1 Comprehensive Demonstration of Spin -Hall Hanle Effect s in Epitaxial Pt Thin Films Jing Li1 Andrew H. Comstock2 Dali Sun2 and Xiaoshan Xu1

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Comprehensive Demonstration of Spin-Hall Hanle Effects in Epitaxial Pt Thin Films

Jing Li1*, Andrew H. Comstock2*, Dali Sun2⊥, and Xiaoshan Xu1⊥

1Department of Physics and Astronomy and Nebraska Center for Materials and Nanoscience,

University of Nebraska, Lincoln, NE 68588, USA

2Department of Physics and Organic and Carbon Electronics Lab (ORaCEL), North Carolina State

University, Raleigh, NC 27695, USA

* Authors with equal contributions

⊥ xiaoshan.xu@unl.edu, dsun4@ncsu.edu

Abstract

We demonstrate a nonlinear Hall effect due to the boundary spin accumulation in Pt films grown

on Al2O3 substrates. This Hall effect and the previously demonstrated Hanle magnetoresistance

provide a complete picture of the spin-precession control of the spin and charge transport at the

boundary of a spin-orbit coupled material, which we refer to as spin-Hall Hanle effects (SHHE).

We also show that the SHHE can be employed to measure the spin diffusion length, the spin-Hall

angle, and the spin relaxation time of heavy metal without the need of magnetic interface or the

input from other measurements. The comprehensive demonstration of SHHE in such a simple

system suggests they may be ubiquitous and needs to be considered for unravelling the spin and

charge transport in more complex thin film structures of spin-orbit coupled materials.

Ever since the discovery of the spin Hall effect (SHE) and the inverse spin Hall effect

(ISHE) [1–5], the strong spin-orbit coupling in materials such as heavy metals has been widely

used for both the generation [6] and detection of pure spin current [7]. Recently, the interaction

between spin current (from the SHE) in a heavy metal and the local spins in an adjacent magnetic

material has been demonstrated to give rise to the spin-Hall magnetoresistance (SMR) [8,9]. This

interfacial mechanism was later employed to effectively detect [10–12] and even manipulate [13]

the antiferromagnetic order which is beyond the capability of bulk magnetometry.

On the other hand, it is difficult to probe SHE and ISHE as bulk effects in heavy metals

using magneto-transport, because the modulation by a magnetic field is limited by the short

timescale of momentum relaxation [14]. In contrast, at the boundary of the heavy metals, the spin

accumulation and diffusion (spin-current reflection) may be more effectively manipulated by a

magnetic field via spin precession according to the Hanle effect [Fig. 1(a)], because it is the spin

relaxation that determines the timescale of the process. Given the relationship between spin

diffusion and charge current as described by the ISHE, Dyakonov predicted a longitudinal

effect [15], which was later observed in Pt and -phase Ta thin films [16,17] and named Hanle

magnetoresistance; one key evidence is the anisotropy since spin precession depends on the angle

between the magnetic field and the initial spin polarization [16].

What’s puzzling is the transverse effect. In principle, spin precession is expected to rotate

the spin polarization and generate a transverse charge current corresponding to a Hall effect.

However, this Hall effect has not been experimentally demonstrated and often overlooked. In

particular, in the previous work [16] where the longitudinal effect was demonstrated, only a linear

field-dependence of the transverse signal was observed and attributed to the ordinary Hall effect

(OHE).

To resolve the puzzle of missing transverse effect, we note that the previous work [16]

may have only probed the weak-precession condition due to the short spin relaxation time τs. In

the weak-precession condition, a linear field dependence of the Hall effect is expected, which

cannot be distinguished from the linear OHE background; meanwhile a quadratic field dependence

of the magnetoresistance is expected which is consistent with the observation [16].

In this work, we perform a magneto-transport study on the interplay of SHE, ISHE, spin

diffusion, and spin relaxation in Pt thin films deposited epitaxially on Al2O3 substrates using pulsed

laser deposition to enhance τs for reaching the strong-precession condition. We observed non-

quadratic and non-linear field dependence for the longitudinal (magnetoresistance) and the

transverse (Hall) effects respectively, indicating the strong precession condition. The dual effects,

which we refer to as the spin-Hall Hanle effects (SHHE), can be fit using the same set of parameters

(spin Hall angle SH, spin diffusion length s, spin relaxation time τs), suggesting that SHHE can

be reliably employed in extracting the spin transport properties without complications from the

magnetic interfaces, such as spin memory loss [18] and proximity-induced magnetism [19,20].

Pt (111) thin films of various thickness were epitaxially grown on Al2O3 (0001) substrates by

pulsed laser deposition with a YAG laser (266 nm wavelength, 70 mJ pulse energy, 3 Hz

repetition rate) in 10-7 torr vacuum at room temperature and subsequently patterned into Hall

bars by photolithography and ion milling. Crystal orientation of the Pt (111) films was confirmed

using X-ray diffraction

(Sec. S1 within the Supplemental Material [21]) while the film thickness was measured using X-

ray reflectivity. Longitudinal and transverse resistivity was measured using the Hall bar (Sec. S2

within the Supplemental Material [21]) in magnetic field along different directions at room

temperature; the field dependence of the longitudinal (



L) and transverse (



T) resistivity was

symmetrized and antisymmetrized respectively to minimize the spurious effects from imperfect

device geometry.

Figure 1(b) shows the change of longitudinal resistivity 



L-



L0 normalized with

respect to the zero-field value



L0 in a 5.2-nm-thick Pt film, where Bx, By and Bz represent the

magnetic field applied along the x, y, and z direction respectively. Overall, 



L increases with the

magnetic field, consistent with the expectation from the SHHE [15]. As illustrated in Fig. 1(a), the

longitudinal charge current ( ||+ ) in the Pt film generates a spin current ( || ) via SHE

toward the Pt/Al2O3 interface with spin polarization  || . The reflected spin current ()

generates a longitudinal charge current    || + via the ISHE before the spin

polarization relaxes, resulting in an overall reduction in the resistivity of the Pt film. The Hanle

effect may be observed when an external magnetic field causes the precession of the spin

polarization of . In this case, the projection of  on  will be reduced, which increases the

longitudinal resistivity, as observed in Fig. 1(b) consistent with that in previous work [16].

The anisotropy in Fig. 1(b) also agrees with SHHE in that L(By)/L0 is smaller than

L(Bx)/L0 and L(Bz)/L0 while the latter two are similar. When the magnetic field is parallel to

the initial polarization direction () of , no spin precession is caused by the external magnetic

field and the SHHE does not contribute to L(By)/L0. As a result, ordinary magnetoresistance

(OMR) is responsible for the non-zero L(By)/L0 observed in Fig. 1(b) which is proportional to

B2; hence the difference 



L(Bz)-



L(By) is attributed to the longitudinal SHHE.

Fig. 1(b) also reveals the strong-precession behavior of the longitudinal SHHE that was

not observed before. Considering both the film-substrate and the film-vacuum boundaries, SHHE

with Bz can be described using [15] (Sec. S3 within the Supplemental Material [21]):



 



 



 

 (1),

where the real and imaginary parts of SHHE are the longitudinal L,SHHE and the transverse

T,SHHE respectively, d is the film thickness,  is a complex quantity with i,

 is the Larmor frequency with  the gyromagnetic factor,  the Bohr magneton,

and  the reduced Planck constant. A numeric simulation is displayed in Fig. 2. According to Eq.

(1) and Fig. 2(a), at low field (weak precession), the longitudinal SHHE is quadratic (Bz2) as

observed previously [16]; at high field (strong precession), the effect saturates when the

precession angle is so large that the projection of  on  cancels, consistent with the reduced

slope L(Bx)/L0 and L(Bz)/L0 at high field in Fig. 1(b).

Figure 1(c) shows the normalized transverse resistivity T/L0, which has non-trivial field

dependence only in Bz. In addition, T(Bz)/L0 exhibits a non-linear relation with a large slope at

low field and a smaller slope at high field. The latter is expected to come from the ordinary Hall

effect (OHE) in a non-magnetic metal. Similar field dependence of the transverse resistivity has

been observed in Pt/ferrimagnetic insulator (FMI) systems, which was explained as anomalous

Hall effect caused by magnetic proximity [19,22]. Here we don’t have the complications from the

magnetic order of the substrate, so the non-linear part of the transverse signal can be directly

ascribed to SHHE as explained in the following.

As illustrated in Fig. 1(a), with Bz, the spin precession leads to non-zero projection of the

spin polarization of  on , which generates a non-zero projection of  on  (Hall signal) via

ISHE. At low field (weak precession), the effect is linear (Bz) according to Eq. (1) and Fig. 2(b).

At high field (strong precession), the transverse effect is expected to vanish because the projection

of  on  cancels due to the large precession angle. This overall nonlinear effect is consistent

with the observation in Fig. 1(c).

The observation of the non-quadratic longitudinal and the non-linear transverse field

dependence in Figs. 1(b) and (c) respectively, suggests the strong-precession condition of SHHE

in Eq. (1). In principle, all the parameters contributing to SHHE, i.e., s, SH, and s can be

extracted by fitting the experimental data using the field dependence in Eq. (1). On the other hand,

a scaling rule pointed out by Dyakonov [15] (Sec. S3 within the Supplemental Material [21]) also

needs to be considered, as described below.

Considering the spin-precession nature, the SHHEs are expected to scale with the spin

precession time 

. For d/s →∞ (thick film limit), 

 is limited by the spin relaxation time s, i.e.,



=s, as illustrated in Fig. 1(a). For d/s →0 (thin film limit), spin precession occurs over the

entire film thickness, so 

 is the same as the spin diffusion time 





, where  





 is the spin diffusion coefficient. Dyakonov then introduced the definition 













 to describe the dependence of 

 on both s and  [15]. As shown in Fig. 2,

L,SHHE and T,SHHE simulated according to Eq. (1) are normalized with the maximum longitudinal

effect L,SHHE(Bz=∞) and plotted as a function of 

. Indeed, the “scaled” field dependence of

SHHE maintains roughly the same curve shape despite that the value of d/s changes over four

orders of magnitude.

The Dyakonov’s scaling rule suggests that it is difficult to unambiguously determine s,

SH, and s altogether by fitting the measured field dependence of SHHE/L0 using Eq. (1)

considering the experimental uncertainty, because it is 

 instead of s that can be directly extracted.

On the other hand, here we notice that s can be estimated based on the thickness dependence of

SHHE, which can then be used to extract  (out of 

) and SH. A close look at Eq. (1) reveals that

the low-field SHHE/L0 has a maximum at an intermediate film thickness because it vanishes in

both the thin and thick film limits: For d/s→0 (thin film limit), SHHE approaches zero because



 →0 means no precession; for d/s →∞ (thick limit), SHHE/L0 also approaches zero because

the effect of the spin precession that occurs at the boundary is unimportant for thick films. It turns

out that the thickness for reaching maximum low-field SHHE/L0 only depends on s, or d/s4.56

and d/s3.28 for L,SHHE/L0 and T,SHHE/L0 respectively (Sec. S3 within the Supplemental

Material [21]), as also given by Eq. S36 and Eq. S38 in ref. [16].

Considering this property, we measured the thickness dependence of SHHE in the epitaxial

Pt films. The experimental L,SHHE/L0 is calculated by subtracting the OMR contribution, i.e.,

[L(Bz) - L(By)]/L0. Fig. 3 shows the thickness dependence of experimental L,SHHE/L0 at 4

T field. Meanwhile, the experimental T,SHHE/L0 is calculated by subtracting the linear OHE

contribution from T(Bz)/L0; the result at 1 T field is displayed in Fig. 3. Fitting the thickness

dependence of both longitudinal and transverse SHHE leads to s=1.63 ± 0.26 nm. The s values

are comparable to the value reported in polycrystalline Pt/sapphire at 300 K [16] and single

crystalline Pt/Fe/MgO [23,24].

With the estimation of s, we may fit the field dependence of SHHE signals using Eq. (1)

and derive the value of SH, τs, and the related diffusion coefficient D=s2/s. Fig. 4 shows fittings

of both longitudinal and transverse SHHE signals from three different Pt/Al2O3 films. For each

film, same set of parameters (SH, s, s) have been used to fit both longitudinal and transverse

SHHE (Sec. S4 within the Supplemental Material [21]). The derived spin transport properties of

Pt are summarized in Table 1 and compared with those from Ref. [16]. One salient difference

between this work and previous work [16] is that the spin relaxation time s is roughly one order

of magnitude longer in the epitaxial Pt films used in this work, which is critical for reaching the

strong-precession condition of SHHE.

As pointed out by Dyakonov [15], for d/s→0, the maximum longitudinal SHHE, i.e.,

ΔL,SHHE()/L0 approaches SH2. As a result, ΔL,SHHE(Bz=)/L0 measured from thin Pt films

generally provides a more precise estimation of SH. Meanwhile, in thick Pt films, the spin

precession time 

 that determines the shape of the field dependence of SHHE is simply s, hence

measurements from thicker Pt films generally provide a more precise estimation of s. Based on

these arguments, we found that SH and s are most likely to be 0.0220.006 and 1.80.9 ps,

respectively in our Pt thin films.

The SH value of our Pt films is lower than the values of 0.0480.015 [23] and

0.0570.014 [24] reported for single crystalline (001) Pt/Fe/MgO measured using spin pumping,

but still lies within the range between 0.01 and 0.1 reported for polycrystalline Pt [25,26]. Crystal

orientation might be responsible for the discrepancy of SH values among different single-

crystalline Pt films. It has been shown that SH of Pt can be tuned from 1% to 10% by varying the

resistivity of polycrystalline Pt films [26]. The SH value of our Pt films is comparable to that of

e-beam evaporated polycrystalline Pt films, while the longitudinal resistivity (20~50 cm) of

our Pt films at 300 K is slightly larger than that (~18 cm) of evaporated Pt in super-clean metal

regime [26]. Considering that the grain size of our (111) Pt films is small (~ 3 nm derived from

x-ray diffraction), it is reasonable to postulate that abundant grain boundaries exist along charge

current flow direction within Pt film plane while there are far fewer grain boundaries hindering

spin current flow along the normal direction of thin film plane, which may explain the similarity

of SH values between epitaxial and polycrystalline Pt films. It is noteworthy that our epitaxial Pt

films do not form interfaces with any magnetic substrates, eliminating the intricacy of separating

spurious contributions, such as spin rectification effect [23,24], from ISHE contribution to the

measured signals.

In conclusion, this work has demonstrated that SHHE emerges as non-linear Hall effect

and non-quadratic magnetoresistance in epitaxial Pt films on Al2O3 substrates at room temperature.

Importantly, we show that SHHE can be employed to reliably measure spin transport properties of

spin-orbit coupled materials, without the complication of magnetic interfaces or the need of other

measurements. The simplicity of SHHE suggests that with a magnitude up to SH2, they are

expected to be ubiquitous in heavy metal thin film systems. Recognition of the contribution of

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1ComprehensiveDemonstrationofSpin-HallHanleEffectsinEpitaxialPtThinFilmsJingLi1*,AndrewH.Comstock2*,DaliSun2⊥,andXiaoshanXu1⊥1DepartmentofPhysicsandAstronomyandNebraskaCenterforMaterialsandNanoscience,UniversityofNebraska,Lincoln,NE68588,USA2DepartmentofPhysicsandOrganicandCarbonElectronicsLab(ORaCE...

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1 Comprehensive Demonstration of Spin -Hall Hanle Effect s in Epitaxial Pt Thin Films Jing Li1 Andrew H. Comstock2 Dali Sun2 and Xiaoshan Xu1

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