# Direct Electrical Detection of Spin Chemical Potential Due to Spin Hall
Effect in $\beta$-Tungsten and Platinum Using a Pair of Ferromagnetic and
Normal Metal Voltage probes

Soumik Aon  Abu Bakkar Miah  Arpita Mandal  Harekrishna Bhunia  Dhananjaya
Mahapatra  Partha Mitra<EMAIL_ADDRESS>Department of Physical
Sciences, Indian Institute of Science Education and Research, Kolkata,
Mohanpur 741246, India

###### Abstract

The phenomenon of Spin Hall Effect (SHE) generates a pure spin current
transverse to an applied current in materials with strong spin-orbit coupling,
although not detectable through conventional electrical measurement. An
intuitive Hall effect like measurement configuration is implemented to
directly measure pure spin chemical potential of the accumulated spins at the
edges of heavy metal (HM) channels that generates large SHE. A pair of
transverse linearly aligned voltage probes in placed in ohmic contact with the
top surface of HM , one being a ferromagnetic metal (FM) with non-zero spin
polarization and other is the reference metal (RM) with zero polarization of
carriers. This combination of FM/RM electrodes is shown to induce an
additional voltage proportional to a spin accumulation potential, which is
anti symmetric with respect to opposite orientations of FM controlled by a 2D
vector magnet. Proof of concept of the measurement scheme is verified by
comparing the signs of voltages for HM channels of Tungsten (W) and Platinum
(Pt) which are known to generate opposite spin accumulation under similar
conditions of applied current. The same devices are also able to detect the
reciprocal effect, inverse spin Hall effect (ISHE) by swapping the current and
voltage leads and the results are consistent with reciprocity principle.
Further, exploiting a characteristic feature of W thin film deposition, a
series of devices were fabricated with W resistivity varying over a wide range
of 10 - 750 $\mu\Omega$-cm and the calculated spin Hall resistivity exhibits a
pronounced power law dependence on resistivity. Our measurement scheme
combined with almost two decades of HM resistivity variation provides the
ideal platform required to test the underlying microscopic mechanism
responsible for SHE/ISHE.

##  I Introduction

Spin Hall effect (SHE) [1, 2, 3] has emerged as a prominent transport
phenomenon arising explicitly due to the spin degree of the carriers in
conductors and relativistic effects. Typically, materials with large spin-
orbit coupling (SOC) strength, e.g. semiconductors, [4] heavy metals (HM), [5,
6] and topological insulators [7] etc. exhibits measurable SHE that results in
the conversion of a charge current density ($\textbf{J}_{c}$) into a
transverse pure spin current density ($\textbf{J}_{s}$) of spin polarization
direction s, described by the phenomenological equation.

$\textbf{J}_{s}=\theta_{SHE}(\frac{\hbar}{2e})\textbf{J}_{c}\times\textbf{s},$
(1)

where, the factor $\hbar/2e$ makes the dimensions of $\textbf{J}_{s}$ same as
that of $\textbf{J}_{c}$. $\theta_{SHE}$ is the spin Hall angle, a
dimensionless material specific parameter whose magnitude quantifies the
efficiency of charge-to-spin current conversion and sign determines the
orientation of deflected spins ($+\textbf{s}$ or$-\textbf{s}$) in a particular
transverse direction. In a conventional Hall effect like geometry with open
circuit condition transverse to current, SHE will lead to accumulation of
opposite spins [2] at the opposite edges of a current carrying conductor. For
normal (paramagnetic) metals with zero spin polarization in the conduction
band, a pure spin current is generated by SHE implying that equal number of
oppositely oriented spins move in opposite directions so that no net charge
current exists. This will lead to opposite spin accumulations at the edges of
equal magnitude and hence no net electric potential difference. Unlike a
charge current, a spin current is not a conserved quantity and decays rapidly
while diffusing through the conductor within a characteristic length known as
the spin diffusion length $\lambda_{sf}$, typically few nano-meters at room
temperature. Thus, only the carriers within $\lambda_{sf}$ distance from the
transverse edges of the conducting channel will contribute to the spin
accumulation. A rigorous expression for spin potential difference at the edges
was derived and in the limit of conducting channel width being much larger
than $\lambda_{sf}$, the maximum limit of the spin potential difference
achieved is shown to be, [2]

$\mu_{\uparrow}-\mu_{\downarrow}=\lambda_{sf}\rho_{SHE}J_{c}$ (2)

Where, $\rho_{SHE}$ is the transverse resistivity due to SHE. The reciprocal
phenomenon of SHE is the inverse spin Hall effect (ISHE) where a transverse
charge current is generated in response to a spin current present in the
paramagnetic conductors with strong SOC, described as

$\textbf{J}_{c}=\theta_{SHE}(-\frac{2e}{\hbar})\textbf{J}_{s}\times\textbf{s}.$
(3)

Figure 1: (a) The schematic diagram of a device for detection of SHE. A
rectangular HM channel (light blue) with long axis chosen to be $x$\- axis is
embedded in an insulating $SiO_{2}$ substrate (not shown). A pair of
transverse (along $y$) linearly aligned FM (light grey) and RM (light golden)
electrodes are in Ohmic contact with on top surface of HM (light blue). Due to
SHE a pure spin current $J_{s}$ flows along $z$, the entire top (bottom)
surface of HM assumes spin accumulation with polarisation along $+(-)y$ shown
by small black arrows. The device is mounted inside a 2D vector electromagnet
(not shown) such that the external magnetic field ($B$) could be varied in the
$xy$-plane making an angle $\phi$ in general with the HM channel. The
magnetization of the FM (deep orange arrow) is shown here to be along the
$y$-direction $\phi=90^{0}$ when the effect explained in (c) is maximum, (b)
FESEM image of a typical device (top view), (c) Illustration of the process of
development of excess voltage in FM due to spin accumulation on top surface of
HM. The band diagram of HM top surface showing non equilibrium spin
accumulation of $+y$ polarization shown as hatched red region (exaggerated) on
an otherwise parabolic band of normal metal with equal occupation of both spin
band (blue (red) for $+(-)y$). An ideal half metallic FM with shallow d-band
of one spin orientation is shown for simplicity of illustration. The
transverse FM and RM electrodes, connected to voltmeter creates an open
circuit condition. As the FM magnetization is flipped with the application of
$B$, the Fermi level of FM needs to align with the corresponding spin band of
the HM which is unaffected by $B$, thus resulting in a measurable voltage
difference proportional to spin accumulation. The side view of the interface
between FM and RM with HM are shown in the bottom for opposite FM
orientations.

The spin dependent transverse deflections are responsible for generating ISHE
is same as that of SHE. However as the incident current is spin polarized,
unequal number of carriers are accumulated at the opposite edges, resulting in
a finite voltage. In ferromagnetic metals (FM), strong exchange interaction
naturally gives rise to a band structure with finite spin polarization of
carriers. Thus the above mechanisms arising from SOC applied to FM leads to a
measurable transverse voltage, which is the phenomenon of Anomalous Hall
effect (AHE), a topic of extensive research spanning over several decades. [8]
The main point of contention was to establish whether the underlying spin
dependent transverse deflections can arises from scattering events (extrinsic
mechanisms) [9, 10, 11] or from band structures effects due to non-zero Berry
phase (intrinsic mechanisms). [12, 13, 14] A scaling relation between
transverse and longitudinal resistivity of the form
$\rho_{xy}\sim\rho_{xx}^{n}$ is found to hold true in general and the values
of the exponent $n$ were associated with specific mechanisms[8]. The generally
accepted view is that all mechanisms could in principle contribute to AHE, but
depending on nature of the material or on the degree of disorder in a given
material one mechanism may be dominant over others. Since the underlying
mechanisms behind SHE (or ISHE) is same as that of AHE, the same concepts can
be put to test, provided one can overcome the experimental challenge posed by
the lack of a straightforward method to directly quantify the presence of spin
chemical potential gradient. In this report we propose a simple intuitive
device structure and measurement scheme that can directly measure the spin
chemical potential of the accumulated spins due to SHE and calculate
$\rho_{SHE}$. Tungsten (W) is a heavy atomic weight metal (HM) a, with highest
reported negative $\theta_{SHE}$, with an useful feature of tunability of
crystal structure and morphology by controlling growth conditions, which in
turn can lead to large variation of resistivity. Thus we have chosen W as the
conducting medium for observing SHE and fabricated series of devices by
varying only the resistivity of W ($\rho_{W}$), keeping other parameters same,
so that the scaling behavior of $\rho_{SHE}$ with $\rho_{W}$ could be
examined.

Figure 2: Hysteresis response of $R_{xy}(B)$ (=$V_{xy}/I$) in the SHE
configuration for $\phi$ = (a) $0^{0}$, (b) $45^{0}$, and (c) $90^{0}$ for a
pure $\beta$-W device with $\rho_{W}$ = 750 $\mu\Omega$-cm and (d) for a Pt
device for $\phi=90^{0}$. Insets show the top view of the device schematic
showing orientation of the FM magnetization (deep orange arrow) with respect
to fixed spin accumulation (black arrow) in the HM top surface. The dashed
lines are eye estimation of approximate saturation values of $R_{xy}$ at high
positive and negative fields for a given $\phi$. The difference in the
saturation values is denoted as $\Delta R_{xy}^{asym}$ which is found to be
(a) zero for $\phi=0^{0}$, (c) maximum for $\phi=90^{0}$ and (b) intermediate
for $\phi=45^{0}$ and (d) opposite sign for Pt device at $\phi=90^{0}$
compared to (c).

###  I.1 Results and Discussion

An elegant and definitive method of detecting spin chemical potential was
demonstrated in the context of lateral spin valve in the so called non-local
measurement geometry [15, 16] where pure spin current was injected into a
normal metal from one FM electrode and a voltage proportional to spin
accumulation was detected by another FM electrode placed within spin diffusion
length of the spin injector. Subsequently, in the seminal work on the
theoretical framework for SHE in diffusive limit, a measurement scheme was
proposed using a FM voltage probe [2] attached to one of transverse edges in
current carrying conductor. The finite spin polarization ($P$) of the FM
conduction band was crucial for such measurements. Early experimental works on
electrical detection of SHE was primarily based on converting the generated
spin current into an electric voltage due to ISHE using complex device
structures[17, 18, 19]. An important step towards direct electrical detection
of SHE was reported [20] using a device featuring two FM electrodes (CoFe) of
different coercive fields placed on a HM channel (Pt), and the voltage
developed between the two FM electrodes were demonstrated to vary with
hysteretically between parallel and anti-parallel states. Under the assumption
that the two FM electrodes are exactly same, for the parallel states the
measured voltage is supposed to give zero SHE contribution , while for anti-
parallel sates the effect of SHE adds up , and hence the half of the width of
the transverse voltage hysteresis loop was identified as the spin Hall
voltage.

Figure 3: (a) The schematic of the device in ISHE configuration. The current
source is connected between FM and RM and voltage is measured between the ends
of the HM channel. The current at the FM/HM interface will inject a spin
current ($J_{s}$) in the HM along the $-z$ direction. Assuming an ideal single
spin band ferromagnetic for simplicity, it is shown that due to ISHE all the
spins are deflected in the $+x$ direction, developing a measurable voltage in
the HM. The hysteresis behaviour of the transverse resistance
$R_{yx}(=V_{yx}/I)$ in the ISHE configuration for the $\beta$-W device with
$\rho_{W}$ = 750 $\mu\Omega$-cm for angles $\phi$ = (b) $0^{0}$, (c) $45^{0}$,
and (d) $90^{0}$. The dashed lines are for eye estimation of approximate
saturation values of $R_{yx}$ at high positive and negative fields for a given
$\phi$. The difference in the saturation values is denoted as $\Delta
R_{yx}^{asym}$ which is found to be (b) zero for $\phi=0^{0}$, (d) maximum for
$\phi=90^{0}$ and (b) intermediate for $\phi=45^{0}$.

We propose a device structure where a transverse voltage in response to a
current in the HM channel is recorded using a combination of one FM electrode
along with a paramagnetic metal electrode, denoted as reference metal (RM) as
shown schematically in Figure 1(a) and explained in the figure caption. A
FESEM image of a typical device is shown in Figure 1(b). Details about the
device fabrication steps can be found in the Materials and Methods section in
Supplementary Information. We highlight a few important refinements
incorporated in our methods. Firstly, steps were taken to ensure that the HM
channel is embedded into insulating the $SiO_{2}$ substrate with only top
surface exposed so that the FM/RM electrodes are in contact only with the spin
accumulation at the top surface of HM. Secondly, we use a 2D vector magnet and
mount the substrate appropriately so that the magnetic field can rotate the FM
magnetization in the plane of the device, while the spin accumulation due to
SHE remains unaffected. The in plane orientation of the magnetic field vector
(B) is determined by the azimuthal angle $\phi$ as shown in Figure 1 (a-b).
The underlying principle for the direct measurement of spin potential is
explained in Figure 1(c) for $\phi=90^{0}$ when the effect is maximum. For HM
materials with positive $\theta_{SHE}$, equation 7 implies that a current in
the HM channel (along $x$) results in a transverse spin current (along $z$)
and the entire top (bottom) surface of the HM assumes spin accumulation in
positive (negative) $y$ direction. The non-equilibrium spin accumulation on
the top surface (along $+y$) is depicted as the hatched red region
(exaggerated for clarity) of the HM band diagram. For simplicity of
understanding, we consider the FM electrode to be a half metal (single spin
sub-band at the Fermi level, P=1). Depending on whether the FM magnetization
is parallel or anti-parallel to the accumulated spin on the top surface of HM,
the Fermi level of the FM will align with the chemical potential of either
$+y$ or $-y$ spin band of HM. In contrast, the Fermi level of a normal metal
reference electrode (RM) electrode (P=0) will align with the average of the
two spin chemical potentials of the HM under all conditions. Thus, a voltmeter
connected between a pair of FM and RM electrode in contact with the HM, will
register a voltage difference when magnetization of FM is reversed (with the
help of an external magnetic field) and can be shown to be proportional to the
difference in spin potentials. A rigorous derivation for a general FM ($P<1$)
shows the voltage [2] to be read in conjunction with equation 2.

$\Delta V=P(\mu_{\uparrow}-\mu_{\downarrow})$ (4)

A proof of concept of the above idea lies in the fact that the sign of $\Delta
V$ deduced above will reverse for HM material with negative $\theta_{SHE}$ and
to test this idea, devices were fabricated with Pt (positive $\theta_{SHE}$)
[21, 22] and W (negative $\theta_{SHE}$) [6, 23, 24] as HM material. The
magnetic field dependence of $R_{xy}=V_{xy}/I$ for the highest resistivity
$\beta$-W device ($\rho_{W}$ = 750 $\mu\Omega$-cm) at three representative
angles of $\phi$ = $0^{0}$, $45^{0}$, and $90^{0}$ are shown in Figure
2(a)-(c). For $\phi=0^{0}$ [Figure 2(a)], corresponding to the configuration
when FM magnetization is always orthogonal to spin accumulation (see inset)
and as expected there is no change in $R_{xy}$ when magnetization is flipped.
In contrast, for $\phi=90^{0}$ [Figure 2(c)], biggest change in $R_{xy}$ is
observed as the magnetization is flipped from being parallel to anti-parallel
to the spin accumulation. For $\phi$ = $45^{0}$ configuration [Figure 2(b)], a
component of total magnetization of FM, changes from being parallel to anti-
parallel to the spin accumulation as the magnetization is flipped, resulting
in an intermediate change in $R_{xy}$. Alternatively, the Pt-based device
depicted in 2(d) exhibits a opposite field dependent transverse resistance as
compared to the W device at $\phi$=$90^{0}$. This contrast could be attributed
to the voltage induced by opposite spin accumulation (compared to W) generated
at the top and bottom surface of the device. The same devices can also be used
for detecting ISHE in the HM by swapping the current and voltage leads as
shown schematically in Figure 3(a). A spin polarized current is injected from
the FM into the HM along $-z$ direction in the overlap region with spin
polarization $\bf{s}$ along $+y$. The SOC driven deflection of injected spins
will result in a transverse charge current [equation 8] towards $-x$ direction
along the HM channel which will thus register a finite transverse voltage
across its ends. Once the magnetization of FM is flipped, the polarization of
the injected spins and hence the sign of voltage is also reversed. The
magnetic hysteresis curves of the transverse resistance in the ISHE
configuration, defined as $R_{yx}=V_{yx}/I$ for the same device of $\rho_{W}$
= 750 $\mu\Omega$-cm as shown in Figure 3(b)-(d). For $\phi$=$0^{0}$, the FM
is always magnetized along the HM channel and thus there is no charge current
and hence voltage along the HM channel. Thus we observe that $R_{yx}$ values
for opposite magnetization orientation is same [Figure 3(b)]. In contrast, for
$\phi$=$90^{0}$ the ISHE charge current generated along the HM channel and the
change in the $R_{yx}$ upon flipping of FM is maximum [Figure 3(d)]. For
$\phi$=$45^{0}$ the change in $R_{yx}$ is intermediate as a component of total
magnetization contributes to a transverse voltage along the HM channel [Figure
3(c)]. The measurements of SHE and ISHE hysteresis scans on each devices at
various angles seems to conform with the Onsager reciprocity principle.

Figure 4: Plot of the asymmetric contribution of the as measured transverse
resistance in (a) SHE configuration $R^{asym}_{xy}(B)$ for data shown in
Figure 2(c) and ISHE configuration $R^{asym}_{yx}(B)$ for data shown in Figure
3(d). Both plots have same scale on both axis. The data for $|B|\geq$ 150 mT
are linearly fitted and shown as dashed lines to take into account possible
artifacts like ordinary Hall effect or AHE of FM. The intercepts of the linear
fits is a measure of the the asymmetric transverse resistance due to SHE at
$B=0$ and denoted as $\Delta R^{asym}_{xy}$ which is almost same in magnitude
for SHE and ISHE in agreement with the reciprocity relation of the two
phenomenon.

The as measured transverse resistance $R_{xy/yx}$ requires three important
corrections so that it can be quantitatively correlated with SHE(ISHE)
resistivity. Firstly, the raw data presented in Figure 2 for SHE and Figure 3
for ISHE, is not purely asymmetric with respect to flipping of magnetization
of the FM, as demanded by the the proposed measurement models. The presence of
a symmetric contribution is typical in any Hall effect measurement, that may
arise in general from misalignment of transverse voltage probes, resulting in
picking up of longitudinal voltage drop and /or in the current scenario from
planner Hall effect [25] of the FM. Following standard procedure, the anti-
symmetric component of transverse voltage can be extracted as
$V_{xy}^{asym}=[V_{xy}(+B)-V_{xy}(-B)]/2$. Secondly, as the HM and FM are in
Ohmic contact, an applied current $I$ will redistribute between HM, FM, and RM
layers in the overlapped regions. The FM is sensing the addition asymmetric
voltage only due to the current flowing through the HM ($I_{HM}$) in contact.
Using parallel resistor model and geometry of the devices (details shown in
SI), we deduce that.

$I_{HM}=\frac{I}{[{{1+\frac{\rho_{HM}}{t_{HM}w_{HM}}(\frac{t_{FM}w_{FM}}{\rho_{FM}}+\frac{t_{RM}w_{RM}}{\rho_{RM}}}})]},$
(5)

where, $\rho_{HM/FM/RM}$, $t_{HM/FM/RM}$ represent the resistivity, thickness
of HM, FM, and RM, respectively. $w_{HM}$ refers to the width of HM and
$w_{FM/RM}$ are overlapped width of FM and RM layers on HM. A plot of the
field dependence of a purely anti-symmetric transverse resistance defined as
$R_{xy}^{asym}=V_{xy}^{asym}/I_{HM}$ is shown in Figure 4. Thirdly, the
calculated $R_{xy}^{asym}$ may still have a component arising from ordinary
Hall effect (OHE) of the HM channel or AHE of the FM electrode that may arise
from slight unavoidable component of $B$ normal to sample plane, both
contribution being linear in $B$. To eliminate such artifacts, linear fitting
are performed on the high field ($|$B$|$ $>$ 150 mT) regions of
$R_{xy}^{asym}(B)$ curves, shown as dashed lines in Figure 4. We identify the
extrapolated values of the fitted lines at $B=0$, as the contribution arising
solely from SHE or ISHE and is denoted as $\Delta R_{xy}^{asym}$ for a given
scan angle $\phi$.

Figure 5: (a) Plots of $\Delta R^{asym}_{xy}$ extracted from individual
hysteresis scans in SHE configuration for a fixed $\phi$ ranging from $0^{0}$
to $360^{0}$ in steps of $10^{0}$, for series of devices with $\rho_{W}$ = 10,
30, 100, 350, 450, 650, 750 $\mu\Omega$-cm and one Pt device with $\rho_{Pt}$
= 34 $\mu\Omega$-cm. Inset shows the low resistivity devices plotted
separately for clarity. Solid lines are fits to the equation of the form
$R_{SHE}sin(\phi-\phi_{0})$, (b) Plot of spin Hall resistivity ($\rho_{SHE}$)
for all devices vs. longitudinal resistivity ($\rho_{W}$) in log-log scale.
The data points for pure $\beta$-W is shown in blue while that for pure
$\alpha$-W and mixed ($\alpha+\beta$)-W are shown as red. Both sets are are
separately fitted to straight lines and the corresponding slopes representing
the power law exponents are indicated.

Our experiment extends over a set of devices with different HM channels that
includes a series of W of resistivity $\rho_{W}$ = 10, 30, 100, 350, 450, 650,
750 $\mu\Omega$-cm, and one Pt of resistivity $\rho_{Pt}$ = 34 $\mu\Omega$-cm,
all having same geometric parameters and FM electrode. For each of our
devices, we recorded $R_{xy}(B)$ hysteresis scans with in-plane $B$ at fixed
$\phi$ ranging from $0^{0}$ to $360^{0}$ in steps of $10^{0}$, in both SHE and
ISHE measurement configurations. $\Delta R^{asym}_{xy}$ is calculated for each
such scan following the analysis mentioned above. The results are summarised
in Figure S5(a). The inset graph highlights the plots for devices with
$\rho_{W}$ = 10 and 30 $\mu\Omega$-cm and $\rho_{Pt}$ = 34 $\mu\Omega$-cm, as
the magnitudes of $\Delta R_{xy}^{asym}$ is comparatively small and
distinguishable in the main plots. The variation of $\Delta
R_{xy}^{asym}(\phi)$ for all the W devices is exactly opposite to that of the
Pt device and is consistent with the opposite signs of $\theta_{SHE}$ of the
materials. It may be that the orientation of the spin accumulation is always
transverse ($\pm y$) to the current irrespective of applied field and the
orientation of the FM with respect to the channel is determined by $\phi$ so
that the component of magnetization along the spin accumulation varies as
$sin(\phi)$. We find that for all the devices, variation of $\Delta
R_{xy}^{asym}$ appears to fit well with curves of the form
$R_{SHE}sin(\phi-\phi_{0})$ [Figure S5(a)], $R_{SHE}$ signifies the maximum
SHE signal when the spin accumulation and FM are parallel and $\phi_{0}$
represents misalignment of the HM channel with the vector magnet $x$-axis,
typically $\sim 5^{0}$. Previous reports on measurement of SHE [18, 26] has
confirmed such angular dependence of measured voltage with detector
magnetization. The values of $R_{SHE}$ extracted from the fits of the W
devices is found to increase with $\rho_{W}$, with highest value of $\sim$ -55
m$\Omega$ observed in the pure $\beta$-W device of resistivity $\rho_{W}$ =
750 $\mu\Omega$-cm. The $R_{SHE}$ for Pt device is significantly low $\sim$
+0.5 m$\Omega$, which is comparable to the value $\sim$ -0.2 m$\Omega$
observed in pure $\alpha$-W with $\rho_{W}$ = 10 $\mu\Omega$-cm. Exact same
analysis performed on the ISHE data for each samples gives $R_{ISHE}$ values
[Figure S5 (a) in SI] same as that of corresponding $R_{SHE}$, which is
expected from the reciprocity relations.

The HM materials in our devices are deposited using magnetron sputtering under
high vacuum conditions on amorphous substrates at room temperatures, resulting
in disordered polycrystalline films with rough morphology which is confirmed
by AFM and SEM images. Hence it is argued that our HM channels are not
conducive to support the intrinsic mechanisms of SHE (ISHE) and only extrinsic
mechanisms are dominant. We further argue that resistivity of the HM channels
is a measure of disorder [27], which implies that in a device with higher
resistivity, the carriers will undergo more scattering and result in higher
transverse spin accumulation in case of SHE configuration or higher charge
current in case of ISHE configuration. This qualitatively explains the
increase of $R_{SHE}/R_{ISHE}$ with the $\rho_{W}$ [Figure S5(a)]. The
extrinsic mechanisms are further classified into two broad categories. For the
skew scattering mechanism[9], the transverse conductivity was shown to be
directly proportional to the mean free path of the material, which translates
into dependence of the form $\rho_{xy}\sim\rho_{xx}$. The side jump mechanism
[10] on the other hand is shown to have a transverse conductivity that is
independent of the mean free path which leads to relations of the form
$\rho_{xy}\sim\rho_{xx}^{2}$. We first calculated the transverse resistivity
following equation 2 and equation4 as,

$\rho_{SHE}=\frac{R_{SHE}w_{HM}t_{HM}}{P\lambda_{sf}}$ (6)

where we have used $P=0.3$ [28] and $\lambda_{sf}$ = 2 nm [29] for all our
devices as we could not independently estimate these parameters. A plot of
$\rho_{SHE}$ as a function of $\rho_{W}$ on a log-log scale is shown in Figure
S5(b) and the slopes of linear fits that gives the exponents of the scaling
relations $\rho_{xy}\sim\rho_{xx}^{n}$. We have further segregated the data
into three regimes of W morphology as indicated by vertical dashed lines: (i)
pure metastable $\beta$-W phase (ii) mixed $(\alpha+\beta)$-W and (iii) pure
$\alpha$-W phase. We find, that if the fitting is restricted to pure $\beta$-W
samples, the slope obtained is 1.9 $\pm$ 0.2 which is indicative of dominant
side jump mechanism. If the linear fit is performed on sample set excluding
the pure $\beta-$W devices, the slope obtained is $1.22\pm 0.05$, which is
indicative of presence of both skew scattering and side-jump mechanism. The
same analysis is performed on the ISHE resistivity (shown in SI) and similar
trend is observed as expected from reciprocity principle. In the context of
AHE, it was established that since transverse conductivity ($\sigma_{xy}$) for
side jump mechanism being independent of the mean free path, is dominant for
more disordered samples with low mean free path. Our results seems to
corroborate the fact in the context of SHE for the highly disordered pure
$\beta$-W samples.

###  I.2 Conclusion

To conclude, we have demonstrated the functioning of an intuitive device
configuration that allows for direct detection of spin potential arising out
of SHE in HM materials using a combination of FM/HM voltage probes. The same
devices were used to detect ISHE and verify the validity of reciprocity
relations. The use of a vector magnet to rotate the magnetization of the FM
provides a accurate measure of the maximum signal arising from SHE/ISHE and
hence resistivity. Further we have exploited the large tunability of
resistivity of W and examined the scaling relations to identify the underlying
microscopic mechanisms and found pronounced evidence for the side jump
mechanism in higher resistivity samples. Our measurement scheme can be further
generalised to study other spintronics phenomenon.

###  I.3 Acknowledgement

We acknowledge the support of IISER Kolkata, Ministry of Education (MoE),
Govt. of India funding the research and providing fellowship to SA and HB. ABM
and DM acknowledge the support of fellowship from CSIR, Govt. of India. PM
acknowledges support from grant SR/FTP/PS-077/2011, SERB, Govt. of India.

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Supplementary Information for ‘Direct Electrical Detection of Spin Chemical
Potential Due to Spin Hall Effect in $\beta$-Tungsten and Platinum Using a
Pair of Ferromagnetic and Normal Metal Voltage probes’

Note 1. Materials and Methods:

Deposition $\&$ Characterization of W-Thin Films of Various Resistivity (HM
layer):

Figure S1: XRD patterns of W thin films under varying $P_{Ar}$. The peak
positions with their relative peak intensities corresponding to the phases of
W are shown as vertical dotted lines. (a) XRD of as-deposited W films with
resistivity ranging from 100 $\mu\Omega$-cm to 750 $\mu\Omega$-cm. The films
at higher $P_{Ar}$ ($\rho_{W}$ = 750, 650, 450, and 350 $\mu\Omega$-cm) are
completely in $\beta$ phase, whereas $\rho_{W}$ = 100, 200 $\mu\Omega$-cm
exhibit a dominant $\beta$ phase mixed with some $\alpha$, characterized by
the shift in peak position from $2\theta$ = $39.7^{0}$ to $40.04^{0}$. (b) As
deposited and annealed W films which are mixed $(\alpha+\beta)$-W and complete
$\alpha$-W respectively. Figure S2: (a) depicts how the the grain size changes
with $\rho_{W}$. As $\rho_{W}$ decreases, the grain size is observed to
increase. The increased magnitude is more pronounced in $\alpha$-W, (b) shows
$\rho_{W}$ as a function of $P_{Ar}$. A monotonic increase in $\rho_{W}$ is
observed in relation to increasing $P_{Ar}$.

We prepared a series of 22 nm thick W-thin films on $Si/SiO_{2}$ (300 nm)
substrates at room temperature by dc magnetron sputtering (Mantis Deposition
LTD) using a combination of scroll and turbo pump achieving a base pressure of
$1\times 10^{-6}$ torr. The W target (Kurt Lesker), measuring 3 inches in
diameter $\times$ 3 mm thickness (99.95% pure), was maintained at a constant
distance of 15 cm from the substrate. A straight deposition setup ensured film
homogeneity. These W films were fabricated under varying argon pressures
($P_{Ar}$), while maintaining a constant sputtering power of 100 watt.
$P_{Ar}$ was systematically varied, ranging from 4.5 mTorr to 8.5 mTorr.
Correspondingly, the deposition rate of W thin films decreased from 1.0 Å/sec
for $P_{Ar}$ = 4.5 mTorr to 0.3 Å/sec for $P_{Ar}$ = 8.5 mTorr, measured by a
crystal monitor. The Crystal structure of deposited W films (22 nm) under
different $P_{Ar}$ were characterized by Bruker x-ray diffraction spectroscope
(XRD) with a resolution of $2^{0}$ as shown in Figure S1. $\rho_{W}$ of W
films exhibited an increase from 10 $\mu\Omega$-cm to 750 $\mu\Omega$-cm as
$P_{Ar}$ was raised from 4.5 mTorr to 8.5 mTorr [Figure S2(b)]. The W films
tend to be more amorphous as $P_{Ar}$ increases. W films grown at higher
$P_{Ar}$ ($\rho_{W}$ ranging from 750 to 350 $\mu\Omega$-cm) have single phase
$\beta$-W which is expected to be A15 type crystal structure [24, 30]. This
phase was characterized by the distinct peaks at 2$\theta$ = $35.6^{0}$,
$39.6^{0}$, $43.8^{0}$ corresponding to $\beta$ (200), (210), (211) planes,
respectively. However, in case of the W films with $\rho_{W}$ = 200, 100
$\mu\Omega$-cm, there was a subtle indication of $\alpha$ phase, evidenced by
a peak shift from 2$\theta$ = $39.6^{0}$ to 2$\theta$ = $40.1^{0}$, which is a
signature of $\alpha$-W (110) peak. These films are dominated by $\beta$ phase
mixed with a minor presence of $\alpha$ as shown in Figure S1(a). As the
$P_{Ar}$ was further reduced ($P_{Ar}$ $\leq$ 5 mTorr), $\alpha$ phase
gradually became dominant. The as-deposited W film ($\rho_{W}$ = 30
$\mu\Omega$-cm) showed a highly crystalline structure dominated by $\alpha$
phase corresponding to $\alpha$ (110), (200), (211) planes at $2\theta$ =
$40.4^{0}$, $58.3^{0}$, $73.6^{0}$ respectively, mixed with weaker $\beta$
peaks along (200), (211) at $2\theta$ = $35.6^{0}$, $43.8^{0}$ respectively,
indicating a mixed ($\alpha$ \+ $\beta$) phase. The transition from mixed
($\alpha$ \+ $\beta$)-W to pure $\alpha$-W occurred when the as-deposited film
($\rho_{W}$=30 $\mu\Omega$-cm) underwent annealing at $280^{0}$ C for 10
minutes, employing a temperature ramp-up duration of 3 hours and followed by a
natural cooling of 6 hours in the vacuum chamber. The absence of $\beta$
diffraction peaks in the annealed film confirmed the existence of a pure
$\alpha$ phase which possesses a bcc crystal structure characterized by the
diffraction peaks corresponding to $\alpha$ (110), (200), and (211) at
$2\theta$ = $40.4^{0}$, $58.3^{0}$, $73.6^{0}$ [Figure S1(b)]. XRD data
provides a rough estimation of the grain size of W-thin films of different
phases. According to Scherrer’s equation [31], average grain size can be
evaluated from the equation expressed as $g=K\lambda/\delta cos(\theta)$,
where, g is the average grain size, K is a shape factor (approximately 0.9),
$\lambda$ is the x-ray wavelength (1.54 ${\AA}$), $\delta$ is width of
diffraction peak at half maxima and $\theta$ is Bragg’s angle [32]. Analyzing
our XRD data, we found that the grain size varied systematically with
$P_{Ar}$, exhibiting a decrease as $P_{Ar}$ increased [Figure S2(a)]. For
further characterization, we measured the resistivity of W-thin films using
the van der Pauw technique. The resistivity $\rho_{W}$ increased with $P_{Ar}$
stabilized with a maximum value of $\rho$ = 750 $\mu\Omega$-cm at 8.5 mTorr,
while maintaining a uniform thickness of 22 nm. It is reported that $\beta$-W
films tend to have notably high resistance because of the A15 crystal
structure which is associated with pronounced electron-phonon scattering [6,
33]. However, $\alpha$-W is recognized for its low resistivity, and in our
study we obtained a minimum of $\rho_{W}$ = 10 $\mu\Omega$-cm which is
identified as pure $\alpha$ phase.

Characterization of Py-Thin Films (FM layer):

Figure S3: (a) XRD image of Py thin film of thickness 30 nm. The observed peak
at $2\theta$ = $44.52^{0}$ confirms the presence of fcc crystal structure, (b)
AFM image captures the surface of Py thin film, covering a scanning area of 2
$\mu m$ $\times$ 2 $\mu m$.

Py films were deposited on top of the HM layer utilizing the thermal
evaporation technique. Prior to each deposition, an ion milling has been
performed using dry argon plasma to clean the surface of the HM layer and to
make the interface transparent. The specific parameters for the permalloy
deposition are: $P_{base}$ $\approx$ $1\times 10^{-7}$ Torr, $P_{dep}$
$\approx$ $1\times 10^{-6}$ Torr, $T_{dep}$ $\approx$ $1300^{0}$C, deposition
rate = (0.8-1.0) A/sec. These deposition parameters remained consistent across
all devices. Characterization of the Py films was carried out using XRD and
AFM techniques. In Figure S3(a), the XRD image of the Py film is presented,
with an observed peak at $2\theta$ = $44.52^{0}$, confirming the presence of
an fcc crystal structure. Figure S3(b) displays the AFM image depicting the
surface morphology of the Py film over a scanning area of 2 $\mu m$ $\times$ 2
$\mu m$. The observed surface roughness was found to be 0.3 nm.

Device Fabrication Procedure:

A four-step deposition process including a contact pad was proposed for device
fabrication, where each step involved photo-lithography (PL), RF ion etching,
material deposition, and lift-off after deposition. (i) Step 1: HM Channel: A
300 $\mu m$ $\times$ 30 $\mu m$ rectangular bar was patterned on $Si/SiO_{2}$
(300 nm) substrates by Photolithography, followed by RF Ar ion etching to
create a trench in the ‘insulating’ $SiO_{2}$ substrate. Etching parameters
were $P_{Ar}$ = $6\times 10^{-4}$ Torr, Power = 80 watt and ion current = 30
mA. Subsequently, HM materials (W and Pt) were deposited by dc magnetron
sputtering, followed by lift off. The etch rates and deposition rates were
calibrated to ensure that only the top surface of the HM channel is exposed
for later steps. A series of W-thin films of thickness $22$ nm were used as HM
channel material with a systematic variation of resistivity ranging from
(10-750) $\mu\Omega$-cm by controlling by the $P_{Ar}$ during deposition,
keeping sputtering power constant at 100 watt. A control device of Pt
($t_{Pt}$ = 37 nm, $\rho_{Pt}$ = 34 $\mu\Omega$-cm) HM channel was deposited
by magnetron sputtering. The parameters used for Pt deposition were $P_{Ar}$ =
$6\times 10^{-3}$ Torr, power = 50 watt. (ii) Step 2: Reference Metal(RM):
Rectangular patterns of dimensions 20 $\mu m$ $\times$ 250 $\mu m$, aligned
perpendicular to the long axis of HM and having overlap of $\sim 5-7\mu m$.
The RM material is essentially same as that of HM but with a constant low
resistivity $\sim 60\mu\Omega$-cm and thickness $t_{RM}$ = 37.5 nm. RM
deposition is preceded by an ion milling was performed to clean the film
surface and enhance interface transparency. (iii) Step 3: Ferromagnetic Metal
(FM) Pattern dimensions are same as that of RM, and aligned linearly with the
RM electrode but on the other side of the HM with overlap $\sim(5-7)\mu m$.
Permalloy is chosen as the FM materials for all the devices, deposited by
thermal evaporation from a RADAK furnace, $P_{dep}$ = $1\times 10^{-6}$ Torr,
rate = 1.0 Å/s. Same thickness and resistivity of FM was maintained for all
devices , $t_{FM}$ = 30nm, $\rho_{FM}\sim 60\mu\Omega$-cm. FM deposition was
preceded by an Ar ion milling. The in plane and out of plane coercive fields
of Py was found to be $\sim 5$mT and $\sim 0.8T$ respective and the hysteresis
curves of AMR and PHE confirms in-plane magnetisation anisotropy, typical of
Py films. (iv) Step 4: Contact pads Cr-Au contact pads was deposited using
electron-beam evaporation, overlapping the two ends of the HM channel and on
free ends of RM and FM electrodes. All the resistivity vales are obtained by
Van der Pauw method on a separate square piece at room temperature.

Measurements

Transport measurements were performed at room temperature using a four pole
vector electromagnet magnet (Dexing, China) (Figure S4). The magnetic field
values were calibrated separately for each pole pair with a DTM-151 Digital
Teslameter and the field variation achieved for each pole were between $\pm
0.5T$ with an accuracy of 0.1mT. The hysteresis scans were limited upto $\pm
0.4T$ and step size of field variation was 1mT. DC current was applied using a
keithley 6221 AC-DC current source and voltages measured using a 2182A Nano-
voltmeter. Each resistance reading was performed with currents of opposite
polarity and averaging the values obtained 20 times. Typical current density
achieved in the devices are found in the range of $(0.5-7.0)\times 10^{9}$
$A/m^{2}$ that limits the joule heating power to $\sim$ 0.5 mW.

Figure S4: Schematic presentation of a spin Hall device loaded inside a vector
magnet.

Note 2. Charge Current Distribution Due to Shunting in FM and RM Electrodes:

In our measurement, we obtain the $R_{xy}$ from the ratio $V_{xy}$/$I$, where,
$I$ represents the total charge current. Notably, the FM and RM electrodes
only partially overlap the HM layer, meaning that only a portion of the total
current passes through the HM layer, contributing to the spin Hall resistance.
As both the FM and RM electrodes form Ohmic contacts with the HM channel, we
must account for the shunting effect in these devices. The resistivity of both
FM and RM electrodes ($\rho_{FM/RM}$ = 60 $\mu\Omega$-cm) is comparable to or
much lower than that of the resistivity of HM channel ($\rho_{HM}$ = 10-750
$\mu\Omega$-cm). Therefore, a significant portion of the current is diverted
through the FM and RM electrodes when it passes through the HM channel in the
region where FM and RM are present. This shunting of current through the FM
and RM electrodes reduces the current flowing through the HM channel.

We calculate the charge current along HM layer, considering the SHE
configuration. However, same calculation is applied to ISHE. The relationship
between the electric field ($\vec{E}$) and the current density ($\vec{J}_{C}$)
is expressed as.

$\vec{E}=\rho\vec{J}_{C}$ (7)

In our device, the electric field at the junctions HM/FM and HM/RM should be
equal. Therefore, the current density through FM and RM can be rewritten using
the equation 7.

$\displaystyle\rho_{HM}J_{HM}=\rho_{FM}J_{FM}:J_{FM}=J_{HM}\frac{\rho_{HM}}{\rho_{FM}}$
(8)
$\displaystyle\rho_{HM}J_{HM}=\rho_{RM}J_{RM}:J_{RM}=J_{HM}\frac{\rho_{HM}}{\rho_{RM}}$
(9)

In the SHE configuration, when a current $I$ is applied along the HM channel,
it is distributed across the FM, HM, and RM as follows:

$\\\ I=I_{HM}+I_{FM}+I_{RM}\\\ I=J_{HM}A_{HM}+J_{FM}A_{FM}+J_{RM}A_{RM}\\\
I=J_{HM}(A_{HM}+\frac{\rho_{HM}}{\rho_{FM}}A_{FM}+\frac{\rho_{HM}}{\rho_{RM}}A_{RM})\\\
$ (10)

Where, $A_{HM}$ = $w_{HM}t_{HM}$, $A_{FM}$ = $w_{FM}t_{FM}$, $A_{RM}$ =
$w_{RM}t_{RM}$. $t_{HM/FM/RM}$ are thicknesses of HM, FM, RM layers,
respectively. $w_{HM}$ is the width of HM layer, and $w_{FM}$, $w_{RM}$ are
overlapped width of FM, RM layers on HM layer. The current density in HM layer
is expressed as.

$\displaystyle
J_{HM}=\frac{I}{A_{HM}(1+\frac{\rho_{HM}}{\rho_{FM}}\frac{A_{FM}}{A_{HM}}+\frac{\rho_{HM}}{\rho_{RM}}\frac{A_{RM}}{A_{HM}})}$
(11)

The current flowing the HM layer is:

$I_{HM}=\frac{I}{[1+\frac{\rho_{HM}}{t_{HM}w_{HM}}(\frac{t_{FM}w_{FM}}{\rho_{FM}}+\frac{t_{RM}w_{RM}}{\rho_{RM}})]}$
(12)

Note 3. Angular Dependence of all W-Devices in Inverse Spin Hall Effect
Configuration:

Figure S5: (a) Shows the plots of $\Delta R^{asym}_{xy}$ as a function of
angle $\phi$ in ISHE configuration for devices with $\rho_{W}$ = 10, 30, 100,
350, 450, 650, 750 $\mu\Omega$-cm. Each graph represents the result of an
individual field scan of $R^{asym}_{xy}$ for a constant $\phi$ range spanning
from $0^{\circ}$ to $360^{\circ}$ in $10^{\circ}$ increments. To enhance
clarity, the inset illustrates the low-resistivity devices plotted separately.
The solid lines depict fits to the equation $R_{SHE}\sin(\phi-\phi_{0})$, (b)
The spin Hall resistivity ($\rho_{ISHE}$) is plotted against the longitudinal
resistivity ($\rho_{W}$) on a logarithmic scale for all devices. Pure
$\beta$-W data points are highlighted in blue, while those for pure $\alpha$-W
and mixed ($\alpha+\beta$)-W are shown in red. Each data set is individually
fitted to straight lines, with the corresponding slopes indicating the power
law exponents.

The angular variation of $\Delta R^{asym}_{yx}$ with different W resistivity
of $\rho_{W}$ = 10, 30, 100, 350, 450, 650, 750 $\mu\Omega$-cm is plotted in
Figure S5(a). The inset graph shows the plots for devices with $\rho_{W}$ = 10
and 30 $\mu\Omega$-cm, as the magnitudes of $\Delta R_{yx}^{asym}$ is
comparatively small and distinguishable in the main plot. Similar to SHE,
variation of $\Delta R_{yx}^{asym}$ is fitted with curves of the form
$R_{SHE}sin(\phi-\phi_{0})$. Figure S5(b) shows the plot of $\rho_{ISHE}$ as a
function of $\rho_{W}$ on log-log scale. The slope extracted for all W-devices
is found to be 1.25 $\pm$ 0.04. When they are plotted separately, the slopes
for pure $\beta$-W and the combined $\alpha$-W, mixed ($\alpha+\beta$)-W are
1.25 $\pm$ 0.06 and 1.9 $\pm$ 0.2, respectively.