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 lara.benfatto_at_roma1.infn.it www
http//www.roma1.infn.it/lbenfat/
29-30 Novembre Conferenza di Progetto
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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

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Basic understanding of many electrons in a solid

• k values are quantized
• Pauli principle N electrons cannot occupy the
  same quantum level
• Fermi-Dirac statistic all level up to the Fermi
  level are occupied
• Excitations unoccupied levels

Quadratic energy-momentum dispersion
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Effect of the lattice

• Allowed electronic states forms energy bands

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Effect of the lattice

• Allowed electronic states forms energy bands and
  have an effective mass

Quadratic energy-momentum dispersion Semiconductor
physics!!
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Dirac fermions from lattice effects graphene

• One layer of Carbon atoms

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Dirac fermions from lattice effects graphene

• One layer of Carbon atoms
• Graphene a 2D metal controlled by electric-field
  effect

Vg
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Dirac fermions from lattice effects graphene

• Carbon atoms many allotropes
• Graphene a 2D metal controlled by electric-field
  effect
• In momentum space

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Dirac fermions from lattice effects graphene

• Carbon atoms many allotropes
• Graphene a 2D metal controlled by electric-field
  effect
• In momentum space Dirac cone

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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
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Bilayer graphene tunable-gap semiconductor
Oostinga et al. arXiv0707.2487 (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)

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Bilayer graphene tunable-gap semiconductor
Oostinga et al. arXiv0707.2487 (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
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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)
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Dirac fermions from interactions d-wave
superconductors

• Example of High-Tc superconductor La1-xSrxCu2O4
• quasi two-dimensional in nature
• CuO2 layers are the key ingredient
• La and Sr supply 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!

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Dirac fermions from interactions d-wave
superconductors
s-wave
Conventional s-wave SC ?const over the Fermi
surface Gapped excitations
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Dirac fermions from interactions d-wave
superconductors
s-wave
¹
d-wave
Conventional s-wave SC ?const over the Fermi
surface Gapped excitations
High-Tc d-wave SC ? vanishes at nodal points
Gapless Dirac excitations
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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.
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Collective phase fluctuations vortices!

• In BCS superconductors superconductivity
  disappears when ? 0 at Tc standard
  paradigm applies
• In HTSC superconductivity is destroyed by phase
  fluctuations where ? remains finite
• Crucial role of vortices

water vortex
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Collective phase fluctuations vortices!

• In BCS superconductors superconductivity
  disappears when ? -gt0 at Tc standard paradigm
  applies
• In HTSC superconductivity is destroyed by phase
  fluctuations where ? 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 2p around the vortex core
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Understanding Kosterlitz-Thouless physics
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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

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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)

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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)
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The absence of the superfluid-density jump

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

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The absence of the superfluid-density jump
L.Benfatto, C. Castellani and T. Giamarchi, PRL
98, 117008 (07)
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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)
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Magnetization above TKT
L.B. et al, PRL (2007)
? diverges at Tc!
No linear M in the range of fields accessible
experimentally
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Magnetization above TBKT
L.B. et al, PRL (2007)
? diverges at Tc!
No linear M in the range of fields accessible
experimentally
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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