Atomic collapse in graphene Andrey V. Shytov (BNL)

Work done in collaboration with:

L.S. Levitov MIT

M.I. Katsnelson University of Nijmegen, Netherlands

* Phys. Rev. Lett. 99, 236801; ibid. 99, 246802 (2007) Outline

1. Atomic collapse and Dirac vacuum reconstruction in high-energy physics

2. Charged impurities in graphene: manifestations of «atomic collapse» a. Local DOS (STM) b. Transport (conductivity) c. Vacuum polarization

3. Conclusion Electrons in graphene

Two sublattices

Pseudo-spin (sublattice) Slow, but «ultrarelativistic» Dirac fermions!

Chirality conservation => no localization (Klein paradox) Why graphene?

High mobility, tunable carrier density

Linear spectrum => high quantization energies, room temperature mesoscopic physics

Realization of relativistic quantum physics in table-top experiments => unusual transport properties

Large fine structure constant => strong field regime, inaccessible in high-energy physics Charged impurities

Dominant contribution to resistivity (RPA screening is not essential)

Nomura, McDonald (2005) Charged impurities

Dominant contribution to resistivity (RPA screening is not essential)

Nomura, McDonald (2005)

Any interesting non-linear effects??? Yes, Dirac vacuum reconstruction Stability of Atom Classical physics: unstable (energy is unbounded) Stability of Atom Classical physics: unstable QM: stable orbits, zero (energy is unbounded) point motion stops the collapse Stability of Atom Classical physics: unstable QM: stable orbits, zero (energy is unbounded) point motion stops the collapse

Relativity: collapsing orbits

v < c Stability of Atom Classical physics: unstable QM: stable orbits, zero (energy is unbounded) point motion stops the collapse

Relativity: collapsing orbits Relativity + QM: ???

? v < c Stability of Atom Classical physics: unstable QM: stable orbits, zero (energy is unbounded) point motion stops the collapse

Relativity: collapsing orbits Relativity + QM: Collapse?

But: position uncertainty: v < c ??? Dirac – Kepler problem Part I (Z < 137) D = 3

Dirac (1929)

What happens at Z > 137? Dirac – Kepler problem Part II (137 < Z < 170)

Pomeranchuk and Smorodinskii (1945) Finite size of nucleus (regularization)

Solution can be continued to Z > 137. Atomic collapse: semiclassical picture

Relativistic particle may fall to the Coulomb center Atomic collapse: semiclassical picture

Relativistic particle may fall to the Coulomb center Dirac – Kepler problem Part II (137 < Z < 170)

Pomeranchuk and Smorodinskii (1945) Finite size of nucleus (regularization)

1S level merges into Dirac sea at Z = 170

Z > 170? Dirac – Kepler problem Part II (137 < Z < 170)

Pomeranchuk and Smorodinskii (1945) Finite size of nucleus (regularization)

Solution can be continued to Z > 137. 1S level merges into hole continuum at Z = 170 Z > 170? Dirac – Kepler problem Part III (Z > Zc = 170)

Gershteyn, Zeldovich (1969) Popov (1970) Resonant electron state in Dirac sea Screening by pair production?

Quasilocalized state Atomic collapse in graphene

Can be modeled by charged impurities:

No mass => no discrete states, continuous spectrum

Manifestations: quasistationary states, resonances

Strong effects in vacuum polarization, no cutoff at Compton wavelength Charged impurity in graphene

Dirac equation:

Ansatz: Two regimes

Subcritical:

Supercritical: Oscillations small r !

Critical value: Exact solution

Subcritical case:

Scale-invariant solution (depends only on kr ) Density of states

●Standing waves ●Bound states Density of states

●Standing waves ●Bound states Density of states

●Standing waves ●Bound states Density of states

Standing waves

Localized state

●Standing waves ●Bound states Quasi-bound states

Type of carriers is reversed at small r

Klein tunneling couples electron-like states at small r to hole-like states at large r Quasi-bound states

Classical motion

Quantization condition:

Transparency:

Exact result near criticality: Transport Drude conductivity

Scattering phases (extracted from exact solution)

Quasi-bound states show up as Fano resonances Vacuum polarization

General form:

RPA (Mirlin et al):

Thomas-Fermi (Katsnelson):

RPA and TF give conflicting results, which is unusual Friedel sum rule

Total charge:

Lin (2006):

Phases extracted from exact solution: Subrcritical case Only deep states contribute to polarization charge (confirmed by a direct calculation) RPA formula is valid. Numericall results:

Terekhov et al: exact formula for polarization charge Supercritical case

(only s-channel is overcritical)

Thomas-Fermi limit

RG flow of the effective charge:

The flow terminates at finite distance:

The charge of an overcritical impurity is always screened to the critical value. Open questions:

Excitons? Effect of vacuum polarization on localized states? Other many-body effects? Kondo? Role of water? Capacitance? Gapped systems (bilayers, epitaxial graphene)? Conclusions:

Two distinct regimes for Coulomb impurities: subcritical and supercritical Local DOS: quasilocalized states, standing waves Fano-like resonances in transport cross-section, conductivity Screening cloud in supercritical regime

Divalent and tri-valent impurities are needed (Ca, Yb, La, Gd can be intercalated, e.g., McChesney et al, (2007)). Alternatively, one can use the charge on STM tip. Signatures of «atomic collapse» can be observed.