17 Axion Electrodynamics in Magnetoelectric Media
487
ment is appropriate for experimental realization only if it were possible to consider
a genuine WSM, for which no external magnetic field is needed.
Scanning SQUID magnetometry
Scanning SQUID (Superconducting Quantum Interference Device) magnetometry
provides an alternative for measuring the induced magnetic field. SQUIDs are very
sensitive magnetometers based on superconducting loops containing Josephson junctions, and they are used to measure extremely low magnetic fields (of the order
of 5 × 10
−18 T [102]). In general terms, these devices measure the magnetic flux
through a loop (parallel to the surface) placed at a fixed distance above the material,
i.e. B =
S
B · dS, where S is the surface of the loop.
In the present case, a simple calculation produces B (R, z) = 2π R ˆ
e ϕ · A z>0
(R, z), where R is the radius of the loop and A z>0 is given by (17.75). Of course,
B = 0 at R = 0. Furthermore, since the magnetic field lines start at the WSM
surface and go back again to the surface (see the right panel of Fig. 17.5), we then
have B → 0 as R → ∞. This behaviour implies the existence of a maximum
flux at a critical radius R c , which can be determined in the usual manner (i.e. by
solving ∂ R B | R=R c = 0 for R c ). For a numerical estimate of the magnetic flux we
consider a charge q = n e |e| placed at a distance z
= 1µm above the surface of the
WSM TaAs and a SQUID of radius R = 10µm located at z = 10µm. We find B ≈
7n e × 10
−14 T · cm
2 , which is measurable with present day attainable sensitivities of
SQUIDs [102]. One of the key challenges for the experimental detection of this flux
profile would be to find a way to fix and localize the charge above the surface.
If this experiment were carried out with a Dirac semimetal instead of a genuine
WSM, the required external magnetic field will overwhelm the topological contribution to the total magnetic flux, as before. Nevertheless, in this case it is still possible
to disentangle these effects by using the fact that the contribution to the flux produced
by the external magnetic field, say
ext
B , is constant in space and time, but not B . A
sensitive magnetometer as the SQUID will be capable to measure small variations
of the flux which amounts to eventually measuring the induced electromotive force
E in the loop. Therefore, this allows for isolating the topological contribution, for
example, by producing a controlled displacement of the SQUID along the z-axis
at speed v z , namely: E = −
d
dt
ext
B + B
= −v z
d B
dz
, where the z-dependence is
read-off from (17.75).
17.7 Conclusions
In this chapter we have given an overview as to how the properties of magnetoelectric
media associated with topological order, that arise from subtleties in their electronic
structure, can be understood as macroscopic optical properties in regards to their
electromagnetic response. We have presented the reader with the theoretical framework that allows to describe the electromagnetic response of magnetoelectric media
487
ment is appropriate for experimental realization only if it were possible to consider
a genuine WSM, for which no external magnetic field is needed.
Scanning SQUID magnetometry
Scanning SQUID (Superconducting Quantum Interference Device) magnetometry
provides an alternative for measuring the induced magnetic field. SQUIDs are very
sensitive magnetometers based on superconducting loops containing Josephson junctions, and they are used to measure extremely low magnetic fields (of the order
of 5 × 10
−18 T [102]). In general terms, these devices measure the magnetic flux
through a loop (parallel to the surface) placed at a fixed distance above the material,
i.e. B =
S
B · dS, where S is the surface of the loop.
In the present case, a simple calculation produces B (R, z) = 2π R ˆ
e ϕ · A z>0
(R, z), where R is the radius of the loop and A z>0 is given by (17.75). Of course,
B = 0 at R = 0. Furthermore, since the magnetic field lines start at the WSM
surface and go back again to the surface (see the right panel of Fig. 17.5), we then
have B → 0 as R → ∞. This behaviour implies the existence of a maximum
flux at a critical radius R c , which can be determined in the usual manner (i.e. by
solving ∂ R B | R=R c = 0 for R c ). For a numerical estimate of the magnetic flux we
consider a charge q = n e |e| placed at a distance z
= 1µm above the surface of the
WSM TaAs and a SQUID of radius R = 10µm located at z = 10µm. We find B ≈
7n e × 10
−14 T · cm
2 , which is measurable with present day attainable sensitivities of
SQUIDs [102]. One of the key challenges for the experimental detection of this flux
profile would be to find a way to fix and localize the charge above the surface.
If this experiment were carried out with a Dirac semimetal instead of a genuine
WSM, the required external magnetic field will overwhelm the topological contribution to the total magnetic flux, as before. Nevertheless, in this case it is still possible
to disentangle these effects by using the fact that the contribution to the flux produced
by the external magnetic field, say
ext
B , is constant in space and time, but not B . A
sensitive magnetometer as the SQUID will be capable to measure small variations
of the flux which amounts to eventually measuring the induced electromotive force
E in the loop. Therefore, this allows for isolating the topological contribution, for
example, by producing a controlled displacement of the SQUID along the z-axis
at speed v z , namely: E = −
d
dt
ext
B + B
= −v z
d B
dz
, where the z-dependence is
read-off from (17.75).
17.7 Conclusions
In this chapter we have given an overview as to how the properties of magnetoelectric
media associated with topological order, that arise from subtleties in their electronic
structure, can be understood as macroscopic optical properties in regards to their
electromagnetic response. We have presented the reader with the theoretical framework that allows to describe the electromagnetic response of magnetoelectric media