RIKEN Accel. Prog. Rep. 50 (2017)

Ⅲ-1. Atomic & Solid State Physics (Ion)

N b/rare-earth bilayers:
RKKY systems in proximity to an s -wave superconductor
H. Yamazaki1We have been studying the superconducting proximity
effects in Nb/rare-earth (RE) bilayers, where RE indicates
Gd, Tb, Dy, or Ho (the first four heavy RE elements in theperiodic table). The RKKY interaction is common
mechanism responsible for the magnetic structures in theRE elements. The Nb/RE bilayers can thus be regarded as
RKKY systems in proximity to an s-wave superconductor(Nb). Recent theoretical studies showed that a new class ofsuperconducting systems based on RKKY interactions
exhibits topological superconductivity and hosts Majoranafermions. This result may unexpectedly shed light on theunderstanding of the superconducting proximity effects inNb/RE bilayers.
The details of the sample preparation have been reported
in the previous APR. The superconducting transition
temperature Tc of each sample was measured, and weperformed a periodicity analysis on the Tc(tRE) data using aquantitative fast Fourier transform (FFT) method, where tRE
is the thickness of the RE layer. In Fig. 1, the vertical bars
indicate the periods of the oscillation components in Tc(tRE).
With the exception of the longest period (~3.5 nm) for Gd,
two types of variations are confirmed in the element
dependence of the period. Here, we refer to the longerperiods (more than 1 nm) as λL and the shorter ones (about1 nm) as λS. The spin modulation period intrinsic to Ho,
λspinHo(=3.4 nm; closed circle: ref. 1), is located within abroad distribution of λL for Ho. We identify a linear changein λL (shown as a broken line) from Gd to Dy. The line isextrapolated to Ho, giving an extrapolated value of 2.45 nm.
The distribution of λL for Ho can be explained by the twounresolved peaks at 3.4 and 2.45 nm.

According to the picture of the RKKY interaction
between conduction electrons and 4f moments, theexchange energy Eex at 0 K scales linearly with the 4f spin
moment S, where S=7, 6, 5, and 4 μB for Gd, Tb, Dy, andHo, respectively. The spatial period of the FFLO-likeoscillation, λFFLO, in the REs therefore increases from Gd toHo, as long as λFFLO=π  vF/2Eex holds and the Fermi
velocity vF is almost unchanged for the REs. The brokenline actually suggests that λL increases as S(∝Eex) decreases.
Further, the values of λL for Ho and Gd are consistent withthe literature data for λFFLO.1, 2) We infer that the broken line
indicates the element dependence of λFFLO.
The Fermi wavelength λF of each RE was calculatedfrom λFFLO (the broken line) and the experimental values ofEex for the Δ2 valence states3) by using the above equationand vF=2π  /meλF (me: electron mass). Note that there islittle experimental data of vF and λF for REs to date. Ourcalculations show that vF=1.25, 1.46, 1.33, and 1.36 (×106m/s) for Gd, Tb, Dy, and Ho, respectively. These values areclose to those predicted by the nearly free-electron model.
The closed squares in Fig. 1 show the calculated results of(λF/2)aliasing, i.e., the λF/2 aliased by discrete-thickness(ΔtRE=0.4 nm) sampling. We recognize that the values of(λF/2)aliasing reproduce λS values very well. Naively,
therefore, λS reflects the formation of quantum well statesin the RE layer, as observed in the superconducting Pbfilm.4)
The above discussion is simple, but provides a reasonable
explanation for the two oscillation components λL and λS
regardless of the RE. When we recall the intricate Fermi
surface (FS) of RE as well as of Nb, however, the basis forthe free-electron-like aspect is not obvious. The Fermi wavevector mismatch between two materials, in general, has alarge influence on the proximity effect, and the FS which
contributes to the proximity effect is mainly determined by
the smaller one among the two materials.5) In our case, itwould be difficult to extract a spherical-like FS from theintricate FSs of RE and Nb. We also have to explain the useof the bare electron mass instead of the effective mass forme, since the mass enhancement of the conduction electrons
due to electron-phonon interaction has been calculated forbulk RE metals.6) This issue deserves further consideration.
ReferencesFig. 1. Summary of the FFT analysis: periods of theoscillation components in Tc(tRE) for Gd, Tb, Dy, and Ho.
The closed circle (λspinHo) shows literature data.1)
11) F. Chiodi et al., Europhys. Lett. 101, 37002 (2013).
2) J. S. Jiang et al., Phys. Rev. Lett. 74, 314 (1995).
3) C. Schüßler-Langeheine et al., Phys. Rev. Lett. 84, 5624 (2000).
4) Y. Guo et al., Science 306, 1915 (2004).
5) Y. Tanaka, M. Tsukada, Phys. Rev. B 42, 2066 (1990).
6) H. L. Skriver, I. Mertig, Phys. Rev. B 41, 6553 (1990).

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