Role of phonon skew scattering in the spin Hall effect of platinum
G. V. Karnad C. Gorini K. Lee T. Schulz R. Lo Conte A. W. J. Wells D. -S. Han K. Shahbazi J. -S. Kim T. A. Moore H. J. M. Swagten U. Eckern R. Raimondi M. Kläui




Role of phonon skew scattering in the spin Hall effect of platinum]Role of phonon skew scattering in the spin Hall effect of platinum

Institut für Physik, Johannes Gutenberg-Universität, Staudinger Weg 7, 55128 Mainz, Germany

cosimo.gorini@physik.uni-regensburg.de
Institut für Theoretische Physik, Universität Regensburg, 93040 Regensburg, Germany


Institut für Physik, Johannes Gutenberg-Universität, Staudinger Weg 7, 55128 Mainz, Germany

Institut für Physik, Johannes Gutenberg-Universität, Staudinger Weg 7, 55128 Mainz, Germany


Institut für Physik, Johannes Gutenberg-Universität, Staudinger Weg 7, 55128 Mainz, Germany
Graduate School of Excellence “Materials Science in Mainz”(MAINZ), Staudinger Weg 9, 55128 Mainz, Germany

School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, United Kingdom

Department of Applied Physics, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands

School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, United Kingdom

[now at: ]DGIST Research Center for Emerging Materials, DIGST, Daegu, 42988, Republic of Korea
Department of Applied Physics, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands



School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, United Kingdom

Department of Applied Physics, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands

Institut für Physik, Universität Augsburg, 86135 Augsburg, Germany

Dipartimento di Matematica e Fisica, Roma Tre University, Via della Vasca Navale 84, 00146 Rome, Italy

klaeui@uni-mainz.de
Institut für Physik, Johannes Gutenberg-Universität, Staudinger Weg 7, 55128 Mainz, Germany
Graduate School of Excellence “Materials Science in Mainz”(MAINZ), Staudinger Weg 9, 55128 Mainz, Germany






             


We measure and analyze the effective spin Hall angle of platinum in the low residual resistivity  regime by second harmonic measurements of the spin-orbit torques for a multilayer of Pt|Co|AlO_x. An angular dependent study of the torques allows us to extract the  effective spin Hall angle responsible for the damping-like torque in the system. We observe a strikingly non-monotonic and reproducible  temperature dependence of the torques. This behavior is compatible with recent theoretical predictions which include both intrinsic and extrinsic (impurities and phonons) contributions to the spin Hall effect at finite temperature. 






[
    M. Kläui
    December 30, 2023
=====================

The demand for high density and efficient data storage devices has driven the research on magnetic memories. Proposals were mainly based on using spin transfer torques for domain wall motion and magnetization switching <cit.>. However, spin transfer torque-based devices are plagued by the requirement of high writing current densities which result in deterioration of the material layers and hence limit the device life. 
Recent demonstration <cit.> of highly efficient current-induced switching via spin orbit torques (SOTs) shows promising results that can overcome these limitations <cit.>. Such devices are made from magnetic multi-layers which consist of a few mono-layers thick ferromagnetic (FM) material sandwiched between a normal metal (NM) and an oxide layer.

The injection of an in-plane current through the normal metal has been shown to efficiently manipulate the magnetization <cit.>. The torques have been attributed to the spin Hall effect (SHE) <cit.> arising from the charge to spin current conversion in the NM and the inverse spin galvanic effect (ISGE) <cit.> arising at the interfaces with the FM.

Firstly, a charge current in the NM underlayer generates via the SHE a spin current flowing towards the FM layer.  
These spins enter the FM and exert by s-d exchange a damping-like torque on the FM magnetization
<cit.>.  
Secondly, due to inversion symmetry breaking at the NM|FM interface, the same charge current gives also rise to an (interfacial) 
non-equilibrium spin polarization via the ISGE <cit.>.  
Such spin polarization results in a field-like torque affecting the FM magnetization.  
It has actually been discussed how the SHE and the ISGE can individually in principle generate both damping-like and field-like torques, due to subtler interfacial spin-orbit effects <cit.>.  
Indeed, we will argue later that in our system the SHE is dominating and responsible for both torques.  
Irrespective of their origin, whether SHE or ISGE, each torque can be described in terms of the corresponding component of an overall,
current-induced spin-orbit field acting on the magnetization: a longitudinal component μ_0 H_DL∼σ̂×m̂,
exerting a damping-like torque ∼m̂× (σ̂×m̂), 
and a transversal one μ_0 H_FL∼σ̂, giving rise to a field-like torque ∼m̂×σ̂.  
Here m̂ is the magnetization direction, and σ̂ that of the spin of the (non-equilibrium) SHE and/or ISGE electrons.  
In our experimental configuration the charge current flows along x̂, while m̂ = ẑ, σ̂ = ŷ, 
see Fig. <ref>.


Efficient magnetization switching using SOTs has motivated studies of various heavy metals, so as to find materials 
with the highest spin Hall angle (SHA).  The SHA characterizes the efficiency of charge-to-spin conversion, 
and is defined as θ_sH = σ_sH/σ = -ρ_sH/ρ, with σ (ρ) and σ_sH (ρ_sH) 
respectively the charge and spin Hall conductivities (resistivities).
There are various mechanisms giving rise to the SHE: extrinsic (side-jump <cit.> 
and skew scattering <cit.>) and Berry curvature-induced intrinsic ones <cit.>.  
In metallic paramagnets it is difficult to identify materials where only one mechanism is at work.  
It is important to rather understand the dominant mechanism in a particular system in order to optimize it for possible technological
applications.  Indeed, phonon skew scattering has been recently suggested to play an important role 
at room temperature <cit.>, and a thorough experimental study of its influence on the SHE is still lacking.


In this Letter we provide such a study, by monitoring the temperature dependence of the SOTs ascribed to the SHE in low-residual 
resistivity Pt (5.3 μΩcm, 6.3 μΩcm).  While the observed non-monotonic behavior is unexpected and cannot be explained following the established phenomenological analysis <cit.>, it is compatible with the temperature scaling derived from the microscopic theory <cit.>. The importance of “ultra-clean" samples is further supported by the simpler monotonic behavior observed in high-residual resistivity Ta.  







Sample characterization. Firstly, we measure the evolution of the magnetic properties with temperature, since the ferromagnetic layers in these stacks are only a few mono-layers thick. The temperature dependence of the saturation magnetization in this thin film structure is measured using SQUID magnetometry. This is plotted in Fig. <ref> and in the relevant temperature range a constant value is found.

The temperature dependence of the effective anisotropy is studied by measuring the extraordinary Hall voltage in individual Hall crosses during the application of a rotating magnetic field <cit.>. This angle dependent variation of Hall voltage (seen in inset of Fig. <ref>) when interpreted in terms of the Stoner-Wolfarth model <cit.> allows us to determine the anisotropy field and hence the effective anisotropy. The effective anisotropy constant is plotted in Fig. <ref> and also shows no significant variation. 
These measurements establish that the multilayer has a Curie temperature much larger than 300K, 
with no significant changes in magnetic properties in the considered temperature range.

Spin-orbit torque measurements. Next, we perform spin orbit torque measurements using the second-harmonics (2ω) technique <cit.> on patterned Hall bars. The material stack that is used <cit.> is: Pt(4.0)|Co(0.8,1.3)|AlO_x(2.0) (all thicknesses in nm).  

The 2ω measurements <cit.> are performed <cit.> by injecting a low frequency (ω/2π = 13.7 Hz) sinusoidal ac  signal through the nano wire (x-axis) while the sample is in a saturated magnetization state (z-axis). The effective fields are measured as a function of temperature, and this is shown in Fig. <ref>. We see a clear non-monotonic dependence that is distinct from the temperature dependence of the magnetic properties shown in Fig. <ref>.  




The observed temperature dependence of the spin-orbit torques can have various origins, such as: change in magnetic properties, spin-mixing conductance, the SHE and ISGE. We individually evaluate all these parameters to pinpoint the origin of the variation of the spin-orbit torques with temperature. 
First, as already shown, the magnetic properties of our sample are effectively 
temperature independent [see Fig. <ref>]. 
Second, experimental results <cit.> show that the spin mixing conductance at the Pt|Co interface is largely temperature independent as well.  We thus attribute any non-monotonic change in the current-induced effective fields to a variation 
of the SHE and/or ISGE and the resulting torques exerted.

At this point we make a crucial observation, namely that damping- and field-like terms change in unison, 
indicating a common origin for the two torques. 
This agrees with our previous conclusions <cit.> of the SHE being the likely cause for both SOTs
in Pt|Co|AlO_x, as well as with results where the SHE was claimed to be the dominant effect 
for the damping-like torque in this system <cit.>.
As mentioned in the introduction, the generation of both torques via the SHE is due to interfacial 
spin-orbit effects <cit.>: 
roughly, part of a σ̂-polarized spin Hall current J_sH incoming on the Co layer is lost and
turned into a σ̂'-polarized spin current δ J_sH by interfacial spin-orbit coupling.  
Once in the FM, the (remaining) J_sH and δ J_sH exert respectively damping-like and field-like torques on the magnetization.  
Since we cannot quantify the loss δ J_sH, we define an effective spin Hall angle
assuming that the remaining spin Hall flux is fully transmitted from Pt to Co
<cit.>,

    θ_sH = 2μ_0H_DLeM_st_FM/ħ j_e,

where M_s is the saturation magnetization and t_FM is the ferromagnetic layer thickness and j_e is the injected charge current.
We plot in Fig. <ref> the temperature evolution of the calculated effective spin Hall angle of Pt.







Theoretical analysis and discussion. We see in Fig. <ref> that the effective SHA varies non-monotonically with temperature. We also observe that there is a significant upturn in its magnitude around 240 - 250 K. This is a distinct temperature in this system, as the spin Hall source in our system, Pt is known <cit.> to have a Debye temperature of T_D = 240 K. The Debye temperature T_D gives 
an estimate of the temperature beyond which all phonon modes are excited and behave classically. 



The temperature dependence of the SHA provides us with the opportunity to gain insight into the relative magnitude of the different mechanisms responsible for the SHE in the system. We will start by considering both intrinsic and extrinsic (skew scattering) mechanisms: The importance of the former in 4d and 5d transition metals is known <cit.>, and so is the fact that in low-impurity samples the dominant extrinsic contribution comes from skew scattering <cit.>.


We now analyze the temperature dependence of the SHA <cit.> 
based on the scaling suggested by Tian et al. <cit.>.
The starting point is

    ρ_sH≈
    -( σ_sH^int + σ_sH^ss) ρ^2
    ≈
    -σ_sH^intρ^2 + ρ_sH^ss,

having assumed that skew scattering (σ_sH^ss) and intrinsic (σ_sH^int) contributions act as parallel channels,
σ_sH=σ_sH^int + σ_sH^ss, and that σ≫σ_sH.  The latter condition implies 
σ_sH≈ρ_sH/ρ^2, with ρ the longitudinal resistance of the Pt layer, and was used in the second passage above.
Then it is argued that the skew scattering resistivity scales as
ρ_sH^ss = θ_sH^ss, impρ^imp + θ_sH^ss, phonρ^phon,
with the spin Hall angle due to phonon skew scattering being temperature independent,
while ρ^imp (ρ^phon) represents the resistivity from impurities (phonons).
The split ρ = ρ^imp + ρ^phon follows Mathiessen's rule.
One thus has

    ρ_sH≈ -σ_sH^intρ^2 + θ_sH^ss, impρ^imp + θ_sH^ss, phon (ρ - ρ^imp).


In the lowest order approximation phonon skew scattering, 
i.e., the last term on the r.h.s. of Eq. (<ref>),
is neglected, and -ρ_sH can be plotted vs. ρ^2.
In this regime, the slope of the linear relation between ρ_sH and ρ^2 
describes the intrinsic contribution to the SHE, indicating the sign of the SHA of the material.
If this analysis yields inconsistent results, the last term in Eq. (<ref>) is also taken into account
<cit.>.  In our case, however, even the full expression (<ref>) is not enough, 
since the relation between ρ_sH and ρ is non-monotonic, see Fig. S5 and S6.  



In the lowest order approximation phonon skew scattering, i.e., the last term on the r.h.s. of Eq. (<ref>),
is neglected, and we plot -ρ_sH vs ρ^2 (T<T_D) as shown in Fig. <ref>. 

The slope of the linear relation between ρ_sH and ρ^2 describes the intrinsic contribution to the SHE, 
indicating the sign of the SHA of the material.  Such a sign is based <cit.> on the number of d electrons, 
and depends on the LS coupling.  Depending on whether the d-shell of the atom is more or less than half filled, 
the expectation value of the LS coupling changes sign, and so does the SHA. 
Pt(5d^96s^1) is expected <cit.> to have a large magnitude with a positive sign, yet
Fig.<ref> shows that -ρ_sH vs ρ^2 has a negative slope, indicating a negative intrinsic spin Hall conductivity.
This furthermore disagrees with experimental observations <cit.>, as well as with our SOT measurements <cit.>
yielding a positive SHA for Pt, where we also studied the ferromagnetic thickness dependence of current induced domain wall motion and SOTs at room temperature.  The lowest order approximation is thus unable to reproduce 
the SHA behavior.  Indeed, the full expression Eq. (<ref>) is not enough either – in particular, it cannot
account for a non-monotonic behavior of the SHA.


This suggests the need to revisit the established temperature scaling relation assumed for the SHA.
The microscopic theory of phonon skew scattering shows indeed that the corresponding SHA
is not temperature independent, but rather a function of the temperature, θ_sH^ss, phon→θ_sH^ss, phon(T). 
As a consequence Eq. (<ref>) is modified to 

    ρ_sH≈ -σ_sH^intρ^2 + θ_sH^ss, impρ^imp + θ_sH^ss, phon(T) (ρ-ρ^imp).


The phononic part of the SHA, θ_sH^ss,phon, increases linearly with T at high temperatures (T≳ T_D), while,  
clearly, it has to vanish at low temperatures, though its precise behavior at T<T_D is not yet known
[The fact that phonon skew scattering arises
from an-harmonic phonon-phonon processes suggests, by analogy with the lattice thermal conductivity, 
a non-monotonic behavior at T<T_D.  Notice also that the current phonon skew scattering theory considers only
leading order 3-phonon processes.].
Its asymptotics can however be used to determine the qualitative behavior of the full SHA in the whole temperature
range.  An order-of-magnitude estimate for a metallic system yields σ^T_D_sH/σ^0_sH≲ 10^-1,
with σ^T_D_sH (σ^0_sH) the spin Hall conductivity at the Debye (zero) temperature <cit.>.
Coupled with the moderate change of the charge conductivity over the same temperature range, σ^T_D/σ^0 ≲ 1, see Fig. S7,
one has θ_sH^ss, phon(T≈ T_D) < θ_sH^ss, imp. 
The resulting qualitative behavior shown in Fig. <ref> is compatible with the measurements in clean Pt samples.
In a clean system the low-T SHE is dominated by skew scattering,
i.e.  -σ_sH^intρ^2(T→0)<θ_sH^ss, impρ^imp.
Thus, the SHA should eventually decrease in magnitude from T=0 down to a minimum at a temperature T_1 given by
-σ_sH^intρ^2(T_1)≈θ_sH^ss, impρ^imp, and increase afterwards.
Furthermore, at a higher temperature T_2≳ T_D such that 
θ_sH^ss,phon(T_2)[ρ(T_2)-ρ^imp]≈θ_sH^ss,impρ^imp
phonon skew scattering would start dominating impurity skew scattering, yielding an asymptotic
linear behavior:

    θ_sH(T≳ T_2) ∼[
    σ_sH^int + σ_sH^ss, T_D
    ] T,

with σ_sH^ss, T_D the saturation value for the phonon skew scattering spin Hall conductivity at 
T≳ T_D <cit.>.  A few comments are in order: (i) The qualitative analysis yields 
T_1 ≲ T_D ≲ T_2, but we cannot give quantitative estimates of 
the temperatures T_1, T_2.  These require precise knowledge of various system parameters,
notably electron-phonon and phonon-phonon coupling strength.  (ii) The non-monotonic behavior
predicted by Eq. (<ref>) for clean samples does not require phonon skew scattering to be
stronger than impurity skew scattering, which happens only at T>T_2: It is enough to recognize
that the phonon SHA is itself a function of the temperature, θ_sH^ss, phon=θ_sH^ss, phon(T),
vanishing at T=0 and (linearly) increasing at high temperatures. (iii) While θ_sH^ss,phon(T→0)=0
has to hold, the approach to zero could be non-trivial and affected e.g. by the interplay between impurity and phonon scattering
<cit.>.

The picture changes in dirty systems with strong intrinsic SHE, such that 
-σ_sH^intρ^2(T→0)>θ_sH^ss, impρ^imp.  
In this case Eq. (<ref>) predicts a simpler monotonic behavior of the SHA,
once more compatible with our measurements in a dirty Ta sample (137 μΩcm), see Fig. S4.
Ta has a strong intrinsic SHE <cit.>, T_D comparable to that of Pt <cit.>,
and our sample is 20 times as resistive as the Pt sample used.
We observe a considerably larger SHA with no minimum, 
indeed roughly constant in the considered temperature range.	

The SHA transition through the intrinsic and extrinsic regimes can be expected due to the low residual resistivity of Pt (5.3 μΩcm, 6.3 μΩcm), as we find <cit.> here. This is essentially because the intrinsic and extrinsic contributions scale differently <cit.> with the resistivity of the metal. 
It was recently shown <cit.> that
tuning the conductivity of Pt leads to a transition of the SHE from a “moderately dirty” to a “super-clean” regime. This corresponds to  a transition from an intrinsic-dominated to an extrinsic-dominated regime. The intrinsic term scales with ∝ ρ^2,
and in the low-resistivity limit is dominated by the phonon contribution, which scales roughly with ∝ (ρ - ρ^imp). 
This could be the reason why other experimental investigations of spin Hall angle of Pt with higher resistivity have observed the dominance of intrinsic contributions, 
while they have observed dominance of extrinsic contribution in case of low resistivity Au <cit.>.


In summary, we used the second harmonic technique to measure the spin orbit torque in Pt|Co|AlO_x and Ta|CoFeB|MgO. 

We deduced the temperature dependent evolution of the SHA of clean Pt and dirty Ta.
The reproducible non-monotonic behavior observed in Pt cannot be accounted for by standard phenomenology,
but is compatible with the temperature scaling derived from microscopic phonon skew scattering theory.  The monotonic behavior observed in dirty Ta
is instead ascribed to the weakness of skew scattering contributions.
Our results reveal that extrinsic contribution to the SHE might
be the key to understanding the temperature dependence in normal metals as well, and
thus provides a more comprehensive understanding of the
spin Hall angle and the resulting spin orbit torques in this
system.




We acknowledge support by the EU
(MAGWIRE, Project No. FP7-ICT-2009-5; Marie Curie ITN WALL, FP7-PEOPLE-2013-ITN 608031); the DFG (KL1811, TRR 173, TRR 80, SFB 689); Graduate School of Excellence Materials Science in Mainz (MAINZ)
GSC 266; Research Center of Innovative and Emerging Materials
at Johannes Gutenberg University (CINEMA); and the Foundation for Fundamental Research on Matter (FOM), which is part of the Netherlands Organization for Scientific Research (NOW). J.-S. Kim acknowledges supports by the Leading Foreign Research Institute Recruitment Program through the NRF funded by the Ministry of Education, Science and Technology (MEST) (2012K1A4A3053565), the DGIST R&D Program of the Ministry of Science, ICT and Future Planning (17-BT-02 and 17-NT-01).






















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[Note1()]Note1
  note The fact that phonon skew scattering arises from an-harmonic
  phonon-phonon processes suggests, by analogy with the lattice thermal
  conductivity, a non-monotonic behavior at T<T_D. Notice also that the
  current phonon skew scattering theory considers only leading order 3-phonon
  processes.Stop
[Schmid(1973)]schmid1973
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[Stewart(1983)]Stewart1983
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