Nonlinear screening in graphene nanostructures

Michael Fogler, UCSD

Acknowledgements

Work in collaboration with:L. Glazman (Yale)D. Novikov (Yale)B. Shklovskii (U MN)Matt Zhang (UCSD)

FundingNSF, ACS UCSD

Quasi-relativistic (“Dirac”) fermions

( ) | |E υ= ±p p

/ 300cυ ≈

0 "electrons"0 "positrons" or "holes"

EE><

300 times slower than light

Fermi surface shrinks to a single point

Graphene: QED in a pencil trace?

SimilaritiesLinear spectrumSpin-like degree of freedomChirality

Differences2+1D instead of 3+1DNo massInteractions are strongNo retardation ( )Doping, disorder, phonons….

cυ <

Klein paradox (1929)

0

electron states

positron states

e p

An example of what “QED in a pencil trace” may test

The result of the correct calculation: transmission < 1 and increases with the barrier height

– Not exactly paradoxical in 2007 but still unusual ...

KatsnelsonNaïve calculation gives transmission > 1 ??

Source of mistake: for the negative-energy states (group velocity) = - (quasi-particle velocity)

Supercritical charge: atomic collapse and creation of antimatter

Energy

22mc

+Z

Schwinger, 1950’sZeldovich, 1970’s

+

+Zsubcritical

CZ >α

1~CZ <α

Fine-structure constant in graphene2 1

137ec

α = ≈h

2

Graphene 0.9

2.3 dielectric constant

eακ υ

κ

= ≈

=h

SiO2

High-k dielectric: weak interactions

constant dielectric,2

== κυκ

αh

e

1,1 If <<>> ακ

κ

HfO2, water, …

Our work – nonlinear screening in graphene

References: arXiv:0707.1023 (PRB 2007), 0708.0892, and 0710.2150

What is screening?

+Z( )extV r

( )V r r++

+

+

+

++

Linear screening

length screening1F

s kr

α=

1)( >>qε

x

Good screening1)( ≅qε

x

Poor screening

External charge ~ sin qxInducedcharge sr

sr…Guinea et al.AndoDas Sarma et al.…

Why would screening be nonlinear in graphene?

No carriersno screening!

Lots of carriersgood screening

Lots of carriersgood screening

Nonlinear screening

Higher electron density,better screening

sr srx

Lower density,worse screening

Summary of possible screening regimes

Screening Poor Poor Good

Valid approximation

N/A(Dirac eq.)

Thomas-Fermi

Classicalelectrostatics

Lengthscaleα

λFsr =Fλ0

1 If <<α

Graphene p-n junction

Electrostatic potential of the gates induces a gradually varying charge density profile, which changes sign.

The back-gate controls the overall carrier densityThe top gate determines the density difference

Recent experimentsResistance of a graphene p-n junction is only a few kΩQualitatively explained by the gapless Dirac spectrumQuantitative theory needed

B. Huard et al., PRL 98, 236803 (2007) B. Ozyilmaz et al., arXiv/cond-mat:0705.3044

Williams et al., Science 317, 638 (2007)

Klein paradox (1929)

0

electron states

positron states

e p

An example of what “QED in a pencil trace” may testNaïve calculation gives transmission > 1 ??

Source of mistake: for the negative-energy states (group velocity) = - (quasi-particle velocity)

The result of the correct calculation: transmission < 1 and increases with the barrier height

– Not exactly paradoxical in 2007 but still unusual ...

Veselago lensing

Cheianov et al. 2007

Electric field in the junctionConductance is determined by the field strength at the interfaceA naïve estimate (Cheianov 06) assumed uniform field:

In reality, electric field issuppressed away from junction where screening is good

enhanced near the interface where screening is poor

V

V

x

x

Nonlinear screening, case α~1Electron density is highly non-uniform, how to compute the screened potential?It is charge density that must be found first!Screened potential V(x) follows from the density of statesThree-line derivation, next

Field enhancement near interface

F

x

Nonlinear screening, α~1Charge density

xx ')( ρρ =

Potential, T-F approx.

xxxxeV ∝== )()()( ρυμ h

Electric fieldxxVF /1)(' ∝−=

ρ

x

Cut the divergence)( ** xx Fλ=

p n

*x− *x x

F

Numerical results

3/16/12 )'(0.1 −−= ρα

ehR

Junction resistance

Results

Previous results for transmission through p-n junction are off by a (parametrically) large factor (~2-10, in practice)Our new results bring theory and experiments in a much better agreementIncluding disorder effects the agreement can be made quantitative (next slide)

Disorder in p-n junction

Coulomb impurity in graphene

Why the Coulomb impurity problem?

Uncontrolled charged impurities in the substrateIntentional doping / gatingIntriguing analogy to atomic collapse and vacuum breakdown in QED

Graphene 0.9 ?cα α≈ >

Supercritical charge: atomic collapse and creation of antimatter

Energy

22mc

1Zα >+Z

1Zα <

Schwinger, 1950’sZeldovich, 1970’s

+

+Zsubcritical

Critical charge in graphene

~ 1/ 2cZ α

Graphene 0.9 ~ 1 ?cZα ≈ ∴

Khalilov (1998)NovikovShytov, LevitovCastro-Neto…

Critical charge in graphene: previous work

DiVincenzo, Katsnelson,Shytov, Castro-Neto

Electron density

r

Zsubcritical

1( ) ~V rr

r

Zsupercritical

1~ln

Vr r

( ) ~ ( )n r rα δ 2 2

1( ) ~ln

n rr r

Electron density

r

2 2

1( ) ~ln

n rr r

Electron density

Problem with previous work: range of validity

Short-range cutoff: bandwidthLong-range: zero massFor αZ~1 both cutoffs ~ lattice constant

Our work – “hyper”critical charge

M.F., Novikov, Shklovskii, PRB (2007)

1>>Zα

New result for “vacuum polarization” around a large

Coulomb chargeNew universal result for the structure of the supercritical coreFor α =1 this core “squeezes out” the regimes discussed in previous literatureSuch regimes return if α << 1 (next slide)

r

Zsupercritical

3/ 2

1~Vr

3

1( ) ~n rr

Electron density

Hypercritical impurity, small α

How to realize large-Z charge?

+Zd

STM / AFMtip