graphene pn junctions

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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

2

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

5

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




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




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




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


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.

51

NEGF simulation of abrupt junctions

x

E x( ) EF = 0.3EPN = −qVJ


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




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




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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graphene pn junctions

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