Supplementary information
Growth of Epitaxial Co2FeSi
25-nm-thick Co2FeSi (CFS) films for a spin injector and detector were grown
on Si(111) templates by low temperature molecular beam epitaxy (LT-MBE) [1]. We
have already established the epitaxial growth for Heusler-compound thin films by
intentionally using (111)-oriented Si or Ge substrates in LT-MBE conditions [2]. It is
essential for obtaining the epitaxial layers to use the atomic matching at the (111)
interfaces between Heusler compounds and Si or Ge. If we use the (100)- or
(110)-oriented substrate, even the epitaxial growth cannot be achieved. Prior to the
growth, surface cleaning of substrates was performed with an aqueous HF solution
(HF : H2O = 1 : 40), and then, they were heat-treated at 450◦C for 20 min in an MBE
chamber with a base pressure of 2 × 10−9 Torr. After the reduction in the substrate
temperature down to 100◦C, we co-evaporated Co, Fe, and Si with stoichiometric
chemical compositions by using Knudsen cells. During the growth, two dimensional
epitaxial growth was confirmed by observing reflection high energy electron diffraction
patterns. The formed epitaxial CFS films were characterized by means of cross sectional
transmission electron microscopy (TEM), nanobeam electron diffraction (ED), and 57Fe
conversion electron Mössbauer spectroscopy. From these detailed characterizations, we
have observed highly ordered L21 structures in the CFS layers[1].
Introducing the resistance area product for nonlocal spin signals
In the lateral spin valve consisting of two ferromagnetic wire bridged by a
nonmagnetic strip as shown in Figs. 2a and 2b, the nonlocal spin signal ΔRS based on a
one-dimensional
Here PF and PI are the bulk and interface spin polarizations of the ferromagnetic
electrode, respectively, and RSFinj , RSFdet and RSN are the spin resistances for the
ferromagnetic injector, detector and the nonmagnetic strip, respectively. Also, RSIinj and
RSIdet are the interface spin resistances for the injecting and detecting junctions,
rescpectively. d and _N are, respectively, the separation distance between the injector
and detector and the spin diffusion length for the nonmagnetic strip. The spin resistance
is defined as 2/((1−P2)S), where ,  and S are the resistivity, the spin diffusion
length and the effective cross section for the spin current, respectively. The interface
spin resistance RSI is defined by 2RA/((1 − PI2 )S), where RA is the resistance area
product. In the nonmagnetic strip with a long spin diffusion length over a few hundred
nanometer, S is given by the cross section of the strip. On the other hand, in the
ferromagnets with a short spin diffusion length less than 10 nm, S is given by the size of
the junction in contact with the nonmagnetic strip because the spin current abruptly
decays in the vicinity of the F/N interface[5]. In the conventional ferromagnetic metals,
RSF is much smaller than RSN because of the short spin diffusion length. However, RSF
increases with P and diverges at P = 1. In the present CFS/Cu LSV, RSF is equal to RSN
at P = 0.97. However, RSF for the CFS is sufficiently smaller than RSN for P < 0:9.
Moreover, RSI for CFS/Cu is also much smaller than RSN. When RSN ≫ RSF;RSI, the
above equation can be simplified as
By introducing the junction sizes (Sinj, Sdet and SN), this equation can be revised as
By defining ΔRSA as ΔRS(SinjSdet/SN), we obtain
In Eq. (4), the influence of the junction sizes on the spin signal can be normalized. Thus,
ΔRSA allows us to fairly evaluate the device performance of the lateral spin valve in
theohmic junctions with the conditions RSF, RSI ≪ RSN for various combinations of a
ferromagnetic metal and a nonmagnetic
Lateral spin valves with interface barrier or highly resistive ferromagnetic electrodes
The backflow of the spin current can also be reducded by inserting the tunnel
burrier at the F/N interface. Relatively large spin accumulations have been reported in
CoFe/Al2O3/Al, Py/Ag and Py/MgO/Ag junctions[6–9]. However, the generation
efficiency of the spin current is limited by the spin polarization of the spin injector.
Moreover, the interface barrier strongly reduces the diffusion of the spin current into
another ferromagnet, giving rise to a poor injection efficiency of the spin current. When
the resistivity of the spin injector is quite large, the similar situation can be realized,
resulting in the large spin accumulation. The large spin signal reported in Ref. [10]
mainly originates from its large resistivity of 420 Ω cm at 77 K, which is ten times
larger than that in the Heusler Co2FeAl film reported by other groups[11, 12]. We also
pointed out that a large interface resistance is intrinsically induced at the interface
between the transition ferromagnetic metal and aluminum [13] although the authors of
Ref. [10] assumed a transparent interface.
Estimation of interface resistance
The interface resistance was estimated by measuring the 4-terminal resistances
with local spin valve configuration. In this configuration, the total resistance consists of
the resistance of the Cu wire and the two interface resistances. Since the resistance of
the Cu wire can be estimated from the resistivity for Cu, we can roughly calculate the
interface resistance by subtracting the resistance for the Cu wire from that in the local
spin valve configuration. By using the relation that the difference in the resistance ΔR is
given by RAF/N(Sinj+Sdet), we can obtain RAF/N. For the CFS/Cu LSVs, ΔR was ∼ 30
mΩ, indicating RACFS/Cu∼ 1 fΩm2. Thus, the value of RACFS/Cu is much smaller than
CuCu, implying that the enhancement of the spin signal is caused by the high spin
polarization, not by the interface resistance.
Estimation of spin polarization
By fitting the experimental data on the distance dependences of the ΔRSA using
Eq. (4), we can estimate the spin polarization of the spin injector in the LSV systems.
For the Py/Cu LSVs, RPy/Cu is less than 0.1 fΩm2, much smaller than PyPy (0.75 fΩm2).
Thus, we can neglect the second term in the numerator of Eq. (3), then obtain PPy ∼ 0.3
at RT and 0.35 at 80 K, respectively with assuming Py;RT = 3 nm and Py;80K = 5 nm[14].
For the CFS/Cu LSVs, as described in the previous section, RCFS=Cu can be
approximately estimated as ∼1 fΩm2. Because of the following reasons, we assumed
that the spin diffusion length for CFS (CFS) is the same order of that for CFSA (CFSA),
which was reported as 2.2 nm at RT and 3 nm at 14 K in recent study of the vertical
magnetoresitance device[15,16]. The spin-diffusion length is proportional to the
magnitude of the spin-orbit interactions. Since the atomic number of Al is close to Si,
we expect that the magnitude of spin-orbit interaction in CFS should be almost same
order of CFSA. From these considerations, we expect that CFS is the same level or
shorter than CFSA. Since the use of the longer CFS prevents an overestimation of PCFS,
we use CFS;RT = 3 nm and CFS;80K = 4 nm, which are slightly longer than the values for
CFSA. We then obtained 0.56 at RT and 0.72 at 80 K with assuming PI = 0.5, which is
typical interface spin polarization between the Co-based alloy and Cu [17, 18].
Switching spin currents for CFS and Py
We roughly compare the switching current of CFS to that of Py. According to
the simple spin-transfer torque model in metallic junctions, the critical current is
proportional to the saturated magnetization MS and the damping constant _. From VSM
measurement, the MS for our CFS is approximately 1000 emu/cc, slightly larger than
that for Py (860 emu/cc). The  values for our CFS and Py are found to be 0.008 and
0.013, respectively, from the FMR measurements. Therefore, the product of MS and 
for CFS is smaller than that for Py. This implies that the critical current for the
magnetization switching is smaller than that for Py. Therefore, assuming the same
switching current both in CFS and Py provides a proper evaluation of the power
consumption.
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