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Cage-like structure and charge hollow in the immiscible Cu–Ta system
Solid State Communications 149 (2009) 1974–1977
Contents lists available at ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Cage-like structure and charge hollow in the immiscible Cu–Ta system Yi Kong a,∗ , Yong Du a , Junqin Li b a
State Key Laboratory of Powder Metallurgy, Central South University, Changsha 410083, Hunan, China
b
Department of Materials Science and Engineering, Shenzhen University, Shenzhen 518060, China
article
info
Article history: Received 9 July 2009 Accepted 19 July 2009 by A.H. MacDonald Available online 24 July 2009 PACS: 61.43.Bn 71.16.Mb 71.23.Cq
abstract Ground states of the immiscible Cu–Ta system are investigated by a hybrid approach of first-principles calculation and cluster expansion method. The obtained pairs, triplets, and tetrads cluster interactions show that the closer-packed cluster may not always contribute to stabilize structures. A CuTa7 phase with a cage-like structure is predicted to be stable at ground state. The obtained spatial valence charge density shows that there exists a charge hollow within this cage-like structure, contributing to stabilize CuTa7 phase. © 2009 Published by Elsevier Ltd
Keywords: A. Cu–Ta B. First principles D. Electronic structure
1. Introduction Over the past decades, progress in modern processing techniques has enabled researchers to artificially create an increasing number of new alloys that are immiscible in thermodynamic equilibrium [1]. In addition to the fundamental exploration of alloying theory and phase transformation, the creation of alloys between immiscible elements has also opened up new opportunities for tailoring materials’ properties. The understanding of these new phases, are clearly of considerable scientific and technologic interest. Up until now, however, convincing evidence is still lacking for establishing the exact extent of the alloying theory for the equilibrium immiscible binary metal systems with positive ∆H [1–3]. As a typical example, despite the immiscibility between the Cu and Ta elements (Ta has been used as a diffusion barrier in very large scale integrated circuits to prevent Cu atoms from diffusing into Si/SiO2 ) [4,5], many experiments have shown that various metastable Cu–Ta phases of either an amorphous or crystalline structure could be formed between Cu and Ta under various nonequilibrium conditions [6–9]. It is, therefore, of both practical and theoretical importance to investigate the interaction between the metals Cu and Ta and reveal the associated underlying physics for the formation of metastable phase. Concerning interaction between immiscible elements, detailed investigations have shown that alloys with positive enthalpy of
formation can show clustering and phase separation or anticlustering and metastable long-range ordering [10,11]. Recently, the calculations performed by Ferrando et al. [12] further confirmed the strong tendency for mixing of the small bimetallic clusters, even for those that presented a very strong tendency against mixing in the bulk phase (such as Ag–Ni, Ag–Co, or Au–Co with positive enthalpy of formation). In the present work, we use a hybrid approach of cluster expansion (CE) method [13] and firstprinciples calculations to investigate the ground state of the immiscible Cu–Ta system. 2. Methodology According to CE method, for a binary alloy with total N atoms, there can be 2N possible number of configurations, which is an astronomically large number when N is large. Searching for the ground states necessitates exploring such a huge configurational space, which is computationally unaffordable when directly using first-principles method. To overcome such difficulties, the CE method is adopted in the present calculation. Detailed description of CE method can be found elsewhere [13–15]. Thus only a brief description of the method is presented here. Under the framework of CE method, the energy of a configuration can be calculated using Ising-like Hamiltonian: ECEM (σ ) =
X f
∗
Corresponding author. Tel.: +86 731 88877300; fax: +86 731 88710855. E-mail address: [email protected] (Y. Kong).
0038-1098/$ – see front matter © 2009 Published by Elsevier Ltd doi:10.1016/j.ssc.2009.07.028
D f Jf
Y
(σ )
(1)
f
where f is a cluster comprised of a group of lattice sites. The simplest cluster is a single site, while pairs (2-body), triplets (3body), and tetrad (4-body) are three important groups of clusters in
Y. Kong et al. / Solid State Communications 149 (2009) 1974–1977
.4
.2
.3
.1 0.0 ECI
.2
CE predicted
1975
.1
-.1 -.2
0.0
with multiplicity without multiplicity zero
-.3 -.1
-.4 -.2 -.2
2 -.1
0.0
.1 .2 First-principles
.3
.4
Fig. 1. Comparison of enthalpies of formation of 76 structures calculated from first principles and predicted from CE method. X: First-principles values, Y: CEM predicted values. The unit is ev/atom.
4
6
8 10 Pair Distance
12
14
Fig. 2. ECIs of pair clusters. The transverse axis is the distance sequence of pair clusters. The larger number means the longer distance between two atoms. The unit for ECI is ev/atom.
a .03
3. Results and discussion For the Cu–Ta system, as a typical example, closed-packed fcc type and open-packed bcc type lattice as parent lattice are chosen to perform CE calculations, respectively. For fcc type lattice, 76 structures are fully relaxed in first-principles calculations, and the obtained enthalpies of formation are used as input parameters in performing CE calculation. The direct comparison of the enthalpies of formation for these 76 structures obtained from first principles and CE predicted are shown in Fig. 1. As shown in this figure, there is a good agreement between these two methods. To further analyze the details of the obtained ECIs, in the following we will show the specific ECI of every pairs and tetrad in Figs. 2 and 3, respectively. The ECIs of triplets are trivial and not shown here. In Fig. 2, the transverse axis is the distance sequence of the pair clusters. The larger number means the longer distance between two atoms. The y-axle is the enthalpy of formation of the pairs. From pair ECI shown in Fig. 2, one can note that there
ECI
.02 .01 0.00 -.01
b ECI
CE method. Df is the multiplicity of a cluster used in a CE calculation to simulate a configuration. Jf is the configuration-independent Ising type interaction for cluster f , and it is called effective cluster interactions (ECI). Once ECIs are known, one can use CE method to predict the energy of any configuration, whatever order or disorder, within the accuracy of first-principles calculations. Consequently, obtaining reliable ECI is the key issue in using CE method to predict ground states. And the CE method mentioned above is implemented in the Alloy Theoretic Automated Toolkit (ATAT) code [16]. With the combination of CE method and first-principles calculations, the output from first-principles calculations is the enthalpy of formation. In the present work, first-principles calculations were performed using the plane wave method with projectoraugmented-wave (PAW) pseudopotentials, as implemented in the highly-efficient Vienna ab initio simulation package (VASP) [17,18]. The k-point meshes for Brillouin zone sampling were constructed using the Monkhorst–Pack scheme and the total number of kpoints per reciprocal atoms was at least 3000 for all the systems. By computing the forces and stress tensor, structural and atomic relaxations were performed and all atoms were relaxed into their equilibrium positions using a conjugate-gradient scheme. The exchange and correlation effects are described by employing generalized gradient approximation (GGA) proposed by Perdew et al. [19], with a plane wave cutoff energy Ecut = 400 eV, and the precision is set to accurate.
.5 .4 .3 .2 .1 0.0 -.1 -.2 -.3 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Fig. 3. ECIs of tetrad clusters. Where the ECI of every tetrad cluster (a) without multiplicity, and (b) with multiplicity. The unit is ev/atom. In (a), the three cluster configurations, which have local minimum formation enthalpies are also shown.
is a periodic phenomenon for the stability of these pair clusters, suggesting that the shorter pair may not always contribute to stabilize the structures in an immiscible system. In the case of tetrad clusters shown in Fig. 3(a), first one can note that the enthalpy of formation for perfect tetrahedron (number 1 cluster) is zero. Second, number 6 cluster has the lowest enthalpy of formation, and the maximum pair distance in this tetrad cluster is 4.654 Å. Finally, number 15 cluster also has local minimum enthalpy of formation. The structures of these three clusters are shown in Fig. 3(a) also. Fig. 3(b) presents the modification of the contribution from these tetrad clusters to the enthalpy of formation for a specific structure, when the multiplicity of the cluster is considered. With the obtained ECI parameters, the enthalpies of formation for total 555 structures are predicted with CE method, and the obtained results are shown in Fig. 4(a) with blue x, while the 76 structures calculated from first principles are also shown in Fig. 4(a) with red squares. From the comparison, the ground state line can be drawn, and the results are shown in Fig. 4(a) with black solid line. For bcc lattice, we also perform the similar calculations by means of CE method and first-principles calculations. With the obtained ECIs, the enthalpies of formation for total 1079 structures are predicted with CE method, and the obtained results are shown
1976
Y. Kong et al. / Solid State Communications 149 (2009) 1974–1977
a
.5
Enthalpy of formation
.4 .3 .2 .1 0.0 -.1 -.2 0.0
b
.2
.4 .6 Ta Content
1.0
.5
CEM First-principles GS Line
.4
Enthalpy of formation
.8
.3 .2 .1 0.0
0.0
.2
.4 .6 Ta Content
.8
1.0
Fig. 4. (color online) Ground state of (a) fcc type lattice, and (b) bcc type lattice, where black solid line represents ground state line, red squares represent structures calculated from first-principles calculation, blue x represent the results predicted from CE method.
in Fig. 4(b) with blue x, while 43 structures calculated from first principles are also shown in Fig. 4(b) with red squares. All of these structures have positive enthalpies of formation, suggesting that there is no any stable phase with bcc type in the Cu–Ta system.
a
From Fig. 4(a), one can note that although Cu–Ta system is predicted to be thermodynamical immiscible according to Miedema theory, there still has one predicted stable phase at the composition of Cu:Ta = 1:7. In Fig. 5(a), close-packed (110) plane of this compound is shown. It is interesting to note that a cage-like configuration can be seen. For such a configuration, there exists a unit cell in which 6 Ta atoms form a nonregular hexagon and one Cu atom occupies the central of this hexagon. A similar cage-like structure is reported in another immiscible Ag–Co system, through empirical tight-binding molecular dynamic simulations [20], indicating this cage-like structure may be universal in immiscible system. Furthermore, it is believed that this cage formation drives dynamical arrest in glass materials, especially at high concentrations [21]. To understand why this cage-like structure can be stabilized in an immiscible system, we further calculate the spatial valence charge density (SVCD) to reveal the bonding nature of this structure [22]. We use slices of (110) plane in the electronic density field to show the details of the SVCD in a quantitative way. The slice is plotted with color filled contour between the lowest charge density (about 0.15 electrons per A3) and maximum charge density (about 10.15 electrons per A). The maxima in the electronic density field correspond to the positions of the atoms that have semicore states, which are explicitly treated as valence. The obtained SVCD is shown in Fig. 5(b). One can note that there exists a charge hollow within this cage-like structure. That is, little charge distributes among 6 surrounding Ta atoms and 1 central Cu atom in this cage-like structure. This little charge distribution could guarantee the net force acting on each Cu atom from the Ta atoms is balanced to be zero in all the directions. At the same time, the Cu–Cu interaction becomes too weak to drive the segregation, as the Cu atoms are kept far apart by the Ta atoms. As a comparison, this charge hollow could be seen as a defect in the electronic structure, just like defect in the crystalline structure. 4. Conclusions In summary, for an immiscible system, to retain metastable phase in a homogeneous structure at an atomic scale, it is necessary
b 0.1500 0.1891 0.2281 0.3063 0.4625 0.7750 1.400 2.650 3.900 5.150 6.400 7.650 8.900 10.15
Fig. 5. (color online) (a) close-packed (110) plane of predicted stable CuTa7 phase, where red ball represents Cu atoms, gray ball represents Ta atoms; (b) slices of (110) plane in the electronic density field.
Y. Kong et al. / Solid State Communications 149 (2009) 1974–1977
to have the atoms somehow organize in a specific configuration (i.e., the proposed cage-like configuration), in which the balanced net force acting on the solute atoms is weakened and actually zero, as well as the interaction of solute atoms becomes too weak to drive phase segregation. Further analysis of the spatial valence charge density shows that the stability of this cage-like configuration lies in the charge hollow forming within it.
[4] [5] [6] [7] [8] [9] [10] [11]
Acknowledgements The financial supports from the National Natural Science Foundation of China (NSFC) (Grant No. 50801069), Creative Research Group of NSFC (Grant No. 50721003), and the Open Project Program of Key Laboratory of Shenzhen University (Grant No. T0803) are acknowledged. References [1] E. Ma, Prog. Mater. Sci. 50 (2005) 413. [2] M.A. Ortigoza, T.S. Rahman, Phys. Rev. B 77 (2008) 195404. [3] B.X. Liu, W.S. Lai, Q. Zhang, Mater. Sci. Eng. R 29 (2000) 1.
[12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
[22]
1977
E.M. Zielinski, R.P. Vinci, J.C. Bravman, J. Appl. Phys. 76 (1994) 4516. K.W. Kwon, C. Ryu, R. Sinclair, S.S. Wong, Appl. Phys. Lett. 71 (1997) 3069. H.J. Lee, K.W. Kwon, C. Ryu, R. Sinclair, Acta Mater. 47 (1999) 3965. L. Chen, N. Magtoto, B. Ekstrom, J. Kelber, Thin Solid Films 376 (2000) 115. K. Sakurai, Y. Yamada, M. Ito, C.H. Lee, T. Fukunaga, U. Mizutani, Appl. Phys. Lett. 57 (1990) 2660. Y. Kong, H.R. Gong, B.X. Liu, Appl. Phys. Lett. 85 (2004) 1517. J.H. He, H.W. Sheng, J.S. Lin, P.J. Schilling, R.C. Tittsworth, E. Ma, Phys. Rev. Lett. 89 (2002) 125507. J.H. He, H.W. Sheng, P.J. Schilling, C.L. Chien, E. Ma, Phys. Rev. Lett. 86 (2001) 2826. R. Ferrando, A. Fortunelli, G. Rossi, Phys. Rev. B 72 (2005) 085449. A. van de Walle, G. Ceder, J. Phase Equilib. 23 (2002) 348. L.G. Ferreira, S.H. Wei, A. Zunger, Int. J. Supercomput. Appl. 5 (1) (1991) 34. C. Jiang, D.J. Sordelet, B. Gleeson, Phys. Rev. B 72 (2005) 184203. A. van de Walle, M. Asta, G. Ceder, CALPHAD 26 (2002) 539. G. Kresse, D. Joubert, Phys. Rev. B 59 (1999) 1758. G. Kresse, J. Hafner, Phys. Rev. B 47 (1993) 558. J.P. Perdew, J.A. Chevary, S.H. Vosko, K.A. Jackson, M.R. Pederson, D.J. Singh, C. Fiolhais, Phys. Rev. B 46 (1992) 6671. H.B. Guo, B.X. Liu, J. Mater. Res. 19 (2004) 1364. C. Mayer, E. Zaccarelli, E. Stiakakis, C.N. Likos, F. Sciortino, A. Munam, M. Gauthier, N. Hadjichristidis, H. Iatrou, P. Tartaglia, H. Lown, D. Vlassopoulos, Nature Mater. 7 (2008) 780. Y. Kong, J.H. Li, L.T. Kong, B.X. Liu, Phys. Rev. B 72 (2005) 024209.
Contents lists available at ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Cage-like structure and charge hollow in the immiscible Cu–Ta system Yi Kong a,∗ , Yong Du a , Junqin Li b a
State Key Laboratory of Powder Metallurgy, Central South University, Changsha 410083, Hunan, China
b
Department of Materials Science and Engineering, Shenzhen University, Shenzhen 518060, China
article
info
Article history: Received 9 July 2009 Accepted 19 July 2009 by A.H. MacDonald Available online 24 July 2009 PACS: 61.43.Bn 71.16.Mb 71.23.Cq
abstract Ground states of the immiscible Cu–Ta system are investigated by a hybrid approach of first-principles calculation and cluster expansion method. The obtained pairs, triplets, and tetrads cluster interactions show that the closer-packed cluster may not always contribute to stabilize structures. A CuTa7 phase with a cage-like structure is predicted to be stable at ground state. The obtained spatial valence charge density shows that there exists a charge hollow within this cage-like structure, contributing to stabilize CuTa7 phase. © 2009 Published by Elsevier Ltd
Keywords: A. Cu–Ta B. First principles D. Electronic structure
1. Introduction Over the past decades, progress in modern processing techniques has enabled researchers to artificially create an increasing number of new alloys that are immiscible in thermodynamic equilibrium [1]. In addition to the fundamental exploration of alloying theory and phase transformation, the creation of alloys between immiscible elements has also opened up new opportunities for tailoring materials’ properties. The understanding of these new phases, are clearly of considerable scientific and technologic interest. Up until now, however, convincing evidence is still lacking for establishing the exact extent of the alloying theory for the equilibrium immiscible binary metal systems with positive ∆H [1–3]. As a typical example, despite the immiscibility between the Cu and Ta elements (Ta has been used as a diffusion barrier in very large scale integrated circuits to prevent Cu atoms from diffusing into Si/SiO2 ) [4,5], many experiments have shown that various metastable Cu–Ta phases of either an amorphous or crystalline structure could be formed between Cu and Ta under various nonequilibrium conditions [6–9]. It is, therefore, of both practical and theoretical importance to investigate the interaction between the metals Cu and Ta and reveal the associated underlying physics for the formation of metastable phase. Concerning interaction between immiscible elements, detailed investigations have shown that alloys with positive enthalpy of
formation can show clustering and phase separation or anticlustering and metastable long-range ordering [10,11]. Recently, the calculations performed by Ferrando et al. [12] further confirmed the strong tendency for mixing of the small bimetallic clusters, even for those that presented a very strong tendency against mixing in the bulk phase (such as Ag–Ni, Ag–Co, or Au–Co with positive enthalpy of formation). In the present work, we use a hybrid approach of cluster expansion (CE) method [13] and firstprinciples calculations to investigate the ground state of the immiscible Cu–Ta system. 2. Methodology According to CE method, for a binary alloy with total N atoms, there can be 2N possible number of configurations, which is an astronomically large number when N is large. Searching for the ground states necessitates exploring such a huge configurational space, which is computationally unaffordable when directly using first-principles method. To overcome such difficulties, the CE method is adopted in the present calculation. Detailed description of CE method can be found elsewhere [13–15]. Thus only a brief description of the method is presented here. Under the framework of CE method, the energy of a configuration can be calculated using Ising-like Hamiltonian: ECEM (σ ) =
X f
∗
Corresponding author. Tel.: +86 731 88877300; fax: +86 731 88710855. E-mail address: [email protected] (Y. Kong).
0038-1098/$ – see front matter © 2009 Published by Elsevier Ltd doi:10.1016/j.ssc.2009.07.028
D f Jf
Y
(σ )
(1)
f
where f is a cluster comprised of a group of lattice sites. The simplest cluster is a single site, while pairs (2-body), triplets (3body), and tetrad (4-body) are three important groups of clusters in
Y. Kong et al. / Solid State Communications 149 (2009) 1974–1977
.4
.2
.3
.1 0.0 ECI
.2
CE predicted
1975
.1
-.1 -.2
0.0
with multiplicity without multiplicity zero
-.3 -.1
-.4 -.2 -.2
2 -.1
0.0
.1 .2 First-principles
.3
.4
Fig. 1. Comparison of enthalpies of formation of 76 structures calculated from first principles and predicted from CE method. X: First-principles values, Y: CEM predicted values. The unit is ev/atom.
4
6
8 10 Pair Distance
12
14
Fig. 2. ECIs of pair clusters. The transverse axis is the distance sequence of pair clusters. The larger number means the longer distance between two atoms. The unit for ECI is ev/atom.
a .03
3. Results and discussion For the Cu–Ta system, as a typical example, closed-packed fcc type and open-packed bcc type lattice as parent lattice are chosen to perform CE calculations, respectively. For fcc type lattice, 76 structures are fully relaxed in first-principles calculations, and the obtained enthalpies of formation are used as input parameters in performing CE calculation. The direct comparison of the enthalpies of formation for these 76 structures obtained from first principles and CE predicted are shown in Fig. 1. As shown in this figure, there is a good agreement between these two methods. To further analyze the details of the obtained ECIs, in the following we will show the specific ECI of every pairs and tetrad in Figs. 2 and 3, respectively. The ECIs of triplets are trivial and not shown here. In Fig. 2, the transverse axis is the distance sequence of the pair clusters. The larger number means the longer distance between two atoms. The y-axle is the enthalpy of formation of the pairs. From pair ECI shown in Fig. 2, one can note that there
ECI
.02 .01 0.00 -.01
b ECI
CE method. Df is the multiplicity of a cluster used in a CE calculation to simulate a configuration. Jf is the configuration-independent Ising type interaction for cluster f , and it is called effective cluster interactions (ECI). Once ECIs are known, one can use CE method to predict the energy of any configuration, whatever order or disorder, within the accuracy of first-principles calculations. Consequently, obtaining reliable ECI is the key issue in using CE method to predict ground states. And the CE method mentioned above is implemented in the Alloy Theoretic Automated Toolkit (ATAT) code [16]. With the combination of CE method and first-principles calculations, the output from first-principles calculations is the enthalpy of formation. In the present work, first-principles calculations were performed using the plane wave method with projectoraugmented-wave (PAW) pseudopotentials, as implemented in the highly-efficient Vienna ab initio simulation package (VASP) [17,18]. The k-point meshes for Brillouin zone sampling were constructed using the Monkhorst–Pack scheme and the total number of kpoints per reciprocal atoms was at least 3000 for all the systems. By computing the forces and stress tensor, structural and atomic relaxations were performed and all atoms were relaxed into their equilibrium positions using a conjugate-gradient scheme. The exchange and correlation effects are described by employing generalized gradient approximation (GGA) proposed by Perdew et al. [19], with a plane wave cutoff energy Ecut = 400 eV, and the precision is set to accurate.
.5 .4 .3 .2 .1 0.0 -.1 -.2 -.3 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Fig. 3. ECIs of tetrad clusters. Where the ECI of every tetrad cluster (a) without multiplicity, and (b) with multiplicity. The unit is ev/atom. In (a), the three cluster configurations, which have local minimum formation enthalpies are also shown.
is a periodic phenomenon for the stability of these pair clusters, suggesting that the shorter pair may not always contribute to stabilize the structures in an immiscible system. In the case of tetrad clusters shown in Fig. 3(a), first one can note that the enthalpy of formation for perfect tetrahedron (number 1 cluster) is zero. Second, number 6 cluster has the lowest enthalpy of formation, and the maximum pair distance in this tetrad cluster is 4.654 Å. Finally, number 15 cluster also has local minimum enthalpy of formation. The structures of these three clusters are shown in Fig. 3(a) also. Fig. 3(b) presents the modification of the contribution from these tetrad clusters to the enthalpy of formation for a specific structure, when the multiplicity of the cluster is considered. With the obtained ECI parameters, the enthalpies of formation for total 555 structures are predicted with CE method, and the obtained results are shown in Fig. 4(a) with blue x, while the 76 structures calculated from first principles are also shown in Fig. 4(a) with red squares. From the comparison, the ground state line can be drawn, and the results are shown in Fig. 4(a) with black solid line. For bcc lattice, we also perform the similar calculations by means of CE method and first-principles calculations. With the obtained ECIs, the enthalpies of formation for total 1079 structures are predicted with CE method, and the obtained results are shown
1976
Y. Kong et al. / Solid State Communications 149 (2009) 1974–1977
a
.5
Enthalpy of formation
.4 .3 .2 .1 0.0 -.1 -.2 0.0
b
.2
.4 .6 Ta Content
1.0
.5
CEM First-principles GS Line
.4
Enthalpy of formation
.8
.3 .2 .1 0.0
0.0
.2
.4 .6 Ta Content
.8
1.0
Fig. 4. (color online) Ground state of (a) fcc type lattice, and (b) bcc type lattice, where black solid line represents ground state line, red squares represent structures calculated from first-principles calculation, blue x represent the results predicted from CE method.
in Fig. 4(b) with blue x, while 43 structures calculated from first principles are also shown in Fig. 4(b) with red squares. All of these structures have positive enthalpies of formation, suggesting that there is no any stable phase with bcc type in the Cu–Ta system.
a
From Fig. 4(a), one can note that although Cu–Ta system is predicted to be thermodynamical immiscible according to Miedema theory, there still has one predicted stable phase at the composition of Cu:Ta = 1:7. In Fig. 5(a), close-packed (110) plane of this compound is shown. It is interesting to note that a cage-like configuration can be seen. For such a configuration, there exists a unit cell in which 6 Ta atoms form a nonregular hexagon and one Cu atom occupies the central of this hexagon. A similar cage-like structure is reported in another immiscible Ag–Co system, through empirical tight-binding molecular dynamic simulations [20], indicating this cage-like structure may be universal in immiscible system. Furthermore, it is believed that this cage formation drives dynamical arrest in glass materials, especially at high concentrations [21]. To understand why this cage-like structure can be stabilized in an immiscible system, we further calculate the spatial valence charge density (SVCD) to reveal the bonding nature of this structure [22]. We use slices of (110) plane in the electronic density field to show the details of the SVCD in a quantitative way. The slice is plotted with color filled contour between the lowest charge density (about 0.15 electrons per A3) and maximum charge density (about 10.15 electrons per A). The maxima in the electronic density field correspond to the positions of the atoms that have semicore states, which are explicitly treated as valence. The obtained SVCD is shown in Fig. 5(b). One can note that there exists a charge hollow within this cage-like structure. That is, little charge distributes among 6 surrounding Ta atoms and 1 central Cu atom in this cage-like structure. This little charge distribution could guarantee the net force acting on each Cu atom from the Ta atoms is balanced to be zero in all the directions. At the same time, the Cu–Cu interaction becomes too weak to drive the segregation, as the Cu atoms are kept far apart by the Ta atoms. As a comparison, this charge hollow could be seen as a defect in the electronic structure, just like defect in the crystalline structure. 4. Conclusions In summary, for an immiscible system, to retain metastable phase in a homogeneous structure at an atomic scale, it is necessary
b 0.1500 0.1891 0.2281 0.3063 0.4625 0.7750 1.400 2.650 3.900 5.150 6.400 7.650 8.900 10.15
Fig. 5. (color online) (a) close-packed (110) plane of predicted stable CuTa7 phase, where red ball represents Cu atoms, gray ball represents Ta atoms; (b) slices of (110) plane in the electronic density field.
Y. Kong et al. / Solid State Communications 149 (2009) 1974–1977
to have the atoms somehow organize in a specific configuration (i.e., the proposed cage-like configuration), in which the balanced net force acting on the solute atoms is weakened and actually zero, as well as the interaction of solute atoms becomes too weak to drive phase segregation. Further analysis of the spatial valence charge density shows that the stability of this cage-like configuration lies in the charge hollow forming within it.
[4] [5] [6] [7] [8] [9] [10] [11]
Acknowledgements The financial supports from the National Natural Science Foundation of China (NSFC) (Grant No. 50801069), Creative Research Group of NSFC (Grant No. 50721003), and the Open Project Program of Key Laboratory of Shenzhen University (Grant No. T0803) are acknowledged. References [1] E. Ma, Prog. Mater. Sci. 50 (2005) 413. [2] M.A. Ortigoza, T.S. Rahman, Phys. Rev. B 77 (2008) 195404. [3] B.X. Liu, W.S. Lai, Q. Zhang, Mater. Sci. Eng. R 29 (2000) 1.
[12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
[22]
1977
E.M. Zielinski, R.P. Vinci, J.C. Bravman, J. Appl. Phys. 76 (1994) 4516. K.W. Kwon, C. Ryu, R. Sinclair, S.S. Wong, Appl. Phys. Lett. 71 (1997) 3069. H.J. Lee, K.W. Kwon, C. Ryu, R. Sinclair, Acta Mater. 47 (1999) 3965. L. Chen, N. Magtoto, B. Ekstrom, J. Kelber, Thin Solid Films 376 (2000) 115. K. Sakurai, Y. Yamada, M. Ito, C.H. Lee, T. Fukunaga, U. Mizutani, Appl. Phys. Lett. 57 (1990) 2660. Y. Kong, H.R. Gong, B.X. Liu, Appl. Phys. Lett. 85 (2004) 1517. J.H. He, H.W. Sheng, J.S. Lin, P.J. Schilling, R.C. Tittsworth, E. Ma, Phys. Rev. Lett. 89 (2002) 125507. J.H. He, H.W. Sheng, P.J. Schilling, C.L. Chien, E. Ma, Phys. Rev. Lett. 86 (2001) 2826. R. Ferrando, A. Fortunelli, G. Rossi, Phys. Rev. B 72 (2005) 085449. A. van de Walle, G. Ceder, J. Phase Equilib. 23 (2002) 348. L.G. Ferreira, S.H. Wei, A. Zunger, Int. J. Supercomput. Appl. 5 (1) (1991) 34. C. Jiang, D.J. Sordelet, B. Gleeson, Phys. Rev. B 72 (2005) 184203. A. van de Walle, M. Asta, G. Ceder, CALPHAD 26 (2002) 539. G. Kresse, D. Joubert, Phys. Rev. B 59 (1999) 1758. G. Kresse, J. Hafner, Phys. Rev. B 47 (1993) 558. J.P. Perdew, J.A. Chevary, S.H. Vosko, K.A. Jackson, M.R. Pederson, D.J. Singh, C. Fiolhais, Phys. Rev. B 46 (1992) 6671. H.B. Guo, B.X. Liu, J. Mater. Res. 19 (2004) 1364. C. Mayer, E. Zaccarelli, E. Stiakakis, C.N. Likos, F. Sciortino, A. Munam, M. Gauthier, N. Hadjichristidis, H. Iatrou, P. Tartaglia, H. Lown, D. Vlassopoulos, Nature Mater. 7 (2008) 780. Y. Kong, J.H. Li, L.T. Kong, B.X. Liu, Phys. Rev. B 72 (2005) 024209.