1 / 22

Quantum Transport in Periodic Quasi-2D Lattices: Solvable Model

Explore transport properties of electrons in quantum networks created on semiconductor surfaces, including quantum wires and wells. The model considers the splitting of variables to modify equations on wires and spectra on infinite wires. The study focuses on the Silicon-Boron two-dimensional structure and its high mobility of charge carriers. Spectral problems with partial quasi-periodic boundary conditions on the lattice elements are examined, leading to dispersion relation analyses. The interaction between boron sublattices through tunneling is investigated in double periodic quasi-2D lattices.

hope
Télécharger la présentation

Quantum Transport in Periodic Quasi-2D Lattices: Solvable Model

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Transport properties of junctions and lattices via solvable model Nikolai Bagraev (impurity.dipole@pop.ioffe.rssi.ru), A.F. Ioffe Physico-Technical Institute,St. Petersburg, Russia Lev Goncharov (Lev.goncharov@mail.ru), Department of Physics of St. Petersburg State University, Russia Gaven Martin (G. J. Martin@massey.ac.nz), New Zealand Institute of Advanced Study, New Zealand Boris Pavlov (pavlovenator@gmail.com) Department of Physics of St. Petersburg State University, Russia Adil Yafyasov (yafyasov@bk.ru), Department of Physics of St. Petersburg State University, Russia

  2. Quantum networks Quasi-1D quantum wires ωiwidth δ • Quantum network constructed on the surface of semiconductor as a union of quantum wells Ωk and quantum wires ωi • Transport of electrons through the network described by the Schrödinger-type equation δ Quantum wells Ωk

  3. Quasi-one-dimensional quantum wires Splitting of variables allows to modify equation on wires Spectra on the semi-infinite wires

  4. Consider the Fermi level inside the first spectral band Δ1 • Assume the temperature to be low, so that essential spectral interval ΔT is inside Δ1 Δ3 Δ2 Δ1 Λ4 Λ3 Λ2 Λ1 λF ΔT

  5. Dirichlet-to-Neumann map Ω Γ2 Γ1 Γ3

  6. Scattering matrix and intermediate DN-map Exact finite-dimensional equation on scattering matrix • Intermediate DN-map DNΛ is a finite-dimensional DN-map of Schrödinger problem with partial-zero boundary condition: Γ2 Γ1 Γ3

  7. Singularities of intermediate DN-map DNΛ may have singularities of two types: • Inherited singularities from DNΓ • Zeros of DN--+K- But singularities of first type compensate each other

  8. Silicon-Boron two-dimensional structure - no boron • Experiment shows high mobility of charge carriers on double-layer quasi-two-dimensional silicon-boron structure • Boron atoms at high concentration form sublattice in silicon matrix - B++B-

  9. Model description Consider the boron sublattice as a periodic quantum network • Elements of the network are connected by aid of rather long and narrow links

  10. Model description • Separation of variables and cross-section quantization on the links generate infinite number of spectral channels • Only finite number of spectral channels is open (oscillating solutions of Schrödinger equation on the links) • Closed channels (exponentially decreasing solutions of Schrödinger equation on the links) could be omitted • Matching on the open channels only allow to reduce the infinite-dimension matching problem to finite-dimensional one

  11. Statement of periodic spectral problem Intermediate problem for single element of the lattice

  12. Spectral problem with partial quasi-periodic boundary condition on the pairs of opposite slots • Now we can exclude links and make respective changes in boundary conditions • Boundary data and boundary currents then are connected by intermediate partial DN-map DNΛ (but not traditional partial DN-map)

  13. Spectral problem with partial quasi-periodic boundary condition on the pairs of opposite slots Γ2+ Excluding links and correct boundary conditions Γ1- Γ1+ Γ2-

  14. Assumption • To simplify following calculations consider a case, when only one spectral channel in cylindrical links is open, so • That simplify above boundary conditions as:

  15. Dispersion relation • Connect boundary data and boundary currents with DNΛ

  16. Double periodic quasi-2D lattice • Assume, that two boron sublattices interact by means of tunneling through the slot Γ0 Ωu is a period of first sublattice and Ωd is a period of the second one

  17. Double-lattices quasi-periodic boundary conditions • These conditions impose a system of homogeneous linear equations on • And we can note the condition of existence of non-trivial Bloch functions

  18. Dispersion relation And dispersion relation is

  19. Dispersion relation • If β→∞, linear system splits in two independent blocks • If β is finite, then intersection of terms transforms to quasi-intersection

  20. N.T. Bagraev, A.D. Bouravleuv, L.E. Klyachkin, A.M. Malyarenko, V.V.Romanov, S.A. Rykov: Semiconductors, v.34, N6, p.p.700-711, 2000.

More Related