close

Вход

Забыли?

вход по аккаунту

?

1361-6463%2Faa8eaf

код для вставкиСкачать
Journal of Physics D: Applied Physics
Related content
PAPER
Topological phase transition in layered XIn2P2 (X
= Ca, Sr)
- Prediction of topological insulators in
supercubane-like materials based on firstprinciples calculations
Guo-Xiang Wang, Shuai Dong and JingMin Hou
To cite this article: Zhenwei Wang et al 2017 J. Phys. D: Appl. Phys. 50 465304
- Topological semimetals predicted from
first-principles calculations
Hongming Weng, Xi Dai and Zhong Fang
- Topological nodal line semimetals*
Chen Fang, Hongming Weng, Xi Dai et al.
View the article online for updates and enhancements.
This content was downloaded from IP address 129.59.95.115 on 29/10/2017 at 02:26
Journal of Physics D: Applied Physics
J. Phys. D: Appl. Phys. 50 (2017) 465304 (5pp)
https://doi.org/10.1088/1361-6463/aa8eaf
Topological phase transition in layered
XIn2P2 (X = Ca, Sr)
Zhenwei Wang, Guangtao Wang , Xianbiao Shi, Dongyang Wang
and Xin Tian
College of Physics and Electronic Engineering, Henan Normal University, Xinxiang, Henan 453007,
People’s Republic of China
E-mail: wangtao@htu.cn
Received 18 July 2017, revised 8 September 2017
Accepted for publication 25 September 2017
Published 26 October 2017
Abstract
Based on fully relativistic first-principles calculations, we studied the topological properties of
layered XIn2P2 (X = Ca, Sr). Band inversion can be induced by strain without SOC, forming
one nodal ring in the kz = 0 plane, which is protected by the coexistence of time reversal and
glide mirror symmetries. Including SOC, a substantial band gap is opened along the nodal line
and the line-node semimetal would evolve into a topological insulator. These results reveal
a category of materials showing quantum phase transition from trivial semiconductors and
topologically nontrivial insulators by tuneable elastic strain engineering. Our investigations
provide a new perspective about the formation of topological line-node semimetal under stain.
Keywords: first-principles calculations, topological insulator, line-node semimetal
(Some figures may appear in colour only in the online journal)
1. Introduction
band structure has been measured by the angle-resolved photoelectron spectroscopy (ARPES) in PbTaSe2 [24] and ZrSiS
[25]. Interestingly, in addition to these materials, researchers
also found node line structures in the photonics crystals [26]
and spin liquids [27]. Their intriguing properties characterizing topological nodal line semimetals include drumhead
like nearly flat surface states [23], unique Landau energy
levels [22], long-range Coulomb interactions [28], which
open an important route to achieve high-temperature superconductivity [29–31].
The topological line-node semimetal could be driven into
3D TI or Dirac semimetal by the SOC and strain. For example,
CaTe [32] is a TSM, possessing the nodal rings without SOC,
exhibiting Dirac semimetal behavior when SOC is included.
When it was applied by appropriate strain, it becomes a strong
topological insulator [32, 33]. However, the CaAgX (X = P,
As) can be driven into a Tl phase from line-node semimetal
by taking into account of SOC [34]. Bulk CaIn2P2 and SrIn2P2
are layered semiconductors with indirect-gap and direct-gap
respectively. They have been investigated with their electronic
and optical properties [35]. Up to now, their topological properties have not been studied. In this study, we find that the
indirect-gap semiconductor can be changed into ‘direct’-gap
Topological insulator (TI) [1, 2] is a new kind of material
possessing gapped bulk states and exotic metallic surface
states. The robust surface state in a 3D system, sheltered
against backscattering from nonmagnetic impurities as long
as the bulk gap remains open and the time-reversal symmetry
is preserved [2–8]. The unique feature of the surface state
has attracted enormous attention in theoretical calculations
and experimental observations [3, 8], not only because of
their theoretical importance but also because of their great
potential applications [9–11]. Recently, the topological properties have been extended into a variety of three dimensional
(3D) topological semimetal (TSM) systems [12–16]. In most
Weyl/Dirac semimetals, conduction bands overlap with
valence bands at certain momentum points. For instance, the
topological properties of Na3Bi [17, 18] and Cd2As3 [19, 20]
semimetals have been experimentally confirmed. However,
in line-node semimetals, the band crossing points around
the Fermi level (EF) form a closed loop. Many systems have
been proposed as line-node semimetals including Mackay–
Terrones crystal (MTC) [21], Bernal stacked graphene bilayer
[22], antiperovskite Cu3PdN [23] and so on. Such line-node
1361-6463/17/465304+5$33.00
1
© 2017 IOP Publishing Ltd Printed in the UK
Z Wang et al
J. Phys. D: Appl. Phys. 50 (2017) 465304
Figure 1. (a) The crystal structure of XIn2P2 with P63/mmc symmetry. (b) Brillouin zone of bulk and the projected surface Brillouin zones
of (0 0 1) plane, as well as high-symmetry points. There is a line-node ring that lies in the kz = 0 plane.
Figure 2. Electronic band structures of unstrained CaIn2P2 with SOC. (b) Electronic band structures of band-inverted CaIn2P2
(a/a0 = 1.05 ) with SOC. The details of band-inverted CaIn2P2 around EF are shown in the inset.
Firstly, we perform density functional calculations by using
the WIEN2K package [37], with the modified Becke–Johnson
exchange potential (mBJ) to get the accurate band gap [38]. The
valence configurations of the Ca, Sr, P and In atoms are as Ca
3s2 3p6 4s2, Sr 4s2 4p6 5s2, P 3s2 3p3 and In 4d10 5s2 5p1, respectively. The plane-wave cutoff parameter RMT kmax is set to 7 and a
12 × 12 × 3 K-mesh is used for the BZ integral. The SOC interaction is included by using the second-order variational procedure.
The tight binding model based on maximally localized Wannier
functions (MLWF) method [39, 40] has been constructed in order
to investigate the surface states. The surface Green’s function of
the semi-infinite system, whose imaginary part is the local density of states to obtain the dispersion of the surface states, can
be calculated through an iterative method [41–43]. To confirm
the compounds are dynamicly stable, the phonon spectrum was
calculated by using the PHONON code [44].
by applying uniaxial strain slightly. Further increasing strain,
the compound undergoes a transition from trivial insulator to
a line-node semimetal with a nodal ring in the kz = 0 plane.
When SOC is taken into account, the compounds become a
nontrivial topological insulator with Z2 index (1000). Our
results provide a new perspective to understand the effect
of strain and SOC on the formation of topological line-node
semimetal and insulator.
2. Crystal structure and method
The ternary compound CaIn2P2 is isostructural with SrIn2P2
and crystallizes in a hexagonal structure with the space
group P63/mmc [36]. The alkaline earth cations X (Ca or
Sr) are located at a site with 3 m symmetry; In and P are
located at sites with 3 m symmetry. The lattice constants of
CaIn2P2 and SrIn2P2 are a = b = 4.022 Å, c = 17.408 Å and
a = b = 4.094 Å, c = 17.812 Å, respectively [24]. The corre­
sponding unit cell of the XIn2P2 (X = Ca, Sr) compounds is
depicted in figure 1(a). It contains two chemical formula units,
2−
the layers of X2+ cations are separated by [In2P2] layers. The
atoms positioned at the following Wyckoff positions: X (Ca,
Sr): 2a (0, 0, 1/2), In: 4f (2/3, 1/3, 0.329) and P: 4f (1/3, 2/3,
0.396). The bulk Brillouin zone (BZ) and projected surface
Brillouin zones of (0 0 1) plane are illustrated in figure 1(b).
3. Results and discussion
The electronic band structures of the unstrained and 5%
strained CaIn2P2 (a/a0 = 1.05 ) with SOC are depicted in
figures 2(a) and (b). The unstrained CaIn2P2 has an indirect
band gap about 0.5 eV, which agrees with previous studies
[35]. In figure 2(a), the Γ8+ band was pushed down, while the
Γ8− was pushed up, when we induced tensile stress in the a-b
2
Z Wang et al
J. Phys. D: Appl. Phys. 50 (2017) 465304
Table 1. The parities of time-reversal invariant k points with SOC
and their corresponding topological invariant.
SOC, strain = 0%
SOC, strain = 5%
Γ
M
A
L
Z2
+
−
+
+
+
+
+
+
0
1
characteristic band structure (around the Γ and near Fermi
energy) of SrIn2P2 under the different strains. To quantify the
effects of strain and SOC on the band inversion in SrIn2P2,
we temporally switch off the SOC in figures 4(a) and (b).
For the unstrained compound (figure 4(a)), there is a direct
band gap about 0.3 eV, with the In-s state above the P-p state,
indicating a trivial topological insulator. When we induce the
strain with a = 5%a0 and without SOC, the In-s state was
pushed down the P-p state, resulting a band inversion. The
band inversion happens along the M–Γ–K direction, resulting
in the nodal ring from the crossing between the valence band
and the conduction band in the kz = 0 plane. The glide symmetry Gz = Mz | (0,0,1/2) is the combination of mirror reflection symmetry Mz: (x, y, z) −→ (x, y, −z) and a translation
by a half lattice vector τz = (0,0,1/2). Thus, the eigenvalue
of Gz is the product of the eigenvalue of Mz and the phase
factor induced by τz . The irreducible representations of the
little group of k points in the kx-ky plane are determined by the
eigenvalue of Mz . As the Hamiltonian without SOC is spinrotation invariant, i.e. Mz2 = 1, the eigenvalue of Mz is +1 or
−1. According to this argument, the two bands with different
representations will not induce a band gap when they cross
each other. Therefore, the appearance of the nodal ring in the
absence of SOC is protected by the glide mirror symmetry.
Therefore, the strained SrIn2P2 formed a line-node band structure (figure 4(b)), which indicates that the band inversion is
caused by the strain rather than the SOC. For the 5% strained
compound, when we include SOC, there is a band gap about
35 meV exactly at the Fermi energy. Comparing the three figures in figure 4, we can draw the conclusion that lattice strain
induces the band inversion while SOC opens the band gap.
To gain further insight into the mechanism of the strain
induced band inversion, we study the electronic structure of
unstrained SrIn2P2 around EF in more detail. Without SOC,
the projected band structure of SrIn2P2 has been shown in
figure 5(a). It shows that the valence bands Γ6 and Γ4 are dominated by the P-pz states, while the conduction bands Γ2 and
Γ3 are mainly composed of In-s state. Both the valence bands
and conduct bands are doubly degenerated at the A-point, and
they split up into Γ− /Γ+ and Γ− /Γ− bands, respectively. As
a result, the Γ+ and Γ− bands give rise to the conduction band
minimum and valence band maximum, respectively. The relative position of the Γ+ and Γ− bands decides whether a band
inversion occurs or not. We have determined the four param­
eters ∆E(A), ∆E (Γ4,6), ∆E (Γ2,3), and ∆Eg as functions of the
uniaxial strain (a − a0 )/a0 (volume is fixed). In the unit cell,
there are two In-P-In-P slabs in XIn2P2. The ‘inter-slab interactions’ come from the van der Waals forces between the two
slabs. The ‘intra-slab interactions’ refers to the covalent bonds
between atoms in one slab. The energy difference ∆E(A) is
determined by the intra-slab covalent s-pz hybridization, while
Figure 3. Phonon dispersion of 5% strained CaIn2P2.
plane and compressive stress along c-axis, i.e. increasing the
a-axis and decreasing the c-axis with fixed cell volume. When
the a 1%a0, the compound becomes a direct band gap insulator. At the critical point a = 2 %a0, the Γ8+ and Γ8− touch
each other. When we further increase the a-axis to a = 5%a0
the Γ8+ dropped below Γ8− band in figure 2(b). Comparing
figures 2(a) and (b), we find that it is the strain rather than
the SOC that induces the ‘band-invertion’ [45]. From the
inset in figure 2(b), we can see a band gap about 35 meV.
It is well known that the inversion of bands with opposite
parity is a strong indication of the formation of topologically
nontrivial phases. This suggests that the strain induces a topological phase transition in CaIn2P2. Additionally, the stability
of unstrained and 5% strained compound CaIn2P2 has been
checked by computing the phonon dispersion, as illustrated in
figure 3. Here, we did not find imaginary frequency neither in
the unstrained nor in the 5% strained CaIn2P2.
To confirm the topological nature of CaIn2P2, all four Z2
topological invariants (v0; v1 v2 v3) [46] were calculated
before (zero strained) and after (5% strained) the band inversion structures. Following the method developed by Fu and
Kane [46], we calculate directly the Z2 topological invariants
from knowledge about the parity of each pair of Kramer’s
degenerate occupied energy bands at the eight time-reversal
momenta (1Γ, 3M, 1A, and 3L), because of the existence
of inversion symmetry in our compounds. Before the band
inversion, the products of parities δi are identical, yielding a
value of zero for all four Z2 topological invariants, i.e. a topologically trivial state (0;000). When the bands with Γ8+ and
Γ8− symmetries switch around EF, see figures 2(a) and (b), δΓ
changes its sign while all other seven δi remain unchanged.
The parities of time-reversal momenta are listed in table 1.
It gives rise to a topological state with Z2 class (1;000) and
confirms that the strain induces a topological phase transition.
Generally, the band inversion is generated by SOC, formatting the topologically nontrivial state, similar to the proto­
typical 3D topological insulator SR2Pb and Bi2Te3 [6, 47].
However, in our study, the band inversion was induced by
the strain only, without considering any SOC. Then, what
role does the strain and SOC play in the topological phase
transformation of XIn2P2? In figure 4, we take SrIn2P2 as the
example to answer the question. Figure 4 shows the orbital
3
Z Wang et al
J. Phys. D: Appl. Phys. 50 (2017) 465304
Figure 4. The orbital characteristic band structures (along the M–Γ–K direction, near Fermi energy) of SrIn2P2 under the different strains.
(a) Zero strain without SOC, (b) 5% of uniaxial strain without SOC, (c) 5% of uniaxial strain with SOC. The weight of atomic orbital In(s)
and P(p) is proportional to the radius of the green (blue) circle. Band inversion could be seen clearly around the Γ point.
Figure 5. (a) The band structure of unstrained SrIn2P2 without SOC, weighted with the In-s and P-Pz characters. In-s and P-Pz states are
distinguished by red and green colors. (b) ∆E(A), ∆E(Γ4,6 ), ∆E(Γ2,3 ) and ∆Eg as functions of the inter-slab tensile strain δ = (a − a0 )/a0
(volume is fixed), respectively.
Figure 6. Surface density of states (SDOS) in SrIn2P2 (a) without and (b) with SOC. The yellow lines highlight the topologically protected
metallic surface states for the (0 0 1) surface.
∆E (Γ4,6) and ∆E (Γ2,3) are related to inter-slab van der Waals
pz–pz and s–s interactions. This observation is in line with
findings for layered transition metal dichalcogenide semiconductors, such as MoS2 [48, 49].
The results are presented in figure 5(b). With increasing
tensile strain in the ab-plane and compressive strain along the
c-axis, the intra-slab interaction decreases, resulting in the
increment of the energy spilt ∆E(Γ4,6 ). However, the interslab interaction increase with the c-axis decreasing, resulting
in a strongly reduced ∆E(A), indicating strongly intra-slab
covalent s-pz hybridization. The energy level EΓ4 (derived
from P-pz orbit) increases with c-axis decreasing. So the band
gap Eg decreases with the strain increment, and the band gap
closes at the critical point δ = 0.02 . With further increasing
the strain, the band inversion takes place and the compound
becomes a topological nontrivial line-node semimetal. SOC
opens the band gap along the nodal ring and drives the compound into topological nontrivial insulator.
4
Z Wang et al
J. Phys. D: Appl. Phys. 50 (2017) 465304
Qi X-L, Li R, Zang J and Zhang S-C 2009 Science 323 1184
[11] Essin A M, Moore J E and Vanderbilt D 2009 Phys. Rev. Lett.
102 146805
[12] Wan X, Turner A M, Vishwanath A and Savrasov S Y 2011
Phys. Rev. B 83 205101
[13] Balents L 2011 Physics 4 36
[14] Ando Y 2013 J. Phys. Soc. Japan 82 102001
[15] Yang K-Y, Lu Y-M and Ran Y 2011 Phys. Rev. B 84 075129
[16] Wehling T O, Black-Schaffer A M and Balatsky A V 2014
Adv. Phys. 63 1
[17] Wang Z, Sun Y, Chen X-Q, Franchini C, Xu G, Weng H,
Dai X and Fang Z 2012 Phys. Rev. B 85 195320
[18] Liu Z K et al 2014 Science 343 864
[19] Wang Z, Weng H, Wu Q, Dai X and Fang Z 2013 Phys. Rev. B
88 125427
[20] Liu Z K et al 2014 Nat. Mater. 13 677
[21] Gao Z, Hua M, Zhang H and Zhang X 2016 Phys. Rev. B 93
205109
[22] Rhim J-W and Kim Y B 2015 Phys. Rev. B 92 045126
[23] Yu R et al 2015 Phys. Rev. Lett. 115 036807
[24] Bian G et al 2015 arXiv:1505.03069
[25] Schoop L M, Ali M N, Straßer C, Topp A, Varykhalov A,
Marchenko D, Duppel V, Parkin S S P, Lotsch B V and
Ast C R 2016 Nat. Commun. 7 11696
[26] Lu L, Fu L, Joannopoulos J D and Soljai M 2013 Nat. Photon.
7 294
[27] Natori W M H, Andrade E C, Miranda E and Pereira R G 2016
Phys. Rev. Lett. 117 017204
[28] Huh Y, Moon E-G and Kim Y B 2016 Phys. Rev. B
93 035138
[29] Kopnin N B, Heikkila T T and Volovik G E 2011 Phys. Rev. B
83 220503
[30] Volovik G E 2015 Phys. Scr. 2015 014014
[31] Heikkila T T and Volovik G E 2015 Basic Physics of
Functionalized Graphite (Berlin: Springer) pp 123–143
[32] Du Y et al 2016 arXiv:1605.07998
[33] Yang K et al 2012 Nat. Mater. 11 614
[34] Yamakage A et al 2015 J. Phys. Soc. Japan 85 013708
[35] Guechi N et al 2013 J. Alloys Compd. 577 587–99
[36] Rauscher J F, Condron C L, Beault T, Kauzlarich S M,
Jensen N, Klavins P, MaQuilon S, Fisk Z and
Olmstead M M 2009 Acta Cryst. C 65 69
[37] Blaha P, Schwarz K, Madsen G K H, Kvasnicka D and
Luitz L 2001 WIEN2k: An Augmented Plane Wave plus
Local Orbitals Program for Calculating Crystal Properties
(Vienna: Technical University of Vienna)
[38] Tran F and Blaha P 2009 Phys. Rev. Lett. 102 226401
[39] Mostofi A A, Yates J R, Lee Y-S, Souza I, Vanderbilt D and
Marzari N 2008 Comput. Phys. Commun. 178 685
[40] Marzari N, Mostofi A A, Yates J R, Souza I and Vanderbilt D
2012 Rev. Mod. Phys. 84 1419
[41] Sancho J M 1984 J. Math. Phys. 25 354
[42] Sancho J M 1985 Phys. Rev. A 31 3523
[43] Winkler G W, Wu Q S, Troyer M, Krogstrup P and
Soluyanov A A 2016 Phys. Rev. Lett. 117 076403
Tamai A et al 2016 Phys. Rev. X 6 031021
[44] Gonze X and Lee C 1997 Phys. Rev. B 55 10355
[45] Zhu Z, Cheng Y and Schwingenschlögl U 2012 Phys. Rev.
Lett. 108 266805
[46] Fu L and Kane C L 2007 Phys. Rev. B 76 045302
[47] Sun Y, Chen X-Q, Franchini C, Li D, Yunoki S, Li Y and
Fang Z 2011 Phys. Rev. B 84 165127
[48] Zhu Z Y, Cheng Y C and Schwingenschlögl U 2011 Phys. Rev.
B 84 153402
[49] Mak K F, Lee C, Hone J, Shan J and Heinz T F 2010 Phys.
Rev. Lett. 105 136805
[50] Wu Q S 2015 https://github.com/quanshengwu/wannier_tools
For the topological materials, the existence of the topological prevent surface states is an important character. So
we calculate the (0 0 1) surface of SrIn2P2 by preforming the
WANNIE R_TOOLS package [50] in a tight-binding (TB)
scheme based on the maximally localized Wannier functions
(MLWFs) [39]. The surface state of SR2In2P2 (0 0 1) surface
without and with SOC have been shown in figures 6(a) and
(b), respectively. When SOC is ignored, the system is a linenode semimetal, with the bulk bands touched along K̄ − Γ̄ ,
Γ̄ − M̄ lines and a surface state lining two touching points.
This is similar to CaAgAs [34], protected by the coexistence
of time-reversal and glide mirror symmetry. When SOC is
included, a gap opened at the two touching points, and the
compound became an topological nontrivial insulator, with
two surfaces states connecting bulk valance and conduction
bands (see figure 6(b)), coinciding with the previous calculated Z2 (table 1).
4. Conclusion
In conclusion, by the first-principles calculations, we find the
quantum phase transition in XIn2P2 from conventional semiconductor into the line node semimetal by strain engineering
only without SOC. While SOC takes the role as opening band
gap along the line node and gives rise to the nontrivial topological insulator state.
Acknowledgments
The authors acknowledge support from the NSF of China
(No.11274095, No.10947001) and the Program for Science
and Technology Innovation Talents in the Universities of
Henan Province (No.2012HASTIT009, No.104200510014,
and No.114100510021). This work is supported by The High
Performance Computing Center (HPC) of Henan Normal
University.
ORCID iDs
Guangtao Wang
https://orcid.org/0000-0001-5331-3754
References
[1] Moore J 2009 Nat. Phys. 5 378
[2] Hasan M Z and Kane C L 2010 Rev. Mod. Phys. 82 3045
[3] Qi X L and Zhang S C 2011 Rev. Mod. Phys. 83 1057
[4] Bernevig B A, Hughes T L and Zhang S-C 2006 Science
314 1757
[5] Kane C L and Mele E J 2005 Phys. Rev. Lett. 95 226801
[6] Zhang H, Liu C-X, Qi X-L, Dai X, Fang Z and Zhang S-C
2009 Nat. Phys. 5 438
[7] König M et al 2007 Science 318 766
[8] Qi X-L and Zhang S-C 2010 Phys. Today 63 33
[9] Fu L and Kane C L 2008 Phys. Rev. Lett. 100 096407
[10] Qi X L, Hughes T L and Zhang S C 2008 Phys. Rev. B
78 195424
5
Документ
Категория
Без категории
Просмотров
0
Размер файла
1 407 Кб
Теги
2faa8eaf, 1361, 6463
1/--страниц
Пожаловаться на содержимое документа