4a-NST-tunneling
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Quantum Tunnelling Prof. Dr. Abdul Majid Friday, July 1, 2016 Prof. Dr. Abdul Majid Friday, July 1, 2016 Is Size mater for Physical Properties ? : Quantum Confinement Quantum Confinement effects in InP nanowires. Figure shows the photoluminescence of InP nanowires as a function of wire diameter, thereby providing direct information on the effective bandgap. As the wire diameter of an InP nanowire is decreased so that it becomes smaller than the bulk exciton diameter of 19 nm, quantum confinement effects set in, and the band gap is increased. This results in an increase in the PL peak energy. The smaller the effective mass, the larger the quantum confinement effects. When the shift in the peak energy as a function of nanowire diameter Fig. is analysed using an effective mass model, the reduced effective mass of the exciton is deduced to be 0.052 m0, which agrees quite well with the literature value of 0.065 m0 for bulk InP. Photo-luminescence of InP nanowires of varying diameters at 7K((b)and (d)) and room temperature ((a) and (c)) showing quantum confinement effects of the exciton for wire diameters of less than 20 nm Prof. Dr. Abdul Majid Although the line widths of the PL peak for the small-diameter nanowires (10 nm) are smaller at low temperature (7K), the observation of strong quantum confinement and bandgap tunability effects at room temperature are significant for photonics applications of nanowires. Friday, July 1, 2016 Tunnelling (Tunneling) Quantum tunneling Quantum tunneling (or tunneling) is the quantum-mechanical effect of transitioning through a classically-forbidden energy state. Prof. Dr. Abdul Majid Friday, July 1, 2016 Tunnelling (Tunneling) Quantum tunneling Quantum tunneling (or tunneling) is the quantum-mechanical effect of transitioning through a classically-forbidden energy state. Prof. Dr. Abdul Majid Friday, July 1, 2016 Tunnelling (Tunneling) Quantum tunneling Quantum tunneling (or tunneling) is the quantum-mechanical effect of transitioning through a classically-forbidden energy state. Prof. Dr. Abdul Majid Friday, July 1, 2016 Tunnelling Quantum tunneling through a barrier. At the origin (x=0), there is a very high, but narrow potential barrier. A significant tunneling effect can be seen. Prof. Dr. Abdul Majid Friday, July 1, 2016 Tunnelling An electron wavepacket directed at a potential barrier. Note the dim spot on the right that represents tunnelling electrons. Prof. Dr. Abdul Majid Friday, July 1, 2016 Tunnelling Prof. Dr. Abdul Majid Friday, July 1, 2016 Tunnelling Scanning Tunneling Microscope (STM) nm 55 nm xenon atoms on on aa nickel nickelsurface surface xenon atoms moving individual atoms around one by one D.M. Eigler, E.K. Schweizer. Positioning single atoms with a STM. Nature 344, 524-526 (1990) Prof. Dr. Abdul Majid Friday, July 1, 2016 Tunnelling Nobel Prize in Physics (1986) "for his fundamental work in electron optics, and for the design of the first electron microscope" Ernst Ruska 1/2 of the prize "for their design of the scanning tunneling microscope" Gerd Binnig Heinrich Rohrer 1/4 of the prize 1/4 of the prize Federal Republic of Federal Republic of Germany Fritz-Haber-Institut der Max-PlanckGesellschaft Berlin, Federal Republic of Germany Prof. Dr. Abdul Majid Germany IBM Zurich Research Laboratory Rüschlikon, Switzerland Switzerland IBM Zurich Research Laboratory Rüschlikon, Switzerland Friday, July 1, 2016 Tunnelling Consider particles of mass m in a quantum state approaching, from the left hand side, a potential barrier of height πΌ ππ π½ and length L. See Figure The state of the particles is that of a Gaussian probability distribution with momentum centered about kinetic energy πΈ = β2 π 2 . 2π Although classically, the particles do not have enough energy to enter the barrier region 0<x<L, quantum mechanically there is a non-zero probability that the particles will appear on the other side of the barrier. This phenomenon is referred to as ``Quantum Tunneling''. Prof. Dr. Abdul Majid Friday, July 1, 2016 Tunnelling Tunnel Effect: - particle with kinetic energy E strikes a barrier with height U0 > E and width L - classically the particle cannot overcome the barrier - quantum mechanically the particle can penetrated the barrier and appear on the other side - then it is said to have tunneled through the barrier examples: Prof. Dr. Abdul Majid - emission of alpha particles from radioactive nuclei by tunneling through the binding potential barrier - tunneling of electrons from one metal to another through an oxide film - tunneling in a more complex systems described by a generalized coordinate varying in some potential Friday, July 1, 2016 Tunnelling Approximate Result: - the transmission coefficient T is the probability of a particle incident from the left (region I) to be tunneling through the barrier (region II) and continue to travel to the right (region III) π = π −2π2 πΏ with π2 = 2π(π−πΈ) β - depends exponentially on width of barrier L and the difference between the particle kinetic energy and the barrier height (U0-E)1/2 and mass of the particle m1/2 example: - An electron with kinetic energy E = 1 eV tunnels through a barrier with U0 = 10 eV and width L = 0.5 nm. What is the transmission probability? π = 1.1 × 10−7 - the probability is small, even for a light particle and a thin barrier - but it can be experimentally observed and used in devices Prof. Dr. Abdul Majid Friday, July 1, 2016 Tunnelling Sketch of calculation of tunnel rate: - Schrödinger equation outside of barrier (regions I and III) β2 π2 2π ππ₯ 2 ππΌ + πΈ π πΌ = 0 same for ψIII β2 π 2 π + πΈ ππΌπΌπΌ = 0 2π ππ₯ 2 πΌπΌπΌ - has solutions ππΌ = π΄π ππ1π₯ +π΅π −ππ1 π₯ ππΌπΌπΌ = πΉπ ππ1 π₯ +πΊπ −ππ1π₯ with - incoming wave ππΌ+ = π΄π ππ1 π₯ - transmitted wave ππΌπΌπΌ+ = πΉπ ππ1 π₯ Prof. Dr. Abdul Majid π1 = 2ππΈ β π =β= 2π π reflected wave ππΌ− = π΅π −ππ1 π₯ incoming flux of particles with group velocity vI+ π = ππΌ+ 2 π£πΌ+ Friday, July 1, 2016 Tunnelling Transmission: - probability T π= ππΌπΌπΌ+ ππΌ+ 2π πΌπΌπΌ+ 2π πΌ+ = β±πΉ ∗ ππΌπΌπΌ+ π΄π΄∗ ππΌ+ - ratio of flux of transmitted particles to incident particles barrier region: - Schrödinger equation for region II β π2 − π + (U − E) ππΌπΌ =0 2π ππ₯ 2 πΌπΌ - solution for U > E ππΌπΌ = πΆπ −π2 π₯ + π·π π2 π₯ With π2 = 2π(π−πΈ) β - exponentially decaying or increasing wave (no oscillations) - does not describe a moving particle - but probability in barrier region is non-zero Prof. Dr. Abdul Majid Friday, July 1, 2016 Tunnelling boundary conditions: - at left edge of well (x = 0) ππΌ = ππΌπΌ πππΌ ππ₯ ; = πππΌπΌ ππ₯ = π ππΌπΌπΌ ππ₯ at right edge of well (X = L) ππΌπΌ = ππΌπΌπΌ - ; πππΌπΌ ππ₯ solve the four equations for the four coefficients and express them relative to A (|A|2 is proportional to incoming flux) - solution Prof. Dr. Abdul Majid Friday, July 1, 2016 Tunnelling Transmission coefficient: equations - find A/F from set of boundary condition π΄ πΉ Prof. Dr. Abdul Majid = 1 π π + ( 2 2 4 π1 − π1 ) π2 π ππ1 +ππ2 πΏ + 1 π π − ( 2 2 4 π1 Friday, July 1, 2016 Tunnelling Transmission coefficient: π΄π΄∗ ππΌπΌπΌ+ T= = πΉπΉ ∗ ππΌ+ 16 4+ π2 π1 2π 2π2 πΏ With π2 = 2π(π−πΈ) β and π1 = Therefore π2 π1 π≈ π 2 = π−πΈ πΈ 2π2 πΏ - T is exponentially sensitive to width of barrier - T can be measured in terms of a particle flow (e.g. an electrical current) through a tunnel barrier - makes this effect a great tool for measuring barrier thicknesses or distances for example in microscopy applications Prof. Dr. Abdul Majid Friday, July 1, 2016 2ππΈ β
