A critical study of the elastic properties and stability of Heusler compounds: Cubic Co$_{2}YZ$ compounds with $L2_{1}$ structure

TL;DR

This study uses first-principles calculations to analyze the elastic properties, stability, and bonding characteristics of various cubic Heusler compounds, revealing correlations with atomic mass and proposing revisions to brittleness criteria.

Contribution

It provides a comprehensive first-principles analysis of elastic and stability properties of Co2-based Heusler compounds, including new insights into their bonding and anisotropy.

Findings

Many properties correlate with the main group element's mass or nuclear charge.

Christensen's criterion effectively describes ductile-brittle transition.

Co2ScAl is predicted to be an isotropic, nearly ideal Cauchy solid.

Abstract

Elastic constants and their derived properties of various cubic Heusler compounds were calculated using first-principles density functional theory. To begin with, Cu$_2$MnAl is used as a case study to explain the interpretation of the basic quantities and compare them with experiments. The main part of the work focuses on Co$_2$-based compounds that are Co$_2$Mn$M$ with the main group elements $M=$~Al, Ga, In, Si, Ge, Sn, Pb, Sb, Bi, and Co$_2TM$ with the main group elements Si or Ge, and the $3d$ transition metals $T=$~Sc, Ti, V, Cr, Mn, and Fe. It is found that many properties of Heusler compounds correlate to the mass or nuclear charge $Z$ of the main group element. Blackman's and Every's diagrams are used to compare the elastic properties of the materials, whereas Pugh's and Poisson's ratios are used to analyze the relationship between interatomic bonding and physical properties. It is found that the {\it Pugh's criterion} on brittleness needs to be revised whereas {\it Christensen's criterion} describes the ductile--brittle transition of Heusler compounds very well. The calculated elastic properties give hint on a metallic bonding with an intermediate brittleness for the studied Heusler compounds. The universal anisotropy of the stable compounds has values in the range of $0.57 <A_U <2.73$. The compounds with higher $A_U$ values are found close to the middle of the transition metal series. In particular, Co$_2$ScAl with $A_U=0.01$ is predicted to be an isotropic material that comes closest to an ideal Cauchy solid as compared to the remaining Co$_2$-based compounds. Apart from the elastic constants and moduli, the sound velocities, Debye temperatures, and hardness are predicted and discussed for the studied systems. The calculated slowness surfaces for sound waves reflect the degree of anisotropy of the compounds.

Tables

Table 1. Table 1: Applied strains to cubic structure are listed here. The isotropic strain type (0) is not volume conserving, it is not directly used but used together with the lattice parameter optimization that yields the bulk modulus. The consequences of the strain types are sketched in Figure  1 .

Type	Strain					$\Delta E/V_0$
(0)	isotropic	$e_1=\delta$	$e_2=\delta$	$e_3=\delta$		$\left(c_{11}+2c_{12}\right){\delta }^2$, see bulk modulus!
(1)	tetragonal	$e_1=\delta$	$e_2=\delta$	$e_3=-\delta \left(2+\delta \right)/{\left(1+\delta \right)}^2$		$\left(c_{11}-c_{12}\right){\delta }^2+O\left({\delta }^3\right)$
(2)	orthorhombic	$e_1=\delta$	$e_2=-\delta$	$e_3={\delta }^2/\left(1-{\delta }^2\right)$		$\left(c_{11}-c_{12}\right){\delta }^2+O\left({\delta }^4\right)$
(3)	monoclinic	$e_1={\delta }^2/\left(1-{\delta }^2\right)$			$e_4=2\delta$	$c_{44}{\delta }^2+O\left({\delta }^4\right)$

Table 2. Table 2: Elastic properties of Cu 2 MnAl. The experimental and optimized lattice parameters ( a 𝑎 a ) are given in Å. Elastic moduli B 𝐵 B (bulk), G 𝐺 G (shear), E 𝐸 E (Young), hardness parameter H 𝐻 H , elastic constants c i ​ j subscript 𝑐 𝑖 𝑗 c_{ij} , C ′ superscript 𝐶 ′ C^{\prime} , and Cauchy pressure p C subscript 𝑝 C p_{\rm C} are given in GPa. k 𝑘 k , ν 𝜈 \nu , ζ 𝜁 \zeta , A e subscript 𝐴 𝑒 A_{e} and A U subscript 𝐴 𝑈 A_{U} are dimensionless quantities. Experimental values for the c i ​ j subscript 𝑐 𝑖 𝑗 c_{ij} are taken from Reference  Michelutti  et al. ( 1978 ) . Values for 0 K are extrapolated from the temperature dependence shown in Figure 3 of Reference  Michelutti  et al. ( 1978 ) .

		Calculated	Experiment
			0 K	300 K
$a$	Å	5.934		5.9615
$c_{11}$	GPa	143.7	128.1	$\left(136.1\pm 0.3\right)$
$c_{12}$	GPa	116.1	101.5	$\left(96.8\pm 0.8\right)$
$c_{44}$	GPa	117.6	104.4	$\left(94.1\pm 0.2\right)$
$C^{\prime }$	GPa	27.6	26.6	39.3
$p_C$	GPa	-1.5	-1.3	2.7
$B$	GPa	125.3	110.4	109.9
$G$	GPa	52.5	47.9	50.9
$E$	GPa	138.0	125.6	132.2
$H$	GPa	6.43	6.06	6.8
$k$		2.41	2.30	2.16
$\nu$		0.32	0.31	0.30
$\zeta$		0.866
$A_e$		9.64	7.85	4.79
$A_U$		7.97	7.17	3.6

Table 3. Table 3: Properties of Cu 2 MnAl. Sound velocities are calculated from elastic constants as given in Table  2 .

		Calculated	Experiment
			0 K	300 K
$a$	Å	5.934		5.9615
$\rho$	kg/m3	6637		6550
$m$	g/mol	209.01
$v_l$	m/s	5421
$v_t$	m/s	2807	see text
$\overline{v}$	m/s	3142
${\Theta }_D^{\mathrm{ac}}$	K	397	372	376
${\zeta }^{\mathrm{ac}}$		1.88
$f_{\nu }$		0.72	0.74	0.76
${\Theta }_D^{\mathrm{qha}}$	K	395
$T_m^c$	K	1402
$H{}_V^C$	GPa	4.38

Table 4. Table 4: QTAIM analysis of Cu 2 MnAl. V 𝑉 V are the basin volumes V 𝑉 V , n e subscript 𝑛 𝑒 n_{e} is the number of electrons in the basin, and q = Z − n e 𝑞 𝑍 subscript 𝑛 𝑒 q=Z-n_{e} is the electron excess/deficiency with respect to the electron occupation in free atoms Z 𝑍 Z . Negative values indicate electron excess, that is, negatively charged ions, and m 𝑚 m is the magnetic moment in the basin in multiples of the Bohr magneton.

	$V$ [Å3]	$n_e$	$q$	$m$ [${\mu }_B$]
Cu	15.547	29.86	-0.86	0.03
Mn	12.228	24.57	0.43	3.45
Al	8.969	11.71	1.28	0.04

Table 5. Table 5: QTAIM critical point analysis of Cu 2 MnAl. p ​ g 𝑝 𝑔 pg is the point group symmetry of the critical point, and mult is the multiplicity of the critical points in the conventional cubic cell; the multiplicities in the primitive cell are mult/4.

$pg$	type	position			mult	name
$O_h$	nucleus	0	0	0	4	Mn
$O_h$	nucleus	1/2	1/2	1/2	4	Al
$T_d$	nucleus	1/4	1/4	1/4	8	Cu
$C_{3v}$	bond	0.397	0.397	0.397	32	$b_1$
$C_{3v}$	bond	0.878	0.878	0.878	32	$b_2$
$D_{2h}$	bond	1/4	1/4	1/2	24	$b_3$
$C_{2v}$	ring	0	0.302	0.302	48	$r_1$
$C_{2v}$	ring	0	0.287	0.213	48	$r_2$
$C_{4v}$	cage	0	0	0.254	24	$c$

Table 6. Table 6: Elastic properties of Co 2 Mn M 𝑀 M ( M = 𝑀 absent M= main group element). The experimental ( a exp subscript 𝑎 exp a_{\rm exp} ) and optimized ( a opt subscript 𝑎 opt a_{\rm opt} ) lattice parameters are given in Å. Elastic moduli B 𝐵 B , G 𝐺 G , E 𝐸 E , hardness parameter H 𝐻 H , elastic constants c i ​ j subscript 𝑐 𝑖 𝑗 c_{ij} , C ′ superscript 𝐶 ′ C^{\prime} , and Cauchy pressure p C subscript 𝑝 C p_{\rm C} are given in GPa. k 𝑘 k , ν 𝜈 \nu , ζ 𝜁 \zeta , A e subscript 𝐴 𝑒 A_{e} , and A U subscript 𝐴 𝑈 A_{U} are dimensionless quantities.

$M$	Al	Ga	In	Si	Ge	Sn	Pb	Sb	Bi
$a_{\exp }$	5.755a	5.770b		5.654b	5.743b	6.000b
$a_{\mathrm{opt}}$	5.700	5.718	5.974	5.643	5.730	5.987	6.102	6.019	6.184
$c_{11}$	267.3	242.2	194.4	310.5	272.8	233.6	200.1	235.3	192.0
$c_{12}$	155.3	167.5	130.6	174.2	160.0	138.3	124.5	133.6	113.2
$c_{44}$	160.4	150.5	131.8	156.9	137.9	125.0	103.8	106.4	72.4
$C^{\prime }$	112.0	74.7	63.8	136.3	112.8	95.3	75.6	101.7	78.8
$p_C$	-5.1	17.0	-1.2	17.3	22.1	13.3	20.7	27.2	40.8
$B$	192.6	192.4	151.9	219.6	197.6	170.0	149.7	167.5	139.5
$G$	105.3	86.6	75.2	112.3	96.4	85.0	69.3	79.1	56.7
$E$	267.2	225.9	193.6	287.9	248.7	218.5	180.1	205.1	149.8
$H$	16.2	11.3	10.7	16.4	13.5	12.1	8.0	10.8	6.8
$k$	1.83	2.22	2.02	1.96	2.05	2.00	2.16	2.12	2.46
$\nu$	0.27	0.30	0.29	0.28	0.29	0.29	0.30	0.30	0.32
$\zeta$	0.692	0.779	0.764	0.676	0.696	0.701	0.767	0.681	0.699
$A_e$	2.86	4.05	4.14	2.30	2.44	2.62	2.75	2.09	1.84
$A_U$	1.46	2.73	2.85	0.88	1.02	1.21	1.88	0.68	0.46

Table 7. Table 7: Elastic properties of Co T 2 ​ M subscript 𝑇 2 𝑀 {}_{2}TM ( T = 𝑇 absent T= Sc to Fe and M = 𝑀 absent M= Al, Si). The experimental ( a exp subscript 𝑎 exp a_{\rm exp} ) and optimized ( a opt subscript 𝑎 opt a_{\rm opt} ) lattice parameters are given in Å. Elastic moduli B 𝐵 B , G 𝐺 G , E 𝐸 E , hardness parameter H 𝐻 H , elastic constants c i ​ j subscript 𝑐 𝑖 𝑗 c_{ij} , C ′ superscript 𝐶 ′ C^{\prime} , and Cauchy pressure p C subscript 𝑝 C p_{\rm C} are given in GPa. k 𝑘 k , ν 𝜈 \nu , ζ 𝜁 \zeta , and A e subscript 𝐴 𝑒 A_{e} are dimensionless quantities.

$M$			Al						Si
$T$	Sc	Ti	V	Cr	Mn	Fe	Sc	Ti	V	Cr	Mn	Fe
$a_{exp}$		5.848c	5.780d	5.726e	5.755a	5.730f		5.740c	5.657g		5.654b	5.640h
$a_{opt}$	5.972	5.837	5.758	5.711	5.700	5.702	5.870	5.758	5.679	5.638	5.643	5.630
$c_{11}$	277.8	286.0	290.0	265.0	267.3	268.2	267.4	295.4	297.4	287.5	310.5	273.6
$c_{12}$	83.8	129.7	150.5	169.2	155.3	150.1	123.5	158.3	183.5	194.2	174.2	168.5
$c_{44}$	105.0	126.7	140.7	156.1	160.4	150.5	111.3	134.8	148.7	162.4	156.9	144.7
$C^{\prime }$	194.0	156.3	139.5	95.8	112.0	118.1	143.9	137.1	113.9	103.3	136.3	105.1
$p_C$	-21.2	3.0	9.8	13.1	-5.1	-0.4	12.2	23.5	34.8	31.8	17.3	23.8
$B$	148.4	181.8	197.0	201.5	192.6	189.5	171.5	204.0	221.5	225.3	219.6	203.5
$G$	101.7	104.4	106.2	97.2	105.3	103.5	93.4	102.8	101.2	98.8	112.3	96.4
$E$	248.4	262.9	270.0	251.3	267.2	262.6	237.2	264.0	263.5	258.6	287.9	249.8
$H$	18.9	16.8	16.2	13.5	16.2	15.9	14.4	14.8	13.4	12.6	16.4	13.1
$k$	1.46	1.74	1.86	2.07	1.83	1.83	1.84	1.99	2.19	2.28	1.96	2.11
$\nu$	0.22	0.26	0.27	0.29	0.27	0.27	0.27	0.28	0.30	0.31	0.28	0.30
$\zeta$	0.449	0.585	0.641	0.738	0.692	0.675	0.593	0.655	0.721	0.767	0.676	0.720
$A_e$	1.08	1.62	2.02	3.28	2.86	2.55	1.55	1.97	2.61	3.48	2.30	2.76
$A_U$	0.01	0.29	0.62	1.88	1.46	1.13	0.23	0.57	1.19	1.75	0.88	1.34

Table 8. Table 8: Physical properties of Co 2 Mn M 𝑀 M ( M = 𝑀 absent M= main group element). The Debye temperature Θ D ac subscript superscript Θ ac 𝐷 \Theta^{\rm ac}_{D} is estimated from the sound velocity v ¯ ¯ 𝑣 \overline{v} .

$M$		Al	Ga	In	Si	Ge	Sn	Pb	Sb	Bi
$\rho$	kg/m3	7166	8617	8959	7424	8665	9023	11110	8972	10726
$m$	g/mol	199.8	242.5	287.6	200.9	245.4	291.5	380.0	294.6	381.8
$v_l$	m/s	6817	5977	5305	7054	6135	5604	4668	5516	4478
$v_t$	m/s	3833	3170	2897	3890	3335	3068	2497	2970	2299
$\overline{v}$	m/s	4265	3543	3231	4335	3720	3421	2789	3316	2575
${\Theta }_D^{\mathrm{ac}}$	K	561	465	406	576	487	429	343	413	312
${\zeta }^{\mathrm{ac}}$		1.59	1.80	1.70	1.66	1.71	1.69	1.77	1.75	1.91
$T_m^c$	K	2133	1984	1702	2388	2165	1934	1736	1944	1688
$H{}_V^C$	GPa	12.0	7.7	8.0	11.4	9.5	8.9	6.7	7.7	4.4

Table 9. Table 9: Physical properties of Co T 2 ​ M subscript 𝑇 2 𝑀 {}_{2}TM ( T = 𝑇 absent T=  Sc, Ti, V, Cr, Mn, Fe; M = 𝑀 absent M= Al, Si). The Debye temperature Θ D ac subscript superscript Θ ac 𝐷 \Theta^{\rm ac}_{D} is estimated from the sound velocity v ¯ ¯ 𝑣 \overline{v} .

$M$				Al						Si
$T$		Sc	Ti	V	Cr	Mn	Fe	Sc	Ti	V	Cr	Mn	Fe
$\rho$	kg/m3	5919	6436	6814	7019	7166	7189	6269	6742	7142	7336	7424	7511
$m$	g/mol	189.8	192.7	195.8	196.8	199.8	200.7	190.9	193.8	196.9	197.9	200.9	201.8
$v_l$	m/s	6928	7062	7049	6868	6817	6748	6872	7112	7064	6978	7054	6649
$v_t$	m/s	4145	4027	3947	3722	3833	3794	3860	3904	3764	3670	3890	3583
$\overline{v}$	m/s	4587	4476	4394	4153	4265	4221	4295	4352	4206	4104	4335	4000
${\Theta }_D^{\mathrm{ac}}$	K	576	575	572	546	561	555	549	567	556	546	576	533
${\zeta }^{\mathrm{ac}}$		1.37	1.54	1.61	1.73	1.59	1.60	1.60	1.68	1.78	1.83	1.66	1.74
$T_m^c$	K	2195	2244	2267	2119	2133	2138	2133	2299	2311	2252	2388	2170
$H{}_V^C$	GPa	16.2	12.9	11.9	9.4	12.0	11.9	11.0	10.5	8.9	8.2	11.4	9.1

Equations

$$\nu =\frac{3k-2}{6k+2}.$$

$$D$$

$$\kappa$$

$${\kappa }_{B/D}=1-\sqrt{1/2},$$

$${\nu }_{B/D}$$

$$k_{B/D}$$

$$\sigma =\mathbf{C}ϵ,$$

$$\mathbf{S}={\mathbf{C}}^{-1}.$$

$${\bm{C}}^{\rm cubic}=\left(\begin{array}[]{ccc ccc}c_{11}&c_{12}&c_{12}&.&.&.\\ c_{12}&c_{11}&c_{12}&.&.&.\\ c_{12}&c_{12}&c_{11}&.&.&.\\ .&.&.&c_{44}&.&.\\ .&.&.&.&c_{44}&.\\ .&.&.&.&.&c_{44}\\ \end{array}\right)$$

$$s_{11}$$

$$s_{12}$$

$$c$$

$$s_{44}$$

$$B=\frac{1}{3}(c_{11}+2c_{12}).$$

$$p_s=c_{11}+2c_{12},$$

$$C^{\prime }=c_{11}-c_{12}.$$

$$p_{\mathrm{C}}=c_{12}-c_{44}.$$

$$G$$

$$k$$

$$\nu$$

$$B$$

$$G_V$$

$$G_R$$

$$\nu$$

$$E$$

$$E^{-1}(\hat{r})=s_{11}-2s{F}_{lmn},$$

$$A_e=\frac{2(s_{11}-s_{12})}{s_{44}}=\frac{2c_{44}}{c_{11}-c_{12}}.$$

$$A_U$$

$$A_U^{cubic}$$

$$G_0^{-1}(\hat{r})$$

$$G^{-1}(\hat{r})$$

$${\bm{C}}^{\rm cubic}_{\rm Cauchy}=\frac{1}{3}\left(\begin{array}[]{lll ccc}3c_{11}&c_{11}&c_{11}&.&.&.\\ c_{11}&3c_{11}&c_{11}&.&.&.\\ c_{11}&c_{11}&3c_{11}&.&.&.\\ .&.&.&c_{11}&.&.\\ .&.&.&.&c_{11}&.\\ .&.&.&.&.&c_{11}.\\ \end{array}\right)$$

$$\beta$$

$$\kappa$$

$$\zeta =\frac{c_{11}+8c_{12}}{7c_{11}+2c_{12}}.$$

$${\Theta }_D^{qha}$$

Full text

Blackman’s and Every’s diagrams are used to compare the elastic properties of the materials, whereas Pugh’s and Poisson’s ratios are used to analyze the relationship between interatomic bonding and physical properties. It is found that the Pugh’s criterion on brittleness needs to be revised whereas Christensen’s criterion describes the ductile–brittle transition of Heusler compounds very well. The calculated elastic properties give hint on a metallic bonding with an intermediate brittleness for the studied Heusler compounds.

The universal anisotropy of the stable compounds has values in the range of $0.57<A_{U}<2.73$. The compounds with higher $A_{U}$ values are found close to the middle of the transition metal series. In particular, Co2ScAl with $A_{U}=0.01$ is predicted to be an isotropic material that comes closest to an ideal Cauchy solid as compared to the remaining Co2-based compounds. Apart from the elastic constants and moduli, the sound velocities, Debye temperatures, and hardness are predicted and discussed for the studied systems. The calculated slowness surfaces for sound waves reflect the degree of anisotropy of the compounds.

††preprint: Wu et al; Elastic properties of cubic Heusler compounds.

I Introduction

There is a broad interest in Heusler compounds owing to the multitude of different thermal, electrical, magnetic, and transport properties that are realized in a rather simple crystalline structure. Owing to both applications and fundamental interests, such as superconductivity, heavy fermions, the Kondo effect, the Hall effect, and half-metallic ferromagnetism, these compounds are among the most studied materials Graf et al. (2011). Regular Heusler compounds crystallize in a cubic fcc lattice with the space group $F\>m\bar{3}m$. In certain cases, the cubic phases of regular Heusler compounds undergo, a martensite-austenite phase transition to a tetragonal lattice, whereby the symmetry changes to $I4/mmm$. In fact, both the cubic and the tetragonal phases have attracted considerable attention owing to their half-metallic ferromagnetic and spin-transfer-torque applications Chen et al. (2006); Winterlik et al. (2008); Sukegawa et al. (2012). Knowledge of the stability of each of these phases is crucial for industrial applications as well as fundamental research.

New Heusler compounds have been suggested to be stable in many theoretical works. However, in many cases it is experimentally found that it is not possible to synthesize these compounds. A possible reason for this is that not all stability criteria are respected in theoretical calculations. In fact, mostly all the used stability criteria are necessary but not sufficient. This implies that a suggested compound that fulfills a particular criterion may not exist as it possibly violates other criteria. One of these necessary, but not sufficient, criteria is the total energy, or the energy of formation, satisfying the condition $E_{\rm compound}<\sum E_{\rm Elements}$ for a compound to be stable. Further, the formation energy of the suggested compound needs to be the minimum on the “convex hull” taking into account all the competing phases. Otherwise, it would decompose into other compounds with lower energies. For example, the appearance of different binaries (XY, XZ, or other similar combinations) may lead to a lower total energy as compared to a single ternary (X2YZ), and thus, hinder the formation of a Heusler compound.

Another important criterion is the mechanical stability of a predicted structure. According to Born Born (1939), a necessary condition for the thermodynamic stability of a crystal lattice is that the crystals have to be mechanically stable against arbitrary (small) homogeneous deformations. In fact, this is the main concept of elastic constants. Elastic constants provide important information concerning the strength of materials, and often act as stability criteria or order parameters in the study of the problem of structural transformations Ayuela et al. (1999); Karki et al. (1997); Wang et al. (1993a). Further physical properties, such as hardness, velocity of sound, Debye temperature, and melting point are also related to the elastic constants Gilman (2001, 2009); Pearson (1997); Fine et al. (1984). The information is not only essential requirements for industrial applications but also for fundamental research. Examples of the latter case are the superconducting and heavy fermion systems, in which a drastic change of elastic constants and related properties have been obtained upon phase transition Bruls et al. (1994); Ledbetter et al. (1989).

There are several reports on calculations of elastic constants and phase stability Kart et al. (2008); Sahariah et al. (2013); Kong et al. (2011). Members of the series of cubic Co${}_{2}TM$ ($T=$ transition element, $M=$ main group element) Heusler compounds have been studied previously to some extent by various authors Chen et al. (2006); Candan et al. (2013); Ouardi et al. (2011); Yin et al. (2013). In many cases, only the three independent elastic constants and the bulk modulus are calculated. In only a few cases, experiments were carried out to measure the hardness and melting temperatures, and to compare them with the calculations. However, most works have been carried out for specific cases, and almost not all relevant properties have been calculated and compared with experiments or properties of other compounds.

The present report is intended to investigate the mechanical properties of a variety of Co2-based Heusler compounds by calculating their elastic constants. To begin with, Cu2MnAl, as a typical Heusler compound, is studied to explain the basic quantities and their interpretation. Results for Co2Mn$M$ ($M=$ main group element), and Co${}_{2}TM$ ($T=3d$-metal, $M=$ Al, Si) are listed and discussed with an in-depth analysis of the physical properties and chemical bonding. As elastic constants are derived from the second derivative of the energy with respect to the lattice displacements, the use of an accurate energy calculator is crucial. Here, the full-potential all-electron method was used to calculate the elastic constants, and related properties. The relationship between the interatomic bonding and the physical properties is considered using Pugh’s and Poisson’s ratios. The Blackman’s diagram provides complementary information about the bonding character of the Heusler compounds. A covalent to metallic bonding with an intermediate ductility or brittleness is found for the studied Heusler compounds. Several other physical properties have been extracted from the elastic constant calculations.

The present work concentrates on the half-metallic Co2-based Heusler compounds that have a high impact on magnetoelectronics. The results for the elastic properties of tetragonal and phase change materials that exhibit magnetic shape memory and magnetocaloric effects will be published elsewhere Wu et al. (2017a). Some basic calculational aspects, including the convergence of the method, are also found in Reference Wu et al. (2017a).

II Computational details

The elastic constants are second derivatives of the total energy with respect to various lattice deformations. Therefore, accurate calculation of the total energy is required. The full-potential linearized augmented plane wave (FLAPW) technique is one such method that provides the required level of numerical accuracy, albeit at the cost of complexity. In particular, in the case of Heusler compounds, FLAPW is a reliable choice as some Heusler compounds are sensitive to the employed method Felser and Fecher (2013), and many Heusler compounds contain atoms from the lanthanide or actinide series with occupied $f$-orbitals. In the present work, the electronic structure was calculated using the full-potential linear augmented plane wave method, as implemented in Wien2k Blaha et al. (2013); Schwarz and Blaha (2003). The details of the calculations are reported in References Kandpal et al. (2007); Fecher et al. (2013) and a forthcoming publication Wu et al. (2017a). The charge density and other site specific properties were analyzed using Bader’s quantum theory of atoms in molecules (QTAIM) Bader (1990) using the built-in routines of Wien2k as well as the Critic2 package of programs Otero-de-la Roza et al. (2009, 2014). We developed our own routines, and used them to determine the elastic constants and to analyze them in detail.

The Perdew–Burke–Ernzerhof implementation of the generalized gradient approximation (PBE-GGA) was used for the exchange-correlation potential. The number of plane waves was defined by $R_{\rm MT}k_{\max}=9.0$ and 8000 $k$ points within the first Brillouin zone were used for integration. The energy convergence criterion was set to 10*-5* Ry and the charge convergence was less than a 10*-3* electronic charge in every case. The convergence of the elastic constants with the parameters of the calculation has been already reported in the publication on tetragonal compounds Wu et al. (2017b).

Most Co2-based Heusler compounds are among the half-metallic ferromagnetic materials, thus only ferromagnetic ordering has been studied here. See References Kandpal et al. (2007); Felser and Fecher (2013) for details of the electronic and magnetic structure of the investigated compounds. The basic equations for the calculations of the elastic constants are discussed in the following. More details are provided in Appendix. The bulk moduli and relaxed lattice parameters are found by fitting the calculated energy–volume relation to the Birch–Murnaghan equation of state Birch (1947); Murnaghan (1944).

There are numerous ways to apply different strains and their combinations to the crystal. For cubic crystals, there are three independent elastic constants and only two more calculations are needed besides the bulk modulus that is found from the equation of state. A necessary supplementary condition in the calculation of the elastic constants is the conservation of volume when strain (or stress) is applied. Table 1 summarizes the applied strains used in the present work to determine the elastic constants. The applied strains are illustrated in Figure 1. More details are found in Reference Wu et al. (2017b) and in the Appendix.

In the present calculations, four distortions of each type in the range of $-3\%\leq x\leq+3\%$ were applied to the relaxed structure with $V_{0}$ from the structural optimization, which is the equation of state fitted to $E_{tot}(V)$. The different distortions are sketched in Figure 1. The energies $E(x)$ of the monoclinic, orthorhombic and tetragonal strain types were fitted to a $4^{th}$ order polynomial $E(x)=E_{0}+a_{2}x^{2}+a_{3}x^{3}+a_{4}x^{4}$. Finally, the $B$ and the $a_{2}$ values were used to determine the elastic constants and their derived quantities. The elastic constants reported below are averaged over the values determined by applying tetragonal or orthorhombic strains. The equations for the properties calculated from the elastic constants are given in detail in Appendix V. It should be noted that models used for the Vicker’s hardness ($H{{}_{V}\rm^{C}}$) and the melting temperatures ($T_{m}^{c}$) are only suitable for cubic structures Chen et al. (2011); Fine et al. (1984) and may be used for the comparison of different compounds rather than yielding absolute values.

III Results

To begin with, the results for the classical Heusler compound Cu2MnAl are presented and compared with experiments, because it is one of the few Heusler compounds for which measured values of the elastic constants are available. Table 2 compares the calculated elastic properties with the experimental work of Michelutti et al Michelutti et al. (1978). As seen in Table 2, the calculated elastic constants agree well with the experiment, where the overestimation of about 10% that is observed could be due to the intrinsic properties of the calculational method Lejaeghere et al. (2014) or due to uncertainties of the experimental set-up. The other calculated properties, such as elastic moduli, Cauchy pressure, and velocity of sound show excellent agreement with those found in the experiment.

The elastic constants of Cu2MnAl are listed in Table 2. $c_{11}$, which represents stiffness against principal strains, is higher than $c_{44}$, which represents shear deformation. The shear ($G$) and tetragonal shear $c^{\prime}$ moduli are also low compared to the bulk modulus. This implies that Cu2MnAl has the lowest resistance against shear deformations. The cross-sections on (110) and (001) crystallographic planes of Young’s moduli of Cu2MnAl are shown in Figure 2(c) and (f). It is seen that the anisotropy of Young’s modulus is noticeable in both the planes. The directions where the maxima appear correspond to the high-fracture energy directions, which are along $\left<111\right>$ in the (110) plane and $\left<110\right>$-direction in the (001) plane.

The Kleinman’s parameter $\zeta$ describes the relative positions of the atoms under strain. The calculated value of $\zeta\approx 0.9$ for Cu2MnAl suggests that the atomic positions are rather rigid against distortions of the lattice. The tetragonal shear modulus $c^{\prime}=C^{\prime}/2\approx(13\ldots 20)$ GPa is the smallest modulus and thus it is the main constraint on stability. Both anisotropies, the Zener ratio $A_{e}$ and universal anisotropy $A_{U}$, are about 8. This is a rather large value and may suggest elastic instability of Cu2MnAl in the $L2_{1}$ structure, as will be discussed in the following.

The use of Blackman’s and Every’s diagrams Ledbetter and Migliori (2008) is an efficient way to compare the elastic properties of cubic materials. In both types of diagrams, dimensionless quantities that are ratios of different moduli are correlated. Figure 3 summarizes the results of the present work in such diagrams. Blackman’s diagram compares in a simple way the ratios $F_{ij}=c_{ij}/c_{11}$ of the elastic constants in a plot of $F_{12}(F_{44})$, whereas Every’s diagram is more complicated. It compares $s_{3}=(1-F_{12}-2F_{44})/(1+2F_{44})$ as a function of $s_{2}=(1-F_{44})/(1+2F_{44})$. Born’s shear criterion restricts $F_{44}$ to positive values and the spinodal and Born criteria restrict $F_{12}$ to the range $-0.5<F_{12}<1$. Therefore, no additional restrictions appear for Blackman’s diagram and all values within the entire range of Figure  3(a) are allowed. In Every’s diagram, the values for stable systems need to fall into the triangle with $(s_{2},s_{3})$ equal to (-1/2,-1)-(1,0)-(1,3/2) as is marked in Figure  3(b).

From both the diagrams shown in Figure 3, it is seen that the Heusler compounds calculated in the present work are close to the Cauchy line, where the Cauchy pressure vanishes. All the studied compounds are in the region where the anisotropy index is positive. Obviously, Cu2MnAl has one of the highest anisotropies and comes close to the line of $2/m,3m$ symmetry breaking phase transitions as marked in Every’s diagram. The Co2-based compounds will be discussed below in more detail.

Table 3 summarizes the physical properties of Cu2MnAl that are related or derived from the elastic properties. The measured Michelutti et al. (1978) sound velocities at room temperature are in the range from 4553 m/s to 6003 m/s for longitudinal modes depending on the direction of propagation and the polarization. The average sound velocity in the transverse modes is about $(3791\pm 3)$ m/s. The values calculated with Anderson’s approach Anderson (1963) are in the same order of magnitude.

The Debye temperature was calculated from the mode averaged sound velocity $\overline{v}$ and in the quasi-harmonic approach. Both methods result in values of about 395 K. This is marginally higher than the experimental value. The slightly larger values are typical for the acoustical approaches that neglect the optical phonons Ouardi et al. (2011); Anderson (1963).

Figure 4 shows the three sheets of the slowness surface of Cu2MnAl. Pronounced extrema, arising from the large anisotropy of the compound, are observed for all the three modes. The pressure ($p$) wave has the highest phase velocity. The minima of its slowness appear along {111}-type directions that is along the space diagonals and the maxima are found along the {001}-type, principle axes. The maxima of the slowness of the fast shear wave ($s_{2}$) are along the {111}-type directions. The slowness surface of the slow shear wave ($s_{1}$) appears to be more complicated; its maxima appear along {110}-type directions. The two shear modes are sixfold degenerate, that is, their slowness is the same at the six $\left[00s\right]$-type points along the three principal axes.

One may roughly categorize materials as ductile (malleable) or as brittle with respect to mechanical characteristics, as for example, machinability. For various applications, the materials need to be malleable, for instance if they are required to be used as wires. Ductile materials usually exhibit metallic bonding, whereas high brittleness indicates a more covalent or ionic character of the bonds. The transition region between these subjective criteria is blurred. Because of the importance in various applications, the malleability criteria are of great significance. Pugh’s and Poisson’s ratios are very helpful mechanical parameters in the characterization of maleability (brittle or ductile).

Pettifor Pettifor (1992) proposed the criterion that a positive Cauchy pressure indicates metallic bonds whereas negative Cauchy pressures are typical in the case of covalent bonds. Another older criterion is based on Pugh’s work Pugh (1954). According to the so-called ”Pugh’s criterion”, many publications Kanchana et al. (2011); Varshney et al. (2011); Niu et al. (2012) indicate that the critical value ($k_{\rm cr}$) that separates brittle ($k\leq k_{\rm cr}$) and ductile ($k\geq k_{\rm cr}$) materials is around $1.75$ or $k^{-1}\approx 0.571$. It is worthwhile to mention that Pugh’s ratio for a cubic, isotropic Cauchy solid is $k_{\rm Cauchy}=5/3=1.6\overline{6}$, as shown in Figure 5(a). The behavior of the Heusler compounds investigated in the present work is summarized in Figure 5 where the Cauchy pressure is plotted as a function of Pugh’s ratio. The two criteria are drawn as vertical and horizontal lines. Following the value for Pettifor’s and Pugh’s criteria, most of the studied compounds should be classified as ductile or metallic materials. There is an obvious contradiction between the empirical rules and the observation that the Co2-based Heusler compounds are all brittle instead of ductile. In particular, it appears that Pugh’s criterion needs to be modified.

Poisson’s ratio $\nu$ is related to Pugh’s ratio by Nye (1985):

[TABLE]

The valid range of Pugh’s ratio ($0<k\leq\infty$) restricts Poisson’s ratio to $-1<\nu\leq 1/2$. The Poisson’ ratio $\nu_{0}=0$ is obtained for $k_{0}=2/3$. Materials with $\nu<0$ are called “auxcetic”. Typically, Poisson’s ratio of covalent materials is small $\nu\approx 0.1$, whereas it is greater than $0.33$ for metallic materials. Poisson’s ratio indicates the degree of directionality of the covalent bonds. Smaller Poisson’s ratios indicate a stronger degree of covalent bonding resulting in higher hardness. The so-called ”Frantsevich rule” is widely used as a criterion for brittleness, which is based on the tables of elastic properties in the book by Frantsevich et al Frantsevich et al. (1983). According to this rule, compounds with a Poisson ratio of $\nu_{cr}\leq 0.33$ are brittle and those with $\nu_{cr}\geq 0.33$ are ductile or malleable. It should be noted that this value is not given explicitly in Reference Frantsevich et al. (1983). It is based on properties reported for the materials tabulated by Frantsevich and was later accepted as empirical rule.

Referring to the original work Pugh (1954), Pugh also did not suggest a criterion. However, Pugh only mentioned that Ir was the least malleable metal ($k=1.74$) and Au was the most malleable metal ($k=6.14$). Based on present knowledge, it is obvious that Ir is hard and brittle Claeys (2011), and hence, the critical value could possibly be between 1.74 and 6.14. Based on the relation between $k$ and $\nu$, it follows that the critical Poisson’s ratio $\nu_{cr}\approx 1/3$ of Frantsevich’s rule corresponds to a critical Pugh’s ratio of $k_{\rm cr}\approx 2.66$ ($k^{-1}_{\rm cr}\approx 3/8$), so that the two empirical rules only differ in the exact number that distinguishes between the two types of behaviors.

Generally, it may be considered that materials with $\nu=0$ are absolutely brittle, whereas those with $\nu=1/2$ are perfectly ductile. Christensen Christensen (2013) used the failure theory to describe the mechanical properties. He introduced a nanoscale variable $\kappa$, which characterizes the relative size of the bond bending and the bond stretching effects. Further, he related it to renormalized Poisson’s or Pugh’s ratios and defined the ductility $D$ by:

[TABLE]

According to this relation, materials with $D=0$ are absolutely brittle, those with $D=1$ are perfectly ductile, and the brittle–ductile transition takes place at $D_{B/D}=1/2$. The latter implies that

[TABLE]

and thus, the critical values of the Poisson’s and Pugh’s ratios defined as,

[TABLE]

$\nu_{B/D}\approx 0.31$ is close to $\nu_{cr}$ of Frantsevich’s rule. Further, it corresponds to a critical Pugh’s ratio of $k_{B/D}\approx 2.3$ ($k_{B/D}^{-1}\approx 0.44$), that is clearly larger than the so-called Pugh’s criterion. It is worthwhile to note that the ductility of a Cauchy solid becomes $D_{\rm Cauchy}=(3/5)^{2}=0.36$, which implies that Cauchy solids should be more brittle than ductile.

The behavior of the Heusler compounds is shown in Figure 6 that relates Poisson’s ratio and Christensen’s ductility to Pugh’s ratio. It is obvious that the compounds —and in particular Cu2MnAl— are far away from the extreme elements, which are elemental Au, and C in the form of diamond, as the most ductile and most brittle elements, respectively.

The above discussion and the experimental observations on various Heusler compounds lead to the conclusion that Frantsevich’s rule (or better Christensen’s ductility criterion) is suitable for the compounds studied here, whose behaviors lie on the border-line between brittleness and ductility.

The crystalline structure of Cu2MnAl is shown in Figure 2(a) and (d). Figure 2(b) and (c) show the charge densities and Young’s modulus of Cu2MnAl in the (110) plane. Figure 2(e) and (f) show the charge densities and Young’s modulus of Cu2MnAl in the (001) plane. Bader’s QTAIM analysis was used to analyze the charge density and magnetic moments. The results of the QTAIM analysis are listed in Table 4, where a charge transfer is observed. On the average, about 0.9 electrons are transferred from the Mn and Al atoms to the Cu atoms with relatively larger contribution from the Al atoms. The Mn atoms carry a magnetic moment of 3.45 $\mu_{B}$, whereas Cu and Al exhibit only a negligible polarization.

The QTAIM critical points of Cu2MnAl and their properties are summarized in Table 5. There are, indeed, three different nuclei that act as attractors. The cage critical point $c$ is found between Mn and Al along the [001] axis and acts as a repeller, which is the absolute minimum of the charge density. Further, two bond critical points $b_{1,2}$ are located between Cu and Al ($b_{1}$), and between Cu and Mn ($b_{2}$). The third bond critical point is located in between the Mn atoms along the [001] direction. When the two ring critical points $r_{1,2}$ are also considered, the Morse sum of the numbers $n_{i}$ of the different critical points vanishes ($n_{n}-n_{b}+n_{r}-n_{s}=0$), as expected for crystals.

The analysis of the bonding type with the properties of the critical points is discussed in Reference Mori-Sanchez et al. (2002). Metallic systems exhibit a flat electron density $\rho$ throughout the valence region. The flatness $f=\rho^{c}_{\min}/\rho^{b}_{\max}$ is a measure of the metallicity of the compounds. $\rho^{c}_{\min}$ is the cage critical point, at which, the density is minimum, and $\rho^{b}_{\max}$ is the highest density among all the bond critical points. For Cu2MnAl, it is $f=0.684$. This is of the same order of magnitude as the flatness in Cu or Fe (both $\approx 0.57$; see Reference Mori-Sanchez et al. (2002)), whereas compounds with covalent bonding typically have $f$ ratios of less than 0.1. From the large electronic flatness, the bonding in Cu2MnAl is clearly metallic.

In the following, the results of the calculations for various Co2-based Heusler compounds are discussed. These compounds, containing Mn in particular, are of much interest in spintronic applications. The Mn containing compounds are discussed in the first part. The second part discusses the variation of the $3d$ transition metal in Co${}_{2}TM$ when the main group element $M$ is attached to Al or Si, and the $3d$ transition metal $T$ ($T=$ Sc, Ti, V, Cr, Mn, Fe) is varied.

III.1 Results for Co2Mn$M$ ($M=$ main group element)

In this section, the elastic and mechanical properties of the Mn containing Heusler compounds Co2Mn$M$ ($M=$ Al, Ga, In, Si, Ge, Sn, Pb, Sb, Bi) are discussed. Table 6 compares the mechanical properties of the Co2Mn$M$ compounds. It may be noted that the compounds with main group heavy elements (In, Pb, Sb, Bi) have not been synthesized up to now and Heusler compounds with this composition most probably do not exist. They are used here to complete the trends when changing the main group elements. It should also be noted that those compounds are stable — at least from the Born-Huang criteria (These criteria are discussed in the Appendix).

The relaxed lattice parameters $a_{\rm opt}$ agree well with the experimental values. They exhibit the typical trend that the lattice parameter increases with the nuclear charge ($Z$) of the main group element. The elastic constants of the Co2Mn$M$ compounds follow the general inequality $B>c_{44}>G>c^{\prime}>0$. Here, the isotropic shear or rigidity modulus $G$ is not the main constraint on stability. The smallest values are obtained for $c^{\prime}=C^{\prime}/2$, that is the tetragonal shear modulus is the limiting parameter for the stability of the cubic $L2_{1}$ structure of the investigated compounds. The values of $c^{\prime}$ are in the range of 37–68 GPa and thus far above the values required to force tetragonal instabilities in the vicinity of ambient pressure. The lowest values for the bulk moduli are obtained for the compounds containing the main group, heavy elements, which exhibit large lattice parameters. The Pb and Bi containing compounds exhibit comparably low values of the rigidity moduli. Within each group, the values of the hardness parameter decrease with increasing $Z$ of the main group elements. As compared to the previous calculations Candan et al. (2013), the elastic constants and bulk modulus fit quite well for most compounds, and the differences are lesser than 20 GPa for each quantity.

Pugh’s ratio $k$ of the Co2Mn$M$ compounds ranges from about 1.83 to 2.46 with a mean value of $\overline{k}\approx 2.1$. Similarly, Poisson’s ratio $\nu$ falls also in a narrow range 0.27–0.32 with a mean value of $\overline{\nu}\approx 0.29$. A convenient way of quantifying the degree of off-axis anisotropy in the elastic constants for a cubic crystal is to use the Zener ratio. Here, Zener ratio ($A_{e}$) exhibits no trend of a dependency on the main group element. It exhibits the smallest value for Co2MnBi (1.84) and the largest value for Co2MnIn (4.14). The latter value is exceptional (compare with the next section) and points to a large elastic anisotropy in the material. Another method is the usage of the so-called universal anisotropy index ($A_{U}$), which shows the same tendency as the Zener ratio. Especially for Co2MnSb, in spite of the value of the anisotropy being in a reasonable range, this compound does not exist, and tested samples exhibit phase separation Ksenofontov et al. (2006). The anisotropy does only judge on the structural stability but not on the chemical stability.

Figure 7 shows the Blackman’s diagram using the elastic data of the Co2Mn$M$ Heusler compounds given in Table 6. All $F$ ratios fall in the allowed range for mechanical stability. The values of $F_{12}$ appear close together in a region around the Cauchy line where the Cauchy pressure vanishes. They also fall in the region of positive Zener ratios ($A_{e}>1$). The figure suggests the type of bonding, covalent or metallic. A positive Cauchy pressure is suggestive of greater degree of metallic bonding. On the contrary, when the Cauchy pressure is negative, there appears to be greater degree of covalent bonding. As discussed in the foregoing and in agreement with the Poisson’s ratio, all the studied compounds are metallic and on the borderline between brittleness and ductility. This is in agreement with experiments, in which all these compounds have a silvery metallic luster.

In fact, directional dependent plots of rigidity $G(\hat{r})$ and Young’s $E(\hat{r})$ moduli are an alternative visual way of showing the Zener anisotropy. The implication of the elastic anisotropy on the elastic moduli will be illustrated for the two borderline cases with the largest (In) and the smallest (Bi) anisotropy. The three dimensional distributions of $G(\hat{r})$ and $E(\hat{r})$ are shown in Figure 8 for Co2MnIn and Co2MnBi.

The anisotropy of Young’s modulus of the In and Bi containing compounds is displayed in Figures 8(a) and (b) that show the three dimensional distribution $E(\hat{r})$. The pronounced anisotropy of the In containing compound is clearly visible. Figures 8(c) and (d) show the three dimensional distribution $G(\hat{r})$ of the rigidity moduli of the two compounds. Again, the differences in the anisotropy of the moduli are clearly visible. Comparing the distribution of the moduli, it is obvious that Young’s modulus is largest in the $\left<111\right>$-type directions whereas the rigidity modulus is largest in the $\left<100\right>$-type directions, that is along the cubic axes. This behavior is generic for all compounds listed in Table 6 and a direct consequence of the condition $A_{e}>1$. These anisotropic compounds exhibit different responses to stress or strain when tested in different directions. The anisotropic property is particularly important for applications where mechanical stress is applied to the materials, directly or by thermal expansion and contraction.

III.2 Results for Co${}_{2}TM$ ($T=3d$-metal, $M=$ Al, Si)

In this section, the influence of the $3d$ transition metal on the elastic and mechanical properties of selected Co2-based Heusler compounds is discussed. Table 7 compares the elastic properties of the Co${}_{2}TM$ ($T=$ Sc, Ti, V, Cr, Mn, Fe and $M=$ Al, Si) compounds. Al and Si were selected as the main group elements, because they exhibit the most complete series over the $3d$ transition metals that exist in reality. The Sc compounds as well as Co2CrSi have not been synthesized up to now and do not possibly exist. They are used here to complete the trends of the properties when changing the transition metal. Similar to the compounds with heavy main group elements reported above, those compounds are stable — at least from the Born-Huang criteria.

The elastic constants of the Co${}_{2}TM$ compounds follow the general inequality $B>c_{44}>G>C^{\prime}/2>0$ as was also observed above for the Mn-containing compounds with varying main group elements. As in the earlier case, the tetragonal shear modulus $c^{\prime}=C^{\prime}/2$ is the most critical of the moduli for crystal stability. The bulk moduli are slightly greater in the Si containing compounds as compared to the Al- containing compounds. The Young’s and rigidity moduli fall in the ranges (237.2–287.9) GPa and (93.4–112.3) GPa, respectively. Our calculated values of the bulk moduli and elastic constants for Co${}_{2}T$Si (except $T$ = Sc) agree well with those reported by Chen et al Chen et al. (2006). Only the value of $c_{11}$ of Co2VSi exhibits a large deviation of 20%, whereas all others deviate by less than 7%.

As observed above for the Mn-containing compounds, all Co${}_{2}TM$ compounds are between brittle and ductile from Frantsevich’s rule based on Poisson’s ratio ($\nu_{cr}\leq 0.33$) and Pugh’s criterion ($k_{cr}\leq 1.75$). The Co2-based compounds synthesized in our laboratories turned out to be mostly brittle in accordance with the prediction of our calculations. Further discussion on the elastic properties will be presented using the Blackman’s diagram that is shown in Figure 9 for the Co${}_{2}TM$ Heusler compounds. The values of $F_{12}$ fall in a very narrow range about the Cauchy line. All Si-containing compounds exhibit a positive Cauchy pressure, whereas $p\rm{{}_{C}}<0$ for the Al compounds with Sc, Mn, or Fe. It is worthwhile to note that the Al-containing Co2 compounds tend to antisite disorder, that is, they exhibit a $B2$ type rather than a $L2_{1}$ type crystalline structure. It is interesting to note that the hypothetical compound Co2ScAl is assumed to be nearly isotropic and very similar to an ideal Cauchy solid. Its universal anisotropy of only 1% is remarkable. Even though the values of $c_{44}/c{11}$, $k$, and $\nu$ still deviate from the ideal Cauchy values by about 10%, out of all the compounds investigated here, it is closest to a Cauchy solid.

The elastic Zener (universal) anisotropy ranges from 1.08 (0.01) for Co2ScAl to 3.48 (1.88) for Co2CrSi (Co2CrAl). Comparing the elastic anisotropy of compounds that are well known to crystallize in an ordered $L2_{1}$ structure and those that are known to tend to disorder or where no successful synthesis is reported up to now, the Zener ratios for the most stable compounds are in the range $1.62<A_{e}<2.86$.

The anisotropy of Young’s moduli and the rigidity moduli of Co2ScAl, Co2CrAl, Co2ScSi and Co2CrSi are displayed in Figures 10 and 11 that show the three dimensional distribution $E(\hat{r})$ and $G(\hat{r})$ as was also plotted above for the Mn containing compounds. The more pronounced anisotropy of the Cr containing compound is clearly visible. The differences in the anisotropy of the moduli between the two compounds are clearly visible, as observed above.

Comparing the distribution of the moduli, the Young’s modulus is largest in the $\left<111\right>$-type directions, whereas the rigidity modulus is largest in the $\left<100\right>$-type directions for most compounds, which are listed in Table 7. The only exception is Co2ScAl, where $A_{e}$ is close to unity and thus the distributions are nearly spherical, which shows a tendency to distortion in the $\left<100\right>$ direction.

IV Derived properties

This section summarizes the physical properties of the compounds that are derived from the calculated elastic constants. Table 8 summarizes the results for the Co2-based, Mn containing Heusler compounds with variation of the main group element Co2Mn$M$ ($M=$ main group element). Properties of the Co2-based Heusler compounds Co${}_{2}TM$ with varying $3d$ transition metals $T$, and Al and Si as the main group elements $T$ are summarized in Table 9

The Debye temperature $\Theta^{\rm ac}_{D}$ and Grüneisen parameters $\zeta^{\rm ac}$ are estimated in the acoustical approximation from the mode averaged sound velocities $\overline{v}$. These quantities depend, in addition to the elastic constants, on the mass density of the materials. The other two properties, melting temperature and hardness are exclusively based on the elastic constants. The melting temperature $T_{m}^{c}$ is roughly estimated from $c_{11}$, and the hardness $H_{V}\rm^{C}$ results from Pugh’s and rigidity moduli. The underlying ideas and equations are given in the Appendix. Table 8 and Table 9 summarize the properties derived for various Co${}_{2}TM$ Heusler compounds. In addition, the density $\rho$ and the molecular mass $m$ are given for completeness. Interestingly, the remaining physical properties do not appear to depend much on the composition. However, some clear trends are recognized on closer inspection.

Table 8 reveals, for the Mn containing compounds, the trend that the sound velocities calculated from the elastic constants and the Debye temperature in the acoustical approach decrease with $Z$ of the main group element, whereas the Grüneisen parameters are all nearly the same for the different compounds. The average values of $\zeta^{ac}$ is about 1.7. The acoustical Debye temperatures are in the range of 312 K–576 K with average values of about $(440\pm 70)$ K. The range of validity for the melting temperature $T_{m}^{c}$ is $\pm 300$ K for the approximation used here, and thus is on the same order as that of the spread of the calculated values. This leads to the estimate that the melting temperatures of the compounds are about ($1960\pm 200$) K. The Vickers’s hardness has average values of the order of ($8.5\pm 1.8)$ GPa.

From Table 9, it is found that the sound velocities calculated from the elastic constants exhibit only small changes among the different transition metals. As a direct consequence, the values for the Debye temperature or Grüneisen parameters in the acoustical approach are all nearly the same for the different compounds. There is no evidence of a distinguishable dependence on the $T$ element, as already mentioned in the previous sections. The $\zeta^{ac}$ values of Co2ScAl, Co2TiAl, Co2TiSi, Co2MnSi and Co2ScSi, however, are slightly below those of the remaining compounds that exhibit average values of 1.57 for Al and 1.72 for Si. The acoustical Debye temperatures are in the range of 533 K–576 K with average values of $(565\pm 10)$ K and $(555\pm 10)$ K for the Al and Si compounds, respectively. The similarity of the acoustical parameters arise from similar masses of Al and Si that determines to a large extent, the vibrational properties of the compounds. The range of validity for the melting temperature $T_{m}^{c}$ is $\pm 300$ K for the approximation used here and is of the same order as the spread of the calculated values. This leads to the estimate that the melting temperatures should be of the order of ($2180\pm 50$) K for the Al containing compounds and ($2260\pm 60$) K for the Si containing compounds. All calculated melting temperatures are consistently larger than the experimental values Yin et al. (2013). On the average, the Al containing compounds exhibit larger hardness values as compared to the Si containing compounds. In both these groups, the Cr containing compounds exhibit clearly lower values for the calculated hardness as compared to the other transition metals. Neglecting the Cr values, the hardness exhibits average values of ($12.4\pm 1.4)$ GPa for the Al and $(9.9\pm 1.1)$ GPa for the Si containing compounds. The only two reported experimental values of hardness for Co2MnGe and Co2MnSi are known to be 7.3 and 7.9, respectively Ouardi et al. (2011). Although the experimental values are smaller than the predicted values, the tendency is the same. This is expected from the approximate nature of the model used.

Figure 12 compares the slowness surfaces of Co2ScAl and Co2MnSi. The latter was considered, since it is known from experiments to be very stable and to exhibit very low disorder. Further, it is the Heusler material with highest tunneling magneto resistance (TMR) ratios. The isotropic elastic behavior of Co2ScAl is reflected in the nearly spherical distributions describing its three slowness surfaces. The shear modes are nearly degenerate and exhibit kiss singularities in the high symmetry directions. The behavior of the slowness surfaces of Co2MnSi is typical for most of the investigated Co2-based Heusler compounds, and its shape is similar to that of Cu2MnAl. Its lower asymmetry as compared to Cu2MnAl results in less pronounced differences between minima and maxima of the slowness.

To use slowness as a parameter for determination of the elastic constants, measurements along different high symmetry directions may be used. For the pressure wave, it may be found, for example, the following slowness eigenvalues $s$: $s_{p}^{001}=\sqrt{\rho/c_{11}}$, $s_{p}^{110}=\sqrt{\rho/c_{44}}$, and $s_{p}^{111}=\sqrt{3\rho/(c_{11}+4c_{44}+2c_{12})}$ (superscript indices indicates the high symmetry direction [hkl], subscript index indicates the direction of polarization). The shear waves are degenerate in the [001] and [111] directions with values $s_{s1,s2}^{001}=\sqrt{\rho/c_{44}}$, and $s_{s1,s2}^{111}=\sqrt{3\rho/(c_{11}+c_{44}-c_{12})}$, respectively. The two non degenerate values for the [110] direction are $s_{s1}^{110}=\sqrt{2\rho/(c_{11}-c_{12})}$ and $s_{s2}^{110}=\sqrt{2\rho/(c_{11}+2c_{44}+c_{12})}$.

V Summary

The elastic and accompanying physical properties of Heusler compounds, Cu2MnAl as well as the Co${}_{2}YZ$ family, have been calculated. The present results for Cu2MnAl are in good agreement with experiments. The directions of the largest Young’s modulus indicates the high-fracture energy directions. According to the Bader’s QTAIM analysis, the bonding in Cu2MnAl is clearly metallic. However, Cu2MnAl has one of the highest anisotropies as compared to the Co2-based compounds. The Debye temperature, where only acoustic vibrational modes contribute, is about 397 K, which is lower than that of most of the studied compounds based on Co2.

Based on the calculation of their elastic properties, the crystalline stability of Co2-based cubic Heusler materials was assessed. The elastic constants of all the studied compounds follow the general inequality $B>c_{44}>G>c^{\prime}>0$ such that the rigidity modulus $G$ is the main constraint on stability. The results of our calculations demonstrate that all the studied compounds are close to the borderline between brittle and ductile. From the elastic point of view, they mainly exhibit bonding behavior between those of covalent and metallic. For most of the studied stable compounds, the universal anisotropy index is in the range $0.57<A_{U}<2.73$. All the studied stable compounds are most stiff in the $\left<111\right>$-type directions. The detailed analysis of all the compounds revealed that Pugh’s criterion for the ductile–brittle transition ($k^{\rm Pugh}_{\rm cr}\approx 1.75$) should be replaced by Christensen’s criterion ($k_{\rm cr}\approx 2.27$ or $k_{\rm cr}^{-1}\approx 0.44$).

For Co2-based compounds, when the nuclear charge $Z$ of the main group element increases, the lattice parameters also increase but the values of the hardness parameter decrease. Here, the Zener ratio ($A_{e}$) and the universal anisotropy index ($A_{U}$) show the same tendency. It should be mentioned however, that the anisotropy does only judge on the structural stability but not on the chemical stability, as is seen especially for the case of Co2MnSb, that does not exist as a pure compound. The Debye temperature in the acoustical approach decreases with $Z$, but there is no evidence for a distinguishable dependence on the $T$ element for the Co${}_{2}TM$ compounds. The Grüneisen parameters in the acoustical approach are all nearly the same for the different compounds. The hardness shows the same tendency. Finally, it is found that Co2ScAl is close to an ideal Cauchy solid and is predicted to be the most hard material in the investigated series.

The calculated material properties can be applied quite reliably to bulk materials. On the other hand, the prediction of stability could be exploited in any compounds in particular to those when the possibility of structural phase transition in crystals is investigated.

Appendix A Cubic elastic constants, elastic moduli, and elastic stability.

The basics of the elastic properties of solids are described in the book by Nye Nye (1985). Here the focus is on the equations for cubic compounds, remarks on tetragonal and hexagonal compounds are found in Reference Wu et al. (2017a). A lattice $A$ can be transformed to a new deformed lattice $A^{\prime}$ by the strain matrix $\epsilon$. The strain matrix $\epsilon$ is symmetric and contains six different strains $e_{i}$. By Hooke’s law, the elastic relation between strain ($\epsilon$) and stress ($\sigma$) is:

[TABLE]

where ${\bm{C}}$ is the elastic stiffness matrices, and the relations between the compliance matrix ($\bm{S}$) and the stiffness matrix is

[TABLE]

In the most general case, the elastic matrix is symmetric and on the order of $6\times 6$ Nye (1985). In triclinic lattices, it contains 21 independent elastic constants. This number is largely reduced in highly symmetric lattices. In cubic lattices, only the three elastic constants $c_{11}$, $c_{12}$, and $c_{44}$ are independent. The elastic matrix for all classes of cubic crystals has the form (zero elements are denoted by dots):

[TABLE]

The matrix has 6 eigenvalues out of which only three are different. These three different values o the eigenvalues are:

• •

$C_{1}^{c}=2c_{12}+c_{11}$ (nondegenerate),

• •

$C_{2,3}^{c}=c_{11}-c_{12}$ (twofold), and

• •

$C_{4,\ldots,6}^{c}=c_{44}$ (threefold degenerate).

They correspond to the bulk, the tetragonal shear, and the shear moduli as will be shown below. The crystal becomes unstable when one of the eigenvalues becomes zero or negative.

For an isotropic system, the elastic matrix $C^{\rm iso}$ contains only the two constants $c_{11}$ and $c_{12}$, whereas the remaining diagonal elements of the matrix are determined by $c_{44}=(c_{11}-c_{12})/2$.

In cubic systems, the relations between the elastic constants $c_{ij}$ and the elements of the compliance matrix $s_{ij}$ are given by,

[TABLE]

The bulk modulus is defined by the elastic constants. For cubic materials it is given by,

[TABLE]

Born and co-workers developed the theory of stability of crystal lattices Born (1940); *Mis40; *BFu40; *BMi40; *Fue41a; *Fue41b. The Born–Huang Born and Huang (1956) elastic stability criteria for a cubic crystal at ambient conditions Wang et al. (1993b) are given by,

• •

$c_{11}+2c_{12}>0$

• •

$c_{44}>0$

• •

$c_{11}-c_{12}>0$

that is, the bulk, $c_{44}$-shear, and tetragonal shear moduli have to be all positive. The criteria are referred to as spinodal, Born’s shear Born (1939) and Born criteria, respectively. The first criterion defines the spinodal pressure,

[TABLE]

whereas the last criterion is often used to define an additional elastic constant,

[TABLE]

This constant is also called tetragonal shear modulus. In some studies, $c^{\prime}=(c_{11}-c_{12})/2$ is used instead, because the tetragonal instability is observed when the hydrostatic pressure becomes $2p>C^{\prime}$, that is $p>c^{\prime}$. In detail, $G_{110}=c^{\prime}$ is the single-crystal shear modulus for the (110) plane along the [110] direction. The single-crystal shear modulus for the (100) plane along the [010] direction is $G_{100}=c_{44}$. It is related to a tetragonal deformation and large values denote high stability of the crystal with respect to tetragonal shear.

The Cauchy pressure for cubic crystals is defined using the Cauchy relation as,

[TABLE]

For single cubic crystals, the shear modulus $G$, Pugh’s ratio $k=B/G$, and the Poisson’s ratio $\nu$ are calculated from the elastic constants using the following relations:

[TABLE]

The first and third Born criteria restrict the range of Poisson’s ratio to $-1\leq\nu\leq 1/2$.

Polycrystalline materials consist of randomly oriented crystals and thus a description of their elastic properties requires only two independent elastic moduli: the bulk modulus ($B$), and the shear modulus ($G$). The relationships between the single-crystal elastic constants and the polycrystalline elastic moduli are given by the Voigt Voigt (1928) or Reuß Reuß (1929) averages. Voigt’s approach uses the elastic stiffnesses $c_{ij}$, whereas Reuß’s approach uses the compliances $s_{ij}$. The bulk moduli in Voigt’s ($B_{V}$) and Reuß’s ($B_{R}$) approach are equal for cubic crystals and given by:

[TABLE]

The isotropic shear or rigidity modulus $G=[G_{V}+G_{R}]/2$ is defined by Voigt’s $G_{V}$ and Reuß’s $G_{R}$ shear moduli, where,

[TABLE]

Accordingly, Poissons’s ratio $\nu$ and Young’s modulus $E$ of polycrystalline cubic materials are calculated from the equations using the averaged bulk and rigidity moduli as,

[TABLE]

In cubic crystals, the bulk modulus is isotropic. However, rigidity and Young’s moduli not isotropic. The directional dependence of Young’s modulus $E(\hat{r})$ is defined by the ratio of longitudinal stress to strain. For cubic systems, the three dimensional distribution is given by,

[TABLE]

where $s=s_{11}-s_{12}-s_{44}/2$, and $F_{lmn}=(\hat{x}^{2}\hat{y}^{2}+\hat{y}^{2}\hat{z}^{2}+\hat{z}^{2}\hat{x}^{2})$ is the orientation function of a cubic single crystal specimen given in terms of the direction cosines ($l:=\hat{x}$, etc.). It is obvious that $E(\hat{r})$ becomes isotropic for $s=0$. Hence, Zener ratio or the elastic anisotropy is defined for cubic crystals as,

[TABLE]

The cubic elastic anisotropy may be used as another important physical quantity for the description of structural stability. Materials exhibiting large $A_{e}$ ratios occasionally show a tendency to deviate from the cubic structure. Materials with negative Zener ratio ($A_{e}<0$) violate at least one of Born’s criteria and are mechanically instable.

Ranganathan and Ostoja-Starzewski Ranganathan and Ostoja-Starzewski (2008) summarized the existing anisotropy theories and developed a so-called universal anisotropy index $A_{U}$ that is calculated for cubic crystals, using the condition $B_{V}=B_{R}$ by the simplified equation,

[TABLE]

Similar to the case of the Young’s modulus, the directional dependence of the rigidity modulus $G(\hat{r})$ is defined by Good (1941),

[TABLE]

where $\chi_{lmn}=\hat{x}^{2}\hat{y}^{2}\hat{z}^{2}$. The last term in Eq.  (20) is the so-called bending–torsion correction (or difference) if $G_{0}$ is defined as the ”true” rigidity modulus Good (1941). $G_{0}$ becomes isotropic for $A_{e}=1$. The bending–torsion correction vanishes for the highly symmetric $\left<100\right>$, $\left<110\right>$, and $\left<111\right>$-type directions.

The Cauchy criterion of vanishing Cauchy pressure for crystals with cubic symmetry is $c_{12}=c_{44}$. The conditions required to satisfy this Cauchy relation are:

• •

Only central forces take part in the interaction between the atoms.

• •

Only harmonic forces exist between the atoms. Anharmonicity will destroy the Cauchy relations.

• •

The atoms are located at the centers of symmetry.

• •

Thermal effects and initial stress are neglected.

From the isotropy ($c_{12}=c_{11}-2c_{44}$) and Cauchy ($c_{12}=c_{44}$) relations, only one independent elastic constant ($c_{11}=3c_{12}=3c_{44}$) would remain for cubic crystals. This has the result that Pugh’s ratio of a cubic, isotropic solid following Cauchy’s relation becomes $k_{\rm Cauchy}=5/3=1.6\overline{6}\approx 1.7$. At the same time, Poison’s ratio simplifies to $\nu_{\rm Cauchy}=1/4$. The elastic matrix of such an ideal Cauchy solid has the form (zero elements are denoted by dots),

[TABLE]

The three different eigenvalues of ${\bm{C}}^{\rm cubic}_{\rm Cauchy}$ are $5c_{11}$, $2c_{11}$, and $c_{11}$, which are nondegenerate, twofold degenerate, and threefold degenerate, respectively.

Apart from the elastic moduli, a few more important physical quantities can be derived from the elastic constants. The volume ($\kappa$) and linear ($\beta$) compressibilities of cubic crystals are isotropic and given by,

[TABLE]

Appendix B Derived physical properties from cubic elastic constants.

In the bond-orbital model, Kleinman’s internal displacement parameter is defined by Harrison (1989):

[TABLE]

It describes the relative positions of atoms in different sublattices under volume conserving strain distortions for which the positions are not fixed by symmetry anymore. $\zeta$ vanishes if no internal displacements appear. $\zeta=1$ when the bond lengths are unchanged and $\zeta=-1/2$ when the bond angles are unchanged, both for linear strain.

In the quasi-harmonic approach, the Debye temperature $\Theta^{qha}_{D}$ depends upon the volume of the crystal. For every volume $V$, $\Theta^{qha}_{D}(V)$ is rigorously defined in terms of the elastic constants through a spherical average of the three components of the sound velocity. The isotropic approximation, allows to evaluate $\Theta^{qha}_{D}$ using the expression Robie and Edwards (1966); Poirier (1991),

[TABLE]

where $\hbar$ is Planck’s constant, $k_{B}$ is Boltzmann’s constant and $n$ is the number of atoms in the primitive cell with volume $V$ unit ($n=4$ in the case of Heusler compounds and the volume of the primitive cell or 16 for the cubic cell with lattice parameter $a$), $B$ is the adiabatic bulk modulus of the crystal and $M$ the mass of the compound corresponding to $V$. Finally, $f_{\nu}$ is a function of the Poisson ratio $\nu$ Francisco et al. (1998):

[TABLE]

Another important mechanical property is the hardness of a material Gilman (2009). Other than for the elastic moduli, there is no straightforward theory to calculate the hardness directly from the elastic constants. However, several models were developed to relate the hardness of a material to the elastic moduli. Pugh Pugh (1954) related the Brinell hardness $H_{B}$ of pure metals to their shear modulus $G$ by $H_{B}=G\>b/c$, where $b$ is the Burger’s vector of the dislocation and $c$ is a constant for all metals of the same structure. Teter Teter (1998) obtained the semi-empirical relation $H_{V}^{\rm T}\approx 0.151G$ between Vicker’s hardness $H_{V}$ and the rigidity modulus $G$. Recently, Chen et al Chen et al. (2011) gave a semi-empirical relation between Vicker’s hardness and the product of the squared Pugh’s modulus ($k=B/G$) ratio and the shear modulus $G$ as,

[TABLE]

Also for cubic metals only, Fine et al. Fine et al. (1984) obtained an approximate linear relationship between the melting temperature $T_{m}$ and the elastic constant $c_{11}$. The $T_{m}^{c}$ of various cubic metals was $\pm 300$ K within a linear dependence when estimated from the following empirical equation:

[TABLE]

The elastic constants and moduli also allow estimation of the averaged sound velocity $\overline{v}$,

[TABLE]

From the longitudinal ($v_{l}$) and transverse ($v_{t}$) elastic wave velocities of isotropic materials the Debye temperature can be estimated, where $v_{l}$ and $v_{t}$ are,

[TABLE]

where, $\rho$ is the mass density of the material. From the average sound velocity at low temperatures, the Debye temperature can be estimated by using the relation Anderson (1963):

[TABLE]

where $N_{A}$ is Avogadoro’s number. Other parameters are the same as in the case of the Debye temperature calculation in the quasi-harmonic approach.

In solids, the Grüneisen parameter $\zeta^{ac}$ is also related to the sound velocities. Belomestnykh Belomestnykh (2004) derived this Grüneisen parameter using,

[TABLE]

The above described acoustical properties concern averages and may be used for polycrystalline materials. Acoustical spectroscopy is used, indeed, also for investigation of the single crystal elastic constants. The directional dependence of the phase velocity $v$ is found from Christoffel’s equation:

[TABLE]

where $\Gamma_{ij}=c_{ijkl}l_{j}l_{l}$ is the Christoffel tensor built from the elastic constants and the direction cosines $l_{i}$ ($i=1\ldots 3$). $\rho$ is the density, $\delta_{ij}$ is the Kronecker symbol and $U$ is the polarization vector.

In cubic systems, only 3 elastic constants are independent and the components of the Christoffel tensor are reduced to,

[TABLE]

with $l_{1}=\sin(\theta)\cos(\phi)$, $l_{2}=\sin(\theta)\sin(\phi)$, and $l_{3}=\cos(\theta)$ in polar co-ordinates. In greater detail, the elements of the Christoffel tensor are (note that: $\Gamma_{ij}=\Gamma_{ji}$),

[TABLE]

In case of an ideal Cauchy solid the Christoffel tensor is further reduced to,

[TABLE]

The results for ideal Cauchy solids are three eigenvalues: $\rho v_{p}^{2}=c_{11}$ for the compression wave, and $\rho v_{s1,s2}^{2}=c_{11}/3$ for the twofold degenerate shear wave. Both the modes, shear $s$ and pressure $p$, are independent of the propagation direction and their slowness surfaces appear spherical with $s_{p}=\sqrt{\rho/c_{11}}$ and $s_{s}=\sqrt{3\rho/c_{11}}$.