Ab Initio Study of Phosphorus Donors Acting as Quantum Bits in
Binghai Yan,*,†Riccardo Rurali,*,‡and Ádám Gali*,§,∥
†Bremen Center for Computational Materials Science, Universität Bremen, Am Fallturm 1, 28359 Bremen, Germany
‡Institut de Ciència de Materials de Barcelona (ICMAB−CSIC), Campus de Bellterra, 08193 Bellaterra, Barcelona, Spain
§Institute for Solid State Physics and Optics, Wigner Research Center for Physics, Hungarian Academy of Sciences, P.O. Box 49, 1525
∥Department of Atomic Physics, Budapest University of Technology and Economics, Budafoki út 8., 1111 Budapest, Hungary
ABSTRACT: A phosphorus (P) donor has been extensively
studied in bulk Si to realize the concept of Kane quantum
computers. In most cases the quantum bit was realized as an
entanglement between the donor electron spin and the
nonzero nuclei spin of the donor impurity mediated by the
hyperfine coupling between them. The donor ionization
energies and the spin−lattice relaxation time limited the
temperatures to a few kelvin in these experiments. Here, we
demonstrate by means of ab initio density functional theory
calculations that quantum confinement in thin Si nanowires
(SiNWs) results in (i) larger excitation energies of donor
impurity and (ii) a sensitive manipulation of the hyperfine
coupling by external electric field. We propose that these
features may allow to realize the quantum bit (qubit) experiments at elevated temperatures with a strength of electric fields
applicable in current field-effect transistor technology. We also show that the strength of quantum confinement and the presence
of strain induced by the surface termination may significantly affect the ground and excited states of the donors in thin SiNWs,
possibly allowing an optical read-out of the electron spin.
KEYWORDS: silicon nanowires, qubits, density-functional theory, electronic structure, excited states, hyperfine coupling
(qubit), an extremely fragile, yet very powerful, elemental piece
of information. Since the first proposal of Kane,1increasing
attention has been devoted in the past decade to the
implementation of quantum computing within the realm of
A qubit is a two-level system whose quantum states replace
the ones and zeros of conventional computers. Tight
requirements must be fulfilled by qubit candidates. In particular
they have to be virtually isolated from the surrounding
environment, to avoid decoherence, yet amenable to quick
read-outs when the state of the qubit must be known, two
conditions that are often difficult to reconcile. The simple and
powerful scheme proposed by Kane consists of using electron
spins for processing, while storing the quantum state in the
nuclear spins, which have much longer coherence times. This
transfer of information is possible by means of the hyperfine
interaction that couples both spin degrees of freedom.
31P donors in an isotopically purified28Si crystal satisfy all of
the conditions for this scheme to be effective: (i) in a neutral
donor the electron wave function is localized around the donor
nucleus; thus, the hyperfine interaction is strong; (ii) all of the
host nuclei has spin I = 0 and are not a source of decoherence
he formidable challenge underlying quantum computation
consists in protecting the coherence of the quantum bit
for the electron spin of the donor. Several demonstrations of
this approach have been reported recently.2−8Two further
requisites, however, would make this approach more efficient:
(i) the donor state should not be too shallow, so that
temperatures at a few kelvin are no longer required for its
electron not to be thermally excited to the valley-orbit excited
state9and (ii) an increased strength of the hyperfine interaction
would make it easier to transfer the quantum state back and
forth from the electron to the nucleus spin. In this letter we
show that both these conditions are fulfilled by a P dopant in Si
nanowires (SiNWs) within the quantum confinement regime.
We further show that a special interplay of quantum
confinement and strain induced by surface termination may
allow to read-out the electron spins optically.
SiNWs are among the most interesting materials for novel
electron devices,10and the possibility to use them as a platform
for quantum computing would be interesting per se. Due to
their potentality to replace conventional solid-state devices in
some applications and to the ideal compatibility with existing
Published: June 13, 2012
February 28, 2012
May 15, 2012
© 2012 American Chemical Society
dx.doi.org/10.1021/nl300816t | Nano Lett. 2012, 12, 3460−3465
CMOS technology, enormous research efforts are being
devoted to the control of the structural and electrical properties
of these materials. The atom-by-atom positioning of dopant
impurities required by Kane’s quantum computer scheme is
challenging by itself, even in bulk Si. Yet, great progress has
been made in SiNWs,11−13and single-dopant experiments have
been reported.14,15When their diameter is reduced below 10
nm, SiNWs enter the quantum confinement regime, whose
major fingerprints are the broadening of the electronic band
gap16−19and the deepening of the impurity levels.20−22Even at
larger diameters, where quantum confinement has vanished, it
has been shown that impurity levels continue to be deeper than
in bulk Si, because of dielectric mismatch.20In this work we
study SiNWs below 3.0 nm where quantum confinement is the
Computational Methods. We perform density-functional
theory (DFT) calculations with the Vienna ab initio simulation
package (VASP) code,23,24using Perdew, Burke, and Ernzerhof
(PBE)25semilocal functional, a plane wave basis set with cutoff
of 450 eV, and standard projector augmented wave (PAW)
potentials26for the nuclei. We consider P substitutional
impurities in 1.0, 1.5, and 2.0 nm SiNWs grown along the
⟨110⟩ axis, considering both cases in which the wire’s surfaces
are H- and CHxpassivated (see discussion below). In some
cases we checked the trend for 3.0 nm SiNWs as well. We
consider a 1 × 1 × 5 supercell which results in an isolated
donor states27and an impurity concentration of about 1019
cm−3, of the order of experimental reports of SiNWs in the high
doping regime.28The energy and the electronic structure of the
excited state are studied by means of constrained DFT
calculations, where the donor electron is promoted from the
occupied Kohn−Sham state to the lowest lying empty state.29It
should be recalled that DFT-PBE suffers from the self-
interaction error which results in an underestimation of the
band gap. Nevertheless, the curvature of the bands is well-
reproduced. In this study, we focus our investigation on the
excitation energy of shallow donors where the total energy
difference between the shallow excited and ground states is
calculated. In this calculation procedure, most of the self-
interaction error is cancelled, and DFT-PBE values provide a
reasonable estimate and can predict the trends as a function of
the size or surface termination of SiNWs well.
Hyperfine interaction between donor electron and nuclei
spins is of prime interest in qubit applications where the so-
called Fermi-contact interaction dominates because the donor
electron ground state sits on s-type hydrogen effective mass
theory (EMT) state.21The Fermi-contact hyperfine interaction
(A) in atomic units reads as
2/3 ( )
where μBis the Bohr magneton, γeand γIare electron and
nuclear magnetogyric ratios with the given nuclei I, and ρspinis
the electron spin density at that nucleus (RI). Equation 1 may
be converted to the most convenient MHz unit by the
where γP= 17.235 and ρspinis in atomic units. We applied the
all-electron spin density at the nucleus position within the PAW
framework to obtain reliable values for the hyperfine constants.
This process is repeated for several values of an external applied
electric field, transverse to the nanowire axis.30
Electronic Structure of the Pristine NW and Surface
Passivation. At first, we focus on the electronic structure of
the pristine SiNW. The nature of conduction band edge of the
pristine system is very important, because the donor state of an
effective mass like impurity originates from split conduction
band states and the valley-orbit splitting can be much larger
than in the bulk counterpart. We consider ⟨110⟩ SiNWs, the
most common growth orientation for wires below 10 nm.31It is
important to note that the local point group symmetry is
significantly reduced in these ultrathin nanowires (C2v)
compared to the Tdsymmetry in bulk counterpart. Hence,
while bulk Si is an indirect semiconductor, ⟨110⟩ SiNWs are
direct semiconductors at the Γ-point with a projection of the Z-
band when the surface is H-terminated (see Figure 1).10,32
The hydrogen-terminated surface is a commonly adopted
model to simulate SiNWs (see refs 10 and 33 and references
therein). Halogens (such as F, Cl, Br, and I)33,34and the
functional groups OH and NH2
the energy gap, relative to the H-passivated nanowires, while
recent models considering Si/SiOx interfaces37,38showed
dramatic modifications to the states near the energy gap.
Although it is widely used in simulations10and experiments,16
however, hydrogen passivation is not very stable in air, and
wires are thermally oxidized even at room temperature.
Recently, it has been shown that much more stable and
oxidation-resistant passivation can be obtained by saturating the
surface dangling bonds with CmHnchains,39−41full coverage
being achieved with the shorter chains, that is, CH3. Thus, we
also studied CHxterminated wires too where x labels the
number of H-atoms needed to make tetrahedral bonds for the
C-atom. CHxtermination will result in charge transfer between
the surface Si and C atoms, and the surface strain will elongate
the Si−Si bonds even in the middle of the wire. As a
consequence, the band structure shows more complex behavior
in CHx-terminated SiNWs than in the H-terminated ones. The
H-terminated wires exhibit a direct energy gap for different
diameters. In contrast, we observed a direct−indirect band gap
transition for CHx-terminated ones: for d < 2 nm wires the
34−36were reported to shrink
Figure 1. Schematic sketch of the band structure of perfect ⟨110⟩
SiNWs near the Fermi-level, azbeing the axial lattice parameter. The
Z-band is a projection of the CBM-related band of bulk Si along the
 direction perpendicular to the axis of the wire while the X−Y
band is the projection along the (001) horizontal plane. Next to the
band structure, the lowest thermal excitations of a P-donor in bulk Si
(left) and in a ⟨110⟩ SiNW (right) are shown, where the values in
parentheses are for H-termination (see Figure 2). The symmetry of
the valley-orbit excited state in the case of the nanowire, that is, A1or
B2, depends on the surface passivation, that is, H or CHx, and on the
diameter of the wire.
dx.doi.org/10.1021/nl300816t | Nano Lett. 2012, 12, 3460−3465
band gap is direct at a Γ-point with the Z-band, while the
conduction band minimum (CBM) built from the X−Y band is
close to the Brillouin-zone edge for d ≥ 2 nm (see Figure 1).
We define the Z-band as a projection of the CBM-related band
of bulk Si along the ⟨001⟩ direction perpendicular to the axis of
the wire, while X−Y band is the projection along the (001)
Phosphorus-Based Qubits. We now move to the analysis
of the SiNW qubits, implemented by substitutional P in the
core of the H-terminated SiNW. To start, it is imperative to
analyze the nature of donor states in detail, of particular
importance, for qubit applications. In bulk Si the CBM is 6×
degenerate and splits to A1, T2, and E symmetry under the Td
point group symmetry of the substitutional EMT donors, with
A1state lowest in energy. In bulk Si the lowest excitation may
occur between A1and T2states with energy difference of about
12 meV which is only weakly allowed, while the dipole allowed
transition occurs between A1(1s) and (2p0) EMT states42with
an excitation energy of about 34 meV. In ⟨110⟩ SiNWs the
EMT donors have C2vsymmetry which is significantly lower
than Td. As a consequence, T2splits to A1, B2, and B1states,
while E state splits to A1and A2(see the sketch in Figure 1).
We are able to identify all of these split states from the
conduction band in SiNWs. In any surface termination or size
A1 symmetry is the ground state for the P donor. The
symmetry of the lowest excited state depends on the band
structure of the host nanowires: in the case of direct-gap SiNWs
it transforms as A1, whereas it transforms as B2in indirect-gap
SiNWs. We notice that both A1→ A1and A1→ B2transitions
are principally allowed in C2v symmetry and might have
nonzero probability because of the low symmetry, in contrast to
their bulk counterpart.
The thermal excitation energy for H-terminated wiresthe
difference E1− E0between the lowest energy of the ground and
valley-orbit excited states (see Figure 2)ranges from 420 to
90 meV, the larger values corresponding to the thinner wires
where quantum confinement is stronger. This energy amounts
to about 12 meV in bulk Si.42One of our purposes here is
increasing the operation temperature of a Kane-like quantum
computing scheme. In these conditions vertical excitations can
also play a role, and the highest probability of the absorption
and emission processes may occur at the structural config-
uration of the initial states. The difference between the vertical
absorption (emission) energy and the minimum thermal
excitation energy discussed above (i.e., the difference between
E′1and E1and between E′0and E0) is a good indicator of how
large the geometry relaxation is upon excitation/de-excitation.
In general, the more the wave function is localized, the larger
the relaxation energy is expected to be. Indeed, this is what we
observe: for the 1.0 nm SiNW, where the donor wave function
is more strongly squeezed by the confinement potential, the
relaxation energy is 130 meV vs 30 and 20 meV for the 1.5 and
2.0 nm wires, respectively.
Due to the potential interest that CHx-termination could
have in future NW-based devices, we have verified to what
extent the results depend on the nature of the different surface
chemistries. Excitation energies are slightly reduced; the
minimal thermal excitations are now 470, 180, and 80 meV
for the diameters investigated, and the overall conclusions
excitation energies about eight times larger than in bulkare
Up to this point we have shown that a donor spin state is
deeper in a strongly confined geometry. Hence, it possible to
envisage quantum computing operations at higher temperatures
than those reported so far because a longer spin−lattice
relaxation time is expected according to our results. The
confinement increases the localization of the donor wave
function, which can be effectively treated as the s-state of a
Figure 2. Cross-section view and sketch of the total energy as a function of the configuration coordinate q for (a) 1.0, (b) 1.5, and (c) 2.0 nm CHx-
passivated SiNWs. Light yellow, dark green, and white spheres represent Si, C, and H atoms, respectively. The values of the thermal excitation E1−
E0and of the vertical adsorption and emission, E′1− E0and E1− E′0, in meV are indicated (in parentheses the values for the same wires with H-
dx.doi.org/10.1021/nl300816t | Nano Lett. 2012, 12, 3460−3465
where a is the effective Bohr radius. The excitation energy Eiis
known to be roughly proportional to 1/a, then the hyperfine
constant is related to Eiin the way,21
= | ∝
This simple model shows that increasing the confinement
leads to an increase of the excitation energy and, accordingly, of
the hyperfine interaction. Both electron spin-resonance experi-
ments43,44and theoretical calculations45,46revealed the
enhanced hyperfine splitting of the P donor in silicon
nanocrystals. Recently ab initio calculations also demonstrated
the size-dependent hyperfine constant for P21,30and Se
dopants47in H-passivated SiNWs.
Here we have further studied the hyperfine interactions for
both H and CHxterminations and explored the possibility of
external field manipulation. To this end, we compute the
hyperfine interaction for P-donors in Si structures. Our method
can reproduce the experimental data in bulk Si for P donors
well: we obtain a 117.0 MHz hyperfine constant in a 512-atom
supercell calculation that should be tentatively compared to
117.53 MHz.48,49In our representative CHx-passivated SiNW
with d = 2.0 nm, we find the hyperfine interaction to be 665
MHz, about six times larger than in bulk Si, for a P substituting
at one of the innermost Si lattice sites. This is a direct
consequence of the confinement of the donor wave function, as
discussed above. Additionally, in CHx-terminated SiNWs the
Si−Si bonds are about 0.05 Å longer than those in H-
terminated SiNWs; thus the P−Si bond lengths are longer than
in H-terminated SiNWs. This results in a larger hyperfine
interaction with P in CHx-terminated than in H-terminated
SiNWs (see Table 1).
What is more important, however, is that the hyperfine
interaction is sensitive to an external applied electric field, thus
making switch on and off the transfer of the quantum state of
the electron to the nucleus easier and vice versa (see Figure 3).
At moderate fields the hyperfine coupling changes more than
1%. The hyperfine coupling could thus effectively be controlled
by an external electric field, for instance through an additional
control gate underneath the P-donor. Multiple control gates
could be fabricated along the quasi 1D chain of P-donors acting
as solid state qubits. Noteworthy, the realization of such a
multigate structure has been recently demonstrated.50We note
that the scalability of quasi 1D chain of qubits is restricted
compared to the original proposal of Kane with quasi 2D planar
configuration of qubits. However, a great advantage of the
former is the ultimate control of hyperfine coupling by allowing
its value to decrease and increase, which adds an additional
degree of freedom to manipulate the qubits with respect to the
original proposal where the hyperfine coupling could be only
decreased to a much less extent.
Substitutional impurities have been reported to have a
tendency to surface segregation, even in the absence of surface
defects.51Therefore, in an effort to describe a more realistic
configuration for the P-based qubit, we have also calculated the
hyperfine coupling for the case of a subsurface substitutional.
When the impurity is closer to the sidewalls of the nanowire, its
wave function is more localized, and a larger hyperfine
interaction is expected. In agreement with this intuitive picture,
we indeed find a value of 807 MHz for our representative CHx-
terminated d = 2.0 nm wire. Again, the most interesting result is
the response to an applied transverse electric field. As shown in
Figure 3, for P-substitutionals closer to the wire surface, the
hyperfine coupling varies about 7 times faster compared to the
innermost substitutional. Therefore, even a a moderate electric
field of 0.05 V/Å = 500 V/μm can induce a significant change
of up to 11% in the hyperfine coupling.
These results can be rationalized in the following way. Under
an electric field ε, the donor excitation energy is Ei= Ei0+
εΔr,30where Ei0is the excitation energy at ε = 0, and Δr is the
real-space distance between the donor state and CBM. The
hyperfine constant can be derived from eq 4 as A ∝ (Ei0+
where A0is the hyperfine constant at ε = 0, and c is a fitting
parameter. This linear relation is in good agreement with the
calculated results shown in Figure 3. We fit our ab initio results
by using Δr = 6.38 Å, for the SiNW with P near the surface and
obtain c = 293.3 MHz/V for the 2.0 nm CHx-terminated wire.
Equation 5 indicates that the response of the hyperfine
coupling to an external electric field can be enhanced by
increasing Δr, i.e. moving the P-donor closer to the surface of
the SiNW. In a recent experiment, the Sb donor electron spin
2εΔr, which can be rewritten as
Table 1. Isotropic Hyperfine Coupling (HF) as a Function of
the Size and Termination of Si Nanowire for a P Impurity
Substituting Si-Site in the Middle of the Wire (See Inset in
diameter (nm)HF (MHz)a1, a2(Å)
aThe corresponding distances between P and first neighbor Si atoms
are also given for inequivalent bond lengths (a1, a2) obtained by DFT-
PBE geometry optimization (see Figure 1).
Figure 3. Hyperfine splitting (HF) as a function of an applied
transverse electric field (E) for two different P substitutionals: at the
innermost part of the wire (circles) and in a subsurface location
(diamonds). We plot the relative variation with respect to the value
without a field, δHF = (HF(E) − HF(E = 0))/HF(E = 0). The
substitutional sites considered are indicated in the inset.
dx.doi.org/10.1021/nl300816t | Nano Lett. 2012, 12, 3460−3465
in silicon was tuned by an electric field (Stark effect) ∼ 0.1 V/
μm, resulting in a hyperfine interaction shift of 0.025 MHz. In
our proposed device with a P-donor at Δr = 6.38 Å the same
shift can be obtained at a very similar ε of 0.13 V/μm. In
another recent work,6a piezoelectric actuator reduced the
hyperfine constant of P by 0.9 MHz. This 0.9 MHz shift makes
quantum computation feasible in the system. Using the
aforementioned SiNW-based architecture with Δr = 6.38 Å,
one can reach the same shift by an electric field 5 V/μm. In
state-of-art field effect devices the applied electric fields are
about 600 V/μm;52thus the values required for the qubit
operation could be easily achieved.
In the previous paragraphs we have proposed that P-donor
qubits may be manipulated much more effectively and operate
at elevated temperatures due to the increased energy scale of P-
donor levels. We have also showed that the subtle interplay
between quantum confinement and surface termination could
lead to direct gap SiNWs. Here, we envisage that the
combination of the unique properties of these SiNWs may
allow the read-out of the P-donor spins optically. A special
excited state of donors can be achieved after optical excitation
which may be described as a complex of an exciton bound by
the neutral defect (see Figure 4). According to our first
principles calculation, the binding energy of the free exciton to
this special excited state is 2−7 times larger than in bulk Si,
depending on the size and/or surface termination, as expected
due to quantum confinement. The nonradiative Auger-
recombination of this special excited state dominates over the
radiative decay in indirect band gap semiconductors, such as
bulk Si. However, this should be not the case for direct gap
SiNWs where an effective luminescence may be expected from
the excited state of P-donors.53By using this advantage of
silicon nanowires, one can use a protocol to initialize and read-
out P-donor spins optically as it has been recently developed
for InGaAs quantum dots.54After applying a small external
magnetic field, the corresponding ground and excited splits due
to the Zeeman effect (see Figure 4). By ignoring the higher
excited state, a so-called λ-type three-level artificial atom is
formed with |0⟩, |1⟩, and |e⟩ sublevels of the donor electron.
Upon continuous-wave irradiation of narrow-band laser
light (Ω) tuned to the transition frequency between states |1⟩
and |e⟩, the electron spin state is finally initialized to the ground
state |0⟩ after several cycles of absorption and spontaneous
emission of photons because of the nonzero spontaneous
emission rate (Γ0) to |0⟩ state. Optical pumping in nanosecond
pumping time is sufficient for such initialization.54This
principle of optical pumping can be used to the measurement
of an electron spin. When a photon with an energy
corresponding to the transition frequency between states |0⟩
and |e⟩ is counted upon the irradiation of pumping light, the
initial state of the electron spin is determined to have been |1⟩.
In contrast, when no photon is counted, the initial state is
determined to have been |0⟩.55Briefly, the increased energy
scale of P-donors in direct-gap silicon nanowires might open
the door to optically read-out the donor spins by a λ-type
transition which could be a robust process.
In summary, we showed by first principles calculations that
donor electrons in SiNWs within the quantum confinement
regime may act as quantum bits and show advantages over their
bulk counterpart. The fine interplay between the effect of
quantum confinement and surface states allows us to design
such quantum bits that have a sufficiently deep donor state and
large hyperfine interaction to control the donor electron and
nuclei spins by nuclear magnetic resonance techniques, and this
may operate at elevated temperatures. In addition, the
hyperfine interaction is remarkably sensitive to external
perturbations like external electric field which may allow to
manipulate the donor spins very efficiently based on Si CMOS
technology. By controlling the orientation, size, and surface
termination of SiNWs one might apply a robust optical read-
out of the donor electron spin that is barely possible56in bulk
Si. Realistic features such as CHxpassivation or subsurface
location of the P impurity have been explicitly included in our
model; they turn out to contribute to an overall out-
performance of SiNW-based qubit over bulk Si, especially
concerning the hyperfine coupling sensitivity. Admittedly, our
proposal requires an ultimate and angstrom-resolution control
of P-doping in silicon nanowire. We note that recent advances
of technologies57,58may realize this idea soon.
*E-mail: email@example.com; firstname.lastname@example.org;
The authors declare no competing financial interest.
R.R. acknowledges funding under Contract Nos. 200950I164,
TEC2009-06986, FIS2009-12721-C04-03, CSD2007-00041
and computational resources at the Centro de Super-
computación de Galicia (CESGA). A.G. thanks the discussion
with A. Morello. B.Y. acknowledges financial support by the
Alexander von Humboldt Foundation of Germany and support
by the Supercomputer Center of Northern Germany.
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