The first principle calculation for g-factors and its application to topological materials

ZhiDa Song, Song Sun, Zhong Fang Institute of Physics, CAS

Xi Dai Hong Kong University of Science and Technology

arXiv:1512.05084

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In memoriam of professor Shoucheng Zhang great scientist, mentor and pioneer of our field

Outline

The definition of g-factor in solid states

How to calculate it from first principle

g-factors for topological insulators with strong SOC

g-factors for topological semimetals

g-factor and field induced topological metals

conclusion

What is g-factor?

Two effects induced by external magnetic field: Zeeman effect and Landau quantization

how to distinguish these two contributions? Intra-band and inter-band effects of vector potential

low and high energy subspaces

the inter low-high subspace coupling caused by the vector potential give rise to Zeeman effect

Why g-factor is important for topological materials?

In 3D TI surface states, both Zeeman effect and Landau quantization can give rise to quantum Hall effect but with very different mechanism and properties, it depends on the sign and amplitude of g-factor

in many topological semimetals, like the Dirac semimetal, node-line semimetal and three fold degeneracy semimetal, the g-factor is crucial to determine the FS topology under the magnetic field.

g-factor for free electrons in vacuum

electron

positron

No g-factor in original Dirac equation it appears when down fold the Ham into the

electron band only

Quadratic region constant g factor

Linear region k-dependent g factor

g-factor in energy bands of solid system: further down folding the Ham. to low energy bands

low energy part

From g0=2 for high eng. mod. to g(k) for effective Low eng. Mod.

Down folding process in the framework of kp model

down fold to low energy subspace labeled by m, m’

Now we consider the external magnetic field by including the vector potential A

Calculate the π-operator in PAW type wave function adopted in VASP

pseudo wave function

projector functions

at the band bottom or top for topological insulators: single band system

Nat. phys. 5, 438, (2009)PRB 80, 085307, (2009)

Results for Bi and Bi2Se3

Dirac semimetal: Na3Bi

- +

-

-

3D Dirac semi-metal generated by band inversion

AlongZ-axiswithextracrystalsymmetryWhichprotectsthebandcrossing

Alonganyotherdirec9onwherethecrossingisnolongerprotected

Kz

Ek

Kx

Ek

band structure of Na3Bi

M Γ K H A Γ L -3

-2

-1

0

1

2En

ergy(e

V)

M Γ K H A Γ L

EF

A Γ-0.8

-0.6

-0.4

-0.2

0

0.2

Energ

y(eV)

EF

P−

P−

P

S

GGA GGA+SOC

Energy (eV)

Dens

ity of

State

s(/eV

/u.c.)

(a)

(b) (c)

0

4

8

12

-3 -2 -1 0 1 2 3 4

TotNa sBi p

Band structure of Na3Bi

•  Nega%vegapatGammapoint•  Thesizeisabout-0.7eV•  Doublecheckedbyhybridfunc%onal(HSE)•  SimilarbandstructureforK3BiandRb3Bi•  Protectedbandcrossingalongthec-axis•  Bandcrossinghappensbetween|±3/2>and|±1/2>

(b)z

y x

Na(1)

Na(1)

Na(2)

Bi

Bi

a=5.448Å b=5.448Å

=9.655Å (a)

PRB 85, 195320, (2012)

g-factors and kp model in Na3Bi

phase diagram of Na3Bi under magnetic field

Zeeman effect on quantum oscillation I: Reduce U(2) Berry’s curvature to U(1), each FS carries a U(1) Berry’s parse

Zeeman effect on quantum oscillation II: split the FS area into two

Spin zeros: when the phase become π*(2n+1)/2!!

Topological insulators close to a phase transition: Bi electron pocket and ZrTe5

Geometrical Structure of ZrTe5

ZrTe3 chain

Te2 zig-zag chain

ZrTe3 chain

Te2 zig-zag chain

ZrTe3 chain

side view

Top view

band structure for single layer ZrTe5

2D QSH insulator

X12-

Y12-

M12-

G11-

parity

PRX 4, 011002, 2014

3D band structure: very close to STI to WTI transition

3D Topological Insulator

PRX 4, 011002, 2014

Zeeman effect generated new topological metal: TaAs2

Band structure and Fermi surfaces

Zeeman effect or g-factor tensor induced Chern number On different Fermi surfaces !

Electron 1 Electron 2 Hole 1

Conclusion

g-factor in band structure is model dependent

It can be calculated from first principle

In many topological materials g-factor is crucial to determine its behavior under field

New type of topological metal: Zeeman generated topological metal