Testing the effects from dark radiation

Yi Zhang a,b,c111Email: zhangyia@cqupt.edu.cn,Yungui Gong c,d222Email:
<EMAIL_ADDRESS>
aCollege of Mathematics and Physics, Chongqing University of Posts and
Telecommunications,
Chongqing 400065, China

bHigh Energy Physics Division, Argonne National Laboratory,
Lemont, IL 60439, USA

cInstitute of Theoretical Physics, Chinese Academy of Science,

Beijing 100190, China

dSchool of Physics, Huazhong University of Science and Technology,

Wuhan 430074, China

Abstract

In this letter, the effects of dark radiation (DR) are tested. Theoretically,
the phase-space analysis method is applied to check whether the model is
consist with the history of our universe which shows positive results.
Observationally, by using the observational data ($SNLS$ (SuperNovae Legacy
Survey) , $WMAP9$(Wilkinson Microwave Anisotropy Probe 9 Years Result),
$PLANCK$ (Planck First Data Release), $BAO$ (Baryon Acoustic Oscillations),
$H(z)$ (Hubble Parameter Data) and $BBN$ (Big Bang Nucleosynthesis)), the dark
radiation is found to have the effect of wiping out the tension between the
$SNLS$ data and the other data in flat $\Lambda CDM$ model. The effects of
dark radiation also make the best fit value of $N_{eff}$ slightly larger than
$3.04$.

##  1 Introduction

The observations hint our universe is accelerating now (e.g. Refs. [1, 2, 3,
4, 5]). The observations also show a nearly flat universe with roughly $72\%$
dark energy, $28\%$ matter and $0.1\%$ radiation (e.g. Refs.[6, 7, 8, 9, 10]).
How to describe these observations by theories? The $\Lambda$CDM model is the
simplest candidate. In $\Lambda$CDM model, the generation of neutrino is
assumed as three. And, the number of the effective neutrino species is
$N_{eff}=3.04$ where the effects from the non-instantaneous neutrino
decoupling from the primordial photon-baryon plasma are taken into account.
However, many theoretical models indicate the existing of extra radiation,
e.g. the FRW model in the Randall-Sundrum scenario [11, 12, 13, 14]; the
Brans-Dicke theory [15, 16]; the Horava-Lifshitz theory [17, Ali:2010sv]; the
decaying vacuum [19, 20, 21]; the negative Casimir effect [22].

Recently, the measurement of the temperature anisotropy of CMB (Cosmic
Microwave Background) shows less power spectrum at small scale, suggesting
that $N_{eff}$ has a bigger value than the one predicted by the standard model
of particle physics, so the existence of “dark radiation”. The results of
$WMAP7$ [10], $ACT$ (the Atacama Cosmology Telescope [23, 24]) and $SPT$
(South Pole Telescope [25]) give out the $1\sigma$ level of the effective
neutrino number are $N_{eff}=4.56\pm 0.75$($WMAP7$), $N_{eff}=2.78\pm
0.55$($WMAP7+ACT$), $2.96\pm 0.44$ ($WMAP7+ACT+SPT$); while the BBN data shows
$N_{eff}=3.24\pm 0.6$ [26]. So many discussions on dark radiation have already
appeared (e.g. Refs.[27, 28, 29, 30]).

In this letter, the $\Lambda CDM$ model with dark radiation will be used to
test the dark radiation effect which could be generated from a electroweak
phase transition [31]. The letter is organized as follows. In Section 2, the
model will be introduced. In Section 3, a phase-space analysis will be
presented to get the evolution of our universe. Then in Section 4, we apply
the observation data to test the model parameter space, including the $SNLS$
complication of supernova Ia data [32, 33], the cosmic microwave background
radiation data from $WMAP9$ [34, 35] and $PLANCK$ [36], the BAO distance
measurements from the oscillations in the distribution of galaxies [37, 38,
39, 40], the Hubble parameter data [41, 42] and the BBN data [43, 44]. We will
show the constraining results in Section 5. Finally, a short summary will be
given out in Section 6.

##  2 The Model

Here, the geometry of space-time is assumed to be described by the FRW (
Friedmann-Robertson-Walker) metric with a non-zero curvature,

$\displaystyle
ds^{2}=-dt^{2}+a^{2}(t)\left(\frac{dr^{2}}{1-kr^{2}}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\varphi^{2})\right),$ (1)

where $a$ is the scale factor, and $k$ is the curvature parameter with the
values of $0,\pm 1$ representing flat, closed and open spatial sector
respectively.

The energy density components in our universe are represented by the
pressureless matter part $\rho_{m}$, the dark energy part $\rho_{d}$, the
ordinary radiation part $\rho_{r}$, the dark radiation part $\rho_{dr}$ and
the curvature part $\rho_{k}$. The Friedmann equation is

$\displaystyle
H^{2}=\frac{1}{3m_{pl}^{2}}(\rho_{m}+\rho_{d}+\rho_{r}+\rho_{dr}+\rho_{k}),$
(2)

where $\rho_{k}=-k/a^{2}$ and $m_{pl}$ is the Planck mass. We call the
$\Lambda$CDM model plus the dark radiation as the flat or curved dark
radiation model. This kind of model could be derived from a quintessence
scenario phenomenologically whose potential includes interactions of the field
with virtual particles and the heat bath. As the potential is similar to the
Higgs potential in the electroweak phase transition, a first-order phase
transition at redshift $z\sim 3$ releases energy in relativistic model (dark
radiation). After that, $\rho_{d}$ becomes a constant; and the dark radiation
appears [31].

After defining $\Omega_{i}=\rho_{i}/8\pi GH^{2}$, $\Omega_{m}$, $\Omega_{d}$,
$\Omega_{r}$, $\Omega_{dr}$ and $\Omega_{k}$ could represent the fractional
energy densities for matter, dark energy, ordinary radiation, dark radiation
and curvature respectively. The energy components are assumed to be conserved
separately. Specially,

$\displaystyle\Omega_{r}+\Omega_{dr}=\left[1+\frac{7}{8}(\frac{4}{11})^{\frac{4}{3}}N_{eff}\right]\Omega_{\gamma},$
(3)

where $h=H_{0}/100Mpc.km.s^{-1}$, the index “0” denotes the present value of
parameter and $\Omega_{\gamma}$ is the energy density of the CMB photons
background at temperature $T_{\gamma}=2.728K$. To represent the dark
radiation, we use the symbols $f=\Omega_{dr0}/\Omega_{d0}$ which represents
the ratio of today’s dark radiation and dark energy. Then, the Friedmann
equation could be rewritten as below

$\displaystyle
H^{2}=H_{0}^{2}[\Omega_{d0}+\Omega_{k0}a^{-2}+\Omega_{m0}a^{-3}+\Omega_{r0}a^{-4}+f\Omega_{d0}a^{-4}].$
(4)

If treating the dark radiation as a signal of dark energy, dark radiation
leads to a characteristical time dependence in the effective EoS (equation of
state) parameter of dark energy,

$\displaystyle\omega(z)=\frac{p_{d}+p_{dr}}{\rho_{d}+\rho_{dr}}=\frac{f(1+z)^{4}/3-1}{f(1+z)^{4}+1},$
(5)

where $z$ is the redshift with the definition $z=a^{-1}-1$. And, the time
derivative of the EoS parameter is

$\displaystyle\omega^{\prime}=-\frac{16}{3}\frac{f(1+z)^{4}}{[f(1+z)^{4}+1]^{2}},$
(6)

where a prime means the derivative with respect to $\ln a$. Based on Eqs.5 and
6, the relations of $\omega-\omega^{\prime}$ , $\omega-z$ and
$\omega^{\prime}-z$ are list in Figure 1. The curves show the deviations from
the $\Lambda$CDM model are tiny with small $f$. Specifically, the knowledge of
$f$ is suffice to know the present value of EoS parameter where
$1+\omega_{0}=+4f/3(1+f)$ and $\omega_{0}^{\prime}=-16f/3(1+f)^{2}$. If $f$ is
at the order of $10^{-5}$, it is not surprising that the EoS parameter is very
close to $-1$ and the derivative of the EoS parameter is tiny.

Figure 1: The left, middle and right panels are the relation of
$\omega-\omega^{\prime}$, $\omega-z$ and $\omega^{\prime}-z$ separately.

##  3 The Phase-Space Analysis

To do phase-space analysis in our model, three dimensionless parameters are
defined firstly,

$\displaystyle u=\sqrt{\frac{H_{0}^{2}\Omega_{m0}a^{-3}}{H^{2}}},$ (7)
$\displaystyle
v=\sqrt{\frac{H_{0}^{2}(\Omega_{r0}+\Omega_{dr0})a^{-4}}{H^{2}}},$ (8)
$\displaystyle w=\sqrt{\frac{H_{0}^{2}\Omega_{k0}a^{-2}}{H^{2}}}.$ (9)

Based the Friedmann equation and the conserved ones 333The conserved equation
for each component is $\dot{\rho}_{i}+3H\rho_{i}(1+\omega_{i})=0$ where
$\omega_{r}=1/3$, $\omega_{dr}=1/3$, $\omega_{m}=0$ and $\omega_{d}=-1$., the
evolutions of $u,v,w$ are

$\displaystyle u^{\prime}=u(-\frac{3}{2}+\frac{3}{2}u^{2}+2v^{2}+w^{2}),$ (10)
$\displaystyle v^{\prime}=v(-2+\frac{3}{2}u^{2}+2v^{2}+w^{2}),$ (11)
$\displaystyle w^{\prime}=w(-1+\frac{3}{2}u^{2}+2v^{2}+w^{2}).$ (12)

When $u^{\prime}$, $v^{\prime}$ and $w^{\prime}$ are all equal to $0$, the
corresponding value of $u,v,w$ gives a critical point. Four points are list in
the Table 1. And, we could put a small perturbation near the critical points’
neighbor. Then, the perturbation equations are gotten. If the real parts of
the eigenvalues of the perturbation equations are all positive, the
corresponding critical point is an unstable fixed point. In contrast, the
negative real parts of the eigenvalues denotes a stable point. Specially if
the real parts of the eigenvalues are mixed with the negative one and the
positive one, the corresponding critical point is an unstable saddle point
[46, 47, 48].

Generally speaking, the model with dark radiation could go through the
unstable radiation dominated phase ($R$), the unstable matter dominated phase
($M$), the stable dark energy dominated phase ($D$) and the unstable curvature
dominated phase ($K$) 444The unstable curvature dominated phase often
represents some very early physics. Here, we ignore this phase on the
discussions of the history of our universe.. The dark energy dominated phase
is stable which means the universe will be dominated by the cosmological
constant in the future. Before that, our universe is supposed to go through
these unstable phases. This process is corresponding to the history of our
universe: the radiation dominated phase at first, then matter dominated phase
and finally the dark energy dominated phase.

Phase | Physical Meaning | $(u,v,w)$ | Existing | $(\lambda_{1},\lambda_{2},\lambda_{3})$ | Stability
---|---|---|---|---|---
$M$ | Matter Dominated | $(1,0,0)$ | Always | $(3,-2,-1)$ | Unstable
$R$ | Radiation Dominated | $(0,\pm 1,0)$ | Always | $(\frac{3}{2},4,-1)$ | Unstable
$K$ | Curvature Dominated | $(0,\pm 1,0)$ | Always | $(-\frac{3}{2},-2,2)$ | Unstable
$D$ | Dark Energy Dominated | $(0,0,0)$ | Always | $(-\frac{3}{2},-2,-1)$ | Stable
Table 1: We list the properties of the critical points: the physical meaning
of the phases, the value of the phases, the existing condition, the
eigenvalues of the points and the stability of the phases.

##  4 The Data and The Method Analysis

Once treating dark radiation as signal of dark energy, the observational
testing method used in the dark energy model could be applied to test the dark
radiation. In this section, the data and the analytical methods will be
presented separately.

###  4.1 The Data Analysis

####  4.1.1 The $SNLS$ data

SNe Ia (supernovae) is used in the standard distance method which measures the
expansion of our universe. For the $SNLS$ data, Ref. [32] gives the apparent
$B$ magnitude $m_{B}$, and the covariance matrix for $\Delta m\equiv
m_{B}-m_{\rm mod}$, with

$\displaystyle m_{\rm mod}=5\log_{10}D_{L}(z|\mbox{ s})-\alpha(s-1)+\beta{\cal
C}_{SN}+{\cal M},$ (13)

where $D_{L}(z|\mbox{ s})$ is the luminosity distance multiplied by $H_{0}$
for a given set of cosmological parameters $s$ 555 $s$ is the stretch measure
of the SNe Ia light curve shape., ${\cal C}_{SN}$ is the color measure for
supernovae and ${\cal M}$ is a nuisance parameter representing some
combination of the absolute magnitude of a fiducial SNe Ia. The time dilation
part of the observed luminosity distance depends on the total redshift $z_{\rm
hel}$ [50]

$\displaystyle D_{L}(z|\mbox{\bf s})\equiv c^{-1}H_{0}(1+z_{\rm
hel})r(z|\mbox{\bf s}),$ (14)

where $c$ is the color index, $z$ and $z_{\rm hel}$ are the CMB rest frame and
heliocentric redshifts of the supernovae. The correlated errors is

$\displaystyle\chi^{2}_{SN}=\Delta m^{T}\cdot\mbox{\ C}_{SN}^{-1}\cdot\Delta
m,$ (15)

where $\mbox{ C}_{SN}$ is the $N\times N$ covariance matrix of the SNe Ia
where $N$ is the number of the components. The nuisance parameter $H_{0}$ is
marginalized over by evaluating $\chi^{2}_{SN}$.

####  4.1.2 The CMB Data

The CMB data is implemented to add distance measurements at higher redshift
($z>10$). We use the derived quantities of the $WMAP9$ and $PLANCK$
measurements [60, 61]: the shift parameter $R(z^{*})$, the acoustic scale
$l_{A}(z^{*})$ at the decoupling redshift and the base parameter $\omega_{b}$
whose definition is $\Omega_{b}h^{2}$ where $\Omega_{b}$ is the fractional
energy densities for baryon. The $\chi^{2}$ of CMB data is

$\chi^{2}_{CMB}(\mathbf{p},\Omega_{b}h^{2},h)=\sum_{i,j=1}^{3}\Delta
x_{i}C_{CMB}^{-1}(x_{i},x_{j})\Delta x_{j},$ (16)

where the three parameters are $x_{i}=(R(z^{*}),l_{A}(z^{*}),\omega_{b})$,
$\Delta x_{i}=x_{i}-x_{i}^{obs}$ and $C_{CMB}(x_{i},x_{j})$ is the covariance
matrix for the three parameters [10, 60, 61]. The shift parameter $R$ is
expressed as $R(z^{*})=\left[\sqrt{\Omega_{m}}\rm
sinn(\sqrt{|\Omega_{k}|}\int_{0}^{z^{*}}dz/E(z))\right]/\sqrt{|\Omega_{k}|}=1.710\pm
0.019$. The acoustic scale is $l_{A}(z^{*})=\pi
d_{L}(z^{*})/(1+z^{*})r_{s}(z^{*})=302.1\pm 0.86$. And the decoupling redshift
$z^{*}$ is fitted by Ref.[51] with
$z^{*}=1048[1+0.00124(\Omega_{b}h^{2})^{-0.738}][1+g_{1}(\Omega_{m}h^{2})^{g_{2}}]=1090.04\pm
0.93$, where
$g_{1}=0.0783(\Omega_{b}h^{2})^{-0.238}[1+39.5(\Omega_{b}h^{2})^{0.763}]$ and
$g_{2}=0.560/[1+21.1(\Omega_{b}h^{2})^{1.81}]$.

####  4.1.3 The $BAO$, $H(z)$ and $BBN$ Data

To produce tightest cosmological constraint, we try to use other cosmological
probes as well.

The $BAO$ data are used as standard rule. Due to the sound waves in the plasma
of the early universe, the wavelength of $BAO$ is related to the co-moving
sound horizon at the baryon drag epoch which is $d_{z}=r_{s}(z_{d})/D_{V}(z)$,
$D_{V}$ is the effective distance with
$D_{V}(z)=\left[zd_{L}^{2}(z)/H(z)(1+z)^{2}\right]^{1/3}$, $z_{d}$ is the drag
redshift defined in [49], and $r_{s}(z)=\int_{z}^{\infty}c_{s}(x)dx/E(x)$ is
the co-moving sound horizon 666 The sound speed is
$c_{s}(z)=1/\sqrt{3[1+\bar{R_{b}}/(1+z)}]$ where
$\bar{R_{b}}=3\Omega_{b}h^{2}/(4\times 2.469\times 10^{-5})$.. For the BAO
data, we use the measurements from the 6dFGS (hereafter Bao1) [40]), the
distribution of galaxies ( hereafter Bao2) [38] and the WiggleZ dark energy
survey (hereafter Bao3) [39]. The 6dFGS (Bao6dF) gives

$\displaystyle\chi^{2}_{Bao1}=\frac{(d_{0.106}-0.336)^{2}}{0.015^{2}}.$ (17)

And, the distribution of galaxies (Bao2) measured the distance ratio at two
redshifts $z=0.2$ and $z=0.35$, whose $\chi^{2}$ is

$\displaystyle\chi^{2}_{Bao2}=\sum_{i,j=1}^{2}\Delta
d_{i}C_{dz}^{-1}(d_{i},d_{j})\Delta d_{j}$ (18)

where $d_{i}=(d_{z=0.2},d_{z=0.35})$, $\Delta d_{i}=d_{i}-d_{i}^{obs}$ and the
covariance matrix $C_{dz}(d_{i},d_{j})$ for $d_{z}$ at $z=(0.2,0.35)$ is taken
from Eq. 5 in Ref.[38]. Furthermore, the WiggleZ dark energy survey measured
the acoustic parameter $A(z)=D_{V}(z)\sqrt{\Omega_{m}H_{0}^{2}}/z$ at three
redshifts $z=0.44$, $z=0.6$ and $z=0.73$, and the results and their covariance
matrix are listed in Table 2 and Table 3 of Ref.[39]. The $\chi^{2}$ is

$\displaystyle\chi^{2}_{Bao3}=\sum_{i,j=1}^{3}\Delta
A_{i}C_{A}^{-1}(A_{i},A_{j})\Delta A_{j},$ (19)

where $A_{i}=[A(0.44),A(0.6),A(0.73)]$, $\Delta
A_{i}=A(z_{i})-A(z_{i})^{obs}$. Then, the total $\chi^{2}$ for $BAO$ is

$\displaystyle\chi^{2}_{Bao}(\mathbf{p},\Omega_{b}h^{2},h)=\chi^{2}_{Bao1}+\chi^{2}_{Bao2}+\chi^{2}_{Bao3}.$
(20)

To alleviate the double integration of the EoS parameter $\omega(z)$, we also
apply the measurements of the Hubble parameter $H(z)$ which could better
constrain $w(z)$ at high redshift. We use the $H(z)$ data at 11 different
redshifts obtained from the differential ages of red-envelope galaxies in
Ref.[41], and three more Hubble parameter data $H(z=0.24)=76.69\pm 2.32$,
$H(z=0.34)=83.8\pm 2.96$ and $H(z=0.43)=86.45\pm 3.27$ determined by Ref.[42].
Then, the $\chi^{2}$ of Hubble parameter data is

$\chi^{2}_{H}(\mathbf{p},h)=\sum_{i=1}^{14}\frac{[H(z_{i})-H_{obs}(z_{i})]^{2}}{\sigma_{hi}^{2}},$
(21)

where $\sigma_{hi}$ is the $1\sigma$ uncertainty in the $H(z)$ data.

Furthermore, the constraint data from $BBN$ is added for this dark radiation
model. The $\chi^{2}$ of the Big Bang Nucleosynthesis ($BBN$) data [43, 44,
45]) is

$\chi_{bbn}^{2}=\frac{\left(\Omega_{b0}h^{2}-0.022\right)^{2}}{0.002^{2}},$
(22)

where $\Omega_{b0}=0.02253h^{2}$ is the present value of dimensionless energy
density for baryon.

####  4.1.4 Data Discussion

To use the data properly, the $SNLS$ data will be used individually at first
and be denoted as “Data i@ ”. To utilize the $WMAP9$ and $PLANCK$ data
seperately, we treat $WMAP9+BAO+H(z)+BBN$ and $PLANCK+BAO+H(z)+BBN$ as “ Data
ii@” and “ Data iii@”. If all the three data are consistent, then we combine
them as “Data iv@: $SNLS+WMAP9+PLANCK+BAO+H(z)+BBN$”.

###  4.2 The Analysis Method

Monte Carlo Markov Chain (MCMC) method is used to compute the likelihood for
the parameters in the model. By using the Metropolis-Hastings algorithm, the
MCMC method randomly chooses values for the parameters, evaluates $\chi^{2}$
and determines whether to accept or reject the set of parameters.

####  4.2.1 The $AIC$ Principle

After combining different data, the total $\chi^{2}$ could be gotten by adding
all the observation’s $\chi^{2}$ together. The model parameters are determined
by minimizing $\chi^{2}$. To value the goodness of fitting, the $AIC$ (Akaike
Information Criterion) principle will be used which is very popular in
Mathematics and Physics [62, 63],

$AIC=\chi_{min}^{2}+2n,$ (23)

where $n$ is the number of parameters and $\chi_{min}^{2}$ is the minimum
value of $\chi^{2}$. The smaller the $AIC$ value is, the better the constraint
is. If the $\chi^{2}$ difference between two models is in a range of
$0<\Delta(AIC)<2$, the constraints of the two models are considered to be
equivalent.

####  4.2.2 The $Om$ Diagnostic

The $Om$ diagnostic [52, 53] is proposed to distinguish dynamical dark energy
from the cosmological constant both with and without the matter density. In
another saying, it is on the basis of observations of the expansion history.
The $Om$ diagnostic could be characterized by

$\displaystyle Om(x)=\frac{H^{2}(x)/H_{0}^{2}-1}{x^{3}-1},\,\,\,\,\,x=1+z.$
(24)

| Flat $\Lambda CDM$ | Curved $\Lambda CDM$ | Flat DR Model | Curved DR Model
---|---|---|---|---
Data i@: $SNLS$
$\Omega_{m0}$ | $0.226^{+0.039+0.070}_{-0.037-0.070}$ | $0.174^{+0.138+0.222}_{-0.154-0.169}$ | $0.297^{+0.306+0.383}_{-0.292-0.292}$ | $0.145^{+0.832+0.835}_{-0.138-0.140}$
$f$ | $-$ | $-$ | $-0.68^{+0.51+0.90}_{-0.12-0.93}$ | $0.22^{0.15+0.19}_{-0.74-0.90}$
$\Omega_{k0}$ | $-$ | $0.150^{+0.41+0.59}_{-0.36-0.56}$ | $-$ | $0.17^{+0.52+0.62}_{-0.82-0.88}$
$N_{eff}(10^{4})$ | $-$ | $-$ | $-0.36_{-1.34-1.65}^{\,\,+1.61+1.91}$ | $0.12_{-2.78-2.78}^{\,\,+1.06+1.54}$
$\chi^{2}$ | $420.10$ | $419.77$ | $418.60$ | $419.78$
$AIC$ | $422.10$ | $423.77$ | $422.60$ | $425.78$
Data ii@: $WMAP9+BAO+H(z)+BBN$
$\Omega_{m0}$ | $0.280_{-0.025-0.036}^{+0.029+0.044}$ | $0.280_{-0.028-0.039}^{+0.034+0.050}$ | $0.286_{-0.033-0.045}^{+0.040+0.059}$ | $0.287_{-0.038-0.051}^{+0.049+0.070}$
$f(10^{-5})$ | $-$ | $-$ | $0.37_{-1.04-1.49}^{+1.13+1.63}$ | $0.45_{-1.60-2.14}^{+1.83+2.63}$
$\Omega_{k0}(10^{-2})$ | $-$ | $0.18_{-0.93-1.31}^{+0.99+1.40}$ | $-$ | $-0.09_{-1.55-2.19}^{+1.46+1.97}$
$H_{0}$ | $70.19_{-2.29-3.40}^{+2.21+3.31}$ | $70.43_{-2.88-4.11}^{+2.86+4.02}$ | $70.73_{-3.05-4.27}^{+3.04+4.35}$ | $70.74_{-3.43-4.67}^{+3.38+4.62}$
$N_{eff}$ | $-$ | $-$ | $3.28_{-0.65-0.92}^{+0.72+1.04}$ | $3.33_{-0.10-1.32}^{+1.16+1.63}$
$\chi^{2}$ | $10.17$ | $9.95$ | $9.55$ | $9.54$
$AIC$ | $12.17$ | $13.95$ | $13.55$ | $15.54$
Data iii@: $PLANCK+BAO+H(z)+BBN$
$\Omega_{m0}$ | $0.290_{-0.020-0.030}^{+0.024+0.036}$ | $0.286_{-0.024-0.333}^{+0.290+0.417}$ | $0.296_{-0.026-0.037}^{+0.030+0.044}$ | $0.292_{-0.039-0.050}^{+0.047+0.068}$
$f(10^{-5})$ | $-$ | $-$ | $0.44_{-0.98-1.42}^{+1.00+1.43}$ | $0.29_{-1.48-2.01}^{+1.78+2.55}$
$\Omega_{k0}(10^{-2})$ | $-$ | $0.30_{-0.71-1.02}^{+0.69+0.97}$ |  | $0.17_{-1.29-1.81}^{+1.05+1.38}$
$H_{0}$ | $69.46_{-1.81-2.67}^{+1.75+2.60}$ | $70.35_{-2.90-4.11}^{+2.76+3.94}$ | $70.31_{-2.80-4.01}^{+2.95+4.15}$ | $70.46_{-3.29-4.54}^{+3.38+4.56}$
$N_{eff}$ | $-$ | $-$ | $3.32_{-0.60-0.84}^{+0.65+0.93}$ | $3.23_{-0.92-1.24}^{+1.09+1.54}$
$\chi^{2}$ | $10.73$ | $9.84$ | $9.71$ | $9.62$
$AIC$ | $12.73$ | $13.84$ | $13.71$ | $15.62$
Data iv@: $SNLS+WMAP9+PLANCK+BAO+H(z)+BBN$
$\Omega_{m0}$ | $-$ | $0.281_{-0.027-0.035}^{+0.029+0.041}$ | $0.286_{-0.025-0.032}^{+0.031+0.042}$ | $0.280_{-0.036-0.045}^{+0.049+0.063}$
$f(10^{-5})$ | $-$ | $-$ | $0.31_{-1.11-1.46}^{+1.10+1.50}$ | $0.08_{-1.53-1.86}^{+1.82+2.43}$
$\Omega_{k0}(10^{-2})$ | $-$ | $0.26_{-0.77-1.07}^{+0.75+0.95}$ | $-$ | $0.32_{-1.40-1.81}^{+0.95+1.24}$
$H_{0}$ | $-$ | $70.65_{-3.10-4.09}^{+3.39+4.25}$ | $70.78_{-3.42-4.49}^{+3.24+4.17}$ | $71.11_{-3.90-5.01}^{-3.45-4.45}$
$N_{eff}$ | $-$ | $-$ | $3.25_{-0.68-0.88}^{+0.74+1.00}$ | $3.09_{-0.97-1.18}^{+1.17+1.53}$
$\chi^{2}$ | $-$ | $433.62$ | $433.86$ | $433.66$
$AIC$ | $-$ | $437.62$ | $437.86$ | $439.66$
Table 2:  The maximum likelihood values with $1\sigma$ and $2\sigma$
confidence ranges are presented for the flat $\Lambda CDM$ model, the curved
$\Lambda CDM$ model, the flat $\Lambda CDM$ model with dark radiation (Flat DR
Model) and the curved $\Lambda CDM$ model with dark radiation (Flat DR Model).

##  5 The Fitting Results

###  5.1 The Flat and Curved $\Lambda$CDM Model

To a certain model, different data may give out very different constraining
results which is called tension. One reason of tension is the system error of
different data. The other reason could be traced to the model which may not
represent the true physics.

For flat $\Lambda CDM$ model, Table 2 shows the $1\sigma$ upper bound of
$\Omega_{m0}$ given by the SNLS data is $0.265$ which is incompatible with the
$1\sigma$ lower bound of $\Omega_{m0}$ given by Data iii@ where
$\Omega_{m0}=0.270$. In another saying, the constraining parameter range from
the two Data sets are not overlapped at $1\sigma$ level. For Data ii@, this
situation are slightly better where the lower bound of $\Omega_{m0}$ is
$0.252$. Anyway, that is just slightly overlapped with the SNLS data. Tension
exists between the $SNLS$ data and the other data (including $WMAP9$,
$PLANCK$, $H(z)$ and $BBN$). And, we could not combine all the data together
for flat $\Lambda CDM$ model.

Fortunately, after adding the curvature, Table 2 shows all the data are
consistent. The $1\sigma$ range values of $\Omega_{m0}$ are overlapped and the
tensions between the $SNLS$ data and the other data are disappeared. Thus, it
is reasonable to combine Data i@, ii@ and iii@ to get tighter constraints for
the curved $\Lambda$CDM model. Comparing the flat and curved $\Lambda CDM$
model, $\Delta(AIC)=1.67<2$; so the constants of the two models are considered
to be equivalent. This tension resolution hints the system error of the data
may not the reason. Refs.[58, 59] show that the assumption of a flat universe
induces critically large errors in reconstructing the dark energy equation of
state even if the true cosmic curvature is very small. As the dark radiation
is also a small component, in the following, we will try to answer the
question that whether the dark radiation part could alleviated the tension
problem or not based on observations777 The effect that extra radiation can
smash off the data tension are reported for other observational data, e.g.
[54, 55, 56, 57]..

###  5.2  The Flat and Curved $\Lambda$CDM Model with Dark Radiation

Figure 2:  The constraining results of the flat and curved dark radiation
models are presented in the left and right panels separately? The results are
given by the $SNLS+WMAP9+PLANCK+BAO+H(z)+BBN$ data (Data iv@).  Figure 3:  The
contours for $H_{0}$ and $N_{eff}$ are given out.

After adding the dark radiation to the $\Lambda$CDM model, Table 2 shows
$SNLS$ data gives a loose parameter range. As we expected, this tension
problem is disappeared. Therefore, it is reasonable to use the combined
$SNLS+WMAP9+PLANCK+BAO+H(z)+BBN$ data. It gives out the tightest constraints.
Then, what results could we get if we add both the curvature and the dark
radiation to the $\Lambda$CDM model? As shown in Figure 2, the parameter
ranges of the curved one are slightly enlarged compared to the flat one.

Generally speaking, the $SNLS$ data gives very poor constraints on the model
parameter compared to other data. Data ii@, iii@ and iv@ present
$\Omega_{k0}\sim 10^{-2}$ and $f\sim 10^{-5}$ which denote the price we paid
for the disappeared tension is reasonable. The AIC analysis also shows the
constraints on both the flat and curved ones are equal because the
$\Delta(AIC)$ is less than $2$. Anyway, the $PLANCK+BAO+H(z)+BBN$ data gives a
tighter constraint than the $WMAP9+BAO+H(z)+BBN$ data.

####  5.2.1 The Dark Radiation

Again, as Table 2 hints, the $SNLS$ data is not sensitive to the effective
neutrino number. In contrast, the constraints from other data are at much
smaller orders. For concise, we only discuss the tightest constraints from
Data<EMAIL_ADDRESS>The combined data favor a positive $f$ which denotes the new
produced dark radiation. Based on our definition, the data gives out
$N_{eff}=3.25_{-0.68-0.88}^{+0.74+1.00}$ in flat dark radiation model and
$N_{eff}=3.09_{-0.97-1.18}^{+1.17+1.53}$ in the curved one. Dark radiation
makes the best fit of $N_{eff}$ slightly larger than $3.04$. We compare the
flat and curved cases by drawing the contours of $H_{0}$ and $N_{eff}$ in
Figure 3 where the ranges of $H_{0}$ are nearly the same, but the range of
$N_{eff}$ is larger in the curved case.

####  5.2.2 The $Om$ Diagnostic

The $Om$ diagnostic is used to distinguish the dark radiation effect. For our
model,

$\displaystyle
Om(x)=\Omega_{m0}+\frac{(\Omega_{r0}+\Omega_{dr0})(x^{2}+1)(x+1)}{x^{2}+x+1}+\frac{\Omega_{k0}(1+x)}{x^{2}+x+1}.$
(25)

Generally, the effect of today’s dark radiation makes $\delta
Om_{dr0}<4\Omega_{dr0}/3$. Meanwhile, the effect of today’s curvature makes
$\delta Om_{k0}<\Omega_{dk0}/2$. As $f$ (or $\Omega_{dr0}$) are relative
small, $\delta Om_{dr0}$ is smaller than $\delta Om_{k0}$. Robustly,
$\Omega_{dr0}$ ($\sim 10^{-5}$) is four orders smaller than $\Omega_{m0}$
($10^{-1}$) and three order smaller than $\Omega_{k0}$ ($10^{-2}$). The
relation of $Om-z$ are drawn out in Figure 4 for the flat and curved dark
radiation cases. In the flat one, the best fitting value of $Om$ is nearly
constant, so does its $1\sigma$ and $2\sigma$ ranges. But the behavior of $Om$
in the curved $\Lambda CDM$ model and the curved dark radiation model show
dynamical signals. Considering the flat $\Lambda CDM$ model which gives a
constant $Om$ [52] as well, the flat dark radiation model could not be
distinguished from the flat $\Lambda CDM$ model while the curved dark
radiation model can.

Figure 4:  The Om diagnostic for the curved $\Lambda CDM$ model, the flat and
curved dark radiation models are give out.

Figure 5: The $1\sigma$ parameter contours given by different data are
presented.

####  5.2.3 Parameter Degeneration

Figure 5 presents the contours of $\Omega_{m0}-f$, $\Omega_{k0}-f$ and
$\Omega_{m0}-\Omega_{k0}$ of the flat and curved dark radiation models given
by different datasets. As we mentioned above, the $SNLS$ data give out loose
constraints. Meanwhile, the three data (Dataii@, iii@ and iv@) give much
tighter constraints which also have the same contour directions. For the
contour of $\Omega_{m0}-\Omega_{k0}$, all the data give the same constrain
direction. In contrast, for $\Omega_{m0}-f$ and $\Omega_{k0}-f$, the SNLS data
and the other data gives contours with different directions. Obviously, the
degeneration between the dark radiation parameter $f$ and the other parameters
need more data to break.

##  6 The Summary

Theoretically, after adding dark radiation, the phase-analysis method proved
that the universe derived from the dark radiation model could go through the
radiation dominated phase, the matter dominated phase and the dark energy
dominated phase sequently. In a conclusion, the model is compatible with the
history of our universe.

Observationally, we use the $SNLS$, $WMAP9$, $PLANCK$, $BAO$, $H(z)$ and $BBN$
data to constrain the dark radiation part. As expected, the dark radiation
wiped out the tension between the $SNLS$ data and the other data in flat
$\Lambda CDM$ model. And, the constraining results are at a reasonable level,
e.g. $f\sim 10^{-5}$. The small dark radiation parameter $f$ give a small
deviation of $\omega_{0}$ and $\omega_{0}^{\prime}$. And, the effect of dark
radiation make the best fit value of $N_{eff}$ slightly larger than $3.04$.
Anyway, the $Om$ diagnostic could extract the curved dark radiation from the
flat $\Lambda CDM$ model, but it has no effect on the flat dark radiation
model. And, more data are needed for dark radiation because of parameter
degenerations.

##  7 Acknowledgements

We are grateful to the useful suggestions from the anonymous referee. YZ
thanks the useful discussion with Dr. Hao Wang, Dr. Hongbo Zhang, Dr. Yu Pan,
Prof. Nana Pan. This work was supported by the Ministry of Science and
Technology of China national basic science Program (973 Project) under grant
Nos. 2010CB833004, the National Natural Science Foundation of China project
under grant Nos. 11175270, 11005164, 11073005 and 10935013, CQ CSTC under
grant No. 2010BB0408, and CQ MEC under grant No. KJTD201016. Part of this
research was supported under the U.S. Department of Energy contract DE-
AC02-06CH11357 and by the DOE under contract W-7405-ENG-36.

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††thanks<EMAIL_ADDRESS>
# Giant anomalous Hall effect in epitaxial Mn3.2Ge films with a cubic kagome
structure

J. S. R. McCoombs  Department of Physics and Atmospheric Science, Dalhousie
University, Halifax, Nova Scotia, Canada B3H 3J5  B. D. MacNeil  Department of
Physics and Atmospheric Science, Dalhousie University, Halifax, Nova Scotia,
Canada B3H 3J5  V. Askarpour  Department of Physics and Atmospheric Science,
Dalhousie University, Halifax, Nova Scotia, Canada B3H 3J5  J. Myra
Department of Physics and Atmospheric Science, Dalhousie University, Halifax,
Nova Scotia, Canada B3H 3J5  H. Herdin  Department of Physics, Simon Fraser
University, Burnaby, British Columbia, Canada, V5A 1S6  M. Pula  Department of
Physics and Astronomy, McMaster University, Hamilton, Ontario, Canada L8P 4N3
M. D. Robertson  Department of Physics, Acadia University, Wolfville, Nova
Scotia, Canada B4P 2R6  G. M. Luke  Department of Physics and Astronomy,
McMaster University, Hamilton, Ontario, Canada L8P 4N3  K. L. Kavanagh
Department of Physics, Simon Fraser University, Burnaby, British Columbia,
Canada, V5A 1S6  J. Maassen  Department of Physics and Atmospheric Science,
Dalhousie University, Halifax, Nova Scotia, Canada B3H 3J5  T. L. Monchesky
Department of Physics and Atmospheric Science, Dalhousie University, Halifax,
Nova Scotia, Canada B3H 3J5

###### Abstract

We report on the first example of epitaxial Mn3+δGe thin films with a cubic
$L1_{2}$ structure. The films are found to exhibit frustrated ferromagnetism
with an average magnetization corresponding to 0.98 $\pm$ 0.06 $\mu_{B}$/Mn,
far larger than the parasitic ferromagnetism in hexagonal Mn3Ge and the
partially compensated ferrimagnetism in tetragonal Mn3Ge. The Hall
conductivity is the largest reported for the kagome magnets with a low
temperature value of $\sigma_{xy}=1587$ S/cm. Density functional calculations
predict that a chiral antiferromagnetic structure is lower in energy than a
ferromagnetic configuration in an ordered stoichiometric crystal. However,
chemical disorder driven by the excess Mn in our films explains why a
frustrated 120$\degree$ spin structure is not observed. Comparisons between
the magnetization and the Hall resistivity indicate that a non-coplanar spin
structure contributes the Hall signal. Anisotropic magnetoresistance and
planar Hall effect with hysteresis up to 14 T provides further insights into
this material.

##  I Introduction

Magnetic topological metals offer opportunities to explore the entanglement of
real-space and momentum space spin-structures. In Weyl semimetals formed by
broken time reversal symmetry, the real-space spin texture provides a way of
manipulating the topological electronic states. In the case of the hexagonal
Mn3Sn and Mn3Ge kagome materials, a rotation of the chiral antiferromagnetic
spin configuration (shown in Fig. 1(b)) with an external magnetic field causes
a rotation of the Weyl nodes Kuroda et al. (2017). The momentum-space Berry
curvature in these structures leads to a large anomalous Hall effect (AHE)
Nakatsuji et al. (2015); Kiyohara et al. (2016) and anomalous Nernst effect
(ANE) Ikhlas et al. (2017); Hong et al. (2020) that have created opportunities
for developing devices. The antiferromagnetic chiral structure of Mn3Ge has
also been used to manipulate spin-triplet Cooper pairs in a Josephson
junction, with potential applications in superconductor logic circuits Jeon et
al. (2023).

The closely related set of cubic kagome antiferromagnets Mn3X, X = {Ir, Pt, Rh
} have a positive vector chirality Tomeno et al. (1999); Yasui et al. (1992);
Krén et al. (1968) (see Fig. 1(e)), opposite to that of Mn3Sn Tomiyoshi and
Yamaguchi (1982) and Mn3Ge Tomiyoshi et al. (1983). Their topological band
structures give rise to large anomalous Hall Liu et al. (2018a); Zuniga-
Cespedes et al. (2023) and spin-Hall effects Zhang et al. (2016). The large
uniaxial anisotropy in the case of Mn3Ir leads to a canting of the spins out
of the kagome planes and to a scalar spin chirality
$\mathbf{S}_{i}\cdot(\mathbf{S}_{j}\times\mathbf{S}_{k})$. However, it is
spin-orbit coupling and broken mirror symmetry of the spin structure that
gives rise to large momentum-space Berry curvature in this material Chen et
al. (2014).

Figure 1: (a) The expected epitaxial relationship of $L1_{2}$ Mn3Ge atop SiC.
(b) The crystal structure and inverse-triangular, AF spin-configuration of
hexagonal $D0_{19}$ Mn3Ge. (c) The crystal structure and DFT calculated
ferromagnetic spin-configuration of cubic $L1_{2}$ Mn3Ge. (d) The crystal
structure and ferrimagnetic spin-configuration of $D0_{22}$ Mn3Ge. (e) The
cubic $L1_{2}$ structure with DFT calculated triangular, AF spin-
configuration.

In this paper we explore the properties of Mn3.2Ge films, which, in fact, has
very similar hexagonal, cubic and tetragonal polytypes with dramatically
different magnetic properties. The phase that is obtained depends sensitively
on the substrate and growth conditions. In bulk crystals, the hexagonal
$\epsilon$-Mn3Ge phase ($D0_{19}$ crystal structure with space group #194,
$P6_{3}/mmc$), is obtained by annealing at temperatures above 500 $\degree$C
Ohoyama (1961); Yamada et al. (1988). The lattice parameters are $a_{h}=0.534$
nm and $c_{h}=0.431$ nm Ohoyama (1961). The Mn atoms occupy the $6h$ Wyckoff
sites and form an AB-stacking of kagome layers with their spins pointing in
the plane at an angle of 120$\degree$ to their neighbors Kádár and Krén
(1971). This order exists below a Curie temperature of 365 K Yamada et al.
(1988). The in-plane easy axis magnetocrystalline anisotropy induces a weak
moment of 0.007 $\mu_{B}$/Mn Yamada et al. (1988)in the kagome plane Tomiyoshi
et al. (1983). Epitaxial Mn3.22Ge(0001) films were achieved on sapphire (0001)
substrates with, a Ru buffer layer, by co-sputtering Mn and Ge at a rate of 10
nm/min and subsequent annealing at 500 $\degree$C Hong et al. (2020), as well
as by molecular beam epitaxy on LaAlO3(111) at a temperature of $570\degree$C
Hong et al. (2022).

Tetragonal $\epsilon_{1}$-Mn3Ge ($D0_{22}$ crystal structure with space group
#139, $I4/mmm$ shown in Fig. 1(d)) is the thermodynamically stable phase below
500$\degree$C, which possesses ferrimagnetic order with spins parallel to the
c-axis Ohoyama (1961) and an estimated Curie temperature of 650$\degree$C
Kádár and Krén (1971). This phase, with lattice parameters $a_{t}=0.3816$ nm
and $c_{t}=0.7216$ nm Kádár and Krén (1971), grows epitaxially on SrTiO3(001),
which has a lattice parameter 2.3% larger than the a-parameter Kurt et al.
(2012). Epitaxial films have also been grown on MgO(001) with a Cr buffer
layer Sugihara et al. (2014). The films have strongly compensated moments with
a net magnetization $M_{s}=73$ kA/m, and a very high uniaxial anisotropy
$K_{u}=0.91$ MJ/m3 despite the absence of heavy elements. The tetragonal phase
can be transformed into the cubic variant by stretching the unit cell by 5%
along the c-axis and interchanging one of the Ge atoms on the $2a$-sites with
one of the Mn atoms on the $2b$-sites. In this way, the tetragonal phase can
be viewed as a distorted ABC-stacked kagome structure. Cubic Mn3Ge structure
($L1_{2}$ structure with space group #221, $Pm\bar{3}m$) differs from the
hexagonal phase only in its ABC-stacking of the kagome layers. The Mn atoms
sit on the $3c$ Wyckoff sites, corresponding to the cubic unit cell faces.
There is only one report to date on the synthesis of this material, which was
achieved by reacting Mn and Ge and high temperatures and pressures Takizawa et
al. (2002). The cubic lattice parameter $a=0.3802(1)$ nm, is close to both the
$a_{t}$ and $a_{h}/\sqrt{2}$ parameters. However, the magnetic properties are
quite different: magnetometry measurements revealed ferromagnetic behavior,
with a saturation magnetization that corresponds to $0.87~{}\mu_{B}$/Mn and a
Curie temperature, $T_{C}$, of approximately 400 K.

Here, we report the first demonstration of the growth of cubic Mn3+δGe films.
We used 4H-SiC(0001) substrates, which have a $-0.75\%$ lattice mismatch.

The rest of this paper is structured as follows: We first describe the growth
procedure and outline the structural experiments we conducted which lead us to
the conclusion that we have an epitaxial, cubic kagome system. We go on to
probe the spin-configuration in this system with complementary in-plane and
out-of-plane magnetometry and DC transport measurements. To partially explain
the interesting results we find, we turn to density functional theory and
discuss how this new system fits into the already rich physics of the Mn3Ge
system.

##  II Growth

We used 4H-SiC(0001)$\pm 0.5\degree$ Si-terminated wafers with a resistivity
of $\approx 1\times 10^{5}~{}\Omega$cm as substrates. Prior to growth, these
substrates were degreased in acetone and methanol, rinsed in deionized water,
dried with nitrogen, then immediately loaded into a VG–V80 SiGe molecular beam
epitaxy (MBE) chamber with a base pressure of less than $4\times 10^{-11}$
Torr. The substrates were then degassed at $600\degree$C for 12 hr, followed
by flashing to $1100\degree$C for 1 hr to remove the SiO2 layer. This process
results in a $\left(\sqrt{3}\times\sqrt{3}\right)R30\degree$ surface
reconstruction free of any oxygen, as is evidenced by in situ reflective high-
energy electron diffraction (RHEED) and Auger electron spectroscopy (AES). The
substrate is then cooled at a rate of approximately $20\degree$C/s to room
temperature, $T_{R}$, for co-deposition of Mn via effusion cell and Ge via
electron beam evaporation. Upon reaching $T_{R}$, Mn and Ge are deposited at
an atomic ratio of 5:1. The Mn rate is monitored with a calibrated ion gauge,
while the Ge rate is monitored with a quartz oscillator. Upon completion of
co-deposition, the sample was heated to $600\degree$C for 1 hr while monitored
with RHEED until diffraction spots are recovered.

Due to Mn evaporation from the film at elevated substrate temperatures, a
ratio Mn:Ge = 5:1 was required to achieve the desired stoichiometery, as is
evidenced by energy dispersive electron spectroscopy (EDS). For this reason,
we use the total deposited thickness of Ge as a calibration of the expected
thickness of our Mn3Ge films. The number density of elemental Ge is
$n_{\text{Ge}}^{\text{Ge}}=0.04412~{}\textup{\AA}^{-3}$, while the number
density of Ge in Mn3Ge is
$n_{\text{Mn}_{3}\text{Ge}}^{\text{Ge}}=0.0182~{}\textup{\AA}^{-3}$. Hence,
for a thickness of $t_{\text{Ge}}$ deposited we expect a thickness of

$t_{\text{Mn}_{3}\text{Ge}}=\frac{n_{\text{Ge}}^{\text{Ge}}}{n_{\text{Mn}_{3}\text{Ge}}^{\text{Ge}}}\frac{t_{\text{Ge}}}{\alpha}$
(1)

for our film, where $\alpha=0.8$ accounts for the remaining excess Mn
occupying Ge sites.

Lastly, before removing the sample from the MBE, it is again cooled to $T_{R}$
for the deposition of a 10 nm protective, amorphous layer of Si.

##  III Structural and chemical characterization

###  III.1 X-ray Diffraction

After aligning the substrate via $\omega$-scan such that the SiC(0001)
direction is exactly along $q_{z}$, traditional $\theta$-$2\theta$ scans are
taken using a Siemens D500 diffractometer using a Cu source and monochrometer
which allows only Cu K-$\alpha$ wavelengths. Figure 2 shows the diffraction
pattern obtained exhibiting only two film peaks with appreciable intensity and
indicating a plane spacing of $2.196~{}\textup{\AA}$ for planes parallel with
the surface of the film.

Figure 2: Log-scale plot of traditional $\theta$-$2\theta$ diffraction pattern
using Cu-K$\alpha$ X-rays. The SiC(0004) and SiC(0008) reflections are
omitted. Solid blue indicates positions of allowed and forbidden
SiC(000$\ell$) reflections, dashed green indicates Mn3Ge (111)-oriented
reflections and dot-dashed red indicates other relatively high structure
factor Mn3Ge peak locations.

Allowed and forbidden SiC$(000\ell)$ reflection angles are highlighted in
blue, Mn3Ge(111) oriented reflections are highlighted in green and other Mn3Ge
reflection locations are shown in red. There is one unaccounted for peak at
$2\theta\approx 46\degree$. This is likely due to another Mn-Ge phase, but the
intensity of this peak is completely insignificant compared to the Mn3Ge(111)
peak. In fact, the only two peaks with any significant intensity are the
Mn3Ge(111) and (222) peaks. This alignment of the film tracks with the
expected epitaxial relationship of Mn3Ge atop SiC. A cubic system with lattice
parameter 0.3803 nm (as obtained by the (111) plane spacing) should have an
in-plane, pseudo-hexagonal lattice parameter of $a_{h}=\sqrt{2}a=0.53356$ nm
Bauer et al. (2001). This unit cell dimension is accommodated on the SiC(0001)
surface by considering a $\left(\sqrt{3}\times\sqrt{3}\right)R30\degree$-type
epitaxy which leads to the expected relationship

$\displaystyle[1\bar{1}0]\text{Mn}_{3}\text{Ge}(111)\|[1\bar{1}00]\text{SiC}(0001).$
(2)

This expected alignment is shown in Fig. 1, which also highlights that there
should be a compressive strain on the film since its in-plane lattice
parameter is 0.80% larger than that of the SiC $R30\degree$ cell.

To verify our expected epitaxial relation and lend credibility to our claim of
a cubic phase of Mn3Ge forming, we explore the reciprocal space map (RSM) of
our films. To distinguish between the two primary, perpendicular, in-plane
directions we will call the direction along SiC$[2\bar{1}\bar{1}0]$
$\hat{\mathbf{x}}$ and the direction along SiC$[10\bar{1}0]$
$\hat{\mathbf{y}}$, with the direction along SiC$[0001]$ reserved for
$\hat{\mathbf{z}}$ of course.

Figure 3 shows a portion of the $(q_{x},q_{z})$ reciprocal space highlighting
the (311) and (211) cubic peaks, which correspond to
$[2\bar{1}\bar{1}]\text{Mn}_{3}\text{Ge}(111)\|[\bar{1}2\bar{1}0]\text{SiC}(0001)$
regions of the film, as well as (022) and (122) peaks, which correspond to
$[\bar{2}11]\text{Mn}_{3}\text{Ge}(111)\|[\bar{1}2\bar{1}0]\text{SiC}(0001)$
regions of the film. The remaining peaks are surmised to be due to a
distortion of the perfect cubic structure along the [100] direction. The
direction of the distortion is consistent with the [001] direction of a
$(112)$-oriented $D0_{22}$ film atop SiC. However, the RSM can only be
explained by a stretching of the cubic cell along [100], with a resulting c/a
ratio of 2.07, which is opposite to the distortion of $D0_{22}$ Mn3Ge with c/a
= 1.90.

Figure 3: Reciprocal space map of the $(q_{x},q_{z})$ plane. Gold, un-starred
labels represent diffraction spots from the perfectly cubic system with
lattice parameter 0.3803 nm. Magenta, starred labels represent diffraction
spots from a cubic system stretched along the [100] direction by 3%.

###  III.2 Microscopy

Scanning transmission electron microscopy (STEM) images were collected with a
200 kV field-emission microscope. Plan-view specimens were prepared by low
angle mechanical polishing and attached to TEM grids with silver epoxy
Robertson et al. (2006). Figure 4(a) shows the presence of holes in the film:
energy dispersive X-ray spectroscopy (EDS) show no Mn or Ge in these regions.
In the other regions of the sample, EDS finds that the film have an excess of
Mn with a ratio of Mn:Ge = 3.2(0.2):1. We have attempted to perform selected
area diffraction. While we do find some randomly distributed diffraction spots
that are not aligned with the substrate, we believe that the majority of the
diffracted intensity from the film is obscured by diffraction spots from the
SiC.

Surface morphology was investigated with atomic-force-microscopy (AFM). The
surface is revealed to be largely continuous with some holes that extend to
the substrate (Fig. 4(b)), as seen in the STEM results. The AFM topographical
map gives root-mean-squared roughness of 31.9 nm and a corresponding nominal
thickness of 43.9 nm. This thickness is consistent with the peak widths of the
(111) and (222) reflections measured in the $\theta$-$2\theta$ X-ray
diffraction scan. This thickness is also consistent with the nominal amount of
Ge deposited as described by Eq. 1 if $\alpha=0.8$, which matches the EDS
measurement.

Figure 4: a) HAADF-STEM image revealing the film island morphology. The arrow
shows one of the holes in the film exposing the bare SiC substrate. A crack
running through the tip of the arrow resulted from the mechanical polishing.
b) AFM of the film surface showing the distribution of holes.

##  IV Magnetometry and Transport

Out-of-plane (OOP) magnetization measurements were performed with a Quantum
Design MPMS SQUID magnetometer in the traditional RSO mode with a maximum
applied field of 7 T. Transport measurements were performed on a Quantum
Design Dynacool PPMS with a maximum applied field of 14T. For transport
measurements, samples were cleaved into $\sim(6\times 1)$ mm rectangles with
constant-current-supplying leads $I^{+}$, $I^{-}$, transverse voltage leads
$V_{xy}^{+}$, $V_{xy}^{-}$ and longitudinal voltage leads $V_{xx}^{+}$,
$V_{xx}^{-}$ wire-bonded to the surface using 20 $\mu$m aluminum wire. Five
constant current leads are used such that the electric field is highly uniform
in the region where the voltage leads are attached, as verified by finite
element analysis of this geometry.

###  IV.1 Out-of-Plane Field

Figure 5 shows $M$-$H$ and the anti-symmetrized $\rho_{xy}$-$H$ hysteresis
loops at various temperatures ranging from 400 K down to 10 K. The linear,
diamagnetic susceptibility of the SiC substrates and the linear, conventional
Hall effect have been subtracted away from the presented data. It should be
noted that although there appears to be saturation in $\rho_{xy}(H)$ for all
temperatures considered, there is a small non-linearity that persists up to 14
T which becomes less prevalent as temperature is reduced.

Figure 5: (a) Magnetization hysteresis loops with external field parallel to
SiC$\left[0001\right]$. The diamagnetic susceptibility of the SiC substrate
has been subtracted. (b) Anti-symmetrized transverse resistivity hysteresis
loops with external field parallel to SiC$\left[0001\right]$. The
conventional, linear Hall effect has been subtracted. Insets show the high
field behaviours of $M$ and $\rho_{xy}$.

Figure 6 shows the difference between the field-decreasing branch of the
hysteresis loops and the field-increasing branch. Clear deviation between the
behaviour of $\rho_{xy}(H)$ and $M(H)$ is observed.

Figure 6: Differences between the field-decreasing and field-increasing (a)
magnetization and (b) transverse resistivity extracted from the isothermal
hysteresis loops presented in Fig. 5.

These differences are summarized in Fig. 7. While the coercive field $H_{C}$,
saturation $M_{S}$ and remanencence $M_{R}$ of the magnetization are all
monotonically increasing with decreasing temperature, the complimentary values
of $\rho_{xy}$ show a more complex dependence. Notably, at $\sim 250$ K, there
is a sharp decrease in both the remanent Hall resistivity $\rho_{xy}^{R}$ and
the field where $\rho_{xy}$ changes sign.

Figure 7: Temperature dependence of (a) saturation magnetization $(M_{S})$ and
saturation transverse resistivity $\rho_{xy}^{S}$, (b) coercive fields of $M$
and $\rho_{xy}$ and (c) magnetization $M_{R}$ and remanent transverse
resistivity $\rho_{xy}^{R}$. All quantities were extracted from the series of
isothemal hysteresis loops shown in Fig. 5.

The significance of this temperature is highlighted in zero-field-warming
(ZFW) measurements of the remanent OOP magnetization. Field cooling at 7T(-7T)
followed by ZFW shows a ferromagnetic-type behaviour up to $T_{C1}=260$ K,
followed by a magnetization reversal above $T_{C1}$. This is contrary to the
strictly positive(negative) remanent moments derived from the isothermal
hysteresis loops above. We attribute this reversal to an antiferromagnetic
coupling between Mn moments located on the central, nominally Ge-occupied
positions and the triangularly coordinated, nominally Mn-occupied positions,
leading to a ferrimagnetic-like behaviour.

Figure 8: (a) ZFW curves and their derivatives after field cooling in a 7 T
and -7 T field. (b) 100 mT FC and FW curves and their derivatives showing
clear thermal hysteresis. (c) 500 mT and 3 T FC curves and their derivatives.
In all plots, the left axis corresponds to the thick $M(T)$ curves while the
right axis corresponds to the thin $dM(T)/dT$ curves.

This thermally onset transition is also observed in field-cooled/field-warmed
scans at various values of $\mu_{0}H$, manifesting as a spike in $dM/dT$ and
moving to slightly higher temperatures as the external field increases. At low
fields, there is substantial thermal hysteresis which recedes as the applied
field increases in magnitude.

A more comprehensive investigation of the temperature dependence of
$\rho_{xy}$ was achieved by field cooling at 5 T and zero-field warming
resulting in a temperature dependence $\rho_{xy}^{0+}(T)$ followed by field
cooling at -5 T and zero-field warming resulting in a temperature dependence
$\rho_{xy}^{0-}(T)$. By examining
$\left(\rho_{xy}^{0+}(T)-\rho_{xy}^{0-}(T)\right)/2$, we obtain the remanent
transverse resistivity as a function of $T$. This temperature dependence is
shown in Fig. 9, along with the temperature dependence of $\rho_{xx}$ at zero
field.

Figure 9: Temperature dependence of (a) $\rho_{xy}^{R}$ and $\rho_{xx}$ upon
warming after field-cooling at 5 T and (b) the derivatives of $\rho_{xy}^{R}$
and $\rho_{xx}$. Vertical lines have been added at the temperatures
$T_{C1}=260$ K and $T_{C2}=377$ K.

Possibly due to the higher maximum temperature of this measurement, the
remanent $\rho_{xy}(T)$ on zero-field-warming exhibits no discernible feature
at $T_{C1}$ and is substantially higher than the remanent values obtained from
the hysteresis loops in Fig. 5. Instead, after peaking at 200 K, it linearly
decreases until it abruptly approaches zero at $T_{C2}=377~{}$K, comparable to
the $T=400$ K reported in bulk crystals Takizawa et al. (2002).

###  IV.2 Field In-Plane

For in-plane measurements, the sample was mounted to a rotating stage in a
manner outlined in the inset of Fig. 11(a). The angle the external field makes
with the direction of the current density, $\mathbf{J}$, and hence the
SiC$[2\bar{1}\bar{1}0]$ direction, will be defined as $\psi$.

Isothermal, longitudinal magnetoresistance (LMR) and planar Hall effect (PHE)
hysteresis loops were conducted at various temperatures, as shown in Fig. 10.
These measurements were performed with $\psi=0\degree$, such that the field
was along the direction of the current density. Negative LMR is observed for
all temperatures, with hysteresis beginning to show at 100 K and growing as
the temperature is reduced to 5 K. There is a small, time-reversal-anti-
symmetric portion of the PHE with a coercive field of around 0.01 T at 300 K
and growing to around 0.08 T at 50 K. This could be attributed to a small
misalignment of the film, leading to a component of the OOP Hall effect
imparting a contribution.

Figure 10: Constant temperature hysteresis loops of (a) LMR and (b) PHE. These
measurements were performed with the applied field parallel to the current
density, $\psi=0\degree$. The data have been offset vertically for clarity.

Anisotropic magnetoresistance (AMR) $\left(\rho_{xx}(\psi)\right)$ and PHE
$\left(\rho_{xy}(\psi)\right)$ were explored by rotating the angle $\psi$
through $380\degree$ (from -100$\degree$ to 280$\degree$) first at various
temperatures with a constant applied field $\mu_{0}H=3$ T. At 300 K, typical
AMR behaviour is observed with $\rho_{xx}\propto\cos^{2}(\psi)$ and
$\rho_{xy}\propto\sin(2\psi)$. However, upon cooling, clear hysteresis arises,
with discontinuous jumps occurring in both $\rho_{xx}$ and $\rho_{xy}$ at
intervals of $\sim 60\degree$. This change in behaviour from high to low
temperature is captured with Fourier analysis of the AMR/PHE as shown in Fig.
12. At higher temperatures, the conventional, 2-fold rotational symmetry is
dominant, but as temperature falls, other even Fourier components contribute,
with 6-fold and 4-fold components the primary among them.

Figure 11: Constant temperature scans of (a) AMR,
$\rho_{xx}(\psi)/\rho_{xx}(\psi=0)$ and (b) PHE,
$\rho_{xy}(\psi)/\rho_{xy}(\psi=0)$ at $\mu_{0}H=3$ T. The inset of (a) shows
the experimental geometry. Figure 12: Relative Fourier components of the
average AMR for angle increasing and angle decreasing as a function of
temperature at a constant field of $\mu_{0}H=3$ T.

To further examine the low temperature, unconventional AMR/PHE, rotation
measurements were performed with varying applied fields at a constant
temperature of 5 K. These are shown in Fig. 13 and Fig. 14. Clear hysteresis
is present for all fields considered, with the discontinuous jumps present at
$60\degree$ intervals appearing to be amplified and smeared out into more
continuous transitions as the field is increased. The difference between the
angle increasing and angle decreasing AMR and PHE shows a very clear 6-fold
rotational symmetry, as is captured by the Fourier analysis shown in Fig. 15.

Figure 13: AMR, $\rho_{xx}(\psi)/\rho_{xx}(\psi=0)$ for (a) $\psi$ increasing,
(b) $\psi$ decreasing, (c) the average of increasing and decreasing and (d)
the difference between increasing and decreasing. Figure 14: PHE,
$\rho_{xy}(\psi)/\rho_{xy}(\psi=0)$ for (a) $\psi$ increasing, (b) $\psi$
decreasing, (c) the average of increasing and decreasing, and (d) the
difference between increasing and decreasing. Figure 15: Relative Fourier
component amplitudes of the difference in the angle increasing AMR and angle
decreasing AMR as a function of the applied field at 5 K.

##  V Density Functional Theory Calculations

###  V.1 Computational Details

The density functional theory (DFT) simulations were carried out with the
Vienna Ab initio Simulation Package (VASP) Kresse and Furthmüller (1996)
within the generalized gradient approximation (GGA) Perdew et al. (1996) and
the projector augmented wave (PAW) method Kresse and Joubert (1999). The
calculations included a non-collinear treatment of the magnetization, spin-
orbit coupling, and the removal of symmetry recognition. The following
simulation parameters were adopted: kinetic energy cut-off of 700 eV,
Monkhorst-Pack Monkhorst and Pack (1976) Brillouin zone k-sampling of
10$\times$10$\times$10 for the cubic phase and 8$\times$8$\times$8 for the
hexagonal phase, Methfessel-Paxton smearing Methfessel and Paxton (1989) of
0.15 eV, electronic energy convergence of 10-8 eV and relaxation energy
convergence of 10-6 eV. Ground state lattice constants were obtained by
relaxing the atomic positions and cell size/shape.

###  V.2 Results

We performed DFT simulations on three ordered phases of Mn3Ge: the $D0_{19}$
structure with AFM order and the cubic $L1_{2}$ phase with both AFM and FM
orders. For the cubic AFM phase, we considered the $T_{1g}$ chiral AFM
structure shown in Fig. 1(e) that is reported in the other $L1_{2}$ cubic AFMs
Krén et al. (1966); Tomeno et al. (1999); Sakuma et al. (2002); Ikeda and
Tsunoda (2003). We found that the local magnetic moment of Mn is 2.446
$\mu_{B}$ with an average angle of 84.36∘ relative to the [111] direction
(small canting out of the (111) plane), as shown in Table 1. The angles of the
Mn moments are found to vary slightly between 84.30$\degree$ and
84.40$\degree$. This out-of-plane tilting leads to a total magnetic moment of
0.76 $\mu_{B}$ aligned 0.04$\degree$ relative to the [111] direction. A
strained cubic lattice, converted into a hexagonal cell, was also tested based
on experimental estimates of the lattice constants $a$ = 5.390 Å and $c$ =
6.285 Å. The calculated local magnetic moments and their average orientation
relative to the $c$-axis (equivalent to the [111] direction in the cubic cell)
are 2.565 $\mu_{B}$/Mn and 88.51∘, which represents an increase of $\approx$
5%. There is also an energy increase of 17 meV/atom with the strained cell.

For the hexagonal phase, we used the $E_{1g}$ antichiral spin structure
determined by spherical neutron polarimetry, Fig. 1(b) Soh et al. (2020). Our
DFT-calculated lattice parameters for the hexagonal AFM phase are within 2% of
the experimental values of $a$ = 5.339-5.347 Å and $c$ = 4.314 Å Yamada et al.
(1988) . The hexagonal AFM structure exhibits a measured local magnetic moment
of 2.65(2) $\mu_{B}$/Mn Soh et al. (2020), which is higher than our value of
2.291 $\mu_{B}$/Mn. In another DFT study Yang et al. (2017), a theoretical
value of 2.7 $\mu_{B}$/Mn was obtained using the experimental lattice
constants and the local density approximation for exchange-correlation. When
adopting the same lattice parameters, we obtain a magnetic moment of 2.569
$\mu_{B}$/Mn, in closer agreement with the experimental value.

Figure 16: The relative energy per atom of the three structures considered in
the DFT simulations.

The spin-configuration used for the cubic phase with a FM order is shown in
Fig. 1(c). The DFT-computed lattice constant is $\approx$ 4% smaller than the
value report in bulk samples, 3.81 Å, and our magnetic moment is not in good
agreement with the value of 0.87$\mu_{B}$/Mn Takizawa et al. (2002). It should
be noted that Ge exhibits a local magnetic moment that is aligned opposite to
that of Mn, suggesting that this phase is ferrimagnetic.

Energetically, the cubic AFM phase is the lowest energy configuration, as
shown in Fig. 16, followed by hexagonal AFM with an additional energy of 13
meV/atom and cubic FM with an additional energy of 54 meV/atom. Finally, we
note that when the cell shape was allowed to change, each phase showed a tiny
structural distortion leading to a negligible change in the properties of
Mn3Ge.

| Lattice constant | Local magnetic | Angle of local | Total magnetic | Angle of total | Energy/atom
---|---|---|---|---|---|---
| (Å) | moment ($\mu_{B}$) | magnetic moment | moment ($\mu_{B}$) | magnetic moment | (eV)
Cubic AFM | 3.707 | 2.446, 0.010 | 84.36∘, 180∘ to [111] | 0.760 | 0.04∘ to [111] | -8.040965
Hexagonal AFM | 5.241, 4.229 | 2.291, 0.000 | 90.00∘, – to [001] | 0.014 | 89.68∘ to [001] | -8.027891
Cubic FM | 3.641 | 1.691, 0.116 | 0∘, 180∘ to [100] | 5.216 | 0∘ to [100] | -7.986951
Table 1: DFT-computed properties of Mn3Ge. For the cubic AFM phase, the angle
of the local moment represents the average with values ranging between
84.30-84.40∘. For the hexagonal phase, the two numbers correspond to the $a$
and $c$ lattice parameters. For the local magnetic moment, the first and
second number correspond to the Mn and Ge components, respectively.

##  VI Discussion

While the $M(H)$ loops in Fig. 5(a) are qualitatively similar to those of the
bulk crystals reported in Ref. Takizawa et al., 2002, with comparable values
of $M_{S}$, our other results do not support the claim that the Mn3.2Ge films
exhibit simple ferromagnetism. The average moment of 0.98 $\mu_{B}$/Mn is only
58% of the DFT calculated value. The reversal of the remanent moment at a
compensation point $T=260$ K indicates that there are at least two different
magnetic sites in the crystal that are coupled antiferromagnetically. The
topological contribution to the Hall signal, evident from the different
hysteretic behaviour of $M(H)$ and $\rho_{(}xy)$ in Fig. 6, indicates that
these spins form a non-collinear configuration. Furthermore, the angular
dependence of the PHE in Fig. 13 shows hysteresis up to at least 14 T, well
above the apparent saturation field in the magnetometry measurements. This may
be related to a competition between the magnetocrystalline anisotropy and the
large exchange field between antiferromagnetically coupled spins. However, the
results of these measurements are difficult to interpret without micromagnetic
modelling and more detailed information about the atomic spin structure.

In addition to scalar spin chirality, there are likely significant
contributions to the Hall effect from band structure topology. The low
temperature longitudinal conductivity $\sigma_{xx}=3\times 10^{4}$ S/cm,
places this material in the the so-called _good metal_ regime, where intrinsic
contributions to the AHE dominate Miyasato et al. (2007). It is interesting to
note that the size of the Hall conductivity $\sigma_{xy}=1587$ S/cm at $T=50$
K is the highest reported for kagome magnets Chen et al. (2021): it is larger
than the kagome ferromagnet Co3Sn2S2, ($\sigma_{xy}=1130$ S/cm Liu et al.
(2018b)) and the frustrated kagome ferromagnet Fe3Sn2, ($\sigma_{xy}=402$ S/cm
Wang et al. (2016)).

To understand the non-collinear structure it is important to consider the
influence of chemical disorder. This disorder could be caused by the excess Mn
in our films. The other report on cubic Mn3Ge does not give any measure of the
composition of the samples for us to compare to Takizawa et al. (2002). To
find a point of reference, we turn to the $D0_{19}$ compounds. Excess Mn is
commonly used to help stabilize the Mn3Ge hexagonal phase, but for low enough
concentrations, this does not destroy the chiral antiferromagnetic structure:
the 120$\degree$ inverse triangular structure was found for $D0_{19}$ Mn3.1Ge
Kádár and Krén (1971) and Mn3.34Sn Zimmer and Krén (1973). At higher
concentrations, glassy magnetic behaviour can appear, as found for $D0_{19}$
Mn3.44Sn Feng et al. (2006), which shows a bifurcation between the zero-field
cooled and field-cooled $M$-vs-$T$ data. This occurs in samples with residual
resistivity ratios lower that 4 Ikhlas et al. (2020), which is comparable to
our films.

Insight into the impact of disorder on the magnetic structure can be gained by
comparing the $L1_{2}$ and $D0_{22}$ Mn3Ge polytypes. In the case of disorder
in a stoichiometric $L1_{2}$ phase, an exchange of Mn and Ge between the $1a$
and $3c$ sites drives the material towards the $D0_{22}$ phase. The Mn anti-
site defects would be expected then to couple antiferromagnetically to the
$3c$ Mn. The change would try to drive a distortion of the structure along the
[100] direction. Surprisingly from Fig. 3, we see a small elongation along the
[100] direction, rather that the contraction expected from the $D0_{22}$
structure. However, the tetragonal phase is not as well matched to SiC(0001),
which may be the reason that the system remains nearly cubic. The fact that
the moment of the $D0_{22}$ phase is far less than what we observe is further
evidence that this phase is not formed. Based on DFT, the Mn on the $3c$ sites
would like to form a 120$\degree$ chiral structure. However, the expected
antiferromagnetic interaction between the Mn3c and Mn1a would cause a large
canting of the Mn away from the (111) plane.

Future studies of the effects of excess Mn should be of great interest. Given
that the DFT simulations predict that the stoichiometric compound should have
a chiral AFM structure, the spin structure should be quite sensitive to
chemical tuning. The material potentially offers the opportunity of creating
frustration-stabilized skyrmions Leonov and Mostovoy (2015), or other textures
such as those found in Fe3Sn2 Hou et al. (2017); Du et al. (2022).

##  VII Summary

We have successfully synthesized Mn3.2Ge films with a cubic kagome structure.
Our magnetometry and magnetotransport measurements indicate that this is an
interesting topological material worthy of future exploration. The giant
anomalous Hall conductivity is the largest reported so far for the kagome
magnets, possibly due to topological features in the underlying kagome crystal
structure. A non-coplanar spin structure is also likely contributing a real-
space Berry curvature component to the AHE signal. Further improvements in
film morphology could make this an attractive material for device applications
given its above-room-temperature Curie temperature and giant Hall response.

##  VIII Acknowledgments

We acknowledge financial support from the Natural Science and Engineering
Research Council (Canada). We would like to thank M. L. Plumer, B. W. Southern
and A. Zelenskiy for helpful discussions. We thank Laurent Kreplak for
assistance with AFM measurements and Christian Lupien for assistance with
transport measurements. Computational resources were provided by the Digital
Research Alliance of Canada.

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##  IX Appendix: DFT-computed atomic structure and magnetic moments

| Atomic position | Local magnetic moment ($\mu_{B}$)
---|---|---
Mn | (0.0 0.5 0.5) | (-1.849 1.132 1.133)
Mn | (0.5 0.0 0.5) | (1.132 -1.850 1.132)
Mn | (0.5 0.5 0.0) | (1.134 1.134 -1.847)
Ge | (0.0 0.0 0.0) | (-0.006 -0.006 -0.006)
Table 2: Cubic AFM Mn3Ge. The lattice vectors are $a$$\cdot$[1 0 0], $a$$\cdot$[0 1 0], $a$$\cdot$[0 0 1], where $a$ = 3.707 Å. The total magnetic moment vector is [0.4387 0.4386 0.4394] $\mu_{B}$. The atomic positions are expressed in terms of the lattice vectors. | Atomic position | Local magnetic moment ($\mu_{B}$)
---|---|---
Mn | (0.16133 0.32266 0.75000) | (0 2.291 0)
Mn | (0.83860 0.67734 0.25000) | (0 2.291 0)
Mn | (0.32266 0.16133 0.25000) | (1.982 -1.150 0)
Mn | (0.83860 0.16133 0.25000) | (-1.983 -1.147 0)
Mn | (0.67734 0.83867 0.75000) | (1.982 -1.150 0)
Mn | (0.16133 0.83867 0.75000) | (-1.983 -1.147 0)
Ge | (0.33333 0.66666 0.25000) | (0 0 0)
Ge | (0.66666 0.33333 0.75000) | (0 0 0)
Table 3: Hexagonal AFM Mn3Ge. The lattice vectors are $a$$\cdot$[1 0 0], $a$$\cdot$[-1/2 $\sqrt{3}$/2 0], $c$$\cdot$[0 0 1], where $a$ = 5.241 Å and $c$ = 4.229 Å. The total magnetic moment vector is [-0.0035 -0.0135 0.0000] $\mu_{B}$. The atomic positions are expressed in terms of the lattice vectors. | Atomic position | Local magnetic moment ($\mu_{B}$)
---|---|---
Mn | (0.0 0.5 0.5) | (1.690 0 0)
Mn | (0.5 0.0 0.5) | (1.692 0 0)
Mn | (0.5 0.5 0.0) | (1.692 0 0)
Ge | (0.0 0.0 0.0) | (-0.116 0 0)
Table 4: Cubic FM Mn3Ge. The lattice vectors are $a$$\cdot$[1 0 0],
$a$$\cdot$[0 1 0], $a$$\cdot$[0 0 1], where $a$ = 3.641 Å. The total magnetic
moment vector is [5.2159 0.0004 -0.0002]. The atomic positions are expressed
in terms of the lattice vectors.