Dangling-bond charge qubit on a silicon surface

Two closely spaced dangling bonds positioned on a silicon surface and sharing an excess electron are revealed to be a strong candidate for a charge qubit. Based on our study of the coherent dynamics of this qubit, its extremely high tunneling rate ~ 10^14 1/s greatly exceeds the expected decoherence rates for a silicon-based system, thereby overcoming a critical obstacle of charge qubit quantum computing. We investigate possible configurations of dangling bond qubits for quantum computing devices. A first-order analysis of coherent dynamics of dangling bonds shows promise in this respect.

fabricated [9]. Other semiconductor charge qubits suffer from large decoherence [10]. Our strategy to reduce decoherence compared to other semiconductor charge qubits is simply to shrink the scale of the constituent quantum dots, and the spacing of quantum dot pairs.
The charge qubit is manifested as a quantum dot pair such that having the excess charge on the 'left' dot corresponds to the logical |0 state and the excess charge on the 'right' dot corresponds to the orthogonal logical |1 state. Coherent tunneling between the two quantum dots yields superpositions of |0 and |1 . The tunneling rate increases exponentially with decreasing inter-dot separation whereas decoherence is expected to be largely independent of dot separation. Therefore, decreasing separation allows many coherent oscillations before the onset of significant decoherence.
While control and read-out operations concomitantly move into the optical frequency regime, at this early stage such technical challenges appear worth considering given the benefits of all-silicon QC that this strategy points toward.
To greatly reduce interdot separation, we abandon heterostructure quantum dots. Our substitute quantum dot is a silicon surface dangling bond (DB) on an H-terminated Si surface. As this quantum dot is of atomic dimensions commensurately close spacing of dots is enabled [9]. Moreover pairs of appropriately separated DBs, share precisely one excess electron per pair (denoted DB-DB − , where DB − is a negatively charged DB). These pairs exhibit tunnel coupling of the electron between the two DBs [9]. Here we show that these atomic-scale tunnel-coupled DB pairs are evolving coherently and hence act as charge qubits. In addition, we discuss how initialization and readout of these charge qubits can be done. These DB-DB − charge qubits could form the basic units of a quantum computer, the dynamics of which we describe with the extended Hubbard model [11].
These DB quantum dots can be separated by subnanometer distances, are almost physically identical, and, as surface entities, are directly amenable to measurement and control. Furthermore, unlike the case for quantum dots composed atoms buried in bulk media, the silicon-based scheme proposed here trades the extraordinarily difficult requirement of precisely positioned single dopant atoms with the challenging but attainable requirement that single H atoms be removed with a scanned probe. Our first aim is to show that all DB-DB − pairs can be initialized such that each excess electron is either in the 'left' or the 'right' DB of each pair; subsequently the potential landscape can be tilted so that all are in the 'left'. Our second aim is to show how these DB pairs could be used in a quantum computer circuit e.g. the flying-qubit circuit model based on a bulksilicon electron-spin qubit version [12] or on measurement-based one-way quantum computing [13].
Dangling bond pairs as charge qubits:-A neutral DB hosts a bound electron within the Si hydrogen monolayer then using a scanning tunneling microscope (STM) tip to remove H atoms at selected sites [9]. The separation between the two DBs forming a pair has a strict lower bound of 3.84Å as determined by the lattice spacing of the Si (100) surface. The DBs must be sufficiently close together (≤ 16Å) such that Coulomb repulsion ensures that a doubly-charged DB − -DB − pair cannot form; such a pair cannot be tunnel-coupled [9] and hence cannot be a charge qubit. In this fashion, DB-DB − pairs are created as closely-spaced DBs, within a tunneling range of about 16Å, so that DBs belonging to distinct pairs are not tunnel coupled. We assert that such DB-DB − pairs can serve as good charge qubits.
The localized nature of the DB wavefunction and its energy level in the band gap allows us to formulate an electron-confinement model corresponding to a potential well accounting for the effect of the environment. Physically, localizing an extra electron at a DB invokes a lattice distortion due to interactions within the crystal yielding: (i) a 0.5eV upward shift of the DB − energy level relative to a neutral DB's to ∼0.85eV above the valence band edge (a change in the potential well resulting in weaker confinement and a lower ionization energy); and (ii) a local lattice deformation where the host Si atom at a DB − is raised by 0.3Å from the plane of the surface. When the electron tunnels out of a DB − , the lattice begins to relax.
In Figs. 1(b,c) we depict a DB pair as an effective double-well potential with (b) an excess electron at the left well immediately after release from a biasing external field, as required for qubit initialization, and (c) after complete lattice equilibration when the potential landscape becomes symmetrical. Due to the localized extra charge, the double-well in case (b) does not exhibit the symmetry of case (c), and the DB energy is shifted upward at the left site. Consequently, during lattice relaxation, the coherent oscillation between the two DBs takes place between two wells of slightly different shapes, resulting in a periodic oscillation that is biased towards the 'left' (excess electron spends more time on the left than on the 'right'). Slow relaxation of the lattice will modify the electron oscillation and cause weak decoherence commensurate with the ratio of relaxation rate to oscillation rate.
To study the effect of relaxation, we calculated tunneling rates for various separations by two different methods. For DB separations of 3.84 and 7.72Å, tunnel splitting is determined to be 307.7 and 88.7meV, respectively, by time-dependent density-functional theory on cluster models [21].
These correspond to tunneling rates of 9. The red curve in Fig. 2 shows the calculated decoherence rate due to LA phonon relaxation of the charge-induced lattice distortion. Our calculation is based on a simple hydrogen-like model using the deformation potential approximation, previously used for charge qubits in bulk silicon [17]. We see that lattice relaxation via this mode occurs over several nanoseconds whereas the tunneling period for the DB-DB − pair with a fewÅ separation is close to 1fs. Other phonon modes both in bulk and at surface [22] are less likely to couple to electron tunneling due to their discrete energy. Therefore, many coherent qubit oscillations will occur before lattice deformation is significant. We thus expect that Rabi type of oscillations will take place over many periods before decoherence sets in, and the advantage of closely spaced quantum dots is evident.
Hamiltonian dynamics:-As decoherence rates are orders of magnitude smaller than tunneling rates, the dynamics of DBs on the surface can be described by a HamiltonianĤ that acts upon the Hilbert space spanned by zero, one, or two electrons at each DB upon the silicon surface. On-site energy, electron tunneling (hopping), intra-and inter-DB Coulomb repulsion between electrons, and potential differences across the surface are all incorporated intoĤ.
We consider any number of DBs on the surface, with each DB at a site labeled i. Let E os be the on-site energy of an electron at any DB, which includes a constant surface chemical potential offset, and η i be a site-dependent energy correction due to local field effects. The slowly changing lattice deformation due to the excess electron and the potential well deformation due to external biasing fields can be incorporated into this η i parameter. The coherent tunneling rate between sites i and j is T ij , which depends on the separation r ij between the two DBs.
For two electrons at the same site, U i denotes the energy cost of putting two electrons of opposite spin at the same site i, including the energy contribution due to screening fields. The cost of putting one electron with spin σ ∈ {↑, ↓} at site i and another electron of spin σ ′ at site j Tunneling between DB sites can be controlled by modifying the surface potential. For example two sites i and j can have a time-dependent potential difference of V ij (t). Forĉ i,σ (ĉ † i,σ ) the annihilation (creation) operator for an electron with spin σ at site i andn i,σ =ĉ † i,σĉ i,σ the number operator for electrons of spin σ at site i, the potential difference operator between sites i and j is 2V ≡ i<j,σ V ij (n i,σ −n j,σ ).
We now have all the terms required to express the Hamiltonian as an extended Hubbard model [11] Decoherence is neglected but could be incorporated by adding to this Hamiltonian coupling terms for various phonon modes such as the LA mode that is responsible for relaxation of the lattice deformation. Here Hamiltonian (1) suffices as the qubit dynamics is much faster than the decoherence processes, and we work in a regime where this Hamiltonian can be simplified to one of coupled qubits with standard descriptions of Markovian qubit decoherence [15]. form a qubit basis [16] with conjugate basis corresponding to |± = (|0 ± |1 ) / √ 2.
Initialization and readout:-For our DB-DB − pair to be an effective charge qubit, initialization to a simple pure state and qubit-specific readout are critical; these are two of DiVincenzo's five criteria (the other criteria are scalability, a universal set of gates, long coherence times, with long coherence times established in the previous section) [18].
Qubits are initialized in the |0 state by temporarily applying an electrostatic potential ∆V ı (t) so that the left DB is lower in energy thus attracting the pair's excess electron [17]. Steps toward this initialization procedure have been achieved experimentally for DB pairs [9]. When initialization is complete, the tilt of the electrostatic potential landscape is eliminated, and tunneling between the two DBs commences. Of course the lattice deformation due to excess charge in Fig. 1(c) is present during subsequent tunneling, but is expected to relax orders of magnitude slower than the tunneling rate hence gradually modify the tunneling dynamics with small decoherence over many cycles.
Advances towards qubit-specific readout were achieved by STM detection of the excess charge [9]. Charge was observed to be preferentially localized at one site in a DB pair, which indicates that both state preparation on one side and readout of 'left' vs 'right' state is feasible.
Both initialization and readout are slow processes. Initializing qubits can be slow so the preparation timescale is not a fundamental problem. As all measurements can be delayed until the end of the quantum computation at an acceptable cost of more qubits and gate operations, slow readout is a technical rather than fundamental obstacle.
Of course fast measurements would be desirable, not only for error correction, but also to measure decoherence. One approach to fast readout is to couple the charge qubit to a nearby molecule or single electron transistor (SET) and detect the changes in those when the qubit is in the |0 and not otherwise. The charging state of a DB was shown to affect the STM current through a nearby molecule implanted in the surface [20]: the molecule's electronic structure can be Stark-shifted by the DB's excess electron.
Universality:-Hamiltonian H q enables a universal set of gates [19]. Single-and two-qubit gates are effected by varying the inter-dot tunneling rate by tilting the potential landscape then rapidly turning off the tilting. Such fast and spatially precise control is beyond the capability of standard electronics but is conceivable by placing a suitable pattern of metallic nanowires near the surface and irradiating it with a laser pulse. The resulting electromagnetic field, created via plasmonic action [23], can bias the surface with a temporal control comparable to the duration of the pulse (as short as femtoseconds). The laser carrier frequency should be low enough to avoid charging and discharging of DBs through excitation processes, thereby causing qubit losses. Different gates are effected via varying the bias duration by controlling the laser pulse. Scalability of our surface charge-qubit quantum computer follows the same arguments as for those cases, but of course better understanding of small-scale devices is required to assess scalability to large devices.
Full universal quantum computing could proceed using a four-rail flying qubit model analogous to the one for electron-spin qubits in bulk silicon [5]. Alternatively the qubits can be stationary, and quantum computing could be implemented as a one-way quantum computer [13].
Summary:-We show that closely-separated DB pairs on the silicon surface should behave as charge qubits, with coherence following from extreme miniaturization of qubits, truly reaching the atomic realm. Excellent coherence arises because the tunneling rate is extremely high due to atomic-scale proximity of DBs, whereas the major source of decoherence scales weakly with separation. The scaling advantage comes at the price of having to achieve rapid gating control.
This fast control is a worthwhile challenge given the advantages of an all-silicon architecture for QC.
Such fast gating control could be achieved by optical-plasmonic interfaces and precise positioning of metallic tips or nanowires.
In the shorter term, experimental characterization of the decoherence for these charge qubits is of paramount importance. Whereas time-domain control is ultimately required, decoherence can be studied soon by fluorescence techniques: charge qubits are dipoles that will fluoresce in the terahertz regime, and decoherence can be learned from linewidths. In addition to weak decoherence, our scheme has another important advantage over other semiconductor charge qubit proposals: the charge qubits are on the surface rather than in the bulk medium, thus enabling more direct preparation, control, and readout. Furthermore steps toward initialization and readout of DB pairs have already been demonstrated so our scheme is leveraging off prior successes with DB quantum dot dynamics. We believe the findings outlined here could reinvigorate charge qubit prospects. Acknowledgments:-This project has been supported by NSERC, MITACS, Quantum-Works, iCORE, and the Southeast University Startup Fund. RAW is a CIFAR Fellow, and BCS is a CIFAR Associate.