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ElelcetrcontircoTrnanicspoTrtrParonpesrtpieosorftFPewr-oLapyeerGrtriaephsenoefMaFteriawls-LayerGrapheneMaterials 1513 80 70 400 300 3 bg V (V) 120 200 E F 100 ) ))V TΩe 100 90 10 / E 2 ( ΩK( m( F 80-10 0 Hqsε 70 -30 R EF δ -100 R 50 1 30 -200 -300 0 -40 -20 0 20 40 -4 0 4 8 a) b) Vtg(V) c) Vbg (V) Fig. 8. (a) shows the Hall coefficient as a function of back gate voltage VBG (open circles) for a fixed perpendicular magnetic field of 9T at 50mK for a trilayer graphene device. The red curve is a fit. The insets depict schematically the position of the Fermi level (εF) at different values of VBG. The graph in (b) shows a 4-terminal resistance measurement of the trilayer device in the inset of (c) versus top gate and for different values of back gate. (c) shows the electric field dependence of the band overlap δε for the same trilayer device. All panels are adapted from Craciun et al. (2009) of few-layer graphene can be achieved on the basis of zone folding of the 3D graphite bands (Mak et al. (2010)). However, so far the electron transport properties of these thicker few-layer graphene materials is largely unexplored, preventing us from identifying the best suited thickness of few-layer graphene for a given application. 4. Landau level structures in few-layer graphene When the charge carriers travelling in few-layer graphene experience a perpendicular magnetic field (B), their trajectories are bent due to the Lorentz force. In the quantum regime, these cyclotron orbits give rise to discrete energy levels known as Landau levels. The precise sequence of these Landau levels reflects the nature of the charge carriers in the few-layer graphene under consideration. In particular, the Landau levels sequence for single layer graphene is intimately related to the mass-less nature of the graphene Dirac fermions (see Eq. 8) and it is very different from what is known in conventional two-dimensional electron gases. The Landau level energies of monolayer graphenes are given by (McClure (1956)): En = h ̄ ωBsgn(n) |n| (n = 0, ±1, ±2, ..), (13) with h ̄ ωB = 2πv2F eB. Each level is four fold degenerate, that is spin and valley degenerate. Due to the linear energy dispersion (see Eq. 8), the energy spacing between the Landau √ levels is proportional to B rather than B as in usual two dimensional systems. At a fixed value of external magnetic field the energy gap between Landau levels in graphene are much larger than the corresponding gaps opened in other 2DEGs (for B=1T the energy gap between n=0 and n=1 in graphene is approximately 35 meV). Another unique feature of the graphene Landau level spectrum is the existence of a zero energy level. This causes the half-integer quantization of Hall conductivity per spin and valley, and is also responsible for the huge diamagnetic susceptibility characteristic of this system. Specifically, the orbital susceptibility has a strong singularity at band touching point (Dirac point), which at zero temperature is expressed as a function of Fermi energy εF as: 2μm A gate gate gate 60 50 40 30 -0.2 -0.1 0.0 0.1 0.2 0.3 Eext(V/nm)PDF Image | GRAPHENE SYNTHESIS CHARACTERIZATION PROPERTIES
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