# Commensurability oscillations in the Hall resistance of unidirectional
lateral superlattices

Akira Endo<EMAIL_ADDRESS>The Institute for Solid State Physics,
The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8581, Japan
Shingo Katsumoto  The Institute for Solid State Physics, The University of
Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8581, Japan  Yasuhiro Iye  The
Institute for Solid State Physics, The University of Tokyo, 5-1-5 Kashiwanoha,
Kashiwa, Chiba 277-8581, Japan

###### Abstract

We have observed commensurability oscillations (CO) in the Hall resistance
$R_{yx}$ of a unidirectional lateral superlattice (ULSL). The CO, having small
amplitudes ($\sim$ 1 $\Omega$) and being superposed on a roughly three-orders
of magnitude larger background, are obtained by directly detecting the
difference in $R_{yx}$ between the ULSL area and the adjacent unmodulated two-
dimensional electron gas area, and then extracting the odd part with respect
to the magnetic field. The CO thus obtained are compared with a theoretical
calculation and turn out to have the amplitude much smaller than the
theoretical prediction. The implication of the smaller-than-predicted CO in
$R_{yx}$ on the thermoelectric power of ULSL is briefly discussed.

##  I Introduction

Commensurability oscillations (CO) have been arguably one of the best-known
magnetoresistance phenomena in mesoscopic systems since their discovery in
1989 [1, 2]. They were uncovered in a unidirectional lateral superlattice
(ULSL), a two-dimensional electron gas (2DEG) subjected to a weak one-
dimensional (1D) periodic modulation $V(x)$ of the electrostatic potential.
The most prominent oscillations were observed in the magnetoresistance
$R_{xx}$ along the modulation, with the minima taking place at the flat-band
conditions,

$\frac{2R_{\mathrm{c}}}{a}=n-\frac{1}{4},\hskip 34.1433pt(n=1,2,3,...),$ (1)

where $a$ is the period of $V(x)$ and $R_{\mathrm{c}}=\hbar
k_{\mathrm{F}}/(e|B|)$ is the cyclotron radius, with
$k_{\mathrm{F}}=\sqrt{2\pi n_{e}}$ the Fermi wavenumber, $n_{e}$ the electron
density, and $e$ the elementary charge. We assume a sinusoidal modulation
$V(x)=V_{0}\cos(2\pi x/a)$ throughout the paper. The magnetic field $B$ is
applied perpendicular ($\parallel z$ axis) to the 2DEG plane ($x$-$y$ plane,
see Fig. 1). Oscillations were also observed in the transverse direction
$R_{yy}$, albeit with much smaller amplitudes and taking maxima instead of
minima at Eq. (1) [1]. Soon after the discovery, a pictorial explanation
invoking the $\bm{E}\times\bm{B}$ drift velocity of semiclassical cyclotron
orbits were presented [3], which captures the physics behind the dominant
mechanism (band contribution, ascribed to the modulation of the Landau-band
dispersion and hence of the group velocity) generating the oscillations in
$R_{xx}$. However, full understanding of CO in a ULSL, including the
oscillations in $R_{yy}$, requires [4] quantum mechanical theories [5, 6, 7,
8], in which additional contribution due to the modulation of the density of
states (collisional contribution) is implemented.

Although occasionally overlooked, the theories [5, 6, 7, 8] predict the
presence of CO also in the Hall resistance $R_{yx}$, resulting from the
collisional contribution as is the case in $R_{yy}$. To the knowledge of the
present authors, however, an unambiguous experimental observation of CO in the
Hall resistance of a ULSL has never been reported for nearly three decades
after the theoretical predictions 111By contrast, CO in the Hall resistance
was reported for two-dimensional antidot square superlattices as early as in
1996 [25]. We surmise that the observation has been hampered mainly by two
obstacles: the smallness of the amplitudes and unintentional mixing of the
$R_{xx}$ component into the measurement. First, the amplitude of the
oscillatory part $\delta R_{yx}$ is predicted to be of the order of 1
$\Omega$, accounting for only $\sim$0.1% of the total $R_{yx}$ $\gtrsim$ 1
k$\Omega$ $\gg$ $R_{xx}$. The signal from $\delta R_{yx}$ can thus readily be
buried in the noise level for the measurement setup with sensitivity adjusted
to measure $R_{yx}$. Second, due to inevitable imperfectness of the Hall bar
device, e.g., the misalignment of voltage probes, a small portion of $R_{xx}$
can inadvertently mix into the measured $R_{yx}$. The effect of the mixed
$R_{xx}$ is totally insignificant in the usual measurement of $R_{yx}$, since
$R_{xx}$$\ll$$|R_{yx}|$ for $B\gtrsim 0.1$ T in high-mobility 2DEGs. Focusing
on the oscillatory parts, however, parasitic $\delta R_{xx}$ component can
easily outweigh the intrinsic $\delta R_{yx}$, since the amplitudes of the
former is about two orders of magnitude larger than the predicted amplitudes
of the latter. Here and in what follows, we denote the oscillatory part of a
quantity $X$ by $\delta X$, and the difference in $X$ with and without $V(x)$
by $\Delta X$. The latter can contain non-oscillatory part induced by $V(x)$
in addition to $\delta X$.

In the present study, we circumvent these problems by employing simple
techniques: directly measuring the excess Hall resistivity $\Delta R_{yx}$
attributable to $V(x)$ and then extracting the antisymmetric part with respect
to the magnetic field. The $\delta R_{yx}$ thus obtained is compared with
$\delta R_{yx}$ numerically calculated from the formula for the conductivity
$\sigma_{yx}$ expressed in terms of summation over the Landau indices given in
[8]. To gain transparent insight into the behavior of $\sigma_{yx}$ and to
efficiently extract the oscillatory part, we also deduce an analytic
asymptotic expression that approximates the $\sigma_{yx}$ quite well. We find
that the observed $\delta R_{yx}$ is much smaller than the theoretical
prediction, even if we consider damping of the oscillations due to small angle
scatterings neglected in the original theory.

The present study is partly motivated by rather counterintuitive isotropic
behavior of the CO in the Seebeck coefficients (diagonal components of the
thermopower tensor) of a ULSL predicted in [8], which we have recently noticed
[10] to be strongly related via the Mott relation to the CO in the Hall
conductivity. As we will see, the smallness of $R_{yx}$ found in the present
study casts doubt on the isotropic behavior.

Figure 1: Schematic drawing of the Hall bar device containing, in series,
areas with (ULSL) and without (p2DEG) the periodic potential modulation
$V(x)$. Wiring for directly measuring the excess Hall resistance, $\Delta
R_{yx}=R_{yx,\mathrm{U}}-R_{yx,\mathrm{P}}$, introduced by $V(x)$ is also
shown.

##  II Experimental details and results

Figure 1 illustrates the schematics of the Hall bar device used in the present
study. The device contains a modulated area (ULSL) and a plain 2DEG (p2DEG)
area in series, with the voltage probes to measure the magnetoresistance
$R_{xx,\mathrm{U/P}}$ and the Hall resistance $R_{yx,\mathrm{U/P}}$ attached
to both areas, where the subscript U and P represent ULSL and p2DEG areas,
respectively. The device was fabricated from a conventional GaAs/AlGaAs 2DEG
wafer having the mobility $\mu=70$ m2/(Vs) and the electron density
$n_{e}=2.1\times 10^{15}$ m-2. Modulation $V(x)$ with the period $a=184$ nm
was introduced by placing a grating of negative-tone electron-beam resist on
the surface of the ULSL area [11], exploiting the strain-induced piezoelectric
effect [11, 12]. All the measurements in this study were performed at 4.2 K.

Figure 2: (a) Hall resistances $R_{yx,\mathrm{U}}$ and $R_{yx,\mathrm{P}}$
(blue lines, left axis), and magnetoresistances $R_{xx,\mathrm{U}}$ and
$R_{xx,\mathrm{P}}$ (red lines, right axis) measured in the ULSL area (solid
lines) and in the p2DEG area (dashed lines). (b) Excess Hall resistance
$\Delta R_{yx}$ obtained by the arrangement depicted in Fig. 1 (blue line,
left axis) and excess magnetoresistance $\Delta R_{xx}$ obtained by taking the
difference between $R_{xx,\mathrm{U}}$ and $R_{xx,\mathrm{P}}$ shown in (a).
Vertical dotted lines indicate the locations of the $n$-th flat-band condition
given by Eq. (1).

In Fig. 2(a), we plot $R_{yx,\mathrm{U/P}}$ and $R_{xx,\mathrm{U/P}}$ measured
employing standard low-frequency ($f=73$ Hz) ac lock-in technique with the
current $I=100$ nA. $R_{xx,\mathrm{U}}$ exhibits prominent CO with the minima
occurring at the positions given by Eq. (1). Small-amplitude oscillations
observed at higher magnetic-field regions ($|B|\gtrsim 0.5$ T) both in
$R_{xx,\mathrm{U}}$ and $R_{xx,\mathrm{P}}$ are the Shubnikov-de Haas (SdH)
oscillations. On the other hand, $R_{yx,\mathrm{U/P}}$ appears as a
featureless line in the plots. To extract the component deriving from $V(x)$,
we take the differences $\Delta R_{yx}=R_{yx,\mathrm{U}}-R_{yx,\mathrm{P}}$
and $\Delta R_{xx}=R_{xx,\mathrm{U}}-R_{xx,\mathrm{P}}$, and plot them in Fig.
2(b). Since the difference is large for $R_{xx}$, $\Delta R_{xx}$ can be
obtained reliably by simply subtracting the two traces in Fig. 2(a)
numerically. We can see that the SdH oscillations are partially cancelled out
in $\Delta R_{xx}$ 222Imperfectness of the cancellation are mainly
attributable to the modulation of the SdH amplitude by $V(x)$. See., e.g.,
[26]. In $R_{yx}$, by contrast, minuscule difference ($\sim$ $\Omega$)
unobservable in Fig. 2(a) needs to be drawn out from orders of magnitude
larger ($\sim$ k$\Omega$) values. To do this with sufficient signal-to-noise
(S/N) ratio, we collect the excess Hall resistance $\Delta R_{yx}$ directly,
employing the arrangement depicted in Fig. 1: the Hall voltages from ULSL and
p2DEG areas are first amplified ($\times$100) by separate differential
preamplifiers 333LI-75A, NF Corporation, and then their outputs are plugged
into differential input of a lock-in amplifier 444LI 5660, NF Corporation. The
input voltage range of the lock-in amplifier can thus be adjusted to the
minimum range that encompasses the small difference voltage, which serves to
significantly improve the S/N ratio. As can be seen in Fig. 2(b), $\Delta
R_{yx}$ obtained by this method clearly shows oscillations corresponding to
both CO and partially cancelled SdH oscillations (or, more precisely,
incipient quantum Hall plateaus). In the present paper, we focus on the CO. A
notable feature is the asymmetry between $B>0$ and $B<0$ regions. In both
regions, maxima are observed roughly at the flatband conditions Eq. (1).
However, the amplitudes of the oscillations are much larger in $B>0$. As
mentioned earlier, the observed CO are considered to be composed of two
components: intrinsic $\delta R_{yx}$ and parasitic $\delta R_{xx}$. Since
$\delta R_{yx}$ is an odd function of $B$ while $\delta R_{xx}$ is an even
function, the two components are superposed either destructively ($B<0$ in the
present case) or constructively ($B>0$), depending on the sign of the magnetic
field. This explains the observed asymmetry in the CO amplitudes. We have
measured several ULSL devices in addition to the one shown in Fig. 2. Similar
asymmetry was observed for all of them.

Figure 3: Even ($\Delta R_{yx,\mathrm{even}}$, thin solid line) and odd
($\Delta R_{yx,\mathrm{odd}}$, thick solid line) parts of the excess Hall
resistance $\Delta R_{yx}$. Linear component $\gamma B$ (with $\gamma=18.8$
$\Omega$/T) is subtracted from $\Delta R_{yx,\mathrm{odd}}$ for clarity.
Linear combination of $R_{xx,\mathrm{U}}$ and $R_{xx,\mathrm{P}}$ (with
$\alpha=-0.0197$ and $\beta=0.0212$), which emulates the parasitic $R_{xx}$
component contained in the measured Hall resistance, is also shown (thin
dashed line). Vertical dotted lines indicate the locations of the $n$-th flat-
band condition given by Eq. (1).

In order to separate the two components, we take the even $\Delta
R_{yx,\mathrm{even}}$ and the odd $\Delta R_{yx,\mathrm{odd}}$ parts of
$\Delta R_{yx}$, $\Delta R_{yx,\mathrm{even}}(B)=[\Delta R_{yx}(B)+\Delta
R_{yx}(-B)]/2$ and $\Delta R_{yx,\mathrm{odd}}(B)=[\Delta R_{yx}(B)-\Delta
R_{yx}(-B)]/2$, corresponding to the parasitic and the intrinsic components,
respectively, and plot them in Fig. 3. Noting that the Hall probes in both
ULSL and p2DEG areas generally can pick up the corresponding parasitic
$R_{xx,\mathrm{U/P}}$ components independently, possibly with differing
weights, we can expect that the parasitic component can be expressed by the
linear combination $\alpha R_{xx,\mathrm{U}}+\beta R_{xx,\mathrm{P}}$ with
small values of $|\alpha|$ and $|\beta|$. By properly selecting $\alpha$ and
$\beta$, with special care to reproduce the oscillatory part due to the CO
(see also Fig. 4(a)), fairly good agreement can be achieved between $\alpha
R_{xx,\mathrm{U}}+\beta R_{xx,\mathrm{P}}$ and the observed $\Delta
R_{yx,\mathrm{even}}$, supporting the interpretation on the origin of $\Delta
R_{yx,\mathrm{even}}$. This confirms that the remnant $\Delta
R_{yx,\mathrm{odd}}$ is the intrinsic CO in $R_{yx}$, the target we are
seeking after in the present study. In the plot of the odd part, we subtracted
a linear term $\gamma B$, which is attributable to the small difference in the
electron density, $\Delta n_{e}\simeq-1.3\times 10^{13}$ m-2, between the ULSL
and the p2DEG areas.

Figure 4: (a) Oscillatory parts $\delta R_{xx,\mathrm{U}}$ (thick solid line)
and $\delta\Delta R_{yx,\mathrm{even}}$ (thin dashed line) obtained by
subtracting the slowly-varying background from $R_{xx,\mathrm{U}}$ and $\Delta
R_{yx,\mathrm{even}}$ shown in Figs. 2(a) and 3, respectively. The latter is
scaled by the factor $1/\alpha=-50.6$. Calculated $\delta R_{xx}$ (Eq. (4)
with $V_{0}=0.35$ meV and $\mu_{\mathrm{w}}=8.2$ m2/(Vs)) is also shown (thin
solid line), exhibiting excellent agreement with $\delta R_{xx,\mathrm{U}}$
(apart from SdH oscillations at $B\gtrsim 0.6$ T). (b) Oscillatory parts
$\delta\Delta R_{yx,\mathrm{odd}}$ (thick solid line), extracted from the
experimentally obtained $\Delta R_{yx,\mathrm{odd}}$ shown in Fig. 3 by
subtracting the slowly-varying background. $\delta R_{yx}$ calculated with Eq.
(16) (thin solid line) and modified Eq. (16) with the damping factor
$A[\pi/(\mu_{\mathrm{w}}B)]$ multiplied to $\delta\sigma_{yx}^{\mathrm{A}}$
(dashed line), using the sample parameters deduced in (a), are also plotted.
Thin dashed curve represents basically the same calculation but with
$\mu_{\mathrm{w}}$ in the damping factor replaced by
$\mu_{\mathrm{w}}^{*}=\mu_{\mathrm{w}}/2$. Vertical dotted lines indicate the
locations of the $n$-th flat-band condition given by Eq. (1).

##  III Comparison with Theoretical Calculations

###  III.1 Deducing superlattice parameters from commensurability oscillations
in the magnetoresistance

The next step is to compare the observed $\delta R_{yx}$ with the theoretical
prediction. Before discussing $\delta R_{yx}$, however, we briefly review
well-documented behavior of $\delta R_{xx}$ [8, 11], from which we draw out
parameters characterizing our ULSL. As mentioned earlier, two different
mechanisms, the band and the collisional contributions, are responsible for
CO. Asymptotic analytic expressions for the oscillatory parts of the
conductivity, valid at low magnetic fields where large numbers of Landau
levels are occupied [8], are given for the two contributions as,

$\delta\sigma_{yy}^{\mathrm{band}}=\frac{\sigma_{0}V_{0}^{2}}{E_{\mathrm{F}}\hbar\omega_{\mathrm{c}}ak_{\mathrm{F}}}A\left(\frac{\pi}{\mu_{\mathrm{w}}B}\right)A\left(\frac{T}{T_{a}}\right)\sin{r_{\mathrm{c}}},$
(2)

and

$\delta\sigma_{xx}^{\mathrm{col}}=-\frac{3\sigma_{0}V_{0}^{2}ak_{\mathrm{F}}}{8\pi^{2}E_{\mathrm{F}}\hbar\omega_{\mathrm{c}}(\mu
B)^{2}}A\left(\frac{\pi}{\mu_{\mathrm{w}}B}\right)A\left(\frac{T}{T_{a}}\right)\sin{r_{\mathrm{c}}},$
(3)

respectively, where $\sigma_{0}=en_{e}\mu$ is the conductivity at $B=0$,
$E_{\mathrm{F}}$ the Fermi energy, $\omega_{\mathrm{c}}=e|B|/m^{*}$ the
cyclotron angular frequency with $m^{*}$ the effective mass,
$r_{\mathrm{c}}\equiv 4\pi R_{\mathrm{c}}/a$,
$T_{a}\equiv\hbar\omega_{\mathrm{c}}ak_{\mathrm{F}}/(4\pi^{2}k_{\mathrm{B}})$,
and $A(x)\equiv x/\sinh x$. Although absent in the original theories [5, 6, 7,
8], an additional damping factor $A[\pi/(\mu_{\mathrm{w}}B)]$ accounting for
the effect of small angle scattering, with the value of $\mu_{\mathrm{w}}$
close to the quantum mobility $\mu_{\mathrm{q}}$ [16, 11], are contained in
Eqs. (2) and (3) in addition to the thermal damping factor $A(T/T_{a})$. More
detailed discussion on the factor $A[\pi/(\mu_{\mathrm{w}}B)]$ will be given
below. The resistivity tensor $\rho_{ij}$ ($i,j=x,y$) is obtained by inverting
the conductivity tensor $\sigma_{ij}$:
$\sigma_{xx}=\sigma_{xx}^{\mathrm{sc}}+\delta\sigma_{xx}^{\mathrm{col}}$,
$\sigma_{yy}=\sigma_{xx}^{\mathrm{sc}}+\delta\sigma_{xx}^{\mathrm{col}}+\delta\sigma_{yy}^{\mathrm{band}}$,
and $\sigma_{yx}=-\sigma_{xy}=\sigma_{yx}^{\mathrm{sc}}+\delta\sigma_{yx}$,
where $\sigma_{xx}^{\mathrm{sc}}=\sigma_{0}/(1+\mu^{2}B^{2})$ and
$\sigma_{yx}^{\mathrm{sc}}=\sigma_{0}\mu B/(1+\mu^{2}B^{2})$ are the
semiclassical conductivities for a p2DEG. Noting that $\mu B\gg 1$ and
$|\delta\sigma_{xx}^{\mathrm{band}}|\gg|\delta\sigma_{xx}^{\mathrm{col}}|$,
and using the relation $R_{xx}/R_{0}=\sigma_{0}\rho_{xx}$ with $R_{0}$
representing $R_{xx}$ at $B=0$, we obtain, to a good approximation,

$\frac{\delta R_{xx}}{R_{0}}=(\mu
B)^{2}\frac{\delta\sigma_{yy}^{\mathrm{band}}}{\sigma_{0}}$ (4)

and likewise $\delta R_{yy}/R_{0}=(\mu
B)^{2}\delta\sigma_{xx}^{\mathrm{col}}/\sigma_{0}$. Eq. (4) has been shown to
describe experimentally obtained CO extremely well [11]. This is confirmed in
Fig. 4(a), which shows $\delta R_{xx,\mathrm{U}}$ extracted from
$R_{xx,\mathrm{U}}$ in Fig. 2(a) by subtracting slowly varying background
following the protocol detailed in [11], along with $\delta R_{xx}$ in Eq. (4)
obtained by the fitting, employing $V_{0}$ and $\mu_{\mathrm{w}}$ as fitting
parameters. The fitting yields $V_{0}=0.35$ meV and $\mu_{\mathrm{w}}=8.2$
m2/(Vs). The value of $\mu_{\mathrm{w}}$ is close to $\mu_{\mathrm{q}}=8.6$
m2/(Vs) deduced from the SdH oscillations in $R_{xx,\mathrm{P}}$ plotted in
Fig. 2(a).

Figure 5: (a) The increment of the Hall conductivity due to $V(x)$, calculated
with the exact (thick solid line, $\Delta\sigma_{yx}$ in Eq. (8)) and the
approximate analytic (thin solid line, $\Delta\sigma_{yx}^{\mathrm{A}}$ in Eq.
(13)) formulas, using sample parameters of the ULSL in the present study. Non-
oscillatory background, $\Delta\sigma_{yx,\mathrm{bg}}^{\mathrm{A}}$ in Eq.
(14), of the approximate formula is plotted with thin dot-dashed line. The
increment deriving only from the first term in Eq. (9), namely,
$\Delta\sigma_{yx}^{\mathrm{A}(1)}=\sigma_{yx,\mathrm{bg}}^{\mathrm{A}(1)}+\delta\sigma_{yx}^{\mathrm{A}(1)}$
in Eq. (10a), is also plotted (dashed line). (b) Oscillatory parts calculated
from exact (thick solid line,
$\delta\sigma_{yx}=\Delta\sigma_{yx}-\Delta\sigma_{yx,\mathrm{bg}}^{\mathrm{A}}$)
and approximate (thin solid line, $\delta\sigma_{yx}^{\mathrm{A}}$ in Eq.
(15)) formulas. Three constituent terms of the oscillatory part are also
plotted separately, with thin dashed, thin dotted, and thick dashed lines
representing $\delta\sigma_{yx}^{\mathrm{A}(1)}$,
$\delta\sigma_{yx}^{\mathrm{A}(21)}$, and $\delta\sigma_{yx}^{\mathrm{A}(22)}$
in Eqs. (10c), (11c), and (12c), respectively. Vertical dotted lines indicate
the locations of the $n$-th flat-band condition given by Eq. (1).

###  III.2 Asymptotic analytic expressions for the Hall conductivity

Now we turn to the Hall component. We start with the expression of the Hall
conductivity in a ULSL presented in [8],

$\sigma_{yx}=\frac{2e^{2}}{h}\
\sum_{N=0}^{\infty}{(N+1)\int_{0}^{1}{\frac{f_{E_{\mathrm{F}}}\left(E_{N,\xi}\right)-f_{E_{\mathrm{F}}}\left(E_{N+1,\xi}\right)}{\left[1+\lambda_{N}\cos{\left(2\pi\xi\right)}\right]^{2}}}d\xi\
}$ (5)

with

$E_{N,\xi}=E_{N}+V_{0}\exp\left(-\frac{u}{2}\right)L_{N}\left(u\right)\cos{\left(2\pi\xi\right)}$
(6)

and

$\lambda_{N}=\frac{V_{0}}{\hbar\omega_{\mathrm{c}}}\exp\left(-\frac{u}{2}\right)L_{N+1}^{-1}(u),$
(7)

where
$f_{E_{\mathrm{F}}}(E)=\\{1+\exp[(E-E_{\mathrm{F}})/k_{\mathrm{B}}T]\\}^{-1}$
is the Fermi-Dirac distribution function, $N$ is the Landau index,
$E_{N}\equiv(N+1/2)\hbar\omega_{\mathrm{c}}$, $\xi\equiv x_{0}/a$ with $x_{0}$
the guiding center, $u\equiv 2\pi^{2}l^{2}/a^{2}$ with $l=\sqrt{\hbar/(e|B|)}$
the magnetic length, and $L_{N}(u)$ and $L_{N}^{M}(u)$ are the Laguerre and
the associated Laguerre polynomials. Note that Eq. (5) is valid only for
$B>0$. Since $\sigma_{yx}$ is an antisymmetric function with respect to $B$,
$\sigma_{yx}$ at $B<0$ is obtained by inverting the sign of Eq. (5). The
increment of $\sigma_{yx}$ introduced by the modulation,

$\Delta\sigma_{yx}=\sigma_{yx}(V_{0})-\sigma_{yx}(V_{0}=0),$ (8)

numerically calculated 555The upper limit of the summation was truncated at
$N=70$, which is large enough for the temperature and magnetic-field range
considered using Eq. (5) with the parameters in the present ULSL is plotted in
Fig. 5(a).

Since the behavior of $\sigma_{yx}$, notably the phase of the oscillations,
are not readily perceived from Eq. (5), we deduce an asymptotic analytic
expression, basically following the prescription taken for $\sigma_{xx}$ and
$\sigma_{yy}$ in [8]. Via the deriving procedure detailed in the Appendix, we
arrive at an approximate formula $\sigma_{yx}^{\mathrm{A}}\simeq\sigma_{yx}$,

$\sigma_{yx}^{\mathrm{A}}=\sigma_{yx}^{\mathrm{A}(1)}+\sigma_{yx}^{\mathrm{A}(21)}+\sigma_{yx}^{\mathrm{A}(22)},$
(9)

with

$\displaystyle\sigma_{yx}^{\mathrm{A}(1)}$
$\displaystyle=\nu\frac{e^{2}}{h}+\Delta\sigma_{yx}^{\mathrm{A}(1)}=\nu\frac{e^{2}}{h}+\sigma_{yx,\mathrm{bg}}^{\mathrm{A}(1)}+\delta\sigma_{yx}^{\mathrm{A}(1)}$
(10a)
$\displaystyle\sigma_{yx,\mathrm{bg}}^{\mathrm{A}(1)}=\nu\frac{e^{2}}{h}\frac{3}{2}{\lambda_{\mathrm{c}}}^{2}$
(10b)
$\displaystyle\delta\sigma_{yx}^{\mathrm{A}(1)}=-\nu\frac{e^{2}}{h}A\left(\frac{T}{T_{a}}\right)\frac{3}{2}{\lambda_{\mathrm{c}}}^{2}\sin{\left(r_{\mathrm{c}}+\delta_{\mathrm{F}}\right)},$
(10c) $\displaystyle\sigma_{yx}^{\mathrm{A}(21)}$
$\displaystyle=\sigma_{yx,\mathrm{bg}}^{\mathrm{A}(21)}+\delta\sigma_{yx}^{\mathrm{A}(21)}$
(11a)
$\displaystyle\sigma_{yx,\mathrm{bg}}^{\mathrm{A}(21)}=-\nu\frac{e^{2}}{h}\frac{ak_{\mathrm{F}}}{\pi}\sin{\left(\frac{\pi}{{ak}_{\mathrm{F}}}\right)}{\lambda_{\mathrm{c}}}^{2}\cos{\left(\frac{\delta_{\mathrm{F}}}{2}+\frac{\pi}{ak_{\mathrm{F}}}\right)}$
(11b)
$\displaystyle\delta\sigma_{yx}^{\mathrm{A}(21)}=-\nu\frac{e^{2}}{h}\frac{ak_{\mathrm{F}}}{\pi}\sin{\left(\frac{\pi}{{ak}_{\mathrm{F}}}\right)}A\left(\frac{T}{T_{a}}\right){\lambda_{\mathrm{c}}}^{2}\sin{\left(r_{\mathrm{c}}+\frac{\delta_{\mathrm{F}}}{2}-\frac{\pi}{{ak}_{\mathrm{F}}}\right)},$
(11c)

and

$\displaystyle\sigma_{yx}^{\mathrm{A}(22)}$
$\displaystyle=\sigma_{yx,\mathrm{bg}}^{\mathrm{A}(22)}+\delta\sigma_{yx}^{\mathrm{A}(22)}$
(12a)
$\displaystyle\sigma_{yx,\mathrm{bg}}^{\mathrm{A}(22)}=-\mathrm{sgn}(B)\frac{e^{2}}{h}\frac{ak_{\mathrm{F}}}{\pi}\cos{\left(\frac{\pi}{{ak}_{\mathrm{F}}}\right)}{\lambda_{\mathrm{c}}}^{2}\sin{\left(\frac{\delta_{\mathrm{F}}}{2}+\frac{\pi}{{ak}_{\mathrm{F}}}\right)}$
(12b)
$\displaystyle\delta\sigma_{yx}^{\mathrm{A}(22)}=\mathrm{sgn}(B)\frac{e^{2}}{h}\frac{ak_{\mathrm{F}}}{\pi}\cos{\left(\frac{\pi}{{ak}_{\mathrm{F}}}\right)}A\left(\frac{T}{T_{a}}\right){\lambda_{\mathrm{c}}}^{2}\cos{\left(r_{\mathrm{c}}+\frac{\delta_{\mathrm{F}}}{2}-\frac{\pi}{{ak}_{\mathrm{F}}}\right)},$
(12c)

where $\nu=n_{e}h/(eB)$ is the Landau-level filling factor ($\nu<0$ for $B<0$
by the definition), $\lambda_{\mathrm{c}}\equiv
2\sqrt{2/\pi}[V_{0}/(\hbar\omega_{c})][\pi/(ak_{\mathrm{F}})]{r_{\mathrm{c}}}^{-1/2}$,
$\delta_{\mathrm{F}}\equiv 2\cot^{-1}{r_{\mathrm{c}}}$, and $\mathrm{sgn}(x)$
represents the sign of $x$. By collecting the corresponding terms, we obtain
the increment of the conductivity, the non-oscillatory background of the
increment, and the oscillatory part,

$\Delta\sigma_{yx}^{\mathrm{A}}=\Delta\sigma_{yx}^{\mathrm{A}(1)}+\sigma_{yx}^{\mathrm{A}(21)}+\sigma_{yx}^{\mathrm{A}(22)},$
(13)
$\Delta\sigma_{yx,\mathrm{bg}}^{\mathrm{A}}=\sigma_{yx,\mathrm{bg}}^{\mathrm{A}(1)}+\sigma_{yx,\mathrm{bg}}^{\mathrm{A}(21)}+\sigma_{yx,\mathrm{bg}}^{\mathrm{A}(22)},$
(14)

and

$\delta\sigma_{yx}^{\mathrm{A}}=\delta\sigma_{yx}^{\mathrm{A}(1)}+\delta\sigma_{yx}^{\mathrm{A}(21)}+\delta\sigma_{yx}^{\mathrm{A}(22)},$
(15)

respectively. Figure 5(a) illustrates that the asymptotic analytic expression
Eq. (13) reproduces Eq. (8) quite well, except for the small-amplitude
oscillations at $B\gtrsim 0.6$ T resulting from the Landau quantization. This
allows us to use the approximate background
$\Delta\sigma_{yx,\mathrm{bg}}^{\mathrm{A}}$ to extract oscillatory part from
$\Delta\sigma_{yx}$ in Eq. (8). The oscillatory part thus obtained,
$\delta\sigma_{yx}=\Delta\sigma_{yx}-\Delta\sigma_{yx,\mathrm{bg}}^{\mathrm{A}}$,
is plotted in Fig. 5(b) along with $\delta\sigma_{yx}^{\mathrm{A}}$ in Eq.
(15).

The analytic expression lets us grasp the outline of the behavior of the
oscillations. Since $\delta\sigma_{yx}^{\mathrm{A}(1)}$,
$\delta\sigma_{yx}^{\mathrm{A}(21)}$, and $\delta\sigma_{yx}^{\mathrm{A}(22)}$
all oscillate with different phases depending on $B$ through
$\delta_{\mathrm{F}}$, the phase of the CO in $\sigma_{yx}$ is expected to
exhibit rather complicated behavior. We note, however, that
$\pi/(ak_{\mathrm{F}})$ ($=$0.148 in the present sample) is generally small
for experimentally achievable values of $a$, and that $\delta_{\mathrm{F}}$ is
also small ($\lesssim 0.4$ in the magnetic-field range where CO is observed)
and approaches 0 with decreasing $B$. Furthermore, at low magnetic fields
where $\nu$ is large, $\delta\sigma_{yx}^{\mathrm{A}(22)}$ becomes much
smaller than the other terms. The dominance of
$\delta\sigma_{yx}^{\mathrm{A}(1)}$ and $\delta\sigma_{yx}^{\mathrm{A}(21)}$,
combined with the smallness of $\pi/(ak_{\mathrm{F}})$ and
$\delta_{\mathrm{F}}$, indicates that the oscillation phase of
$\delta\sigma_{yx}$ is close to that of $\delta\sigma_{xx}^{\mathrm{col}}$ and
thus takes maximum at the flat band conditions Eq. (1), or equivalently, at
$r_{\mathrm{c}}=2\pi n-\pi/2$. The calculated $\delta\sigma_{yx}$ plotted in
Fig. 5(b) are seen to actually take maxima at the flatband conditions at low
magnetic fields. With the increase of the magnetic field, slight deviation of
the peak positions becomes apparent, mainly due to the increase of
$\delta_{\mathrm{F}}$ and of the relative importance of the third term
$\delta\sigma_{yx}^{\mathrm{A}(22)}$, whose oscillation phase differs from
$\delta\sigma_{yx}^{\mathrm{A}(21)}$ by $\pi/2$. Noting that
$(ak_{\mathrm{F}}/\pi)\sin{[\pi/(ak_{\mathrm{F}})]}\sim$ 1, the two dominant
terms are expected to have comparable oscillation amplitudes, which can also
be confirmed in Fig. 5(b).

###  III.3 Comparison between experimental and calculated commensurability
oscillations
in the Hall resistance

We obtain the Hall resistivity $\rho_{yx}$ ($=R_{yx}$ in a 2DEG) by inverting
the conductivity tensor, and find that the oscillatory part of the Hall
resistance $R_{yx}$ is given, considering
$|\delta\sigma_{yx}^{\mathrm{A}}|\ll|\sigma_{yx}|$, by

$\delta R_{yx}=\frac{1}{{\sigma_{0}}^{2}}\\{[(\mu
B)^{2}-1]\delta\sigma_{yx}^{\mathrm{A}}+\mu
B(2\delta\sigma_{xx}^{\mathrm{col}}+\delta\sigma_{yy}^{\mathrm{band}})\\}.$
(16)

The oscillations are dominated by the first term. The second term is
negligibly small since
$|\delta\sigma_{xx}^{\mathrm{col}}|\ll|\delta\sigma_{yy}^{\mathrm{band}}|$.
The third term, having the phase roughly opposite to the first term, serves to
reduce the oscillation amplitude. In Fig. 4(b), we compare experimentally
obtained oscillatory part $\delta\Delta R_{yx,\mathrm{odd}}$, extracted from
$\Delta R_{yx,\mathrm{odd}}$ shown in Fig. 3 by subtracting the slowly varying
background [11], with $\delta R_{yx}$ calculated by Eq. (16) using the sample
parameters deduced above from the analysis of $\delta R_{xx}$. The figure
shows that the observed amplitude of the CO is much smaller than the
theoretical prediction especially at lower magnetic fields, while the phase of
the oscillations is roughly in agreement.

It is well known that the scattering in a GaAs/AlGaAs 2DEG is predominantly
caused by remote ionized donors, for which scattering angles are generally
small [18]. Although the momentum relaxation is not significant for small
scattering angles, cyclotron orbits are disturbed regardless of the scattering
angle and thus the CO amplitudes are severely diminished even by the small-
angle scattering [16, 11]. The damping of the CO is more prominent for lower
magnetic fields where the circumference of the cyclotron orbit $2\pi
R_{\mathrm{c}}$ becomes large. The effect of small-angle scattering, which has
not been considered thus far for $\sigma_{yx}^{\mathrm{A}}$, can be
implemented by multiplying $A[\pi/(\mu_{\mathrm{w}}B)]$, following the recipe
applied for $\delta\sigma_{yy}^{\mathrm{band}}$ and
$\delta\sigma_{xx}^{\mathrm{col}}$ described above. Figure 4(b) reveals,
however, that the discrepancy between the amplitudes of the observed and the
calculated CO is still large even with the inclusion of the effect of the
small-angle scattering. Apparently, the theory overestimates the CO amplitudes
in the Hall resistance, possibly because $\delta R_{yx}$ is much more
vulnerable to the scattering compared to $\delta R_{xx}$ and thus its damping
cannot be described by the factor $A[\pi/(\mu_{\mathrm{w}}B)]$ with the same
value of $\mu_{\mathrm{w}}$. Note that the damping factor
$A[\pi/(\mu_{\mathrm{w}}B)]$ with $\mu_{\mathrm{w}}\simeq\mu_{\mathrm{q}}$ is
firmly established theoretically [16] and experimentally [11] only for
$\delta\sigma_{yy}^{\mathrm{band}}$ and thus may not be applicable to
$\delta\sigma_{xx}^{\mathrm{col}}$ and $\sigma_{yx}^{\mathrm{A}}$ without
modification 666We have also found that $\delta R_{yy}$ calculated assuming
the same damping factor as $\delta\sigma_{yy}^{\mathrm{band}}$ for
$\delta\sigma_{xx}^{\mathrm{col}}$ significantly exceeds (falls below) the
experimentally observed CO amplitudes at low (high) magnetic fields [10]. As
demonstrated in Fig. 4(b), heavier damping of $\sigma_{yx}^{\mathrm{A}}$
achieved by halving the $\mu_{\mathrm{w}}$ roughly reproduces the experimental
CO amplitudes, albeit without solid theoretical underpinnings.

##  IV Possible anisotropy in the Seebeck Coefficient

Finally, we briefly discuss the effect of $\delta\sigma_{yx}$ on the CO of the
Seebeck coefficients $S_{xx}$ and $S_{yy}$, the diagonal components of the
thermopower tensor $S_{ij}$. The theory [8] predicts that $S_{xx}$ and
$S_{yy}$ are almost identical and thus the Seebeck coefficient accommodates CO
isotropically. $S_{ij}$ can be written as the product of the resistivity
tensor $\rho_{ij}$ and the thermoelectric conductivity tensor
$\varepsilon_{ij}$. The latter is related to the conductivity tensor by the
Mott formula [20],
$\varepsilon_{ij}=-L_{0}eT(d\sigma_{ij,T=0}/dE)|_{E=E_{\mathrm{F}}}$ with
$L_{0}=\pi^{2}{k_{\mathrm{B}}}^{2}/(3e^{2})$ the Lorenz number, at low
temperatures 777More precisely, we should use the exact formula
$\varepsilon_{ij}=(1/eT)\int
dE[df_{E_{\mathrm{F}}}(E)/dE](E-E_{\mathrm{F}})\sigma_{ij,T=0}$, but this does
not alter the qualitative argument in this section. We thus have
$S_{xx}=\rho_{xx}\varepsilon_{xx}-\rho_{yx}\varepsilon_{yx}$ and
$S_{yy}=\rho_{yy}\varepsilon_{yy}-\rho_{yx}\varepsilon_{yx}$, where we made
use of the relations $\rho_{xy}=-\rho_{yx}$ and
$\varepsilon_{xy}=-\varepsilon_{yx}$. The corresponding oscillatory parts due
to the CO are given, to a good approximation, by $\delta
S_{xx}\simeq(\delta\varepsilon_{xx}+\mu B\delta\varepsilon_{yx})/\sigma_{0}$
and $\delta S_{yy}\simeq(\delta\varepsilon_{yy}+\mu
B\delta\varepsilon_{yx})/\sigma_{0}$ 888We can readily show numerically that
$\delta
S_{xx}\simeq\rho_{xx,\mathrm{bg}}\delta\varepsilon_{xx}-\rho_{yx,\mathrm{bg}}\delta\varepsilon_{yx}+\varepsilon_{xx,\mathrm{bg}}\delta\rho_{xx}-\varepsilon_{yx,\mathrm{bg}}\delta\rho_{yx}\simeq\rho_{xx,\mathrm{bg}}\delta\varepsilon_{xx}-\rho_{yx,\mathrm{bg}}\delta\varepsilon_{yx}$
and $\delta
S_{yy}\simeq\rho_{xx,\mathrm{bg}}\delta\varepsilon_{yy}-\rho_{yx,\mathrm{bg}}\delta\varepsilon_{yx}+\varepsilon_{xx,\mathrm{bg}}\delta\rho_{yy}-\varepsilon_{yx,\mathrm{bg}}\delta\rho_{yx}\simeq\rho_{xx,\mathrm{bg}}\delta\varepsilon_{xx}-\rho_{yx,\mathrm{bg}}\delta\varepsilon_{yx}$,
where the subscript “bg” signifies the non-oscillatory background. With
$\rho_{xx,\mathrm{bg}}=1/\sigma_{0}$ and $\rho_{yx,\mathrm{bg}}=-\mu
B/\sigma_{0}$, we arrive at the approximate equations presented here.. Since
$\mu B\gg 1$ for a high-mobility 2DEG in the magnetic field range where CO can
be observed, both $\delta S_{xx}$ and $\delta S_{yy}$ are dominated by the
identical second term if the magnitude of $|\delta\varepsilon_{yx}|$ is
comparable to those of $|\delta\varepsilon_{xx}|$ and
$|\delta\varepsilon_{yy}|$ as predicted in the theory [8], leading to the
rather counterintuitive isotropic behavior $\delta S_{xx}\simeq\delta S_{yy}$.
However, the small $|\delta\sigma_{yx}|$ we have experimentally found in the
present study, combined with the Mott formula, implies that
$|\delta\varepsilon_{yx}|$ is much smaller than the theoretical prediction.
The resulting enhancement in the relative importance of the first terms can
lead to anisotropic behavior $\delta S_{xx}\neq\delta S_{yy}$. This, however,
needs to be verified experimentally 999Measurements of $S_{xx}$ in a ULSL have
been reported in [27], but to the knowledge of the present authors, no attempt
to measure $S_{yy}$ has been made thus far. We have also made measurements of
the thermopower in ULSLs, and have found it difficult to obtain $S_{xx}$ and
$S_{yy}$ correctly, owing to the dominance of the $S_{xy}$ component and the
tilting of the temperature gradient caused by the magnetic field [10, 28].

##  V Summary

To summarize, we have experimentally captured the CO in $R_{yx}$ of a ULSL,
theoretically predicted some 30 years ago, by employing the measurement
arrangement designed to efficiently pick out the extra component of $R_{yx}$
introduced by $V(x)$ and further by eliminating the parasitic component due to
an unintentionally mixed $R_{xx}$ distinguishable as an even function of the
magnetic field. The amplitude of the CO thus observed is found to be much
smaller than the theoretical prediction. We have also deduced an asymptotic
analytic expression for CO in the Hall conductivity $\delta\sigma_{yx}$ to
facilitate the comparison between the theory and the experiment and to clarify
the oscillation phase of $\delta\sigma_{yx}$. The smallness of
$\delta\sigma_{yx}$ demonstrated in the present experiment suggests the
possibility of considerable anisotropy in the Seebeck coefficient, contrary to
the theoretical prediction.

###### Acknowledgements.

This work was supported by JSPS KAKENHI Grant Numbers 20K03817 and 19H00652.

*

## Appendix: Derivation of the Asymptotic Analytic Expressions

In this Appendix, we describe the derivation of the asymptotic analytic
expression $\sigma_{yx}^{\mathrm{A}}$ given by Eq. (9) from $\sigma_{yx}$ in
Eq. (5). Noting that $\lambda_{N}\lesssim 0.1$ for practical values of
$V_{0}$, $a$, and $B$, we obtain, up to $O(\lambda_{N}^{2})$,

$\displaystyle\sigma_{yx}$ $\displaystyle\simeq$
$\displaystyle\displaystyle\frac{2e^{2}}{h}\
\sum_{N=0}^{\infty}\left(N+1\right)\int_{0}^{1}{d\xi\
\left\\{f_{E_{\mathrm{F}}}\left(E_{N}\right)-f_{E_{\mathrm{F}}}\left(E_{N+1}\right)+\left[\frac{df_{E_{\mathrm{F}}}\left(E_{N}\right)}{dE}L_{N}\left(u\right)-\frac{df_{E_{\mathrm{F}}}\left(E_{N+1}\right)}{dE}L_{N+1}\left(u\right)\right]V_{0}e^{-\frac{u}{2}}\cos{\left(2\pi\xi\right)}\right\\}}$
(17)
$\displaystyle\displaystyle\times\left[1-2\lambda_{N}\cos{\left(2\pi\xi\right)}+3{\lambda_{N}}^{2}\cos^{2}{\left(2\pi\xi\right)}\right]$
$\displaystyle=$
$\displaystyle\displaystyle\sigma_{yx}^{(1)}+\sigma_{yx}^{(2)}$

with

$\sigma_{yx}^{(1)}=\frac{2e^{2}}{h}\
\sum_{N=0}^{\infty}\left(N+1\right)\left[f_{E_{\mathrm{F}}}\left(E_{N}\right)-f_{E_{\mathrm{F}}}\left(E_{N+1}\right)\right]\left(1+\frac{3}{2}{\lambda_{N}}^{2}\right)$
(18)

and

$\sigma_{yx}^{(2)}=\frac{2e^{2}}{h}\
\sum_{N=0}^{\infty}\left(N+1\right)\left[-\frac{df_{E_{\mathrm{F}}}\left(E_{N}\right)}{dE}L_{N}\left(u\right)+\frac{df_{E_{\mathrm{F}}}\left(E_{N+1}\right)}{dE}L_{N+1}\left(u\right)\right]V_{0}e^{-\frac{u}{2}}\lambda_{N},$
(19)

where we performed the integration with respect to $\xi$. In the asymptotic
limit of many filled Landau levels ($N\gg 1$), we can make the replacements,
$e^{-u/2}L_{N}(u)$ $\rightarrow$ $(\pi^{2}Nu)^{-1/4}\cos{(2\sqrt{Nu}-\pi/4)}$
and $e^{-u/2}L_{N}^{-1}(u)$ $\rightarrow$
$u(4\sqrt{\pi})^{-1}(Nu)^{-5/4}$$\left[4\sqrt{Nu}\cos{\left(2\sqrt{Nu}+\pi/4\right)-\sin{\left(2\sqrt{Nu}+\pi/4\right)}}\right]$,
and take the continuum limit, $E_{N}\rightarrow E$, $\sum_{N}^{\infty}$
$\rightarrow$
$\int_{\hbar\omega_{\mathrm{c}}/2}^{\infty}dE/(\hbar\omega_{\mathrm{c}})$. The
latter can be replaced by
$\int_{-\infty}^{\infty}dE/(\hbar\omega_{\mathrm{c}})$ at low temperatures. We
further make an approximation
$f_{E_{\mathrm{F}}}\left(E_{N}\right)-f_{E_{\mathrm{F}}}\left(E_{N+1}\right)\simeq\hbar\omega_{\mathrm{c}}(-\partial
f/\partial E)|_{E_{\mathrm{F}}-\frac{\hbar\omega_{c}}{2}}$. By performing the
energy integral, we get

$\sigma_{yx}^{(1)}\simeq\nu\frac{e^{2}}{h}\left[1+\frac{3}{2}{\lambda_{\mathrm{c}}}^{2}-A\left(\frac{T}{T_{a\mathrm{H}}}\right)\frac{3}{2}{\lambda_{\mathrm{c}}}^{2}\sin{\left(r_{\mathrm{c}}+\delta_{\mathrm{F}}\right)}\right]$
(20)

and
$\sigma_{yx}^{(2)}=\sigma_{yx}^{(2\mathrm{a})}+\sigma_{yx}^{(2\mathrm{b})}$
with

$\sigma_{yx}^{(2\mathrm{a})}\simeq\left(\nu+1\right)\frac{e^{2}}{h}\sqrt{\frac{2}{\pi}}\frac{V_{0}}{\hbar\omega_{c}}\frac{\lambda_{\mathrm{c}}}{\sqrt{r_{\mathrm{c}}^{-}}}\
\left[A\left(\frac{T}{T_{a\mathrm{H}}^{+}}\right)\cos{\left(\frac{r_{\mathrm{c}}^{+}+r_{\mathrm{c}}^{-}}{2}+\frac{\delta_{\mathrm{F}}^{+}}{2}\right)}-A\left(\frac{T}{T_{a\mathrm{H}}^{\delta}}\right)\sin{\left(\frac{\delta_{\mathrm{F}}^{+}}{2}+\frac{r_{\mathrm{c}}^{+}-r_{\mathrm{c}}^{-}}{2}\right)}\right]$
(21)

and

$\sigma_{yx}^{(2\mathrm{b})}\simeq\left(\nu-1\right)\frac{e^{2}}{h}\sqrt{\frac{2}{\pi}}\frac{V_{0}}{\hbar\omega_{c}}\frac{\lambda_{\mathrm{c}}}{\sqrt{r_{\mathrm{c}}}}\left[A\left(\frac{T}{T_{a\mathrm{H}}^{-}}\right)\cos{\left(r_{\mathrm{c}}^{-}+\frac{\delta_{\mathrm{F}}^{-}}{2}\right)}-\frac{1}{\sqrt{1+{r_{\mathrm{c}}}^{2}}}\right],$
(22)

where $\sigma_{yx}^{(2\mathrm{a})}$ ($\sigma_{yx}^{(2\mathrm{b})}$) derives
from the first (second) term in Eq. (19), $\lambda_{\mathrm{c}}\equiv
4\sqrt{2/\pi}[V_{0}/(\hbar\omega_{c})]u\sqrt{(1+r_{\mathrm{c}}^{2})/r_{\mathrm{c}}^{5}}$,
$T_{a\mathrm{H}}\equiv T_{a}(r_{\mathrm{c}}^{2}+1)/(r_{\mathrm{c}}^{2}-1)$,
$r_{\mathrm{c}}^{\pm}\equiv r_{\mathrm{c}}\sqrt{1\pm\nu^{-1}}$,
$\delta_{\mathrm{F}}^{\pm}\equiv 2\cot^{-1}{r_{\mathrm{c}}^{\pm}}$,
$T_{a\mathrm{H}}^{+}\equiv T_{a}\cdot
2\sqrt{1-\nu^{-2}}({r_{\mathrm{c}}^{+}}^{2}+1)/[\sqrt{1+\nu^{-1}}({r_{\mathrm{c}}^{+}}^{2}+1)+\sqrt{1-\nu^{-1}}({r_{\mathrm{c}}^{+}}^{2}-1)]$,
$T_{a\mathrm{H}}^{-}\equiv
T_{a}\cdot\sqrt{1-\nu^{-1}}({r_{\mathrm{c}}^{-}}^{2}+1)/{r_{\mathrm{c}}^{-}}^{2}$,
and $T_{a\mathrm{H}}^{\delta}\equiv T_{a}\cdot
2\sqrt{1-\nu^{-2}}({r_{\mathrm{c}}^{+}}^{2}+1)/[\sqrt{1+\nu^{-1}}({r_{\mathrm{c}}^{+}}^{2}+1)-\sqrt{1-\nu^{-1}}({r_{\mathrm{c}}^{+}}^{2}-1)]$.

Noting that $\nu$, $r_{\mathrm{c}}$, $r_{\mathrm{c}}^{\pm}$ $\gg 1$ in the
range of the magnetic field where CO is observed, we may neglect the
difference between $T_{a}$ and $T_{a\mathrm{H}}$, $T_{a\mathrm{H}}^{\pm}$ to a
good approximation. We can also make an approximation
$\lambda_{\mathrm{c}}\simeq
4\sqrt{2/\pi}[V_{0}/(\hbar\omega_{c})]ur_{\mathrm{c}}^{-3/2}$ to attain the
definition of $\lambda_{\mathrm{c}}$ presented in the main text. With these
approximations, Eq. (20) becomes equivalent to Eq. (10a) in the main text. It
can also readily be found that $A(T/T_{a\mathrm{H}}^{\delta})\simeq 1$ at the
cryogenic temperatures where CO is observed. After approximating
$\lambda_{\mathrm{c}}/\sqrt{r_{\mathrm{c}}^{-}}$ in Eq. (21) by
$\lambda_{\mathrm{c}}/\sqrt{r_{\mathrm{c}}}$ for the sake of simplicity, we
expand $(\nu+1)$ and $(\nu-1)$ in Eqs. (21) and (22), respectively. Then we
collect the terms containing (not containing) the factor $\nu$, which yields
Eq. (11a) (Eq. (12a)) by further using the approximations
$r_{\mathrm{c}}^{\pm}\simeq r_{\mathrm{c}}(1\pm\nu^{-1}/2)$ and
$\delta_{\mathrm{F}}^{\pm}\simeq\delta_{\mathrm{F}}\mp\nu^{-1}r_{\mathrm{c}}/(1+{r_{\mathrm{c}}}^{2})$
within the $\cos$ and $\sin$ terms. The factor $\mathrm{sgn}(B)$ is
incorporated in Eqs. (12b) and (12c) to ensure the antisymmetry with respect
to $B$. (See the caveat to Eq. (5) in the main text.)

We note in passing that the expression
$\sigma_{yx}=(2e^{2}/h)(N+1)(1+3{\lambda_{N}}^{2}/2)$ presented just below Eq.
(28) in [8] 101010We have found that a factor 2 was missing in [8], which we
resumed here. corresponds to the low temperature limit of $\sigma_{yx}^{(1)}$
in the present study. As mentioned in the main text, $\sigma_{yx}^{(1)}$ only
accounts for roughly half of the CO in $\sigma_{yx}$ (see also Fig. 5(a)).

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  * Note [8] We can readily show numerically that $\delta S_{xx}\simeq\rho_{xx,\mathrm{bg}}\delta\varepsilon_{xx}-\rho_{yx,\mathrm{bg}}\delta\varepsilon_{yx}+\varepsilon_{xx,\mathrm{bg}}\delta\rho_{xx}-\varepsilon_{yx,\mathrm{bg}}\delta\rho_{yx}\simeq\rho_{xx,\mathrm{bg}}\delta\varepsilon_{xx}-\rho_{yx,\mathrm{bg}}\delta\varepsilon_{yx}$ and $\delta S_{yy}\simeq\rho_{xx,\mathrm{bg}}\delta\varepsilon_{yy}-\rho_{yx,\mathrm{bg}}\delta\varepsilon_{yx}+\varepsilon_{xx,\mathrm{bg}}\delta\rho_{yy}-\varepsilon_{yx,\mathrm{bg}}\delta\rho_{yx}\simeq\rho_{xx,\mathrm{bg}}\delta\varepsilon_{xx}-\rho_{yx,\mathrm{bg}}\delta\varepsilon_{yx}$, where the subscript “bg” signifies the non-oscillatory background. With $\rho_{xx,\mathrm{bg}}=1/\sigma_{0}$ and $\rho_{yx,\mathrm{bg}}=-\mu B/\sigma_{0}$, we arrive at the approximate equations presented here.
  * Note [9] Measurements of $S_{xx}$ in a ULSL have been reported in [27], but to the knowledge of the present authors, no attempt to measure $S_{yy}$ has been made thus far. We have also made measurements of the thermopower in ULSLs, and have found it difficult to obtain $S_{xx}$ and $S_{yy}$ correctly, owing to the dominance of the $S_{xy}$ component and the tilting of the temperature gradient caused by the magnetic field [10, 28].
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# Emergent anisotropy in magnetic metamaterials

Nanny Strandqvist  Björn Erik Skovdal  Merlin Pohlit  Henry Stopfel  Vassilios
Kapaklis  Björgvin Hjörvarsson  Department of Physics and Astronomy, Uppsala
University, Box 516, SE-75120 Uppsala, Sweden

###### Abstract

We demonstrate directional dependence of the self-modification of internal
mesospin textures in magnetic metamaterials, arising from the coupling of the
atomic- and mesoscopic length-scales. Dressing the mesospins in different
directions enables the quantification of the degree of texture in the internal
magnetization and its impact on the interaction energy of the mesospins. The
emerging anisotropy is manifested in a directional dependence of the remanent
magnetization with temperature.

††preprint: APS/123-QED

##  I Introduction

Magnetic metamaterials can be used for investigations of phase transitions,
with a control of system parameters only surpassed by numerical methods [1, 2,
3, 4, 5, 6, 7, 8]. In these studies, the magnetic islands - mesospins \- have
been treated as indivisible building blocks without inner structure, with a
classical order-disorder transition of the mesoscopic elements. It is only
recently their inner degrees of freedom have been included in the description
[9, 10, 11]. For disk-shaped mesospins, a collection of works addressed
dynamics involving bistability of magnetization textures within, depending on
geometry and temperature [12, 13]. It was further shown that the size and
shape of the mesospins determines also to which extent static internal
magnetic textures, such as S-, C-, O- and vortex states, prevail [14, 15, 16,
17]. A more detailed understanding of the role of thermal excitations in the
magnetic texture was only recently developed [9, 11]. In the light of this,
systems allowing for the study of the effect of internal mesospin degrees of
freedom on the collective order and thermal dynamics, have been shown to
exhibit exotic order-disorder transitions [9, 18]. In all of these cases,
circular magnetic islands effectively act as XY spins with variable spin
lengths.

The emergence of an anisotropy in magnetic metamaterials has previously been
demonstrated and attributed to geometrical parameters: mesospin shape and
lattice symmetry [19, 20, 21]. Here, we show that the internal magnetic
textures of disk-shaped mesospins, along with their local magnetization
configurations also play an essential role. The directional dependence is
shown to originate from an interplay between the extension of the elements and
the symmetry of the lattice, giving rise to a self induced-anisotropy for the
inter-island interactions. Thus, the internal degrees of freedom lead to an
emergent magnetic anisotropy on the mesoscale, which is neither present in the
parent material nor in the individual mesospins. Furthermore, we show that the
strength of the anisotropy can be altered by the size of the mesospin,
effectively allowing for the choice of fourfold or eightfold rotational
degeneracy of metastable states with collinear magnetization components.

##  II Materials and methods

The samples were produced by post-patterning a polycrystalline three-layer
thin film using electron beam lithography. The thin films consisted of Pd (40
nm) - Fe13Pd87 (10 nm) - Pd (2 nm) and were deposited by dc magnetron
sputtering on a fused silica substrate in an ultrahigh-vacuum (base pressure
below $2\times 10^{-7}$ Pa). A detailed description of the complete process
can be found in Skovdal  _et al._ [9]. The periodic square arrays of circular
islands were fabricated with a fixed edge-to-edge distance of 20 nm and
diameters $D=$ 250, 350, and 450 nm. Each array contains 1 – 34 $\times
10^{7}$ mesospins.

The temperature dependence of the magnetization was determined using standard
magneto-optical Kerr effect (MOKE) in a longitudinal mode [9]. The incident
wavelength of the $p$-polarized incident laser was set to 659 nm with a spot
size of 1 mm2, giving the response of approximately million discs. An external
field with an amplitude of 40 mT was applied along the [10] and [11]
directions, with a sweep rate of 0.11 Hz.

The interplay of coupling and inner magnetization texture states, was explored
by performing micromagnetic simulations using MuMax3 [22]. Disks, mimicking
the actual mesospincs, were placed on a 5$\times$5 square lattice with
periodic boundary conditions in lateral directions (PBCx = PBCy = 5, PBCz =
0). A saturation magnetization of $M_{\mathrm{sat}}=3.5\cdot 10^{5}$ A/M, and
exchange stiffness of $A_{\mathrm{ex}}=3.36\cdot 10^{-12}$ J/m, were chosen
based on previous work [13, 23], using the same alloys. The in-plane cell size
was defined as 0.5$\cdot l_{\mathrm{ex}}$, were the exchange length,
$l_{\mathrm{ex}}$ is a material parameter defined by $M_{\mathrm{sat}}$ and
$A_{\mathrm{ex}}$ [24]. The average magnetization $|\langle\mathbf{m}\rangle|$
of the magnetic states in Table 1 were obtained by saturating disks placed on
a 5$\times$5 grid along [10] and [11] direction. This was followed by relaxing
the systems to its minimum free energy, leading to a mixture of different
states. Thereafter, each island was extracted, and the net moment along one
axis was used as a base for calculating $|\langle\mathbf{m}\rangle|$ and is,
therefore, an average of several disks.

##  III Results and discussion

###  III.1 Thermally induced transition

Figure 1:  Top: Magnetization loops at 83 K and 276 K for mesospins with a
diameter of 350 nm and magnetic field applied along [11] direction. The curves
have been normalized to the net magnetization at 0 K, obtained by fitting the
magnetization at 40 mT. Illustrations represent possible internal
magentization textures at 40 and 0 mT, respectively. Bottom: Temperature
dependence of magnetization $M_{\mathrm{40mT}}$ and $M_{\mathrm{0mT}}$ (left
axis) and the ratio $M_{\mathrm{m}}=M_{\mathrm{0mT}}/M_{\mathrm{40mT}}$ (right
axis). The inset shows a scanning electron microscopy image with arrows
indicating the principal lattice directions [10] and [11].

Representative hysteresis loops measured along [11] direction are shown in the
upper half of Fig. 1 for disk-shaped mesospins with a diameter of 350 nm.
Applying a field of 20 mT results in a state with a net normalized
magnetization of 0.94(1) at 83 K. When reducing the applied field, the
magnetization decreases, and in a zero field, a remanence of 0.52(1) is
obtained. When increasing the temperature, the hysteresis loops change form.
At $e.g.$, 276 K, zero magnetization is observed at remanence while saturation
is obtained already at 7 mT, with a net magnetization of 0.37(1). Hence, the
hysteresis loops obtained at elevated temperatures are representative of
vortex states [25], while the hysteresis loops obtained at low temperatures
have a clear ferromagnetic component [9].

The hysteresis loops described above are clearly affected by the mesospin
magnetic textures. For example, results obtained at 276 K are representative
for a response of a magnetic vortex state, which is removed by applying a
small external field (7 mT). Small fields like this, do little to the
thermally induced reduction of the magnetization. Consequently, the
magnetization (at these modest fields) can be viewed as representing the
intrinsic magnetization of the material ($M_{\mathrm{40mT}}$). A modified
Bloch law is used to describe the temperature dependence:
$M_{\mathrm{40mT}}(T)=M_{\mathrm{40mT}}(0)(1-(T/T^{\star})^{\alpha})$, where
$\alpha=$ 1.89(4) and $T^{\star}=$ 352(1) K, represented by a black solid line
in bottom panel of Fig. 1.

The temperature dependence of the remanent magnetization, $M_{\mathrm{0mT}}$,
is symbolised by red squares in Fig. 1. The remanence vanishes at 273 K, well
below the ordering temperature of the material. We can extract the
contribution from the magnetic texture to the changes in magnetization by
dividing by the intrinsic magnetization of the material
($M_{\mathrm{m}}$=$M_{\mathrm{0mT}}$/$M_{\mathrm{40mT}}$) [1, 9]. Therefore,
the reduction of the relative magnetization arising from the presence of inner
magnetic textures can be written as $M_{\mathrm{T}}=1-M_{\mathrm{m}}$. A
plateau with $M_{\mathrm{m}}=0.54(1)$ is observed at temperatures below 245 K.
Consequently, almost half ($M_{\mathrm{T}}=$0.46(1)) of the moment is not
contributing to the obtained magnetization. $M_{\mathrm{m}}$ vanishes in a
relatively narrow temperature range, and at 273 K, the transition to vortex
states is completed. Above this temperature, all the moment is in the magnetic
textures giving rise to zero magnetization up to the ordering temperature of
the material.

###  III.2 Size dependence

Figure 2:  Temperature dependence of
$M_{\mathrm{m}}=M_{\mathrm{0mT}}/M_{\mathrm{40mT}}$ along the [10] direction
for mesospins with $D=$ 250, 350 and 450 nm. Error functions (solid lines)
were chosen in order to determine the transition temperatures. Illustrations
exemplify the possible internal texture of a mesospin.  Figure 3:  The free
energy density of vortex, intermediate and collinear state as a function of
vortex core displacement for $D=[150,250,350,450]$ nm dressed along the [10]
lattice direction. The size dependence of $E_{\mathrm{v}}-E_{\mathrm{c}}$,
display the boundary between collinear (blue shade) and vortex (red shade)
states is shown in the inset. The top of the figure illustrates how the core
is moved with a distance of $2r$, leading to C-states with varying degree of
magnetic texture until the collinear state is reached at infinity.

The influence of size on the transition of the mesospins is illustrated in
Fig. 2, where the temperature dependence of $M_{\mathrm{m}}$ for mesospin
diameters $D=[250,350,450]$ nm, with an external field applied along the [10]
direction is shown. All samples display a error function-like transition, with
a plateau value of $M_{\mathrm{m}}$ in the range 0.68(1) to 0.75. Here we note
that the smaller the diameter is, the lower the temperature is for the onset
of the transition. No clear trends are observed with respect to the
temperature when the transition is completed.

To rationalize these results, we need to get a handle on different
contributions to the energy landscape, including the effect of interactions on
the inner texture of the mesospins. Therefore we performed micromagnetic
calculations using MuMax [22]. The total magnetic energy of the arrays can be
defined as: $E_{\mathrm{tot}}=E_{\mathrm{t}}+E_{\mathrm{s}}+E_{\mathrm{j}}$,
where $E_{\mathrm{t}}$ is the energy cost of magnetic texture,
$E_{\mathrm{s}}$, the magnetostatic energy arising from the stray field, and
$E_{\mathrm{j}}$, the energy associated with magnetostatic coupling between
the islands. $E_{\mathrm{tot}}$ has been calculated as the vortex cores have
been moved along paths, resembling one of many possible trajectories arising
from the application of an external magnetic field as in the experimental
procedure illustrated at the top of Fig. 3.

From the expression described above, the energy density of the magnetic
texture for $D=[150,250,350,450]$ nm was calculated. In Fig. 3 we show the
results corresponding to applying the field along the [10] direction. In a
vortex state, the energy ($E_{\mathrm{v}}$) is only dependent on the cost of
texture since $E_{\mathrm{s}}\approx E_{\mathrm{j}}\approx 0$. The results
reveal a size dependence, where the cost of the vortices decreases with
increasing size. Moreover, moving the vortex core from the center results in a
collinear magnetic component with a corresponding stray field, leading to a
coupling between the mesospins. Consequently, the energy increases, reaching a
maximum at $2r/D\approx 0.9$. Continuing moving the vortex core, results in an
energy decrease. When the core has been annihilated at the rim, $2r/D=1$, all
mesospins possess a magnetization texture resembling a C-state. The degree of
texture continues to vary until the vortex cores have reached infinity and all
disks are in a collinear state. Here the energy ($E_{\mathrm{c}}$) decreases
with size. Energy barriers separate the vortex and collinear state. The height
of the barrier changes with size and depends on whether it is passed from the
collinear or vortex state.

The inset in Fig. 3 shows the boundary between the collinear and vortex state
as a function of $1/D$. As the energy difference
($E_{\mathrm{v}}-E_{\mathrm{c}}$) increases with size, the linear fit reveals
that for $D\lesssim$ 189 nm, the collinear state is the ground state, while a
vortex state is favored when the diameter is larger. This is consistent with
PEEM results discussed by Skovdal  _et al._ [9], where a mesospin diameter $>$
189 nm was concluded to preferably have a vortex texture. Interacting
mesospins with $D=$ 150 nm displayed a collinear state and can therefore be
viewed as two-dimensional XY-rotor. Thus, the vortex is the state with the
lowest energy for all samples investigated experimentally in this work.
Consequently, the elements can be viewed as being “dressed” by an external
field, inducing a change in energy and giving rise to a ferromagnetic signal
in the measurements. Before addressing the importance of these two
contributions, we need to investigate what happens when the direction of the
net magnetization of the elements changes.

Figure 4:  Directional dependence of
$M_{\mathrm{m}}=M_{\mathrm{0mT}}/M_{\mathrm{40mT}}$ as a function of
temperature with external field applied along [10] and [11] direction for
mesospins with a $D=$ 250 nm. Error functions (solid lines) were chosen in
order to determine the transition without assigning any physical
interpretation to it. Yellow dots represent the transition temperature
$T_{\mathrm{m}}$ separating the collinear-like and vortex state, and
illustrations exemplify possible internal states.

###  III.3 Directional dependence

Figure 5:  Qualitative trends for a) the extent of magnetic texture,
$M_{\mathrm{m}}(T=0\>\mathrm{K})$, and b) the transition temperature
$T_{\mathrm{m}}$ separating the dressed and vortex state for $D=[250,350,450]$
nm along [10] and [11] direction. The data error bars are smaller than the
symbols and are therefore not visible. The shaded areas represent the
estimated uncertainty for the respective data sets.

The directional dependence in the magnetization for mesospins with a diameter
of 250 nm is displayed in Fig. 4. A plateau value of $M_{\mathrm{m}}=$ 0.68(1)
is obtained when the field has been applied along the [10] direction, whilst
$M_{\mathrm{m}}=$ 0.63 is obtained along the [11] direction. The lower
magnetization of the array obtained when dressed along [11], is accompanied by
a more stable configuration, manifested in a higher transition temperature. To
allow for quantitative comparison, we define a transition temperature $T_{m}$
defined as $M_{\mathrm{m}}(T_{m})=\frac{1}{2}M_{\mathrm{m}}\>(T=0$ K). By this
definition, $T_{\mathrm{m}}=$ 193(3) K is obtained when the external field has
been applied along [10] while $T_{\mathrm{m}}=$ 274(1) K along the [11]
direction.

Fig. 5 shows the effect of the chosen direction of the applied field on
$M_{\mathrm{m}}$ and $T_{\mathrm{m}}$ for all sample sets. The fits reveal
linear relations with inverse mesospin diameter, $1/D$. Positive slopes are
observed when the external field has been applied along [11]. This is
consistent with a decrease in magnetic texture (increasing $M_{\mathrm{T}}$)
with decreasing mesospin size and an increase in the transition temperature.
When the field has been applied along the [10] direction, a decrease of
$M_{\mathrm{m}}$ is obtained, accompanied by a profound decrease in
$T_{\mathrm{m}}$. Hence, the direction of the applied field results in a
significant difference in the degree of magnetic texture and in the onset of
the transition. The difference in response to an external field vanishes at
diameters around 212 nm.

We now turn our attention to the impact of the direction of the magnetization
on the energy landscape. Fig. 6 shows the normalized energy
($E/E_{\mathrm{v}}-1$) as a function of vortex core displacement for
$D=[150,200,350]$ nm dressed along the [10] (filled symbols) and the [11]
direction (empty symbols). The ground state for mesospins with $D=$ 350 nm is
the vortex state, while the collinear state is metastable and independent of
orientation. However, when inner magnetic textures are allowed, [10] and [11]
are no longer equivalent, and the energy is observed to be lowered when the
net magnetization is along the [11] direction. Since the degree of magnetic
texture is equivalent for the two directions, the energy gap can only be
associated with an energy gain mediated by magnetostatic coupling. This can be
related to the experimental results observed in Fig. 5 where the transition
temperature is higher when the external field has been applied along [11]
direction. For $D=$ 200 nm, the lowest energy is achieved at an intermediate
state along the [11] direction. Implying that the energy landscape can be
modified, where rotation of the net magnetization leads to an altered
degeneracy of the metastable states. When the diameter is 150 nm, the
collinear state is energetically favored, and minor effects are observed from
internal magnetic texture on the interaction of the elements.

Figure 6:  Energy landscape for the transition between vortex and collinear
state for $D=[150,200,350]$ nm. The energy has been normalized with respect to
the vortex state. Hence, $E/E_{\mathrm{vortex}}-1=0$ when the vortex core is
in the middle of the disks. Numerical calculations were generated by moving
the vortex core along a path corresponding to [10] (empty symbols) or [11]
(filled symbols) direction in an alternating manner, either up or down for
every other disk. At the top of the figure, illustrations show how the core
moves with a distance of $2r$, leading to C-states with varying degrees until
the collinear state is reached.

###  III.4 Interplay of magnetic interactions and texture

Table 1:  Magnetic texture and topology of V-, O-, S- and C-state depicting
the nonuniformity of the mesospins. The average magnetization,
$|\langle\mathbf{m}\rangle|$, is extracted along the horizontal axis of the
illustrated states.

The experimental results obtained for $M_{\mathrm{m}}$ and $T_{\mathrm{m}}$
illustrated in Fig. 5, reveal the existence of magnetic textures with
transverse components and an altered onset of the transition. The extent of
this is clearly dependent on the size of the mesospins and the direction of
the externally applied field. To understand this, we need to take a closer
look at different magnetic states and the nature of the interaction between
the mesospins when the external field is applied along different directions.

Even though the ground state is a vortex, the circular shape can accomodate
internal magnetic textures such as O-, S-, and C- states, illustrated in Table
1 [16, 17]. The average magnetization, $|\langle\mathbf{m}\rangle|$, hereafter
referred to as the mesospin moment, has been extracted along the horizontal
axis of the illustrated textures shown in the table. The results from the
simulations imply that the mesospin moment along the horizontal axis decrease
with increasing diameter.

Spatial maps of the amplitude of the demagnetizing field for mesospins with a
diameter of 350 nm are shown in Fig. 7. The system was initially saturated
along the [10] and [11] axis (illustrated by $H_{\mathrm{ext}}$), followed by
reducing the field to zero. This results in different magnetic textures
stabilizing, indicated by the letters at the center of the disks, with the
arrows specifying the rotation of $|\langle\mathbf{m}\rangle|$. The lighter
regions in these magnetostatic field maps represent a bond between the
islands.

The results displayed in Fig. 7a arise from dressing the mesospins with a
field along the horizontal axis. The overall magnetic texture consists of
O-states, with a mesospin moment along the direction of the externally applied
field. As seen in the figure, the attractive interaction is confined to the
nearest neighboring mesospins along [10]. Comparing
$|\langle\mathbf{m}\rangle|$ for the O-state in Table 1 and $M_{\mathrm{m}}$
in Fig. 5a, reveals a lower net moment in the experiments as compared to the
fully dressed mesospins. Capturing these changes therefore requires a rotation
of the mesospins and/or a presence of other states, resulting in a reduced net
moment along the [10] direction. In Fig. 7b the external field has been
applied along [11], resulting in islands with both C- and S-states with
rotated mesospin moments. Furthermore, for this applied field direction,
mesospins can link up to four neighbors. This can be seen for the case of the
center island, having an S texture and strong amplitudes of the magnetostatic
field towards its four nearest neighbors. The possibility of mixing mesospin
textures of more than just one type, having further considerable transverse
magnetization components, explains the consistently lower values of
$M_{\mathrm{m}}$ shown in Fig. 5a and the [11] direction compared to the [10]
direction.

Figure 7:  Spatial distribution maps of demagnetizing field in relaxed state
obtained from micromagnetic simulations for mesospins with a diameter of 350
nm. Initially the disks were saturated along [10] and [11]
($H_{\mathrm{ext}}$) followed by relaxing in a external applied field of 0 mT.
The lighter regions along the disk perimeters illustrate the strength of the
interaction of the disks. The letters represent the magnetic texture and
arrows the orientation of the mesospin moments.

##  IV Conclusions

We provide experimental evidences of a configurational dependence in the
energy of mesospins forming a square lattice. An emerging anisotropy is
obtained, in line with previous findings [26, 27, 23, 28, 29]. We demonstrate
the effects of the anisotropy in the interaction energy by determining the
directional dependence of the thermally induced transition from dressed
collinear states to vortex states of the elements. The key component required
for rationalizing the results is an interplay between the interaction of the
mesospins and their magnetic texture: When a mesospin and its neighbors have a
net magnetization along [10] in the square lattice, the interaction can be
viewed as being one dimensional with the interaction restricted to two
neighbors along the [10] axis. When the net moment is along the [11], the
elements can have interactions with four neighbors. These differences are
caused by self-induced modification of the inner magnetic texture, resulting
in changes of the interaction potential of the mesospins. The spontaneous
interactions of elements below the critical limit are therefore expected to
bias the [11] configuration, while a mixed order with different range of
correlations is expected to emerge at finite temperatures.

## Acknowledgements

B.H. and V.K. acknowledge financial support from the National Research Council
(VR) (Project No. 2019-03581 and 2019-05379). We acknowledge Myfab Uppsala for
providing facilities and experimental support. Myfab is funded by the Swedish
Research Council (2019-00207) as a national research infrastructure.

## Author Declarations

### Data availability

The data that support the findings are available from the corresponding
authors upon reasonable request.

### Conflict of Interest

The authors have no conflicts to disclose.

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