Caustics, focusing and Veselago lens for electrons in graphene PN junctions

Vladimir FalkoVadim Cheianov

Centre for Nanoscale Dynamics and Mathematical Physics

Boris Altshuler - Columbia U, NY

nvppvH rrrr⋅=⋅= σσˆ

Chiral electrons:‘isospin’ direction is linked to

the axis determined by the electron momentum:

Bloch function amplitudes on the AB sites - ‘isospin’ ⎟⎟

⎠

⎞⎜⎜⎝

⎛=

B

A

ϕϕ

ψ1=⋅nrrσ

vp=ε

xpyp

Fε

DoS

gatecarriers Vn ∝

holes electrons

Graphene: 2D zero-gap

semiconductor 2

4

0

ρ(kΩ

)

6

4-4 0n (1012 cm-2)

280K

4K

140K

Novoselov et al - Science 306, 666 (2004)

1939 - Jack Scaff and Henry Theurer discovered p-type and n-type regions in silicon at Bell Labs, Holmdel NJ

typenUpositive −, typepUnegative −,'

kkv

kv

vkkcondr

rr

r

=∂∂

=

=

ε

ε )(

kkv

kv

vkkvalr

rr

r

−=∂∂

=

−=

ε

ε )(

vk

Fermi momentum

ck

Fermi momentum

π/2ce

c

kN

vkeU

=

−=

π/

'2vh

v

kN

vkeU

=

=

The PN junction in graphene

kkv

kv

vkkcondr

rr

r

=∂∂

=

=

ε

ε )(

kkv

kv

vkkvalr

rr

r

−=∂∂

=

−=

ε

ε )(

vk

Fermi momentum

ck

Fermi momentum

π/2ce

c

kN

vkeU

=

−=

π/

'2vh

v

kN

vkeU

=

=

conduction band electrons

valence band

1=⋅nrrσ

1−=⋅nrrσ

pr

pr Due to the isospin conservation, A-B symmetric potential of PN junction cannot backward scatter chiral electrons,

Ando, Nakanishi, Saito, J. Phys. Soc. Jpn 67, 2857 (1998)Cheianov, VF - PR B 74, 041403 (2006)

1=⋅nrrσ

kkvvvkcond

rr== ;ε

)sin,cos( cccc kkk θθ=r

kr

cθ

ccc Vvvr

=)sin,cos( θθ nkk

c

v

v

c =−=θθ

sinsin

Snell’s law

n-type

''

'

kkvv

vkvalr

r−=

−=ε

)sin,cos(' vvvv kkk θθ −−=rvθ

)sin,cos( vvv vvV θθ=r

vvccyy kkkk θθ sinsin' −=⇒=

p-type

PN junction

θθ 2cos)( =wstepsharp

vc kk =

θθ 2cos)( =wstepsharp

vc kk =

)(ˆ xUivH +∇−= σr

)(),(][ 00 rrrrGUiv rrrrr−=+∇− δσ

0)( =+= xUE ε Electrons at electro-chemical potential, E=0

)(),(0,0,

axkxGxkxk

ki yv

cyyxx +=⎥

⎦

⎤⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

><−

++∂− δσσ

),(),( 0 yyik

y kxGedkrrG y∫=rr

)(),(0,0,

axikxGxkxk

ik xyv

cxyzx +=⎥

⎦

⎤⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

><−

+−∂ δσσσ

);,()(),( ykyxiyy ekMdkyxG φ∫=

yyvycy ykkkxkkakyx +−−−= 2222);,(φ

0=ydk

dφ Classical trajectory n

NN

kk

e

h

c

v

v

c =−=−=θθ

sinsin

vc xay θθ tantan +=

0)( =+= xUE ε Electrons at electro-chemical potential, E=0

cv kk 8.0=

nkk

c

v

v

c =−=θθ

sinsin

Snell’s law

vc xay θθ tantan +=

0=cd

dyθ

02

2

==cc dyd

ddy

θθ

caustic

cusp

⎟⎟⎠

⎞⎜⎜⎝

⎛== 02

2

yy dkd

dkd φφ

⎟⎟⎠

⎞⎜⎜⎝

⎛=== 03

3

2

2

yyy dkd

dkd

dkd φφφ

⎟⎟⎠

⎞⎜⎜⎝

⎛= 0

ydkdφ

Rene Thom Michael Berry Vladimir Arnold

general catastrophe

theory

nkk

c

v

v

c =−=θθ

sinsin

Snell’s law

vc xay θθ tantan +=

0=cd

dyθ

02

2

==cc dyd

ddy

θθ

caustic

cusp

⎟⎟⎠

⎞⎜⎜⎝

⎛== 02

2

yy dkd

dkd φφ

⎟⎟⎠

⎞⎜⎜⎝

⎛=== 03

3

2

2

yyy dkd

dkd

dkd φφφ

⎟⎟⎠

⎞⎜⎜⎝

⎛= 0

ydkdφ

cv kk 8.0=

cv kk 8.0=

vc kk =

Veselago Lens

Veselago, Sov. Phys.-Usp., 10, 509 (1968) Pendry, Phys. Rev. Lett., 85, 3966 (2000)

Graphene bipolar transistor: Veselago lens for electrons

w2

w

Focusing beam-splitter for electrons

Wishful thinking about graphene microstructures:Focusing/caustics and Veselago lens

in graphene PN junctions.Cheianov, VF, Althsuler (2006)

)()(1̂ˆ yaxu δδ +=

)()(ˆ yaxu z δδσ += )()(ˆ yaxu z δδσ +=

)()(1̂ˆ yaxu δδ +=

Mirage of a perturbation u on the other side of a sharp PN junctionrepresented by the local density of states image (STM).

Image of a bilayer island (‘dot’) embedded in

monolayer graphene: images show the

modulation of the difference between tunnelling currents

to the A and B sites.

Image of a charged tip on the other side of the junction: ‘teleported’Friedel oscillations.

typenUpositive −, typepUnegative −,'

kkv

kv

vkkcondr

rr

r

=∂∂

=

=

ε

ε )(

kkv

kv

vkkvalr

rr

r

−=∂∂

=

−=

ε

ε )(

vk

Fermi momentum

ck

Fermi momentum

π/2ce

c

kN

vkeU

=

−=

π/2vh

v

kN

vkeU

=

=

A smooth ‘high density’ PN junction graphene, d>>λF

Transmission of chiral electrons through the PN junction in graphene

Due to the isospin conservation, electrostatic potential cannot scatter chiral fermions in backward direction.

Cheianov, VF - PR B 74, 041403 (2006)

θθ θπ 2sin cos)(2dkFew −=

θ

1−

1=⋅nrrσ

μθ =++ )(sin222 xupp Fx0=u

dk

heg F

np π

22=

eIII )1( 21−=⋅

Due to selective transmission of electrons with a small incidence angle, an PN junction in grapheneshould display a finite conductance per unit length (no pinch-off) and a characteristic Fano factor in the shot noise.

Cheianov, VF - PR B 74, 041403 (2006) Smooth junction, d>>λF

Transmission of chiral electrons through the PN junction in graphene

contRR = )(2)(*B

Bfg

aRBRnp

contπ

+=

)0()( RBR −

*B↑

B

weak-field magnetoresitance of ballistic P-N junctions

θπθ2sin)( dkFew −=

1|| 0 <<=< dkFπθθ

1−

0*)(

' θθ =≈Brr

c

*BB < *BB >

∫∞

∞−+ −+

=1

2*

2 )/( BBzz

npnpn

eedzg

gπ

Selective transmission in a smooth junction, d>>λF