#  Pressure controlled trimerization for switching of anomalous Hall effect in
triangular antiferromagnet Mn3Sn

Charanpreet Singh  School of Physical Sciences, National Institute of Science
Education and Research, HBNI, Jatni-752050, India  Vikram Singh  School of
Physics, Indian Institute of Science Education and Research
Thiruvananthapuram, Kerala-695551, India  Gyandeep Pradhan  School of Physical
Sciences, National Institute of Science Education and Research, HBNI,
Jatni-752050, India  Velaga Srihari  High Pressure and Synchrotron Radiation
Physics Division, Bhabha Atomic Research Centre, Mumbai 400085, India
Himanshu Kumar Poswal  High Pressure and Synchrotron Radiation Physics
Division, Bhabha Atomic Research Centre, Mumbai 400085, India  Ramesh Nath
School of Physics, Indian Institute of Science Education and Research
Thiruvananthapuram, Kerala-695551, India  Ashis K. Nandy<EMAIL_ADDRESS>School of Physical Sciences, National Institute of Science Education and
Research, HBNI, Jatni-752050, India  Ajaya K. Nayak<EMAIL_ADDRESS>School
of Physical Sciences, National Institute of Science Education and Research,
HBNI, Jatni-752050, India

###### Abstract

Here, we present a detailed theoretical and experimental study on the pressure
induced switching of anomalous Hall effect (AHE) in the triangular
antiferromagnetic (AFM) compound Mn3Sn. Our theoretical model suggests
pressure driven significant splitting of the in-plane Mn bond lengths $i.e.$
an effective trimerization, which in turn stabilizes a helical AFM ground
state by modifying the inter-plane exchange parameters in the system. We
experimentally demonstrate that the AHE in Mn3Sn reduces from 5 $\mu\Omega$ cm
at ambient pressure to zero at an applied pressure of about 1.5 GPa.
Furthermore, our pressure dependent magnetization study reveals that the
conventional triangular AFM ground state of Mn3Sn systematically transforms
into the helical AFM phase where the symmetry does not support a non-vanishing
Berry curvature required for the realization of a finite AHE. The pressure
dependent x-ray diffraction (XRD) study rules out any role of structural phase
transition in the observed phenomenon. In addition, the temperature dependent
in-plane lattice parameter at ambient pressure is found to deviate from the
monotonic behavior when the system enters into the helical AFM phase, thereby,
supporting the proposed impact of trimerization in controlling the AHE. We
believe that the present study makes an important contribution towards
understanding the stabilization mechanism of different magnetic ground states
in Mn3Sn and related materials for their potential applications pertaining to
AHE switching.

Anomaous Hall Effect, electrical switching, triangular antiferromagnts,
Heusler compounds

###### pacs:

75.50.Gg, 75.50.Cc, 75.30.Gw, 75.70.Kw

##  I INTRODUCTION

Effective manipulation of magnetic structure can decisively control the
electrical AHEferroReview ; AHEreviewNagaosa ; AHEtheorynonC ;
AHEmn3SnNakatsuji ; AHEmn3GeAjaya , thermal THEtheory  and optical
opticskerrtheory ; optickerrTopo ; AFopticsreviw  properties in magnetic
materials. In case of electrical transport in a magnetic system, the spins of
the conduction electrons exhibit a strong Hund’s coupling with the underlying
magnetic structure. The fictitious magnetic field arising from such an
interaction can significantly modify the path of the conduction electrons,
leading to the observation of anomalous Hall effect (AHE) and topological Hall
effect (THE) in ferromagnets AHEferroReview /antiferromagnets AHEtheorynonC ;
AHEmn3SnNakatsuji ; mn3snPredeep . A lot of effort has been put forward to
study the possibility of switching these transport signals to facilitate
device applications. Although the AHE in ferromagnets has a great potential
for its utilization as a memory element in spintronic devices, the only viable
mechanism seems to be the reversal of underlying magnetic structure. In this
regard, antiferromagnets with diverse magnetic structures are excellent
candidates to achieve a control over the AHE by tuning the magnetic ground
state. The fact that antiferromagnets are extremely stable against external
perturbations makes them attractive candidates for spintronics. Although
electrical switching of antiferromagnets have been demonstrated recently
EswitchJung ; Leandro , yet the realization of such a scenario in real devices
remains afar.

In particular, the non-collinear antiferromagnets are extensively pursued as
their magnetic states can be easily manipulated by spin transfer torque
related mechanisms. Recently, the hexagonal crystal structure based triangular
antiferromagnets Mn3Sn and Mn3Ge have received a special attention due to
their various anomalous transport properties, both in bulk systems
AHEmn3SnNakatsuji ; AHEmn3GeAjaya ; teraAnomaly ; invSpinMn3sn  and thin films
mn3snfilm1 ; mn3snfilm2 ; mn3snfilm3  . In the hexagonal structure of Mn3Sn,
the Mn atoms form a kagome-type lattice with Sn atoms residing at the center
of the hexagon. The geometrical frustration arising due to the AFM coupling of
Mn moments and the kagome type of arrangement of Mn atoms incite a
$120^{\circ}$ magnetic ground state mn3snNutron . In this structure, two
layers of Mn atoms in a unit cell together form an inverse triangular spin
structure as depicted in Fig. 1(a). The topological Weyl semi-metallic nature
of these materials has been demonstrated by theoretical calculations
theoryWeylmn3snge_Kubler ; theoryWeylmn3snge_BYan , as well as experimental
studies expWeylmn3sn . The presence of Weyl nodes near the Fermi level is
believed to be the source of large AHE in these materials AHEweylTheory ;
expWeylmn3sn ; theoryWeylmn3snge_BYan . These Weyl nodes occur in pairs with
opposite chirality and act as magnetic monopoles in the momentum space. An
electron crossing between two Weyl nodes of opposite chirality picks up an
extra Berry phase, and consequently a large AHE has been observed in case of
Mn3Sn and Mn3Ge. Since the topological Weyl nature, and hence the AHE of these
materials is critically inherited from the special kind of magnetic structure,
a precise control over the magnetic state is essential for any possible
switching of the AHE.

Mn3Sn exhibits distinct AFM ground states in different temperature regimes
depending upon the Mn concentration in the system diffMncomposition . While
most of the samples show an in-plane triangular AFM ordering at room
temperature, the presence of a slightly canted triangular AFM structure has
also been found at very low temperatures Fengcaxis ; Tomicaxis . The recent
observation of large THE below 50 K in Mn3Sn corroborates the existence of
noncoplanar magnetic structure at low temperatures mn3snPredeep . While the
samples used by Kuroda et al. expWeylmn3sn  display an additional magnetic
transition below 50 K, the presence of a sharp transition around 240 K related
to the helical magnetic ordering has also been observed in other systems
mn3sndiffphase . Irrespective of the nature of the low temperature
transitions, all the samples display a large AHE at room temperature. However,
the AHE ceases to zero below 240 K when the system enters into a spin spiral
(SS) state mn3sndiffphase . Using $ab~{}initio$ calculations and Monte Carlo
simulations, Park et al. have demonstrated the presence of an incommensurate
helical modulation of the triangular spin structure that propagates along the
$c$-direction mn3qm . Interestingly, no Weyl nodes were found in the
electronic band structure for the SS magnetic state, explaining the absence of
AHE in the helical magnetic phase mn3qm . Therefore, it is important to
investigate the correlation between the two observed magnetic ground states of
Mn3Sn to explore the possibility of controlling the transition between them.
Motivated by the aforementioned discussion, herein, we utilize hydrostatic
pressure as a control parameter to probe the switching mechanism of AHE in
case of the non-collinear antiferromagnetic Mn3Sn.

##  II Methods

The density functional theory (DFT) calculations were employed to study the
magnetic properties of the present system. The spin-polarized calculations was
carried out within the projector augmented wave (PAW) method paw  as
implemented in Vienna Ab initio Simulation Package (VASP) V1 ; V2 ; V3 ; V4 .
The generalized gradient approximation (GGA) was used as the exchange-
correlation potential in the form of Perdew-Burke-Enzerhof (PBE) pbe . For the
Brillouin-Zone (BZ) integration, a $\Gamma$-centered $k$-point grid of
8$\times$8$\times$9 was used with a plane-wave cut-off energy of 500 eV. Note
that the noncollinear version of the code was utilized for determining the
conventional magnetic ground state in this type of hexagonal system. The
experimentally obtained lattice constants were kept fixed in our calculations
where the internal coordinates were relaxed till the forces become less than 1
meV/Å. Further, the spin-spiral calculations were performed using the full
potential linearized augmented plane wave (FLAPW) method implementing the
noncollinear version of the FLEUR code fleur1 . The energy of the SS state
characterized by wave vector q was calculated using the generalized Bloch
theorem fleur2 . We have also employed PBE exchange-correlation functional as
the GGA approximation. We have considered 16$\times$16$\times$18 mesh for
$k$-points in the full-BZ and a cutoff of $K_{max}$ = 4.1 $a.u.^{-1}$ to
expand the LAPW basis functions. The q =0 state was considered as a perfect
$120^{\circ}$ triangular spin configuration on each Kagome sublattice.

Polycrystalline ingots of Mn3+xSn1-x with $x=$ 0.05, 0.04, and 0.03 were
prepared by arc melting stoichiometric amount of high pure Mn and Sn metals
under argon atmosphere. The samples were melted multiple times to ensure a
better homogeneity. The as-prepared ingots were annealed for eight days in
evacuated quartz tubes at a temperature of 1073 K, followed by slow cooling to
room temperature. Room temperature x-ray powder diffraction measurements were
performed using a Rigaku SmartLab x-ray diffractometer with a Cu-K$\alpha$
source. Low temperature XRD measurements were carried out by utilizing a low
temperature attachment (Oxford Pheonix) to the x-ray diffractometer
(PANalytical). To investigate the crystal structure with applied pressure, XRD
measurements were performed using a diamond anvil cell at the beamline BL-11,
INDUS-II synchrotron radiation source (Raja Ramanna Centre for Advanced
Technology, Indore, India). Magnetic and transport measurements were carried
out using a Quantum Design (QD) SQUID magnetometer and QD Physical Property
Measurement System (PPMS), respectively. Pressure dependent magnetic and
transport studies were conducted using respective pressure cells provided by
Quantum Design.

Figure 1: (Color online) (a) Unit cell for hexagonal Mn3Sn and stacking of
three triangular layers (Kagome layers) of Mn moments with $120^{\circ}$ AFM
spin structure. Red and blue balls represent different layers of Mn atoms,
whereas, the Sn atoms are shown in gray balls. $J_{1}$, $J_{2}$, and $J_{3}$
represent first three out of plane (interlayer) neighboring exchange
interactions between the Mn moments. (b) Top view of Mn3Sn where two ideal
Kagome sublattice of Mn atoms are stacked. The inplane bond lengths $l_{1}$
and $l_{2}$ of the equilateral triangles shown in dashed boxes $`$1’ and $`$2’
and here $\delta$ = $\frac{l_{2}-l_{1}}{l_{2}+l_{1}}$ = 0. (c) Top view of the
distorted (relaxed) Kagome sublattice stacking in Mn3Sn with two different
inplane bond lengths ($l_{2}>l_{1}$) for the equilateral triangles as shown in
dashed boxes $`$1’ and $`$2’ with $\delta$ $\neq$ 0\. Splitting of $J_{2}$ to
$J_{2}^{l}$ and $J_{2}^{s}$ is shown in green lines. (d) Phase diagram in the
$J_{2}$ \- $J_{3}$ plane for ideal Kagome layer stacking. $J_{2}$ and $J_{3}$
are measured in the unit of the magnitude of J1. The color code indicates the
angle of rotation of Mn triangle spins between two layers due to the helical
modulation. The deep blue region indicates the $120^{\circ}$ triangular AFM
state. (e) Phase diagram in the $J_{2}^{l}$ \- $J_{2}^{s}$ plane where the
nonzero $\delta$ value results in splitting of $J_{2}$ to $J_{2}^{l}$ and
$J_{2}^{s}$. The values marked by the down triangles on the $J_{3}$ axis in
(d) are used for the phase diagrams shown in Fig. 8 (Appendix A). (f) Ab
initio calculated spin spiral energy as a function of the spin spiral vector q
= $(0,0,q_{z})$ where $120^{\circ}$ triangular AFM state corresponds to q = 0.
Energies of the spin spiral state are calculated with respect to the
$120^{\circ}$ AFM state at zero pressure for $\delta$= 0 (filled triangles),
$\delta$= 2.4 (open squares), and $\delta$= 12.0 (filled diamonds). The data
for $\delta$= 0 at the zero pressure (filled triangles) are divided by a
factor of ten for better visibility and comparison in the plot. The q
dependence of the spiral energy at pressure 5 GPa in case of ideal ($\delta$ =
0) and relaxed ($\delta$ = 10.34) Kagome structure are also shown with filled
and open circles, respectively. The symbols are calculated data points and the
solid lines are to guide the eye. The negative slope in the dispersion curves
indicates the helical modulation of the $120^{\circ}$ triangular AFM state
along $z$.

##  III RESULTS AND DISCUSSION

It is very important to understand the underlying magnetic states which govern
the control and switching of AHE in Mn3Sn. The primary contributions to the
total energy of the magnetic state can be narrowed down to the exchange
interactions, Dzyaloshinskii-Moriya interaction (DMI), and anisotropy energy.
Since the inverse triangular spin structure in Mn3Sn is mainly stabilized by
geometrical frustration, the role of DMI to influence the long-range magnetic-
order in the present centrosymmetric system is extremely limited moriyaDMI ,
except breaking the degeneracy of the sense of rotation. The contribution from
the anisotropy energy can also be safely ruled out for the helical modulation
of an in-plane triangular spin structure along $z$-axis. Thus, the competing
out of plane exchange interactions primarily determine the helical modulation
in the system. To understand the origin of the helically modulated magnetic
ground state, we first analyze the role of different exchange parameters
within classical Heisenberg model,
$H=\sum_{i>j}J_{ij}\hat{n}_{i}\cdot\hat{n}_{j}$, where the exchange constants
$J_{ij}$s determine the strength and nature of interactions between Mn moments
pointing along the unit vector $\hat{n}$.

In the hexagonal Mn3Sn structure each Mn atom in the Kagome plane has four
nearest neighbors, where two equilateral triangles are connected by a common
Mn atom [see the lattice with red/blue Mn balls in Fig 1(b)]. However, it has
been reported that Mn3Sn forms a distorted Kagome lattice where one of the Mn
triangle shrinks while the other expands as shown in Fig. 1(c) ANEmn3sn .
Here, we first consider the case of an ideal Kagome sublattice structure and
try to find out the role of frustration in the exchange parameters in
stabilizing the SS ground state. For this purpose, we made a reasonable
assumption that the in-plane exchange constants do not play any role in
stabilizing the helical order along the $z$-direction. Therefore, we fixed the
in-plane magnetic structure to a perfect $120^{\circ}$ triangular AFM
configuration. The first three inter-plane nearest-neighbor exchange
parameters, $J_{1}$, $J_{2}$, and $J_{3}$ as labeled in Fig. 1(a), are taken
into consideration in order to study the transition from noncollinear AFM
state to helically modulated magnetic state. In the inelastic neutron
scattering experiment, $J_{1}$ and $J_{3}$ were reported to be AFM in nature
while $J_{2}$ was found to be ferromagnetic (FM) mn3qm . The energy of the
magnetic state per Mn atom is now translated into the following classical
exchange Hamiltonian,

$H=2J_{1}\cos(\frac{2\pi}{3}+\theta)+2J_{1}\cos(\frac{2\pi}{3}-\theta)+6J_{2}\cos\theta\\\
+2J_{3}\cos(2\theta).$ (1)

Here, $\theta$ is the measure of helical modulation i.e. the angle of rotation
between two successive layers of spin-triangles connected with the SS vector q
and $\frac{2\pi}{3}$ in $J_{1}$ is the initial phase between two consecutive
Kagome sublattices. The value of $\theta$ varies from 0 to $180^{\circ}$ and
the corresponding lowest energy state is the ground state in the
$J_{2}$-$J_{3}$ plane. Here, both $J_{2}(\leq 0)$ and $J_{3}(\geq 0)$ are
relative quantity with respect to the magnitude of $J_{1}$. The results
presented in Fig. 1(d) show a sharp boundary between the noncollinear AFM
ordering with and without the helical modulation along $z$. We find that the
competing $J_{2}$ and $J_{3}$ drives the system into the helical SS state,
indicating that the exchange frustration is the key route for achieving the
spiral modulation along $z$. It is important to mention here that the SS
ground states stabilized by competing exchange interactions have also been
found in hexagonal femono  and square NandyPRL  lattices.

The ideal Kagome sublattice in Mn3Sn is expected to be distorted in two
unequal equilateral triangles as discussed above. Therefore, we define an
effective trimerization parameter, $\delta$ as
$\frac{l_{2}-l_{1}}{l_{2}+l_{1}}$, where $l_{1}$ and $l_{2}$ are the bond
lengths corresponding to two equilateral triangles formed by Mn atoms in the
Kagome sublattice [see dashed square $`$1’ and $`$2’ in Fig. 1(b-c)]. One can
consider $\delta$ as the effective trimerization parameter where the inplane
exchange coupling within two unequal triangles can be changed significantly
mn3qm ; gd3trimer . Here we note that $\delta=0$ defines the ideal Kagome
lattice. Since a nonzero value of $\delta$ causes bond length mismatch related
to the exchange constant $J_{2}$, corresponding large and small out of plane
exchange constants $J_{2}^{l}$ and $J_{2}^{s}$ are expected to change their
strength significantly and even their sign too. We, therefore, extend our
analyses by constructing phase diagrams in the $J_{2}^{l}$ \- $J_{2}^{s}$
plane for several values of $J_{3}$ marked by arrows in Fig. 1(d). Figure.
1(e) shows the $J_{2}^{l}$ \- $J_{2}^{s}$ phase diagram for $J_{3}=0.35$.
Similar phase diagrams for $J_{3}=0.2$ (inside non-helical AFM phase) and
$J_{3}=0.6$ (inside the helical AFM phase) with all possible $`$sign’
combinations of $J_{2}^{l}$ and $J_{2}^{s}$ are shown in Fig. 8 (Appendix A).
It is evident that the area of helical magnetic state increases with
increasing $J_{3}$ value i.e. the $\frac{J_{3}}{J_{1}}$ ratio. It is therefore
remarkable that the control in $\delta$ values plays a crucial role in
determining the magnetic state of the hexagonal systems like Mn3Sn.

Table 1:  $\delta$, the measure of effective trimerization at different pressure values in case of ideal and relaxed structures along with Mn Wyckoff position. Pressure (GPa) |  $x$ (6c: $x$,$2x$,0.25) |  $\delta$ (10-2)
---|---|---
0.0 (ideal) | 0.1666 | 0.0
5.23 (ideal) | 0.1666 | 0.0
0.0 (forced) | 0.1466 | 12.0
0.0 (relaxed) | 0.1626 | 2.4
2.75 (relaxed) | 0.1605 | 3.7
5.23 (relaxed) | 0.1493 | 10.34

The conventional $120^{\circ}$ triangular AFM ground state can undergo a phase
transition into the helical SS state by means of effective trimerization,
which may bring significant changes in $J$’s due to the changes in bond
lengths. Hence, the effect of pressure $P$ in the hexagonal Mn3Sn system has
been carried out within ab initio electronic structure calculations where
hydrostatic pressure $P$ is used as a parameter. The last column in Table 1
lists the $\delta$ values for various pressures. The corresponding Wyckoff
position of Mn atom at the middle column determines the Kagome sublattice
structures. Interestingly, the application of pressure, chemical as well as
hydrostatic, results in a nonzero $\delta$ value that keeps on increasing with
increasing pressure. Figure 1(f) summarizes the calculated energy
corresponding to the noncollinear magnetic SS states characterized by q. In
order to understand the role of pressure on $\delta$, and hence the magnetic
ground state, we have taken $120^{\circ}$ inverse triangular AFM state as the
initial state corresponding to q = 0. Subsequently, the triangular spin state
is helically modulated for $q_{z}\neq$ 0, as can be seen by comparing the spin
configurations in Fig. 1(b) and (c). The ideal Kagome sublattice stacked Mn3Sn
shows noncollinear $120^{\circ}$ inverse triangular AFM structure as the
ground state up to a hydrostatic pressure $P=5.23$ GPa. The SS dispersion does
not show any significant changes like softening or hardening near the ground
state (see Fig. 9 (Appendix A)). However, the forced structure with $\delta$ =
0.12 at ambient pressure ($P$ = 0) shows a SS ground state with a period of
about 16.3 Å. Such a helically modulated spin structure is indeed stabilized
by the effective trimerization of the system which is fostered by pressure.

To interpret our calculations, we map the results onto a classical Heisenberg
model which allows us to determine the first three/four inter sublattice
exchange constants as discussed earlier. A good fit is obtained by using
$J_{1}$, $J_{2}$, and $J_{3}$ for the ambient pressure ideal structure where
the nature of exchange constants is similar to the experimental results mn3qm
. In case of forced structure at ambient pressure, the four exchange constants
which include $J^{s}_{2}$ and $J^{l}_{2}$ give good fitting to the results
with SS ground state. Details of the calculation is provided in Appendix A. We
find that the ratio of $\frac{J_{3}}{J_{1}}$ does not change significantly
when we move from the ideal to the forced trimerization case. Interestingly,
$J^{s}_{2}$ changes sign to become antiferromagnetic while $J^{l}_{2}$ remains
ferromagnetic. For the ambient pressure ideal structure, the relative values
of $J_{2}$ and $J_{3}$ measured in the unit of $J_{1}$ are about 0.445 and
0.340, respectively. The corresponding point lies in the dark blue region in
Fig. 1(d). In case of the forced trimer system, the $J_{3}$ value is about
0.336, which is slightly smaller than the value used for constructing Fig.
1(e). This is indeed remarkable as our calculated $J^{s}_{2}$ and $J^{l}_{2}$
(in the unit of $J_{1}$) lie in the region of SS phase in Fig. 1(e) [see table
2 (Appendix A)]. Now, the relaxed structure at ambient pressure changes
$\delta$ value from 0 (ideal) to 0.024 due to the chemical pressure effect.
The SS energy dispersion calculated up to q = (0,0,0.125) shows significant
softening, possibly due to the effective trimerization. The relaxed structure
at $P$= 5.23 GPa shows significantly large $\delta$ value of about 0.10, which
further establishes the role of pressure in stabilizing the helical ground
state. Interestingly, the dispersion shows negative slope near q = 0,
indicating the ground state at finite $(0,0,q_{z})$ value. Hence, the pressure
induced effective trimerization in the Kagome sublattices modifies the
frustrated exchange constants which ultimately determine the magnetic ground
state.

Figure 2: (Color online) Field dependence of Hall resistivity for Mn3.05Sn0.95
at different hydrostatic pressures measured at (a) $T=100$ K and (b) $T=300$
K. Figure 3: (Color online) Temperature dependent zero field longitudinal
resistivity [$\rho_{xx}(T)$] measured at ambient and 1.5 GPa pressure for
Mn3.05Sn0.95. The measurement is performed both in cooling and heating modes
as marked by arrows. The inset shows an amplified view of the 1.5 GPa
$\rho_{xx}(T)$ data around the phase transition region. A small kink at 35 K
present in the zero pressure data corresponds to the instrumental artifact
arising from the inconel used in the PPMS.

Guided by our theoretical results, we set out to explore the possibility of
controlling the magnetic ground state and its effect on the AHE in Mn3Sn with
the help of hydrostatic pressure as a control parameter. The structural
characterization for the samples studied in the present manuscript can be
found in Appendix B. Figure 2 shows the field dependence of anomalous Hall
resistivity ($\rho_{xy}$) measured at 100 K and 300 K at different applied
pressures. As one can see, the $\rho_{xy}$ value at 100 K monotonically
decreases from a large value of 5 $\mu\Omega$ cm at ambient pressure to almost
zero by increasing the pressure to 1.5 GPa. It is worth mentioning that the
magnitude of AHE at ambient pressure matches well with the previously reported
value AHEmn3SnNakatsuji . A further increase in pressure does not affect the
anomalous component of $\rho_{xy}$ that remains at zero. Contrary to the
$\rho_{xy}$ behavior at 100 K, the anomalous Hall resistivity at 300 K remains
almost constant when the pressure is changed from 0 GPa to 1.45 GPa, before
showing a slight increasing trend at higher pressures. To further support the
observed pressure induced switching of AHE in the present system, we have
measured temperature dependent longitudinal resistivity [$\rho_{xx}(T)$] at
ambient and 1.5 GPa pressures (Fig. 3). The ambient pressure $\rho_{xx}(T)$
curves measured both in cooling and heating modes exhibit a metallic type of
temperature dependency without the signature of any transition, as reported
before AHEmn3SnNakatsuji . In contrast, the $\rho_{xx}(T)$ data measured at
1.5 GPa display a transition like anomaly with the appearance of a small
hysteretic behavior between the cooling and heating $\rho_{xx}(T)$ curves at
about 150 K. The first-order nature of this transition suggests the presence
of a coupled magnetic and structural transition might be responsible for the
pressure induced switching of AHE in the present system.

Figure 4: (Color online) Temperature dependence of zero field cooled (ZFC)
magnetization [$M(T)$] for Mn3.05Sn0.95 sample measured in an applied pressure
of (a) $P=0$ GPa, (b) $P=0.72$ GPa, and (c) $P=1.92$ GPa. $M(T)$ curves for
Mn3.04Sn0.96 sample measured in (d) $P=0$ GPa and (e) $P=1.03$ GPa. (f) $M(T)$
curve for Mn3.03Sn0.97 measured in ambient pressure. All the magnetization
measurements were performed in an applied field of 0.1 T.

So far all the Hall effect measurements are carried out in the sample with
chemical composition Mn3.05Sn0.95. To understand the contrasting nature of the
pressure dependent AHE at 100 K and 300 K, we have carried out a thorough
magnetization study in the presence of hydrostatic pressure for three
different samples with a small variation in the Mn/Sn ratio. The temperature
dependence of magnetization [$M(T)$] curves measured at three different
pressures and temperature up to 390 K are shown in Figs. 4(a)-(c). The zero
pressure $M(T)$ curve exhibits almost a temperature independent behavior down
to 50 K, below which the system undergoes a transition to the canted AFM state
as reported earlier Tomicaxis . Since we mainly focus on the switching of AHE
in the triangular AFM phase, the rest of our discussion will be concentrated
on the results above 50 K. The $M(T)$ curve measured at a pressure of 0.72 GPa
develops a small step-like feature at about 150 K, marked by an arrow in Fig.
4(b). This additional transition at 150 K becomes very prominent when the
pressure is further increased to 1.13 GPa [see Fig. 4(c)]. To probe the origin
of this new transition with pressure we have collected $M(T)$ data for
Mn3.04Sn0.96 which consists of a slightly lower Mn concentration, as depicted
in Fig. 4(d). Interestingly, the ambient pressure $M(T)$ data for Mn3.04Sn0.96
display a similar transition as that is observed at 1.13 GPa for Mn3.05Sn0.95.
The additional transition in Mn3.04Sn0.96 shifts to about 200 K at 1.03 GPa
[Fig. 4(e)]. Similar pressure dependent shifting of the low temperature
transition has also been reported earlier IEEE2017 . By further reducing the
Mn concentration in Mn3.03Sn0.97, the ambient pressure transition in the
$M(T)$ appears at around 220 K [Fig. 4(f)]. Here we note that lke the $M(T)$
data, the ambient pressure $\rho_{xx}(T)$ data for Mn3.03Sn0.97 also display a
transition around 220 K Suppl. As mentioned in the introduction section, Mn3Sn
exhibits multiple magnetic phases depending on the Mn concentration. The extra
transition found in case of Mn3.04Sn0.96 and Mn3.03Sn0.97 at ambient pressure
corresponds to the magnetic phase transition from the high temperature
triangular AFM state to the helical AFM phase at low temperatures. The
application of hydrostatic pressure shifts this transition to higher
temperatures. Hence the appearance of the extra transition in case of
Mn3.05Sn0.95 actually represents a pressure induced magnetic phase transition
from the conventional triangular AFM to the helical AFM phase. Since the
helical AFM structure does not support any AHE, the observation of switching
of anomalous Hall resistivity in Fig. 2(a) indeed corresponds to the magnetic
phase transition in the sample.

Figure 5: (Color online) Field dependent isothermal magnetization loops
[$M(H)$] for Mn3.05Sn0.95 sample measured at different hydrostatic pressures
and at two temperatures (a) $T=100$ K and (b) $T=300$ K. Insets: the magnified
magnetization data in the low field regime. Figure 6: (Color online) (a)
Pressure dependence of lattice parameters $a$ (left ordinate) and $c$ (right
ordinate) for the hexagonal Mn3.05Sn0.95 calculated at room temperature.
Inset: The unit cell volume as a function of hydrosatic pressure. Temperature
dependence of lattice parameters $a$ and $c$ for (b) Mn3.05Sn0.95 and (c)
Mn3.03Sn0.97 in ambient pressure. Insets of (b) and (c) present the unit cell
volume as a function of temperature. The temperature at which the change of
lattice parameters/unit cell volume occurs is marked by a downward arrow.

Our pressure dependent $M(T)$ measurements for different Mn3Sn samples provide
an excellent correlation between the magnetic structures and the observed AHE.
To further establish the connection, we have carried out isothermal
magnetization measurements [$M(H)$] at different pressures at $T=$ 100 K and
300 K, as shown in Fig. 5. The $M(H)$ loops measured at 100 K exhibit a nearly
linear behavior up to 5 T and remains almost unchanged with applied pressure
up to 1.2 GPa [Fig 5(a)]. However, a close look to the low field regime [see
the inset of Fig. 5(a)] reveals a unique pressure dependency of the
magnetization. It is worth mentioning that the triangular AFM phase in Mn3Sn
displays a small spontaneous magnetization owing to a tiny deviation from the
$120^{\circ}$ structure due to the alignment of Mn moments with their
respective local easy axes localeasytilt . In the present case, this residual
spontaneous magnetization can even be seen for pressure of 0.6 GPa. When the
system transforms to the helical AFM phase at higher pressure, the magnitude
of the spontaneous magnetization starts decreasing. Finally, a nearly linear
kind of $M(H)$ loop is obtained for $P=1.2$ GPa. The disappearance of the
residual spontaneous magnetization at higher pressures directly supports our
previous finding of pressure induced helical magnetic state, where the
uncompensated in-plane magnetic vector sums up to zero when it is rotated by
$360^{\circ}$ in a single helical pitch. In corroboration with the 100 K Hall
resistivity data, no pressure variation in the spontaneous magnetization is
found for the $M(H)$ loop measured at 300 K, as depicted in Fig. 5(b). These
findings thoroughly establish the correlation between the pressure induced
switching of the AHE and magnetic ground states.

Finally, pressure dependent XRD measurement is carried out at room temperature
to find out the role of pressure on the crystal structure. Figure 6(a) shows
the variation of lattice parameters $a$ and $c$ with hydrostatic pressure for
Mn3.05Sn0.95. Both $a$ and $c$ decrease linearly with increasing pressure
without showing any signature of structural phase transition up to a pressure
of 10 GPa (see the XRD pattern in Appendix B). The unit cell volume ($V$) also
changes linearly with pressure [see the inset of Fig. 6(a)]. Therefore, we
rule out any role of structural phase transition in the observed switching of
AHE. We have also performed temperature dependent XRD study at ambient
pressure for Mn3.05Sn0.95 to find out any structural anomaly at low
temperatures [Fig. 6(b)]. Both $a$ and $c$ as well as $V$ [inset of Fig. 6(b)]
show a monotonic decrease with decreasing temperature, ruling out any
temperature dependence of structural phase transition. It is also important to
find out the relation between the crystal and magnetic structure in the
helical AFM phase. Since our XRD measurements under pressure are limited to
room temperature, we have selected Mn3.03Sn0.97 for the temperature dependent
XRD study as this sample exhibits ambient pressure helical phase transition
around 220 K. The temperature variation of lattice parameters for Mn3.03Sn0.97
are shown in Fig. 6(c). Interestingly, the in-plane lattice parameter $a$ and
$V$ display a dip-type feature at the helical phase transition. It is to be
noted here that our theoretical study emphasizes the role of pressure induced
trimerization of Mn atoms, where the in-plane bond length between neighbouring
Mn triangles changes, in stabilizing the helical AFM phase. Hence, the
observed anomaly in the temperature dependent $a$ and $V$ [inset of Fig. 6(c)]
with almost no change in the out-of-plane lattice parameter $c$ corroborate
our theoretical proposition.

Although hydrostatic pressure stabilizes the helical phase at low temperature
for Mn3.05Sn0.95, a similar transition is also found at ambient pressure for
Mn3.03Sn0.97 with a larger unit cell volume. Here we note that irrespective of
its exact composition, Mn3Sn intrinsically exhibits some amount of
trimerization of the Mn atoms even at ambient pressure ANEmn3sn , consistent
with our theoretical calculations. The stability of the helical phase depends
on the relative strength of $J_{3}$ in the presence of competing $J_{2}^{l}$
and $J_{2}^{s}$ exchange constants. In the ideal structure of Mn3Sn, there are
six nearest neighbours for the Mn-Mn exchange constants connected by $J_{2}$
[three above and three below, see Fig 1(b)]. The trimerization process splits
the $J_{2}$ into a group of four $J_{2}^{l}$ and two $J_{2}$ [see Fig. 1(c)].
Hence, an overall decrease in the strength of $J_{2}$ is expected upon
trimerization. As we can see from the phase diagram presented in Fig 8
(Appendix A), the $J_{3}$ value is very sensitive in stabilizing the helically
modulated phase. It is also evident from Fig. 1(d) that an overall decrease in
the strength of $J_{2}$ increases the helical phase region. Therefore, the
helically modulated phase is a result of the subtle balance between $J_{3}$
and the splitting of $J_{2}$ due to the trimerization. The higher cell volume
in case of Mn3.03Sn0.97 may intrinsically give lower effective $J_{2}$ values
in comparison to that of Mn3.05Sn0.95. In addition, different concentration of
Mn atoms in these two samples also modifies the exchange interactions in the
system. Hence, the observed different pressure effects in Mn3.05Sn0.95 and
Mn3.03Sn0.97 depends not only on the volume of the unit cell, also on the
position of extra Sn and the effective exchange interaction strength in the
system. Therefore, both positive and negative pressure may cause the helical
AFM transition.

The present study exemplifies the mechanism for the switching of AHE in Mn3Sn
system by establishing a route to control the modulation by means of pressure.
As the exchange interaction between the Mn moments depends directly on the
bond length, the switching of ground state by exchange parameter variation can
be achieved by control over the lattice parameters by means of
pressure/strain. Though chemical pressure can bring about the required change
in the lattice parameters but it cannot be used for the switching purpose in
device applications. Hydrostatic pressure, on the other hand, has been widely
used to manipulate the lattice parameters and even to bring about a structural
transition in a system. The present mechanism to manipulate AHE differs
completely from the recent reports of AHE switching in antiferromagnets
natelecSwitch ; mn3gePressureSwitch ; mn3snEswitch_nakatsuji . Switching of
AHE in thin films of Mn3Pt relies on the structural distortion controlled
through electrical response in a piezo-electrical substrate, whereas, in case
of bulk Mn3Ge the underlying mechanism for the pressure induced sign change of
AHE is still unknown. In the present work, we have uncovered the process of
trimerization in Mn3Sn that was not taken into consideration till now. In such
a scenario, three nearest-neighbour Mn moments collectively respond to any
external stimuli and in turn can greatly affect the magnetic properties at
higher trimerization levels gd3trimer ; Gd3helicaltrimer . Now that we
understand the ingredients for the stabilization of different ground states,
the effect of Mn doping can also be explored to manipulate exchange parameters
that will help to achieve switching at room temperature.

##  IV CONCLUSION

In conclusion, we have successfully demonstrated the switching of anomalous
Hall effect in antiferromagnetic Mn3Sn using hydrostatic pressure. The
theoretical analyses presented in this work establish a first hand link
between different magnetic ground states of Mn3Sn. Our experimental results
convincingly manifest the role of helical modulation of the inverse triangular
spin structure to manipulate the AHE in the system. The pressure induced
effective trimerization of the Mn atoms controls the exchange frustration
which finally leads to the helical modulation. The present work could
immensely help in the greater understanding of extensive range of properties
demonstrated by Mn3Sn and related materials.

A.K.Nayak acknowledges the support from Department of Atomic Energy (DAE), the
Department of Science and Technology (DST)-Ramanujan research grant (Grant No.
SB/S2/RJN-081/2016), SERB research grant (Grant No. ECR/2017/000854) and
Nanomission research grant [Grant No. SR/NM/NS-1036/2017(G)] of the Government
of India. A.K.Nandy acknowledges the support of DAE, SERB research grant
(Grant No. SRG/2019/000867) of the Government of India. A.K.Nandy thanks Prof.
P. M. Oppeneer for the Swedish National Infrastructure for Computing (SNIC)
facility and Dr. Gustav Bihlmayer for the computational support, JURECA at
JSC, Forschungszentrum, Jülich, Germany. VS and RN acknowledge BRNS, India,
for financial support bearing sanction Grant No. 37(3)/14/26/2017.

##  V APPENDIX A: Theoretical Model

To understand the role of competing exchange interaction in stabilizing the
helical modulation, we write down the classical Heisenberg Hamiltonian for a
single Mn atom considering the inter Kagome sub lattice coupling J1, J2 and J3
(see Eq. (1)). As J1 is the strongest among three exchange parameters mn3qm ,
we varied J2 and J3 in the units of J1 over a range to obtain the ground state
for all combinations. For every particular combination, we change the value of
the angle of rotation ($\theta$) between two layers from 0 to 90 degree in
order to determine the lowest energy state. After performing such
calculations, two prototypical plots are shown in Fig 7. The value of $\theta$
corresponding to the energy minima for these plots is assigned as the magnetic
ground state. Here, Fig 7 (a) represents an unmodulated inverse triangular
ground state while Fig 7 (b) shows the spin-spiral (SS) energy for a helically
modulated ground state. Similar calculations were performed for all the
combinations and the corresponding phase diagrams are constructed.

In case of trimerization with unequal equilateral triangles in the Kagome
sublattices, the bond corresponding to J2 now splits into two unequal lengths.
The Hamiltonian for this case is written as

$H=2J_{1}\cos(\frac{2\pi}{3}+\theta)+2J_{1}\cos(\frac{2\pi}{3}-\theta)+2J_{2}^{l}\cos(0+\theta)+2J_{2}^{l}\cos(0-\theta)+J_{2}^{s}\cos(0+\theta)+J_{2}^{s}\cos(0-\theta)\\\
+J_{3}\cos(0+2\theta)+J_{3}\cos(0-2\theta)$ (2)

where, J${}_{2}^{l}$ (J${}_{2}^{s}$) is the exchange constant corresponding to
the larger (smaller) bond length. Similar to the previous calculations, here,
we have performed the energy calculations for all sign and strength
combinations of J${}_{2}^{l}$ and J${}_{2}^{s}$ where J1 and J3 are fixed at
constant values. Main text includes the phase diagram in the J2-J3 plane
measured in the unit of J1. In case of effective trimerization, Fig 8 contains
the phase diagrams for various relative values of J3 ranging from J3/J1 = 0.2
to 0.6. From Fig 8, we conclude that the helical ground state can be achieved
for various relative values of J3. Importantly, the area of the SS phase in
the J${}_{2}^{l}$ -J${}_{2}^{s}$ plane is increasing with the relative
strength of J3. Here, we mention that these values of J3 are from different
regime of the phase diagram calculated for ideal Kagome structure (marked in
Fig 1 (d)). J3 = 0.2, 0.35, 0.6 are taken outside the helical phase boundary,
close to the phase boundary and deep inside the helical phase, respectively.
We also note that the sign change of either J${}_{2}^{l}$ or J${}_{2}^{s}$
favors the helically modified ground state, supporting our exchange constant
values extracted from the DFT calculations. An increase in J3 moves the phase
boundary between zero and finite helical modulation towards higher J2 and vice
versa (Figure 8).

Figure 7: Two prototype plots as obtained for (a) exchange parameters
supporting a non-modulated structure and (b) exchange parameters supporting a
modulated structure.

###  V.1 Spin-spiral calculations

The SS calculations were performed for different level of trimerization due to
nonzero pressure in the system. Fig. 9 (a) shows the energy as a function of
spin spiral vector q $=$ $(00q_{z})$ plot close to q $=$ 0 (main text includes
the plot for full range of q). Note, the above $\theta$ value can be
translated to $q_{z}$ in the SS calculations. From Fig 9 (a), we clearly see
that in case of ideal Kagome sublattice stacking, the magnetic state does not
show any helical modulation even at high pressure till 5 GPa. After structure
relaxation, the energy curve at ambient pressure gets softer than that for
ideal structure, but q $=$ 0 still remains the magnetic ground state. However,
the softening of the dispersion displays the importance of effective
trimerization which is originating from the size mismatch of Mn and Sn atoms
i.e. the chemical pressure effect. Further increase in the trimerization at
ambient pressure ($\delta$ $=$ 0.12) leads to a helical ground state
corresponding to the $q_{z}$ $\neq$ 0 state. For the relaxed structure at 5
GPa pressure, where the effects of trimerization is higher ($\delta$ $=$
0.103) than that at ambient pressure ($\delta$ $=$ 0.024), the energy curve
shows a large negative slope. This establishes a helically modulated SS to be
the ground state with applying hydrostatic pressure as our experimental
observations suggest. The SS curves were fitted using the Hamiltonian as in
Eq. (1) for ideal case, and Eq. (2) for the trimerized case. Fitted profiles
are shown in Fig. 9 (b-c). The fitted exchange parameters are tabulated in
Table-2.

Table 2:  Out-of-pane exchange parameters as extracted by fitting the $E({\bf q})$ vs $q$ curves with the usual exchange Hamiltonian. | J1 (meV) | J2 (meV) [-J2/J1] | J3 (meV) [J3/J1]
---|---|---|---
|  | J${}_{2}^{l}$ (meV) [-J${}_{2}^{l}$/
---
J1]
J${}_{2}^{s}$ (meV) [-J${}_{2}^{s}$/J1]
0 GPa (Ideal) | 20.83 | -9.29 [0.445] | 7.09 [0.340]
0 GPa (Forced) | 13.12 | -0.32 [0.024] | 2.59 [-0.197] | 4.41 [0.336]
Figure 8: Phase diagrams for the angle of rotation between two adjacent Kagome
layers in the J${}_{2}^{s}$-J${}_{2}^{l}$ plane with J3 fixed at (a) 0.2 J1,
(b) 0.35 J1, (c) 0.5 J1 and (d) 0.6 J1.  Figure 9: (a) Spin spiral energy
$E({\bf q})$ vs ${\bf q}(00q_{z})$ as calculated for various pressure values
with and without effective trimerization. $E(q_{z})$ vs $q_{z}$ data points
along with fitted curves for (b) Ideal and (c) forced trimerize Kagome
sublattice at ambient pressure.

##  VI APPENDIX B: X-ray Diffraction (XRD) Measurements

XRD measurements were performed on polycrystalline samples using a Rigaku
X-ray diffractometer. Figure 10(a)-(c) shows the fitted profile for different
composition of Mn3+xSn1-x samples. Fittings were performed taking into care
the extra Mn concentration. Occupancy of respective sites were modified
according to extra Mn. No extra impurity peaks were observed for any sample.
Refined lattice parameters are plotted in Fig. 10(d). We find a small
monotonic change in the lattice parameters with Mn composition.

Figure 10: XRD pattern with Rietveld refinement for (a) Mn3.05Sn0.95, (b)
Mn3.04Sn0.96, and (c) Mn3.03Sn0.97 sample. (d) Calculated $a$ and $c$ values
from the fitting.

###  VI.1 Pressure dependent XRD measurements

Pressure dependence of XRD measurements were performed using ECXRD (BL-11)
beamline at synchrotron radiation source beam facility at RRCAT, Indore,
India. Figure 11 (a) shows the ambient pressure XRD pattern with fitted
profile for Mn3.05Sn0.95 sample. XRD pattern performed at such high fluence
again establishes the single phase nature of sample. Fig. 11 (b)-(c) show the
fitted profile at finite pressure values. The presence of some additional
peaks marked inside the figure originate from the high pressure gasket
material used in diamond anvil cell. We find that with the pressure cell in
place, the intensity mismatch between the experimental and simulated patterns
increases for some reflections. This may be due to some preferred orientation
of grains inside the pressure cell. This mismatch also explains the presence
of larger error bars in lattice parameters obtained in presence of pressure in
the main text.

Figure 11:  XRD patterns with Rietveld refinement measured at (a) 0 GPa, (b)
0.78 GPa, and (c) 5.23 GPa.

###  VI.2 Low temperature XRD measurements

Low temperature XRD measurements were performed using a low temperature
(Oxford Phenix) attachment to the X-Ray diffractometer (PANalytical). Fig. 12
and Fig. 13 show the XRD patterns with fitting at different temperatures. No
structural transition is visible down to 13 K in both the samples. Lattice
parameters were first refined for room temperature XRD. Then, the refined
parameters were used as a starting point for the refinement of the subsequent
lower temperature. Similar process was followed for all the temperatures.
Fitted lattice constants $a$ and $c$ are presented in the main text.

Figure 12:  Profile fitting graphs of XRD for Mn3.03Sn0.97 sample at various
temperatures. Red circles represent the experimental data. Black line shows
the calculated fitting. Difference between calculated and experimental graphs
is shown as blue line. Straight blue ticks mark various Bragg’s positions.
Figure 13:  Profile fitting graphs of XRD for Mn3.05Sn0.95 sample at various
temperatures. Red circles represent the experimental data. Black line shows
the calculated fitting. Difference between calculated and experimental graphs
is shown as blue line. Straight blue ticks mark various Bragg’s positions.

##  VII APPENDIX C: Resistivity measurements

Temperature dependence of resistivity measured at ambient pressure for
Mn3.03Sn0.97 is shown in Fig. 14.

Figure 14:  Temperature dependence of resistivity measured at ambient pressure
for Mn3.03Sn0.97. The inset shows the amplified view of the resistivity data
around 200 K. The presence of hysteresis between the cooling and heating
resistivity data signifies the presence of a first-order phase transition.

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# A theory of inductive loops in electrochemical impedance spectroscopy

Yury Grabovsky Department of Mathematics, Temple University, Philadelphia, PA
19122, USA. Jacob Guynee Department of Mathematics, Georgia Institute of
Technology, Atlanta, GA 30332, USA.

###### Abstract

We demonstrate that failure of time-invariance assumption in the modeling of
electrochemical systems by equivalent circuits can lead to the formation of
low frequency “inductive loops” that manifest themselves as positive imaginary
parts of the impedance function. Assuming that the properties of the
equivalent circuits change slowly in time we perform an asymptotic analysis
and obtain a new integral representation of the impedance function that
reduces to the standard one at high frequencies, while exhibiting inductive
loops at low frequencies.

##  1 Introduction

Electrochemical impedance spectroscopy (EIS) is an indispensable tool to
describe complex electrochemical systems in a unified and graphical way. In
this approach any system is described by a single complex-valued impedance
function $Z(\omega)$. The basic theory of EIS [19, 3, 4] says that this
function possesses special analytic properties that are most concisely
expressed by the representation

$Z(\omega)=\displaystyle\frac{1}{iC_{0}\omega}+\int_{0}^{\infty}\frac{d\sigma(\tau)}{1+i\omega\tau},\qquad\int_{0}^{\infty}\frac{d\sigma(\tau)}{1+\tau}<+\infty,\quad
0<C_{0}\leq\infty.$ (1)

Here $\sigma$ is a positive Borel-regular measure on $[0,+\infty)$. It is
often convenient to approximate such measures by linear combinations of delta-
functions, resulting in rational approximations of $Z(\omega)$ [1]. Such
approximations can be interpreted as impedances of electrical circuits made of
resistors and capacitors only. They are called the equivalent circuit models
(ECM). One easily verifiable feature of the representation (1) is the
negativity of the imaginary part of the impedance function.

The experimentally observed arcs in a Nyquist plot of the impedance with
positive imaginary part in the low frequency part of the spectrum received the
unfortunate moniker of an “inductive loop” [12, 8, 23, 21, 10]. It is
generally agreed that magnetic effects play no significant role in the
functioning of most electrochemical systems and devices, and even when they
do, the inductive effects manifest themselves at the high end of the frequency
spectrum [7, 16, 6]. The explanation of inductive loops in the current
literature [16, 6] by means of ECMs with negative resistances and capacitances
creates more problems than it solves. If one permits negative resistances and
capacitances, then _arbitrary_ “experimental data” can be matched by such an
impedance function with any degree of precision due to the Riesz theorem (see
e.g. [17, 14, 15]). This is not surprising, since it is the passivity of the
system [22, 9, 20], violated by the the negative resistances and capacitances
in ECMs, that are responsible for the possibility of stable reconstruction of
the impedance function.

A far more reasonable explanation of the experimental appearance of low
frequency inductive loops is the violation of the time-invariance assumption
at time scales commensurate with the inverses of low frequencies. This
explanation is semi-explicit in [21, 16, 18], where specific electrochemical
processes altering the properties of the system on slow time scales are
identified.

When the time-invariance assumption is violated the input-output behavior of
the system can no longer be mathematically described by a well-defined
impedance function at low frequencies. In this paper we will show that if one
applies a particular impedance measurement procedure, delivering the correct
impedance for any time-invariant system, the inductive loops could be observed
in non time-invariant systems.

The goal of this paper is to extend the classical EIS/ECM approach to
electrochemical systems with slowly “drifting” properties. In Section 3 we
demonstrate that a single Voigt element with drifting properties may exhibit
an inductive loop. In Section 4 we use asymptotic analysis to extend the
EIS/ECM approach to non time-invariant systems with slowly varying properties.
The resulting impedance representation formula, though more complicated, may
still be used to reconstruct the impedance function values within the
frequency band containing experimental measurements by performing the
classical Kramers-Kronig analysis [2, 5, 11, 20] for high frequency data,
while fitting the additional “drift” parameters using the inductive loop low
frequency data.

The paper is organized as follows. In Section 2 we discuss a measurement
technique that gives a good approximation of the true impedance function for
time-invariant systems. We then show in Section 3 that the same measurement
technique for non time-invariant Voigt elements can produce “inductive loop”
data. Representing a general non time-invariant electrochemical system by an
ECM with non time-invariant Voigt elements connected in series, gives us a
mathematical model that can be analyzed. An asymptotic analysis applied to
each non time-invariant Voigt element leads us in Section 4 to a
generalization of the EIS theory that reduces to classical formulas at high
frequencies and is capable of modeling low frequency inductive loops.

##  2 Impedance measurements

Let us assume that the input signal is the current

$I(t)=I_{0}\sin(\omega t),\quad t\in[0,T(\omega)],\quad T(\omega)=\frac{2\pi
n(\omega)}{\omega},$ (2)

where $n(\omega)\in\mathbb{N}$ should be as large as possible, so that the
time $T(\omega)$ it takes to make a measurement is still acceptable to whoever
makes the measurements. Of course, this is an issue only for low frequencies
$\omega$. Thus, we can measure the system response only for frequencies
$\omega\geq\omega_{\min}=2\pi/T_{\max}$, where $T_{\max}$ is the maximal
admissible time for making low frequency measurements. Hence, in the low
frequency regime we always choose $n(\omega)=1$. At high frequencies we use a
different measurement strategy. We choose a fixed measurement time $T_{0}$ and
set $n(\omega)=$round$(T_{0}\omega/2\pi)$. Hence, for all
$\omega\geq\omega_{\min}$ we define

$n(\omega)=\max\left\\{1,\texttt{round}\left(\frac{T_{0}\omega}{2\pi}\right)\right\\},\qquad\omega\geq\omega_{\min}.$
(3)

The Fourier transform of $I(t)$ can be computed explicitly, but is an unwieldy
expression, except at the frequency $\omega$ of the input sinusoid:

$\widehat{I}(\omega)=I_{0}\int_{0}^{T(\omega)}\sin(\omega t)e^{-i\omega
t}dt=\frac{\pi I_{0}n}{i\omega}=\frac{I_{0}T(\omega)}{2i}.$ (4)

For linear, time-invariant (LTI) systems the output voltage $U(t)$ must
satisfy $\widehat{U}(\omega)=Z(\omega)\widehat{I}(\omega)$, where the complex
factor $Z(\omega)$ is called the impedance. Theoretically, when $I(t)=0$ for
$t>T(\omega)$, the output voltage is still non-zero. However, in most cases it
decays exponentially fast and can therefore be neglected. In this case we have
the approximation

$\widehat{U}(\omega)\approx\int_{0}^{T(\omega)}U(t)e^{-i\omega t}dt.$ (5)

Since $U(t)$ is measured, formula (5) can be used to estimate
$\widehat{U}(\omega)$ and thus, we have the formula for estimating the
impedance from the experimental data:

$Z_{0}(\omega)=\frac{2i}{T(\omega)}\int_{0}^{T(\omega)}U(t)e^{-i\omega t}dt.$
(6)

Figure 1: Comparison of $Z_{0}(\omega)$ and $Z(\omega)$ for an elementary
Voigt circuit with $R=2k\Omega$, $C=200\mu F$.

If we apply formula (6) to an elementary Voigt circuit, consisting of a
resistor $R$ and a capacitor $C$ connected in parallel, we _will not_ get the
correct answer

$Z_{V}(\omega)=\frac{R}{i\theta\omega+1},\qquad\theta=RC.$ (7)

Figure 1 shows that $Z_{0}$ is not a very good approximation for $Z_{V}$,
especially in the intermediate frequency range. The remedy is to understand
this discrepancy and then devise a way to correct for it. Using the explicit
expression for $Z_{0}(\omega)$ (which we don’t display here), we discover that
for an elementary Voigt circuit we have

$Z_{0}(\omega)=Z_{V}(\omega)-\left(\frac{1}{T\omega}\mathfrak{Im}(Z_{V}(\omega))+\frac{i}{T}Z^{\prime}_{V}(\omega)\right)\left(1-e^{-\frac{T}{\theta}}\right),$
(8)

where $Z^{\prime}_{V}(\omega)$ denotes the derivative of $Z_{V}(\omega)$ with
respect to $\omega$. Since relation (8) between the measured and the true
impedance function of an elementary Voigt circuit is linear, it extends to
_all LTI systems_.

We recall that in an LTI system the output (voltage $U(t)$) depends on the
input (current $I(t)$) via

$U(t)=\rho_{0}I(t)+\int_{-\infty}^{t}I(\tau)K(t-\tau)d\tau,$ (9)

where the function $K(s)$ is called a memory kernel.

######  Theorem 2.1.

Suppose that the memory kernel decays exponentially:

$|K(s)|\leq\frac{R_{0}}{\theta}e^{-s/\theta},\qquad s>T_{0}.$ (10)

for some $T_{0}<T(\omega)$ for all $\omega$. Then

$\left|Z_{0}(\omega)-Z(\omega)+\frac{1}{T\omega}\mathfrak{Im}(Z(\omega))+\frac{i}{T}Z^{\prime}(\omega)\right|\leq
R_{0}\left(1+\frac{\theta}{T}\right)e^{-T/\theta},$ (11)

where

$Z(\omega)=\rho_{0}+\int_{0}^{\infty}K(s)e^{-i\omega s}ds$ (12)

is the true impedance of the system.

The proof can be found in Appendix A

Thus, we are lead to a simple method for estimating the true impedance by
making measurements over a longer time $T_{2}=2\pi
n_{2}(\omega)/\omega>T_{1}=2\pi n_{1}(\omega)/\omega$. Then, up to an
exponentially small error we must have

$Z(\omega)\approx Z_{\rm
exp}(\omega)=\frac{T_{2}Z_{0}(\omega;T_{2})-T_{1}Z_{0}(\omega;T_{1})}{T_{2}-T_{1}}.$
(13)

We still want to require that at high frequencies the time it takes to make a
an impedance measurement is fixed. Let us call it $T_{0}^{\prime}$, since we
still want $n_{1}(\omega)$ to be given by (3). In that case

$n_{2}^{\min}(\omega)=n_{1}(\omega)+1\leq
n_{2}(\omega)\leq\max\left\\{2,\texttt{round}\left(\frac{T^{\prime}_{0}\omega}{2\pi}\right)\right\\}=n_{2}^{\max}(\omega).$
(14)

Thus, for measuring very low frequencies we set $n_{1}(\omega)=1$ and
$n_{2}(\omega)=2$. When $\omega=4\pi/T_{0}$ we have $n_{1}(\omega)=2$, in
which case we must require that $n_{2}(\omega)\geq 3$. It follows that
$T^{\prime}_{0}\geq 3T_{0}/2$, and therefore,
$n_{2}^{\max}(\omega)>n_{1}(\omega)$. In practice, the experimenter can choose
any integer $n_{2}(\omega)$ satisfying $n_{2}^{\min}(\omega)\leq
n_{2}(\omega)\leq n_{2}^{\max}(\omega)$, once the constant $T^{\prime}_{0}\geq
3T_{0}/2$ has been set.

The correction method (13) can be interpreted as a directive to ignore the
transient response over the time interval $[0,T_{1}]$, effectively taking data
during time interval $[T_{1},T_{2}]$. Indeed, using formula (6), we have

$Z_{\rm
exp}(\omega)=\frac{2i}{T_{2}(\omega)-T_{1}(\omega)}\int_{T_{1}(\omega)}^{T_{2}(\omega)}U(t)e^{-i\omega
t}dt,\qquad T_{1,2}(\omega)=\frac{2\pi n_{1,2}(\omega)}{\omega}.$ (15)

One convenient choice is $n_{1}=n(\omega)$ and $n_{2}=2n(\omega)$, where
$n(\omega)$ is given by (3), so that the length of the time interval over
which the data is taken is still $T(\omega)=2\pi n(\omega)/\omega$.

######  Theorem 2.2.

Under assumptions of Theorem 2.1 we have the estimate

$|Z_{\rm
exp}(\omega)-Z(\omega)|\leq\frac{T_{1}}{T_{2}-T_{1}}R_{0}\left(1+\frac{\theta}{T_{1}}\right)e^{-T_{1}/\theta}+R_{0}e^{-T_{2}/\theta}.$
(16)

The proof can be found is in Appendix A.

To see how formula (15) improves the evaluation of the impedance we take the
same Voigt circuit with $R=2k\Omega$, $C=200\mu F$, shown in Figure 1, and
compare $Z_{\rm exp}(\omega)$ and $Z(\omega)=Z_{V}(\omega)$. The difference
between the Nyquist plots of the two functions can no longer be visualized as
in Figure 1, since it is less that $0.01$%, according to numerics, when we
choose $n_{1}=n(\omega)$ and $n_{2}=2n(\omega)$.

##  3 Non time-invariant elementary Voigt circuits

In this section we examine elementary Voigt circuits made of a resistor and a
capacitor connected in parallel. Our main assumption is that the parameters
$R$ and $C$ of the circuit do not stay constant, but slowly change in time,
instead. We will show that if we apply the impedance measurement recipe from
the previous section to such non time-invariant systems we may obtain
impedance curves with inductive loops.

###  3.1 General theory

Let us start by considering an elementary Voigt circuit consisting of a
resistor $R=R(t)$ and a capacitor $C=C(t)$ connected in parallel. In such a
circuit we have the Ohm’s laws for each of the elements

$I_{R}(t)=\frac{U(t)}{R(t)},\qquad I_{C}(t)=C(t)\dot{U}(t).$

Combining this with the Kirchhoff’s law $I(t)=I_{R}(t)+I_{C}(t)$ we obtain the
constitutive relation (i.e. dependence of $I(t)$ on $U(t)$) in the form

$I(t)=\frac{U(t)}{R(t)}+C(t)\dot{U}(t).$ (17)

Since we use the current as the input we need to solve (17) for $U(t)$:

$U(t)=\int_{-\infty}^{t}\frac{I(s)}{C(s)}\exp\left\\{-\int_{s}^{t}\frac{dx}{\theta(x)}\right\\}ds,\qquad\theta=RC.$
(18)

The quantity $\theta$ is called the relaxation time of the Voigt circuit.
Formula (15) gives the experimentally measured “impedance” of a simple Voigt
circuit with non time-invariant elements:

$Z_{\rm exp}(\omega)=\frac{2i}{\Delta T}\int_{T_{1}}^{T_{2}}e^{-i\omega
t}\int_{0}^{t}\frac{\sin(\omega
s)}{C(s)}\exp\left\\{-\int_{s}^{t}\frac{dx}{\theta(x)}\right\\}dsdt,\quad\Delta
T=T_{2}-T_{1}.$ (19)

For future reference we also have the following formula for the impedance of a
non time-invariant capacitor and resistor

$Z^{C}_{\rm exp}(\omega)=\frac{2i}{\Delta T}\int_{T_{1}}^{T_{2}}e^{-i\omega
t}\int_{0}^{t}\frac{\sin(\omega s)}{C(s)}dsdt=\frac{2}{\Delta
T\omega}\int_{T_{1}}^{T_{2}}(e^{-i\omega t}-1)\frac{\sin(\omega t)}{C(t)}dt,$
(20) $Z^{R}_{\rm exp}(\omega)=\frac{2i}{\Delta
T}\int_{T_{1}}^{T_{2}}e^{-i\omega t}R(t)\sin(\omega t)dt.$ (21)

###  3.2 Explicit non time-invariant models

Let us now examine a particular model of the time dependence of $R(t)$ and
$C(t)$. We assume that $R$ and $C$ undergo an “exponential drift” from $R_{-}$
(resp. $C_{-}$) at $t=-\infty$ to $R_{+}$ (resp. $C_{+}$) at $t=+\infty$:

$R(t)=\frac{aR_{-}+R_{+}e^{t/\tau}}{a+e^{t/\tau}},\quad
C(t)=\frac{bC_{-}+C_{+}e^{t/\tau}}{b+e^{t/\tau}},\quad a,b>0.$ (22)

The evolution law for $R(t)$ is governed by two parameters: the time scale
$\tau$ and the “current position” $a>0$ that tells us how far along $R(t)$ is
on the way from $R_{-}$ to $R_{+}$. It is in 1-1 correspondence with $R(0)$.
The value $a=1$ means $R(0)$ is exactly half-way between $R_{+}$ and $R_{-}$.
Exact same comments apply to $C(t)$. It is easy to calculate explicitly

$-\int_{s}^{t}\frac{dx}{\theta(x)}=\frac{\tau}{\theta_{+}}\left(\frac{s-t}{\tau}+\kappa_{C}\ln\left(\frac{\rho_{C}e^{-s/\tau}+1}{\rho_{C}e^{-t/\tau}+1}\right)+\kappa_{R}\ln\left(\frac{\rho_{R}e^{-s/\tau}+1}{\rho_{R}e^{-t/\tau}+1}\right)\right),$
(23)

where

$\kappa_{C}=\frac{(a-\rho_{C})(b-\rho_{C})}{\rho_{C}(\rho_{C}-\rho_{R})},\quad\kappa_{R}=\frac{(a-\rho_{R})(b-\rho_{R})}{\rho_{R}(\rho_{R}-\rho_{C})},\quad\rho_{R}=\frac{aR_{-}}{R_{+}},\quad\rho_{C}=\frac{bC_{-}}{C_{+}},\quad\rho_{R}\not=\rho_{C}.$

We note a relation

$\kappa_{C}+\kappa_{R}=1-\frac{\theta_{+}}{\theta_{-}},\qquad\theta_{\pm}=C_{\pm}R_{\pm}.$
(24)

If $\rho_{R}=\rho_{C}=\rho$, then

$-\int_{s}^{t}\frac{dx}{\theta(x)}=\frac{\tau}{\theta_{+}}\left(\frac{s-t}{\tau}+\alpha\ln\left(\frac{\rho
e^{-s/\tau}+1}{\rho
e^{-t/\tau}+1}\right)+\frac{\beta(e^{-s/\tau}-e^{-t/\tau})}{(\rho
e^{-s/\tau}+1)(\rho e^{-t/\tau}+1)}\right),$ (25)

where

$\alpha=1-\frac{\theta_{+}}{\theta_{-}},\qquad\beta=\frac{(a-\rho)(b-\rho)}{\rho}.$

A different drift model

$R(t)=R_{0}e^{-t/\tau}+R_{\infty}(1-e^{-t/\tau}),\qquad
C(t)=C_{0}e^{-t/\tau}+C_{\infty}(1-e^{-t/\tau})$ (26)

is a limiting case of our model (22) with $R_{+}=R_{\infty}$,
$C_{+}=C_{\infty}$, and

$a,b\to 0,\quad R_{-},C_{-}\to\infty,\quad aR_{-}\to R_{0}-R_{\infty},\quad
bC_{-}\to C_{0}-C_{\infty}.$

In this limit, however, the restriction that $R_{-}$ and $C_{-}$ are positive
are no longer required, since $R(t)$ and $C(t)$ are required to be positive
only for $t\geq 0$. In this drift model we can still use (23) with

$\rho_{R}\to\frac{R_{0}}{R_{\infty}}-1,\quad\rho_{C}\to\frac{C_{0}}{C_{\infty}}-1,\quad\kappa_{R}\to\frac{\rho_{R}}{\rho_{R}-\rho_{C}},\quad\kappa_{C}\to\frac{\rho_{C}}{\rho_{C}-\rho_{R}},$

provided $\rho_{R}\not=\rho_{C}$. If $\rho_{R}=\rho_{C}=\rho$, then we use
formula (25) with $\alpha=1$ and $\beta=\rho$.

Once, $\int_{s}^{t}\theta(x)^{-1}dx$ has been evaluated, we compute

$U(t)=\frac{I_{0}}{C_{+}}\int_{0}^{t}\sin(\omega
s)\frac{1+be^{-s/\tau}}{\rho_{C}e^{-s/\tau}+1}\exp\left\\{-\int_{s}^{t}\frac{dx}{\theta(x)}\right\\}ds.$
(27)

The experimenatally measured impedance $Z_{\exp}(\omega)$ is then computed by
means of formula (15). The numerical evaluation of $Z_{\exp}(\omega)$ presents
several challenges, since parameter $\tau$ must be very large, relaxation
times $\theta_{\pm}$ are typically very small and $\omega$ can range from
$10^{-6}$Hz to $10^{6}$Hz, making some integrands highly oscillatory. These
issues are addressed in Appendix B.

Figure 2: An inductive loop in a non time-invariant Voigt element.

Figure 2 shows an inductive loop in the Nyquist impedance plot of a non time-
invariant Voigt element, with $R(t)$ and $C(t)$ given by (22), where
$R_{+}=0.4k\Omega$, $R_{-}=2k\Omega$, $a=10$, $C_{+}=1.6mF$, $C_{-}=0.8mF$,
$b=6$, $\tau=600s$. The figure shows that at high frequencies the impedance is
very well captured by the impedance of the time-invariant Voigt element with
$R=R(0)$, and $C=C(0)$.

##  4 Slow parameter drift asymptotics

Assume that $1/\tau=\epsilon$ is a small parameter and that
$\omega=\epsilon\Omega$, where the rescaled frequency $\Omega$ is assumed to
be fixed. We need to find the asymptotics of $Z(\epsilon\Omega)$ as
$\epsilon\to 0$ and see if $\Im(Z(\epsilon\Omega))$ can indeed be positive. We
now assume that $R(t)=R_{0}(\epsilon t)$ and $C(t)=C_{0}(\epsilon t)$, where
$R_{0}(x)$ and $C_{0}(x)$ are strictly positive, bounded smooth functions with
bounded first derivatives on $\mathbb{R}$. Our goal is not only to obtain the
asymptotics of $Z(\omega)$, defined by (6), as $\epsilon\to 0$, but estimate
the error between the actual impedance and its asymptotic approximation. We
obtain for the elementary Voigt circuit, using (18)

$Z_{\epsilon}(\omega)=\frac{2i}{\Delta T}\int_{T_{1}}^{T_{2}}e^{-i\omega
t}\int_{0}^{t}\frac{\sin(\omega s)}{C_{0}(\epsilon
s)}\exp\left\\{-\int_{s}^{t}\frac{d\tau}{\theta(\epsilon\tau)}\right\\}dsdt,\quad\Delta
T=T_{2}-T_{1}.$

When the drift time scale $\tau=1/\epsilon$ is large, the formula for
$Z_{\epsilon}(\omega)$ above can be simplified:

$Z_{\epsilon}(\omega)=\frac{i\Omega}{\pi\Delta n(\omega)}\int_{2\pi
n_{1}(\omega)/\Omega}^{2\pi
n_{2}(\omega)/\Omega}e^{-i\Omega\eta}\mathfrak{Im}\left(\frac{R_{0}(\eta)e^{i\Omega\eta}}{1+i\omega\theta_{0}(\eta)}\right)d\eta+O(\epsilon),$
(28)

where the order $\epsilon$ error $O(\epsilon)$ is _uniform over the entire
frequency spectrum_.

Formula (28) shows that if the functions $\theta_{0}(\eta)$ and $R_{0}(\eta)$
are constants, then the approximation becomes exact, as it reduces to the
classical impedance of the elementary Voigt circuit. One more simplification
is possible in the asymptotics $\epsilon\to 0$. Specifically, we can replace
$\theta_{0}(\eta)$ by $\theta(0)$ in (28). Our final asymptotics is

$Z^{*}_{\epsilon}(\omega)=Z(\omega,\Omega)=\frac{i\Omega}{\pi\Delta
n(\omega)}\int_{2\pi n_{1}(\omega)/\Omega}^{2\pi
n_{2}(\omega)/\Omega}e^{-i\Omega\eta}\mathfrak{Im}\left(\frac{R_{0}(\eta)e^{i\Omega\eta}}{1+i\omega\theta_{0}(0)}\right)d\eta.$
(29)

Figure 3: The quality of the asymptotics of an inductive loop.

For numerical evaluation of (29) we choose $n_{1}=n(\omega)$,
$n_{2}=2n(\omega)$ and change variables of integration $\eta=sn/\Omega$. Then,
using the representation $\mathfrak{Im}(z)=(z-\overline{z})/2i$ we obtain the
formula we use in the Matlab code to make Fig. 3:

$Z_{\varepsilon}^{*}(\omega)=\displaystyle\frac{1}{2\pi}\int_{2\pi}^{4\pi}\frac{R_{0}\left(\frac{sn}{\Omega}\right)}{1+i\omega\theta_{0}(0)}ds-\displaystyle\frac{1}{2\pi}\int_{2\pi}^{4\pi}\frac{R_{0}\left(\frac{sn}{\Omega}\right)e^{-2ins}}{1-i\omega\theta_{0}(0)}ds.$
(30)

For the model (22) we compute

$\displaystyle\frac{1}{2\pi}\int_{2\pi}^{4\pi}R_{0}\left(\frac{sn}{\Omega}\right)ds=R_{+}+\frac{(R_{+}-R_{-})\Omega}{2\pi
n}\ln\left(1+\frac{a(e^{-\frac{4\pi n}{\Omega}}-e^{-\frac{2\pi
n}{\Omega}})}{1+ae^{-\frac{2\pi n}{\Omega}}}\right).$

In the high frequency regime

$R_{0}\left(\frac{sn}{\Omega}\right)=R_{0}\left(\frac{\epsilon
sT_{0}}{2\pi}\right)$

We can therefore, use a linear approximation $R_{0}(x)\approx b_{R}+m_{R}x$,
when $x\in[\epsilon T_{0},2\epsilon T_{0}]$, where the parameters $b_{R}$ and
$m_{R}$ are found from the least squares fit. In that case

$\displaystyle\frac{1}{2\pi}\int_{2\pi}^{4\pi}R_{0}\left(\frac{sn}{\Omega}\right)e^{-2ins}ds\approx\frac{im_{R}}{2\Omega}.$

Figure 3 shows the Nyquist plot of $Z_{\epsilon}^{*}(\omega)$ superimposed on
the computed “measured” impedance corresponding to the model (22). The
difference between the two graphs is not detectable at the scale of the entire
Nyquist plot, shown in the left panel of the figure. The right panel shows the
blown-up portion of the plot containing the inductive loop. The overall
relative error of the asymptotic approximation (29) is about 0.1%. In the
figure we used (22) with the same parameter values as in Figure 2.

Now, if we have a general electrochemical system exhibiting parameter drift,
then the measured impedance can be represented as

$Z_{\rm
exp}(\omega)=\frac{i\Omega}{\pi(n_{2}(\omega)-n_{1}(\omega))}\int_{2\pi
n_{1}(\omega)/\Omega}^{2\pi
n_{2}(\omega)/\Omega}e^{-i\Omega\eta}\mathfrak{Im}\left(Z(\omega,\eta)e^{i\Omega\eta}\right)d\eta+O(\epsilon),$
(31)

where for each $\eta\in\mathbb{R}$ the function $\omega\mapsto Z(\omega,\eta)$
is a classical (instantaneous) impedance function, provided there is no
leading capacitance in the ECM. In the presence of the leading capacitance,
the term

$Z_{\rm
exp}^{C}(\omega)=\frac{i\Omega}{\pi(n_{2}(\omega)-n_{2}(\omega))}\int_{2\pi
n_{1}(\omega)/\Omega}^{2\pi
n_{2}(\omega)/\Omega}(e^{-i\Omega\eta}-1)\frac{\sin(\Omega\eta)}{i\omega
C_{0}(\eta)}d\eta$

must be added to the right-hand side of (31).

If we approximate the instantaneous impedance function $Z(\omega,\epsilon t)$
by a finite Voigt circuit ECM [1]

$Z(\omega,\eta)=R_{0}(\eta)+\sum_{j=1}^{N}\frac{R_{j}(\eta)}{1+i\omega\theta(\eta)},$

then, up to a uniform, over the entire spectrum, order $\epsilon$ error, we
have

$Z_{\rm
exp}(\omega)=\frac{i\Omega}{\pi(n_{2}(\omega)-n_{1}(\omega))}\int_{2\pi
n_{1}(\omega)/\Omega}^{2\pi
n_{2}(\omega)/\Omega}e^{-i\Omega\eta}\mathfrak{Im}\left(Z^{0}(\omega,\eta)e^{i\Omega\eta}\right)d\eta+O(\epsilon),$
(32)

where

$Z^{0}(\omega,\eta)=R_{0}(\eta)+\sum_{j=1}^{N}\frac{R_{j}(\eta)}{1+i\omega\theta(0)}.$
(33)

If $R_{j}(\eta)$ in (33) are modeled according to (26),

$R_{j}(\eta)=R_{j}(0)+\Delta R_{j}\left(1-e^{-\mu_{j}\eta}\right),\quad
j=0,\ldots,N,$ (34)

then the integral in (32) can be computed explicitly:

$Z_{\rm exp}(\omega)=\sum_{j=0}^{N}\left\\{\frac{R_{j}(0)+\Delta
R_{j}}{1+i\omega\theta_{j}(0)}+\Delta
R_{j}F\left(\frac{\epsilon\mu_{j}}{\omega},\omega\theta_{j}(0)\right)\right\\}+O(\epsilon),$
(35)

where $\theta_{0}=0$, and

$F(W,v)=i\frac{e^{-2\pi n_{1}W}-e^{-2\pi
n_{2}W}}{\pi(n_{2}-n_{1})W}\frac{v(W+i)-1}{(1+v^{2})(W+2i)}.$

Typically, one would choose $n_{1}(\omega)=n(\omega)$ and
$n_{2}(\omega)=2n(\omega)$, where $n(\omega)$ is given by (3). At low
frequencies $n(\omega)=1$, while at high frequencies $2\pi n(\omega)/\omega$
can be replaced with a constant $T_{0}$.

##  5 Conclusions and discussion

In this paper we have proposed an explanation of the inductive loop phenomena
observed in experiments. The hypothesis is that the act of the impedance
measurement of an electrochemical system might speed up slow processes, such
as corrosion and charge diffusion, altering the properties of the system on
time scales commensurate with the inverses of the frequencies at which the
impedance is measured. The modeling and analysis of these processes can lead
to specific drift laws that can be used to validate our general theory.

A natural question is whether our theory makes it apparent how the time-
dependent nature of the system causes the occurrence of inductive loops. By
way of the answer we refer to formula (30) and set $\omega=0$, $n(\omega)=1$.
Then we obtain

$\mathfrak{Im}(Z(\epsilon\Omega))\approx\displaystyle\frac{1}{2\pi}\int_{2\pi}^{4\pi}R_{0}\left(\frac{s}{\Omega}\right)\sin(2s)ds$
(36)

While $R_{0}(u)$ is a strictly positive, smooth real function, the sign of the
integral in (36) is the outcome of balancing positive and negative
contributions of the oscillatory integrand. For example, if $R_{0}(u)$ is a
decreasing function, then positive contributions will always be larger than
the negative ones and inductive loops will be present. Conversely, if
$R_{0}(u)$ is an increasing function, no inductive loops will be produced. If
the slowly oscillating sinusoidal current input causes a corresponding in-
phase oscillation of $R_{0}$, the inductive loop effect could be several times
stronger.

One open question, not directly addressed in the paper is capturing the time
scale $\tau$ over which the system’s properties change. A crude estimate would
be the inverse frequency at which the imaginary part of the impedance hits
zero. For example, in the simulation corresponding to Figure 2 the time scale
$\tau=600s$, while $2\pi/\omega_{0}=610s$, where
$\mathfrak{Im}(Z(\omega_{0}))=0$. Our simulation of the time-dependent
elementary Voigt circuit shows that the deviations from the classical model
start at frequencies $\omega$ as large as two orders of magnitude over $\tau$.
They become vividly pronounced at frequencies on the order of $1/\tau$ and
persist over frequencies an order of magnitude lower.

Finally, whether or not the inductive loop data can give additional
information about the system beyond the time scale $\tau$ depends on whether
the drift model (34) is acceptable. In this case one should be able to use the
algorithm in [13] applied to the high frequency data to compute parameters
$R_{j}(0)$, $\theta_{j}(0)$ and $N$. The intermediate and low frequency data
can then be used in a non-linear least squares fit to to estimate drift
parameters $\mu_{j}$ and $\Delta R_{j}$. Future research into the inductive
loops should address these questions.

Acknowledgments. This material is based upon work supported by the National
Science Foundation under Grant No. DMS-2005538.

##  Appendix A Mathematical proofs

###  A.1 Proof of Theorem 2.1 and 2.2

For $I(t)=\chi_{[0,T(\omega)]}(t)\sin(\omega t)$ we have, making a change of
variables $s=t-\tau$,

$U(t)=\rho_{0}\sin(\omega t)+\int_{0}^{t}K(s)\sin(\omega(t-s))ds,\qquad 0\leq
t\leq T(\omega).$

We can then write

$U(t)=\rho_{0}\sin(\omega
t)+\int_{0}^{\infty}\chi_{(s,+\infty)}(t)K(s)\sin(\omega(t-s))ds,$

so that we can substitute this into (6) and switch the order of integration.
We obtain

$Z_{0}(\omega)=\rho_{0}+\frac{2i}{T(\omega)}\int_{0}^{T(\omega)}K(s)\left(\int_{s}^{T(\omega)}\sin(\omega(t-s))e^{-i\omega
t}dt\right)ds.$

Computing the inner integral we obtain

$Z_{0}(\omega)=\rho_{0}+\int_{0}^{T(\omega)}K(s)\left(e^{-i\omega
s}-\frac{\omega se^{-i\omega s}-\sin(\omega s)}{2\pi n}\right)ds.$

Using formula (12) for the true impedance, we compute

$Z(\omega)-\frac{1}{2\pi
n(\omega)}\mathfrak{Im}(Z(\omega))-\frac{i\omega}{2\pi
n(\omega)}Z^{\prime}(\omega)=\rho_{0}+\int_{0}^{\infty}K(s)\left(e^{-i\omega
s}+\frac{\sin(\omega s)-\omega se^{-i\omega s}}{2\pi n}\right)ds.$

Therefore

$Z(\omega)-\frac{\mathfrak{Im}(Z(\omega))}{\omega
T(\omega)}-\frac{i}{T(\omega)}Z^{\prime}(\omega)-Z_{0}(\omega)=\int_{T(\omega)}^{\infty}K(s)\left(e^{-i\omega
s}+\frac{\sin(\omega s)-\omega se^{-i\omega s}}{2\pi n}\right)ds.$ (37)

For $s\geq T(\omega)=2\pi n/\omega$ we have $\omega s\geq 2\pi n$. Denoting
$x=\omega s$ and $A=1/(2\pi n)$, we estimate

$|e^{-ix}+A(\sin x-xe^{-ix})|\leq|1-Ax|+A=Ax-1+A\leq Ax,$

since $Ax\geq 1$ and $A\leq 1/(2\pi)<1$. Hence,

$\left|Z_{0}(\omega)-Z(\omega)+\frac{\mathfrak{Im}(Z(\omega))+i\omega
Z^{\prime}(\omega)}{2\pi
n(\omega)}\right|\leq\displaystyle\frac{1}{T(\omega)}\int_{T(\omega)}^{\infty}s|K(s)|ds.$

Using the exponential decay (10) of the memory kernel we obtain the estimate

$\left|Z_{0}(\omega)-Z(\omega)+\frac{\mathfrak{Im}(Z(\omega))+i\omega
Z^{\prime}(\omega)}{2\pi n(\omega)}\right|\leq
R_{0}\left(1+\frac{\theta}{T(\omega)}\right)e^{-\frac{T(\omega)}{\theta}},$
(38)

proving (11).

Let us now prove Theorem 2.2. We compute,

$Z_{\rm
exp}(\omega)-Z(\omega)=\frac{T_{1}}{T_{2}-T_{1}}\int_{T_{1}}^{T_{2}}K(s)\left(e^{-i\omega
s}+\frac{\sin(\omega s)-\omega se^{-i\omega s}}{2\pi
n_{1}}\right)ds-\int_{T_{2}}^{\infty}K(s)e^{-i\omega s}ds.$

Uisng estimate (38) we obtain

$|Z_{\rm
exp}(\omega)-Z(\omega)|\leq\frac{T_{1}(\omega)}{T_{2}(\omega)-T_{1}(\omega)}R_{0}\left(1+\frac{\theta}{T_{1}(\omega)}\right)e^{-\frac{T_{1}(\omega)}{\theta}}+R_{0}e^{-\frac{T_{2}(\omega)}{\theta}}.$

###  A.2 Proof of the asymptotic foormula (31)

Let us analyze the asymptotic behavior of $Z_{\epsilon}(\omega)$. We first
change variables in the innermost integral $\sigma=\epsilon\tau$:

$Z_{\epsilon}(\omega)=\frac{2i}{\Delta T}\int_{T_{1}}^{T_{2}}e^{-i\omega
t}\int_{0}^{t}\frac{\sin(\omega s)}{C_{0}(\epsilon
s)}\exp\left\\{-\displaystyle\frac{1}{\epsilon}\int_{\epsilon s}^{\epsilon
t}\frac{d\sigma}{\theta(\sigma)}\right\\}dsdt.$

Next we change variables $\xi=\epsilon s$ in the integral with respect to the
$s$ variable:

$Z_{\epsilon}(\omega)=\frac{2i}{\Delta
T\epsilon}\int_{T_{1}}^{T_{2}}e^{-i\omega
t}\int_{0}^{t\epsilon}\frac{\sin(\omega\xi/\epsilon)}{C_{0}(\xi)}\exp\left\\{-\displaystyle\frac{1}{\epsilon}\int_{\xi}^{\epsilon
t}\frac{d\sigma}{\theta(\sigma)}\right\\}d\xi dt.$

Finally, we change variables in the outermost integral $\eta=\epsilon t$:

$Z_{\epsilon}(\omega)=\frac{2i}{\Delta
T\epsilon^{2}}\int_{T_{1}\epsilon}^{T_{2}\epsilon}e^{-i\Omega\eta}\int_{0}^{\eta}\frac{\sin(\Omega\xi)}{C_{0}(\xi)}\exp\left\\{-\displaystyle\frac{1}{\epsilon}\int_{\xi}^{\eta}\frac{d\sigma}{\theta(\sigma)}\right\\}d\xi
d\eta,$

where $\Omega=\omega/\epsilon$ could be large, when $\omega$ is not very
small, but could also be of order 1, when $\omega$ is of order $\epsilon$.

The main approximation idea is to realize that

$E_{\epsilon}(\xi,\eta)=\exp\left\\{-\displaystyle\frac{1}{\epsilon}\int_{\xi}^{\eta}\frac{d\sigma}{\theta(\sigma)}\right\\}$

is exponentially small when $\eta-\xi$ is not very small. However, when
$\eta\approx\xi$, then $\theta(\sigma)\approx\theta(\eta)$, which means that

$E_{\epsilon}(\xi,\eta)\approx
E^{0}_{\epsilon}(\xi,\eta)=\exp\left\\{-\frac{\eta-\xi}{\theta(\eta)\epsilon}\right\\}.$

To make this approximation quantitative we use the inequality

$\frac{e^{x_{2}}-e^{x_{1}}}{x_{2}-x_{1}}<e^{\max\\{x_{1},x_{2}\\}},$

which is a consequence of convexity of the exponential function. Hence,

$|E_{\epsilon}(\xi,\eta)-E^{0}_{\epsilon}(\xi,\eta)|\leq\frac{L_{\theta^{-1}}(\eta-\xi)^{2}}{2\epsilon}\exp\left\\{-\frac{\eta-\xi}{M_{\theta}\epsilon}\right\\}=\epsilon
LB\left(\frac{\eta-\xi}{\epsilon}\right),$

where

$M_{\theta}=\max_{t\in\mathbb{R}}\theta(t),\qquad
L_{\theta^{-1}}=\max_{t\in\mathbb{R}}\left|\left(\displaystyle\frac{1}{\theta(t)}\right)^{\prime}\right|,\qquad
B(x)=\frac{x^{2}}{2}\exp\left\\{-\frac{x}{M_{\theta}}\right\\}.$

Thus, we can replace $Z_{\epsilon}(\omega)$ with its approximation

$Z_{\epsilon}^{(1)}(\omega)=\frac{2i}{\Delta
T\epsilon^{2}}\int_{T_{1}\epsilon}^{T_{2}\epsilon}e^{-i\Omega\eta}\int_{0}^{\eta}\frac{\sin(\Omega\xi)}{C_{0}(\xi)}E_{\epsilon}^{0}(\xi,\eta)d\xi
d\eta.$

Moreover, we also have

$|Z_{\epsilon}(\omega)-Z_{\epsilon}^{(1)}(\omega)|\leq\frac{2L_{\theta^{-1}}}{m_{C}\Delta
T\epsilon}\int_{T_{1}\epsilon}^{T_{2}\epsilon}\int_{0}^{\eta}B\left(\frac{\eta-\xi}{\epsilon}\right)d\xi
d\eta,$

where

$m_{C}=\min_{t\in\mathbb{R}}C(t).$

Changing variables in the inner integral $x=(\eta-\xi)/\epsilon$ we obtain

$|Z_{\epsilon}(\omega)-Z_{\epsilon}^{(1)}(\omega)|\leq\frac{2L_{\theta^{-1}}\epsilon}{m_{C}}\int_{0}^{\infty}B(x)dx=\frac{\epsilon
M_{\theta}^{3}L_{\theta^{-1}}}{m_{C}}.$

The function $E^{0}_{\epsilon}(\xi,\eta)$ is exponentially small when $\xi$ is
not very close to $\eta$. However, when $\xi$ is very close to $\eta$ we can
replace $C_{0}(\xi)$ with $C_{0}(\eta)$. Hence we have

$Z_{\epsilon}^{(1)}(\omega)\approx Z_{\epsilon}^{(2)}(\omega)=\frac{2i}{\Delta
T\epsilon^{2}}\int_{T_{1}\epsilon}^{T_{2}\epsilon}\frac{e^{-i\Omega\eta}}{C_{0}(\eta)}\int_{0}^{\eta}\sin(\Omega\xi)E_{\epsilon}^{0}(\xi,\eta)d\xi
d\eta.$

Moreover,

$|Z_{\epsilon}^{(1)}(\omega)-Z_{\epsilon}^{(2)}(\omega)|\leq\frac{2L_{C^{-1}}}{\Delta
T\epsilon^{2}}\int_{T_{1}\epsilon}^{T_{2}\epsilon}\int_{0}^{\eta}(\eta-\xi)\exp\left\\{-\frac{\eta-\xi}{\epsilon
M_{\theta}}\right\\}d\xi d\eta,$

where

$L_{C^{-1}}=\max_{t\in\mathbb{R}}\left|\left(\displaystyle\frac{1}{C(t)}\right)^{\prime}\right|.$

Changing variables $x=(\eta-\xi)/\epsilon$ in the inner integral we obtain the
bound

$|Z_{\epsilon}^{(1)}(\omega)-Z_{\epsilon}^{(2)}(\omega)|\leq\frac{2L_{C^{-1}}}{\Delta
T}\int_{T_{1}\epsilon}^{T_{2}\epsilon}\int_{0}^{\infty}x\exp\left\\{-\frac{x}{M_{\theta}}\right\\}dxd\eta=2L_{C^{-1}}M_{\theta}^{2}\epsilon.$

Now, the inner integral in $Z_{\epsilon}^{(2)}(\omega)$ can be evaluated
explicitly:

$\int_{0}^{\eta}\sin(\Omega\xi)\exp\left\\{-\frac{\eta-\xi}{\theta(\eta)\epsilon}\right\\}d\xi=\frac{\epsilon\theta\sin(\Omega\eta)-\epsilon^{2}\theta^{2}\Omega\cos(\Omega\eta)+\epsilon^{2}\theta^{2}\Omega
e^{-\eta/(\epsilon\theta)}}{1+\epsilon^{2}\theta^{2}\Omega^{2}}.$

We observe that the first two terms can be combined nicely, and

$Z_{\epsilon}^{(2)}(\omega)=\frac{2i}{\Delta
T\epsilon}\int_{T_{1}\epsilon}^{T_{2}\epsilon}e^{-i\Omega\eta}\mathfrak{Im}\left(\frac{R_{0}(\eta)e^{i\Omega\eta}}{1+i\omega\theta_{0}(\eta)}\right)d\eta+\Delta_{\epsilon}(\omega)=Z_{\epsilon}^{(3)}(\omega)+\Delta_{\epsilon}(\omega),$

where

$\Delta_{\epsilon}(\omega)=\frac{2i\Omega}{\Delta
T}\int_{T_{1}\epsilon}^{T_{2}\epsilon}e^{-i\Omega\eta}\frac{\theta^{2}e^{-\eta/(\epsilon\theta)}}{C_{0}(\eta)(1+\epsilon^{2}\theta^{2}\Omega^{2})}d\eta.$

We estimate

$|\Delta_{\epsilon}(\omega)|\leq
2\Omega\epsilon\max_{\eta\in[T_{1}\epsilon,T_{2}\epsilon]}\frac{\theta(\eta)^{2}e^{-\eta/\epsilon\theta(\eta)}}{C_{0}(\eta)(1+\epsilon^{2}\theta(\eta)^{2}\Omega^{2})}\leq
M_{R}e^{-T_{1}/M_{\theta}}.$

$Z_{\epsilon}^{(3)}(\omega)$ can be written as a two-scale impedance function
(28).

In order to prove a simplified formula (29) we estimate

$|Z^{(3)}_{\epsilon}(\omega)-Z^{*}_{\epsilon}(\omega)|\leq\frac{\Omega}{\pi\Delta
n(\omega)}\frac{M_{R}L_{\theta}}{1+m_{\theta}^{2}\omega^{2}}\int_{2\pi
n_{1}(\omega)/\Omega}^{2\pi n_{2}(\omega)/\Omega}\omega\eta
d\eta=\frac{2\pi\epsilon(n_{1}+n_{2})M_{R}L_{\theta}}{1+m_{\theta}^{2}\omega^{2}}.$

If we choose $n_{1}=n$ and $n_{2}=2n$, then in the regime $n(\omega)=1$ we get
the bound

$|Z^{(3)}_{\epsilon}(\omega)-Z^{*}_{\epsilon}(\omega)|\leq 6\pi\epsilon
M_{R}L_{\theta}.$

If $n(\omega)>1$, then $n(\omega)=T_{0}\omega/2\pi$, and we obtain

$|Z^{(3)}_{\epsilon}(\omega)-Z^{*}_{\epsilon}(\omega)|\leq\frac{3\epsilon
T_{0}\omega M_{R}L_{\theta}}{1+m_{\theta}^{2}\omega^{2}}\leq\frac{3\epsilon
T_{0}M_{R}L_{\theta}}{2m_{\theta}}.$

Hence, we obtain a uniform in the entire spectrum bound

$|Z_{\epsilon}(\omega)-Z^{*}_{\epsilon}(\omega)|\leq M\epsilon$

for some constant $M$ that depends on the parameter functions $R_{0}(s)$ and
$C_{0}(s)$, and that scales like $R\theta$.

##  Appendix B Numerical evaluation of the impedance in explicit drift models

The key to the effective numerical computation of $Z_{\exp}(\omega)$ for
explicit drift models (22) and (26) is a non-dimensionalization of all
quantities. We therefore begin by rescaling the variable of integration in the
definition (19) of $Z(\omega)$, $t=n\hat{t}/\omega$.

$Z(\omega)=\frac{i}{I_{0}\pi}\int_{2\pi}^{4\pi}U\left(\frac{n\hat{t}}{\omega}\right)e^{-in\hat{t}}d\hat{t},$

where we have chosen $n_{1}=n(\omega)$, $n_{2}=2n(\omega)$, and where
$n(\omega)$ is given by (3). We then observe that $U(t)$ has the form

$U(t)=\frac{I_{0}}{C_{+}}\int_{0}^{t}\sin(\omega
s)f\left(\frac{s}{\tau}\right)e^{\frac{\tau}{\theta_{+}}\left[g\left(\frac{s}{\tau}\right)-g\left(\frac{t}{\tau}\right)\right]}ds,$
(39)

where

$f(u)=\frac{be^{-u}+1}{\rho_{C}e^{-u}+1},\qquad
g(u)=u+\kappa_{R}\ln(\rho_{R}e^{-u}+1)+\kappa_{C}\ln(\rho_{C}e^{-u}+1),$

provided $\rho_{C}\not=\rho_{R}$. If $\rho_{C}=\rho_{R}=\rho$, then we use

$g(u)=u+\alpha\ln(\rho e^{-u}+1)-\frac{\beta}{\rho(\rho e^{-u}+1)}.$

We therefore, rescale the variable of integration $s=n\hat{s}/\omega$ in (39):

$U\left(\frac{n\hat{t}}{\omega}\right)=\frac{I_{0}n}{C_{+}\omega}\int_{0}^{\hat{t}}\sin(n\hat{s})f\left(\frac{n\hat{s}}{\omega\tau}\right)e^{\frac{\tau}{\theta_{+}}\left[g\left(\frac{n\hat{s}}{\omega\tau}\right)-g\left(\frac{n\hat{t}}{\omega\tau}\right)\right]}d\hat{s}$

Hence, we obtain the formula we use in our Matlab code

$Z(\omega)=\frac{in}{\pi
C_{+}\omega}\int_{2\pi}^{4\pi}e^{-in\hat{t}}\left(\int_{0}^{\hat{t}}\sin(n\hat{s})f\left(\frac{n\hat{s}}{\omega\tau}\right)e^{\frac{\tau}{\theta_{+}}\left[g\left(\frac{n\hat{s}}{\omega\tau}\right)-g\left(\frac{n\hat{t}}{\omega\tau}\right)\right]}d\hat{s}\right)d\hat{t}.$
(40)

In the frequency band where $n(\omega)/\omega\approx T_{0}$, we approximate,
assuming that $\tau$ is large,

$f\left(\frac{n\hat{s}}{\omega\tau}\right)\approx
b_{f}+m_{f}\frac{n\hat{s}}{\omega\tau},\qquad
g\left(\frac{n\hat{s}}{\omega\tau}\right)\approx
b_{g}+m_{g}\frac{n\hat{s}}{\omega\tau},$

where the slopes $m_{f}$ and $m_{g}$, and intercepts $b_{f}$ and $b_{g}$ are
found from linear least squares fits of $f(x)$ and $g(x)$,
$x\in[T_{0}/\tau,2T_{0}/\tau]$. Then, all integrals in (40) can be computed
explicitly. Up to exponentially small terms

$Z(\omega)\approx\frac{b_{f}R_{+}}{m_{g}+i\omega\theta_{+}}+\displaystyle\frac{1}{2\omega\tau}\frac{m_{f}R_{+}}{m_{g}+i\omega\theta_{+}}\left(3T_{0}\omega+3i-\frac{4im_{g}^{2}}{m_{g}^{2}+\omega^{2}\theta_{+}^{2}}\right),\quad
n(\omega)>1,$

where in the final expression we replaced $n(\omega)$ with
$T_{0}\omega/(2\pi)$, which is valid exactly in the high frequency regime.

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