Diffusive Charge Transport in Graphene

Michael S. Fuhrer KITP Graphene Week University of Maryland

Diffusive Charge Transport in Graphene

Michael S. FuhrerMichael S. FuhrerDepartment of Physics andDepartment of Physics and

Center for Nanophysics and Advanced MaterialsCenter for Nanophysics and Advanced MaterialsUniversity of MarylandUniversity of Maryland


OutlineOutlineI. Introduction to Graphene

“Massless” electronsPseudospin and Berry’s phase

II. Fabrication and Characterization of Graphene on SiO2

Micro-Raman spectroscopyCleaning graphene

III. Diffusive Transport in GrapheneBoltzmann Transport

Charged impuritiesCharged impurities – minimum conductivityPhonons Dielectric Environment – Tuning Fine Structure ConstantCorrugationsLattice defects


Carbon and Carbon and GrapheneGraphene

C-

--

-

Carbon Graphene

4 valence electrons

1 pz orbital

3 sp2 orbitals

Hexagonal lattice;1 pz orbital at each site


GrapheneGraphene Unit CellUnit Cell

Two identical atoms in unit cell: A B

Two representations of unit cell:

1/3 each of 6 atoms = 2 atoms

Two atoms2av1av


Band Structure of Band Structure of GrapheneGrapheneTight-binding model: P. R. Wallace, (1947)(nearest neighbor overlap = γ0)

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+±=

2cos4

2cos

23cos41)( 2

0

akakakEE yyxF γk

kx

ky

E


Bonding vs. AntiBonding vs. Anti--bondingbonding

00

0 0

0γ

γγ

±=⎥⎦

⎤⎢⎣

⎡−

−= EH

ψ “anti-bonding”anti-symmetric wavefunction

“bonding”symmetric wavefunction

022 11

21 γψ −=⎥

⎦

⎤⎢⎣

⎡= E

011 1

12

1 γψ +=⎥⎦

⎤⎢⎣

⎡−

= E

γ0 is energy gained per pi-bond


Bloch states:

AB

AB

⎟⎟⎠

⎞⎜⎜⎝

⎛01

⎟⎟⎠

⎞⎜⎜⎝

⎛10

FA(r), or

FB(r), or

“anti-bonding”E = +3γ0

“bonding”E = -3γ0

⎟⎟⎠

⎞⎜⎜⎝

⎛−11

21

⎟⎟⎠

⎞⎜⎜⎝

⎛11

21

Γ point:k = 0

Band Structure of Band Structure of GrapheneGraphene –– ΓΓ point (point (kk = 0)= 0)


34

32

1

π

π

i

i

e

e

λλ

λ

K

K

K

⎟⎟⎠

⎞⎜⎜⎝

⎛01FA(r), or ⎟⎟

⎠

⎞⎜⎜⎝

⎛10FB(r), or

Phase:

K 23a

=λa3

4π=K

Band Structure of Band Structure of GrapheneGraphene –– K pointK point


34

32

0 1

π

π

i

i

i

e

e

e =

Phase:

Bonding is Frustrated at K pointBonding is Frustrated at K point

32

02

π

γi

eE −=

001

ieE γ−=

34

03

π

γi

eE −=

0

034

32

00 =⎟⎟

⎠

⎞⎜⎜⎝

⎛++−=

ππ

γiii eeeE

Re

Im

E1

E2

E3


34

32

0 1

π

π

i

i

i

e

e

e =

Phase:


0δ ⎟⎠⎞

⎜⎝⎛ +

−=δπ

γ 32

02

ieE

( )δγ +−= 001

ieE

⎟⎠⎞

⎜⎝⎛ +

−=δπ

γ 34

03

ieE

( ) 0034

32

00 ==⎟

⎟⎠

⎞⎜⎜⎝

⎛++−=

⎟⎠⎞

⎜⎝⎛ +⎟

⎠⎞

⎜⎝⎛ +

+ δδπδπ

δγ iii

i eeeeE

Re

Im

E1

E2

E3


⎟⎟⎠

⎞⎜⎜⎝

⎛01

FA(r), or

⎟⎟⎠

⎞⎜⎜⎝

⎛10

FB(r), or

K

23a

=λa3

4π=K

0π/3

2π/3π

5π/3

4π/3

“anti-bonding”

E = 0!

“bonding”

E = 0!

⎟⎟⎠

⎞⎜⎜⎝

⎛−11

21

⎟⎟⎠

⎞⎜⎜⎝

⎛11

21

K point:Bonding and anti-bonding

are degenerate!



)()()( rrvF FFkσ ε=⋅h

kvbeibe

ek Fi

ii

k

k

h=⎟⎟⎠

⎞⎜⎜⎝

⎛−=

−⋅ εθ

θ

;2

12/

2/rk

θk is angle k makes with y-axisb = 1 for electrons, -1 for holes

Eigenvectors: Energy:

Hamiltonian:

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

−

)()(

)()(

00

rFrF

rFrF

ikkikk

vB

A

B

A

yx

yxF εh

electron has “pseudospin”points parallel (anti-parallel) to momentum

K’

K

linear dispersion relation“massless” electrons

Band Structure of Band Structure of GrapheneGraphene: : kk··pp approximationapproximation


Visualizing the Visualizing the PseudospinPseudospin0

π/3

2π/3π

5π/3

4π/3


30 degrees

390 degrees

Visualizing the Visualizing the PseudospinPseudospin0

π/3

2π/3π

5π/3

4π/3


PseudospinPseudospin

K

K’

kvH

ikkikk

vkvH

tFK

yx

yxFFK

vvh

hvv

h

⋅=

⎟⎟⎠

⎞⎜⎜⎝

⎛+

−=⋅=

σ

σ

'

00σ || k

σ || -k

• Hamiltonian corresponds to spin-1/2 “pseudospin”Parallel to momentum (K) or anti-parallel to momentum (K’)

• Orbits in k-space have Berry’s phase of π


K’ K

K: k||-x K: k||xK’: k||-x

real-spacewavefunctions(color denotesphase)

k-spacerepresentation

bondingorbitals

bondingorbitals

anti-bondingorbitals

PseudospinPseudospin: Absence of Backscattering: Absence of Backscattering

bonding

anti-bonding


““PseudospinPseudospin””: Berry: Berry’’s Phase in IQHEs Phase in IQHE

π Berry’s phase for electron orbits results in ½-integer quantized Hall effect

-80 -60 -40 -20 0 20 40 60 800

5

10

15

20

-34-30-26-22-18-14-10-6-22610141822263034

σxy (e

2/h)

QHE at T=2.3K, B=7.94T

Rxx

(kΩ

)

Vg (V)

⎟⎠⎞

⎜⎝⎛ +==

214

2

nhe

xy ννσ

422 =×=vs gg Berry’s phase = π

holes

electr

ons









GrapheneGraphene –– FabricationFabrication

• Starting material is single-crystal Kish graphite• Mechanically exfoliate on 300 nm SiO2/Si chips

single layer

two layersseveral layers

Optical micrograph (layer thickness verified by AFM)

Single layer device after e-beam lithography

Method adapted from Novoselov, et al. PNAS 102 10341 (2005)


Raman spectroscopy of Raman spectroscopy of graphenegraphene

1200 1350 1500 1650 2400 2550 2700 2850

After Ne+ Irradiation

Ram

an In

tens

ity [a

.u.]

b)

G'

D

D

G

G

Pristine Graphene

Wave Number [cm-1]

G'

a)

K K’

Einc

Escatt

q

q

K K’

Einc

Escatt

q

defect

K K’

Einc

Escatt

q≈0

2D or G’ band: ΔERaman = 2ħω(K)

G band: ΔERaman = ħω(Γ)D band: ΔERaman = ħω(K)


GrapheneGraphene fingerprint in Microfingerprint in Micro--RamanRaman

• Raman G’ band is two-photon/two-phonon resonant excitation; sensitive to electronic structure of graphene

1550 1600 2600 27000

1000

2000

3000 Single Layer Lorentz Fit

Ram

an In

tens

ity

Wavenumber [cm-1]

Ferrari, et al., PRL 97, 187401 (2006)

Fuhrer group sample

single Lorentzian G’ peak indicates single-layer graphene


500 nm

300 nm

300 nm

Novel photoresist residue removal processAnneal in flowing H2 at 400°C

Residues from PMMA/MMA photoresist

Complete removal of photoresist residues Atomically clean STM images

Removing Removing PhotoresistPhotoresist Residue from Residue from GrapheneGrapheneIshigami, et al., Nano Letters 7, 1643 (2007)









Electrical Characterization of Electrical Characterization of GrapheneGraphene

• Ambipolar, symmetric conduction• Finite minimum conductivity ~ [4-10]e2/h• Field-effect mobility up to 20,000 cm2/Vs

-100 -80 -60 -40 -20 0 20 40 60 80 1000

1

2

3

4

5

T=2.1K

ρ(kΩ

)

Vg(V)

holes electrons


Electrical Characterization of Electrical Characterization of GrapheneGraphene

• Ambipolar, symmetric conduction• Finite minimum conductivity ~ [4-10]e2/h• Field-effect mobility up to 20,000 cm2/Vs

-100 -80 -60 -40 -20 0 20 40 60 80 1000

2

4

6

8

10

12

14

T=2.1K

σ(kΩ

-1)

Vg(V)

σmin ≈ 6e2/h

holes electrons

ggFE dV

dcdn

de

σσμ 11==


Boltzmann TransportBoltzmann Transport

-100 -80 -60 -40 -20 0 20 40 60 80 1000

2

4

6

8

10

12

14

T=2.1K

σ(kΩ

-1)

Vg(V)

τσ )(2

22

EDve F=D(E) is density of statesτ is momentum relaxation timevF is Fermi velocity

E

D(E)

EF

22

2)(F

F

vEEDhπ

=Graphene:

But: Fermi’s Golden Rule:

constant!)(

)('21 2

∝∴

∝

τ

πτ

ED

EDkVkh

σ is independent of EF!True for point defects, phononssee e.g. Pietronero (1980), T. Ando (1996)


How to explain linear How to explain linear σσ((VVgg)?)?

-100 -80 -60 -40 -20 0 20 40 60 80 1000

2

4

6

8

10

12

14

T=2.1K

σ(kΩ

-1)

Vg(V)

)('21 2EDkVk

h

πτ

∝

σ ~ EF1/2 ~ n ~ Vg

See:Ando, J. Phys. Soc. Jpn. 75, 074716 (2006)Nomura & MacDonald PRL 98, 076602 (2007)Cheianov & Fal'ko PRL 97, 226801 (2006)Hwang, Adam, & Das Sarma, PRL 98, 186806 (2007)

Interaction must be q-dependent

Coulomb interaction: qeVCoulomb κ

π 22=

q = |k – k’| ~ kF

N.B. In graphene, screenedCoulomb interaction remains ~1/kF


Minimum Conductivity of Minimum Conductivity of GrapheneGraphene

e h

• At minimum conductivity point, graphene breaks into electron and hole “puddles”• Minimum conductivity decreases with increasing impurity concentration

μσ en*min =Residual density in puddles

]20[1

)]4,(2[* 02

he

n

dkrCrnn

imp

FsRPA

simp

=

=

μ

Weak function of nimp, rs, d

Adam, et al., PNAS 104, 18392 (2007)









Charged Impurity Scattering: Potassium Doping in UHVCharged Impurity Scattering: Potassium Doping in UHVJ. H.J. H. Chen, et al. Chen, et al. Nature Physics Nature Physics 44, 377 (2008), 377 (2008)

-80 -60 -40 -20 0 200

20

40

60

Con

duct

ivity

[e2 /h

]

Gate Voltage [V]

Doping time 0 s 6 s 12 s 18 s

Upon doping with K:1) mobility decreases2) σ(Vg) more linear3) σmin shifts to negative Vg

4) plateau around σmin broadens5) σmin decreases (slightly)

All these feature predicted for Coulomb scattering in grapheneAdam, et al., PNAS 104, 18392 (2007)

• Clean graphene in UHV at T = 20 K• Potassium evaporated on graphene from getter

impnVs5x1011

=μMagnitude of scattering in quantitative agreement with theory:









e h

MagnetoresistanceMagnetoresistance at Minimum Conductivity Pointat Minimum Conductivity PointS. Cho and M. S. Fuhrer, S. Cho and M. S. Fuhrer, PRB PRB 7777, 084102(R) (2008), 084102(R) (2008)

-8 -6 -4 -2 0 2 4 6 8

0.2

0.4

0.6

0.8

B (T)

ρ xx (h

/e2 )

Expt. T = 300 K Two-fluid model α = 0.4 EMT model σxx,1 = 0.88

Functional form of ρ(B):effective medium theory for inhomogeneous e/h regions[Guttal and Stroud, PRB 71 201304 (2005)]

0 5 10 15 20 25 300

20

40

60

80

100

120

0

1

2

3

4

5

d2ρ

xx /dH2 (he

-2T-2)

σ xx (e

2 /h)

V g (V)

Large spike in magnetoresistance at Dirac point

• At minimum conductivity point, graphene breaks into electron and hole “puddles”Hwang, et al., PRL 98, 186806 (2007); Adam, et al., PNAS 104, 18392 (2007)


Charged Impurity Scattering: Minimum ConductivityCharged Impurity Scattering: Minimum ConductivityJ. H.J. H. Chen, et al. Chen, et al. Nature Physics Nature Physics 44, 377 (2008), 377 (2008)

e h

• At minimum conductivity point, graphene breaks into electron and hole “puddles”• Minimum conductivity decreases with increasing impurity concentration

0 2 4 6 8 10 12

2

4

6

8Run 1: σminRun 2: σminRun 3: σminRun 4: σminTheory: d=0.3nm d=1nm

Con

duct

ivity

[e2 /h

]

1/μe [Vs/m2] ~ nimp-80 -60 -40 -20 00

4

8

12

16

20

Con

duct

ivity

[e2 /h

]

Gate Voltage [V]


σmin

Adam, et al., PNAS 104, 18392 (2007)









Linear T-dependenceat low T

Longitudinal acousticphonons in graphene

ρ = ρ0 + ATA = 0.1 Ω/K

A is independent of charge carrier density, as predicted

0 200 4000.006

0.008

0.010

0.012

0.014

0.016

200 400

ρ[h/

e2 ]

T [K]

Sample 1 Run 1 Run2 Vg=

20V 20V 30V 30V 40V 40V 50V 50V 60V 60V

Sample 2 Vg=

20V 30V 40V 48V 57V

2222

2

2 16)(

Fss

BAA vve

TkDehT

ρρ

h⎟⎠⎞

⎜⎝⎛= → DA = 18 ± 1 eV

(good agreement w/CNT, graphite)

ElectronElectron--Phonon ScatteringPhonon ScatteringJ. H. Chen, et al. J. H. Chen, et al. Nature Nanotechnology Nature Nanotechnology 33, 206 (2008), 206 (2008)


0 200 400

0.01

0.1

0 200 400

ρ [h

/e2 ]

T [K]

Sample 1 Sample 2Activated T-dependence at high T

Consistent with:

⎟⎠⎞

⎜⎝⎛

−+

−

++=

−

15.6

11

),(

/)155(/)59(

0

TkmeVTkmeV BB eeBn

ATTn

α

ρρ

LA phonons in graphene

polar optical surface phonons in SiO2(see Fratini and Guinea, PRB 77, 195415 (2008))

Potential due to polar optical phonons is long-ranged; leads to density-dependent resistivity

(3 global fit parameters for all curves)

ElectronElectron--Phonon ScatteringPhonon ScatteringJ. H. Chen, et al. J. H. Chen, et al. Nature NanotechnologyNature Nanotechnology 33, 206 (2008), 206 (2008)


Mobility Limits in Mobility Limits in GrapheneGrapheneJ. H. Chen, et al. J. H. Chen, et al. Nature Nanotechnology Nature Nanotechnology 33, 206 (2008), 206 (2008)

Room Temperature Limits:

Currently:μRT ~ 10,000 cm2/Vs(charged impurities)

Substrate-limited:SiO2 surface phonons:μRT ~ 40,000 cm2/Vs

Intrinsic:acoustic phonons:μRT ~ 200,000 cm2/Vs@ n = 1012 cm-2

Room temperature mobility of 200,000 cm2/Vs possible!Ballistic transport over >2 microns

10 100

104

105

106

μ [c

m2 /V

s]

T [K]

Graphene: Sample 1 Sample 2

LA phonons SiO2 phonons

Impurities: Sample 1 Sample 2

Total implied mobilities: Sample 1 Sample 2

At room temp.:Si:

InSb:

μ = 1,500 (e)μ = 450 (h)

μ = 77,000 (e)









rrss in in graphenegraphene??

2/122/3

2

4*

nem

aar

bs

hεπ== Problem: what is mass?

(no characteristic length ab for massless particles)

Fs v

erhπε4

2

=

avkvK

aeU

FF

hh ==

=πε4

2

Independent of density!

KUrs =

U = potential energy of two electrons at distance a

K = kinetic energy of electron with wavelength λ = 2πa

There still exists a unitless quantity:

bs a

aKUr ==

Massive particles:


GrapheneGraphene’’ss Fine Structure Constant?Fine Structure Constant?

Fs v

erhπε

α4

2

=≡For graphene, define:Fine structure constant, with

c → vFε0 → ε

describes strength of Coulomb interaction

For:vF = 108 m/s = c/300

ε = 2.5ε0

1≈α Graphene is:weakly interacting for condensed matter,strongly interacting for relativistic Fermions

Interesting opportunities:Atomic collapse of hypercritical nuclei: Zc = 1/α = 137 (difficult to achieve in nuclear physics),Possible in graphene where Zc ≈ 1

036.1371

4 0

2

≈=c

ehπε

αα is fine structure constant

“coupling constant” – describes relative strength of Coulomb

interaction


Tuning the Tuning the ““Fine Structure ConstantFine Structure Constant””C. Jang, et al. C. Jang, et al. Physical Review Letters Physical Review Letters 101101, , 146805 (2008) 146805 (2008)

2/122/3

2

4*

nemrshεπ

=

Conventional 2D electron system:

Tune rs thru density n

FF ve

ve

hh 0

22

44 πκεπεα ==

Graphene:

Graphene’s “Fine Structure Constant”α independent of n

Fveh0

2

4πεα =

Fveh0

2

21 42

πεκκα ⎟⎟

⎠

⎞⎜⎜⎝

⎛+

=

But, can tune κ!


Two Effects of Dielectric ScreeningTwo Effects of Dielectric ScreeningC. Jang, et al. C. Jang, et al. Physical Review Letters Physical Review Letters 101101, , 146805 (2008) 146805 (2008)

Reducing α:• Reduces interaction of carriers with charged impurities

– Dominant effect for charged-impurity scattering• Reduces screening by carriers

– Dominant effect for short-range scattering

...3

322

)( ;)(

0 +−==απα

ασσ SS

S FF

Short-range scattering increased:Conductivity σS decreases

Coulomb scattering reduced:Mobility μL increases

...)( ;)(

12 22

+== πααα

σ LLimp

L FFn

nhe

Within RPA:

1~ ;~* ;* 22

min αμαμσ LL nen=

e-h puddle density decreased,mobility increased:Min. conductivity σmin constant


-30 -20 -10 0 10 20 30 40 500

20

40

60

80

Expt. Vacuum Ice

Fit Eqn. 2 Vacuum Ice

σ (

e2 /h)

Vg (V)

Add ice to clean graphene in UHV:α (SiO2/vacuum) = 0.81 α (SiO2/ice) = 0.56

-30 -20 -10 0 10 20 30 40 500

20

40

60

80

Expt. Vacuum Ice

σ (

e2 /h)

Vg (V)-30 -20 -10 0 10 20 30 40 50

0

20

40

60

80

Expt. Vacuumσ

(e2 /h

)

Vg (V)

Fit:

σ-1 = (neμL)-1 + σS-1

Coulomb (long-range) scattering

Short-range scattering

(slight asymmetry in Coulomb scattering; take symmetric component of each)

Effects of Dielectric ScreeningEffects of Dielectric ScreeningC. Jang, et al. C. Jang, et al. Physical Review Letters Physical Review Letters 101101, , 146805 (2008) 146805 (2008)


8000

10000

12000

14000

μ L (c

m2 V

-1s-1

)160

200

240

280

σ S (e

2 /h)

0 1 2 3 4 5 6 7

5

6

7

8

σ min

(e2 /h

)

Number of Ice Layers

Coulomb scattering reduced:Mobility μL increases

Short-range scattering increased:Conductivity σS decreases

e-h puddle density decreased,mobility increased:Min. conductivity σmin constant

Dielectric Screening: Theory and Expt.Dielectric Screening: Theory and Expt.C. Jang, et al. C. Jang, et al. Physical Review Letters Physical Review Letters 101101, , 146805 (2008) 146805 (2008)


GrapheneGraphene in highin high--K liquids K liquids –– a mystery?a mystery?

T. M. Mohiuddin et al. Arxiv:0809.1162 (Manchester group)• Mobility increases <50% in ethanol (κ = 25) and liquid water (κ = 80)• Concluded that charged impurities NOT dominant scatterers in graphene


GrapheneGraphene in highin high--K liquids K liquids –– our groupour group

-20 -10 0 10 200

20

40

60

80

100

120

σ (e

2 /h)

Vg (V)

vacuum

μ = 5500 cm2/Vs



-20 -10 0 10 200

20

40

60

80

100

120

μ = 28,000 cm2/Vs

σ (e

2 /h)

Vg (V)

vacuum isopropanol

μ = 5500 cm2/Vs

Isopropanolκ = 19.9α = 0.167

Mobility up 510%



-20 -10 0 10 200

20

40

60

80

100

120

μ = 28,000 cm2/Vs

μ = 45,000 cm2/Vs

σ (e

2 /h)

Vg (V)

vacuum isopropanol water

μ = 5500 cm2/Vs

DI Waterκ = 80

α = 0.047Mobility up 820%

Isopropanolκ = 19.9α = 0.167

Mobility up 510%

Possibly reaching limit set by substrate polar optical

phonons


What is the difference?What is the difference?

Au

SiO2

Si

graphene

+ + + + + + + + + + + + + + + + + + + +

- -- -

- -

- -- -

- -

Vg

Au

SiO2

Si

graphene

+ + + + + + + + + + + + + + + + + + + +

Vg

- - - - - - - - - - - - - - - - - - - - - - - - - - - -

-20 -10 0 10 200

20

40

60

80

100

120

μ = 28,000 cm2/Vs

μ = 45,000 cm2/Vs

σ (e

2 /h)

Vg (V)

vacuum isopropanol water

μ = 5500 cm2/Vs

Mohiuddin et al. :- Used electrolyte as gate- Gate charges are ions → scatterers!

Our work :- Used SiO2 back-gate- No add’l scatterers









GrapheneGraphene Corrugation Corrugation -- ScatteringScattering

)('21 2EDkVk

h

πτ

∝ q-dependent interaction→ carrier-density dependent σ(n)

q = |k – k’| ~ kF

τσ FEhe22

=

1) Coulomb interaction: σ ~ n

[ ] Hrhrh 22)0()( ∝− σ ~ n2H-1

height-height correlation function

2) Corrugated graphene†:

†Katsnelson & Geim, Phil. Trans. R. Soc. A366, 195-204 (2008)

What is exponent 2H?


Model 1: Intrinsic graphene bending constrained via interface confining potential

Model 2: Corrugations determined by strong direct interaction governed by height variations of the substrate

Substrate

grapheneV

h

ho

F =12

κ ∇2h(x, y)[ ]2+

12

Vh2 x,y( )

h(r) − h(0)( )2 ~ r2

σ(n) ~ n(mimics Coulomb scatting)

Height-height correlations will match those of the substrate.

Typical non-equilibrium surfaces show: with 2H ≈ 1.

( ) Hrhrh 22 ~)0()( −

grapheneho

Substrate

σ(n) ~ constant(mimics short range scattering)

GrapheneGraphene CorrugationCorrugation


200 nm-1.0 -0.5 0.0 0.5 1.0

0.00

0.05

SiliconGraphene

Prob

abili

ty

Height [nm]

Non-contact AFM image in UHV

M. Ishigami et al., Nano Letters 7, 1643 (2007)

SiO2

graphene

G ~ r1.1

GrapheneGraphene Corrugations on SiOCorrugations on SiO22


200 nm-1.0 -0.5 0.0 0.5 1.0

0.00

0.05

SiliconGraphene

Prob

abili

ty

Height [nm]

• σoxide = 3.1 Å and σgraphene = 1.9 Å

• Graphene 60% smoother than SiO2


Oxide-graphene boundary


SiO2

graphene

G ~ r1.1



200 nm


Oxide-graphene boundary


SiO2

graphene10

0.01

0.1

GrapheneSilicon Oxide

G(x

) [nm

2 ]Distance [nm]

Height-height correlations function

with 2H ≈ 1

( ) Hrhrh 22 ~)0()( −σ(n) ~ constant

(mimics short range scattering)



STM vs. NCSTM vs. NC--AFM topographyAFM topography

STM: 1V, ~50 pA NC-AFM: 4.6 Hz Δf

• Both images acquired from same area, on 1-layer graphene device.• Why does STM measure topography so differently? • STM more strongly interacting – electro-mechanical effect

Similar to Morgenstern group (preprint) Reproduces our earlier workIshigami, et al. Nano Letters 7, 1643 (2007)

Same area of graphene!









Lattice defect scattering in Lattice defect scattering in graphenegrapheneCharged-impurity scattering:

Linear σ(Vg)

Intravalley scatteringNo backscatteringweak anti-localization

Metallic

Defect scattering:

Constant σ(Vg)? [Shon, & Ando, (1998)]Linear σ(Vg)? [Hentschel (2007);

Stauber (2007)]

Expect intravalley and intervalley scatteringBackscattering allowedweak anti-localization or weak localization?

Metallic or insulating?

X X


Inducing lattice defects in Inducing lattice defects in graphenegraphene

• Sample is cleaned in H2/Ar at 300 °C [Ishigami, Nano Letters 7, 1643 (2007)]

• Sample baked in UHV at 220 °C overnight

• Ne+ or He+ ion irradiation at 500 eV via sputter gun• Dose given by current collected by Faraday cup

• Sample annealed at 220 °C overnight between ion irradiation runs; mobility partially recovers on annealing

Expect:• One ion → one defect consisting of multi-atom vacancy

See e.g. G. M. Shedd and P. E. Russell, JVSTA 9, 1261 (1991)J. R. Hahn, et al., PRB 53, R1725 (1996)


Raman D peak - intervalley scattering

1200 1350 1500 1650 2400 2550 2700 2850

After Ne+ Irradiation

Ram

an In

tens

ity [a

.u.]

b)

G'

D

D

G

G

Pristine Graphene

Wave Number [cm-1]

G'

a)

μ = 8600 cm2/Vs

μ = 1300 cm2/Vs

[ ]1

4310 nm2.4x10−

=−⎟⎟⎠

⎞⎜⎜⎝

⎛=

G

Da I

IL λGraphitic particles of grain size La:Cancado, et al.,

APL 88, 163106 (2006).

Point defects: identify La with defect scattering length.Our samples: La = 70 nm

μ = 1300 cm2/Vs; n ≈ 1013 cm-2 (in ambient) → lmfp ≈ 50 nm

Defect scattering lengths from Raman and transport agree

0

20

40

60

σ (e

2 /h)

-40 -20 0 20 400

20

40

60

Vg (V)

Mobility


Defects in Defects in graphenegraphene

Fit each curve to: σ-1 = (neμ)-1 + ρS

“Long-range scattering”Constant mobility

“Short-range scattering”Constant resistivity

-50 -25 0 25 500

30

60

90

120Ne+ Dosage (1010 cm-2)

0 3.8 8.2 18.2 38.7 72.2

σ [e

2 /h]

Vg [V]

T = 41 K



-50 -25 0 25 500

20

40

60

80Ne+ Dosage (1010 cm-2)

0 3.8 8.2 18.2 38.7 72.2

Con

duct

ivity

[e2 /h

]

Gate Voltage [V]

T = 41 K

-80 -60 -40 -20 0 20 400

20

40

60

80

Con

duct

ivity

[e2 /h

]

Gate Voltage [V]

K dose (1010 cm-2) 0 160 370 570

Ne+ irradiation(lattice defects)

K doping(charged impurities)



0 2 4 6 8 10 120

2

4

6

8

10

12

14

Ne+ irradiation He+ irradiation

Charged Impurities (K)

1/μ

[Vs/

m2 ]

Dosage [1011 cm-2]

0 2 4 6 8 10 120

1

2

3

4

5

6


ρ shor

t-ran

ge [h

/e2 /1

000]

Dosage [1011 cm-2]

Defects:• Defects change the linear term in σ(Vg)

Like charged impurities!

• Linear σ(Vg) scattering 4x stronger than for same concentration of charged impurities

Defects:• Carrier-density-independent ρs scattering does not change

• ρs corresponds to lmfp ~2 microns

→ ρs cannot be the scattering seen in Raman D band


Defects in Defects in graphenegraphene –– Minimum conductivityMinimum conductivity

0.0 0.2 0.4 0.6 0.80

2

4

6

Ne+ irradiation He+ irradiation K Dosing

σ min [e

2 /h]

μ [m2/Vs]

Minimum conductivity:

n* is carrier density in “puddles”n* is function of charged impurity density

μσ en*min =

Charged impurities:nimp increases: μ decreases, n* increases

→ σmin changes very weakly

Defects:ndefect increases: μ decreases, n* constant

→ σmin proportional to μμσ en*min =

μσ en*min =n* ↑μ ↓

n* const.μ ↓

increasing irradiation


Defects in Defects in graphenegraphene –– Metal or Insulator?Metal or Insulator?Theory:Graphene with only intravalleyscattering is metallic (weak anti-localization)

Graphene with intervalley scattering is insulator (weak localization)[Bardarson, et al. PRL 99, 106801 (2007)]

Experiment:Graphene with charged impurities shows metallic ρ(T) at low T[Novoselov, Nature 438, 197 (2005)][Chen, Nature Nano 3, 206 (2008)]

Graphene with defects shows diverging ρ(T) at low T even for modest mobilities (~2,000 cm2/Vs)!

10 1000.01

0.1

ρ [h

/e2 ]

Temperature [K]

Ne+ irradiatedVg-Vg,min =

0 V 14V 30V

Pristine Vg-Vg,min =

0 V 14V 30V

μRT = 2,500 cm2/Vs

μRT = 13,000 cm2/Vs


MidgapMidgap states states -- TheoryTheory

( )Rknn

hene F

ddd

22

ln2== μσ

[Hentschel & Guinea, PRB 76, 115407 (2007); Stauber, Peres, & Guinea, PRB 76, 205423 (2007)]

• Defect potential modeled as circular well of radius R, depth ε0, intervalley scattering Δ.• Spectrum inside the potential well is gapped by Δ; has bound midgap states.• Conductivity is:

0 2 4 6 8 10 120

2

4

6

8

10

12

14


Charged Impurities (K)

1/μ

[Vs/

m2 ]

Dosage [1011 cm-2]

Experimentally, μd ≈ [1.2 × 1015 V-1s-1]/nd

For n = 2×1012 cm-2 → R ~ 8 ÅReasonable value for 500 eV irradiation(multiple-atom vacancies)

ln2(kFR) dependence not observed, but kF only varies by factor of ~3 outside puddle regime (n > n*)


ConclusionsConclusions• Mobility of graphene on SiO2 limited by charged impurities

– Charged impurities give linear σ(Vg) – Minimum conductivity determined by density in e-h puddles

– Addition of dielectric layer increases mobility

• Room temperature intrinsic mobility ~200,000 cm2/Vs– Remote interfacial phonon scattering from SiO2 limits to ~40,000 cm2/Vs

• Corrugations– Graphene corrugations follow SiO2 substrate roughness

• Graphene with lattice defects– Linear σ(Vg) with 4x lower mobiltiy compared to charged impurities– Consistent with midgap states, R = 2-3 Å– Depressed σmin~ μ; can be less than 4e2/πh– Intervalley scattering gives insulating ρ(T); Raman D band


Prof. Michael S. Fuhrer’s GroupSungjae ChoChaun Jang

Shudong XiaoAlexandra Curtin

Prof. Ellen D. Williams’ Group:Dr. William Cullen

Prof. Masa Ishigami (now @ UCF)Daniel Hines Jianhao Chen

grapheneFabry-Pérot

transfer-printing

STM

MR in graphene

Fuhrer group:Fuhrer group:Williams group:Williams group:

Das Sarma group:Das Sarma group:More Info:More Info:

wwww.physics.umd.edu/mfuhrerww.physics.umd.edu/mfuhrerwwww.physics.umd.edu/spgww.physics.umd.edu/spgwww.physics.umd.edu/cmtcwww.physics.umd.edu/cmtc

-80 -60 -40 -20 0 200

20

40

60

Con

duct

ivity

[e2 /h

]

Gate Voltage [V]


UHVdoping,

dielectric expts.

Prof. Sankar Das Sarma’s Group Dr. Shaffique Adam Dr. Euyheon Hwang

Dr. Enrico Rossi Wang-Kong TseTheory

Funding:Funding:

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Diffusive Charge Transport in Graphene

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