Practical design and simulation of silicon-based quantum-dot qubits

Physical Review B - Condensed Matter and Materials Physics

Volumn 67, Issue 12, 2003, Pages 4-

  Savage, Donald E ^a   Friesen, Mark ^a   Lagally, Max G ^a   Eriksson, Mark A ^a   Joynt, Robert ^a   Rugheimer, Paul ^b   van der Weide, Daniel W ^b  

^a UNIVERSITY OF WISCONSIN MADISON   (United States)

^b University of Wisconsin Madison   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 85038910362     PISSN: 10980121     EISSN: 1550235X     Source Type: Journal    
DOI: 10.1103/PhysRevB.67.121301     Document Type: Article

References (28)

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3. For donor-bound electrons in bulk Si, for example, the spin-lattice relaxation-time (Formula presented) can be greater than 3000 s at low temperatures and (Ref. 22) fields. When uniaxial strain is applied, (Formula presented) grows by many orders of (Ref. 23) magnitude. The transverse spin relaxation-time (Formula presented) is typically shorter, and can be enhanced through the use of isotopically (Formula presented)Si to minimize the effect of nuclear (Ref. 24) spins. It is generally expected that decoherence times in confined structures like quantum dots will exceed the bulk values.
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11. The electrostatic and Hartree-Fock calculations were performed using finite-element software, (Formula presented). Image potentials arising from the nontrivial gate structure were calculated self-consistently using a numerical Green’s function technique.(Test charges were first introduced into the heterostructure. The direct contributions from the test charges were then subtracted from the solutions to obtain exact image potentials.) The quantum-mechanical problem was approximated as a single envelope (Ref. 16) function. A basis set of 18 single-electron Hartree-Fock wave functions was obtained in real space on an adaptive finite-element mesh. A basis of about 50 two-electron wave functions was then constructed in the configuration interaction approach. The Hamiltonian matrix was computed and diagonalized, giving an essentially exact result for the envelope function. For the four-qubit simulation [Fig. 11(b)], the wave functions of the outer two electrons were found to have an insignificant overlap with the two center electrons. Their couplings were, therefore, treated as purely Coulombic. It was not possible to calculate directly the tiny values of J corresponding to high potential barriers between the quantum dots. In this regime, we made use of the nearly exponential dependence of J on the gate voltage to obtain extrapolations.
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16. Due to strain effects, the six-fold degeneracy of the silicon conduction band in the quantum well is lifted, so that only the (Formula presented) bands are populated. The envelope function approximation can be used to describe lateral variations of the wave function when the confinement potential varies slowly with respect to atomic distances; for quantum dots of size (Formula presented) the approximation should be reasonable. In the vertical direction, however, the band edge has sharp discontinuities at the quantum well interfaces. For this case, an extension of Ref. 25 to quantum wells shows that the envelope function approximation remains accurate, although the fine structure of the wave function (i.e., the Bloch function) should be modified to accomodate intervalley scattering. Symmetry allows two possible valley-coupled wave functions: (Formula presented) where (Formula presented) and (Formula presented) are the appropriately modified Bloch functions at the (Formula presented) and (Formula presented) band minima. The envelope functions in these two directions are equal, (Formula presented) Perturbation theory gives the energy splitting for the two eigenstates (Formula presented) although a complete treatment of the interface physics is still lacking. Using the estimate given in Ref. 25 (Formula presented) we predict a valley splitting of (Formula presented) (0.7 K) for our heterostructure—more than 10 times our dilution fridge temperature. By optimizing the heterostructure to increase the electric field in the quantum well, it is possible to increase (Formula presented) significantly. At low temperatures then, all electrons should be in the valley-split ground state, with no interference effects of the type discussed in Ref. 26.
17. The (Formula presented) estimate assumes two-qubit operations between any pair of qubits. In a linear qubit array with only nearest-neighbor couplings, a more restrictive threshold may be appropriate.
18. Using the estimate (Formula presented) for bulk (Ref. 24) silicon, the fault-tolerant error threshold suggests a maximum pulse length of (Formula presented) For a flat-top pulse of width (Formula presented) we estimate a minimum pulse edge of 10 ps to satisfy adiabatic gating requirements (Refs. 2728). Such a pulse must be produced with less than (Formula presented) relative error in its duration. Such accuracy is currently beyond the limits of commercial pulse generation technology.
19. Estimates are based on specifications for pulse amplitude jitter in PB-4 and PB-5 sub-MHz pulse generators from Berkeley Nucleonics Corporation (http://www.berkeleynucleonics.com).
20. Specifications from the Agilent Technologies 8133 and 81100 families of GHz pulse generators are listed as (Formula presented) (http://www.agilent.com).
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