千葉大学理学部集中講義 2015年7月9日-10日

バルクエッジ対応の物理の多様性と普遍性

Electronic structure of silicenein the extended Weaire-Thorpe model
Flat bands and Dirac cones
Y. Hatsugai
Univ. Tsukuba
Y.Hatsugai, K.Shiraishi, H. Aoki, New J. Phys. 17, 025009 (2015)
arXiv:1410.7885

PlanSingular dispersions & silicene“Topological” deformation of Takeda-Shiraishi’s.
Flat bands in materials: counting dimensions

Overlapping molecular orbitals & flat bands
Without translational invariancePhysical origin of (nearly) flat bands

Weaire-Thorpe model and extension
3D to 2D
Hydrogen terminationBuckling

Plan
Singular dispersions & silicene“Topological” deformation of Takeda-Shiraishi’s.
Flat bands in materials: counting dimensions

Overlapping molecular orbitals & flat bands
Without translational invariancePhysical origin of (nearly) flat bands

Weaire-Thorpe model and extension
3D to 2D
Hydrogen termination
Buckling

Silicene as a silicon analogue of graphene
One line history of singular dispersions (Dirac cones)
Graphene

Predicted in 1947, then, realized in 2004
Novoselov-Geim et al.
WallaceWallace, Phys. Rev.71, 622 (1947)

Silicene

Predicted in 1994, then, realized (??) in 2012
Takeda-Shiraishi
Lalmi et al./Vogt et al.

Takeda-Shiraishi,Phys. Rev. B50, 14916 (1994)

History repeats itself
Dirac cones & something else, what ??

EF-5

Energy (eV)0

Revisiting Takeda-Shiraishi’s

Γ

XS

Γ

Revisiting Takeda-Shiraishi’s

EF-5

Energy (eV)0“Topological” deformation of the bands

Γ

X

S

Classify into two: bands of Dirac fermions & elseΓRevisiting Takeda-Shiraishi’s

EF-5

Energy (eV)0“Topological” deformation of the bands

ΓX

SClassify into two: bands of Dirac fermions & elseΓRevisiting Takeda-Shiraishi’s

EF-5

Energy (eV)0“Topological” deformation of the bands

Γ

XS

Classify into two: bands of Dirac fermions & elseΓRevisiting Takeda-Shiraishi’s

EF-5

Energy (eV)0“Topological” deformation of the bands

Γ

X

S

Classify into two: bands of Dirac fermions & elseΓRevisiting Takeda-Shiraishi’s

EF-5

Energy (eV)0“Topological” deformation of the bands

ΓX

SClassify into two: bands of Dirac fermions & elseΓRevisiting Takeda-Shiraishi’s

EF-5

Energy (eV)0“Topological” deformation of the bands

Γ

X

S

Classify into two: bands of Dirac fermions & elseΓRevisiting Takeda-Shiraishi’s

EF-5

Energy (eV)0“Topological” deformation of the bands

Γ

X

S

Classify into two: bands of Dirac fermions & elseΓRevisiting Takeda-Shiraishi’s

EF-5

Energy (eV)0“Topological” deformation of the bands

ΓX

SClassify into two: bands of Dirac fermions & elseΓRevisiting Takeda-Shiraishi’s(3)
EFdegeneracy

(1)

-5

Energy (eV)0“Topological” deformation of the bands

ΓX

SClassify into two: bands of Dirac fermions & elseΓSimplified silicene : Dirac cones & Flat bands(3)
EFdegeneracy

(1)

-5

Energy (eV)0“Topological” deformation of the bands

ΓXSΓ

Dirac cones : Symmetry protected
Flat bands : Due to multi-orbital character !!

“Flat” bands in 1,2,3 dimensions
dp (Lieb) model
Mielke-Tasaki,Comm.Math.Phys.
158, 341 (1993)

Fujita states 1D

JPSJ 65, 1920 (1996)

d=3, Weaire-Thorpe model
d=2 Kagome
d=3 Pyrochlore

Y. Hatsugai, I. Maruyama,
EPL 95, 20003 (2011)

Ring states on Penrose tiling
Kohmoto-Sutherland,
Phys. Rev. Lett. 56, 2740(1986)

dice lattice

B.Sutherland, Phys. Rev. B34, 5208(1986)

Weaire-Thorpe,Phys. Rev. B4, 2508 (1971)
Dagotto, Fradkin,Moreo,
Phys. Lett.B 172, 383 (1986)

d=2, Silicene

Singular dispersions & silicene“Topological” deformation of Takeda-Shiraishi’s.
Flat bands in materials: counting dimensions

Overlapping molecular orbitals & flat bands
Without translational invariancePhysical origin of (nearly) flat bands

Weaire-Thorpe model and extension
3D to 2D
Hydrogen termination
Buckling

Overlapping MO’sY. Hatsugai, I. Maruyama,
EPL 95, 20003 (2011)

MO’s

c1

c1c2
c4

MO annihilation op.

C†p†††= (c1 + c2 + c4 )/p 1
1/p3
B 1/ 3 C
BCB 0 CB p C† † † † †C= (c1 , c2 , c3 , c4 , c5 , · · · ) B
1/3BC
B 0 C@A††..
=c= 1normalized.

3

C C= (c†1 c2 + · · · )/3 = c† P c , P =H

µN =m=1c20†

MXc3††
Em Cm Cm = c hc ,Pm Pn 6= 0, (m 6= n)†h==P2 projection

MXE m Pm

Sum of projections
m=1Itinerancy : NON orthogonality of MO’sMO’sOverlapping MO’s
1235416107823849765Z

108 = 2 At least 2 zero energies !

M

Z: # of zero eigen states
N: # of sites

TheoremZN

M: # of MO’s
Do NOT need translational invariance
If translationally invariant, use in momentum space

PROOFN dimensional Hamiltonian = Sum of M projections
Diagonalizable within M dimensional linear space
Non zero energy bands are at most M

Rest is null, N-M zero energy “flat” bands in N-dim.
L? : null

ex: 3D (3sites)

1D is null (1 zero energy)

L12D...

LMFlat bands are stable for perturbation(deformation of MO’s )Cpp= (c1 + c2 + c3 )/ 3 ! (⇤c1 + ⇤c2 + ⇤c3 )/ ⇤MO’sOverlapping MO’s
12345

1
6782

1038749Z

108=2Z

NM

65

N : number of sites (degree of freedom)M : number of MO’s (itinerancy by overlapping)Slightly extended theorem M =Xdim Pm

ex. P =1

†1+m2

†2,dim P = 2

If the itinerancy is not enough, some states are localized
Physical meaning of flat bands

Singular dispersions & silicene“Topological” deformation of Takeda-Shiraishi’s.
Flat bands in materials: counting dimensions

Overlapping molecular orbitals & flat bands
Without translational invariancePhysical origin of (nearly) flat bands

Weaire-Thorpe model and extension
3D to 2D
Hydrogen termination
BucklingWeaire-Thorpe model Weaire-Thorpe, Phys. Rev. B4, 2508 (1971)
3D Multiorbital (sp3) tight binding hamiltonian

Gapless points !
flat bands !
Unit cell & 3 primitive translations
Simple but 8×8 : need some work to diagonalize

HW T (k) =2V16 V1
66 V1
66 V1
66 V2
66 0
64 00

V1
V1
V1
V10
V200

V1
V1
V1
V100
V20

V1
V1
V1
V1000
V2

V2000
V1
V1 e i(ky +kz )V1 e i(kz +kx )V1 e i(kx +ky )0
V20000V20

V1 ei(ky +kz )V1
V1 e i(kx ky )V1 ei(kz kx )V1 ei(kz +kx )V1 ei(kx ky )V1V1 e i(ky kz )000V23777777i(kx +ky ) 7
V1 e7i(kz kx ) 7
V1 e7
V1 ei(ky kz ) 5
V1

Weaire-Thorpe model

Weaire-Thorpe,Phys. Rev. B4, 2508 (1971)

3D Multiorbital (sp3) tight binding hamiltonian
2HW T (k) =Unit cell & 3 primitive translations= 4V1projections

V1
V1
V1
V10V200Co0V1
V1
V1
V100
V20

V1
V1
V1
V1000V2unt†0V2000V1ing

V1 e i(ky +kz )V1 e i(kz +kx )V1 e i(kx +ky )00k,0V20000V20

V1 ei(ky +kz )V1
V1 e i(kx ky )V1 ei(kz kx )V1 ei(kz +kx )V1 ei(kx ky )V1V1 e i(ky kz )dim 0 E+ V en
E si 0 ,
ons
2†k4

4000
V2777777i(kx +ky ) 7
V1 e
7i(kz kx ) 7
V1 e7
V1 ei(ky kz ) 532V1

116e ik1 776k =ik24 e 2 5e ik3= 2V2 E8 + 4V1 P1 + 4V1 P2 + 2V2 P!3+= +2V2 E8 + 4V1 P1 + 4V1 P2 2V2 P3flat bands !

Pi =V16 V1
66 V1
66 V1
66 V266 0
64 00i†i

2Pi = Pi1 =002 =0k,

13± = p2

written by 2 ways

E4±E4

P1 P3± 6= 0 Non orthogonal
P1 P2 = 0dim P1 = dim P2 = 1 , dim P3± = 4

1+1+4=686=2

Flat bands at

3±2V2 !!

Extended Weaire-Thorpe model for silicene
with hydrogen termination (✏H )2D Multiorbital (sp3) tight binding hamiltonian✏H
HSilicene (k) =HV (0) ✏H E
V2 E4CCV2 E 4

HV (k)✏H E✏HCHSilicene ± V2 E8 = 4V1 P1 + 4V1 P2 ± 2V2 P3± ± (V2 ⌥ ✏H )P5

dimP1 + dimP2 + dimP3± + dimP5 = 1 + 1 + 3 + 2 = 7✏H Blue bonds are special

2D array of the unit cells

8-7=1 flat band at±V2

Extended Weaire-Thorpe model for silicene
with hydrogen termination (✏H )✏HHSilicene (k) =Energy (eV)
02D Multiorbital (sp3) tight binding hamiltonian(3)CV2 E 4

HV (0) E ✏H E
V2 E4CFHV (k)✏H E(1)-5✏HCHSilicene ± V2 E8 = 4V1 P1 + 4V1 P2 ± 2V2 P3± ± (V2 ⌥ ✏H )P5

dimP1 + dimP2 + dimP3± + dimP5 = 1 + 1 + 3 + 2 = 7 5
ΓBlue bonds are special

XSΓ

When

✏H = V2 < 0✏H
8-5=3 flat bands at

V2

triply degenerate
8-7=1 flat band at
2D array of the unit cells+V2

Various band ordering by changing ✏H(hydrogen termination)“Blue bonds are special”

Buckling can be included partlyhlocal (✓) =⇢X†
Vij ci cj + h.c.,hi,ji

V1 (hi, ji = h01i, h12i, h20i)
Vij =V10 (hi, ji = h03i, h13i, h23i)
cos ✓1V10=, cos ✓0 =V1
cos ✓03✏H ,cos ✓HSilicene (k) =HV✓ (0) ✏✓H E
V2 E4C

HV✓ (k) = 4V
✓1V2 E4C
HV✓ (k) ✏✓H E✓✓ †(kk) ,cos ✓, 1, 1, 1
cos ✓0◆2✓cos ✓✓✏H✏H =cos ✓0◆✓k = diagk,,Energy (eV)0Energy (eV)0first principle

SummaryEF

analytic(3)EF(1)-5-5?!
ΓXSΓΓXS

Less dispersive bands: due to multi-orbital character
Possible instability (ferromagnetic/structure)Γ