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Chaotic Dirac Billiard in Graphene Quantum Dots L. A. Ponomarenko, et al. Science 320, 356 (2008) Presented by Suprem R. Das PHYS570X (02/25/2009). Important findings of the paper: 0D graphene Nanoelectronics/transport study All graphene SET/QD first report
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L. A. Ponomarenko, et al. Science 320, 356 (2008)
Suprem R. Das
0D graphene Nanoelectronics/transport study
All graphene SET/QD first report
Both SET and QD regime studied
Graphene QD Dirac fermion statistics study (Q cap 1/D for D > 100 nm but 1/D2 for smaller dots, quantum chaos and neutrino billiards)
Very small Dot (~ 1nm) Room temperature Graphene QD transistor action – possibility of graphene based molecular electronics (top down approach)
Size ~ 100 nm or less (> 100 nm: SET)
Semiconducting materials, small molecular clusters, small metallic grains – all having similar transport properties
# of free e in the dot – very small (~ 1 to few hundred)
Few characteristics of QDs (otherwise called artificial atoms)
Coupling of S/D contacts with the iland through particle exchange and coupling of G with the iland through electrostatic or capacitive coupling
Formation of a well defined iland, such that total charge inside the iland is an integer (Ne, say), when no coupling to S/D.
Tunneling to S/D => N adjusts itself until the energy of the whole circuit gets minimized, charge on the iland suddenly changes by the quantized amount e
Change in the Electrostatic potential energy/ Coulomb energy of the iland is called charging energy EC = e2/C, where C is the capacitance of the iland.
EC (charging energy) becomes important when EC > kBT
The barriers are sufficiently opaque such that the electrons are located either in S,D, or the CI => Quantum fluctuations in N due to tunneling through the barriers << 1/(time scale of measurements), where time scale of measurement = e/current.
Above requirement sets a lower limit for the tunnel resistances Rt of the barriers
t = charging/discharging time for the dot = RtC
E t > h => (e2/C).(RtC) > h
=> Rt >> h/e2, the resistance quantum (25.813k)
If Rt ~ h/e2, then energy uncertainty will not be smaller than charging energy to see the effect.
Achieved by weakly coupling the dot to S/D (using side gates)
Achieved by making the dot smaller, since C R, the radius of dot (the dot can be squeezed by applying potential to the central pair of gates)
Note of T regimes:
Ref: L. P. Kouwenhoven, C.M. Marcus et al., Proc of Adv Study Institute on Mesoscopic Electron Transport
Tunneling changes the iland’s charge by an integer while the gate voltage VG induces an effective continuous charge q=CGVG
If we sweep VG, the build up of the induced charge will be compensated in periodic intervals by tunneling of discrete charges onto the dot
This competition between the continuously induced charge and discrete compensation leads to Coulomb Oscillations (usually, plotted between conductance G (= I/VDS) in units of e2/h vs VG)
Plot of I vs VDS for particular VG is called Coulomb staircase. A new current step occurs at a threshold voltage (~e2/C) at which an extra electron is energetically allowed to enter into the iland
M. Kastner, MIT, Physics Today article
Patterned on Si/SiO2 (300 nm) using EBL
D > 100 nm: Conventional SET, characterized by periodic Quolomb Blockade peaks
100 nm < D < few nm: A QD having quantum confinements characterized by strong non-periodic peaks, random peak spacings ---statistics fitted to chaotic neutrino billirds
Short constrictions ~ few nm: Conductive with confinement gap of up to 0.5eV – possibility of top down approach of graphene molecular electronics
by regions of zero conductance
(D = 250 nm, T = 0.3K, Measurements around VG~ 15V)
High T: peaks become broader and overlap, gradually
transforming into CB oscillations
(D = 250 nm, T = 4K, Measurements over large VG)
D = 40 nm
C: Variation of VG over 140 n.n. peaks obtained over a wide range of VG for 40 nm dot biased at zero Vb
E = EC + E and we know Ec 1/C(geomet) and E C(Quantum)
And Geometrical Cap D and Quantum capacitance 1/D => E or < VG> 1/D
But analysis of Q.Cap. Vs D shows the following
Blue (solid line): <VG> 1/D : fits well above 100nm but not below
Red (dashed line): <VG> 1/D2
Green (dotted): <VG> 1/D with = 1.25: Fits well in all D
Another Observation: Notable change in shape of the spectral distribution below D ~ 100nm
Statistical analysis of peak spacing for four QDs with different D