GRAPHENE: A NEW STAR IN MATERIAL SCIENCE S. Sahoo 1 & A. K. Dutta 2 Department of Physics, National Institute of Technology Durgapur-713209, West Bengal, India. 1 E-mail: sukadevsahoo@yahoo.com 2 E-mail: atanu.30255@gmail.com 1. Introduction 33 Graphene is the recently discovered two-dimensional (2D) allotrope of carbon [1,2]. It is a monolayer of carbon atoms packed into a dense honeycomb crystal structure. Graphene 2 sheets are one-atom thick, 2D layers of s p -bonded carbon. It is a 2D nanomaterial. The name comes from graphite + -ene ; graphite itself consists of many graphene sheets coupled by weak Vanderwaal forces. The carbon-carbon bond length in graphene is about 0.142 nm. Graphene has 2 atoms per unit cell. It is the thinnest and strongest material tested till now. It gives nonzero electrical conductivity even when charge concentration is zero. Its charge carriers 6 1 (electrons) can travel with Fermi velocity ( v F ~ 10 m s ) with very large mobility and zero effective mass. These particles are called Dirac fermions and obey the relativistic physics. Graphene creates a new branch of physics known as relativistic condensed matter physics. This peculiar property makes graphene as a new star of modern science and technology. Fig 1: Honeycomb structure of graphene [1] Fig 2: Electron microscope image 2. Existence is a miracle According to Landau and Peierls atoms in 2D crystals are displaced from its equilibrium position due to the thermal fluctuations [3,4] and this displacement is comparable with
the interatomic distance at finite temperature. Moreover experimentally it is prove that the melting temperature of thin films rapidly decreases with the decreasing thickness. So in a film when there exist near about 12 layers [5] it becomes unstable, so they should not exit. But in 2004, physicists Andre Geim and Kostya Novoselov from Manchester University, UK, extracted single layer of graphite [6] which is known as graphene from 3D graphite. They used top down approach starting from large graphite finally produced high quality graphene crystal. Electrons can travel thousand of interatomic distance 6 1 without scattering with a velocity of 10 m s. Fig 3: Atomic force microscopy picture of a graphene on top of an oxidized Si substrate [7]. 3. Mother of all graphitic materials Fig 4 Graphene is the building block for carbon materials of all other dimensions therefore it is known as the mother of all graphitic materials [8]. Graphite is obtained by the stacking of graphene layers. Diamond can be obtained from graphene under extreme pressure and
temperatures by transforming the 2-dimensional sp 2 bonds into 3-dimensional sp 3 bonds. Carbon nanotubes are synthesized from rolled up graphene. Fullerenes can also be obtained from graphene by modifying the hexagons into pentagons and heptagons in a systematic way (Fig 4). 4. Is it a metal or a semiconductor? Graphene has properties like both a normal metal and a semiconductor. Like metal it is strong (strongest material tested till today) in terms of young modulus and elastic stiffness. It can conducts electricity even better than copper. Usually for metals we require only one energy band to describe them and for semiconductor we need two energy bands (conduction and valance band). Graphene has two bands one for particle which is empty and other for antiparticles (holes) which is filled, but there is no gap between the two bands [9]. Hence we can say graphene is a semi metal or a zero gap semiconductor or a hybrid between a metal and a semiconductor (Fig 5). But in many applications where a large on and off current ratio is needed, this zero gap is become a drawback. So research is going on to generate gap between the two bands. There are many ways to generate the gap. Theoretically the simplest way to do this is like following. If we consider that the honeycomb lattice is made by two identical interpenetrating triangular sub lattices there will be no energy band gap and if two sub lattices are different (Fig 6), then a gap will be generated. Fig 5: Energy bands for different materials [9].
5. Some peculiar properties a. Massless charge particle Fig 6: Honeycomb lattice of graphene In ordinary metal or semiconductor the electronic energy can be written as, 2 2 * E = h k / 2M, where h = h / 2π, h is the Planck s constant, k is the wave vector and M * is the effective mass of the electron. But in case of graphene electrons are obeying a linear dispersion relation (i.e. the electron energy is linearly proportional to the wave vector, E = h k vf ) and behave as massless relativistic particles, called Dirac fermions [10,11]. Here v F is the Fermi velocity of electron in the graphene. This property implies that the speed of electrons in graphene is a constant, independent of momentum, like the speed of photons is a constant c. Recently it is found that the velocity of electrons in 6 1 graphene is about10 m s. This velocity is large but still 300 times slower than the velocity of light in vacuum c. Since the electrons are sluggish compared to the speedy photons they exchange when interacting, the physics of electron-electron interaction in graphene is different from that of photon-mediated interactions between fermions in quantum electrodynamics (QED). In graphene the interactions among electrons are 2 extremely strong and graphene s dimensionless coupling constant α G R = e / hvf 1 is 2 larger than the dimensionless coupling constant of QED, α = e / hc 1/ 137. The large difference between c and v F implies that the interacting electrons in a graphene sheet is not like the 2D version of QED. b. Giant mobility and lowest resistivity Graphene has a very high electron mobility at room temperature, with values of 15,000 cm 2 V -1 s -1 [12] and it can be increased upto 200,000 cm 2 V -1 s -1 at a carrier density of 10 20 cm -2. The corresponding resistivity of the graphene sheet would be 10-6 Ω cm, less than the resistivity of silver, the lowest resistivity substance known at room temperature. c. Non-zero conductivity with zero charge concentration Graphene exhibits a minimum conductivity of the order of the quantum unit e 2 /h when the carrier charge concentration is zero. But in case of ordinary system it is zero when the charge concentration is zero. The origin of this peculiar property is still unclear.
d. Anomalous quantum Hall effect Graphene shows very interesting behavior in the presence of a magnetic field at very low temperature [13], typically below 243 0 C. Graphene shows an anomalous quantum Hall effect with the sequence shifted by 1/2 with respect to the standard sequence. The quantum Hall effect is one the most remarkable phenomena in condensed matter physics discovered in the second half of the 20 th century. The basic fact characterizing quantum Hall effect is that the diagonal electric conductivity of a two-dimensional electron system in a strong magnetic field is vanishingly small σ xx 0, while the non-diagonal conductivity is quantized in multiples of e 2 2 /h : σ xy = p e / h, where p is an integer (the integral quantum Hall effect, IQHE) [14]. When p is a fractional number, it is known as fractional quantum Hall effect (FQHE). The authors have discussed both IQHE and FQHE in 2D electron gas briefly in ref. 15. In recent experiments, the quantum Hall effect is observed in graphene. It is found [16-18] that the Hall conductivity 1 σ = ± 4 2 xy e / h N +, where N is the Landau level index and the factor 4 accounts 2 for graphene s double spin and double band (valley) degeneracy. That is why; it is characterized as half-integer quantum Hall effect. The first plateau occurs at 2e 2 / h. This anomalous QHE is the direct evidence for Dirac fermions in graphene. 6. Other properties The near-room temperature thermal conductivity of graphene lies between (4.84±0.44) 10 3 to (5.30±0.48) 10 3 Wm -1 K -1 [19] which is 100 times larger than the graphite. Till today graphene appears as the strongest material ever tested in nature. Measurements have shown that graphene has a breaking strength 200 times greater than steel [20]. Its spring constant lies in the range 1-5 N/m and the Young's modulus is 0.5 TPa [21], which differs from that of the bulk graphite. These high values make graphene very strong and rigid. 7. Applications Graphene has 2D structure so its entire volume is exposed to the surrounding. Hence it can be used as a very good gas detector and also can be used to make excellent transistor which can run at higher frequency and more efficient than silicon transistor. It is due to the fact that in graphene the charge carriers move very fast. Graphene can be used as a coating against acid and alkalis such as hydrofluoric acid and amonia. The Massachusetts Institute of Technology built an experimental graphene hip known as frequency multiplier which can produce multiple of the incoming frequency. It is expected that the graphene microprocessor can appear within 20 years [12]. Graphene powder may also be used in battery.
8. Conclusions Graphene is a monoatomic layer of graphite with carbon atoms arranged in a twodimensional honeycomb lattice configuration. The electronic structure of graphene can be modelled by two-dimensional massless relativistic fermions. This property gives rise to numerous applications both in applied science and in theoretical physics. Graphene research is one of the fastest growing areas in material science, but it is still a young field. There are many challenges and opportunities for investigation, because graphene is not a standard solid state material. It is a new star in material science. Graphene has some peculiar properties which is not matched with the ordinary metal and semiconductor. So it is necessary to establish a new generalized theory for it. Graphene has potential for serving as an excellent electronic material that can be used in place of silicon for making ultrafast and stable transistors. It is considered as a promising candidate for electronics and spintronics applications. It provides a bridge between condensed matter physics and quantum electrodynamics. References 1. wikipedia.org/wiki/graphene. 2. S. Sahoo and A. K. Dutta, Emerging Science, 2(2), 16 (2010). 3. R. E. Peierls, Ann. I. H. Poincare, 5, 177 (1935). 4. L. D. Landau, Phys. Z. Sowjetunion, 11, 26 (1937). 5. J. A. Venable, G. D. T. Spiller and M. Hanbuken, Rep. Prog. Phys. 47, 399 (1984). 6. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science, 306, 666 (2004). 7. M. I. Katsnelson, Materialstoday, 10(1-2), 20 (2007). 8. C. Srinivasan. Current Science, 92, 1338 (2007). 9. Antonio H. Castro Neto, Materialstoday, 13(3). 1 (2010) 10. S. Sahoo and S. Das. Indian J. pure & Appl. Phys. 47, 186 (2009). 11. S. Sahoo and S. K. Sahoo, Indian J. Sci. & Tech., 2(12), 74 (2009). 12. A. K. Geim and K. S. Novoselov, Nature Mater., 6, 183 (2007). 13. B. Basu, Science Reporter, 45(7), 33 (2008). 14. V. P. Gusynin and S. G. Sharapov, Phys. Rev. Lett., 95, 146801 (2005) [arxiv: cond-mat/0506575]. 15. S. Sahoo and M. Goswami, IAPT Bulletin, 24(12), 388 (2007). 16. A. K. Geim and A. H. MacDonald, Physics Today, 60, 35 (August 2007). 17. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, Nature, 438, 197 (2005). 18. Y. Zhang, Y. W. Tan, H. L. Stormer, and P. Kim, Nature, 438, 201 (2005). 19. A. A. Balandin et al., Nano Letter, 8(3), 902 (2008). 20. C. Lee et al., Science, 320(5887), 385 (2008). 21. I. W. Frank, D. M. Tanendaum, A. M. Van der Zande and P. L. McEuen, J. Vac. Sci. Technol. B25, 2558 (2007).