# First-principles theory of intrinsic spin and orbital Hall and Nernst
effects in metallic monoatomic crystals

Leandro Salemi<EMAIL_ADDRESS>Peter M. Oppeneer  Department of
Physics and Astronomy, Uppsala University, P.O. Box 516, SE-75120 Uppsala,
Sweden

(August 27, 2024)

###### Abstract

The generation of spin and orbital currents is of crucial importance in the
field of spin-orbitronics. In this work, using relativistic density functional
theory and the Kubo linear-response formalism, we systematically investigate
the spin Hall and orbital Hall effects for 40 monoatomic metals. The spin Hall
conductivity (SHC) and orbital Hall conductivity (OHC) are computed as a
function of the electrochemical potential and the influence of the spin-orbit
interaction strength is also investigated. Our calculations predict a rather
small OHC in $sp$ metals, but a much larger OHC in $d$-band metals, with
maximum values [$\sim 8000\,(\hbar/e)\Omega^{-1}{\rm cm}^{-1}$] near the
middle of the $d$ series. Using the Mott formula, we evaluate the thermal
counterparts of the spin and orbital Hall effects, the spin Nernst effect
(SNE) and the orbital Nernst effect (ONE). We find that the as-yet unobserved
ONE is significantly larger ($\sim 10\times$) than the SNE and has maximum
values for group 10 elements (Ni, Pd, and Pt). Our work provides a broad
overview of electrically- and thermally-induced spin and orbital transport in
monoatomic metals.

###### pacs:

Valid PACS appear here

††preprint: APS/123-QED

##  I Introduction

The ability to control, generate and detect spin currents is a central issue
to design efficient spintronics devices. Spin-polarized currents have already
become exploited in spin-transfer torque magnetic random access memories (STT-
MRAM) (see, e.g., [1]). In the past ten years the spin-orbit torque (SOT) has
come in the research focus, when it was discovered that the SOT leads to very
efficient magnetization switching in magnetic elements [2, 3]. This drew the
attention to the generation and utilization of pure spin currents that are
understood to be behind the efficient switching caused by the SOT [4].

One of the key phenomenon in the efficient generation of pure spin currents is
the spin Hall effect (SHE), where a transverse spin current arises from a
longitudinal charge current [5, 6]. Initially predicted based on the
assumption of extrinsic electron scattering [7], impurities-independent
intrinsic SHEs in various systems such as p-type semiconductors [8] or a two-
dimensional electron gas with a Rashba-type spin-orbit coupling (SOC) [9] were
later theoretically predicted. The intrinsic contribution originates from the
Berry curvature associated to the band structure of the material [8, 9, 10,
11] while extrinsic mechanisms, such as skew-scattering and side jumps,
originate from spin-dependent scattering on defects [12, 13]. It also became
clear, via indirect measurements of the inverse SHE (ISHE), that metals
exhibit orders of magnitude larger spin Hall conductivities (SHC) than
semiconductors [14, 15, 16].

Extended studies on metallic alloys [17, 18, 19, 20, 21, 22, 23, 24, 25, 26,
27, 28, 29, 30] have showed that, while some systems mainly show extrinsic
mechanisms [30, 25, 21] others are mainly dominated by intrinsic
contributions[26, 24, 21]. The crossover between extrinsic and intrinsic
contribution have been recently investigated in Au/Cu alloys, where varying
the concentration of Cu can lead to either intrinsic-dominated or extrinsic-
dominated SHE [31].

Further support for the existence of a large intrinsic SHE in metals came from
ab initio calculations of the SHC in Pt that predicted a huge SHC $\sim
2000\,(\frac{\hbar}{e})(\Omega\,\textrm{cm})^{-1}$ [32]. This quickly raised
the question of the physical origin of this huge SHE in metallic systems.
Kontani et al. [33] studied the particular case of Pt and using a tight-
binding Hamiltonian and found that there exists in fact a huge orbital Hall
effect (OHE) which arises from a phase factor analogous to the Aharonov-Bohm
phase factor, without requiring any SOC. This phase factor is induced by an
effective magnetic flux due to the angular dependence of the $d$-orbitals.
Other theoretical works on the OHE were conducted on various systems such as
4$d$ and 5$d$ transition metals [10, 34], Sr2MO4 (M = Ru, Rh, Mo) [35] and
heavy-fermion systems [36].

Recently, a renewed interest has emerged for the OHE as it might offer an
intriguing way to generate orbital currents and utilize these to perform
magnetization switching [37]. Recent works re-investigated the OHE in
$d$-transition metals systems as well as $sp$ metals such as Li or Al [38,
39]. These reached similar conclusions: the intrinsic OHE is a generic
quantity that is bigger than the SHE and arises without the interplay of SOC
while the intrinsic SHE arises as a result of the OHE and SOC.

Those works offer new perspective on orbital-related phenomena, which often
tend to be neglected due to the well known orbital-quenching in periodic
solids. Other recent investigations in closely related fields support the
notion of orbital-driven physics in more exotic systems such as
antiferromagnets [40] or in chiral structures [41]. Efforts are actually being
devoted to unify spin and orbital dynamics for a more accurate modeling of
SOTs [42].

The analogy between transverse charge and spin transport in the normal Hall
effect and SHE has been extended to thermally-driven transport in the last
years. Thermally-driven spin transport phenomena such as the spin-dependent
Seebeck effect [43, 44, 45] and spin Peltier effect [46, 47] have been
observed. Transverse thermally-induced charge transport, given by the von
Ettingshausen-Nernst effect [48], was predicted to have a spin analog, the
spin Nernst effect (SNE) [49]. The SNE was initially investigated
theoretically [50, 51, 52, 53], and, recently, it was observed in Pt and W
thin films [54, 55, 56].

The SNE can be seen as the thermal counterpart of the SHE. Pushing the analogy
a bit further, we propose the concept of the orbital Nernst effect (ONE),
where a longitudinal temperature gradient induces transverse orbital angular
momentum thermal transport. There is however not much known about the behavior
of the SNE in metals and even less about the ONE.

To analyze trends in electrically- and thermally-driven transverse spin and
orbital transport, we compute the intrinsic contribution of SHE, OHE, SNE, and
ONE in 40 monoatomic crystals.

Our calculations are performed within the framework of relativistic density
functional theory (DFT) and Kubo linear-response theory. The main focus in set
on $3d$, $4d$, and $5d$ transitions metals but metals from the first and
second columns (e.g., Li, Na) as well as $sp$ metals (e.g., Al, In, and Pb)
are also considered. Compared to previous works which used tight-binding
Hamiltonians [33, 10, 34, 35, 36, 38, 39], the use of all-electron, full-
potential relativistic DFT allows in general for a more precise description of
the electronic structure.

This paper is organized as follow. First, we introduce the concepts of spin
and orbital currents in Sec. II, looking at electrically and thermally driven
transport coefficients. The Mott formula is used to link those two. The
quantum description of the systems is discussed and the linear response
formula is presented. Second, using our previously defined transport
coefficients as well as the linear response framework, we compute in Sec. III
the SHC and orbital Hall conductivity (OHC) as a function of the
electrochemical potential for the 40 monoatomic crystals considered, with a
specific focus on the $3d$, $4d$, and $5d$ series. The influence of SOC
strength is investigated. Finally, we discuss the SNC and ONC throughout the
elements considered. As we show later, the ONC is found to be the dominating
quantity, being about 10 times bigger than the SNC.

##  II Theory

###  II.1 Spin and orbital transport

The spin current density $\bm{J}^{S^{k}}$ is a $3$-dimensional vector
describing the flow of spin angular momentum polarized along the $k$-direction
($k=x,y,z$) and can be related to the the external electric field $\bm{E}$ as

$J^{S^{k}}_{i}=\sigma^{S^{k}}_{ij}E_{j},\hskip 14.22636pt(i,j=x,y,z)$ (1)

where we have assumed the Einstein summation notation. The quantity
$\sigma^{S_{k}}_{ij}$ is the $ij^{\rm th}$ component of the
$2^{\text{nd}}$-rank spin conductivity tensor $\bm{\sigma}^{S^{k}}$.
Analogously, we can define the orbital current density $\bm{J}^{L^{k}}$ as
well as the orbital conductivity tensor $\bm{\sigma}^{L^{k}}$ with

$J^{L^{k}}_{i}=\sigma^{L^{k}}_{ij}E_{j}.$ (2)

In the presence of a thermal gradient, thermally induced spin and orbital flow
can also occur. Extending Eqs. (1) and (2), we can write

$\displaystyle J^{S^{k}}_{i}$
$\displaystyle=\sigma^{S^{k}}_{ij}E_{j}-\Lambda^{S^{k}}_{ij}\frac{\partial}{{\partial}r_{j}}T,$
(3) $\displaystyle J^{L^{k}}_{i}$
$\displaystyle=\sigma^{L^{k}}_{ij}E_{j}-\Lambda^{L^{k}}_{ij}\frac{\partial}{\partial
r_{j}}T,$ (4)

where $\Lambda^{S^{k}}_{ij}$ ($\Lambda^{L^{k}}_{ij}$) is the $ij^{\rm th}$
component of the $2^{\text{nd}}$-rank spin (orbital) magneto-thermal
conductivity tensor $\bm{\Lambda}^{S^{k}}$ ($\bm{\Lambda}^{L^{k}}$), and $T$
the temperature. Note that the electrical term of Eqs. (3) and (4) can be
expressed in a similar way as the thermal term using the Cartesian spatial
derivative of $E_{j}=-\frac{\partial}{\partial r_{j}}V$, where $V$ is the
electric potential.

Thermally induced transport can be related to electrically induced transport
through

$\bm{\Lambda}^{S^{k}(L^{k})}=\frac{\pi^{2}k_{B}^{2}T}{-3e}\Big{(}\frac{d}{dE}\bm{\sigma}^{S^{k}(L^{k})}\Big{)}_{\\!E=E_{F}},$
(5)

where $k_{B}$ is the Boltzmann constant and $e$ the elementary charge.
Equation (5) is the famous Mott formula [57] applied to metallic systems. The
derivative on the right-hand side is taken with respect to the electronic
potential $E$ and evaluated at the Fermi energy $E_{F}$. Explicit calculations
have proven that the Mott formula gives exact results for small temperature
excursions [58]. Note that our focus here is on “pure” thermal transverse spin
and orbital transport. An additional effect can occur when a longitudinal
thermal electric current, described by the Seebeck coefficient $S$, is
converted to a transverse spin (or orbital) current through the SHE (or OHE),
which will give a contribution $\sim\sigma_{ij}^{S^{k}(L^{k})}S\partial
T/\partial r_{j}$ with $i$, $j$, and $k$ all different. This effect is not
considered here, but it can be evaluated from the calculated transverse spin
(orbital) conductivities and literature values for the Seebeck coefficient.

The expressions (3) and (4) for the spin and orbital current density contain
both longitudinal and transverse transport quantities. Our focus is here on
the transverse conductivities. These are, for nonmagnetic metals, given by the
spin Hall and orbital Hall conductivities, $\sigma_{ij}^{S_{k}}$ and
$\sigma_{ij}^{L_{k}}$, which are nonzero and identical for all indices such
that the Levi-Civita tensor $\epsilon_{ijk}\neq 0$.

We can thus express the tensorial SHE and OHE by a single spin Hall
conductivity (SHC) $\sigma_{\text{\tiny SH}}$ and an orbital Hall conductivity
(OHC) $\sigma_{\text{\tiny OH}}$, respectively, which can be defined as
$\sigma_{\text{\tiny SH}}\equiv\sigma^{S^{z}}_{xy}$, and $\sigma_{\text{\tiny
OH}}\equiv\sigma^{L^{z}}_{xy}$. In a similar way, the SNE and ONE can be
related to their respective spin Nernst thermal conductivity (SNC)
$\sigma_{\text{\tiny SN}}$ and orbital Nernst conductivity (ONC)
$\sigma_{\text{\tiny ON}}$, both obtained using Eq. (5), i.e.,

$\displaystyle\sigma_{\text{\tiny SN}}$
$\displaystyle\equiv\frac{\pi^{2}k_{B}^{2}T}{-3e}\Big{(}\frac{d}{dE}\sigma_{\text{\tiny
SH}}\Big{)}_{E=E_{F}},$ (6) $\displaystyle\sigma_{\text{\tiny ON}}$
$\displaystyle\equiv\frac{\pi^{2}k_{B}^{2}T}{-3e}\Big{(}\frac{d}{dE}\sigma_{\text{\tiny
OH}}\Big{)}_{E=E_{F}}.$ (7)

The ab initio calculation of the SHC and OHE is detailed in the following
subsection.

###  II.2 First-principles linear response

To evaluate the response quantities we consider the influence of an external
electric field $\bm{E}$ which leads to an additional term $\hat{V}$ to the
unperturbed Hamiltonian $\hat{H}_{0}$ that can be written as
$\hat{V}=-e\hat{\bm{r}}\cdot\bm{E}$, where $e$ is the electron charge and
$\hat{\bm{r}}$ the position operator. For the unperturbed Hamiltonian we adopt
the relativistic Kohn-Sham Hamiltonian $\hat{H}_{0}$. Using the DFT package
WIEN2k [59], we solve the eigenvalue equation
$\hat{H}_{0}|n\bm{k}\rangle=\epsilon_{n\bm{k}}|n\bm{k}\rangle$, where
$|n\bm{k}\rangle$ is the single-electron state at band-index $n$ and
reciprocal wavevector $\bm{k}$ with eigenenergy $\epsilon_{n\bm{k}}$. To
compute the spin and orbital conductivity tensors, we first define their
respective quantum mechanical operators $\hat{J}^{\hat{S}^{k}}_{i}$ and
$\hat{J}^{\hat{L}^{k}}_{i}$ as

$\displaystyle\hat{J}^{\hat{S}^{k}}_{i}$
$\displaystyle=\frac{\\{\hat{S}^{k},\hat{p}_{i}\\}}{2Vm_{e}},$ (8)
$\displaystyle\hat{J}^{\hat{L}^{k}}_{i}$
$\displaystyle=\frac{\\{\hat{L}^{k},\hat{p}_{i}\\}}{2Vm_{e}},$ (9)

where $\\{\hat{A},\hat{B}\\}=\hat{A}\hat{B}+\hat{B}\hat{A}$ is the anti-
commutator, $\hat{S}^{k}$ ($\hat{L}^{k}$) the $k^{\text{th}}$ component of the
spin (orbital) angular momentum operator, $\hat{p}_{i}$ the $i^{\text{th}}$
component of the momentum operator, $V$ the volume of the unit cell, and
$m_{e}$ the electron mass.

Using the Kubo linear-response formalism [60, 61, 62], $\bm{\sigma}^{S^{k}}$
and $\bm{\sigma}^{L^{k}}$ can then be computed using

$\begin{split}\mathcal{A}&=-\frac{ie}{m_{e}}\int_{\Omega}\frac{d\bm{k}}{\Omega}\sum_{n\neq
m}\frac{f_{n\bm{k}}-f_{m\bm{k}}}{\hbar\omega_{nm\bm{k}}}\leavevmode\nobreak\
\frac{A_{mn\bm{k}}\leavevmode\nobreak\
p_{j,nm\bm{k}}}{-\omega_{nm\bm{k}}+i\tau_{\text{inter}}^{-1}}\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
-\frac{ie}{m_{e}}\int_{\Omega}\frac{d\bm{k}}{\Omega}\sum_{n}\frac{\partial
f_{n\bm{k}}}{\partial\epsilon}\leavevmode\nobreak\
\frac{A_{nn\bm{k}}\leavevmode\nobreak\
p_{j,nn\bm{k}}}{i\tau_{\text{intra}}^{-1}}\,,\end{split}$ (10)

where $f_{n\bm{k}}$ is the occupation of Kohn-Sham state $|n\bm{k}\rangle$
with energy $\epsilon_{n\bm{k}}$, $\Omega$ is the Brillouin zone volume,
$p_{j,nm\bm{k}}$ the $j^{\text{th}}$ component of the momentum-operator
($\hat{\bm{p}}$) matrix element, and
$\hbar\omega_{nm\bm{k}}=\epsilon_{n\bm{k}}-\epsilon_{m\bm{k}}$.
$A^{i}_{mn\bm{k}}$ stands for a generic matrix element of a generic operator
$\hat{A}$; if it is $A_{mn\bm{k}}=\hat{J}^{\hat{S}^{k}}_{i,mn\bm{k}}$ then
$\mathcal{A}=\sigma^{S^{k}}_{ij}$ while if it is
$A_{mn\bm{k}}=\hat{J}^{\hat{L}^{k}}_{i,mn\bm{k}}$ then
$\mathcal{A}=\sigma^{L^{k}}_{ij}$. The parameters $\tau_{\text{inter}}$ and
$\tau_{\text{intra}}$, which are lifetime parameters, are set to $0.4$ eV, a
value that is reasonable for metals [61]. They are effective decay constants
modelling interactions with external baths (e.g., phonons).

It is important to understand that our calculations give the intrinsic parts
of the SHE, OHE, SNE, and ONE in the sense that they focus on the Berry
curvature related spin and orbital transport coefficients. They are not
intrinsic in the sense that the system is considered impurity-free as we use
finite lifetime parameters $\tau_{\text{inter}}$ and $\tau_{\text{intra}}$. A
further appropriate note at this point is that for the transverse spin and
orbital conductivities the intraband part does not contribute. Also, we note
that our formulation [Eq. (10)] differs from other formulations in which the
lifetime broadening appears also in the denominator, i.e.,
$(f_{n\bm{k}}-f_{m\bm{k}})/[\omega_{nm\bm{k}}-i\tau_{\rm inter}^{-1}]$, giving
thus a denominator of the form
$[\epsilon_{n\bm{k}}-\epsilon_{m\bm{k}}+i\delta]^{2}$, with
$\delta=\hbar\tau^{-1}_{\rm inter}$. The inclusion of the lifetime broadening
$\tau_{\text{inter}}$ and $\tau_{\text{intra}}$ in our formulation is done
from the beginning of the derivation of the Kubo linear-response formalism as
an effective non-Hermitian decay of the first-order correction to the density
matrix. Therefore, we argue that the formulation of Eq. (10) is exact in the
linear-response framework.

Figure 1: Calculated spin Hall conductivity $\sigma_{\text{SH}}$ (top) and
orbital Hall conductivity $\sigma_{\text{OH}}$ (bottom) for a range of
nonmagnetic monoatomic metallic materials. For the ferromagnetic 3$d$ elements
Fe, Co, and Ni, the values for their magnetic phase are shown as red
triangles.

The conductivities $\sigma_{\text{\tiny SH}}$ and $\sigma_{\text{\tiny OH}}$
are computed as a function of the electrochemical potential $E$ where $E=0$
corresponds to the Fermi level. This is facilitated by the occupation numbers
$f_{n\bm{k}}$ in Eq. (10), given by the Fermi-Dirac distribution

$f_{n\bm{k}}(E)=\frac{1}{\exp\Big{(}\frac{\epsilon_{n\bm{k}}-(E_{F}+E)}{k_{B}T}\Big{)}+1},$
(11)

where the electronic potential $E$ is then treated as a variable, allowing us
to compute $\sigma_{\text{\tiny SH}}(E)$ and $\sigma_{\text{\tiny OH}}(E)$.
The SNE and ONE transport coefficients $\sigma_{\text{\tiny SN}}$ and
$\sigma_{\text{\tiny ON}}$ can then be computed using the Mott formula [Eq.
(5)].

###  II.3 Computational Details

The required equilibrium electronic structures of the monoatomic systems are
computed using the full-potential, all-electron WIEN2k method [59]. In this
method, the product between the smallest muffin-tin radius $R_{MT}$ and the
largest reciprocal vector $K_{\text{max}}$, $RK_{\text{max}}=R_{MT}\times
K_{\text{max}}$ is set to appropriate values for each system (see Table 1 in
the Appendix A for more details). The $k$-meshes contain at least $2\,10^{4}$
$k$-points. Second, a single-shot calculation using a refined $k$-mesh of at
least $2\,10^{5}$ $k$-points is performed and $\sigma_{\text{\tiny SH}}(E)$
and $\sigma_{\text{\tiny OH}}(E)$ are computed according to Eq. (10). In a
post processing step, Eq. (5) is used to compute $\sigma_{\text{\tiny SN}}(E)$
and $\sigma_{\text{\tiny ON}}(E)$. Structural information on the systems
computed in this work can be found in Table 1.

##  III Results

###  III.1 SHE and OHE

We first focus our discussion on the SHE and OHE. Figure 1 shows the
$\sigma_{\text{\tiny SH}}$ (top panel) and $\sigma_{\text{\tiny ON}}$ (bottom
panel) values, calculated at the Fermi level for all the materials considered.
The blue squares show $\sigma_{\text{\tiny SH}}$ and $\sigma_{\text{\tiny
OH}}$ for the nonmagnetic phase while the red triangle shows the results for
ferromagnetic Fe, Co, and Ni. For both $\sigma_{\text{\tiny SH}}$ and
$\sigma_{\text{\tiny OH}}$, we observe a clear trend, where each transition
metal row of the Periodic Table (3$d$, 4$d$, and 5$d$) shows a similar
pattern.

Figure 2: (a) Spin Hall conductivity $\sigma_{\text{SH}}$ and (b) orbital Hall
conductivity $\sigma_{\text{OH}}$ computed as a function of the
electrochemical potential $E$ for the 3$d$ series. The equilibrium Fermi level
is at $E=0$. Note that the ordinate scales in (a) are different for the two
rows, reflecting that $\sigma_{\text{SH}}$ increases with the atomic number
$Z$. For Fe, Co, and Ni, results are shown for both the ferromagnetic (red)
and nonmagnetic (blue) phases.

For the SHE, a local maximum is reached for Ni, Pd, and Pt, respectively, for
the 3$d$, 4$d$, and 5$d$ series. Those three elements are located in the same
column, the group 10 elements of the Periodic Table. These metals have similar
electron configurations with 10 electrons in their $s$ and $d$ shells. Their
$d$ shells are filled to around half of the middle of the second half of the
$d$ series (about $d^{8}$). Going from the 3$d$ to the $4d$ and to the $5d$
series, one notices that the $\sigma_{\text{\tiny SH}}$ increases, which was
to be expected since the SHE is known to be a SOC dependent property [5, 6]
(and the SOC strength increases with an increasing atomic number $Z$). The
largest SHC value of 1800 $(\hbar/e)\,(\Omega\,\textrm{cm})^{-1}$ is found for
Pt, consistent with previous theoretical work [32, 63]. The SHC of W is
negative, but small, as the calculation here is done for the bcc ($\alpha$)
phase and not for the metastable $\beta$ phase, which has a higher SHC [64].

It can furthermore be seen that the SHC is very small in the light $sp$ metals
(Li to Ca). Elements with a filled $d$ shell, such as Zn and Cd, also have a
very small SHC. The SHC of $p$-band metals as In, Sn, and Pb is quite small,
too, and only Tl with a large SOC has a somewhat larger SHC (-210
$(\hbar/e)\,(\Omega\,\textrm{cm})^{-1}$). Together, this underlines that, to
obtain a large SHC, one needs strong SOC as well as $d$-band electrons at the
Fermi energy.

For the OHE, a local maximum is reached around the middle of the three $d$
series, specifically, for Mn, Ru, and Os. Mn is located in the middle of the
3$d$ series (group 7 of the Periodic Table, with five $d$ electrons), but Ru
and Os are both located in group 8. There is thus a difference for the OHE
between the $3d$ series and the $4d$ and $5d$ series, which could be due to
the increased spin-orbit coupling when the atomic number $Z$ increases. This
gives a larger splitting of $d_{3/2}$ and $d_{7/2}$ subshells and therefore a
somewhat different $d$ band filling. Also, while for the SHE maxima the metals
Ni, Pd, and Pt have the same crystal structure (face-centered cubic), Mn, Ru,
and Os do not have the same crystal structure. Mn is body-centered cubic while
Ru and Os are hexagonal close-packed. Importantly, as opposed to the SHE, the
OHE does not seem to scale as we proceed from one $d$ series to the other,
suggesting that $\sigma_{\text{\tiny OH}}$ barely depends on the SOC strength.
We will address this later on.

It is instructive to compare our calculations with previous work. Jo et al.
[38] computed the OHC of several $3d$ elements. They obtained a maximum OHC of
about $9\,10^{3}$ $(\hbar/e)(\Omega\,$cm)-1 for Mn; our results are in good
agreement with their calculations. Tanaka et al. [10] computed the OHC of 4$d$
and 5$d$ elements. They obtained however a different trend across the series,
with a maximum OHC for the 4$d$ series at Mo and for the 5$d$ series at Ir.
Also their values for the OHC are in general smaller (about $4\,10^{3}$
$(\hbar/e)(\Omega$ cm)-1 and less). These differences pinpoint the need for ab
initio calculations to achieve precise SHC and OHC values.

Most of the investigated elements are nonmagnetic, with the exception of Fe,
Co, and Ni that are well-known ferromagnets. We see that ferromagnetism (red
data points in Fig. 1) has a tendency of reducing the magnitude of both the
SHE and OHE, the exception being the OHE for Ni that is slightly higher in the
ferromagnetic phase than in the nonmagnetic phase. As discussed below, this
effective reduction is due to the spin spitting of the $d$ bands around the
Fermi energy, caused by the Stoner instability.

Figure 3: Electrochemical potential dependence of (a) the spin Hall
conductivity $\sigma_{\text{\tiny SH}}$ for the 4$d$ series, (b) the orbital
Hall conductivity $\sigma_{\text{\tiny OH}}$ for the 4$d$ series, (c) the spin
Hall conductivity $\sigma_{\text{\tiny SH}}$ for the 5$d$ series, and (d) the
orbital Hall conductivity $\sigma_{\text{\tiny OH}}$ for the 5$d$ series. For
the 5$d$ series, Hg, which is liquid at room temperature, has not been
considered.

We now compute $\sigma_{\text{\tiny SH}}(E)$ and $\sigma_{\text{\tiny OH}}(E)$
where $E$ denotes the electrochemical potential (ECP), and $E=0$ is the
equilibrium Fermi energy (see Fig. 2). We consider $E$ within the range
$E\in[-10;10]$ eV. For both SHE and OHE, we observe that the shape of
$\sigma_{\text{\tiny SH}}(E)$ and $\sigma_{\text{\tiny OH}}(E)$ is essentially
the same across the series, but the central part of the spectrum shifts toward
lower energies as the atomic number $Z$ is increased. This clearly shows that
the SHE and OHE depend on the amount of $d$-shell filling. For the
$\sigma_{\text{\tiny SH}}(E)$, the amplitude of the maximum increases as the
atomic number $Z$ increases but such behavior is not observed for
$\sigma_{\text{\tiny OH}}(E)$ for which the maximum seems to be higher when it
is close to $E=0$ and smaller when it occurs at higher or lower energies.

For the spontaneous ferromagnets Fe, Co, and Ni the $\sigma_{\text{\tiny SH}}$
and $\sigma_{\text{\tiny OH}}$ have been computed both for the nonmagnetic and
magnetic phases, shown by blue and red curves, respectively. The emergence of
ferromagnetism in those metals is well understood: a high density of states
around the Fermi energy in the nonmagnetic phase gives rise to a Stoner
instability, which in turn leads to an exchange splitting of the $d$ bands in
spin-up and spin-down states. This influence of this splitting of the $d$
bands can be observed directly for both the $\sigma_{\text{\tiny SH}}(E)$
[Fig. 2(a)] and the $\sigma_{\text{\tiny OH}}(E)$ [Fig. 2(b)], where the
nonmagnetic curves become split in the spectral range around the Fermi level.
This suggests that inherent magnetism arising from a high-density instability
at the Fermi energy tends to play against large values of the SHE and OHE.

In a similar fashion, we compute $\sigma_{\text{\tiny SH}}(E)$ and
$\sigma_{\text{\tiny OH}}(E)$ for the 4$d$ and 5$d$ series. As shown in Fig.
3, the same kind of observations as for the 3$d$ series can straightforwardly
be made: the shape of both the SHE and OHE is similar for all elements while
the position of the maximum tends to be shifted toward lower energies as $Z$
increases. The SHE increases in magnitude as $Z$ increases and the OHE tends
to be stronger when the maximum of the spectrum is near the Fermi energy.
Particular attention can be paid to Pt, often considered as the best metallic
candidate for transverse spin-current generation [2, 65, 66]. The fact that
the SHE is largest in Pt stems from the location of the maximum of
$\sigma_{\text{\tiny SH}}(E)$, which occurs at the Fermi level. Therefore,
electronic-structure engineering, such as doping, is not required to achieve a
maximal $\sigma_{\text{\tiny SH}}(E)$ for Pt. It deserves nonetheless to be
noted that Ir and Au have quite similar $\sigma_{\text{\tiny SH}}$ maxima,
which suggests that doping these metals could bring their ECP to the maximal
SHC position and hence they should yield similar performances as Pt.

###  III.2 SOC scaling of SHE and OHE

Our results, consistent with the literature [7, 32, 6], suggest that the
strength of SOC plays an important role, especially in the emergence of the
SHE. To obtain a better understanding of the interplay of SOC and SHE/OHE, we
introduce in our calculation a parameter $\alpha$ ($\alpha\in\Re^{+}$), the
SOC scaling parameter, which is artificially inserted in the Kohn-Sham
Hamiltonian $\hat{H}$ such that

$\hat{H}=\hat{H}_{\text{scRel}}+\alpha\,\hat{H}_{\text{SOC}},$ (12)

where $\hat{H}_{\text{scRel}}$ is the scalar-relativistic part of the
Hamiltonian and $\hat{H}_{\text{SOC}}$ is the SOC part of $\hat{H}$, i.e.
$\hat{H}_{\text{SOC}}\propto\hat{\bm{L}}\cdot\hat{\bm{S}}$. The SOC scaling
parameter $\alpha$ controls how strong the SOC is; $\alpha=1$ corresponds to
the real SOC in the material. Our calculations are done fully self-
consistently, that is, the electronic density $n(\bm{r})$ is computed self-
consistently for each $\alpha$ value considered.

Figure 4: (a) Spin Hall conductivity $\sigma_{\text{SH}}$ and (b) orbital Hall
conductivity $\sigma_{\text{OH}}$ computed for different SOC scaling $\alpha$
for Cu, ferromagnetic Ni, Ag, and Pt. The spin Hall conductivity vanishing
when we suppress the SOC ($\alpha=0$), while the orbital Hall conductivity
remains finite. Increasing $\alpha$ increases $\sigma_{\text{SH}}$ and
somewhat decreases $\sigma_{\text{OH}}$ for Ag and Pt. While the scaling
appears linear for lighter elements at low $\alpha$, non-linear behavior can
clearly be observed for Pt. Figure 5: Scaling of the maximum of
$\sigma_{\text{\tiny SH}}$ (red triangles) and the maximum of
$\sigma_{\text{\tiny OH}}$ (blue triangles) as a function of the SOC scaling
factor $\alpha$ at the peak of the effect ($\max_{E}\sigma_{\text{\tiny
SH}}(E)$ and $\max_{E}\sigma_{\text{\tiny OH}}(E)$). Figure 6: Scaling of (a)
the maximum of $\sigma_{\text{\tiny SH}}$ and (b) the maximum of
$\sigma_{\text{\tiny OH}}$ as a function of $|\Delta E_{\alpha}|$ for Cu
(yellow squares), Ni (red circles), Ag (green triangles) and Pt (blue
diamonds). The points corresponding to $\alpha=1$ (“true SOC strength”) are
circled in black. For the SHE, the scaling of the effect as a function of
$|\Delta E_{\alpha}|$ follows a similar trend, qualitatively but also
quantitatively, for all the elements considered, suggesting that the magnitude
of $\sigma_{\text{\tiny SH}}$ is mainly determined by the SOC strength rather
than by the band structure. For the OHE the considered elements follow
qualitatively a similar trend.

We consider four materials, Ni, Cu, Ag, and Pt, such that each row of the $d$
series is considered, taking care of including a magnetic (Ni) and a
nonmagnetic (Cu) material for the $3d$ series. The SHE and OHE spectra,
$\sigma_{\text{\tiny SH}}(E)$ and $\sigma_{\text{\tiny OH}}(E)$, for varying
$\alpha$ are shown in Fig. 4. For all considered materials, no SHE can be
observed in the case $\alpha=0$, which is of course to be expected: the lack
of coupling between the real-space and the spin-space in
$\hat{H}_{\text{scRel}}$ forbids any coupling between the electron momentum
(which couples to the external electric field) and its spin angular momentum.
For the OHE, in contrast, a finite effect is observed in all cases even when
$\alpha=0$.

The $\alpha$ dependencies of the SHE and the OHE are different: while the SHE
increases as $\alpha$ increases, the OHE has a weak tendency to decrease in
amplitude. The decreasing trend for the OHE is not negligible for Ag and Pt.
This observation is different from previous works where it was suggested that
the OHE virtually did not depend on SOC strength, or even increased with the
SOC strength [10]. We suspect the reason for this difference to be two-fold.
First, it is common to include the SOC in a one-shot non-selfconsistent
manner, which may shadow any influence of the SOC onto the orbital character
of electronic states in the proximity of the Fermi energy. Second, if we were
to focus at the variation around $E=0$, we would indeed observe a negligible
influence of the SOC strength for all material considered here. The largest
variations, both for $\sigma_{\text{\tiny SH}}(E)$ and $\sigma_{\text{\tiny
OH}}(E)$, are observed close to their maximum.

We compute $\max_{E}\sigma_{\text{\tiny SH}}(E)$ and
$\max_{E}\sigma_{\text{\tiny OH}}(E)$ as a function of $\alpha$. As shown in
Fig. 5, one can see that for the considered materials, the OHE follows an
opposite trend to the SHE. The trend appears to be linear at low $\alpha$
while larger $\alpha$ leads to saturation and nonlinear scaling. The critical
$\alpha$ for which the linear behavior starts to become nonlinear is different
for the considered materials. Whereas for Cu and Ni the change of
$\sigma_{\text{\tiny SH}}$ and $\sigma_{\text{\tiny OH}}$ mostly follows a
linear trend, non-linear behavior is seen for $\alpha>1$ for Ag and
$\alpha>0.5$ for Pt. Interestingly, the saturation value for
$\sigma_{\text{\tiny SH}}$ seems to lie around $\sigma_{\text{\tiny SH}}\sim
2000\,(\frac{\hbar}{e})(\Omega\,\textrm{cm})^{-1}$, independent of the
material considered.

Since the intrinsic SOC tends to scale as $Z^{2}$ to $Z^{4}$ [67] we consider
the quantity $\Delta E_{\alpha}$, defined as

$\Delta
E_{\alpha}=\langle\hat{H}_{\text{scRel}}+\alpha\hat{H}_{\text{SOC}}\rangle_{n(\bm{r},\alpha)}-\langle\hat{H}_{\text{scRel}}\rangle_{n(\bm{r},0)}$
(13)

where $\langle.\rangle_{n(\bm{r},\alpha)}$ refers to the expectation value
with respect to the self-consistent density $n(\bm{r},\alpha)$. Note that this
definition of $\Delta E_{\alpha}$ is not equivalent to
$\langle\alpha\hat{H}_{\text{SOC}}\rangle_{n(\bm{r},\alpha)}$ because we take
the self-consistent effect of the SOC on the electronic structure into
account.

Figure 7: Spin Nernst conductivity $\sigma_{\text{SN}}$ (top) and orbital
Nernst conductivity $\sigma_{\text{ON}}$ (bottom) computed for a temperature
of $300$ K for a range of materials. For the ferromagnetic 3$d$ elements Fe,
Co, and Ni, results for the ferromagnetic phase are shown as red triangles.

The scaling of $\max_{E}\sigma_{\text{\tiny SH}}(E)$ and
$\max_{E}\sigma_{\text{\tiny OH}}(E)$ with respect to $\Delta E_{\alpha}$ is
shown in Fig. 6. For Cu and Ni, Ag, and Pt, $\Delta E_{\alpha}$ is of the
order of $10^{-2}$ eV, $10^{-1}$ eV, and $10^{0}$ eV, respectively, i.e.,
increasing by an order of magnitude when going from one of the $d$-series to
the next one, consistent with the increase of the SOC with $Z$.

For the SHE [Fig. 6(a)], the $\sigma_{\text{\tiny SH}}$ tends to increase
rapidly up to $\leavevmode\nobreak\ 1$ eV and then starts to saturate. A
remarkable point here is that the magnitude of $\sigma_{\text{\tiny SH}}$ at a
given $\Delta E_{\alpha}$ is similar for all materials considered, with a
slightly higher value for Cu. For Ni, something peculiar happens: for $\Delta
E_{\alpha}<1$ eV, the behavior of $\sigma_{\text{\tiny SH}}$ is similar to Pt
and Ag, while for $\Delta E_{\alpha}>2$ eV it follows closely the one of Cu.
This change of behavior for Ni can be correlated to the decrease of the
magnetic moment as a function of $\Delta E_{\alpha}$. We have observed that
for $\Delta E_{\alpha}\gtrsim 2$ eV, magnetism in Ni disappears.

For $\max_{E}\sigma_{\text{\tiny OH}}(E)$ [Fig. 6(b)], the behavior is
opposite to the one of $\max_{E}\sigma_{\text{\tiny SH}}(E)$. For $\Delta
E_{\alpha}<0.1$ eV, the $\sigma_{\text{\tiny OH}}$ is virtually independent to
the SOC strength. Beyond this threshold, $\max_{E}\sigma_{\text{\tiny OH}}(E)$
decreases monotonically. Here, Ni’s $\max_{E}\sigma_{\text{\tiny OH}}(E)$ does
not join the curve of Cu for large $\Delta E_{\alpha}$ suggesting that
presence or absence of magnetism has little influence on the
$\sigma_{\text{\tiny OH}}$.

###  III.3 SNE and ONE

From the calculated $\sigma_{\text{\tiny SH}}(E)$ and $\sigma_{\text{\tiny
OH}}(E)$ we can compute the thermal $\sigma_{\text{\tiny SN}}$ and
$\sigma_{\text{\tiny ON}}$ using the Mott formula, Eq. (5). Note that since
the Mott formula is valid only around the Fermi level, one cannot extract the
ECP profile of the SNE and ONE. We choose $T=300$ K and evaluate the SNE and
ONE for this temperature. In Fig. 7, we display the computed
$\sigma_{\text{\tiny SN}}$ and $\sigma_{\text{\tiny ON}}$ for all the elements
considered in this work. Similarly to the $\sigma_{\text{\tiny SH}}$ and
$\sigma_{\text{\tiny OH}}$, we find that the orbital Nernst values are
systematically much larger than their spin counterparts. The effect of
ferromagnetism on the $\sigma_{\text{\tiny SN}}$ and $\sigma_{\text{\tiny
ON}}$ is also quite similar to their Hall counterparts, with a substantial
increase of the $\sigma_{\text{\tiny SN}}$ and $\sigma_{\text{\tiny ON}}$ when
magnetism is artificially turned off for Fe, Co, and Ni. A further similarity
with the Hall conductivities is that the Nernst conductivities are very small
for $sp$ metals.

Differences with respect to the Hall conductivities are also noticeable.
First, as opposed to the $\sigma_{\text{\tiny SH}}$ and $\sigma_{\text{\tiny
OH}}$, the maximum value of the $\sigma_{\text{\tiny SN}}$ is obtained for Pd,
a $4d$ element. This shows that although the maximum values of the
$\sigma_{\text{\tiny SH}}$ relate directly to the increase of the SOC, there
is no clear scaling of the $\sigma_{\text{\tiny SN}}$ as it relates to the
derivative of $\sigma_{\text{\tiny SH}}$. Conversely, the ONE does reveal a
trend across the $d$ series. The maximum of the (nonmagnetic) ONE occurs for
the group 10 elements, Ni, Pd, and Pt. Also, there is a change in sign that
occurs around the middle of each $d$ series, with the early $d$ elements
having negative ONEs. Like the OHE, the ONE does not depend on the SOC, but in
addition, the narrowness of the $d$ bands plays a role, and Ni has the most
correlated 3$d$ bands of the group 10 elements. Although the orbital Nernst
effect has not yet been observed, our results suggest that giant ONEs could be
present in correlated materials having narrow bands with a strong variation of
the $d$ or $f$ states density close to the Fermi energy.

Experimental and theoretical investigations of the spin Nernst effect were
reported for several materials [51, 50, 68, 54, 55, 56], see Ref. [69] for a
recent review. Our results are in overall agreement with those of Géranton et
al. [68], who gave computed values of the intrinsic Nernst coefficient for Rh,
Pd, Ir and Pt, but our values are roughly a factor of two smaller. Our value
for Pt is however in good agreement with the calculated value of Meyer et al.
[54]. Differences between ab initio calculated values could be due to the
choice of lifetime broadening. A smaller lifetime broadening will lead to
sharper features in the SHC and OHC spectra and thus modify the SNC and ONC.

##  IV Conclusions

In this work, we investigated the intrinsic $\sigma_{\text{\tiny SH}}$ and
$\sigma_{\text{\tiny OH}}$ as a function of the electrochemical potential $E$
for 40 monoatomic elements. We showed that for the $d$-elements, the
qualitative shapes of the $\sigma_{\text{\tiny SH}}(E)$ and
$\sigma_{\text{\tiny OH}}(E)$ spectra are similar for the elements in a
specific $d$ series, with the relative position of $E=0$ for each element
strongly dependent on the filling of the $d$-shell. For the
$\sigma_{\text{\tiny OH}}(E)$ spectra, maximum values are obtained when the
$d$-band is roughly half filled. The $\sigma_{\text{\tiny SH}}$ becomes
maximal when the Fermi level falls in the middle of the second half of the $d$
series, which happens for (nonmagnetic) Ni, Pd, and Pt. We also found that
magnetism (in Fe, Co, and Ni) tends to reduce $\sigma_{\text{\tiny SH}}(E)$
and $\sigma_{\text{\tiny OH}}(E)$, by splitting the $d$-states away from the
Fermi level.

We furthermore considered the influence of SOC on $\sigma_{\text{\tiny
SH}}(E)$ and $\sigma_{\text{\tiny OH}}(E)$. The $\sigma_{\text{\tiny OH}}(E)$
is obtained even when SOC is turned-off, consistent with previous reports [10,
39]. We also showed that while it is true that $\sigma_{\text{\tiny OH}}(E)$
depends less on the SOC than the $\sigma_{\text{\tiny SH}}(E)$, nonetheless, a
non-negligible influence is observed. Analyzing the SOC influence using
$\Delta E_{\alpha}$ allowed us to compared elements from the $3d$, $4d$, and
$5d$ series on an equivalent footing. Artificially increasing the SOC
increases the $\sigma_{\text{\tiny SH}}$, but this scaling saturates as
$\Delta E_{\alpha}$ increases, suggesting some kind of limit to the intrinsic
Hall effect.

For the $\sigma_{\text{\tiny SN}}$ and $\sigma_{\text{\tiny ON}}$, we
presented a survey of their theoretical values for the here considered
materials. As we showed, the orbital Nernst effect is about one order of
magnitude larger than the spin Nernst effect. This is notably for the lifetime
broadening adopted here ($\hbar\tau^{-1}_{\rm inter}=0.4$ eV) and even larger
differences can be expected for smaller lifetime broadening.

Our work emphasizes that the orbital contributions, both for the Hall and
Nernst effect, should be quite important. Encouraging reports of observations
of orbital currents, orbital torque, and the OHE have appeared recently [70,
71, 72]. The orbital Nernst effect in metals has so far not been detected. A
magnonic equivalent of the orbital Nernst effect was proposed recently to
exist for magnetic insulators [73]. There are thus still questions about the
nature of orbital transport and its direct experimental observation that
remain. It is however interesting to see that the orbital part, which has
previously been discarded, is not only far from being negligible but actually
appears dominant. Since the orbital Hall and Nernst effects are present
without SOC, but not the SHE and SNE, the latter quantities arise from the
former ones through the spin-orbit interaction. Large orbital effects could
thus be harvested for lighter 3$d$ and 4$d$ metals and compounds [10, 38], in
place of the heavy metals Pt, Ta, and W that are favorable for large SHE.
Hence, we anticipate that our work will contribute and stimulate research in
the emergent field of orbitronics.

###### Acknowledgements.

We thank Marco Berritta for valuable discussions. This work has been supported
by the Swedish Research Council (VR) and the Swedish National Infrastructure
for Computing (SNIC) (Grant No. 018-05973). This work has been funded by the
European Union’s Horizon2020 Research and Innovation Programme under FET-OPEN
Grant agreement No. 863155 (s-Nebula). The calculations were performed at the
PDC Center for High Performance Computing and the Uppsala Multidisciplinary
Center for Advanced Computational Science (UPPMAX).

##  Appendix A Table of lattice parameters

Table 1: List of elements considered in this work with their lattice structure
and lattice parameters. The structures considered are either face-centered
cubic (fcc, $a=b=c$ and $\alpha=\beta=\gamma=\frac{\pi}{2}$), body-centered
cubic (bcc, $a=b=c$ and $\alpha=\beta=\gamma=\frac{\pi}{2}$) or hexagonal
close-packed (hcp, $a=b\neq c$ and
$\alpha=\beta=\frac{\pi}{2},\gamma=\frac{2\pi}{3}$). The column
$RK_{\text{max}}$ refers to the product between the smallest muffin-tin radius
$R_{MT}$ and the largest reciprocal vector $K_{\text{max}}$, and is an
important parameter for WIEN2k [59] calculations.
Element |  $a$ [Å] |  $c$ [Å] |  Structure | $RK_{\text{max}}$ |  |  | Element |  $a$ [Å] |  $c$ [Å] |  Structure | $RK_{\text{max}}$
---|---|---|---|---|---|---|---|---|---|---|---
Li | 3.51 |  | bcc | 4.5 |  |  | Mo | 3.15 |  | bcc | 7.5
Be | 2.29 | 3.58 | hcp | 5.0 |  |  | Tc | 2.73 | 4.39 | hcp | 8.0
Na | 4.29 |  | bcc | 6.5 |  |  | Ru | 2.71 | 4.28 | hcp | 8.0
Mg | 3.21 | 5.21 | hcp | 6.5 |  |  | Rh | 3.80 |  | fcc | 8.0
Al | 4.05 |  | fcc | 6.5 |  |  | Pd | 3.89 |  | fcc | 8.0
K | 5.33 |  | bcc | 6.5 |  |  | Ag | 4.09 |  | fcc | 8.0
Ca | 5.59 |  | fcc | 6.5 |  |  | Cd | 2.98 | 5.62 | hcp | 8.0
Sc | 3.31 | 5.27 | hcp | 7.5 |  |  | In | 3.25 | 4.95 | hcp | 8.0
Ti | 2.95 | 4.69 | hcp | 7.5 |  |  | Sn | 5.83 | 3.18 | hcp | 8.0
V | 3.03 |  | bcc | 7.5 |  |  | Lu | 3.50 | 5.55 | hcp | 8.0
Cr | 2.91 |  | bcc | 7.5 |  |  | Hf | 3.20 | 5.05 | hcp | 8.0
Mn | 3.51 |  | fcc | 8.0 |  |  | Ta | 3.30 |  | bcc | 8.0
Fe | 2.87 |  | bcc | 8.0 |  |  | W | 3.17 |  | bcc | 8.0
Co | 2.51 | 4.07 | hcp | 8.0 |  |  | Re | 2.76 | 4.46 | hcp | 8.0
Ni | 3.52 |  | fcc | 8.0 |  |  | Os | 2.73 | 4.32 | hcp | 8.5
Cu | 3.61 |  | fcc | 8.0 |  |  | Ir | 3.84 |  | fcc | 8.5
Zn | 2.66 | 4.95 | hcp | 8.0 |  |  | Pt | 3.92 |  | fcc | 8.5
Y | 3.65 | 5.73 | hcp | 7.5 |  |  | Au | 4.08 |  | fcc | 8.5
Zr | 3.23 | 5.15 | hcp | 7.5 |  |  | Tl | 3.46 | 5.52 | hcp | 8.5
Nb | 3.30 |  | bcc | 7.5 |  |  | Pb | 4.95 |  | fcc | 8.5

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