TOPOLOGICAL INSULATORS AND DIRAC
SEMIMETALS – SYNTHESIS AND
CHARACTERIZATION
Filip Orbanić
Faculty of science, Department
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
topological invariant (nontrivial).
There is an energy gap.
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
• Examples of TI and topological invariant ?
Z2 TI (TRS & IS)
Quantum Hall effect
 xy
n
e2
 n , n N
h
Time reversal operator:
4
TRS, 2D
  0,1
3D :
M 2  1
Nontrivial topological invariant if there is an band
inversion at some 𝜆 !
N
2D : (1)    2 n (i )
i 1 n 1
𝜈 = 0, topological invariant is not
determined by TRS but with crystal
simmetries (mirror symmetry).
 2  1
Edge states
topological invariant
(Chern number or TKNN)
Topological crystalline insulator
parity
Z2 topological invariant
 0 , 1 , 2 , 3
SOC – spin orbit coupling
4
TOPOLOGICAL INSULATORS
• Edge/surface states properties.
Dirac dispersion in low-energy excitations
Spin-momentum locking
high mobility!
forbidden backscattering!
5
Downloaded from
Phys. Rev. Lett. 113,
027603 (2014)
DIRAC SEMIMETALS
Dirac semimetal
Dirac dispersion in
the touch points.
Valence and conduction band touching in
discrete points.
• Dirac dispersion in 3D!
• 2D – graphene: 𝐻 𝑘 = 𝑣 𝑘𝑥 𝜎𝑥 + 𝑘𝑦 𝜎𝑦
• 3D: 𝐻 𝑘 = 𝑣𝑖𝑗 𝑘𝑖 𝜎𝑗 ,
𝑗 = 𝑥, 𝑦, 𝑧
SOC ~ 𝜎𝑧 opens the gap.
Robust against perturbations!
6
DIRAC SEMIMETALS
Downloaded from Nat.
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
• Dirac semimetals
2D Dirac dispersion.
3D Dirac dispersion.
Possibility for investigating Dirac’s fermion physics!
Evidence of Dirac fermions?
8
QUANTUM OSCILLATIONS
• Electrons in strong B-field
Landau levels.
2 2
1

  kz
E   N    
2

 2m
For Dirac dispersion 𝐸± (𝑘) = ±𝑣𝑓 𝑘
for 𝐸 𝑘 =
ℏ2 𝑘 2
2𝑚
Periodic behavior of DOS.
E ( N )   (2ev B / c) N
2
f
Oscillations of physical quantities in 1/B!
𝑀
𝜎
de Haas van Alphen oscillations.
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)
 F 1 1

M  ART RD Rs sin 2      

 B 2 8
 xx
 F 1 1

 ART RD Rs cos 2      

 B 2 8
T
B
RT 
 T 
sinh 

 B 
RD  e

TD
  14.69
F
 2
kF
2e
TD 

2k B
B
 m 

RS  cos g
2
m
0 

m
TK 1
m0
Informations about
carrier density and
Fermi surface snape.
Quantum scattering time
2𝜋𝛽 = 𝛾 Berry phase!
  i  d k  (k )  k  (k )
C
For Dirac fermions
𝛾=𝜋
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 𝑚𝑏𝑎𝑟)
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? (𝑥 ≈ 0.18)
𝑥 = 0.17, 0.18
15
PbSnSe
• Magnetization (SQUID) and magnetoresistance measurements in Pb0.82Sn0.18Se.
16
PbSnSe
• By subtracting the background we get a pure oscillatory part.
Zeeman splitting of
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
RT 
B
 T 
sinh 

 B 
RD  e

TD
B
1
F 1 1


2          2 N  
2
B 2 8


18
Cd3As2
• Cd3As2 is an intrinsic (nontrivial) Dirac semimetal.
• Very stable material, except toxicity ideal for application and experiment.
• High mobility ~106 𝑐𝑚2 𝑉 −1 𝑠 −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
Cd3As2
B perpendicular to ab plane.
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
topological insulator.
Metallic behavior
because off surface
states.
Typical semiconducting
behavior
Succsesfully observed quantum oscillations.
21
CONCLUSION
• Topological insulators.
• symmetry protected surface states
• Dirac semimetals.
spin locking and robustness at nonmagnetic impurities.
An insight into the physics of Dirac ’s fermions.
• 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.
22