Caustics, focusing and Veselago lens for electrons in...
Transcript of Caustics, focusing and Veselago lens for electrons in...
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