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Y. Hatsugai
Univ. Tsukuba
Electronic structure of silicene in the extended Weaire-Thorpe model
Y.Hatsugai, K.Shiraishi, H. Aoki, New J. Phys. 17, 025009 (2015) arXiv:1410.7885
Flat bands and Dirac cones
バルクエッジ対応の物理の多様性と普遍性千葉大学理学部集中講義 2015年7月9日-10日
Plan
Singular dispersions & silicene “Topological” deformation of Takeda-Shiraishi’s.
Flat bands in materials: counting dimensions
Overlapping molecular orbitals & flat bands Without translational invariance
Physical origin of (nearly) flat bands
Weaire-Thorpe model and extension 3D to 2D
Hydrogen termination
Buckling
Plan
Singular dispersions & silicene “Topological” deformation of Takeda-Shiraishi’s.
Flat bands in materials: counting dimensions
Overlapping molecular orbitals & flat bands Without translational invariance
Physical origin of (nearly) flat bands
Weaire-Thorpe model and extension 3D to 2D
Hydrogen termination
Buckling
Silicene as a silicon analogue of grapheneOne line history of singular dispersions (Dirac cones)
Graphene
Silicene
Predicted in 1947, then, realized in 2004Wallace Novoselov-Geim et al.
Predicted in 1994, then, realized (??) in 2012 Takeda-Shiraishi Lalmi et al./Vogt et al.
History repeats itself
Wallace, Phys. Rev.71, 622 (1947)
Takeda-Shiraishi, Phys. Rev. B50, 14916 (1994)
Dirac cones & something else, what ??
Revisiting Takeda-Shiraishi’s
Ener
gy (e
V)-5
0EF
X SΓ Γ
Revisiting Takeda-Shiraishi’s
Ener
gy (e
V)-5
0EF
X SΓ Γ
“Topological” deformation of the bands
Classify into two: bands of Dirac fermions & else
Revisiting Takeda-Shiraishi’s
Ener
gy (e
V)-5
0EF
X SΓ Γ
Classify into two: bands of Dirac fermions & else
“Topological” deformation of the bands
Revisiting Takeda-Shiraishi’s
Ener
gy (e
V)-5
0EF
X SΓ Γ
Classify into two: bands of Dirac fermions & else
“Topological” deformation of the bands
Revisiting Takeda-Shiraishi’s
Ener
gy (e
V)-5
0EF
X SΓ Γ
Classify into two: bands of Dirac fermions & else
“Topological” deformation of the bands
Revisiting Takeda-Shiraishi’s
Ener
gy (e
V)-5
0EF
X SΓ Γ
Classify into two: bands of Dirac fermions & else
“Topological” deformation of the bands
Revisiting Takeda-Shiraishi’s
Ener
gy (e
V)-5
0EF
X SΓ Γ
Classify into two: bands of Dirac fermions & else
“Topological” deformation of the bands
Revisiting Takeda-Shiraishi’s
Ener
gy (e
V)-5
0EF
X SΓ Γ
Classify into two: bands of Dirac fermions & else
“Topological” deformation of the bands
Revisiting Takeda-Shiraishi’s
Ener
gy (e
V)-5
0EF
X SΓ Γ
Classify into two: bands of Dirac fermions & else
“Topological” deformation of the bands
Revisiting Takeda-Shiraishi’s
Ener
gy (e
V)-5
0EF
X SΓ Γ
Classify into two: bands of Dirac fermions & else
“Topological” deformation of the bands
(3)
(1)degeneracy
Simplified silicene
Ener
gy (e
V)-5
0EF
X SΓ Γ
: Dirac cones & Flat bands“Topological” deformation of the bands
(3)
(1)degeneracy
Dirac cones : Symmetry protectedFlat bands : Due to multi-orbital character !!
Fujita states 1DJPSJ 65, 1920 (1996) Kohmoto-Sutherland,
Phys. Rev. Lett. 56, 2740(1986)
Ring states on Penrose tiling
dice lattice B.Sutherland, Phys. Rev. B34, 5208(1986)
“Flat” bands in 1,2,3 dimensions
d=2, Silicene
d=3, Weaire-Thorpe modelWeaire-Thorpe, Phys. Rev. B4, 2508 (1971)Dagotto, Fradkin,Moreo, Phys. Lett.B 172, 383 (1986)
dp (Lieb) modelMielke-Tasaki, Comm.Math.Phys. 158, 341 (1993)
d=2 Kagomed=3 PyrochloreY. Hatsugai, I. Maruyama, EPL 95, 20003 (2011)
Singular dispersions & silicene “Topological” deformation of Takeda-Shiraishi’s.
Flat bands in materials: counting dimensions
Overlapping molecular orbitals & flat bands Without translational invariance
Physical origin of (nearly) flat bands
Weaire-Thorpe model and extension 3D to 2D
Hydrogen termination
Buckling
Overlapping MO’s MO’s
Itinerancy : NON orthogonality of MO’s
c1
c2c4
MO annihilation op.
= (c†1c2 + · · · )/3
† = (c†1, c†2, c
†3, c
†4, c
†5, · · · )
c3
c1
c2
0
BBBBBBB@
1/p3
1/p3
01/
p3
0...
1
CCCCCCCA= c†
C
= c†Pc
† = 1normalized
projectionP = † = P 2
Sum of projections
C†C
Y. Hatsugai, I. Maruyama, EPL 95, 20003 (2011)
= (c†1 + c†2 + c†4)/p3
H � µN =MX
m=1
EmC†mCm = c†hc ,
PmPn 6= 0, (m 6= n)
,
h =MX
m=1
EmPm
Overlapping MO’s MO’s1
2
6
5 78
9
103
12
3
4
56
7
8
Z � N �M
Theorem Z: # of zero eigen statesN: # of sitesM: # of MO’s
Do NOT need translational invarianceIf translationally invariant, use in momentum space
Z � 10� 8 = 2 At least 2 zero energies !
4
Diagonalizable within M dimensional linear space
N dimensional Hamiltonian = Sum of M projections
Non zero energy bands are at most M
L? : nullRest is null, N-M zero energy “flat” bands in N-dim.
Flat bands are stable for perturbation
L1
LM
...
PROOF
ex: 3D (3sites)
2D
1D is null (1 zero energy)
(deformation of MO’s )
C = (c1 + c2 + c3)/p3 ! (⇤c1 + ⇤c2 + ⇤c3)/
p⇤
Overlapping MO’s MO’s1
4
2
6
5 78
9
103
12
3
4
56
7
8
Z � N �M
Z � 10� 8 = 2
If the itinerancy is not enough, some states are localized
NM
: number of sites (degree of freedom): number of MO’s (itinerancy by overlapping)
Physical meaning of flat bands
M =X
m
dimPm P = 1 †1 + 2
†2, dimP = 2ex.Slightly extended theorem
Singular dispersions & silicene “Topological” deformation of Takeda-Shiraishi’s.
Flat bands in materials: counting dimensions
Overlapping molecular orbitals & flat bands Without translational invariance
Physical origin of (nearly) flat bands
Weaire-Thorpe model and extension 3D to 2D
Hydrogen termination
Buckling
Unit cell & 3 primitive translations
Weaire-Thorpe model Weaire-Thorpe, Phys. Rev. B4, 2508 (1971)
3D Multiorbital (sp3) tight binding hamiltonian
2
66666666664
V1 V1 V1 V1 V2 0 0 0V1 V1 V1 V1 0 V2 0 0V1 V1 V1 V1 0 0 V2 0V1 V1 V1 V1 0 0 0 V2
V2 0 0 0 V1 V1ei(ky
+kz
) V1ei(kz
+kx
) V1ei(kx
+ky
)
0 V2 0 0 V1e�i(ky
+kz
) V1 V1ei(kx
�ky
) V1e�i(kz
�kx
)
0 0 V2 0 V1e�i(kz
+kx
) V1e�i(kx
�ky
) V1 V1ei(ky
�kz
)
0 0 0 V2 V1e�i(kx
+ky
) V1ei(kz
�kx
) V1e�i(ky
�kz
) V1
3
77777777775
HWT (k) =
flat bands !
Simple but 8×8 : need some work to diagonalize
Gapless points !
Unit cell & 3 primitive translations
Weaire-Thorpe model Weaire-Thorpe, Phys. Rev. B4, 2508 (1971)3D Multiorbital (sp3) tight binding hamiltonian
k =1
2
2
664
1e�ik1
e�ik2
e�ik3
3
775 ,
2
66666666664
V1 V1 V1 V1 V2 0 0 0V1 V1 V1 V1 0 V2 0 0V1 V1 V1 V1 0 0 V2 0V1 V1 V1 V1 0 0 0 V2
V2 0 0 0 V1 V1ei(ky
+kz
) V1ei(kz
+kx
) V1ei(kx
+ky
)
0 V2 0 0 V1e�i(ky
+kz
) V1 V1ei(kx
�ky
) V1e�i(kz
�kx
)
0 0 V2 0 V1e�i(kz
+kx
) V1e�i(kx
�ky
) V1 V1ei(ky
�kz
)
0 0 0 V2 V1e�i(kx
+ky
) V1ei(kz
�kx
) V1e�i(ky
�kz
) V1
3
77777777775
HWT (k) =
= 4V1
0
†0 0
0 k †k
�+ V2
0 E4
E4 0
�
= �2V2E8 + 4V1P1 + 4V1P2 + 2V2P3+
= +2V2E8 + 4V1P1 + 4V1P2 � 2V2P3�flat bands !
Pi = i †i
projections
1 =
0
0
� 2 =
0 k
� , 3± =
1p2
E4
±E4
� ,
P 2i = Pi P1P2 = 0 P1P3± 6= 0 Non orthogonal
written by 2 ways
dimP3± = 4dimP1 = dimP2 = 1 ,
1 + 1 + 4 = 6 8� 6 = 2 Flat bands at !! ±2V2
Counting dimensions !
Extended Weaire-Thorpe model for silicene with hydrogen termination
2D Multiorbital (sp3) tight binding hamiltonian
2D array of the unit cells
H✏HSilicene(k) =
HV (0)� ✏HE V2E
C4
V2EC4 HV (k)� ✏HE
�
H✏HSilicene ± V2E8 = 4V1P1 + 4V1P2 ± 2V2P
C3± ± (V2 ⌥ ✏H)P5
Blue bonds are special✏H
(✏H)
dimP1 + dimP2 + dimP3± + dimP5 = 1 + 1 + 3 + 2 = 7
8-7=1 flat band at ±V2
Extended Weaire-Thorpe model for silicene with hydrogen termination
2D Multiorbital (sp3) tight binding hamiltonian
2D array of the unit cells
H✏HSilicene(k) =
HV (0)� ✏HE V2E
C4
V2EC4 HV (k)� ✏HE
�
H✏HSilicene ± V2E8 = 4V1P1 + 4V1P2 ± 2V2P
C3± ± (V2 ⌥ ✏H)P5
Blue bonds are special ✏H
(✏H)
dimP1 + dimP2 + dimP3± + dimP5 = 1 + 1 + 3 + 2 = 7
8-7=1 flat band at
When
5
8-5=3 flat bands at triply degenerate
✏H = V2 < 0
�V2
+V2
Ener
gy (e
V)-5
0
EF
X SΓ Γ
(3)
(1)
Various band ordering by changing ✏H(hydrogen termination)
“Blue bonds are special”
Buckling can be included partly
hlocal
(✓) =X
hi,ji
Vijc†i cj + h.c.,
Vij =
⇢V1
(hi, ji = h01i, h12i, h20i)V 01
(hi, ji = h03i, h13i, h23i)V 01
V1
=
cos ✓
cos ✓0
, cos ✓0
= �1
3
H✏H ,cos ✓Silicene
(k) =
H✓
V (0)� ✏✓HE V2
EC4
V2
EC4
H✓V (k)� ✏✓HE
�,
H✓V (k) = 4V1
✓k(
✓k)
†,
✓k = diag
✓cos ✓
cos ✓0, 1, 1, 1
◆ k,
✏✓H =
✓cos ✓
cos ✓0
◆2
✏H
Summary
?!
Less dispersive bands: due to multi-orbital character Possible instability (ferromagnetic/structure)
Ener
gy (e
V)-5
0
EF
X SΓ Γ
(3)
(1)
Ener
gy (e
V)-5
0
EF
X SΓ Γ
analyticfirst principle
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