electronic structure of silicene in the extended weaire...

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

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


Ener

gy (e

V)-5

0EF

X SΓ Γ

“Topological” deformation of the bands

Classify into two: bands of Dirac fermions & else


Ener

gy (e

V)-5

0EF

X SΓ Γ




Ener

gy (e

V)-5

0EF

X SΓ Γ



(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)







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







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 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 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

electronic structure of silicene in the extended weaire...

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