Modular Approach for MetalâSemiconductor Heterostructures with

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Modular Approach for Metal-semiconductor Heterostructures with Very-large Interface Lattice Misfit: A First-principles Perspective Weiyu Xie, Michael Lucking, Liang Chen, Ishwara Bhat, Gwo-Ching Wang, Toh-Ming Lu, and Shengbai Zhang Cryst. Growth Des., Just Accepted Manuscript • DOI: 10.1021/acs.cgd.6b00118 • Publication Date (Web): 18 Mar 2016 Downloaded from http://pubs.acs.org on March 21, 2016

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Crystal Growth & Design

Modular Approach for Metal-semiconductor Heterostructures with Very-large Interface Lattice Misfit: A First-principles Perspective Weiyu Xie, Michael Lucking, Liang Chen, Ishwara Bhat, Gwo-Ching Wang, Toh-Ming Lu, and Shengbai Zhang* Department of Physics, Applied Physics, and Astronomy, Rensselaer Polytechnic Institute, Troy, New York, 12180, USA.

ABSTRACT: Realizing high-quality heteroepitaxy of a wide variety of films of very large lattice misfit, f ≥10%, with the substrate is a great challenge, but also a potential advancement because the films may be made threading-dislocation-free as all the dislocations will be confined at the interface. In spite of the numerous experimental findings in the literature, first-principles theory for such systems is virtually non-existing due to their intrinsic heterogeneity, namely, away from the interface, the film is strain free, but at the interface, not only strain but also misfit dislocation develop. Here, a modular approach is proposed to study such heterogeneous films by a combined firstprinciples and elasticity theory method to predict, for example, their epitaxial relationship. Four representative metal-semiconductor interfaces, Al(111)/Si(111), Cu(111)/Si(111), Cu(001)/Si(001), and CaF2(111)/Ni(001), are considered. By taking 1 ACS Paragon Plus Environment


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into account the chemical bonding information at the interface by first-principles theory, our results show good agreement with experiments. Moreover, by constructing the electron localization function (ELF) that utilizes the first-principles results, we are able to demonstrate the formation of interfacial covalent bonds between Si and metal atoms.

INTRODUCTION Heteroepitaxial film growth and misfit defect control are an actively area of research.1-3 To date, most crystal films are grown on substrates with small lattice misfit, but one may also grow films on substrates with large or even very large lattice misfit. As a matter of fact, going beyond the small lattice misfit can be extremely useful as it would release the tremendous potential of combining dislike materials. The recent interests in and drive to realizing van der Waals epitaxy is just one such example.4 To design high-efficiency solar cells, for instance, one would only have to consider the band gap difference for optimal solar light absorption, not whether one can grow epitaxial films.5 One can further engineer the interfacial properties between any two materials to obtain type-I, II, or III band alignment. A type-II alignment can promote photoelectron and photohole separation, thereby increasing solar cell efficiency.6 Such an alignment between a normal insulator and a topological insulator can also lead to a modified spin texture at the interface and a spin bottleneck.7 A type-III alignment, on the other hand, can create interesting topological physics at the interface between two ordinary semiconductors.8 Currently, this great opportunity has be severely hindered by the restrictive small latticemisfit requirement for epitaxy. For a film grown on a substrate with small lattice misfit and the same crystal structure, usually it is expected that the growth is pseudomorphic and the misfit is accommodated 2 ACS Paragon Plus Environment

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by strains up to a critical film thickness hc where the interface misfit dislocations are formed to relieve the strain.9 In this case, a one-to-one lattice matching (pseudomorphic growth) is often assumed for the modeling of the film/substrate interfacial structure and for the calculation of the interfacial cohesive energy.10,11 On the other hand, for epitaxy between materials with large misfit, e.g., f = |ds − df |⁄df ≥ 10%, where df and ds are the planar spacing of the film and the substrate, respectively, along their in-plane alignment directions, the pseudomorphic growth picture no longer holds because of the extremely small critical thickness hc. For instance, with a misfit of f = 22% for TiN epitaxy on Si(100),12,13 hc would only be about 1.38 Å according to the expression hc ≈ b⁄9.9f,14 in 1

which the Burgers vector b = 2 [110]aTiN ≈ 3.0 Å . Instead of having a one-to-one matching, the film and its substrate may have “domain matching epitaxy”13 with a m-to-n matching, where m and n are positive integers. This way, the lattice misfit strain f is greatly reduced to the superlattice residual strain ɛ , which is defined as ɛ = |nds − mdf |⁄mdf . As long as the crystal symmetries are the same between the film and the substrate at the interface, their crystal structures could be quite different without affecting the epitaxial quality of the film. Previously, a simple geometric superlattice matching model has been proposed for the large-misfit epitaxy based on two simple geometric factors: (1) minimizing the interface superlattice area A within a prescribed constraint in the superlattice residual strain (ɛ ≤ ɛmax )15,16 and (2) minimizing the misfit area ∆A within a prescribed constraint in superlattice area (A ≤ Amax ).16 The misfit area ∆A is expressed by ∆A = A∆a⁄a + ∆b⁄b + ∆θ cot θ,

(1)

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where a and b are the magnitudes of the in-plane lattice vectors for the superlattice, and θ is the angle between them. Between the film and substrate, ∆a, ∆b, and ∆θ are the misfit of a, b and θ, respectively. The model has been successfully applied to many large mismatch epitaxial systems such as GaN(0002)101 0/Mo(110)001 17 (which is an abbreviation

for

GaN(0002)/Mo(110)

in

the

out-of-plane

orientation

GaN101 0/Mo001 in the in-plane orientation), Al(111)11 0/Si(111)11 0

16

and and

CaF2 (001)110/Ni(001)010 (unpublished results). However, despite the success and simplicity of the superlattice matching model, its pure geometric nature determines that there is no information on the chemistry of the interface, which could sometimes result in the discrepancy between theory and experiment. For example, although Cu has the same crystal structure as Al, the superlattice matching model cannot explain the experimentally observed epitaxial relationship of Cu(111)12 1/Si(111)11 0 18 (see Table 1). Firstprinciples calculations for large misfit epitaxy, on the other hand, usually uses strained supercells, as required by the periodic boundary condition, which may endure large unphysical strain to overshadow the meaningful results. In this paper, we propose a modular approach that combines first-principles calculations and elasticity theory to determine the epitaxial relationship at interfaces. Key development is to divide a lateral superlattice into multi modules, the proportions of which meet the requirement that the overall residual strain in the film can be fully relaxed, leaving the subsequent film growth strain-free. Experiments13 have shown that for a large lattice misfit system like ZnO(0001) on α-Al2O3(0001) with f ≈ 15% and hc less than one monolayer (ML), a full relaxation occurs as early as at six ZnO MLs. Such a thickness will be defined as H, starting from which one-dimensional (1D) 5/6 and 6/7


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domains will form in alternation. Strictly speaking, 1D domains do not exist. What exist at the interface are two-dimensional (2D) modules, whose projection in either x or y direction yields the 1D domains. We note that, while the residual strain at each individual module can still be sizable near the interface, it quickly diminishes as the film thickness approaches H. The use of the supercell approximation, on the other hand, even in the modular approach, will prevent the strain from being fully relaxed. This problem can, however, be circumvented by evaluating the strain energy through the elasticity theory. Here, the modular approach is applied to four different metal-semiconductor heterostructures

with

markedly

different

structural

and

chemical

identities:

Al(111)/Si(111), Cu(111)/Si(111), Cu(001)/Si(001), and CaF2(001)/Ni(001), as metal-onsemiconductor is the foundation for Ohmic contact and Schottky barrier device fabrication for optoelectronic and electronic applications. In all cases, the calculations yield good agreement with experiments.16,18,19 To understand the physics at the metalsemiconductor interface, an electron localization function (ELF) analysis is developed based on first-principles results, which reveals interestingly the covalent nature of interfacial bonding, despite the marked differences in their electronic properties. THEORETICAL BASIS 1. From 1D Lattice to 2D Module. The general expression for large misfit 1D lattice matching is (from Ref.13): m + αdf = n + αds ,

(2)

where m and n are positive integers. To minimize the misfit dislocation density at the interface, usually, m = n – 1 when df ≥ ds and m = n + 1 when df < ds . Defining x = df ⁄|df − ds | , then n = x is the floor and α = (x mod 1) is the remainder for x. The



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foundation of the modular approach is that there are always two lattice domains, m⁄n and m + 1⁄n + 1 , along each matching direction, one with compressive residual strain whereas the other with tensile residual strain, alternating with each other at frequency 1 − α and α , respectively. By placing the two lattice domains together in such a proportion, the overall residual strain vanishes. Figure 1 illustrates how to use the above expressions for real systems. Here, we choose appropriate lattice constants to satisfy 4.5df = 3.5ds so that a simple example can be generated with x = 3.5 , n = x =3 , m = n + 1 = 4, and α = (x mod 1) = 0.5. The two domains along the 1D matching direction are: (1) m⁄n = 4⁄3 with a compressive residual strain [Fig. 1(a)] and (2) m + 1⁄n +1 = 5⁄4 with a tensile residual strain [Fig. 1(b)]. Then, the proportionality between

the

two

lattice

domains

can

be

determined

by

Pm⁄n : Pm + 1⁄n + 1 = 1 − α : α = 50% : 50% [Fig. 1(c)]. The above discussion is for 1D lattice matching. As mentioned earlier, however, the interface is always 2D, as can be seen in Fig. 1(d). As a result, there exist not only diagonal 2D modules [4/3, 4/3], [5/4, 5/4], but also off-diagonal modules [4/3, 5/4], [5/4, 4/3].


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Figure 1. A schematic illustration of the modular approach when x = 3.5 and α = 0.5. 1D lattice matching: (a) m⁄n = 4⁄3 (compressive strain, indicated by blue arrows), and (b) m + 1⁄n + 1 = 5⁄4 (tensile strain, indicated by red arrows). (c) In this simple example, of all lattice matchings, 4/3 and 5/4 take 50% and 50%, respectively. (d) 2D matching exhibiting three 2D modules. Film atoms are solid diamonds and substrate atoms are open squares. Our modules include both diagonal [4/3, 4/3], [5/4, 5/4] and off-diagonal terms [4/3, 5/4], [5/4, 4/3]. Figure 2 illustrates, schematically, how to divide an interface superlattice [Fig. 2(a)] into

three

types

of

modular

superlattices

m⁄n , m⁄n

,

m + 1⁄n + 1 , m + 1⁄n + 1 , and m⁄n , m + 1⁄n + 1 [Figs. 2(b)-(d)] at a proper proportion to fully relax the film. For the ith module, its area Ai and proportion Pi are obtained from its 1D lattice matching counterpart as follows: Module 1 = m⁄n , m⁄n, P1 = 1 − α2 , A1 = n2 d2s sin θ, Module 2 = m⁄n , m + 1⁄n + 1, P2 = 2α1 − α, A2 = nn + 1d2s sin θ,

(3a) (3b)

Module 3 = m + 1⁄n + 1 , m + 1⁄n + 1, P3 = α2 , A3 = n + 12 d2s sin θ, (3c) where θ is the angle between the in-plane axes of each individual module. Note that each module in Fig. 2 is still under a superlattice residual strain (ɛi ), which will be discussed later.



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Figure 2. A schematic model for the modular approach to demonstrate how to divide (a) the interface superlattice into (b)-(d) three modular superlattices. In modules 1 and 3, the residual strains on their two sides are the same: ɛ1 and ɛ3 , while in module 2, a mixed strain is applied. In this schematic model, ɛ1 is compressive (blue arrows) and ɛ3 is tensile (red arrows). Above a thickness H, the epilayer film in the interface is fully relaxed, of which the lattice constant restores its bulk value. Here, we propose that the total interfacial formation energy ( Eform ) may be approximated by a weighted average over those of individual modules (Eform,i ), i.e., Eform = ∑3i=1 Pi Ai Eform,i ⁄∑3i=1 Pi Ai ,

(4)

where Eform,i can be further expressed as the sum of two components: the interface s-l

formation energy in the superlattice model by first-principles calculations Eform,i and the strain energy by the elasticity theory Estr,i , s-l

Eform,i Eform,i + Estr,i ,

(5) 8

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where all the ’s are in units of energy per area. The reason to divide the total formation energy into two parts is the following: While the first-principles approach can capture the interfacial chemistry, in a supercell calculation, the system can be made either strained or not strained, but cannot have a gradual change with film thickness as in an experiment. On the other hand, the elasticity theory can properly handle such a stain energy. A combined approach complementing the first-principles results by the elasticity theory can therefore circumvent the long-lasting problem of how to model systems with large lattice misfit. 2. Formation Energy from DFT Calculations. Our first-principles calculations are carried out by employing the density functional theory (DFT) within the revised generalized-gradient approximation by Perdew, Burke, and Ernzerhof for solids (PBEsol)20 using projector-augmented-wave potentials, as implemented in the VASP codes.21,22 The plane wave cutoff is 400 eV and the structures are optimized until forces on each atom are less than 0.025 eV/Å. For Al and Cu films on Si substrates, the interfaces are modeled by a slab of four [for Al(111)/Si(111) and Cu(111)/Si(111)] or five [for Cu(001)/Si(001)] layers of metal film on a six-layer semiconductor substrate with the bottom-most substrate layer being chemically passivated and fixed in position. For CaF2 film on Ni substrate, the interfaces are modeled by a slab of eight layers of semiconductor substrate on a six-layer metal substrate with the bottom-most substrate layer being fixed in position and the top-most film layer being chemically passivated. Previous study of epitaxial growth of Fe film on GaN substrate suggested that a reasonable convergence of the interface formation energy can be obtained when the film thickness is four layers or thicker.23



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The first term in Eq. (5) can be further written as s-l

Eform,i = −Ebind,i + Efilm-surf,i + C,

(6)

where Ebind,i is the binding energy and C is a constant independent of in-plane alignment and i (see Supporting Information for details). The second term in Eq. (6), the film surface energy in the ith module Efilm-surf,i, on the other hand, can be obtained by a linear regression of the total energy of a strained film as a function of the film thickness.24 3. Residual Strain Energy Evaluation through the Elasticity Theory. As discussed above, above the film thickness H, the residual strain can be fully relaxed as long as conditions in Eq. (3) are met. What left is the residual strains, which are the largest at the interfacial region but gradually relieve as the film thickness approaches H. To evaluate the strain energy, we make the linear-scaling assumption, ε1 z= nds + mdf − nds z⁄H − mdf ⁄mdf ,

(7a)

ε3 z= n + 1ds + m + 1df − n + 1ds z⁄H − m + 1df ⁄m + 1df ,

(7b)

where z is the distance to the interface. The strain energies in Eq. (5) are thus (see Supporting Information for details), H G

Estr,1 = 0 H G

Estr,2 = 0

1 − v

21 + vε21 dz, G

1 − v

ε21 + ε23 + 2vε1 ε3 + 2 ε3 − ε1 cot θ2 dz, H G

Estr,3 = 0

1 − v

21 + vε23 dz.

(8a) (8b) (8c)

where G and v are the shear modulus and Poisson’s ratio for the film material, respectively. As can be seen in Fig. 2, for all the cases studied here, strain is isotropic except for module 2. This is because strains in module 2 involve both ε1 and ε3 , resulting in an additional shear-strain term in Eq. (8b).


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4. Electron Localization Function Analysis. In the geometric superlattice matching model, regardless of the epitaxial relationship, only one kind of lattice site, namely, the coincident site of the coincident site lattice (CSL),25 is in an ideal position for chemical bond formation at the interface. As such, smaller superlattice area A is always preferred. An important difference between the superlattice matching model and first-principles calculations is thus the formation of interfacial chemical bonds at non-coincident sites. A good description for such bonds is the electron localization function (ELF),26 defined as ELF =

1 1 + Dσ ⁄D0σ

2

,

(9)

where Dσ is a function of the charge density calculated by first-principles method, which physically provides a measure of the Pauli repulsion between electrons and D0σ is a normalization parameter to reproduce the homogeneous electron gas limit. A higher ELF indicates more localized electrons, and the upper bound for ELF is 1, corresponding to a perfect electron localization, i.e., no Pauli repulsion. RESULTS AND DISCUSSION 1. Interfacial Structures.

Here, we apply the theory above to four metal-

semiconductor heterostructures. In each case, we consider two in-plane structures, one is parallel aligned and one is rotated, based on experimental observations and the geometric matching predictions. Table 1, columns 1-5, summarizes the geometric parameters for these interfacial structures (A and ∆A calculated based on the module with the largest Pi ). In the first-principles calculation, we have performed interfacial structural optimization by displacing the overlayer in the x-y plane. Table 1. Epitaxial Relationships and Calculated Results.



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The experimental lattice constants, aAl (4.05 Å), aCu (3.61 Å), aSi (5.43 Å), aCaF2 (5.46 Å) and aNi (3.52 Å),27 are used in the calculations for the lattice misfit f (%), the suprelattice area A (Å2), and the mismatch area ∆A (Å2). All energies, including the s-l

formation energy by DFT using superlattice model (Eform ), the strain energy by the elasticity theory (Estr ), the combined formation energy (Eform ), are given relative to the parallel in-plane alignment in units of meV/Å2. The check marks in the last column indicate the experimental epitaxial relationship. Bold font is used for the relatively smaller value. Film/substrate

In-plane align.

f

A

∆A

Es-l form

Estr

Eform

Expt.

0

0

0

√

Al11 0/Si11 0

34.1 114.9 1.3

Al12 1/Si11 0

22.6 255.3 16.5 2.0

1.3

3.3

Cu11 0/Si11 0

50.4 51.1

0

0

Cu12 1/Si11 0

13.2 715.0 14.8 −1.3

−2.0 −3.3

Cu110/Si110

50.4 59.0

0.3

0

0

Cu010/Si110

6.4

14.7

1.9

−47.4 33.7

−13.8 √

CaF2(001)/Ni(001) CaF2 110/Ni110 35.5 55.9

3.6

0

0

2.2

−44.5 −8.5 −53.0 √

Al(111)/Si(111)

Cu(111)/Si(111)

Cu(001)/Si(001)

CaF2 110/Ni010 8.8

12.4

0.3

0

0

√

0

a. Al(111)/Si(111). (Experiment from Ref.16.) Fig. 3(a) shows the modules for parallel epitaxy,

Al11 0/Si11 0

.

Along

the

in-plane

matching

direction,

1 1 dAl = 2 [11 0]aAl = 2.86 Å and dSi = 2 [11 0]aSi = 3.84 Å, so the lattice misfit f = 34.1%.

Equation (2) yields m = 3, n = 2, and α = 0.93. The three corresponding modules are [3/2, 3/2], [3/2, 4/3], and [4/3, 4/3]. The third module, with the smallest residual strain (0.6%), dominates with P3 = 87.4% . Fig. 3(b) shows the modules for 30° epitaxy Al12 1/Si11 0 , namely 30° in-plane rotation around the [111]-axis. With in-plane 1 1 dAl = 2 [12 1]aAl = 4.96 Å and dSi = 2 [11 0]aSi = 3.84 Å , the lattice misfit is now

f = 22.6% . Equation (1) yields m = 3 , n = 4 , and α = 0.43 , instead. The three


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corresponding modules are [3/4, 3/4], [3/4, 4/5], and [4/5, 4/5]. The second module, with residual strain (3.2%, −3.2%), dominates with P2 = 48.9%.

Figure 3. Top view of Al(111)/Si(111) modules at interface: (a) Parallel epitaxy with x//Al11 0//Si11 0 and y//Al101 //Si101 ; (b) 30° epitaxy, namely 30° in-plane rotation around the [111]-axis, with x//Al12 1//Si11 0 and y//Al21 1 //Si101 . For clarity, only two atomic layers at the interface are shown. Blue Si hexagon and red Al hexagon are used to illustrate the relative in-plane alignment. b. Cu(111)/Si(111). (Experiment from Ref.18.) Fig. 4(a) shows the modules for parallel epitaxy,

Cu11 0/Si11 0

.

With

in-plane

1 dCu = 2 [11 0]aCu = 2.55 Å

and

1 dSi = 2 [11 0]aSi = 3.84 Å, the lattice misfit is f = 50.4%. Equation (2) yields m = 2, n = 1,

and α = 0.98. The three corresponding modules are [2/1, 2/1], [2/1, 3/2], and [3/2, 3/2]. The third module, with smallest residual strain (0.3%), dominates with P3 = 96.7%. Fig. 4(b) shows the modules for 30° epitaxy, Cu12 1/Si11 0 . With in-plane 1 1 dCu = 2 [12 1]aCu = 4.42 Å and dSi = 2 [11 0]aSi = 3.84 Å , the lattice misfit is now

f = 13.2% . Equation (2) yields m = 6 , n = 7 , and α = 0.60 , instead. The three corresponding modules are [6/7, 6/7], [6/7, 7/8], and [7/8, 7/8]. The second module, with 13 ACS Paragon Plus Environment


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residual strain (1.3%, −0.8%), dominates with P2 = 48.0%. Al(111) and Cu(111) are isostructural, but with different lattice parameters. In both a and b cases above, due to the six-fold rotational symmetry of the fcc crystal along [111], the angle between in-plane axes is θ = 60°.

Figure 4. Top view of Cu(111)/Si(111) modules at interface: (a) Parallel epitaxy with x//Cu11 0//Si11 0 and y//Cu101 //Si101 ; (b) 30° epitaxy with x//Cu12 1//Si11 0 and y//Cu21 1 //Si101 . For clarity, only two atomic layers at the interface are shown. Blue Si hexagon and orange Cu hexagon are used to illustrate the relative in-plane alignment. c. Cu(001)/Si(001). (Experiment from Ref.18,19.) Fig. 5(a) shows the modules for 1

parallel epitaxy, Cu110/Si[110] . With in-plane dCu = [110]aCu = 2.55 Å and 2

1

dSi = 2 [110]aSi = 3.84 Å, the lattice misfit is f = 50.4%. Equation (2) yields m = 2, n = 1, and α = 0.98. The three corresponding modules are [2/1, 2/1], [2/1, 3/2], and [3/2, 3/2]. The third module, with the smallest residual strain (0.3%), dominates with P3 = 96.7%. Fig. 5(b) shows the modules for 45° epitaxy, Cu010/Si[110] . With in-plane


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1

dCu = [010]aCu = 3.61 Å and dSi = [110]aSi = 3.84 Å, the lattice misfit is only f = 6.4%. 2

Hence, the epitaxial relationship is expected to be one-to-one.

Figure 5. Top view of Cu(001)/Si(001) modules at interface: (a) Parallel epitaxy with x//Cu110//Si110 and y//Cu1 10//Si1 10; (b) 45° epitaxy with x//Cu010//Si110 and y//Cu1 00//Si1 10. For clarity, only two atomic layers at the interface are shown. Blue Si square and orange Cu square are used to illustrate the relative in-plane alignment. d. CaF2(001)/Ni(001). (Experiment from the authors group’s unpublished results.) Fig. 6(a)

shows

CaF2 110/Ni110

the .

With

modules in-plane

for

parallel 1

dCaF2 = 2 [110]aCaF2 = 3.86 Å

epitaxy, and

1

dNi = 2 [110]aNi = 2.49 Å, the lattice misfit is f = 35.5%. Equation (2) yields m = 1, n = 2, and α = 0.82. The three corresponding modules are [1/2, 1/2], [1/2, 2/3], and [2/3, 2/3]. The third module, with the smallest residual strain (−3.2%), dominates with P3 = 66.8%. Fig. 6(b) shows the modules for 45° epitaxy, CaF2 110/Ni010 . With in-plane 1

dCaF2 = 2 [110]aCaF2 = 3.86 Å and dNi = [010]aNi = 3.52 Å , the lattice misfit is again relatively small, f = 8.8%, and a one-to-one epitaxial relationship can be assumed. In both



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cases c and d, due to the four-fold rotational symmetry of the fcc crystal along the [001] direction, the angle between in-plane axes is θ = 90°.

Figure 6. Top view of CaF2(001)/Ni(001) modules at interface: (a) Parallel epitaxy with x//CaF2 110//Ni110

and

y//CaF2 1 10//Ni1 10 ;

(b)

45°

epitaxy

with

x//CaF2 110//Ni010 and y//CaF2 1 10//Ni1 00. For clarity, only two atomic layers at the interface are shown. Grey Ni square and green CaF2 square are used to illustrate the relative in-plane alignment. 2. Interfacial Formation Energies and ELF. Table 1, columns 6-8, summarizes the calculated energies by first-principles methods, as well as those by the elasticity theory, s-l

in particular, the superlattice formation energy (Eform ), the strain energy (Estr ), and the combined formation energy (Eform ) that determines the ultimate stability between the two competing epitaxial relationships. It appears that all the results agree with experiments.16,18,19 s-l

a. Al(111)/Si(111). Parallel epitaxy is favored both because of a lower Eform and a lower Estr . In this case, both A and ∆A are smaller ones, so the result agrees with the superlattice matching model. However, the agreement is only a coincidence, because interfacial


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bonds are formed, not only on the coincident lattice sites but on every lattice site across the interface. To see this, Fig. 7 shows an isosurface for ELF = 0.75 for the [3/4, 3/4] module in 30° epitaxy. The ELF value at such a high percentage is an indication of strong chemical bond formation, as can be seen for the interior Si-Si bonds. The nearly as large ELF value between Al and Si at the interface signals the formation of strong and directional Si-Al bonds. In fact, every interfacial Si atom forms at least one bond across the interface, but some with two Al atoms of about equal distance.

Figure 7. Top view of [3/4, 3/4] module at Al(111)12 1/Si(111)11 0 interface. For clarity, only one Al (red) layer and two Si (blue) layers are shown. The yellow lobes correspond to the isosurface of ELF that has a magnitude of 0.75. The motif with one SiAl bond (or two Si-Al bonds) is enlarged in the black (or green) square. b. Cu(111)/Si(111). Al and Cu have the same crystal structure and, in both cases, A and ∆A are noticeably smaller for parallel epitaxy. Yet, the 30° epitaxy is favored, both s-l

because of a lower Eform and a lower Estr . Hence, the result disagrees with the superlattice matching model. Since the only difference between Cu and Al is in their lattice constants with Cu being 11% smaller than Al, the failure of the model signals the importance of correctly account for the interfacial bond energy.



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c. Cu(001)/Si(001). Here, the 45° epitaxy is pseudomorphic with a critical thickness hc = 4 MLs. It is known that below hc, the strain is unrelaxed. Above hc, on the other hand, the strain is quickly relaxed. In the calculation, we consider a fixed residual strain up to hc. Despite the larger strain, the 45° epitaxy is still energetically favored due to the even larger interfacial binding energy. This is confirmed by the ELF plot in Fig. 8, in which we see that each surface Si atom forms two directional Si-Cu bonds at where the Si dangling bonds would be. The interfacial binding is strong because of the undistorted Si sp3-hybridization with 109.47° bond angles. In comparison, here the superlattice matching model is indecisive for Cu(001)/Si(001), as A favors 45° epitaxy, but ∆A favors parallel epitaxy.

Figure 8. Side view of [1/1, 1/1] module at Cu(111)010/Si(111)110 interface. The yellow lobes correspond to the isosurface of ELF that has a magnitude of 0.70. The interfacial motif, enlarged in the black square, indicates the formation of only two Si-Cu bonds, although there are four nearest-neighboring Cu atoms around the Si atom. d. CaF2(001)/Ni(001). Again, the 45° epitaxy is pseudomorphic with a slightly smaller critical thickness hc = 2 MLs . Similar to case c above, strained 45° epitaxy is 18 ACS Paragon Plus Environment

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energetically favored because of the large interfacial binding energy. Different from case c. Cu(001)/Si(001), however, here the superlattice matching model works again as both A and ∆A favor the 45° epitaxy. CONCLUSION In summary, we have developed a modular approach that enables theoretical study of large-misfit epitaxy by a combined first-principles methods and elasticity theory. Applications to metal-semiconductor interfaces, namely, Al(111) and Cu(111) on Si(111), Cu(001) on Si(001), and CaF2(001) on Ni(001), show good agreement with experiments. This may be contrasted to available models based on pure geometric considerations, which may not agree with experiments. An ELF analysis is also developed to show the formation of directional chemical bonds at the interfaces and their importance in determining the epitaxial relationship. We expect our new approach will guide experiments in realizing high-quality epitaxy between large lattice misfit materials with drastically different physical properties for novel heterostructure devices. ASSOCIATED CONTENT Supporting Information Derivation of 1. Interface Formation Energy of the Module i; 2. Elastic Strain Energies of the Modules. This material is available free of charge via the Internet at http://pubs.acs.org. AUTHOR INFORMATION Corresponding Authors *(Shengbai Zhang) E-mail: [email protected]. Note 19 ACS Paragon Plus Environment


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The authors declare no competing financial interest. ACKNOWLEDGMENT We thank Dr. Xianqiang Liu in Beijing University of Technology for valuable discussions. WYX was supported by the NSF under Award No. 1305293, ML and SBZ were supported by the DOE under Grant No. DE-SC0002623. The supercomputer time was provided by the Center for Computational Innovations (CCI) of Rensselaer Polytechnic Institute and by National Energy Research Scientific Computing Center (NERSC) supported by the DOE Office of Science under Grant No. DE-AC0205CH11231. REFERENCES (1)

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Seixas, L.; West, D.; Fazzio, A.; Zhang, S. B. Nat. Commun. 2015, 6, 7630. 20 ACS Paragon Plus Environment

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For Table of Contents Use Only Modular Approach for Metal-semiconductor Heterostructures with Very-large Interface Lattice Misfit: A First-principles Perspective Weiyu Xie, Michael Lucking, Liang Chen, Ishwara Bhat, Gwo-Ching Wang, Toh-Ming Lu, and Shengbai Zhang* Table of Contents Graphic:

Synopsis: Based on the experimental findings of different alternating matching domains at heterostructures with very-large interface lattice misfit, we propose an atomistic simulation approach including multiple interfacial modules. To determine the epitaxial relationship properly, first-principles calculations and elasticity theory are combined. The electron localization function demonstrates the formation of covalent bonds at metal-Si interfaces.


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