Tuning the topological nontrivial nature in a family of alkali-metal-based inverse Heusler compounds: A first-principles study

Accepted Manuscript Tuning the topological nontrivial nature in a family of alkali-metal-based inverse Heusler compounds: A first-principles study Tingzhou Li, Yang Wu, Yilin Han, Xiaotian Wang PII: DOI: Reference:

S0304-8853(18)30854-0 https://doi.org/10.1016/j.jmmm.2018.05.025 MAGMA 63937

To appear in:

Journal of Magnetism and Magnetic Materials

Received Date: Revised Date: Accepted Date:

22 March 2018 5 May 2018 9 May 2018

Please cite this article as: T. Li, Y. Wu, Y. Han, X. Wang, Tuning the topological nontrivial nature in a family of alkali-metal-based inverse Heusler compounds: A first-principles study, Journal of Magnetism and Magnetic Materials (2018), doi: https://doi.org/10.1016/j.jmmm.2018.05.025

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Tuning the topological nontrivial nature in a family of alkali-metal-based inverse Heusler compounds: A first-principles study Tingzhou Lia, Yang Wub, Yilin Hana and Xiaotian Wanga,* School of Physical Science and Technology, Southwest University, Chongqing 400715, PR China;

a

School of Materials Science and Engineering, Guilin University of Electronic Technology, Guilin

b

541004, PR China [email protected] or [email protected]

Abstract Based on the first-principles calculations, the total energy and electronic structures of some new inverse Heusler-based ternary intermetallic compounds X2YZ (X = alkali metals: Li, Na, K, Rb; Y = Ag, Pd, Cu; Z = Sb, Te, As) have been investigated in detail. Our results reveal that most of them are stable in inverse Heusler structure and naturally exhibit band-inversion nature without the effect of spin-orbit coupling (SOC). That is to say, most of our studied compounds are newly designed topological semi-metals. We also study the effects of the uniform strain and the spin-orbit coupling on the band inversion. Moreover, for our predicted compounds, all of them have a negative formation energy, which makes them possible in material growth. Keywords: Topological semi-metals; Heusler compound; First-principles

calculation; electronic structures.

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1. Introduction Semi-metals have attracted more and more attention in condensed physics and materials science & engineering in the past few years due to their interesting physical phenomena issuing from the distinctive Fermi-surface (FS) and bulk topological nontrivial nature, which opens exciting opportunities to design new materials with novel behaviors [1, 2]. Till now, some new semi-metallic systems have been predicted, such as bismuth [1], WTe2 [2], NbSb2 [3], some heavy and light half-Heusler compounds [4-6], Cd3As2 [7] and Na3Bi [8] with Dirac nodes in their bulk FS). Among them, lots of heavy half-Heusler semi-metals with nontrivial band inversion has been recently reported in theoretical regions as well as in experimental regions in very recent year. And these type materials exhibit many

special

physical

properties,

such

as

heavy

fermions,

non-centrosymmetric superconductors and possible topological ordering behavior [9-11]. For example, recent ARPES works [12] have confirmed the existence of the predicted topological surface states in LuPtBi, although they largely overlap with the bulk valence bands. Moreover, magneto-transport investigations on LuPtBi indicate very large field-induced MR and ultrahigh carrier mobilities [5]. The scope of heavy Heusler semi-metals has also been extended to the case of inverse Huesler compounds. Namely, Zhang et al. [13] have investigated the topological band structures of the inverse Heusler

2

compounds X2YZ (X = Sc, Y, La, R; Y = Ru, Re, Os; Z = Sb, Pb, Bi) by the first-principle calculations and they found that most of the studied inverse Heusler bulks naturally show topological insulating nature in their work. We should noted that almost all the motioned heavy Heusler compounds contain heavy elements (i.e., atomic number Z>60) and have 18 valence electrons. In 2014, Wang et. al [14] predicted a new series of light half-Heusler semi-metals with 8 valence electrons and without heavy elements (i.e., atomic number Z<60). Also, they found that, for the bulk electronic structures, the s-p band inversion occurred without SOC. In this paper, by means of first-principles calculation, we performed a complete study on the band topological ordering of some newly deigned alkali metal-based ternary intermetallic compounds X2YZ (X = alkali metals: Li, Na, K, Rb; Y = Ag, Pd, Cu; Z = Sb, Te, As). The effects of the SOC and uniform strain on the topological nontrivial nature for these compounds X2YZ have also been discussed in detail.

2. Computational method In this study, electronic structure calculations were performed using the full-potential linearized augmented plane-wave (FP-LAPW) [15] code implemented in the package Wien2k [16] on basis of the density function theory. A converged ground state was obtained using 5000 k points in the first Brillouin zone with Kmax×Rmt = 9.0, where Kmax is the maximum size of the

3

reciprocal-lattice vectors and Rmt is the smallest atomic sphere radius in the unit cell. Moreover, wave functions and potentials inside the atomic sphere are expanded in spherical harmonics up to l = 10 and 4, respectively. The correlation potential of the local-density approximation and a combination of modified Becke-Johnson exchange potential [17] are used to obtain the band structures. The effect of SOC is also added by package Wien2k using a basis of fully relativistic eigenfunctions [18].

3. Results and discussion It is well known [19] that there are two possible different types of atom arrangement in the ternary Heusler compounds X2YZ: for type (I), named Hg2CuTi (cubic XA)-type structure, with X atoms place in the A (0, 0, 0) and B (1/4, 1/4, 1/4) Wyckoff positions, Y in C (1/2, 1/2, 1/2) and Z in D (3/4,3/4,3/4), respectively. The structure of the Hg2CuTi-type compounds is also called inverse Heusler in crystallography. For type (II), named Cu2MnAl (cubic L21) -type structure, the X atoms occupy the A (0,0,0) and C (1/2,1/2,1/2) sites, Y atoms occupy the B (1/4,1/4,1/4) site and Z atoms occupy the D (3/4,3/4,3/4) site. In Fig. 1, we show the crystal structures of both Heusler types. According to the site preference rule in full-Heusler compounds [13]. For the compounds X2YZ (X = Li, Na, K, Rb; Y = Ag, Pd, Cu; Z = Sb, Te, As), the inverse Heusler structure is more stable than the L21 one, and therefore, we performed all of the calculations at their more stable atomic arrangement, i.e., inverse Heusler

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structure. The band structures of the trivial CdTe compound and the topologically nontrivial HgTe compound [20, 21, 22] have been widely studied, and thus, we use them as comparison to discuss and deduct on the band topological ordering in our new material systems. The distinctive inverted band order at the Γ point of HgTe is that the s-like Γ6 state sits below the p-like Γ8 state. Simultaneously, the valence and conduction bands away from the Γ point and well separated without overlapping. The double degenerate symmetry bands Γ6 are below Γ8 and EF, in which the fourfold degenerate Γ8 are divided into two parts: one is above the EF and another is below the EF. Such band inversion only occurs once throughout the Brillouin zone. Therefore, one can see that compounds with such inverted band order belong to topologically nontrivial materials, also called topological semi-metals. Nevertheless, the band structure of CdTe is trivial band ranking, i.e., Γ6 states lies above the Γ8 states, and opens a direct gap at the Γ point, reflecting it is just a trivial semiconductor/insulator. In Fig. 2, we computed the band structures of compounds X2YZ (X = Li, Na, K, Rb; Y = Ag, Pd, Cu; Z = Sb, Te, As) with SOC. Form Fig. 2, for Na2CuAs compound, the distinctive inverted band order at the Γ point can be obviously observed, namely, the s-like Γ6 state sits below the p-like Γ8 state and the EF. Therefore, Na2CuAs is a topologically nontrivial compound in its ground state. Also, the band structures of other compounds X2YZ (except for Li2CuAs

5

compound) in our current study are similar to those of Na2CuAs. So they are all the candidates of topological semi-metals. Nevertheless, the band structures of Li2CuAs are extremely similar to those of CdTe, which show normal band ranking (Γ6 states lies above the Γ8 states and the Ef). Also, at the Γ point, a direct gap can be found, revealing it is just a trivial semiconductor/insulator. Noted that, for the well-known (Bi,Sb)2(Se,Te)3 family with band inversion, the mechanism produces the inversion is obvious [23, 24]. Namely, the bulk energy gap usually opens without SOC. By theoretically switching on SOC in calculations, the original gap closes, and a new energy gap is re-opened by inverting the conduction and valence bands. In this study, we also computed the band structures of compounds X2YZ (X = Li, Na, K, Rb; Y = Ag, Pd, Cu; Z = Sb, Te, As) without SOC. The results have been given in Fig. 3. We can see that, for most of the compounds, their band structure calculations commonly show an interesting s-p inverted band structure without SOC. For example, for X2CuAs (X = Na, K, Rb), X2AgSb (X = Na, K, Rb) and X2PdTe (X = Li, Na, K, Rb), the s-like Γ1 state sits below the p-like Γ5 state. Therefore, they are all topologically nontrivial compounds even the effect of SOC is not taken into account. The s-p band inversion in our compounds is mainly due to relativistic effects [25], just as the effect of SOC in the [Bi,Sb]-[Se,Te] families. The effects of relativistic can be divided into two parts: (i) direct relativistic effect. The inner electrons in an atom of the heavy element move very fast and

6

their speeds approach the speed of light. According to the Einstein’s theory of relativity, the fast speed results in a mass increase and this leads to a smaller Bohr radius and then brings about a relativistic orbital shrinkage and energy decrease of all s and most of the p orbitals; (ii) indirect relativistic effect. The expansion and destabilization of the outside d and f shells lead to decrease the shielding of the nuclear charge but the contraction of the inner s and p shells generate strong nuclear shielding charge. As a result the modified nuclear shielding stabilizes the outermost valence s shell [26]. Thus, these two factors may decrease the energy of the valence s electrons in heavy elements, resulting to the nontrivial band inversion. However, for Li2AgSb compound, by theoretically removing SOC from calculations, this topologically nontrivial compound becomes a trivial semiconductor. That is to say, the SOC effect has not done nothing for the s-p band inversion. In order to further analyze the effect of the SOC, we calculated the band inversion strength (EBIS) of all the compounds X2YZ (X = Li, Na, K, Rb; Y = Ag, Pd, Cu; Z = Sb, Te, As). The band structure specific of SOC topological nontrivial compounds can be characterized by a function of EBIS noSOC = EΓ6 − EΓ8 ( EBIS = EΓ5 − EΓ1), in other words, topological band inversion

strength. The compounds with a negative EBIS show that they are with a topological nontrivial nature, however, a positive EBIS indicates that the compounds are in a trivial phase. Next, we use the EBIS to perform a systematic investigation of the band

7

topology of the compounds X2YZ (X = Li, Na, K, Rb; Y = Ag, Pd, Cu; Z = Sb, Te, As), and we have summarized the EBIS as functions of their equilibrium lattice constants and the average nuclear charge number for these compounds in Figs. 4 a-c. In Figs. 4 a and c, the SOC is added and in Fig. 4 b the SOC effect is noSOC removed. In Fig. 4 a, we can see that most of them have a negative EBIS ,

demonstrating their topological nontrivial nature. However, for Li2CuAs and noSOC Li2AgSb, they are trivial semiconductors with a positive EBIS . In Fig. 4 b,

we found that a majority of the Heusler compounds (except for Li2CuAs) are SOC with the topological nontrivial phase due to their negative EBIS . By

comparison, we can see that whether with the effect of SOC or not, the topological nontrivial phase of most of compounds still maintain, however, SOC noSOC the EBIS is more lower than the EBIS . Therefore, we can judge that the SOC

can promote the degree of band inversion and it can make the EBIS more negative. From Fig. 4c, one can see that the average nuclear charge numbers of our predicted compounds are all less than 45. Previously, topological nontrivial compounds with less than 60 are rarely studied in the inverse or half Heusler compounds. Importantly, for our studied inverse compounds, the SOC effect is feeble due to they contain less heavy elements. Moreover, not all of the Heusler compounds can be synthesized and form stable phase. So in order to examine the phase stability of the compounds X2YZ (X =Li, Na, K, Rb; Y = Ag, Pd, Cu; Z = Sb, Te, As) with inverse Heusler

8

structure (in Fig. 1 a), we have calculated the formation energy using the formula [27, 28]: Ef =(Etotal - m × EX - n × EY - p × EZ) / (m + n + p), in which the Ef is the formation energy, Etotal is the energy of the inverse Heusler compounds under their equilibrium lattice constants, and EX, EY and EZ, respectively, represents the energy when X, Y, Z crystallizes in pure metals. The coefficients m, n, p show the proportions of X, Y, Z in the inverse Heusler formula. The coefficients m : n : p = 50 % : 25 % : 25 %, m + n + p = 100 %. In Fig. 4 d, we display the calculated formation energies of the compounds at their own equilibrium lattice constants. We can see that all the calculated formation energies are negative for our predicted compounds, reflecting that they are promising to be synthesized for experimental characterization. Also, as a typical example, the phonon calculations of inverse-type Li2AgSb compound has been performed to further examine the dynamic stability of the proposed structure. The results containing the calculated phonon dispersion curve and phonon DOS for Li2AgSb compound have been given in Fig. 5. One can see that the phonon dispersion spectrum has no imaginary frequencies, reflecting the dynamical stability of Li2AgSb compound. Finally, we want to tune the topological nontrivial band ordering for Li2CuAs compound. Therefore, the effect of uniform strain on the band inversion has been examined in this work, and in Fig. 6, we use the EBIS as a function of the a/a0 ratio (strained lattice constant/equilibrium lattice constant ratio) for Li2CuAs compound. The E SOC and E noSOC refer to EBIS with and BIS BIS 9

without the effect of SOC. We define ΔEBIS = E SOC − E noSOC as the energy BIS BIS discrepancy to reveal the strength of band inversion merely produced by the SOC. It can be found that all the ΔEBIS are transferred from positive to negative with the extent of lattice expansion. When a/a0 ratio is up to 1.022 and 1.027, respectively, the E SOC and E noSOC become negative. That is to say, BIS BIS the uniform strain can continuously tuning the band inversion strength, however, as shown in Fig. 6, the SOC is not the root cause but can diminish the energy difference EBIS.

4. Conclusions In the present work, we studied the band structures of the inverse Heusler compound X2YZ (X = Li, Na, K, Rb; Y = Ag, Pd, Cu; Z = Sb, Te, As) by using the full potential linearized augmented plane wave method based on the density function theory. We found that most of them naturally exhibit topological band-inversion feature without the effect of the SOC. Moreover, our results show that the SOC is not a leading cause to the band inversion mechanism, and the topological band order can be turned by uniform strain alone. We hope our newly predicted topological semi-metals can be prepared in the future. Acknowledgments: Funding for this research was proved by Doctoral Fund Project of Southwest University, Grant No. 117041. References

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Figure captions Fig. 1. Two crystal structures of full-Heusler compounds X2YZ. (a) type (I) X (0, 0, 0), X (1/4, 1/4, 1/4), Y (1/2, 1/2, 1/2), and Z (3/4, 3/4, 3/4); (b) type (II) X (0, 0, 0), X (1/2, 1/2, 1/2), Y(1/4, 1/4, 1/4), and Z (3/4, 3/4, 3/4). Fig. 2. Band structures of X2YZ (X = Li, Na, K, Rb; Y = Ag, Pd, Cu; Z = Sb, Te, As). In this case, the effect of SOC has been taken into account during the calculations. Fig. 3. Band structures of X2YZ (X = Li, Na, K, Rb; Y = Ag, Pd, Cu; Z = Sb, Te, As). We should point that the effect of SOC has not been take into consideration. Fig. 4. (a)-(c) Band inversion strength (EBIS) of the calculated inverse Heusler compounds as functions of the lattice constant and the average nuclear charge number . (a) and (c) calculated without SOC, (b) calculated with SOC; (d) Calculated formation energy of the newly designed inverse Heusler compounds as a function of their equilibrium lattice constants. Fig. 5. Calculated phonon dispersion curve and phonon DOS of inverse Li2AgSb compound. For the phonon spectrum of Li2AgSb compound, the finite displacement method as implemented in the VASP code based on the first-principle and the PAW method with GGA-PBE. Fig. 6. The EBIS as a function of the a/a0 ratio for Li2CuAs. The a/a0 is uniform strained/equilibrium lattice constant. E

SOC BIS

and E

noSOC BIS

are the calculated EBIS

with and without SOC, respectively. The difference of ΔEBIS = E

15

noSOC BIS

- E

SOC BIS

is

also shown.

16

Figures

(a)

(b)

Fig. 1. Two crystal structures of full-Heusler compounds X2YZ. (a) type (I) X (0, 0, 0), X (1/4, 1/4, 1/4), Y (1/2, 1/2, 1/2), and Z (3/4, 3/4, 3/4); (b) type (II) X (0, 0, 0), X (1/2, 1/2, 1/2), Y(1/4, 1/4, 1/4), and Z (3/4, 3/4, 3/4).

1

Fig. 2. Band structures of X2YZ (X = Li, Na, K, Rb; Y = Ag, Pd, Cu; Z = Sb, Te, As). In this case, the effect of SOC has been taken into account during the calculations. 2

Fig. 3. Band structures of X2YZ (X = Li, Na, K, Rb; Y = Ag, Pd, Cu; Z = Sb, Te, As). We should point that the effect of SOC has not been take into consideration. 3

Fig. 4. (a)-(c) Band inversion strength (EBIS) of the calculated inverse Heusler compounds as functions of the lattice constant and the average nuclear charge number . (a) and (c) calculated without SOC, (b) calculated with SOC; (d) Calculated formation energy of the newly designed inverse Heusler compounds as a function of their equilibrium lattice constants.

4

Fig. 5 Calculated phonon dispersion curve and phonon DOS of inverse

Li2AgSb compound. For the phonon spectrum of Li2AgSb compound, the finite displacement method as implemented in the VASP code based on the first-principle and the PAW method with GGA-PBE.

5

Fig. 6. The EBIS as a function of the a/a0 ratio for Li2CuAs. The a/a0 is uniform strained/equilibrium lattice constant. E

SOC BIS

and E

noSOC BIS

are the calculated EBIS

with and without SOC, respectively. The difference of ΔEBIS = E also shown.

6

noSOC BIS

- E

SOC BIS

is

Highlights 1. Prediction of topological nontrivial behavior in alkali metal-based ternary intermetallic compounds from first principles; 2. The SOC is not a leading cause to the band inversion mechanism; 3. The topological band order can be turned by uniform strain alone.

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