TOPOLOGICAL INSULATORS AND DIRAC Filip Orbanić SEMIMETALS – SYNTHESIS AND Faculty of science, Department CHARACTERIZATION of phyisics, Zagreb 1 OVERVIEW

• What are topological insulators and Dirac semimetals?

• Quantum transport and magnetic properties (quantum oscillations).

• Synthesis

• Some concrete materials and measurements.

2 TOPOLOGICAL INSULATORS (TI)

Topological insulator(TI)

Quantum state characterized by There is an energy gap. topological invariant (nontrivial).

Defined over the wave functions (that give the energy gap) Invariant under adiabatic change of Hamiltonian. The same as long as there is a gap!

• Closing of the energy gap at the boundary of topological and normal insulator (vacuum) conductive edge/surface

3 TOPOLOGICAL INSULATORS Topological crystalline insulator 휈 = 0, topological invariant is not • Examples of TI and topological invariant ? determined by TRS but with crystal simmetries (mirror ). Z TI (TRS & IS) 2 M 2  1 2 Quantum Hall effect Time reversal operator:   1

e2  xy  n , n N Edge states h Nontrivial topological invariant if there is an band n topological invariant 4 N  inversion at some 휆 ! (Chern number or TKNN) 2D : (1)  2n (i ) i1 n1 parity TRS, 2D

  0,1 Z2 topological invariant

3D :  0 ,1, 2 , 3 SOC – spin orbit coupling 4 TOPOLOGICAL INSULATORS

• Edge/surface states properties.

Dirac dispersion in low-energy excitations high mobility!

Spin-momentum locking forbidden backscattering!

5 Downloaded from Phys. Rev. Lett. 113, DIRAC SEMIMETALS 027603 (2014)

Dirac semimetal

Dirac dispersion in Valence and conduction band touching in the touch points. discrete points.

• Dirac dispersion in 3D!

• 2D – : 퐻 푘 = 푣 푘푥휎푥 + 푘푦휎푦 SOC ~ 휎푧 opens the gap.

• 3D: 퐻 푘 = 푣푖푗푘푖휎푗, 푗 = 푥, 푦, 푧 Robust against perturbations!

6 Downloaded from Nat. DIRAC SEMIMETALS Comm. 5, 4896 (2014).

• Dirac semimetals come in two topological clasces too.

• Consequence of topological phase transition NI – TI.

• Intrinsic as a result of additional symmetries (rotational symmetry).

7 TOPOLOGICAL INSULATORS & DIRAC SEMIMETALS

• Topological insulators 2D Dirac dispersion. • Dirac semimetals 3D Dirac dispersion.

Possibility for investigating Dirac’s physics!

Evidence of Dirac ?

8 QUANTUM OSCILLATIONS

• Electrons in strong B-field Landau levels.

2 2  1  2k 2 ℏ 푘 E   N     z for 퐸 푘 =  2  2m 2푚 For Dirac dispersion 퐸±(푘) = ±푣푓푘

Periodic behavior of DOS. 2 E (N)   (2ev f B / c)N

Oscillations of physical quantities in 1/B!

푀 de Haas van Alphen oscillations. 휎 Shubnikov de Haas oscillations. Downloaded from J. Phy. Soc. Jap. 82, 102001 (2013).

9 QUANTUM OSCILLATIONS

Effective mass • de Haas van Alphen and Shubnikov de Haas oscillations (for 3D)

T m 1   F 1 1  B  14.69 TK M  AR R R sin 2     RT  m T D s    T  0 Informations about   B 2 8  sinh   B   2 carrier density and F  kF T  D 2e Fermi surface snape. R  e B   F 1 1  D   AR R R cos 2      xx T D s      m  T    B 2 8  R  cos g  D 2k  S  2 m  B  0  Quantum scattering time 2휋훽 = 훾 Berry phase!

For Dirac fermions   i dk  (k) k  (k) C 훾 = 휋

10 SYNTHESIS

• Aim: high quality monocrystal samples. • The fewer impurities and defects minimize the influence of bulk states and increase mobility.

Sealing the quartz tube. High vacuum in ampoule • clean atmosphere • volatile elements

Material in vacuum seald quartz ampoule (vacuum ~10−6 푚푏푎푟)

11 SYNTHESIS

• Modified Bridgman method.

Temperature gradient is achieved by two-zone tube furnaces.

• (Chemical) vapor deposition. Synthesis parameters: temperature, gradient, heating/cooling rate, growth time, amount and shape of material, ampoule dimensions. Optimization of parameters!

12 SYNTHESIS

Cd3As2 • Results of synthesis:

5 mm

BiSbTeSe2 PbSnSe

SnTe

TaP 13 SAMPLE PREPARATION

1mm • For transport measurements good contacts are crucial (~ Ω).

Samples of Cd3As2 with contacts.

1mm

Spot welding

14 PbSnSe

• Pb1-xSnxSe is a topological crystalline insulator for x > 0.23. • Known for ages energy gap depends on the T and x. Topological phase transition with T or x. Downloaded from Phys. Rev. 157, 608-611 (1967)

• What happens at the transition point? (푥 ≈ 0.18) 푥 = 0.17, 0.18

15 PbSnSe

• Magnetization (SQUID) and magnetoresistance measurements in Pb0.82Sn0.18Se.

16 PbSnSe

Zeeman splitting of • By subtracting the background we get a pure oscillatory part. Landau levels.

17 PbSnSe

• How to get physical values from quantum oscillations?

  F 1 1  M  ART RD Rs sin2         B 2 8 

  F 1 1   xx  ART RD Rs cos2         B 2 8 

T R  B T T  sinh   B  T  D  F 1 1   1  R  e B 2         2N   D  B 2 8   2 

18 Cd3As2

• Cd3As2 is an intrinsic (nontrivial) Dirac semimetal. • Very stable material, except toxicity ideal for application and experiment. • High mobility ~106푐푚2푉−1푠−1.

• A pair of Dirac points in the direction of rotational symmetry axis (kz). • Anisotropy of Fermi surface? Different frequencies for different direction of magnetic field.

Downloaded from Phys. Rev. Lett. 113, 027603 (2014)

Downloaded from Phys. Rev. Lett. 113, 027603 (2014). 19 B perpendicular to ab plane. Cd3As2

F = 60 T

• Magnetization and magnetoresistance are measured.

Superposition of frequencies.

B in ab plane.

20 OTHER MATERIALS

• Other materials we have succsesfully synthesized:

TaP Weyl semimetal candidate. BiSbTe2S .

Metallic behavior because off surface states. Typical semiconducting behavior

Succsesfully observed quantum oscillations.

21 CONCLUSION

• Topological insulators. • symmetry protected surface states spin locking and robustness at nonmagnetic impurities.

An insight into the physics of Dirac ’s fermions. • Dirac semimetals. • 3D analogue of graphene. Symmetry protected Dirac points. • The idea is to synthesize (determination of synthesis parameters) and characterize the obtained materials. • Examine the consequences of the Dirac nature of cariers in magnetization and transport.

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