3.1. Magnetoresistance effect: theoretical framework and applications
The MR can be measured in different geometric configurations determined by the relative angle between the electric current
I and the magnetic field
. Also for the case of semimetals the most common MR measurement is performed in the Hall configuration with
oriented out-of-plane and perpendicular to the direction of the current. The main source of MR in semimetals is related to the deviation of the charge carrier (both electrons and holes) trajectory due to the Lorentz force (Hall effect). During the last years, great attention has been devoted to the MR effect in high-mobility semimetals and it has been found that topological semimetals exhibit a huge MR effect [
126,
127,
128,
129,
130,
131,
132,
133] as occurs for topological insulators [
134]. Quantitatively, and analogously to other types of materials, the MR can be expressed also for topological semimetals in percentage units as:
where
is the material electrical resistivity at
T and
is the material electrical resistivity at
T, i.e.,
(expressed in Ω m or Ω cm). For both electrons and holes (
), the field-dependent electrical resistivity
and electrical conductivity
at low and intermediate magnetic fields can be written as [
135]:
with
the electrical conductivity at
T. Specifically, for fixed mobility
, at low
B the conductivity
, namely it linearly increases with the external magnetic field, while at intermediate and high fields
, i.e., the electrical conductivity rapidly diminishes with increasing
B.
The MR is, in general, huge even for small external magnetic fields in high-mobility semimetals especially at low temperatures at which the carrier mobility and the electrical conductivity are remarkable. Topological semimetals, both of the Dirac and Weyl type, show high MR due to a balanced hole-electron resonance effect resulting from their peculiar band structure, therefore there is not only the contribution to MR from electrons but also from holes. As a consequence, the MR does not saturate at low and intermediate B, as occurs for semiconductors and graphene.
In
Figure 4 the general trend of measured MR for some representative topological semimetals is displayed.
Figure 4(a) graphically expresses the relationship between the measured MR (%) and electrical conductivity
. This dependence is shown for MR (%) probed at
K and at
T [
126]. The highest MR(%) (almost
) is measured in WP
2 (a Weyl II semimetal) at a conductivity of about
cm
−1 and a slightly lower value is exhibited by MoP
2, another Weyl II semimetal. However, also NbP, LaSb and PtBi
2, which are Weyl I semimetals, exhibit a MR (%) of about
at lower conductivities. Lower but still considerable values of MR (%) (about
) are typical of TaAs, the transition-metal dichalcogenides WTe
2, NbSb
2, PtSn
4 representing Dirac nodal semimetals and the Cd
3As
2 Dirac semimetal.
It is well-known that MR is proportional to resistivity but is inversely proportional to conductivity so that the MR effect is significant in low-conductivity materials. However, looking at
Figure 4(a), Weyl II semimetals WP
2 and MoP
2 simultaneously exhibit large MR and high values of conductivity larger than those of the Cd
3As
2 Dirac semimetal and with
of the same order of magnitude as K and one order of magnitude smaller than Cu. Therefore, Weyl II semimetals are materials having large carrier mobilities comparable to those of the typical metals. This means that they can be regarded as materials very similar to metals and with electrical transport properties not only determined by electrons but also by holes.
Figure 4(b-e) display MR measurements of typical high-mobility semimetals as a function of the applied external magnetic field at different temperatures in the Hall configuration, i.e. the out-of-plane external magnetic field
is perpendicular to the in-plane current (
). The two general observations for all the semimetals, which can be drawn looking at
Figure 4(b-e), are: 1) MR does not reach a saturation value but tends to increase with increasing
B for low and intermediate
B and 2) MR decreases with increasing
T as in III-V semiconductors but at room temperature it is still huge. Regarding 1), the contribution to MR due to holes in high-mobility semimetals is more relevant with respect to the one present in III-V semiconductors giving rise to a more accentuated electron-hole resonance effect which leads to a larger MR value in semimetals.
Concerning 2), also in high-mobility semimetals carrier mobility decreases with increasing temperature due to the huge scattering of electrons and holes with acoustic and optical phonons, especially at intermediate and high
T close to room temperature, but the effect of scattering is partially masked by the appreciable contribution of holes to the mobility. One notes that there are some quantitative discrepancies among the measured MR (%) values depending on the type of semimetal. In particular, the highest values at any temperature are exhibited by the NbP type-I Weyl semimetal, but also the WTe
2 type-II Weyl semimetal shows big values of MR(%) especially at large applied external magnetic fields. On the other hand, at fixed external magnetic field, the MR (%) for the representative Dirac semimetal is at least one order of magnitude less than the one exhibited by Weyl semimetals, while the nodal-line semimetal has a MR (%) comparable to the one of type II Weyl semimetal. The reason of these discrepancies is attributed to the higher carrier mobility characterizing Weyl semimetals which leads to a more accentuated increase of the field-dependent resistivity with increasing
B as expressed by Eq. (
2).
MR is measured also for other geometries (
) and in the so-called longitudinal configuration with
. As shown in
Figure 4(f-h) [
107,
121,
137,
138] the MR vs.
is negative and the minimum deepens with decreasing
T. When
is not orthogonal to the current (and, as a result, to the electric field
) there is not anymore the quantum mechanical conservation of the particle number, the so called Adler–Bell–Jackiw anomaly or chiral anomaly which manifests itself in the presence of well-defined chirality appearing when the Fermi level is close to the Weyl nodes [
137]. It can be proved that chiral anomaly in Weyl semimetals leads to negative MR vs.
with MR absolute value increasing as temperature decreases and this effect is maximum when
. In the presence of chiral anomaly, the external magnetic field suppresses the phase coherence of the back-scattered electron and hole waves and destroys the weak localization effect. The more
deviates from the collinear configuration with the current, the more the Lorentz force gives a positive contribution which cancels the negative MR effect.
Considering that the MR absolute value is much lower with respect to that in the Hall configuration at fixed B intensity, the design of MR sensors based on this effect could be useful but less exploitable if compared to that in the Hall geometry.
The mobilities of representative topological semimetals are summarized in
Table 1 together with their MR(%) at low temperatures and at given values of the external magnetic fields [
20]. Among the different types of semimetals, Weyl II semimetals are the ones exhibiting the highest MR(%) but also Weyl I semimetals are characterized by high mobility especially at low temperature and, as a result, also by a huge value of MR(%). Looking at Table (
Table 1) one notes that there is a strict relationship between the two physical quantities as for other types of high-mobility materials such as graphene or some types of III-V semiconductors.
3.2. Novel type of quantum Hall effect in 3D topological
semimetals
Very recently, a 3D quantum Hall effect has been reported in some types of high-mobility topological semimetals [
125,
148,
149]. It can be considered the equivalent in three dimensions [
150] of the usual 2D quantum Hall effect [
151,
152]. This effect, also called integer quantum Hall effect to distinguish it from the fractional quantum Hall effect, is the corresponding quantum version of the Hall effect. It consists of the quantization of the Hall conductance (or equivalently of the Hall resistance) in 2D (
x-
y plane) electron systems brought to low temperature (liquid helium temperature
K) in the presence of a strong external magnetic field
B (several Tesla) applied perpendicularly to the plane of the charge carrier system. The Hall quantized electrical conductance
with
I the electrical current and
the Hall potential can be expressed as
, where
is a non-negative integer expressing the
quantization in terms of Landau levels lying below the Fermi energy
,
e is the electron charge and
h is the Planck constant. In 2D systems
has the dimensions of
in Siemens (S), while in 3D systems
has the dimensions of
over length in S/m or S/cm.
The main observed feature is the exhibition of the quantum Hall conductance plateaus at
either as a function of
or of the carrier density which persist by varying widely
B or the carrier density. Equivalently, the quantum Hall resistance
shows plateaus at
) as a function of
B or of the electron density. This result looks surprising because quantum Hall conductance does not depend explicitly on carrier density. The family of TaAs Weyl semimetals together with the Cd
3As
2 and Na
3Bi Dirac semimetals are high-mobility materials and, therefore, satisfy the main requirement necessary to exhibit the 3D quantum Hall effect. The main requirement is the presence of the Fermi arcs placed on the Fermi surface between Weyl nodes and formed by the topologically protected states. In this way, by exploiting the tunnel effect, the electrons can pass through the Fermi arcs placed at opposite surfaces via a "wormhole" tunneling supported by the Weyl nodes [
125]. In addition, to have the 3D quantum Hall effect: 1) the Fermi arcs should also make closed Fermi loops; 2) bulk carriers must be depleted by tuning the Fermi energy to the Weyl nodes and 3) there should be band anisotropy so that the Fermi arcs give rise to a 2D gas [
125]. Due to the time-reversal symmetry, a complete Fermi loop representing one of the requirements to observe a 3D quantum Hall effect occurs on a single surface of the Cd
3As
2 and Na
3Bi Dirac semimetals. In particular, the 3D quantum Hall effect based on Weyl orbits has been recently realized in the 3D Dirac Cd
3As
2 high-mobility semimetal in nanostructured form [
149]. It is crucial to understand the behavior of Weyl orbit under an external magnetic field. There are two pairs of Weyl nodes of opposite chiralities acting like "wormholes" connected by Fermi arcs lying on two opposite surfaces in the
-
plane, the top and the bottom Fermi arc. The propagation of the electrons occurs along the
z-direction through the bulk chiral Landau levels in order to complete the cyclotron motion.
The quantized 3D Hall conductivity for topological Weyl and Dirac semimetals can be calculated starting from the Kubo formula
with
Weyl, Dirac,
the eigenstate corresponding to the energy
when the external magnetic field is applied along the
y direction,
with
the velocity operator,
is the Fermi distribution,
the effective volume of the slab or the areas hosting the Fermi arcs and
expresses the level broadening.
means that disorder is weak and does not affect the width of plateaus. The sheet 2D quantum Hall conductivity having dimensions of
(S) is obtained as
with
L the slab thickness. First, let’s deal with the quantum Hall conductivity
(with the current density along
x and the induced electric field along
z) of the Weyl topological semimetal. From Eq. (
4) it turns out to be at zero temperature [
125]
with
,
,
,
and .
Let’s now calculate the quantum Hall conductance of Fermi arc I at the upper slab surface (
). The effective Hamiltonian of the Fermi arc in a 3D Weyl topological semimetal having the form of a slab of thickness
L (see
Figure 5) can be derived from the Hamiltonian of a 3D Weyl topological semimetal and parametrized in the form
for the
surface with
being
,
,
A and
M model parameters. The quantum Hall conductance of arc I
for a Weyl topological semimetal slab (for
) obtained from the 2D electron gas quantum Hall conductivity
(with
the number of Landau levels below the Fermi energy) is [
125]
with
R =
,
the area filled by arc I in the momentum space and
denoting rounding down. The sign of the quantum Hall conductance depends on the signs of
R and of
in turn depending on the carrier type, charge type and on the direction of the external magnetic field. Similar conclusions can be drawn by considering Fermi arc II at the lower slab surface (
).
Analogously, also the quantum Hall conductivity of a Dirac semimetal slab can be calculated from (
4) at zero temperature. It turns out to be [
125]
with
,
being
the uniform magnetic field applied in the
-
plane,
V volume of the slab,
being
the
x-component of the wave vector,
and
the effective masses of the electrons,
,
.
Figure 5 shows the 3D quantum Hall effect observed both in the TaAs family topological Weyl semimetal (
Figure 5 (a)-(d)) and in the Cd
3As
2 topological Dirac semimetal ((
Figure 5 (e)-(l))).
Figure 5(a)-(b) display the topological features of 3D TaAs in the form of a slab occurring in the presence of a 3D quantum Hall effect including the energy dispersion and the topology of the Fermi arcs at
. In
Figure 5(c) the corresponding calculated 2D sheet quantum Hall conductivity vs. the inverse of the magnetic field shows the distinctive plateaus expression of the Landau level quantization of Hall conductivity. An analogous behavior is depicted in
Figure 5(d) for the quantum Hall conductance of Fermi arc I.
In
Figure 5(e)-(h) the behaviors of the measured longitudinal resistance
and of the transverse Hall resistance
of Cd
3As
2 films lying on the
plane of different thicknesses as a function of the magnetic field
B are shown. All films are characterized by the same carrier density
cm
−3. As soon as the external magnetic field is turned on, Shubnikov-de Haas (SdH) oscillations of the longitudinal resistance begin to appear exhibiting a decreasing frequency with increasing the magnitude of the magnetic field. The vanishing of
occurs for
T, especially for the smaller thicknesses (
Figure 5(g) and (h)).
By observing the trend of the transverse Hall resistance , the quantum Hall states begin to appear analogously to what occurs for the calculated conductivity. In the special case, the quantized values can be expressed in the form , with an integer number depending in this case not only on but also on the degeneracy factor s. One notes that the integer quantum Hall states clearly emerge for for all the thicknesses considered. Moreover, the degeneracy factor drastically changes from s = 2 to s = 1 for the thickness passing from t = 14 nm to t = 16 nm. This change is due to spin splitting and is not related to other types of degeneracies, such as the lifting of valley or surface states. Interestingly, by performing the Fourier transform of the SdH oscillations the area of the Fermi surface can be calculated via the Onsager relation as where is the primary oscillation frequency. For example, for the t = 12 nm film .
Figure 5(i) illustrates the 3D quantum Hall modelization for a Cd
3As
2 slab (coordinate system
-
-
) grown along the [112] and [110] crystallographic directions and the corresponding calculated sheet Hall conductivity as a function of
for different orientations of the magnetic field marked by the angle ξ for
(
Figure 5(l)). Conductivity plateaus appear for any orientation of
and their widths are of different sizes for any
investigated.