Accepted Manuscript Newly synthesized MgAl2Ge2: A first-principles comparison with its silicide and carbide counterparts A.M.M. Tanveer Karim, M.A. Hadi, M.A. Alam, F. Parvin, S.H. Naqib, A.K.M.A. Islam PII:
S0022-3697(18)30067-2
DOI:
10.1016/j.jpcs.2018.02.037
Reference:
PCS 8452
To appear in:
Journal of Physics and Chemistry of Solids
Received Date: 9 January 2018 Revised Date:
7 February 2018
Accepted Date: 15 February 2018
Please cite this article as: A.M.M. Tanveer Karim, M.A. Hadi, M.A. Alam, F. Parvin, S.H. Naqib, A.K.M.A. Islam, Newly synthesized MgAl2Ge2: A first-principles comparison with its silicide and carbide counterparts, Journal of Physics and Chemistry of Solids (2018), doi: 10.1016/j.jpcs.2018.02.037. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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ACCEPTED MANUSCRIPT
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10 8
4 2 0
-2 -4 -6 -8
-10
MgAl2Ge2
Γ A
H
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Energy, E - EF (eV)
6
MgAl2Si2
K
Γ
M
L
H Γ A
H
K
EP AC C
Γ
M
High symmetry points
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High symmetry points
L
EF
MgAl2C2
H Γ A
H
K
Γ
M
High symmetry points
L
H
ACCEPTED MANUSCRIPT Newly synthesized MgAl2Ge2: A first-principles comparison with its silicide and carbide counterparts A. M. M. Tanveer Karim1, M. A. Hadi2,3*, M. A. Alam4, F. Parvin2, S. H. Naqib2 and A. K. M. A. Islam2,5 1
Department of Physics, Rajshahi University of Engineering and Technology, Rajshahi-6204, Bangladesh Department of Physics, University of Rajshahi, Rajshahi-6205, Bangladesh 3 Department of Physics, Govt. BMC Women’s College, Naogaon, Naogaon-6500, Bangladesh 4 Department of Physics, Mawlana Bhashani Science and Technology University, Santosh,Tangail-1902, Bangladesh 5 International Islamic University Chittagong, Kumira, Chittagong-4318, Bangladesh
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2
Abstract
Using plane-wave pseudopotential density functional theory (DFT), the first-principle
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calculations are performed to investigate the structural aspects, mechanical behaviors and electronic features of the newly synthesized CaAl2Si2-prototype intermetallic compound,
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MgAl2Ge2 for the first time and the results are compared with those calculated for its silicide and carbide counterparts MgAl2Si2 and MgAl2C2. The calculated lattice constants agree fairly well with their corresponding experimental values. The estimated elastic tensors satisfy the mechanical stability conditions for MgAl2Ge2 along with MgAl2Si2 and MgAl2C2. The level
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of elastic anisotropy increases following the sequence of X-elements Ge → Si → C. MgAl2Ge2 and MgAl2Si2 are expected to be ductile and damage tolerant, while MgAl2C2 is a brittle one. MgAl2Ge2 and MgAl2Si2 should exhibit better thermal shock resistance and low
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thermal conductivity and accordingly these can be used as thermal barrier coating (TBC) materials. The Debye temperature of MgAl2Ge2 is lowest among three intermetallic
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compounds. MgAl2Ge2 and MgAl2Si2 should exhibit metallic conductivity; while the dual characters of weak-metals and semiconductors are expected for MgAl2C2. The values of theoretical Vickers hardness for MgAl2Ge2, MgAl2Si2, and MgAl2C2 are 3.3, 2.7, and 7.7 GPa, respectively, indicating that these three intermetallics are soft and easily machinable. Keywords: Intermetallics, Structural properties, Mechanical behaviors, Electronic features
*
Corresponding author email:
[email protected]
ACCEPTED MANUSCRIPT 1. Introduction Compounds with the so-called AB2X2 (122) stoichiometry are divided into three main different chemical classes: firstly, the chalcogenides group, which includes oxides, sulfides, selenides and tellurides; secondly, the pnictides group, which includes nitrides, phosphides,
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arsenides, antimonides and bismuthides; and lastly, compounds (carbides, silicides, and germanides) of CaAl2Si2-prototype containing an element of groups 13−14 of the periodic table. In the AB2X2 stoichiometry, A is an alkali, alkaline-earth, or rare earth element, B refers
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to a transition metal or main-group metalloid, and X is an element from groups 13−16 of the periodic table [1]. This large family of intermetallics includes over 700 members and exhibits
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an exceptional collection of chemical and physical properties. A large array of fascinating physical phenomena, such as superconductivity at high temperature and large magnetoresistance, has been observed in such crystal systems [2−5]. Because of the multifaceted coupling among spin, charge as well as lattice degrees of freedoms in these crystals, a slight
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change of temperature, pressure, magnetic field, etc., may cause a remarkable modification of their physical properties [6].
Among different structures in this large family, the compounds with the ThCr2Si2 structure
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have been investigated widely [7−12] from the time when the Fe-based superconductors
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AFe2As2 (A = K, Ca, Sr, Ba, etc.) have been discovered [13,14]. This structure is famous for hosting many other fascinating quantum electronic states, for example, heavy fermions [15,16] and quantum criticality [17−19]. The ThCr2Si2–type compound crystallizes in the tetragonal system and is a member of the I4/nmm space group. The B2X2 layers in the structure, apart from the A-site atoms, consist of an edge-sharing network of BX4 tetrahedra. The CaAl2Si2-type structure in the AB2X2 stoichiometry is also layered and the B2X2layers are separated by the A-site atoms [20]. The CaAl2Si2–prototype compounds crystallize in the trigonal structure with a 31space group instead of tetragonal system of I4/nmm
ACCEPTED MANUSCRIPT space group for the ThCr2Si2–type compounds. The cations within each B2X2-layer (ab-plane) form a triangle lattice, as shown in Figure 1. The prototype itself (CaAl2Si2) is an extensively studied compound [21−25]. For the past 10 years, several ternary intermetallics with CaAl2Si2-type structure have been synthesized and studied broadly because of their promising
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transport, superconducting, thermoelectric and magnetic properties [26−32]. Recently, Pukas et al. synthesized MgAl2Ge2 and studied its crystal structure with X-ray powder diffraction [33]. Their study confirms that the new compound crystallizes in the trigonal space group
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31 with a CaAl2Ge2-type structure. The synthesized germanide is the third and last member in the Mg-Al-X (X = C, Si and Ge) systems with the CaAl2Si2-type structure. The
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carbide and silicide members in this sub-family have already been synthesized [34,35]. The present study also includes the carbide and silicide members to make a comparison for the newly synthesized germanide with them. On the basis of studied properties, we also justify the title compounds for using as thermal barrier coating (TBC) materials. TBC is a two-
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layered system made of a ceramic top-coat and an underlying metallic bond-coat. To be a top coat material, it should have relatively high coefficients of thermal expansion, high thermal shock resistance and low thermal conductivity. Conversely, good oxidation resistance, high
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thermal expansion coefficient, low thermal conductivity, slower growth rate, adherent to the
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thermally grown oxide (TGO), and stability, matching adequately with the substrate are required for a metallic bond-coat material. In this paper, the first-principle calculations are employed to explore the structural aspects, mechanical behaviors and electronic features of the newly synthesized intermetallic compound MgAl2Ge2 for the first time along with its silicide and carbide counterparts. The rest of the paper is arranged as follows: The method of calculations is described briefly in Section 2. Section 3 includes analysis of the results. The concluding remarks are presented in section 4.
ACCEPTED MANUSCRIPT 2. Methods of Calculations The first-principles calculations are carried out by means of pseudopotential plane-wave method as implemented in the CASTEP computer simulation program within Materials Studio [36]. The generalized gradient approximation introduced by Perdew-Burke-Ernzerhof
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known as GGA-PBE [37] is employed for evaluating the electronic exchange-correlation energy. GGA with PBE functional works for a wide range of systems and it is assumed as a universal approximation for evaluating electronic exchange-correlation potential. The
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advantage of this approximation is that it is free from parameters. It is better to use a nonhybrid functional GGA with good performance for metallic and low band gap materials as in
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our cases. The electron-ion interaction is treated with the Vanderbilt-type ultrasoft pseudopotential [38]. To search the ground state of the crystal system, the BFGS (BroydenFletcher-Goldferb-Shanno) minimization technique is applied [39]. The number of plane waves is kept limited in expansion by using an energy cutoff of 750 eV. Monkhorst-Pack
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(MP) grid size for self consistent field (SCF) calculation is chosen as 16×16×8 k-points mesh for sampling the Brillouin zone [40], which results in a high degree of convergence. The self consistent computations converged while the difference in the total energy of the crystal
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remains less than 5×10-6 eV/atom. In this process, the system geometry is fully relaxed until the ionic forces turn out to be less than 0.01 eV/Å, maximum ionic displacement less than
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5×10-4 Å and maximum stress less than 0.02 GPa. The computations are carried out allowing lattice constants, volume and atomic positions to be fully relaxed. The CASTEP code allows us to calculate the elastic properties of crystalline solids using the finite-strain method [41] based on density functional theory (DFT). In this method, a set of finite identical strains (deformations) of preset value is applied and then the resulting set of stress values is determined allowing the internal degrees of freedom. This method has become successful for calculating the elastic properties of different kinds of crystalline solids
ACCEPTED MANUSCRIPT [42−54]. The stress tensor σij under a set of applied strain δj provides the elastic constants Cij via the Equation, =
(1)
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The polycrystalline elastic properties are calculated by means of Voigt-Reuss-Hill approximations [55−57]. The bulk modulus BV and shear modulus GV in the Voigt approximation for the trigonal lattice are expressed as:
and = 1/30( + − 4 + 12 + 12 )
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= 1/9[(2( + ) + 4 + )]
(2)
(3)
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In the Reuss approximation, the bulk modulus BR and shear modulus GR are defined as: ! = [(2" + " ) + 2(" + 2" )]# and
! = 15[4(2" + " ) − 4(" + 2" ) + 3(2" + " )]#
(4)
(5)
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where, the elastic compliance constants " = # and C66 = (C11 – C12)/2. In the Hill approximation, the bulk modulus B and shear modulus G are given by B = (BV+ BR)/2
and
G = (GV + GR)/2
(6)
Y = 9BG/(3B + G)
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The Young’s modulus Y and Poisson’s ratio v are calculated using B and G via the Equations: and
v = (3B – 2G)/(6B + 2G)
(7)
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To quantify the elastic anisotropy of studied crystals, the anisotropic factors A, AB, AG, AU and kc/ka are calculated as follows: A = 4C44/(C11 + C33 – 2C13)
(8)
%& = [( − ! )/( + ! )] × 100%
(9)
%) = [( − ! )/( + ! )] × 100%
(10)
%* = 5( / ! ) + ( /! ) − 6 ≥ 0
(11)
-. /-/ = ( + − 2 )/( − )
(12)
ACCEPTED MANUSCRIPT The Debye temperature is calculated using the Anderson’s method [58], which is based on the average sound velocity, vm given as: 01 = [1/3(1/02 + 1/03 )#/
(13)
In which vl and vt are the longitudinal and transverse velocity of sound travelling through a
02 = [(3 + 4 )/34]/
03 = [ /4]/
and
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crystal and can be expressed with the bulk modulus B and shear modulus G as follows:
(14)
where ρ is the mass-density of the material. The average sound velocity then links to the
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Anderson’s method for calculating the Debye temperature through the Equation: 56 = ℎ/-& [389: 4/4;? = 740B@ − @ C(0D )#E/ ′
(16)
In this formalism, Pµ is the Mulliken population of a bond of type µ, @ is the metallic @
′
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population, and 0D is the volume of a µ-type bond. Again, the metallic population is the
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number of free electrons in a cell per unit volume and can be expressed as: MN
@ = (1/F)8GHII = (1/F) J 9(K)LK ′
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(17)
where, EP and EF are, respectively, the energy at the pseudogap and Fermi level. The volume @
of a µ-type bond 0D can be calculated in terms of the bond length L @ of type µ and the number of ν-type bonds 9DP per unit volume via the Equation: @
0D = (L @ ) / [(L Q ) 9DQ ] Q
(18)
ACCEPTED MANUSCRIPT The hardness of a crystal with complex multiband is a geometric average of all bond hardnesses and can be determined as [61,62]: µ
µ
H V = [Π( H vµ ) n ]1 / Σn
(19)
µ
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where nµ is the number of µ-type bonds of the multiband crystal. 3. Results and discussions 3.1. Structural properties
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The newly synthesized MgAl2Ge2 and its silicide and carbide counterparts MgAl2Si2, MgAl2C2 crystallize in the CaAl2Si2-prototype structure that belongs to the trigonal 31
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space group (No. 164). In the optimized structure shown in Figure 1, the Mg atoms reside on the one-fold 1a Wyckoff position with the fractional coordinates (0, 0, 0) and correspond to the Ca atoms in the structure of CaAl2Si2. The Al and Ge atoms orderly occupy two two-fold sites in 2d Wyckoff position with the fractional coordinates (1/3, 2/3 0.6303) and (1/3, 2/3,
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0.2404), respectively. The optimized lattice constants a and c, axial ratio c/a, internal parameter zX and unit cell volume V of the newly synthesized MgAl2Ge2 are tabulated in Table 1 along with those found in present calculations and in the experiments for the
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isostructural and isoelectronic silicide and carbide counterparts [34,35]. The calculated structural parameters of three CaAl2Si2-prototype compounds are found to be consistent with
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their corresponding experimental values. From experimental results, the present values deviate not more than 0.97, 0.78, 1.69, 2.34, and 1.41% for a, c, c/a, zX, and V, respectively, indicating that the present study is reliable. From the Figure 2, it is observed that a gradual increase in lattice constants as well as in unit cell volume occurs in the Mg-Al-X (X = Ge, Si, C) systems with the CaAl2Si2 structure following the sequence of C→Si→Ge, which also follows the order of atomic radius of C, Si, and Ge. The axial ratio c/a follows the reverse of this trend.
ACCEPTED MANUSCRIPT 3.2. Mechanical properties We have calculated the single-crystal elastic tensors Cij to provide the essential information on the mechanical stability and the stiffness of the newly synthesized MgAl2Ge2 and its carbide and silicide counterparts MgAl2C2 and MgAl2Si2. The six independent elastic
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constants (C11, C12, C13, C14, C33, C44) for the three studied trigonal crystals have been investigated. The calculated elastic properties are listed in Table 2 and shown in Figure 3. The mechanical stability of the trigonal crystals depends on the fulfillment of the conditions
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derived from the independent elastic constants: C11 > 0; C33 > 0; C44 > 0; (C11 + C12) C33 > 2C13C13; (C11 - C12)C44 > 2C14C14 [63]. The newly synthesized MgAl2Ge2 and its isostructural
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carbide and silicide counterparts, MgAl2C2 and MgAl2Si2, are mechanically stable due to obeying the above conditions though C14 has negative values for MgAl2Ge2 and MgAl2Si2. It should be remembered that C14 is not required to be positive for mechanical stability. Natural quartz and corundum crystallize in trigonal structure like the title compounds and also
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possess negative values for elastic constant C14 [64-70].
It is observed that the unidirectional elastic constants C11 and C33 are greater than the pure shear elastic constant C44, suggesting that the shear deformation for the three
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isostructural MgAl2Ge2, MgAl2Si2 and MgAl2C2 should be easier than the linear compression in the crystallographic a- and c-directions with an order of MgAl2C2 > MgAl2Si2 >
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MgAl2Ge2. Additionally, these three intermetallics MgAl2C2, MgAl2Si2 and MgAl2Ge2 are estimated to be more incompressible along the a-axis compared with that along the c-axis as their elastic tensor C11 is greater than C33. It means that the atomic bonding between nearest neighbors in the (100) planes is stronger than that in the (001) planes. The rank of incompressibility along the a-axis for the three studied compounds are MgAl2C2 > MgAl2Si2 > MgAl2Ge2. The difference between C11 and C33 also indicates a significant anisotropy in the elastic properties of the compounds. The differences between these two elastic constants
ACCEPTED MANUSCRIPT are 26, 31 and 109 GPa for MgAl2Ge2, MgAl2Si2 and MgAl2C2, respectively, indicating that the newly synthesized MgAl2Ge2 is comparatively elastically less anisotropic. Due to the combination of C12 and C13, a functional stress component in the crystallographic a-direction exists, while uniaxial strains along the crystallographic b- and c-axes are present. The small
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values of these elastic tensors suggest that the intermetallic compounds MgAl2Ge2, MgAl2Si2 and MgAl2C2 should be able to accept shear deformation along the crystallographic b- and caxes, when a sufficient stress is applied to the crystallographic a-axis.
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The difference (C12 – C44) is known as the Cauchy pressure, which is frequently used to assess the nature of chemical bonding in a solid crystal [71]. Its negative value corresponds
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to the directional covalent bonding with angular character, whereas its positive value is a sign of metallic bonding. The newly synthesized intermetallic MgAl2Ge2 and its silicide counterpart MgAl2Si2 are characterized by a positive Cauchy pressure and therefore, their chemical bonding is dominated by metallic bonding, whereas the carbide phase has a
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negative Cauchy pressure with a small value, indicating its weak directional covalent bonding. The Cauchy pressure can also be used to predict the failure mode (ductile/brittle nature) of solids. A negative Cauchy pressure is associated with brittle materials, whereas a
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positive Cauchy pressure is characteristic of ductile materials. Therefore, the ternary
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germanides and its silicide counterpart can be classified as ductile materials, while the carbide is brittle in nature.
The elastic constant C14 is negative with small values for the newly synthesized MgAl2Ge2 and its silicide counterpart MgAl2Si2, while C14 is positive with a large value for the isostructural carbide phase. The small C14 leads to a symmetry increase to hexagonal (transverse isotropic) and therefore, the trigonal MgAl2Ge2 and MgAl2Si2 seem to be hexagonal systems.
ACCEPTED MANUSCRIPT Mechanical behaviors of crystals can be judged with the bulk and shear modulus. The bulk modulus B assesses the crystals’ ability to resist volume deformation. Equally, the shear modulus G predicts the ability to improve the resistance to shape change i.e., plastic deformation. The low values of B and G (see Table 3) suggest that the CaAl2Si2-prototype
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compounds MgAl2Ge2 and its carbide and silicide counterparts MgAl2C2 and MgAl2Si2 should be soft materials that can be easily mechinable. Their hardness values are expected to be low as well.
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From the ratio of B to G taken as an essential tool, Pugh [72] has successfully explained the failure mode of solids. He predicts that brittle failure is associated with a solid for which
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the B/G ratio has a value less than 1.75 and that ductile failure happens for a material that possesses a value greater than 1.75. The newly synthesized MgAl2Ge2 and its silicide counterpart MgAl2Si2 should behave as ductile materials due to having a B/G value greater than 1.75, whereas the carbide phase should exhibit brittleness.
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The Young’s modulus is calculated from B and G via the Equation 7. This elastic parameter assesses the stiffness as well as influences on the thermal shock resistance of the materials. The low Young’s modulus (see Table 3) implies that the recently synthesized MgAl2Ge2 and
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its silicide counterpart MgAl2Si2 are not stiff enough materials compared with the carbide
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phase. In fact, MgAl2Ge2 and MgAl2Si2 are softer materials than their carbide counterpart MgAl2C2. The critical thermal shock resistance R varies inversely with the Young’s modulus Y [73]. Therefore, the low Y-value gives a high R-value and the materials MgAl2Ge2 and MgAl2Si2 are expected to show better thermal shock resistance. Based on high thermal shock resistance a material can be selected as thermal barrier coating (TBC) substance. Therefore, MgAl2Ge2 and its silicide counterpart MgAl2Si2 can be used as TBC materials. The Poisson’s ratio can be used to assess the mechanical behaviors of crystals. The Poisson’s ratio can predict the crystal stability against shear [74]. A crystalline solid achieves
ACCEPTED MANUSCRIPT structural stability against shear if it is characterized by a low Poisson’s ratio. The new compound MgAl2Ge2 and its carbide and silicide counterparts MgAl2C2 and MgAl2Si2 are expected to be stable against shear due to their low Poisson’s ratios. The Poisson’s ratio also serves as an essential tool to evaluate the nature of forces among atoms in a crystalline solid
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[75]. A material whose Poisson’s ratio lies between 0.25 and 0.50 is expected to be stable due to central forces among its constituent atoms. The Poisson’s ratio lies outside of this range when a crystal system is stabilized with the non-central forces. For the newly
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synthesized compound and its silicide counterpart, the central forces should be responsible for their structural stability, while the carbide phase is stabilized due to non-central forces.
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The Poisson’s ratio is another essential tool in engineering science to predict the failure mode (ductile/brittle nature) of solids with a critical value of v = 0.26 [76,77]. A crystal with Poisson’s ratio v > 0.26 is observed to show ductile nature and with v < 0.26 is characterized as a brittle material. In the case of MgAl2Ge2 and MgAl2Si2, the v-values are 0.277 and
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0.275, which correspond to their ductile nature, whereas the v-value of MgAl2C2 equal to 0.218 implies that it is a brittle material. The Poisson’s ratio also plays a crucial role to assess the nature of chemical bonding in crystals [78]. A Poisson’s ratio of v = 0.1 indicates an ideal
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covalent crystal and a value of v = 0.33 corresponds to a purely metallic compound. The
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Poisson’s ratios of three intermetallics lie within these two characteristic values, indicating a combination of covalent and metallic nature in the chemical bonding of MgAl2Ge2, MgAl2Si2 and MgAl2C2.
The elastic anisotropy has significant implications in engineering science due to correlation with the possibility to introduce microcracks in the crystals. The calculated elastic anisotropy factors are listed in Table 4 and shown in Figure 4. The shear anisotropy factor A defined with Equation 8 for the {100} shear planes between the 〈011〉 and 〈010〉 directions is calculated to shed light on elastic anisotropy of three intermetallics MgAl2Ge2, MgAl2Si2 and
ACCEPTED MANUSCRIPT MgAl2C2. The unit value of this factor corresponds to a complete isotropic nature in elastic properties and other than the unit value signifies anisotropy in elastic behaviors. The large deviation of A from unit value indicates significant shear elastic anisotropy in these compounds and their degrees of anisotropy are almost similar due to similar A-values. The
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percentage anisotropy in compression and shear is also calculated using Equations 9 and 10. The obtained values indicate that the newly synthesized MgAl2Ge2 is less anisotropic in both compression and shear compared with its silicide and carbide counterparts. The so-called
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universal anisotropy index AU is calculated in accordance with the Equation 11 to be either zero or a positive value. The zero value represents completely isotropic nature and positive
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values indicate a level of anisotropy possessed by the crystals. The level of elastic anisotropy increases following the sequence of X-elements Ge → Si → C (see Figure 4). Another anisotropy factor kc/ka is commonly used to quantify the elastic anisotropy with ka and kc being the linear compressibility coefficients along the a- and c-axes, respectively. It is
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calculated using Equation 12 and found that all values of kc/ka are greater than unity, indicating higher compressibility along the c-direction than that along the a-direction for all
c-axis.
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compounds. Among the three intermetallics, the silicide phase is more compressible along the
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3.3. Elastic Debye temperature
The Debye temperature is a distinctive temperature of solids that makes feasible the estimation of many physical properties of crystalline solids such as the thermal expansion and conductivity, lattice vibration, specific heat and melting temperature. In addition, it is related to the electron-phonon coupling constant and superconducting transition temperature in superconducting materials. Moreover, the energy of vacancy formation in metals generally depends on their Debye temperatures. In the present study, the Debye temperatures are calculated using the bulk and shear moduli via the elastic wave velocities applying in
ACCEPTED MANUSCRIPT Equations 13 – 15. The calculated Debye temperatures and elastic (sound) wave velocities are listed in Table 5 and shown in Figure 5. The average sound velocity follows the trend of transverse sound velocity, while the longitudinal sound velocity shows the reverse trend. The Debye temperature decreases as the X-element moves from C to Ge across the periodic table.
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The Debye temperature of newly synthesized MgAl2Ge2 and its silicide counterpart MgAl2Si2 is small compared with that of MgAl2C2. As a general rule, a low Debye temperature is a sign of low phonon thermal conductivity. Therefore, MgAl2Ge2 as well as MgAl2Si2 is thermally
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low conductive and as a result these should be promising TBC materials. In fact, a conventional TBC is composed of a ceramic top coat, a metallic bond coat, and a thermally
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grown oxide (TGO) layer that forms due to oxidation of the bond coat during the service. A dense and continuous alumina layer (Al2O3) is favorable for TGO layer as it has a low growth rate [79]. The studied compounds MgAl2Ge2 and MgAl2Si2 can be used as metallic bond coat due to their expected behaviour of low thermal conductivity and high thermal shock
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resistance (see section 3.2) along with partial metallic conductivity. In addition, as a result of oxidation an Al2O3-layer is likely to form which acts a shielding layer [80] and prevents further oxidation at higher temperature (as in the case of Ti3AlC2 in air at 1300 K [81]).
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Further, the isostructural compounds CaAl2Si2 (B = 58.9 GPa, G = 40.4 GPa, Y = 98.6 GPa)
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and SrAl2Si2 (B = 58.9 GPa, G = 37.8 GPa, Y = 93.4 GPa) have physical properties similar to MgAl2Ge2 and MgAl2Si2 and also have high melting points, 1248 and 1293 K, respectively [82,83]. So, we can expect such melting temperatures for the compounds under study with the consequence that these would possess high oxidation resistance [80]. 3.4. Electronic structures The electronic band structures of three CaAl2Si2-prototype crystals are investigated along the high symmetry directions in the first Brillouin zone and shown in Figure 6. The bands profile of the carbide phase MgAl2C2 is quite different from those of the other two phases MgAl2Ge2
ACCEPTED MANUSCRIPT and MgAl2Si2, which are almost identical to each other. In these two phases, the Fermi surfaces lie just below the valence band (colored in cyan) with a maximum near the Γ point and some valence bands cross (colored in blue) the Fermi level and overlap with the conduction bands (colored in green). Consequently, no band gap appears at the Fermi level
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and the compounds should exhibit metallic conductivity. Conversely, in the carbide phase, the Fermi surface lies just over the valence bands maximum around the Γ point and a few valence bands touch the Fermi level and only one of them just crosses this level. A band gap
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of width 1.7 eV appears above the Fermi level. Due to these reasons, the dual characters of weak-metal and semiconductor are expected for MgAl2C2.
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The total and partial densities of states calculated for MgAl2Ge2, MgAl2Si2, and MgAl2C2 are shown in Figure 7. The DOS for the germanide and silicide phases are very similar and the finite values of DOS at the Fermi level, EF indicate the metallic nature of these two intermetallics. The DOSs at EF for these two compounds are also the same with a value of
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~1.0 states/eV/unit cell. Both s- and p-states of all atoms contribute almost equally to the DOS at the Fermi energy. The main difference in DOS of the carbide phase from those of the germanide and silicide phases is that there is a band gap of width 1.7 eV slightly above the
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Fermi level and no valence bands overlap with the conduction bands as well as the DOS at EF
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is only 0.4 states/eV/unit cell. As a result, the carbide phase is expected to show dual character of weak-metal and semiconductor. The lowest valence bands in all the compounds are mainly composed of s-electrons of X-elements. Only ~18% contributions come from Als+p states. The contributions in the lowest valence bands from Mg-atoms are 8.2, 2.3, and 1.1% for germanide, silicide, and carbide phases, respectively. The lowest valence bands indicate the strong covalent Al-X bonds in three intermetallics. The lowest valence bands in germanide and silicide phases are situated within around the same energy ranges, but in the carbide phase it is positioned at a deeper energy range, indicating that the covalent Al-Ge and
ACCEPTED MANUSCRIPT Al-Si bonds have almost same strength and the Al-C bond is strongest among all Al-X bonds. The higher valence bands in the three studied compounds are separated by a band gap. The width of the band gap is large in the carbide phase and smaller in the silicide phase. The higher valence bands are mainly attributed to the p-states of X-elements and s+p-states of Al-
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atoms. Some contributions also come from Mg-s+p states. The bonding peaks situated around -5.7 eV in MgAl2Ge2 and MgAl2Si2 and at -5.1 eV in MgAl2C2 signify the strong hybridization of Al-s and X-p states. The bonding peak, at about -1.32 eV, in germanide and
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silicide phase and at -1.1 eV in carbide phase correspond to the hybridization between Mg-p and X-p states. The Fermi levels of MgAl2Ge2 and MgAl2Si2 are situated at a dip in their total
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DOS but it is placed on the slope of a peak of total DOS for MgAl2C2, indicating that the germanide and silicide phases are more stable structurally compared with the carbide phase. 3.5. Mulliken atomic population analysis
Mulliken atomic population analysis gives a deep understanding on bonding characteristics in
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crystals. This scheme allocates charge to wave functions presentable in linear combination of atomic orbitals (LCAO) basis sets. Sanchez-Portal et al. [84] introduce a technique to apply this scheme with a projection of the plane wave states onto a localized basis for
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implementation to the CASTEP module. The analysis of atomic populations with resultant
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projected states is subsequently performed using the Mulliken formalism [85].This scheme offers the bond overlap populations and the Mulliken charges, which lead to the presumption of ionicity/covalency of a chemical bond and the calculation of effective valence. The effective valence is a difference between the formal ionic charge and the Mulliken charge assigned to an anion species. Effective valence can be used to determine the strength of a chemical bond with the level of ionicity or covalency. Pure ionic bond is expected to exist while the effective valence has a zero value. Conversely, a covalent bond appears following a non-zero effective valence. Positive value of effective valence corresponds to the level of
ACCEPTED MANUSCRIPT covalency of a chemical bond. Table 6 lists the calculated effective valence for atoms in the three intermetallic compounds of CaAl2Si2-proptotype structures. From which, we may conclude that the atomic binding in the three compounds considered here are significantly covalent. The charge transfer from atom to atom can be predicted by Mulliken atomic
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populations. The charge transfer to Ge, Si and C from Mg and/or Al is 0.56, 0.81 and 1.42e in MgAl2Ge2, MgAl2Si2 and MgAl2C2, respectively. The bond overlap population is an essential tool for predicting the character of atomic bonding in crystalline solids. An overlap
SC
population with a value close to zero is an indication of a weak interaction of electronic populations between two atoms. The chemical bonding between such two atoms plays no role
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in materials’ hardness. Small overlap population is associated with an ionic bonding. Conversely, high value of overlap population indicates the increasing level of covalency of a chemical bond. Bonding and antibonding states in a chemical bond arise due to positive and negative overlap populations, respectively. Table 7 lists the bond overlap populations
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calculated for MgAl2Ge2, MgAl2Si2 and MgAl2C2.
The covalent first Al-Ge bond in MgAl2Ge2 is expected to be as strong as the similar Al-Si bond in MgAl2Si2 but weaker than the Al-C bond in MgAl2C2. The second Al-Ge bond in
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MgAl2Ge2 appears with a large negative population, while the equivalent Al-Si and Al-C
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bonds in MgAl2Si2 and MgAl2C2 arise with small populations of 0.06 and 0.03, respectively. The Mg-Ge bond in MgAl2Ge2 has a small negative population, but the analogous Mg-Si bond in MgAl2Si2 carries a positive value and Mg-C bond in MgAl2C2 possesses a large negative value. The Al-Al bonds in the three intermetallics have positive populations with different values, indicating different levels of covalency. 3.6. Theoretical hardness Both the intrinsic and extrinsic properties control the hardness of a material. Bond strength, cohesive energy and crystal structure are the main intrinsic properties of solids, which
ACCEPTED MANUSCRIPT influence on the materials’ hardness. Equally, the extrinsic properties such as defects, stress fields and morphology manipulate the hardness of a crystal. The hardness value obtained in experiment is observed to be different for different methods, temperature, etc. In the same way, the theoretical values depend on the formalism designed for the calculations. Moreover,
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the theory modeled with a single crystal cannot predict accurately the hardness of a crystalline solid. Theoretically modeled single crystals in most cases exclude defects, impurities, fracture, slip systems which accommodate dislocation motion during indentation,
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etc. The formalism used in this study is described in Section 2 with Equation 16 – 19 and calculated theoretical Vickers hardness with relevant quantities is listed in Table 8. The
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values of theoretical Vickers hardness for MgAl2Ge2, MgAl2Si2, and MgAl2C2 are 3.3, 2.7, and 7.7 GPa, respectively, indicating that these three intermetallic compounds should be soft and easily machinable in their ideal crystal states. The rank of softness and machinability are
4. Conclusions
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expected to be MgAl2Si2 > MgAl2Ge2 > MgAl2C2.
In summary, using first-principle methods, we have investigated the structural properties,
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mechanical behaviors and electronic features of three CaAl2Si2-prototype intermetallics MgAl2Ge2, MgAl2Si2 and MgAl2C2. The calculated lattice parameters show fair agreement
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with the measured values. The mechanical stability conditions for MgAl2Ge2 together with MgAl2Si2 and MgAl2C2 are fulfilled. For three intermetallic compounds the order of incompressibility along a-axis are MgAl2C2 > MgAl2Si2 > MgAl2Ge2. The silicide phase is more compressible along c-axis compared with germanide and carbide phase. MgAl2Ge2 and MgAl2Si2 are ductile and damage tolerant, while MgAl2C2 exhibits brittle nature. MgAl2Ge2 and MgAl2Si2 can be used as metallic bond coating thermal barrier materials. Newly synthesized MgAl2Ge2 and its silicide counterpart achieve their structural stability due to the central forces, whereas the carbide phase is stabilized by non-central forces. MgAl2Ge2 has
ACCEPTED MANUSCRIPT the lowest Debye temperature among the three intermetallic compounds. The rank of elastic anisotropy of these three intermetallics follows the order of X-elements as Ge → Si → C. Metallic conductivity is expected for MgAl2Ge2 and MgAl2Si2, while MgAl2C2 exhibits the dual characters of weak-metal and semiconductor. The theoretical hardness for MgAl2Ge2,
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MgAl2Si2, and MgAl2C2 are found to be 3.3, 2.7, and 7.7 GPa, respectively, signifying that all these three CaAl2Si2-prototype intermetallics are soft and easily machinable. References
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ACCEPTED MANUSCRIPT Mg
Al
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Ge
4.8
6.0 5.5 5.0
4.0
4.5
3.6
4.0
3.2
C
Si X-elements
Ge
1.80 1.75
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4.4
1.85
3.5
Calc. c/a Calc. V Expt. c/a Expt. V
90 75 60 45
1.70
30
1.65
15
1.60
C
3
1.90
Calc. a Calc. c Expt. a Expt. c
5.2
105
Si X-elements
Ge
Unit cell volume, V (Å )
1.95
6.5
SC
7.0
5.6
Hexagonal ratio, c/a
6.0
Lattice constant, c (Å)
Lattice constant, a (Å)
Figure 1. Crystal structure of MgAl2Ge2 as a structural model of CaAl2Si2-prototype compounds.
0
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C11 C12 C13 C14 C44
275 200 125 50 -25
C
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Elastic constants, Cij (GPa)
350
Elastic moduli, B, G, Y (GPa)
Figure 2. Structural properties of MgAl2C2, MgAl2Si2, and MgAl2Ge2 as a function of X-element.
Ge
Si X-elements
250
B G Y
200 150 100 50 0
C
Ge
Si X-elements
1.6 1.4
0.27
B/G v
1.2 1.0
0.29
C
Si X-elements
Ge
0.25 0.23 0.21
6
A AB AG AU kc/ka
5 4
15 10 5
3
0
2
-5
1
-10
0
C
Si X-elements
Ge
-15
Figure 4. Pugh’s and Poisson’s ratio and elastic anisotropy factors as a function of X-elements.
Anisotropy factors AG and AU
1.8
0.31
Poisson's ratio, v Anisotrpy factors A, AB and kc/ka
Pugh's ratio, B/G
2.0
AC C
Figure 3. Elastic properties of MgAl2C2, MgAl2Si2, and MgAl2Ge2 as a function of X-element.
12
800 vl vt vm
10
640
θD
8
480 320
4
160
2
C
Ge
Si X-elements
0
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6
Debye temperature, θD (K)
Sound velocities, vl , vt , vm (km/s)
ACCEPTED MANUSCRIPT
Figure 5. Sound velocities, Debye temperature and melting temperature as a function of X-elements. 10
SC
8
4 2 0
-2 -4 -6 MgAl2Ge2
-8 -10
Γ A
H
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Energy, E - EF (eV)
6
MgAl2Si2
Γ
K
M
L
H Γ A
High symmetry points
H
Γ
K
M
L
EF
MgAl2C2
H Γ A
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High symmetry points
H
Γ
K
M
L
H
High symmetry points
Figure 6. Electronic band structures of MgAl2Ge2, MgAl2Si2, and MgAl2C2.
MgAl2Ge2
2
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EF
3 1 0 0.6 0.4
EF
MgAl2Si2
EF
MgAl2C2
Mg-3s Mg-2p
Mg-3s Mg-2p
Mg-3s Mg-2p
Al-3s Al-3p
Al-3s Al-3p
Al-3s Al-3p
Ge-4s Ge-4p
Si-3s Si-3p
C-2s C-2p
AC C
Electronic densities of states (States/eV/Unit cell)
4
0.2 0.0 1.2 0.8 0.4 0.0 1.5 1.0 0.5 0.0
-12 -10
-8
-6
-4
-2
0
Energy, E - EF (eV)
2
4 -12 -10
-8
-6
-4
-2
0
Energy, E - EF (eV)
2
4 -12 -10
-8
-6
-4
-2
0
2
Energy, E - EF (eV)
Figure 7. Electronic total and partial densities of states of MgAl2Ge2, MgAl2Si2, and MgAl2C2.
4
ACCEPTED MANUSCRIPT Table 1. The structural properties of MgAl2Ge2 along with its isostructural silicide and carbide counterparts. Compounds
a (Å)
c (Å)
c/a
zX
V(Å3)
Remarks
MgAl2Ge2
4.130
6.839
1.656
0.2404
101.0
This calc.
4.117
6.787
1.649
0.2419
99.6
Expt. [33]
MgAl2C2
0.316
0.766
0.424
0.6201
1.4
Dev. (%)
4.089
6.688
1.636
0.2436
96.8
This calc.
4.050
6.740
1.664
0.2463
95.7
Expt. [34]
0.963
0.772
1.683
1.0962
1.1
Dev. (%)
3.390
5.849
1.725
0.2716
58.2
This calc.
3.377
5.817
1.723
0.2654
57.5
Expt. [35]
0.385
0.550
0.116
2.3361
1.2
Dev. (%)
RI PT
MgAl2Si2
C11 113 127 325
C12 51 57 79
C13 28 30 46
C14 -2 -4 50
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Compounds MgAl2Ge2 Mg2Al2Si2 Mg2Al2C2
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Table 2. The calculated elastic constants (Cij in GPa) of MgAl2Ge2, MgAl2C2 and MgAl2Si2. C33 87 96 216
C44 26 29 81
Table 3. The calculated polycrystalline bulk (BV, BR, B in GPa), shear (GV, GR, G in GPa), and Young’s (Y in GPa) modulus and Pugh’s (G/B), Poisson’s (v) ratio of MgAl2Ge2 and its counterparts MgAl2C2 and MgAl2Si2. BV 58.3 65.1 134.2
BR 55.9 62.2 127.3
B 57.1 63.6 130.7
GV 30.3 34.3 103.6
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Compounds MgAl2Ge2 Mg2Al2Si2 Mg2Al2C2
GR 29.5 33.0 78.3
G 29.9 33.6 90.9
Y 76.4 85.7 221.4
B/G 1.91 1.89 1.44
v 0.277 0.275 0.218
Table 4. Elastic anisotropy factors A, AB, AG, AU and kc/ka of MgAl2C2, MgAl2Si2, and MgAl2Ge2. AB (%) 2.102 2.278 2.639
EP
A 0.722 0.712 0.722
AC C
Compounds MgAl2Ge2 Mg2Al2Si2 Mg2Al2C2
AU 0.179 0.244 1.670
AG (%) 1.338 1.932 13.909
kc/ka 1.831 1.879 1.835
Table 5. Calculated density (ρ in gm/cm3), longitudinal, transverse, and average sound velocities (vl, vt, and vm in km/s), and Debye temperature (θD in K). Compounds
ρ
vl
vt
vm
θD
MgAlGe2 MgAl2Si2 MgAl2C2
3.675 2.306 2.918
5.137 6.856 6.799
2.852 3.817 5.581
3.177 4.250 5.889
347 471 774
ACCEPTED MANUSCRIPT Table 6. Calculated Mulliken atomic populations, Mulliken charge and effective valence for MgAl2Ge2, MgAl2Si2 and MgAl2C2.
MgAl2Si2
MgAl2C2
d 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
Total 7.19 2.84 4.56 6.96 2.71 4.81 6.61 2.28 5.42
Charge (e) -0.81 -0.16 -0.56 -1.04 -0.29 -0.81 -1.39 -0.72 -1.42
Effective valence (e) 1.19 2.84 ---0.96 2.71 ---0.61 2.28 ----
RI PT
MgAl2Ge2
Mulliken atomic populations Species s p Mg 0.68 6.52 Al 1.12 1.72 Ge 1.31 3.25 Mg 0.54 6.41 Al 1.03 1.68 Si 1.53 3.28 Mg 0.38 6.22 Al 0.78 1.50 C 1.54 3.87
SC
Compounds
Table 7. Calculated Mulliken bond number nµ, bond length dµ, and bond overlap population Pµof µ-type bond for MgAl2Ge2, MgAl2Si2, and MgAl2C2. µ
d (Å) 2.55162 2.66857 2.88692 2.97264
µ
P -2.15 -0.62 -0.07 -0.74
MgAl2Si2 Bond nµ Al-Si 2 Al-Si 2 Mg-Si 2 Al-Al 1
MgAl2C2 Bond nµ Al-C 2 Al-C 2 Mg-C 2 Al-Al 1
M AN U
MgAl2Ge2 Bond nµ Al-Ge 2 Al-Ge 2 Mg-Ge 2 Al-Al 1
µ
d (Å) 2.49765 2.61376 2.86819 2.96711
µ
P 2.15 0.06 0.10 0.31
dµ (Å) 2.01060 2.21166 2.52103 2.62675
Pµ -2.42 -0.03 -0.42 -0.07
Bond
MgAl2Ge2
Al-Ge Al-Al Al-Si Al-Al Al-C Al-Al
MgAl2Si2
AC C
MgAl2C2
nµ 2 1 2 1 2 1
dµ (Å) 2.55162 2.97264 2.49765 2.96711 2.01060 2.62675
EP
Compounds
TE D
Table 8.Calculated Vickers hardness Hv of MgAl2Ge2, MgAl2Si2, and MgAl2C2 with the relevant quantities such as metallic population Pµ′, bond volume vbµ and bond hardness H vµ . Pµ 2.15 0.74 2.15 0.31 2.42 0.07
Pµ′ 0.00246 0.00246 0.00137 0.00137 0.00610 0.00610
3 vbµ (Å )
28.2 44.6 26.3 44.1 13.8 10.7
H vµ
(GPa) Hv (GPa)
6.08 0.97 6.84 0.41 22.50 0.91
3.3 2.7 7.7
ACCEPTED MANUSCRIPT Highlights: 1. The structural, mechanical and electronic properties of three intermetallics are calculated. 2. MgAl2Ge2 and MgAl2Si2 are ductile and damage tolerant, while MgAl2C2 is a brittle one.. 3. MgAl2Ge2 and MgAl2Si2 can be used as thermal barrier coating (TBC) materials.
AC C
EP
TE D
M AN U
SC
RI PT
4. The studied three intermetallics are soft and easily machinable.
