Atomic clock transitions in silicon-based spin qubits

Abstract

A major challenge in using spins in the solid state for quantum technologies is protecting them from sources of decoherence. This is particularly important in nanodevices where the proximity of material interfaces, and their associated defects, can play a limiting role. Spin decoherence can be addressed to varying degrees by improving material purity or isotopic composition, for example, or active error correction methods such as dynamic decoupling (or even combinations of the two). However, a powerful method applied to trapped ions in the context of atomic clocks is the use of particular spin transitions that are inherently robust to external perturbations. Here, we show that such 'clock transitions' can be observed for electron spins in the solid state, in particular using bismuth donors in silicon. This leads to dramatic enhancements in the electron spin coherence time, exceeding seconds. We find that electron spin qubits based on clock transitions become less sensitive to the local magnetic environment, including the presence of (29)Si nuclear spins as found in natural silicon. We expect the use of such clock transitions will be of additional significance for donor spins in nanodevices, mitigating the effects of magnetic or electric field noise arising from nearby interfaces and gates.

Full text

Atomic clock transitions in silicon-based spin qubits
Gary Wolfowicz,
1, 2, ∗
Alexei M. Tyryshkin,
3
Richard E. George,
1
Helge Riemann,
4
Nikolai V. Abrosimov,
4
Peter Becker,
5
Hans-Joachim Pohl,
6
Mike L. W. Thewalt,
7
Stephen A. Lyon,
3
and John J. L. Morton
1, 8, †
1
London Centre for Nanotechnology, University College London, London WC1H 0AH, UK
2
Dept. of Materials, Oxford University, Oxford OX1 3PH, UK
3
Dept. of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA
4
Institute for Crystal Growth, Max-Born Strasse 2, D-12489 Berlin, Germany
5
Physikalisch-Technische Bundesanstalt, D-38116 Braunschweig, Germany
6
Vitcon Projectconsult GmbH, 07745 Jena, Germany
7
Dept. of Physics, Simon Fraser University, Burnaby, British Columbia V5A 1S6, Canada
8
Dept. of Electronic & Electrical Engineering, University College London, London WC1E 7JE, UK
(Dated: January 10, 2014)
A major challenge in using spins in the solid state for quantum technologies is protecting them
from sources of decoherence. This can be addressed, to varying degrees, by improving material
purity or isotopic composition [1, 2] for example, or active error correction methods such as dynamic
decoupling [3, 4], or even combinations of the two [5, 6]. However, a powerful method applied to
trapped ions in the context of frequency standards and atomic clocks [7, 8], is the use of particular
spin transitions which are inherently robust to external perturbations. Here we show that such ‘clock
transitions’ (CTs) can be observed for electron spins in the solid state, in particular using bismuth
donors in silicon [9, 10]. This leads to dramatic enhancements in the electron spin coherence time,
exceeding seconds. We find that electron spin qubits based on CTs become less sensitive to the local
magnetic environment, including the presence of
29
Si nuclear spins as found in natural silicon. We
expect the use of such CTs will be of additional importance for donor spins in future devices [11],
mitigating the effects of magnetic or electric field noise arising from nearby interfaces.
Out of the various candidates for solid state qubits,
spins have been of particular interest due to their rel-
ative robustness to decoherence compared to other de-
grees of freedom such as charge. So far, the most coher-
ent solid state systems investigated have been the spins
of well-isolated donors in bulk 28-silicon, with coherence
times (T
2
) of up to seconds (extrapolated) for the electron
spin [1] and minutes for the nuclear spin [5], comparable
to those of ion trap qubits [12, 13]. However, in practical
devices, spin coherence times are likely to be limited by
factors such as coupling to nearby qubits and magnetic
or electric field noise from the environment. For exam-
ple, cross-talk with other donors 100 nm away limits the
electron spin T
2e
to a few milliseconds [1], while a nearby
interface can limit the donor electron spin T
2e
to 0.3 ms
at 5.2 K [14]. Finally, without isotopic enrichment, the
5% natural abundance of
29
Si limits the electron spin
T
2e
to less than 1 ms [9, 10].
An approach to creating more robust qubits is to tune
free parameters of the system Hamiltonian to obtain in-
sensitivity to specific sources of decoherence. This has
been extensively used in ion trap qubits to protect against
magnetic field fluctuations [12, 13], building on work on
atomic clocks where hyperfine states, used as frequency
standards, must remain stable against such variations.
These so-called “clock transitions” (CTs) have a transi-
tion frequency (f ) which is insensitive to magnetic field
(B) variations, at least to first-order (in other words
df/dB = 0). More recently, superconducting circuit
qubits have also taken advantage of a tuned Hamiltonian
to remain immune to charge, flux or current noise [15, 16].
Nuclear spin CTs in rare-earth dopants (nuclear spins
I > 5/2) have been studied in the context of optical quan-
tum memories [17, 18] leading to a 600-fold improvement
of the coherence times to 150 ms, limited by second-order
effects, while recent experiments on phosphorus donor
nuclear spins also exploited a CT [5]. For electron spins
in the solid-state, CTs remain relatively unused due in
part to the requirement of a spin Hamiltonian of sufficient
complexity. One of the richest single-defect spin systems
is the bismuth donor in silicon (Si:Bi), which possesses an
electron spin S = 1/2 coupled to a nuclear spin I = 9/2.
The electron spin decoherence rates for Si:Bi have been
found to follow df/dB in both natural silicon [9], and
isotopically enriched
28
Si [19]. These results, combined
with the identification of a number of CTs in the spin
Hamiltonian of Si:Bi [20, 21], motivate the study of spin
coherence times around CTs in Si:Bi, where df/dB → 0.
In this Letter, we investigate one such CT in Si:Bi, at
7.0317 GHz, using both natural silicon and
28
Si.
When describing the states of coupled electron and nu-
clear spins, two basis conventions are typically used: in
the high magnetic field limit, the electron and nuclear
spin projections m
S
and m
I
are good quantum numbers,
while in the zero-field limit, the total spin F (= I ± S)
and its projection m
F
(= m
S
+ m
I
) are used. CTs are
often found in an intermediate regime [22], nevertheless
it is possible to categorize them as nuclear magnetic reso-
nance (NMR)- or electron spin resonance (ESR)-type, on
the basis of whether the transition couples primarily to
S
x
or I
x
, where these are the electron and nuclear spin op-
erators perpendicular to the applied magnetic field. The
arXiv:1301.6567v1 [quant-ph] 28 Jan 2013

2
-10
10
0
Energy (GHz)
A B
|df/dB| (γ
e
)
0 100 200 300 400 500 600
0
0.2
1.0
0.4
0.6
0.8
0
20
100
40
60
80
|df/dB| (γ
n
)
70 74 78 82 86 90
7.0315
7.0320
7.0325
7.0330
7.0335
Transition frequency (GHz)
Transition
frequency (GHz)
5
10
–1 –2
F = 5
F = 4
m
F
=
ESR-type CTs
NMR-type CTs
X-band
FIG. 1. Electron spin resonance (ESR)-type clock transitions (CTs) of Si:Bi. A, The eigenstate energies (top) of
Si:Bi as function of magnetic field, the ESR- (black) and NMR-type (grey) transition frequencies between these states (middle),
and the first-order magnetic field dependence (df/dB) of these transition frequencies (bottom). ESR-type CTs (blue lines and
open circles) are found at 27, 80, 133 and 188 mT, and appear in the spectrum as doublets ∆F ∆m
F
= ±1 separated by up to
3 MHz. NMR-type CTs are found above 300 mT (red lines and open circles). B, Electron spin echo-detected magnetic field
sweeps around the 80 mT CT measured at microwave frequencies ≥ 7.0315 GHz. The transition probablities for ∆F∆m
F
= +1
(dark blue) and −1 (light blue) transitions are equal near the CT.
ESR-type CTs which we investigate in this manuscript
involve states which are close to pure in the |F, m
F
i ba-
sis and hence for convenience we label them according to
the dominant |F, m
F
i component (full details are given
in the Supplementary Material and in Ref [20]).
For bismuth donors in silicon, NMR-type CTs can
be found at high field (> 350 mT) with frequencies
around 1 GHz as shown in red in Figure 1A. At low
field (< 200 mT), four ESR-type CTs are present with
frequencies in the range 5.2 to 7.3 GHz as shown in blue
in the same figure. We will focus here on the ESR-type
CTs, which possess only slightly reduced spin manipu-
lation time compared to free electron spins as well as a
large energy splitting even at low magnetic field (which
has interesting applications for use in hybrid supercon-
ducting circuits [9, 23, 24]).
In the silicon samples we study here, Bi donors were
introduced during crystal growth using the method de-
veloped in Ref [25], with concentrations ranging from
3.6 × 10
14
cm
−3
to 4.4 × 10
15
cm
−3
. Pulsed-ESR experi-
ments were performed using a spectrometer based around
a modified Bruker Elexsys E580 system with a ∼7 GHz
loop-gap cavity (for the CT) and 9.75 GHz dielectric res-
onator.
Figure 1B shows ESR spectra measured using mi-
crowave frequencies between 7.031 and 7.034 GHz, by
plotting electron spin echo intensity as a function of mag-
netic field. The spectra show two transitions correspond-
ing to [{∆F, ∆m
F
} = {±1, ±1}] and [{∆F, ∆m
F
} =
{±1, ∓1}]; for brevity, these transitions can be distin-
guished by the value of the product ∆F ∆m
F
= ±1.
Together, they offer a controllable two-qubit subsystem
with low sensitivity to magnetic field fluctuations (see
inset of Figure 1B).
We model the ESR spectra using an isotropic spin
Hamiltonian common for group V donors in silicon:
H
0
= B
0
(γ
e
S
z
⊗ 1 − γ
n
1 ⊗ I
z
) + A
~
S.
~
I (1)
where the two first terms correspond to the electronic
(S) and nuclear (I) spin Zeeman interactions with an
external field B
0
and the last term corresponds to the
hyperfine coupling A. A common way to estimate Hamil-
tonian parameters such as the electron and nuclear gyro-
magnetic ratios (γ
e
and γ
n
) and the hyperfine constant is
by measuring the magnetic field dependences of the spin
transition frequencies. We use the opportunity provided
by the CT (with df /dB → 0) to extract a measure of the
hyperfine constant A = 1.47517(6) GHz with high preci-
sion, because uncertainties in the magnetic field become
irrelevant. In our simulations, we additionally use the
previously reported value of γ
e
= 27.997(1) GHz/T [26]
and the generic value of γ
n
= 7 MHz/T for
209
Bi [9].
Figure 1B shows that the ESR linewidth in the mag-
netic field domain increases around the CT: the deriva-
tive df /dB tends to zero hence its inverse, dB/df, di-
verges until it becomes limited by the non-linear terms

3
Resonant Bi
Central Bi
O-resonant Bi
209
Bi
SD (ID)dFF
{
29
Si
{
SD
{
z
S
z
S,
y
S
y
S+
x
S
x
S
z
I
z
S
{
SD (T
1e
, iFF)
z
S
z
S
T
2e
(s)
1
0.1
0.01
110
-1
10
-2
10
-3
|df/dB| (γ
e
)
10
3.6×10
14
cm
-3
2.0×10
15
cm
-3
4.4×10
15
cm
-3
dFF
iFF
ID
9.75 GHz
7 GHz
A
B
1 20 3 4 5
6
4
2
0
1/T
2e
(s
-1
)
Concentration (10
15
cm
-3
)
SD: spectral diusion
ID: instantaneous diusion
dFF/iFF: direct/indirect ip-op
FIG. 2. Decoherence mechanisms of Bi donors in sil-
icon and their dependence on df/dB. A, In the central
spin representation, a Bi donor is coupled to neighbouring Bi
donors as well as
29
Si spins. At the ESR CT, all spectral dif-
fusion (SD) contributions to decoherence are essentially elim-
inated, leaving only the direct flip-flop term (dFF) between
the central spin and a neighbouring, resonant Bi spin. B, T
2e
measurements at 4.8 K show a strong dependence on df/dB,
as shown for 3 different donor concentrations in
28
Si:Bi. Mea-
surements close to df /dB = γ
e
were taken using the ten X-
band ESR transitions, while the remaining points were taken
close to the CT. For each concentration, the dependence on
df/dB is modeled using contributions from ID, FF and iFF,
as shown separately in dashed lines for the lowest concen-
tration. Inset shows the limit of 1/T
2e
when approaching to
the exact CT as a function of donor concentration, showing a
nearly linear dependence, as expected for dFF.
in f(B
0
). These spectra are all well fit assuming a con-
stant linewidth in the frequency domain of 270 kHz. This
linewidth can be attributed to a distribution in the hy-
perfine constant of around 60 kHz, using ∆f =
df
dA
∆A
at the CT. Fourier-Transform ESR performed at a range
of frequencies confirmed that the ESR linewidth in fre-
quency domain is indeed magnetic field independent (see
Supplementary Material).
We now examine the decoherence mechanisms which
affect the electron spin of donors in silicon. At sufficiently
low temperature (< 5 K), spin-lattice relaxation T
1e
can
be mostly neglected, and dipolar interactions (∼ 2S
z
S
z
−
(S
x
S
x
+ S
y
S
y
)) with neighbouring spins are the primary
source of decoherence. In a central spin representation,
as shown in Figure 2A, the surrounding spins can be
divided into three categories: i) resonant spins affected
by microwave excitation; ii) off-resonant spins of the same
species, i.e. Bi spins in m
F
levels not addressed by the
microwaves; and iii) other spin species such as
29
Si. Away
from CTs, the limiting factor for electron spin coherence
times is spectral diffusion (SD) from the S
z
S
z
term of the
dipolar interaction. This term can be assimilated into
effective fluctuations in the magnetic field environment
of the central spin. SD is independent of any frequency
detuning between spins and thus is valid between the
central spin and any others.
In the static case, dipolar couplings to (ii) and (iii)
can be refocused with a microwave π-pulse such as in
the Hahn echo sequence. However, this does not cor-
rect for the dipolar coupling between resonant spins (i)
as both spins are simultaneously flipped by the π-pulse.
This is called “instantaneous diffusion” (ID) and lim-
its T
2e
to ∼ 10 − 100 ms for typical donor concentra-
tions (> 10
14
cm
−3
) [1, 19] [27]. Furthermore, dynamic
changes from spin flips in the environment cannot be refo-
cused. At high temperature, such flips arise from phonon
scattering but at low temperature, this is due to flip-flops
(FF) from the S
x
S
x
+ S
y
S
y
term of the dipolar interac-
tion. FF are energy conserving and as such are only rel-
evant between spins that have similar transition frequen-
cies. In natural silicon, the dominant decoherence mech-
anism is SD from
29
Si FF, while in isotopically enriched
28
Si, it arises from FF between resonant Bi spin pairs. In
the latter case, we distinguish between FF which involve
the central spin (direct FF, dFF), and those which do
not (indirect FF, iFF).
We begin by discussing results on samples of isotopi-
cally enriched
28
Si (100 ppm
29
Si). At the CT the transi-
tion frequency is insensitive to magnetic field fluctuations
in first order, so we expect SD to have little effect, leav-
ing only the dipolar coupling between resonant spin pairs.
With reference to Figure 2A, this implies then that all
terms apart from dFF vanish. In Figure 2B, measure-
ments of electron spin coherence times (T
2e
) are shown
for three different concentrations over a wide range of
df/dB. The data includes values measured at X-band as
well as those near the CT (∆F ∆m
F
= +1) at 79.8 mT,
7.0317 GHz. Measurements at the CT shown here were
taken at 4.8 K where T
1e
= 9 s, however no increase in
T
2e
was seen at lower temperature.
For each sample, enhancements of about two orders of
magnitude are seen at the CT, compared to the case for a
free electron g-factor, such as that of phosphorus donors.
As shown in Figure 2B, the dependence of the measured
T
2e
on df/dB arises from two factors: the effect on ID,
and on iFF. ID has a known quadratic dependence on the

4
420 6
0
1
Echo signal (a.u.)
Time, 2τ (s)
0.2
0.4
0.6
0.8
T
2e
= 2.7 s
0.20.10 0.3
T
2e
= 93 ms
28
Si:Bi
nat
Si:Bi
A B
π
2
π
ττ
Time, 2τ (s)
FIG. 3. Hahn echo decay at the CT. A,
28
Si:Bi at 4.3 K
with a Bi concentration of 3.6 × 10
14
cm
−3
. B,
nat
Si:Bi at
4.8 K with a Bi concentration of 10
15
cm
−3
. The decay in
natural Si is a stretched exponential, and therefore T
2e
is de-
fined as the time when the amplitude reaches 1/e. Magnitude
detection was used to eliminate instrumental noise, likely due
to phase noise in the microwave source.
gyromagnetic ratio of the central spin [19, 28], and be-
comes a negligible effect for df/dB < 0.1γ
e
. Indirect FF
dephase the central spin through the S
z
S
z
term, giving
a linear dependence of T
2e
on df /dB. Direct FF, on the
other hand, are not eliminated at the CT, and provide
an upper bound on T
2e
for a given donor spin concentra-
tion, as plotted in the inset of Figure 2B. For the lowest
concentration sample, electron spin coherence times of
up to 2.7 s were measured from simple two-pulse Hahn
echo decays, as shown in Figure 3A.
We now turn to measurements on Bi-doped natural sil-
icon (
nat
Si:Bi), which has 5%
29
Si. Away from the CT
the effect of the
29
Si (I = 1/2) is both to broaden the
ESR linewidth to about 0.4 mT (equivalent to 12 MHz
in the frequency domain for a free electron) due to unre-
solved
29
Si hyperfine, as well as to limit the T
2e
to about
0.8 ms due to SD [9]. At the CT we find that the ESR
linewidth reduces to 500 kHz (see Supplementary Ma-
terial), within a factor of two of the value for enriched
28
Si material, while T
2e
increases by over two orders of
magnitude to about 90 ms (Figure 3B). The effect of
the suppression of SD around the CT has been simu-
lated for
nat
Si:Bi using cluster expansion methods [29],
though further refinements are required in the simula-
tion before a quantitative comparison can be made. The
stretched-exponential decay implies that T
2e
is still lim-
ited at 93 ms by SD from
29
Si due to the second order
term (d
2
f/dB
2
6= 0). For modest
28
Si enrichment (e.g.
[
29
Si] ≈ 1000 ppm), T
2e
should already exceed seconds,
and indeed there may be an optimal
28
Si purity above
which T
2e
at the CT drops, due to the role of
29
Si or
30
Si
in detuning otherwise-identical spins [30].
We have shown how CTs in Si:Bi can be used to pro-
duce magnetic field-insensitive spin qubits with directly
measured coherence times of several seconds. Such qubits
would be insensitive to magnetic field noise arising, for
example, from fluctuating dangling-bond spins at the
Si/SiO
2
interface. Conversely, if electric field noise is
dominant, this can couple to donor spins via the hyper-
fine interaction and cause decoherence. Again, CTs can
be designed to be immune from electric charge noise by
selecting points where df/dA → 0 (see Supplementary
Material). Through the use of CTs, it is likely that the
seconds-long electron spin coherence times measured in
the bulk can be harnessed for spins in practical quantum
devices.
We thank Stephanie Simmons, Tania Monteiro and Se-
trak Balian for fruitful discussions. This research is sup-
ported by the EPSRC through the Materials World Net-
work (EP/I035536/1) and a DTA, as well as by the Eu-
ropean Research Council under the European Commu-
nity’s Seventh Framework Programme (FP7/2007-2013)
/ ERC grant agreement no. 279781. Work at Princeton
was supported by NSF through Materials World Net-
work (DMR-1107606) and through the Princeton MR-
SEC (DMR-0819860), and also by NSA/LPS through
LBNL (6970579). J.J.L.M. is supported by the Royal
Society.
∗
gary.wolfowicz@materials.ox.ac.uk
†
jjl.morton@ucl.ac.uk
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Supplementary Material
ESR- AND NMR-TYPE MAGNETIC FIELD “CLOCK” TRANSITIONS (CT)
A B
F + 1
F
m
F
m
F
-1
ESR-type CTs
Magnetic eld (mT)
0 50 100 150 200 250
-5
5
0
Energy (GHz)
-2
-4
-6
-8
ESR-type CTs
L-Z anticrossings
= +1
F
m∆F∆
1−=
F
m∆F∆
FIG. 1. Description of ESR-type CTs. A, The eigenstate energies of Si:Bi as function of
magnetic field. The color scale shows the logarithmic distance to pure Bell states (Landau-
Zener (L-Z) anticrossings) in the |m
S
, m
I
i basis, defined as log(|θ − π/4|) for an eigenstate
|Φi = cos(θ)


∓
1
2
, m
I
±
1
2

± sin(θ)


±
1
2
, m
I
∓
1
2

. The Bell state at 0 mT is barely visible here
due to degeneracy. B, ESR-type CTs with ∆F ∆m
F
= +1 in dark blue and −1 in light blue. The
four involved eigenstates, which are two pairs of hyperfine coupled states in the |m
S
, m
I
i basis,
form a subspace of the Hilbert space.
Below we discuss the general requirements for ESR and NMR-type CTs for systems with
electron spin S = 1/2 and nuclear spin I and assuming an isotropic hyperfine coupling,
with a particular focus on Group V donors in silicon. This letter is the first measurement of
ESR-type CTs to our knowledge, though they have been theoretically described by Moham-
mady et.al. [1, 2] for donors in Si, in particular Bi. On the other hand, NMR-type CTs have
been used in various systems in the past [3, 4], including in phosphorus donors in silicon [5],
to reduce sensitivity to magnetic field inhomogeneities (i.e. increase frequency resolution)
or to increase nuclear coherence times.
In the basis of the electron and nuclear spin |m
S
, m
I
i, the isotropic hyperfine interaction
(A
~
I ·
~
S) couples pairs of states within the Hilbert space such that [∆m
S
= ±1, ∆m
I
= ∓1].
In the strongly coupled electron-nuclear spin basis |F, m
F
i (F = I ±S, m
F
= m
S
+m
I
), these
pairs of states share the same m
F
value. When the static magnetic field is increased, the
Zeeman energy rises to the same order of magnitude as the hyperfine interaction, resulting
1

in avoided Landau-Zener crossings between states with m
F
≤ 0 as shown in Figure 1A.
ESR-type CTs are located between pairs of these avoided crossings.
At the intermediate fields relevant to CTs and Landau-Zener anticrossing, the eigenstates
can be expressed either as being close to Bell states in the |m
S
, m
I
i basis, or still quite pure
in the |F, m
F
i basis. For example, the CT at 7.0317 GHz, explored in the main text,
connects the following pairs of states:
−0.74




1
2
, −
5
2

+ 0.67




−
1
2
, −
3
2

⇔ 0.74




1
2
, −
3
2

+ 0.67




−
1
2
, −
1
2

in the |m
S
, m
I
i basis
0.99 |4, −2i + 0.15 |5, −2i ⇔ −0.15 |4, −1i + 0.99 |5, −1i in the |F, m
F
i basis
Hence, for convenience, we refer to these states by the dominant term in the |F, m
F
i basis
(i.e. |4, −2i and |5, −1i in the example above). The ∆F ∆m
F
= +1 and ∆F ∆m
F
= −1 CTs
are each transitions between one state of the first Landau-Zener crossing to a second state
of the second crossing, forming a 4-dimensional subspace of the Hilbert space (Figure 1B).
For non-integer [integer] values of I, there are I + 1/2 [I] Landau-Zener anticrossings and
consequently 2(I − 1/2) [2I] ESR-type CTs. The minimum complexity required is thus a
nuclear spin I ≥ 1 to have at least two pairs of hyperfine coupled states, which is not the
case for phosphorus donors (
31
P has I = 1/2).
Arsenic (
75
As, I = 3/2) and antimony (
121
Sb, I = 5/2 and
123
Sb, I = 7/2) have sufficient
nuclear spin to permit ESR-type CTs, however the hyperfine coupling is relatively weak
(A ∼ 198, 186 and 101 MHz, respectively), so the avoided crossings are found at low
magnetic field and transition frequencies (see Table I). Bismuth
209
Bi with I = 9/2 and
A = 1.475 GHz is thus the optimal Group V donor in silicon from the point of view of CTs,
possessing four of them at GHz frequencies.
75
As (I = 3/2)
121
Sb (I = 5/2)
123
Sb (I = 7/2)
209
Bi (I = 9/2)
∆F = +1, m
F
= −1 ↔ 0 −1 ↔ 0 −2 ↔ −1 −1 ↔ 0 −2 ↔ −1 −3 ↔ −2 −1 ↔ 0 −2 ↔ −1 −3 ↔ −2 −4 ↔ −3
Magnetic field (mT) 3.8 3.4 10.4 1.8 5.5 9.3 26.6 79.8 133.3 187.8
Frequency (GHz) 0.384 0.552 0.482 0.403 0.376 0.314 7.338 7.032 6.372 5.214
TABLE I. Summary of ESR-type magnetic-field CTs in donors in silicon. CTs exist in
pairs (∆F ∆m
F
= ±1) separated by less than 0.15 mT in magnetic field and 3 MHz in frequency.
Phosphorus does not possess any CT due to its small nuclear spin (I = 1/2).
As df/dB → 0, the next figure of merit is the electron transition probability amplitude,
2

which is always 50% of the high field limit for ESR-types. This means that manipulation
times are only slightly reduced while electron coherence times T
2e
are drastically increased.
Conversely, as flip-flops between two donor electron spins follow the square of this probability
amplitude, the flip-flopping rate remains strong, limiting T
2e
as observed and explained in
the main text.
NMR-type CTs occur at strong magnetic field (in Si:Bi, 0.3 T < B
0
< 5 T, from m
I
=
−9/2 at low field to m
I
= 9/2 at high field) and as such are transitions between quasi-
pure states in the (m
S
, m
I
) basis. NMR-type CTs possess a change in nuclear spin state
of ∆m
I
= 1 and can be manipulated in a conventional electron nuclear double resonance
(ENDOR) or NMR experiment. Amongst the NMR-type CTs, those at higher magnetic
fields have a smaller electron spin component. This leads to a reduced coupling to the
environment, but also increased spin manipulation times, converging to that of a regular
NMR transition (typically ∼ 10 µs).
ESR LINEWIDTHS
Measurements of spin linewidths provide important details regarding the spin environ-
ment, yielding information on crystalline defects and strains amongst other properties. Line
broadening arises from a variation in either the hyperfine interaction (∆A) or the magnetic
field (∆B) (or g-tensor) across the sample. They can either be measured in a magnetic
field-swept spectrum where (to first order):
∆B
T otal
= ∆B +
dB
df
df
dA
∆A (1)
or in Fourier-Transform (FT) ESR where the FT of the free induction decay after a π/2
rotation gives the spectrum in the frequency domain. In this case, (to first order):
∆f =
df
dB
∆B +
df
dA
∆A (2)
At the CT where dB/df → ∞, the linewidth in the magnetic field domain broadens
strongly until it becomes limited by the second order dependence on magnetic field. Both
∆B and ∆A can be identified by varying the transition frequency and fitting numerically
knowing
dB
df
df
dA
. To confirm that this change in linewidth is strictly related to the term df/dB,
we can make use of the FT ESR technique which simplifies at the CT to ∆f =
df
dA
∆A, where
3

0.5 1 1.5 2 2.5
0.2
0.4
0.6
0.8
Frequency (MHz)
FT−ESR intensity (a.u.)
80.4 mT, 7.0306 GHz
45.7 mT, 7.0970 GHz
0
1
= +1
F
m∆F∆
1−=
F
m∆F∆
0.2
0.4
0.6
0.8
FT−ESR intensity (a.u.)
0
1
2 4 6 80
Frequency (MHz)
0 3
28
Si:Bi
nat
Si:Bi
80.4 mT, 7.0270 GHz
= +1
F
m∆F∆
1−=
F
m∆F∆
BA
= 0µ∆
= 1µ∆
= 2µ∆
Sum
Model (normalized):
= 3µ∆
FIG. 2. FT-ESR around the CT. Each spectrum is the FT of the free induction decay taken
using a microwave frequency slightly below resonance (values given in legend). A, For the case of
28
Si:Bi, we observe two peaks in the ESR spectrum around the CT, corresponding to the transitions
∆F ∆m
F
= ±1. Spectra are shown as measured at two settings of magnetic field/microwave
frequency. In the magnetic field domain, the ESR linewidths in these two cases are 1.6 mT close to
the CT and 0.07 mT farther away (see Figure 1 of the main manuscript), however in the frequency
domain as shown above, the ESR linewidths are constant. B, In
nat
Si:Bi, these two primary ESR
transitions are further split into sub-peaks, corresponding to a mass-effect from nearest neighbour
Si atoms. Each shift of one neutron mass (∆µ) yields a shift of −1.7 MHz in transition frequency
(or 0.024% change in the hyperfine coupling A). Dashed lines show simulated peaks whose intensity
is calculated from a trinomial distribution of
30
Si,
29
Si and
28
Si isotopes in
nat
Si (with respective
concentration 3.1%, 4.7% and 92%). As the FT-ESR is derived from the free induction decay, the
intensities are normalized by the FT of the inhomogeneous decay T
2e
∗
(Lorentzian) and the cavity
bandwidth.
df/dA is quasi-constant around the CT. In Figure 2A, the linewidth is indeed constant about
270 kHz.
In natural silicon, the elimination of the ∆B term dramatically reduces the ESR
linewidth. Away form the CT (e.g. at X-band), ∆B is normally around 4 G due to unre-
solved coupling to
29
Si nuclear spins. In the frequency domain, this would be equivalent to
nearly 12 MHz, hiding multiple spectral features. First, the two transitions ∆F ∆m
F
= ±1
would not be resolvable. Second, as show in Figure 2B, we observe several other peaks (ab-
sent in isotopically pure
28
Si samples) which arise from variations in the hyperfine coupling
4

due to the total mass of nearest-neighbour silicon atoms (
28
Si,
29
Si and
30
Si). This effect is
described in full detail, including ENDOR experiments, in a forthcoming work [6].
ELECTRIC FIELD CLOCK TRANSITIONS
-10
10
0
Energy (GHz)
5
5
15
-15
Magnetic eld (mT)
0 200 400 600 800 1000
ESR-type CTs
NMR-type CTs

0→
dA
df

FIG. 3. Clock transitions in Si:Bi where df/dA = 0 which should be robust to electric
field noise. Both ESR- and NMR-type CTs can be observed, as for magnetic-field CTs. One
further ESR-CT is found at higher magnetic fields (2.6 T, not shown).
In the experiments reported in the main text, CTs were used to reduce the sensitivity
of electron spin to magnetic field variations, as quantified by df/dB. While magnetic field
noise is indeed the main decoherence mechanism in bulk materials, this may not be the case
in nanoscale devices where the electric field at interfaces could couple strongly with both
the donor electron and nuclear spins through the hyperfine interaction (and also, to lesser
extent, through a modulation in the electron spin g-factor). The sensitivity of a spin to
this effect can be quantified by the gradient of the frequency with respect to the hyperfine
constant df/dA, combined with values for the DC Stark effect for donors in silicon (which
for Group V donors is in the order of 10
−3
µm
2
/V
2
, as a fractional change in the hyperfine
coupling [7, 8]). Those CTs which will be most robust to electric field noise (df/dA → 0)
are identified in Si:Bi in Figure 3 and in Table II for all Group V donors in silicon.
In practice, both magnetic and electric field fluctuations will participate to the donor spin
decoherence. There will thus be an optimal CT, at a specific magnetic field and frequency,
where the coherence time would be maximum. For example, the magnetic field CT near
188 mT in Bi has the lowest value of df /dA out of the four possible CTs. In other scenarios,
it might be advantageous to minimise the inhomogeneous broadening as much as possible
5

75
As (I = 3/2)
121
Sb (I = 5/2)
123
Sb (I = 7/2)
209
Bi (I = 9/2)
∆m
S
= +1, m
I
= −1/2 −1/2 −3/2 −1/2 −3/2 −5/2 −1/2 −3/2 −5/2 −7/2
Magnetic field (mT) 53 117 39 114 38 23 2607 868 519 369
Frequency (GHz) 1.43 3.21 0.92 3.17 0.98 0.49 72.64 23.18 12.57 7.30
TABLE II. Summary of ESR-type electric-field CTs in donors in silicon. At the given
magnetic fields, the electron and nuclear spins are weakly coupled and the eigenstates must thus
be expressed in the |m
S
, m
I
i basis. The [∆m
S
= ±1, ∆m
I
= ∓2] (∆F ∆m
F
= −1) transitions
are nearly completely forbidden here; they would have been found at the same magnetic field as
the [∆m
S
= ±1, ∆m
I
= 0] (∆F ∆m
F
= +1) transitions, but separated by less than 40 MHz in
frequency.
(e.g. for coupling a spin ensemble to a microwave resonator), and this would also require
different optimal operating points within the Hilbert space of the bismuth electron and
nuclear spins.
[1] M. Mohammady, G. W. Morley, and T. S. Monteiro, Phys. Rev. Lett. 105, 067602 (2010).
[2] M. H. Mohammady, G. W. Morley, A. Nazir, and T. S. Monteiro, Phys. Rev. B 85, 094404
(2012).
[3] W. Hardy, A. Berlinsky, and L. Whitehead, Physical Review Letters 42, 1042 (1979).
[4] J. Longdell, A. Alexander, and M. Sellars, Physical Review B 74, 195101 (2006).
[5] M. Steger et al., Journal of Applied Physics 109, 102411 (2011).
[6] A. Tyryshkin et al., in preparation (2013).
[7] F. Bradbury et al., Physical Review Letters 97, 176404 (2006).
[8] R. Rahman et al., Physical Review Letters 99, 36403 (2007).
6