Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
  • Editors' Suggestion

Splitting valleys in Si/SiO2: Identification and control of interface states

Amintor Dusko, A. L. Saraiva, and Belita Koiller
Phys. Rev. B 89, 205307 – Published 13 May 2014
PDFHTMLExport Citation
Abstract
Authors
Article Text
  • INTRODUCTION
  • FORMALISM
  • INTERFACE STATE ENERGY AND LOCALIZATION…
  • ELECTRIC FIELD BOUND STATES
  • FIELD CONTROL OF THE SPLITTING
  • CAPACITANCE AND INTERFACE STATES
  • DISCUSSIONS AND CONCLUSIONS
  • ACKNOWLEDGEMENTS
  • APPENDICES
    • References

    Abstract

    Interface states in a silicon/barrier junction break the silicon valley degeneracy near the interface, a desirable feature for some Si quantum electronics applications. Within a minimal multivalley tight-binding model in one dimension, we inspect here the spatial extent of these states into the Si and the barrier materials, as well as favorable conditions for its spontaneous formation. Our approach—based on Green's-function renormalization-decimation techniques—is asymptotically exact for the infinite chain and shows the formation of these states regardless of whether or not a confining electric field is applied. The renormalization language naturally leads to the central role played by the chemical bond of the atoms immediately across the interface. In the adopted decimation procedure, the convergence rate to a fixed point directly relates the valley splitting and the spread of the wave function, consequently connecting the splitting to geometrical experimental parameters such as the capacitance of a two-dimensional electron gas—explicitly calculated here. This should serve as a probe to identify such states as a mechanism for enhanced valley splitting.

    • Figure
    • Figure
    • Figure
    • Figure
    • Figure
    • Figure
    • Figure
    3 More
    • Received 22 October 2013
    • Revised 31 January 2014

    DOI:https://doi.org/10.1103/PhysRevB.89.205307

    ©2014 American Physical Society

    Authors & Affiliations

    Amintor Dusko, A. L. Saraiva, and Belita Koiller

    • Instituto de Física, Universidade Federal do Rio de Janeiro, Caixa Postal 68528, 21941-972 Rio de Janeiro, Brazil

    Article Text (Subscription Required)

    Click to Expand

    References (Subscription Required)

    Click to Expand
    Issue

    Vol. 89, Iss. 20 — 15 May 2014

    Reuse & Permissions
    Access Options
    Author publication services for translation and copyediting assistance advertisement

    Authorization Required

    Log In
    Other Options
    ×

    Images

    • Figure 1
      Figure 1

      Schematic representation of the tight-binding one-dimensional model for the Si/barrier interface. The labels A, B, and I refer to Si, barrier, and interface regions, respectively. Dark sites are labeled by s=−∞,...,−2,−1 and represent Si occupation, while light sites indicate the effective barrier species, at s=1,2,...,∞. The junction A – B corresponds to bond (−1) – (1), and the I region includes the range −2≤s≤2. The label s=0 is discarded so that A⟷B symmetry is obtained by changing −s⟷s along the decimation procedure (see Appendix pp1).

      Reuse & Permissions
    • Figure 2
      Figure 2

      Diagrammatic structure of the renormalized hoppings explicitly appearing in the formalism after n≥1 decimation steps. At each stage, alternate dimers are projected out from Dyson's equations (see Appendix pp1). For the original chain with hoppings up to second nearest neighbors, the decimated configurations involve hoppings ranging from first up to fourth neighbors.

      Reuse & Permissions
    • Figure 3
      Figure 3

      (a) Energy dispersion E(k) for the bulk materials as indicated, with a=aSi (aSiO2) for the dispersion of Si (SiO2). (b) Local density of states (LDOS) in bulk Si and oxide. (c) LDOS at the interface sites with α=0.5. (d) LDOS at the interface sites with α=1.0: here an interface state splits from the lower band into the gap, as indicated by the arrow. The vertical dotted lines mark the Van Hove singularities of the one-dimensional tight-binding model for bulk Si and bulk oxide parameters.

      Reuse & Permissions
    • Figure 4
      Figure 4

      Energy eigenvalue of the interface state as a function of the interpolation parameter α. We identify the valley splitting as the energy gap between the two lowest-energy eigenstates, i.e., the band edge and bound state eigenvalue.

      Reuse & Permissions
    • Figure 5
      Figure 5

      Properties of the intrinsic interface state as functions of the valley splitting (ΔVS). (a) Electronic probability density (|Ψs|2) at the Si (s=−1) and barrier (s=1) junction sites. (b) Localization length (ℓ) into Si (s≤−1) and barrier (s≥1) regions. The inset shows length scales characteristic of the TS and FH states (see Sec. 4) as a function of ΔVS.

      Reuse & Permissions
    • Figure 6
      Figure 6

      Approximate electronic density of the TS state extracted from the envelope-function asymptotic behavior along the chain as a function of the energy of the ground state (ΔVS), where z≤−1 corresponds to A sites (Si), and z≥1 corresponds to B sites (barrier). Values for the interface sites in Si and the barrier were obtained from direct calculation, and we assume an exponential decay beyond these sites, which should apply to |s|≫1.

      Reuse & Permissions
    • Figure 7
      Figure 7

      Electric field dependence of the valley splitting for different values of α, as indicated. Linear fits for each α are also given. Supercell: 67.9 nm Si layer adjacent to a 6.8 nm oxide layer.

      Reuse & Permissions
    • Figure 8
      Figure 8

      (a) Electric field dependence of ground and first excited energy levels calculated within tight binding for an electric field bound state (α=0). The inset shows the energy difference between theses states, i.e., the ΔVS. (b) Same as (a) for an intrinsic interface state (α=0.5). Supercell: 67.9 nm Si layer adjacent to a 9.5 nm oxide layer.

      Reuse & Permissions
    • Figure 9
      Figure 9

      (a) Relative capacitance for TS and FH interface states as a function of the valley splitting. The lower-left inset shows a schematic representation of the capacitor structure discussed here; δ is the effective thickness of the equivalent capacitor. The length Λ is large enough to include the total charge bound near the interface. (b) Percent difference between capacitances in the presence of a TS or FH state as a function of the valley splitting ΔVS and the barrier thickness d.

      Reuse & Permissions
    • Figure 10
      Figure 10

      (a) Original chain (prior to decimation), considering the nearest-neighbor dimer coupling. (b) Intercalated dimers are decimated each cycle, and renormalized hoppings connect the next-nearest dimers. (c) Final product of the decimation process, i.e., a single dimer with negligible coupling with nearest dimers. After the process of decimation, the original system is represented by two effective atoms with complex tight-binding parameters.

      Reuse & Permissions
    ×

    Sign up to receive regular email alerts from Physical Review B

    Log In

    Cancel
    ×

    Search

    Article Lookup

    Paste a citation or DOI
    Enter a citation
    ×