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# Superexchange coupling of donor qubits in silicon

Mushita M. Munia  Serajum Monir  Edyta N. Osika  Michelle Y. Simmons  Rajib
Rahman  School of Physics, University of New South Wales, Sydney, NSW 2052,
Australia.  Silicon Quantum Computing Pty Ltd., Level 2, Newton Building, UNSW
Sydney, Kensington, NSW 2052, Australia

###### Abstract

Atomic engineering in a solid-state material has the potential to
functionalize the host with novel phenomena. STM-based lithographic techniques
have enabled the placement of individual phosphorus atoms at selective lattice
sites of silicon with atomic precision. Here, we show that by placing four
phosphorus donors spaced 10-15 nm apart from their neighbours in a linear
chain, it is possible to realize coherent spin coupling between the end
dopants of the chain, analogous to the superexchange interaction in magnetic
materials. Since phosphorus atoms are a promising building block of a silicon
quantum computer, this enables spin coupling between their bound electrons
beyond nearest neighbours, allowing the qubits to be spaced out by 30-45 nm.
The added flexibility in architecture brought about by this long-range
coupling not only reduces gate densities but can also reduce correlated noise
between qubits from local noise sources that are detrimental to error
correction codes. We base our calculations on a full configuration interaction
technique in the atomistic tight-binding basis, solving the 4-electron problem
exactly, over a domain of a million silicon atoms. Our calculations show that
superexchange can be tuned electrically through gate voltages where it is less
sensitive to charge noise and donor placement errors.

superexchange, silicon qubits, NEMO, FCI, donors, STM

##  I Introduction

Donor qubits in silicon are promising candidates for encoding quantum
information in the solid state due to their long coherence times [1, 2, 3] and
their technological link to the silicon platform of the electronics industry.
Experimental advancements in the past decades have enabled the precision
placement of phosphorus donors in silicon [4, 5, 6, 7, 8]. The platform of
phosphorus donor based quantum computing has been bolstered by key milestone
achievements over the last decade, including single-shot spin-readout [9], the
realization of single electron and nuclear spin qubits [10, 11], and more
recently, two-qubit SWAP gates [12] and three-qubit donor quantum processor
with universal logic operation [13]. Exchange coupling between the electronic
spins of donors remains a key mechanism for fast coupling of two qubits [12,
14]. The exchange interaction depends on the overlap between the electronic
wavefunctions and ultimately limits the separation of donor qubits to about
10-15 nm in silicon devices. Long-range coupling schemes through resonators
and cavities have recently been explored in donor qubits [15, 16, 17],
however, these typically require additional fabrication and integration steps
adding complexity to the overall manufacturing process. Spacing out the qubits
is beneficial from an architectural point of view in fault-tolerant quantum
computing [18] as correlations between the qubits due to local noise sources
can be minimised. An increase in separation also relaxes stringent gate
density requirements and offers more independent electrostatic control of the
qubits by reducing their capacitive cross-talk. For STM-patterned donors with
phosphorus doped in-plane gates, the density is already low [19], so this
technique is particularly appealing.

In this work, we study long-range exchange coupling between the end spins of
four single donor (1P) quantum dots in a linear chain. With each donor
containing a single electron, a superexchange coupling is found to emerge
between the donors at the end of the chain. This third nearest neighbour
interaction enables the qubits to be separated by 30-45 nm. Using atomistic
full configuration interaction calculations, we study the eigenvalues and
eigenvectors of four electron spins across four 1P atoms in silicon. We
investigate the regime of superexchange where the distant qubits can be
coherently manipulated and provide guidelines on the appropriate donor
placement to achieve this. We also investigate the role of the conduction band
valleys on superexchange for donor separation along different crystallographic
directions. We simulate the system using realistic electrostatic potentials
produced by the surrounding in-plane STM-patterned gates where we demonstrate
tunability of superexchange with gate voltages, a crucial requirement for the
realization of electrically-controlled singlet-triplet oscillations. Finally,
we comment on the sensitivity of superexchange to charge noise and donor
placement errors, as well as the role of nuclear spins in singlet-triplet
oscillations induced by superexchange.

Figure 1: Schematic representation of a simulation using four single
phosphorus donors each with an electron. (a) Four phosphorus donors placed in
the silicon crystal (grey) where the electrons localized in the two middle
donors (blue) are mediators ($M_{1}$ and $M_{2}$) for the electron spins
localized in the end donors ($Q_{1}$ and $Q_{2}$) (red). The separation of the
blue mediator spins is $r_{M}$ and their exchange coupling is $j_{M}$. The
first qubit $Q_{1}$ is separated from the first mediator $M_{1}$ by $r_{1}$
($\sim$ 10 nm) and the second qubit $Q_{2}$ is separated from the second
mediator $M_{2}$ by $r_{2}$ ($\sim$ 10 nm). The corresponding exchange
coupling between them is $j_{1}$ and $j_{2}$ respectively. The total
separation of the qubits $Q_{1}$ and $Q_{2}$ is $R$ ($\sim$ 30 nm). The
probability density of the lowest single electron valley-orbital (b) bonding
and (c) anti-bonding state in a logarithmic scale calculated using an
atomistic tight-binding method.

Recent experiments and theoretical calculations have shown indirect coupling
of distant electrons in quantum dots via a central empty mediator [20, 21, 22,
23, 24], multi-electron quantum dot [21, 24, 25, 26, 27, 28, 29] and linear
chain of singly-occupied quantum dots [20, 30, 31]. However, to date,
comprehensive studies have not been performed on long-range indirect coupling
for donor qubits in silicon. Compared to electrostatically defined quantum
dots, donor quantum dots in silicon typically have atomic-scale properties
with wavefunction length scales an order of magnitude less and large quantity
of orbital-valley energy splittings [32]. A single phosphorus donor can bind
at most one or two electrons with the nucleus having a net $1/2$ spin. Early
papers on single donor qubits showed how the exact position and axis of
separation of donor qubits can significantly affect their direct exchange,
including such effects as exchange oscillations due to valley interference
[33, 34]. More recent work has shown that these effects can be mitigated by
separating the donors along specific crystallographic directions, by using
multi-donor quantum dots [35] or by applying strain or placing the donors
close to the surface [33]. Highly tunable nearest-neighbour exchange has also
been predicted [35] and demonstrated [12] in asymmetric donor quantum dots,
showing that multi-electron molecular physics can be tuned by gate voltages.
However, the atomistic character of these donors and donor dot systems needs
to be accounted for when considering indirect couplings like superexchange
since the phenomenon emerges from individual nearest-neighbour exchange
couplings.

##  II Methods

The exact calculation of a multi-electron system is challenging because of the
complicated and numerically intensive electron-electron interaction term in
the Schrödinger equation [36]. The accuracy relies both on the quality of the
basis states and the multi-electron approximation. In semiconductor materials
such as GaAs, eigenstates from effective Hubbard Hamiltonian [20, 25, 37] or
simple Fock-Darwin states are often used as basis states [21]. These are
however unsuitable for silicon due to the multi-valley states. The effective
mass approximation has also been used for calculating single electron basis
states for donors in silicon [38, 39]. However, the theory does not provide a
complete description of the band structure and can be inaccurate at higher
energy levels. On the contrary, the full-band atomistic tight-binding model
has the potential to capture all intricate nuances of the wavefunction of
phosphorus donors in silicon hence its use in this work.

The most common approximation for multi-electron calculations is the Hartree-
Fock method which neglects the electron-electron correlations to minimize the
complexity of the calculations. Configuration interaction (CI), on the other
hand, enhances the accuracy in the treatment of many-body interactions but can
be computationally intensive. This is particularly true for a multi-valley
material like silicon where only a few electron calculations are present in
the literature [33, 40]. In this paper, we have performed state-of-the-art
investigation combining atomistic basis states with a full configuration
interaction method to calculate the energy levels of a four-electron donor
system without compromising accuracy.

###  II.1 Atomistic Full Configuration Interaction

Full configuration interaction (FCI) is a method to obtain the exact numerical
solution of a many-body Schrödinger equation, limited by the number and
quality of single electron basis states. The single electron basis states used
here are calculated using a 10-band $sp^{3}d^{5}s^{*}$ atomistic tight-binding
(TB) method in NEMO3D [41, 42]. This approach uses a localized atomic orbital-
based method with nearest-neighbour interactions. The TB parameters are
optimized to reproduce the bulk silicon band structure [43]. The phosphorus
donors are represented using a Coulomb potential with a central cell
correction at the donor site which can successfully determine the
experimentally measured energy spectra of donors in silicon [44, 45].

The schematic representation of the simulations is illustrated in Figure 1(a).
There are four phosphorus donors placed in a chain, each with an electron. The
end electron spins, $Q_{1}$ and $Q_{2}$ are the qubits separated by R. The
middle spins, $M_{1}$ and $M_{2}$ work as mediators with corresponding donors
being separated by $r_{M}$. The separation between $Q_{1}$ and $M_{1}$ is
$r_{1}$ and the separation between $M_{2}$ and $Q_{2}$ is $r_{2}$. The
simulation domains used in the calculations entail $\sim 1.14$ million atoms
which account for $\sim 60$ nm of silicon in the direction of separation and
$\sim 20$ nm in the other two directions. The single electron basis states,
solved from a parallel Block Lanczos eigensolver, are used to calculate the
anti-symmetric Slater determinants of the multi-electron problem. The lowest
two single electron valley-orbital states of the system are shown in Figure
1(b) and (c). Here we see bonding and anti-bonding state formation in the
middle two donors $M_{1}$ and $M_{2}$ of a four-donor chain. The rest of the
single electron molecular basis states are shown in Figure 8 in the
Supplementary Information where we see the next two valley-orbital states are
mainly localized in the outer donor dots.

The four-electron wavefunction is a superposition of various symmetry-
permitted configurations of the Slater determinants. All possible integrals
between the Slater Determinants with pairwise electronic interaction operators
are computed to capture Coulomb, exchange, and higher-order correlations. The
four electron Hamiltonian constructed from the Slater determinants is solved
using the block Krylov-Schur algorithm within the Trilinos framework [46]. The
number of single electron basis states in FCI calculations is increased until
the eigenvalues of the four-electron system converge within a chosen tolerance
– see Supplementary Information. On average, 56 single-electron basis states
are sufficient to reach convergence. The eigenvalue of the ground state is
separated from the triply degenerate excited states by the indirect exchange
coupling. We place the donor atoms in different separations and orientations
in our simulations and calculate this exchange coupling.

###  II.2 Effective spin Hamiltonian

To analyze and interpret our FCI results, we compare them with an effective
Hamiltonian model [30]. We can represent the four-spin system with a spin
Hamiltonian such as -

$H_{eff}=\frac{j_{1}}{4}\sigma_{1}.\sigma_{2}+\frac{j_{M}}{4}\sigma_{2}.\sigma_{3}+\frac{j_{2}}{4}\sigma_{3}.\sigma_{4}$
(1)

where $\sigma_{i}$ is a Pauli matrix corresponding to an electron spin located
on $i^{th}$ donor in the chain. $j_{M}$ is the exchange coupling between the
middle two spins, $M_{1}$ and $M_{2}$. $j_{1}$ is the exchange coupling
between $Q_{1}$ and $M_{1}$ and $j_{2}$ is the exchange coupling between
$Q_{2}$ and $M_{2}$ – see Figure 1(a) for the schematic of the system.

We only consider the subspace with spin-zero states ($S_{z}=0$) to construct
the Hamiltonian as we are interested in the singlet-triplet oscillations in
the qubits $Q_{1}$ and $Q_{2}$. When $j_{1},j_{2}\ll j_{M}$, we can isolate
the low energy states of the four-electron spin Hamiltonian using a
Schrieffer-Wolff transformation. In that case, the effective Hamiltonian in a
Heisenberg exchange form, $H_{SW}$ [30] now becomes -

$\begin{split}H_{SW}&=\frac{J_{SW}}{4}\sigma_{1}.\sigma_{4}\\\
J_{SW}&=\frac{j_{1}j_{2}}{2j_{M}}[1+\frac{3(j_{1}+j_{2})}{4j_{M}}]\end{split}$
(2)

In the regime where $j_{1},j_{2}\ll j_{M}$, the lowest energy eigenstates are
characterized by the singlet state formed within the two middle dots and
singlet or triplet states formed within the outer dots [30]. The energy
difference, $\Delta E$ between the lowest long-distance singlet and triplet
states, i.e.

$\begin{split}S^{l}\approx 1/\sqrt{2}(\ket{\uparrow
S\downarrow}-\ket{\downarrow S\uparrow})\\\ T^{l}_{0}\approx
1/\sqrt{2}(\ket{\uparrow S\downarrow}+\ket{\downarrow S\uparrow})\end{split}$
(3)

is what we call superexchange ($\Delta E$) in this paper – see Figure 3(a) for
the energy level diagram of the system. For $j_{1},j_{2}\ll j_{M}$ the middle-
dot singlet manifold is well separated from all the middle-dot triplet states
desirable for coherent coupling of the outer spins only, without any
interference from the middle-dot spins.

##  III Results and Discussions

###  III.1 Equal nearest-neighbour separation of donors

We first analyze the equidistant case, where the separation between each
neighbouring pair of donors is equal, $r_{1}=r_{2}=r_{M}=R/3$ – see Figure 1.
Here we have approximately equal values of exchange coupling between each pair
of donors, i.e. $j_{1}\approx j_{2}\approx j_{M}$. In Figure 2 (with red
dots), we show the values of the indirect exchange coupling for these
equispaced donors (4P chain) with constant but increasing separation between
the dots along the $[100]$ (left) and $[110]$ directions (right) in the
silicon crystal, calculated using FCI. For comparison, we also plot (with blue
dots) the direct exchange values for two donors, i.e. 1P-1P system, separated
by R, as previously calculated in [35]. Looking at the fitted dashed lines
between the 4P chain and 1P-1P cases, we can see that the presence of the
mediator donors dramatically increases the exchange coupling between the two
outer spins. The indirect exchange coupling for a donor chain of $R$ of about
25 nm reaches $\sim 10^{2}$ GHz ($10^{-1}$ meV), while for direct exchange for
the same separation without the presence of mediators would fall below
$10^{-4}$ GHz ($10^{-7}$ meV) (extrapolated from the dataset in [35]).

Figure 2: Comparison of indirect and direct exchange coupling of 1P donors in
silicon calculated using atomistic FCI (a) Comparison of indirect exchange (4P
chain) of the end-spins (red dots) with nearest-neighbour exchange coupling
(1P-1P) (blue dots, replicated from [35]) along the $[100]$ direction. Dashed
lines provide linear fits to both data sets. Here we see that the indirect
exchange is higher than the nearest-neighbour exchange, with the exponential
dependence with separation being less steep for indirect coupling. (b) Same as
(a) but along the $[110]$ direction. Here we see a similar trend in terms of
the comparison between direct and indirect exchange as for the $[100]$
direction. However, we also observe oscillations in the exchange coupling
along this crystalline orientation due to valley quantum interference.

We also see that the direct exchange decays much faster than the indirect
exchange by comparing the slope of these two plots, true for both [100] and
[110] crystal directions. For the [100] case in Figure 2(a), the slope of the
fitted dashed line for the direct exchange as a function of qubit separation
is $\sim-0.38$/nm whereas the slope for the indirect exchange is
$\sim-0.12$/nm. Similarly in the [110] case in Figure 2(b), the slope of the
fitted line for the direct exchange is $\sim-0.36$/nm whereas the slope for
the indirect exchange is $\sim-0.11$/nm. In general, the indirect exchange
decays almost three times less with donor separation than the direct exchange.
This dependency with distance lets the qubits be far separated in the device
and allows less correlated noise between them while maintaining strong spin
coupling. Since the nearest neighbour exchange coupling has an exponential
dependence with increasing separation of the donors, one might expect
superexchange to change as an exponential function of $r_{1}+r_{2}-r_{M}$ –
see Equation 2. On the contrary, the nearest neighbour exchange coupling
changes as an exponential function of the qubit separation
$R=r_{1}+r_{2}+r_{M}$ which results in a faster decay of direct exchange than
the indirect one.

We observe no oscillations in the exchange coupling when the donors are
separated in the [100] direction whilst we see oscillations in the exchange
energy for separation in the [110] orientation, a signature of inter-valley
quantum interference arising from the band structure of silicon [33, 35, 47].

Figure 3: Estimate of exchange coupling strength from the spin Hamiltonian.
(a) Energies of 4-electron eigenstates calculated with the effective
Hamiltonian $H_{eff}$ as a function of $j_{1,2}/j_{M}$. The two lowest states
(spanned by $\ket{\uparrow S\downarrow}$ and $\ket{\downarrow S\uparrow}$) are
well separated from the higher states when $j_{1,2}/j_{M}$ is smaller. The
separation of these two energy levels is the superexchange $\Delta E$. The
labels on the right refer to the dominant contribution of the corresponding
energy states when $j_{1,2}/j_{M}$ $\sim$ $0$. (b) A comparison of the
exchange coupling, $\Delta E$ calculated from the effective spin and the
Schrieffer-Wolff Hamiltonian shows that up to approximately $j_{1,2}/J_{M}$
$=$ $0.3$, the superexchange from $H_{eff}$ and $H_{SW}$ is the same since the
Schrieffer-Wolff Hamiltonian is a valid approximation of the spin Hamiltonian.
Beyond this regime, their behaviour diverges significantly because the
Schrieffer-Wolff transformation no longer holds indicating the development of
an admixture in the singlet-like state of the middle two spins. (c)
Contribution of the inner-singlet and inner-triplet basis states to the ground
state of $H_{eff}$ as a function of $j_{1,2}/j_{M}$. Here we see that, as
$j_{1,2}/j_{M}$ increases, the singlet contribution from the inner mediator
spins decreases, and the triplet contribution increases. At $j_{1,2}/j_{M}$
$=$ $1$, the ground state has equal contributions from singlet and triplet
mediator spin states. The contribution of the inner singlet to the ground
state is more than 90% for $j_{1,2}/J_{M}<0.3$ (shaded region).

###  III.2 Different regimes of superexchange from effective spin Hamiltonian

To obtain superexchange and coherent control between the electron spin qubits
at the end of the chains $Q_{1}$ and $Q_{2}$, it is essential that the middle
spins $M_{1}$ and $M_{2}$ form a singlet-like state. Otherwise, there will be
additional oscillations originating from the middle spins impeding coherent
manipulation of the qubits. Thus we are looking for an operational regime
where the ground state and the first excited state are well separated from the
higher energy states in which the indirect coupling can be termed
superexchange. In Figure 3(a) we plot the energies of all the eigenstates of
the effective Hamiltonian $H_{eff}$ as a function of $j_{1,2}/j_{M}$ varying
from 0 to 1. When the outer spins are weakly coupled with the middle spins
($j_{1,2}/j_{M}\sim 0$), we see two clear branches of energy states separated
by $d$ but the separation between the branches decreases as $j_{1,2}/j_{M}\sim
1$.

In Figure 3(b), we plot the energy difference between the two lowest states,
as a function of $j_{1,2}/j_{M}$ as calculated from $H_{eff}$ and $H_{SW}$.
These two lowest states are the long-distance singlet and triplet states,
namely $S^{l}$ and $T^{l}_{0}$. Here we see that, in the regime where
$j_{1,2}/j_{M}\sim 0$, the solutions from the effective spin and the
Schrieffer-Wolff Hamiltonian are the same since the assumption of the
transformation is valid here. As $j_{1,2}/j_{M}$ increases, the two solutions
start to diverge suggesting the Schrieffer-Wolff transformation no longer
holds. The lowest two energy states (spanned by $\ket{\uparrow S\downarrow}$
and $\ket{\downarrow S\uparrow}$) are no longer well separated from the
excited states and there are now triplet-like admixtures in the singlet-like
state of the middle two spins.

In Figure 3(c), we can see the contributions in the ground state $S^{l}$ of
the inner-dot singlet states (i.e. $\ket{\uparrow S\downarrow}$ and
$\ket{\downarrow S\uparrow}$) and inner-dot triplet states (all the remaining
basis states). When $j_{1,2}/j_{M}=1$ (i.e. the equidistant case where all
donors are equally separated), the inner-dot singlet and triplet contributions
in $S_{l}$ are both approximately 50% and we no longer have a well-defined
two-level system of $S^{l}$ and $T^{l}_{0}$, but now have contributions from
the four higher states shown in 3(a). As $j_{1,2}/j_{M}$ decreases, the
contribution of the inner-dot singlet reaches more than $90\%$ where
$j_{1,2}<\sim 0.3j_{M}$ (shaded region). This regime can be considered as a
threshold for coherent operation of the qubits. However, we note that the
threshold we mention here ($j_{1,2}/j_{M}\sim 0.3$) is not the upper limit of
the coherent regime. The ratio should ideally be very close to $0$.

Figure 4: Exchange coupling as a function of different middle donor
separations, $r_{M}$. The exchange coupling $\Delta E$ is calculated using
atomistic FCI for different separations of the middle donors $r_{M}$ in the
(a) [100] and (b) [110] direction while keeping the total outer qubit
separation fixed (R = 29.3274 nm for [100] and R = 27.648 nm for [110]). The
system is symmetric ($r_{1}$ $=$ $r_{2}$) to the end donors. The results show
that the dependence of exchange energy changes on either side of the
equidistant separation (dashed grey line) for all donors. The dashed line
indicates where $r_{1}$ = $r_{2}$ = $r_{M}$ and the left region of this line
where $r_{1}$ $=$ $r_{2}$ $>$ $r_{M}$ is where the two middle donors form a
singlet state and the exchange coupling between the end donors is termed
superexchange. We compare our FCI results with the effective spin Hamiltonian,
$H_{eff}$ of Eq. 1 with $j_{1,2}$ and $j_{M}$ extracted from [35]. For both
cases, the results show reasonable agreement with each other. We attribute any
differences mainly to the fact that the total confinement potential of 4
donors is deeper than that of 2 donors which results in a slightly higher
exchange coupling $\Delta E$ in our atomistic FCI calculations.

###  III.3 Modulating superexchange by changing the middle donor separation

To explore the coherent-control regime of the 4-donor chain using our FCI
calculations, we switch from the equidistant case discussed in Figure 2 to a
chain with varied $r_{1,2}/r_{M}$ ratios, see Figure 4. Here, we present
values of the indirect exchange coupling for chains oriented along the [100]
and [110] direction in 4 (a), (b) respectively, as a function of middle donor
separation while keeping the outer donor separation $R$ constant. The lower
middle donor separation corresponds to a higher exchange coupling $j_{M}$ and
lower $j_{1}$ and $j_{2}$ couplings, moving the system closer to the regime
well described by the Schrieffer-Wolff approximation. In general, we see from
Figure 4 that the exchange coupling exponentially decreases as we decrease the
middle donor separation. The grey dashed line represents the separation where
$r_{1}=r_{2}=r_{M}$. The exchange coupling shows two different dependencies on
either side of this point. The first is where the distance between the middle
donors is smaller than the values of $r_{1}$ or $r_{2}$ and the second is
where it is large. For donors separated along the [110] direction, we see
clear oscillations in $\Delta E$ as a function of $r_{M}$.

As we have discussed in the previous section the equidistant donor chain is
not suitable for coherent manipulation of distant spins due to the admixture
of states. The right side of the grey dashed line where $r_{1}=r_{2}\leq
r_{M}$ is therefore not valid for operating qubits. To specify the limits of
these different regimes, we use the values of direct exchange from the 1P-1P
results in Ref. [35] to find $r_{1,2}$ and $r_{M}$ and the corresponding
threshold. Decreasing the separation of the middle dots gives rise to a value
of the superexchange that decreases exponentially while increasing the inner
singlet contributions to the ground state – see Table 1 in the supplementary
information. So there is a trade-off between the strength of coupling and
operation fidelity. However, we see even small changes in the middle donor
separation, of the order of $\sim$ 2 nm, can increase the inner singlet
contributions to the ground state from $\sim 49\%$ to $\sim 98\%$ while
keeping the value of superexchange still high enough ($\sim 100$ MHz or $\sim
1\times 10^{-3}$ meV) for use in realistic devices.

###  III.4 Electrical control of superexchange

Figure 5: Modulating superexchange using voltage biases on phosphorus-doped
top gates above the donor plane. Electrostatic potential of the device in the
(a) Schematic representation of the simulation domain. The qubit donor atoms
($Q_{1}$ and $Q_{2}$) and the mediator donor atoms ($M_{1}$ and $M_{2}$) are
placed in-plane between the source and drain (S, D) electrodes. Three
phosphorus-doped top gates ($G_{1}$, $G_{2}$, and $G_{M}$) are placed 30 nm
above the donor plane. (b) The total potential of the device in the x-z plane
(cut taken at donor location). Here the voltage of gates $G_{1}$ and $G_{2}$
are $V_{1}$ = $V_{2}$ = $-0.1V$ whilst the middle gate $G_{M}$ has a voltage
$V_{M}$ = $0.5V$ and S and D are grounded. The red and blue dots represent the
location of the donors and are not to scale. 1D potentials with cuts taken at
donor location when $V_{1}$ = $V_{2}$ = 0V and (c) $V_{M}$ = 0V, (d) $V_{M}$ =
0.3V and (e) $V_{M}$ = 0.5V respectively as we deplete the barrier between the
donors. The dashed lines represent the potential from the gates and the solid
lines represent the individual donor potentials combined with the
electrostatic potential from the gates. As the voltage of the top gate
increases, the curvature of the potential also increases. Same as (c-e) when
$V_{M}=0.5V$ and (f) $V_{1}=V_{2}$ = -0.1V, (g) $V_{1}=V_{2}$ = -0.3V and (h)
$V_{1}=V_{2}$ = -0.5V respectively. (i) Superexchange as a function of $V_{M}$
when $V_{1}$ = $V_{2}$ = 0V. We see that the magnitude of the superexchange is
increasing as a positive bias is applied to $G_{M}$. The increased curvature
of the applied potential of $V_{M}$ = $0.5V$ (comparing (c) and (e)) gives
rise to the modulation of the magnitude of superexchange by a factor of 10.
(j) Superexchange as a function of $V_{1}$ = $V_{2}$ when $V_{M}$ = $0.5V$.
Here we see the superexchange is decreasing as more negative bias is applied
as a result of decreased potential curvature (comparing (f) and (h))

To address the tunability of superexchange, we have applied an electric field
along the direction of the donor chain in our simulations where all donors are
separated by $9.7758$ nm ($r_{1}$ = $r_{2}$ = $r_{M}$ = 9.7758 nm). For a
realistic applied electric field of $2$ MV/m, we observe an exchange coupling
of 18.09 GHz ($74.83\,\mu$eV) compared to 11.01 GHz ($45.556\,\mu$eV) under no
electric field. The electric field detunes the qubits and increases the
virtual tunnelling between them which results in slightly increased (less than
a factor of 2) superexchange. However, when the donors are all charge neutral
with one electron on each phosphorus atom, it is difficult to reach the
(2,1,1,0) charge regime from a (1,1,1,1) regime without applying a very large
bias. A similar challenge has been observed in [35] where applying an electric
field of 2 MV/m the exchange coupling was modulated by 5 times. In the case of
superexchange, the sensitivity of the coupling with an applied electric field
is even less, hence, the distanced qubits are less susceptible to charge noise
caused by electric field fluctuations compared to the nearest neighbour ones.
To explore different levels of control over superexchange, we apply a J-gate
bias on a system with a
phosphorus-doped silicon top gate based on the original Kane architecture
[14]. We place gates 30 nm above the donor plane to control the potential
barrier between donor atoms and the nearest neighbour exchange and observe
their effect on superexchange. Similar three-dimensional tuning of the
potential barrier between in-plane phosphorus-doped gates has been recently
experimentally demonstrated [48] in precision engineered STM tunnel junctions
where the gates were degenerately phosphorus-doped silicon layers. The
schematic of the system is illustrated in Figure 5(a). The device consists of
five quasi-metallic gates (source S, drain D, and three top gates - $G_{1}$,
$G_{2}$ and $G_{M}$). $G_{1}$ and $G_{2}$ are used to tune $j_{1}$ and $j_{2}$
and $G_{M}$ is used to tune $j_{M}$. The total electrostatic potential profile
of the device in the x-z plane is shown in Figure 5(b) where a cut is taken at
the donor location. Here we see that, in the donor plane, the top gates create
a parabolic potential profile. This total electrostatic potential is added to
the tight-binding Hamiltonian while solving the energy levels of the donor
quantum dots. In Figure 5(c-h), 1D cuts are taken at the donor locations shown
for different $V_{M}$ and $V_{1}$(=$V_{2}$), i.e. for different voltages
applied at gates $G_{M}$ and $G_{1}(G_{2})$, respectively. Here S and D are
grounded, i.e. $V_{S}=V_{D}=0$. We note that just the presence of the
electrostatic leads around the donors also modifies the total electrostatic
potential of the device, even without an applied voltage. This is due to the
band structure mismatch between the leads and the surrounding material, and
the band-bending caused by the leads [49]. We calculate the total
electrostatic potential profile of the device using a multi-scale modelling
technique that combines the atomistic calculation of the band structure of the
gates with a non-linear Poisson solution of the entire device [50].

####  III.4.1 Applying voltage to the middle top gate, $G_{M}$

Since superexchange $\Delta E$ is a function of nearest-neighbour exchanges
(i.e. $j_{1}$, $j_{2}$ and $j_{M}$), it is necessary to understand how these
exchange couplings change under an applied bias. When we apply a positive
voltage to the middle gate $G_{M}$, the potential barrier between the
mediators $M_{1}$ and $M_{2}$ decreases but $j_{M}$ would increase - see
Figure 5(c-e) for increasing values of $V_{M}$. However, from the dashed line
in Figure 5(e), we see that the applied bias in $G_{M}$ detunes the qubits
$Q_{1}$ and $Q_{2}$ with respect to $M_{1}$ and $M_{2}$ so $j_{1}$ and $j_{2}$
would also increase. As we can see from Equation 2, the superexchange strongly
depends on the interplay between $j_{M}$ and $j_{1},j_{2}$. In our case, here
the superexchange increases overall with $V_{M}$ since the effect caused by
detuning between the qubits and the mediator donors is stronger than from
lowering the middle barrier – see the potential profile in Figure 5(e). We
observe this effect on the value of $\Delta E$ in Figure 5(i), where we can
see the superexchange as a function of $V_{M}$ for $V_{1}=V_{2}=0$ (case
corresponding to 5(c-e)). By changing the middle gate potential from 0 to
0.5V, we modulate the magnitude of the superexchange by a factor of 10. The
increase of superexchange is due to a relatively higher increase of
$j_{1},j_{2}$ than $j_{M}$, resulting in lesser singlet contribution from the
mediator dots, as discussed in the previous section. In our case, the singlet
contribution in the middle dot drops from 92.2% ($V_{M}=0$) to 67%
($V_{M}=0.5V$).

####  III.4.2 Applying voltage to all three top gates, $G_{1}$, $G_{2}$ and
$G_{M}$

We can minimize the detuning of the qubits $Q_{1}$ and $Q_{2}$ by applying a
negative voltage to the left and right gates. Then the superexchange would
decrease back - here from 65.42 $\mu$eV to 6 $\mu$eV when changing
$V_{1}=V_{2}$ from 0 to -0.5 V for $V_{M}=0.5$ V – see Figure 5(j). The
positive bias applied to $G_{M}$ increases $j_{M}$ and the negative bias
applied to $G_{1,2}$ decreases $j_{1,2}$, so the $j_{1,2}/j_{M}$ ratio becomes
smaller. The potential profile at the donor location for $V_{M}=0.5V$ and
$V_{1}$ = $V_{2}$ = -0.1V, -0.3V and -0.5V is shown in Figure 5(f-h). If we
look at the inner singlet contributions on the ground state, we see that the
contribution has increased up to 94.05% when $V_{1}$ = $V_{2}$ = -0.5V. Tuning
superexchange by using three gates might be useful in increasing the fidelity
of the operation, otherwise, the tuning shows a similar range of superexchange
even with one top gate. With these simulations, we have shown that it is
possible to manipulate superexchange in a 4P chain by purely electrostatic
means. Applying voltage through top gates exhibits better tunability of
superexchange than a tilt voltage along the donor separation. We have shown
that it is possible to modulate superexchange by a factor of 10 with modest
gate potentials of 0.5 V. We note that even higher tunability might be desired
to turn the coupling on and off for qubit operations. Such high tunability of
the exchange coupling can also be achieved by depleting the middle donors.
Without electrons in the middle donors, the qubits will be coupled by direct
exchange which is significantly low at these separations. On the other hand,
loading electrons in the middle donors will introduce a high value of
superexchange. This mechanism can be crucial for the realization of singlet-
triplet oscillations in distanced qubits.

###  III.5 Effect of asymmetric separation of the donors

Current STM lithography techniques allow us to place donor atoms in silicon
with the precision of a single lattice constant [51, 52, 53, 54, 55]. As a
consequence, we can investigate how small shifts in the exact P atom location
within the four donor chain can affect the value of superexchange available.
For that purpose, we analyze a chain oriented along the [100] direction with
constant middle dot separation $r_{M}=5.431$ nm but with variable positions of
the outer donors, described by $r_{1}$ and $r_{2}$ where $r_{1},r_{2}\geq$ 7
nm. We have deliberately chosen a smaller value of $r_{M}$ so that
$r_{1},r_{2}>r_{M}$. We do not consider nearest neighbour separations less
than 7 nm (qubit separation $R<$ 20 nm) since we are interested in the long-
distance qubit coupling regime and want to exceed the distance achievable by
direct exchange.

Figure 6: Effect of donor placement on the magnitude of superexchange
calculated using atomistic FCI along the [100] direction. Keeping the
separation of the middle donors fixed ($r_{M}$ = $5.431nm$), we change
$r_{1,2}$ to account for any deviation in donor placement. The superexchange
$\Delta E$ is maximum when $r_{1}$ = $r_{2}$. We also see that $\Delta E$
remains constant when $r_{1}+r_{2}$ is constant – see dashed lines.

The results are shown in Figure 6. Here we can see the superexchange $\Delta
E$ is maximum (11.53 GHz) for the smallest values of $r_{1}$ and ($r_{1}$ =
$r_{2}$ = $7.6$ nm). This result arises both from minimizing the total chain
separation, $R$ as well as maximizing the $r_{M}/r_{1,2}$ ratio – as observed
in Figure 4. The value of $\Delta E$ decreases exponentially if either of the
outer donors is moved outwards, i.e. if either $r_{1}$ or $r_{2}$ increases.
However, at the same time, we can see $\Delta E$ does not change if both of
the outer donors are shifted simultaneously in the same way – see the dashed
lines where $r_{1}+r_{2}$ is constant. This result is consistent with the
Schrieffer-Wolff approximation for superexchange in Eq. 2, where the
dominating contribution to $\Delta E$ comes from $j_{1}j_{2}/2j_{M}$. While
$j_{1}$ and $j_{2}$ change approximately exponentially with distance, their
product will not change when $r_{1}+r_{2}$ is kept constant.

Thus we can conclude that any small shifts in the location of the donors
within the [100]-oriented chain can result in large changes in total $\Delta
E$. However, interestingly, a simultaneous shift of both outer (or both
middle) donors in the same direction would not have any impact on the
superexchange (indicated by the black dashed lines). The situation is
completely different for the [110] crystalline direction due to the
oscillations of two-donor exchange with distance which lifts the smooth
$j_{1}j_{2}/2j_{M}$ dependency for varied $r_{1},r_{2}$ when $r_{1}+r_{2}$ is
constant. Here $\Delta E$ will change in an oscillatory manner if one or both
middle donors are shifted.

###  III.6 Impact of donor nuclear spins on the superexchange

Finally, we comment on the presence of nuclear spins in the donor system and
how they can also impact superexchange and coherent electron spin
manipulation. The nuclear spins of phosphorus donor atoms couple with their
electron spins through the hyperfine interaction. From the perspective of the
electron spin, this can be treated as a small additional magnetic field
dependent on the nuclear spin polarization. In the case of an electron
singlet-triplet spin qubit, localized within the middle double donor dot, this
hyperfine interaction can create a magnetic field gradient that mixes singlet
and triplet states [17, 56]. For a 1P-1P system, the gradient is $\sim$ few
Milli Tesla which gives rise to a Zeeman energy difference, $\Delta E_{z}$
$\sim$ $5\times 10^{-4}$ meV between the donors when the two nuclear spins are
oriented antiparallel ($\Uparrow\Downarrow$) and $\Delta E_{z}$ $\sim 0$ when
the two nuclear spins are aligned in the same direction ($\Uparrow\Uparrow$).

To create well-defined eigenstates as well as singlet and triplet states, the
exchange coupling $\Delta E$ needs to dominate over $\Delta E_{z}$. We can
consider the 4-electron system discussed in this paper essentially as two
pairs of singlet-triplet qubits – one defined between the two inner dots and
one between the two outer dots. To avoid mixing of the singlet and triplet
states it is desirable for both $j_{M}$ and $\Delta E$ to be significantly
greater than $\Delta E_{z}$. From Figure 2 this is satisfied when the middle
donor separation $r_{M}$ is smaller than 10-15 nm, which includes most of the
results presented in this work. The value of superexchange $\Delta E$
ultimately depends both on $r_{M}$ and $R$. From Figure 4 we can see it is
possible to design 4P configurations which belong to the regime where
$j_{1},j_{2}/j_{M}<0.3$ and at the same time satisfy $\Delta E>\Delta E_{z}$.
These considerations are however no longer relevant if we deterministically
initialize the nuclear spins to all-parallel before any operation using a
local Nuclear Magnetic Resonance (NMR) [10]. In this case, there is no
magnetic field gradient and thus no limitations on the minimum values of
$j_{M}$ and $\Delta E$.

##  IV Conclusion

In this work, we have presented the results of strong non-nearest neighbour
exchange coupling over a long distance (20-35 nm) with a possible extension of
up to 45 nm, which has not been explored in donor quantum dots before. Using
an atomistic full configuration interaction technique, we have calculated the
eigenvalues of a 4-electron system with high-quality atomistic basis states
and showed a comparison with an effective spin Hamiltonian to determine the
viability of our intensive numerical calculations. We have shown that by
placing the mediator donors one atomic position inwards compared to the
symmetric chain, it is possible to enable coherent control of the qubits
separated in [100] and [110] directions. Similarly to previous works on direct
exchange, we observe a monotonic dependence of superexchange versus distance
for donors separated in [100] and oscillatory dependence in [110].
Importantly, we demonstrate that the superexchange can be modulated by an
order of magnitude by purely electrostatic means using realistic experimental
gate voltages. These results pave the way for the realization of fast singlet-
triplet control for distanced qubits. The calculations support the
experimental realization of long-distance coupling of donor qubits in silicon.

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##  Appendix A Convergence of multi-electron states

The convergence of multi-electron states calculated using FCI depends on the
number of single-electron states taken into the basis. As in this work, we
investigate the effect of superexchange, i.e. the energy difference $\Delta E$
between outer-dot singlet and triplet states, $S^{l}$ and $T^{l}_{0}$
respectively, we use $\Delta E$ also to determine the convergence of our
calculated four-electron states. We gradually increase the number of single-
electron basis states in FCI calculations and observe when the changes in
$\Delta E$ are negligible within some numerical tolerance. In Figure 7 we show
examples of such analysis for donors separated in both [100] and [110] crystal
directions. We find that in most of the cases considered in the paper, it is
sufficient to use 56 basis states, i.e. 28 valley-orbital states, with two-
fold spin degeneracy. In a few cases, we have used up to 72 basis states to
reach convergence in the energy difference.

Figure 7: Convergence of superexchange as a function of a number of single-
electron basis states. Plots (a,c) show results for equidistant dots with
$r_{1}=r_{2}=r_{M}=18a_{0}$ along the [100] direction where $a_{0}=0.5431$ nm
and $r_{1}=r_{2}=r_{M}=24a$ along the [110] direction where
$a=a_{0}/\sqrt{2}$, respectively. Plots (b,d) show results for non-equidistant
cases $r_{1}=r_{2}=20a_{0}$, $r_{M}=14a_{0}$ in [100] and $r_{1}=r_{2}=26a$,
$r_{M}=20a$ in [110], respectively.

##  Appendix B Analysis of four-electron eigenstates

Figure 8: Single electron basis states calculated using an atomistic tight-
binding approach. The wavefunctions of the first 16 valley-orbit molecular
states are plotted here on a logarithmic scale. The titles indicate the
binding energy of each of the states.

In this section, we analyze the 4-electron states in terms of their
contributions from different Slater determinants, i.e. consisting of different
1-electron atomistic tight-binding basis states. This will help us to
determine the regimes of $j_{1},j_{2},j_{M}$ where the eigenstates can be
confidently described in terms of singlet-triplet symmetries. First, we focus
on [100] crystal direction and the equidistant case
$r_{1}=r_{2}=r_{M}=18a_{0}$. In Figure 8 we plot several of the lowest-energy
single electron states.

  * •

We can see that the ground and first-excited orbitals (states numbered 1 and 3
in the figure) are localized mainly in the two middle dots. This is because
superposing the four donors confining potentials in TB calculations results in
the total potential being slightly deeper within the middle dots than the
outer ones.

  * •

The next two excited orbitals (states 5 and 7) are localized mainly within two
outer dots.

  * •

We consider the even-numbered states (2,4,6,8…) to be of the same orbital as
odd (1,3,5,7…) but of opposite spin.

  * •

The 4-electron ground state, labelled by us as $S_{l}$, has main contributions
from following configurations: 14.7% $\ket{1,3,6,8}$, 14.7% $\ket{2,4,5,7}$,
10.7% $\ket{1,2,5,6}$, 9.5% $\ket{1,2,7,8}$, 8.9% $\ket{3,4,7,8}$, 7.8%
$\ket{3,4,5,6}$, and smaller contributions from other configurations.

  * •

The first two contributions can be interpreted as inner-triplet outer-triplet
states ($\ket{1,3,6,8}$ is $T_{-}$ in inner dots and $T_{+}$ in outer dots,
$\ket{2,4,5,7}$ is opposite to that).

  * •

Other configurations are inner-singlet outer-singlet states.

  * •

We can then see that a significant part of the ground state is described as an
inner-dot triplet, which is specific for $j_{1,2}\approx j_{M}$ regime and has
been expected by us following the above discussion on effective Hamiltonian
states.

  * •

To obtain a rough estimate of the percentage of singlet and triplet
contributions within the middle dots we sum up contributions from states
$\\{\ket{1,2,i,j},\ket{3,4,i,j}\\}$ for singlet and
$\\{\ket{1,3,i,j},\ket{1,4,i,j}$, $\ket{2,3,i,j}$, $\ket{2,4,i,j}\\}$ for
triplet.

We show the results in Table 1. As expected from the effective Hamiltonian
model here both contributions are close to 50%. The presented numbers are,
however, just an estimate, not a precise evaluation of singlet and triplet
admixtures in the middle dots as i) we do not sum up all the higher-order
Slater determinant probabilities and ii) the orbitals 1 and 3 have small but
non-zero probabilities also in the outer dots.

Next, we look at the non-equidistant case, with $r_{1}=r_{2}>r_{M}$ which
guarantees $j_{1}=j_{2}<j_{M}$. In Table 1 we compare inner-singlet and inner-
triplet contributions in $S_{l}$ for 3 values of $r_{M}$, i.e. $18a_{0}$,
$16a_{0}$ and $14a_{0}$, while keeping the total four-donor separation $R$
constant at $54a_{0}$. We can see the inner-singlet (inner-triplet)
contribution is significantly increasing (decreasing) when the middle donors
are pulled closer together.

Separation ($a_{0}$) | Inner singlet contribution | Inner triplet contribution
---|---|---
18 | 0.4927 | 0.483
16 | 0.9209 | 0.06706
14 | 0.97904 | 0.00238
Table 1: Approximate contributions of the inner-singlet and inner-triplet
configurations in the FCI ground state of a 4-donor chain separated along the
[100] direction Here we show 3 different middle donor separations $18a_{0}$,
$16a_{0}$ and $14a_{0}$ with a constant outer donor separation of $54a_{0}$.
We see that the inner singlet contribution to the ground state dramatically
increases from $\sim 49\%$ to $\sim 98\%$ as the separation decreases by
$4a_{0}$.