Supporting Information for

Strong anisotropy nodal lines in TiBe family

Z. C. Zou,1 P. Zhou,2, ∗ Z. S. Ma,2 and L. Z. Sun1, †

1Hunan Provincial Key laboratory of Thin Film Materials and Devices,School of Material Sciences and Engineering, Xiangtan University, Xiangtan 411105, China

2Key Laboratory of Low-dimensional Materials and Application Technology,School of Material Sciences and Engineering, Xiangtan University, Xiangtan 411105, China

(Dated: March 10, 2019)

I. BAND STRUCTURE OF OTHER TIBE FAMILY MATERIALS

We calculate the band structures of other TiBe family materials and present in Fig.S1, similar nodal lines semimetalcharacter can be found for all these compounds around Fermi level.

-2

-1

0

1

2(b)

Energy

(eV)

M X R M

ZrZn

-2

-1

0

1

2(g)

Energy

(eV)

M X R M

YAg

-2

-1

0

1

2(d)

Energy

(eV)

M X R M

ScZn

-2

-1

0

1

2(f)En

ergy

(eV)

M X R M

LaZn

-2

-1

0

1

2(i)

Energy

(eV)

M X R M

LaCd

-2

-1

0

1

2(h)

Energy

(eV)

M X R M

ScCd

-2

-1

0

1

2(e)

Energy

(eV)

M X R M

YZn

-2

-1

0

1

2(c)

Energy

(eV)

M X R M

HfZn

-2

-1

0

1

2

Energy

(eV)

M X R M

(a) TiZn

Figure S1: (color online)(a) Band structures of other materials of TiBe family (a) TiZn, (b) ZrZn, (c) HfZn, (d) ScZn, (e) YZn,(f) LaZn (g) YAg ,(h)ScCd ,(i) LaCd. Similar bands features as TiBe can be observed around Fermi level.

Electronic Supplementary Material (ESI) for Physical Chemistry Chemical Physics.This journal is © the Owner Societies 2019

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II. TIBE’S BAND STRUCTURE WITH HSE06 FUNCTION

The 3D bands in the planes of (a) kz = 0.95 π/a and (b) kz = 1.05 π/a(a is the value of crystal constant) areplotted in Fig.S2. We found the band gap has been opened between the two bands around Fermi level in these twoplanes.

ky

kx

ky

Energy(eV)

Energy(eV)

(b)(a)

kx

Figure S2: (color online) 3D bands in the planes of (a) kz = 0.95 π/a and (b) kz = 1.05 π/a.

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III. 3D ENERGY BAND DISPERSION OF LOW-ENERGY K·P MODEL

3D energy bands of the low-energy k·p model is shown in Fig.S3, the nodal line is formed with the crossing line ofthe orange surface E1(k) and blue surface E2(k).

Energy(eV)

ky

kx

Figure S3: (color online)3D energy bands of the low-energy k·p model as a function of kx and ky.

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IV. TIBE’S BAND STRUCTURE WITH HSE06 FUNCTION

We plot the TiBe’s band structure with HSE06 hybrid function in Fig.S4. We found no obvious changes have beenhappened comparing the results of PBE.

-2

-1

0

1

2En

ergy(eV)

M X R M

HSE

Figure S4: (color online)(a)TiBe’s band structure with the help of HSE06 hybrid function.

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V. TIBE BAND STRUCTURE WITH SPIN-ORBITAL COUPLING(SOC)

Band structure of TiBe with SOC as show in Fig.S5(a)-(c). (b) and (c) denote the enlarged band structure aroundgreen and red circles in (a), the energy gap only 21.6 meV and 13.2 meV for C1 and C2 in main manuscript, respectively.We also plot the energy difference distribution for the two bands around Fermi level in Fig.S5(d). We found no energyband crossing point exist around X points when considering SOC.

-2

-1

0

1

2

Energy

(eV)

M X R M

k y(p

/a)

kx(p/a)

(0.0,-0.6,1.0)(-0.6,-0.6,1.0)

(-0.6,0.6,1.0)

(meV)(0.6,-0.6,1.0)

(a)

0.00

0.02

0.04

0.06

Energy

(eV)

X R

13.2meV0

10

20

30

50

40

(d)

0.08

0.10

0.12

0.14

Energy

(eV)

M X

21.6meV

(c)(b)

Figure S5: color online)(a) Band structure of TiBe along high symmetry lines with SOC. (b) and (c) donote the enlarged bandstructure surround by green and red circles in (a). (d) The energy difference distribution for the two bands around Fermi levelin the plane of kz = π/a.

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Table S1: Parity product of occupied states at the TRIM points. The Z2 indices are (1;111).

TRIM point R Γ M (×3) X (×3)Parity product + - - -

VI. TABLE OF THE PARITIES PRODUCT FOR TIME-REVERSAL INVARIANT POINTS.

The parities product of the occupied states at eight time-reversal invariant momenta (TRIM) point are shown inTab.S1. According to topological index calculations of inversion symmetry materials, The Z2 indices are (1;111), thefirst index is 1 and it indicates TiBe is a strong three dimensional topological insulator if disregard the metallizationaround M points1.

∗ Electronic address: zhoupan71234@xtu.edu.cn† Electronic address: lzsun@xtu.edu.cn1 L. Fu and C. L. Kane, Phys. Rev. B 76, 045302 (2007).