Dimensionality and Valency Dependent Quantum Growth of Metallic

We confirmed that, for Au and Pb wires, the second derivative of the edge energy Δ2E as a function of wire length changes little between the 1 × 1 Ã...
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Dimensionality and Valency Dependent Quantum Growth of Metallic Nanostructures: A Unified Perspective Chenhui Li,† Seho Yi,‡ Congxin Xia,§ Ping Cui,∥ Chunyao Niu,† Jun-Hyung Cho,*,‡,∥,† Yu Jia,*,†,⊥ and Zhenyu Zhang∥ †

International Laboratory for Quantum Functional Materials of Henan, and School of Physics and Engineering, Zhengzhou University, Zhengzhou 450001, China ‡ Department of Physics and Research Institute for Natural Sciences, Hanyang University, 17 Haengdang-Dong, Seongdong-Ku, Seoul 133-791, Korea § College of Physics and Materials Science, Henan Normal University, Xinxiang 453000, China ∥ International Center for Quantum Design of Functional Materials (ICQD), Hefei National Laboratory for Physical Sciences at the Microscale, and Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, P. R. China ⊥ Key Laboratory for Special Functional Materials of Ministry of Education, and School of Physics and Electronics, Henan University, Kaifeng 475004, China S Supporting Information *

ABSTRACT: Quantum growth refers to the phenomena in which the quantum mechanically confined motion of electrons in metallic wires, islands, and films determines their overall structural stability as well as their physical and chemical properties. Yet to date, there has been a lack of a unified understanding of quantum growth with respect to the dimensionality of the nanostructures as well as the valency of the constituent atoms. Based on a first-principles approach, we investigate the stability of nanowires, nanoislands, and ultrathin films of prototypical metal elements. We reveal that the Friedel oscillations generated at the edges (or surfaces) of the nanostructures cause corresponding oscillatory behaviors in their stability, leading to the existence of highly preferred lengths (or thicknesses). Such magic lengths of the nanowires are further found to depend on both the number of valence electrons and the radial size, with the oscillation period monotonously increasing for alkali and group IB metals, and monotonously decreasing for transition and group IIIA-VA metals. When the radial size of the nanowires increases to reach ∼10 Å, the systems equivalently become nanosize islands, and the oscillation period saturates to that of the corresponding ultrathin films. These findings offer a generic perspective of quantum growth of different classes of metallic nanostructures. KEYWORDS: Friedel oscillations, valence electrons, radial size, quantum growth

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theoretically and experimentally studied during the last several decades. For example, Na, Al, B, and C nanoclusters are anomalously stable with their magic numbers of 20, 13, 12, and 60 atoms, respectively;8,25−27 However, for one-dimensional (1D) and 2D nanostructures, the intriguing characteristics are the “oscillatory properties” of transport conductivity,10 thermal stability,7 superconducting transition temperature,5,28 work

uantum growth, or the formation of low-dimensional nanostructures due to quantum size effect (QSE), is closely associated with the quantization of the kinetic energy of confined electrons, giving rise to an influence of the structural morphology and stability.1−6 Experimentally, the dimensionality of these nanostructures grown on various substrates can range from zero-dimensional (0D) to two-dimensional (2D), such as cluster,7,8 nanowire,9−13 nanotube,14 nanoribbon,15 nanomeasa,16−18 nanopillar,19 nanoring,20 nanofilms,21−24 and so on. For 0D nanostructures, the distinct characteristic is the “magic size” in structural morphology, which has been © XXXX American Chemical Society

Received: August 10, 2016 Revised: September 29, 2016

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ultrathin films, the reason for which has, however, remained a long-standing puzzle. Since nanowires, nanomesas, nanoislands, nanopillars, and ultrathin films can be utilized as building blocks of electrical nanodevices including extremely dense logic and memory circuits,44−47 it becomes urgent to explore the underlying mechanism of their formations. So far, despite a number of theoretical studies for the stability of single-atom wires or ultrathin films, there has not yet been a systematic firstprinciples study that investigates the stability of the abovementioned various types of nanostructures with respect to the dimensionality, size, and valency of constituent atoms. Such a unified theory for the growth of metallic wires, islands, and films is indispensable for not only understanding the underlying mechanism of quantum growth but also designing future nanostructure devices. Here, we present comprehensive firstprinciples density functional theory (DFT) calculations for the stability of metallic nanostructures of Au, Ag, Cu, Na, Zn, Al, Ir, Ti, Pb, and Bi elements, as a function of the wire length or film thickness as well as the radial size. By calculating the edge or surface energy of such metallic nanostructures with varying their length or thickness, we find that the magic lengths or thicknesses dramatically change depending on the radial size as well as the valency. More importantly, we reveal that the stability of nanowires exhibits a rapid convergence of the separation between the consecutive magic lengths even at small diameters of less than ∼10 Å, thereby approaching the difference of consecutive magic thicknesses in the corresponding ultrathin films. Our findings provide a generic view of the important QSEs in the stabilities of nanowires, nanosize islands, and films, as demonstrated by the observed magic lengths of Au and Ir nanowires or magic thicknesses of Pb nanosize islands and films. Dimension Dependent Quantum Growth. We systematically investigate the quantum growth of several representative metallic nanostructures of Au, Ag, Cu, Na, Zn, Al, Ir, Ti, Pb, and Bi elements with different valencies. Here we consider metallic nanowires by varying both their length and radial size. In Figure 1a−f, we display the nanowires containing single, double, triple, septuple, and quattuordecuple atoms in the cross section, together with nanofilms. For some nanowires or nanoislands, we also consider different geometrical arrangements of the cross section (see Figure 1c−f). We find that such a variation of the cross section little changes the oscillatory

function,2 and chemical reactivity,29 with respect to their length or thickness. It was reported that Au nanowires exhibit oscillatory charge-density waves (CDWs) with increasing wire length,11 whereas Pb(111) ultrathin films display a periodic oscillation of stability with increasing thickness.21−23 Obviously, the size dependent physical and chemical properties of metallic nanostructures are of great interest because of their promising prospects in both fundamental and applied research. As one type of 1D nanostructures, nanowires have drawn a great deal of attention in the past decades. Nanowires of different radial sizes, i.e., single-atom, double-atom, triple-atom, and column nanowires, have been experimentally fabricated, some of which exhibit size dependent structural and electronic properties. The strong QSE in metallic nanowires has been demonstrated by a number of experiments for various metallic nanowires grown on semiconductor or metal substrates.9−12,30−32 For example, alkali metals, noble metals, Al, and Bi wires were observed to exhibit an oscillatory behavior of electrical conductivity as a function of their radial diameter. It was also observed that Au nanowires formed on the Si(553) surface have not only the oscillation of length distributions with a period of 7.68 Å but also the CDWs with a periodicity of either the three-atom or four-atom length, while Au nanowires on Si(111) exhibit the oscillation of conductivity with a periodicity of the four-atom length.10 Most recently, Ir doubleatom wires on the Ge(110) surface were observed to exhibit a strong preference of the wire lengths with integer multiples of the six-atom length.12 Generally, ultrathin films with a thickness less than ∼30 atomic monolayers (ML) are the typical 2D nanostructures, which can be fabricated by employing various epitaxial technology. For such 2D ultrathin films, QSEs lead to a rather striking growth mode with specific preferred thicknesses determined by the characteristic relationship between the Fermi wave vector and interlayer spacing. The quantum growth of Ag ultrathin films was experimentally realized in 1996, where atomically flat 2 and 5 ML Ag(100) ultrathin films on GaAs(100) or Fe(100) substrate were formed.4 By contrast, Ag(110) ultrathin films on NiAl(110) were observed to display QSEs with strong bilayer oscillations.33 Especially, the most intensively studied 2D nanostructure is Pb(111) ultrathin films, where the Fermi wavelength is nearly commensurate with the interlayer spacing, resulting in the bilayer-by-bilayer growth mode up to more than 40 ML.17,22 Other metal films such as Al(111), Mg(0001), and Bi(111) ultrathin films were also found to dramatically exhibit QSEs34−39 of many other physical properties including work function,2 superconducting transition temperature,5,28 and electron-photon coupling,40 which are strongly dependent on the film thickness. Nanomesa or nanoisland is another type of 2D nanostructures formed by the epitaxial growth processes, which can be regarded as a transition nanostructure between 1D nanowires and 2D ultrathin films. Indeed, it is often difficult that nanomesas or nanorings can be fabricated with the layer-bylayer growth mode on semiconductor or insulator substrates, but their growth is usually formed with magic thicknesses. For instances, Ag, Au, and Pt islands were observed to be fabricated on Ge(111), Si(100), and GaAs(110) substrates with diameters less than ∼50 Å;41−43 the growth of Pb atoms on the Si(111) or Cu(111) substrate was also observed in the formation of mesas with the consecutive thicknesses of double layer.16−18 Interestingly, Pb islands on the Si(111) substrate were observed to exhibit the same growth mode as the corresponding Pb

Figure 1. Structural models of (a) single-atom, (b) double-atom, (c) triple-atom, (d) septuple-atom, and (e) quattuordecuple-atom nanowires and (f) films, together with the views of cross section. In the bottom panel, the triple-atom, septuple-atom, and quattuordecupleatom nanowires with different cross section are also given. B

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corresponding ultrathin films (see Figure 2), which are 4.5a0, 5a0, 3a0, and 2a0 for Au, Zn, Al, and Pb films, respectively. To understand the underlying physics of the abovementioned oscillatory behaviors of wire stability, we examine the quantum well states (QWSs) of various single-atom nanowires of Au, Zn, Al, and Pb, as well as their ultrathin films. It has been well-known that quantum confinement of electrons within metal nanowires is of importance for the determination of their physical properties.32,48 As the wire length increases, such confined electrons produce the subbands that sequentially cross the Fermi level (EF). Consequently, the stability of nanowires as well as their physical properties should oscillate with increasing the wire length: i.e., QWSs crossing EF periodically give rise to the oscillations of wire stability. Figure 3a−d shows the energy levels of the QWSs at the Γ point for

behaviors of stability (see Figure S1 of the Supporting Information). Therefore, we hereafter only consider the circular-shaped arrangement in the cross section. In order to examine the stability of nanowires, we calculate the second derivative of edge energy Δ2E with respect to wire length: see the section of the calculation method. Figure 2a−d

Figure 2. Second derivative of the edge energy of (a) Au, (b) Zn, (c) Al, and (d) Pb nanowires as a function of wire length. In panel (a), a1− a5 correspond to single-atom, double-atom, triple-atom, septuple-atom, and quattuordecuple-atom nanowires, respectively. The same indices in panels (b), (c), and (d) are used. The second derivative of the surface energy of Au, Zn, Al, and Pb films as a function of film thickness is given in a6, b6, c6, and d6, respectively.

displays the results of Δ2E for Au, Zn, Al, and Pb nanowires up to the length of 15 atomic spacing, respectively, together with those of the corresponding ultrathin films. According to the definition of the stability of nanostructures, the positive (negative) extremum value of Δ2E indicates that the system is the most stable (unstable). It is seen that all of Au, Zn, Al, and Pb nanowires (films) exhibit the oscillating behaviors of Δ2E up to the length (thickness) of 15 atomic spacing (layers), indicating a strong manifestation of QSE on the nanowire (film) stability. It is noticeable that the magnitude of oscillation period depends on the nanowire radial size, demonstrating that the magic length or magic thickness can be varied with the radial size. For Au nanowires, the oscillation period increases as 2.5a0, 3a0, 3.5a0, 4a0, and 4.5a0 for the single-atom, double-atom, triple-atom, septuple-atom, and quattuordecuple-atom wires, respectively (a0, interatomic spacing of the nanowire). However, for the cases of Zn, Al, and Pb, the oscillatory period decreases from 10a0 (Zn), 7a0 (Al), and 4a0 (Pb) at the single-atom nanowire to 5a0 (Zn), 3a0 (Al), and 2a0 (Pb) at the quattuordecuple-atom nanowire, respectively. It is remarkable that the oscillation periods of quattuordecuple-atom nanowires considered in the present study converge to those of their

Figure 3. Calculated energy levels of the QWSs at the Γ point for single-atom wires (left panel) and ultrathin films (right panel) of (a) Au, (b) Zn, (c) Al, and (d) Pb as a function of wire length and film thickness. The green dotted line represents EF, and the blue lines indicate the evolution of QWSs around EF.

the single-atom wires (films) of Au, Zn, Al, and Pb as a function of the wire length (film thickness), respectively. The blue lines in Figure 3a−d show the evolution of QWSs around EF with the wire length (film thickness). Consequently, the oscillatory period of QWSs crossing EF gives the oscillation of stability as a function of the wire length (film thickness). These results manifest that the stability oscillation of various nanowires or ultrathin films is driven by quantum confinement of electrons along the wire direction or the direction perpendicular to the surface of films. C

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Nano Letters Valence-Electron Number Dependent Quantum Growth. To have more information for the quantum growth of 1D and 2D nanostructures with respect to the valenceelectron number (NV), we perform additional DFT calculations for a variety of metal nanowires and ultrathin films with NV ranging from 1 to 5. Figure 4a,b shows the oscillation periods of

nearly insensitive to the radial size, as discussed below. It is noted that the spin−orbit coupling (SOC) effects in Bi, Sb, and Te wires or films should lift the spin degeneracy of the surface states, but it would not influence the wire or film stability because the degenerate surface bands are equally shifted upward or downward. Indeed, earlier first-principles DFT calculations for the In wires on the Si(111) surface49 demonstrated that the SOC produces a large spin splitting of the In-related surface bands, but such a Rashba spin splitting barely affects the energetics of the systems. Similarly, DFT calculations for the Pb/Si(111) thin films50 also showed that the SOC marginally influences the quantum size effects. For all the considered elements, we find that, as the radial size increases up to ∼10 Å, the oscillation period of stability converges to that of the corresponding ultrathin films, as observed by Pb nanostructures where the nanoislands exhibit the same growth mode as the film.16−18 Radial Size Dependent Fermi Wave Vector in Metallic Nanowires. It was reported that the underlying physical origin of the oscillatory stability in metal ultrathin films is associated with the interference of surface-induced Friedel oscillations.6 For Pb films, the strong Fermi-surface nesting in the Brillouin zone (BZ) along the [111] direction results in the Friedel oscillations that decay as 1/r, where r is the distance from the surface. Such an intriguing aspect of Pb films lead to the presence of QSE up to 40 ML in Pb(111) films. Note that the Friedel oscillations in most of the metallic films decay as ∼1/r2, while those in nanowires are expected to decay as ∼1/r. Therefore, 1D nanowires tend to have a long-range, robust quantum growth compared to 2D films. As shown in Figure 2a−d, the oscillation period of wire stability depends on the radial size, reflecting that the Fermi wave vector of nanowires would change with the radial size. To clarify this fact, we investigate the relation between the Fermi wavelength λF and the so-called magic length of nanowires. Here, the magic length nda0 (nd is the wire length divided by a0) is well fitted to the integer n times the half of Fermi wavelength, leading to a relation of λF = 2nda0/n. Thus, the Fermi wave vectors producing the magic lengths or magic thicknesses are n given by kF = n kBZ , where kBZ = π/a0 is the length of the BZ

Figure 4. Calculated oscillation periods of the stability of (a) alkali metals and group IB transition metals and (b) other metals (Zn, Al, Ir, Pb, Ti, and Bi) as a function of the radial size. The results of metals with different valencies are plotted in different colors.

the stability of Ag, Cu, Na, Ir, Ti, and Bi nanowires and the corresponding films as a function of the radial size, together with Au, Zn, Al, and Pb ones. We find that the oscillation periods of all the metal nanowires vary depending on the radial size. Interestingly, such an oscillation period of stability is dramatically sensitive to the valency of metals. Based on the calculated oscillation periods of various metal nanowires, we can classify the quantum growth modes into three categories with respect to the valency. The first category includes alkali metals and group IB transition metals with NV = 1, where the oscillation period of stability increases monotonously with increasing the radial size, as shown in Figure 4a. We note that, although Ag and Cu equally have one valence electron, the oscillation period of their wire (film) stability is different from each other as 2a0 (2.5a0) and 2.5a0 (5a0), respectively. These different features between the Ag and Cu nanowires or films can be explained in terms of the different Fermi wave vectors in such free-electron-like metals. The second category covers most of metals including transition metals and group IIIA and IVA metals, where the oscillation period of stability decreases monotonously with increasing the radial size, as shown in Figure 4b. For instance, the oscillation period of Zn decreases from 10a0 at a singleatom nanowire to 5a0 at the ultrathin film, while that of Pb decreases from 4a0 at a single-atom nanowire to 2a0 at the ultrathin film. Here, it is noteworthy that, as valency increases, the oscillation period of stability decreases. For the elements with the same valency, the rather delocalized sp electron elements have lager oscillation periods compared to sd electron elements. Specifically, Al (Pb) wires with three (four) sp electrons have larger oscillation periods than Ir (Ti) wires with three (four) sd electrons. The third category includes Bi, Sb, and Te nanowires, where the oscillation period of the wire stability hardly changes with respect to the radial size (see Figure 4b). This constant oscillation period is associated with the fact that the Friedel oscillations for finite-length wires are

d

along the wire direction. Since subtracting the integer times kBZ from kF does not influence the oscillation period of QSE, we take kF′ within the reduced BZ to estimate the magic lengths or magic thicknesses (see Figure 5b). Figure 5a(a1−a6) presents the band structures of the infinitely long single-atom, double-atom, triple-atom, septuple-atom, and quattuordecuple-atom Au nanowires as well as the corresponding ultrathin films, respectively. We find that there are several electronic states crossing EF, which may play a role in the formation of Friedel oscillations (FOs) at the edges or surfaces. For Au wires of finite length or semi-infinite Au surfaces, the dominant Fermi wave vector kF can be obtained by taking the Fourier transform of the FOs of charge density. For an infinitely long single-atom Au nanowire, we find that only one quantum state dominantly crosses EF, producing a welldefined peak of the Fourier transform of the FOs at 0.73 Å−1 [see Figure 5a(a1)]. This Fermi wave vector is close to 3/5kBZ in magnitude, thus leading to ∼2.5a0 oscillation period of stability. This estimated oscillating period of stability is well consistent with the DFT result of Δ2E, as shown in Figure 2a. As the radial size increases, the multiple energy bands of double-atom, triple-atom, septuple-atom, and quattuordecupleD

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Figure 5. (a) Energy band structures and Fourier transforms of the Friedel oscillations for (a1) single-atom, (a2) double-atom, (a3) triple-atom, (a4) septuple-atom, and (a5) quattuordecuple-atom Au wires. The corresponding ones for Pb nanowires are given in (b). For comparison, the results for the Au and Pb films are also given in (a6) and (b6), respectively. The Fermi wave vectors of Au and Pb nanowires are given in (a7) and (b7), respectively, as a function of the radial size. In (b7), the triangles represent the values of kF′ folded back within the first BZ.

nanowires and film are located in the second BZ. By folding back to the first BZ, we can estimate the periods of nanowire stability as 4a0, 3a0, 2.5a0, 2a0, and 2a0 for the single-atom, double-atom, triple-atom, septuple-atom, and quattuordecupleatom Pb nanowires, respectively, therefore converging to a 2 ML-period for the Pb ultrathin film. It is noted that the dominant Fermi wave vector for Bi nanowires is nearly insensitive to the radial size, thereby leading to a constant oscillation period of the wire stability (see Figure 4b). This feature may reflect a strongly covalent bonding nature in the Bi, Sb, and Te nanowires. In Figure 5(a7,b7), we summarize the Fermi wave vectors of Au and Pb nanowires as a function of the radial size, respectively. It is interesting to notice that the values of kF in Au and Pb nanowires show a drastic difference as a function of the radial size. These different behaviors of kF between Au and Pb nanowires can be explained in terms of the free electron model within the nanowires. For 1D electron systems, kF(1D)

atom nanowires cross EF. Consequently, the Fourier transform of the FOs in each wire shows multiple peaks. In Figure 5a(a2− a5), the dominant Fermi wave vectors are located at 0.92, 1.09, 1.16, and 1.19 Å−1 for double-atom, triple-atom, septuple-atom, and quattuordecuple-atom Au nanowires, respectively. Thus, we estimate the periods of wire stability as 3a0, 3.5a0, 4a0, and 4.5a0, respectively, corresponding to the values obtained using the DFT calculations [see Figure 2a]. For the semi-infinite Au ultrathin film, we obtain the Fermi wave vector of 1.2 Å−1, corresponding to the 4 ML- or 5 ML-oscillating period of stability [see Figure 5a(a6)]. Meanwhile, Figure 5b(b1−b6) shows the band structures and the Fourier transforms of the FOs for Pb nanowires and ultrathin films. We find that the dominant Fermi wave vectors for the single-atom, double-atom, triple-atom, septuple-atom, and quattuordecuple-atom Pb nanowires and Pb film are 1.94, 1.84, 1.79, 1.68, 1.61, and 1.58 Å−1, respectively. Unlike the cases in Au nanowires and film, these Fermi wave vectors for Pb E

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function because the films usually grown on stepped surfaces are composed of a series of Pb islands with different heights.2 Since all nanowires were grown on substrates, the substraterelated parameters such as the orientation, reconstruction, and step or terrace are most likely to influence the quantum growth of nanowires. Therefore, it is necessary to properly understand the substrate effect on quantum growth. Experimentally, metallic nanowires have been fabricated on metallic or semiconductor substrates. It is noted that metallic substrates tend to weakly hybridize the electronic states with the attached nanowires. The resulting marginal charge transfer between nanowires and metallic substrates may give rise to a little change of the Fermi wave vector compared to the case of freestanding nanowires. However, for metallic nanowires formed on semiconductor substrates, the rather strong bonding between nanowires and substrates largely affects the density states of free electrons in metallic nanowires with a charge spilling into the substrates, thereby producing a large change of the Fermi wave vector. To illustrate these different features of metallic nanowires on metal and semiconductor substrates, we perform additional DFT calculations for Cu single-atom nanowires on both Cu(111) and Si(111) substrates. For Cu wire on the Cu(111) substrate, we find that the oscillation period of stability is 2.5a0, which is the same as that of the freestanding Cu nanowire. Meanwhile, for Cu nanowire on the Si(111) substrate, the oscillation period of stability is found to be 4a0 (see Figure S2 of the Supporting Information). Conclusion. Using first-principles DFT calculations, we have performed a systematic theoretical study of the quantum growth of metal nanowires, nanoislands, and ultrathin films with various structural parameters. We found that all metallic nanowires exhibit the periodic oscillation of stability with respect to their length, and the resulting preferred lengths depend on both the radial size of wires and the valency of constituent atoms. Especially, alkali and group IB metals showed a monotonous increase in the oscillation period of stability with increasing the radial size, whereas the other nanowires including transition and group IIIA-IVA metals showed the reversed behaviors. We also found that, as the radial size of nanowires approaches ∼10 Å, the oscillation period of stability rapidly converges to that of the corresponding ultrathin films. Our findings not only shed light on a unified mechanism of quantum growth in the stability of nanowires, nanoislands, and ultrathin films, but also provide an explanation for the magic lengths or magic thicknesses recently observed from Au and Ir nanowires and Pb nanosize islands. Methods. Our first-principles DFT calculations were performed by using Vienna ab initio simulation package (VASP) with the projector-augmented wave method.55 For the treatment of exchange-correlation energy, we employed the generalized-gradient approximation (GGA) functional of Perdew, Burke, and Ernzerhof.56 In the Brillouin zone sampling, Monkhorst−Pack scheme was employed57 and Fermi-level smearing approach of Methfessel and Paxton with proper parameters is applied to accelerate electronic relaxation.58 The cutoff energy for plane wave basis is set to be 250 eV. In order to simulate the electronic growth of nanowires and nanoislands, we construct a series of nanowires with different atom numbers: see Figure 1. Ultrathin films, which can be viewed as nanowires with an infinite radius, are also considered. A larger supercell simulating an isolated wire of finite length was employed with the vacuum distances larger than ∼20 Å along the x, y, and z directions. We confirmed that,

is given by πN/(2a0), where N is the number of valence electrons. However, as the radial size increases to infinite, the Fermi wave vector can be approximated to the bulk value, kF(3D) = (3πN/a03)1/3.51 On the basis of this simple picture, kF(3D) is likely larger than kF(1D) for N = 1, while kF(3D) becomes smaller than kF(1D) for N > 1. This aspect of kF with respect to N is consistent with the present DFT results: i.e., monovalent Au nanowires increase kF with increasing the radial size, while quartvalent Pb nanowires decrease kF with increasing the radial size. It is noteworthy that, if the radial size becomes larger than ∼10 Å, various nanowires exhibit the same oscillation period of stability, but their oscillation phases can be different from each other. For instance, the septuple-atom and quattuordecupleatom Pb nanowires have the same oscillation period of 2a0, but their oscillatory patterns are totally out of phase. Since such a phase shift in the stability is determined by the different charge spillage at the edge,52−54 the oscillation pattern of stability is likely to depend on the radial size of nanowires. We anticipate that the different phase shifts of the oscillatory patterns between Pb islands and Pb film will be observed in future experiments. Unified Understanding of Quantum Growth in Metallic Nanostructures. Based on our DFT calculations for various nanowires and ultrathin films, we present a unified picture of quantum growth with their different dimensions and radial sizes. Experimentally, Au, Ag, Cu, Ir, and Bi nanowires were fabricated with the forms of single-atom wire, doubleatom wire, or triple-atom wire, which exhibited the oscillatory distributions with certain wire lengths. For instance, Au doubleatom nanowires formed on Si(553)9 showed an oscillatory length distribution with a period of 7.68 Å, close to our DFT prediction with a periodic distance of 3a0 = 7.08 Å. Meanwhile, Au triple-atom wires formed on Si(111)10 exhibited the oscillation period of conductivity with 4a0, corresponding to our predicted oscillation period of the stability of the freestanding Au triple-atom wire. We also find that the magic length of Ir double-atom wire is l = 5a0 (equivalent to ∼40 Å: see Figure 4), close to the experimentally observed magic length of ∼48 Å.12 Here, a somewhat deviation between our DFT prediction and the experiment may be caused by the charge spillage to the substrate in experimental sample, as discussed below. It is noticeable that, when the radial size is larger than ∼10 Å, nanoislands tend to have the same oscillation period of stability as the corresponding ultrathin films. This finding provides an explanation for the experimental observations that Pb islands and films exhibit an identical oscillation period of their stabilities, which has been a long-standing puzzle.16−18 Interestingly, although the same even−odd oscillation periods are predicted in Pb islands and films, the phase of their oscillations can be changed due to the mismatch between the Fermi wavelength and lattice interlayer spacing (see Figure 2d(d5−d6)). For Pb films, both the free electron model and the DFT calculations have predicted the π/2 phase shift between the oscillations of surface energy and work function, which can be obtained from the locations of their oscillation crossovers.52 It is noted that the phase of the edge energy of nanowires could be somewhat different from that of the surface energy of the corresponding films, possibly resulting in a different phase relation between the edge energy and work function. This may raise an interesting issue of experimental observations for the precise phase relation between the surface energy and work F

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Letter

Nano Letters

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for Au and Pb wires, the second derivative of the edge energy Δ2E as a function of wire length changes little between the 1 × 1 × 1 and 3 × 3 × 1 k-point grids: see Figure S3 of the Supporting Information. We also found that increasing the energy cutoff to 300 eV has no effect on Δ2E (Figure S3 of the Supporting Information). Considering the fact that nanostructures ranging from the 0D to 2D dimensionality are experimentally grown on various substrates, we only relaxed the atomic positions in the direction along the nanowire, fixed along the perpendicular directions. The characterization of nanowire stability is determined by taking into account the edge energy, which is similar to the surface energy of thin films. Edge energy Ee(N) of a finitely long nanowire is defined as Ee = [E(N) − E∞(N)]/2, where E(N) is the total energy of a finitely long wire containing N atoms; E∞(N) is the total energy of an infinitely long wire with N atoms. Similar to the second derivative of surface energy that can determine the stability of ultrathin films,16 we generalize the second derivative of edge energy to describe the stability of a certain length of nanowires.13 Therefore, the second derivative of edge energy Δ2E for a certain length of nanowire containing N atoms is defined as Δ2E(N) = Ee(N + 1) + Ee(N − 1) − 2Ee(N). Here, the positive value of Δ2E means that the nanowire containing N atoms is relatively more stable than its neighbors. Meanwhile, the negative values correspond to unstable nanowires.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.nanolett.6b03351. Supporting figures (PDF)

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Y.J. and C.L. are supported by the National Basic Research Program of China (No.2012CB921300), the National Natural Science Foundation of China (Grant Nos.11274280 and 11504332), and Innovation Scientists and Technicians Troop Construction Projects of Henan Province. J.-H.C. is supported by the National Research Foundation of Korea (NRF) grant funded by the Korea Government (MSIP) (No. 2015R1A2A2A01003248). Z.Z. and P.C. are supported by the National Basic Research Program of China (No.2014CB921103) and the National Natural Science Foundation of China (Grant Nos. 11634011, 61434002, and 11374273).

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DOI: 10.1021/acs.nanolett.6b03351 Nano Lett. XXXX, XXX, XXX−XXX

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DOI: 10.1021/acs.nanolett.6b03351 Nano Lett. XXXX, XXX, XXX−XXX