Dirac fermions with zero effective mass in condensed matter new perspectives
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Dirac fermions with zero effective mass in condensed matter: new perspectives. Lara Benfatto* Centro Studi e Ricerche “Enrico Fermi” and University of Rome “La Sapienza”. *e-mail: [email protected] www: http://www.roma1.infn.it/~lbenfat/. 29-30 Novembre Conferenza di Progetto.

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Dirac fermions with zero effective mass in condensed matter: new perspectives

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Dirac fermions with zero effective mass in condensed matter: new perspectives

Lara Benfatto*

Centro Studi e Ricerche “Enrico Fermi”

and University of Rome “La Sapienza”

*e-mail: [email protected]

www: http://www.roma1.infn.it/~lbenfat/

29-30 Novembre Conferenza di Progetto


Outline

  • Why Dirac fermions? Common denominator in emerging INTERESTING new materials

  • Dirac fermions from lattice effect: the case of graphene

    • Bilayer graphene: “protected” optical sum rule

  • Dirac fermions from interactions: d-wave superconductivity

    • Collective phase fluctuations: Kosterlitz-Thouless vortex physics

  • Acknowledgments: C. Castellani, Rome, Italy

    T.Giamarchi, Geneva, Switzerland

    S. Sharapov, Macomb (Illinois), USA

    J. Carbotte, Hamilton (Ontario), Canada


Basic understanding of many electrons in a solid

  • k values are quantized

  • Pauli principle: N electrons cannot occupy the same quantum level

  • Fermi-Diracstatistic:all level up to the Fermi level are occupied

  • Excitations: unoccupied levels

Quadratic energy-momentum dispersion


Effect of the lattice

  • Allowed electronic states forms energybands


Effect of the lattice

  • Allowed electronic states forms energybands and have an “effective mass”

Quadratic energy-momentum dispersion

Semiconductor physics!!


Dirac fermions from lattice effects: graphene

  • One layer of Carbon atoms


Au contacts

SiO2

GRAPHENE

Si

Dirac fermions from lattice effects: graphene

  • One layer of Carbon atoms

  • Graphene: a 2D metal controlled by electric-field effect

Vg


Dirac fermions from lattice effects: graphene

  • Carbon atoms: many allotropes

  • Graphene: a 2D metal controlled by electric-field effect

  • In momentum space


Dirac fermions from lattice effects: graphene

  • Carbon atoms: many allotropes

  • Graphene: a 2D metal controlled by electric-field effect

  • In momentum space: Dirac cone


Universal conductivity

  • Despite the fact that at the Dirac point there are no carriers the system has a finite and (almost) universal conductivity!!

Dirac fermions are “protected” against disorder

Deviations: charged impurities, self-doping,

Coulomb interactions, vertex corrections


Bilayer graphene: tunable-gap semiconductor

Oostinga et al.

arXiv:0707.2487 (2007)

LARGE gap (a fraction of the Fermi energy)

Does it affect the total spectral weight of the system?

Ohta et al.

Science 313, 951 (2006)


Bilayer graphene: tunable-gap semiconductor

Oostinga et al.

arXiv:0707.2487 (2007)

LARGE gap (a fraction of the Fermi energy)

Does it affect the total spectral weight of the system?

Ohta et al.

Science 313, 951 (2006)

Analogous problem in oxides: electron correlations

decrease considerably the carrier spectral weight


“Protected” optical sum rule

The optical sum rule is almost constant despite the large gap opening: large redistribution of spectral weight is expected

(a prediction to be tested experimentally)

L.Benfatto, S.Sharapov and J. Carbotte, preprint (2007)


Cooper pair

Dirac fermions from interactions: d-wave superconductors

  • Example of High-Tc superconductor La1-xSrxCu2O4:

    • quasi two-dimensional in nature

    • CuO2layers are the key ingredient

    • LaandSrsupply “doping”

  • Superconductivity: formation of Cooper pairs

    which “Bose” condense

    High Tc: not explained within standard BCS theory

    for “conventional” low-Tc superconductors

    New quasiparticle excitations!

    New “collective” excitations!


Dirac fermions from interactions: d-wave superconductors

s-wave

Conventional s-wave SC:

Δ=const over the Fermi surface

Gapped excitations


massless

Dirac

fermions

vF

vD

Dirac fermions from interactions: d-wave superconductors

s-wave

¹

d-wave

Conventional s-wave SC:

Δ=const over the Fermi surface

Gapped excitations

High-Tc d-wave SC:

Δ vanishes at nodal points

Gapless Dirac excitations


vF

vD

Measuring Dirac excitations

Gomes et al. Nature 447, 569 (2007)

Dirac fermions are “protecetd” against disorder

Low-energy part does not depend on the position

High-energy part is affected by position, disorder, etc.


Collective phase fluctuations: vortices!

  • In BCS superconductors superconductivity disappears when |Δ| 0 at Tc: standard paradigm applies

  • In HTSC superconductivity is destroyed by phase fluctuations where |Δ| remains finite

    Crucial role of vortices

water vortex


Collective phase fluctuations: vortices!

  • In BCS superconductors superconductivity disappears when |Δ| ->0 at Tc: standard paradigm applies

  • In HTSC superconductivity is destroyed by phase fluctuations where |Δ| remains finite

    Crucial role of vortices

    Kosterlitz-Thouless like physics

J.M.K. and D.J.T. J. Phys. C

(1973, 1974)

Superconducting hc/2e vortex

Superconducting vortex is a topological

defect in phase .

 winds by 2π around the vortex core


Understanding Kosterlitz-Thouless physics


Understanding Kosterlitz-Thouless physics

  • Need of a new theoretical approach to the Kosterlitz-Thouless transition

    • Mapping to the sine-Gordon model

  • Crucial role of the vortex-core energy

    • “Non-universal” jump of the superfluid density

      L.Benfatto, C.Castellani and T.Giamarchi, PRL 98, 117008 (07)

      L.Benfatto, C.Castellani and T.Giamarchi, in preparation


Understanding Kosterlitz-Thouless physics

  • Need of a new theoretical approach to the Kosterlitz-Thouless transition

    • Mapping to the sine-Gordon model

  • Crucial role of the vortex-core energy

    • “Non-universal” jump of the superfluid density

      L.Benfatto, C.Castellani and T.Giamarchi, PRL 98, 117008 (07)

      L.Benfatto, C.Castellani and T.Giamarchi, in preparation

    • Non-linear field-induced magnetization

      L.Benfatto, C.Castellani and T.Giamarchi, PRL 99, 207002 (07)


The absence of the superfluid-density jump

  • In pure 2D superfluid/superconductors Js jumps discontinuously to zero, with an universal relation to TKT

4He films

McQueeney et al.

PRL 52, 1325 (84)


YBCO D.Broun et al, cond-mat/0509223

The absence of the superfluid-density jump

  • In pure 2D superfluid/superconductors Js jumps discontinuously to zero, with an universal relation to TKT


The absence of the superfluid-density jump

L.Benfatto, C. Castellani and T. Giamarchi,

PRL 98, 117008 (07)


Non-linear magnetization effects

  • Field-induced magnetization is due to vortices but one does not recover the LINEAR regime as T approaches Tc

Tc

Correlation length

(diverges at Tc)

M=-a H

L. Li et al, EPL 72, 451 (2005)


Magnetization above TKT

L.B. et al,

PRL (2007)

ξ diverges at Tc!

No linear M

in the range of

fields accessible

experimentally


Magnetization above TBKT

L.B. et al,

PRL (2007)

ξ diverges at Tc!

No linear M

in the range of

fields accessible

experimentally


Conclusions

  • New effects in emerging low-dimensional materials

  • Need for new theoretical paradigms: quantum field theory for condensed matter borrows concepts and methods from high-energy physics

Dirac cone!!

Einstein cone


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