# Giant spin Hall angle in the Heusler alloy Weyl ferromagnet Co2MnGa

L. Leiva  Department of Electronic Science and Engineering, Kyoto University,
Kyoto 615-8510, Japan  S. Granville  Y. Zhang  Robinson Research Institute,
Victoria University of Wellington, Wellington 6140, New Zealand  The
MacDiarmid Institute for Advanced Materials and Nanotechnology, Wellington
6011, New Zealand  S. Dushenko  Department of Electronic Science and
Engineering, Kyoto University, Kyoto 615-8510, Japan  Institute for Research
in Electronics and Applied Physics, University of Maryland, College Park, MD
20742, USA  Physical Measurements Laboratory, National Institute of Standards
and Technology, Gaithersburg, MD 20899, USA  E. Shigematsu  T. Shinjo  R.
Ohshima  Y. Ando  M. Shiraishi  Department of Electronic Science and
Engineering, Kyoto University, Kyoto 615-8510, Japan

###### Abstract

Weyl semimetals are playing a major role in condensed matter physics due to
exotic topological properties, and their coexistence with ferromagnetism may
lead to enhanced spin-related phenomena. Here, the inverse spin Hall effect
(ISHE) in the ferromagnetic Weyl-semimetal Heusler alloy Co2MnGa was
investigated at room temperature by means of electrical spin injection in
lateral spin valve structures. Spin transport properties such as spin
polarization and spin diffusion length in this material were precisely
extracted in order to estimate the spin Hall angle $\theta_{\textrm{SH}}$,
which was found to be $-0.19\pm 0.04$ and is among the highest reported for a
ferromagnet. Although this value is on the same order of magnitude of known
heavy metals, the significantly higher resistivity of Co2MnGa implies an
improvement on the magnitude of detection voltages, while its ferromagnetic
nature allows controlling the intensity of SHE through the magnetization
direction. It was also shown that Onsager’s reciprocity does not hold for this
system, which is in part attributable to a different spin-dependent Hall
conductivity for spin-up and spin-down carriers.

††preprint: APS/123-QED

Weyl semimetals have been attracting significant attention since the discovery
of a nonmagnetic Weyl material, TaAs [1], because of its band-crossing points
that give rise to plenty of unique physical properties, such as the Fermi arc
surface states, the chiral anomaly coming from Nielsen-Ninomiya theorem [2]
and monopole-like Berry curvature [3]. Furthermore, a prominent class of Weyl
semimetals is Weyl magnetic materials such as the ferromagnetic Heusler alloy
Co2MnGa [4, 5] and antiferromagnetic Mn3Sn [6]. These two materials are
playing pivotal roles in condensed-matter physics because of the recent
discoveries of gigantic anomalous Hall effect (AHE) [7, 8], magnetic spin Hall
effect (a novel family of Hall effects) [9] and spin caloritronics phenomena
such as a large anomalous Nernst effect (ANE) [4, 10]. Additionally, magnetic
Heusler alloys have emerged as promising materials in the field of spintronics
due to their either half-metallic or semimetallic nature, which would lead to
a high spin polarization [11, 12], as has been reported in Co-based full
Heusler compounds [13, 14]. Combining the remarkable spin transport properties
of Heusler alloys with the unique band structure of Weyl semimetals may lead
to new exotic phenomena for topologically driven spintronic applications.

In the quest for these novel phenomena, an old acquaintance has emerged, the
spin Hall effect (SHE). Together with its reciprocal version, the inverse spin
Hall effect (ISHE), these two are utilized as essential methods for the
generation and detection of pure spin currents in spintronics devices, since
they enable the conversion of a charge current into a transversal spin current
and vice versa [15]. These spin-orbit coupling phenomena have been widely
observed in many non magnetic heavy metals (HMs) [16, 17, 18, 19, 20], but it
is thought that replacing a HM with a ferromagnet (FM) offers potential
advantages such as precise control of the spin current through the
magnetization direction, which could be applied to spin-transfer torque
devices [21]. However, measurements on only a few ferromagnetic materials have
been reported to date, most of them exhibiting spin Hall angles of just a few
percentage points [22, 23, 24, 25, 26, 27, 28]. To overcome this limitation,
materials exhibiting a large AHE are required, since it is understood that
both the AHE and the SHE are driven by the same intrinsic and extrinsic
scattering mechanisms, related to the spin-orbit interaction [29, 30]. The
large Berry curvature distribution around the Fermi level in Co2MnGa, thought
to be responsible for its large AHE [31], and ab initio calculations that
predict a strong intrinsic spin Hall effect in other Weyl semimetals [32],
indicate Co2MnGa is a strong candidate to observe large spin Hall voltages. In
addition, significant spin polarization [33, 34], strong resistance to
oxidation [7] and a higher resistivity than conventional metals [35] suggest
that Co2MnGa is a suitable platform to study the spin transport by means of
electrical spin injection. In this work, the ISHE of Co2MnGa is investigated,
wherein a giant spin Hall angle of $-0.19\pm 0.04$ is found, which is among
the highest reported for a FM so far.

Figure 1: (a) XRD pattern obtained for an out-of-plane $\theta$-2$\theta$ scan
of a sample consisting of a 30 nm-thick Co2MnGa film grown on a single-crystal
MgO substrate. Only the reflections due to the main X-ray source (Co-Kα) are
indexed. Inset shows the rocking curve for the (111) peak. (b) Scanning
electron microscope (SEM) image of a 2-wire LSV. In an ISHE/SHE measurement,
the magnetic field $H$ is applied parallel to the copper channel (Colored in
edition). (c) SEM image of a close view to the electrodes in a LSV with an
absorption middle wire (Colored in edition). (d) Schematics of the nonlocal
ISHE measurement. A charge current $I_{c}$ is injected from $F1$ to $N$,
producing spin accumulation at the $F1/N$ interface and a consequent spin
current $I_{s}$ diffuses along $N$. Part of this spin current is absorbed by
the detector $D$ in negative $z$ direction. Then, a charge current in positive
$y$ direction is induced in $D$ due to the inverse spin Hall effect, and a
voltage can be measured in equilibrium open circuit condition. Figure 2: (a)
Schematics of ISHE setup for a ferromagnetic detector. In this kind of samples
both FM electrodes are made of the same material, either Co2MnGa or Py. For
SHE setup, current source and voltmeter are exchanged preserving the polarity.
(b) Schematics of ISHE setup for a non-magnetic detector. In this case Py was
used for FM electrodes, but one of them is not used in this measurement. SHE
setup is obtained by exchanging the current source with the voltmeter while
preserving the polarity. (c) Nonlocal resistance for the ISHE setup and for
its reciprocal (direct) SHE setup for Co2MnGa, measured as a function of in-
plane external magnetic field $H$ at room temperature. Bottom panel shows a
typical AMR curve for Co2MnGa electrode $F1$. (d) Equivalent nonlocal ISHE and
SHE resistances measured for Pt and Py at room temperature. Bottom panel shows
the AMR curve for the electrode $F1$ made of Py in both cases.

The samples consisted of lateral spin valve (LSV) structures that were
fabricated starting from 30-nm-thick films of Co2MnGa epitaxially grown on a
(001)-oriented MgO single-crystal substrate. An X-ray diffraction (XRD)
$\theta$-2$\theta$ (out-of-plane) scan was performed to check the crystal
structure in the films. The presence of (002) and (004) peaks of Co2MnGa
confirms $B2$ ordering in the thin films, as shown in Fig. 1(a), meanwhile the
presence of (111) peak, as verified by the rocking curve in the inset,
confirms the $L2_{1}$ ordering in the samples. The FM electrodes were
patterned on the sub-micron scale using conventional electron-beam lithography
and Ar ion milling techniques. Two types of devices were fabricated, one with
two parallel FM electrodes (Fig. 1(b)) with various channel lengths $L$ (edge-
to-edge separation) while the other with three electrodes (Fig. 1(c)) with a
fixed distance of 600 nm between $F1$ and $F2$ and various widths of the
middle wire $M$, in order to estimate the spin resistance through the
absorption technique [19, 26, 36, 37, 38, 39, 40]. Finally, copper channels
perpendicularly connecting the ferromagnetic electrodes, as well as the
macroscopic connection pads, were patterned using electron-beam lithography
and thermal evaporation techniques. It is worth mentioning that, in order to
obtain Ohmic transparent interfaces, a low-acceleration-voltage in-situ Ar ion
milling was performed prior to copper deposition. Cu was selected for the non-
magnetic channel because of its long spin diffusion length and long spin
relaxation time [41], which make it a typical material for nonlocal signal
measurements. For reference, devices with the same geometry but with Py
ferromagnetic electrodes were fabricated on a thermally oxidized Si substrate.
In some of the 2-wire Py devices, an absorption middle wire of Pt was
deposited as a reference non-magnetic material, as shown in the scanning
electron microscope image of Fig. 1(c). All the transport measurements were
performed at room temperature in a commercial physical property measurement
system (PPMS) using a DC technique that consists of averaging the absolute
value of the voltage measured with positive and negative DC currents, which is
equivalent to an AC lock-in technique [42]. Further details of the fabrication
procedure, as well as testing of the transparent interfaces and the
determination of the spin diffusion length of Co2MnGa are presented in the
Supplemental Material in Secs. A, B and C [43].

Figure 1(d) shows the scheme of a typical ISHE measurement setup in a LSV. An
electric current $I_{c}$ is injected from the $F1$ electrode into the left
side of the $N$ channel (terminals $I^{+}$ and $I^{-}$ in Figs. 1(b) and 1(c))
producing spin accumulation at the $F1/N$ interface that induces a diffusive
pure spin current along the $N$ wire [44]. Part of the spin current is then
absorbed vertically (negative $\bm{z}$ direction) in a detection electrode $D$
($D$ can be $F2$ or $M$ depending on the device). Since the spin orientation
$\bm{\hat{s}}$ of the conduction electrons is given by the magnetization of
$F1$, which is fixed by the external magnetic field applied along the $N$
channel, a charge current density $\bm{j_{c}}$ given by
$\bm{j_{c}}=\theta_{SH}\bm{\hat{s}}\times\bm{j_{s}}$ [45] (where $\theta_{SH}$
is the spin Hall angle and $\bm{j_{s}}$ is the spin current density) is
generated along the $D$ electrode length. In the equilibrium state in open
circuit condition, a voltage $V_{\textrm{ISHE}}$ is generated to suppress the
charge current in $D$, measured between terminals $V^{+}$ and
$V^{-}_{\textrm{SHE}}$. When the magnetization of the injection electrode is
switched, by sweeping the external magnetic field, the sign of the generated
voltage is also expected to switch, as it is shown in the ISHE labeled
measurements in Fig. 2(c) for Co2MnGa and Fig. 2(d) for Pt and Py as detector
electrodes, where $R_{\textrm{ISHE}}=V_{\textrm{ISHE}}/I_{c}$. The difference
between the saturation values of $R_{\textrm{ISHE}}$ for positive and negative
field is defined as $2\Delta R_{\textrm{ISHE}}$. Reciprocally, if the probe
setup is inverted, by exchanging $V^{+}$ with $I^{+}$ and
$V^{-}_{\textrm{SHE}}$ with $I^{-}$, a charge current along $D$ will inject a
pure spin current in the $D/N$ interface by means of the (direct) SHE. This
spin current will diffuse again across the $N$ channel to be detected as a
nonlocal voltage $V_{\textrm{SHE}}$ between the electrodes $F1$ and $N$. The
SHE setup nonlocal resistance $R_{\textrm{SHE}}$ (defined as
$V_{\textrm{SHE}}/I_{c}$) is also shown in the SHE labeled measurements on
Fig. 2(c) for Co2MnGa and Fig. 2(d) for Pt and Py, where $2\Delta
R_{\textrm{SHE}}$ can be compared with $2\Delta R_{\textrm{ISHE}}$. Note that
the magnitudes of ISHE and SHE signals for Co2MnGa are not the same, which is
discussed later, and they are substantially large, roughly 20 times greater
than for Pt and Py in very similar geometries. Anisotropic magnetoresistance
(AMR) curves for the injection electrode $F1$ are shown to prove that the
saturation of the ISHE and SHE signals correspond to the saturation of the
magnetization of $F1$, where it is worth noting that the AMR of Co2MnGa was
found to be negative, as previously reported [46].

For the ISHE measurement setup, the ISHE resistance $\Delta R_{\textrm{ISHE}}$
can be expressed as [45, 37]:

$\Delta R_{\textrm{ISHE}}=\frac{\langle
I_{s}\rangle}{I_{c}}\theta_{SH}\frac{x}{W_{D}},$ (1)

where $\langle I_{s}\rangle$ is the spatial average of the absorbed spin
current along the $\bm{z}$ direction, $W_{D}$ is the width of the detector
wire and $x$ is the shunting factor that takes into account that part of the
generated charge current is being shunted by the copper wire. By applying a
one dimensional spin diffusion model [47] for transparent interfaces and
considering $t_{D}>>\lambda_{D}$, where $t_{D}$ the thickness of the detector
electrode and $\lambda_{D}$ its spin diffusion length, $\Delta
R_{\textrm{ISHE}}$ can be written as:

$\Delta
R_{\textrm{ISHE}}^{2W}=\frac{\theta_{SH}xW_{N}}{t_{F}}\frac{(1-\alpha_{F}^{2})2\alpha_{F}R_{F}^{2}R_{N}e^{-L/\lambda_{N}}}{(2R_{F}+R_{N})^{2}-R_{N}^{2}e^{-2L/\lambda_{N}}}$
(2)

for a 2-wire LSV using $F2$ as detector and

$\displaystyle\Delta R_{\textrm{ISHE}}^{3W}$ $\displaystyle=$
$\displaystyle\frac{\theta_{SH}xW_{N}(1-\alpha_{M}^{2})}{t_{M}}$
$\displaystyle\times$
$\frac{2\alpha_{F}R_{F}R_{M}R_{N}e^{-L/2\lambda_{N}}}{(2R_{F}+R_{N})(2R_{M}+R_{N})-(2R_{M}-R_{N})R_{N}e^{-L/\lambda_{N}}}$

for a 3-wire LSV using $M$ as detector, where $W_{N}$ is the width of $N$
wire, $\lambda_{N}$ is the spin diffusion length of Cu, $\alpha_{F}$ and
$\alpha_{M}$ are the spin polarizations of the ferromagnetic electrodes and
the middle wire $M$ material, respectively, and $R_{N}$, $R_{F}$ and $R_{M}$
are respectively the spin resistances of $N$, $F1$ (considered to be the same
as $F2$) and $M$ wires, defined in Ref. [43]. For Eq. Giant spin Hall angle in
the Heusler alloy Weyl ferromagnet Co2MnGa it was considered that the middle
wire $M$ was located at the middle of the gap distance $L$ between $F1$ and
$F2$ electrodes.

Figure 3: (a) Channel length dependence of the nonlocal signal measured for
several LSV devices at room temperature. The inset shows how is $\Delta
R_{\textrm{NL4T}}$ determined in an example of the acquired spin signal as a
function of in-plane longitudinal magnetic field. (b) Hanle effect nonlocal
signal as a function of the external out-of-plane magnetic field, for two
different channel lengths. A simultaneous fitting with common parameters is
shown by the solid red line. In both figures, insets show a schematic of the
measurement setup in each case.

In the case when a FM material is used as a detector electrode, $\Delta
R_{\textrm{ISHE}}$ can be written in terms of the conventional
nonlocal-4-terminal (NL4T) resistance $\Delta R_{\textrm{NL4T}}$, which can be
measured by connecting $V^{-}_{\textrm{NL4T}}$ terminal instead of
$V^{-}_{\textrm{SHE}}$ in Fig. 1(b) and sweeping the external magnetic field
in a direction parallel to the ferromagnetic electrodes length. A typical NL4T
signal is shown in the inset of Fig. 3(a), where two different voltage levels
are measured depending on whether the relative configuration of the
magnetization of the ferromagnetic electrodes is parallel or anti-parallel.
The difference between these two voltages, normalized by the injection
current, define $\Delta R_{\textrm{NL4T}}$ that is a direct measure of the
amount of spin current being absorbed by the detection wire. Then, Eqs. 2 and
Giant spin Hall angle in the Heusler alloy Weyl ferromagnet Co2MnGa are
simplified into:

$\Delta
R_{\textrm{ISHE}}=\frac{\theta_{SH}xW_{N}}{t_{D}}\frac{(1-\alpha_{D}^{2})}{2\alpha_{D}}\Delta
R_{\textrm{NL4T}},$ (4)

where the only unknown spin transport parameter is the spin polarization of
the detector electrode $\alpha_{D}$. In order to determine $\alpha_{D}$,
$\Delta R_{\textrm{NL4T}}$ was measured in 2-wire LSVs for several different
channel lengths $L$. This gap dependence is shown in Fig.3(a) where the data
has been fitted according to the equation [47]:

$\Delta
R_{\textrm{NL4T}}=\frac{4\alpha_{F}^{2}R_{F}^{2}R_{N}e^{-L/\lambda_{N}}}{(2R_{F}+R_{N})^{2}-R_{N}^{2}e^{-2L/\lambda_{N}}},$
(5)

where the only three unknown parameters are $\alpha_{F}$, $\lambda_{N}$ and
$\lambda_{F}$, the spin diffusion length of the FM. Since $\alpha_{F}$ and
$\lambda_{F}$ can not be extracted independently, additional spin absorption
measurements were performed to determine $\lambda_{F}$ self-consistently,
yielding $\lambda_{F}=(3.1\pm 1.1)$ nm for Co2MnGa [43]. The spin polarization
$\alpha_{F}$ was also determined via Hanle effect nonlocal measurements, which
have the same geometry and sample structure as in NL4T setup except for an
out-of-plane external magnetic field instead of in-plane. The data for two
different channel lengths is shown in Fig.3(b), where the fitting was
performed according to Eqs. (1), (2) and (3) in Ref. [48] for the case of
transparent interfaces, to consistently obtain $\alpha_{F}=0.15\pm 0.03$.

The only unknown parameter left to estimate $\theta_{\textrm{SH}}$ is the
shunting factor $x$, which was determined experimentally with devices
specially designed to that end, as detailed in Sec. D of the Supplemental
Material [43]. Finally, a linear fitting, shown in Fig. 4, can be performed
from the linear relation between $\Delta R_{\textrm{NL4T}}$ and $\Delta
R_{\textrm{ISHE}}$ (Eq. 4) to extract a large spin Hall angle
$\theta_{\textrm{SH}}=(-19\pm 4)\%$, which is simply ascribable to the sizable
Berry curvature as aforementioned. This value is significantly larger than
conventional FMs such as Py ($\sim 1\%$ [22, 23]), CoFe ($\sim 5\%$ [28]) and
CoFeB ($\sim 14\%$ [27]).

Figure 4: Data of $-2\alpha_{F}t_{F}\Delta
R_{\textrm{ISHE}}/(1-\alpha_{F}^{2})W_{N}x$ as a function of $\Delta
R_{\textrm{NL4T}}$ obtained for Co2MnGa at room temperature for several
devices. The spin Hall angle can be obtained from the slope of the linear
fitting. Data points correspond to devices with different channel lengths $L$.
Figure 5: Anomalous Hall resistance for Co2MnGa (top panel) and Py (bottom
panel) measured at room temperature. Inset of bottom panel shows a schematic
of the measurement setup, where the magnetic field is applied out-of-plane in
a Hall bar structure.

According to Onsager’s reciprocal relation [49], the resistance obtained for
the SHE setup and ISHE setup should be the same, as it is the case for the Py
and Pt control samples in Fig. 2(b). This was first experimentally
demonstrated by Kimura et al. [50] for the case of Pt and later verified for
many other materials [37, 38, 26]. However, it was not the case for Co2MnGa
samples, where ISHE and SHE signals are clearly different, as shown in Fig.
2(c). It is important to note that there is no effect of the geometry of the
devices on whether the reciprocity holds or not, as verified in the control
experiments detailed in Sec. H of Supplemental Material [43].

While the origin of the reciprocity is well understood for nonmagnetic
materials [51], it is still unclear why the relation should hold in FM
systems, since there is a breaking of the time-reversal symmetry. Furthermore,
recent advances in the field suggest that nonreciprocal transport can exist in
chiral materials such as Weyl semimetals [52].

In case of Co2MnGa, the non-reciprocity could be explained through a
phenomenological picture of the Hall phenomena, by introducing the spin-
dependent spin Hall angles
$\theta_{\uparrow}=\sigma_{xy}^{\uparrow}/\sigma_{xx}^{\uparrow}$,
$\theta_{\downarrow}=\sigma_{xy}^{\downarrow}/\sigma_{xx}^{\downarrow}$ and
the polarization of the spin Hall angle $p_{\textrm{SH}}$ through the
relations $\theta_{\textrm{SH}}=(\theta_{\uparrow}+\theta_{\downarrow})/2$ and
$p_{\textrm{SH}}=(\theta_{\uparrow}-\theta_{\downarrow})/(\theta_{\uparrow}+\theta_{\downarrow})$,
where $\sigma_{xy}^{s}$ $(s=\uparrow,\downarrow)$ is the spin-dependent Hall
conductivity and $\sigma_{xx}^{s}$ $(s=\uparrow,\downarrow)$ is the spin-
dependent normal conductivity [34, 26]. In the conventional case where
$p_{\textrm{SH}}$ is assumed to be zero, the anomalous Hall angle
$\theta_{\textrm{AHE}}$ is expected to be related to $\theta_{\textrm{SH}}$
via the spin polarization $\alpha_{F}$ in the form
$\theta_{\textrm{AHE}}=\alpha_{F}\theta_{\textrm{SH}}$ [22], as demonstrated
for Py [26]. However, the value of $\theta_{\textrm{AHE}}$ observed for
Co2MnGa and Py have the same sign, as shown in Fig. 5, while
$\theta_{\textrm{SH}}$ exhibits different signs in both materials, as shown in
Figs. 2(c) and 2(d). If a finite $p_{\textrm{SH}}$ is considered, the relation
$\theta_{\textrm{AHE}}=(\alpha_{F}+p_{\textrm{SH}})\theta_{\textrm{SH}}$
should hold [43], indicating $p_{\textrm{SH}}$ should be negative and
$|p_{\textrm{SH}}|>\alpha_{F}$. In addition, the relation between ISHE and SHE
resistances obtained with the one dimensional spin diffusion model [43]:

$-\frac{\Delta R_{\textrm{ISHE}}}{\Delta
R_{\textrm{SHE}}}=\frac{1-\alpha_{F}^{2}}{1+\alpha_{F}p_{\textrm{SH}}},$ (6)

indicates $|\Delta R_{\textrm{ISHE}}|>|\Delta R_{\textrm{SHE}}|$, which is
clear in Fig. 2(c). Refer to Sec. G of Supplemental Material for a more
detailed description of this spin-dependent spin Hall angle-based approach
[43].

In conclusion, a direct and effective method to determine the spin Hall angle
in ferromagnets was introduced, to obtain a significantly large value of
$\theta_{\textrm{SH}}=(-19\pm 4)\%$ for Heusler alloy Co2MnGa, a Weyl
semimetal. Combined with the ability to control the intensity of ISHE through
the magnetization direction and the high resistivity of the compound [43],
this result situates Co2MnGa as a robust platform for detection and generation
of spin currents in future spintronic devices. Furthermore, a lack of
reciprocity between ISHE and SHE resistances was observed and attributed to a
negative polarization of the spin Hall angle.

This research was supported in part by a Grant-in-Aid for Scientific Research
from the Ministry of Education, Culture, Sports, Science and Technology (MEXT)
of Japan (Innovative Area “Nano Spin Conversion Science” KAKENHI No.
26103003), Grant-in-Aid for Young Scientists (A) No. 16H06089 and Grant-in-Aid
for Scientific Research (S) ”Semiconductor Spincurrentronics” No. 16H06330. L.
L. acknowledges support from MEXT doctoral scholarship. S. G. acknowledges
financial support from the New Zealand Science for Technological Innovation
National Science Challenge. The MacDiarmid Institute is supported under the
New Zealand Centres of Research Excellence Programme.

## References

  * Lv  _et al._ [2015] B. Q. Lv, H. M. Weng, B. B. Fu, X. P. Wang, H. Miao, J. Ma, P. Richard, X. C. Huang, L. X. Zhao, G. F. Chen, Z. Fang, X. Dai, T. Qian, and H. Ding, Experimental discovery of Weyl semimetal TaAs, Phys. Rev. X 5, 031013 (2015).
  * Nielsen and Ninomiya [1983] H. Nielsen and M. Ninomiya, The Adler-Bell-Jackiw anomaly and Weyl fermions in a crystal, Physics Letters B 130, 389 (1983).
  * Yang  _et al._ [2017] H. Yang, Y. Sun, Y. Zhang, W.-J. Shi, S. S. Parkin, and B. Yan, Topological Weyl semimetals in the chiral antiferromagnetic materials Mn3Ge and Mn3Sn, New J. Phys. 19, 015008 (2017).
  * Sakai  _et al._ [2018] A. Sakai, Y. P. Mizuta, A. A. Nugroho, R. Sihombing, T. Koretsune, M.-T. Suzuki, N. Takemori, R. Ishii, D. Nishio-Hamane, R. Arita, _et al._ , Giant anomalous nernst effect and quantum-critical scaling in a ferromagnetic semimetal, Nature Physics 14, 1119 (2018).
  * Chang  _et al._ [2017] G. Chang, S. Y. Xu, X. Zhou, S. M. Huang, B. Singh, B. Wang, I. Belopolski, J. Yin, S. Zhang, A. Bansil, H. Lin, and M. Z. Hasan, Topological Hopf and chain link semimetal states and their application to Co2MnGa, Phys. Rev. Lett. 119, 156401 (2017).
  * Matsuda  _et al._ [2020] T. Matsuda, N. Kanda, T. Higo, N. Armitage, S. Nakatsuji, and R. Matsunaga, Room-temperature terahertz anomalous Hall effect in Weyl antiferromagnet Mn3Sn thin films, Nat Commun 11, 909 (2020).
  * Ludbrook  _et al._ [2017] B. M. Ludbrook, B. J. Ruck, and S. Granville, Perpendicular magnetic anisotropy in Co2MnGa and its anomalous Hall effect, Appl. Phys. Lett. 110, 062408 (2017).
  * Nakatsuji  _et al._ [2015] S. Nakatsuji, N. Kiyohara, and T. Higo, Large anomalous Hall effect in a non-collinear antiferromagnet at room temperature, Nature 527, 212 (2015).
  * Kimata  _et al._ [2019] M. Kimata, H. Chen, K. Kondou, S. Sugimoto, P. K. Muduli, M. Ikhlas, Y. Omori, T. Tomita, A. H. MacDonald, S. Nakatsuji, _et al._ , Magnetic and magnetic inverse spin Hall effects in a non-collinear antiferromagnet, Nature 565, 627 (2019).
  * Reichlova  _et al._ [2018] H. Reichlova, R. Schlitz, S. Beckert, P. Swekis, A. Markou, Y.-C. Chen, D. Kriegner, S. Fabretti, G. Hyeon Park, A. Niemann, _et al._ , Large anomalous nernst effect in thin films of the weyl semimetal co2mnga, Applied Physics Letters 113, 212405 (2018).
  * Coey and Venkatesan [2002] J. M. D. Coey and M. Venkatesan, Half-metallic ferromagnetism: Example of CrO2 (invited), J. Appl. Phys. 91, 8345 (2002).
  * Felser and Fecher [2013] C. Felser and G. H. Fecher,  _Spintronics: from materials to devices_ (Springer Science & Business Media, 2013).
  * Kimura  _et al._ [2012] T. Kimura, N. Hashimoto, S.Yamada, M. Miyao, and K. Hamaya, Room-temperature generation of giant pure spin currents using epitaxial Co2FeSi spin injectors, NPG Asia Materials 4, e9 (2012).
  * Shirotori  _et al._ [2015] S. Shirotori, S. Hashimoto, M. Takagishi, Y. Kamiguchi, and H. Iwasaki, All-metallic nonlocal spin valves using polycrystalline Co2 (FeMn) Si Heusler alloy with large output, Appl. Phys Express 8, 023103 (2015).
  * Sinova  _et al._ [2015] J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back, and T. Jungwirth, Spin hall effects, Rev. Mod. Phys. 87, 1213 (2015).
  * Saitoh  _et al._ [2006] E. Saitoh, M. Ueda, and H. Miyajima, High spin polarization of the anomalous hall current in co-based heusler compounds, Appl. Phys. Lett. 88, 182509 (2006).
  * Wang  _et al._ [2014] H. L. Wang, C. H. Du, Y. Pu, R. Adur, P. C. Hammel, and F. Y. Yang, Scaling of spin hall angle in 3d, 4d, and 5d metals from ${\mathrm{y}}_{3}{\mathrm{fe}}_{5}{\mathrm{o}}_{12}$/metal spin pumping, Phys. Rev. Lett. 112, 197201 (2014).
  * Tao  _et al._ [2018] X. Tao, Q. Liu, B. Miao, R. Yu, Z. Feng, L. Sun, B. You, J. Du, K. Chen, S. Zhang, _et al._ , Self-consistent determination of spin hall angle and spin diffusion length in pt and pd: The role of the interface spin loss, Science advances 4, eaat1670 (2018).
  * Morota  _et al._ [2011] M. Morota, Y. Niimi, K. Ohnishi, D. H. Wei, T. Tanaka, H. Kontani, T. Kimura, and Y. Otani, Indication of intrinsic spin hall effect in $4d$ and $5d$ transition metals, Phys. Rev. B 83, 174405 (2011).
  * Mosendz  _et al._ [2010] O. Mosendz, J. E. Pearson, F. Y. Fradin, G. E. W. Bauer, S. D. Bader, and A. Hoffmann, Quantifying spin hall angles from spin pumping: Experiments and theory, Phys. Rev. Lett. 104, 046601 (2010).
  * Taniguchi  _et al._ [2015] T. Taniguchi, J. Grollier, and M. D. Stiles, Spin-transfer torques generated by the anomalous hall effect and anisotropic magnetoresistance, Phys. Rev. Applied 3, 044001 (2015).
  * Tsukahara  _et al._ [2014] A. Tsukahara, Y. Ando, Y. Kitamura, H. Emoto, E. Shikoh, M. P. Delmo, T. Shinjo, and M. Shiraishi, Self-induced inverse spin hall effect in permalloy at room temperature, Phys. Rev. B 89, 235317 (2014).
  * Miao  _et al._ [2013] B. F. Miao, S. Y. Huang, D. Qu, and C. L. Chien, Inverse spin hall effect in a ferromagnetic metal, Phys. Rev. Lett. 111, 066602 (2013).
  * Seki  _et al._ [2015] T. Seki, K.-i. Uchida, T. Kikkawa, Z. Qiu, E. Saitoh, and K. Takanashi, Observation of inverse spin hall effect in ferromagnetic fept alloys using spin seebeck effect, Applied Physics Letters 107, 092401 (2015).
  * Seki  _et al._ [2019] T. Seki, S. Iihama, T. Taniguchi, and K. Takanashi, Large spin anomalous Hall effect in $l{1}_{0}$-FePt: Symmetry and magnetization switching, Phys. Rev. B 100, 144427 (2019).
  * Omori  _et al._ [2019] Y. Omori, E. Sagasta, Y. Niimi, M. Gradhand, L. E. Hueso, F. Casanova, and Y. Otani, Relation between spin hall effect and anomalous hall effect in $3d$ ferromagnetic metals, Phys. Rev. B 99, 014403 (2019).
  * Iihama  _et al._ [2018] S. Iihama, T. Taniguchi, K. Yakushiji, A. Fukushima, Y. Shiota, S. Tsunegi, R. Hiramatsu, S. Yuasa, Y. Suzuki, and H. Kubota, Spin-transfer torque induced by the spin anomalous hall effect, Nature Electronics 1, 120 (2018).
  * Wimmer  _et al._ [2019] T. Wimmer, B. Coester, S. Geprägs, R. Gross, S. T. Goennenwein, H. Huebl, and M. Althammer, Anomalous spin hall angle of a metallic ferromagnet determined by a multiterminal spin injection/detection device, Applied Physics Letters 115, 092404 (2019).
  * Zimmermann  _et al._ [2014] B. Zimmermann, K. Chadova, D. Ködderitzsch, S. Blügel, H. Ebert, D. V. Fedorov, N. H. Long, P. Mavropoulos, I. Mertig, Y. Mokrousov, and M. Gradhand, Skew scattering in dilute ferromagnetic alloys, Phys. Rev. B 90, 220403 (2014).
  * Hoffmann [2013] A. Hoffmann, Spin hall effects in metals, IEEE Transactions on Magnetics 49, 5172 (2013).
  * Markou  _et al._ [2019] A. Markou, D. Kriegner, J. Gayles, L. Zhang, Y.-C. Chen, B. Ernst, Y.-H. Lai, W. Schnelle, Y.-H. Chu, Y. Sun, _et al._ , Thickness dependence of the anomalous hall effect in thin films of the topological semimetal co 2 mnga, Physical Review B 100, 054422 (2019).
  * Sun  _et al._ [2016] Y. Sun, Y. Zhang, C. Felser, and B. Yan, Strong intrinsic spin hall effect in the taas family of weyl semimetals, Phys. Rev. Lett. 117, 146403 (2016).
  * Kolbe  _et al._ [2012] M. Kolbe, S. Chadov, E. A. Jorge, G. Schönhense, C. Felser, H. J. Elmers, M. Kläui, and M. Jourdan, Test of band structure calculations for Heusler compounds by spin-resolved photoemission spectroscopy, Phys. Rev. B 86, 024422 (2012).
  * Tung and Guo [2013] J.-C. Tung and G.-Y. Guo, High spin polarization of the anomalous hall current in co-based heusler compounds, New Journal of Physics 15, 033014 (2013).
  * Holmes and Pepper [2002] S. Holmes and M. Pepper, Magnetic and electrical properties of co 2 mnga grown on gaas (001), Applied physics letters 81, 1651 (2002).
  * Laczkowski  _et al._ [2015] P. Laczkowski, H. Jaffrès, W. Savero-Torres, J.-C. Rojas-Sánchez, Y. Fu, N. Reyren, C. Deranlot, L. Notin, C. Beigné, J.-P. Attané, _et al._ , Evaluation of spin diffusion length of auw alloys using spin absorption experiments in the limit of large spin-orbit interactions, Physical Review B 92, 214405 (2015).
  * Niimi  _et al._ [2011] Y. Niimi, M. Morota, D. H. Wei, C. Deranlot, M. Basletic, A. Hamzic, A. Fert, and Y. Otani, Extrinsic spin hall effect induced by iridium impurities in copper, Phys. Rev. Lett. 106, 126601 (2011).
  * Niimi  _et al._ [2012] Y. Niimi, Y. Kawanishi, D. H. Wei, C. Deranlot, H. X. Yang, M. Chshiev, T. Valet, A. Fert, and Y. Otani, Giant spin hall effect induced by skew scattering from bismuth impurities inside thin film cubi alloys, Phys. Rev. Lett. 109, 156602 (2012).
  * Sagasta  _et al._ [2017] E. Sagasta, Y. Omori, M. Isasa, Y. Otani, L. E. Hueso, and F. Casanova, Spin diffusion length of permalloy using spin absorption in lateral spin valves, Applied Physics Letters 111, 082407 (2017).
  * Zahnd  _et al._ [2018] G. Zahnd, L. Vila, V. Pham, M. Cosset-Cheneau, W. Lim, A. Brenac, P. Laczkowski, A. Marty, and J. Attané, Spin diffusion length and polarization of ferromagnetic metals measured by the spin-absorption technique in lateral spin valves, Physical Review B 98, 174414 (2018).
  * Jedema  _et al._ [2003] F. J. Jedema, M. S. Nijboer, A. T. Filip, and B. J. van Wees, Spin injection and spin accumulation in all-metal mesoscopic spin valves, Phys. Rev. B 67, 085319 (2003).
  * Casanova  _et al._ [2009] F. Casanova, A. Sharoni, M. Erekhinsky, and I. K. Schuller, Control of spin injection by direct current in lateral spin valves, Phys. Rev. B 79, 184415 (2009).
  * [43] See Supplemental Material at [url will be inserted by publisher] for details on sample preparation, determination of relevant parameters, characterization of magnetic electrodes, angle dependence of ISHE signal and discussion on the validity of reciprocal relations, .
  * Valenzuela [2009] S. O. Valenzuela, Nonlocal electronic spin detection, spin accumulation and the spin Hall effect, International Journal of Modern Physics B 23, 2413 (2009).
  * Takahashi and Maekawa [2008] S. Takahashi and S. Maekawa, Spin current, spin accumulation and spin hall effect, Science and Technology of Advanced Materials 9, 014105 (2008), pMID: 27877931, https://doi.org/10.1088/1468-6996/9/1/014105 .
  * Sato  _et al._ [2018] T. Sato, S. Kokado, S. Kosaka, T. Ishikawa, T. Ogawa, and M. Tsunoda, Large negative anisotropic magnetoresistance in Co2MnGa Heusler alloy epitaxial thin films, Appl. Phys. Lett. 113, 112407 (2018).
  * Takahashi and Maekawa [2003] S. Takahashi and S. Maekawa, Spin injection and detection in magnetic nanostructures, Phys. Rev. B 67, 052409 (2003).
  * Rojas Sánchez  _et al._ [2013] J.-C. Rojas Sánchez, P. Laczkowski, W. Savero Torres, M. Cubukcu, V. Nguyen, L. Notin, C. Beigné, C. Vergnaud, A. Marty, M. Jamet, _et al._ , In-plane and out-of-plane spin precession in lateral spin-valves, Applied Physics Letters 102, 132408 (2013).
  * Onsager [1931] L. Onsager, Reciprocal relations in irreversible processes. II., Phys. Rev. 38, 2265 (1931).
  * Kimura  _et al._ [2007] T. Kimura, Y. Otani, T. Sato, S. Takahashi, and S. Maekawa, Room-temperature reversible spin Hall effect, Phys Rev Lett 98, 156601 (2007).
  * Jacquod  _et al._ [2012] P. Jacquod, R. S. Whitney, J. Meair, and M. Büttiker, Onsager relations in coupled electric, thermoelectric, and spin transport: The tenfold way, Phys. Rev. B 86, 155118 (2012).
  * Tokura and Nagaosa [2018] Y. Tokura and N. Nagaosa, Nonreciprocal responses from non-centrosymmetric quantum materials, Nat Commun 9, 3740 (2018).
  * Hu and Kimura [2014] S. Hu and T. Kimura, Significant modulation of electrical spin accumulation by efficient thermal spin injection, Physical Review B 90, 134412 (2014).
  * Pedersen and Vernon [1967] R. J. Pedersen and F. L. Vernon, Effect of film resistance on low-impedance tunneling measurements, Appl. Phys. Lett. 10, 29 (1967).
  * Fenn  _et al._ [1998] M. Fenn, G. Akuetey, and P. E. Donovan, Electrical resistivity of Cu and Nb thin films, J. Phys.: Condens. Matter 10, 1707 (1998).
  * Damsgaard  _et al._ [2009] C. D. Damsgaard, M. C. Hickey, S. N. Holmes, R. Feidenhans’l, S. O. Mariager, C. S. Jacobsen, and J. B. Hansen, Interfacial, electrical and spin-injection properties of epitaxial Co$\\_2$MnGa grown on GaAs(100), J. Appl. Phys. 105, 124502 (2009).
  * Hamaya  _et al._ [2012] K. Hamaya, N. Hashimoto, S. Oki, S. Yamada, M. Miyao, and T. Kimura, Estimation of the spin polarization for Heusler-compound thin films by means of nonlocal spin-valve measurements: Comparison of Co2FeSi and Fe3Si, Phys. Rev. B 85, 100404 (2012).
  * Zhang  _et al._ [2011] H. Zhang, F. Freimuth, S. Blügel, Y. Mokrousov, and I. Souza, Role of spin-flip transitions in the anomalous hall effect of fept alloy, Phys. Rev. Lett. 106, 117202 (2011).

Supplemental Materials: Giant spin Hall angle in the Heusler alloy Weyl
ferromagnet Co2MnGa

##  .1 Sample preparation

Co2MnGa films were grown using an ultra-high vacuum DC magnetron sputtering
system with a base pressure of 2 $\times$10-8 Torr at a temperature of 400 ∘C,
with a subsequent annealing at 550 ∘C during 60 minutes. Since the growth of
the films and the microfabrication process were made at different places, a 40
nm-thick copper capping layer was grown on top of the Heusler alloy layer
without breaking the vacuum but after cooling to room temperature. This
capping layer was removed right before making the lateral spin valves.
Injection and detection ferromagnetic electrodes ($F1$ and $F2$ in Fig. 1(b))
width varied between 150 nm and 200 nm. Different switching fields were
achieved by changing the shape of the electrodes, favoring the domain rotation
with a flag structure in case of $F1$, so the same width (at the interfaces
with the nonmagnetic channel) could be used for both electrodes in a same
device. The edge-to-edge separation, also known as channel length $L$, between
the ferromagnetic electrodes varied from 150 nm to 1500 nm. In case of middle
wire devices (Fig. 1(c) on main text) the width of the middle wire was between
60 nm and 150 nm, while $L$ was fixed at 600 nm. For Cu channels, a width
$W_{N}$ of 200 nm and a thickness $t_{N}$ of 100 nm were used.

##  .2 Interface resistance measurement

Obtaining transparent interfaces is important to reliably extract spin
transport parameters such as spin polarization $\alpha_{F}$ and spin diffusion
length $\lambda_{F}$ from the nonlocal resistance measurements. To this end, a
500V acceleration voltage Ar ion milling step is performed in-situ prior to Cu
deposition. The interface resistance was measured at every device using the
cross configuration shown in Fig. S1, where typical I-V curves for the
interfaces of $F1$ and $F2$ with $N$ are shown. A significant quadratic
contribution was observed, attributed to thermally originated electromotive
forces due to the temperature gradient in the interface [53]. Apart from that,
a clear negative linear contribution of around -0.05 $\Omega$ is obtained from
the polynomial fitting, as expected for a transparent interface. This measured
negative resistance is an artifact that occurs when the resistance of the
metal electrodes along the junction is similar or higher than the interface
resistance of the junction itself [54]. To estimate the real magnitude of the
interface resistances, a finite element method simulation was performed,
where, for a fixed current, the interface resistance was systematically
changed until the voltage at the measurement points matched the experimental
result. Calculation yield to a product of the interface resistance with the
junction area $R_{i}A_{Fi}\lesssim 2$ f$\Omega$m2 in all reported cases.

Figure S1: Schematics of conventional cross configuration for interface
resistance measurement for (a) the F1/N interface and (b) the F2/N interface.
(c) Typical I-V curves where a quadratic fitting of the data was performed.

##  .3 Determination of the spin diffusion length of absorption wire

To estimate the spin diffusion length of Co2MnGa, the spin absorption
technique was used. The experiment setup is equivalent to that for the
nonlocal 4 terminal (NL4T) resistance measurement, where $\Delta R_{NL4T}$ is
obtained, as detailed in the main text. Electrodes $F1$ and $F2$ were used as
injector and detector, respectively (see Fig. 1(c)), in devices with different
widths of the middle wire $M$, obtaining $\Delta R_{\textrm{NL4T}}^{M}$.
Additionally, devices with the same separation between $F1$ and $F2$ but no
middle wire were used as reference samples, obtaining $\Delta
R_{\textrm{NL4T}}^{ref}$. The signal is expected to decrease with the presence
of the middle wire, since part of the spin current is absorbed there. The
quotient between these two signals can be calculated using the 1D spin
diffusion model, leading to:

$\frac{\Delta R_{\textrm{NL4T}}^{M}}{\Delta
R_{\textrm{NL4T}}^{ref}}=\frac{2R_{M}(R_{N}+e^{L/\lambda_{N}}(2R_{F}+R_{N}))}{(2R_{M}-R_{N})R_{N}+e^{L/\lambda_{N}}(2R_{F}+R_{N})(2R_{M}+R_{N})},$
(S1)

where it was assumed $R_{F1}=R_{F2}=R_{F}$ and the position of the middle wire
$M$, in a point contact model, is right at the middle of the gap. The spin
resistance $R_{X}$ is defined as
$\lambda_{X}\rho_{X}/(1-\alpha_{X}^{2})A_{X}$, where $\lambda_{X}$,
$\rho_{X}$, $\alpha_{X}$ and $A_{X}$ denotes the electrical resistivity, the
spin diffusion length, the spin polarization and the effective cross sectional
area, respectively, for an specific “X” material. A typical measurement is
shown in Fig. S2, where $M$ is the same material as $F1$ and $F2$, and thus it
has the same spin polarization $\alpha_{F}$ and spin diffusion length
$\lambda_{F}$. In order to determine $\lambda_{F}$, equation S1 should be used
together with Eq. (5) and the data of Fig. 3 of main text, where the $L$
dependence of $\Delta R_{\textrm{NL4T}}$ for devices with no middle wire was
studied. Unknown parameters are $\lambda_{N}$, $\lambda_{F}$ and $\alpha_{F}$,
since the resistivities were determined with a 4-probe method obtaining
$\rho_{\textrm{Cu}}=(2.8\pm 0.1)$ $\mu\Omega$cm and
$\rho_{\textrm{Co${}_{2}$MnGa}}=(220\pm 30)$ $\mu\Omega$cm, which agree with
reports in the literature [55, 56], and the effective cross sectional areas
depend on the sample geometry that was determined with Atomic Force Microscopy
technique and SEM imaging. By iterating between equations S1 and 5, the three
unknown parameters where extracted consistently obtaining $\alpha_{F}=0.15\pm
0.03$, $\lambda_{N}=(410\pm 80)$ nm and $\lambda_{F}=(3.1\pm 1.1)$ nm. This
latter value is in the order of the reported spin diffusion lengths for
similar Heusler compounds [57].

Figure S2: Nonlocal 4 terminal resistances for Co2MnGa LSVs as a function of a
magnetic field applied parallel to FM electrodes, for a reference device with
no middle wire and a another with a 60 nm middle wire.

##  .4 Determination of the shunting factor $x$

Figure S3: (a) SEM image of a device with a single Cu shunting wire used for
the estimation of the shunting factor, with schematics of the voltmeters
connections. (b) SEM image of a similar device used for the estimation of the
shunting factor but with several (15) Cu shunting wires.

In the ISHE setup, a spin current is absorbed by the ferromagnet F2 from the
nonmagnetic channel N. Since $\lambda_{F}$ is, as previously calculated, in
the order of a few nm, the spin to charge conversion will occur mostly very
close to the N/F2 interface. Therefore, it is expected that some of the ISHE-
generated charge current at the $FM$ will shunt through the much more
conductive Cu channel. This will produce a decrease of the generated voltage,
in a fraction $x$ ($0<x<1$) of the total expected voltage if no shunting
existed. This can also be understood as $x$ being the fraction of the total
current that will go through the FM. In a similar way, in the SHE setup, a
charge current $I_{c}$ is injected at the FM, and it is also expected that, at
the N/F2 interface, part of the current will shunt through the Cu wire
meanwhile a fraction $x$ will remain in the FM. This will produce a decrease
of the voltage drop across the FM wire, compared to when no shunting happens.
In order to estimate the magnitude of this effect, that is quite critical for
the spin Hall angle calculation, multi-terminal devices such as those of Fig.
S3 were fabricated. The experiment is described in Fig. S3(a), where a charge
current is injected through the Co2MnGa electrode terminals. The voltage
$V_{1}$ is expected to be less than $V_{2}$ due to the additional shunting
through the Cu middle wire, following the relation:

$\frac{V1}{V2}=\frac{2W_{2}+2W_{1}x}{2W_{2}+W_{1}(x+1)},$ (S2)

where $W_{1}$ is the width of the Cu wires and $W_{2}$ is the separation
between adjacent Cu wires.

Since there was a significant dispersion on the $x$ values extracted by this
method, due to process variations, devices with several shunting middle wires
were designed, as shown in Fig. S3(b), in order to average the shunting effect
through the multiple wires. The widths of the Cu and Co2MnGa wires were
selected to replicate those of the lateral spin valves used for the SHE
measurements. Finally, the reciprocal of the shunting factor was found to be
$x^{-1}=(2.63\pm 0.37)$.

##  .5 Angular dependence of inverse spin Hall effect voltage

Figure S4: (a) Schematics of the geometry for applying the external magnetic
field $H$. To note is that in principle $\theta_{H}\neq\theta_{M}$. (b)
Magnetization angle dependence of ISHE voltage. Bottom panel shows the AMR
curve obtained with an external magnetic field of 60 kOe.

In order to prove that the magnitude of the inverse spin Hall effect signals
can be controlled through the magnetization orientation, the dependence of the
ISHE voltage with the angle of the magnetization was studied. To this end, the
angle of the external magnetic field $\theta_{H}$ was varied, as shown in Fig.
S4(a). Since a magnetic anisotropy exists due to the electrode shape,
$\theta_{H}$ in principle will not coincide with the angle of the
magnetization $\theta_{M}$. AMR measurements for $F1$ and $F2$ confirmed that
both magnetizations $M1$ and $M2$ remain approximately parallel to each other
as they rotates due to the rotation of the external field. Since the maximum
value used for the sweep of the magnetic field in the ISHE measurements
($\sim$4 kOe) was not enough to assure $H$ and $M$ were parallel, a correction
was made to convert $\theta_{H}$ to the real $\theta_{M}$. To this end, the
AMR curve of $F1$ was measured for a sufficiently high magnetic field of 60
kOe so $\theta_{H}=\theta_{M}$, shown in Fig. S4(b). The magnetic field angle
dependence of the resistance was also obtained with a field of 4kOe to
determine $\theta_{M}$ by matching the resistances with the high field curve.
The ISHE voltage, for a same injection current of 1 mA, as a function of
$\theta_{M}$ is shown in Fig. S4(b), where the expected cosine behavior
confirms the signal magnitude is being controlled through the magnetization
direction.

##  .6 Characterization of ferromagnetic electrodes

Before patterning of ferromagnetic electrodes, the magnetization of Co2MnGa
thin films was measured using a SQUID magnetometer along the crystalline
directions [100], [110] and [001]. As shown in the XRD pattern of Fig. 1(a) in
the main text, the [001] axis is out of the substrate plane, and therefore the
saturation occurs at a high magnetic field of 10 kOe (not shown) due to shape
anisotropy. Concerning the in-plane directions, it was found the existence of
magneto-crystalline anisotropy with easy axis along the [110] direction (Fig.
S5(a)), which was verified to be parallel to the substrate sides. However,
after the patterning of FM electrodes, the shape anisotropy of these thin
wires was found to be much more significant than the original magneto-
crystalline anisotropy of the films. In Fig. S5(b), it can be seen that for a
high enough magnetic field of 60 kOe, magnetization and magnetic field are
parallel, showing the AMR a $\cos^{2}\theta$ behavior. However, for a smaller
magnetic field of 3 kOe, it is clear how the shape anisotropy makes the
magnetization to be closer to the direction parallel to electrode length. When
the in-plane magnetic field direction is fixed perpendicular to ferromagnetic
electrodes length ($\theta=0^{\circ}$), this is, the same condition as in SHE
and ISHE measurements, it can be seen that the AMR of both F1 and F2
electrodes saturates for a field around 2000 Oe. The relative amplitude of AMR
is smaller in F1 electrode, but this is due to its different shape, as shown
in Fig. 1(b) in the main text, where part of the electric current is not
parallel to the electrode length. However, the AMR curves for both F1 and F2
are the same except for a scale factor. The conclusion is that the
magnetization of both ferromagnetic electrodes rotate coherently when the
magnetic field is swept, probably because F1 and F2 have the same width.

Figure S5: (a) Magnetization hysteresis loop of a Co2MnGa film measured for
three different crystalline directions at room temperature. (b) AMR of a
Co2MnGa ferromagnetic electrode, F2 in this case, as a function of the angle
of the applied magnetic field in the plane of the substrate, for three
different magnetic fields. (c) AMR of both ferromagnetic electrodes of a
Co2MnGa LSV as a function of the intensity of magnetic field for a fixed angle
$\theta=0^{\circ}$, this is, the same condition of ISHE/SHE experiments.

##  .7 On the validity of Onsager’s reciprocal relation

In order to explain the origin of the breaking of reciprocity observed in the
ISHE and SHE experiments on Co2MnGa, a phenomenological picture is presented.
Assuming the validity of the two channel model, the total charge current
density $\bm{j_{c}}$ is decomposed into the spin components as
$\bm{j_{c}}=\bm{j_{\uparrow}}+\bm{j_{\downarrow}}$, meanwhile the spin current
(with charge current units) is defined as
$\bm{j_{s}}=\bm{j_{\uparrow}}-\bm{j_{\downarrow}}$. The spin Hall effect-
generated currents $\bm{j_{\uparrow}}^{H}$ and $\bm{j_{\downarrow}}^{H}$ can
be described for each spin-component of the current as
$\bm{j_{\uparrow}}^{H}=\theta_{\uparrow}\bm{\hat{s}}\times\bm{j_{\uparrow}}$
and
$\bm{j_{\downarrow}}^{H}=-\theta_{\downarrow}\bm{\hat{s}}\times\bm{j_{\downarrow}}$,
where $\theta_{\uparrow}=\sigma_{xy}^{\uparrow}/\sigma_{xx}^{\uparrow}$ and
$\theta_{\downarrow}=\sigma_{xy}^{\downarrow}/\sigma_{xx}^{\downarrow}$ are
the spin-dependent spin Hall angles, defined in terms of the spin-dependent
Hall $\sigma_{xy}^{s}$ $(s=\uparrow,\downarrow)$ and normal $\sigma_{xx}^{s}$
$(s=\uparrow,\downarrow)$ conductivities. The spin Hall angle is defined as
$\theta_{\textrm{SH}}=(\theta_{\uparrow}+\theta_{\downarrow})/2$ and the
polarization of the spin Hall angle is introduced as
$p_{\textrm{SH}}=(\theta_{\uparrow}-\theta_{\downarrow})/(\theta_{\uparrow}+\theta_{\downarrow})$.

Figure S6: Schematics of electron motion when a pure spin current induces a
spin polarized charge current via the spin dependent ISHE.

The ISHE case is shown schematically in Fig. S6, where the direction of
electron motion is that of the charge flow, for simplicity. In this kind of
experiment, the source is a diffusive pure spin current $\bm{j_{s}}$ that is
absorbed by the FM material, so $j_{\uparrow}=-j_{\downarrow}=j_{s}/2$ along
the $x$ direction. Then, the ISHE-generated charge current along $y$ direction
is $j_{c}^{\textrm{ISHE}}=\theta_{\textrm{SH}}j_{s}$, meanwhile the generated
spin current is $j_{s}^{\textrm{ISHE}}=\theta_{\textrm{SH}}p_{SH}j_{s}$. Since
the measured ISHE voltage is proportional to $j_{c}^{\textrm{ISHE}}$, this
method allows to extract $\theta_{\textrm{SH}}$ regardless of the value of
$p_{SH}$.

Figure S7: Schematics of electron motion when a spin polarized charge current
induces both a charge current, known as anomalous Hall effect, and a spin
current via the spin dependent SHE.

In contrast, the SHE case, shown schematically in Fig. S7, is essentially
different since the source is not a pure charge current (as it would be in a
nonmagnetic material) but a spin polarized charge current, because of the spin
polarization $\alpha_{F}$ of the FM, defined as
$\alpha_{F}=(\sigma_{\uparrow}-\sigma_{\downarrow})/(\sigma_{\uparrow}+\sigma_{\downarrow})$.
Therefore the spin components of the current along $x$ direction are
$j_{\uparrow}=j_{c}(1+\alpha_{F})/2$ and
$j_{\downarrow}=j_{c}(1-\alpha_{F})/2$. Then, the SHE-generated charge current
along the $y$ direction, also known as anomalous Hall effect current, is
$j_{c}^{\textrm{AHE}}=\theta_{\textrm{SH}}(\alpha_{F}+p_{SH})j_{c}$ and the
anomalous spin Hall angle $\theta_{\textrm{AHE}}$ can be expressed as
$\theta_{\textrm{AHE}}=\theta_{\textrm{SH}}(\alpha_{F}+p_{\textrm{SH}})$,
where it is clear that $\theta_{\textrm{AHE}}$ and $\theta_{\textrm{SH}}$ may
have different sign if $p_{SH}$ is negative. In Fig. S7 diagram, it would mean
a larger number of spin-down (blue) electrons will scatter to left. If this
number is high enough, the direction of $j_{c}^{\textrm{AHE}}$ can be inverted
while the direction of $j_{s}^{\textrm{SHE}}$ remains the same. On the other
hand, the SHE-generated spin current is
$j_{s}^{\textrm{SHE}}=\theta_{\textrm{SH}}(1+\alpha_{F}p_{SH})j_{c}$, where a
non-zero $p_{SH}$ would imply an asymmetry with respect to the ISHE case.

To calculate the SHE nonlocal resistance measured in the SHE setup for a
2-wire device, as described in the main text, a similar derivation as used for
the NL4T resistance (Eq. 5 on main text) can be done. While the detection
mechanism is essentially the same, the spin injection is different in SHE and
NL4T cases. In electrical spin injection, the source of spin accumulation is
an electric-field-driven drift spin current in the ferromagnet, while in SHE-
driven spin injection, the source of spin accumulation is also a drift spin
current but generated via the SHE. Then, the boundary conditions are modified
at the injection interface. This imply that one of the $\alpha_{F}$ factors in
Eq. 5 of main text ends up being replaced by
$\theta_{\textrm{SH}}(1+\alpha_{F}p_{SH})xW_{N}/t_{F}$, being the final
expression for $\Delta R_{\textrm{SHE}}$:

$\Delta
R_{\textrm{SHE}}=\frac{\theta_{SH}xW_{N}}{t_{F}}\times\frac{(1+\alpha_{F}p_{SH})2\alpha_{F}R_{F}^{2}R_{N}e^{-L/\lambda_{N}}}{(2R_{F}+R_{N})^{2}-R_{N}^{2}e^{-2L/\lambda_{N}}}.$
(S3)

By comparing Eq. S3 with the expression for $\Delta R_{\textrm{ISHE}}$ of Eq.
2 in main text, the Eq. 6 of the main manuscript is obtained, where the lack
of reciprocity is exposed. This model consistently explains that a different
sign for $\theta_{SH}$ and $\theta_{\textrm{AHE}}$ implies the SHE resistance
is reduced with respect to the ISHE resistance, which agrees with the
experimental results in Fig. 2(c) of main text.

It should be pointed out that the above decomposition of the spin Hall effect
in two spin-conserving effects for up-spin and down-spin carriers is not a
fully precise picture, since it is known the spin-orbit coupling generally
mixes spin-up and spin-down states through spin flip processes. However, for
light elements such as 3d transition metals, the perfect two-current model is
expected to be still reasonably valid [58].

##  .8 Checking the influence of sample geometry on the reciprocity of
transport signals

Figure S8: ISHE and SHE signals measured for an injection current of 1mA, at
several different configurations changing the injection and detection
electrodes, detailed in the schematics.

As shown in the SEM image of Fig. 1(b) of the main text, as well as in the
schematics of Fig. S8, the ferromagnetic electrodes F1 and F2 have different
shape. This design is in order to achieve different switching fields when the
magnetic field is parallel to the electrode length in the nonlocal-4-terminal
setup. Therefore, F1 and F2 are not equivalent, and the measured signals when
using F1 as injector and F2 as detector and vice versa were checked, both for
ISHE and SHE setups, as shown in Fig. S8. It can be seen that, regardless of
which electrodes are used for injection and detection, the measured amplitudes
are the same, either for the SHE or the ISHE setup. Thus, any possible
influence of the non-symmetrical geometry on the non-reciprocity is discarded.
At the same time, the breaking of the reciprocity, this is, different
amplitudes for $\Delta V_{\textrm{ISHE}}$ and $\Delta V_{\textrm{SHE}}$ (for
the same injection current), were consistently observed again in random device
tests made for this check.


# Variation after full projection with triaxially deformed nuclear mean field

Zao-Chun Gao  China Institute of Atomic Energy, Beijing 102413, China  State
Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,
Chinese Academy of Sciences, Beijing 100190, China.  Mihai Horoi  Department
of Physics, Central Michigan University, Mount Pleasant, Michigan 48859, USA
Y.S. Chen  China Institute of Atomic Energy, Beijing 102413, China

###### Abstract

We implemented a variation after projection (VAP) algorithm based on a
triaxially deformed Hartree-Fock-Bogoliubov vacuum state. This is the first
projected mean field study that includes all the quantum numbers (except
parity), i.e., spin ($J$), isospin ($T$) and mass number ($A$). Systematic VAP
calculations with $JTA$-projection have been performed for the even-even
$sd$-shell nuclei with the USDB Hamiltonian. All the VAP ground state energies
are within 500 keV above the exact shell model values. Our VAP calculations
show that the spin projection has two important effects: (1) the spin
projection is crucial in achieving good approximation of the full shell model
calculation. (2) the intrinsic shapes of the VAP wavefunctions with spin
projection are always triaxial, while the Hartree-Fock-Bogoliubov methods
likely provide axial intrinsic shapes. Finally, our analysis suggests that one
may not be possible to associate an intrinsic shape to an exact shell model
wave function.

###### pacs:

21.60.Jz,21.60.Cs,21.10.Hw

##  I Introduction

Hartree-Fock-Bogoliubov (HFB) method has been very successful in describing
the global properties of the ground states throughout the whole nuclear
region. As a mean field method, HFB breaks the symmetries of the nuclear
system, and can be used to study the intrinsic shapes. The HFB calculations
with Gogny force show that almost all the calculated 1712 nuclei have axially
symmetric HFB minimaDE10 .

Projection can be done on a HFB vacuum to recover the symmetries that the
Hamiltonian obeys. To test the quality of the projected wavefunctions, one can
compare them with the exact shell model ones using a common Hamiltonian. HFB
and variation after projected HFB calculations with shell model Hamiltonians
have been reported by several authors MA11 ; RO08 ; RB11 ; Schmid04 . For
those calculations without projection, the HFB vacuum states are often assumed
to be axially symmetric RB11 . Indeed, we will see below that all the
calculated HFB minima in $sd$-shell nuclei, except 24Mg, are exactly axial
with the USDB Hamiltonian usdb .

However, if one performs the variation of the projected HFB vacuum, usually
called variation after projection (VAP) Ring80 , it is likely that the
intrinsic shape may changes due to the inclusion of beyond mean field
correlations. One typical example is the ground state of 32Mg, which is
predicted to be spherical at the mean field level Reinhard99 , but it turns
out to have a quadrupole deformation when the correlations associated with the
restoration of the broken rotational symmetry are considered Rodr00 . Another
example is 56Ni, whose ground state is spherical at the mean field level, but
is slightly deformed when performing the projected energy surface calculation
Gao09b .

Moreover, the triaxial ($\gamma$) degree of freedom plays important roles on
the low-lying collective dynamics in this mass region Kurath72 . In 24Mg the
possibility of the triaxial deformation in the ground state was discussed for
decades Bonche87 ; Koepf88 ; Sheline88 , and it is still being used as the
testing ground for modern theories involving angular momentum (spin)
projection Bender08 ; Yao10 ; Rodr10 .

In this work, we perform VAP calculations of the even-even $sd$-shell nuclei
using the USDB Hamiltonian. Here, we allow the $\gamma$ degree of freedom in
the HFB transformation. The shell model Hamiltonian conserves the spin ($J$),
isospin ($T$), as well as the mass number ($A$). Hence a complete projection
should recover all $J,\ T$, and $A$ quantum numbers. This is generally very
much time-consuming because of the 7-dimensional integration ( 3 for $J$, 3
for $T$, and 1 for $A$). Presently, we can only carry out such extensive
studies in the $sd$ shell. For efficiency, we use the new techniques of Refs.
gao14 ; hu14 ; wang14  to evaluate the kernels for projections.

##  II The VAP method

From a randomly chosen HFB vacuum state $|\Phi_{0}\rangle$, one can construct
a new HFB vacuum state $|\Phi\rangle$ using the Thouless theorem Ring80 .
Namely,

$\displaystyle|\Phi\rangle=\mathcal{N}e^{\frac{1}{2}\sum_{\mu\nu}d_{\mu\nu}a^{\dagger}_{\mu}a^{\dagger}_{\nu}}|\Phi_{0}\rangle,$
(1)

where $d$ is a skew symmetric matrix, and $\mathcal{N}$ is the normalization
factor. The triaxiality of the HFB vacuum can be treated similar to Ref.
Hara95  so that the $Q_{2\pm 1}$ components of the quadrupole moment vanish.

Projecting $|\Phi\rangle$ onto good quantum numbers $J$, $T$, and $A$ one gets
the so called $JTA$-projection (similarly, $TA$-projection for $T,A$, etc.).
One can evaluate the $JTA$-projected energy,

$\displaystyle
E_{JTA}=\sum_{MKM_{T}K_{T}}f^{*}_{MM_{T}}f_{KK_{T}}{\langle\Phi|\hat{H}P^{J}_{MK}P^{T}_{M_{T}K_{T}}P^{A}|\Phi\rangle},$
(2)

where $P^{J}_{MK}$, $P^{T}_{M_{T}K_{T}}$ and $P^{A}$ are the spin, isospin and
mass number projection operators, respectively. The isospin projection
operator is similar to the spin projection operator but in the isospin space.
$E_{JTA}$ and the corresponding coefficients $f_{KK_{T}}$ are obtained by
solving

$\displaystyle\sum_{KK_{T}}\langle\Phi|(\hat{H}-E_{JTA})P^{J}_{MK}P^{T}_{M_{T}K_{T}}P^{A}|\Phi\rangle
f_{KK_{T}}=0,$ (3)

with $f_{KK_{T}}$ satisfying

$\displaystyle\sum_{MKM_{T}K_{T}}f^{*}_{MM_{T}}f_{KK_{T}}\langle\Phi|P^{J}_{MK}P^{T}_{M_{T}K_{T}}P^{A}|\Phi\rangle=1.$
(4)

One can also calculate $E_{TA}$ with $TA$-projection by simply removing the
spin projection from Eqs. (2-4). For the $A$-projection, the corresponding
energy, $E_{A}$, is reduced to

$\displaystyle
E_{A}=\frac{\langle\Phi|\hat{H}P^{A}|\Phi\rangle}{\langle\Phi|P^{A}|\Phi\rangle}.$
(5)

Without any projection, we define

$\displaystyle E_{HFB}=\langle\Phi|\hat{H}|\Phi\rangle.$ (6)

It is natural that one may consider the neutron ($N$) and proton ($Z$)
projection, as has been done in Refs. MA11 ; Schmid04 . However, this is
essentially the same as the $T_{Z}A$-projection ($T_{Z}=(N-Z)/2$). Here, we
prefer to take $TA$-projection to recover the total isospin symmetry.

VAP calculations can be performed by changing the $d$ matrix in Eq.(1). Here,
we impose the following restrictions for the $d$ matrix: (1) $d$ is real, (2)
keeping the time reversal symmetry, and (3) no mixing between neutron and
proton in the HFB transformation. Therefore the total number of free VAP
parameters for $sd$-shell is reduced to $N_{VAP}=42$. In practice we start
with $d=0$ and with Nilsson+BCS vacuum states $|\Phi_{0}\rangle$ obtained with
randomly chosen quadrupole parameters Gao09b .

To extract the intrinsic shape, the quadrupole moment and the triaxial degree
of freedom, $Q$ and $\gamma$, are defined such that

$\displaystyle Q\cos\gamma$ $\displaystyle=$
$\displaystyle\langle\Psi|\sqrt{\frac{16\pi}{5}}\frac{r^{2}}{b^{2}}Y_{20}|\Psi\rangle,$
(7) $\displaystyle Q\sin\gamma$ $\displaystyle=$
$\displaystyle\langle\Psi|\sqrt{\frac{16\pi}{5}}\frac{r^{2}}{b^{2}}\frac{1}{\sqrt{2}}(Y_{22}+Y_{2-2})|\Psi\rangle,$
(8)

where $b$ is the harmonic oscillator length. $|\Psi\rangle$ refers to an
intrinsic state, which may have different forms. Explicitly, we define,

(1) $Q_{HFB}$ and $\gamma_{HFB}$ for $|\Psi\rangle=|\Phi\rangle$,

(2) $Q_{A}$ and $\gamma_{A}$ for
$|\Psi\rangle=\frac{P^{A}|\Phi\rangle}{\sqrt{\langle\Phi|P^{A}|\Phi\rangle}}$,
and

(3) $Q_{TA}$ and $\gamma_{TA}$ for
$|\Psi\rangle=\sum_{K_{T}}f_{K_{T}}P^{T}_{M_{T}K_{T}}P^{A}|\Phi\rangle$,

##  III  VAP calculations for 24Mg

When performing the energy variation, one may find that there might be more
than one energy minima. Therefore, the energy variation should be calculated
several times with different starting $|\Phi_{0}\rangle$ states which are
randomly chosen. We then identify the lowest minimum, and denote it with
$E^{*}$. Here and below, we only discuss the results corresponding to $E^{*}$.

In the present work, we adopt the USDB Hamiltonian usdb . The HFB energy for
24Mg is $E^{*}_{HFB}=-80.965$ MeV with the constraints
$\langle\Phi|\hat{N}|\Phi\rangle=N$ and $\langle\Phi|\hat{Z}|\Phi\rangle=Z$.
This is the only $sd$-shell nucleus for which the HFB calculation gives a non-
axial shape with $Q^{*}_{HFB}=18.659$ and $\gamma^{*}_{HFB}=11.96^{\circ}$
(here and below the $Q^{*}$ and $\gamma^{*}$ are the shape parameters that can
be associated with the absolute minimum for some VAP choice). Let’s first do
the simplest VAP with only $A$-projection (called VAP-A). Since the particle
number is already projected out, it might be unnecessary to impose a
constraint to the average particle number of the HFB vacuum. To check this
conjecture, we start from several different $|\Phi_{0}\rangle$ states and
perform VAP-A. The results for few selected $\left|\Phi_{0}\right>$ choices
are shown in Table 1. One can see that the VAP-A energies are identical
($E^{*}_{A}=-81.358$ MeV). However, the corresponding $E_{HFB}$, $Q_{HFB}$,
$\gamma_{HFB}$ and
$\langle\hat{A}\rangle\equiv\langle\Phi|\hat{A}|\Phi\rangle$ appear randomly.
This means that the converged vacua $|\Phi\rangle$ are not unique, but
correspond to the same $A$-projected state. The $Q_{A}$ values are the same,
and although the $\gamma_{A}$ values look different, they actually represent
the same shape but with different orientations. Therefore, one can adopt the
values $Q^{*}_{A}=18.284$ and $\gamma^{*}_{A}=10.05$ to define the shape of
the VAP-A minimum. If one imposes $\langle\hat{A}\rangle=A=8$, we still have
$E^{*}_{A}=-81.358$ MeV, now the converged $|\Phi\rangle$ vacuum becomes
unique, with $E_{HFB}=-79.720$, $Q_{HFB}=17.905$, and
$\gamma_{HFB}=9.05^{\circ}$ (see the last line in Table 1). However, for the
VAP with $TA$-projection, the situation becomes a little different.

Table 1: Results of the VAP-A calculations for 24Mg. We perform the VAP calculations for several times. Each time we start with different $|\Phi_{0}\rangle$ states. The numbers in the first column denote different $|\Phi_{0}\rangle$ states. The second column shows the converged energy $E^{*}_{A}$. Quantities in other columns are calculated with the converged $|\Phi\rangle$ vacua. Energies are in MeV. $|\Phi_{0}\rangle$ | $E^{*}_{A}$ | $Q_{A}$ | $\gamma_{A}(^{\circ})$ | $E_{HFB}$ | $Q_{HFB}$ | $\gamma_{HFB}(^{\circ})$ | $\langle\hat{A}\rangle$
---|---|---|---|---|---|---|---
1 | -81.358 | 18.284 | 10.05 | -81.008 | 18.005 | 9.46 | 8.110
2 | -81.358 | 18.284 | 130.05 | -90.178 | 18.371 | 128.94 | 9.013
3 | -81.358 | 18.284 | -109.95 | -82.684 | 18.120 | -110.61 | 8.259
4 | -81.358 | 18.284 | 10.05 | -79.720 | 17.905 | 9.05 | 8.000

VAP calculations with $TA$-projection (called VAP-TA) are listed in Table 2.
Unlike VAP-A, even if one imposes $\langle\hat{A}\rangle=A=8$ for 24Mg, the
converged $|\Phi\rangle$ is still not unique as the $E_{HFB}$ energy appears
randomly. Moreover, the $E_{A}$ energy is not unique either. Interestingly,
after $TA$-projection, those different $|\Phi\rangle$ vacuum states have
exactly the same projected energy $E^{*}_{TA}=-82.831$(MeV) and the same
$Q_{TA}=17.295$. Therefore, we can associate the shape parameter corresponding
$Q^{*}_{TA}=Q_{TA}=17.295$ to this projected minimum. Similarly, we found
(after rotation) $\gamma^{*}_{TA}=\gamma_{TA}=0.09^{\circ}$, which describes
an almost axial-shape. One can conclude that only $Q_{TA}$ and $\gamma_{TA}$
are meaningful in describing the shape of the VAP-TA projected state.

Table 2: Similar to Table. 1 but for the VAP-TA calculations. $\langle A\rangle=8$ is imposed. $|\Phi_{0}\rangle$ |  $E^{*}_{TA}$(MeV) | $Q_{TA}$ | $\gamma_{TA}(^{\circ})$ |  $E_{A}$(MeV) | $Q_{A}$ | $\gamma_{A}(^{\circ})$ |  $E_{HFB}$(MeV) | $Q_{HFB}$ | $\gamma_{HFB}(^{\circ})$ | $\langle A\rangle$
---|---|---|---|---|---|---|---|---|---|---
1 | -82.831 | 17.295 | -119.91 | -75.826 | 16.376 | -118.62 | -74.921 | 15.755 | -118.23 | 8.000
2 | -82.831 | 17.295 | 0.09 | -74.402 | 16.167 | 2.47 | -73.909 | 15.563 | 3.06 | 8.000
3 | -82.831 | 17.295 | 120.09 | -76.633 | 16.526 | 120.09 | -75.525 | 15.897 | 120.08 | 8.000

A complete symmetry restoration is the $JTA$-projection. VAP results with
$JTA$-projection (called as VAP-JTA) are shown in Table 3. All the converged
$E^{*}_{JTA}$ energies are $-86.919$ MeV, significantly closer to the shell
model result $E_{SM}=-87.105$ MeV. Again, both $E_{A}$ and $E_{HFB}$ in Table
3 can not be uniquely determined, even if one enforces the
$\langle\hat{A}\rangle=A$ constraint. Fortunately, with the additional spin
projection, all $E_{TA}$ values are found to be $-79.879$ MeV, and similarly
the corresponding shape is described by $Q_{TA}=19.057$ and
$\gamma_{TA}=16.96^{\circ}$. Therefore, the quantities that can be associated
with the shape of VAP-JTA wavefunction should also be
$Q^{*}_{JTA}=Q_{TA}=19.057$ and $\gamma^{*}_{JTA}=\gamma_{TA}=16.96^{\circ}$.

Table 3: Similar to Table. 1 but for the VAP-JTA calculations. $\langle A\rangle=8$ is imposed.  $|\Phi_{0}\rangle$ |  $E^{*}_{JTA}$(MeV) |  $E_{TA}$(MeV) | $Q_{TA}$ | $\gamma_{TA}(^{\circ})$ |  $E_{A}$(MeV) | $Q_{A}$ | $\gamma_{A}(^{\circ})$ |  $E_{HFB}$(MeV) | $Q_{HFB}$ | $\gamma_{HFB}(^{\circ})$ | $\langle A\rangle$
---|---|---|---|---|---|---|---|---|---|---|---
1 | -86.919 | -79.879 | 19.057 | -16.964 | -75.600 | 17.482 | -20.225 | -73.781 | 16.230 | -23.772 | 8.000
2 | -86.919 | -79.879 | 19.057 | -16.963 | -75.641 | 17.510 | -20.119 | -73.830 | 16.264 | -23.604 | 8.000
3 | -86.919 | -79.879 | 19.057 | -16.963 | -75.644 | 17.520 | -20.068 | -73.845 | 16.281 | -23.506 | 8.000

One can study the shape evolution of 24Mg from HFB to VAP-JTA. In VAP-TA,
$Q^{*}_{TA}$ looks smaller than $Q^{*}_{HFB}$ in HFB, and $\gamma^{*}_{TA}$
tends to be close to zero (axial shape). However in VAP-JTA, $Q^{*}_{JTA}$ is
larger than the $Q^{*}_{HFB}$ in HFB, and $\gamma^{*}_{JTA}$ tends to describe
a triaxial shape. This triaxiality in VAP-JTA, in comparison with VAP-TA, is
likely caused by the spin projection. To determine if this phenomenon is more
general, we performed systematic VAP calculations for a larger number of even-
even $sd$-shell nuclei.

##  IV  VAP calculations for even-even $sd$-shell nuclei

VAP calculations have been performed for the ground states of even-even
$sd$-shell nuclei. The calculated energies relative to the shell model ones
are shown in Figure 1a. The numerical results are given in Table 4. Here, we
didn’t include the Oxygen isotopes and the $N=20$ isotones because their VAP-
JTA energies are exactly the same as the shell model results ($E_{SM}$). This
special case is discussed below. The VAP-JTA energies are much lower than
those of HFB and VAP-TA. Moreover, The VAP-JTA energies for 20Ne, 28Ne, and
36Ar nuclei are exactly the same as the shell model results (see also Figure
1b). This can be understood by comparing the number of VAP parameters,
$N_{VAP}$, with the shell model dimension, $N_{JT}$ (the total number of the
independent basis states with good $JT$). Here, $N_{VAP}=42$. The $N_{JT}$
values with $J=0$ and $T=0$ for both 20Ne and 36Ar are only $21$. For 28Ne,
$N_{JT}$ for $J=0$ and $T=4$ is 43. It looks that when $N_{JT}$ is less than,
or close to $N_{VAP}$, then the VAP-JTA energy is likely to be the same as the
shell model one. Indeed, for all even-even oxygen isotopes and for the $N=20$
isotones, for which $N_{JT}\leq N_{VAP}$, we have obtained
$E^{*}_{JTA}=E_{SM}$. In Figure 1b, one can also see that the energy
difference $E^{*}_{JTA}-E_{SM}$ increases with $N_{JT}$. The largest
$E^{*}_{JTA}-E_{SM}=0.446$ MeV is obtained for 26Mg, corresponding to the
largest $N_{JT}=1132$.

Table 4:  Converged energies and associated shape parameters for even-even $sd$-shell nuclei calculated with the USDB Hamiltonian. Nucleus |  |  | VAP-JTA | VAP-TA | HFB | VAP-HF
---|---|---|---|---|---|---
| $N_{JT}$ | $E_{SM}$ | $E^{*}_{JTA}$ | $Q^{*}_{JTA}$ | $\gamma^{*}_{JTA}$ | $E^{*}_{TA}$ | $Q^{*}_{TA}$ | $\gamma^{*}_{TA}$ | $E^{*}_{HFB}$ | $Q^{*}_{HFB}$ | $\gamma^{*}_{HFB}$ | $E^{*}_{PHF}$ | $Q^{*}_{PHF}$ | $\gamma^{*}_{PHF}$
20Ne | 21 | -40.472 | -40.472 | – | – | -37.069 | 14.7 | 0.0 | -36.404 | 15.3 | 0.0 | -40.265 | 13.861 | 3.551
22Ne | 148 | -57.578 | -57.501 | 12.1 | 13.8 | -54.572 | 15.8 | 0.0 | -53.474 | 16.5 | 0.0 | -56.958 | 15.675 | 8.632
24Ne | 287 | -71.725 | -71.570 | 11.0 | 30.1 | -68.084 | 10.1 | 60.0 | -66.402 | 12.0 | 0.0 | -71.037 | 13.449 | 32.786
26Ne | 191 | -81.564 | -81.465 | 9.2 | 28.4 | -78.949 | 8.6 | 0.0 | -77.518 | 8.3 | 0.0 | -80.988 | 9.760 | 17.265
28Ne | 43 | -86.543 | -86.543 | – | – | -84.920 | 7.0 | 60.0 | -83.949 | 7.1 | 0.0 | -86.294 | 9.848 | 23.934
24Mg | 325 | -87.105 | -86.919 | 19.1 | 17.0 | -82.831 | 17.3 | 0.0 | -80.965 | 18.7 | 12.0 | -86.636 | 19.165 | 16.427
26Mg | 1132 | -105.521 | -105.075 | 15.8 | 28.7 | -100.648 | 13.8 | 25.7 | -98.992 | 15.9 | 60.0 | -104.264 | 16.238 | 32.331
28Mg | 874 | -120.500 | -120.205 | 14.5 | 20.2 | -117.091 | 14.4 | 0.0 | -115.625 | 15.1 | 0.0 | -119.354 | 16.306 | 19.835
30Mg | 191 | -130.474 | -130.400 | 10.4 | 20.0 | -128.035 | 10.3 | 0.0 | -126.735 | 10.9 | 0.0 | -129.926 | 11.864 | 27.322
28Si | 839 | -135.860 | -135.539 | 16.1 | 58.6 | -131.501 | 17.8 | 60.0 | -130.021 | 19.8 | 60.0 | -134.617 | 17.116 | 58.038
30Si | 1132 | -154.754 | -154.402 | 14.3 | 47.0 | -150.380 | 10.6 | 60.0 | -148.475 | 14.5 | 60.0 | -153.777 | 14.633 | 46.167
32Si | 287 | -170.519 | -170.373 | 12.5 | 58.2 | -167.721 | 10.6 | 60.0 | -166.344 | 12.4 | 60.0 | -169.996 | 12.175 | 52.023
32S | 325 | -182.452 | -182.234 | 15.1 | 33.3 | -179.925 | 0.6 | 60.0 | -176.393 | 0.0 | 0.0 | -181.856 | 14.681 | 33.188
34S | 148 | -202.504 | -202.380 | 10.6 | 53.0 | -200.331 | 0.0 | 0.0 | -198.493 | 0.0 | 0.0 | -202.039 | 11.496 | 50.992
36Ar | 21 | -230.277 | -230.277 | – | – | -228.355 | 0.0 | 0.0 | -226.611 | 13.2 | 60.0 | -230.112 | 12.068 | 52.293

Figure 1:  (Color online) (a) Calculated ground state energies relative to the
shell model results, $E_{SM}$. (b) Relative VAP-JTA energy,
$E^{*}_{JTA}-E_{SM}$, versus the shell model dimension, $N_{JT}$, in $JT$
subspace.
Figure 2: (Color online) Quadrupole moments $Q$ and $\gamma$ values for (a)
HFB vacuum states, (b) VAP-TA states, (c) VAP-JTA states and (d) VAP-HF states
which is based on a slater determinant.

The quadrupole moment and the $\gamma$ degree of freedom can be extracted
using Eqs. (7) and (8). In Fig. 2a, the $\gamma^{*}_{HFB}$ values in HFB are
either $0^{\circ}$ or $60^{\circ}$, except $\gamma^{*}_{HFB}=12^{\circ}$ for
24Mg, thus supporting the conclusion that HFB likely presents axially deformed
shapes. In Fig. 2b, the shapes in VAP-TA calculations still remain axially
symmetric, except for 26Mg, which has $\gamma^{*}_{TA}=25.7^{\circ}$. Quite
differently, the $\gamma^{*}_{JTA}$ values in the VAP-JTA calculations (Fig.
2c) show that all these nuclei are non-axial without exception. Comparing
these results with those of Fig. 2a, one can conclude that the triaxiality in
VAP-JTA is definitely a beyond mean-field effect, which is likely to be a
universal phenomenon. Fig. 2b, however, excludes the possibility that the
isospin projection and the mass projection lead to triaxiality. Thus, the only
possible cause of the triaxiality is the beyond mean-field spin projection.

To study directly the effect of spin projection, one can start from a Hartree-
Fock (HF) Slater determinant (SD) and perform VAP calculations with only spin
projection (called VAP-HF). The converged energies, $E^{*}_{PHF}$, relative to
$E_{SM}$, are shown in Fig. 1a. The results show that VAP-HF is better than
VAP-TA, and quite close to the VAP-JTA. The quadrupole moment $Q^{*}_{PHF}$
and $\gamma^{*}_{PHF}$ corresponding to $E^{*}_{PHF}$ can be calculated using
Eqs. (7) and (8) with $|\Psi\rangle$ replaced by the converged SD. These
quantities are uniquely determined, and are shown in Fig. 2d. Again, all the
$\gamma^{*}_{PHF}$ values are distributed in the interval
$(0^{\circ},60^{\circ})$, which is very similar to Fig. 2c. Therefore, we
could conclude that VAP results that include spin projection can always be
associated with intrinsic states having triaxial deformation.

One more interesting phenomenon, however, is related to the VAP-JTA
calculations for 20Ne, 28Ne, and 36Ar. We have shown above that the
$E^{*}_{JTA}$ energies of these nuclei are the same as the exact shell model
results. Surprisingly, the corresponding $Q_{TA}$ and $\gamma_{TA}$ values are
not unique, which is quite different from other nuclei with
$E^{*}_{JTA}>E_{SM}$. For example, the results for 20Ne are shown in Table 5.
With the same converged $E^{*}_{JTA}=-40.472$MeV, one can clearly see that
starting with different initial states $|\Phi_{0}\rangle$, the result for
$Q_{TA}$ and $\gamma_{TA}$ could be different. These results indicate that it
may not be possible to associate an unique intrinsic deformation with an exact
eigenstate of the Hamiltonian.

Table 5: VAP results with JTA projection for 20Ne. $|\Phi_{0}\rangle$ |  $E^{*}_{JTA}$(MeV) |  $E_{TA}$(MeV) | $Q_{TA}$ | $\gamma_{TA}(^{\circ})$
---|---|---|---|---
1 | -40.472 | -28.284 | 6.314 | -45.134
2 | -40.472 | -30.468 | 11.873 | -124.746
3 | -40.472 | -27.932 | 9.876 | 2.592

##  V  Summary

We implemented an algorithm that performs variation after projection (VAP) on
spin, isospin, and mass number of a triaxially deformed Hartree-Fock-
Bogoliubov vacuum state. This is the first projected mean field study that
includes all these quantum numbers.

We start from a randomly chosen HFB vacuum state and carry out VAP
calculations for 24Mg in $sd$-shell with various projections. In the VAP-A
case the converged solution is independent of the Fermi level (chemical
potential). Although the associated HFB vacuum does not have definite
quadrupole moment $Q_{HFB}$ and triaxial deformation parameter $\gamma_{HFB}$,
one can use the unique $Q_{A}$ and $\gamma_{A}$ to describe the intrinsic
deformation of the VAP-A state. Similarly, in the VAP-TA calculations, $Q_{A}$
and $\gamma_{A}$ can not be uniquely determined, but $Q_{TA}$ and
$\gamma_{TA}$ are unique and can be associated with the intrinsic deformation
of the VAP-TA state. It is not possible to directly define deformation
parameters $Q$ and $\gamma$ for the VAP-JTA wave function, which has the
symmetries fully restored, but the $Q_{TA}$ and $\gamma_{TA}$ calculated with
the VAP-JTA vacuum state $|\Phi\rangle$ are also unique, and can be associated
with the intrinsic deformation of the VAP-JTA state.

Systematical VAP calculations of even-even $sd$-shell nuclei have been
performed using the USDB Hamiltonian. The VAP-JTA energies, $E^{*}_{JTA}$, are
very close to the shell model results, $E_{SM}$. Moreover, the relative
energy, $E^{*}_{JTA}-E_{SM}$, increases with the shell model dimension
$N_{JT}$. The shapes described by the HFB minima are always axial. However,
with spin projection VAP calculations always produce triaxial shapes. We
believe that such triaxiality is an universal phenomenon caused by the beyond
mean-field dynamic correlations. Finally, we show that those VAP-JTA states
reaching the exact shell model results do not have clearly defined intrinsic
shapes.

###### Acknowledgements.

Z-C. Gao thanks Prof. G. F. Bertsch for fruitful discussions, and Prof. Y. Sun
for reading the manuscript. This work is partly supported by the National
Natural Science Foundation of China under Contract No.s 11175258, 11575290,
11321064 and 11275068; the CUSTIPEN (China-U.S. Theory Institute for Physics
with Exotic Nuclei) under DOE Grant No. DE-FG02-13ER42025; the Open Project
Program of State Key Laboratory of Theoretical Physics, Institute of
Theoretical Physics, Chinese Academy of Sciences, China (No.Y5KF141CJ1). M.H.
acknowledges the U.S. NSF Grant No. PHY-1404442.

## References

  * (1) J.-P. Delaroche et al., Phys. Rev. C 81, 014303 (2010).
  * (2) I. Maqbool, J. A. Sheikh, P. A. Ganai and P. Ring, J. Phys. G38 045101 (2011).
  * (3) R. Rodríguez-Guzmán, Y. Alhassid, and G. F. Bertsch, Phys. Rev C 77, 064308 (2008).
  * (4) L. M. Robledo and G. F. Bertsch, Phys. Rev. C 84, 014312 (2011).
  * (5) K.W. Schmid, Prog. Part. Nucl. Phys. 52, 565 (2004).
  * (6) B. A. Brown and W. A. Richter, Phys. Rev. C 74, 034315 (2006).
  * (7) P. Ring and P. Schuck, The Nuclear Many-Body Problem (Springer Verlag, New York, Heidelberg, Berlin, 1980).
  * (8) P.-G. Reinhard, D. J. Dean, W. Nazarewicz, J. Dobaczewski, J. A. Maruhn, and M. R. Strayer, Phys. Rev. C 60, 014316 (1999).
  * (9) R. R. Rodríguez-Guzmán, J. L. Egido, and L. M. Robledo, Phys. Lett. B474, 15 (2000); Phys. Rev. C 62, 054319 (2000).
  * (10) Zao-Chun Gao, Mihai Horoi, and Y. S. Chen, Phys. Rev. C80, 034325 (2009).
  * (11) D. Kurath, Phys. Rev. C 5, 768 (1972).
  * (12) W. Koepf and P. Ring, Phys. Lett. B 212, 397 (1988).
  * (13) P. Bonche, H. Flocard, and P. H. Heenen, Nucl. Phys. A 467, 115 (1987).
  * (14) R. K. Sheline, I. Ragnarsson, S. Åberg, and A.Watts, J. Phys. G 14, 1201 (1988).
  * (15) Michael Bender and Paul-Henri Heenen Phys. Rev. C78, 024309 (2008).
  * (16) J. M. Yao, J. Meng, P. Ring, and D. Vretenar Phys. Rev. C81, 044311 (2010).
  * (17) Tomás R. Rodríguez and J. L. Egido Phys. Rev. C81, 064323 (2010).
  * (18) Z.-C. Gao, Q.-L. Hu, Y.S.Chen Phys. Lett. B732, 360 (2014).
  * (19) Q.-L. Hu, Z.-C. Gao, Y.S.Chen Phys. Lett. B734, 162 (2014).
  * (20) L.-J. Wang, F.-Q. Chen, T. Mizusaki, M. Oi, Y. Sun, Phys. Rev. C 90 (2014) 011303(R)
  * (21) K. Hara, Y. Sun, Int. J. Mod. Phys. E 4, 637 (1995).