys. Rev. B, 89, 054310, 2014.

7. D. Raymand, A.C. van Duin, M. Baudin, K. Hermansson, A reactive force field (reaxff) for zinc oxide, Surf. Sci. 602, pp. 1020–1031, 2008.

8. G. R. Berdiyorov and F. M. Peeters. Influence of vacancy defects on the thermal stability of silicene: a reactive molecular dynamics study. RSC Adv., 4, pp. 1133-1137, 2014.

9. V. Bocchetti, H. T. Diep, H. Enriquez, H. Oughaddou, and A. Kara. Thermal stability of standalone silicene sheet. Journal of Physics: Conference Series, 491:012008, 2014.