Topological Kondo Insulator SmB6 - 東京大学 物性研究所 · 2014. 8. 21. · Kondo effect...
Transcript of Topological Kondo Insulator SmB6 - 東京大学 物性研究所 · 2014. 8. 21. · Kondo effect...
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Topological Kondo Insulator SmB6
Tetsuya Takimoto
Department of Physics, Hanyang University
J. Phys. Soc. Jpn. 80 123720, (2011).
Collaborator: Ki-Hoon Lee (POSTECH)
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Content
1. Introduction of SmB6
in-gap state
2. Topological insulator
3. Model Hamiltonian of SmB6
relevant orbitals
4. Topological index <= parity eigenvalue
topological index (1:111) for SmB6
5. Necessary and sufficient conditions for TKI
6. Surface state
7. Recent experimental data
8. Summary
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Kondo effect
Kondo lattice (strong correlation):
Lattice of localized spin
+
Sea of conduction electrons
Conduction
electron spinLocalized
spin
JK
JK: exchange interaction
Spin-singlet for T<TK
Logarithmic increase
of resistivity below TK
Due to the coherence of Kondo states, heavy fermion are formed in the Kondo lattice
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Heavy fermion system and Kondo insulator
Another picture from weak correlation (Anderson lattice)
Renormalization of hybridization
between conduction and localized electrons
Ef
(instead of exchange interaction)
T>TK T<TK
eF
k k
Ef
T>TK T<TK
eF
k k
Heavy fermion
system
Kondo insulator
metallic
metallic
metallic
insulator
Anderson lattice
Kondo lattice
Strong coupling limit
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IntroductionSmB6: Kondo insulator
in-gap state below 5K
periodic Anderson model (s+f)
SmB6
f: jz=-1/2 and +1/2 (G8 of Oh)
Cooley et al. PRL 74 1629 (1995)
point group : Oh
Dgap=15 meV
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Topological Insulator
・two copies of integer quantum Hall system
・time reversal invariant
two-dimensional case (Kane & Mele PRL 95)
L
circumferential electric field E
can be induced by a threading flux
flux
h/e
k k+2p/L
E
trivial insulator (n0=0) no effect
topological insulator (n0=1) spin Hall
・metallic edge (surface) state
particle-hole excitation around EF on the edge
gapped around EF
spin-direction is different between particle and hole
Dirac cone on a surface of Bi2Se3
Wray et al. Nature Phys. 7 32
Hasan & Kane RMP 82
Qi & Zhang RMP
ref.
・large spin-orbit coupling
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Topological number in noninteracting system
Chern # = 0 (cancellation in Kramers doublet)
In TRI compound (2D square lattice)
topological # n0
calculated by choosing one from a doublet
gauge invariance (phase of wave function)
time reversal symmetry H(-k)=QH(k)Q-1 (Q=syK)
determine gauge potential topological #
kx
ky
(0,0)
(p,p)
Fu and Kane: PRB 74 195312
F=dA
trivial case
n0=0A is continuous
in Brillouin zone.
topological case
n0=odd #A has one or three singular points
at (0,0), (p,0), (0,p), or (p,p).
field strengthn0
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Bulk-edge correspondence
electronic states in bulk topological number n0
TI (n0=1) usual insulator (n0=0)
a junction between TI and usual insulator in real space
n0=1: topological
n0=0: trivial
bulk gap due to spin-orbit bulk gap due to hybridization
between electrons with the same spin
different origin of bulk gap
necessary close the gap to move to the other side!
TI (n0=1) vacuum (n0=0)
edge
edge
gapless states must appear on the edge
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relevant orbitals
dominant electronic states
(4f)5(5d)1 and (4f)6
band structure calculation
Yanase and Harima:
Prog. Theor. Phys. Suppl. 108 19, (1992).
Antonov et al. : PRB 66 165209, (2002).
4f and 5d electrons are from Sm. EF
Sm3+: (4f)5
angular momentum: J=5/2
CEF ground state: G8
~ one hole in G8 states
in j-j coupling scheme
j=5/2
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effective modeltight-binding
5d-electron term (NN+NNN) 4f-electron term (NN+NNN)
hybridization term (NN+NNN)
: orbital in eg
s : real spin
g : orbital in G8
t : pseudo spin in a Kramers doublet
n.n. hopping
n.n.n. hopping
n.n. hopping
n.n.n. hopping
n.n. hopping
5d
5d5d
4f 4f
4f
The seed of singularity for A
n.n.n. hopping
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renormalization of interaction term
・Gutzwiller projection
・slave boson
interactions between 4f
interactions between 5d
interactions between 4f and 5d
z : renormalization factor in Fermi liquid theory
localization character of 4f-electron
renormalization of interactions between 4f
map into effective Hamiltonian
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electronic structure
Antonov et al.: PRB 66 165209, (2002).
band mixing around X point
5d-bands
4f-bands
half-filling case provides insulator
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topological index(n0: n1,n2,n3)
parity operator
time reversal invariant point
dn,m: parity eigenvalue
of n-th two-fold degenerate band at km*
: 2X2 matrix between d- and f-
electronic states
hybridization term is formed by sx and sy
hybridization term is irrelevant at k* for dn,m
with even parity
with odd parity
is the highest energy
our model
(1: 1,1,1)
n0=0 WTI
n0=1 STI
Fu and Kane: PRB 74 195312 (2006)
PRB 76 045302 (2007)
eF
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The seed of singularity for A at k*
Necessary condition for TKI
~ k’.s at k=k*+k’ (|k’|<<p)
k’-linear: different parities
s: large spin-orbit coupling
hybridization gap between d- and f-bands
around k=k*
Necessary and sufficient conditions for TKI
Nontrivial topological #
Determined by Hamiltonian without Hhyb
Sufficient condition for TKI
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Spectral function of (001)-surface
Intensity around X is stronger
than that around G
There are Dirac cones
at G and X points.
There is a v.-H. singularity
along G-X.
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Recent ARPES measurement for (001) surface
Nat. Commun. 4, 2991 (2013).
M. Neupane et al. (M.Z.Hasan)
Fermi surface around G and X
topology of Fermi surface
is the same as the theoretical result
T=6K
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only bulk states at 12K
development of in-gap states by decreasing T
T=6K
D~15 meV
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NMR in SmB6
・insulator behavior down to ~20K
・contribution from the in-gap state
below 20K
・magnetic field suppresses
the in-gap state
The magnetic field breaks TRS.
The metallic surface states of TI is protected by TRS.
The surface states should be gapped by the magnetic field
in-gap state~surface state of topological insulator
Caldwell et al. PRB 75 075106 (2007)
Why the value of 1/T1T is so large at the low temperature limit?
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Summary
We study the possibility of the topological insulator for SmB6,
calculating the topological index and spectral function of [001] surface,
based on a realistic model.
The topological index is (1: 1,1,1).
In the spectral function, surface states appear with odd number of
Dirac cones.
Based on the calculation, SmB6 is a topological Kondo insulator.
Recent ARPES measurements show Fermi surfaces surrounding the Dirac points
consistent with theoretical suggestions.
experiment: measurements of Dirac points, spin texture
theory: clarification of electron correlation in the surface states
next step