Nano by Numbers M. Stopa Outline 1. Introduction to semiconductor quantum dots. 2. The Coulomb blockade and Coulomb oscillations: manipulating the charge on a quantum dot.
Download ReportTranscript Nano by Numbers M. Stopa Outline 1. Introduction to semiconductor quantum dots. 2. The Coulomb blockade and Coulomb oscillations: manipulating the charge on a quantum dot.
Nano by Numbers M. Stopa Outline 1. Introduction to semiconductor quantum dots. 2. The Coulomb blockade and Coulomb oscillations: manipulating the charge on a quantum dot one electron at a time. 3. Double quantum dots: honeycomb stability diagram. 4. Example from fluid dynamics: non-linearity and the importance of computation. 5. Electronic structure of quantum dots: selfconsistent solution of Schrödinger and Poisson equations. 6. Arrays of quantum dots, basic results. 7. Another nanoscale system. What is an artificial atom ? How do you make a semiconductor quantum dot ? crystal growth and surface lithography Quantum dots: growth and lithography V(z) air or vacuum AlGaAs +++++++++ 2DEG Donor layer (e.g. silicon) 1000 Angstroms POINT: High mobility several microns GaAs Conduction band profile substrate 2DEG Two Dimensional Electron Gas E-beam lithography plunger Apply negative voltage to gates air or vacuum AlGaAs +++++++++ 2DEG GaAs substrate 2DEG Later: capacitance model for quantum dot energy Vertical quantum dots GaAs AlGaAs Vg Seigo Tarucha and co-workers, circa 1997. What are the Coulomb blockade and Coulomb oscillations ? Inducing charge with gate Vary “plunger” gate voltage Effective continuous positive charge is induced by gate “helium” (N=2,N=3) Like increasing nuclear charge ! (N=2,N=3) (N=2,N=3) (N=1,N=2) (N=1,N=2) (N=1,N=2) N.B. “small” sourcedrain bias At some specific voltage, we have a bi-stable point where either N=2 or N=3 is allowed energetically Coulomb oscillations The gate induces a fictitious charge ρ ρ = CgVg The total charge is the N electrons plus this fictitious charge Q = ρ - Ne The total energy is just E = Q2/2C Where C is the dot self-capacitance Q=0 here N 3 2 1 0 Coulomb diamond: beyond linear source-drain wyala Small source-drain bias s X d Increase source-drain bias s d (N,N+1) wyala double dots series one current path parallel two current paths double dot circuit Vg1 Vg2 effective charges Q1 C1Vg1 N1e 1 N1e Q2 C2Vg 2 N2e 2 N2e Energy of double dot U Q1 , Q2 1Q 2Q Q1Q2 2 1 First: inter-dot capacitance set to zero wyala 2 2 Depends on capacitance between dots Cint Vg1 “stability diagram”(N(n1,1,Nn22)) Stability diagram Going across diagonal changes total number NT= N1+ N2 by two Vg2 Double dot “honeycomb stability” diagram Vg1 NT 3 ? 2 1 0 For series double dot, (linear) current only flows at triple points Current through a series double dot Inter-dot capacitance increases from A to F. noninteracting (1,1) (0,0) C. Livermore et al. Science 274, 1332 (1996) dots coupled into a single dot Q. Can you make a “vertical” double dot and if so, what is it called ? Quantum Big Mac Computational Nanoscience: parallel with fluid dynamics UL Re ratio of inertial effects to viscous effects in the flow Slides courtesy of Howard Stone Osborne Reynolds (1842–1912) Increasing complexity of flows with increasing Reynolds number: flow past a circular cylinder Re=1 Re=26 ~ laminar Re=41 Re=10 Re=2000 turbulence Point: transition to turbulence computed not “derived.” Reference: Van Dyke, Album of Fluid Motion Why is computational fluid dynamics important ? (First, fluid dynamics of enormous practical importance). Transition from “laminar flow” to turbulence occurs in Couette flow and Poiseuille flow when Reynolds number becomes high enough. Observed in experiment and verified by numerous numerical studies. Not “understood” according to any theory (or, more exactly, understood differently according to many theories). u 2 u.u g -p u t Navier-Stokes equation is non-linear Basic idea of non-linearity: a quantity depends on itself. “Most” equations studied in physics are linear, e.g.: Schrödinger equation Poisson equation Maxwell’s equations my focus In electronic structure, non-linearity arises from the ability of the electron to influence its surroundings (recall introduction): the potential in the gap changes when the electron enters due to polarization response of the medium. N.B. the electron does not directly interact with itself. Electronic structure of lateral semiconductor heterostructures: quantum dots Basic real atom Artificial atom Self-consistency = non-linearity 2 e i A e r Vxc r n r n n r c r 2 4e r N 2 r r leads r donorsr n1 Nothing new about self-consistent electronic structure. What is new is the ability to manipulate nanometer scale objects like quantum dots. Some introduction…. Electronic structure of lateral semiconductor heterostructures Inputs: 1. Wafer profile 2. Donor profile/concentration 3. Gate pattern/voltages 4. Temperature 5. Magnetic field band offset effective mass dielectric constant g factor spin-orbit coupling Outputs: 1. electrostatic potential (r) 2. charge density (r) 3. for a confined region (i.e. a dot) 1. eigenvalues Ei 2. eigenfunctions i 3. tunneling coefficients i 4. total free energy F(N,Vg,T,B) EXAMPLE I Chaotic quantum dot: eigenfunctions and periodic orbits Note: here treating donor layer as “jellium.” Periodic orbits calculated by solving Hamilton’s equations in the effective 2D selfconsistent potential. EXAMPLE 2,3,4 Examples of SDFT results Triple dot rectifier Single photon detector 1 M. Stopa, PRL 88, 146802 (2002) 3 Influence of disorder on statistics of quantum dot level spacings 2 Evolution of magnetic field induced compressible and incompressible strips in a quantum dot S. Komiyama et al., in preparation M. Stopa, Superlattices and Microstructures, 21, 493-500 (1997). EXAMPLE 5 Modelling an STM tip probing a 2DEG heterostructure Vacuum/AlGaAs Surface gate pattern donor layer 50 nm 50 nm AlGaAs/GaAs 1 aB*=10 nm, 1 Ry*=5.8 meV More complicated still Array of quantum dots Vg=0 for all “gates” Vth=e/2Ceff Comparison with gedanken experiment What’s this ? Transport in ion channels protein 2 H O 80 2 Basic channology • Channels are proteins • ~2000 types • control flow of Na+, K+, Cl-, Ca+ • channel types selective • channels open and close (a.k.a. gating) • measured in the laboratory • complex noise behavior Open questions • Channel gating mechanism ? • Microscopic basis for channel selectivity ? • Sources of current noise ? • Folding or bending in channel protein ? • Molecular structure ? (cf. Roderick MacKinnon et al.) Course notes from Univ. Calif. Nice intro to at San Francisco channels http://www.keck.ucsf.edu/~hackos/channels.htm