In this work, using the density functional theory in the framework of the spin non-collinear generalized gradient approximation, the behavior of the magnetic moments of B:Si hole center qubits was investigated. The spin dynamics, the charge density transfer, and the influence of these parameters on the total electronic energy of the system are calculated. The results obtained are promising for the design of a two-qubit quantum system as a logical gate for future quantum computers.
Due to the strong spin-orbit interactions of holes in a silicon system [
In this paper, using quantum mechanical modeling, we investigate the regularities of the behavior of the magnetic moments of the qubits of the B:Si hole centers. We study in detail their spin dynamics and the transformation of the charge density in their environment. The results obtained will help develop recommendations for experimenters and technologists engaging in the production of future quantum computers.
Optimization (ESPRESSO) software code [
2021 Copyright for this paper by its authors. silicon atoms. Then, the 72-atom model was placed in a 23.070 × 23.070 × 12.490 Å cell. A special kpoint set of 2 × 2 × 1, with an energy cutoff of 476.20 eV, was used to calculate the silicene characteristics.
To study the behavior of boron atom qubits in silicene, two silicon atoms were substituted by boron impurity atoms. For this, we chose two hexagonal rings located at the edges of the silicene monolayer so that the distances between the boron atoms were no less than 10 Å. For the purity of the computational experiment, the boron impurity atom was alternately placed in all six positions in the ring, and, after complete relaxation of the system, the total energy of the system was calculated for each position. All positions turned out to be energetically approximately equivalent to each other, so any of them could be chosen for further research. In this work, we present the calculation results for the B1 and B2 positions of the boron atom in silicene (Figure 1). Upon the incorporation of boron impurity atoms into silicene, an even greater distortion of its structure is observed. This is clearly demonstrated by the value of the angle Si – B – Si, which significantly increases in comparison with the system without impurities and was computed to be 119.55°. The length of the interatomic bond d(B-Si), on the contrary, decreased in comparison with the system without impurities and amounted to 1.949 Å. With increasing distance from the boron atom, the disturbances in the silicene structure begin to smooth out. Thus, if in the hexagonal ring closest to the boron atom, the angle Si – Si – Si averages to 118°, then in the next nearest ring, the angle is equal. The Si – Si – Si angle, on average, is already equal to 116.97°, which practically corresponds to the undoped structure.
At the next stage of the work, we needed to investigate how the change in the angle, denoted θ, in the Bloch sphere between the z-axis and the magnetic moments of the boron atom affects the total energy, denoted Etotal, of the system. For this, the magnetic moments on the B atoms were simultaneously deflected by equal angles of θ from the 0° base position (quantum state |0>) to 180° (quantum state |1>) with a step of 10°. It should also be noted that the initial angle θ and the angle obtained after the calculation differed from each other on average by a factor of two. So, for example, when setting the initial angle to θ = 10°, at the output, we arrived at θ = 5°. As a result of these calculations, curves were obtained showing the dependence, Etotal(θ) (Figure 2).
Thus, it can be seen that the curves of the dependence of the total energy on the angle of rotation
of the magnetic moments of B1 and B2 boron atoms in silicene behave in the same way, with the only
remark being that the curve for B2 is wider. In the sections from 0° to 5° and from 175° to 180°, we
observe a mirror-like abrupt growth of Etotal, and then, in the sections from 5° to 55° and from 125° to
175°, we observe a further gradual increase in the total energy. Attempts to increase the value of the
angle θ to 90° were unsuccessful. We believe that the fact is that in the region from 55° to 125°, the
so-called spin blockade occurs, which is observed in double quantum dots [
Next, we investigated how the charge distribution density in the Si70B2 system was changed depending on the position of the magnetic moments of the boron atoms. For this, the differences in the charge density distribution at the main reference points of 0°, 5°, 55°, and 180° were calculated. These differences in charge density are shown in Figure 3, and they were obtained by finding the difference in charge density for each of the corresponding states. A detailed analysis of the spatial localization of the charge density showed a significant redistribution in the environment of the B1, B2, and neighboring silicon atoms due to the exchange interactions between the atoms.
Indeed, during the simultaneous rotation of the magnetic moments on the B atoms through an angle from 0° to 110°, the charge density increases on the B1 atom, on the B1 – Si bond, and on the silicon atoms closest to it. However, on the second B2 atom, the charge density decreases. It turns out that the B1 atom and the nearest Si atom pull down the electron density from the B2 atom. With the rotation of the magnetic moments on each boron atom, in the range from 130° to 180° (where the angle between the magnetic moments of the B1 and B2 qubits varies from 260° to 360°), the opposite situation is observed. The charge decreases on the B1 atom and the B1 – Si bond with the nearest Si atom, and on the B2 qubit, the charge density increases. Thus, in this case, the B1 atom and the nearest Si atom give a charge to the B2 atom.
Figure 4 shows the change in the magnetization on each of the boron qubits depending on the
angle. With the rotation of the magnetic moments on the boron atoms in the range from 0° to 110°,
the magnetization decreases on the B atoms but increases on the B-Si bond and slightly increases on
the Si atom. However, the opposite is observed when turning from 260° to 360°, where the
magnetization increases on the B atoms and decreases on the Si atom and in the interatomic space on
the B-Si bond. It can be seen that the magnetization on the left (B1) qubit, with an increase in the
angle from 0° to 55°, first decreases from 0.019 to 0.013 μB/cell, and then, with an increase in the
angle from 125° to 180°, it increases from 0.013 to 0.019 μB/cell. At the same time, on the second
(B2) qubit, within the angle from 0° to 55°, the magnetization first falls from 0.023 to 0.018 μB/cell,
and then, with an increase in the angle from 125° to 180°, it increases from 0.018 to 0.023 μB/cell. We
see that the magnetization on the left qubit is about 20-30% less than on the right qubit. In our
opinion, such a difference in the values of the magnetic moments is observed due to the anisotropy of
the electron shells and the interference of the valleys in the Brullian zone [
Thus, our results for the studied two-dimensional B:Si system are interesting and promising and can be used by technologists and experimenters to evaluate, design, and predict the physical properties of a two-qubit quantum system as a logical gateway for quantum computers.
The studies were carried out using the resources of the Center for Shared Use of Scientific Equipment "Center for Processing and Storage of Scientific Data of the Far Eastern Branch of the Russian Academy of Sciences", funded by the Russian Federation represented by the Ministry of Science and Higher Education of the Russian Federation under project No. 075-15-2021-663. The authors would like to thank them for providing access to the HPC-cluster at the Joint Supercomputer Center of the Russian Academy of Sciences.