## Abstract

Although aluminum has a smaller modulus in {111}〈112̄〉 shear than that of copper, we find by first-principles calculation that its ideal shear strength is larger because of a more extended deformation range before softening. This fundamental behavior, along with an abnormally high intrinsic stacking fault energy and a different orientation dependence on pressure hardening, are traced to the directional nature of its bonding. By a comparative analysis of ion relaxations and valence charge redistributions in aluminum and copper, we arrive at contrasting descriptions of bonding characteristics in these two metals that can explain their relative strength and deformation behavior.

The minimum shear stress necessary to cause permanent deformation in a material without imperfections is fundamental to our concept of materials strength and its theoretical limits under large strains (1, 2). With the possible exception of recent nanoindentation measurements (3), it has not been feasible to directly measure the ideal shear strength of crystals. The demonstration that this property can be reliably determined by first-principles calculations therefore would have important implications for the understanding of the behavior of solids at the limit of structural stability. Results on stress-strain behavior of Al and Cu in {111}〈112̄〉 shear, calculated with density functional theory (DFT) and accounting for full atomic relaxation, have been reported (4), where Cu was found to have a higher ideal shear strength than that of Al. Using various DFT methods and systematically cross-checking the results, we further investigated the shear strength and deformation of Al and Cu and found instead that Al has the higher strength. Here, we report and substantiate our findings by detailing the energetics of shear deformation, the pressure-hardening behavior, and valence charge redistribution during deformation. These considerations show that the ideal shear strength and related properties such as stacking fault energies of Al and Cu can be accurately calculated and that the results can be rationalized by the underlying electronic structure. We suggest that bonding in Al is much more like a “hinged rod,” and we emphasize the importance of the breaking and reformation of directional bonds as compared to the isotropic “sphere-in-glue”–like behavior in Cu.

The intrinsic stacking fault energy, a measure of the energy penalty when two adjacent atomic planes in a crystal lattice are sheared relative to each other, is known to play an important role in the structure and energetics of dislocations formed by slip processes. Although it is known experimentally that the intrinsic stacking fault energy is much larger in Al than in Cu, this finding has not been related to their ideal shear strengths. For this purpose, we introduce a general function (Fig. 1A)(1)where *x* is the relative displacement in the slip direction between two adjacent atomic planes (we focus on {111}〈112̄〉 slip here),*E _{n}
*(

*x*) is the increase in total energy relative to its value at

*x*= 0, with

*n*+ 1 being the number of planes involved in the shearing and

*S*

_{0}being the cross-sectional area at

*x*= 0. The series of functions γ

_{1}(

*x*), γ

_{2}(

*x*), …, γ

_{∞}(

*x*) may be called the multiplane generalized stacking fault energy, with γ

_{1}(

*x*) being the conventional generalized stacking fault energy (GSF) (5) and γ

_{∞}(

*x*) being the affine strain energy. The intrinsic stacking fault energy γ

_{sf}is γ

_{1}(

*b*

_{p}), whereb⃗

_{p}= [112̄]

*a*

_{0}/6 is the partial Burgers vector (

*a*

_{0}, equilibrium lattice constant). The unstable stacking energy γ

_{us}, an important parameter in determining the ductility of the material (6), is γ

_{1}(

*x*

_{0}), where

*d*γ

_{1}/

*dx*(

*x*

_{0}<

*b*

_{p}) = 0. It is instructive to compare different γ

_{n}(

*x*) of the same slip system as

*n*varies. The difference should be relatively small from a local “glue” (shaded region in Fig. 1A) viewpoint where we take the valence electron cloud to be the glue. We also have the asymptotic behavior at large

*n*(2)where γ

_{twin}(

*b*

_{p}) is the unrelaxed twin boundary energy. The rate of convergence to Eq. 2 reflects the localization range of metallic bonding in a highly deformed bulk environment.

The DFT calculations, following the same procedure previously described (4), were performed with our own plane wave code and four other packages, Vienna ab Initio Simulation Package (VASP) (7, 8), Cambridge Serial Total Energy Package (CASTEP) (9), WIEN2k (10), and ABINIT (11), with different setups used to cross-check each other. The results reported here are primarily those obtained from VASP with Perdew-Wang (PW91) generalized gradient approximation (GGA) exchange-correlation density functional (12), ultrasoft (US) pseudopotential (8), and Methfessel-Paxton smearing method (13) with 0.3-eV smearing width, and the cell being oriented as in Fig. 1A. The cutoff energies for the plane wave basis set for Al and Cu are 162 and 292 eV, respectively. Table 1 shows the agreement of our results with experimental and other calculations. To compute the equilibrium lattice constant*a*
_{0}, as well as relaxed and unrelaxed {111}〈112̄〉 shear moduli (*G*
_{r}′ and *G*
_{u}′, respectively) (4), we use a six-atom supercell of three {111} layers. After relaxation, all stress components other than σ_{13} (=σ_{31}) are reduced to <0.1 GPa. For the intrinsic and unstable stacking energies γ_{sf} and γ_{us}, respectively, we use a 24-atom supercell of 12 layers for Al (10 layers for Cu) with layers 1 and 12 facing vacuum and shearing between layers 6 and 7. Relaxation of all layers along the <111> direction is terminated when the force on each atom is <0.01 eV/Å. Al has a much larger γ_{sf} value than that of Cu, yet their γ_{us} values are quite close.

For affine deformation calculations, we consider pure shear (σ_{ij} = 0, except σ_{13}) and simple shear (*x* ≠ 0 with no relaxations). The corresponding stress-displacement curves are shown in Fig. 1, B and D, respectively. The stress values are obtained from analytical expressions; they have been checked against numerical energy derivatives at several values of strain. After analyzing the effects of smearing width, energy cutoff, and Brillouin zone integration*k⃗*-point convergence, we estimate that the maximum stress values in Table 2 have an uncertainty of <0.1 GPa within each method used. ABINIT uses the Perdew-Burke-Ernzerhof GGA functional (14) and norm-conserving Troullier-Martins (TM) pseudopotential (15); CASTEP uses PW91-GGA/US; our own plane wave code uses PW91-GGA/TM; and WIEN2k is a full-potential augmented plane wave plus local orbitals method, where the core and valence states are treated by the Dirac equation and the scalar-relativistic approximation, respectively (10). Furthermore, the full-potential projector augmented-wave (PAW) method (16) and the local density approximation (LDA) options of VASP also have been included in the cross-check. The difference between US and PAW is <5% in the maximum stress values, and LDA consistently gives a stress that is 10 to 20% higher than that of GGA. With all methods used, Al is found to have ideal simple shear and pure shear strengths that are higher than those of Cu (Table 2) (17).

At equilibrium, Cu is considerably stiffer than Al; its bulk, simple, and pure shear (along {111}〈112̄〉) moduli are greater than those of Al by 80, 65, and 25%, respectively. However, Al has an ideal pure shear strength that is 32% larger than that of Cu because it has a longer range of elastic strain before softening (Fig. 1B):*x*
_{max}/*b*
_{p} = 0.28 or*γ*
_{max} = 0.20 in Al, which are the displacement and the engineering shear strain at the maximum shear stress, respectively, versus*x*
_{max}/*b*
_{p} = 0.19 or*γ*
_{max} = 0.13 in Cu. The ion relaxations in these two metals are different (Fig. 1C). In Al, when the top atom slides over the bottom atoms, the top atom hops in the*z* direction, and the bottom atoms contract in the*y* direction (relaxation in *x* is almost zero). In Cu, there is almost no relaxation in the *z* direction; the top atom translates essentially horizontally, and the bottom atoms expand and contract in the *y* and *x* directions, respectively.

The difference in relaxation patterns has important implications for the shear strength–hardening behavior (Table 3), which also has been noted and discussed in terms of third-order elastic constant (18). When pressurized in the 〈110〉 direction, Cu hardens, whereas Al softens substantially; however, if pressurized in the 〈111〉 direction, Al hardens substantially, whereas Cu softens slightly. These results show that the pressure-hardening effect is highly dependent on orientation. A rough estimate of the stress state at the displacement burst observed in nanoindentation experiments (3) shows that the pressure components are at the level indicated in Table 3. Thus, a large effect on the shear strength is to be expected. However, because the actual stress state is a complicated triaxial condition and given that the pressure-hardening behavior is very anisotropic, one cannot ascertain its real effect without an accurate stress analysis. For this purpose, we apply a method combining atomistic and finite-element calculations (19).

Because Al has no core d states, its partially occupied valence d bands are abnormally low in energy, which gives rise to directional bonding. At the six-atom interstice in Al, the pocket of charge density has cubic symmetry and is very angular in shape, with a volume comparable to the pocket centered on every ion (Fig. 2A). In Cu, there is no such interstice pocket, and the charge density is nearly spherical about each ion (Fig. 2B). Thus, Al has an inhomogeneous charge distribution in the interstitial region because of bond covalency (20) and directional bonding (21), whereas Cu has relatively homogeneous distribution and little bond directionality. To probe these bonding characteristics further, we look at how the valence charge density ρ(**r**,*x*)*V*
_{cell}(*x*) varies along a path in cell, a-b-c, during pure shear, as atom b moves away from its initial nearest neighbor atom a (at *x* =*x*
_{1}) and takes on a new nearest neighbor atom c (at *x* = *x*
_{2}). The Δ[ρ(**r**,*x*)*V*
_{cell}(*x*)] patterns, with *V*
_{cell} being the cell volume, for Al and Cu again show substantial contrast. In Al (Fig. 2C), the maximum change occurs halfway between the two nearest ions, which indicates that when atoms change neighbors, the breaking and reformation of directional bonding is an important activity. There is little such activity in Cu (Fig. 2D). Δ[ρ(**r**,*x*)*V*
_{cell}(*x*)] mainly reflects an accommodation process, like soft spheres squeezing past each other by distorting their own shape. A similar attempt to connect stacking fault energy with redistribution and topological properties of charge density was made recently (22).

The charge-density behavior just discussed, along with the relaxation patterns seen in Fig. 1C, suggest a hinged-rod model to describe the shear strength for Al, in contrast to the conventional “muffin-tin” or sphere-in-glue model for Cu. It is reasonable to think that when the bonding is directional (rodlike), a longer range of deformation can be sustained before breaking than when the bonding is spherically symmetric, because of different geometrical factors of charge-density decay with bond length. In covalent systems like Si (23) and SiC, we verified that during shear, the bonds generally do not break until the engineering shear strain reaches 25 to 35%, which is substantially larger than those of metallic systems. Conversely, when the bonds do break, a directionally bonded system can be expected to be more frustrated and less accommodating, as manifested in a larger intrinsic stacking fault energy, for example.

To quantify our interpretation, we return to the behavior of the multiplane generalized stacking fault energies in the form of stress-displacement functions*d*γ_{1}(*x*)/*dx* and*d*γ_{∞}(*x*)/*dx* (Fig. 1D). First, we note that for Cu,*d*γ_{1}(*x*)/*dx* and*d*γ_{∞}(*x*)/*dx* are not very different across the entire range of shear, so the local glue picture is appropriate. The fact that the sliding of a layer is effectively decoupled from that of adjacent layers indicates that bonding in Cu has nearly no bond-angle dependence. On the other hand, the same functions behave much more differently in Al, especially when *x*> *x*
_{max}, at which the gradient reaches a maximum. Even in the range of *x* <*x*
_{max}, the relative magnitudes of*d*γ_{1}(*x*)/*dx* and*d*γ_{∞}(*x*)/*dx* are opposite in order in Al as compared to those in Cu, suggesting a possibly different nature of bonding. Second, the value of*x*
_{max} is almost identical between*d*γ_{1}(*x*)/*dx* and*d*γ_{∞}(*x*)/*dx* in both Al and Cu, with Al having the larger *x*
_{max} value, implying that the longer range directional bonding in Al could be a more general feature than being specific to the affine strain energy γ_{∞}(*x*). Third, we see that when *x*≫ *x*
_{max} and the directional bonds in Al are broken (confirmed by a depleted charge at the interstice in Fig. 2C),*d*γ_{1}(*x*)/*dx* in Al stays positive for an extended range, whereas*d*γ_{1}(*x*)/*dx* in Cu becomes negative quickly. Thus, although Al and Cu have approximately the same unstable stacking energy (Table 1), we see that when the displacement *x* reaches *b*
_{p} and the configuration becomes an intrinsic stacking fault, Cu has recovered most of its losses in the sense of a low value of γ_{sf}, whereas Al has recovered very little as its γ_{sf} value remains close to the γ_{us} value. The implication is that when a directional bond is broken, it is more difficult for the electrons to readapt. In contrast, for sphere-in-glue–type systems, even if the bond angles are wrong, as long as the volumes fit as in the intrinsic stacking fault, the electrons can redistribute well, and the system does not incur a large energy penalty.

In this work, we exploit the connection between the generalized stacking fault energy and the stress-strain response to show that the abnormally high ideal shear strength and intrinsic stacking fault energy of Al have the same electronic-structure origin (namely, directional bonding). On one hand, directional bonds give rise to a relatively longer shear deformation range, which accounts for the larger ideal shear strength of Al in relation to that of Cu; on the other hand, once the existing bonds are broken and new bonds are formed with unfavorable bond angles, the electrons cannot readjust easily, resulting in an anomalous intrinsic stacking fault energy for Al. Our findings are supported by the detailed behavior of the valence charge density obtained from first-principles calculations that have been systematically cross-checked. We studied the pressure dependence of the shear strength and have shown its very anisotropic character. These results suggest that conventional crystal plasticity notions such as a scalar or pressure-independent yield criterion based on critical resolved shear stress, although successful for macroscopic face-centered cubic metals, should be viewed cautiously when interpreting nanoindentation experiments (3). Contemporary empirical potentials (24) may be useful for providing a qualitative description of the nonlinear, anisotropic stress distribution under the nanoindenter and for ascertaining the likely site and character of the instability; the quantitative importance of these results remains to be scrutinized by more accurate ab initio calculations.

↵* To whom correspondence should be addressed. E-mail: syip{at}mit.edu