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Journal of Magnetism and Magnetic Materials 468 (2018) 279–286
Contents lists available at ScienceDirect
Journal of Magnetism and Magnetic Materials
journal homepage: www.elsevier.com/locate/jmmm
Research articles
Hydrostatic strain-induced topological phase of KNa2Sb
⁎
T
Shahram Yalameha, Zahra Nourbakhsh , Aminollah Vaez
Department of Physics, Faculty of Sciences, University of Isfahan, Isfahan, Iran
A R T I C LE I N FO
A B S T R A C T
Keywords:
Topological insulators
Band inversion
Density functional theory
Elastic properties
Electronic properties
Topological insulators are novel state of quantum matter that have a bulk band gap like an ordinary insulator,
but have protected conducting states by time reversal symmetry on their edge or surface. The spin-orbit coupling
can play an important role in these materials, resulting in a band inversion at time reversal invariant momenta
(TRIM) points. The topological phase and the effect of the hydrostatic pressure on the electronic structure and
topological phase of the KNa2Sb compound are investigated by using both first-principles calculations and abinitio based tight-binding computations. Under hydrostatic lattice strain until 5.6%, the KNa2Sb compound is
semimetal with zero energy band gap and has an inverted band order. In this pressure, the Z2 invariants of this
compound are calculated using the parity analysis at TRIM points and evolution of wannier charge centers at the
¯ line at
six TRIM plane. The calculated surface states at (0 0 1) surface show a single Dirac cone exists on the X¯ Γ̄W
the surface Brillouin zone. To investigate the stability of KNa2Sb compound the phonon dispersions and elastic
tensors of this compound in the cubic structure are calculated.
1. Introduction
In 2007, it was theoretically realized that the Z2 topological invariant in a Quantum spin Hall insulator can be generalized to threedimensions, thus realizing the first three dimensional topological phase
of matter [1–3]. Although discovered only in the recent few years, topological insulators (TIs) have attracted intensive interest among the
community of condensed matter physics and material science. TIs are
insulating in the bulk but have conductive gapless edge or surface states
on the boundaries, which have their origin in the nontrivial bulk band
topology that is induced by the strong spin-orbital interactions in the
materials. Time-reversal symmetry is required to specify the class of
topological insulators. It can be visualized in the surface state point of
view, if this symmetry is broken, the surface state degeneracy at the
Kramers point can be lifted. In three dimensions, there are eight Kramers points, which leads to four independent Z2 topological invariants
(ν0; ν 1,ν2, ν3), where ν0 is the strong invariant and (ν1, ν2, ν3) are the
weak invariants. Liang Fu and C.L. Kane [4] showed that the presence
of inversion symmetry simplifies the calculation of topological invariants, which can be determined from the knowledge of the parity of
the occupied electronic states at the time reversal invariant momenta
(TRIM) points in the Brillouin zone (BZ). A band inversion is a key
ingredient of a topologically nontrivial material and it is needed to
change the topological band order [4]. The band inversion is usually
result from the band splitting at the high symmetry points in BZ caused
⁎
by SOC. Zhiyong Zhu et al., determined that scalar relativistic, nonrelativistic and fully-relativistic (FR) effects are contributing to the
band inversion [5]. Although the topological band inversion is usually
due to SOC, it has been specified that the band inversion of HgTe does
not depend on the SOC strength [5,6]. In recent years, to control topological band order, the electromagnetic fields [7,8], composition
[9,10], pressure [11–14], strain [15–18] and temperature effects [19]
are used. In this paper, we study the topological band order of KNa2Sb
under hydrostatic lattice strain.
The alkali antimonides compounds, such as X3Sb (X = Na, K, Cs, Li),
KNa2Sb, RbNa2Sb, CsNa2Sb, RbK2Sb, CsK2Sb and CsRb2Sb, are widely
used as photocathode materials [20,21]. Their characteristics such as
photon absorption and work function make these compounds suitable
candidates for electron emission devices applications [22]. In the alkali
antimonide compounds there isn't any comprehensive study to examine
the topological band order characteristics, regardless of their prospective technological applications [23,24]. Consequently, the focus on
topological phase of these materials can lead to various emergent
photonics applications due to creation of surface states [25]. Also, the
bi-alkali antimonides compounds are highly quantum efficient semiconductors [26]. Unlike the alkali bismuthides compounds, A3Bi
(A = Na, K, Rb), where the valence and conduction bands touch at
discrete points at the Fermi level [27], the alkali and bi-alkali antimonides compounds usually have non-zero gaps. In the alkali bismuthides, KNa2Bi, theoretically demonstrated that this compound is a
Corresponding author.
E-mail address: z.nourbakhsh@sci.ui.ac.ir (Z. Nourbakhsh).
https://doi.org/10.1016/j.jmmm.2018.07.086
Received 4 May 2018; Received in revised form 15 June 2018; Accepted 30 July 2018
Available online 01 August 2018
0304-8853/ © 2018 Elsevier B.V. All rights reserved.
Journal of Magnetism and Magnetic Materials 468 (2018) 279–286
S. Yalameha et al.
Table 1
Positions of atoms, experimental and calculated lattice parameter and elastic constants of the cubic KNa2Sb compound.
Position
Lattice constant (Å)
Elastic constants (GPa)
This work
Other
Na1
Na2
K
Sb
a0
aexpt
C11
C12
C44
C11
C12
C44
(1/4, 1/4, 1/4)
(3/2, 3/3, 3/2)
(1/2, 1/2, 1/2)
(0, 0, 0)
7.79
7.74a
30.18
12.82
20.27
22.28b
19.3b
17.59b
a
b
Ref. [47] Experimental.
Ref. [26] Ab initio.
electron charge density and the maximum angular momentum are optimized as Gmax = 12 Ry1/2 and lmax = 10 Ry1/2, respectively. In addition, 9 × 9 × 9 k-point meshes are employed in the reciprocal space.
To identify the Z2 topological invariant, the Wannier center evolutions is calculated in the whole BZ for six TRIM planes (k1 = 0.0,
k1 = 0.5; k2 = 0.0, k2 = 0.5; k3 = 0.0, k3 = 0.5) [33]. The Bloch spectral functions (BSFs) defined by the imaginary part of Green’s function,
is diagonal in the momentum space. They have been carried out using
the fully-relativistic Korringa-Kohn-Rostoker (KKR) Green’s function
method [34]. The topological surface states were calculated using ab
initio based tight-binding formalism [35,36] after obtaining the tightbinding Hamiltonian from the MLWF [37] as implemented in the
Wien2k package.
The phonon spectrum is calculated using the all-electron FHI-aims
code [38] with the phonopy code [39] within the GGA as parametrized
by Perdew-Burke-Ernzerhof (PBE). The reciprocal space integration
was carry out over an 8 × 8 × 8 Monkhorst-Pack grid. In additions, the
FHI-aims “tight” default option has been used, which specifies a preconstructed high-accuracy all-electron basis set of numerical atomic
orbitals. The convergence criteria of total atomic forces for the supercells, that made by the phonopy code, was less than 10−5 eV/Å.
zero-gap semimetal with an inverted band structure at the BZ center. It
can be driven into nontrivial topological phases by applying hydrostatic
pressure and uniaxial strain [13].
In this paper, we show that the ternary alkali antimonides KNa2Sb
compound under hydrostatic strain until 5.6%, a zero-gap semimetal
with an inverted band structure at the BZ center, because the inverted
Γ8 and Γ6 bands in near the Γ point. In addition, after applying pressure,
the parity eigenvalues in the Γ point are changed with and without
considering spin-orbit coupling (SOC). Spin-orbit coupling and relativistic effects are enhanced in materials with heavy elements [5,6].
Nevertheless, the Bloch spectral functions (BSFs) of KNa2Sb were obtained from FR calculations show that, both FR effects and SOC are play
a central role in band inversion. Therefore, to identify the topological
properties, we have to calculate the Z2 topological invariants. The Z2
topological invariants for 3D bulk system can be obtained from the
calculation of the wannier charge center (WCC) for the six time-reversal
invariant momentum plane; we have compared the topological invariants with and without SOC. Also, with respect to two important
symmetries, time reversal symmetry (TRS) and inversion symmetry
(IS), we calculated the topological invariants (ν0;ν1,ν2,ν3) with determined from the knowledge of the parity of the occupied band eigenstates at the eight TRIM points in the BZ. The results of both
methods show that the most important invariant is ν0; ν0 = 1 representing a strong topological insulator (STI), while weak invariants
are ν1 = ν2 = ν3 = 0. Thus, an STI display topological surface states
(TSS) with odd number of Dirac cones on any surface, while a weak
topological insulator (WTI) determines TSS with even number of Dirac
cones only at specific surface planes [28]. These results and the TSS
calculated using the maximally localized wannier functions (MLWFs) on
(0 0 1) surface, clearly confirms the Bulk-Boundary Correspondence
(BBC) [29].
The first good of this paper is to investigate the stability of KNa2Sb
compound using the calculated elastic constants (which show mechanical stability criteria) and phonon spectrum (which show the dynamical stability) and to study the electronic band structure and energy
band gap of this compound. The second good of this paper is to study
the topological phase of KNa2Sb compound using the calculated band
inversion strength between s and p orbitals, Z2 topological invariants
and topological surface states. The third good of this paper is to investigate the effect of pressure on the topological phase and band order
of this compound.
3. Results and discussion
3.1. Structural and electronic properties
The atomic positions and the experimental and calculated optimal
lattice parameter (a0) of the KNa2Sb cubic structure with Fm3̄m space
group are given in Table 1. The calculated lattice parameter is in acceptable agreement with the experimental value [39]. Fig. 1 (a) displays the band structures of KNa2Sb compound using GGA in the presence of SOC; the calculated direct energy band gap (0.51 eV) is smaller
than the experimental value (1.0 eV) [40]. The DFT with GGA approach
usually underestimate the energy band gap. But, the EV-GGA approach
with optimize exchange potential improves some electronic properties,
such as energy band gap. Thus, to improve the calculated energy band
gap, the EV-GGA approach is used. The calculated energy band gap of
this compound, within EV-GGA (0.9 eV) is in good agreement with
experimental value.
To investigate the dynamical stability of KNa2Sb in the cubic
structure, the phonon dispersion curve at the equilibrium volume (V0)
is calculated (Fig. 1(b)). Overall, the acoustic phonon frequencies of this
compound over the whole BZ are positive that showing the dynamical
and the internal stability of the atomic positions. There is no imaginary
frequency at hydrostatic lattice strain, which reveals the stability of this
compound.
The mechanically stability can be described by the elastic constants
(Cij). The KNa2Sb compound with cubic structure has three independent
elastic constant components (C11, C12 and C44). The calculated elastic
constant of this compound are given in Table 1. The mechanical stability criteria for cubic phase are given by Eq. (1) and according to the
calculated results of Table 1, this stability is well represented.
2. Method and computational details
The structural and electronic properties and the elastic constants
calculations were performed using full potential linearized augmented
plane-wave (FP-LAPW) method [30] within the generalized gradient
approximation (GGA) and Englo–Vosko GGA (EV-GGA) [31], as implemented in the Wien2k package [32]. In the calculations, an energy
threshold of -7 Ry is used to separate the valence and the core states.
The wave functions in the interstitial region were expanded in the terms
of plane waves with a k vector cut-off Kmax = 9/RMT (a.u.)−1, where
RMT denotes the smallest muffin-tin sphere radius and Kmax is the largest
k vector in the plane wave expansion. The Fourier expansion of the
C11 > 0, C11 > |C12|, (C11 + 2C12 ) > 0, C44 > 0 .
280
(1)
Journal of Magnetism and Magnetic Materials 468 (2018) 279–286
S. Yalameha et al.
Fig. 1. (a) The dispersion of the energy bands and (b) the phonon band structure of KNa2Sb using GGA. Orientation dependence of Poisson’s ratio in (c) (1 0 0) and (d)
(0 1 1) planes. Blue color is maximum, green color is minimum positive value of Poisson’s ratio and red color is minimum when it is negative. (For interpretation of
the references to color in this figure legend, the reader is referred to the web version of this article.)
their interior but can support the flow of electrons on their surface,
while conventional insulators do not have this feature. Band order is the
most important condition for identifying topological phase of materials,
which is the most practically useful guideline for the initial screening of
possible topological phase transitions of materials. The band inversion
is a phenomenon in the electronic structure where the relative energy
positions between the conduction and valence bands around a certain
TRIM point in the BZ are reversed. From this point of view, to investigate the topological phase of this compound, the band structure in
the presence of SOC and description distribution of s and p electron at Γ
symmetry point are shown in Fig. 2(a-d). To study strain effects on the
band structure and band order of this compound, the external hydrostatic strain which is defend as ε = (a − a0)/a0, (where a and a0 are
cubic lattice constants with and without strain, respectively) is applied.
The distribution of s and p electrons of Bi atoms on the band structure at
Γ symmetry point around the Fermi energy is only focused in Fig. 2. The
p-type bands of K and Na atoms have negligible distributions around the
Fermi level. In the equilibrium state, Fig. 2(a), the s-type band Γ6 (red
points) sits above the p-type Γ8 (green points) of valence band. Therefore, this compound has normal band order with 0.51 eV direct energy
band gap and it is a trivial topological phase. As the hydrostatic strain
increases, the band splitting between Γ8 as the valance band and Γ6 as
the conduction band decreases (Fig. 2(b, c)). When V/V0 = 1.18, the Γ6
state shifts downward with respect to the Γ8 state, where the direct
band energy gap disappear and a new band gap between Γ6 and Γ8 is
created, as illustrated in Fig. 2 (d). Therefore, a topological phase
transition from a trivial insulator (TI) to a non-trivial phase topological
is induced.
To confirm this topological phase transition, we illustrate the parities of the Bloch states at Γ point before and after the topological
phase transition pressure in the presence and absence of SOC in
Unfortunately, no pervasive experimental and theoretical study has
been devoted to examine the elastic properties, especially Poisson’s
ratio, ν, of the KNa2Sb compound. Some textbooks have expressed that
negative values in Poisson’s ratio are impassible, while Negative
Poisson’s ratio (NPR) is theoretically possible [41]. In a solid, ν has
defined as the transverse contraction strain (εtransverse ) divided by the
longitudinal extension strain (εlongitudinal ). Poisson’s ratio is important
because it influenced on the strain concentration, wave speeds in a
material and interatomic forces [42,43]. For metrical with cubic lattice
structure the anisotropy factor, A, is defined as:
A=
2C44
(C11−C12 )
(2)
For isotropic materials, A = 1. Using the calculated elastic constants, the anisotropy factor of KNa2Sb compound is calculated (2.33),
which shows that this compound is anisotropic. Indeed, for isotropic
solids the range of ν is from −1 to 0.5, while in anisotropic solids is not
as restricted as it is in isotropic solids. The calculated ν of this compound in different planes, (1 0 0) and (0 1 1) are shown in Fig. 1(c-d),
that performed using the Elam code [41]. The minimum (green color)
and maximum (blue color) positive values of the ν in the (1 0 0) and
(0 1 1) planes are 0.31 and 0.47, respectively. It shows that when the
stress is applied in the direction perpendicular to the (1 0 0) and (0 1 1)
plane, the minimum strain is emerged in the [0 0 1] (or [0 1 0]) direction
and the maximum strain in the [0 1 1] direction, while in this direction,
the ν has appeared negative (−0.12) (red color). This feature is also
exists in the (0 1 1) plane.
3.2. Topological phase of KNa2Sb
3.2.1. Strain-induced s-p band-inversion
Topological insulators are materials that are normal insulator in
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Journal of Magnetism and Magnetic Materials 468 (2018) 279–286
S. Yalameha et al.
Fig. 2. (a-h) Energy band structures, (i) the distributions of s and p orbitals of KNa2Sb at Γ symmetry point around the Fermi energy as a function of pressure. The red
points shows s orbit composition and the green points shows p orbit. The symbols “+ ”and “−” label the parities of the bands at the Γ point. (For interpretation of the
references to color in this figure legend, the reader is referred to the web version of this article.)
(s−) with different parities. So, the system is an insulator and the band
gap is Eg > 0.51 eV, as shown the pink area in stage (I). When the SOC
is taken into account in stage (II) of pink area, the p− state splits into
the |p , ± 1/2〉 with a total angular momentum of j = 3/2 and j = 1/2.
Also, the energy band gap decreases, which is converted into a semiconductor system with the band gap of 0.51 eV. It is observed that the
crossing between the s+ and p− level do not occurs at V/V0 = 1, due to
the sizeable energy band gap. When considering the hydrostatic strain
(V/V0 = 1.18) as can be seen in stage (I) of the orange area, the energy
gap is decreased. Thus, in the presence of SOC, the s+ and p−
3/2 levels
crossing in stage (II), leads to a parity exchange near the Fermi energy
and band inversion.
Furthermore, we have found that FR effects can also induce the
band inversion to the KNa2Sb. The band structure based on BSFs is
presented in Fig. 3. As we already mentioned, as the pressure increases,
the energy band gap reduces significantly and the topological phase
transition conditions is provided. Fig. 3(a) shows BSFs at the equilibrium state include the FR effects. The red and blue color are corresponding to BSF values greater and less than 300 and 0.1 a.u., respectively. Thus, the FR effects using the BSF calculations show that the
most electron distributions around the Γ symmetry point that is related
to the conduction bands (circle in Fig. 3(a)). Under hydrostatic pressure, the energy band splits due to the FR effects and the most contribution of BSFs are related to the valence bands (circle in Fig. 3(b)),
Fig. 2(e-h). When V/V0 = 1, in the absence of SOC, the parities of Bloch
states of the conduction and valence bands at Γ are “+” and “−”, respectively. In the presence of SOC, the p− state splits into the
|p , j = ± 3/2〉 (green bands) and |p , j = ± 1/2〉 (blue band) with odd
parities, while the s+ state has an even parity (Fig. 2(e, f)). At phase
transition pressure (V/V0 = 1.18) in the absence of SOC, only the energy band gap is reduced (Fig. 2(g)). Nevertheless, in the presence of
SOC, KNa2Sb compound converts into a zero-gap semimetal, in which
the degenerate and parities of conduction and valence bands at the Γ
point are changed (Fig. 2(h)). As a result, the s-type state (Γ6) is lower
in energy than the p-type state (Γ8); show the topologically non-trivial
band order. The behavior is similar to some half-heuslers TIs [44] and
KNa2Bi [13].
To explain the physical mechanism underlying the band inversion
and the parity exchange of this compound, the schematically evolution
of the energy levels of the atomic levels at Γ point under hydrostatic
strain, in the presence and absence of SOC is shown in Fig. 2(i).
Fig. 2(i), divided into two regions: the pink and orange areas that displayed before and after the hydrostatic strain including, respectively.
According the normal band flling order near the Fermi energy level, the
valence band should be formed of p-Sb orbitals and the conduction
band should be composed of s-type states, which hybridise and split
into the bonding and antibonding states in the absence of SOC (because
of formation of chemical bond in this compound), i. e., p+ (s+) and p−
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Journal of Magnetism and Magnetic Materials 468 (2018) 279–286
S. Yalameha et al.
Fig. 3. The Bloch spectral functions (BSF) of KNa2Sb in high symmetry directions ( X −Γ−L ); (a)/(b) In the absence / present of pressure. The red color corresponds to
BSF values greater than 120 a.u.. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
points, at the zero pressure (V/V0 = 1) and topological phase transitions pressure (V/V0 = 1.18). As shown in Table 2, at the zero pressure
the PIs at Γ, X and L are −1, −1 and +1 TRIM points, respectively. In
the case, the topological invariants are (0; 0, 0, 0), which corresponds
to a conventional insulator.
As expected, at topological phase transitions pressure of KNa2Sb (V/
V0 = 1.18), the parities of valence bands at the Γ symmetry point
change, due to the s-p band inversion. Thus, the sign of PI at Γ point
changes from −1 to +1. As a result, the topological invariant indexes
of this compound are (1; 0, 0, 0), which shows the strong topological
insulator.
However, the parity invariants method is efficient method for topological phase investigation, but for comprehensive cases where inversion symmetry is absent, the evaluation of the topological invariants
is difficult. The Wannier charge centers (or Wilson loop) method, proposed by Soluyanov and Vanderbilt is better than the parity invariants
method. The Wilson loop method not require a gauge-fixing condition
so that greatly simplifying the calculation. In addition, it can be used to
a system with or without IS.
Fig. 4 shows the evolution lines of WCCs along ky of KNa2Sb compound at six TRIM planes at normal conditions (V/V0 = 1) and topological phase transitions pressure (V/V0 = 1.18). Z2 topological invariants can be obtained using the numbers of crossing between any
arbitrary horizontal reference line (purple line) and evolution of the
WCCs. As shown in Fig. 4(a) in the equilibrium state (V/V0 = 1), it can
be seen that the WCC lines cross the reference line an even number of
times for all six planes (kx = 0, ky = 0, kz = 0, kx = 0.5, ky = 0.5 and
kz = 0.5). So, this compound is topological trivial and its topological
invariants are (0; 0, 0, 0), which confirms the previous results.
Corresponding to above analysis at topological phase transitions
pressure V/V0 = 1.18, the WCC lines cross the reference line an odd
number of times for all three planes (kx = 0, ky = 0, kz = 0) containing
(0, 0, 0) point and even number of times for all three planes (kx = 0.5,
ky = 0.5, kz = 0.5) containing (0.5, 0.5, 0.5) point (Fig. 4(b)). Therefore, under hydrostatic strain the KNa2Sb compound is a nontrivial
topological insulator with (1; 0, 0, 0).
that show a band inversion. Therefore, FR effects like the SOC have a
central role on the band order of in the KNa2Sb compound.
3.2.2. Z2 topological invariants
After the topological phase investigations of materials, the classification of topological insulator must be determined. The classification of
topological insulator can be identified by its Z2 topological invariants
that are determined by four indexes. It is note that time-reversal symmetry requires in this classification. In this subsection, two methods
have been widely used to investigate the topological invariants. One is
to calculate the topological invariants proposed by Fu and Kane [4] and
another method is demonstrated the topological invariants proposed of
materials by Soluyanov and Vanderbilt [33,45]. In the first method,
with consider two TRS and crystal-inversion symmetry (IS), the topological invariants can be determined by the parity eigenvalues of topmost isolated valence bands at the eight TRIM points. The face centered
cubic structure of KNa2Sb with the Fm3̄m group, has three nonequivalent TRIM points (Γi) in the momentum space. These points are
one Γ, three X (3X) and four L (4L) which defined by −Γi = Γi + G,
where G is a bulk reciprocal lattice vector. Accordingly, the strong topological invariant, ν0, is determined by [4]
8
N
δi =
∏ ξ2m (Γ),
i
m=1
(−1) ν0 =
∏ δi
(3)
i=1
where the ξ2m (Γ)
i = ± 1 are the parity eigenvalues of the N occupied
bands at three dimensions (3D) TRIM points Γi and δi are parity invariants (PIs) for each occupied bands for each TRIM points. The other
three weak topological invariant, (ν1, ν2, ν3) are given by
∏
(−1) vl =
δi = (n1 n2 n3) , l = 1, 2, 3
nl = 1
nj ≠ 0,1
(4)
Table 2 shows the parity eigenvalues of the Bloch wave function of 4
occupied bands of KNa2Sb compound at three non-equivalent TRIM
Table 2
Parities of 4 occupied bands and parity invariants at the TRIM points (δ(Γi)) of
the KNa2Sb for V/V0 = 1, V/V0 = 1.18. Positive and negative signs explain
even and odd parities, respectively.
V/V0 = 1
3.2.3. Topological surface states Calculations.
After the investigation of the band inversion and topological invariants of this compound using three different methods, the calculation of bound states (surface or edge states) of KNa2Sb compound becomes necessary. The bound states calculations are useful for
experiments, including scanning tunneling microscope and angle-resolved photoemission spectroscopy (ARPES) which it furnish the essential information for direct comparison. Before discussing the bound
states calculations, the theoretical predictions of the TI surface state
V/V0 = 1.18
Γi
Parity of 4 occupied bands
δ(Γi)
Γi
Parity of 4 occupied bands
δ(Γi)
G
3X
4L
−−+−
−−−+
++++
−1
−1
+1
G
3X
4L
+−+−
−−+−
++−−
+1
−1
+1
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Journal of Magnetism and Magnetic Materials 468 (2018) 279–286
S. Yalameha et al.
Fig. 4. Evolution of the wannier charge centers (WCCs) along ky at six TRIM palnes (a) at normal conditions (V/V0 = 1) and (b) at topological phase transitions
pressure (V/V0 = 1.18). Each of the six planes passing through four TRIM points. The purple lines are arbitrary reference lines. (For interpretation of the references to
color in this figure legend, the reader is referred to the web version of this article.)
Fig. 5. (a, b) The projected surface density of states (PSDS) and (c, d) band dispersions using the ab initio TB model with 41 atomic layers of [0 0 1] surface in
consideration of SOC. (a) and (c), when V/V0 = 1.81 and (c) and (d), when V/V0 = 1. (e) Brillouin Zone of bulk and the projected surface Brillouin Zones of (0 0 1)
plane.
π (Λ a) = (−1)nc δ (Γ)
i δ (Γj)
electronic structure proposed by Teo et al [46] are explained. Accordingly, the TSS is dependence to the values of the surface fermion parity
π(Λa) at the TRIMs surface. π(Λa) can be determined from the PIs (δ (Γi))
as
(5)
where nc is the number of occupied bulk bands, which is 4.
Whenπ (Λ a) = 1, none or an even number of enclosing contours is
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Journal of Magnetism and Magnetic Materials 468 (2018) 279–286
S. Yalameha et al.
References
anticipated, while forπ (Λ a) = −1, an odd number of enclosing contours
will result. Note that frequently, instead of expressing that there is a
closed Fermi contour around a TRIM, one finds the expression that
there is a Dirac point at this TRIM. Fig. 5(e) illustrates these predictions
for the (0 0 1) surfaces of KNa2Sb. At normal conditions (V/V0 = 1),
according to Table 2, the bulk parity inversions at the bulk Γ, L and X
points areδ (Γ) = −1, (L) = 1 and δ (X) = −1, respectively. Therefore,
the bulk parity inversion lead to a positive surface fermion parity on the
Γ̄ and X̄ TRIM surface,
π (Γ̄) = (−1) 4 δ (Γ) δ (X ) = +1 , π (X¯ ) = (−1) 4 δ (L) δ (L) = +1 .
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(6)
So, this case predicts that there is not closed Fermi contours around
the Γ̄ and X̄ TRIMs surface. However, after the topological phase transition (V/V0 = 1.18) the surface fermion parity on the Γ̄ and X̄ TRIM
surface isπ (Γ̄) = −1 andπ (X¯ ) = 1, which predicts the existence of an
odd number of closed Fermi contours or Dirac cone around the surface
Γ̄ TRIM point; this is symbolized by the blue circle around the Γ point as
display in Fig. 5(e).
The calculated projected surface density of states (PSDS) at [0 0 1]
surface using the Green’s function is displayed in Fig. 5(a) and (b). From
Fig. 5(b), it can be seen that bulk states not connected by surface states
at Γ̄ point at zero pressure. In fact, surface states there is not appeared
in this case. Also, the band dispersions of this surface using the ab initio
TB model with 41 atomic layers confirms this issue (Fig. 5(d)). To
confirm the pressure induced topological phase transition of KNa2Sb
compound, the PSDS and band dispersions of this compound using the
ab initio TB model with 41 atomic layers of [0 0 1] surface are calculated
and shown in Fig. 5(c) and (d). From this figure, we can see that bulk
states are connected by the TSSs at Γ̄ point in this surface, which lead to
the appearance of the Dirac cone. This results are confirmed the BBC,
which relates the boundary topological invariants to the difference in
the bulk topological invariants [29]. As expected, the theoretical predictions discussed before, clearly show these results.
4. Conclusion
The calculated phonon dispersions, elastic constants and anisotropy
factor show that KNa2Sb compound is mechanically and dynamically
stable and is anisotropic. The hydrostatic strain and the relativistic effects such as SOC and FR effects can lead to a topological phase transition. The FR effects can be necessary for the band inversion mechanism rather than the SOC. The band inversion in the band structure,
the evolution lines of the WCCs, Z2 topological invariants (ν0=1) and
topological surface states confirm the topological phase transition of
KNa2Sb compound under hydrostatic pressure. The first-principles
calculations show that the KNa2Sb compound can be tuned to a 3D TI
by imposing hydrostatic strain until 5.6%. The KNa2Sb is a strong topological insulator with (1; 0, 0, 0) Z2 topological class. Interestingly,
KNa2Sb compound has normal band order using density functional
theory in the absence of spin-orbit coupling (SOC), while in the presence of SOC this compound possesses an inverted band order between
the s-type Γ6 and p-type Γ8 states. The calculated band structure of this
compound using the Bloch spectral functions within the fully relativistic calculation without SOC (i.e. SOC is excluded) shows band
inversion at Γ point. The topological surface states of this compound
show a Dirac cone at Γ̄ symmetry point. In sum, photocathode properties, along with a striking feature of mechanical properties (i.e. negative-Poisson’s ratio or auxetic materials) and topological features under
presser, make this compound suitable for practical applications.
Acknowledgement
We would like to thank Parviz Saeidi and Mohammad hossein
Shahidi kaviyany.
285
Journal of Magnetism and Magnetic Materials 468 (2018) 279–286
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