Quantum stabilities and growth modes of thin metal films

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Surface Science 602 (2008) 2532–2540

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Quantum stabilities and growth modes of thin metal ﬁlms: Unsupported and NiAl-supported Ag(1 1 0) and Ag(1 0 0) Yong Han a,*, J.W. Evans b,c, Da-Jiang Liu c a

Institute of Physical Research and Technology, Iowa State University, Ames, IA 50011, United States Department of Mathematics, Iowa State University, Ames, IA 50011, United States c Ames Laboratory-USDOE, Iowa State University, Ames, IA 50011, United States b

a r t i c l e

i n f o

Article history: Received 20 March 2008 Accepted for publication 28 May 2008 Available online 12 June 2008 Keywords: Metal ﬁlms Ag ﬁlms NiAl Quantum size effect Surface energy Epitaxial growth Density functional theory Free-electron model

a b s t r a c t We present density functional theory (DFT) analyses of the stability of Ag thin ﬁlms versus ﬁlm thicknesses for various surface orientations. We include benchmark results for freestanding ﬁlms, but consider in detail Ag(1 1 0) ﬁlms supported on a NiAl(1 1 0) substrate, and Ag(1 0 0) ﬁlms supported on a NiAl(1 0 0) substrate. The supported ﬁlms exhibit an almost perfect lattice-match between ﬁlm and substrate surface unit cells, so one can assess ﬁlm stability in the absence of signiﬁcant lateral mismatch strain. We also provide a characterization of ﬁlm growth modes for these NiAl-supported Ag ﬁlms based on DFT results for the relevant energetics. Ó 2008 Elsevier B.V. All rights reserved.

1. Introduction Thin metal ﬁlms with thicknesses on the order of nanometers, as well as other metal nanostructures, can have quite different physical properties from their bulk counterparts due to quantum size effects (QSE). These effects are associated with electron conﬁnement, which for thin ﬁlms occurs just in the direction orthogonal to the substrate. Such QSE are of considerable theoretical and technological interest, especially with regard to achieving desired size-dependent functionality in nanodevice fabrication [1,2]. During the last 20 years, QSE in Pb(1 1 1) ﬁlms have been studied intensively both theoretically and experimentally, perhaps most frequently for Si(1 1 1) substrates [3–14], but also for Cu(1 1 1) [15–21], Ge(0 0 1) [22], and Ge(1 1 1) [9,21] substrates. All of these systems exhibit bilayer oscillations in various physical quantities often attributed to the feature that each bilayer increment in height of the ﬁlm can accommodate almost exactly 3 half Fermi wavelengths of the conﬁned electrons. Of particular interest for Pb/Si(1 1 1) is the feature that the bilayer oscillations are modiﬁed by a remarkable beating effect (switching stability from ﬁlms of even to odd thickness) which persists even up to more than 30 monolayers (ML) [7,13]. These experimental observations have prompted theoretical analysis by many groups. These often con* Corresponding author. E-mail address: [email protected] (Y. Han). 0039-6028/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.susc.2008.05.040

sider only unsupported or freestanding Pb ﬁlms [23], given the complexity of treating the ﬁlm-substrate interface for Pb/Si(1 1 1) and the effects of strain. These calculations have applied both jellium models and ab initio density functional theory (DFT). Behavior for unsupported ﬁlms actually describes well observations for Pb ﬁlms with stable heights of 2, 4, 6, . . . layers on 6H–SiC(0 0 0 1) due to a weak ﬁlm-substrate interaction [24]. However, in other cases, the effect of the ﬁlm-substrate interface is important, shifting stability to ﬁlms of height 3, 5, 7, . . . layers, so calculations for supported ﬁlms are desired. In addition to simpler one-dimensional pseudopotential calculations [19,20], there exist recent DFT analyses for Pb on Ge(1 1 1) and Cu(1 1 1) substrates where a reasonable lattice-match between suitably chosen larger unit-cell of the Pb(1 1 1) overlayer and the substrate was possible [21]. Despite the focus in theoretical studies on Pb(1 1 1) ﬁlms, there have actually been several observations of QSE for Ag ﬁlms which we now enumerate. A remarkably strong height selection has been observed of 7-layer high Ag(1 1 1)-like ﬁlms on GaAs(1 1 0) [25,26], and similar behavior occurs on numerous other III–V substrates [27–29]. Selection of bilayer height Ag(1 1 1)-like islands and ﬁlms occurs on top of a single wetting layer of Ag on Si(1 1 1) [30], and preference for thicker Ag(1 1 1)-like ﬁlms with even-layer heights (above the wetting layer) has recently been reported [31]. Particularly stable 1-, 2- and 5-layer Ag(1 0 0)-like ﬁlms have been found on Fe(1 0 0) [32–34]. Initial bilayer-by-bilayer growth of Ag(1 1 0)-like ﬁlms has been demonstrated on

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NiAl(1 1 0) [35]. Finally, selection of 0.84 nm 3-layer high mesalike Ag islands has been observed for deposition on 5-fold icosohedral AlPdMn quasicrystalline substrates [36]. Even with the above extensive set of observations for Ag ﬁlms, there have been relatively few theoretical analyses relevant for these systems. There has not been available a comprehensive set of DFT calculations for freestanding Ag ﬁlms with various orientations which address the following basic questions: (i) Do there exist regular bilayer oscillations in some systems as suggested by certain observations above? (ii) Can beating effects be observed? Of course, one would ideally prefer high-level DFT analysis for supported metal ﬁlms where results could be directly compared with experiment. This requires carefully selecting systems where there is a reasonable lateral lattice-match between substrate and overlayer to minimize complicating strain effects and to ensure a simple substrate-ﬁlm interface structure which can be described reliably. It should be noted that one such case for which a DFT analysis has already been performed is Ag on Fe(1 0 0) [33,37], where the fcc Ag(1 0 0) overlayer with surface lattice constant of 0.289 nm matches well with the bcc Fe(1 0 0) substrate with lattice constant of 0.287 nm. In this work, we will ﬁrst provide a series of ‘‘benchmark” DFT calculations for unsupported or freestanding Ag ﬁlms with both Ag(1 1 0) and Ag(1 0 0) structure, and also brieﬂy comment on the Ag(1 1 1) case. However, motivated by recent experiments and with the intention of prompting additional new experiments, we will also provide DFT analysis for supported ﬁlms in suitably selected systems, speciﬁcally Ag ﬁlms on NiAl. Our key motivation and goal is to provide insight into the growth modes which are or should be observed in these systems. Recent deposition studies of Ag on NiAl(1 1 0) [35] were performed noting a remarkable near-perfect lattice-match between the NiAl(1 1 0) surface unit cell of 0.2887 0.4083 nm2 and the Ag(1 1 0) surface unit cell of 0.2889 0.4086 nm2. Scanning Tunneling Microscopy (STM) analysis revealed initial bilayer-by-bilayer Ag(1 1 0) ﬁlm growth mode which would be effectively free of lateral strain. The substrate-ﬁlm interface structure is simple in this system with the lowest-layer Ag adatoms located at the Ni short-bridge sites according to DFT [35] and atomically-resolved STM [38] studies. The current work will extend previous DFT analysis which was limited to thin ﬁlms up to just 3 bilayers [35]. Our new DFT calculations for unsupported as well as NiAl(1 1 0)-supported Ag(1 1 0) ﬁlms reveal a strong bilayer oscillation pattern with prominent beating, providing another example of this type of behavior beyond the commonly studied Pb(1 1 1) ﬁlms. In addition, we will analyze supported Ag(1 0 0) ﬁlms on NiAl(1 0 0) substrates with either Al- or Ni-termination, noting again an nearperfect lattice-match between the surface unit cells. On the NiAl(1 0 0) substrate, the lowest layer Ag adatoms are naturally located at four-fold hollow sites, so again the substrate-ﬁlm interface structure is simple. Behavior will be quite different from Ag(1 0 0) on Fe(1 0 0) due to substrate effects. There is a large lateral pﬃﬃﬃmismatch between NiAl(1 1 1) and Ag(1 1 1) (by a factor of 2), so we do not consider this case. In Section 2, we provide a brief but instructive overview for the stability oscillation behavior of thin metal ﬁlms (taking fcc metals as the example) using a free-electron model [39]. In Section 3, we describe some details of the DFT calculation procedures used in this work, and the determination of ﬁlm stability from total energies. In Section 4, we present our DFT results focusing on the stability of both unsupported and NiAl-supported Ag(1 1 0) and Ag(1 0 0) ﬁlms, and discuss in some detail the ramiﬁcations for ﬁlm growth modes. Section 5 presents a brief discussion of other aspects of QSE for Ag(1 1 0) ﬁlms and provides a summary. The Appendix includes additional results related to QSE in thin NiAl slabs used to represent the substrate in our analyses of supported ﬁlms.

2. Brief overview of stability oscillations: Schulte analysis In 1976, Schulte ﬁrst demonstrated QSE in thin metal ﬁlms from self-consistent calculations for jellium slabs [39]. A physical quantity will have oscillations with period near kF/2, where kF is the Fermi wavelength, as a function of the ﬁlm thickness if it is regarded as a continuous variable. When the physical interlayer spacing d is commensurate with this oscillation, so that it satisﬁes the condition

jd ¼ i

kF ; 2

i ¼ 1; 2; 3; . . . ;

ð1Þ

then the ﬁlm will exhibit a j-layer oscillation in stability when j is an integer and j 6¼ i (where j and i have no common factor). No oscillation will occur when j ¼ ið¼ 1Þ. For a speciﬁc metal ﬁlm, generally speaking, j is never exactly equal to but sometimes close to an integer, resulting in a beating-like pattern. If we simply take kF from the free-electron gas model, then Eq. (1) becomes

p 1=3 jd ¼ i 3ne

ð2Þ

where ne is the free-electron density. For metal ﬁlms, n1=3 has the e same order of magnitude as the interlayer spacing d, so that stability oscillation in the metal ﬁlm can be observable if j is close to a small integer. For other materials, e.g., semiconductor or insulator ﬁlms, the free-electron density is very small or close to zero due 1=3 to the formation of chemical bonds, and then ne and thus j will be very large, so that such stability oscillations are unobservable. For fcc metal ﬁlms, Eq. (2) is equivalent to

j ¼ ibZ 1=3 ;

ð3Þ

where Z is the valence electron number of a metal atom. Here, b is purely numerical and is determined by the relationship between the interlayer spacing for the selected surface orientation and the bulk unit cell lattice constant. Speciﬁcally, one has

pﬃﬃﬃ b ¼ ð2 2p=3Þ1=3 1:809; ð2p=3Þ1=3 pﬃﬃﬃ 1:279; and ð 3p=4Þ1=3 1:108;

ð4Þ

for (1 1 0), (1 0 0), and (1 1 1) ﬁlms, respectively. Analogous expressions can be obtained for other crystal structures. Eq. (3) shows that the j values depend only on the valence electron number and surface orientation. Thus, it is possible to construct a useful table characterizing stability oscillations for all fcc metals. Table 1 lists all the j values when i = 1–5 for Z = 1–4, fcc metal (1 1 1), (1 1 0), and (1 0 0) ﬁlms with a few key values in bold

Table 1 j values from Eqs. (3) and (4) for fcc metal ﬁlms Z

1

2

3

4

Typical metals

Cu, Ag, Au

Ca, Sr

Al

Pb

(1 1 1) ﬁlm

i=1 i=2 i=3 i=4 i=5

1.108 2.216 3.324 4.432 5.540

0.879 1.759 2.638 3.518 4.397

0.768 1.537 2.305 3.073 3.841

0.698 1.396 2.094 2.792 3.490

(1 1 0) ﬁlm

i=1 i=2 i=3 i=4 i=5

1.809 3.619 5.428 7.238 9.047

1.436 2.872 4.308 5.744 7.181

1.255 2.509 3.764 5.018 6.273

1.140 2.280 3.420 4.559 5.699

(1 0 0) ﬁlm

i=1 i=2 i=3 i=4 i=5

1.279 2.559 3.838 5.118 6.397

1.015 2.031 3.046 4.062 5.077

0.887 1.774 2.661 3.548 4.436

0.806 1.612 2.418 3.224 4.030

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font. Table 1 can be readily extended to larger i values if useful. For Pb(1 1 1) ﬁlms where Z = 4, j = 2.094 for i = 3 is very close to 2, explaining the robust bilayer oscillation. Note that for Pb(1 1 1), the free-electron Fermi sphere lies outside the ﬁrst Brillouin zone and there is some ambiguity in determining the relevant Fermi wavelength if one takes the lattice structure into account. If we fold the Fermi wave vector to the ﬁrst Brillouin zone, then measure from the L point along the (1 1 1) direction [23], one has kF/ (2d) = 2.31. Alternatively, measured from the C point [6], one has kF/(2d) = 1.76. Nonetheless, all choices are consistent with bilayer oscillation. While small j values close to integers indicate strong oscillation period, larger j values close to integers represent an approximate beating-like periodicity due to the ‘‘phase-mismatch” between kF/2 and d, as now discussed in more detail. It can be shown from a consideration of phase-mismatch accumulation that the ‘‘beating period” can be expressed as

jbeat ¼

kF ; 2jj ½j jd

ð5Þ

where j* is the smallest j value close to an integer, and [ j*] represents the integer nearest to j*. In other words, Eq. (5) indicates that the [ j*]-layer oscillation is modulated by a jbeat -layer oscillation. However, it should be noted that the multiples of jbeat are generally not exactly equal to any j values, but around each multiple of jbeat , there are always two j values nearest to two integers that have a difference of [ j*]. For example, j* = 2.094 2 and kF/d = 1.396 for Pb(1 1 1) ﬁlms, so from Eq. (5), jbeat ¼ 7:423. There are two j values, 6.980 (for i = 10) and 9.074 (for i = 13), nearest to jbeat, and there are two j values, 13.960 (for i = 20) and 16.054 (for i = 23), nearest to 2jbeat, and so on. Thus, the ‘‘physical” beating period could be 7–9 Pb(1 1 1) layers. These predictions are basically consistent with the results from experiments and DFT calculations [6,8,10,21]. As an aside, we note that j* = 3.073 3 and kF/d = 1.537 for Al(1 1 1) (where Z = 3) suggesting the possibility of growth with trilayer oscillations with jbeat ¼ 10:516 11. For Ag(1 1 0) ﬁlms where Z = 1, one has j = 1.809 2 for i = 1 suggesting the possibility of bilayer oscillations. From Eq. (5) using kF/d = 3.619, we get jbeat ¼ 9:493, indicating the ‘‘physical” beating period to be 10 Ag(1 1 0) layers. Our DFT calculations for Ag(1 1 0) ﬁlms in this work will reveal such an oscillation pattern. For Ag(1 0 0) ﬁlms, there is no indication of bilayer or trilayer oscillations but, e.g., j = 3.838 4 for i = 3, and j = 5.118 5 for i = 4, suggests the possibility of less ordered longer range variation, as will be seen in our DFT calculations. Although the free-electron model is very successful in predicting the oscillation period and even the beating-like behavior for the above mentioned cases, it might not be very useful to predict the magnitudes and the phase of the oscillations of the stabilities of ﬁlm thickness, which are the focus of this paper. Thus, the above analysis provides only a rough guide in the search for QSE in speciﬁc systems. Relevant DFT calculations are needed for a comprehensive analysis. 3. DFT analysis and ﬁlm stability criterion DFT calculations are performed using the plane-wave-based Vienna ab initio simulation package (VASP) [40]. We use the PBE form of the generalized gradient approximation (GGA) [41] due to its better overall agreement with experiments compared with the local density approximation (LDA). The electron-ion interactions are described by the projector augmented-wave (PAW) approach [42]. The converged magnitude of the forces on all relaxed atoms is always less than 0.1 eV/nm. To prevent spurious interactions between adjacent replicas of the thin ﬁlm system, we use a vacuum layer that is 1.5 nm thick in the direction perpen-

dicular to the surface. The optimized lattice constants are aAg = 0.4166 nm for Ag, and aNiAl = 0.2896 nm for NiAl, to be compared with the experimental values of 0.4086 nm and 0.2887 nm, respectively. These theoretical lattice constants were used in all subsequent calculations. In all calculations of surface free energies, the surface size of periodic slab (supercell) is 1 1. For Ag(1 1 0), Ag(1 0 0), NiAl(1 0 0), and Ag/NiAl(100), the k mesh is chosen as 20 20 1; for NiAl(1 1 0) and Ag/NiAl(1 1 0), the k mesh is chosen as 15 21 1. Ag ﬁlms are adsorbed on one side of the substrate, except for the work function calculation where they are adsorbed on both sides. For the calculations of freestanding ﬁlms, we relax all atoms in the supercell. For a supported Ag(1 1 0) [Ag(1 0 0)] ﬁlms, we always use 11 layers of NiAl(1 1 0) [NiAl(1 0 0)] as the substrate. The bottom 6 layers of the substrate are ﬁxed, while other layers of the substrate and all Ag adlayers are entirely relaxed. The reason for this selection and treatment of the substrate will be explained in detail in the next section. In calculations of other quantities, e.g., adsorption energies for isolated adatoms, the supercell size, k-points, and substrate choices will be speciﬁed below for speciﬁc cases. Generally, the surface free energy can be used to judge the stability of a ﬁlm, lower energies corresponding to more stable ﬁlms. For a freestanding metal ﬁlm, the surface free energy as a function of thickness, L (in layers), is calculated as

cL ¼

EL N L Ec ; 2A

ð6Þ

where EL is the total energy for the supercell, NL is the total number of atoms in the supercell, Ec is the cohesive energy per atom for the bulk metal, and A is the area of the bottom or top surface of the supercell. Thus, by calculating EL and Ec, the surface free energy cL can be obtained. For a supported metal ﬁlm, the surface free energy cannot be simply calculated from Eq.(6) because of the existence of a substrate, and instead we consider

aL ¼ ct þ cb þ ci ct;0 cb;0 ¼

EL E0 N L Ec ; A

ð7Þ

corresponding to the relative surface free energy of the ﬁlm. Now, EL is the total energy of the supercell including the substrate, and NL is the total atom number in the added L layer metal ﬁlm. The script ‘‘0” corresponds to no metal layers on the substrate. Here, ct, cb, and ci are the free energies of top surface, bottom surface, and interface, respectively, and generally speaking, all three energies are functions of metal ﬁlm thickness. In terms of the ﬁrst-order discrete difference function, DQL = QL+1 QL, where Q is the studied physical quantity, it is useful to deﬁne chemical potential type quantities as lL = 2DcL1 for unsupported ﬁlms and lL = DaL1 for supported ﬁlms. Then, the second-order discrete difference of EL/A, can be expressed as

DlL ¼

ELþ1 þ EL1 2EL ; A

ð8Þ

which reduces to a second-order difference of the 2cL (aL) for unsupported (supported) ﬁlms. If DlL P 0 (the chemical potential increases or does not change going from thickness L to L + 1), the ﬁlm of thickness L is identiﬁed as ‘‘stable”, and if DlL < 0, this ﬁlm is identiﬁed as ‘‘unstable”. Therefore, the second-order discrete difference function, DlL, can be described as a ‘‘stability index”. Another perspective on this stability criterion comes from the observation that DlL P 0 is needed to avoid an energetic preference for bifurcation of an L layer thick ﬁlm into ﬁlms of thickness L ± 1. An advantage of using the second difference function, DlL, to judge the ﬁlm stability is that this quantity is independent of the choice of Ec. From Eqs. (6) and (7), cL and aL sensitively depend on the Ec value requiring stringent selection of the value of Ec beyond the accuracy of DFT calculations. Thus, when calculating cL and aL, we always

Y. Han et al. / Surface Science 602 (2008) 2532–2540

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take Ec as an adjustable parameter with an uncertainty of roughly ±2 meV about the DFT value. Finally, before we present our DFT results, we comment brieﬂy on previous relevant theoretical calculations regarding Ag surface free energies. For the most stable Ag(1 1 1) surface, the embedded atom method (EAM) in Ref. [43] predicts a value of 620 mJ/m2. DFT calculations with LDA generally predict a much higher value around 1230 mJ/m2, which is very close to the experimental value. On the other hand, using the popular PBE form of GGA, the DFT prediction is 780 mJ/m2. However, it is worth noting that a recently proposed functional [44] signiﬁcantly improve this apparent drawback of GGA. See Refs. [45–47] for more discussion. Our DFT calculations, as will be discussed below, show that, relative to Ag(1 1 1) surface, the surface free energies of Ag(1 0 0) and Ag(1 1 0) are 10% and 16% higher, respectively. This is in general agreement with previous calculations. The focus of this paper is on the relative stability of Ag ﬁlms of different thickness, rather than the absolute surface free energy and its orientation dependence, and we believe that the method employed here is reliable for this purpose.

Ag(1 1 0) ﬁlm. Results exhibit a regular bilayer oscillation with a salient beating effect. According to the ﬁlm stability criterion discussed in Section 3, from Fig. 1b, the stable ﬁlm thicknesses are L = 2, then 5, 7, 9, 11, 13, and then 16, 18, 20, 22, 24, 26, 28, and 30, with other thicknesses being unstable. The prominent bilayer oscillations correspond to j* = 1.809 2 for i = 1 in Table 1. The beating nodes at L = 4, 15, and 23 are consistent with the suggested beating period around 10 layers predicted by the free-electron model in Section 2. We also note that the beating node at L = 23 does not correspond to an odd–even switch point. The beating pattern is somewhat sensitive to the choice of k points, and therefore requires a large enough number of k points. So, we also perform a test with a larger number of k points (28 28 1) for unrelaxed bulk-terminated Ag(1 1 0) ﬁlms, and the results (not shown) are basically consistent with Fig. 1, including the position and the lack of odd–even switch feature of the beating node at L = 23. Similar behavior has been observed for freestanding Pb(1 1 1) ﬁlms through DFT calculations and the nonspherical shape of Fermi surface is invoked as the mechanism for the complex beating pattern [13].

4. DFT results and ﬁlm growth modes

4.2. Ag(1 1 0) ﬁlms supported on NiAl(1 1 0)

4.1. Freestanding Ag(1 1 0) ﬁlms

Fig. 2a–d show schematics of the structure of the NiAl(1 1 0) and Ag(1 1 0) surfaces, and of a supported single Ag(1 1 0) layer and a supported bilayer Ag(1 1 0) island. As noted in Section 1, there is an almost perfect match between the lateral unit cell for NiAl(1 1 0) and Ag(1 1 0). Fig. 3 shows DFT values for the relative surface free energy, aL, and stability index, DlL, versus the thickness L of a Ag(1 1 0) ﬁlm supported on an 11-layer NiAl(1 1 0) slab. Results exhibit a strong bilayer oscillation pattern again with salient beating. Note that relative to freestanding Ag(1 1 0) ﬁlm shown in Fig. 1, there is an apparent shift in the stability pattern toward larger L direction, i.e., the stability of any thickness L in Fig. 3 can be obtained from that for L + D in Fig. 1. A shift of D = 3 seems to work better for smaller L’s, while a shift of D = 1 seems to work better

Fig. 1 shows DFT values for the surface free energy, cL, and stability index, DlL, versus the thickness L of the freestanding

a

900 Freestanding

890

γL [mJ/m2]

880 870 860 850 780 770 760 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 Ag(110) film thickness L [ML]

ΔμL [mJ/m2]

b

Freestanding

420 90 60 30 0 -30 -60 -90 -120 -150 -180 -210 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 Ag(110) film thickness L [ML]

Fig. 1. (a) Surface free energy, cL, and (b) second-order discrete energy difference, DlL, versus ﬁlm thickness, L, for freestanding Ag(1 1 0) ﬁlms. Note the persistent bilayer oscillations with beating (maximum in DlL and minima in cL reﬂecting more stable ﬁlms).

Fig. 2. Schematics showing: (a) NiAl(1 1 0) surface and unit cell, (b) Ag(1 1 0) surface and unit cell, (c) single-layer Ag(1 1 0) ﬁlm on NiAl(1 1 0) with Ag at preferred Nishort bridge adsorption sites and (d) Ag(1 1 0) island on NiAl(1 1 0) which has primarily bilayer structure, also illustrating the process with upward transport of a ﬁrst-layer atom at a peripheral kink site.

Y. Han et al. / Surface Science 602 (2008) 2532–2540

a 10 0 -10 -20 -30 -40 -50 -60 -70 -80 -90 -100

αL [mJ/m2]

Supported on NiAl(110)

0

ΔμL [mJ/m2]

b

2

180 150 120 90 60 30 0 -30 -60 -90 -120 -150 -180

4 6 8 10 12 14 16 18 20 Ag(110) film thickness L [ML]

Supported on NiAl(110)

0

2

4 6 8 10 12 14 16 18 20 Ag(110) film thickness L [ML]

Fig. 3. (a) Surface free energy, cL, and (b) second-order discrete energy difference, DlL, versus ﬁlm thickness, L, for Ag(1 1 0) ﬁlms supported on NiAl(1 1 0) substrate. Note the persistent bilayer oscillations with beating (maximum in DlL and minima in cL reﬂecting more stable ﬁlms).

for larger L’s. The stable thicknesses for supported ﬁlms become L = 2, 4, 6, 8, 10, and then 11, 13, 15, 17, 19, and other thicknesses are unstable with the odd–even switch points at L = 1 (trivial), and 12 (within the ﬁrst 21 layers). The NiAl(1 1 0) substrate plays a role modifying behavior relative to that for the freestanding ﬁlm, leading to the odd–even exchange in stability. It is appropriate to consider whether the above results for supported Ag(1 1 0) ﬁlms are sensitive to the thickness of the NiAl(1 1 0) substrate used in the calculations. In fact, we have used a thick 11-layer substrate as analysis in the Appendix indicates signiﬁcant QSE for thinner NiAl(1 1 0) slabs. However, even if QSE are present for the substrate slab, these effects should not necessarily impact in our results for the relative stability of Ag ﬁlms with different thicknesses calculated using the same substrate. Nonetheless, we have also performed DFT calculations for 4- and 5-layer NiAl(1 0 0) slabs as substrates with the bottom-most NiAl(1 1 0) layer ﬁxed. We recover oscillation pattern of Fig. 3 with only small changes in the absolute values of aL and DlL. 4.3. Discussion of Ag/NiAl(1 1 0) ﬁlm growth modes

ever, that experimental growth behavior is complicated for thicker ﬁlms: the step heights of islands formed in higher levels progressively decrease from values compatible with bilayer Ag(1 1 0) islands to lower values comparable to single layer Ag(1 1 1)-like islands. This transition in growth mode likely has its origins in observed deviations from perfect Ag(1 1 0) bilayer structure even for lowest level islands. These deviations presumably grow in higher layers facilitating transformation to a Ag(1 1 1)-like structure driven by the lower surface free energy for Ag(1 1 1) (see Section 5). The origin of the deviations is unclear. Perhaps, they are intrinsic to the system, or they could be due to non-ideality in the substrate (e.g., due to surface segregation of one component, sub-surface atoms at anti-sites, etc.). In any case, reﬁned growth procedures which stabilize the Ag(1 1 0) structure to higher layers on NiAl(1 1 0) or on other substrates would be needed to probe experimentally the more subtle beating phenomena exhibited in Figs. 1 and 3. It is instructive to also provide some remarks on the QSE-mediated kinetics of the initial bilayer-by-bilayer growth of Ag(1 1 0) on NiAl(1 1 0). The thermodynamic driving force is reﬂected in the bilayer oscillations in surface energy described above. From the perspective of growth kinetics, it perhaps more instructive to regard this driving force in terms of preference for Ag adatoms to populate the top of a single Ag layer on the substrate rather than the substrate, and also the tops of higher odd-layer rather than even-layer ﬁlms. This preference is quantiﬁed in terms of the absorption energy of an isolated Ag on top of Ag(1 1 0) layers of various thickness, L, deﬁned as Ea = (Etot Eslab). Here, Etot is the total energy of the supported ﬁlm plus the adatom, and Eslab is the total energy of supported ﬁlm without the adatom. In calculating Ea, we use a 2 3 supercell with 4 4 1 k mesh, and 4-layer NiAl(1 1 0) slab with the bottom-most layer ﬁxed. Results shown in Fig. 4 reveal a 0.05 eV preference to absorb on top of layer 1 relative to layer 0, and 0.1 eV preference to adsorb on top of layer 2n + 1 relative to layer 2n for n = 1, 2, 3, and 4. Note also that both the upper and lower envelopes of the Ea versus L curve decrease with L at least up to L = 8. This is consistent with smooth bilayer-by-bilayer growth observed in experiment [35], i.e., bilayer islands in each level grow and coalesce to form a nearly complete ﬁlm of even thickness before signiﬁcant population of the next level. Observation of QSE in thin ﬁlm growth requires facile mass transport kinetics which can respond to weak QSE-mediated thermodynamic driving forces even at low temperatures (T). This feature is manifest for supported Pb(1 1 1) ﬁlms resulting in

2.78 2.76

Supported on NiAl(110)

2.74 2.72

Ea [eV]

2536

2.70 2.68 2.66 2.64 2.62 2.60

Recent STM experiments of Ag deposition on NiAl(1 1 0) revealed initial bilayer-by-bilayer growth of Ag(1 1 0)-like ﬁlms [35]: ﬁlm thicknesses of L = 2, 4, and 6 are stable, while L = 1, 3, and 5 are unstable. The process involves nucleation and growth of bilayer islands in each level, where a schematic of such a bilayer island on the substrate is shown in Fig. 2d. These observations are consistent with our DFT results in Fig. 3. It should be noted, how-

2.58 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Ag(110) film thickness L [ML] Fig. 4. Adsorption energy, Ea, for an isolated Ag adatom on top of an L-layer Ag(1 1 0) ﬁlm on NiAl(1 1 0). Persistent bilayer oscillations occur with stronger binding on top of ﬁlms of odd thickness (consistent with even thickness ﬁlms being more stable).

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4.4. Freestanding Ag(1 0 0) ﬁlms As discussed in Section 2, one does not expect bilayer or other strong short-period stability oscillations for Ag(1 0 0) ﬁlms, in contrast to Pb(1 1 1) and Ag(1 1 0) ﬁlms. This is conﬁrmed by the DFT calculations for the surface free energy, cL, and stability index, DlL, shown in Fig. 5, which do not exhibit the regular or large amplitude oscillation. Apart from strong variations in surface free energy for ﬁlms of thickness L = 1 and L = 2, we note that ﬁlms with L = 5 appear to be particularly stable. Finally, it should be noted that variations in cL and DlL persist (i.e., QSE persists) even for very thick ﬁlms with L 30. Here it should be mentioned that the DlL curve shown in Fig. 5 is basically consistent with the previous ultrasoft-pseudopotential DFT result [33], where only L = 1–12 are calculated. As mentioned in Section 4.1, the stability pattern is sometimes sensitive to the choice of k points, and therefore as a test, we also use a denser k mesh of 28 28 1 to unrelaxed bulk-terminated Ag(1 0 0) ﬁlms. Results (not shown) are consistent with Fig. 5 for L < 6. For L > 5, the oscillation feature with the k mesh of 28 28 1 is a little different from that with the k mesh of 20 20 1 (used in Fig. 5). Also, the results with the denser k mesh show a seemingly more robust periodicity with a period 5 layers, and this is consistent with the free-electron model anal-

60 30 0 -30 -60 -90 -120 -150 -180 -210 -240 -270 -300 -330

840 830 820 γL [mJ/m2]

dramatic mesa-like island morphologies around 200 K. Why is the bilayer growth mode for Ag/NiAl(1 1 0) facile at 200 K [35], and likely also at signiﬁcantly lower T [48]? Terrace diffusion on the substrate is isotropic with a barrier of Ed 0.27 eV [35]. Terrace diffusion in higher layers should be similar to that for Ag/ Ag(1 1 0) which is anisotropic with a barrier of Ed 0.28 eV for easier in-channel diffusion [49], so these processes are operative well below 200 K. However, bilayer island formation also requires facile upward mass transport. For Ag/NiAl(1 1 0), bilayer islands observed in each level have elongated rectangular shapes indicative of active edge diffusion and a strong anisotropy in adatom nearest-neighbor (NN) interactions. Weaker (stronger) interactions occur in the direction orthogonal to (parallel to) the long sides of the islands. Indeed, detailed DFT analysis together with atomistic simulation indicates values of Ebs 0.09 eV and Ebw 0.04 eV for ﬁrst-layer of Ag/NiAl(1 1 0) [48], and preliminary calculations indicate that these interactions in higher layers are close to the values of Ebs 0.16 eV and Ebw 0.02 eV for Ag/Ag(1 1 0) [49]. The weaker NN interactions of Ebw 0.02–0.04 eV facilitate upward transport at these long sides, as previously noted for metal(1 1 0) homoepitaxy [50]. The stronger NN interactions, Ebs, listed above imply that island formation will be effectively irreversible (i.e., Ag dimers do not dissociate on the time scale of aggregation) [49] below T 140 K for the ﬁrst layer and below T 165 K in higher layers for the experimental deposition ﬂux of 6 103 ML/s. Nonetheless, bilayer island formation is expected even at these lower T in part since Ag adatoms can still escape from kink sites in lower layers (see Fig. 2d) with a low barrier of Ed + Ebs + Ebw 0.40–0.46 eV on the time-scale of island growth (which is longer than the time-scale for aggregation). This further enables upward mass transport. This contrasts behavior on isotropic surfaces where atoms are bound at kink sites by multiple strong bonds. At very low T, say around 130 K, only partial bilayer islands of a few 10’s of atoms would likely form in the initial stages of population of any level, and this would lead to rough morphologies for prolonged deposition. However, if deposition is terminated just after formation of such islands, then even at 130 K, they would convert to complete bilayer islands by upward hopping of atoms at kink sites at the island edge (with a barrier of 0.40–0.46 eV and rate of 102 s1) within 10 min.

ΔμL [mJ/m2]

Y. Han et al. / Surface Science 602 (2008) 2532–2540

810 800 790 780 770 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

Freestanding 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 Ag(100) film thickness L [ML]

Fig. 5. Second-order discrete energy difference, DlL, versus ﬁlm thickness, L, for freestanding Ag(1 0 0) ﬁlms. Inset: surface energy, cL, versus L. No oscillations occur with a well-deﬁned period, but ﬁlms of L = 5, and 11 are particularly stable (maximum in DlL, and minima in cL).

ysis in Section 2. In principle, the DFT results with a denser k mesh are more reliable, but the computation efﬁciency becomes worse. From the above test, the results with the k mesh of 20 20 1 for L < 6 have converged. In the following calculations, our discussion is mainly focused on the very thin ﬁlms, so we choose the k mesh to be 20 20 1. 4.5. Ag(1 0 0) ﬁlms supported on Al-terminated NiAl(1 0 0) As noted in Section 1, there is an almost perfect match between the lateral unit cell for Al-terminated NiAl(1 0 0) and Ag(1 0 0). Fig. 6a shows the top view of an Al-terminated NiAl(1 0 0) surface, and Fig. 6b shows a complete Ag monolayer on that surface. The black solid square with a side length of aNiAl = 0.2887 nm marks the surface unit cell for NiAl used in our calculations. The white dashed square (rotated by 45° with respect to the NiAl unit cell) marks a face of the non-primitive fcc unit cell for Ag with a side length of aAg = 0.4086 nm almost perfectly matching the diagonal pﬃﬃﬃ ð 2 aNiAl = 0.4083 nm) of the NiAl unit cell. Thus, lateral surface strain due to mismatch should be negligible. Fig. 7 shows calculated relative surface free energy, aL, and stability index, DlL, as the functions of thickness L of Ag(1 0 0) ﬁlm supported by an 11-layer NiAl(1 0 0) slab terminated by Al on both sides. By comparison with Fig. 4 for freestanding Ag(1 0 0), substrate has a strong effect: ﬁlms of thickness L = 1 and 3 become very stable, and L = 2 becomes very unstable.

Fig. 6. Schematics showing: (a) Al-terminated NiAl(1 0 0) surface and unit cell, and (b) single layer of Ag on Al-NiAl(1 0 0), also indicating one face of the non-primitive fcc Ag unit cell.

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Y. Han et al. / Surface Science 602 (2008) 2532–2540

-660

1000 900 800 700 600 500 400 300 200 100 0 -100 -200 -300

-680 -700

ΔμL [mJ/m2]

αL [mJ/m2]

-720 -740 -760 -780 -800 -820 -840 -860 2

4

6

8

10 12 14 16 18 20

Supported on Al-terminated NiAl(100)

0

2

4 6 8 10 12 14 16 18 20 Ag(100) film thickness L [ML]

Fig. 7. Second-order discrete energy difference, DlL, versus ﬁlm thickness, L, for supported Ag(1 0 0) on Al-terminated NiAl(1 1 0) ﬁlms. Inset: surface free energy cL versus L. Films of L = 1 and 3 are particularly stable (maximum in DlL, and minima in cL).

4.6. Ag(1 0 0) ﬁlms supported on Ni-terminated NiAl(1 0 0) Again, there is of course also a near perfect lattice-match for Niterminated NiAl(1 0 0) and Ag(1 0 0), where the Ag adatoms sit at four-fold hollow sites. Fig. 8 shows calculated relative surface free energy, aL, and stability index, DlL, as the functions of thickness L of Ag(1 0 0) ﬁlm supported by an 11-layer NiAl(1 0 0) slab terminated by Ni on both sides. Note that the basic features of the stability curve for Ag(1 0 0) on Ni-terminated NiAl(1 0 0) for L = 1–8 in Fig. 8 appear to be obtained from a 2-layer shift of the stability curve for Ag(1 0 0) on Al-terminated NiAl(1 0 0) for L = 3–10 in Fig. 7. 4.7. Discussion of Ag/NiAl(1 0 0) ﬁlm growth modes The above analyses of QSE for lateral strain-free Ag(1 0 0) ﬁlms on NiAl(1 0 0) are intended to motivate or provide guidance for experimental studies of Ag deposition on NiAl(1 0 0). Here, we comment further on the expected ﬁlm growth morphologies. Fig. 7 shows that for Ag deposition on Al-terminated NiAl(1 0 0), there is a strong thermodynamic driving force for completion of the ﬁrst Ag(1 0 0) layer before populating higher layers. However, once completed, there is a strong preference to form bilayer Ag(1 0 0) islands on top of this ﬁrst layer. These features are also reﬂected in the dependence shown in Fig. 9 of the adsorption ener-

gies for isolated Ag adatoms on top of Ag(1 0 0) ﬁlms of various thicknesses L. These energies are calculated using a 3 3 supercell with 4 4 1 k mesh, and 7-layer NiAl(1 0 0) slab with the bottom two layers ﬁxed. They reveal the expected strong adsorption on the substrate and on top of the 2nd layer relative to adsorption on top of the 1st, 3rd, ... layers. Note that the variation in the adsorption energies (0.6–0.8 eV stronger binding to the substrate than to higher layers, and 0.2 eV stronger binding on top of the second layer relative to the ﬁrst) is far larger than seen for Ag(1 1 0) ﬁlms on NiAl(1 1 0). We now discuss the kinetics of this ﬁlm growth process. Nucleation and growth of single-layer two-dimensional islands during formation of the ﬁrst layer will be controlled primarily by isotropic diffusion of Ag on Al-terminated NiAl(1 0 0) and by the NN Ag–Ag attraction. These quantities are calculated from DFT using a 4 4 supercell with 4 4 1 k mesh, and 7-layer NiAl(1 0 0) slab with the bottom two layers ﬁxed. We obtain a diffusion barrier of Ed 0.45 eV and NN interaction of Eb 0.26 eV. It follows that effectively irreversible island formation [49] should occur at T 300 K and below for a typical experimental deposition ﬂux of 102 ML/s. Diffusion along close-packed step edges in this system should be facile as for fcc(1 0 0) homoepitaxy since atoms in the transition state are strongly interact with two edge atoms lowering the transition state energy. Thus, islands should exhibit compact near-square shapes prior to coalescence. During continued deposition beyond a complete ﬁrst Ag layer on Al-terminated NiAl(1 0 0), presumably single-layer islands ﬁrst nucleate in the 2nd layer. Values of diffusion barriers and interaction energies for Ag adatoms in higher layers are expected to shift from the above values towards similar values of Ed 0.42–0.45 eV and Eb 0.22 eV for Ag/Ag(1 0 0) [49]. Thus, one expects irreversible island formation at T 300 K and below. These islands would have, e.g., a total of 1000 atoms for deposition of 0.4 ML at typical ﬂuxes [49]. To form the thermodynamically-favored bilayer Ag(1 0 0) islands, it is necessary for Ag to climb up on top of these 2nd layer islands. One does not expect this process to be inhibited by a large step edge barrier, based on the low value of this barrier for Ag(1 0 0) homoepitaxy [49,51]. However, it is necessary for Ag adatoms at the edge of islands in the 2nd layer to overcome strong bonds to climb up, particularly those at kink sites for which the detachment barrier is Ed + 2Eb 0.78 eV. Consequently, this upward mass transport process would be far less facile than for Ag/ NiAl(1 1 0). Even for deposition at 300 K, partial bilayer islands would result, aided by direct deposition on top of 2nd layer islands. If deposition is terminated just after formation of such islands, then at 300 K, they would convert to complete bilayer islands

3.1

Supported on Ni-terminated NiAl(100)

1200

-820

2.9

-840

600

-860

Ea [eV]

800

αL [mJ/m2]

ΔμL [mJ/m2]

Supported on Al-terminated NiAl(100)

3.0

-800

1000

-880 -900 -920 -940

400

2.7 2.6

-960 -980

200

2.8

2.5 2

4

6

8

10 12 14 16 18 20

2.4

0 0

2

4 6 8 10 12 14 16 18 20 Ag(100) film thickness L [ML]

Fig. 8. Second-order discrete energy difference, DlL, versus ﬁlm thickness, L, for supported Ag(1 0 0) on Ni-terminated NiAl(1 1 0) ﬁlms. Inset: surface free energy cL versus L. Other than L = 1, there are no particularly stable ﬁlm thicknesses.

0

1

2 3 4 5 6 7 8 Ag(100) film thickness L [ML]

9

10

Fig. 9. Adsorption energy, Ea, for isolated Ag on top of a supported L-layer Ag(1 0 0) ﬁlm on Al-terminated NiAl(1 0 0). Stronger binding on the substrate and on ﬁlms of L = 2 are consistent with the results in Fig. 7.

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5. Discussion and summary We have provided a fairly comprehensive analysis of quantum stabilities due to QSE in unsupported and NiAl-supported Ag thin ﬁlms with both (1 1 0) and (1 0 0) surface orientations. In closing we make a few additional comments. First, for completeness, we brieﬂy describe an additional DFT analysis that we performed for the surface free energy, cL, versus thickness L of freestanding Ag(1 1 1) ﬁlms. For the k mesh of 20 20 1, we ﬁnd that the surface free energy cL=19 = 582, 716, 726, 733, 728, 732, 727, 732, 729 mJ/m2, respectively) initially increases, and then exhibits weak bilayer oscillations, which disappear around L = 10–11, and then reemerge beyond L = 11. However, we ﬁnd a change in the odd– even stability upon increasing the number of k mesh to 28 28 1, so these results are not fully converged. Thus, in order to obtain a more reliable stability analysis for Ag(1 1 1) ﬁlms, it is necessary to further increase the number of k points. Second, while we have noted the utility of the Schulte analysis in providing a simple prediction of stability oscillations, it is appropriate to mention analyses of stability oscillations for other systems which go beyond jellium or free-electron models. It should be noted that fcc metal (1 1 0) and (1 0 0) ﬁlms exhibit AB-stacking, so freestanding ﬁlms of odd versus even thickness have different symmetries, a feature which could impact bilayer oscillations [53]. Also, it has been shown that bilayer oscillations can reﬂect details of the crystal band structure, e.g., arising from beating of quantum well states associated with different points in the twodimensional Brillouin zone, where individual periodicity is consistent with the bulk Fermi wave vector [54]. Third, it is also appropriate to note that many studies of QSE in thin ﬁlms consider a variety of quantities exhibiting strong variations with thickness such as interlayer relaxations and the work function, and furthermore assess the behavior of individual quantum well states versus thickness. In this spirit, we have also performed a comparative analysis of the dependence of the work function on thickness for both freestanding and NiAl(1 1 0)-supported Ag(1 1 0) ﬁlms. Results shown in Fig. 10 reveal some tendency for initial bilayer oscillation with a shift in the locations of maxima for freestanding versus supported ﬁlms. We have also analyzed the energies of quantum well state versus ﬁlm thickness (not shown), ﬁnding a clear correlation between states crossing the Fermi level roughly every 2 layers for unsupported ﬁlms and bilayer oscillations described above. Behavior is less clear for the supported ﬁlm due to crowding of the energy levels.

5.0

Freestanding Supported on NiAl(110)

4.8

Work function [eV]

by upward hopping of atoms at kink sites at the island edge (with a barrier of 0.8 eV and rate of 1 s1) within 10 min. Although the barrier for upward hopping and the number of atoms to be transferred are far greater than for the example given for Ag/ NiAl(1 1 0) in Section 4.3, to compensate T 300 K is also far higher. For Ag deposition on Ni-terminated NiAl(1 0 0), again there is a strong driving force to complete the 1st layer before populating higher layers. However, there is no strong preference for speciﬁc ﬁlm thickness of higher layers. Likewise, once Ag(1 0 0) bilayer islands merge on top of the 1st layer of Ag on Al-terminated NiAl(1 0 0) to form a complete 3-layer ﬁlm, there is no strong preference for speciﬁc larger thicknesses. Thus, after the initial growth in both these systems, behavior and growth morphologies should be quite similar to those for Ag/Ag(1 0 0) homoepitaxy. Previous detailed characterization of Ag/Ag(1 0 0) homoepitaxy, especially for deposition below 300 K [51,52], suggests that despite a low step edge barrier, mound formation with complex coarsening dynamics occurs for extended Ag deposition on NiAl(1 0 0).

4.6 4.4 4.2 4.0 3.8 0

1

4 5 2 3 6 Ag(110) film thickness L [ML]

7

Fig. 10. Work function versus ﬁlm thickness L for freestanding and NiAl(1 1 0)supported Ag(1 1 0) ﬁlms. Substrate atoms were ﬁxed, while Ag atoms were fully relaxed.

In summary, we have utilized density functional theory to analyze quantum stabilities due to QSE in Ag thin ﬁlms with various surface orientations. Our analysis above has focused on the energetics relevant to ﬁlm stability and growth modes. Benchmark calculations for freestanding ﬁlms are compared with results for Ag ﬁlms supported in NiAl. Bilayer stability oscillations are particularly dramatic for freestanding and NiAl(1 1 0)-supported Ag(1 1 0) ﬁlms. The substrate has a signiﬁcant impact on behavior shifting the stable thicknesses for freestanding relative to supported Ag(1 1 0) ﬁlms. In addition, we observe a signiﬁcant inﬂuence of the substrate in determining stable thicknesses for Ag(1 0 0) ﬁlms on Al- versus Ni-terminated NiAl(1 0 0). This is reasonable since the different substrate terminations will lead to, e.g., different charge spilling especially for the small L. Our comprehensive analysis of quantum stability for these supported ﬁlms, together with some insights into and DFT analysis of diffusion and interaction energies in these systems, allows us to provide detailed predictions for ﬁlm growth modes. These predictions are either supported by existing studies, or can motivate new experiments. Acknowledgments This work was supported by NSF Grants CHE-0414376 and CHE0809472. DJL was supported by the Division of Chemical Sciences, Basic Energy Sciences, US Department of Energy (USDOE). Computational support at NERSC was provided through the USDOE CMSN Project on Surface-based Nanostructures. We also acknowledge discussions with Cai-Zhuang Wang on electronic structure, and with Baris Unal on the Ag/NiAl(1 1 0) experiments. The work was performed at Ames Laboratory which is operated for the USDOE by Iowa State University under Contract No. DE-AC02-07CH11358. Appendix QSE. for freestanding NiAl(1 1 0) and NiAl(1 0 0) slabs Table 2 shows calculated surface free energy cL and stability index, DlL, versus the thickness L of freestanding NiAl (1 1 0) slabs. In calculating cL for the alloy NiAl(1 1 0) slab, Eq. (6) can be still used, but now NL is understood as the total number of Ni–Al atom pairs in the supercell, and Ec is understood as the cohesive energy per Ni–Al atom pair in the bulk metal. As L increases, cL quickly converges (by L 11) towards the constant value of 1.57 J/m2, which is, as might be expected, comparable to but lower than the value of 1.65 J/m2 obtained from a previous DFT analysis of an unrelaxed substrate [55]. In these studies, we ﬁnd noticeable relaxations of interlayer spacing among the top 5 layers NiAl(1 1 0) with the rip-

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Y. Han et al. / Surface Science 602 (2008) 2532–2540

Table 2 Surface free energy and stability index versus L for NiAl slabs L

NiAl(1 1 0) 2

1 2 3 4 5 6 7 8 9 10 11

NiAl(1 0 0) 2

cL (J/m )

DlL (mJ/m )

c2L (J/m2)

1.612 1.625 1.573 1.566 1.573 1.569 1.576 1.576 1.572 1.570 1.571

129.361 89.104 27.538 21.678 22.405 12.575 10.564 6.652 4.063 0.386

2.044 2.263 2.304 2.291 2.295 2.297 2.295 2.296 2.294 2.298 2.296

For NiAl(1 1 0), also cL=12–21 = 1.572, 1.572, 1.572, 1.571, 1.569, 1.570, 1,572, 1.571, 1.571, and 1.571 J/m2, respectively.

ple structure of topmost Al and Ni atoms [35]. From the 6th ML toward the center of the slab, interlayer spacing relaxations have become negligibly small [35]. Thus, in the calculations in Section 4 for supported Ag ﬁlms on NiAl(1 1 0), we take an 11-layer NiAl(1 1 0) slab as the substrate, and only relax top 5 layers of NiAl(1 1 0) as well as all Ag atoms, but ﬁx the bottom 6 layers of NiAl(1 1 0). Table 2 also reports results for surface energies for NiAl(1 0 0) slabs. Such slabs are composed of alternating Ni and Al monolayers, each having a simple square-lattice unit cell. Thus, the two faces of a NiAl(1 0 0) slab can be either Ni-terminated or Al-terminated, so we distinguish three possibilities: both Al-terminated, both Ni-terminated, and one of each. It is not possible to obtain surface free energy of an odd-L Ni–Ni or Al–Al terminated NiAl(1 0 0) ﬁlm by the total-energy method from a simple expression like Eq. (6). However, for the even-L Ni–Al terminated ﬁlm, Eq. (6) can still be used, but NL is understood as the total number of Ni–Al atom pairs in the supercell, and Ec is understood as the cohesive energy per Ni–Al atom pair in the bulk metal. Because of different terminations for two faces, the surface free energy calculated from Eq. (6) should be an average quantity, which is deL for an L-layer Ni–Al terminated ﬁlm. Table 2 also noted as c L as the function of L, indicating convergence at L = 10 or shows c 12. In these studies, we ﬁnd noticeable relaxations of interlayer spacing among the top 5 layers NiAl(1 0 0), but those from the 6th ML toward the center of the slab become negligibly small. Thus, in the calculations in Section 4 for supported Ag ﬁlms on NiAl(1 0 0), we always take an 11-layer NiAl(1 0 0) slab as the substrate, and only relax top 5 layers of NiAl(1 0 0) as well as all Ag atoms, but ﬁx bottom 6 layers of NiAl(1 0 0). References [1] [2] [3] [4]

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Magnetic Properties of (FePt)100-xCux Thin Films