Universal conductance fluctuations in a MnBi2Te4 thin film

Molly P. Andersen1,2, Evgeny Mikheev3,2,4, Ilan T. Rosen5,2,6, Lixuan Tai7,
Peng Zhang7, Kang L. Wang7, Marc A. Kastner3,2,8, and David Goldhaber-
Gordon3,2,a

1Department of Materials Science and Engineering, Stanford University,
Stanford, California 94305, USA
2Stanford Institute for Materials and Energy Sciences, SLAC National
Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, California 94025, USA
3Department of Physics, Stanford University, Stanford, California 94305, USA
4Department of Physics, University of Cincinnati, Cincinnati, Ohio 45221, USA
5Department of Applied Physics, Stanford University, Stanford, California
94305, USA
6Research Laboratory of Electronics, Massachusetts Institute of Technology,
Cambridge, MA, 02139
7Department of Electrical and Computer Engineering, Department of Physics and
Astronomy, University of California, Los Angeles, California 90095, USA
8Department of Physics, Massachusetts Institute of Technology, Cambridge,
Massachusetts 02139, USA
aTo whom correspondence should be addressed; E-mail: goldhaber-
<EMAIL_ADDRESS>
Quantum coherence of electrons can produce striking behaviors in mesoscopic
conductors, including weak localization and the Aharonov-Bohm effect. Although
magnetic order can also strongly affect transport, the combination of
coherence and magnetic order has been largely unexplored. Here, we examine
quantum coherence-driven universal conductance fluctuations in the
antiferromagnetic, canted antiferromagnetic, and ferromagnetic phases of a
thin film of the topological material MnBi2Te4. In each magnetic phase we
extract a charge carrier phase coherence length of about 100 nm. The
conductance magnetofingerprint is repeatable when sweeping applied magnetic
field within one magnetic phase, but changes when the applied magnetic field
crosses the antiferromagnetic/canted antiferromagnetic magnetic phase
boundary. Surprisingly, in the antiferromagnetic and canted antiferromagnetic
phase, but not in the ferromagnetic phase, the magnetofingerprint depends on
the direction of the field sweep. To explain these observations, we suggest
that conductance fluctuation measurements are sensitive to the motion and
nucleation of magnetic domain walls in MnBi2Te4.

The van der Waals material MnBi2Te4 (MBT) has drawn recent excitement based on
its topological properties and multiple magnetic phases. Below the Néel
temperature $\sim 20$ K, within each van der Waals layer magnetic moments on
the Mn sites order ferromagnetically with an out-of-plane easy axis. Depending
on external magnetic field and film thickness a variety of interlayer
orderings have been observed, some associated with quantized Hall resistance
or axion insulating states [1, 2, 3, 4, 5, 6, 7, 8, 9]. The well-defined
stoichiometry implied by the chemical formula suggests that this should be a
cleaner system than the more widely studied quaternary alloy
Cr_x(Bi,Sb)_2-xTe3 [10, 11, 12]. However, in practice disorder is ubiquitous
in both thin-film and bulk single-crystal samples of MBT. Sources of defects
include antisite defects [13, 14, 15], impurity phases [16, 17, 18, 19], and
magnetic domains [20, 21]. These have hindered experimental validation of
striking predictions for behavior near $B=0$ such as quantization of Hall
resistance and persistence of topological phenomena at elevated temperatures
[22].

Regardless of a system’s topology, electrical transport measurements are
sensitive to charge carrier scattering through the material’s disorder
landscape. When device features become similar in size or smaller than the
phase coherence length, wavelike interference between different trajectories
can have dramatic effects on electronic transport measurements [23, 24].
Magnetism is known to suppress charge carrier coherence through scattering off
of magnetic disorder, domain walls, or spin waves [25, 26, 27]. Few works have
explored residual coherence-driven phenomena in magnetically ordered materials
[28, 29, 30, 31, 32, 33, 34]. Fewer still have explored charge carrier
coherence within and between different magnetic phases of the same material.

In this work, we study low-temperature electronic transport through Hall bars
patterned from a thin film of MBT grown by molecular beam epitaxy [19]. We
observe Universal Conductance Fluctuations (UCFs) [35], a familiar
manifestation of coherent transport through a disordered material, in all
three magnetic phases of MBT. We analyze the UCFs as a function of device
geometry, magnetic field, and temperature to gain insight into charge carrier
scattering mechanisms, phase coherence lengths, and timescales for
rearrangement of the elastic scatterer disorder landscape. Surprisingly, our
UCF data appear sensitive to the presence and motion of magnetic domains in
MBT’s anti-ferromagnetic (AFM) and canted anti-ferromagnetic (cAFM) phases, as
well as the nucleation of new domain patterns at the AFM/cAFM magnetic phase
transition. We note that we do not observe quantized resistivity in our
devices, implying that neither Chern nor axion insulating states are
established, presumably due to doping or disorder [17, 36, 19, 37].

Figure 1: (a) The x-ray diffraction pattern of the 6 SL MBT thin film on a
GaAs substrate. Peaks associated with MBT (GaAs substrate [19]) are indicated
with black stars (grey circles). (b) Topography of the MBT thin film, acquired
with atomic force microscopy. Scale bar, 200 nm. (c) Line cut of topography
data along the cyan line in (b). Data were averaged across 15 parallel traces
spanning 60 nm in the transverse direction, indicated by the extent of the
crosspieces appended to the tails of the cyan line.

The MnBi2Te4 film used in this work [19] was approximately 6 van der Waals
layers (called septuple layers (SLs)) thick and grown on a semi-insulating
GaAs (111)B substrate via molecular beam epitaxy [38]. X-ray diffraction was
used to characterize the film’s crystal structure (Figure 1(a)). Three (00l)
(l = 9, 21, 24) reflections from the MBT film are apparent, as are two high-
intensity (111) and (222) peaks from the GaAs substrate, confirming epitaxial
growth with (001)R of the MBT film parallel to (111)C of the GaAs substrate.
From the (00l) XRD reflections, the out-of-plane lattice constant of MBT is
$c=40.876\pm 0.003$ Å, in agreement with other reports of MBT lattice
parameters [39, 13, 17, 40]. Since one unit cell of MBT is composed of 3 SLs,
our data indicate 1 SL = 13.63 Å. Topography of the film, as measured by
atomic force microscopy several months after film growth, is shown in Figure
1(b). A linecut across a step edge (Figure 1(c)) yields a height 1.24 nm,
reasonably consistent with a single SL step. Steps with smaller heights are
also visible in Figure 1(b); these may reflect phases besides the MnBi2Te4
stoichiometry produced during growth or through surface reconstruction [16,
17, 18, 19], or may result from post-growth sample aging.

Two standard Hall bar devices were fabricated on a single chip of this film.
Throughout device fabrication, care was taken to ensure minimal thermal or
chemical degradation of the sample [38]. Each Hall bar featured two pairs of
voltage taps separated by two squares of material; in Device A (B), one square
was $10\times 10$ ($20\times 20$) $\mu$m2. These were larger than measured
electronic phase coherence lengths in the material, as discussed below.
Optical images of both devices are shown in the supplemental materials.

Figure 2: (a) Longitudinal resistance $R_{xx}$ as a function of out-of-plane
applied magnetic field, swept from positive to negative values. Field values
at which magnetoresistance qualitatively changes, corresponding to magnetic
phase transitions identified in previous works, are marked by dashed vertical
lines. The corresponding orientation of Mn magnetic moments within successive
layers of MBT (grey boxes) is indicated by black arrows. Inset: $R_{xx}$ near
zero applied field for each field sweep direction. (b) Conductance
fluctuations $\Delta G_{xx}$ as a function of magnetic field for $B=10-14$ T,
extracted by subtracting a polynomial background from the same measurements
shown in (a) and from a corresponding upward field sweep. The red box on the
field axis in (a) indicates the subset of data plotted in (b). (c) Correlation
of $\Delta G_{xx}(B)$ and $\Delta G_{xx}(B+\Delta B)$ as a function of $\Delta
B$, calculated for the data shown in (b) acquired between 10–14 T. For all
plots, red (black) indicates field sweeping up (down).

Electronic transport measurements were taken on Device A and Device B
simultaneously in a dilution refrigerator with a base temperature of 35 mK
using 5–8 Hz few-nA current excitations and standard lock-in techniques. Data
from the top longitudinal voltage pair of Device A are presented in the main
text. Data from Device B, presented in the supplemental materials, show no
substantial differences from those from Device A beyond an expected geometric
scaling. Although devices feature electrostatic top gates, for all
measurements presented in the main text the electrostatic top gates were
explicitly grounded to avoid time-dependent conductance fluctuations
associated with changes in charge trap population as and after the gate
voltage is swept [38]. Where measurements were acquired as a function of
applied external magnetic fields, the field was incremented in either 2 or 5
mT steps. We term positive field values to be parallel to the upward-facing
sample normal. Full details of device fabrication, measurement techniques, and
measurement parameters are presented in the supplemental materials.

Figure 2(a) plots longitudinal resistance $R_{xx}=V_{xx}/I$ [38] as a function
of applied magnetic field at base temperature of about 35 mK. Below the Néel
temperature of about 20 K [41], MBT orders into an out-of-plane
antiferromagnetic (AFM) phase for weak externally applied magnetic field. The
spin arrangement is schematically illustrated within the figure for this and
other phases that arise as the external magnetic field is increased: a canted
antiferromagnet (cAFM), followed by ferromagnetic (FM) interlayer coupling at
high fields [7]. Though our electronic transport measurements give no direct
insight into the exact magnetic ordering at any applied field, vertical dashed
lines mark qualitative changes in magnetoresistance that agree with magnetic
phase boundaries observed in other studies [5, 7, 19, 18, 40]. Hall
measurements acquired simultaneously reveal a charge carrier density of about
$4\times 10^{12}$ cm-2 electrons, and permit analysis of the charge carrier
mobility as a function of applied field (Figure S3) [38].

Background magnetoresistance is roughly independent of field sweep direction
at fields $>\absolutevalue{1}$ T. Across 0 T, however, a hysteretic dip in
magnetoresistance occurs (Figure 2(a) left inset). This low-field butterfly
hysteresis could be caused by MnTe_x or Mn-doped Bi2Te3 magnetic impurities
that often condense with MBT during growth or form upon air exposure and aging
[16, 17, 18, 19], although the XRD data from our film present no evidence of
these phases. Other possible explanations for the butterfly hysteresis include
flipping of uncompensated magnetic moments in odd-layer regions of the MBT
film or surface spin flop transitions which have been observed to occur at low
fields, prior to the bulk spin flop transition [21]. Regardless of its origin,
this butterfly hysteresis prevents a standard analysis of (anti-)localization
magnetoresistance peaks near $B=0$ T.

We focus instead on fluctuations in $R_{xx}$ apparent in Figure 2(a) across
the entire magnetic field range studied. As discussed below, these
fluctuations were highly reproducible upon sweeping field up and down over
several Tesla (Figure 2(b)) but apparently random (uncorrelated beyond a small
field scale, Figure 2(c)). The phenomenology of these fluctuations is
consistent with universal conductance fluctuations, motivating the analysis
below, and is inconsistent with other known mechanisms of quantum oscillations
(Section S4) [38].

As a charge carrier propagates, wavelike constructive and destructive
interference between different paths affects transmission probabilities,
modulating the local conductance of the sample. In macroscopic structures, the
charge carrier phase coherence length $\ell_{\phi}$ is far smaller than sample
dimensions, so quantum coherence-driven conductance changes average out.
However, when $\ell_{\phi}$ exceeds a sample’s dimensions measured conductance
depends on details of the disorder landscape, and exhibits fluctuations as a
function of any control parameter that adjusts the relative phases of
different paths. These conductance fluctuations have a characteristic scale of
$e^{2}/h$ independent of sample details (hence the term “universal”), where
$e$ is electron charge and $h$ is Plank’s constant [35, 42]. Magnetic field
modulates the relative phase of different paths, so a hallmark feature of UCF
is conductance variations that look like random noise but are in fact a
consistent function of applied magnetic field so long as the disorder
landscape remains stable.

Figure 2(b) shows $\Delta G_{xx}$ – fluctuations in conductance
$G_{xx}=1/R_{xx}$ after subtracting a best-fit fourth-order polynomial to
remove smooth background magnetoresistance. These conductance fluctuations
within the high-field FM phase have root-mean-squared magnitude $\Delta
G_{\mathrm{rms}}=0.0045~{}e^{2}/h$. Measurements of $\Delta
G_{\mathrm{rms}}\ll e^{2}/h$ are well-established in UCF literature [43, 44]:
when $\ell_{\phi}$ is smaller than the device dimensions, transport is
coherent within local portions of the device but incoherent averaging between
regions separated by more than $\ell_{\phi}$ reduces the measured $\Delta
G_{\mathrm{rms}}$. For our device dimensions of width $w=10~{}\mu$m and length
$l=20~{}\mu$m, our observation of 200-fold reduction of $\Delta
G_{\mathrm{rms}}$ compared to a fully coherent scenario suggests
$\ell_{\phi}\approx(l,w)/200\approx 100$ nm.

A thermal spread of charge carrier energy and thus momentum can also suppress
$\Delta G_{\mathrm{rms}}$, since carriers of different momentum accumulate
different phase along the same path [35, 45, 42]. Based on our measured
conductivity and estimated Fermi velocity we calculate that phase is thermally
randomized over a length $\ell_{T}\approx 2~{}\mu$m (Section S3 [38]). Since
$\ell_{T}\gg\ell_{\phi}$, thermal smearing can be neglected at the electron
temperatures attained in our experiments (discussed below and in Section S6
[38]).

Where thermal averaging can be neglected,

$\Delta
G_{\mathrm{rms}}=\frac{e^{2}}{h}c_{d}\sqrt{\frac{ks^{2}}{\beta}}\sqrt{\frac{\ell_{\phi}^{2}}{wl}}.$

Constants $c_{d}$ and $\beta$ depend on sample dimensionality ($c_{d}=0.862$
for two dimensions [35]) and symmetries of the ensemble of disordered
wavefunctions ($\beta=2$ when time-reversal symmetry is broken [45]),
respectively. Parameters $s$ and $k$ account for multiplicity of spin and
other degeneracies. These degeneracies are challenging to determine for the
MBT thin film used in this work. Even uniform MBT has a complicated band
structure–featuring multiple topological surface states in addition to
topologically trivial bulk states [2, 22]–and sample-to-sample variations [1,
13, 22, 46, 15] only complicate the picture. In our disordered film with
varying layer number and no specific knowledge of the location of the Fermi
level relative to band structure features, we use $k=s=1$ and acknowledge that
the absolute magnitude of $\ell_{\phi}$ we determine may accordingly be off by
a constant multiple. With these constants, we find $\ell_{\phi}=104$ nm in the
FM phase.

The Pearson product-moment correlation coefficient $\mathscr{R}$ quantifies
the degree of correlation between a pair of field sweeps $i,j$. Calculated as

$\mathscr{R}=\frac{\sum(\Delta G_{i}(B)-\overline{\Delta G_{i}})(\Delta
G_{j}(B)-\overline{\Delta G_{j}})}{\sqrt{\sum(\Delta G_{i}(B)-\overline{\Delta
G_{i}})^{2}(\Delta G_{j}(B)-\overline{\Delta G_{j}})^{2}}},$

$\mathscr{R}=1$ for a pair of perfectly correlated sweeps and $\mathscr{R}=0$
for a pair of perfectly uncorrelated sweeps. For the pair of field sweeps
shown in Figure 2(b), $\mathscr{R}=$ 0.73, in accord with the strong visible
similarity, despite the $\sim 25$ hour separation between the acquisition of
the two field sweeps. A discussion of repeatability of measured UCFs over
varied time scales is presented in Section S5 [38].

Correlation of UCFs as a function of separation in applied field rather than
in time provides direct insight into $\ell_{\phi}$ [35, 44]. Changing field by
an amount sufficient to thread a single flux quantum through a single coherent
area largely randomizes the relative phases of different trajectories. This
“coherence field” scale $B_{c}$ is therefore related to $\ell_{\phi}^{-2}$ and
can be extracted as the half-width at half-max of the correlation function
$C(\Delta B)=\langle\Delta G_{xx}(B)\Delta G_{xx}(B+\Delta B)\rangle$. In two
dimensions,

$B_{c}=0.49\frac{h}{e}\frac{1}{\ell_{\phi}^{2}},$

provided that thermal averaging can be neglected [35, 47]. Figure 2(c) shows
$C(\Delta B)$ for the $\Delta G_{xx}$ traces plotted in Figure 2(b). The
corresponding $B_{c}$ is 40 mT, yielding $\ell_{\phi}\approx 240$ nm, within a
factor of 3 of the estimate obtained via $\Delta G_{\mathrm{rms}}$. Given the
uncertainty associated with $B_{c}$ as a metric for phase coherence in two
dimensions [47], the two methods produce acceptable agreement. Due to the
limited field domains over which most data were acquired, further discussion
of $B_{c}$ is confined to the supplemental materials. Additional estimates of
$\ell_{\phi}$ below instead use the $\Delta G_{\mathrm{rms}}$ method.

Since MBT is a magnetic material, one might question whether its magnetic
disorder has an impact on UCF measurements. Magnetic force microscopy
measurements on single-crystal MBT have revealed spatial domains of
magnetization that evolve with magnetic field within each magnetic phase and
across phase boundaries [20]. To study the impact of magnetic disorder on our
UCFs, we next present measurements of $\Delta G_{xx}$ upon repeated sweeps
within each magnetic phase. Then we examine how this changes if we cross a
magnetic phase boundary before returning for a subsequent sweep in an initial
phase. These UCF measurements appear sensitive to the motion and nucleation of
MBT’s magnetic domains.

Figure 3: Repeatability of UCF, with attention to field sweep direction. (a-c)
$\Delta G_{xx}$ as a function of applied field for sweeps up (black) and down
(grey) in the (a) AFM ($\Delta G_{\mathrm{rms}}=0.0049~{}e^{2}/h$; value
averaged across all sweeps), (b) cAFM ($\Delta
G_{\mathrm{rms}}=0.0043~{}e^{2}/h$), and (c) FM ($\Delta
G_{\mathrm{rms}}=0.0050~{}e^{2}/h$) phases. For each magnetic phase, field was
swept back and forth across a 1 T range in immediately subsequent runs. (d-f)
Correlation $\mathscr{R}$ between each separate pair of $\Delta G_{xx}$ traces
within the (d) AFM, (e) cAFM, and (f) FM phases. In the AFM and cAFM phases,
runs with the same field sweep direction are markedly more highly correlated
than runs with opposite field sweep directions. The slight reduction in
correlation between the initial (“Up 1”) sweep and subsequent sweeps in the
AFM and cAFM phases can likely be attributed to slight cooling over time; all
three datasets were acquired after measurements at elevated temperatures and
allowing the system to return to base temperature.

UCFs occur in each magnetic phase of MBT. Figures 3(a-c) present changes in
longitudinal conductance $\Delta G_{xx}$ while applied magnetic field was
swept continuously back and forth across a 1 T range. Data acquired while
sweeping the field up (down) are shown in black (grey). Repeatable conductance
fluctuation patterns indicative of UCF are apparent, particularly in the cAFM
and FM phases. In each magnetic phase, $\Delta
G_{\mathrm{rms}}=0.004-0.005~{}e^{2}/h$, indicating $\ell_{\phi}\geq 90$ nm
across the entire magnetic phase diagram. The specific magnitude of $\Delta
G_{\mathrm{rms}}$ and therefore $\ell_{\phi}$ is sensitive to the size of the
current excitation, likely due to current-driven heating as discussed below
and Section S6 [38].

Figures 3(d-f) plot the correlation between each pair of field sweeps within
each magnetic phase. In the FM phase $\mathscr{R}>0.95$ for nearly all pairs
of field sweeps. In contrast, in the AFM and cAFM phases the magnitude of
$\mathscr{R}$ strongly depends on whether or not magnetic field was swept in
the same direction for both $\Delta G_{i}$ and $\Delta G_{j}$; correlation
between runs with opposite field sweep direction is clearly reduced. This
effect is not typically observed in measurements of UCFs and is likely related
to magnetic domain reconfiguration, as discussed below.

Figure 4: Repeatability of UCF pattern with intervening crossing of magnetic
phase boundaries. (a) $\Delta G_{xx}$ as a function of magnetic field.
Measurements were taken in order from top to bottom. Each sweep is labelled
with a letter on the right side of the plot, and presented in a unique color.
Corresponding arrows indicate field sweep direction. (b-d) Repeatability
$\mathscr{R}$ of $\Delta G_{xx}$ between field sweeps presented in (a) for the
(b) AFM, (c) cAFM, and (d) FM phases. Black dashed lines indicate where the
AFM/cAFM phase boundary was crossed.

Figure 4(a) presents $\Delta G_{xx}$ as magnetic field was swept continuously
across specific magnetic phase transitions. Data collection started at 12 T at
the top of Figure 4(a) and proceeded from top to bottom, with the direction of
the seven field sweeps (lettered “A” through “G”) indicated by the
corresponding arrows. Field sweep ranges were chosen to isolate behavior in
each magnetic phase before and after crossing each magnetic phase boundary.
Figure 4(b-d) shows $\mathscr{R}$ for the pairs of field sweeps acquired
within each magnetic phase. In the cAFM and FM phases, we observe that the
repeatability between traces is generally lower when the AFM/cAFM phase
boundary (at which the magnetic ordering undergoes a spin-flop transition) was
crossed between the two sweeps; for example, in the cAFM phase,
$\mathscr{R}_{\text{D-E}}<\mathscr{R}_{\text{E-F}}$, although in one case this
trend is not followed ($\mathscr{R}_{\text{E-G}}>\mathscr{R}_{\text{E-F}}$).
Below, we attribute this reduced correlation to reconfiguration of magnetic
domains.

Figure 5: Temperature dependence of UCF. (a-c) $\Delta G_{xx}$ as a function
of applied magnetic field in the (a) AFM, (b) cAFM, and (c) FM phases.
Temperature is indicated by the color scale, and increases from bottom to top.
(d) Phase coherence length $\ell_{\phi}$ as a function of temperature, fit to
$\ell_{\phi}^{-2}=a+bT^{p}$ with free parameters $a$, $b$, and $p$ in each
magnetic phase. For the AFM phase, fitting produces $\ell_{\phi}^{-2}\sim
T^{2.4}$. For the cAFM phase, $\ell_{\phi}^{-2}\sim T^{2.3}$. For the FM
phase, $\ell_{\phi}^{-2}\sim T^{2.4}$. (e) Longitudinal resistance $R_{xx}$ as
a function of temperature extracted from the same measurements as used to
generate (a-d). Values of $R_{xx}$ are plotted for external magnetic field 1.5
T (AFM, black), 5.5 T (cAFM, blue) and 11.5 T (FM, red).

To complete our characterization of the phenomenology of UCFs in MBT, we next
examine the temperature dependence of $\Delta G_{xx}$. As temperature
increases, coherence-breaking scattering mechanisms are activated. The
temperature scaling of the UCF-determined $\ell_{\phi}$ can be used to
determine the dominant coherence-breaking scattering mechanism. Figure 5(a-c)
shows $\Delta G_{xx}$ traces as a function of applied field at various
temperatures for each magnetic phase. Increasing temperature decreased the
magnitude of the fluctuations as expected.

In literature, temperature scaling relationships are often expressed in terms
of the dephasing rate $\tau_{\phi}^{-1}=(D\ell_{\phi})^{-2}\propto T^{p}$ for
temperature $T$, diffusivity $D$, and coherence time $\tau_{\phi}$ [48, 49,
50, 51]. Common scaling relationships include $p=1$, which indicates electron-
electron scattering is dominant [48, 49], and $p$ between 2 and 4, which
indicates electron-phonon scattering is dominant [50]. Figure 5(d) plots
$\ell_{\phi}$, as determined from $\Delta G_{\mathrm{rms}}(T)$, as a function
of temperature. Fitting $\ell_{\phi}^{-2}\propto T^{p}$ yields $p$ between 2
and 3 in all magnetic phases, consistent with electron-phonon scattering.

Power-law exponents extracted from data ranging over a single decade in
temperature likely have significant quantitative uncertainties. This challenge
is exacerbated by the low-temperature saturation in $\ell_{\phi}$ apparent in
Figure 5(d). Similar behavior is also visible in Figure 5(e), which shows
$R_{xx}$ as a function of temperature at 1.5 T (AFM phase), 5.5 T (cAFM
phase), and 11.5 T (FM phase). The saturation of transport behavior below
about 100 mK was likely caused by current-driven heating resulting in electron
temperature decoupling from the thermometer-measured lattice temperature. We
explore this further in Section S6 [38].

In each magnetic phase, Figure 5(e) reveals $R_{xx}\propto-\log(T)$ for
temperatures above this saturation. Common causes of a low-temperature
logarithmic increase in $R_{xx}$ as temperature decreases include weak
localization [48] or Kondo [52] effects. Each of these mechanisms is
suppressed by magnetic field: weak localization at a scale $B\sim B_{c}$ and
Kondo at a scale $B\sim k_{B}T_{K}/(g\mu_{B})$ (here $k_{B}$ is the Boltzmann
constant, $g$ is the $g$-factor, and $\mu_{B}$ is the Bohr magneton). All the
data of Figure 5(e) were acquired at $B\gg B_{c}$, excluding weak localization
as a possible explanation for the temperature scaling. As for Kondo, we see no
significant field dependence for temperature scaling of resistivity out to our
maximum field of 12 T. To avoid suppression of Kondo physics across this field
range, the Kondo temperature $T_{K}$ would need to be at least several times
larger than $12~{}\mathrm{T}*g\mu_{B}/k_{B}\approx 16$ K. But scattering from
Kondo impurities shows logarithmic scaling only for $0.1T_{K}\lessapprox
T\lessapprox 10T_{K}$, so explaining our resistivity data with such a large
Kondo temperature is not internally consistent with our observation of
logarithmic scaling down to a fraction of a Kelvin. Instead, we attribute this
scaling to another well-established phenomenon: the low-temperature
suppression of the density of states near the Fermi level caused by electron-
electron interactions [53, 54].

Having established the basic phenomenology of UCFs in our MBT device, we next
explore the possible relationship between features of our data and domains in
the magnetization of MBT. Magnetic domains in the AFM and cAFM phases of
exfoliated flakes of MBT have been studied by magnetic force microscopy [20,
21]. Antiferromagnetic domains result from intralayer regions of opposite
magnetization; canted antiferromagnetic domains result from intralayer regions
of opposite canting [20]. In contrast, in the FM phase domains are eliminated
by the high external magnetic fields and MBT is found to be uniformly
magnetized [20]. Although the presence of domains is well-established through
these magnetic force microscopy measurements, their impact on transport
measurements has yet to be studied.

We suggest that the field-sweep-direction-dependent changes to the UCF
patterns in the AFM and cAFM phases (Figure 3) indicate reversible motion of
magnetic domain walls as a function of applied field. Such an interaction
between magnetic domain reconfiguration and UCF patterns should be expected.
Variations in both elastic and inelastic charge carrier scattering upon
changing the magnetic domain structure are well-known theoretically [55, 56,
57] and experimentally [58, 59, 60, 61, 28]. Since UCFs are sensitive to
changes in the scattering landscape seen by charge carriers, the
magnetofingerprint should therefore be sensitive to changes in the magnetic
domain structure. Consistent with this interpretation, micron-scale field-
driven domain wall motion has been observed via magnetic force microscopy
[20], although correlations across repeated field sweeps or as sweep direction
is reversed have not previously been examined.

Further signatures of interaction between magnetic domain walls and UCF
patterns can be seen in changes to the UCF pattern upon crossing magnetic
phase boundaries, as shown in Figure 4. A spin-flop transition occurs at the
AFM/cAFM phase boundary [21, 40, 37]. The number and location of magnetic
domain walls need not be preserved through this phase transition, and
different domain patterns on either side of this phase transition have been
observed in magnetic force microscopy [20]. We suggest the reduction in
$\mathscr{R}$ upon crossing the AFM/cAFM magnetic phase boundary indicates the
formation of new domains, and therefore a new scattering landscape. In
contrast, crossing the cAFM/FM phase boundary does not appear to substantially
perturb the UCF pattern, consistent with reports that crossing this boundary
brings not a sudden rearrangement of magnetization but rather a gradual
reduction of the in-plane canting angle [40].

Taken together, these observations suggest measurements of UCF serve as a
transport-based probe of magnetic domain wall reconfiguration in MBT thin
films. Though our experiment cannot directly measure correlation between
domain wall positions and UCF magnetofingerprints, we are aware of no
alternative mechanism by which the scattering landscape of MBT could exhibit
the observed field-sweep-direction-dependence in the AFM and cAFM phases, but
not the FM phase. Future simultaneous scanning probe and transport
measurements could more rigorously test the relationship between domain wall
motion and UCF measurements. Sensitivity to magnetic ordering on a 100 nm
length scale while in a high magnetic field is within the capabilities of
magnetic force microscopy and (at least for the AFM phase) of nanoSQUID [62].

In this work, we present measurements of universal conductance fluctuations in
a thin film of the magnetic topological material MnBi2Te4. Analysis of the
UCFs in three magnetic phases of MBT reveal information on phase coherence
lengths and coherence-breaking scattering mechanisms. We observe strong
indications that magnetic domain wall motion and domain rearrangement are
expressed in UCF measurements. Our work establishes UCFs as a transport-based
probe of magnetization heterogeneities and spin-flop transitions in a
topological magnet, a first step toward the critical experimental need for
practical probes relating topological and magnetic order in disordered
materials.

The authors thank D. Natelson, B. Shklovskii, and L. K. Rodenbach for useful
discussions. M. P. A., E. M., I. T. R., M. A. K., and D. G.-G. were supported
by the U.S. Department of Energy, Office of Science, Basic Energy Sciences,
Materials Sciences and Engineering Division, under Contract DE-AC02-76SF00515.
M. P. A. additionally acknowledges support from Gordon and Betty Moore
Foundation through Grant No. GBMF9460 and Air Force Office of Scientific
Research (AFOSR) Multidisciplinary Research Program of the University Research
Initiative (MURI) under grant number FA9550-21-1-0429 during the later stages
of this work. L.T., P.Z., and K.L.W. acknowledge support from the National
Science Foundation (NSF) under grant numbers 2125924 and 1936383 and the Army
Research Office Multidisciplinary University Research Initiative (MURI) under
grant number W911NF-19-S-0008. Infrastructure and cryostat support were funded
in part by the Gordon and Betty Moore Foundation through Grant No. GBMF3429.
We thank NF Corporation for providing low-noise, high-input-impedance voltage
preamplifiers. We acknowledge measurement assistance from colleagues at the
National Institute of Standards and Technology. Part of this work was
performed at the nano@Stanford labs, supported by the National Science
Foundation under award ECCS-2026822.

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Supplemental information for: Universal conductance fluctuations in a MnBi2Te4
thin film

Molly P. Andersen1,2, Evgeny Mikheev3,2,4, Ilan T. Rosen5,2,6, Lixuan Tai7,
Peng Zhang7, Kang L. Wang7, Marc A. Kastner3,2,8, and David Goldhaber-
Gordon3,2,a

1Department of Materials Science and Engineering, Stanford University,
Stanford, California 94305, USA
2Stanford Institute for Materials and Energy Sciences, SLAC National
Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, California 94025, USA
3Department of Physics, Stanford University, Stanford, California 94305, USA
4Department of Physics, University of Cincinnati, Cincinnati, Ohio 45221, USA
5Department of Applied Physics, Stanford University, Stanford, California
94305, USA
6Research Laboratory of Electronics, Massachusetts Institute of Technology
7Department of Electrical and Computer Engineering, Department of Physics and
Astronomy, University of California, Los Angeles, California 90095, USA
8Department of Physics, Massachusetts Institute of Technology, Cambridge,
Massachusetts 02139, USA
aTo whom correspondence should be addressed; E-mail: goldhaber-
<EMAIL_ADDRESS>

###### Contents

  1. S1 Details of experimental methodology
  2. S2 Additional linecuts of atomic force microscopy images
  3. S3 Density, mobility, and thermal diffusion length
  4. S4 Correlation in magnetic field
  5. S5 Repeatability over long timescales
  6. S6 Varied ac current bias
  7. S7 Electrostatic gating
  8. S8 Additional device

## S1 Details of experimental methodology

The MnBi2Te4 film used in this work [S1] was nominally 6 septuple layers (SLs)
thick and was grown in a Perkin-Elmer ultra-high vacuum molecular beam epitaxy
(MBE) system. Throughout the process, growth was monitored by in situ
reflection high-energy electron diffraction (RHEED), with RHEED images
captured by a KSA400 system from K-space Associates, Inc. The growth substrate
was epi-ready semi-insulating GaAs (111)B; after loading the substrate into
the MBE chamber, substrates were pre-annealed at 630 °C in a Te-rich
environment. The substrate was kept at 200 °C during growth. To grow the film,
high-purity Mn, Bi and Te were evaporated simultaneously from standard Knudsen
cells. The film was annealed in a Te-rich environment for 2 minutes at 290 °C
after deposition to improve crystallinity.

Two Hall bar devices were fabricated on a single chip of this film. To define
device mesas, the chip was first cleaned in acetone and isopropanol. After
patterning mesa geometries with photolithography, the mesas were etch-defined
with 35 s Ar ion milling with a 400 V accelerating voltage. The chip was
cleaned with 30 s sonication in acetone and an isopropanol rinse to remove
resist and sidewalls from ion milling. Current and voltage contacts were then
defined with photolithography, and 5/85 nm Ti/Au was deposited with electron
beam (e-beam) evaporation after a 10 s in situ Ar pre-etch. Contacts were
lifted off with a 5 minute soak in acetone and 10 s sonication followed by a
solvent rinse. An alumina electrostatic top gate was then added. First, a 1 nm
seed layer of aluminum was globally deposited by e-beam evaporation and
allowed to oxidize in air. Next, 400 cycles of atomic layer deposition (ALD)
was used to deposit alumina with tetramethylammonium and water precursors at
60 °C, where the reduced temperature was chosen to avoid thermal degradation
of the sample. The chip was next rinsed with acetone and isopropanol. A
photolithographically-masked 150 s wet etch with Microposit MF-CD-26 developer
was used to remove alumina over the contact pads. The chip was again rinsed in
acetone and isopropanol to remove the photoresist. Top gate electrodes were
patterned with photolithography; at this point optical micrographs of the
devices were taken (Figure S1). E-beam evaporation was then used to deposit
5/95 nm Ti/Au after a 10 s in situ Ar pre-etch. Metal liftoff was performed
with sonication in acetone and a solvent rinse.

All photolithography steps were performed as follows: The sample was spin-
coated at 5,500 rpm with a hexamethyldisilazane adhesion layer and SPR 3612
photoresist. The resist was then baked at 80 °C for 5 minutes; this lower-
than-typical bake temperature was chosen to avoid thermal decomposition of the
film. Patterns were exposed on a MicroWriter Direct Write tool with a 385 nm
wavelength and a 115 mJ/cm2 dose. Exposed regions of the resist were developed
with 35 s/20 s/20 s rinses in Microposit Developer CD-30/deionized (DI)
water/DI water.

Figure S1: Optical micrographs of (a) Device A, with two 10 $\mu$m squares
between longitudinal voltage taps, and (b) Device B, with two 20 $mu$m squares
between longitudinal voltage taps. Electrical measurement setup is indicated
in (a): Current $I_{src}$ is sourced in from the left, measured ($I$), and
drained to ground out the right contact. All combinations of longitudinal
($V_{xxt}$, top longitudinal pair; $V_{xxb}$, bottom longitudinal pair) and
Hall ($V_{xyl}$, left Hall pair; $V_{xyr}$, right Hall pair) voltages were
measured as indicated. Device B was measured in the same manner. Components of
the optical micrographs are described in (b); both images were taken after the
device mesa was defined (bare MBT region), contacts were added (gold region),
a global ALD dielectric was deposited, and the top gate electrodes were
patterned with photolithography. Photoresist from the top gate electrode
lithography step is still in place in both images (resist on GaAs). Scale bars
in both images: 20 $\mu$m.

Electronic transport measurements were performed in a Leiden Cryogenics
dilution refrigerator with a base temperature of about 35 mK. Measurement
lines were filtered with low-pass RF filters and discrete RC filters at the
mixing chamber stage to ensure charge carrier thermalization. A 5-8 Hz ac
current was sourced separately into each device by a Stanford Research 830
lock-in amplifier (SR830) applying a 2 or 5 V signal across a 1 G$\Omega$
resistor to produce a 2 or 5 nA bias current for main text data. Voltage
measurements were acquired by additional SR830s, as well as Stanford Research
860 lock-in amplifiers, after $10^{2}$ V/V amplification by either separate NF
LI-75A voltage preamplifiers or a single NF multi-channel preamplifier.
Currents were measured on SR830s after amplification by an Ithaco 1211 current
preamplifier. External out-of-plane magnetic fields were applied using a 14 T
superconducting magnet. Measurements at elevated temperatures were taken by
heating the sample with a heater on the mixing chamber stage of the dilution
refrigerator. Measurements were taken between two separate cool-downs. Table
S1 shows the magnitude of current excitations and step sizes in applied field
for main-text Figures 2–5. Larger currents were used during measurements
acquired in the first cool-down, before the impact of the current excitation
magnitude had been realized. Larger step sizes in applied field were used for
main-text Figure 2 to limit measurement time, but smaller step sizes were used
for all other measurements.

Main-text figure | Current excitation (nA) | Applied field step size (mT)
---|---|---
2 | 5 | 5
3 | 2 | 2
4 | 5 | 2
5 | 2 | 2

## S2 Additional linecuts of atomic force microscopy images

Figure S2 shows three additional linecuts across step edge features in the
topographic map of the MBT thin film. Two of these steps were about 0.9 nm,
consistent with Bi2Te3. One step is 0.5 nm, possibly resulting from MnTe_x.

Figure S2: (a) Reproduction of Figure 1(b), showing a topographic map of the
MBT thin film acquired with atomic force microscopy. (b-d) Additional linecuts
with extracted step heights. The location of each linecut in (a) is are
indicated by (b) cyan, (c) fuchsia, and (d) chartreuse lines. Data were
averaged across several parallel traces in the transverse direction, indicated
by the extent of the crosspieces appended to the tails of the lines in (a).

## S3 Density, mobility, and thermal diffusion length

Hall resistance measured as a function of external magnetic field on the left
voltage contact pair on Device A is shown in Figure S3(a). Electron density
$n$, extracted from the slope of the Hall resistance over a 2.5 T sliding
window, is shown in Figure S3(b). At lower external magnetic fields, a higher
Hall slope (and correspondingly lower calculated charge carrier density) is
observed. These changes in Hall signal are likely related to MBT’s magnetic
phase transitions, as discussed in the context of longitudinal resistance
measurements in the main text. We therefore opt to use densities extracted
near zero external magnetic field ($2.5*10^{12}$ cm$-2$) for our calculations.
Mobility $\mu$, calculated via $\sigma_{xx}=ne\mu$ for longitudinal
conductivity $\sigma_{xx}$, is shown in Figure S3(c).

Using the calculated mobility, $l_{T}=\sqrt{hD/k_{B}T}$ can be approximately
determined. Diffusivity $D$ can be calculated as
$D=v_{F}\ell_{el}/4=v_{F}^{2}\mu m^{*}/4e$, where $l_{el}$ is the elastic
scattering length. With a minimum mobility $\mu\approx 200$ cm2/Vs (Figure
S3(b)), electron charge $e$, and estimates for effective mass $m^{*}=0.25$
electron mass and surface state Fermi velocity $v_{F}=3*10^{5}$ m/s taken from
theory and other experiment [S2,S3], $l_{T}\approx 2$ $\mu$m at 30 mK for our
MBT thin film. We note the band structure details used to calculate
diffusivity are only approximate, but our estimation of $l_{T}$ is likely
correct to within an order of magnitude.

Figure S3: (a) Hall measurements acquired simultaneously with the data of
Figure 2. (b) Field-dependent density $n$ extracted from the Hall measurements
of (a) over a 2.5 T sliding window. (c) Mobility $\mu$ derived from the
longitudinal and Hall measurements of Figure 2 and (a). For all data, red
(blue) points were acquired with field sweeping up (down)

## S4 Correlation in magnetic field

As described in the main text, an analysis of the field scale over which
correlations persist in UCFs can provide insight into $l_{\phi}$. This method
provides a sensible estimate of $l_{\phi}$ when applied to large-field-domain
data in Figure 2. However, most data acquired in this work was taken over a 1
T field range. Calculated $C(\Delta B)$ plots for these 1 T-range datasets
reveal a central coherence peak surrounded by reasonably high-intensity quasi-
periodic oscillations instead of minimal noise about $\Delta B=0$ T as is
typical. These oscillations appear to affect the central coherence peak and
therefore prevent the analysis of $B_{c}$ described in the main text. A few
examples are shown in Figure S4.

Figure S4: Correlation $C(\Delta B)$ calculated from conductance fluctuations
measured across a 1 T range with 2 mT step size and a 2 nA bias current in the
(a) AFM, (b) cAFM, and (c) FM phases. Conductance fluctuation data from which
these plots were calculated are shown in “Down 1” sweeps of Figure 2.

One possible explanation for these qualitative features is an actual periodic
signal is obscured below the apparent random conductance fluctuations. Such a
periodic signal could arise from the Aharanov-Bohm effect [S4], caused by
quantum interference of charge carriers traversing rings with flux threaded
through the ring. If the Aharanov-Bohm effect were present in our data,
different analysis techniques than are presented in the main text would be
required.

### i Fast Fourier transform analysis

To show the low frequency quasi-periodic oscillations in the correlation
functions did not arise from currents forming Aharanov-Bohm loops, we compare
the fast Fourier transforms (FFTs) of UCFs over subsets of the field range
plotted in Figures 2 and S3. The full dataset was acquired in 5 nA steps while
sweeping magnetic field from 14 T$\rightarrow$-14 T$\rightarrow$14 T at base
temperature. Here, we limit our analysis to the $|10-14|$ T ranges for both
field sweep directions, where it can be assumed there is no impact from
magnetic domain wall motion. Different quasi-periodic oscillations in
different field ranges indicates that these oscillations are not associated
with specific Aharanov-Bohm loops but rather an artifact of a limited field
range.

Figures S5 and S6 plot conductance fluctuations, calculated correlation, and
FFTs of the conductance fluctuations for 1 T and 4 T field ranges. Both field
sweep directions and field orientations are included in each plot, resulting
in four unique traces per plot. Conductance fluctuations and correlation
functions are calculated from measurements as discussed in the main text. The
FFTs are calculated from $\Delta G_{xx}$ traces as follows: (1) windowing the
data with a Hamming window; (2) assuming only real inputs, calculating a
discrete one-dimensional FFT with Numpy’s built-in routine; (3) calculating
the frequency-dependent magnitude of the FFT by combining the real and
imaginary parts of the FFT; (4) normalizing the magnitude by a constant to
bring the maximum FFT amplitude close to unity.

For the smallest field domain (1 T, Figure S5), some low-frequency quasi-
periodic oscillations are clear in the correlation plots (for example, black-
boxed regions). Importantly, different field ranges result in apparently
distinct oscillations. If Aharanov-Bohm loops were present one expects the
same underlying oscillations to persist at all field ranges. Additionally, as
the field domain is increased, the magnitude of any specific oscillations are
suppressed and the correlation function appears random outside of the central
coherence peak (Figure S6). If oscillations persisted throughout the whole
field range, their magnitude would not be suppressed by extending the field
domain. This again suggests transport around Aharanov-Bohm loops are not the
source of the quasi-periodic oscillations in some correlation field plots.

However, although visual inspection of the correlation analyses appears to
rule out Aharanov-Bohm effects, noise data requires more careful examination.
Here, we turn to the FFTs of the $\Delta G_{xx}$ data. Across the field range
and domains studied here no consistent low-frequency structure is apparent.
Substantial variation is clear even between data related by Onsager relations
(i.e. traces within the same plot). This supports the conclusion that the
conductance fluctuations we observe are caused by universal conductance
fluctuations and not by currents circulating around a network of loops.

Figure S5: Conductance fluctuations, correlation function, and fast Fourier
transform magnitude for data taken over 1 T field ranges between 10 T and 14
T. (a-c) 10-11 T; (d-f) 11-12 T; (g-i) 12-13 T; (j-l) 13-14 T. Blue (red)
traces were acquired with external field sweeping down (up). Dashed (solid)
lines were acquired with a positive (negative) out-of-plane external magnetic
field. Figure S6: Conductance fluctuations, correlation function, and fast
Fourier transform magnitude for data taken over a 4 T field range between 10 T
and 14 T. Blue (red) traces were acquired with external field sweeping down
(up). Dashed (solid) lines were acquired with a positive (negative) out-of-
plane external magnetic field.

## S5 Repeatability over long timescales

Figure S7 demonstrates the repeatability of the UCFs as a function of external
magnetic field. To acquire the conductance fluctuation data shown in (a), the
magnetic field was swept 11 T$\rightarrow$14 T$\rightarrow$11 T. Each 3 T
field sweep took about 100 minutes. After a 2 hour delay at 11 T, the pair of
field sweeps were repeated. After a second 2 hour delay, the pair of field
sweeps were repeated again. All data was acquired with a 5 nA current bias.
The correlation between pairs of field sweeps is presented in Figure S7(b).
For successive runs, $\mathscr{R}>0.8$. For runs separated by about 11 hours,
$\mathscr{R}>0.5$.

We note that the highest values of $\mathscr{R}$ are smaller in the data shown
in Figure 4 than that shown in Figure 3. We attribute this reduction to the
longer data collection time (limited by the magnet’s ramp rate) of
measurements shown in Figure 4: the data shown each sub-panel of Figure 3 were
collected in about 3.5 hours, while the complete measurement shown in Figure
4(a) took about 29 hours. We believe rearrangement of disorder on the
timescale of the latter measurement reduces the measured correlation.
Additionally, measurement parameters like current bias, domain in field, and
step size in field can affect $\mathscr{R}$.

Figure S7: Repeatability of conductance fluctuations in the FM phase across a
14 hour time window. (a) Conductance fluctuations acquired between 11 and 14 T
with a 2 mT step size and a 5 nA bias current. Traces shown in black (grey)
were acquired while sweeping field up (down). Traces were acquired
sequentially starting from the top and are offset for clarity. (b)
Repeatability $\mathscr{R}$ for the data shown in (a).

## S6 Varied ac current bias

A low-temperature saturation in both $R_{xx}$ and $\Delta G_{xx}$ is observed
in the main-text data (see Figure 4(d,e)). Several possible explanations exist
for this behavior, including scattering lengths exceeding device dimensions
(unlikely, given that the device is 100 times larger than the largest
$l_{\phi}$ seen here) or Kondo physics (previously excluded due to the lack of
magnetic field dependence). Instead, we observe the saturation to be an
artifact of ac current-driven Joule heating of the sample. Figure S8 shows
similar measurements to those in main text Figure 4, taken within the FM phase
with ac current biases ranging from 1-50 nA. As current bias is increased, the
magnitude of lowest-temperature $\Delta G_{\mathrm{rms}}$ clearly decreases,
indicating heating of the sample and therefore a smaller phase coherence
length. By 50 nA, increasing temperature from 35-680 mK causes only minimal
changes in the UCF pattern, which suggests the effective sample temperature is
at least several hundred milliKelvin.

Figure S8: Artificially offset traces of $\Delta G_{xx}$ from the top contact
pair of Device A, acquired as field is swept between 11 and 12 T at varied
temperatures in the same manner as the data presented in Figure 4 (2 nA data
shown here is reproduced in Figure 4). Different subplots were acquired with
differing ac current biases: (a) 1 nA; (b) 2 nA; (c) 5 nA; (d) 10 nA; (e) 20
nA; (f) 50 nA. As current bias is increased, the magnitude of UCF at low
temperatures decreases.

Figure S9(c,f) shows analogies of the main text Figure 4(d,e) for the FM phase
and varied current bias. Low-temperature saturation in both $R_{xx}$ and
$\Delta G_{\mathrm{rms}}$ clearly increases with the ac current bias, a clear
indication of current-driven Joule heating. Similar measurements at a few low
current biases demonstrate the same behavior in both the AFM and cAFM phases;
a summary of data in the AFM and cAFM phases is shown in Figure S9(a,b,d,e).

We note that our main text data was acquired with either a 2 or 5 nA current
bias. At these currents, this current-driven heating effect is apparent at
lowest temperatures, resulting in suppressed measurements of $\Delta
G_{\mathrm{rms}}$. It is therefore likely we underestimate $\ell_{\phi}$ in
our lowest-temperature calculations. A similar measurement suite with ac
currents below 1 nA is necessary to more accurately determine $\ell_{\phi}$.

Additionally, this current-driven heating decouples the lattice temperature
from the electron temperature. The thermometer on the mixing chamber stage of
the probe therefore does not accurately reflect the temperature of the charge
carriers. Effective electron temperature can be estimated from Figure S9 as
the temperature at which $\Delta G_{xx}$ and $R_{xx}$ are no longer saturated
and begin changing with increasing temperature. For 2 and 5 nA bias currents,
this appears to be 80-120 mK. We note that the elevated electron temperature
relative to lattice temperature does not affect our argument that
$\ell_{T}>>\ell_{\phi}$. Since $\ell_{T}\propto T^{-1/2}$, we expect at most a
$\sqrt{3}$ reduction in $\ell_{T}$ compared to what we calculated above. In
this case, $\ell_{T}\approx 1$ $\mu$m, which still exceeds our highest
estimates of $\ell_{\phi}$.

Figure S9: Temperature dependence of (a-c) $\Delta G_{\mathrm{rms}}$ and (d-f)
$R_{xx}$ in the (a,d) AFM (b,e) cAFM and (c,f) FM magnetic phases with varied
ac current bias. All data was acquired with the top contact pair of Device A.

## S7 Electrostatic gating

As mentioned above, Hall bars presented here featured electrostatic top gates
to allow experimental control over charge carrier density. However, long
timescales for equilibration of charge traps after changes to the applied gate
voltage restrict this work to data acquired exclusively with the top gate
grounded. Figure S10 shows measured longitudinal conductance $G_{xx}$ over
time after the top gate voltage is changed from -4 V $\rightarrow$ 4 V.
Persistent random fluctuations in $G_{xx}$ on the order of 0.01 $e^{2}/h$ are
obvious. We attribute these fluctuations to slow (de-)population of charge
traps within the gate dielectric (likely, since the dielectric was deposited
at low temperatures (see above) and therefore contains residual carbon
contamination and other defects) or elsewhere in the system. Since the
electrostatic landscape contributes to the elastic scattering landscape charge
carriers see, these charge trap (de-)population events affect the device
conductance.

Figure S10: Longitudinal conductance of the 10 $\mu$m Hall bar as a function
of time after last gate voltage excursion.

Figure S11 shows UCFs measured upon repeated field sweeps after changing the
top gate voltage from -4 V $\rightarrow$ 4 V over the course of several hours.
Each pair of field sweeps took a total of 12 minutes to acquire. Even on such
short time scales, differences between the measured conductance fluctuations
are clear. While changes to UCF with charge carrier density would be
interesting to study, to avoid conflating conductance changes from UCF with
conductance changes from time-dependent charge trap (de-)population, all data
presented elsewhere in this paper were acquired with the gate grounded and at
least 12 hours after the last gate voltage change.

Figure S11: Conductance fluctuations as a function of external magnetic field
with varied start times after the last gate voltage excursion. Solid (dotted)
lines correspond to field sweeps up (down). Curves are offset for clarity.

## S8 Additional device

All data presented so far was measured at the top (longitudinal) or left
(Hall) voltage pairs on Device A. Simultaneously, the bottom and right voltage
pairs on Device A, as well as all four different voltage pairs on Device B,
were also measured. We observed no unexpected differences between these
additional datasets and those presented so far; as an example, the same plots
as those shown in the main text Figure 2 are plotted below for the additional
device (Figures S12). In all cases, the same dependence of correlation on
field sweep direction is observed, and the overall magnitude of the
correlation is similar. The one notable expected distinction between the
Device A measurements and Device B is caused by geometric differences; since
UCF are geometrically averaged out as the device size spans higher multiples
of $l_{\phi}$, we expect a $\sqrt{4}$ reduction in the UCF magnitude in Device
B. This reduction in $\Delta G_{\mathrm{rms}}$ is roughly observed: in the AFM
phase, $\Delta G_{\mathrm{rms}}=0.0037\>e^{2}/h$ (Device A,
$0.0048\>e^{2}/h$); in the cAFM phase, $\Delta
G_{\mathrm{rms}}=0.0032\>e^{2}/h$ (Device A, $0.0043\>e^{2}/h$); in the FM
phase, $\Delta G_{\mathrm{rms}}=0.0030\>e^{2}/h$ (Device A,
$0.0049\>e^{2}/h$).

Figure S12: Reproduction of main text Figure 3 for data acquired on Device B’s
top longitudinal contact pair $V_{xxt}$. This data was acquired simultaneously
to that shown in Figure 2. (a-c) $\Delta G_{xx}$ as a function of applied
field for sweeps up (black) and down (grey) in the (a) AFM, (b) cAFM, and (c)
FM phases. For each magnetic phase, field was swept back and forth across a 1
T range with 2 mT steps in immediately subsequent runs. Data was acquired with
a 2 nA ac bias current. (d-f) Correlation $R$ between each separate pair of
$\Delta G_{xx}$ traces within the (d) AFM, (e) cAFM, and (f) FM phases.

Supplemental References
[S1] L. Tai et al., ACS Nano, 2022, 16, 17336.
[S2] Q. Jiang et al., Phys. Rev. B, 2021, 103, 20, 205111.
[S3] C. X. Trang et al., ACS Nano, 2021, 15, 8, 13444–13452.
[S4] Y. Aharonov and D. Bohm, Phys. Rev., 1959, 115, 3, 485–491.


$\mathrm{Sh}_{K}=\mathrm{Sh}_{K}(\widetilde{G},\\{h_{\widetilde{G}}\\})$ is of
dimension $n-1$ and has a canonical model over $\mathrm{Spec}\,E$. It has the
following moduli interpretation, see [RSZ21, Definition 3.3] or [LZ22a,
§11.2]. For a locally noetherian $E$-scheme $S$, $\mathrm{Sh}_{K}(S)$ is the
groupoid of tuples $(A_{0},\iota_{0},\lambda_{0},A,\iota,\lambda,\bar{\eta})$
satisfying

  1. (1)

$(A_{0},\iota_{0},\lambda_{0})\in\mathcal{M}_{0}^{\xi}(S)$, see (H1) below;

  2. (2)

$A$ is an abelian scheme over $S$;

  3. (3)

$\iota:F\rightarrow\operatorname{End}^{0}(A)$ is an action of $F$ on $A$ up to
isogeny satisfying the Kottwitz signature condition
$\\{(r_{\phi},r_{\bar{\phi}})_{\phi\in\Phi}\\}$:

(10.1)
$\mathrm{charpol}(\iota(a)\mid\operatorname{Lie}A)=\prod_{\phi\in\Phi}(T-\phi(a))^{r_{\phi}}\cdot(T-\bar{\phi}(a))^{r_{\bar{\phi}}}$

for any $a\in F$;

  4. (4)

$\lambda:A\rightarrow A^{\vee}$ is a quasi-polarization whose Rosati
involution induces the automorphism given by the nontrivial Galois
automorphism of $F/F_{0}$ via $\iota$;

  5. (5)

$\bar{\eta}$ is a $K$-orbit of isometries of
$\mathbb{A}_{F,f}/\mathbb{A}_{F_{0},f}$-Hermitian modules

$\bar{\eta}:\hat{V}(A_{0},A)\rightarrow V\otimes_{F}\mathbb{A}_{F,f}.$

Here

$\hat{V}(A_{0},A)=\operatorname{Hom}_{\mathbb{A}_{F,f}}(\hat{V}(A_{0}),\hat{V}(A)),$

where $\hat{V}(A)$ (resp. $\hat{V}(A_{0})$) is the rational Tate-module of $A$
(resp. $A_{0}$) and $\hat{V}(A_{0},A)$ is endowed with a Hermitian form:

$h(x,y)=\lambda_{0}^{-1}\circ y^{\vee}\circ\lambda\circ
x\in\operatorname{End}_{\mathbb{A}_{F,f}}(\hat{V}(A_{0}))=\mathbb{A}_{F,f}.$

###  10.2. Integral model

Assume that $K_{Z^{\mathbb{Q}}}\subseteq Z^{\mathbb{Q}}(\mathbb{A}_{f})$ is
the unique maximal open compact subgroup. Assume that
$K_{G}=\prod_{v}K_{G,v}\subset G(\mathbb{A}_{f})$ where $v$ runs over finite
places of $F_{0}$. Let $(m_{v})_{v}$ be a collection of integers $m_{v}\geq 0$
indexed by finite places of $F_{0}$ such that $m_{v}=0$ for all but finitely
many places and $m_{v}=0$ for all nonsplit places $v$. Let $\Lambda$ be an
$O_{F}$-lattice of $V$. Assume that for any finite place $v$ of $F_{0}$ (write
$p$ its residue characteristic), the following conditions are satisfied.

  1. (G0)

If $p=2$, then $v$ is unramified in $F$.

  2. (G1)

If $v$ is inert in $F$ and $V_{v}$ is split, then $\Lambda_{v}\subset V_{v}$
is self-dual and $K_{G,v}$ is the stabilizer of $\Lambda_{v}$. If $v$ is
further ramified over $p$, then the subset $\\{\phi\in\Phi:\phi\text{ induces
}w\\}\subseteq\operatorname{Hom}(F_{w},\overline{\mathbb{Q}}_{p})$ is the
pullback of a CM type
$\Phi^{\mathrm{ur}}\subseteq\operatorname{Hom}(F_{w}^{\mathrm{ur}},\overline{\mathbb{Q}}_{p})$
of $F_{w}^{\mathrm{ur}}$. Here $w$ is the place of $F$ above $v$ and
$F_{w}^{\mathrm{ur}}$ is the maximal subfield of $F_{w}$ unramified over
$\mathbb{Q}_{p}$.

  3. (G2)

If $v$ is inert in $F$ and $V_{v}$ is nonsplit, then $v$ is unramified over
$p$ and $\Lambda_{v}\subset V_{v}$ is almost self-dual, i.e.,
$\Lambda_{v}^{\sharp}/\Lambda_{v}$ is a $1$ dimensional space over the residue
field of $F_{w}$. Moreover $K_{G,v}$ is the stabilizer of $\Lambda_{v}$.

  4. (G3)

If $v$ is ramified in $F$, then $v$ is unramified over $p$ and
$\Lambda_{v}\subseteq V_{v}$ is unimodular.

  5. (G4)

If $v$ is split in $F$, then
$\mathrm{U}(V)(F_{0,v})\cong\mathrm{GL}_{n}(F_{0,v})$ and we assume
$\Lambda_{v}\subset V_{v}$ is self-dual. If $m_{v}=0$, let $K_{G,v}$ be the
stabilizer of $\Lambda_{v}$. If $m_{v}>0$, further assume that all places of
$E$ above $v$ satisfies condition D in [LZ22a, §11.5] and let $K_{G,v}$ be the
principal congruence subgroup modulo $\pi_{v}^{m_{v}}$ inside
$\mathrm{GL}_{n}(F_{0,v})$ where $\pi_{v}$ is a uniformizer of $F_{0,v}$.

Assume that $K=K_{Z^{\mathbb{Q}}}\times K_{G}\subset\widetilde{G}$. Define the
moduli functor $\mathcal{M}_{K}$ as follows. For a locally noetherian
$O_{E}$-scheme $S$, $\mathcal{M}_{K}(S)$ is the groupoid of tuples
$(A_{0},\iota_{0},\lambda_{0},A,\iota,\lambda,\mathcal{F},\eta)$ such that

  1. (1)

$A$ (resp. $A_{0}$) is an abelian scheme over $S$;

  2. (2)

$\iota$ (resp. $\iota_{0}$) is an action of $O_{F}$ on $A$ (resp. $A_{0}$)
satisfying the Kottwitz condition (10.1) for signature
$\\{(r_{\phi},r_{\bar{\phi}})_{\phi\in\Phi}\\}$ (resp.
$\\{(1,0)_{\phi\in\Phi}\\}$);

  3. (3)

$\lambda$ (resp. $\lambda_{0}$) is a polarization of $A$ (resp. $A_{0}$) whose
Rosati involution induces the automorphism given by the nontrivial Galois
automorphism of $F/F_{0}$ via $\iota$ (resp. $\iota_{0}$);

  4. (4)

$\mathcal{F}$ is locally a direct summand $\mathcal{O}_{S}$-submodule of
$\operatorname{Lie}A$ which is stable under the $O_{F}$-action. Moreover
$O_{F}$ acts on $\mathcal{F}$ by the structural morphism and on
$\operatorname{Lie}A/\mathcal{F}$ by the Galois conjugate of the structural
morphism.

  5. (5)

$\eta=\prod_{w}\eta_{w}$ where $w$ runs over all places of $F$ above a split
place $v$ of $F_{0}$ such that $m_{v}>0$. For each such $w$, $\eta_{w}$ is a
finite $\mathcal{O}_{F,w}$-linear homomorphisms of finite flat group schemes

$\eta_{w}:\pi_{w}^{-m_{v}}\Lambda_{w}/\Lambda_{w}\rightarrow\operatorname{Hom}_{\mathcal{O}_{F,w}}(A_{0}[w^{m_{v}}],A[w^{m_{v}}]),$

where $\pi_{w}$ is a uniformizer of $F_{w}$. See [RSZ20, §4.3] for more
details.

We further require the following conditions to be satisfied.

  1. (H1)

$(A_{0},\iota_{0},\lambda_{0})\in\mathcal{M}_{0}^{\xi}$ where $M_{0}^{\xi}$ is
an integral model of
$\mathrm{Sh}_{K_{Z^{\mathbb{Q}}}}(Z^{\mathbb{Q}},h_{Z^{\mathbb{Q}}})$
depending on the choice of a similarity class $\xi$ of $1$ dimensional
$F/F_{0}$-Hermitian spaces, see [HSY21, §12.2].

  2. (H2)

For each finite place $v$ of $F_{0}$, $\lambda$ induces a polarization
$\lambda_{v}$ on the $p$-divisible group $A[v^{\infty}]$. We require
$\ker\lambda_{v}\subseteq A[\iota(\varpi_{v})]$ and is of rank equal to the
size of $\Lambda_{v}^{\sharp}/\Lambda_{v}$, where $\varpi_{v}$ is a
uniformizer of $F_{0,v}$.

  3. (H3)

For each place $v$ of $F_{0}$, we require the _sign condition_ and _Eisenstein
condition_ as explained in [RSZ20, §4.1]. We remark that the sign condition
holds automatically when $v$ is split in $F$, and the Eisenstein condition
holds automatically when the places of $F$ above $v$ are unramified over $p$.

  4. (H4)

We impose the _Krämer condition_ on $\mathcal{F}$ as explained in [HSY21,
Condition (iii) §12.2].

A morphism
$(A_{0},\iota_{0},\lambda_{0},A,\iota,\lambda,\mathcal{F},\eta)\rightarrow(A^{\prime}_{0},\iota^{\prime}_{0},\lambda^{\prime}_{0},A^{\prime},\iota^{\prime},\lambda^{\prime},\mathcal{F}^{\prime},\eta^{\prime})$
is a pair $(f_{0}:A_{0}\rightarrow A^{\prime}_{0},f:A\rightarrow A^{\prime})$
of $O_{F}$-linear isomorphism of abelian schemes over $S$ such that
$f^{*}(\lambda^{\prime})=\lambda$,
$f^{*}_{0}(\lambda^{\prime}_{0})=\lambda_{0}$,
$f_{*}(\mathcal{F})=\mathcal{F}^{\prime}$ and $f_{*}(\eta)=\eta^{\prime}$. Let
$\mathcal{V}_{\mathrm{ram}}$ (resp. $\mathcal{V}_{\mathrm{asd}}$) be the set
of finite places $v$ of $F_{0}$ such that $v$ is ramified in $F$ (resp. $v$ is
inert in $F$ and $\Lambda_{v}$ is almost self-dual). The moduli problem
$\mathcal{M}_{K}$ is representable by a Deligne-Mumford stack over $O_{E}$
which is regular and is semistable at all places of $E$ above
$\mathcal{V}_{\mathrm{ram}}\cup\mathcal{V}_{\mathrm{asd}}$. The generic fiber
of $\mathcal{M}_{K}$ is the Shimura variety $\mathrm{Sh}_{K}$.

###  10.3. Global main theorems

For a locally noetherian $O_{E}$-scheme $S$ and a point

$(A_{0},\iota_{0},\lambda_{0},A,\iota,\lambda,\mathcal{F},\eta)\in\mathcal{M}_{K}(S),$

define the _space of special homomorphisms_ to be

$\mathbb{V}(A_{0},A)\coloneqq\operatorname{Hom}_{O_{F}}(A_{0},A),$

a locally free $O_{F}$-module of finite rank. Then $\mathbb{V}(A_{0},A)$
carries a $O_{F}$-valued hermitian form: for $x,y\in\mathbb{V}(A_{0},A)$, the
pairing $(x,y)\in O_{F}$ is given by

$(A_{0}\xrightarrow{x}A\xrightarrow{\lambda}A^{\vee}\xrightarrow{y^{\vee}}A_{0}^{\vee}\xrightarrow{\lambda_{0}^{-1}}A_{0})\in\operatorname{End}_{O_{F}}(A_{0})=\iota_{0}(O_{F})\simeq
O_{F}.$

Let $m\geq 1$. Given an $m$-tuple
$\mathbf{x}=[x_{1},\ldots,x_{m}]\in\mathbb{V}(A_{0},A)^{m}$, define its
_fundamental matrix_ to be

$T(\mathbf{x})\coloneqq((x_{i},x_{j}))_{1\leq i,j\leq
m}\in\mathrm{Herm}_{m}(O_{F}),$

an $m\times m$ hermitian matrix over $O_{F}$.

Let $\mathbb{V}$ be the _incoherent_
$\mathbb{A}_{F}/\mathbb{A}_{F_{0}}$-hermitian space associated to $V$, namely
$\mathbb{V}$ is totally positive definite and $\mathbb{V}_{v}\cong V_{v}$ for
all finite places $v$. Let $\varphi_{K}\in\mathscr{S}(\mathbb{V}^{m}_{f})$ be
a $K$-invariant (where $K$ acts on $\mathbb{V}_{f}$ via the second factor
$K_{G}$) Schwartz function. We say $\varphi_{K}$ is admissible if
$\varphi_{K,v}=\mathbf{1}_{(\Lambda_{v})^{m}}$ at all $v$ nonsplit in $F$. Let
$T\in\mathrm{Herm}_{m}(F_{0})$ be a nonsingular $F/F_{0}$-hermitian matrix of
size $m$. Associated to $(T,\varphi_{K})$ we can define the _arithmetic
special cycles_ $\mathcal{Z}(T,\varphi_{K})$ over $\mathcal{M}_{K}$ ([LZ22a,
§14.3]) generalizing the $\mathcal{Z}(T)$ in [KR14] as follows. First we
consider a special Schwartz function of the form

(10.2)
$\varphi_{K}=(\varphi_{i})\in\mathscr{S}(\mathbb{V}^{m}_{f}),\quad\varphi_{i}=\mathbf{1}_{\Omega_{i}},\quad
i=1,\ldots,m,$

where $\Omega_{i}\subseteq\mathbb{V}_{f}$ is a $K$-invariant open compact
subset such that $\Omega_{i,v}=\Lambda_{v}$ for all finite $v$ such that $v$
is nonsplit in $F$. Given such a special Schwartz function $\varphi_{K}$ and
$T\in\mathrm{Herm}_{m}(O_{F})$, define $Z(T,\varphi_{K})$ over
$\mathrm{Sh}_{K}$ as follows. For a locally noetherian $E$-scheme $S$, define
$Z(T,\varphi_{K})(S)$ to be the groupoid of tuples
$(A_{0},\iota_{0},\lambda_{0},A,\iota,\lambda,\bar{\eta},\mathbf{x})$ where

  1. (1)

$(A_{0},\iota_{0},\lambda_{0},A,\iota,\lambda,\bar{\eta})\in\mathrm{Sh}_{K}(S)$,

  2. (2)

$\mathbf{x}=[x_{1},\ldots,x_{m}]\in\mathbb{V}(A_{0},A)^{m}$ with fundamental
matrix $T(\mathbf{x})=T$.

  3. (3)

$\eta(\mathbf{x}_{*})\in(\Omega_{i})\subseteq(\mathbb{V}_{f})^{m}$. Here
$\mathbf{x}_{*}\in\operatorname{Hom}_{F}(\hat{V}(A_{0}),\hat{V}(A))^{m}$ is
the element induced by $\mathbf{x}$.

Let $\mathcal{Z}(T,\varphi_{K})$ be the Zariski closure of $Z(T,\varphi_{K})$
in $\mathcal{M}_{K}$. The functor $S\mapsto\mathcal{Z}(T,\varphi_{K})(S)$ is
represented by a (possibly empty) Deligne–Mumford stack which is finite and
unramified over $\mathcal{M}$ ([KR14, Proposition 2.9]), and thus defines a
cycle $\mathcal{Z}(T,\varphi_{K})\in\mathrm{Z}^{*}(\mathcal{M}_{K})$. For a
general admissible Schwartz function
$\varphi_{K}\in\mathscr{S}(\mathbb{V}^{m}_{f})$, by extending
$\mathbb{C}$-linearly we obtain a cycle
$\mathcal{Z}(T,\varphi_{K})\in\mathrm{Z}^{*}(\mathcal{M})_{\mathbb{C}}$.

Now we assume $m=n$. Analogous to the local situation (1.1), we can define its
local arithmetic intersection numbers
$\operatorname{\mathrm{Int}}_{T,\nu}(\varphi_{K})$ for any place $\nu$ of $E$.
Let $v$ be the place of $F_{0}$ under $\nu$. By [HSY21, Lemma 15.3] (see also
[KR14, Proposition 2.22]), it suffices to consider the case when $v$ is
nonsplit in $F$. When $\varphi_{K}$ is of the form (10.2) and $v$ is finite,
define

(10.3)
$\operatorname{\mathrm{Int}}_{T,\nu}(\varphi_{K})\coloneqq\chi(\mathcal{Z}(T,\varphi_{K}),\mathcal{O}_{\mathcal{Z}(t_{1},\varphi_{1})}\otimes^{\mathbb{L}}\cdots\otimes^{\mathbb{L}}\mathcal{O}_{\mathcal{Z}(t_{n},\varphi_{n})})\cdot\log
q_{\nu},$

where $q_{\nu}$ denotes the size of the residue field $k_{\nu}$ of $E_{\nu}$,
$\mathcal{O}_{\mathcal{Z}(t_{i},\varphi_{i})}$ denotes the structure sheaf of
the Kudla–Rapoport divisor $\mathcal{Z}(t_{i},\varphi_{i})$,
$\otimes^{\mathbb{L}}$ denotes the derived tensor product of coherent sheaves
on $\mathcal{M}_{K}$, $\chi$ denotes the Euler–Poincaré characteristic. For a
general admissable function $\varphi_{K}$, we can extend the definition
$\mathbb{C}$-linearly. Using the star product of Kudla’s Green functions, we
can also define its local arithmetic intersection number
$\operatorname{\mathrm{Int}}_{T,v}(\mathsf{y},\varphi_{K})$ at infinite places
([LZ22a, §15.3]), which depends on a parameter
$\mathsf{y}\in\mathrm{Herm}_{n}(F_{0,\infty})_{>0}$ where
$F_{0,\infty}=F_{0}\otimes_{\mathbb{Q}}\mathbb{R}$. Combining all the local
arithmetic numbers together, define the _global arithmetic intersection
number_ , or the _arithmetic degree_ of the special cycle
$\mathcal{Z}(T,\varphi_{K})$ in the arithmetic Chow group of
$\mathcal{M}_{K}$,

$\widehat{\deg}_{T}(\mathsf{y},\varphi_{K})\coloneqq\sum_{\nu\nmid\infty}\operatorname{\mathrm{Int}}_{T,v}(\varphi_{K})+\sum_{\nu\mid\infty}\operatorname{\mathrm{Int}}_{T,v}(\mathsf{y},\varphi_{K}).$

######  Theorem 10.1.

Let $\mathrm{Diff}(T,\mathbb{V})$ be the set of places $v$ such that
$\mathbb{V}_{v}$ does not represent $T$. Let $T\in\mathrm{Herm}_{n}(F_{0})$ be
nonsingular such that $\mathrm{Diff}(T,\mathbb{V})=\\{v\\}$ where $v$ is
nonsplit in $F$ and not above 2. Then

$\widehat{\deg}_{T}(\mathsf{y},\varphi_{K})q^{T}=c_{K}\cdot\partial\mathrm{Eis}_{T}(\mathsf{z},\varphi_{K}),$

where $q^{T}\coloneqq\psi_{\infty}(\operatorname{Tr}T\mathsf{z})$,
$c_{K}=\frac{(-1)^{n}}{\operatorname{vol}(K)}$ is a nonzero constant
independent of $T$ and $\operatorname{vol}(K)$ is the volume of $K$ under a
suitable Haar measure. Finally,
$\partial\mathrm{Eis}_{T}(\mathsf{z},\varphi_{K})$ is the $T$-the coefficient
of the modified central derivative of Eisenstein series in (1.11)

###### Proof.

When $v$ is finite and
$v\notin\mathcal{V}_{\mathrm{ram}}\cup\mathcal{V}_{\mathrm{asd}}$, the theorem
is proved in [LZ22a, Theorem 13.6]. For $v\in\mathcal{V}_{\mathrm{asd}}$, our
definition of $\operatorname{\mathrm{Int}}_{T,\nu}(\varphi_{K})$ differs from
that of [LZ22a, (13.5.0.14)]. Correspondingly on the analytic side, our
definition of $\partial\mathrm{Eis}_{T}(\mathsf{z},\varphi_{K})$ is also
modified, see (1.9) and (1.10). However using [LZ22a, Theorem 10.5.1] instead
of [LZ22a, Theorem 10.3.1], the proof of [LZ22a, Theorem 13.6] works the same
way in this case. When $v$ is infinite, the theorem is proved in [Liu11,
Theorem 4.17,4.20] and independently in [GS19, Theorem 1.1.2]. When $v$ is
finite and $v\in\mathcal{V}_{\mathrm{ram}}$, the theorem is a corollary of
Theorem 9.7 and can be proved in the same way as [HSY21, Theorem 15.3]. ∎

We say $\varphi_{v}\in\mathscr{S}(\mathbb{V}_{v}^{n})$ is _nonsingular_ if its
support lies in $\\{\mathbf{x}\in\mathbb{V}_{v}^{n}:\det T(\mathbf{x})\neq
0\\}$, see [LZ22a, §12.3] or [Liu11, Proposition 2.1].

######  Theorem 10.2.

Assume that $F/F_{0}$ is split at all places above 2. Further assume that
$\varphi_{K}$ is nonsingular at two places split in $F$. Then

$\widehat{\deg}(\mathsf{z},\varphi_{K})=c_{K}\cdot\partial\mathrm{Eis}(\mathsf{z},\varphi_{K}),$

where $\widehat{\deg}(\mathsf{z},\varphi_{K})$ is defined in (1.12). In
particular, $\widehat{\deg}(\mathsf{z},\varphi_{K})$ is a nonholomorphic
hermitian modular form of genus $n$.

###### Proof.

The theorem can be derived from Theorem 10.1 by the same way as [LZ22a,
Theorem 15.5.1]. ∎

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