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Orientation-Dependent Structural Transition and Melting of Au Nanowires Yu-Hua Wen,*,† Yang Zhang,† Jin-Cheng Zheng,† Zi-Zhong Zhu,† and Shi-Gang Sun‡ Department of Physics and Institute of Theoretical Physics and Astrophysics, Xiamen UniVersity, Xiamen 361005, China, and State Key Laboratory of Physical Chemistry of Solid Surfaces, Department of Chemistry, College of Chemistry and Chemical Engineering, Xiamen UniVersity, Xiamen 361005, China ReceiVed: July 7, 2009; ReVised Manuscript ReceiVed: October 10, 2009
Using molecular dynamics simulations with the quantum corrected Sutton-Chen type many-body potential, we have studied the thermal stability of Au nanowires along the [100], [110], and [111] crystallographic orientations during continuous heating. The bond pair analysis and Lindemann index are used to characterize the structural and thermal evolution of these nanowires. The results show that the critical temperatures of structural transition, melting, and fracture are dependent on the crystallographic orientation of Au nanowires. It is found that all the Au nanowires exhibit an inhomogeneous melting behavior from the surface into the interior. The structural transition from a fcc to hcp structure prior to surface premelting is closely associated with the activities of Shockley partial dislocations driven by the internal stress because of the thermal expansion of the nanowires with increased temperature. A comparison of the results of three types of nanowires indicates that the [110] nanowire possesses a better thermal and structural stability compared with other oriented nanowires, which helps to explain why Au nanowires possess a [110] preferred orientation during the experimental growth procedure. 1. Introduction Over the past decade, materials based on nanometer-sized structures have attracted a great deal of interest because of the potential applications of nanostructures in electronic, optoelectronic, and electromechanical systems.1 As one of the most important one-dimensional (1-D) nanostructures, metal nanowires are particularly interesting because they possess unique physical properties associated with their highly anisotropic geometry as well as finite size effects when compared with their macroscopic counterpart.2 These physical properties have motivated the development of novel nanowire-based nanoelectromechanical systems (NEMS) that have been proposed for nanoscale interconnects, sensors, transparent, conductive electrodes in photovoltaic and optoelectronic devices, and usage as tips for scanning electron microscopy (SEM) and atomic force microscopy (AFM).3-6 There are numerous studies dedicated to the preparation, structural characterization, and properties of metal nanowires.7-16 In these studies, one of the most challenging focuses is to control the size and morphology of metal nanowires.11,12,16 Knowledge of thermal properties of metal nanowires and the effects of size and shape are essential for the preparation, fabrication, and functionality of nanowires in their various areas of applications. The melting behaviors of nanowires have exhibited dramatically different characteristics from their bulk counterpart with the melting temperatures of nanowires decreasing with the reduction of their diameters, leading to strong size effects.17-21 The majority of experimental studies and theoretical calculations show that the melting begins preferentially at the surface in the melting process of nanowires and nanorods because surface * To whom correspondence should be addressed. E-mail: yhwen@ xmu.edu.cn (Y.-H.W.),
[email protected] (S.-G.S.). † Department of Physics and Institute of Theoretical Physics and Astrophysics. ‡ State Key Laboratory of Physical Chemistry of Solid Surfaces, Department of Chemistry, College of Chemistry and Chemical Engineering.
atoms have fewer nearest neighbors and weaker bonding that could lead to an earlier surface melting behavior.17,19,20 An abnormal behavior, however, was discovered in molecular dynamics investigations of the melting of Au, Pd, and Zr nanowires in which the melting started from the interior atoms, and the surface melting was representative of the overall melting temperature of wires.22-26 Structural transition from an initial fcc to hcp structure prior to the melting transition and solid-liquid coexistence during the heating process were also observed in Ti, Pd, Pd-Cu, and Pd-Rh nanowires.21,23,27 These studies have prompted the development of metal nanowires. Recently, gold nanowires have been the subject of intense theoretical and experimental studies.7,10,11,15,22,28-34 This interest has mainly risen after a “neck” of atoms just a few atomic diameters long were formed under tensile stress of Au nanowires in both experiments and theoretical simulations and displayed quantized conductance, together with the good stability of very thin Au nanowires.28-34 These nanowires could be grown in different crystallographic orientations, such as [100] and [110].11,28-31 Experimental and theoretical studies show that the mechanical properties of Au nanowires are strongly dependent on their crystallographic orientations.11,31-35 However, less is known about the thermal properties and melting of these nanowires with different crystallographic orientations. To date, the orientation dependence of the thermal properties and melting process of Au nanowires remains unclear. In this paper, we will employ molecular dynamics simulations to investigate the structural evolution and dynamics associated with the melting of Au nanowires oriented in the [100], [110], and [111] directions and address the orientation-dependent effects on their thermal properties. A brief description of the simulation methods is given in the following section (section 2). We present the calculated results, discussion, and comparisons with other results in the third section (section 3). The main conclusions are summarized in the fourth section (section 4).
10.1021/jp906393v CCC: $40.75 2009 American Chemical Society Published on Web 11/10/2009
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Figure 1. Schematic illustration of the [100], [110], and [111] Au nanowires. The corresponding cross sections are illustrated with their crystallographic directions. Red spheres denote the boundary atoms, and blue spheres denote moveable atoms.
2. Simulation Methodology In atomistic computer simulations, we use the quantum corrected Sutton-Chen (Q-SC) type potentials17 to describe interatomic interactions for gold nanowires. These potentials represent many-body interactions, and their parameters are optimized to describe the lattice parameter, cohesive energy, bulk modulus, elastic constants, phonon dispersion, vacancy formation energy, and surface energy, leading to an accurate description of many properties of metals and their alloys.19,21,36 For the Q-SC type potential, the total potential energy for a system of atoms can be written as
U)
[
∑ Ui ) ∑ ε 21 ∑ V(Rij) - c√Fi i
i
j*i
]
(1)
Here, V(Rij) is a pair interaction function defined by the following equation
V(Rij) )
() a0 Rij
n
(2)
accounting for the repulsion between the i and j atomic cores; Fi is a local electron density accounting for cohesion associated with atom i defined by
Fi )
∑ j*i
() a0 Rij
m
(3)
In eqs 1-3, Rij is the distance between atoms i and j, a0 is a length parameter scaling all spacings (leading to dimensionless V and F), c is a dimensionless parameter scaling the attractive terms, ε sets the overall energy scale, and n and m are integer parameters such that n > m. Given the exponents (n and m), c is determined by the equilibrium lattice parameter, and ε is determined by the total cohesive energy. For the Q-SC type potential of gold, the parameters are given as follows: n ) 11, m ) 8, ε ) 7.8052 meV, c ) 53.581, and a0 ) 4.0651 Å. Three types of Au nanowires, that is, [100], [110], and [111] Au nanowires, are constructed from a large cubic fcc single crystal of gold, as shown in Figure 1. The [100] Au nanowire is oriented in the [100] direction with a polyhedral cross section with (010) and (011) side surfaces, and the [110] Au nanowire is oriented in the [110] direction with a polyhedral cross section
with (001) and (1-11) side surfaces. The last [111] Au nanowire is oriented in the [111] direction with a polyhedral cross section and (1-10) and (0-11) side surfaces. All the three types of nanowires have a diameter of about 2.29 nm and length of about 16.43 nm. The constructed nanowires contain 3977, 3855, and 3857 atoms for the [100], [110], and [111] ones, respectively. Upon starting the MD simulations, Au nanowires are first quasi-statically relaxed to a local minimum configuration through the conjugate gradient method.36 After full relaxation, these nanowires are subjected to a heat processing. To make the simulations more realistic, we employ constant temperature molecular dynamics (NVT-MD) to allow energy fluctuations, which may be critical to the resulting dynamics. The equations of atomic motion are integrated by the Verlet-velocity algorithm37 with a time step of 1.0 fs. These three types of nanowires undergo the heating process from 0 to 1400 K with temperature increments of 50 K. However, a smaller increment of 10 K is used to determine the melting point more precisely when the heating process is near the melting range. The simulations are carried out for 200 ps of the relaxation time at each temperature. The desired temperature is maintained by a Nose-Hoover thermostat.38 During the heating process, the boundary atoms, residing within four atomic layers at both ends, are clamped and not allowed to move from their initial positions. The percentage of fixed atoms is about 9.8, 10.4 and 11.2% for the [100], [110], and [111] nanowires, respectively. 3. Results and Discussion 3.1. Melting Point Identification. Important information concerning the thermodynamic properties, the characteristics, and extent of melting during the heating process can be obtained from data records calculated by molecular dynamics simulations. The transition temperature from the solid to liquid phase is usually identified by investigating the variation in the thermodynamic properties, such as potential energy and specific heat capacity.17,21,39 The melting point is defined as a temperature at which the heat capacity reaches its maximum. We calculate the average potential energy during the heating process as a function of temperature and then obtain the heat capacity from the following equation
Cp(T) )
dU 3 + Rgc dT 2
(4)
where Rgc ) 8.314 J/mol K is the ideal gas constant. This equation was also used in previous studies.17,40 Because the melting temperature is sensitive to the potential type adopted in molecular dynamics simulations, it is necessary for us to test the validity of the Q-SC potential. Therefore, the heating process of bulk Au without a free surface is first simulated. The calculated results are illustrated in Figure 2, showing that the surface-free Au transforms spontaneously into a liquid at 1380 K. Considering that superheating to temperatures above the equilibrium melting points has been confirmed in a surface-free perfect crystal,41,42 here, the simulated melting point should be closer to the experimental value of Tm ) 1338 K for pure Au,43 implying that the Q-SC potential is suitable for investigation of the thermodynamic properties of Au metal. Figure 2 also illustrates the temperature dependence of the potential energy as well as the specific heat capacity for Au nanowires. In the same way, the overall melting point can be deduced to be 930, 980, and 920 K for the [100], [110], and [111] Au nanowires, respectively. It shows that the overall melting temperature of a Au nanowire is strongly dependent
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δi )
1 N-1
∑
√〈Rij2〉 - 〈Rij〉2 〈Rij〉
j*i
(5)
and the system-averaged Lindemann index is given by
δ)
Figure 2. Potential energy and heat capacity of Au nanowires with three different orientations as a function of temperature. Solid lines denote the potential energy, and dashed lines denote the heat capacity.
on its crystallographic orientation. The comparison of the average potential energy curves during the heating process reveals that, in three types of nanowires, the [110] nanowire is the most energetically favorable, which is in excellent agreement with experimental observations and the previous simulated results.11,35 In comparison to the bulk value, the melting points of Au nanowires are significantly reduced by 400-460 K. The lowering of the melting temperature with decreasing size is predicted by the Gibbs-Thompson equation that was derived within the frame of classical thermodynamics.44 Subsequently, such dependence of melting point on nanowire size was also verified by molecular dynamics simulations in previous studies.17-21 The reduction of overall melting point could be associated with a high surface-volume ratio in nanometer-sized wires and low surface premelting temperature, which was reported by investigating the diffusive behavior of atoms in nanowires.17 The potential energy per atom of Au nanowires monotonically increases with the increasing temperature before the occurrence of their overall melting, followed with a jump in the potential energy at the overall melting point (see Figure 2). It should be noted that there is a decrease in the potential energy curve beyond the melting temperature. As a result, the specific heat capacity becomes negative and reaches its minimum. This feature could be attributed to the fracture of Au nanowires at high temperature. The critical temperature for the fracture is 960, 1020, and 980 K for the [100], [110], and [111] Au nanowires, respectively. Moreover, the persistent range from the overall melting to fracture is 30, 40, and 60 K for the [100], [110], and [111] Au nanowires, respectively. High melting and fracture temperatures indicate that the [110] Au nanowire exhibits a better thermal stability compared with other oriented nanowires. 3.2. Mechanism of Melting in Nanowires. The melting mechanism of materials can be detected through analysis of the structural evolution, diffusion coefficients, root-mean-square displacement (rmsd) of atoms and so on. However, the Lindemann index is a simple but effective measurement of thermally driven disorder.45 It is often used to characterize the thermal evolution of a system. For a system of N atoms, the local Lindemann index for the ith atom in the system is defined as the root-mean-squared (rms) bond length fluctuation as46,47
1 N
∑ δi
(6)
i
where Rij is the distance between the ith and jth atoms and N is the number of atoms in the system. The Lindemann index was originally developed to study the melting behavior of bulk crystals. The Lindemann criterion says that the melting occurs when the index is in the range of 0.1-0.15, depending on materials.48 However, a much smaller criterion index of 0.03 was recently proposed for the melting of clusters and homopolymers because of the relaxed constraint of the surface atoms.46 Figure 3 shows the temperature dependence of the calculated Lindemann indices of Au nanowires. It can be seen that a jump occurs in Lindemann index curves when the nanowire is melted completely. Clearly, 0.03 is not suitable as the critical value for the nanowires and a smaller critical one is necessary. Generally, this critical value is dependent on materials and their environment. Here, we find that the value of 0.016, which is slightly larger than the Lindemann index at the overall melting point, can be a reasonable Lindemann criterion, as denoted by the dashed line in Figure 3. The Lindemann index of a Au nanowire is increased with the increasing temperature before melting. Different from the continuously increasing Lindemann index of metal nanoparticles after melting,39 we find that these Au nanowires present a remarkable reduction of the Lindemann index beyond their melting, followed with a continuous increase during the heating process. The reason for such a reduction is associated with the decrease of atom fraction with high mobility in the center of nanowires during the process of necking and breaking of nanowires right after melting. For investigation of the melting process, the Lindemann atom is introduced in this paper. It is defined as the atom whose Lindemann index is larger than the critical value of the Lindemann criterion. The percentage of Lindemann atoms in the nanowires is shown in the inset of Figure 3. It can be seen from the inset that no Lindemann atom occurs in three types of Au nanowires below 650 K. They begin to appear and have more and more percentage with further increased temperature. At the melting point, the percentage of Lindemann atoms increases abruptly. It exceeds 95% after the overall melting and is close to 100% with further heating. It seems that there is no significant effect of breaking on the percentage of Lindemann atoms, implying that all the atoms belong to the Lindemann ones during the breaking process in Au nanowires. Figure 4 further shows the distribution of Lindemann atoms for Au nanowires. It can be clearly seen that the Lindemann atoms locate on the surface of Au nanowires before their overall melting, meaning that the premelting first occurs on the surface. When the temperature exceeds the melting point, the atoms located in the core of the nanowire are also transformed into Lindemann atoms. Meanwhile, Figure 4 also shows that the surface premelting is inhomogeneous. It preferentially appears in the region far from the ends of nanowires, indicated by the fact that those atoms located in the end regions have a lower Lindemann index than those in other regions. As mentioned in the Simulation Methodology, the boundary atoms at both ends
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Figure 3. Temperature dependence of the Lindemann indices of Au nanowires. The inset shows the percentage of Lindemann atoms in these nanowires as a function of temperature.
are fixed and not allowed to move from their initial positions during the heating process. The inactive atoms at both ends act actually as two “solid surfaces”, which can result in the stronger confined behavior for those atoms near the “solid surfaces”. Hence, those atoms located in the end regions have lower mobility than those in other regions. The inhomogeneous premelting suggests that the melting does not occur all at once throughout a nanowire but rather initiates via local instabilities. Beyond the melting point, non-Lindemann atoms are few and all of them are concentrated in both ends of nanowires. This behavior could be associated with those atoms clamped at both ends. For the [110] nanowire, interestingly, we observe that it breaks into three clusters after two sequential fractures. However, only one fracture case is observed in the [100] and [111] nanowires during the heating process. By capturing the breaking process of Au nanowires, we find that the [110] nanowire prefers to form two-atom thick chains during the heating process but other nanowires prefer to form one-atom thick chains. Our recent study also shows that the [110] Au nanowire is easier to form an atom thick chain nanostructure under stress than the [100] and [111] ones, indicating that it exhibits a better ductibility. Therefore, our simulation results have verified that the [110] Au nanowire is a preferential candidate when considering the formation of an atom chain structure through the stretching and heating of the nanowire. Besides, it should be noted that the [111] nanowire experiences 70 K from melting to breaking, whereas the [100] nanowire only experiences 30 K, though they both form one neck during the heating process. It may indicate that the bonding between atomic layers along the axis is stronger for the former than for the latter in the presence of the liquid phase. It should also be noted that the boundary condition of nanowires plays an important role in the simulated results. If the nanowires are free-standing and their lengths are short, they will directly collapse into a cluster beyond their overall melting point. However, they will first break and then evolve into clusters after overall melting if their lengths are too long or infinite by applying the periodic boundary condition along their axis, as observed in our previous studies on infinite Ni and Au nanowires.19,49 3.3. Structural Transition in Nanowires. To further characterize the structural evolution of Au nanowires in the heating process, we use the common neighbor analysis (CNA) proposed by Honeycutt and Andersen50 to monitor the local structure of nanowires. The CNA method has already been used successfully to analyze the structural evolution during the deformation process.51-53 This analysis assigns four indices, ijkl, to each pair of atoms that have common neighbors and provides a description
Figure 4. Snapshots of Au nanowires taken at four temperatures. The calculated Lindemann index distributions of atoms at these temperatures are presented at right. Note that red, blue, and green spheres denote the boundary atoms, non-Lindemann atoms, and Lindemann atoms, respectively.
of the local environment of the pair. In this analysis, the bonds between an atom and its nearest neighbors are examined to determine the crystal structure. The different types of pairs are associated with different types of local order. All bonded pairs in the fcc crystal are of type 1421, whereas the hcp crystal has equal numbers of types 1421 and 1422. Because the pairs beside types 1421 and 1422 do not reveal some useful information, here, we have used CNA to classify atoms into three categories. Atoms in a local fcc order are considered to be fcc atoms. Atoms in a local hcp order are classified as hcp atoms whose occurrence in an fcc crystal is regarded as the structure of stacking faults. Atoms in all other local orders are considered to be “other” atoms. The temperature dependence of the percentages of the three categories of atoms is shown in Figure 5a for the three Au
Structural Transition and Melting of Au Nanowires
Figure 5. (a) Percentages of three categories of atoms are shown as a function of temperature. Black, red, and green points denote the CNA results for the [100], [110], and [111] Au nanowires, respectively. Solid square, circle, and triangle denote the fcc, hcp, and “other” atom, respectively. (b) Temperature dependence of the normal stress along the axis of the nanowire.
nanowires. The considerable proportion of “other” atoms (about 40% of all atoms) at low temperature below 600 K indicates that the configurations of nanowires are not in perfect fcc structure due to the presence of surface. Their interiors are still closer to an ideal fcc structure. No hcp atoms can be found below 600 K for the [100] and [111] nanowires as well as 750 K for the [110] nanowire. Comparison of the percentages of fcc and “other” atoms in Figure 5a indicates that only a minor part of fcc atoms transform into “other” ones with increased temperature. Therefore, these nanowires basically retain their initial fcc structure. However, hcp atoms occur in the [100] and [111] Au nanowires at 650 K as well as in the [110] Au nanowire at 800 K. These hcp atoms form local hcp structures. Their occurrences mean that the structural transition happens at these critical temperatures. Part of the hcp structures remains until the overall melting of nanowires while the rest of them disappear before melting. At the melting point, all fractions of fcc and hcp atoms show a sudden decrease to near zero, indicative of the phase transition to liquid. These are in good agreement with the evolution of the simulated potential energy and specific heat capacity. After the overall melting, all fcc and hcp atoms in nanowires transform into “other” atoms, suggesting the full loss of initial nanowire structure. The aforementioned results show that the solid-solid transition from fcc to hcp occurs prior to the melting of nanowires. The existence of these metastable hcp structures endure a broad temperature range of 290, 190, and 280 K for the [100], [110], and [111] Au nanowires, respectively. The structural transition has been verified in Ti, Pd, Pd-Cu, and Pd-Rh nanowires.21,23,27 Besides, the structural changes during the heating process have also been observed in metallic and bimetallic clusters.54-56 The existence of a low-temperature hcp phase is characteristic of fcc metal nanowires and might be closely associated with the presence of internal stress in these nanowires during the heating temperature. To explore the reason for the occurrence of the hcp phase, we calculate the normal stress of nanowires at different temperatures,36 as shown in Figure 5b. The normal stress along the axis of the nanowire is negative and increased with enhanced temperature before the occurrence of structural transition. The negative stress means that the nanowire endures
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Figure 6. Temperature dependence of the percentage of pairs for the (a) [100] Au nanowire, (b) [110] Au nanowire, and (c) [111] Au nanowire.
a compressive stress when the temperature is increased, which is mainly attributed to the thermal expansion of the nanowire. Once the stress reaches the negative maximum, further increased temperature will initiate the structural transition from fcc to hcp in nanowires. These hcp atoms are regularly arranged in a series of two adjacent {111} planes through the CNA analyses of atomic arrangement of nanowires. As we know, intrinsic stacking faults appear as two adjacent {111} planes of hcp atoms and extrinsic stacking faults are two {111} planes of hcp atoms separated by a single {111} plane of fcc atoms, whereas twin boundaries will be seen as a single {111} plane of hcp atoms.52 In terms of theory of crystal dislocations, two adjacent {111} planes of hcp atoms will appear when a Shockley partial dislocation nucleates and propagates through an fcc crystal. Therefore, we may deduce that the activities of Shockley partial dislocations occur in {111} planes during the heating process. Once a partial dislocation propagates through the nanowire, the internal stress will partly release, which is verified by Figure 5b. With the activities of partial dislocations, the normal stress is gradually decreased and passes over zero after the melting point. This trend continues until the breaking of nanowires. After the nanowire breaks into nanoclusters, the normal stress is near to zero. For further investigation on the structural evolution of Au nanowires during the heating process, a detailed bond pair analysis is introduced to characterize the local atomic structures in this paper. The bond pair analysis technique is an effective method to describe various clusters in the liquid or solid states.23-26 Accordingly to the bond pair analysis, all bonded pairs in the bulk fcc crystal are of type 1421, whereas the bulk hcp crystal has equal numbers of type 1421 and 1422. The 1441 and 1661 are typical pairs of the bulk bcc crystals. The 1201 and 1311, 1321, and 1331 bond pairs represent the rhombus symmetrical features of short-range order. The 1551 bond pair, corresponding to a pentagonal bipyramid, is characteristic of icosahedral order, and the 1541 bond pair is a distorted 5-fold symmetric structure. As a typical bond pair in liquid, the 1431 bond pair corresponds to tetrahedral structures. The 1541 and 1431 pairs represent the short-range ordering in liquid and amorphous states. Figure 6a-c illustrates how the percentage of each bond pair mentioned above in all the bond pairs of our simulated nanowires does change with temperature during the heating process. Three dashed lines denote the critical temperatures of the structural transition, overall melting, and fracture. It is seen
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that the 1421 bond pair occupies 80% or so of all the bond pairs at low temperature, indicating the good fcc structure of Au nanowires. The rest of bond pairs belonging to 1201, 1211, 1301, and 1311 are mainly distributed on the surface region of nanowires. The percentages of all bond pairs remain unchanged before the structural transition. When the temperature exceeds the first critical temperature, the 1422 bond pairs occur in the nanowires and increase with increased temperature. Meanwhile, the number of 1421 bond pairs decreases significantly but the numbers of 1201, 1301, and 1311 increase. All the above indicate that the 1421 bond pairs partly transform into the 1422 bond pairs, and the rest of them transform into these disordered bond pairs. At the melting point, the proportion of 1421 bond pairs decreases dramatically to 23% or so. These reserved 1421 bond pairs are mainly distributed near both ends of nanowires. Some bond pairs that characterize the disordered structures, such as 1201, 1211, 1301, 1311, 1321, 1541, and 1551, present a sharp increase. It is suggested that the presence of the 1331, 1321, 1301, and 1211 is an indication of disordered structures in solids.23-26 This means that the fcc structure does not remain dominant in the nanowires, which is in agreement with the results of the potential energy and Lindemann index. As the temperature is increased, the numbers of both 1421 and 1422 bond pairs decrease monotonically toward zero, while the numbers of 1201 and 1211 bond pairs increase continuously, as seen from Figure 6. It seems that the distribution of various bond pairs has no change in the breaking of nanowires, unlike those in the overall melting process. After the formation of the clusters, the main bond pairs are 1201, 1211, 1301, and 1311, indicative of the disordered structures in liquid. Interestingly, the numbers of both 1541 and 1551 bond pairs, corresponding to icosahedral clusters, have a continuous reduction as the temperature increases. The number of 1431 bond pairs remains basically zero. Because the liquid clusters cannot be composed of only one kind of local atomic structure, the existence of various local atomic structures in liquid could just meet the packing demand.23 The above analyses show that the [110] Au nanowire presents higher critical temperatures of structural transition, melting, and fracture, indicating its better thermal and structural stability compared with other oriented nanowires. Why does it possess an excellent stability? This should be closely associated with surface atomic configurations, which is commonly determined by the surface Miller index and cross-sectional shape. As we know, the surface energy of different crystalline planes of an fcc metal is increased in the order of γ{111} < γ{100}
