Atomically defined tips in scanning probe microscopy - McGill Physics

2.2 METAL NANOPARTICLES AS QUANTUM DOTS 19a Gateb Gate cGateq Gq GC GC G- q G- q GNPNPq TV Bq T+qNPV BC T- q T- q TC SELFC TSourceelectrodeSourceelectrodeSourceelectrodeFigure 2.4: (a) A single-electron box that consists of a nanoparticle (NP) separated from asource electrode by a tunnel barrier. Another electrode that is is only capacitively coupled,called the gate, can be used to apply an external electric field and induce charging of thenanoparticle. The polarization charges on the tunnel junction, q T , and the gate capacitor,q G , can take continuous values. (b) If a charged defect happens to be close to the nanoparticle,it will polarize its surface by creating an image charge. (c) The self-capacitance C SELF ,understood as a capacitance with reference to an infinitely distant grounded plane, can beconceptually added to the capacitance of either the tunnel barrier or the gate by groundingthe source electrode or the gate, respectively.related to the size of the nanoparticle. If the size is sufficiently small, E add canexceed the thermal energy, k B T , and cause the number of electrons on the nanoparticleto be fixed for a given gate bias voltage. Additional charging of the nanoparticlewill then require a sufficiently high bias applied to the gate in order toattract an electron to the nanoparticle and increase the the number of electrons byone. This effect is known as the Coulomb blockade. Depending on the electricfield applied by the gate, an electron can be also removed from the nanoparticle.In order to observe such single-electron tunneling there is also an additionalcondition to be met. Observation of Coulomb blockade effects requires that theelectron wave function is localized on the nanoparticle.Using the uncertaintyprinciple one can obtain a requirement for the resistance of the tunnel barrier thathas to be larger than the resistance quantum h/e 2 (h is the Planck’s constant) 3 . Thisallows for the charge to be well localized on the nanoparticle.For a two-terminal capacitor with capacitance C, the charging energy is e22C .3. From ∆E∆t > h, where ∆E ∼ e 2 /C is the characteristic energy of adding one electron tothe nanoparticle with capacitance C, and ∆t = RC is the characteristic time of this process, whereR is the resistance of the tunnel barrier, then e 2 R > h.