Weak localisation magnetoresistance in graphene Edward McCann - - PowerPoint PPT Presentation

Weak localisation magnetoresistance in graphene

Edward McCann Lancaster University, UK

with

• K. Kechedzhi, V.I. Fal’ko,
• H. Suzuura, T. Ando,

B.L. Altshuler

Outline

• 1. Introduction – chiral particles in graphene, Berry’s

phase π π π π, absence of backscattering, antilocalisation (?)

• 2. Weak localisation in graphene – trigonal warping and

“hidden” valley symmetry [high density ε ε ε εFτ τ τ τ >>1]

• 3. Weak localisation in bilayer graphene

J.Slonczewski and P.Weiss, Phys. Rev. 109, 272 (1958)

i

e

3 / 2π i

e

3 / 2π i

e−

p

• π

+

π

• 

       =

B A

ψ ψ ψ

( )

p v p p v v ip p ip p v H

y y x x y x y x

• .

σ σ σ π π = + =         =         + − =

+

For one K point (e.g. ξ ξ ξ ξ=+1) we have a 2 component wave function, with the following effective Hamiltonian: Bloch function amplitudes on the AB sites (‘pseudospin’) mimic spin components of a relativistic Dirac fermion.

• n

vp p v v H

• ⋅

= ⋅ =         =

+

σ σ π π

vp E =

x

p

y

p

Chiral electrons

pseudospin direction is linked to an axis determined by electronic momentum. for conduction band electrons, valence band (‘holes’)

1 = ⋅n

• σ

1 − = ⋅n

• σ

p

• p

• ϕ

ϕ ϕ ϕ = 0

( )

        = ⇔ =         =         =

− − + 2 / 2 / 2 1

;

ϕ ϕ ϕ ϕ

ϕ ψ π π

i i i i

e e vp E e e vp v H

( ) ( ) ( )

2 / cos

2 2

ϕ ϕ ψ ϕ ψ = =

• page 5098

• article 266603

!""#

• !" #$%&'()$& *

'+%,$%-./0-.12--/3 %'* ( '+%,$%-4/0-212--/3 " 5$$$6 &# $#*$57-/-4809 : $ ; $"#<=&$$>%&'+%,$%.?/0-@12--/3 $$ "%& $ >$57-/-82-- %$$$ &&$$57-/-82?8

        =

+

π π v H

&π π π π

• '

&

' () '

*

*

        −         =

+ +

) (

2 2 1

π π µ π π ξv H

1 1 − = + =               ξ ξ A B B A

+ξ ξ ξ ξ,-..

φ φ

π π

i y x i y x

pe ip p pe ip p

− +

≡ − = ≡ + =

*

        −         =

+ +

) (

2 2 1

π π µ π π ξv H

1 1 − = + =               ξ ξ A B B A

1 / ; 3 cos 2

2 2 2 4 2 3 2 2 2

<< + − = v p p vp p v µ µ φ ξµ ε

• *

( ) ( ) [ ]

F j j j j

v l p p

∑

− − =

• ε

ε δ

( ) ( )

[ ]

F j j j j

v l p p

• ∑

− − = ε ε δ

*

• ( )

1 ~ / ~

2 2 2

• τ

δ

φt

p h Tr

w

( )

( )

2 2 2 2 2 1

/ 2 ~ / ~ v p h Tr

w w

• µ

ε τ τ τ − τ

F j

v l ~ τ / t τ >> t

/ * +

*

        −         =

+ +

) (

2 2 1

π π µ π π ξv H

1 1 − = + =               ξ ξ A B B A

1 / ; 3 cos 2

2 2 2 4 2 3 2 2 2

<< + − = v p p vp p v µ µ φ ξµ ε

• +

              = 1 1 1 1 ) ( ) ( ˆ r u r V

• (

)

' ) ' ( ) (

2

r r u r u r u

• −

= δ

• ))
• 12 τ

τ τ τ"

.

132

Usually write 4 by 4 matrix using two sets of Pauli matrices:

              − − = Π               − − = Π               = Π 1 1 1 1 ; ; 1 1 1 1

z y x

i i i i

              − − =               − − =               = 1 1 1 1 ; ; 1 1 1 1

z y x

i i i i σ σ σ

[ ]

3 3 2 1 2 1

2 ,

l l l l l l

i Π = Π Π ε

valley: lattice:

[ ]

3 3 2 1 2 1

2 ,

s s s s s s

i σ ε σ σ =

[ ]

, = Πl

s

σ

two sets commute

132

Instead, we introduce two sets of 4 by 4 Hermitian matrices:

              − − = Λ               − − = Λ               − − = Λ 1 1 1 1 ; ; 1 1 1 1

z y x

i i i i

              − − = Σ               − − = Σ               − − = Σ 1 1 1 1 ; ; 1 1 1 1

z y x

i i i i

[ ]

3 3 2 1 2 1

2 ,

l l l l l l

i Λ = Λ Λ ε

valley ‘pseudospin’ lattice ‘isospin’

[ ]

3 3 2 1 2 1

2 ,

s s s s s s

i Σ = Σ Σ ε

[ ]

, = Λ Σ

l s

two sets commute

132

We introduce two sets of 4 by 4 Hermitian matrices:

; ; σ σ σ ⊗ Π = Λ ⊗ Π = Λ ⊗ Π = Λ

z z z y y z x x

z z y z y x z x

σ σ σ ⊗ Π = Σ ⊗ Π = Σ ⊗ Π = Σ ; ;

[ ]

3 3 2 1 2 1

2 ,

l l l l l l

i Λ = Λ Λ ε

valley ‘pseudospin’ lattice ‘isospin’

[ ]

3 3 2 1 2 1

2 ,

s s s s s s

i Σ = Σ Σ ε

[ ]

, = Λ Σ

l s

two sets commute

132

Why?

; ; σ σ σ ⊗ Π = Λ ⊗ Π = Λ ⊗ Π = Λ

z z z y y z x x

z z y z y x z x

σ σ σ ⊗ Π = Σ ⊗ Π = Σ ⊗ Π = Σ ; ;

l y y l y y

Λ − = Λ Σ Λ Λ Σ

∗

they all change sign under time inversion

s y y s y y

Σ − = Λ Σ Σ Λ Σ

∗

z z z y z x y z y y y x x z x y x x

I Λ Σ Λ Σ Λ Σ Λ Σ Λ Σ Λ Σ Λ Σ Λ Σ Λ Σ , , , , , , , , , ˆ

basis for non-magnetic, static disorder

16 possible Hermitian matrices: 10 time-reversal invariant 6 not time-reversal invariant

z y x z y x

Λ Λ Λ Σ Σ Σ , , , , ,

4&

• l

s z y x l s sl r

u r u I r V Λ Σ + =

∑

= , , ,

) ( ) ( ˆ ) ( ˆ

• (

)

' ) ' ( ) (

2

r r u r u r u

• −

= δ

• 12

τ τ τ τ"

.

z y z x

Λ Σ Λ Σ ,

y z y y y x x z x y x x

Λ Σ Λ Σ Λ Σ Λ Σ Λ Σ Λ Σ , , , , ,

z zΛ

Σ

( ) ( ) ( )

' '

' ' 2 ' '

r r u r u r u

ll ss sl l s sl

• −

= δ δ δ

tr

v D τ

2 2 1

=

• D

e gxx υ

2

4 =

• τ

τ τ τ ,!τ τ τ τ"

5

ξ ξ ξ ξ ,662 α α α α ,&

( ) ( )

' ' ' ' ' ' , ' ' , ' , ' , , ' , ' , , 4 1

2 2 1 1 2 1 2 1

ξ µ α β µ ξ ξµ β α αβ ξµ αβ µ ξ µ ξ β α β α y l y s l y s y l l s s

C C Λ Λ Σ Σ Λ Λ Σ Σ =

∑ ∑

5

. ,"+7 . ,"+7 8 9Σ Σ Σ Σ: 9Λ Λ Λ Λ: 9":9+7:

z y x s z y x l C C

ll ss l s

, , , ; , , , ; = = ≡

.#

8"

5

; 1 ; 1 ; 1 ; 1

1 1 1 1

              − = ↓               = ↑               = ↓               = ↑

− − + +

( ) ( )

ξ ξ ξ ξ ξ ξ

φ φ φ φ

ψ ψ ↓ + ↑ − = ↓ + ↑ =

− − − −

2 2 2 2

. 2 , . 2 1 , i i r p i i p i i r p i p

e e e e e e

• '

' ' ' , ' ,

~

ξ ξ φ ξ ξ φ ξ ξ ξ ξ ξ ξ ψ

ψ ↓ ↓ + ↑ ↑ − ↑ ↓ − ↓ ↑

− − i i p p

e e

• φ

φ φ φ

;;+

• <

;

5

l

C0

1

1 1 2 1 2 1 2 1 2 2 2 2 2 2 1

=               − Γ +

− − − − − − −

C vq vq vq vq i q v

y i x i y i x i l

τ τ τ ω τ

( )

1 :

2 2 2 2 1

= Γ + − = =

l l tr

C i Dq v v D ω τ τ

*

=

l y

C

l z

C

l

C0

l x

C

0 =

Γl

1 −

= Γ τ

l z 1 2 1 −

= Γ = Γ τ

l y l x

;

• l

z l y l x l

C C C C , , ,

>

5

~ C C C C g

z y x

− + + δ

?8

• l

C0

l

C0

0 =

Γl

dressing of Hikami boxes leads to a reduction by factor ½

What happens to the four gapless modes when there is trigonal warping and symmetry breaking disorder? >

5

~ C C C C g

z y x

− + + δ

l

C0

l

C0

0 =

Γl

132

leading terms do not contain valley operators Λ, so they remain invariant with respect to valley transformations

( ) ( )

( ) ( )

l s z y x l s sl x x z x

r u r u I p p p v H Λ Σ + + Σ Σ Σ Λ Σ Σ − Σ =

∑

=

• ,

, , 1

ˆ . . . ˆ µ

; ≠ Γ = Γ = Γ = Γ ⇒ Λ Σ

y x z z s

warping term is invariant with respect to valley transformation Λz only intravalley disorder intervalley disorder ; ≠ Γ = Γ = Γ = Γ ⇒ Λ Σ

z y x x s

; ≠ Γ = Γ = Γ = Γ ⇒ Λ Σ

z x y y s

; ≠ Γ = Γ = Γ = Γ

y x z

= Γ = Γ = Γ = Γ

z y x

• ~

C C C C g

z y x

− + + δ

• @+

• y

x 1 *

Γ = Γ =

−

τ

• @+
• (

)

                      + −             + =

i tr

e B g τ τ τ τ τ τ π δ

φ φ φ

2 1 ln 1 / ln 2 2 ~

* 2 2

• y

x 1 *

Γ = Γ =

−

τ

• ε

ε ε ε>τ τ τ τ AA.)B = =B 6&@4"#"%!""

( )

                      + −             + =

i tr

e B g τ τ τ τ τ τ π δ

φ φ φ

2 1 ln 1 / ln 2 2 ~

* 2 2

Bilayer [Bernal (AB) stacking]

Bilayer [Bernal (AB) stacking]

Bilayer [Bernal (AB) stacking]

• E. McCann and V.I. Fal’ko

PRL 96, 086805 (2006)

y x

ip p + = π

• E. McCann and V.I. Fal’ko

PRL 96, 086805 (2006)

• ϕ

ϕ ϕ ϕ = 0

( )

( )

        = ⇔ =         − =         − =

− − + ϕ ϕ ϕ ϕ

ϕ ψ π π

i i i i

e e m p E e e m p m H

2 1 2 2 2 2 2 2

2 ; 2 2 1

( ) ( ) ( )

ϕ ϕ ψ ϕ ψ

2 2

cos = =

&

&!π π π π C

2 2 1

τ v D =

• D

e gxx υ

2

4 =

&

• τ

τ τ τ ,τ τ τ τ"

• E. McCann and V.I. Fal’ko

PRL 96, 086805 (2006)

        +         − =

+ +

) ( 2 1

3 2 2 2

π π ξ π π v m H

1 1 ~ ~ − = + =               ξ ξ A B B A

*

2 2 3 3 3 2 2 2

3 cos 2 p v m p v m p + −         = φ ξ ε

= 9 4:

2 12

10 4 ~

−

× cm

2 11

10 ~

−

cm

Berry phase π suppressed backscattering weak anti-localisation ? Berry phase 2π weak localisation ?

        =

+ 1

π π v H         − =

+

) ( 2 1

2 2 2

π π m H

Berry phase romantics

higher order expansion

‘trigonal warping’: valley symmetry of wave vector K is lower than the hexagonal symmetry

• ff-diagonal

interlayer hopping

) ( ˆ ) (

2 2 1

r V v H

• +

        −         =

+ +

π π µ π π ξ ) ( ˆ ) ( 2 1

3 2 2 2

r V v m H

• +

        +         − =

+ +

π π ξ π π

1 1 − = + =               ξ ξ A B B A 1 1 ~ ~ − = + =               ξ ξ A B B A

B A~

' ' ' ' 1 K K KK antisymm KK symm KK

C C C C g + + + − =

− −

δ

' ' ' ' 2 K K KK antisymm KK symm KK

C C C C g − − + − =

− −

δ

Weak localisation correction

may be suppressed by intervalley scattering due to atomically sharp scatterers

• r edges

i

τ

can only be suppressed by decoherence Berry phase π killed by trigonal warping reflecting the asymmetry in each valley Berry phase 2π

) ( ) ( p H p H

• ≠

−

1 >> τ εF

High electron (hole) density and remote Coulomb scatterers

Berry phase π ‘slow’ inter-valley scattering: neither WL nor WAL magnetoresistance ‘fast’ inter-valley scattering: usual WL magnetoresistance cut at

' ' ' ' 1 K K KK antisymm KK symm KK

C C C C g + + + − =

− −

δ

' ' ' ' 2 K K KK antisymm KK symm KK

C C C C g + − + − =

− −

δ

Weak localisation magnetoresistance

ϕ

τ τ >

i

) ( ) ( R B R −

ϕ

τ τ <

i

i

B

B

ϕ ϕ

τ φ D B ~

i i

D B τ φ0 ~

• K. Kechedzhi, V.I. Falko,
• E. McCann, B.L. Altshuler,

2006

• E. McCann, K. Kechedzhi,

V.I. Falko, H. Suzuura,

• T. Ando, B.L. Altshuler,

PRL 97, 146805 (2006)

1 >> τ ε F

Weak localisation magnetoresistance in graphene

DE 7F EG076EC0=6>EB/H6I /?B)$%".#<".9!""#:

> E /

• =

8

• Weak localisation magnetoresistance in graphene

DE 7F EG076EC0=6>EB/H6I /?B)$%".#<".9!""#:

nm ltr 80 ≈

1 1 1 − − −

>> >>

φ

τ τ τ

w

ps 50 ≈

φ

τ ps

w

1 ≈ τ 12 / ≈

• τ

ε F ps 04 .

0 ≈

τ meV

F

200 ≈ ε

2 12

10 3

−

× ≈ cm n

Weak localisation magnetoresistance in graphene

DE 7F EG076EC0=6>EB/H6I /?B)$%".#<".9!""#:

E

8 4 *

• E. McCann, K. Kechedzhi, V.I. Fal’ko, H. Suzuura, T. Ando, B.L. Altshuler,

Phys Rev Lett. 97, 146805 (2006)

• K. Kechedzhi, V.I. Fal’ko, E. McCann, B.L. Altshuler (2006)