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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

- 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

- 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

- Allowed electronic states forms energybands

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

Quadratic energy-momentum dispersion

Semiconductor physics!!

- One layer of Carbon atoms

Au contacts

SiO2

GRAPHENE

Si

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

Vg

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

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

- 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

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)

≈

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

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

- 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!

s-wave

Conventional s-wave SC:

Δ=const over the Fermi surface

Gapped excitations

massless

Dirac

fermions

vF

vD

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

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.

- 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

- 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

- 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-universal” jump of the superfluid density

- 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)

- “Non-universal” jump of the superfluid density

- 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

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

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

PRL 98, 117008 (07)

- 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)

L.B. et al,

PRL (2007)

ξ diverges at Tc!

No linear M

in the range of

fields accessible

experimentally

L.B. et al,

PRL (2007)

ξ diverges at Tc!

No linear M

in the range of

fields accessible

experimentally

- 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