Topological insulators

http://www.physik.uni-regensburg.de/forschung/fabian
Topological insulators
Jaroslav Fabian
Institute for Theoretical Physics
University of Regensburg
Stara Lesna, 21.8.2012 DFG SFB 689

what are topological insulators?
gapped bulk states +
conducting (gapless) edge (surface) states due to topology
M. Z. Hasan and C. L. Kane, Topological insulators, Rev. Mod. Phys. 82, 3045 (2010)

:outline:
• mesoscopics for pedestrians
• topological insulators
• edge states in HgTe quantum wells
• magnetism of HgTe edge states
B. Scharf, A. Matos-Abiague, and J. Fabian,
Phys. Rev. B 86, 075418 (2012)

mesoscopics for pedestrians
bulk states
quantum point contacts
current states

How much current per spin flows in a quantum channel?
…….. conductance quantum
…….. resistance quantum

conductance quantization: n-channel transport
…….. conductance quantization

conductance quantization
B. J. van Wees, Phys. Rev. Lett. 60, 848 (1988).
D. A. Wharam et al. J. Phys. C 21, L209 (1988).
e
n=1 (2 spins)
GaAs/AlGaAs QPC
lead resistance subtracted
B. J. van Wees, Phys. Rev. Lett. 60, 848 (1988)
picture from C. W. J. Beenakker and H. van Houten, Solid State Physics, 44, 1 (1991).

integer quantum Hall effect: topological edge states
bulk states
bulk Landau levels
edge states
edge states
no backscattering

integer quantum Hall effect: topological edge states
voltage probe

integer quantum Hall effect
K. von Klitzing, G. Dorda, M. Pepper, Phys. Rev. Lett. 45, 494 (1980)
Si MOSFET
Original MOSFET
von Klitzing’s web site
SdH IQHE
GaAs/AlGaAs
Cl
picture from http://www.ptb.de/en/org/2/Inhalte/qhe/E-quantenhalleffekt.htm

topological nature of the integer
quantum Hall effect
1 st Chern number
Blount-Berry curvature
Blount-Berry phase
Bloch state
D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Phys. Rev. Lett. 49, 405 (1982)
Q. Niu, D.J. Thouless, and Y.-S. Wu, Phys. Rev. B 31, 3372 (1985)

topological insulators
another look at the quantum Hall edge states
quantum Hall edge states come in spin pairs:
time reversal symmetry is broken
skipping orbits
bulk orbits (Landau levels)

topological insulators
edge states with time reversal preserved: spin-orbit coupling
spin-orbit edge states come in spin pairs, but move opposite:
time reversal symmetry is preserved
edge states
no backscattering !!!
the edge states are topologically protected against TR scattering

emergence of spin-orbit fields
space-inversion symmetry breaking

Time reversal points
spin degeneracy preserved

Z 2 invariance
stable to continuous change of band parameters
even number of crossings
odd number of crossings

(non-exotic) materials classes of
topological insulators
• Graphene
Kane and Mele, 2005, spin quantum Hall effect
• 2d topological insulators
Zhang and co., 2006, HgTe quantum wells (see later)
• 3d topological insulators (Zhang and Co)
BiSe, BiTe, BiSb …
yet to be experimentally confirmed
first 3d experimental TI
• other special materials structures …

Electronic structure of CdTe and HgTe
CdTe
HgTe
normal band ordering
1.6 eV gap
narrow-gap semiconductor
inverted band structure
“negative band gap” -0.3 eV

HgTe/CdTe quantum wells
CdTe
CdTe
CdTe
CdTe
HgTe
HgTe
2d topological insulator

HgTe/CdTe quantum wells: TI states
B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Science 314, 1757 (2006).
CdTe
HgTe
2d topological insulator
“trivial” interface states
CdTe
B. A. Volkov and O. A. Pankratov, JETL Lett. 42, 178 (1985)
M. I. Dyakonov and A. V. Khaetskii, JETP Lett. 33, 110 (1981).

experimental evidence of TI states
mesoscopic transport
tunable gate and width
Konig et al. (Molenkapm group, Wurzburg), Science 318, 766 (2007)

experimental evidence of TI states
mesoscopic transport
(d)
R14,23 (Ω)
G = .3 e 2 /h
G = 2 e 2 /h
V g -V thr (V)
Konig et al. (Molenkapm group, Wurzburg), Science 318, 766 (2007)

magnetism of the TI edge states in HgTe
how the edge states evolve with B-field?
CdTe
CdTe
HgTe
B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Science 314, 1757 (2006).

magnetism of the TI edge states in HgTe
how the edge states evolve with B-field?
B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Science 314, 1757 (2006).

magnetism of the TI edge states in HgTe
B = 0
B. Scharf, A. Matos-Abiague, and J. Fabian, Phys. Rev. B 86, 075418 (2012)
E [meV]
20
10
0
-10
-20
(a)
Fig. (b) Fig. (c)
2
10.78 1
10.76
10.74
-2 -1 0 1 2
k [10 6 1/m]]
-2 -1 0 1
k [10 6 1/m]
(b)
v k
0
bulk states
v k
0
5
4
3
0
5
0
2 -100 -50 0 50 100
y [nm]
4
3
2
1
ρ [10 13 1/m 2 ]
ρ [10 13 1/m 2 ]
E [meV]
20
10
0
-10
-20
(a)
8
7.6
7.2
Fig. (b)
-2 -1 0 1
k [10 6 1/m]
Fig. (c)
-2 -1 0 1 2
k [10 6 1/m]
(b)
(c)
v k
0
TI states
1.5
1
ρ [10 14 1/m 2 ]
0
1.5
0
2 -100 -50 0 50 100
y [nm]
1
ρ [10 14 1/m 2 ]

magnetism of the TI edge states in HgTe
B =10 T
B. Scharf, A. Matos-Abiague, and J. Fabian, Phys. Rev. B 86, 075418 (2012)
E [meV]
100
50
0
-50
(a)
Fig. (b)
Fig. (c)
(b) v k
0
QH edge states
5
4
3
2
1
0
5
4
3
2
1
ρ [10 14 1/m 2 ]
ρ [10 14 1/m 2 ]
E [meV]
100
50
0
-50
(a)
Fig. (b)
Fig. (c)
(b) v k
0
QH edge states
5
4
3
2
1
0
5
4
3
2
1
ρ [10 14 1/m 2 ]
ρ [10 14 1/m 2 ]
-100
-2 -1 0 1
k [10 9 1/m]
0
2 -100 -50 0 50 100
y [nm]
-100
-2 -1 0 1
k [10 9 1/m]
0
2 -100 -50 0 50 100
y [nm]

magnetism of the TI edge states in HgTe
how the edge states evolve with B-field?
B. Scharf, A. Matos-Abiague, and J. Fabian, Phys. Rev. B 86, 075418 (2012)
100
E [meV]
50
0
-50
SQH
QH
-100
0 2 4 6 8 10
B [T]

magnetism of the TI edge states in HgTe
bulk magnetic susceptibility
B. Scharf, A. Matos-Abiague, and J. Fabian, Phys. Rev. B 86, 075418 (2012)
χ [10 17 J/(Tm) 2 ]
8
6
4
2
0
-2
-4
-6
-8
T = 1 K
T = 10 K
T = 100 K
μ = 20 meV
0.25 5 10 15 20
1/B [1/T]
χ [10 17 J/(Tm) 2 ]
8
6
4
2
0
-2
-4
-6
-8
T = 1 K
T = 10 K
T = 100 K
μ = 20 meV
0.25 5 10 15 20
1/B [1/T]

Conclusion
topological insulators are a new playground for
(not only*) solid state physicists
*X. Qi, E. Witten, and S. Zhang, Axion topological field theory of topological insulators, arXiv: 1206.1407