Gate tunable spin-orbit coupling and weak antilocalization effect in an epitaxial
La
2/3Sr
1/3MnO
3thin film
Shao-Pin Chiu,1,*Michihiko Yamanouchi,2,3Tatsuro Oyamada,3Hiromichi Ohta,2,3and Juhn-Jong Lin1,4
1Institute of Physics, National Chiao Tung University, Hsinchu 30010, Taiwan
2Research Institute for Electronic Science, Hokkaido University, N20W10, Kita, Sapporo 001-0020, Japan 3Graduate School of Information Science and Technology, Hokkaido University, N14W19, Kita, Sapporo 060-0814, Japan
4Department of Electrophysics, National Chiao Tung University, Hsinchu 30010, Taiwan (Received 24 April 2017; revised manuscript received 12 July 2017; published 28 August 2017) Epitaxial La2/3Sr1/3MnO3 (LSMO) films have been grown on SrTiO3 (001) substrates via pulsed laser deposition. In a 22-nm-thick LSMO film with a low residual resistivity of ρ0≈ 59 μ cm, we found a zero-field dip in the magnetoresistance (MR) below 10 K, manifesting the weak antilocalization (WAL) effect due to strong spin-orbit coupling (SOC). We have analyzed the MR data by including the D’yakonov-Perel’ spin-relaxation mechanism in the WAL theory. We explain that the delocalized spin-down electron sub-band states play a crucial role for facilitating marked SOC in clean LSMO. Moreover, we find that the SOC strength and gate voltage tunability are similar to those in a two-dimensional electron gas at the LaAlO3/SrTiO3interface, indicating the presence of an internal electric field near the LSMO/SrTiO3interface. In a control measurement on a 5-nm-thick high resistivity (ρ0≈ 280 μ cm) LSMO film, we observe only a small zero-field peak in MR from weak localization effect, indicating negligible SOC.
DOI:10.1103/PhysRevB.96.085143
I. INTRODUCTION
The perovskite manganite La2/3Sr1/3MnO3 (LSMO) has
recently been exploited for its possible use in spintronics. Researchers have found a nearly 100% spin polarization (P ) in LSMO films from the experiments of spin-resolved photoemission spectroscopy [1] and magnetoresistance (MR) in magnetic tunnel junctions [2]. They have ascribed the results to intrinsic half metallicity and classified LSMO as a traditional, type-I half metal. On the other hand, point-contact Andreev reflection (PCAR) studies showed a broad range of
P (58–92%) in electrical current [3]. Moreover, the P and residual resistivity revealed a correlation demonstrating that highly resistive samples exhibited higher P values [3]. At first glance, this is a surprising correlation opposite to what people would expect for a traditional half metal. This puzzle can be understood in terms of the energy band structure of a type-IIIAhalf metal, as schematically depicted in Fig.1(a)
[4]. The energy band structure of Fig.1(a)is constituted of a up sub-band of relatively mobile holes and a spin-down sub-band of comparatively heavy electrons. There are theoretical calculations [5,6] and experiments [7,8] on LSMO, which are supportive of this kind of band structure. In this type of half metals, there is no energy gap between the two 3d spin sub-bands. In the presence of strong disorder, the spin-down sub-band states can become (largely) localized, and hence the electrical transport properties are governed entirely by the spin-up sub-band states [6]. In a type-I half metal, scattering between spin-up carriers and spin-down carriers will mostly be frozen out at low temperatures, due to an energy gap between the two sub-bands. Therefore, one does not expect a (strong) spin-relaxation process to occur even if the spin-orbit coupling (SOC) strength is finite. On the other hand, in a type-IIIA half metal, spin-relaxation
processes can become marked if the spin-down sub-band states are delocalized. Then, interaction between spin-up holes and spin-down electrons can take place, which may foster fast spin relaxation of the charge carriers. Microscopically, the spin-relaxation interaction can be substantiated by a finite SOC which prevails in the material/device under study. A clean LSMO thick film with delocalized spin-down sub-band states provides an opportunity to test this concept. The SOC originates from an internal electrical field which is induced at the LSMO/SrTiO3(LSMO/STO) interface.
In low-dimensional systems, the low-field MR due to the weak localization (WL) and weak antilocalization (WAL) effect provides a powerful tool for extracting the spin-orbit scattering time, τso, and the corresponding SOC splitting so
(defined below) [9,10]. In the presence of SOC, the spin-part wave function will change sign over a characteristic length scale called the spin-orbit scattering length Lso=
√
Dτso,
where D is the charge-carrier diffusion constant. When Lso
is much shorter than the electron (hole) dephasing length
Lϕ=
Dτϕ, where τϕis the electron (hole) dephasing time,
the WAL effect with a zero-field dip in MR is expected. In the opposite limit of negligible SOC (Lso Lϕ), the WL effect
with a zero-field peak in MR is expected. Thus far, only the WL effect has been observed in a 10-nm-thick epitaxial LSMO film by Niu et al. [11]. In this paper, we report our experimental realization of the WAL effect in a 22-nm-thick epitaxial LSMO film. Our film is relatively clean and has a residual resistivity 30 times lower than that of the 10-nm sample reported in Ref. [11]. In particular, the backgate-voltage tunability of so
found in this film indicates that the SOC originates from an interfacial Rashba-type interaction. Therefore, the two spin sub-bands nearby the LSMO/STO interface split along the momentum direction owing to the Rashba effect [12], apart from the magnetism induced double exchange splitting along the energy direction [Fig.1(a)]. Away from the interface, only the exchange splitting remains effective in the bulk of LSMO film.
FIG. 1. (a) Schematic energy band structure for a type-IIIA half metal. EFdenotes the Fermi energy. (b) Temperature dependence of resistivity ρ(T ) for a 22- and a 5-nm-thick epitaxial LSMO film, as indicated.
II. EXPERIMENTAL METHOD
LSMO films with nominal thickness of 5 and 22 nm were heteroepitaxially grown on (001) SrTiO3 single-crystalline
substrates by pulsed laser deposition (PLD) at the substrate temperature of 700◦C under oxygen atmosphere (PO2=
25 Pa). A KrF excimer laser (λ= 248 nm, pulse duration ∼20 ns, fluence ∼1.6 J cm−2pulse−1, 1 Hz, COMPex 102)
was used to ablate the ceramic target of LSMO. After the deposition, the LSMO films were annealed for 20 min under the identical conditions for the growth. The thickness of the resultant films was measured using x-ray reflectivity (XRR, ATX-G, Rigaku Co.) with monochromated Cu Kα1
radiation. The resistance and MR were measured with ac resistance bridges (Linear Research model LR700 or LR400 operating at 16 Hz), by employing the van der Pauw electrode configuration. The backgate voltage was applied by a Keithley model 2635A sourcemeter. Low-temperature measurements were performed on an Oxford Heliox3He cryostat equipped with a 2-T superconducting magnet. In the following, we focus mostly on the 22-nm-thick epitaxial LSMO film.
III. RESULTS AND DISCUSSION
Figure 1(b) shows the temperature dependence of resis-tivity, ρ(T ), for the 22-nm-thick LSMO film between 0.36 and 300 K. [The ρ(T ) for the 5-nm-thick film is also
plotted for comparison.] A large relative resistance ratio of
ρ(290 K)/ρ(10 K)= 17 indicates the sample being a good metal. The residual resistivity is ρ0= 58.7 μ cm. This ρ0
value is as low as that of the optimal samples fabricated by PLD method [13] and close to that of the cleanest samples used in PCAR experiments [3]. Therefore, the lattice structure of this film is of high quality, with a low defect number density and minimal grain boundaries.
It has previously been found that, in LSMO films grown on various substrates, an interfacial, insulating (“dead”) layer often exists. The thickness of the insulating layer varies from 2.4 to 5 nm [14,15]. If taking this matter into account, the effective thickness (and the ρ0value given above) of our film
will be reduced by 10–20%.
Figure 2(a) shows the sheet resistance Rs as a function
of magnetic field H at several temperatures, as indicated. The magnetic field was applied perpendicular to the film plane. We first examine the overall MR behavior in the wide magnetic field range of|Hc| 1.2 T. Previously, a large
negative MR due to grain boundary scattering has been found in polycrystalline LSMO films [16,17]. In contrast, here we observe a small positive MR which is similar to that seen in thick, clean epitaxial LSMO in Ref. [17]. In the field range |H| < 0.7 T, hysteretic behavior is observed, which stems from the alignment processes of magnetic domains in ferromagnetic LSMO. In Fig.2(b), Hc+ and Hc− denote the coercive fields, with the value |Hc| ≈ 0.27 T. In the rest of
this paper, we shall focus on the low magnetic field regime of
H |Hc| to address the WAL MR. In Fig.2(b), the Rs(H )
curve reveals an evident zero-field dip, or a zero-field peak in the sheet magnetoconductance Gs(H )= 1/Rs(H ), as shown
in the inset. The magnitude of the zero-field peak in Gs(H )
amounts to about one-half of the quantum conductance e2/ h, where e is the electronic charge, and h is the Planck constant. Figure2(a)demonstrates that the zero-field dip decreases with increasing temperature T , and it vanishes at≈ 10 K [see also Fig. 4(a)]. These temperature-dependent positive MRs are the quantum-interference manifestations of the WAL effect induced by marked SOC [18].
We evaluate the charge-carrier elastic mean free time to be τe≈ 2.9 fs, using the Drude model with our measured ρ0 value, a carrier density n extracted from the Hall effect
FIG. 2. (a) MR at several T values, as indicated. Black (red) curves correspond to the MR measured with magnetic field sweeping from +1.2 to −1.2 T (−1.2 to +1.2 T). (b) MR at T = 0.36 K. Hc+and Hc−denote the coercive fields. Inset: Sheet magnetoconductance Gs(H )
measurement (1.25× 1022cm−3 in a 25-nm-thick LSMO on STO) [19], and an effective mass of the majority carrier (hole) m∗= 0.6m0[6,19], where m0is the free-electron mass.
From the free-electron model, we estimate the carrier elastic mean free path to be le≈ 4 nm, the diffusion coefficient D= vFle/3≈ 18.5 cm2/s, and the product kFle≈ 28, where vFis the Fermi velocity, and kFis the Fermi wave number.
In the measurement scheme shown in the upper panel of Fig. 3(a), a backgate voltage Vbg is applied to induce a
band bending near the LSMO/STO interface, which in turn modifies the charge-carrier density near the interface. Because the carrier (hole) density is fairly high in this film, the sheet resistance reveals only a small variation (≈ 0.02%) as Vbg
is swept from −40 to +40 V, Fig. 3(b). Nevertheless, the variation of Rs with Vbg reveals a noticeable nonmonotonic
feature. We can explain the nonmonotonic behavior in terms of the presence of two conduction sub-bands, as depicted in Fig. 1(a). Because our film is clean, the heavy electrons in the spin-down sub-band are delocalized. The two conduction channels (light holes and heavy electrons) respond in an opposite way to the applied Vbg, resulting in a nonmonotonic
characteristic of Rsversus Vbg. This interpretation is supported
by the Hall effect studies of LSMO, where a two-band model is needed to explain the data [19]. Moreover, this interpretation is supported by the angle-resolved photoemission studies [8]. In the latter experiment, a Fermi surface (FS) of an electron pocket centered around the point was observed in metallic LSMO films, whereas the FS gradually diminished in those highly resistive samples lying close to the metal-insulator transition boundary [8]. The presence of delocalized spin-down electron states plays a crucial role for facilitating the WAL effect, as emphasized above [18].
Being a quantum-interference effect, the WAL MR is only important in the low magnetic field regime of H |Hc|.
To minimize any possible hysteretic effect on the measured WAL MR, we have symmetrized our data by taking the even part of Rs(H ), i.e., Rs(H )even= [Rs(H )+ Rs(−H )]/2.
[For simplicity, in the following discussion we shall use the same notation Rs(H ) to denote the symmetrized data.] The
positive parabolic background MR due to the Lorentz force (in high magnetic fields) has also been subtracted. Figure3(c)
shows the symmetrized, normalized MR measured at 0.36 K and at several Vbg values. Note that the zero-field MR dip
is sensitive to Vbg, reflecting a significant interfacial SOC
effect due to a broken inversion symmetry at the LSMO/STO interface. (The hysteretic part of MR is also affected by Vbg.
This issue requires further investigations.) The MR dip is the largest at Vbg= 0 V, with its magnitude decreasing with
increasing|Vbg|. A gate tunable MR dip immediately implies
that the SOC is of the Rashba type [20,21]. The Rashba SOC can nurture the D’yakonov-Perel’ (DP) spin-relaxation processes [22,23], where spin relaxation arises from the spin precession between two consecutive elastic-scattering events. The spin-relaxation, i.e., spin-orbit scattering, rate τso−1 is
predicted to vary linearly with τe. Therefore, the cleaner
the system is made, the higher the τso−1 scattering rate will
be [24,25]. Iordanskii, Lyanda-Geller, and Pikus (ILP) have theoretically calculated the WAL MR by explicitly taking the DP spin-relaxation mechanism into account [10,26]. Their prediction for a quasi-two-dimensional system in the presence
of a perpendicular magnetic field can be expressed in a compact form in terms of τϕand τso(Ref. [27]):
Rs(H )− Rs(0) R2 s(0) = − e2 2π2¯h 1 2+ Hϕ+ Hso H − lnHϕ+ Hso H + 1 2 1 2+ Hϕ+ 2Hso H −1 2ln Hϕ+ 2Hso H − 1 2 1 2 + Hϕ H +1 2ln Hϕ H (1) with Hi≡ ¯h 4eDτi , Li = Dτi,i= ϕ, so
where (x) is the digamma function, and Hϕis a characteristic
scattering field. One can explicitly write Hϕ = ¯h/(4eDτϕ)=
¯h/(4eDτin)+ 2¯h/(4eDτs), where τin−1 is the total inelastic
scattering rate, and τs−1 is the spin-spin (spin-flip) scattering
rate due to magnetic impurities [18]. In practice, τs−1 is
essentially temperature independent and can be extracted from the measured τϕ−1(T → 0 K). We reiterate that in performing
least-squares fits to Eq. (1), we have included only the MR data measured at H < 0.5|Hc|. Therefore, any possible hysteresis
effect can largely be ignored. Also, in this low magnetic field regime, one does not need to consider the MR due to the many-body electron-electron interaction (EEI) effect [28].
The red curves in Fig.3(c)are fitted curves using Eq. (1). Charge-carrier dephasing length Lϕ and spin-orbit scattering
length Lso obtained from the fits are plotted as a function of
Vbgin Fig.3(d). We find that Lϕ(Vbg) takes a maximum value
of ≈ 347 nm at Vbg = 0, corresponding to a charge-carrier
dephasing time of τϕ= 65 ps and a characteristic scattering
field of Hϕ= 1.4 mT. Therefore, the WAL MR manifests at H |Hc|. The obtained result Lϕ t justifies the application
of Eq. (1), where t is the film thickness. We find that Lϕ (τϕ)
decreases with increasing|Vbg|. This kind of Vbg dependence
is similar to what has previously been observed in topological insulators Bi2Te3 [29] and Bi2Se3 [30]. The applied gate
voltage creates (enhances) an internal electrical field (which already exists at Vbg= 0 V) between the LSMO film and the
backgate electrode, inducing a thin depletion (accumulation) layer by positive (negative) Vbg for the spin-up conduction
holes. Simultaneously, the applied gate voltage induces a thin accumulation (depletion) layer by positive (negative) Vbg for
the spin-down conduction electrons. Heuristically, we may envision the sample as constituted of a LSMO thick film (a bulk) and a thin interfacial LSMO/STO regime. Both are metallic and conduct in parallel. An applied Vbg will have
little effect on the electrical properties of the former, but can affect the latter regime markedly. Thus, the total Rs of the
sample, which is dominated by the bulk LSMO, shall depend only weakly on Vbg, as is seen in Fig. 3(b). On the other
FIG. 3. (a) Upper panel: A schematic of electrical measurement configuration. Lower panel: A schematic of electron band bending under a positive backgate voltage Vbg. (b) Rsas a function of Vbg. The black (red) symbols are measured by sweeping Vbgfrom+40 to −40 V (−40 to +40 V). (c) Symmetrized and normalized MR measured at 0.36 K and at several Vbgvalues, as indicated. Data are offset for clarity. The solid curves are least-squares fits to Eq. (1). (d) Extracted dephasing length Lϕand spin-orbit scattering length Lsoas a function of Vbg. The bottom panel shows the variation of SOC spin splitting energy sowith Vbg.
the interfacial regime hundreds or thousands of times over τϕ,
the quantum-interference quantities shall thus be modified by
Vbg. Especially, if the dephasing processes are notably stronger in the interfacial regime than in the bulk LSMO, Lϕand Lso
will reveal appreciable variations with Vbg. This is the case
illuminated in Fig. 3(d). A stronger dephasing rate in the interfacial regime arises from the fact that this regime, being in proximity to a dead layer, is much less conductive compared to the bulk LSMO.
In weakly disordered metals (kFle>1), the magnitude
and temperature dependence of Lϕ are determined by the
responsible inelastic electron scattering processes. According to the current understanding, the carrier dephasing in two dimensions is governed by the Nyquist electron-electron (e-e) scattering at low temperatures, while the electron-phonon scattering can become important at somewhat higher temper-atures [31,32]. In the EEI theory, the Coulomb interaction in a low-conductivity sample will be enhanced due to a suppression of the screening effect through the reduced density of states (or carrier concentration n) [28]. The e-e scattering rate is then increased, leading to a decreased Lϕ at low temperatures. In
our case, upon the application of Vbg, a measurable increase
in Rs implies that the conductivity of the interfacial layer is
drastically decreased. The phase-coherent charge carriers that traverse through this regime will then encounter a reduced
n and D and undergo enhanced e-e (hole-hole) scattering. This picture explains why we have observed the Lϕ value to
decrease with increasing|Vbg|.
Figure3(d)shows that Lso decreases monotonically from
72 to 59 nm as Vbg increases from −20 to +40 V. The
value of Lso(Vbg= 0 V) = 68.5 nm corresponds to a spin-orbit
scattering (spin-relaxation) rate of τso−1≈ 3.9 × 1011s−1. In
the DP mechanism, τso−1= (so2τe)/ ¯h2, where so is the
energy band splitting due to SOC [22]. We obtain so(Vbg =
0 V)≈ 7.7 meV from the above τso−1value. The lower panel
of Fig. 3(d) shows a monotonic increase of so with Vbg.
From the relation so= 2kFα [33], we obtain the Rashba
SOC coefficient α≈ 5.4 × 10−13eV m. For comparison, our
so(Vbg= 0 V) value is on the same order of magnitude as
that (≈3.3 meV) found in the two-dimensional electron gas (2DEG) at the LaAlO3/SrTiO3 (LAO/STO) interface [24]
and that (≈ 5.4 meV) in the InGaAs/InAlAs heterostructure [20]. Moreover, our value of so(Vbg = +40 V) ≈ 8.9 meV is
comparable to that [so(Vbg = +50 V) ≈ 7 meV] found in the
LAO/STO interface [24]. Due to possible uncertainties in the evaluations of τeand D, we estimate our extracted sovalue
to be accurate to within a factor of∼2.
Our observation of backgate tunable sostrongly suggests
the presence of an internal electric field near the LSMO/STO interface. The electric field stems from a discontinuity in the
FIG. 4. (a) Symmetrized and normalized MR measured at Vbg= 0 V and at several T values, as indicated. Data are offset for clarity. The solid curves are least-squares fits to Eq. (1). (b) Extracted dephasing length Lϕas a function of temperature. The dashed straight line indicates
T−1/2temperature dependence and is a guide to the eye. The red horizontal line represents the spin-orbit scattering length Lso. (c) Dephasing rate τϕ−1as a function of temperature. The black solid curve is a least-squares fit to τϕ−1= 2τs−1+ AeeT (see text). The dashed straight line
is the theoretical prediction of 2D e-e scattering rate τee−1(T ), using the measured Rsvalue. The red horizontal line represents the spin-orbit
scattering rate τso−1.
layer-by-layer ionic structure of these two complex oxide materials, as has recently been theoretically predicted [34–36] and experimentally confirmed [37]. In this situation, the band structure in the interfacial regime will be different from that [Fig.1(a)] in bulk LSMO. The interfacial band structure is to be modified by double exchange interaction [5,38] together with Rashba splitting, i.e., the two spin sub-bands split in both energy and momentum directions (see, for example, a schematic in Fig. 9 of Ref. [12]). Consequently, the sovalue
should vary with Vbg, which adjusts the internal electric field
near the interface.
Figure 4(a) shows the symmetrized, normalized MR at
Vbg = 0 V and at several T values between 0.36 and 10 K,
as indicated. (The positive parabolic background MR has been subtracted, as mentioned.) The solid curves represent fitted curves using Eq. (1) and taking a constant Lso= 68.5 nm
together with Lϕ as an adjusting parameter. The extracted Lϕ as a function of temperature is plotted in Fig.4(b). The
dashed straight line indicates a T−1/2temperature dependence and is a guide to the eye. At 10 K, the MR dip gradually evolves into a small MR peak (not shown), corresponding to a crossover from the strong SOC regime Lϕ> Lsoto the weak
SOC regime Lϕ < Lso at∼ 8 K. In short, the WAL effect is
observed in our film which is clean and possesses a relatively large|Hc| Hϕ. A clean film with finite so(τso−1) renders
a long dephasing length at low temperatures, and hence the criterion Lϕ> Lsois achieved.
Figure4(c)shows the variation of τϕ−1with temperature.
The black solid curve is a least-squares fit and will be discussed below. The dashed straight line is the theoretical prediction of the 2D Nyquist e-e scattering rate τee−1(T )= (Aee)thT,
where (Aee)th= (e2kBRs/2π ¯h2)× ln(π ¯h/e2Rs), and kBis the
Boltzmann constant [32]. From our Rs value, we estimate
the scattering strength to be (Aee)th= 8.5 × 108s−1K−1 for
this sample. This theoretical value is more than one order of magnitude lower than the experimental value. However, the Rsvalue in the interfacial regime should be much higher
than the measured value. Therefore, the above calculation of (Aee)th is an underestimate (see below). Up to our highest
measurement temperature of 10 K, no signature of electron-phonon scattering is observed, which should cause a τϕ−1∝ Tp dependence with the exponent p 2 [31].
For comparison with the MR in the clean LSMO film, we have measured the MR in a thick film. The 5-nm-thick film has a relatively high resistivity compared with that in the 22-nm-thick film [see Fig. 1(b)]. Similar to previous results found in Refs. [11,17], our measurements reveal that the overall MR is negative in the wide magnetic field range|H| 1.2 T [Fig.5(a)]. Note that we do not observe any signature of the WAL effect even at T < 2 K. Instead, a zero-field peak in MR, i.e., the WL effect, is observed. Figure5(b)shows the zero-field peaks for the 5-nm-thick LSMO film at several T values, as indicated. The magnitudes of the WL MR are small and nearly disappear at 5 K. The solid curves are the theoretical predictions of Eq. (1) with the fitting parameter Lso→ ∞, i.e.,
negligible SOC. The fitted Lϕ(T ) values are 91, 85, 78, 64,
and 35 nm at 0.36, 0.66, 1.0, 2.0, and 5.0 K, respectively. The inset of Fig.5(b) shows the variation of the extracted τϕ−1
with T for this film. The solid curve is a least-squares fit to be discussed below.
To extract the spin-spin scattering rate due to magnetic impurities, we have fitted the measured τϕ−1(T ) data in
Figs. 4(c) and 5(b) to the form of τϕ−1(T )= 2τs−1+ AeeT
[31]. The fitted values are τs−1= 4.1 × 109s−1 and Aee =
3.1× 1010s−1K−1for the 22-nm-thick film, and τ−1
s = 1.5 ×
1010s−1 and Aee = 3.8 × 1010s−1K−1 for the 5-nm-thick
film. The 22-nm sample is much cleaner than the 5-nm sample in terms of magnetic impurities. The magnetic impurities
FIG. 5. (a) MR at T = 0.36 K for a 5- and a 22-nm-thick epitaxial LSMO film, as indicated. The MR for the 5-nm-thick film was measured with magnetic field sweeping from+1.2 to −1.2 T. The data for the 22-nm-thick film are taken from Fig.2(a). (b) Low-field MR for the 5-nm-thick LSMO film at several T values, as indicated. The solid curves are least-squares fits to Eq. (1) with τso−1= 0. The inset shows the variation of τϕ−1with temperature. The solid curve is a least-squares fit to τϕ−1= 2τs−1+ AeeT (see text).
may (partly) originate from nonuniform magnetization near the LSMO/STO interface where the insulating dead layer can be antiferromagnetic [14,39]. With weak magnetic spin-spin scattering, the spin-relaxation processes are mainly dominated by the SOC processes in the 22-nm sample, with τso−1/τs−1≈
100. On the other hand, in the 5-nm-thick sample, besides a larger τs−1, the spin-down sub-band states are probably
(mostly) localized. Therefore, the SOC splitting so in this
sample, if any exists, will be ineffective in causing spin relaxation, leading to τs−1 τso−1and WL MR. Theoretically,
Dugaev and coworkers have extended their original work [12] to show that WAL effect can happen in a ferromagnetic conductor under the condition τs−1< τso−1 [40]. They applied
this new theory to explain the experimental observations by Neumaier et al. in (Ga,Mn)As nanostructures [41,42]. Our results for the 22-nm-thick film also support this theoretical condition.
In the 5-nm-thick film, we have Rs = ρ0/t= 560 , giving
rise to the e-e scattering strength (Aee)th ≈ 9 × 109s−1K−1.
This is a factor of ∼ 4 lower than the experimental value. Considering that there very likely exists a thin dead layer which will reduce the effective conduction thickness of the film, the discrepancy between experimental and theoretical values would be smaller. This level of consistency is satisfactory and supports that the inelastic dephasing processes in LSMO/STO are dominated by the 2D Nyquist e-e scattering. The fact that the fitted Aee value for the 22-nm film is similar to that for
the 5-nm film suggests that the dephasing processes are much
stronger in the interfacial LSMO/STO regime than in bulk LSMO.
IV. CONCLUSION
We have studied a 22-nm-thick epitaxial LSMO film with a low resistivity. We have observed the WAL MR, suggesting a marked SOC effect in this clean sample. The observations of nonmonotonic gate-voltage dependent sheet resistance suggests an important contribution from the minority-spin-down electrons to total electrical transport. The gate-voltage tunable SOC strength indicates the existence of an internal electric field near the LSMO/STO interface. Our extracted SOC splitting energy so is comparable to that found in the
2DEG at the LAO/STO interface. The gate tunability of SOC may have potential applications in spintronic devices.
ACKNOWLEDGMENTS
J.J.L. was supported by Taiwan Ministry of Science and Technology through Grant No. MOST-103-2112-M-009-017-MY3 and the Ministry of Education ATU Plan. M.Y. was supported by a Grant-in-Aid for Young Scientists A (Grant No. 15H05517) from the Japan Society for the Promotion of Science (JSPS). H.O. was supported by a Grant-in-Aid for Scientific Research on Innovative Areas (Grant No. 25106007) from JSPS. A part of this work was supported by Dynamic Alliance for Open Innovation Bridging Human, Environment and Materials.
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