Post on 07-Apr-2015
description
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NCN@Purdue Summer School, July 20-24, 2009
Tony Low and Mark LundstromNetwork for Computational Nanotechnology
Discovery Park, Purdue UniversityWest Lafayette, IN
Graphene PN Junctions
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acknowledgments
Supriyo Datta and Joerg Appenzeller
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outline
1) Introduction
2) Electron optics in graphene
3) Transmission across NP junctions
4) Conductance of PN and NN junctions
5) Discussion
6) Summary
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PN junctions: semiconductors vs. graphene
EC x( )∝ −qV x( )
EV x( )
x
E x( )
EF1
In
I0 ∝ ni2 ∝ e− EG kbT
q Vbi − VA( )
+−
I = I0eqVA kbT
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experimental observation
ID
VA
GNP
GN+ N
GNP < GN+ N
B. Huard, J.A. Sulpizo, N. Stander, K. Todd, B. Yang, and D. Goldhaber-Gordon, Transport measurements across a tunable potential barrier in graphene,” Phys. Rev. Lett., 98, 236803, 2007.
electron “optics” in graphene
Semiclassical electron trajectories are analogous to the rays in geometrical optics.
If the mfp is long, one may be able to realize graphene analogues of optical devices.
Snell’s Law
θ1
n1
θ2
n2
n1 sinθ1 = n2 sinθ2 sinθ2 =n1
n2
sinθ1
y
x
negative index of refraction
θ1
n1
θ2 < 0
n2 < 0
n1 sinθ1 = n2 sinθ2 sinθ2 =n1
n2
sinθ1
y
x
Veselago lens
n1 > 0
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n2 = −n1
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theoretical prediction
Science, 315, 1252, March 2007 N-type P-type
electron trajectories
making graphene PN junctions
From: N. Stander, B. Huard, and D. Goldhaber-Gordon, “Evidence for Klein Tunneling in Graphene p-n Junctions,” PRL 102, 026807 (2009)
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band diagrams: conventional PN junctions
N+ P
EV x( )
x
E x( )
Fn
I +
q Vbi − VA( )
Fp
EI x( )∝ −qV x( )
EC x( )
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band diagrams: graphene PN junctions
ENP x( )∝ −qV x( )
x
E
EF1
E
kx
E
kx
E
kx
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an abrupt graphene N+N junction
x
E x( )EF1
ENP x( )
Apply a low bias. Conduction occurs through states near the Fermi level.
kx
E υx = +υFE
kx
υx = +υF
qVJ < 0
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an abrupt graphene PN junction
kx
x
E x( )
EF1
E
ENP x( )E
kx
Apply a low bias. Conduction occurs through states near the Fermi level.
υx = +υF
υx = +υF
Note that kx – kx0 changes sign!
qVJ
ID
VA
GN+ N
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conductance of graphene junctions
x
E x( )EF1
ENP x( )kx
E υx = +υFE
kx
υx = +υF
qVJ
N+N
kx
x
E x( )
EF1
E
ENP x( )E
kx
υx = +υF
υx = +υF
qVJ
NP
kx
x
E x( )
EF1
E
ENP x( )
E
kx
υx = +υF
υx = +υF
qVJ
NIGNP
GNI
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objectives
To understand:
1) Electron “optics” in NP junctions
2) The conductance of NP and NN junctions
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about graphene
kx
E x( )
EF
gV = 2
“neutral point”
“Dirac point”
( ) 2 2F F x yE k k k kυ υ= ± = ± +
( ) 1F
Ekk
υ υ∂= =
∂
2 2( ) 2 FD E E π υ=
υF ≈ 1× 108 cm/s
( ) 1 cosx Fx
Ekk
υ υ θ∂= =
∂
( ) 1 siny Fy
Ekk
υ υ θ∂= =
∂
( ) 2 FM E W E π υ=
ky
electron wavefunction in graphene
y
x
ψ x, y( )=1A
ei kx x+ kyy( )
ψ x, y( )= 1seiθ
ei kx x+ kyy( )
s = sgn E( )
θ = arctan ky kx( )
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absence of backscattering
kx
E x( )
EF
ψ x, y( )= 1seiθ
ei kx x+ kyy( )
s = sgn E( ) θ = arctan ky kx( )
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1−1
kx
ky
k
′k
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outline
1) Introduction
2) Electron optics in graphene
3) Transmission across NP junctions
4) Conductance of PN and NN junctions
5) Discussion
6) Summary
a graphene PN junction
kx
x
E x( )
EF1
E
ENP x( )E
kx
υx = +υF
υx = +υF
N-type P-type
kx
ky
υparallel to kυ
anti-parallel to kυ
kx
ky
υ
υx = −υF
y
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group velocity and wavevector
kx
E x( )
ky
group velocity parallel to k
( ) 1g F
E kkk k
υ υ∂= =
∂
group velocity parallel to -k
( ) 1g F
E kkk k
υ υ∂= = −
∂
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what happens for parabolic bands?
kx
ky
( ) ( )1g kk E kυ = ∇
conduction band
( ) *x
g xkk
mυ = +
valence band
( ) *x
g xkk
mυ = −
E k( )
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optics
Snell’s Law
θ1
1θ ′ θ2
ki
kr
kt
n1 n2
n1 sinθ1 = n2 sinθ2
θr = θi
θ1
′θ1 θ2
υi
υr
υt
n1 n2
group velocity parallel to wavevector
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electron trajectories in graphene PN junctions
rays in geometrical optics are analogous to semiclassical electron trajectories
θ1
1θ ′ θ2
ki
kr
kt
N-type P-type
( )e
d kF q
dt= = −
E
x
y1) ky is conserved
kyi = ky
r = kyt
2) Energy is conserved
Ei = Er = Et
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on the N-side…
θ1
1θ ′ θ2
ki
kr
kt
N-type P-type
i r ty y yk k k= =
Ei = Er = EF
kyi = kF sinθ1 = ky
i = kF sin ′θ1
1 1θ θ ′=
angle of incidence = angle of reflection
x
y
F F FE kυ=
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a symmetrical PN junction
kx
x
E x( )
EF
E
ENP x( )E
kx
υx = +υF
υx = +υF
N-type P-type
υx = −υF
( )F NPF
F
E E xk
υ−
=
Symmetrical junction:
EF − ENP 0( )= ENP L( )− EF
−qVJ = 2 EF − ENP 0( ) kF
i = kFt
ε1
ε2
One choice for ky, but two choices for kx.
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wavevectors
θ1
′θ1
ki
kr
N-type P-type
1) transverse momentum (ky) is conserved
2) transmitted electron must have a positive x velocity
anti-parallel to kυ
kx
ky
υkyi = ky
r = kyt
kxi = −kx
r = −kxt
The sign of the tangential k-vector (ky) stays the same, and the normal component (kx) inverts.
x
y
θ2
kt
parallel to kυ
kx
υ
ky
υ
wavevectors and velocities
θ1
′θ1
θ2
ki
kr
ktN-type P-type
x
y
kyi = ky
r = kyt
kxi = −kx
r = −kxt
θ1
′θ1
θ2
υi
υr
υtN-type P-type
x
y
θ2 = −θ1
“negative index of refraction”
θ1 = ′θ1
n1 sinθ1 = n2 sinθ2
1 2n n= −
more generally
θ1
1θ ′
θ2
υi
υr
υtN-type P-type
1) y-component of momentum conserved
2) energy conserved
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ε1 sinθ1 = ε2 sinθ2
critical angle for total internal reflection
θC = sin−1 ε2 ε1( ) ε1 > ε2( )
ε = EF − ENP( )1 1θ θ ′=
reflection and transmission
θ1
1θ ′
θ2
υi
υr
υtN-type P-type
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We know the direction of the reflected and transmitted rays, but what are their magnitudes?
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outline
1) Introduction
2) Electron optics in graphene
3) Transmission across NP junctions
4) Conductance of PN and NN junctions
5) Discussion
6) Summary
reflection and transmission
N-type P-type
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x
y
R θi( )∝ r 2 T θi( )∝ t 2
θ1
ki
1) incident wave:
ψ i x, y( )= 1seiθ
ei kx x+ kyy( )
1θ ′
kr
2) reflected wave:
ψ r x, y( )= r 1seiθ
ei − kx x+ kyy( )
θ2
kt
3) transmitted wave:
ψ x, y( )= t 1seiθ
ei − kx x+ kyy( )
s = sgn E( ) θ = arctan ky kx( )
−90−60
−30
1.00
transmission: abrupt, symmetrical NP junction
perfect transmission for = 0
kx
ky
υ
kx
ky
υincident
transmitted
T θi( )= cos2 θi
conductance of abrupt NN and NP junctions
T θi( )= cos2 θi
This transmission reduces the conductance of NP junctions compared to NN junctions, but not nearly enough to explain experimental observations.
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a graded, symmetrical PN junction
kx
x
E x( )
EF
E
ENP x( )E
kx
υx = +υF
N-type P-type
ε1
−ε1
d
E
The Fermi level passes through the neutral point in the transition region of an NP junction. This does not occur in an NN or PP junction, and we will see that this lowers the conductance of an NP junction.
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treat each ray (mode) separately
x
y
θ
ki ky = kF sinθ
sinF FE k υ θ=
sinF FE k υ θ= −
2 2 2sinF x FE k kυ θ= +
ky
E
kx = 0kx
E
2 sinG F FE kυ θ=
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for each ky (transverse mode)
kx
x
E x( )
EF
E
ENP x( )E
kx
υx = +υF
N-type P-type
ε1
−ε1
d
E
2 sinG F FE kυ θ=
band to band tunneling (BTBT)
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NEGF simulation
T. Low, et al., IEEE TED, 56, 1292, 2009
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propagation across a symmetrical NP junction
x
E x( )
EF
ENP x( )= ε1
−ε1
d
ENP x( )= −ε1
( ) 2 2F x yx k kε υ= +
( ) 2 2 2sinF x Fx k kε υ θ= +
( ) ( )( )22 2 2sinx F Fk x x kε υ θ= −
E
kx
ε1
kx > 0kx
E
kx < 0
kx2 < 0
E
kx
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WKB tunneling
( ) ( )( )22 2 2sinx F Fk x x kε υ θ= −
For normal incidence, kx is real perfect transmission
ψ x( ) : eikx x For any finite angle, there will a region in the junction where kx is imaginary evanescent
T : e−2i kx x( )dx
− l
+ l
∫ Slowly varying potential no reflections.
( ) 2sinFk dT e π θθ −
2 sinG F FE kυ θ=
12 2 F Fkqd qdε υ
= =E
( )2 2G FE qT e π υ−
E
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transmission vs. angle
More generally, to include the reflections for abrupt junctions (small d):
T θ( ) : cos2 θ e−π kF d sin2 θ
−30
−60−90
T θ( )= e−π kF d sin2 θ
T θ( )= cos2 θ
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outline
1) Introduction
2) Electron optics in graphene
3) Transmission across NP junctions
4) Conductance of PN and NN junctions
5) Discussion
6) Summary
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graphene junctions: NN, NP, PP, PN
kx
x
E x( )
EF
E
ENP = −qVJ
E
kx
qVJ < 0
two independent variables: 1) EF and 2) VJ
↑EF > 0
N
PEF < 0
↓
↑−qVJ > EF
P
N−qVJ < EF
↓
NN
qVJ > EF, right side N-type
PNNP
PP
qVJ < EF, right side P-type
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graphene junctions
↑EF
qVJ → −qVJ = EF
0
EF > 0, left side N-type
EF < 0, left side P-type
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conductance of graphene junctions
G =2q2
hM EF( ) G W =
2q2
h1
Wmin M1, M2( )
kx
x
E x( )
EF
E
ENP = −qVJ
E
kx
i) EF > 0,qVJ < EF ⇒ NP
M2 < M1
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conductance vs. EF and VJ
x
E x( ) EF = 0.3EPN = −qVJ
x
E x( )
EF
EPN = 0.6
T. Low, et al., IEEE TED, 56, 1292, 2009
measured transport across a tunable barrier
B. Huard, J. A. Sulpizio, N. Stander, K. Todd, B. Yang, and D. Goldhaber-Gordon, “Transport Measurements Across a Tunable Potential Barrier in Graphene, Phys. Rev. Lett., 98, 236803, 2007.
EF
experimental resistance
B. Huard, J. A. Sulpizio, N. Stander, K. Todd, B. Yang, and D. Goldhaber-Gordon, “Transport Measurements Across a Tunable Potential Barrier in Graphene, Phys. Rev. Lett., 98, 236803, 2007.
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NEGF simulation of abrupt junctions
x
E x( ) EF = 0.3EPN = −qVJ
T. Low, et al., IEEE TED, 56, 1292, 2009
GPN GNN
GPN < GNN
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conductance of abrupt graphene junctions
N-type P-type N-type N+ -type
GNN W =2q2
hM EF( )
GNP W =23
2q2
hM EF( )
( )22
y
NP yk
qG T kh
= ∑
GNN
GPN
T θ( )= cos2 θ =kx
kF
2
= 1−ky
kF
2
GNP =23
GNN
EF = 0.3 EF = 0.3
GNP < GNN
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conductance of graded graphene junctions
N-type P-type N-type N+ -type
GNN W =2q2
hM EF( )
GNP = GNN kF d
GNP =2q2
hT ky( )
ky
∑
EF = 0.3 EF = 0.3
GNP << GNN
kF d >> 1→ λF << dT. Low, et al., IEEE TED, 56, 1292, 2009
T θ( )= e−π kF d sin2 θ = e−π kF d ky kF( )2
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outline
1) Introduction
2) Electron optics in graphene
3) Transmission across NP junctions
4) Conductance of PN and NN junctions
5) Discussion
6) Summary
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conductance vs. gate voltage measurements
Back gate
(doped Si)
grapheneSiO2
L
VG
either N-type or P-type depending on the metal workfunction
either N-type or P-type depending on the back gate voltage
G
′VG
B. Huard, N. Stander, J.A. Sulpizo, and D. Goldhaber-Gordon, “Evidence of the role of contacts on the observed electron-hole asymmetry in graphene,” Pbys. Rev. B., 78, 121402(R), 2008.
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outline
1) Introduction
2) Electron optics in graphene
3) Transmission across NP junctions
4) Conductance of PN and NN junctions
5) Discussion
6) Summary
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conclusions
1) For abrupt graphene PN junctions transmission is reduced due to wavefunction mismatch.
2) For graded junctions, tunneling reduces transmission and sharply focuses it.
3) Normal incident rays transmit perfectly
4) The conductance of a graphene PN junction can be considerably less than that of an NN junction.
5) Graphene PN junctions may affect measurements and may be useful for focusing and guiding electrons.
questions
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