equations can be significantly simplified since most of the terms on the
right-hand side are normally suppressed. Indeed, considering the case of
axions and assuming $E\simeq M\simeq|D|$, the axion-photon couplings satisfy
$g_{a\mbox{\tiny{AA}}}/g_{a\mbox{\tiny{BB}}}\simeq(e/g_{0})^{2}\lesssim 2\cdot
10^{-4}$ and $|g_{a\mbox{\tiny{AB}}}|/g_{a\mbox{\tiny{BB}}}\simeq
e/g_{0}\lesssim 10^{-2}$. Moreover, the cosmic axions that form dark matter
have typical velocities $v_{a}\sim 10^{-3}$, so that the gradient of the
oscillating axion field is suppressed with respect to its time derivative:
$|\boldsymbol{\nabla}a|\sim 10^{-3}\,\dot{a}$.

Leaving only the first three dominant terms, we obtain the following
simplified axion Maxwell equations:

$\displaystyle\boldsymbol{\nabla}\\!\times\\!\mathbf{B}_{a}-\dot{\mathbf{E}}_{a}=0\,,$
(96)
$\displaystyle\boldsymbol{\nabla}\\!\times\\!\mathbf{E}_{a}+\dot{\mathbf{B}}_{a}=-g_{a\mbox{\tiny{BB}}}\left(\mathbf{B}_{0}\\!\times\\!\boldsymbol{\nabla}a+\dot{a}\mathbf{E}_{0}\right)+g_{a\mbox{\tiny{AB}}}\dot{a}\mathbf{B}_{0}\,,$
$\displaystyle\boldsymbol{\nabla}\\!\cdot\\!\mathbf{B}_{a}=0\,,$
$\displaystyle\boldsymbol{\nabla}\\!\cdot\\!\mathbf{E}_{a}=0\,,$

where we included the dominant effect arising from an external electric field
$\mathbf{E}_{0}$. Note that all the existing haloscopes use an external
magnetic field $\mathbf{B}_{0}$ instead, partly because the dominant effect
for the usually considered $g_{a\mbox{\tiny{AA}}}$ coupling is due to the term
with an external magnetic, but not electric, field and partly because it is
technologically challenging to sustain a large enough electric field in a big
enough volume. It is clear from Eq. (96) that if the latter technological
problem is solved, so that $E_{0}\gtrsim 10^{-2}\left(|D|/\zeta M\right)B_{0}$
in the CP-violating case or $E_{0}\gtrsim 10^{-3}\,B_{0}$ in the CP-conserving
case, the $g_{a\mbox{\tiny{BB}}}\dot{a}\mathbf{E}_{0}$ term will allow one to
search for dark matter axions in an external electric field with the
sensitivity which is not worse than the one of the conventional searches
conducted in an external magnetic field.

Returning to the case of the existing haloscopes where $\mathbf{E}_{0}=0$,
$\mathbf{B}_{0}\neq 0$, we see that the axion Maxwell equations (96)–(96) have
significantly different structure compared to the conventional axion Maxwell
equations which take into account solely the $g_{a\mbox{\tiny{AA}}}$ coupling.
While in the latter case axions generate an effective electric current
$\mathbf{j}^{e}_{\text{eff}}=-g_{a\mbox{\tiny{AA}}}\dot{a}\mathbf{B}_{0}$, in
the former case an effective magnetic current is generated:

$\mathbf{j}^{m}_{\text{eff}}=g_{a\mbox{\tiny{BB}}}\mathbf{B}_{0}\\!\times\\!\boldsymbol{\nabla}a-g_{a\mbox{\tiny{AB}}}\dot{a}\mathbf{B}_{0}\,.$
(97)

Note that in the case $M\simeq|D|$, the term proportional to the CP-violating
$g_{a\mbox{\tiny{AB}}}$ coupling is dominant. To the contrary, if the
underlying UV theory is CP-conserving, then $D=0$ and
$g_{a\mbox{\tiny{AB}}}=0$, so that only the term proportional to the
$g_{a\mbox{\tiny{BB}}}$ coupling contributes.

Let us now find experimental implications of the magnetic current (97).
Applying the curl differential operator to the Eqs. (96) and (96), and using
the other equations from the system (96)–(96), we obtain:

$\displaystyle\Delta\mathbf{E}_{a}-\ddot{\mathbf{E}}_{a}=\boldsymbol{\nabla}\\!\times\\!\mathbf{j}^{m}_{\text{eff}}\,,$
(98)
$\displaystyle\Delta\mathbf{B}_{a}-\ddot{\mathbf{B}}_{a}=\partial\mathbf{j}^{m}_{\text{eff}}/\partial
t\,.$ (99)

The leading terms contributing to the right-hand side are:

$\displaystyle\boldsymbol{\nabla}\\!\times\\!\mathbf{j}^{m}_{\text{eff}}=g_{a\mbox{\tiny{BB}}}\left(\boldsymbol{\nabla}a\\!\cdot\boldsymbol{\nabla}\right)\mathbf{B}_{0}-g_{a\mbox{\tiny{AB}}}\dot{a}\,\boldsymbol{\nabla}\\!\times\\!\mathbf{B}_{0}\,,$
(100) $\displaystyle\partial\mathbf{j}^{m}_{\text{eff}}/\partial
t=g_{a\mbox{\tiny{BB}}}\mathbf{B}_{0}\\!\times\\!\boldsymbol{\nabla}\dot{a}-g_{a\mbox{\tiny{AB}}}\ddot{a}\mathbf{B}_{0}\,.$
(101)

The axion dark matter field is given by the following expression:
$a(t,\mathbf{r})=a_{0}\cos\\!\left(\omega_{a}t-\mathbf{k}_{a}\\!\\!\cdot\\!\mathbf{r}\right)$,
where $\omega_{a}=m_{a}$ and $\mathbf{k}_{a}=m_{a}\mathbf{v}_{a}$. The leading
CP-conserving effect then depends on the direction of the axion wind and thus
experiences modulations with the periods of one sidereal day $T_{d}$ and one
sidereal year $T_{y}$ due to the rotation of the Earth around its axis and
around the Sun, respectively. To find the axion-induced $\mathbf{E}_{a}$ and
$\mathbf{B}_{a}$ fields, Eqs. (98) and (99) have to be solved for a particular
geometry of a given haloscope experiment.

Let us illustrate the general features of the solution by considering the
example of a very long solenoid of radius $R$ with magnetic field
$\mathbf{B}_{0}$ directed along the z-axis. In this case, Eqs. (98) and (99)
become:

$\displaystyle\Delta\mathbf{E}_{a}-\ddot{\mathbf{E}}_{a}=-\left(g_{a\mbox{\tiny{BB}}}\,\partial_{\rho}a\,\mathbf{e}_{z}+g_{a\mbox{\tiny{AB}}}\dot{a}\,\mathbf{e}_{\phi}\right)B_{0}\,\delta\\!\left(\rho-R\right)\,,$
(102)
$\displaystyle\Delta\mathbf{B}_{a}-\ddot{\mathbf{B}}_{a}=g_{a\mbox{\tiny{BB}}}\mathbf{B}_{0}\\!\times\\!\boldsymbol{\nabla}\dot{a}-g_{a\mbox{\tiny{AB}}}\ddot{a}\mathbf{B}_{0}\,,$
(103)

where we work in cylindrical coordinates $(\rho,\phi,z)$ with unit vectors
$(\mathbf{e}_{\rho},\mathbf{e}_{\phi},\mathbf{e}_{z})$. Let us parameterize
the direction of the axion wind
$\boldsymbol{\hat{v}}_{a}=\left(\sin\theta\cos(\phi-\xi),\,-\sin\theta\sin(\phi-\xi),\,\cos\theta\right)$
in cylindrical coordinates by two angles $\theta$ and $\xi$. Assuming
$T_{d}\gg 2\pi/\omega_{a}$, which corresponds to $m_{a}\gg 5\cdot
10^{-20}~{}\text{eV}$, we neglect the terms proportional to $\dot{\xi}$ and
$\dot{\theta}$ in the Eqs. (102) and (103). It is then straightforward to
obtain the solutions of these equations in terms of Bessel functions. All we
need however are these solutions in the limit $\omega_{a}R\ll 1$, as we are
interested in axions with large Compton wavelengths. In the latter limit,
solutions to Eqs. (102), (103) with physical boundary conditions are:

$\displaystyle\mathbf{E}_{a}=\left\\{\begin{array}[]{l}\frac{1}{2}\,a_{0}\,\omega_{a}\rho\,B_{0}\Bigl{(}g_{a\mbox{\tiny{AB}}}\,\mathbf{e}_{\phi}-g_{a\mbox{\tiny{BB}}}\,v_{a}\sin\theta\,\cos(\phi-\xi)\,\mathbf{e}_{z}\Bigr{)}\sin\omega_{a}t\,,\;\;\rho<R\\\
\frac{1}{2}\,a_{0}\,\omega_{a}\frac{R^{2}}{\rho}B_{0}\Bigl{(}g_{a\mbox{\tiny{AB}}}\,\mathbf{e}_{\phi}-g_{a\mbox{\tiny{BB}}}\,v_{a}\sin\theta\,\cos(\phi-\xi)\,\mathbf{e}_{z}\Bigr{)}\sin\omega_{a}t\,,\;\;\rho>R\end{array}\right.\,,\qquad\qquad$
(106)
$\displaystyle\mathbf{B}_{a}=\left\\{\begin{array}[]{l}\frac{1}{2}\,a_{0}\left(\omega_{a}R\right)^{2}B_{0}\Bigl{(}g_{a\mbox{\tiny{AB}}}\,\mathbf{e}_{z}+g_{a\mbox{\tiny{BB}}}\,v_{a}\sin\theta\,\bigl{\\{}\cos(\phi-\xi)\,\mathbf{e}_{\phi}+\bigr{.}\Bigr{.}\\\
\qquad\qquad\qquad\qquad\Bigl{.}\bigl{.}\sin(\phi-\xi)\,\mathbf{e}_{\rho}\bigr{\\}}\Bigr{)}\Bigl{(}\ln\omega_{a}R+\frac{\rho^{2}}{2R^{2}}\Bigr{)}\cos\omega_{a}t\,,\;\;\rho<R\\\
\frac{1}{2}\,a_{0}\left(\omega_{a}R\right)^{2}B_{0}\Bigl{(}g_{a\mbox{\tiny{AB}}}\,\mathbf{e}_{z}+g_{a\mbox{\tiny{BB}}}\,v_{a}\sin\theta\,\bigl{\\{}\cos(\phi-\xi)\,\mathbf{e}_{\phi}+\bigr{.}\Bigr{.}\\\
\qquad\qquad\qquad\qquad\qquad\qquad\Bigl{.}\bigl{.}\sin(\phi-\xi)\,\mathbf{e}_{\rho}\bigr{\\}}\Bigr{)}\ln\omega_{a}\rho\,\cos\omega_{a}t\,,\;\;\rho>R\end{array}\right.\,.$
(111)

It is immediately clear that ${E}_{a}\gg{B}_{a}$, which means that in an
experiment with the geometry of our example one has to search for axion-
induced electric, but not magnetic, fields. Moreover, it turns out that this
feature persists for any other possible geometry as well. Indeed, our Eqs.
(98) and (99) can be rendered equivalent to the equations studied in Ref.
Ouellet:2018nfr  by substituting $\mathbf{E}_{a}\rightarrow\mathbf{B}_{a}$,
$\mathbf{B}_{a}\rightarrow-\mathbf{E}_{a}$ and
$\mathbf{j}^{m}_{\text{eff}}\rightarrow\mathbf{j}^{e}_{\text{eff}}$. In the
latter work, it was found that for any haloscope geometry with characteristic
length scale $L$, the equation involving the time derivative of the effective
current yields solutions which are suppressed by powers of $\omega_{a}L\ll 1$
with respect to the solutions of the equation involving the curl of this
current. In our case, this means that in any haloscope probing
$m_{a}\ll\mu\text{eV}$, the axion-induced magnetic field $\mathbf{B}_{a}$ is
suppressed with respect to the axion-induced electric field $\mathbf{E}_{a}$.

Note however that all the existing as well as many projected haloscopes
searching for such light axions – such as ABRACADABRA, ADMX SLIC, DM Radio,
SHAFT, … – aim to measure only the axion-induced magnetic, but not electric,
fields. Thus, the constraints on the conventional $g_{a\mbox{\tiny{AA}}}$
coupling obtained by these experiments do not hold for the dominant axion-
photon couplings $g_{a\mbox{\tiny{AB}}}$ and $g_{a\mbox{\tiny{BB}}}$. The
latter couplings can be probed by future haloscopes equipped with electric
field sensors191919While this work was being prepared for publication, the new
search strategies were discussed in more detail in Ref. Li:2022oel .. One
haloscope of such kind has already been proposed, see Refs. McAllister:2018ndu
; Tobar:2020kmz . We hope that our work will encourage more experimental
effort in this direction, as we provided a sound theoretical motivation for
such an endeavor. Since the electromagnetic fields generated in a haloscope by
the $g_{a\mbox{\tiny{BB}}}$ and $g_{a\mbox{\tiny{AB}}}$ couplings are
qualitatively different from the fields generated by the conventional
$g_{a\mbox{\tiny{AA}}}$ coupling, for which $B_{a}\gg E_{a}$ Ouellet:2018nfr ,
the first detection of cosmic axions with electric, but not magnetic, sensor
haloscope would not only constitute the discovery of axions and dark matter,
but also provide a circumstantial experimental evidence for the existence of
heavy magnetically charged particles. Furthermore, as one can see from Eq.
(106), the direction of the detected electric field could allow one to infer
the ratio $g_{a\mbox{\tiny{AB}}}/g_{a\mbox{\tiny{BB}}}=2\alpha(D/\zeta M)$ and
thus get information about the spectrum of dyons in the UV.

| Axions | Gravitational waves
---|---|---
$\mathbf{P}_{e}$ | $\quad-g_{a\mbox{\tiny{AA}}}\,a\mathbf{B}_{0}+g_{a\mbox{\tiny{AB}}}\,a\mathbf{E}_{0}\quad$ | $\mathbf{h}\\!\times\\!\mathbf{B}_{0}+H\mathbf{E}_{0}-\left(h_{00}+\frac{1}{2}\,h\right)\\!\mathbf{E}_{0}$
$\mathbf{M}_{m}$ | $g_{a\mbox{\tiny{BB}}}\,a\mathbf{B}_{0}+g_{a\mbox{\tiny{AB}}}\,a\mathbf{E}_{0}$
$\mathbf{M}_{e}$ | $-g_{a\mbox{\tiny{AA}}}\,a\mathbf{E}_{0}-g_{a\mbox{\tiny{AB}}}\,a\mathbf{B}_{0}$ | $\quad-\mathbf{h}\\!\times\\!\mathbf{E}_{0}+H\mathbf{B}_{0}-\left(h_{00}+\frac{1}{2}\,h\right)\\!\mathbf{B}_{0}\quad$
$\mathbf{P}_{m}$ | $g_{a\mbox{\tiny{BB}}}\,a\mathbf{E}_{0}-g_{a\mbox{\tiny{AB}}}\,a\mathbf{B}_{0}$
Table 1: Comparison between axion and gravitational wave electrodynamics in
external electric $\mathbf{E}_{0}$ and magnetic $\mathbf{B}_{0}$ fields in
terms of the effective induced polarization and magnetization vectors, defined
by Eqs. (115)–(115); $\mathbf{h}$, $H$, $h_{00}$ and $h$ are characteristics
of the gravitational wave defined in Ref. Sokolov:2022dej .

Finally, to compare different extensions of electrodynamics which could be
probed by haloscope experiments, it is convenient to reexpress the axion
Maxwell equations in terms of the axion-induced polarization and magnetization
vectors Tobar:2018arx ; Ouellet:2018nfr , which are defined as follows:

$\displaystyle\boldsymbol{\nabla}\\!\times\\!\mathbf{B}_{a}-\dot{\mathbf{E}}_{a}=\frac{\partial\mathbf{P}_{\text{eff}}^{e}}{\partial
t}+\boldsymbol{\nabla}\\!\times\\!\mathbf{M}_{\text{eff}}^{e}\,,$ (115)
$\displaystyle\boldsymbol{\nabla}\\!\times\\!\mathbf{E}_{a}+\dot{\mathbf{B}}_{a}=-\frac{\partial\mathbf{P}_{\text{eff}}^{m}}{\partial
t}+\boldsymbol{\nabla}\\!\times\\!\mathbf{M}_{\text{eff}}^{m}\,,$
$\displaystyle\boldsymbol{\nabla}\\!\cdot\\!\mathbf{B}_{a}=-\boldsymbol{\nabla}\\!\cdot\\!\mathbf{P}_{\text{eff}}^{m}\,,$
$\displaystyle\boldsymbol{\nabla}\\!\cdot\\!\mathbf{E}_{a}=-\boldsymbol{\nabla}\\!\cdot\\!\mathbf{P}_{\text{eff}}^{e}\,.$

Note that along with the ordinary effective electric polarization and
magnetization vectors $\mathbf{P}_{\text{eff}}^{e}$ and
$\mathbf{M}_{\text{eff}}^{e}$, we introduced effective magnetic polarization
and magnetization vectors $\mathbf{P}_{\text{eff}}^{m}$ and
$\mathbf{M}_{\text{eff}}^{m}$, which describe electromagnetic properties of an
effective medium consisting of magnetically charged particles. In Table 1, we
give explicit expressions for the effective polarization and magnetization
vectors corresponding to a generic axion in an external electromagnetic field.
The fact that it is possible to bring the axion Maxwell equations (85)–(88)
into the form (115)–(115) suggests that the effects of axions in external
electromagnetic fields are analogous to the effects of a certain medium
consisting of both electrically and magnetically charged particles. From what
has been discussed before, it is clear that for a generic axion, effects of
the magnetic polarization (magnetization) vectors
$\mathbf{P}_{\text{eff}}^{m}$ ($\mathbf{M}_{\text{eff}}^{m}$) in an external
magnetic field dominate the effects of the electric polarization
(magnetization) vectors $\mathbf{P}_{\text{eff}}^{e}$
($\mathbf{M}_{\text{eff}}^{e}$). Such asymmetry between the effects exhibited
by electric and magnetic constituents of the effective medium is to be
contrasted with the case of gravitational wave electrodynamics Sokolov:2022dej
, where there exists an electric-magnetic $U(1)$ duality symmetry rendering
electric and magnetic variables equivalent. This difference in the symmetry
properties of the two theories can be easily understood from the fact that the
axion-photon interactions are fundamentally mediated by heavy charged
particles which break the duality symmetry, while in General Relativity, the
interaction between gravity and electromagnetic field is direct and
independent of any charges. For an experimentalist, this distinction signifies
a substantial difference in the distribution of the induced electromagnetic
fields inside the haloscope. Indeed, as it was discussed before, for a generic
light axion ($m_{a}\ll\mu\text{eV}$,
$g_{a\mbox{\tiny{BB}}}\gg|g_{a\mbox{\tiny{AB}}}|\gg g_{a\mbox{\tiny{AA}}}$) in
an external magnetic field one expects
$\mathbf{j}^{m}_{\text{eff}}\gg\mathbf{j}^{e}_{\text{eff}}$ and thus
${E}_{a}\gg B_{a}$, while for a gravitational wave in an external magnetic
field, plugging the expressions given in Table 1 into the Eqs. (115)–(115),
one obtains $\mathbf{j}^{m}_{\text{eff}}\sim\mathbf{j}^{e}_{\text{eff}}$ and
thus ${E}_{a}\sim B_{a}$.

##  7 Conclusion

As it was asserted by J. Polchinski Polchinski:2003bq , “the existence of
magnetic monopoles seems like one of the safest bets that one can make about
physics not yet seen”. Indeed, in the beginning of this article, we reviewed a
number of theoretical arguments which together provide an overwhelming
theoretical evidence for magnetic monopoles, like there probably exists for no
other kind of hypothetical particles. There are nevertheless multiple
practical challenges behind the experimental study of monopoles. While some of
the previously mentioned arguments for the existence of monopoles do not
restrict their masses, another part of these arguments suggests that monopoles
are super heavy, with masses well beyond the energy reach of the present-day
collider experiments. Thus, although monopoles almost certainly exist from the
theoretical point of view, it might be difficult to directly probe them with
existing and near-future experiments.

In this work, we proposed experiments which could probe magnetic monopoles
indirectly, in particular through the influence virtual monopoles would
exhibit on the interactions of axions with an electromagnetic field. To study
the effects of virtual monopoles, we classified all independent marginal
operators preserving the symmetries and degrees of freedom of QEMD. Then, we
built a generic axion-photon EFT, which holds for the QCD axion as well as for
other ALPs. We showed that this EFT features previously unknown couplings
which arise from UV models containing magnetic charges. We then found that
these new couplings give rise to unique experimental signatures in LSW
experiments and in some kinds of axion searches, such as haloscopes searching
for low-mass axions ($m_{a}\ll\mu\text{eV}$). In the latter case, we showed
that the best sensitivity to the new electromagnetic couplings of axions could
be achieved by measuring an induced oscillating electric field, instead of an
induced oscillating magnetic field, contrary to the setup of existing
experiments. The case of low mass axions is particularly interesting, since,
as it can be seen from Fig. 1, the simplest axion models predict rather large
$g_{a\mbox{\tiny{BB}}}$ and $g_{a\mbox{\tiny{AB}}}$ couplings, which are not
excluded and which can be probed by many projected experiments. Thus, we
encourage the development of electric sensor haloscopes which would search for
axion dark matter in the corresponding parameter region.

Apart from unveiling these intriguing experimental applications, our work
reconsidered several questions in axion theory. First, we showed that contrary
to the existing statements, the main contribution to the axion-photon coupling
need not be quantized in units proportional to $e^{2}$. Indeed, we established
that the standard axion-photon coupling $g_{a\gamma\gamma}\equiv
g_{a\mbox{\tiny{AA}}}$ and the Witten-effect induced coupling
$g_{a\mbox{\tiny{Aj}}}$ are two different couplings and therefore the previous
quantization arguments based on the Witten effect apply only to the
$g_{a\mbox{\tiny{Aj}}}$ interaction. While the other couplings
$g_{a\mbox{\tiny{AA}}}$, $g_{a\mbox{\tiny{AB}}}$ and $g_{a\mbox{\tiny{BB}}}$,
as Chern-Simons-type couplings, have to be quantized in the case of a non-
trivial topology of the base manifold, the corresponding quantization units
are proportional to $e^{2}$ only in the case of the $g_{a\mbox{\tiny{AA}}}$
coupling, whereas the $g_{a\mbox{\tiny{AB}}}$ and the $g_{a\mbox{\tiny{BB}}}$
couplings are quantized in the units of $eg$ and $g^{2}$, respectively.
Moreover, in the case of the trivial topology of the base manifold, there is
no a priori reason for the latter three couplings to be quantized. Second,
contrary to what has been advocated recently in the literature, we found that
magnetic monopoles of an Abelian gauge field need not give mass to axions
coupled to this gauge field. We showed that the axions do get mass in theories
with magnetic monopoles only if these monopoles carry an additional instanton
degree of freedom interacting with the axion $a$, e.g. in the case of the
spontaneously broken symmetry phase of a non-Abelian gauge theory with
$aGG^{d}$ term in the Lagrangian, where $G$ ($G^{d}$) is the (dual) field
strength tensor of the non-Abelian gauge field. This axion mass is generated
via the conventional mechanism through instantons of the non-Abelian theory,
which however live on the ’t Hooft-Polyakov monopoles in the low energy phase.
Finally, we found that the interaction between axions and an electromagnetic
field need not preserve CP. In particular, as long as one has heavy dyons in
the high energy theory, there exists a natural source of CP-violation which
can be transferred to low energy physics through axion-photon interactions. We
discussed experiments which can probe this new CP-violating axion coupling.

## Acknowlegments

We thank P. Quilez for bringing the argument about the quantization of the
axion-photon coupling to our attention as well as for numerous valuable
discussions at the beginning stages of this project. A.R. acknowledges support
and A.S. is funded by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) under Germany’s Excellence Strategy – EXC 2121 Quantum
Universe – 390833306. This work has been partially funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) - 491245950.

##  Appendix A Calculation of the anomaly coefficients

Let us calculate the anomaly coefficients $E$, $M$ and $D$, which enter the
expressions (65), (66) and (68) for the axion-photon couplings. We start with
a high energy QEMD theory and integrate out heavy fermions $\psi$ carrying PQ
charges. In the PQ-symmetric phase, the Lagrangian includes the following
terms for each of the fermions $\psi$:

$\mathcal{L}\;\supset\;i\bar{\psi}\gamma^{\mu}\mathcal{D}_{\mu}\psi+y\left(\Phi\,\bar{\psi}_{L}{\psi}_{R}+\text{h.c.}\,\right)\,,$
(116)

where $y$ is a dimensionless Yukawa constant,
$\mathcal{D}_{\mu}=\partial_{\mu}-e\,q_{\psi}A_{\mu}-g_{0}\,g_{\psi}B_{\mu}$
is a covariant derivative including both magnetic and electric four-potentials
multiplied by the corresponding electric and magnetic charges, and $\Phi$ is
the PQ complex scalar field.

Below the PQ symmetry breaking scale, one can expand
$\Phi=(v_{a}+\rho)\exp{\left(ia/v_{a}\right)}/\sqrt{2}$, where $\rho$ is a
heavy radial mode and $a$ is an axion. The terms in the resulting Lagrangian
which are relevant for the low energy phenomenology are:

$\mathcal{L}\;\supset\;i\bar{\psi}\gamma^{\mu}\mathcal{D}_{\mu}\psi+\frac{yv_{a}}{\sqrt{2}}\left\\{\exp{\left(\frac{ia}{v_{a}}\right)}\,\bar{\psi}_{L}\psi_{R}+\text{h.c.}\right\\}.$
(117)

We perform an axial rotation of the fermion
$\psi\rightarrow\exp{(ia\gamma_{5}/2v_{a})}\cdot\psi$, after which there arise
an anomalous contribution $\mathcal{L}_{\text{F}}$ from the transformation of
the measure of the path integral and a derivative coupling of $a$ to the axial
current of $\psi$:

$\mathcal{L}\;\supset\;i\bar{\psi}\gamma^{\mu}\mathcal{D}_{\mu}\psi+\frac{yv_{a}}{\sqrt{2}}\,\bar{\psi}\psi-\frac{\partial^{\mu}a}{2v_{a}}\,\bar{\psi}\gamma_{\mu}\gamma_{5}\psi-\mathcal{L}_{\text{F}}\,.$
(118)

The fermion $\psi$ gets its mass $m=yv_{a}/\sqrt{2}$, which we assume to be
large compared to the energy scales probed in experiments. In the large $m$
limit, the derivative axion coupling from Eq. (118) does not contribute to the
low energy Lagrangian of axion-photon interactions due to the Sutherland-
Veltman theorem Sutherland:1967vf ; Veltman67 . The axion-photon couplings are
thus given by the anomalous terms $\mathcal{L}_{\text{F}}$ which can be
calculated using the Fujikawa method Fujikawa:1979ay . In particular,
following Fujikawa, we apply the gauge invariant heat kernel regularization to
the path integral measure which yields:

$\mathcal{L}_{\text{F}}\;=\;-\frac{a}{v_{a}}\cdot\lim_{\begin{subarray}{c}\Lambda\to\infty\\\
x\to
y\end{subarray}}\mathrm{tr}\left\\{\gamma_{5}\exp{\left({\ooalign{$\mathcal{D}$\cr$/$\cr}}^{2}/\Lambda^{2}\right)}\,\delta^{4}(x-y)\right\\}.$
(119)

The expression for ${\ooalign{$\mathcal{D}$\cr$/$\cr}}^{2}$ is:

${\ooalign{$\mathcal{D}$\cr$/$\cr}}^{2}=\mathcal{D}^{2}-i\gamma_{\mu}\gamma_{\nu}\left(eq_{\psi}\,\partial^{\mu}\\!A^{\nu}+g_{0}g_{\psi}\,\partial^{\mu}\\!B^{\nu}\right).$
(120)

It is then convenient to express the delta-function as a superposition of
plane waves: ${\delta^{4}(x-y)=\int
d^{4}k\,e^{ik(x-y)}/\left(2\pi\right)^{4}}$, each of which shifts the
derivative operator $\mathcal{D}_{\mu}\rightarrow\mathcal{D}_{\mu}+ik_{\mu}$.
Taking into account Eq. (120), we obtain:

$\displaystyle\mathcal{L}_{\text{F}}\;=\;-\frac{a}{v_{a}}\cdot\lim_{\Lambda\to\infty}\int\frac{d^{4}k}{\left(2\pi\right)^{4}}\,\mathrm{tr}\left\\{\gamma_{5}\exp{\Bigl{(}-i\gamma_{\mu}\gamma_{\nu}\left(eq_{\psi}\,\partial^{\mu}\\!A^{\nu}+g_{0}g_{\psi}\,\partial^{\mu}\\!B^{\nu}\right)/\Lambda^{2}\Bigr{.}}\right.+\qquad\quad$
$\displaystyle\left.\left.\left(\mathcal{D}+ik\right)^{2}/\Lambda^{2}\right)\right\\}\,.\quad\;$
(121)

Any terms in the integrand which are $o\\!\left(1/\Lambda^{4}\right)$ vanish
after performing the integration and sending $\Lambda\to\infty$. Taylor
expanding the exponent, we are then left with the finite number of terms,
which after taking the trace, the integral and the limit simplify into:

$\mathcal{L}_{\text{F}}\;=\;\frac{a\,d\\!\left(C_{\psi}\right)}{8\pi^{2}v_{a}}\cdot\epsilon_{\mu\nu\lambda\rho}\Bigl{(}eq_{\psi}\,\partial^{\mu}\\!A^{\nu}+g_{0}g_{\psi}\,\partial^{\mu}\\!B^{\nu}\Bigr{)}\left(eq_{\psi}\,\partial^{\lambda}\\!A^{\rho}+g_{0}g_{\psi}\,\partial^{\lambda}\\!B^{\rho}\right),$
(122)

where $d\\!\left(C_{\psi}\right)$ is the dimension of the color representation
of $\psi$. Using the notations of sec. 4, Eq. (122) can be rewritten as
follows:

$\displaystyle\mathcal{L}_{\text{F}}\;=\;-\frac{a\,d\\!\left(C_{\psi}\right)}{16\pi^{2}v_{a}}\cdot\left(q_{\psi}^{2}e^{2}\,\mathrm{tr}\left\\{\left(\partial\wedge
A)\,(\partial\wedge
A\right)^{d}\right\\}+g_{\psi}^{2}g_{0}^{2}\,\mathrm{tr}\left\\{\left(\partial\wedge
B)\,(\partial\wedge B\right)^{d}\right\\}+\right.\qquad$
$\displaystyle\left.2\,q_{\psi}g_{\psi}\,eg_{0}\,\mathrm{tr}\left\\{\left(\partial\wedge
A)\,(\partial\wedge B\right)^{d}\right\\}\right).\quad$ (123)

If there are multiple dyons $\psi$, each of them gives a similar contribution
to the resulting axion-photon Lagrangian. Thus, the expressions for the axion-
photon couplings
$g_{a\mbox{\tiny{AA}}},g_{a\mbox{\tiny{BB}}},g_{a\mbox{\tiny{AB}}}$ and the
corresponding anomaly coefficients $E,M,D$ are:

$\displaystyle g_{a\mbox{\tiny{AA}}}=\frac{Ee^{2}}{4\pi^{2}v_{a}}\,,\quad
E=\sum_{\psi}q_{\psi}^{2}\cdot d\\!\left(C_{\psi}\right)\,,$ (124)
$\displaystyle g_{a\mbox{\tiny{BB}}}=\frac{Mg_{0}^{2}}{4\pi^{2}v_{a}}\,,\quad
M=\sum_{\psi}g_{\psi}^{2}\cdot d\\!\left(C_{\psi}\right)\,,$ (125)
$\displaystyle g_{a\mbox{\tiny{AB}}}=\frac{Deg_{0}}{4\pi^{2}v_{a}}\,,\quad
D=\sum_{\psi}q_{\psi}g_{\psi}\cdot d\\!\left(C_{\psi}\right)\,.$ (126)

In Ref. Sokolov:2023pos , we obtained the same results for the axion-photon
couplings (124)-(126) directly integrating over the fermion fields in the path
integral of the theory and discussing all the subtleties associated to the
non-locality and $n_{\mu}$-dependence in detail. The associated calculation
can be considered as a proof of the validity of the Fujikawa method presented
in this Appendix.

## References

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  * October 2022

# Exploring room temperature spin transport under band gap opening in bilayer
graphene

Christopher R. Anderson  Department of Physics and Astronomy, University of
Manchester, M13 9PL, Manchester, UK<EMAIL_ADDRESS>Noel
Natera-Cordero  Department of Physics and Astronomy, University of Manchester,
M13 9PL, Manchester, UK  Consejo Nacional de Ciencia y Tecnología (CONACyT),
México<EMAIL_ADDRESS>Victor H. Guarochico-Moreira
Department of Physics and Astronomy, University of Manchester, M13 9PL,
Manchester, UK  Facultad de Ciencias Naturales y Matemáticas, Escuela Superior
Politécnica del Litoral, ESPOL, Campus Gustavo Galindo, Km. 30.5 Vía
Perimetral, P.O. Box 09-01-5863, 090902 Guayaquil, Ecuador  Center of
Nanotechnology Research and Development (CIDNA), Escuela Superior Politécnica
del Litoral, ESPOL, Campus Gustavo Galindo Km 30.5 Vía Perimetral, Guayaquil,
Ecuador<EMAIL_ADDRESS>Irina V. Grigorieva  Department of Physics and
Astronomy, University of Manchester, M13 9PL, Manchester, UK  National
Graphene Institute, University of Manchester, M13 9PL, Manchester, UK
<EMAIL_ADDRESS>Ivan J. Vera-Marun  Department of Physics
and Astronomy, University of Manchester, M13 9PL, Manchester, UK  National
Graphene Institute, University of Manchester, M13 9PL, Manchester, UK
<EMAIL_ADDRESS>
###### Abstract

We study the room-temperature electrical control of charge and spin transport
in high-quality bilayer graphene, fully encapsulated with hBN and contacted
via 1D spin injectors. We show that spin transport in this device architecture
is measurable at room temperature and its spin transport parameters can be
modulated by opening of a band gap via a perpendicular displacement field. The
modulation of the spin current is dominated by the control of the spin
relaxation time with displacement field, demonstrating the basic operation of
a spin-based field-effect transistor.

##  1 Concept and novelty

The state variable in digital devices has been realised, to date, by
measurement of charge. Spintronics offers a new paradigm, whereby the quantum
spin states of one or more electrons could be used as an alternative;
improving performance by quick and efficient manipulation of that spin state
[1]. Improvements in transport and control of spin are necessary for spin
logic to be fully realised. Graphene offers desirable properties including
theoretically large, although experimentally small spin ‘quality’ parameters
[2, 3]. Bilayer graphene (BLG) presents the opportunity to electrically
control its band gap [4, 5], $E_{g}$, (see Figure 1.a.) and, as a result, the
spin transport parameters in the channel. Graphene based spin-based field
effect transistors have been previously attempted: Materials with large and
electrostatically tunable spin-orbit coupling (SOC) have been brought into
contact with a graphene channel, enabling spins to be absorbed [6, 7]; this
precludes the fabrication of a fully encapsulated homogeneous transport
channel. A combination of injector current and back gate voltage can be tuned
to create a spin transistor-like action by reversing the polarity of the
contact spin polarisation [8, 9], however, the latter varied between contacts
and consequently so did the spin signal, making this approach not scalable.
High-quality graphene devices with reproducible contact spin polarisation can
overcome this limitation [10]. Alternatively, the electrostatic control of the
graphene channel’s band gap [11, 12, 13] offers a small, but promising effect,
which has the potential to be scalable whilst maintaining a simple
architecture, with excellent transport properties.

Spin transport has previously been observed in single layer (SLG) and BLG
devices which have transparent (invasive) contacts (exhibiting the spin
resistance mismatch problem [14, 15]) and a SiO2 substrate [16, 17, 18] which
brakes graphene’s inversion symmetry causing Rashba fields, increased SOC and
spin relaxation. Alternatively, devices have been fabricated with an hBN
substrate but are not fully encapsulated [12], leading to a heterogeneous
channel consisting of sections with and without encapsulation that suffer from
environmental degradation. Quantum tunnelling contacts have been shown to
solve the resistance mismatch problem [17, 11, 12, 19, 20] improving the
measured spin transport parameters. However, such tunneling contacts are
difficult to fabricate due to troublesome selection of atomically thin hBN
[21] or inconsistency of the growth of an atomically thin barrier [17].

The effect which we have observed, where we propose that spin anisotropy is
induced in the BLG by a small but tuneable band gap, has previously been seen
but at high carrier densities and low temperatures [16]. In contrast, here we
experimentally study a bilayer device at room temperature (RT) and show how,
close to the Dirac point, the transport parameters are effected by an applied
perpendicular displacement field which induces the band gap to open (Figure
1.a). In this regime, however, it is typically difficult to measure the
device’s spin transport parameters, as a consequence of the low signal to
noise ratio and an induced band gap commensurate with the thermal activation
energy. Conversely, at the Dirac point the effects of the band gap opening on
spin transport are more readily observed, with the spin splitting being more
prominent at the band edge. Consequently, the quality of the homogeneous
channel is of paramount importance, which allows the observation of a
modulation of the spin relaxation time, $\tau_{s}$, which has not previously
been shown close to the Dirac point and at RT. Here this is made possible by
the fabrication of a BLG transport channel which is fully encapsulated, where
the effect of the contacts and the resistance mismatch problem are minimised
by the use of nanoscale 1D edge contacts [9, 10].

##  2 Results

Figure 1:  Bilayer graphene transport channel and device a. Band structure of
pristine BLG without (left) and with (right) an applied perpendicular electric
displacement field b. Optical micrograph of of our $\sim$1 µm wide BLG
graphene transport channels (blue) with contacts and top gate (in the
measurement region). The charge (spin) transport current injection, $I$, and
local (non-local) potential difference, $V$, measurement configuration. Inset:
Schematic of the device heterostructure showing the Co/Au 1D edge contacts,
with the graphene represented by the balls and sticks.

###  2.1 Device architecture and fabrication

The full encapsulation of the BLG with hBN leaves only the sandwiched
graphene’s edges exposed; on these edges 1D contacts are formed (Figure 1.b).
The device was designed to enable electrostatic gating of the whole device
using the Si substrate as a back gate and of a region between two contacts
using a microfabricated top gate. The design facilitates the fabrication of
multiple fully encapsulated regions with top gate electrodes. The 1D edge
magnetic contacts are fabricated from gold-capped cobalt, whilst the
electrostatic top gates are gold, with a thin chromium adhesion layer.

###  2.2 Charge transport measurements

Our local measurements of the top-gated and hBN-encapsulated BLG, using the
configuration shown in Figure 1.b, explored the charge transport by varying
the back gate, $V_{bg}$, and top gate, $V_{tg}$, voltages and measuring the
BLG resistance, $R$. From the latter the sheet resistance, $\rho$, is
calculated and exhibits a modulation as a function of the gate voltages as
shown in Figure 2.a. The gate voltages are used to calculate the carrier
density, $n$, and electrical displacement field, $D$, which enable re-plotting
the sheet resistance in terms of $n$ and $D$. Figure 2.a and b show this
transformed map at RT, along with symbols indicating a selection of ($n,D$)
pairs where spin transport measurements were performed. The inset of Figure
2.b shows a near linear relationship between the transport channel’s sheet
resistance and the applied displacement field, close and parallel to the Dirac
ridge (the path of zero carrier density as the displacement field is
modulated), exhibiting a monotonic increase in $\rho$ as $D$ increases in
magnitude.

Figure 2: a. A 2D charge transport measurement map at RT showing the effect on
the sheet resistance, $\rho$, of applying a back, $V_{bg}$, and top gate
voltage, $V_{tg}$. b. The same charge transport measurements transformed into
a map as a function of carrier density, $n$, and electric displacement field,
$D$. In both maps the symbols are shown where the spin transport measurements
were made. b. (inset) Sheet resistance close and parallel to the Dirac ridge
($n\approx 0$)

The device shows a carrier field effect mobility, $\mu$, of up to 40,000
cm2/Vs at 20 K and 11,000 cm2/Vs at RT; where $\mu=(d\sigma/{dn})/e$ and
$\sigma=1/\rho$ and are evaluated at a moderate carrier density, $|n|\sim
1\times 10^{12}$ cm-2. These values are comparable with those achieved in
fully encapsulated BLG, tunnel barrier contacted devices [11, 19, 20] and
substantially higher than non-encapsulated BLG devices [12, 16]. The BLG
transport channel does not suffer from any significant contact doping due to
the small contact area (a few carbon atoms depth from the edge of the device)
and small contact width ($\leq 400$ nm) [10] and minimal bubbles within the
heterostructure. We propose that even higher mobilities may be obtained with
the fabrication of a wider channel and eradication of any residual bubbles in
the channel, potentially improving the spin transport parameters of the
device, however, this is dependent on the cause of the scattering [11]. We
have confirmed that the top gate voltage has little effect on the resistance
of the regions either side of the top gated region.

###  2.3 Spin transport measurements

To characterise spin transport we first measured a series of spin valves,
using the non-local configuration shown in Figure 1.b, at RT. These are
measurements where we sweep an in-plane magnetic field, $B_{\parallel}$,
applied along the direction of contacts, which reverses the magnetisation of
the 1D contacts and enables us to obtain either a parallel or antiparallel
magnetic alignment between the injector and detector contacts. The trace and
retrace scans of magnetic field are then subtracted from one another to obtain
a purely spin-dependent signal. Such measurements were repeated across a
carrier density range ($n$ = 0.45 to 0.8 ×10 12 cm-2), with the top gate
grounded, the results of which are shown in Figure 3.a. The vertical bands of
colour seen to the left (red) and right (darker blue) of zero magnetic field
strength, $B=0$ T, are the distinctive spin-valve signals as the magnetic
injector/detector configurations change from parallel to anti-parallel. The
average magnitude of the spin signal, the difference in non-local resistance
between parallel and anti-parallel magnetic configurations, $\Delta R_{nl}$,
across this range of $V_{bg}$ is $\sim$10 m. A close examination of the 2D map
shows that the spin signal and the signal to noise ratio (SNR) diminish as the
back gate approaches the charge neutrality point (CNP, $n$ = 0) from the
electron carrier regime. This is a characteristic of 1D edge contacts, whose
spin polarisation approaches zero near the CNP [9].

The changes in spin signal as a function of carrier density were further
studied by performing spin precession measurements. In doing so, we sweep a
magnetic field strength perpendicular to the plane of the graphene (see Figure
1.b), $B_{\bot}$, across a range of $\pm$200 mT, which causes the diffusing
electronic spins to experience Larmor precession, and fit the spin signal with
the Hanle equation [22]. The spin relaxation time, the spin diffusion
coefficient and the corresponding spin relaxation length, are extracted from
the Hanle curve fits (examples shown in Figure 4). Uncertainties in
determining these parameters are realised by the error bars in Figures 3.b, c
and 5. We use the spin, rather than the charge diffusion coefficient in our
analysis, as the measurements are made close to the neutrality point, whilst
also opening a band gap, consequently it is inappropriate to use the Einstein
relation to extract the charge diffusion coefficient from charge transport
measurements. We select five pairs of ($n,D$) values for spin transport
measurements in the electrons regime, at fixed top gate voltage and $-7$ Vbg,
where carrier density variation dominates over variation in displacement
field. The selected five points are shown in Figure 2 (A-B path). The
corresponding spin transport measurements and Hanle fits are shown in 4 (A and
B) and the extracted spin transport parameters are shown in Figure 3.b and c.
Here we note that $\tau_{s}$ and ${\lambda_{s}}$ decrease with increasing
carrier density (at low carrier density) which, although small, is
commensurate with that typically observed for BLG spin devices [11].

Figure 3: a. 2D map of spin valves at RT as the electron carrier density is
increased. The anti-parallel magnetic alignments are clearly visible as the
vertical red and darker blue bands. As the CNP is approached the signal
becomes increasingly weak. b. and c. Spin transport parameters ($\tau_{s}$ and
$\lambda_{s}$, respectively) extracted from spin precession measurements at
RT, for the 5 points along the A-B path shown in Figure 2.

Next, we explore spin transport near charge neutrality under the effect of a
displacement field. For this purpose we measured spin precession at five
different ($n,D$) values indicated by the C-D path shown in Figure 2.a and b.
Here we fixed a carrier density close to the CNP, $n<2\times 10^{11}$ cm-2,
and varied $D$ from $-0.04$ V nm-1 to $-0.12$ V nm-1. Examples of the spin
precession measurements are shown in Figure 4 C and D. A wider peak is seen at
the high end of the displacement field range (C) and a narrower peak at the
low end of the range (D). These changes in lineshape correspond to a
modulation in the spin transport parameters extracted from the Hanle analysis.
The spin transport parameters: spin relaxation time, $\tau_{s}$, spin
relaxation length, $\lambda_{s}$, and spin signal (at $B$ = 0), are shown in
Figure 5.a, b and c, respectively. As the magnitude of $D$ is increased there
is a threefold decrease in $\tau_{s}$, and a corresponding modulation of
$\lambda_{s}$.

Figure 4: Spin precession measurements at RT. The lineshape of the Hanle curve
changes with the change in $n$, shown in A and B, whilst C and D (at
$n\approx$ 0) shows the lineshapes when the displacement field changes. The
labels correspond to the same ($n,D$) pairs labelled in Figure 2. The Hanle
equation fits are shown in red.  Figure 5: Spin transport measurement and
analysis close and parallel to the Dirac ridge at RT. Spin relaxation time
(a), spin relaxation length (b) and spin signal (c) for the five points along
the C-D path shown in Figure 2. d. Band gap energy, $E_{g}$, vs displacement
field, $D$, from Arrhenius analysis of charge transport measurements.

##  3 Discussion

The spin signal, $\Delta R_{nl}$, measured along the Dirac ridge ($n\sim 0$)
at RT, shows a decrease with increasing displacement field (Figure 5.c), with
a modulation of approximately half.

To understand the role of the applied displacement field, $D$, we measured a
series of maps of local resistance versus $n$ and $D$ from 20 K up to RT. This
enables the study of thermal excitation of carriers for a given $D$, with a
fixed $n\approx 0$. Arrhenius analysis [23, 24, 16] of these measurements
yielded the value of the band gap in the encapsulated BLG channel, exhibiting
band gap values of up to $\sim 19$ meV as the magnitude of the displacement
field increases to 0.2 V nm-1 (see Figure 5.d). A finite band gap at zero
displacement field applied is noted, along with a non-linear relationship
between the magnitude of $D$ and $E_{g}$, which may be caused by the presence
of the hBN encapsulation. These results are commensurate with previous work on
BLG [4, 16].

Having evidenced the role of the displacement field in opening a band gap in
the BLG channel, we proceed to consider how this relates to the changes
observed in spin transport at RT. We note that the dependence of $\tau_{S}$,
$\lambda_{S}$ and $\Delta R_{nl}$ on $D$ is correlated with the shape of the
relationship between $E_{g}$ and $D$. At low $D$ there is little change in
$E_{g}$; the larger changes in $E_{g}$ only being seen at higher $D$ (Figure
5.d). Therefore, within the transport parameter error bars shown (Figure
5.a-c), we propose that for $|D|\lesssim$ 0.08 V.nm-1 reveal no effect of the
displacement field, however, for $|D|$ ¿ 0.08 V.nm-1 we begin to see an
effect. A displacement field, beyond the range shown, was limited by potential
damage to the device.

At RT and close to the Dirac point it is typically difficult to measure the
device’s spin transport parameters, as a consequence of the low signal to
noise ratio, the resistance mismatch problem [25] and an induced band gap
which is commensurate with the thermal activation energy. However, at the
Dirac point the effects of the band gap are more readily observed, with the
spin splitting being more prominent at the band edge in BLG [26].

It should be noted that our measurement configuration is designed to evaluate
the difference in energy between the spin up and spin down directly, rather
than attempting to measure the spin up and spin down accumulations
independently. Consequently, it is sensitive to small net spin accumulations,
of the order of $\sim$ µeV. Also, the quality of our homogeneous channel
allows us to show a modulation of the spin relaxation time, $\tau_{s}$, which
has not previously been shown. This is made possible by the fabrication of a
BLG transport channel which is fully encapsulated, where the effect of the
contacts and the resistance mismatch problem are minimised by the use of
nanoscale 1D edge contacts [9, 10].

A trivial mechanism by which opening a band gap can modulate spin transport is
via the modulation of the charge resistance. An increase of the sheet
resistance, $\rho$, in the region where the displacement field is applied,
between the injector and detector contacts, would inhibit carrier diffusion
and prevent the flow of spin current across this region [11]. We indeed
observe a modulation of charge transport at RT, as shown in the inset of
Figure 2.b, confirming that the band gap is being electrically opened.
Nevertheless, at RT this only accounts for a change of 5% in $\rho$ for the
spin transport measurements along the C-D path. The spin diffusion between
injector and detector contacts, at a distance $L=3.0$ µm apart, is described
by the following relation [17],

$\Delta R_{nl}\propto\rho\lambda_{S}e^{-\frac{L}{\lambda_{S}}},$ (1)

from which we would expect, to first order, the spin signal, $\Delta R_{nl}$,
to only increase by 5% with increasing displacement field, as it is
proportional to $\rho$. On the contrary, we observe that $\Delta R_{nl}$
decreases as the displacement field increases (see Figure 5.c), with an on/off
ratio of $\lesssim 2$ from D to C.

A more significant mechanism by which the displacement field can modulate spin
transport is via changes in the spin relaxation length, $\lambda_{S}$. As
shown in Equation (1), the factors containing $\lambda_{S}$ have the effect of
decreasing $\Delta R_{nl}$ as $\lambda_{S}$ decreases, which is consistent
with the observed dependence of $\lambda_{S}$ on displacement field (see
Figure 5.b). Therefore the effect of $\lambda_{S}$ overwhelms the small effect
of $\rho$. Note that ideally, based on a twofold decrease in $\lambda_{S}$,
one would expect a $\Delta R_{nl}$ on/off ratio of up to 200, larger than that
observed. We find that this expected modulation in $\Delta R_{nl}$ is,
however, muted in our measurements, due to a simultaneous increase in the
contact polarisation from 0.5% to 5%, which partially counteracts the effect
of $\lambda_{S}$. The latter is a limitation of our current device design,
where the contacts are under the influence of the back gate, and is consistent
with a strong dependence of the polarisation of 1D contacts as a function
carrier density [9]. Modification of the device design, by introducing a local
back gate that does not act on the contacts, would enable a larger on/off
ratio due to only the effect of opening a band gap. Given that the charge
transport parameters, resistivity and diffusion coefficient, vary only by 5%
across our range of displacement field, the variation in the spin relaxation
length is given by the dependence $\lambda_{S}\propto\sqrt{\tau_{s}}$.

We observe that as the displacement field is increased, the spin relaxation
time, $\tau_{s}$, decreases by up to a threefold, and correspondingly the
length, $\lambda_{s}$ decreases too. This observation is commensurate with the
increasing width and decreasing height of the central peak in the spin
precession measurements (Figure 4) as $D$ increases. Previous work has
demonstrated that in BLG the spin lifetime anisotropy, the ratio of the out-
of-plane to the in-plane spin relaxation time, is modulated by a perpendicular
displacement field, due to the resulting out-of-plane spin-orbit fields [16].
This previous work showed that the anisotropy increases with increasing
displacement field in BLG, as a result of both the increase in out-of-plane
spin relaxation time and the decrease in the in-plane spin relaxation time.
However, unlike our experiment, this was carried out at high carrier densities
and low temperatures.

Given our measurement geometry determines the transport of in-plane spins
[17], such a change in anisotropy due out-of-plane spin-orbit fields in BLG is
commensurate with our observations shown in Figure 5.a. Note that this result
of a perpendicular displacement field is opposite to that present in SLG,
where it results in an effective in-plane Rashba spi-orbit field that causes
dephasing of the out-of-plane spins, i.e. the opposite effect on spin
anisotropy [27]. On the other hand, in BLG the dominant phenomena is that of
the change in band structure due to opening the band gap, which allows spin
splitting at the band edges and gives rise to the observed modulation of spin
relaxation time with displacement field [16]. The spin transport measurements
and analysis described above, presented in Figures 5.b, c and d, show that it
is possible to modulate the spin signal in a BLG spin transistor operating at
RT, by virtue of the resulting modulation in $\tau_{s}$ when opening the band
gap.

##  4 Conclusion

We have shown that in a fully hBN encapsulated, 1D edge contacted BLG at RT,
it is possible to electrostatically modulate a nonlocal spin current purely by
a displacement field. The latter contributes to opening a band gap, with only
a small modulation of the charge transport, but a sizeable modulation of the
spin relaxation time and corresponding spin signal.

This work, using a van der Waals device architecture which ensures minimally
invasive contacts and scalable design of electrostatic gates, paves the way
towards further exploration of bilayer graphene as a prototypical spin-FET at
RT, which has application in spin logic [28]. SOC effects are now of
significant interest in technological areas of spintronics such as for memory
devices [29]. Further, investigation of the combination of proximity induced
SOC with the tuneable band gap SOC afforded by BLG and/or a drift current [30]
may produce an enhanced on/off ratio and spin current guiding.

##  5 Methods

###  5.1 Sample Fabrication

BLG samples were prepared by mechanical exfoliation of natural graphite [31],
using the so called Scotch tape method and transferred onto a Si/SiO2 (290 nm)
substrate. BLG candidates were initially found using optical microscopy and
the number of layers confirmed with Raman spectroscopy. Similarly hBN
candidates were identified, confirming their thickness with a combination of
AFM and profilometry. The bottom hBN flake of our devices has a thickness of
12.5 nm and the top flake 5.5 nm. Using the dry transfer technique, a Poly
Methyl Methacrilate (PMMA) membrane ($\sim$500 nm thick) lifted the top hBN
flake, assisted by a transfer rig equipped with micromanipulators and a hot
plate, which improved the membrane stickiness. This top hBN, on the membrane,
was used to lift the BLG with van der Waals forces. Finally both flakes were
deposited on top of the bottom hBN, already on the Si/SiO2 substrate. The
remaining PMMA membrane was removed by soaking the sample in acetone and IPA
for $\sim$5 minutes each. The stack was annealed in a H/Ar atmosphere for 3
hours at 300 °C to remove PMMA residues and contamination. AFM imaging
qualitatively determined the cleanest areas of the stack (clean, bubble free)
where the graphene spin channel could be patterned. A double PMMA electron
beam lithography (EBL) resist layer (solutions of 3% of each of 495 and 950
molecular weight) was spin-coated onto the sample. The channel was patterned
with EBL and etched with an O2/CHF3 plasma, leaving the graphene’s edges
exposed, ready to accept 1D edge contacts. Finally, contacts were patterned
using EBL and Cr/Au evaporated to form non-magnetic contacts, whilst Co/Au was
used to form ferromagnetic contacts with a range of aspect ratios.

###  5.2 Transport Measurement

All spin transport measurements were made in a standard non-local lateral 4
point configuration. Spin precession measurements were fitted with the Hanle
equation and spin parameters extracted, the results of which are shown in
Figures 5.b, c and d. The measurements were made by setting the contact
magnetic configuration to parallel (P) and anti-parallel using an in-plane
magnetic field in a non-local spin valve configuration. In each of the
parallel and anti-parallel configurations an out of plane magnetic field was
swept and the non-local resistance, $R_{nl}$, recorded. The top and back gate
voltages, $V_{tg}$ and $V_{bg}$, were stepped to achieve the required carrier
density, $n$, and electric displacement field, $D$, according to Equations (2)
and (3) [16],

$n=\epsilon_{0}\epsilon_{r_{bg}}\frac{V_{bg}-V_{D_{bg}}}{et_{bg}}+\epsilon_{0}\epsilon_{r_{tg}}\frac{V_{tg}-V_{D_{tg}}}{et_{tg}}$
(2)

$D=\epsilon_{r_{bg}}\frac{V_{bg}-V_{D_{bg}}}{2t_{bg}}+\epsilon_{r_{tg}}\frac{V_{tg}-V_{D_{tg}}}{2t_{tg}}$
(3)

where $V_{D}$ is the top gate, $tg$, and back gate, $bg$, voltage required to
overcome residual channel doping and achieve charge neutrality. $t$ is the
thickness of the gate dielectric and $\epsilon_{r}$ is the relative
permittivity of the gate dielectric, which can be calculated from the
respective gate’s capacitance, $C$, equations,

$C_{bg}=\frac{\epsilon_{r_{bg}}}{t_{bg}}A=\frac{\epsilon_{r_{SiO_{2}}}\epsilon_{r_{bghBN}}}{\epsilon_{r_{SiO_{2}}}t_{bghBN}+\epsilon_{r_{gbhBN}}t_{SiO_{2}}}A$
(4)
$C_{tg}=\frac{\epsilon_{r_{tg}}}{t_{tg}}A=\frac{\epsilon_{r_{tghBN}}}{t_{tghBN}}A$
(5)

where $A$ is the area of the gate, ${\epsilon_{r_{SiO_{2}}}}$ = 3.9 and
${\epsilon_{r_{bghBN}}}$ = ${\epsilon_{r_{tghBN}}}$ = 3.2 are the relative
permittivities of the SiO2 substrate and hBN, respectively. $t_{tghBN}$ and
$t_{bghBN}$ are the thicknesses of the two hBN flakes, described earlier. The
device presented here has a SiO2 thickness, ${t_{SiO_{2}}}$ = 290 nm.

## Data availability

The datasets used and/or analysed during the current study are available from
the corresponding author on reasonable request.

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## Acknowledgments

Financial support by the Engineering and Physical Science Research Council
(EPSRC) for a PhD studentship and Doctoral Prize Fellowship is gratefully
acknowledged, as well as support from the Graphene Flagship Core3 project.
Additionally, the financial support of Consejo Nacional de Ciencia y
Tecnología (CONACyT), México for a PhD studentship is also gratefully
acknowledged.

## Author Contributions

C.R.A. and N.N. carried out the research. V. H. G. provided assistance with
the device fabrication. The work was supervised by I.V.G. and I.J.V. All
authors contributed to the preparation of the manuscript.

## Additional Information

Competing Interests: The authors declare no competing interests.