Anisotropic Elastic and Thermal Properties of M₂InX (M = Ti, Zr and X = C, N) Phases: A First-Principles Calculation

Round 1

Reviewer 1 Report

The authors performed first-principles calculations for (Ti/Zr)2In(C/N) to evaluate the elastic and thermal properties. The results contain interesting results which merit publication in crystals. However, there are several problems for publication as listed below.

The formation enthalpy, defined by Eq. (2), was calculated with reference to the energy of Ebulk. Ebulks are described as the steady-state energies of solitary M, In, and X atoms. The formation enthalpy strongly depends on the choice of the phase of the elements. The expression of “the steady state” is ambiguous. Please describe the phase of each element used in Eq. (2).

The authors list the lattice parameters, cohesive energy and formation enthalpy of the present work with the values of the previous works in Table 1. However, there are many mistakes in Refs. In the column of Refs., the label of “Exp.” is used for all references listed in Table 1. The authors do not mention about the label of “Exp.”. In most cases, the label of “Exp.” is used for experimental values. However, as far as I have checked, the values in Ref. [12] and [23] are not experimental but calculated results. In addition, the lattice parameters of Ref. [12] are referenced from other papers. Furthermore, I cannot find the formation enthalpy of -0.669 in Ref. [22] and -1.158 in Ref. [23]. The authors should carefully check all references in Table 1.

In the Line 127-128, the authors state that “This is consistent with Ti2lnC having the strongest chemical bond and Ti2lnN having the weakest chemical bond [13,22]”. In Ref. [13], the superconductivity in Ti2lnN was reported. In Ref. [22], the electrical and thermal properties of Ti2lnC was reported. I have checked Ref. [13] and [22], but I cannot find any statements about the strength of the chemical bonding in these references. In Ref. [13], it is stated that the chemical bond between Ti-N is more polarized than Ti-C. However, this statement does not relate to the strength of the chemical bonding. Please describe the reason why the Ti2lnC and Ti2lnN has the strongest and the weakest chemical bonding, respectively, in more detail.

Author Response

Response to Reviewer 1 Comments

metals-1785495

<Metals>

<Anisotropic elastic and thermal properties of M₂InX (M=Ti, Zr and X = C, N) phases: A first-principles calculation>

Reviewer comments:

Reviewer #1: The authors performed first-principles calculations for (Ti/Zr)₂In(C/N) to evaluate the elastic and thermal properties. The results contain interesting results which merit publication in crystals. However, there are several problems for publication as listed below.

Point 1: The formation enthalpy, defined by Eq. (2), was calculated with reference to the energy of E_bulk. E_bulks are described as the steady-state energies of solitary M, In, and X atoms. The formation enthalpy strongly depends on the choice of the phase of the elements. The expression of “the steady state” is ambiguous. Please describe the phase of each element used in Eq. (2).

Response: Thanks for your professional advice. Statement on formation enthalpy of in the manuscript. Here, I express my sincerest apologies to the reviewers for misinterpreting what I meant by using the wrong grammar. Secondly, I have reworked the subsection formation enthalpy in the manuscript with grammatical corrections. The expression "the steady state" is wrong and should be "stable state", and the expression "solitary" is also wrong and should be changed to "isolated". The specific modifications are as follows:

Here, E is the total energy of M₂lnX. E_bulk(M)、E_bulk(ln) and E_bulk(X) represent the energies of a single M、ln and X atoms in a stable state, respectively. E_iso(M)、E_iso(ln) and E_iso(X) are the energies of isolated M、ln and X atoms, respectively.

Point 2: The authors list the lattice parameters, cohesive energy and formation enthalpy of the present work with the values of the previous works in Table 1. However, there are many mistakes in Refs. In the column of Refs., the label of “Exp.” is used for all references listed in Table 1. The authors do not mention about the label of “Exp.”. In most cases, the label of “Exp.” is used for experimental values. However, as far as I have checked, the values in Ref. [12] and [23] are not experimental but calculated results. In addition, the lattice parameters of Ref. [12] are referenced from other papers. Furthermore, I cannot find the formation enthalpy of -0.669 in Ref. [22] and -1.158 in Ref. [23]. The authors should carefully check all references in Table 1.

Response: Thanks for your professional advice. First of all, I express my sincere apologies for my mistakes. Second, I have rechecked and corrected the references in Table 1 and revised the data in Table 1. Thanks again for your suggestion.

Point 3: In the Line 127-128, the authors state that “This is consistent with Ti2lnC having the strongest chemical bond and Ti2lnN having the weakest chemical bond [13,22]”. In Ref. [13], the superconductivity in Ti2lnN was reported. In Ref. [22], the electrical and thermal properties of Ti2lnC was reported. I have checked Ref. [13] and [22], but I cannot find any statements about the strength of the chemical bonding in these references. In Ref. [13], it is stated that the chemical bond between Ti-N is more polarized than Ti-C. However, this statement does not relate to the strength of the chemical bonding. Please describe the reason why the Ti2lnC and Ti2lnN has the strongest and the weakest chemical bonding, respectively, in more detail.

Response: Thanks for your professional advice. Regarding your question "describe in detail why Ti₂lnC and Ti₂lnN have the strongest and weakest chemical bonds, respectively", we will explain the strong chemical bonds by analyzing the electronic properties of the compounds (electron density of states and population analysis) ties.

The electronic properties (density of states) of M₂InX (M = Ti, Zr and X=C, N) MAX phases were studied to better understand chemical bonding and bond behaviors. Fig.11 depicts the M₂InX phases' total density of states (TDOS) and partial density of states (PDOS). As a result, Fig.11 provides the following information. To begin, it is clear that DOS has a significant finite value at the Fermi level, indicating that these compounds exhibit metallic conductivity. Fig.11 shows that the total density of states (E_f) value of Ti₂InN and Zr₂lnN are greater than that of Ti₂InC and Zr₂lnC, indicating that Ti₂InN and Zr₂lnN are more conductive than Ti₂InC and Zr₂lnC. Secondly, the peak topologies and relative heights of the peaks around E_f in the TDOS plot are highly comparable, indicating the presence of similar chemical bonds in Zr₂AN. The time difference around the Fermi level is mostly made up of ln-p and M-d states. The time difference below the Fermi energy is caused mostly by the X-s, M-s, and M-p states, whereas the time difference above the Fermi energy is caused primarily by the ln-s and X-p states. As shown in Fig.11, the ln-p, M-d, and X-p states exhibit substantial hybridization, allowing M-C and M-N chemical bonds to form, resulting in the high elastic modulus of M₂InX. PDOS exhibits multiple hybridizations of the electronic states M, ln, and X. The valence band of M₂InX in Fig.11 displays substantial hybridization of the M-d and X-p states, as predicted for covalent compounds. The d–p hybrid state corresponding to the M-ln bond is discovered to be in a greater energy range than the M-X bond. As a result, the M-X d-p hybridization helps to maintain the crystal structure. Finally, it is demonstrated that M's electronic charge density almost overlaps that of ln, indicating that the bonding between M and ln is quite weak. These findings are consistent with the observation that the biggest phase features very strong M-X bonds and very weak M-ln bonds.

As can be seen from Table 9, the M atom loses electrons, while the ln and X atoms gain electrons. Among them, for the Ti₂InX system, the Ti-C bond has the largest BP value, indicating that the Ti-C bond has a strong chemical bond. Therefore, it is proved that Ti₂lnC has the strongest chemical bond and Ti₂lnN has the weakest chemical bond. For the Zr₂InX system, it can also be stated that Zr₂InC has the strongest chemical bond and Zr₂InN has the weakest chemical bond. The M and X atoms form a strongly directed M–X covalent bond originating from the hybrid M d–X p state. These results are also consistent with the finding that the largest phase typically has very strong M–X bonds and relatively weak M–A bonds.

Fig. 11. The calculated total and partial density of states for (a) Ti₂InC, (b) Ti₂InN, (c)Zr ₂InC and (d) Zr ₂InN.

Table 9. Calculate Mulliken charge and bond population (BP) analysis of M₂InX (M = Ti, Zr and X=C, N) MAX phases.

Author Response File: Author Response.pdf

Reviewer 2 Report

Methodology properly described,
Systems are of interest,
Results look reasonable.
In general, I recommend publishing this paper in Metals,
Small point:
Table 3 shows approx. 3% difference with data from ref 45, despite it looks to me that extremely similar methodology (functional, cutoffs, k-points, and even software) were used. Maybe a short comment on this inconsistency would be beneficial. Also, reference 45 seems to miss a few last authors.
Also the phrase "(M = Ti, Zr and X=C, N)" is repeated in the manuscript 23 times, not counting subparts of it. It is not improving readability, and I would recommend repeating it not more than 2 or 3 times. 

Author Response

Response to Reviewer 2 Comments

metals-1785495

<Metals>

<Anisotropic elastic and thermal properties of M₂InX (M=Ti, Zr and X = C, N) phases: A first-principles calculation>

Reviewer comments:

Reviewer #2: Methodology properly described, Systems are of interest, Results look reasonable. In general, I recommend publishing this paper in Metals,

Small point:

Table 3 shows approx. 3% difference with data from ref 45, despite it looks to me that extremely similar methodology (functional, cutoffs, k-points, and even software) were used. Maybe a short comment on this inconsistency would be beneficial. Also, reference 45 seems to miss a few last authors.

Also the phrase "(M = Ti, Zr and X=C, N)" is repeated in the manuscript 23 times, not counting subparts of it. It is not improving readability, and I would recommend repeating it not more than 2 or 3 times.

Response: Thanks for your professional advice. Regarding the 3% difference between the data in Table 3 and reference 45. Although very similar methods were used in the calculations (both using the Perdew-Wang generalized gradient approximation (PW91) in the generalized gradient approximation (GGA), the maximum stress was 0.02 GPa, the plane wave cutoff energy was 450 even. The k point was taken as 10×10×2). First of all, we need to consider the different versions of the software (we use Materials Studio 2017 for this calculation), which will cause difference in the calculation results. Secondly, when calculating the elastic constant, we set the number of steps for each strain to 7 and the maximum strain amplitude to 0.003, the choice of these parameter values is different, and the accuracy of the calculation will also be different, which also makes a small gap between our calculated value and the theoretical value.

I have revised reference 45 in the manuscript and marked it in red in the manuscript. For the expression "(M = Ti, Zr and X=C, N)", I have edited it in the manuscript. Thanks again for your suggestion.

Author Response File: Author Response.pdf

Reviewer 3 Report

MAX phase compounds have an excellent properties, such as machinability, promising  electrical and thermal conductivity. Thus the current manuscript addressing to studying Ti₂lnX (X=C, N) and Zr₂lnX (X=C, N) M₂AX phases will be interesting to the research community in the field of ceramics. I would like the authors to clarify some points in their study.

11.     Section “Methods”. Please explain exactly which ultrasoft pseudopotentials (USPPs) were used in your calculations.

22.     Line 55 of the manuscript. “The reactions between …” is misprint. “The interactions between …” is correct.

33.     What about the electrical properties of studied Ti₂lnX (X=C, N) and Zr₂lnX (X=C, N) compounds? I am sure the authors have no difficulty to present  the DOS figures for the compounds studied and then to comment in general terms on the expected electrical properties.

Author Response

Response to Reviewer 3 Comments

metals-1785495

<Metals>

<Anisotropic elastic and thermal properties of M₂InX (M=Ti, Zr and X = C, N) phases: A first-principles calculation>

Reviewer comments:

Reviewer #3: MAX phase compounds have an excellent properties, such as machinability, promising electrical and thermal conductivity. Thus the current manuscript addressing to studying Ti₂lnX (X=C, N) and Zr₂lnX (X=C, N) M₂AX phases will be interesting to the research community in the field of ceramics. I would like the authors to clarify some points in their study.

11. Section “Methods”. Please explain exactly which ultrasoft pseudopotentials (USPPs) were used in your calculations.

Response: Thanks for your professional advice. Generally speaking, supersoft pseudopotentials can be subdivided into two types: supersoft and on-the-fly generation (OTFG) supersoft. However, in the calculations in this manuscript, we use the OTFG ultrasoft pseudopotential to calculate the reaction between electrons and ion nuclei, and use the Perdew-Wang generalized gradient approximation (PW91) method in the generalized gradient approximation (GGA) to model the exchange correlation potential.

Firstly, with regard to ultrasoft pseudopotentials (USPPs), it can facilitate calculations with the lowest possible cutoff energy in the plane-wave basis set. Since it can be understood that there is an inherent limit to the convergence of the optimal norm-conserving pseudopotential, a completely different approach is devised. The rationale behind USPPs is that, in most cases, high cutoff energies are required for plane-wave basis sets only when tightly bound orbitals are present and most of their weight is within the core region of the atom. In these cases, the only way to reduce the basis set is to violate the gauge conservation condition by removing the charges associated with these orbitals from the core region. Therefore, the pseudo-wave function is allowed to be as soft as possible within the core, thereby significantly reducing the cutoff energy.

And for on-the-fly generation (OTFG) ultrasoft pseudopotentials (USPPs), the latest set of settings for OTFG ultrasoft pseudopotentials has been developed to minimize errors in fully converged all-electronic DFT calculations. The set achieved an error of 0.4 meV/atom, making CASTEP one of the most accurate pseudopotential codes available.

Therefore, we use the OTFG ultrasoft pseudopotentials in this calculation. Compared with the ultrasoft pseudopotentials (USPPs), using these pseudopotentials can make the calculation result more accurate and the calculation error smaller.

22. Line 55 of the manuscript. “The reactions between …” is misprint. “The interactions between …” is correct.

Response: Thanks for your professional advice. I have reworked the errors in the manuscript and marked them in red. Thanks again for your suggestion.

33. What about the electrical properties of studied Ti₂lnX (X=C, N) and Zr₂lnX (X=C, N) compounds? I am sure the authors have no difficulty to present the DOS figures for the compounds studied and then to comment in general terms on the expected electrical properties.

Response: Thanks for your professional advice. I have re-supplemented the electronic properties of the compounds in the manuscript at subsection 3.7 and performed the analysis, which are marked in red in the text. Thanks again for your useful advice.

The electronic properties (density of states) of M₂InX (M = Ti, Zr and X=C, N) MAX phases were studied to better understand chemical bonding and bond behaviors. Fig.11 depicts the M₂InX phases' total density of states (TDOS) and partial density of states (PDOS). As a result, Fig.11 provides the following information. To begin, it is clear that DOS has a significant finite value at the Fermi level, indicating that these compounds exhibit metallic conductivity. Fig.11 shows that the total density of states (E_f) value of Ti₂InN and Zr₂lnN are greater than that of Ti₂InC and Zr₂lnC, indicating that Ti₂InN and Zr₂lnN are more conductive than Ti₂InC and Zr₂lnC. Secondly, the peak topologies and relative heights of the peaks around E_f in the TDOS plot are highly comparable, indicating the presence of similar chemical bonds in Zr₂AN. The time difference around the Fermi level is mostly made up of ln-p and M-d states. The time difference below the Fermi energy is caused mostly by the X-s, M-s, and M-p states, whereas the time difference above the Fermi energy is caused primarily by the ln-s and X-p states. As shown in Fig.11, the ln-p, M-d, and X-p states exhibit substantial hybridization, allowing M-C and M-N chemical bonds to form, resulting in the high elastic modulus of M₂InX. PDOS exhibits multiple hybridizations of the electronic states M, ln, and X. The valence band of M₂InX in Fig.11 displays substantial hybridization of the M-d and X-p states, as predicted for covalent compounds. The d–p hybrid state corresponding to the M-ln bond is discovered to be in a greater energy range than the M-X bond. As a result, the M-X d-p hybridization helps to maintain the crystal structure. Finally, it is demonstrated that M's electronic charge density almost overlaps that of ln, indicating that the bonding between M and ln is quite weak. These findings are consistent with the observation that the biggest phase features very strong M-X bonds and very weak M-ln bonds.

Fig. 11. The calculated total and partial density of states for (a) Ti₂InC, (b) Ti₂InN, (c)Zr ₂InC and (d) Zr ₂InN.

Author Response File: Author Response.pdf