Evidence of 3D Dirac dispersion in PbSnSe by

the de Haas - van Alphen oscillations

Mario NovakDepartment of Physics

University of Zagreb

TOPOLOGICAL INSULATORSGRAPHENE vs

𝐻2𝐷 = ℏ𝑣𝐹𝝉 ∙ 𝒌 𝐻2𝐷 = ℏ𝑣𝐹𝝈 ∙ 𝒌

Latice pseudospin Pseudospin – momentum

locking - helicity

Real spin Spin – momentum

locking: 𝒌 ⊥ 𝝈 in plane

2D systemWeak S-O interactionLattice symmetry protection

By Google 107

𝒌𝝉𝒌 𝝉

𝒌

𝝈𝒌

𝝈

2D QSH3D Z2 invariantsStrong S-O interactionTRS protection

By Google 105

What are Topological insulators

Special materials with insulating balk and conducting surface

Origin of metallic surface lies special properties of bulk wave functionBulk surface correspondence

Bi2Se3

Rep. Prog. Phys. 75 096501 (2012)Binghai Yan and Shou-Cheng Zhang

Band inversion is curtail – NONTRIVIAL TOPOLOGYDirac cones at TRS pointsOdd number of Dirac cones in the band gap

InsulatororVacuum

Interesting field for testing Elemental Particle Physics concepts

- Dirac, Weyl and Majorana fermion, magnetic monopoles and axion dynamics

PbSnSe and Topological Crystalline Insulators

Topological crystalline insulator

Ordinary insulator

x

Topological Crystalline Insulators 𝑻𝑹𝑺 ⟷ 𝑪𝒓𝒚𝒔𝒕𝒂𝒍 𝑺𝒚𝒎𝒆𝒕𝒓𝒚

𝑃𝑏1−𝑥𝑆𝑛𝑥𝑆𝑒

For x=0.17 critical point- band inversion pintI. Zeljkovic et al.

Nature Materials 14, 318 (2015)

Topological SS@ x= 0.17 -Topological phase transition- 3D Dirac dispersion?

Band inversion

3D Dirac semimetals

3D equivalent of graphene

𝐻3𝐷 = ℏ𝑣𝐹𝝈 ∙ 𝒌

𝜎𝑥 , 𝜎𝑦, 𝜎𝑧 Spin Pauli matrix

𝑘𝑥 , 𝑘𝑦, 𝑘𝑧 kristal momentum

Robust to perturbation Dirac cone is spin degenerated Helicity

3D Dirac semimetals have nontrivial topology – due to band inversion- strong S-O interaction

Band gap is not open to crate Topological Insulator

𝜒 =𝝈 ∙ 𝒌

𝜎𝑘

𝒌𝝈

𝒌𝝈

Intrinsic 3D Dirac semimetal

- Crystal symmetry protection from gap opening

Cd3As2, Na3Bi, BaAgBi….

Q. D Gipson et al. arXiv:1411.0005

Accidental band touching

- Nontrivial topology- CB and VB touching by composition tuning- Always at Topological critical point

Cd3As2

Pb1-xSnxSe, Pb1-xSnxTe

Na3Bi

- High mobility 107 cm2/Vs- Very strong MR- Giant diamagnetism

TlBiSSe

Beyond 3D Dirac systems:

Weyl semimetals

New state of matter – Chiral Dirac particles

Left and right Dirac electrons

𝐻3𝐷 = ±ℏ𝑣𝐹𝝈 ∙ 𝒌 ′

Spin degenerate 3D Dirac

𝒌𝝈

𝒌𝝈

Dirac point

𝒌 ′𝝈 𝒌 ′𝝈

DP

Weyl point Weyl point arXiv: 1501.00755

Symmetry breaking

TaAs, TaP, NbAs, NbP

Family of recently predicted WSM

How to get information on type dispersion by magnetic and transport measurements?

∆𝑀~𝐴 𝑇, 𝐵 sin(2𝜋 𝐹

𝐵+ ∅(𝛾))

Lanadu quantization and de Haas von Alphon oscillations (dHvA)

Phase factor𝜙𝐵 = 2𝜋1

2− 𝛾

𝛾 =1/20

Berry phase

𝐴𝒌 𝜖𝒌 = 𝑛 + 𝛾 2𝜋𝑒𝐵/ℏ Osanger’s relation

Parabolic

Linear

∅ 𝛾 = 2𝜋(1

4±1

4− 𝛾 ±

1

8)

dHvA

Sample: Monocrystal of Pb0.83Sn0.17Se with reduced charge carriers ~1017 cm3

Sample information

20 40 60 800.0

0.2

0.4

0.6

0.8

1.0

(422)(420)(222)

(220)

(600)

(400)

Inte

nsity (

a.u

.)

2

(200)

Samples were grown by modified Bridgman method

Solid solution with cubic structure

0.2 0.4 0.6 0.8 1.0-0.2

-0.1

0.0

0.1

0.2 5K 7K

8K 10K

13K 15K

17K 20K

30K

M

(em

u/m

ol)

1/B (1/T)

5K

0 20 40 60 80

Am

pli

tud

e

F(T)

F=8.7 T

SQUID measurements of dHvA oscillations

dHvA oscillations FFT of the oscillations

Landau level indexing

Phase determination

Quadratic dispersion: Schrodinger electrons

Linear dispersion: Dirac electrons

0 1 2 3 4 5 6 7 80.0

0.2

0.4

0.6

0.8 MAX

MIN

QUADRATIC

LINEAR

1/B

n (

1/T

)

n

0.2 0.4 0.6 0.8 1.0-0.2

-0.1

0.0

0.1

0.2

M

(em

u/m

ol)

1/B (1/T)

∅ 𝛾 = 2𝜋(1

4±1

4− 𝛾 ±

1

8)

Spin splitting

0 5 10 15 20 25 300

1

Am

pli

tud

e

T(K)

m*=0.04m

e

Effective mass

0.0 0.2 0.4 0.6 0.8 1.0

-2.1

-1.8

-1.5

-1.2

-0.9

ln(

M B

1/2si

nh(

T/B

))

1/B (T-1)

dHvA

=5´-13

s

Scattering rate

Much stronger than form transport measurements

Thank you for your

attention