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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