Transport properties of junctions and lattices via solvable model

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

21. 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.