# High-precision star formation efficiency measurements in nearby clouds

Zipeng Hu,${}^{\href https://orcid.org/0000-0002-3758-552X\,1}$ Mark R.
Krumholz,${}^{\href https://orcid.org/0000-0003-3893-854X\,1,2}$ Riwaj Pokhrel
${}^{\href https://orcid.org/0000-0002-0557-7349\,3}$ and Robert A. Gutermuth
${}^{\href https://orcid.org/0000-0002-6447-899X\,4}$
1Research School of Astronomy and Astrophysics, Australian National
University, Canberra, ACT 2611, Australia
2ARC Centre of Excellence for Astronomy in Three Dimensions (ASTRO-3D),
Canberra, ACT 2611, Australia
3Ritter Astrophysical Research Center, Department of Physics and Astronomy,
University of Toledo, Toledo, OH 43606, USA
4Department of Astronomy, University of Massachusetts, 710 North Pleasant
Street, Amherst, MA 01003, USA
<EMAIL_ADDRESS>(ZPH)

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

On average molecular clouds convert only a small fraction $\epsilon_{\rm ff}$
of their mass into stars per free-fall time, but differing star formation
theories make contrasting claims for how this low mean efficiency is achieved.
To test these theories, we need precise measurements of both the mean value
and the scatter of $\epsilon_{\rm ff}$, but high-precision measurements have
been difficult because they require determining cloud volume densities, from
which we can calculate free-fall times. Until recently, most density estimates
assume clouds as uniform spheres, while their real structures are often
filamentary and highly non-uniform, yielding systematic errors in
$\epsilon_{\rm ff}$ estimates and smearing real cloud-to-cloud variations. We
recently developed a theoretical model to reduce this error by using column
density distributions in clouds to produce more accurate volume density
estimates. In this letter, we apply this model to recent observations of 12
nearby molecular clouds. Compared to earlier analyses, our method reduces the
typical dispersion of $\epsilon_{\rm ff}$ within individual clouds from 0.35
dex to 0.31 dex, and decreases the median value of $\epsilon_{\rm ff}$ over
all clouds from $\approx 0.02$ to $\approx 0.01$. However, we find no
significant change in the $\approx 0.2$ dex cloud-to-cloud dispersion of
$\epsilon_{\rm ff}$, suggesting the measured dispersions reflect real
structural differences between clouds.

###### keywords:

stars: formation – ISM: structure.

††pubyear: 2021††pagerange: High-precision star formation efficiency
measurements in nearby clouds–High-precision star formation efficiency
measurements in nearby clouds

##  1 Introduction

Star formation is inefficient. A star forming region with little resistance to
self-gravity will convert its gaseous mass to stars within a single free-fall
time $t_{\rm ff}=\sqrt{3\pi/32G\rho}$, where $\rho$ is the volume density; the
ratio of the actual star formation rate to this maximum is called the
efficiency per free-fall time $\epsilon_{\rm ff}$ (Krumholz & McKee, 2005).
Zuckerman & Evans (1974) were the first point out that the observed star
formation rate of the Milky Way as a whole implies that, averaged over all
molecular clouds, $\epsilon_{\rm ff}\sim 0.01$. Krumholz & Tan (2007) extended
this conclusion to the denser parts of molecular clouds traced by HCN, and
more recent work has obtained similar means both for clouds traced by dust
(e.g., Heyer et al., 2016) and for $\sim 100$ pc-scale patches in nearby
galactic discs (e.g., Utomo et al., 2018). The study-to-study dispersion in
$\epsilon_{\rm ff}$ is $\approx 0.3$ dex, while the dispersion within any
single study is about 0.3 – 0.5 dex (Krumholz et al., 2019).

Theoretical efforts to explain the origin of observed low $\epsilon_{\rm ff}$
values can be categorized into two main types. Some theories focus on galactic
scale physical processes (e.g. Kim et al., 2011; Ostriker & Shetty, 2011;
Faucher-Giguère et al., 2013), while others are developed from internal star
formation regulation processes within individual molecular clouds (e.g.
Elmegreen & Parravano, 1994; Krumholz et al., 2011a; Federrath & Klessen,
2012). Despite predicting similarly-low mean $\epsilon_{\rm ff}$ values, the
two types of models yield significantly different estimates for the dispersion
of $\epsilon_{\rm ff}$ – models regulated on the cloud scale generally predict
much smaller dispersions than those regulated on the galactic scale (Lee et
al., 2016; Krumholz & McKee, 2020). Therefore, it is important to measure
cloud-scale $\epsilon_{\rm ff}$ values with enough fidelity to extract both
its mean value and dispersion. The most accurate measurements to date are
those of Pokhrel et al. (2021), who determine a median value
$\log\epsilon_{\rm ff}=-1.59$ with a dispersion of 0.18 dex in a sample of 12
nearby molecular clouds.

However, most previous $\epsilon_{\rm ff}$ measurements, including those of
Pokhrel et al. (2021), incur a substantial error when calculating the volume
density, which is required for the free-fall time. The fundamental challenge
is that the volume density is a three-dimensional quantity, which is not
directly accessible in a two-dimensional observation. The most common practice
in literature is to estimate the density by assuming that the area of interest
is the projection of a uniform sphere, whose radius is equal to the mean
radius of the projected shape. For a cloud with a total projected area $A$ and
a total mass $M$, this approximation gives a density estimate $\rho_{\rm
sph}=3M/4\sqrt{A^{3}/\pi}$. This method has been used by a number of authors
in the Milky Way (e.g., Krumholz et al., 2011b; Lada et al., 2013; Evans et
al., 2014; Pokhrel et al., 2021) and in the Large Magellanic Cloud (Ochsendorf
et al., 2017). While simple, this procedure likely introduces significant
systematic errors, coming from two primary sources. One is that in the past
decades, it has become clear that the interstellar medium is characterised
mainly by filamentary structures (e.g. Schneider & Elmegreen, 1979; Dobashi et
al., 2005; Arzoumanian et al., 2011; André et al., 2014; Kainulainen et al.,
2016), which results in elongated contours identified from column density
maps. The mean density of such structure is likely to be different from that
calculated with spherical assumption. Second, the free-fall time computed from
the mean volume density is not identical to the mean free-fall time, because
the relationship between the two, $t_{\rm ff}\propto\rho^{-1/2}$, is non-
linear. The quantity of interest for $\epsilon_{\rm ff}$ is the mass-weighted
mean of $1/t_{\rm ff}$, and this mean can differ substantially from the
overall mean density (e.g. Hennebelle & Chabrier, 2011; Federrath & Klessen,
2012; Federrath, 2013; Salim et al., 2015).

Hu et al. (2021) recently proposed an alternative approach to estimate the
free-fall time weighted mean density $\rho_{\rm eff}$, which immediately
yields the correct free-fall time for the purposes of estimating
$\epsilon_{\rm ff}$. Hu et al. analyse star formation simulations from
Cunningham et al. (2018), and show that one can estimate $\rho_{\rm eff}$ with
higher accuracy than is obtained from the simple spherical assumption by
making use of the full two-dimensional (2D) column density distribution,
rather than simply its mean. Applied to simulations, their method corrects a
$\approx 0.13$ dex overestimate and removes a $\sim 0.25$ dex scatter in
$\epsilon_{\rm ff}$ caused by spherical assumption. In this paper, we apply
this model to the observations of Pokhrel et al. (2020, 2021) in order to
derive higher-accuracy $\epsilon_{\rm ff}$ measurements than have previously
been possible. We summarise the observations in Section 2, describe how we
apply the Hu et al. method to them in Section 3, present the results of the
analysis in Section 4, and draw conclusions in Section 5.

##  2 Observations

Our data reduction and analysis method is described in Pokhrel et al. (2020),
and full details are provided there. Here we simply summarise for convenience.
This study analyses 12 nearby star forming regions: Ophiuchus, Perseus,
Orion-A, Orion-B, Aquila-North, Aquila-South, NGC 2264, S140, AFGL 490, Cep
OB3, Mon R2, and Cygnus-X. For each region we have a matched protostellar
catalogue and cloud column density map. The latter are derived from
Herschel/PACS and Herschel/SPIRE imaging at 160, 250, 350, and 500 $\mu$m,
convolved to a common resolution (André et al., 2010). To obtain the column
density in each pixel, we fit the spectral energy distribution with a dust
emission model where the only two free parameters are the temperature and the
column density. The best-fitting column density can be equivalently expressed
in column of $\rm H_{2}$ molecules, $N(\rm H_{2})$, or column of gas mass
$\sigma_{\rm gas}$, which are related by $\Sigma_{\rm gas}=2m_{\rm H}/{X}N(\rm
H_{2})$, where $m_{\rm H}$ = $1.67\times 10^{24}$ g is the hydrogen atom mass
and $X=0.71$ is the hydrogen mass fraction of the local interstellar medium
(Nieva & Przybilla, 2012). On the obtained column density map, we first mask
pixels with the best-fitting dust temperature exceeds the Rayleigh–Jeans (R-J)
limit, since such pixels are in the R-J limit for all the Herschel bands, and
thus the fits are highly uncertain. The exact temperature limits are provided
in Table 2 of Pokhrel et al. (2020). Second, we mask pixels with derived
column densities $N(\rm H_{2})>10^{23}\>\rm cm^{-2}$, because for these dust
optical depth effects can be significant and thus fitted $N(\rm H_{2})$ values
may only represent lower limits. The effect of both masks is negligible in our
results, since the number of masked pixels is very small, and, for many
clouds, zero.

The SESNA catalog (R. Gutermuth et al., in preparation) we use for protostars
is a combination of Spitzer and Two Micron All-Sky Survey observations (2MASS;
Skrutskie et al. 2006), spanning about 90 $\rm deg^{2}$. For the farthest
target Cygnus-X, the deeper UKIDSS (Lawrence et al., 2007) near-IR Galactic
Place Survey (Lucas et al., 2008) is used. We mask parts of column density
maps outside SESNA coverage. After subtracting field stars, sources with
excess IR emission are classified into different types of young stellar object
(YSO) and contaminant by using flux selections and a series of reddening-safe
color (Gutermuth et al., 2009). Hence the total number of protostars $N_{\rm
PS}$ in any enclosed contour on the column density map can be calculated after
contamination correction, whose detailed procedure is provided in Pokhrel et
al. (2020).

##  3 Methods

The first step of our analysis is to generate column density contours across
the full range of $N(\rm H_{2})$ covered by the observations. For each of the
12 regions, we start by finding the lowest $N(\rm H_{2})$ value such that all
pixels with column density above it are inside the SESNA coverage;
equivalently, we set the minimum value of $N(\rm H_{2})$ to be the lowest
possible choice such that the contour sits entirely within the SENSA
footprint. We define $M_{0}$ as the total gas mass enclosed by this contour.
We then draw 100 $N(\rm H_{2})$ contours at higher $N(\rm H_{2})$, with the
levels chosen such that the total mass above each level is equally spaced from
$M_{0}$ to $M_{0}/100$ with a step of $M_{0}/100$. We show an example of
Ophiuchus cloud column density map with $N(\rm H_{2})$ contours, protostar
positions, and SESNA coverage area in Figure 1.

For every contour, we measure four values: total gas mass $M_{\rm gas}$, total
area $A$, completeness-corrected total protostar number $N_{\rm PS}$ and the
Gini coefficient $g$ of the surface density of all pixels within the contour
(see Hu et al. 2021 for details). From these parameters we derive 4 additional
quantities: mean gas surface density $\Sigma_{\rm gas}=M_{\rm gas}/A$, star
formation surface density $\Sigma_{\rm SFR}$, the free-fall time derived from
the spherical assumption $t_{\rm ff,sph}$ (see Section 1), and the
corresponding star formation efficiency per free fall time $\epsilon_{\rm
ff,sph}$. To calculate $\Sigma_{\rm SFR}$, we assume the mean mass of
protostars in SESNA to be $M_{\rm PS}\approx 0.5M_{\rm\odot}$ (Evans et al.,
2009), and the mean duration of protostellar phase included in SESNA
observations to be $t_{\rm PS}\approx 0.5$ Myr (Dunham et al., 2014, 2015).
Thus, $\Sigma_{\rm SFR}=N_{\rm PS}M_{\rm PS}/At_{\rm PS}$, and $\epsilon_{\rm
ff,sph}=\Sigma_{\rm SFR}/(\Sigma_{\rm gas}/t_{\rm ff,sph})$.

Our next step is to apply the correction obtained by Hu et al. (2021), who
find a relation between free-fall time weighted mean density $\rho_{g}$ and
$\rho_{\rm sph}$:

$\rho_{g}=10^{kg-b}\rho_{\rm sph},$ (1)

with $k=4.6$ and $b=0.93$. Replacing $\rho_{\rm sph}$ with $\rho_{g}$ gives us
the corrected values of free-fall time $t_{\rm ff,g}$ and star formation
efficiency per free-fall time $\epsilon_{\rm ff,g}$. In order to accomplish
our goal of obtaining high-accuracy estimates of $\epsilon_{\rm ff}$, it is
important to understand the uncertainties in this relation. Although the
values of $k$ and $b$ are fitted from exact simulation data without error
bars, we can obtain estimate of the errors on these quantities from
bootstrapping. We randomly choose half of the simulation sample to fit
equation 1, repeating this process $10^{4}$ times. We take our uncertainties
to be the 16th and 84th percentiles of the fitted $k$ and $b$ values, which
give $k=4.6\pm 0.1$ and $b=0.93\pm 0.03$. We show below that these
uncertainties are small enough that they contribute negligibly to the overall
error budget.

Figure 1: A column density map of the Ophiuchus Cloud derived from Herschel
observations. The green contour is the Spitzer coverage area, and we we
illustrate the largest contour that fits within this footprint in purple; the
yellow contour is set at a level that encloses half as much mass as the purple
one. The red dots are the positions of YSOs.

We next remove from our sample contours small enough that beam-smearing
precludes an accurate estimate of $g$. To do so, for each contour we compute
the ratio of the mean contour radius $r=\sqrt{A/\pi}$ to the resolution $R$ on
the Herschel maps (expressed as the effective beam FWHM, taken from Table 1 of
Pokhrel et al. 2020). To check when beam smearing becomes significant, we
artificially blur our data: for each cloud we construct 8 column density maps
smeared by Gaussian beams with sizes uniformly spaced between $R$ and $20R$,
and place contours on these smeared maps exactly as we do for the original
maps. We plot the distribution of $g$ values for all contours with
$\Sigma_{\rm SFR}>0$ in Figure 2. The figure clearly shows the expected
effect: contours with small $r/R$ have systematically smaller values of $g$
than larger ones. To quantity this, we fit the distribution of $(\log(r/R),g)$
values with a piece-wise linear function $g=k[\log(r/R)-x_{0}]+y_{0}$ for
$\log(r/R)<x_{0}$, and $g=y_{0}$ otherwise, where $k$, $x_{0}$, and $y_{0}$
are free parameters. The green line shows the best fit, $k=0.47$,
$x_{0}=0.95$, and $y_{0}=-0.65$, which illustrates the effect in which we are
interested: the distribution of $g$ values becomes independent of resolution
when $\log(r/R)\>\geq 0.95$, and drops below this threshold. We therefore
remove from our analysis of the original, unsmeared maps any contour with
$\log(r/R)<0.95$. We also include only contours that contain at least one
protostar, since we obviously cannot compute $\epsilon_{\rm ff}$ values for
those that contain none. After this down-select we obtain 3410 contours that
form the data set for further analysis.

Figure 2: 2D distribution plot of $g$ versus $\log(r/R$). The values are
determined from the smeared contours, and the color shows the number of
contours in each bin (one black dot represents one single contour). The green
dash line shows a piecewise linear fit to this distribution.

##  4 Results

With the properties determined from the unsmeared contours on different
surface density levels, we first investigate how the triplet ($g$,
$\epsilon_{\rm ff,sph}$, $\epsilon_{\rm ff,g}$) changes with $\Sigma_{\rm
gas}$. In Figure 3 we show the scatter plot of $g$ versus log ($\Sigma_{\rm
gas}$), which presents a clear trend of $g$ decreasing with $\Sigma_{\rm
gas}$. It is important to note that this is not a resolution effect, since we
have removed contours small enough that lack of resolution leads to them being
blurred; instead, it is a real physical trend. The median value of $g$ is
$g_{\rm median}=0.24$, which is above 0.2, the value for a uniform-density
sphere (Hu et al., 2021). This indicates that the free-fall time of most
contours will be overestimated by the spherical assumption. For a typical Gini
coefficient $g=0.24$, our estimated uncertainties on $k$ and $b$ in equation 1
translate to an increase in the dispersion of $\epsilon_{\rm ff}$ by 0.01 dex,
which is negligible. For this reason, we will simply treat $k$ and $b$ as
constants fixed to their central values for the remainder of this analysis.

Figure 3: 2D scatter plot of $g$ versus log ($\Sigma_{\rm gas}$). The values
are determined from the unsmeared contours, and the color shows the number of
contours in the scatter circle. The grey dash line shows the $g$ = 0.2
reference line for the projection of a uniform sphere.

To study the difference between estimates of $\epsilon_{\rm ff}$ derived using
the spherical assumption and our improved method, we define two quantities:
the mean star formation efficiency for an individual cloud
$\langle\epsilon_{\rm ff}\rangle$, and the mean dispersion in star formation
efficiency $\sigma$. We compute these as follows: for each cloud, we sort the
contours by $\Sigma_{\rm gas}$ and place them in 10 bins of equal size, or
fewer if that leaves a cloud with $<20$ contours per bin. (Recall that
$\Sigma_{\rm gas}$ is the mean surface density inside a given contour, not the
contour level itself, so two contours at the same level still generally have
different $\Sigma_{\rm gas}$.) Within each bin, we denote the 16th, 50th, and
84th percentiles of $\epsilon_{\rm ff}$ (for both $\epsilon_{\rm ff,sph}$ and
$\epsilon_{\rm ff,g}$) as $\epsilon_{\rm ff,16}$, $\epsilon_{\rm ff,50}$, and
$\epsilon_{\rm ff,84}$. We plot these quantities as a function of $\Sigma_{\rm
gas}$ in Figure 4. We then define the mean star formation efficiency for one
cloud $\langle\epsilon_{\rm ff}\rangle$ as

$\langle\epsilon_{\rm ff}\rangle=\frac{\int_{\log\Sigma_{\rm
gas,min}}^{\log\Sigma_{\rm max}}\epsilon_{\rm ff,50}(\Sigma_{\rm
gas})\,d(\log\Sigma_{\rm gas})}{\log\Sigma_{\rm gas,max}-\log\Sigma_{\rm
gas,min}},$ (2)

where $\Sigma_{\rm gas,min}$ and $\Sigma_{\rm gas,max}$ are the minimum and
maximum contour surface density available for a given cloud; in terms of
Figure 4, $\langle\epsilon_{\rm ff}\rangle$ is simply the mean value of the
coloured line for each cloud. We evaluate this integral by approximating it as
a finite sum over our groups. Similarly, we define the mean star formation
efficiency dispersion $\sigma$ as

$\sigma=\frac{\int_{\log\Sigma_{\rm gas,min}}^{\log\Sigma_{\rm
gas,max}}(\log\epsilon_{\rm ff,84}(\Sigma_{\rm gas})-\log\epsilon_{\rm
ff,16}(\Sigma_{\rm gas}))\,d(\log\Sigma_{\rm gas})}{\log\Sigma_{\rm
gas,max}-\log\Sigma_{\rm gas,min}},$ (3)

where we again evaluate numerically as a finite sum over our bins of
$\Sigma_{\rm gas}$. In terms of Figure 4, $\sigma$ is simply the mean width of
the grey band that surrounds each of the coloured lines.

Figure 4: Distributions of log $\epsilon_{\rm ff,sph}$ (top) and log
$\epsilon_{\rm ff,g}$ (bottom) as a function of log $\Sigma_{\rm gas}$. For
each cloud, the coloured line and grey band show the 50th percentile and
16th-84th percentile range of $\epsilon_{\rm ff}$ in a bin of $\log\Sigma_{\rm
gas}$. The black dashed lines show the median values $\langle\epsilon_{\rm
ff,sph}\rangle$ and $\langle\epsilon_{\rm ff,g}\rangle$ over all clouds.

We report the $\langle\epsilon_{\rm ff}\rangle$ and $\sigma$ values we measure
using the spherical assumption (denoted by subscript sph) and with equation 1
(subscript g) for all 12 clouds in Table 1. After applying our model, the
median value of $\log\langle\epsilon_{\rm ff}\rangle$ decreases from
log$\langle\epsilon_{\rm ff,sph}\rangle=-1.80$ to $\log\langle\epsilon_{\rm
ff,sph}\rangle=-1.91$. This is consistent with the prediction in Hu et al.
(2021) that use of the spherical assumption leads to a $\sim 0.13$ dex
overestimate of $\epsilon_{\rm ff}$. We also measured the difference in
dispersion $\Delta\sigma=\sigma_{\rm sph}-\sigma_{\rm g}$ derived using the
spherical assumption versus using equation 1 for each cloud. We find that 8 of
the 12 studied clouds yield positive $\Delta\sigma$, corresponding to a
reduction in the dispersion; the median reduction is $\Delta\sigma_{\rm
median}=0.03$ dex. This demonstrates that our model does decrease the
dispersion, but less than the $\sim 0.15$ dex found when testing the method on
simulated data in Hu et al. (2021). This is likely due to the difference
between the simulated data and observations. Hu et al. (2021) calibrate their
method based on simulations from Cunningham et al. (2018) that use periodic
boundary conditions, so the column density maps used in the calibration are
from infinitely large self-similar clouds. The observed clouds, however, are
from of finite size, so, for example, they can contain large-scale density
gradients that are absent in periodic boxes. This suggests that we might
obtain an improved version of equation 1 by analyzing a zoom-in galactic
simulation.

For all 12 clouds, we determine the standard deviations (STD) of both types of
$\langle\epsilon_{\rm ff}\rangle$ values: $\text{STD}_{\rm sph}=0.19$, and
$\text{STD}_{\rm g}=0.21$. The difference is not significant, but this is
perhaps not surprising because even with the spherical assumption the
dispersion of $\approx 0.2$ is very small, and may well reflect real physical
differences between clouds, as we discuss below. Given the relatively small
$\Delta\sigma_{\rm median}$ we obtain, it is also interesting to ask whether
we could forgo individualized corrections altogether, and simply adopt the
median value $g=0.24$ for all contours. Doing so would still produce a 0.1 dex
median value decrease in $\epsilon_{\rm ff}$, while leaving the dispersion
unchanged. However, such an approach would miss an important subtlety: while
$g=0.24$ is the median value for all contours on all scales, the value of $g$
also changes systematically with size scale: on the largest scales of the 12
clouds we study, $g_{\rm median}=0.35$. Properly accounting for this is
crucial to obtaining the correct changes in $\epsilon_{\rm ff}$ versus
$\Sigma_{\rm gas}$, and thus the correct $\Delta\sigma$ values within
individual clouds. For this reason, we prefer to use individual-contour
corrections when possible.

Cloud | log$\langle\epsilon_{\rm ff,sph}\rangle$ | log$\langle\epsilon_{\rm ff,g}\rangle$ | $\sigma_{\rm sph}$ | $\sigma_{\rm g}$ | $\Delta\sigma$ | $\log\Sigma_{\rm gas}$
---|---|---|---|---|---|---
|  |  | (dex) | (dex) | (dex) | ($\rm M_{\odot}/\rm pc^{2}$)
Ophiuchus | -1.44 | -1.57 | 0.35 | 0.25 | 0.10 | (2.05, 2.79)
Perseus | -1.80 | -1.96 | 0.56 | 0.62 | -0.06 | (1.85, 2.67)
Orion-A | -2.12 | \- 2.27 | 0.32 | 0.31 | 0.01 | (2.24, 3.00)
Orion-B | -1.87 | -2.03 | 0.38 | 0.31 | 0.07 | (2.10, 2.64)
Aquila-N | -1.84 | -2.02 | 0.56 | 0.44 | 0.12 | (2.02, 2.56)
Aquila-S | -1.71 | -1.86 | 0.30 | 0.23 | 0.07 | (1.92, 2.78)
NGC 2264 | -1.94 | -2.19 | 0.06 | 0.11 | -0.05 | (1.97, 2.77)
S140 | -1.62 | -1.77 | 0.07 | 0.07 | 0.00 | (1.95, 2.36)
AFGL 490 | -1.45 | -1.55 | 0.02 | 0.01 | 0.01 | (2.14, 2.32)
Cep OB3 | -1.86 | -1.86 | 0.36 | 0.30 | 0.06 | (1.92, 2.41)
Mon R2 | -1.70 | -1.97 | 0.17 | 0.20 | -0.03 | (1.77, 2.48)
Cygnus-X | -1.80 | -1.77 | 1.02 | 0.89 | 0.13 | (1.90, 2.78)
Median | -1.80 | -1.91 | 0.34 | 0.28 | 0.03 |
Mean | -1.76 | -1.90 | 0.35 | 0.31 | 0.03 |
STD | 0.19 | 0.21 |  |  |  |
Table 1: Estimates of $\langle\epsilon_{\rm ff}\rangle$ and $\sigma$ for
individual clouds, using both the spherical assumption (values with subscript
“sph”) and the Gini model equation 1 (values with subscript “g”);
$\Delta\sigma=\sigma_{\rm sph}-\sigma_{\rm g}$. The column $\log\Sigma_{\rm
gas}$ reports the (min, max) surface density measured for each cloud. Finally,
the last three rows list the median, mean, and standard deviation values (STD)
of the corresponding columns.

##  5 Conclusions

We use a new method proposed by Hu et al. (2021) to combine column density
maps derived from Herschel with young stellar objects from the SESNA catalog
to determine the star formation efficiency per free-fall time $\epsilon_{\rm
ff}$ in 12 nearby clouds. Our method provides a more realistic estimate of the
mean volume densities of clouds seen in projection, substantially reducing the
error incurred by assuming that projected clouds are spherical, and allowing
higher-precision estimates of $\epsilon_{\rm ff}$ than previously possible. We
find that the spherical assumption leads to $\sim 0.1$ dex overestimate of
$\log\langle\epsilon_{\rm ff}\rangle$, and also increases the estimated intra-
cloud dispersion in $\log\langle\epsilon_{\rm ff}\rangle$ by $\sim 0.03$ dex
on average. With our new method, we find that our sample of 12 clouds has a
median star formation efficiency per free-fall time $\log\langle\epsilon_{\rm
ff}\rangle=-1.9$, and the median spread in $\log\langle\epsilon_{\rm
ff}\rangle=0.28$ dex within a single cloud. The inter-cloud dispersion in
$\log\langle\epsilon_{\rm ff}\rangle$ is nearly identical, at 0.21 dex, and
this value is, within the uncertainties, unaffected by the use of the Hu et
al. (2021) model for the gas density. This strongly suggests that the intra-
cloud dispersion we are measuring reflects a real variation in cloud
properties, not an observational error.

Our results confirm the existence of a universal $\epsilon_{\rm ff}\sim 0.01$
value, and, importantly, let us identify a real $\approx 0.2$ dex spread from
cloud to cloud with 3D cloud geometry considered for the first time. As
discussed in Pokhrel et al. (2021), such a small spread is in tension with
models where star formation is regulated mainly by galactic-scale processes,
but individual molecular clouds undergo rapid collapse. These models predict a
much larger dispersion. Conversely, however, our measured spread in
$\epsilon_{\rm ff}$ can be used to evaluate the spread in parameters that
enter models for cloud-scale regulation of star formation, which do predict
dispersions comparable in size to the observed one. For example, in the
turbulence regulated star formation of Krumholz & McKee (2005), a $\sim 0.2$
dex spread in $\epsilon_{\rm ff}$ could naturally be explained by a
$\sigma_{\alpha}\sim 0.3$ dex spread in cloud virial parameters (or a
$\sigma_{\mathcal{M}}\sim 0.7$ dex spread in Mach number), while in the
similar model of Hennebelle & Chabrier (2011) the required dispersion is
$\sigma_{\alpha}\sim 0.8$ dex ($\sigma_{\mathit{M}}\sim 0.6$ dex), and for the
model of Padoan et al. (2012) would require $\sigma_{\alpha}\sim 0.5$ dex. For
comparison, Lee et al. (2016) study 195 star forming giant molecular clouds
and find a scatter of $0.32$ dex in the virial parameter. Thus observed clouds
have approximately the level of dispersion in virial parameter required to
reproduce the spread we see in $\epsilon_{\rm ff}$. In future work, we can use
the same technique of high-precision estimates of $\epsilon_{\rm ff}$ deployed
here to search not just for the dispersion in $\epsilon_{\rm ff}$, but to look
for systematic variations with virial parameter or other cloud properties,
thereby opening up a new method for testing theories of star formation.

## Acknowledgements

MRK acknowledges funding from Australian Research Council awards DP190101258
and FT180100375. RP and RAG acknowledge support from NASA ADAP awards
NNX15AF05G, 80NSSC18K1564 and NNX17AF24G. RP acknowledges funding from NASA
ADAP award 80NSSC18K1564, and RAG acknowledges funding from NASA ADAP awards
NNX11AD14G and NNX13AF08G. We further acknowledge high-performance computing
resources provided by the Australian National Computational Infrastructure
(grants jh2 and ek9) through the National and ANU Computational Merit
Allocation Schemes, and by the Leibniz Rechenzentrum and the Gauss Centre for
Supercomputing (grant pr32lo).

This research has made use of data from the Herschel Gould Belt survey (HGBS)
project, a Herschel Key Programme jointly carried out by SPIRE Specialist
Astronomy Group 3 (SAG 3), scientists of several institutes in the PACS
Consortium (CEA Saclay, INAF-IFSI Rome and INAF-Arcetri, KU Leuven, MPIA
Heidelberg), and scientists of the Herschel Science Center (HSC).

## Data Availability

The data underlying this article will be shared upon reasonable request to the
corresponding author.

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# Injection of anomalous Hall current: the role of impedance matching

D. Lacour  Institut Jean Lamour UMR 7198 CNRS, Université de Lorraine,
Vandoeuvre les Nancy France  M. Hehn  Institut Jean Lamour UMR 7198 CNRS,
Université de Lorraine, Vandoeuvre les Nancy France  Min Xu  Institut Jean
Lamour UMR 7198 CNRS, Université de Lorraine, Vandoeuvre les Nancy France
J.-E. Wegrowe<EMAIL_ADDRESS>Laboratoire des Solides
Irradiés, Ecole polytechnique, CNRS, CEA, Université Paris-Saclay, F 91128
PALAISEAU, France

(August 28, 2024)

###### Abstract

The electric-power of the anomalous-Hall current injected into a lateral load
circuit is studied. The anomalous-Hall current is generated by a
$\mathrm{Co_{75}Gd_{25}}$ ferrimagnetic Hall bar and injected into lateral
contacts lithographied at the two edges. The current, the voltage and the
power injected in the lateral circuit are studied as a function of the
magnetization state, the load resistance $R_{l}$, and the temperature. The
power efficiency shows a sharp maximum as a function $R_{l}$, which
corresponds to the condition of the resistance matching of the two sub-
circuits. The maximum power efficiency is of the order of the square of
anomalous-Hall angle. The observations are in agreement with recent
predictions based on a non-equilibrium variational approach.

###### pacs:

72.25.Mk, 85.75.-d

The search for low power consumption electronic devices is one of the main
motivation for the development of spintronics. Indeed, the spins attached to
the charge carriers allow a direct manipulation of the magnetization states.
Accordingly, the power used is that of the spin degrees-of-freedom, and not
directly the electric power. Yet the transport of the spins attached to the
charge carriers follows the thermodynamic rules that determines Joule
dissipation. In the case of Hall effect (HE), anomalous Hall effect (AHE) AHE
, or spin-Hall effect (SHE) SHE_Review , the Joule dissipation is minimized
due to the presence of an effective magnetic field that breaks the time-
invariance symmetry at the microscopic scaleOnsager . The effect of the
effective magnetic field on the electric carriers is then taken into account
by a typical supplementary “Hall-like” term in the Ohm’s lawJAP1 ; Madon ;
Matrix ; SHE  (see Eq.(1) below).

It is well-known that the force associated with a magnetic field - typically
the Lorentz force for the HE - cannot produce mechanical work in vacuum (the
Lorentz force is always perpendicular to the trajectory). More generally, due
to the Onsager reciprocity relations Onsager ; Supp , it is often assumed that
the Hall-current produced by Hall-like effects is dissipationlessScience04 ;
Sinova ; Onose ; Meng ; X_Jin . However, it is not necessarily the case:
typically, the power associated with the Hall voltage in a perfect Hall bar is
indeed null, but the power associated with a Corbino disk Supp  (all other
things being equivalent) dissipates Benda ; Madon ; JAP3 . This question is
the object of the present study.

The understanding of the injection of spin-current produced by SHE or AHE has
attracted much attention in the context of spin-orbit torque (SOT) effects, or
charge-to-spin conversion mechanisms. Indeed, SOT allows a ferromagnetic layer
to be reversed by the injection of spin-polarized current from an adjacent
non-magnetic Hall bar (for the SHE) or from an adjacent magnetic Hall bar (for
the AHE)Stiles1 ; Mishra ; Stiles2 ; Wu ; SOT_Review1 ; SOT_Review2 ;
SOT_Review3 ; Buhrman . These SOT effects hence render possible the direct
manipulation of magnetic memory units at nanometric scales on the basis of the
spin degrees-of-freedom only. The spin-to-charge conversion would then be
effective with minimum Joule dissipation, or even with a totally
dissipationless Hall currentDissipation .

The present study focuses on the physical conditions that govern the electric-
power injection and the efficiency of the anomalous-Hall current. In the
experimental protocol proposed here (see Fig.1), the load circuit is devoted
to measure the amount of Joule power injected from the ferrimagnetic layer due
to the AHE, regardless of the spin properties Rque ; Otani .

The measurements show that the Joule power consumption is indeed very small:
it is of the order of the square of the anomalous Hall angle
$\theta_{AH}^{2}\sim 10^{-5}$. Furthermore, a sharp maximum of the power-
efficiency of the current injection is defined by the impedance matching
between the magnetic layer and the load circuit. This observation shows that -
like for direct spin-injection into semiconductors Schmidt ; Fert  \- the
impedance matching could also be a crucial issue for metallic interfaces in
the Hall configuration.

Figure 1:  : (a) Picture of the $CoGd$ Hall bar, with the Au contact pads. The
current is injected through the vertical bar (longitudinal current). (b)
Circuit with the lateral load resistance $R_{l}$, the longitudinal-current
generator and the position of the ammeter and the voltmeters.

The samples is a $\mathrm{Co_{75}Gd_{25}}$ layer of thickness $t=30$ nm,
sputtered on a glass substrate and the buffer layer. The choice of the
ferrimagnet $\mathrm{Co_{75}Gd_{25}}$ is motivated by its high amplitude of
$AHE$, its negligible planar Hall effect (i.e. its negligible anisotropic
magnetoresistance) and its negligible magnetocrystalline anisotropy Madon .
The magnetic layer is sandwiched between $5$ nm thick Ta buffer and $3$ nm
thick Pt cap. As shown in Fig.1, the length of the Hall bar is
$L_{xx}=400\,\mu m$ and the width $W_{xx}=200\,\mu m$. The magnetic and
transport properties of the thin layers have been previously studied (see
Supplementary Materials Supp ). The magnetization is uniform for the quasi-
static states under consideration in this study. The out-of-plane shape
anisotropy corresponds to a field of about $0.8$ $T$ defined by the
magnetization at saturation. The structure of $\mathrm{CoGd}$ is amorphous and
there is no textured.

At room temperature the longitudinal voltage is $\Delta V=0.148$ $V$ and the
resistance of the $CoGd$ layer is $R_{CoGd}\approx 344$ $\Omega$. The
anomalous Hall voltage per Tesla is $V_{xy}^{0}=0.886$ $mV$ $T^{-1}$ (the
index 0 sands for the open circuit). The anomalous Hall angle per Tesla
$\theta_{AH}=\frac{V_{xy}^{0}}{\Delta V}=6$ $10^{-3}$ $T^{-1}$ is deduced,
leading the anomalous resistivity per Tesla of about $\rho_{AH}\approx
1.02\,10^{-2}$ $\mu\Omega.m.T^{-1}$.

Gold contact pads are formed thanks to a standard 2 steps UV lithographic
process. As shown in Fig.(1), two longitudinal pads allow the
$\mathrm{Co_{75}Gd_{25}}$ Hall bar to be contacted to the electric generator,
and two opposite lateral pads at the edges ($L_{xy}=450$ $\mu$m and
$W_{xy}=200$ $\mu m$) define the lateral terminals for the load resistances
$R_{l}$, range between $1\Omega$ and $100k\Omega$ (decade resistance box).
Voltmeters and ampermeter allow the lateral voltage $V_{xy}$ and the lateral
current $I_{xy}$ to be measured while injecting a DC longitudinal current of
$0.45$ $mA$ through the $\mathrm{Co_{75}Gd_{25}}$ layer (see Figure 1). An
external magnetic field $H_{app}$ varying between $\pm 1.5$ $T$ is applied
with from an electromagnet at an angle $\Phi_{\mathrm{Happ}}$ defined with
respect to the direction of the injected current $I_{xx}$.

Figure 2:  : (a) Voltage $V_{xy}$ and (b) current $I_{xy}$ measured at the
terminals of the load resistance $R_{l}$ as a function of the magnetic field
applied perpendicular to the ferrimagnetic layer. (c) The Joule Power
$P=I_{xy}V_{xy}$ measured as a function of the load resistance $R_{l}$.
Current $I_{xy}$ measured as a function of the angle $\Phi_{\mathrm{Happ}}$ of
the applied field at load resistance $R_{l}=270\,\Omega$ for three values of
the amplitude of the applied field. The continuous lines are calculated from
the AHE.

Figure 2 shows the lateral voltage $V_{xy}$ (Fig.2a), the lateral current
$I_{xy}$ (Fig.2b) and the power $P=V_{xy}I_{xy}$ (Fig.2c) measured on the load
circuit and plotted as a function of the amplitude of the magnetic field at
$\Phi_{\mathrm{Happ}}=90^{o}$ (applied normal to the layer) at room
temperature. The load resistance $R_{l}$ is used as a parameter, depicted in
the color code shown in the right (green for 1 $\Omega$ up to red for 8000
$\Omega$).

Figure 2d shows the profiles of the anomalous Hall current as a function of
the angle $\Phi_{\mathrm{Happ}}$ in a load resistance of $R_{l}=270\,\Omega$
(close to the maximum of the power in the profile of Figure 3b), for three
different amplitudes of the applied field. At $H_{app}=1.5$ $T$ (at
saturation), the magnetization direction $\vec{m}$ follows approximately the
applied field. Below $1$ Tesla, the angle of the magnetization and that of the
magnetic field are significantly different due to the shape anisotropySupp .

The continuous lines in Fig.2d corresponds to the well-known angular profile
of the AHE, $V_{xy}(\vec{H}_{app})\propto\vec{m}(\vec{H}_{app}).\vec{e}_{z}$
where $\vec{m}$ is the magnetization direction and $\vec{e}_{z}$ gives the
direction normal to the plane of the Hall bar. The contribution of the Planar
Hall effect (PHE) of the $CoGd$ layer is about two orders of magnitude below
the AHE and can be neglected Supp . The important point is that the observed
profiles show a typical signature of the AHE. It is worth pointing-out that
the analysis about AHE in Fig.2d is performed on the measurement of the
anomalous-Hall current injected in the Hall circuit, and not on the voltage of
the open circuit: such measurements have not been reported in the literature
so far (to the best of our knowledge).

The calculation of $\vec{m}(\vec{H}_{app})$ and the details of the magnetic
simulation are presented in the supplementary materials Supp  and in
references Madon . The excellent agreement between the fit and the data in
Fig.2d - together with the specificity of these profiles - confirm that the
current measured in the load circuit is due to the anisotropic Hall current
generated by the ferrimagnetic Hall bar.

As can be seen in Fig.2a, when a load resistance $R_{l}$ is contacted to the
edges, the amplitude of the anomalous Hall voltage $V_{xy}$ is a monotonous
decreasing function of $R_{l}$. In contrary, the anomalous hall current
injected into the lateral circuit is a monotonous increasing function of
$R_{l}$. This is compatible with the intuitive interpretation that the
electric charges accumulated at the edges of the Hall bar - as a consequence
of the anomalous Hall effect - are extracted and injected into the load
circuit in proportion of the load resistance.

Note that the sign of the anomalous-Hall current is inverted when the
direction of the magnetization is changed. This is a consequence of the change
of the sign of the accumulated charges when the magnetization is rotated from
up to down direction (this sign is equally inverted when direction of the
longitudinal current is reversed). The profile of power shown in Fig.2d is
well-characterized by a quadratic shape interrupted by the two horizontal
lines corresponding to the up and down saturation states of the magnetization
Rque2 . In Fig.3a the voltage variation (in Volt per Tesla) is plotted as a
function of the load resistance $R_{l}$. On the other hand, Fig.3b shows the
profile of the Joule power dissipated in the load resistance. The profile of
the power is no longer a monotonous function of the load resistance and a
sharp maximum appears for a well-defined resistance. The two typical profiles
shown in Fig.3a and Fig.3b are analyzed below.

Figure 3:  : (a) Voltage $V_{xy}$ per unit of magnetic field plotted as a
function of the load resistance $R_{l}$. The line is a one parameter fit from
Eq.(2), with the geometric adjustable parameter $\alpha$. (b) Measured
electric power $P$ as a function of the load resistance (squares). The lines
are the calculation deduced from Eq.(3). The quantitative shift in pink can be
attributed to the curvature of the current lines due to the lateral contacts

The transverse Hall-current is zero for the open circuit. In contrast, if the
Hall bar is in contact to the load circuit, an anomalous-Hall current is
generated, that injects the charge carriers accumulated at the edges into the
load circuit. The anomalous-Hall voltage decreases accordingly, as seen in
Fig.2a. At stationary regime, the out-of-equilibrium balance between the
charge accumulation and current injection into to the load circuit is
determined by the minimum power dissipation under the contraints imposed to
the system OnsagerBis ; DeGroot .

This scenario is investigated in the reference JAP3  for the usual Hall effect
in a perfect Hall bar (assuming the invariance by translation along the
longitudinal direction $x$).

The generalized Ohm’s law relates the electric current density
$\vec{J}=\\{J_{x},J_{y},J_{z}\\}$ to the electrochemical potential $\mu$. The
gradient $\vec{\nabla}\mu$ of the electrochemical potential is used instead of
the electric field in order to include the effect of the electric screening
JAP3 . The Hall-device is defined in the plane $\\{x,y\\}$, in which $x$ is
the longitudinal direction defined by the current injection from the
generator, $y$ is the transverse direction, and $z$ is the direction
perpendicular to the plane of the layer. The Ohm’s law then reduces to:

$\vec{J}=-\sigma\left(\vec{\nabla}\mu-\theta_{AH}\,\vec{e_{z}}\times\vec{\nabla}\mu\right),$
(1)

where $\times$ denotes the cross product,
$\vec{\nabla}=\\{\partial_{x},\partial_{y},\partial_{z}\\}$ is the gradient,
$\sigma=1/\left(\rho\left(1+\theta_{AH}^{2}\right)\right)$ is the conductivity
of the $CoGd$ ferrimagnetic layer Supp  while $\theta_{AH}$ is the anomalous
Hall angle introduced above. The total electric power dissipated in the device
is given by $P=\int\vec{J}.\vec{\nabla}\mu\,dv$ (integrated over the volume of
the device). After performing the functional minimization of the electric
power $P$ under the two constraints of global charge conservation and global
current conservation, the Hall voltage at the edges is given by the expression
(for constant temperature and small charge accumulation) JAP3 :

$V_{xy}(R_{l})=V_{xy}^{0}\,\frac{1}{1+\frac{\rho}{\alpha\,R_{l}}},$ (2)

where $\rho$ is the resistivity of the $CoGd$ layer and the unknown
geometrical parameter $\alpha$ is such that the ratio $\rho/\alpha$ defines
the resistance $R$ of the active region of the ferrimagnetic Hall cross. The
fit of the data in Fig.3a with the adjustable parameter $\alpha=0.93\,10^{-8}$
$m$ is given by the red curve. This coefficient
$\alpha\approx(\ell\,t)/W_{xx}$ defines the geometry of the lateral current
injection through the active part of the sample, as shown in Fig.1a. Indeed,
the anomalous-Hall current is injected through the effective section $\ell.t$
where $t=3$ $10^{-8}$ m is the thickness of the sample, $\ell$ is the size of
the lateral pad in contact to the Hall bar, and $W_{xx}=3\ell$.

On the other hand, the expression of the power readsJAP3 :

$P(R_{l})=P_{0}\,\theta_{AH}^{2}\frac{\frac{\rho}{\alpha
R_{l}}}{\left(1+\frac{\rho}{\alpha R_{l}}\right)^{2}}$ (3)

where $P_{0}\approx 64\,\mu W$ is the input power injected into the CoGd Hall
bar. This value takes into account the correction due to the current flowing
through the buffer layer and through the cap layer. The profile of the power
$P(R_{l})$ (Eq.(3)) is calculated and plotted in Fig.3b (continuous line),
together with the measured power (squares).

As can be seen, the measurements are qualitatively in agreement with the
model. A sharp maximum is measured for a load resistance equal to the
resistance $R_{l}=\rho/\alpha$. The pink zone between the calculated curve and
the experimental points shows a shift between the measurements and the theory.
It is surprising to see that the calculated profile is below the measured
profile, since the model is based on an optimized ideal Hall-bar (invariant by
translation). We ascribe the shift to the under-estimation of the measured
anomalous Hall angle $\theta_{AH}$, which is due to the inhomogeneity of the
current lines near the lateral contacts. This effect has been discussed in the
context of spin-Hall measurements Neumann , and is studied in the
supplementary materials Supp .

This observation hence validates qualitatively and quantitatively the
prediction about the maximum transfer theorem derived in reference JAP3 , and
the low power dissipated into the load circuit, of the order of
$\theta_{AH}^{2}$ times the power injected in the Hall bar. These measurements
confirm the general assumption that the anomalous-Hall current (like the Hall
current) is very sober in terms of power consumption, but invalidate the claim
that it is dissipationless.

Before concluding it is important to point-out that - as shown in Fig.4 - the
same measurements performed at different temperatures, from $T=30$ $K$ to
$T=295$ $K$, does not change the profiles (described by Eqs.(2) and Eq.(3)).
There is no qualitative change in the dissipation regime for the temperature
range under consideration. In our context, suffice it to say that the
temperature dependence of the power is defined by the temperature dependence
of the parameters $\theta_{AH}$ and $\rho$ in Eq.(3), whatever the mechanism
responsible for the presence of the effective magnetic field : either spin-
orbite coupling or Berry curvature (see the discussion of references Luttinger
; Kondo ; Berger ; Nozieres ; Bruno ; Haldane ).

Figure 4:  : (a) Voltage $V_{xy}$ per unit of magnetic field and (b)
Normalized electric power $P/P_{0}$ as a function of the load resistance
$R_{l}$ for six different temperatures.

In conclusion, the anomalous-Hall current and the electric power carried by
this current have been studied as a function of the magnetization states
$\Phi_{\mathrm{Happ}}$, the load resistance $R_{l}$ and the temperature $T$.
The observations show that the maximum electric-power dissipated in the load
circuit is indeed very small, of the order of the square of the anomalous-Hall
angle $\theta_{AH}^{2}$. Furthermore, the profile shows a sharp maximum
corresponding the resistance matching. The matching occurs when the resistance
of the load circuit $R_{l}$ equals the resistance $R$ of the active part of
the device, defined by the geometrical parameter $\alpha\approx\ell t/W_{xx}$.

Last but not least, the presence of a sharp peak of the power injected could
have importante consequences for the optimization of SOT. Indeed, the power
carried by the spins in a spin-current is supposed to be controlled by the
Joule power carried by the electric charges. The resistance $R_{l}$ is
suspected to play a crucial role in the efficiency of the spin-injection,
which could be drastically reduced in case of impedance mismatch. This study
suggests that for the search of a maximum power efficiency, the ensemble of
processes responsible for the SOT could be separated into an upstream
anomalous-Hall transducer that transforms a part of the electric power of the
generator into the Hall current (Eq.(3)), and a downstream charge-to-spin
converter that uses the spin-polarization of the current in order to switch a
magnetic layer located at nanoscopic distance from the Hall bar.

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# Metal-free Stoner and Mott-Hubbard magnetism in 2D polymers with honeycomb
lattice

Hongde Yu  Faculty of Chemistry and Food Chemistry, TU Dresden, Bergstrasse
66c, 01069 Dresden, Germany  Thomas Heine<EMAIL_ADDRESS>Faculty
of Chemistry and Food Chemistry, TU Dresden, Bergstrasse 66c, 01069 Dresden,
Germany  Helmholtz-Zentrum Dresden-Rossendorf, Institute of Resource Ecology,
Permoserstrasse 15, 04318 Leipzig, Germany  Department of Chemistry, Yonsei
University, Seodaemun-gu, Seoul 120-749, Republic of Korea

###### Abstract

We computationally demonstrate Stoner-ferromagnetic half-metals and
antiferromagnetic Mott-Hubbard insulators in metal-free 2D polymers. Coupling
radicaloid (hetero)triangulene monomers via strong covalent bonds preserving
the in-plane conjugation of the electronic $\pi$ system yields 2D crystals
with long-range magnetic order and magnetic couplings above the Landauer
limit. Dual-site honeycomb lattices produce both flat bands and Dirac cones.
Depending on the monomers, electron correlations lead to either a bandgap at
the Dirac points for antiferromagnetic Mott insulators, or Stoner
ferromagnetism with both spin-polarized Dirac cones and flat bands at the
Fermi level. These results pioneer a new type of Stoner and Mott-Hubbard
magnetism emerging in the electronic $\pi$ system of crystalline conjugated 2D
polymers.

Long-range magnetic ordering in metal-free materials is one of the unicorns of
materials science. While early investigations into magnetic carbon derived
from pressurized fullerenes garnered considerable interest, they also raised
skepticism; subsequent studies attributed observed phenomena to impurities or
defects Makarova  _et al._ (2001). The foundation of metal-free magnetism lies
in both stable paramagnetic centers and the ability to control magnetic
coupling. Half-filled flat bands in two-dimensional (2D) materials, such as
magic-angle twisted bilayer graphene Bultinck  _et al._ (2020); Sharpe  _et
al._ (2021); Tschirhart  _et al._ (2021), zigzag carbon nanoribbons Son  _et
al._ (2006); Fujita  _et al._ (1996) and 2D polymers with Lieb or kagome
lattice Springer  _et al._ (2020); Thomas  _et al._ (2019a); Jiang  _et al._
(2019); Jin  _et al._ (2017); Galeotti  _et al._ (2020), offer the
exploitation of structural topology to generate paramagnetic centers,
partially with long-range ordering. These spin-polarized flat bands harbor a
variety of exotic quantum phenomena, ranging from fractional quantum Hall
effect Tang  _et al._ (2011); Neupert  _et al._ (2011), unconventional
superconductivity Cao  _et al._ (2018); Balents  _et al._ (2020); Peri  _et
al._ (2021), non-Fermi liquid Sayyad  _et al._ (2020) to high-order
topological insulators Ni  _et al._ (2022a), spin frustration Alcon  _et al._
(2023), and Mott insulators Thomas  _et al._ (2019a); Wu  _et al._ (2018).
However, controlling the magnetic coupling in such systems remains
particularly challenging, especially above cryogenic conditions and when
aiming for metal-free ferromagnetism (FM). In this letter, we propose a novel
design concept for 2D polymers, encompassing diverse magnetic orderings and
exhibiting robust coupling constants. The concept is based on stable
paramagnetic building blocks that are coupled via covalent bonds to form 2D
polymers with strong in-plane conjugation. Leveraging strategic lattice
topology design, we observed both spin-polarized linear band crossings and
flat bands.

Figure 1:  Chemical structures of (hetero)triangulene monomers and linkages
(a) and 2D polymer (b). The red line in b shows the honeycomb lattice and the
blue line illustrates the unit cell.

Triangulene (TRI), first proposed by Erich Clar in 1953 Clar and Stewart
(1953); Clar and Schoental (1964), is recognized as the smallest nanographene
featuring a triplet ground state. This biradical with a unique non-Kekulé
structure has been synthesized both in solution and on the surface (Figs. 1a)
Mishra  _et al._ (2020); Pavliček  _et al._ (2017); Arikawa  _et al._ (2021).
In addition to pristine carbon-centered triangulene, hetero-triangulenes, such
as nitrogen-centered triangulene (TRI(N)) Wang  _et al._ (2022); Wei  _et al._
(2022) and boron-centered triangulene (TRI(B)) Hirai  _et al._ (2019), and
smaller variant phenalenyl (PLY) (i.e., [2] triangulene) Goto  _et al._
(1999); Krane  _et al._ (2023); Turco  _et al._ (2023), have also been
realized, owing to rapid advancements in precision synthesis of planar organic
radicals (Fig. 1). Notably, the TRI monomer possesses two energy-degenerate
orthogonal radical orbitals (i.e. singly occupied molecular orbitals (SOMOs))
(Fig. 2a) Pavliček  _et al._ (2017), differing from many mono-radicals with a
single SOMO, such as PLY, perchlorotriphenylmethyl (PTM), triarylmethyl (TAM),
and trioxotriangulene (TOT) Thomas  _et al._ (2019a); Wu  _et al._ (2018).
These triangular-shaped monomers can be integrated into 2D polymers via well-
established Yamamoto or Ullmann coupling reactions Mishra  _et al._ (2020,
2021); Niu  _et al._ (2022); Liu  _et al._ (2015). Here, we create 15 novel 2D
polymers, using these radicals as building blocks, linked directly or bridged
by common linear linkages, including -C$\equiv$C-C$\equiv$C- (CCCC) and phenyl
(Ph) (Figs. 1 and S1-S5).

As shown in Fig. 1b, 2D polymers made of triangulene present a honeycomb
lattice similar to graphene Novoselov  _et al._ (2004) (Fig. 1b and S6).
However, unlike graphene and many mono-radical-based 2D polymers with only one
half-filled orbital per lattice site, such as PLY, TOT, TAM, and PTM Thomas
_et al._ (2019a); Novoselov  _et al._ (2004); Neto  _et al._ (2009), the
triangulene-based 2D polymers host two orthogonal orbitals at each site
(denoted as $\pi_{x}$ and $\pi_{y}$), originating from the degenerate SOMO
orbitals of its triplet monomer (Fig. 2a). We first explore the electronic
structure of this four-orbital-based 2D lattice by employing a tight-binding
(TB) model with a ($\pi_{x}$, $\pi_{y}$) basis and the Slater-Koster scheme
Slater and Koster (1954). The corresponding effective Hamiltonian representing
single-electron properties is expressed as
$H_{eff}=\left(\begin{array}[]{cc}\begin{array}[]{cc}\varepsilon_{0}&0\\\
0&\varepsilon_{0}\\\ \end{array}&\begin{array}[]{cc}V_{xx}&V_{xy}\\\
V_{xy}&V_{yy}\\\ \end{array}\\\
\begin{array}[]{cc}V_{xx}^{\ast}&V_{xy}^{\ast}\\\
V_{xy}^{\ast}&V_{yy}^{\ast}\\\
\end{array}&\begin{array}[]{cc}\varepsilon_{0}&0\\\ 0&\varepsilon_{0}\\\
\end{array}\end{array}\right)$, where
$V_{xx}=(\frac{3}{4}pp\sigma+\frac{1}{4}pp\pi)(e^{i\mathit{k}\mathit{a}_{1}}+e^{i\mathit{k}\mathit{a}_{2}})+pp\pi\
e^{i\mathit{k}(\mathit{a}_{1}+\mathit{a}_{2})}$, $\
V_{yy}=(\frac{1}{4}pp\sigma+\frac{3}{4}pp\pi)(e^{i\mathit{k}\mathit{a}_{1}}+e^{i\mathit{k}\mathit{a}_{2}})+pp{\sigma\
e}^{i\mathit{k}(\mathit{a}_{1}+\mathit{a}_{2})}$,
$V_{xy}=\frac{\sqrt{3}}{4}(pp\sigma-
pp\pi)(e^{i\mathit{k}\mathit{a}_{1}}-e^{i\mathit{k}\mathit{a}_{2}})$,
$pp\sigma$ and $pp\pi$ are the Slater-Koster parameters Ni  _et al._ (2022a);
Jiang  _et al._ (2021a).

Figure 2:  (a) Chemical structure, the singly occupied molecular orbitals
(SOMOs) and the spin density distribution of triangulene monomer.
Illustrations of honeycomb 2D polymer with a ($\pi_{x},\pi_{y}$) basis (b) and
corresponding band structure (c) with Dirac point flanked by two flat bands.
Upon tuning the electron filling, the systems potentially change from
antiferromagnetic (AFM) to ferromagnetic (FM).

As illustrated by Fig. 2b and 2c, the ($\pi_{x}$, $\pi_{y}$)-TB model,
representing the 2D polymers of triangulenes with doubly degenerate frontier
orbitals, yields a distinctive four-band structure: a Dirac point flanked by
two flat bands. In contrast, graphene only shows a Dirac point near the Fermi
level Neto  _et al._ (2009). This four-orbital honeycomb lattice is different
from many typical hexagonal lattices, such as one-orbital honeycomb, kagome
and honeycomb-kagome lattices Springer  _et al._ (2020); Thomas  _et al._
(2019b) (Fig. S7-S9). Such an unconventional electronic structure stems from
the number of on-site orbitals and has seldomly been observed in mono-radical-
based 2D polymers, such as those made of TOT, TAM and PTM Thomas  _et al._
(2019a) (Fig. S10-S12). Beyond triangulene-based 2D polymers, this ($\pi_{x}$,
$\pi_{y}$)-TB model can also represent many 2D polymers structured with a
honeycomb lattice and comprised of monomers with doubly degenerate orbitals.
For instance, it can delineate the band structure of diamagnetic 2D covalent
organic framework (COF) with benzene nodes that also possesses energy-
degenerate highest occupied molecular orbitals (HOMOs) Ni  _et al._ (2022a)
(Fig. S13).

Under half-filling conditions, the Fermi level of carbon-centered triangulene
2D polymer, [TRI], intersects the Dirac point resembling graphene (Fig. 2c and
3a). Conversely, by adjusting the electron number of the system, it is
feasible to shift the Fermi level to align with the flat band, for example,
via introducing nitrogen or boron in [TRI(N)] and [TRI(B)]. Within the flat
bands, the many-body effects typically dominate over the kinetic energy,
resulting in a high state density of strongly-correlated electrons due to the
interference effect, despite the presence of substantial orbital overlap (Fig.
2c and 3d). This differs from the atomic flat bands observed at the core level
or in the presence of dangling bonds, which originate from the lack of overlap
between adjacent atomic wavefunction overlap Regnault  _et al._ (2022). These
flat bands can provoke many intriguing physical phenomena, such as fractional
quantum Hall effects and unconventional superconductivity, as observed in
twisted bilayer graphene and 2D kagome lattices Cao  _et al._ (2018); Sheng
_et al._ (2011); Torma  _et al._ (2022); Yankowitz  _et al._ (2019). In
addition, the high density of states near the Fermi level corresponding to the
flat bands could induce spontaneous spin-polarization according to the Stoner
model Coey (2010), where $U/t>1$ is a critical value for phase transition.
However, in most metal-free systems, introducing spin-polarization is of
fundamental challenge due to the absence of degenerate orbitals coupled with
the presence of strong electronic coupling ($t$) in $\pi$-orbitals. This
contrasts sharply with inorganic compounds featuring transition metals and
energy-degenerate d-orbitals

As carbon-centered triangulene 2D polymers, including [TRI] and [PLY], possess
balanced sublattices, Ovchinnikov’s rule is applicable and predicts that they
have a singlet ground state, i.e. $S=0$ (Fig. S14-S16). However, it is unclear
whether it is a closed-shell (diamagnetic) or open-shell (antiferromagnetic,
AFM) singlet state. In order to understand the ground state and the spin-
polarization in these polymers, we employed first-principles calculations as
implemented in Vienna Ab initio Simulation Package Kresse and Furthmuller
(1996) (VASP 5.4.4) and CRYSTAL17 Dovesi  _et al._ (2018) (see computational
details for more information). As illustrated in Fig. 3a, for the diamagnetic
state, [TRI] shows a band structure featuring a Dirac point and twin flat
bands, which can be well-represented by the four-orbital TB model with
$t=0.16$ eV. As in graphene, its Fermi level intersects the Dirac point (Fig.
3a). However, unlike graphene’s diamagnetic and semi-metallic ground state,
the spin-polarized singlet state in [TRI] is 2.17 eV more stable than the
diamagnetic one. This is in accord with the non-Kekulé structure and
corresponding unpaired electrons as featured by its radical building blocks.
This is further manifested by a substantial $U/t$ ratio of 16.6 as per the
Stoner model (Table 1). Hereafter, to evaluate the stability of spin-polarized
states, the spin-polarization energy ($\Delta E_{\text{spin}}$) is defined by
the energy difference between spin-polarized ground state ($E_{\text{GS}}$)
and diamagnetic state ($E_{\text{CS}}$), $\Delta
E_{\text{spin}}=E_{\text{GS}}-E_{\text{CS}}$.

We further find for the [TRI] that the AFM state is more stable than the FM
state, with a magnetic coupling of -32 meV between adjacent sites, according
to the Heisenberg–Dirac–van Vleck (HDVV) Hamiltonian, $\hat{H}=-\sum_{<i,\
j>}{{J\hat{S}}_{i}{\hat{S}}_{j}}$ (Fig. 3b, 3c and Table 1). From a spin-
exchange perspective, the strong electronic coupling of 0.16 eV between
neighbouring sites results in a predominant super-exchange (kinetic exchange)
of -38 meV, favoring AFM interaction and anti-parallel spin alignment (Table
1). As a consequence of the pronounced on-site Coulomb repulsion, [TRI]
becomes a Mott insulator with a bandgap of 2.66 eV, thus diverging from the
“gapless” semi-metallic nature of graphene (Fig. 3b). Indicated by the
degenerate AFM bands of [TRI], the spin-up and spin-down electrons are
energetically equivalent and delocalized in momentum space, whereas, in real
space, they are highly localized within adjacent monomers owning to the on-
site Coulomb repulsion (Fig. 3b and 3c). Similar phenomena are also observed
in metal-oxide complexes and 2D polymers made of mono-radicals, such as PTM-
COF, TOT-COF, and TAM-COF Coey (2010); Wu  _et al._ (2018); Thomas  _et al._
(2019a). Analogously, other half-filled 2D polymers, including [PLY] and
[TRI(B)-TRI(N)], also obey Ovchinnikov’s rule and exhibit AFM ground states
with $J$ values of -98 and -125 meV. These $J$ values significantly exceed the
Landauer limit for minimum energy dissipation (18 meV), indicating their
potential as promising candidates for spin logic operations at room
temperature Landauer (1961); Lambson  _et al._ (2011). Additionally, the
magnetic coupling $J$ decreases as the spacer lengthens from direct linkage to
-CCCC- and -Ph- spacers, arising from the reduction in the overlap between
localized spin-orbitals (Fig. 4e and S17-S26).

Besides Ovchinnikov’s rule, for a spin-polarized system, the Goodenough-
Kanamori rule can be employed to predict and understand the magnetic
interactions by analyzing the angle and type of adjacent spin-polarized
orbitals. Generally, within the flat-bands, the limited dispersion ($t\approx
0$) will neutralize superexchange interaction ($-4t^{2}/U$), resulting in
”orthogonal” extended wavefunctions Mori (2016). Therefore, the potential
exchange, favoring parallel spin alignments, prevails, leading to flat-band
ferromagnetism. Despite substantial observations of flat bands in diamagnetic
kagome 2D polymers, reports on metal-free ferromagnetism are still very rare
Anindya and Rochefort (2022); Ortiz  _et al._ (2022).

Table 1: Magnetic properties of the triangulene-based 2D polymers, including magnetic coupling ($J$), the magnetic moment per unit cell ($M$), electronic coupling ($|t|$), on-site Coulomb repulsion ($U$), $U/t$, and spin-polarization energy ($\Delta E_{\text{spin}}$). | Systems | $J\text{ (meV)}$ | $M\ (\mu B)$ | $U\text{ (eV)}$ | $|t|\text{ (eV)}$ | $U/t$ | $\Delta E_{\text{spin}}\text{ (eV)}$
---|---|---|---|---|---|---|---
| [TRI] | -31.8 | 0.00 | 2.66 | 0.16 | 16.6 | -2.17
| [TRI(N)] | 59.2 | 2.00 | 0.35 | 0.01 | 35.00 | -0.09
| [TRI(B)] | 55.9 | 2.00 | 0.38 | 0.01 | 38.00 | -0.09
| [TRI(B)-TRI(N)] | -124.8 | 0.00 | 0.27 | 0.12 | 2.25 | -0.07
| [PLY] | -97.6 | 0.00 | 2.61 | 0.11 | 23.7 | -1.04
Figure 3: Band structures and density of states of diamagnetic (a, d, g, j, m)
and spin-polarized ground states (b, e, h, k, n) and spin density
distributions (c, f, i, l, o) of triangulene-based 2D polymers, including
[TRI] (a, b, c), [TRI(N)] (d, e, f), [TRI(B)] (g, h, i), [TRI(B)-TRI(N)] (j,
k, l), and [PLY] (m, n, o). Blue dash lines in diamagnetic band structures are
calculated by the tight-binding model. For [TRI], [PLY], [TRI(B)-TRI(N)], the
ground states are AFM states, while they are FM states for [TRI(N)] and
[TRI(B)].

Different from half-filled triangulene 2D polymers, such as [TRI] and [PLY],
employing electron-sufficient building units as in [TRI(N)] will introduce
additional p-electrons to the systems and elevate the Fermi level to the flat
bands, while maintaining the stoichiometric and charge neutrality of the
system Jing and Heine (2018). The corresponding high density of states near
the Fermi level stimulates spontaneous spin-polarization, as demonstrated by
the large $U/t$ ratio of 35, according to the Stoner model Stoner (1938) (Fig.
3d and 3e and Table 1). Furthermore, the FM state is energetically more stable
than the AFM state with a $J$ value of 59 meV, resulting in parallel spin
alignment (Table 1 and Fig. 3f). This metal-free FM and corresponding high-
spin ground state appear to be beyond the scope of Ovchinnikov’s rule.
However, this behavior can be understood from the Goodenough-Kanamori rule.
Since the kinetic exchange $-4t^{2}/U$, stemming from the virtual hopping of
antiparallel spins, is suppressed by the limited band dispersion of the flat
band ($t=0.01$ eV), the direct exchange (i.e., potential exchange) between
neighboring parallel spins governs the magnetic interaction, leading to the
overall FM coupling (Table 1). Similarly, 2D polymers made of the electron-
deficient TRI(B) monomer, i.e. [TRI(B)], shift the Fermi level down to the
flat band as the p-electron is diminished compared to [TRI] (Fig. 3g).
Consequently, [TRI(B)] also emerges as a 2D FM polymer with a magnetic
coupling of 56 meV, arising from the reduced electronic coupling of 0.01 eV
and kinetic exchange of -1 meV (Table 1).

The Curie temperatures $T_{\text{c}}$ for [TRI(N)] and [TRI(B)] are predicted
using Monte Carlo (MC) simulations, demonstrating spontaneous FM phase
transitions occurring at about 260 K and 250 K, respectively. This indicates
the possibility of forming long-range parallel spin ordering near room
temperature (Fig. 4b and 3c). Remarkably, these $T_{\text{c}}$ surpass many
inorganic 2D van der Waals FM magnets, such as
$\text{Cr}_{2}\text{Ge}_{2}\text{Te}_{6}(J\approx 4\ \text{
meV},T_{\text{c}}=66\text{ K}$) and $\text{CrI}_{3}(J\approx 4\text{
meV},T_{\text{c}}=45\ \text{ K}$) Huang  _et al._ (2017, 2018); Gong  _et al._
(2017), which can be ascribed to the significantly stronger magnetic coupling
in these 2D polymers with spin-polarized flat bands . We notice the long-
standing controversy about the existence of magnetism in low-dimensional
systems Jiang  _et al._ (2021b), especially for the metal-free systems
featuring small spin-orbit coupling and magnetic anisotropic energy, as
Mermin-Wagner theorem prevents the long-range magnetic order at any finite
temperature as consequences of thermal fluctuations Mermin and Wagner (1966).
However, long-range spin order has been observed even in 1D systems, such as
monatomic metal chains of cobalt and zigzag edges of narrow graphene
nanoribbon Magda  _et al._ (2014); Gambardella  _et al._ (2002), in contrast
to the Ising model. Recent research shows that, at finite system size that is
over several millimeters and exceeds the scale of typical nanotechnology
devices, magnetic ordering can be stabilized by short-ranged interactions
without any magnetic anisotropy, although for infinite-size system, it will be
disturbed at non-zero temperature Jenkins  _et al._ (2022). Our findings
demonstrate that the spin-polarized flat bands at the Fermi level and metal-
free FM can be achieved by manipulation of topology and chemical components in
the non-Kekulé 2D polymers without the need for external doping or high
magnetic fields.

In contrast to the AFM Mott insulators of carbon-centered triangulene 2D
polymers and the semi-metal behavior of graphene, both [TRI(N)] and [TRI(B)]
distinctly exhibit FM half-metallic properties, where one spin channel is
(semi-)metallic and the other is single-band semiconductor (Fig. 3e, 3h, 4a
and 4b). For [TRI(N)], the Fermi level intersects the Dirac point of the spin-
down channel, indicating massless down-spin electrons, where the wavefunction
is fully extended (Fig. 3e, 4a and S27). As the FM breaks the time-reversal
symmetry and two spin-polarized bands have a linear crossing, these 2D
polymers should host Weyl fermions in comparison to Dirac fermions that
typically have four-component spinors and time-reversal symmetry Yan and
Felser (2017). Further validation of the emergence of Weyl fermions will be
investigated in future work. Conversely, the spin-up channel shows unusual
spin-polarized single-band semiconducting characteristics with a bandgap of
0.96 eV (Fig. 3e and 4a). This single-band feature provides an opportunity to
disentangle the motions of charge carriers, owning to the mobile electrons at
the parabolic conduction band maximum (CBM) and the heavy holes at the flat
valence band. [TRI(B)] also demonstrates a semi-metallic feature for the spin-
up channel, while the spin-down channel is semiconducting with a bandgap of
0.94 eV (Fig. 3h and 4b). These characteristics imply that [TRI(N)] and
[TRI(B)] can potentially serve as spin filters in organic spintronics for
down-spin and up-spin electrons, respectively Ouardi  _et al._ (2013); Wang
(2008). Moreover, due to the linear dispersion at the Dirac point, spin-
polarized currents can be transported with remarkably high mobility. Unlike
inorganic systems containing heavy elements, organic materials composed of B,
C, and N atoms generally possess smaller spin-orbit coupling, which impedes
spin relaxation, thereby maintaining spin polarization over a larger diffusion
length and longer relaxation time Dai  _et al._ (2018); Yu (2012); Alcon  _et
al._ (2023).

Figure 4:  Illustration of half-metallic band structures of [TRI(N)] (a) and
[TRI(B)] (b). Normalized spin ($S$) per site and electronic heat capacity
($C_{\text{v}}$) with the temperature from the Monte Carlo (MC) simulations of
[TRI(N)] (c) and [TRI(B)] (d). (e) Correlations between linkages and strength
of magnetic coupling ($|J|$).

The binary 2D polymer, [TRI(B)-TRI(N)], composed of alternating nitrogen- and
boron-centered triangulene monomers, is also half-filling similar to its
isoelectronic counterparts [TRI] (Fig. 3j). However, distinct from [TRI], the
different chemical potential in TRI(B) and TRI(N) induces a bandgap opening of
0.12 eV at the Dirac point for the diamagnetic state (Fig. 3j). This behavior
parallels that observed in hexagonal boron nitride (hBN) relative to graphene
and can be recapitulated by the TB model with an on-site energy difference of
0.1 eV (Fig. 3j). [TRI(B)-TRI(N)] also exhibits an AFM ground state as per
Ovchinnikov’s rule analogous to [TRI] (Table 1 and Fig. 3l). Interestingly,
different from most AFM 2D polymers with degenerate spin-up and spin-down
bands, for example, [TRI], [PLY], PTM-COF, TOT-COF, and TAM-COF Thomas  _et
al._ (2019a), [TRI(B)-TRI(N)] is a spin-polarized half-semiconductor. In
particular, [TRI(B)-TRI(N)] shows a narrow bandgap of 0.037 eV for spin-down
electrons and a substantially larger one of 0.81 eV for spin-up electrons
(Fig. 3k). This unprecedented half-semiconductor property in 2D polymers
offers immense possibilities for controlling spin conductance through external
fields. For instance, the application of laser light of a specific wavelength
or an electric field could lead to the generation of a spin-polarized current
via excitation or injection.

In summary, we proposed a spin-polarized flat band approach to realize metal-
free ferromagnetism in 2D polymers. Using first-principles calculations, we
examined 15 triangulene-based 2D polymers, where a distinct sandwiched
electronic structure with a Dirac point and twin flat bands was unveiled. We
demonstrated that half-filled systems, such as [TRI], [PLY], and
[TRI(B)-TRI(N)], had AFM magnetic couplings of -32, -98 and -125 meV,
respectively. Beyond half-filling, [TRI(N)] and [TRI(B)], showed metal-free FM
with $J$ values of 59 and 56 meV. These FM couplings were due to the
intersection of the Fermi level and the flat band, suppressing the electronic
coupling and the AFM kinetic exchange, as per the Goodenough-Kanamori rule. MC
simulations predicted Curie temperatures of 260 K and 250 K for these systems,
implying the stability of long-range spin ordering at near-room temperatures.
Compared to the AFM Mott insulators of [TRI] and [PLY], [TRI(N)] and [TRI(B)]
were half-metals. In these 2D polymers, one spin channel was semi-metallic
owing to the linear band crossing at the Fermi level, while the other spin
channel was a single-band semiconductor. Additionally, we presented
unprecedented AFM half-semiconductor properties in [TRI(B)-TRI(N)], where
spin-polarized bands are energetically non-degenerate. These spin-dependent
band gaps furnish considerable opportunities for the fine-tuning and control
of spin-conducting behavior via external fields. Our findings not only enrich
the fundamental understanding of the electronic and magnetic properties of 2D
polymers, but also provide promising candidates for the realization of metal-
free ferromagnetism and advancement of organic spintronics, particularly as
spin filters or generators.

###### Acknowledgements.

This project has been funded by the Alexander von Humboldt Foundation and
Deutsche Forschungsgemeinschaft within CRC 1415 and RTG 2861. The authors
would like to thank Dr. Thomas Brumme, Dr. Rico Friedrich, Dr. Ji Ma, Tsai-
Jung Liu, Fupeng Wu and Xingyi Wang for helpful discussions. The authors
gratefully acknowledge the computing time provided to them on the high-
performance computers Noctua 2 at the NHR Center PC2. These are funded by the
Federal Ministry of Education and Research and the state governments
participating on the basis of the resolutions of the GWK for the national
high-performance computing at universities.

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