Influence of tetragonal distortion on the topological electronicstructure of the half-Heusler compound LaPtBi from first principles
ABSTRACT The electronic structures of tetragonally distorted half-Heuselr compound
LaPtBi in the C1b structure are investigated in the framework of density
functional theory using the full potential linearized augmented plane with
local spin density approximation method. The calculation results show that both
the band structures and the Fermi level can be tuned by using either
compressive or tensile in-plane strain. A large bulk band gap of 0.3 eV can be
induced through the application of a compressive in-pane strain in LaPtBi with
the assumption of a relaxed volume of the unit cell. Our results could serve as
a guidance to realize topological insulators in half-Heusler compounds by
Influence of tetragonal distortion on the topological electronic structure
of the half-Heusler compound LaPtBi from first principles
X. M. Zhang,1,3 W. H. Wang,1, a) E. K. Liu,1 G. D. Liu,3 Z. Y. Liu,2 and G. H. Wu1
1Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese
Academy of Sciences, Beijing 100080, P. R. China
2 State Key Laboratory of Metastable Material Sciences and Technology, Yanshan University
Technology, Qinhuangdao 066004, P. R. China
3 School of Material Sciences and Engineering, Hebei University Technology, Tianjin 300130, P.
PACS numbers: 71.20.Gj; 71.15.-m; 73.20.At
The electronic structures of tetragonally distorted half-Heuselr compound LaPtBi in
the C1b structure are investigated in the framework of density functional theory using
the full potential linearized augmented plane with local spin density approximation
method. The calculation results show that both the band structures and the Fermi level
can be tuned by using either compressive or tensile in-plane strain. A large bulk band
gap of 0.3 eV can be induced through the application of a compressive in-pane strain
in LaPtBi with the assumption of a relaxed volume of the unit cell. Our results could
serve as a guidance to realize topological insulators in half-Heusler compounds by
a) Electronic mail: firstname.lastname@example.org
Topological insulator (TI) is of a new class of materials, which has a bulk band
gap generated by strong spin-orbit coupling but contains gapless surface states. 1-3
These surface states are chiral and inherently robust to external perturbations because
they are protected by time-reversal symmetry. Based on such unique electronic
surface states, TI is supposed to open up innovative directions for future technological
applications in spintronics and quantum computing as well. 4,5
Since the first discovery of a two dimensional TI in an HgTe based quantum
well,6,7 several other families of materials for three-dimensional (3D) TI have been
proposed theoretically and recently studied experimentally.8-16 For example,
tetradymite semiconcuctors, such as Bi2Te3, Bi2Se3, and Sb2Te3 are confirmed to be
3D TI with a single Dirac-cone on the surface, where the bulk band gap is as large as
0.3 eV, making the room temperature application possible.8-12 However, a clear
shortcoming of Bi2Te3 family is that these materials cannot be made with coexisting
magnetism, a much desired property for spintronic applications. Although doping can
be used to achieve the magnetically ordered behavior,17 this creates extra complexity
in material growth and could introduce detrimental effects upon doping.
Recently, a new family of ternary half-Heusler compounds with 18 valence
electrons has been predicted to be 3D TI with proper stain engineering.18-21 Most
importantly, it was proposed that in the half-Heusler family the topological insulator
allows the incorporation of new properties such as superconductivity or magnetism.
By assuming a constant volume of the unit cell, Xiao et al.20 have calculated the band
structure of half-Heusler compound LaPtBi under uniaxial strain. They find that
LaPtBi under proper uniaxial strain is a topological insulating phase. In order to
simulate at best the experimental condition for thin film growth, in the present work,
we have made use of an accurate full-potential density-functional method to study
systematically the band structures for half-Heusler compound LaPtBi with the C1b
structure under tetragonal distortions without the assumption of a constant volume.
We will show that LaPtBi in the compressive in-plane strain state is an excellent
topological insulator with a very large band gap of 0.3 eV. In addition, they will be
proved to be mechanically stable and approximately 25-30 meV per formula unit
higher in energy than the corresponding ground-state phases and, therefore, would be
grown epitaxially in the form of thin films for spintronic applications.
The band-structure calculations in this work were performed using full-potential
linearized augmented plane-wave method,22 implemented in the package WIEN2K.23
The exchange correlation of electrons was treated within the local spin density
approximation (LSDA) including Spin-orbital coupling (SOC). Meanwhile, a
17×17×17 k-point grid was used in the calculations, equivalent to 5000 k points in the
first Brillouin zone. Moreover, the muffin-tin radii of the atoms are 2.5 a.u, which are
generated by the system automatically.
In order to determine the equilibrium lattice constant and find how the total
energy varies with the lattice constant, we have performed structural optimizations on
LaPtBi with C1b structure. In Figure 1, we show the total energy as a function of
lattice constant. The equilibrium lattice constant aeq was determined by total energy
minimization. It is found that LaPtBi with the lattice constant of 6.90Å shows the
lowest energy, which is a little larger than that of the experimentally reported lattice
parameter of 6.83 Å for bulk LaPtBi.24 The inset of Fig. 1 shows the fully relativistic
energy band structure of LaPtBi, which features a distinctive inverted band order at
the Γ point, where the s-like Γ6 states lie below the fourfold degenerate p-like Γ8
states. Away from the Γ point, the valence band and conduction bands are well
separated without crossing each other. Since the band inversion occurs only once
throughout the Brillouin zone and therefore, LaPtBi is a topologically nontrivial phase
in its unstrained state. Our result is consistent with previous calculations of ref. 20.
Moreover, we should point out that the fourfold degeneracy of the Γ8 states is
protected by the cubic symmetry of the C1b structure.
For the investigation of the effects of tetragonal distortion on the topological
band structures of LaPtBi, we next carried out a series of calculations of the
compressive (tensile) strained LaPtBi using an reduced (increased) theoretical
in-plane lattice constant a of a calculation unit cell with respect to the theoretical
equilibrium lattice constant of LaPtBi, by leaving the c-axis unconstrained (free to
relax). Here, in-plane strain was simulated by tetragonal lattice distortions using a =λ⋅
aeq with λ=0.94, 0.96,…,1.04, 1.06. The equilibrium value c was then determined by
total energy minimization with fixed a. The total energies as a function of equilibrium
c obtained from the calculations are shown in Figure 2(a). It is immediately clear that
the cubic (λ=1.00) situation is the most stable state with minimum energy. Moreover,
as shown in Fig. 2(b), the total energy of relaxed-volume calculations for the case of
in-pane strain of ±2% is only increased by 25-30 meV, i.e., about the typical thermal
energy at room temperature. Compared to the energies involved with homogenous
lattice compression/expansion, such an energy scale is rather small. In the case of a
Heusler thin film growing on a single crystalline substrate, in fact, the structure of
film has to be considered as very soft with respect to small non-relaxed tetragonal
distortions. According to our results, a change in the lattice parameters of LaPtBi thin
film with varying growth conditions or annealing temperature can be caused by
epitaxial matching with the substrates.
For comparison, In Fig. 2(b), we also show the total energy as a function of the
distortion parameter λ for the constant-volume (filled symbols) calculations. We find
that the total energy of the fully relaxed-volume calculations is lower than that of the
constant-volume one, which in turn strongly indicates that the assumption of a
relaxed-volume is more reasonable. Moreover, as shown in Fig. 2 (c), the
relaxed-volume is not constant upon distortion and increases monotonic with the
increasing of λ. The change rate of volume is only about 1.25%, which means the
contribution to the volume change would be more prominent from ab plane than c.
Further insight into the influence of tetragonal distortion on the band
structures of LaPtBi is shown in Fig. 3. For clarity, we only give out the cases of λ
from 0.96 to 1.04. As expected, the in-plane strain lifts the degeneracy of the Γ8 states
and opens up an energy gap around Γ point, while leaving the inverted band order
intact. The most striking property here is the different behavior if the lattice is
compressed (λ<1) or expanded (λ>1). With increasing compression, a global band
gap is opened by the in-plane strain when the lattice is compressed with the Fermi
energy lies in between (λ=0.96). However, in the expansion case, although a global
band gap is opened, the Fermi energy now shifts to the valance bands, making the
material effectively a topologically trivial phase (λ=1.04). Our results for the strained
LaPtBi clearly suggest that epitaxial strain encountered during epitaxial growth of
films can result in electronic topological transition from zero-gap semiconductors or
topologically trivial phases to topological insulator states.
From the unstrained cubic structure to the in-plane strained tetragonal structure,
the original Γ8 states split into states with Γ7 and Γ6 symmetry, which typically form
the top set of the valence bands and the bottom set of the conduction bands. We
therefore define the strain-induced bulk band strength ΔE as the energy difference
between the conduction band minimum and the valence band maximum at the Γ point.
In Fig. 4 (a) and (b), we show ΔE and the inversion strength, define as energy
difference between the valence band maximum and the s-orbital originated Γ6 at the Γ
point, as a function of the distortion parameter λ. We find that the compressive
in-pane strain (λ<1) seems to have higher efficiency on opening the bulk band than
that of tensile one (λ>1), while the inversion strength increases with the increasing λ.
For the case of LaPtBi with relaxed-volume (λ=0.94), a strain-induced bulk band gap
of 0.3 eV can be realized in LaPtBi with relaxed-volume. This is significantly larger
than typical band gaps opened by the uniaxial strain in LaPtBi with a constant volume,
20 and is also well above the energy scale of room temperature. Thus, when dealing
with gap-engineering technique by applying strain in half-Heusler compounds, it is
necessary to consider relaxed geometries.
In the present study, we have shown by ab-initio calculations that the cubic
crystal structure of LaPtBi is easily for tetragonal distortion, which we therefore
expect to occur in epitaxially grown thin films. In comparison with calculations that
assume a constant-volume, we find that the assumption of a relaxed-volume is more
reasonable due to the energetically favorable. It is shown, that the bulk band gap as
well as the Fermi level can be tuned by small lattice distortions in LaPtBi. We hope
that the results presented here will motivate further experimental investigation of the
electronic topological transition in LaPtBi or other half-Heusler compounds.
This work was supported by National Natural Science Foundation of China (Grant
Nos. 51071172, 51021061, and 51025103).
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FIG. 1. (Color on line) Total energy as a function of the relaxation of the cubic lattice
constant a. For the minimum determination more points were calculated, which are
not shown here. The inset shows the band structures of LaPtBi. The lattice constants
are taken from the optimized values, i.e., at the total energy of the relaxed cubic
configuration with aeq=6.90Å.
FIG. 2. (Color on line) (a) Total energy as a function of the tetragonal lattice constant
c, indicated for the different distortion parameters λ (λ=a/aeq). (b) Total energy for
relaxed (opened circles) and constant (filled circles) unit cell volume calculations. (c)
Relaxed unit cell volume as a function of the distortion parameter λ.
FIG. 3. (Color on line) Band structures of LaPtBi in different λ-values with relaxed c;
The inset in (a) shows the energy gap induced by tetragonal distortion and ΔE
indicates the opened band strength.
FIG. 4. (Color on line) (a) The opened band strength ΔE and (b) topological band
inversion strength as a function of the distortion parameter λ. Here, the inversion
strength is defined as the energy difference between the valence band maximum and
the s-orbital originated Γ6 at the Γ point.
6.4 6.66.87.07.2 7.4 7.6
The lattice constant a (A)
Energy( -1319132+ eV)
220.127.116.11.8 7.0 7.27.4
0.94 0.96 0.98 1.00 1.02 1.04 1.06300
The distortion parameter (λ)
The lattice constant of c(A)
Energy( -1319132+ eV)
Energy( -1319132+ eV)
0.94 0.960.98 1.00 1.02 1.041.06
Inversion strength (eV)
The distortion parameter (λ)