presention on topological insulator

8/15/2019 Presention on Topological Insulator

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Heavy Adatoms on MagneticSurfaces: Chern insulator search

Kevin F. Garrity

and

David Vanderilt

!lectronic Structure" #$%&

'SF DM()%$)$*+&+


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,o-ological Materials ,()invariant/

&D: Strong ,o-ological 0nsulator 1i

#Se

&

2.3. Chen et. al.Science *" %4+ #$$5/

#D: 6uantum S-in HallHg,e)Cd,eM. Koning et. al.Science &%+" 477 #$$4/


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%5++: 6AH insulator ,()ro8en/

t%

t#eiφ

6uantum Anomalous Hall 0nsulator 9 Chern 0nsulator


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utline● 0ntroduction to Chern insulators

− 1erry curvature

− ;revious searches

● ur search strategy

− Directly comine s-in)orit < magnetism

− ;roduces many non)trivial and structures

● First -rinci-les verification

− Several Chern insulators● Conclusions


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'ormal/ Hall !ffect

● !lectrons feel 3orent= force

●

Charge uilds u- on sides

BextEext

'ormal Metal

+ + + + + + + + + +

– – – – – – – – – –

e>


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Anomalous Hall !ffect

● 'o e?ternal magnetic field.

● Ferromagnetic metal net M" rea8s ,(/

● 0ntrinsic contriution single and/: 1erry Curvature

Karplus and Luttinger; Sundaram and Niu

Ferromagnetic Metal

Bext 9 $Eext

+ + + + + + + + + +

– – – – – – – – – –


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(evie@: 1erry -hase and curvature

|uk >

k x

k y

0 2π /a

Bloch function

mod #π

Sto8es theorem:

1erry curvature:

1erry -hase:

1erry -otential:

ϕ


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k x

k y

0

Brillouin zone

e!ie"# $hern %heorem

Brillouin zone is closed manifold

B&

$hern theorem# ' 2π

C

Chern 'umer


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1erry curvature and AHC

k x

k y

0 2π /a

(ermi Sea

)etal

*nomalous +all conducti!ity#


For Metals

Ω


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k x

k y

0 2π /a

*nomalous +all conducti!ity#


(ermi Sea

Ω


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k x

k y

0 k x

k y

0

,nsulator

B&

' 2π C

-$hern num.er or -%KNN in!ariant

uantum *nomalous +all#


Ω

2π /a


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#D Chern 0nsulators

● Requirements:

− 1ul8 and ga-

− 1ro8en time)reversal ,(/

− S-in)orit cou-ling

●

Features:− S-in)-olari=ed edge state

□ Dissi-ationless trans-ort

− Magnetoelectric effects

● Questions:

− Ho@ to constructB

− 3arge ga-B room tem-eratureB/

C 9 $

C 9 %

!dge 1and Structure

0mage: Hasan et. al. (M; 80" E* #$%$/


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;revious ;ro-osals

● Haldane Model

● Magnetically do-e 8no@n to-ological system:%

− Mn in Hg,e#)&" CrFe in 1i#Se&E)4" Fe on gra-hene+" 1i)ilayers5

●

Stoichiometric com-ounds:−Gd1i,e&

%$" HgCr #Se

E%%

● Difficult to achieve e?-erimentally

−

3ea8y" small ga-s− Align s-ins

%iang et. al. PRB +* $E*EE* #$%#/#

3iu et. al. PRL %$% %E7+$# #$$+/&1uhmann. March Meeting ;#4.% #$%#/E2u et. al. Science 9% #$%$/*Kou et. al. J. Appl. Phys. %%# $7&5%# #$%#/7u et. al. Nat. Phys. + #$%#/4'iu et. al. APL 55 %E#*$# #$%%/+6iao et. al. PRB +# %7%E%E #$%$/5hang et. al. ariv:%&$%EE#7 #$%&/%$hang et. al. Ariv:%%$+.E+*4 #$%%/%%u et. al. PRL %+7+$7 #$%%/


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Magnetic do-ing: Claim for 6AH

1.ser!ed.elo" 3K

1i"S/#,e

& do-ed @ith Cr


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Aside on chemistry

● 'eed strong magnetism" s-in)orit" insulating

● Hard to find all three together.

● e comine on surface.

Most magnetic

Most Anionic

Strongest spin-orbit

nteraction


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Magnetic 0nsulator

M

ur Strategy

Magnetic nsu!ator

) 1rea8s time reversal) FM or A)ty-e AFM) ,o-ologically trivial

Comine: heavy atoms < magnetic insulator

"ea#$ Atoms ) Au to 1i

) 3arge s-in)orit

A%#antages:) S-ins align automatically

) 'o do-ing) 3arge ga- insulators) 3arge s-in)orit

&isa%#antages:) Hard to -re-are surface


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First ;rinci-les Init Cell● Sustrates:

– Mn,e" MnSe A)ty-e AFM/

– !uS FM/

● $)% M3 heavy atom

– Au to 1i

● ;olar surface

– Cleave surface in vacuum

– Heavy atoms self)organi=e

● Strategy doesnt de-end ona -articular surfacesymmetry

'b

Mn

(e

+2

-2

+2

-2

+2

-2

+2

-2

+2

-2

+2

-1

-1

3ayer Charge


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Calculation Details

● 6uantum !s-resso @ith ;0IM norm)conserving -otentials

● VAS; ;As

● 3DA < I

−I9* eV for Mn" 7 eV for !u● @annier5$ J 0nter-olation to com-ute Chern numers

● @annier5$ -ost)-rocessing code AHC%)#

%

ang et. al. ;(1 4E %5*%%+ #$$7/#ang et. al. ;(1 47 %5*%$5 #$$4/

% M3 ,l M ,


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% M3 ,l on Mn,e

Γ M K Γ K

!n

ergy

.eV/

!F

● 0solated surface ands

Surface 1and Structure

% M3 ,l M ,


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% M3 ,l on Mn,e

!nergy

.eV/

!F

● 0solated surface ands● 'on)trivial Chern numers● Metallic

Surface 1and Structure oomed 0n

C 9 $

C 9 $

) *

Γ M K Γ K


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servation % 'o degeneracies

#D J 0solated ands

− S < ro8en ,( 9 no high symmetry degeneracies

− 'o accidental degeneracies:

● 'ear Crossing

● Degeneracy J h? 9 h

y 9 h

= 9 $

● nly vary 8?" 8

y/

;auli matrices


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servation # Many Chern Ls

● 0f s-in)orit" magnetic e?change" ho--ings similar strength

● ,hen: many non)trivial Chern numers

● !?am-le random ')orital tight)inding model

−(andom com-le? on)site ho--ing -arameters−%st n.n." diagonal ho--ings

t%

t#t&

Chern 'umer

Histogram'97 Model" ho--ings 9 onsite

L o f e ? a m

- l e s

'N' on)site Hermetian matri?

'N' com-le? ho--ing matrices


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'eed Chern L < Ga- across 1

● Com-etition et@een Chern numers and ga-s

Ho--ing Strength

F r a c t i o

n

'9E ,ight)1inding Model


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'eed Chern L < Ga- across 1

● Com-etition et@een Chern numers and ga-s

● 'eed to reduce and dis-ersion.

− 0dea: lo@er adatom coverage

Ho--ing Strength

'9E ,ight)1inding Model

F r a c t i o

n

S


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O M K O K

Attem-t 00: %& M3 1i on MnSe

E –

E F

, e - .

) * 0

● ,oo little ho--ing" no non)trival Chern numers.


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Attem-t 000 ) Honeycom● 0dea: tune surface ho--ing

● #& M3 honeycom

– ,ri-les u.c.

– Doules L adatom ands

#& M3 ; on Mn,e z s-ins


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O M K O K

#& M3 ; on Mn,e z s-ins

E –

E F

, e - .

) * -

) * -) * 0

! !F eV/

A H C

. e # h /

● 0solated surface ands● 'on)trivial Chern numers● 0nsulating

#& M3 ; on Mn,e x s-ins


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#& M3 ; on Mn,e x s-ins

● 0solated surface ands● 'on)trivial Chern numers● Metallic

E –

E F

, e - .

) * 0

) * 0) * -

O M K O K

! !F eV/

A H C

. e # h /

Search for Chern 0nsulators


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● ngre%ients:

– Sustrates:

● Mn,e" !uS

● MnSe )#P e-ita?ial strain to align s-ins along //

– %)# Heavy adatoms -er tri-led u.c.

– $)# Anion:

● 1r" 0" Se" ,e

● AdQust Fermi level

●

Screening 'roce%ure:– 0nitial calculations @ith # M3 sustrate.

● R*$P non)trivial" R#$P Chern insulators

– 0f interesting" final calculations E M3.

S h f Ch 0 l t


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Strained )#P


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1i1r on MnSe%E# meV ga-

E –

E

F

, e - .

) * +

) * 0

E

– E

F ,

e - .

) * -

) * -) * -

;;0 on MnSe

*7 meV ga-

O M K O K

O M K O K

Future or8: !?-erimentaltheoretical


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Future or8: !?-erimentaltheoreticalsearch

● Heavy atoms < Magnetic 0nsulator 9 Many C0s

● ,hese e?am-les for com-utational convenience

● hich surfaces can e -re-aredB

– Many magnetic insulators– De-osit lo@ coverages heavy atoms

● Characteri=e surface

● 3oo8 for large AHC

– ,heoretical in-ut to modify surface

● Ma8e insulating non)trivial


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Conclusions

● Heavy atoms < magnetic sustrates

– 0solated ands

– ,y-ically have Chern numers

– Find gloal ga-

● First -rinci-les verification

– Ga-s at least $.%E eV

● Future @or8

– ,heory e?-erimental collaoration

– hat surfaces are achievale in -ractice

Phys. Rev. Lett.110, 1160! "!01#$


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#D Chern 0nsulators

● 0nteger uantum Hall effect %5+$s/− Strong magnetic field

− 6uanti=ed conductance

● 6uantum anomalous Hall 6AH/

− 'o e?ternal field

− Haldane model %5++/

− 6AH reuirements:

□ 1ro8en time reversal

□ S-in)orit cou-ling

Chern ,heorem


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Chern ,heorem

egion * egion B

Stokes applied to *#

Stokes applied to B#

4ni5ueness of mod 2#

* ' 6 B 7 2$

* *

BB

B&

$hern theorem# ' 2 $

'ormal/ Hall !ffect


36/40

'ormal/ Hall !ffect

● !lectrons feel 3orent= force

● Charge uilds on to-ottom

EextBext into sceen

'ormal Metal

e>

– – – – – – – – – –

+ + + + + + + + + +



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● 'o e?ternal magnetic field.

● Ferromagnetic metal rea8s ,(/

● 0ntrinsic contriution single and/:

1erry Curvature


Eext

Ferromagnetic Metal

e>

– – – – – – – – – –

+ + + + + + + + + +

Bext 9 $

Com-uting Chern L


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1erry Connection

1erry Curvature

1

$ 1? #T

#T

1$

$

● Continuous)1:

@here

g

Com-uting Chern L


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● Continuous)1:

● Discrete)1:

– 'eed 1errys ;hase around each discrete loo-.

● or8s @ith random -haseeigenvectors

$ 1? #T

#T

1$

$

Ise annier Functions


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Ise annier Functions

● Gives real)s-ace tight)inding Hamiltonian

– Chea- 8)s-ace inter-olation

– !asy to calculate overla-s

● @annier5$ ma?imally)locali=ed annier functions

– Disentangle conduction and states

● Also calculate anomalous hall conductivity AHC/for metals. ang et. al. ;hys. (ev. 1 4E" %5*%%+ #$$7//

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