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1. EARLIER STUDIES 2. PHYSICAL PROPERTIES: BINDING ENERGIES, ANYSOTROPIES, COMPRESSIBILITY,..

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### 2.Physical Propertiesb)Calculation binding energies from Lennard-Jones 6-12potentials

Transporte Ohmico y Ballístico en HOPG: Camino Libre Medio, Movilidades y Efectos Superconductores.

- 1. EARLIER STUDIES
- 2. PHYSICAL PROPERTIES: BINDING ENERGIES, ANYSOTROPIES,
- COMPRESSIBILITY,..
- 3. IS GRAPHITE A METAL OR AN INSULATOR?, IS IT A GOOD
- CONDUCTOR?
- 4. STRUCTURE OF THE HOPG
- 5. POTENTIAL AND DENSITY OF CARRIERS FLUCTUATIONS
- 6. MAGNETIC FIELD EFFECTS: MAGNETORESISTANCE (OMR) AND
- METAL INSULATOR TRANSITION, SENSOR APPLICATIONS, FINITE
- SIZE EFFECTS (REDUCTION OF OMR), QUANTUM EFFECTS.
- 7. GRAPHENE: STABILITY, ELECTRICAL CONDUCTION AND MAGNE-
- TORESISTANCE.
- 8. WHAT ABOUT SUPERCONDUCTIVITY ?

Work done in collaboration with P. Esquinazi (U. Leipzig)

EARLIER STUDIES

- a) R.R. Haering and P.R Wallace (J.Chem. Phys. Solids,1957) proved that for
- graphene the dispersion relation of energy is?: at what energies?
- E(k)=hv.k/2p (2) Esto no está probado a bajas densidades,
- A densidad alta, mayor que 10^10, puede ser.
- Linear in k, this implies no mass for the carriers LIKE PHOTONS AND SATISFY DIRAC EQUATION.(Gonzalez, Guinea and Vozmediano PRL 1992; Kopelevich, PRL 2004)?.
- Also zero Fermi Energy. It is a zero gap semiconductor (no sabemos)??.
- Notice: (h2/2m)k2 (2p)-2 for massive carriers.
- b) Then to form graphite one has to couple graphene planes. McClure, PR 1957, and Slonczewski and Weiss PR 1858 coupled the graphene planes by a chemical tight binding model. This gives a γ(perpendicular to planes) of 340meV and aγ(parallel to planes) of 4000meV. WHAT ARE THE IMPLICATIONS OF THESE VALUES?.

=compresibility

C3=900meV/A^3, too , factor of 1.66, binding energy 54meV/C. Experimental 42mev/C

Medidas de compresibilidad prueban que la interacción es vdW

Calculation Girifalco and Lad

C3=900meV/Â^3, Girifalco and Lad,

EB(onelayer-semeinfinite bulk)=60meV/C

EB(semi-semi)=68meV/C); Compre=3.1*10^132 cm^2/dyn

Calculation binding energy using polarizability of Graphite 0.62A3

C3=520meV/Â3,Garcia 2006,

EB(onelayer-semeinfinite bulk)=42meV/C

EB(semi-semi)=38meV/C); Compre=3.93*10-12 cm2/dyn

C=520meV/A3, obtained from

Polarizability and Clausius-Mossotti law.

EB(Exp)=42.5meV/C

Calculation van der Waals (eV/A3) as a function of polarizability- xobtained from Clausius –Mossoti law

free C atom

graphene

graphite

ANISOTROPIES

- a) Binding energy in plane sp2 orbitals

have energies of 3eV

- b) Interplane binding energies are 42meV
- THEREFORE THE BINDING ENERGIES RATIO IS 10-2.
- c) The electrical conductivity measured (Esquinazi y Kopelevich 2003) has an anisotropy of 10-4– 10-6 with rplane(300K)=40mW.cm,

for Cu=1.7mW.cm, Ti=43.1mW.cm.

- d) Magnetoresistance (OMR) has anisotropies

of 10-3.

Clarification of the role theγ´s and the current anysotropy and binding energies between planes

- A) There has been discussions identifying γ(perp. to plane) with the binding energy between planes as well as giving the current anysotropy between planes. Recent work (Arovas; Esqunazi, Garcia and Guinea):
- Eγ (binding)= γ(perp)2 /(4γ(para)= 7meV
- Eexp(binding)=42.5meV=
- Evdw+Eγ =38+7=45meV ( so the system is vdw).
- B) γ(perp) does not carry current perpendicular because connect unequivalent atoms. Therefore neither contradicts anysotropy.
- C)γ(perp) only defines curvature of bands and Fermi surface.

3.Metal or Insulator?

- a) The in-plane resistivity is of the order of electrical resistivities of ordinary metals but very large, semiconducting at low T perpendicular to plane.
- b) However a metal is not characterized by being a good conductor but by being a good screener of the electric field.
- Metals have negative dielectric constant at optical frequencies, however graphite dielectric constant is:
- eplane=5.6 + 7.0i , ez=2.25. Hoinkes, Rev. Mod. Phys.52, 933(1980);
- Greenway et al, Phys. Rev. 178, 1340 (1969).
- So, it is an insulator in z (perpendicular to plane), weak screening. Also an insulator in plane but with large absorption (large imaginary part)
- This implies large charge and potential fluctuations and large field penetration. Then we propose to study graphite with electric field microscopy.

GRAPHITE REAL LIFE STRUCTURE

The images show the in plane (a) and the c-axis orientation (b) of the crystallites of the HOPG sample. Note that the scan size is 600 x 200 µm and the crystallites observed here are in the 10 µm range of elongation, matching the results of EFM measurements.

D. Spoddig, Leipzig

Laboratorio de Física de Sistemas Pequeños y Nanotecnología

Bias Voltage

V

I

Diagram of the electrostatic force microscopic measurement.

GRAPHITE POTENTIAL FLUCTUATION metallic-insulating

No structure in the surface AFM

REDUCTION OMR WITH CONSTRICTION SIZE AND LOCALIZATION OF CARRIERS

CHANGE IN SLOPE AS A FUNCTION OF FIELD IS DUE TO LOCALIZATION.

AT SMALL FIELDS CARRIERS LOCALIZE

Current versus constriction width

- Rs2D=2/π(ρ/t)ln(Ω/W)+ (ρ/t)L/W.
- Rs3D = ρ/W Maxwell´s result.

L=1 micron

3D

2D

2D

L=4microns

Mean Free Path and Fermi WL

- Nota: valores de 10 y 1 micron para l y λ

Densidad y Movilidad

- Movilidad= e/h( l(t).λ(T))=
- 2.42*10^14(1/V.s)( l(t).λ(T)cm^2)

Movilidad: Grafito-Grafeno suspendido

Las densidades del grafeno mayores que 2*10^10, el punto de

Dirac todavía esta lejos ~ 10^8

grafito

grafeno

Superconductividad?Metal-Aislante: Experimento para prácticas

- Efecto campo, la resistencia crece mucho

Fig.2: The same as in Fig. 1 but the resistance is measured between 7 and 11 electrodes.Muestrads pequeñas de grafito

QUANTUM OSCILLATIONS IN GRAPHENE

QUANTUM OSCILLATIONS WITH THE DENSITY OF CARRIERS (OR THE APPLIED VOLTAGE Vg)

Novoselov et al., Science 2004)

QUANTUM OSCILLATIOJN IN THE MAGNETORESISTANCE OF GRAPHENE

Zhang et al., Nature 2005

SdH oscillations have a peculiar behaviour

No SdH in the multigraphene!!!

2nd

5th

4rd

3rd

Magnetic field

Fig. 3: Usual ferromagnetic loop for a material with positive magnetoresistance. Note that the minima that occur at the coercive fields (zero or minimum of magnetization) are located at the third and fifth quartals, as expected. This is clearly qualitative different from the hysteresis loops fluxons in Josephson-coupled granular superconducting domains produce.

Ciclo de Histeresis con HResistance

Histeresis magnetica

3rd

5th

4th

2nd

1st

Fig. 5: Surface resistance vs. external field (Fig.2 from Ref.[5] of our manuscript) of a granular YBCO sample. Note the difference in the hysteresis to the ferromagnetic case (Fig. 3), in particular the quartals where the minima are located, i.e. second and fourth quartals. This behavior is identical to that observed in our sample (3), see Fig. 3 in our manuscript.

Esta histeresis corresponde a Josephson coupling entre fluxones- Histeresis en grafito

Fig.2.

However bombarded and non-bombarded graphite are magnetic

Magnetization of virgin graphite. Linear with T implies 2D

Linear T

Exactly linear T

Bombarded spot

Bombarded spots

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