text: 5 Next, we discuss the case where a ﬂat or nearly ﬂat- on the RSOC λ, the ﬂat-band ZESABS splits into two band ZESABS appears for a suﬃciently large λ value [Figs. 9(b), (f)]. The LDOS in the DN still has a ZEP [Figs. 6, 7 , and 8]. Owing to the rotational symmetry [Fig. 9(j)]. For λ/∆ = 1 and 50, a helical edge mode is 0 of the spin for the p (x)-wave case, at λ = 0, Figs. 6(a), generated [Figs. 9(c) and (d)]. Owing to this helical edge x (e), and (i) are the same as Figs. 3(a), (e), and (i). For mode, the LDOS at the surface of the SC at E = 0 has a λ/∆ = 0.1, ZESABS splits into two [Fig. 6(b)]; however, nonzero value, as shown in Figs 9(g) and (h). Although 0 the splitting is very small compared to the p (z) case. the height of the ZEP of the LDOS is suppressed, the x The LDOS at the surface of the SC with λ/∆ = 0.1 LDOS in the DN has a ZEP [Figs 9(k) and (l)]. The 0 has a double peak structure [see the inset in Fig. 6(f)]. anomalous proximity eﬀect is suppressed by the RSOC; Thus, the height of the ZEP of the LDOS in the DN is however, it is not completely destroyed in contrast to the suppressed [Fig. 6(j)]. For λ/∆ = 1, the bulk energy gap case where the d-vector is parallel with the z-direction, 0 closes only at k = 0 [Fig. 6(c)], and the LDOS at the as shown in Fig 3. y surface of the SC at E = 0 is greatly suppressed, as shown For the case of the chiral p(x)-wave [Fig. 10] without in Fig. 6 (g). Consequently, the height of the ZEP for the RSOC, Figs. 10(a), (e), and (i) are the same to those LDOS in the DN is also suppressed, as shown in Fig. 6(k). in the chiral p(y) and chiral p(z) cases. The momentum However, for λ/∆0 = 50, a Fermi surface splitting occurs resolved LDOS≥for λ = 0 and λ/∆0 = 0.1 are similar, as indicated by the arrows in Fig. 6(d). E| ve|ntually, a but for λ/∆0 1 the bulk energy gap closes at ky = 0 ﬂat-band ZESABS appears between k1 < ky < k2 [k1 [Figs. 10(c) and (d)]. The LDOS at the surface of th≤e SC and k2 are shown in Fig. 6(d)]. Reﬂecting this ﬂat-band is nearly constant as a function of energy for λ/∆0 0.1 ZESABS, the LDOS has a ZEP at the surface of the SC [Figs. 10(e) and (f)], and a double peak structure with ≥ [Fig. 6(h)]. It should be noted that the LDOS in the DN zero-energy minimum appears for λ/∆ 1 [Figs. 10(g) 0 has a sharp ZEP [Fig. 6(l)]. and (h)]. The ZESABS for λ/∆ = 0 and 0.1 is located 0 For the helical p -wave case, helical edge modes exist, only at k = 0. However, for λ/∆ = 1 and 50, it is also II y ± 0 as shown in Figs. 7 (a)–(c). These edge modes are unaf- located at k = 0 and k = k [Figs. 10(a)–(d)]. The ≤ y y 1 fected by λ for λ ∆ . It should be noted that although LDOS in the DN always has a ZEP and its height is in 0 a SABS exists, the resulting LDOS at the surface of the the same order, independent of λ [Figs. 10(i)–(l)]. SC has a gap-like structure. As shown is Figs. 7(e), (f), Last group includes helical f- and helical p -waves. I and (g), the line shape of the LDOS is nearly the same These two types of pairings have helical edge modes at and independent of λ. The resulting LDOS in the DN λ = 0 and the direction of the d-vector is parallel to that has a broad ZEP, as shown in Figs. 7(i), (j), and (k). of the RSOC. The p -wave is a typical helical p-wave pair- I For λ/∆ = 50, a Fermi surface splitting occurs, as indi- ing realized in the thin ﬁlm of the Helium B-phase and 0 cated by the arrows in Fig. 7(d). A ﬂat-band ZESABS it belongs to the D-III class79. As shown in Figs 11(a)– | | appears between k < k < k [k and k are shown (d) and Figs 12(a)–(d), the line shape of this helical edge 1 y 2 1 2 in Fig. 7(d)]. Owing to these ﬂat-bands, the LDOS at mode is practically unaﬀected by the RSOC, as the di- the surface of the SC has a ZEP [Fig. 7(h)]. The result- rection of the d-vector and that of the RSOC is parallel ing height of the ZEP of the LDOS in the DN [Fig. 7(l)] with each other. In addition, the LDOS at the surface ≤ becomes larger than that for λ/∆ 1. of the SC [Figs. 11(e)–(h) and Figs. 12(e)–(h)] and the 0 The qualitative features of the chiral p(y)-wave case are LDOS in the DN [Figs. 11(i)–(l) and Figs. 12(i)–(l)] does slightly diﬀerent from those for the p (x) and the helical not change qualitatively by λ. x p cases. For λ = 0, owing to the spin-rotational symme- In order to understand the anomalous proximity ef- II try, the momentum resolved LDOS, shown in Fig. 8(a), fect in more detail, we calculated the odd-frequency spin- (e), and (i), is the same as in Figs. 5(a), (e) ,and (i), re- triplet even-parity (OTE) s-wave pair amplitude, shown spectively. With the increase of λ, the chiral edge mode in Figs. 13–18. For the p (z)- and f(z)-wave supercon- x splits into two and the one of the slope of it is reduced, ductor junctions, the magnitude of the OTE s-wave pair as shown in Figs. 8(b), (c), and (d). For λ/∆ = 50 amplitude, which is very large at λ = 0, decreases with − 0 the slope of the SABS from k to k [k is shown in the increase of λ. Its value is seriously suppressed for 1 1 1 Fig. 8(d)] is nearly zero and the zero-energy LDOS at λ/∆ = 50 where the ZEP in the DN disappears, as 0 the surface of the SC greatly increases [Fig. 8(h)]. The shown in Figs. 3(i)–(l) and Figs. 4(i)–(l). resulting LDOS in the DN has a clear ZEP, as shown in In the case of the p (x)-wave superconductor junction x Fig. 8(l). In these three cases, the anomalous proximity [Fig. 15], the OTE s-wave pair amplitude decreases with eﬀect is prominent for large RSOC values. λ; however, its value at λ/∆ = 50 is larger than that for 0 Next, we discuss the group of those pairings, which the px(z) junction. have helical or chiral edge modes for large RSOC val- For the helical p -wave superconductor junction [Fig. 16], II ues. The resulting anomalous proximity eﬀect is not very the OTE s-wave pair amplitude increases as λ increases, signiﬁcant in this limit. First, we focus on the p (y)- which is consistent with the height of the ZEP shown in x wave case. For λ = 0, due to the spin-rotational sym- Figs. 7(i)–(l). For the chiral p(y)-wave case [Fig. 16], the metry, Figs. 9(a), (e), and (i) are the same as those of OTE s-wave pair amplitude also increases as λ increases, the cases for p (x)- and p (z)-waves. After switching and its value at λ/∆ = 50 is much larger than that for x x 0

text: Next,

text: we

text: discuss

text: the

text: case

text: where

text: a

text: ﬂat

text: or

text: nearly

text: ﬂat-

text: on

text: the

text: RSOC

text: λ,

text: the

text: ﬂat-band

text: ZESABS

text: splits

text: into

text: two

text: band

text: ZESABS

text: appears

text: for

text: a

text: suﬃciently

text: large

text: λ

text: value

text: [Figs.

text: (f)].

text: The

text: LDOS

text: in

text: the

text: DN

text: still

text: has

text: a

text: ZEP

text: [Figs.

text: and

text: Owing

text: to

text: the

text: rotational

text: symmetry

text: [Fig.

text: For

text: λ/∆

text: and

text: a

text: helical

text: edge

text: mode

text: is

text: of

text: the

text: spin

text: for

text: the

text: p

text: (x)-wave

text: case,

text: at

text: λ

text: Figs.

text: generated

text: [Figs.

text: and

text: (d)].

text: Owing

text: to

text: this

text: helical

text: edge

text: x

text: (e),

text: and

text: (i)

text: are

text: the

text: same

text: as

text: Figs.

text: (e),

text: and

text: (i).

text: For

text: mode,

text: the

text: LDOS

text: at

text: the

text: surface

text: of

text: the

text: SC

text: at

text: E

text: has

text: a

text: λ/∆

text: ZESABS

text: splits

text: into

text: two

text: [Fig.

text: however,

text: nonzero

text: value,

text: as

text: shown

text: in

text: Figs

text: and

text: (h).

text: Although

text: the

text: splitting

text: is

text: very

text: small

text: compared

text: to

text: the

text: p

text: (z)

text: case.

text: the

text: height

text: of

text: the

text: ZEP

text: of

text: the

text: LDOS

text: is

text: suppressed,

text: the

text: x

text: The

text: LDOS

text: at

text: the

text: surface

text: of

text: the

text: SC

text: with

text: λ/∆

text: LDOS

text: in

text: the

text: DN

text: has

text: a

text: ZEP

text: [Figs

text: and

text: (l)].

text: The

text: has

text: a

text: double

text: peak

text: structure

text: [see

text: the

text: inset

text: in

text: Fig.

text: anomalous

text: proximity

text: eﬀect

text: is

text: suppressed

text: by

text: the

text: RSOC;

text: Thus,

text: the

text: height

text: of

text: the

text: ZEP

text: of

text: the

text: LDOS

text: in

text: the

text: DN

text: is

text: however,

text: it

text: is

text: not

text: completely

text: destroyed

text: in

text: contrast

text: to

text: the

text: suppressed

text: [Fig.

text: For

text: λ/∆

text: the

text: bulk

text: energy

text: gap

text: case

text: where

text: the

text: d-vector

text: is

text: parallel

text: with

text: the

text: z-direction,

text: closes

text: only

text: at

text: k

text: [Fig.

text: and

text: the

text: LDOS

text: at

text: the

text: as

text: shown

text: in

text: Fig

text: y

text: surface

text: of

text: the

text: SC

text: at

text: E

text: is

text: greatly

text: suppressed,

text: as

text: shown

text: For

text: the

text: case

text: of

text: the

text: chiral

text: p(x)-wave

text: [Fig.

text: without

text: in

text: Fig.

text: (g).

text: Consequently,

text: the

text: height

text: of

text: the

text: ZEP

text: for

text: the

text: RSOC,

text: Figs.

text: (e),

text: and

text: (i)

text: are

text: the

text: same

text: to

text: those

text: LDOS

text: in

text: the

text: DN

text: is

text: also

text: suppressed,

text: as

text: shown

text: in

text: Fig.

text: in

text: the

text: chiral

text: p(y)

text: and

text: chiral

text: p(z)

text: cases.

text: The

text: momentum

text: However,

text: for

text: λ/∆0

text: a

text: Fermi

text: surface

text: splitting

text: occurs

text: resolved

text: LDOS≥for

text: λ

text: and

text: λ/∆0

text: are

text: similar,

text: as

text: indicated

text: by

text: the

text: arrows

text: in

text: Fig.

text: E|

text: ve|ntually,

text: a

text: but

text: for

text: λ/∆0

text: the

text: bulk

text: energy

text: gap

text: closes

text: at

text: ky

text: ﬂat-band

text: ZESABS

text: appears

text: between

text: ky

text: [Figs.

text: and

text: (d)].

text: The

text: LDOS

text: at

text: the

text: surface

text: of

text: th≤e

text: SC

text: and

text: are

text: shown

text: in

text: Fig.

text: Reﬂecting

text: this

text: ﬂat-band

text: is

text: nearly

text: constant

text: as

text: a

text: function

text: of

text: energy

text: for

text: λ/∆0

text: ZESABS,

text: the

text: LDOS

text: has

text: a

text: ZEP

text: at

text: the

text: surface

text: of

text: the

text: SC

text: [Figs.

text: and

text: (f)],

text: and

text: a

text: double

text: peak

text: structure

text: with

text: [Fig.

text: It

text: should

text: be

text: noted

text: that

text: the

text: LDOS

text: in

text: the

text: DN

text: zero-energy

text: minimum

text: appears

text: for

text: λ/∆

text: [Figs.

text: has

text: a

text: sharp

text: ZEP

text: [Fig.

text: and

text: (h)].

text: The

text: ZESABS

text: for

text: λ/∆

text: and

text: is

text: located

text: For

text: the

text: helical

text: p

text: -wave

text: case,

text: helical

text: edge

text: modes

text: exist,

text: only

text: at

text: k

text: However,

text: for

text: λ/∆

text: and

text: it

text: is

text: also

text: II

text: y

text: as

text: shown

text: in

text: Figs.

text: (a)–(c).

text: These

text: edge

text: modes

text: are

text: unaf-

text: located

text: at

text: k

text: and

text: k

text: k

text: [Figs.

text: The

text: y

text: y

text: fected

text: by

text: λ

text: for

text: λ

text: It

text: should

text: be

text: noted

text: that

text: although

text: LDOS

text: in

text: the

text: DN

text: always

text: has

text: a

text: ZEP

text: and

text: its

text: height

text: is

text: in

text: a

text: SABS

text: exists,

text: the

text: resulting

text: LDOS

text: at

text: the

text: surface

text: of

text: the

text: the

text: same

text: order,

text: independent

text: of

text: λ

text: [Figs.

text: SC

text: has

text: a

text: gap-like

text: structure.

text: As

text: shown

text: is

text: Figs.

text: (f),

text: Last

text: group

text: includes

text: helical

text: f-

text: and

text: helical

text: p

text: -waves.

text: I

text: and

text: (g),

text: the

text: line

text: shape

text: of

text: the

text: LDOS

text: is

text: nearly

text: the

text: same

text: These

text: two

text: types

text: of

text: pairings

text: have

text: helical

text: edge

text: modes

text: at

text: and

text: independent

text: of

text: λ.

text: The

text: resulting

text: LDOS

text: in

text: the

text: DN

text: λ

text: and

text: the

text: direction

text: of

text: the

text: d-vector

text: is

text: parallel

text: to

text: that

text: has

text: a

text: broad

text: ZEP,

text: as

text: shown

text: in

text: Figs.

text: (j),

text: and

text: (k).

text: of

text: the

text: RSOC.

text: The

text: p

text: -wave

text: is

text: a

text: typical

text: helical

text: p-wave

text: pair-

text: I

text: For

text: λ/∆

text: a

text: Fermi

text: surface

text: splitting

text: occurs,

text: as

text: indi-

text: ing

text: realized

text: in

text: the

text: thin

text: ﬁlm

text: of

text: the

text: Helium

text: B-phase

text: and

text: cated

text: by

text: the

text: arrows

text: in

text: Fig.

text: A

text: ﬂat-band

text: ZESABS

text: it

text: belongs

text: to

text: the

text: D-III

text: As

text: shown

text: in

text: Figs

text: appears

text: between

text: k

text: k

text: k

text: [k

text: and

text: k

text: are

text: shown

text: (d)

text: and

text: Figs

text: the

text: line

text: shape

text: of

text: this

text: helical

text: edge

text: y

text: in

text: Fig.

text: Owing

text: to

text: these

text: ﬂat-bands,

text: the

text: LDOS

text: at

text: mode

text: is

text: practically

text: unaﬀected

text: by

text: the

text: RSOC,

text: as

text: the

text: di-

text: the

text: surface

text: of

text: the

text: SC

text: has

text: a

text: ZEP

text: [Fig.

text: The

text: result-

text: rection

text: of

text: the

text: d-vector

text: and

text: that

text: of

text: the

text: RSOC

text: is

text: parallel

text: ing

text: height

text: of

text: the

text: ZEP

text: of

text: the

text: LDOS

text: in

text: the

text: DN

text: [Fig.

text: with

text: each

text: other.

text: In

text: addition,

text: the

text: LDOS

text: at

text: the

text: surface

text: becomes

text: larger

text: than

text: that

text: for

text: λ/∆

text: of

text: the

text: SC

text: [Figs.

text: and

text: Figs.

text: and

text: the

text: The

text: qualitative

text: features

text: of

text: the

text: chiral

text: p(y)-wave

text: case

text: are

text: LDOS

text: in

text: the

text: DN

text: [Figs.

text: and

text: Figs.

text: does

text: slightly

text: diﬀerent

text: from

text: those

text: for

text: the

text: p

text: (x)

text: and

text: the

text: helical

text: not

text: change

text: qualitatively

text: by

text: λ.

text: x

text: p

text: cases.

text: For

text: λ

text: owing

text: to

text: the

text: spin-rotational

text: symme-

text: In

text: order

text: to

text: understand

text: the

text: anomalous

text: proximity

text: ef-

text: II

text: try,

text: the

text: momentum

text: resolved

text: LDOS,

text: shown

text: in

text: Fig.

text: fect

text: in

text: more

text: detail,

text: we

text: calculated

text: the

text: odd-frequency

text: spin-

text: (e),

text: and

text: (i),

text: is

text: the

text: same

text: as

text: in

text: Figs.

text: (e)

text: ,and

text: (i),

text: re-

text: triplet

text: even-parity

text: (OTE)

text: s-wave

text: pair

text: amplitude,

text: shown

text: spectively.

text: With

text: the

text: increase

text: of

text: λ,

text: the

text: chiral

text: edge

text: mode

text: in

text: Figs.

text: For

text: the

text: p

text: (z)-

text: and

text: f(z)-wave

text: supercon-

text: x

text: splits

text: into

text: two

text: and

text: the

text: one

text: of

text: the

text: slope

text: of

text: it

text: is

text: reduced,

text: ductor

text: junctions,

text: the

text: magnitude

text: of

text: the

text: OTE

text: s-wave

text: pair

text: as

text: shown

text: in

text: Figs.

text: (c),

text: and

text: (d).

text: For

text: λ/∆

text: amplitude,

text: which

text: is

text: very

text: large

text: at

text: λ

text: decreases

text: with

text: the

text: slope

text: of

text: the

text: SABS

text: from

text: k

text: to

text: k

text: [k

text: is

text: shown

text: in

text: the

text: increase

text: of

text: λ.

text: Its

text: value

text: is

text: seriously

text: suppressed

text: for

text: Fig.

text: is

text: nearly

text: zero

text: and

text: the

text: zero-energy

text: LDOS

text: at

text: λ/∆

text: where

text: the

text: ZEP

text: in

text: the

text: DN

text: disappears,

text: as

text: the

text: surface

text: of

text: the

text: SC

text: greatly

text: increases

text: [Fig.

text: The

text: shown

text: in

text: Figs.

text: and

text: Figs.

text: resulting

text: LDOS

text: in

text: the

text: DN

text: has

text: a

text: clear

text: ZEP,

text: as

text: shown

text: in

text: In

text: the

text: case

text: of

text: the

text: p

text: (x)-wave

text: superconductor

text: junction

text: x

text: Fig.

text: In

text: these

text: three

text: cases,

text: the

text: anomalous

text: proximity

text: [Fig.

text: the

text: OTE

text: s-wave

text: pair

text: amplitude

text: decreases

text: with

text: eﬀect

text: is

text: prominent

text: for

text: large

text: RSOC

text: values.

text: λ;

text: however,

text: its

text: value

text: at

text: λ/∆

text: is

text: larger

text: than

text: that

text: for

text: Next,

text: we

text: discuss

text: the

text: group

text: of

text: those

text: pairings,

text: which

text: the

text: px(z)

text: junction.

text: have

text: helical

text: or

text: chiral

text: edge

text: modes

text: for

text: large

text: RSOC

text: val-

text: For

text: the

text: helical

text: p

text: -wave

text: superconductor

text: junction

text: [Fig.

text: II

text: ues.

text: The

text: resulting

text: anomalous

text: proximity

text: eﬀect

text: is

text: not

text: very

text: the

text: OTE

text: s-wave

text: pair

text: amplitude

text: increases

text: as

text: λ

text: increases,

text: signiﬁcant

text: in

text: this

text: limit.

text: First,

text: we

text: focus

text: on

text: the

text: p

text: (y)-

text: which

text: is

text: consistent

text: with

text: the

text: height

text: of

text: the

text: ZEP

text: shown

text: in

text: x

text: wave

text: case.

text: For

text: λ

text: due

text: to

text: the

text: spin-rotational

text: sym-

text: Figs.

text: For

text: the

text: chiral

text: p(y)-wave

text: case

text: [Fig.

text: the

text: metry,

text: Figs.

text: (e),

text: and

text: (i)

text: are

text: the

text: same

text: as

text: those

text: of

text: OTE

text: s-wave

text: pair

text: amplitude

text: also

text: increases

text: as

text: λ

text: increases,

text: the

text: cases

text: for

text: p

text: (x)-

text: and

text: p

text: (z)-waves.

text: After

text: switching

text: and

text: its

text: value

text: at

text: λ/∆

text: is

text: much

text: larger

text: than

text: that

text: for

text: x

text: x


text: 6 − − the helical p -wave case. The p (y)-wave case is similar (cos k cos k )(sin k eˆ sin k eˆ ), respectively. This II x x y y 1 x 2 to that of the p (x)-wave case. The magnitude of OTE helical p -wave pair potential is proposed at the oxide x I s-wave pair amplitude decreases as λ increases. interfaces68,82 where a Majorana fermion appears, when B. Spin-singlet case a Fermi surface splitting occurs83,84. We have shown that the helical edge modes are quite robust against the In this subsection, the spin-singlet d -wave supercon- RSOC. It is remarkable that we can expect anomalous xy ductor case is presented. Firstly, the momentum resolved proximity eﬀect in interface-induced superconducting LDOS [Figs. 19(a)–(d)] is discussed. There is ﬂat-band systems. ZESABS, independent of the amplitude of λ, as it is It should be also noted that the anomalous proxim- known that the ZESABS generated from an even-parity ity eﬀect is switched on by the RSOC in spin-singlet pair potential is robust against the RSOC17,74. It can be dxy-wave superconductor junctions. The resulting zero- seen at λ/∆ = 50, that the Fermi surface splits into two, energy LDOS and the magnitude of the odd-frequency 0 as indicated by the arrows in Fig. 19(d), and a ﬂat-band spin-triplet s-wave pair amplitude is increased with the ZESABS exists as well. At λ = 0, the LDOS at the sur- increase of the magnitude of λ. This indicates that high face of the dxy-superconductor has a ZEP80 [Fig. 19(e)] TC cuprate junctions can be used to detect the anomalous similar to the p (i)-wave case (i = x, y.z). In the DN re- proximity eﬀect. Our obtained results provide a guide to x gion, the LDOS is nearly constant as a function of E and detect the anomalous proximity eﬀect in unconventional x [Fig. 19(i)]. It is known from the theory of proximity superconductor junctions and topological superconduc- eﬀects, that the LDOS is reduced to that in the normal tors. state and the proximity eﬀect is absent73,81; thus, there is In this paper, we assume that a Rashba spin–orbit no anomalous proximity eﬀect. The slight energy depen- coupling exists in the two-dimensional superconductor dence is due to the ﬁnite size eﬀect. A similar behavior uniformly along the z-direction, as superconductor is as- also appears for λ/∆ = 0.1 and λ/∆ = 1. However, sumed to be thin. In addition to this eﬀect, spin–orbit 0 0 for large RSOC values (λ/∆ = 50), the LDOS has a coupling also exists near the interface, on the plane par- 0 ZEP in the DN region [Fig. 19(l)], as the OTE s-wave allel with the interface85,86. In this case, the direction of pair amplitude becomes suﬃciently large, as shown in the Rashba vector is diﬀerent. It is also interesting to Fig. 20. study this type of spin–orbit coupling. IV. SUMMARY In this paper, we focus on the localized odd-frequency pairing accompanied by the SABS, which is generated In this paper, we have studied the proximity eﬀect by the symmetry breaking from the bulk conventional in the DN/superconductor junctions, focusing on the even-frequency pair potential. On the othe hand, sev- anomalous proximity eﬀect, where the LDOS in the DN eral studies have been pursuing the realization of a bulk has a zero-energy peak. For the spin-triplet superconduc- odd-frequency superconductor so far18,87–104. Although tor cases, when the direction of the d-vector is parallel it is diﬃcult to express the bulk odd-frequency supercon- with the z-direction, the ZESABS is suppressed by the ducting states consistent with various conditions105,106, RSOC λ and the anomalous proximity eﬀect disappears a recently proposed odd-frequency gap function in the for large λ values, as shown by the p (z)-, chiral p(z)-, two channel Kodo lattice model (TCKL) is a promis- x and f(z)-wave superconductor junctions, where the d- ing system107. The calculation of charge transport and vector is given by d(k) = sin k eˆ , d(k) = (sin k + Josephson eﬀect has already started108. It is interesting x 3− x i sin k )eˆ , and d(k) = sin k (cos k cos k )eˆ , respec- to study proximity eﬀects in the DN/TCKL system. y 3 x x y 3 tively. If the d-vector is on the xy-plane, the condition ACKNOWLEDGMENTS becomes more complicated. An additional ZESABS ap- pears for large λ values for p (x)-, chiral p(y)-, and helical Y. Asano and S. Kobayashi are thanked for useful x p -wave superconductor junctions, where the d-vector discussions. This work was supported by a Grant-in- II is given by d(k) = sin k eˆ , d(k) = (sin k + i sin k )eˆ , Aid for Scientiﬁc Research on Innovative Areas Topo- x 1 − x y 2 and d(k) = sin k sin k (sin k eˆ sin k eˆ ), respectively. logical Material Science JPSJ KAKENHI (Grants No. x y y 1 x 2 Another interesting case is the helical p - and heli- JP15H05851,No. JP15H05853, and No. JP15K21717), I cal f-wave superconductor junction where the d-vector and a Grant-in-Aid for Scientiﬁc Research B (Grant No. − is given by d(k) = (sin k eˆ sin k eˆ ) and d(k) = JP15H03686 and JP18H01176). y 1 x 2 ∗ 4 S. Kashiwaya and Y. Tanaka, Rep. Prog. Phys. 63, 1641 (2000). † 5 T. Lo¨fwander, V. S. Shumeiko, and G. Wendin, Supercond. Sci. Technol. 14, R53 (2001). 6 Y. Tanaka and S. Kashiwaya, Phys. Rev. Lett. 74, 3451 (1995). 1 L. J. Buchholtz and G. Zwicknagl, Phys. Rev. B 23, 5788 (19817).M. Covington, M. Aprili, E. Paraoanu, L. H. Greene, F. Xu, J. Zhu, and C. A. Mirkin, Phys. Rev. Lett. 79, 277 (1997). 2 J. Hara and K. Nagai, Prog. Theor. Phys. 76, 1237 (1986). 8 L. Alﬀ, H. Takashima, S. Kashiwaya, N. Terada, H. Ihara, Y. Tanaka, M. Koyanagi, and K. Kajimura, Phys. Rev. B 55, 3 C. R. Hu, Phys. Rev. Lett. 72, 1526 (1994). R14757 (1997). ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: the

text: helical

text: p

text: -wave

text: case.

text: The

text: p

text: (y)-wave

text: case

text: is

text: similar

text: (cos

text: k

text: cos

text: k

text: )(sin

text: k

text: eˆ

text: sin

text: k

text: eˆ

text: respectively.

text: This

text: II

text: x

text: x

text: y

text: y

text: x

text: to

text: that

text: of

text: the

text: p

text: (x)-wave

text: case.

text: The

text: magnitude

text: of

text: OTE

text: helical

text: p

text: -wave

text: pair

text: potential

text: is

text: proposed

text: at

text: the

text: oxide

text: x

text: I

text: s-wave

text: pair

text: amplitude

text: decreases

text: as

text: λ

text: increases.

text: where

text: a

text: Majorana

text: fermion

text: appears,

text: when

text: B.

label: section

text: Spin-singlet

text: case

text: a

text: Fermi

text: surface

text: splitting

text: We

text: have

text: shown

text: that

text: the

text: helical

text: edge

text: modes

text: are

text: quite

text: robust

text: against

text: the

text: In

text: this

text: subsection,

text: the

text: spin-singlet

text: d

text: -wave

text: supercon-

text: RSOC.

text: It

text: is

text: remarkable

text: that

text: we

text: can

text: expect

text: anomalous

text: xy

text: ductor

text: case

text: is

text: presented.

text: Firstly,

text: the

text: momentum

text: resolved

text: proximity

text: eﬀect

text: in

text: interface-induced

text: superconducting

text: LDOS

text: [Figs.

text: is

text: discussed.

text: There

text: is

text: ﬂat-band

text: systems.

text: ZESABS,

text: independent

text: of

text: the

text: amplitude

text: of

text: λ,

text: as

text: it

text: is

text: It

text: should

text: be

text: also

text: noted

text: that

text: the

text: anomalous

text: proxim-

text: known

text: that

text: the

text: ZESABS

text: generated

text: from

text: an

text: even-parity

text: ity

text: eﬀect

text: is

text: switched

text: on

text: by

text: the

text: RSOC

text: in

text: spin-singlet

text: pair

text: potential

text: is

text: robust

text: against

text: the

text: It

text: can

text: be

text: dxy-wave

text: superconductor

text: junctions.

text: The

text: resulting

text: zero-

text: seen

text: at

text: λ/∆

text: that

text: the

text: Fermi

text: surface

text: splits

text: into

text: two,

text: energy

text: LDOS

text: and

text: the

text: magnitude

text: of

text: the

text: odd-frequency

text: as

text: indicated

text: by

text: the

text: arrows

text: in

text: Fig.

text: and

text: a

text: ﬂat-band

text: spin-triplet

text: s-wave

text: pair

text: amplitude

text: is

text: increased

text: with

text: the

text: ZESABS

text: exists

text: as

text: well.

text: At

text: λ

text: the

text: LDOS

text: at

text: the

text: sur-

text: increase

text: of

text: the

text: magnitude

text: of

text: λ.

text: This

text: indicates

text: that

text: high

text: face

text: of

text: the

text: dxy-superconductor

text: has

text: a

text: [Fig.

text: TC

text: cuprate

text: junctions

text: can

text: be

text: used

text: to

text: detect

text: the

text: anomalous

text: similar

text: to

text: the

text: p

text: (i)-wave

text: case

text: (i

text: x,

text: y.z).

text: In

text: the

text: DN

text: re-

text: proximity

text: eﬀect.

text: Our

text: obtained

text: results

text: provide

text: a

text: guide

text: to

text: x

text: gion,

text: the

text: LDOS

text: is

text: nearly

text: constant

text: as

text: a

text: function

text: of

text: E

text: and

text: detect

text: the

text: anomalous

text: proximity

text: eﬀect

text: in

text: unconventional

text: x

text: [Fig.

text: It

text: is

text: known

text: from

text: the

text: theory

text: of

text: proximity

text: superconductor

text: junctions

text: and

text: topological

text: superconduc-

text: eﬀects,

text: that

text: the

text: LDOS

text: is

text: reduced

text: to

text: that

text: in

text: the

text: normal

text: tors.

text: state

text: and

text: the

text: proximity

text: eﬀect

text: is

text: thus,

text: there

text: is

text: In

text: this

text: paper,

text: we

text: assume

text: that

text: a

text: Rashba

text: spin–orbit

text: no

text: anomalous

text: proximity

text: eﬀect.

text: The

text: slight

text: energy

text: depen-

text: coupling

text: exists

text: in

text: the

text: two-dimensional

text: superconductor

text: dence

text: is

text: due

text: to

text: the

text: ﬁnite

text: size

text: eﬀect.

text: A

text: similar

text: behavior

text: uniformly

text: along

text: the

text: z-direction,

text: as

text: superconductor

text: is

text: as-

text: also

text: appears

text: for

text: λ/∆

text: and

text: λ/∆

text: However,

text: sumed

text: to

text: be

text: thin.

text: In

text: addition

text: to

text: this

text: eﬀect,

text: spin–orbit

text: for

text: large

text: RSOC

text: values

text: (λ/∆

text: the

text: LDOS

text: has

text: a

text: coupling

text: also

text: exists

text: near

text: the

text: interface,

text: on

text: the

text: plane

text: par-

text: ZEP

text: in

text: the

text: DN

text: region

text: [Fig.

text: as

text: the

text: OTE

text: s-wave

text: allel

text: with

text: the

text: In

text: this

text: case,

text: the

text: direction

text: of

text: pair

text: amplitude

text: becomes

text: suﬃciently

text: large,

text: as

text: shown

text: in

text: the

text: Rashba

text: vector

text: is

text: diﬀerent.

text: It

text: is

text: also

text: interesting

text: to

text: Fig.

text: study

text: this

text: type

text: of

text: spin–orbit

text: coupling.

text: IV.

text: SUMMARY

text: In

text: this

text: paper,

text: we

text: focus

text: on

text: the

text: localized

text: odd-frequency

text: pairing

text: accompanied

text: by

text: the

text: SABS,

text: which

text: is

text: generated

text: In

text: this

text: paper,

text: we

text: have

text: studied

text: the

text: proximity

text: eﬀect

text: by

text: the

text: symmetry

text: breaking

text: from

text: the

text: bulk

text: conventional

text: in

text: the

text: DN/superconductor

text: junctions,

text: focusing

text: on

text: the

text: even-frequency

text: pair

text: potential.

text: On

text: the

text: othe

text: hand,

text: sev-

text: anomalous

text: proximity

text: eﬀect,

text: where

text: the

text: LDOS

text: in

text: the

text: DN

text: eral

text: studies

text: have

text: been

text: pursuing

text: the

text: realization

text: of

text: a

text: bulk

text: has

text: a

text: zero-energy

text: peak.

text: For

text: the

text: spin-triplet

text: superconduc-

text: odd-frequency

text: superconductor

text: so

text: Although

text: tor

text: cases,

text: when

text: the

text: direction

text: of

text: the

text: d-vector

text: is

text: parallel

text: it

text: is

text: diﬃcult

text: to

text: express

text: the

text: bulk

text: odd-frequency

text: supercon-

text: with

text: the

text: z-direction,

text: the

text: ZESABS

text: is

text: suppressed

text: by

text: the

text: ducting

text: states

text: consistent

text: with

text: various

text: RSOC

text: λ

text: and

text: the

text: anomalous

text: proximity

text: eﬀect

text: disappears

text: a

text: recently

text: proposed

text: odd-frequency

text: gap

text: function

text: in

text: the

text: for

text: large

text: λ

text: values,

text: as

text: shown

text: by

text: the

text: p

text: (z)-,

text: chiral

text: p(z)-,

text: two

text: channel

text: Kodo

text: lattice

text: model

text: (TCKL)

text: is

text: a

text: promis-

text: x

text: and

text: f(z)-wave

text: superconductor

text: junctions,

text: where

text: the

text: d-

text: ing

text: The

text: calculation

text: of

text: charge

text: transport

text: and

text: vector

text: is

text: given

text: by

text: d(k)

text: sin

text: k

text: eˆ

text: d(k)

text: (sin

text: k

text: Josephson

text: eﬀect

text: has

text: already

text: It

text: is

text: interesting

text: x

text: 3−

text: x

text: i

text: sin

text: k

text: )eˆ

text: and

text: d(k)

text: sin

text: k

text: (cos

text: k

text: cos

text: k

text: )eˆ

text: respec-

text: to

text: study

text: proximity

text: eﬀects

text: in

text: the

text: DN/TCKL

text: system.

text: y

text: x

text: x

text: y

text: tively.

text: If

text: the

text: d-vector

text: is

text: on

text: the

text: xy-plane,

text: the

text: condition

text: ACKNOWLEDGMENTS

text: becomes

text: more

text: complicated.

text: An

text: additional

text: ZESABS

text: ap-

text: pears

text: for

text: large

text: λ

text: values

text: for

text: p

text: (x)-,

text: chiral

text: p(y)-,

text: and

text: helical

text: Y.

text: Asano

text: and

text: S.

text: Kobayashi

text: are

text: thanked

text: for

text: useful

text: x

text: p

text: -wave

text: superconductor

text: junctions,

text: where

text: the

text: d-vector

text: discussions.

text: This

text: work

text: was

text: supported

text: by

text: a

text: Grant-in-

text: II

text: is

text: given

text: by

text: d(k)

text: sin

text: k

text: eˆ

text: d(k)

text: (sin

text: k

text: i

text: sin

text: k

text: )eˆ

text: Aid

text: for

text: Scientiﬁc

text: Research

text: on

text: Innovative

text: Areas

text: Topo-

text: x

text: x

text: y

text: and

text: d(k)

text: sin

text: k

text: sin

text: k

text: (sin

text: k

text: eˆ

text: sin

text: k

text: eˆ

text: respectively.

text: logical

text: Material

text: Science

text: JPSJ

text: KAKENHI

text: (Grants

text: No.

text: x

text: y

text: y

text: x

text: Another

text: interesting

text: case

text: is

text: the

text: helical

text: p

text: and

text: heli-

text: and

text: No.

text: I

text: cal

text: f-wave

text: superconductor

text: junction

text: where

text: the

text: d-vector

text: and

text: a

text: Grant-in-Aid

text: for

text: Scientiﬁc

text: Research

text: B

text: (Grant

text: No.

text: is

text: given

text: by

text: d(k)

text: (sin

text: k

text: eˆ

text: sin

text: k

text: eˆ

text: and

text: d(k)

text: and

text: y

text: x

text: S.

text: Kashiwaya

text: and

text: Y.

text: Tanaka,

text: Rep.

text: Prog.

text: Phys.

text: T.

text: Lo¨fwander,

text: V.

text: S.

text: Shumeiko,

text: and

text: G.

text: Wendin,

text: Supercond.

text: Sci.

text: Technol.

text: Y.

text: Tanaka

text: and

text: S.

text: Kashiwaya,

text: Phys.

text: Rev.

text: Lett.

text: L.

text: J.

text: Buchholtz

text: and

text: G.

text: Zwicknagl,

text: Phys.

text: Rev.

text: B

text: Covington,

text: M.

text: Aprili,

text: E.

text: Paraoanu,

text: L.

text: H.

text: Greene,

text: F.

text: Xu,

text: J.

text: Zhu,

text: and

text: C.

text: A.

text: Mirkin,

text: Phys.

text: Rev.

text: Lett.

text: J.

text: Hara

text: and

text: K.

text: Nagai,

text: Prog.

text: Theor.

text: Phys.

text: L.

text: Alﬀ,

text: H.

text: Takashima,

text: S.

text: Kashiwaya,

text: N.

text: Terada,

text: H.

text: Ihara,

text: Y.

text: Tanaka,

text: M.

text: Koyanagi,

text: and

text: K.

text: Kajimura,

text: Phys.

text: Rev.

text: B

text: C.

text: R.

text: Hu,

text: Phys.

text: Rev.

text: Lett.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: I. INTRODUCTION Magnetic skyrmions are nano-sized noncollinear magnetization conﬁgurations character- ized by the topological skyrmion number,1–3 which can exist in the noncentrosymmetric helimagnet with ﬁnite Dzyaloshinskii-Moriya interaction (DMI).4,5 Due to their topologi- cal stability and ﬂexible controllablity by spin-polarized current, magnetic skyrmions show great advantages as information carriers to develop spintronics devices,6 such as memories,7 logic gates,8 transistors,9 etc. For the successful operations of these skyrmion-based devices, the ﬁrst step is to create isolated magnetic skyrmions in a simple and controllable way. Physically, the generation of an isolated mgnetic skyrmion implies the topological transition of magnetization conﬁgurations, which is accompanied by overcoming a potential barrier. Up to date, several methods to create isolated magnetic skyrmions from the uniform fer- romagnetic(FM) states have been proposed and realized, including pulsed magnetic ﬁeld,10 spin wave,11 local heating,12, laser pulse,13 electric ﬁeld,14 and spin-polarized current.15–22 To avoid a brute-force destruction of the FM state, the eﬀect of geometry conﬁnement on the nucleation dynamics of magnetic skyrmions can be utilized.17 Furthermore, it has been shown that the isolated magnetic skyrmion can be converted from either a domain-wall(DW) pair23 or a chiral stripe domain24 by spin torque in a junction geometry. In this paper, we report a novel mechanism to continuously generate magnetic skyrmions in the T-shaped helimagnetic nanojunction based on micromagnetic simulations. When the nanojunction is driven far from the equilibrium by applying a spin-polarized current, the periodic magnetization dynamics can arise for certain simulation parameters, which is known as “temporal dissipative structure”.25 Since one magnetic skyrmion can be formed during each cycle of the dynamics, the magnetic skyrmions will be continuously generated by the temporal dissipative structure realized here. Unlike the previous mechanisms to generate magnetic skyrmions in the junction structure,23,24 where a DW pair or a stripe domain should be prepared in priori, the ingredients in our mechanism here only include the special geometry shape of the nanojunction and the suitable spin-polarized current. II. STRUCTURE AND METHOD The T-shaped helimagnetic nanojunction considered here is schematiclaly shown in Fig. 1, which consists of a narrow part and a wide part. The length and width of the wide(narrow) 2

text: I.

text: INTRODUCTION

text: Magnetic

text: skyrmions

text: are

text: nano-sized

text: noncollinear

text: magnetization

text: conﬁgurations

text: character-

text: ized

text: by

text: the

text: topological

text: skyrmion

text: which

text: can

text: exist

text: in

text: the

text: noncentrosymmetric

text: helimagnet

text: with

text: ﬁnite

text: Dzyaloshinskii-Moriya

text: interaction

text: Due

text: to

text: their

text: topologi-

text: cal

text: stability

text: and

text: ﬂexible

text: controllablity

text: by

text: spin-polarized

text: current,

text: magnetic

text: skyrmions

text: show

text: great

text: advantages

text: as

text: information

text: carriers

text: to

text: develop

text: spintronics

text: such

text: as

text: logic

text: etc.

text: For

text: the

text: successful

text: operations

text: of

text: these

text: skyrmion-based

text: devices,

text: the

text: ﬁrst

text: step

text: is

text: to

text: create

text: isolated

text: magnetic

text: skyrmions

text: in

text: a

text: simple

text: and

text: controllable

text: way.

text: Physically,

text: the

text: generation

text: of

text: an

text: isolated

text: mgnetic

text: skyrmion

text: implies

text: the

text: topological

text: transition

text: of

text: magnetization

text: conﬁgurations,

text: which

text: is

text: accompanied

text: by

text: overcoming

text: a

text: potential

text: barrier.

text: Up

text: to

text: date,

text: several

text: methods

text: to

text: create

text: isolated

text: magnetic

text: skyrmions

text: from

text: the

text: uniform

text: fer-

text: romagnetic(FM)

text: states

text: have

text: been

text: proposed

text: and

text: realized,

text: including

text: pulsed

text: magnetic

text: spin

text: local

text: laser

text: electric

text: and

text: spin-polarized

text: To

text: avoid

text: a

text: brute-force

text: destruction

text: of

text: the

text: FM

text: state,

text: the

text: eﬀect

text: of

text: geometry

text: conﬁnement

text: on

text: the

text: nucleation

text: dynamics

text: of

text: magnetic

text: skyrmions

text: can

text: be

text: Furthermore,

text: it

text: has

text: been

text: shown

text: that

text: the

text: isolated

text: magnetic

text: skyrmion

text: can

text: be

text: converted

text: from

text: either

text: a

text: domain-wall(DW)

text: or

text: a

text: chiral

text: stripe

text: by

text: spin

text: torque

text: in

text: a

text: junction

text: geometry.

text: In

text: this

text: paper,

text: we

text: report

text: a

text: novel

text: mechanism

text: to

text: continuously

text: generate

text: magnetic

text: skyrmions

text: in

text: the

text: T-shaped

text: helimagnetic

text: nanojunction

text: based

text: on

text: micromagnetic

text: simulations.

text: When

text: the

text: nanojunction

text: is

text: driven

text: far

text: from

text: the

text: equilibrium

text: by

text: applying

text: a

text: spin-polarized

text: current,

text: the

text: periodic

text: magnetization

text: dynamics

text: can

text: arise

text: for

text: certain

text: simulation

text: parameters,

text: which

text: is

text: known

text: as

text: “temporal

text: dissipative

text: Since

text: one

text: magnetic

text: skyrmion

text: can

text: be

text: formed

text: during

text: each

text: cycle

text: of

text: the

text: dynamics,

text: the

text: magnetic

text: skyrmions

text: will

text: be

text: continuously

text: generated

text: by

text: the

text: temporal

text: dissipative

text: structure

text: realized

text: here.

text: Unlike

text: the

text: previous

text: mechanisms

text: to

text: generate

text: magnetic

text: skyrmions

text: in

text: the

text: junction

text: where

text: a

text: DW

text: pair

text: or

text: a

text: stripe

text: domain

text: should

text: be

text: prepared

text: in

text: priori,

text: the

text: ingredients

text: in

text: our

text: mechanism

text: here

text: only

text: include

text: the

text: special

text: geometry

text: shape

text: of

text: the

text: nanojunction

text: and

text: the

text: suitable

text: spin-polarized

text: current.

text: II.

text: STRUCTURE

text: AND

text: METHOD

text: The

text: T-shaped

text: helimagnetic

text: nanojunction

text: considered

text: here

text: is

text: schematiclaly

text: shown

text: in

text: Fig.

text: which

text: consists

text: of

text: a

text: narrow

text: part

text: and

text: a

text: wide

text: part.

text: The

text: length

text: and

text: width

text: of

text: the

text: wide(narrow)


text: 8 X. Zhang, M. Ezawa, and Y. Zhou, Sci. Rep. 5, 9400 (2015). 9 X. Zhang, Y. Zhou, M. Ezawa, G. P. Zhao, and W. Zhao, Sci. Rep. 5, 11369 (2015). 10 O. Dzyapko, I. Lisenkov, P. Nowik-Boltyk, V. E. Demidov, S. O. Demokritov, B. Koene, A. Kir- ilyuk, T. Rasing, V. Tiberkevich, and A. Slavin, Phys. Rev. B 96, 064438(R) (2017). 11 Y. Liu, G. Yin, J. Zang, J. Shi, and R. K. Lake, Appl. Phys. Lett 107, 152411 (2015). 12 W. Koshibae and N. Nagaosa, Nat. Commun. 5, 5148 (2014). 13 M. Finazzi, M. Savoini, A. R. Khorsand, A. Tsukamoto, A. Itoh, L. Du`o, A. Kirilyuk, T. Rasing, and M. Ezawa, Phys. Rev. Lett. 110, 177205 (2013). 14 P.-J. Hsu, A. Kubetzka, A. Finco, N. Romming, K. von Bergmann, and R. Wiesendanger, Nat. Nanotechnol. 12, 123 (2017). 15 N. Romming, C. Hanneken, M. Menzel, J. E. Bickel, B. Wolter, K. von Bergmann, A. Kubetzka, and R. Wiesendanger, Science 341, 636 (2013). 16 J. Sampaio, V. Cros, S. Rohart, A. Thiaville, and A. Fert, Nat. Nanotechnol. 8, 839 (2013). 17 J. Iwasaki, M. Mochizuki, and N. Nagaosa, Nat. Nanotechnol. 8, 742 (2013). 18 Y. Tchoe and J. H. Han, Phys. Rev. B 85, 174416 (2012). 19 S. Lin, C. Reichhardt, and A. Saxena, Appl. Phys. Lett. 102, 222405 (2013). 20 G. Yin, Y. Li, L. Kong, R. K. Lake, C. L. Chien, and J. Zang, Phys. Rev. B 93, 174403 (2016). 21 H. Y. Yuan and X. R. Wang, Sci. Rep. 6, 34898 (2016). 22 W. Legrand, D. Maccariello, N. Reyren, K. Garcia, C. Moutaﬁs, C. Moreau-Luchaire, S. Collin, K. Bouze-houane, V. Cros, and A. Fert, Nano. Lett. 17, 2703 (2017). 23 Y. Zhou and M. Ezawa, Nat. Commun. 5, 4652 (2014). 24 W. Jiang, P. Upadhyaya, W. Zhang, G. Yu, M. B. Jungﬂeisch, F. Y. Fradin, J. E. Pearson, Y. Tserkovnyak, K. L. Wang, O. Heinonen, S. G. E. te Velthuis, and A. Hoﬀmann, Science 349, 283 (2015). 25 D. Kondepudi and I. Prigogine, Modern Thermodynamics:From Heat Engines to Dissipative Structures, 2nd ed. (Wiley, Chichester, United Kingdom, 2015). 26 A. V. Khvalkovskiy, V. Cros, D. Apalkov, V. Nikitin, M. Krounbi, K. A. Zvezdin, A. Anane, J. Grollier, and A. Fert, Phys. Rev. B 87, 020402 (R) (2013). 27 M. Donahue and D. Porter, OOMMF User’s Guide, Version 1.0 Interagency Report NISTIR 6376, (National Institute of Standards and Technology, Gaithersburg, MD, USA, 1999). 13

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text: te

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text: and

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text: Hoﬀmann,

text: Science

text: D.

text: Kondepudi

text: and

text: I.

text: Prigogine,

text: Modern

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text: Heat

text: Engines

text: to

text: Dissipative

text: Structures,

text: ed.

text: (Wiley,

text: Chichester,

text: United

text: Kingdom,

text: A.

text: V.

text: Khvalkovskiy,

text: V.

text: Cros,

text: D.

text: Apalkov,

text: V.

text: Nikitin,

text: M.

text: Krounbi,

text: K.

text: A.

text: Zvezdin,

text: A.

text: Anane,

text: J.

text: Grollier,

text: and

text: A.

text: Fert,

text: Phys.

text: Rev.

text: B

text: (R)

text: M.

text: Donahue

text: and

text: D.

text: Porter,

text: OOMMF

text: User’s

text: Guide,

text: Version

text: Interagency

text: Report

text: NISTIR

text: (National

text: Institute

text: of

text: Standards

text: and

text: Technology,

text: Gaithersburg,

text: MD,

text: USA,


text: (a) (b) 35 2.6 30 2.4 25 20 10nm 20nm 2.2 15 30nm 10 T- junction 2.0 5 1.8 0 -5 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8 3.4 3.6 3.8 4.0 4.2 4.4 (c) D(mJ/ m2) (d) D(mJ /m2) 130 130 3.6mJ/m2 125 3.8mJ/m2 125 4.0mJ/m2 120 4.2mJ/m2 120 115 115 110 110 1.85u(cid:20)(cid:19)(cid:20)(cid:21)A/m2 105 2.00u(cid:20)(cid:19)(cid:20)(cid:21)A/m2 105 2.15u(cid:20)(cid:19)(cid:20)(cid:21)A/m2 100 100 1.8 1.9 2.0 2.1 2.2 3.4 3.6 3.8 4.0 4.2 4.4 J(u101 2A/m2) D(mJ/m2) FIG. 3. (a) The D-J phase diagram. The yellow region gives the parameter space for which the continuous nucleation dynamics of magnetic skyrmions can happen. (b) The critical current densities to generate magnetic skyrmions in the current-perpendicular-to-plane (CPP) conﬁguraton and the T-shaped nanojunction (T-junction). Three diﬀerent diameters 10, 20, 30 nm have been set for the contact region in CPP conﬁguration. (c) The dependence of nucleation period T on the 0 injected current density J for diﬀerent DMI strength D. (d) The dependence of nucleation period T on the DMI strength D for diﬀerent injected current density J. 0 critical current density J to create a magnetic skyrmion in the wide region with the CPP c method. We see that J will gradually decrease when the DMI strength D is increased for c ﬁxed contact area. While for given D, the critical density J will signiﬁcantly decrease when c the diameter of the contact region is increased from 10 nm to 30 nm. For comparison, we also show the critical current density J to continuously create magnetic skrymions in the T- c1 shaped junction, which is lower than the critical current densities in CPP conﬁguration here. Therefore, the mechanism proposed here has the advantage to create magnetic skyrmions with lower current density. Based on the D-J phase diagram, we further investigate the eﬀect of current density J and the DMI strength D on the period T of the continuous nucleation dynamics in 0 8 )2m/A2101u(J )sp( T 0 )2m/A2101u(J )sp( T 0 ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: (a)

font: default

text: (b)

text: T-

text: junction

text: (c)

text: D(mJ/

text: (d)

text: D(mJ

text: FIG.

text: (a)

text: The

text: D-J

text: phase

text: diagram.

text: The

text: yellow

text: region

text: gives

text: the

text: parameter

text: space

text: for

text: which

text: the

text: continuous

text: nucleation

text: dynamics

text: of

text: magnetic

text: skyrmions

text: can

text: happen.

text: (b)

text: The

text: critical

text: current

text: densities

text: to

text: generate

text: magnetic

text: skyrmions

text: in

text: the

text: current-perpendicular-to-plane

text: (CPP)

text: conﬁguraton

text: and

text: the

text: T-shaped

text: nanojunction

text: (T-junction).

text: Three

text: diﬀerent

text: diameters

text: nm

text: have

text: been

text: set

text: for

text: the

text: contact

text: region

text: in

text: CPP

text: conﬁguration.

text: (c)

text: The

text: dependence

text: of

text: nucleation

text: period

text: T

text: on

text: the

text: injected

text: current

text: density

text: J

text: for

text: diﬀerent

text: DMI

text: strength

text: D.

text: (d)

text: The

text: dependence

text: of

text: nucleation

text: period

text: T

text: on

text: the

text: DMI

text: strength

text: D

text: for

text: diﬀerent

text: injected

text: current

text: density

text: J.

text: critical

text: current

text: density

text: J

text: to

text: create

text: a

text: magnetic

text: skyrmion

text: in

text: the

text: wide

text: region

text: with

text: the

text: CPP

text: c

text: method.

text: We

text: see

text: that

text: J

text: will

text: gradually

text: decrease

text: when

text: the

text: DMI

text: strength

text: D

text: is

text: increased

text: for

text: c

text: ﬁxed

text: contact

text: area.

text: While

text: for

text: given

text: D,

text: the

text: critical

text: density

text: J

text: will

text: signiﬁcantly

text: decrease

text: when

text: c

text: the

text: diameter

text: of

text: the

text: contact

text: region

text: is

text: increased

text: from

text: nm

text: to

text: nm.

text: For

text: comparison,

text: we

text: also

text: show

text: the

text: critical

text: current

text: density

text: J

text: to

text: continuously

text: create

text: magnetic

text: skrymions

text: in

text: the

text: T-

text: shaped

text: junction,

text: which

text: is

text: lower

text: than

text: the

text: critical

text: current

text: densities

text: in

text: CPP

text: conﬁguration

text: here.

text: Therefore,

text: the

text: mechanism

text: proposed

text: here

text: has

text: the

text: advantage

text: to

text: create

text: magnetic

text: skyrmions

text: with

text: lower

text: current

text: density.

text: Based

text: on

text: the

text: D-J

text: phase

text: diagram,

text: we

text: further

text: investigate

text: the

text: eﬀect

text: of

text: current

text: density

text: J

text: and

text: the

text: DMI

text: strength

text: D

text: on

text: the

text: period

text: T

text: of

text: the

text: continuous

text: nucleation

text: dynamics

text: in

text: )sp(

text: T

text: )sp(

text: T

text: ##LTFigure##

text: ##LTFigure##

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text: 27. Barés, J., Hattali, M. L., Dalmas, D. & Bonamy, D. Fluctuations of global energy release and crackling in nominally brittle heterogeneous fracture. Physical Review Letters 113, 264301 (2014). 28. Kanomori, H. The energy release in great earthquakes. Journal of Geophysical Research 82, 2981–2987 (1977). 29. van Stiphout, T., Zhuang, J. & Marsan, D. Seismicity declustering, community online resource for statistical seismicity analysis (corssa), doi: 10.5078/corssa-52382934 (2012). 30. Kagan, Y. Y. & Knopoff, L. Stochastic synthesis of earthquake catalogs. Journal of Geophys- ical Research: Solid Earth 86, 2853–2862 (1981). 31. Helmstetter, A. & Sornette, D. Bath’s law derived from the Gutenberg-Richter law and from aftershock properties. Geophysical Research Letters 30, 2069 (2003). 32. Luo, J. & Zhuang, J. Three regimes of the distribution of the largest event in the critical etas model. Bulletin of the Seismological Society of America 106, 1364–1369 (2016). 33. Corral, A. Universal earthquake-occurrence jumps, correlations with time, and anomalous diffusion. Physical Review Letters 97, 178501 (2006). 34. Stojanova, M., Santucci, S., Vanel, L. & Ramos, O. High frequency monitoring reveals after- shocks in subcritical crack growth. Physical Review Letters 112, 115502 (2014). 35. Kagan, Y. Y. Short term properties of earthquake catalogs and models of earthquake source. Bulletin of the Seismological Society of America 94, 1207–1228 (2004). 23

text: Barés,

text: J.,

text: Hattali,

text: M.

text: L.,

text: Dalmas,

text: D.

text: Bonamy,

text: D.

text: Fluctuations

text: of

text: global

text: energy

text: release

text: and

text: crackling

text: in

text: nominally

text: brittle

text: heterogeneous

text: fracture.

text: Physical

text: Review

text: Letters

text: Kanomori,

text: H.

text: The

text: energy

text: release

text: in

text: great

text: earthquakes.

text: Journal

text: of

text: Geophysical

text: Research

text: van

text: Stiphout,

text: T.,

text: Zhuang,

text: J.

text: Marsan,

text: D.

text: Seismicity

text: declustering,

text: community

text: online

text: resource

text: for

text: statistical

text: seismicity

text: analysis

text: (corssa),

text: doi:

text: Kagan,

text: Y.

text: Y.

text: Knopoff,

text: L.

text: Stochastic

text: synthesis

text: of

text: earthquake

text: catalogs.

text: Journal

text: of

text: Geophys-

text: ical

text: Research:

text: Solid

text: Earth

text: Helmstetter,

text: A.

text: Sornette,

text: D.

text: Bath’s

text: law

text: derived

text: from

text: the

text: Gutenberg-Richter

text: law

text: and

text: from

text: aftershock

text: properties.

text: Geophysical

text: Research

text: Letters

text: Luo,

text: J.

text: Zhuang,

text: J.

text: Three

text: regimes

text: of

text: the

text: distribution

text: of

text: the

text: largest

text: event

text: in

text: the

text: critical

text: etas

text: model.

text: Bulletin

text: of

text: the

text: Seismological

text: Society

text: of

text: America

text: Corral,

text: A.

text: Universal

text: earthquake-occurrence

text: jumps,

text: correlations

text: with

text: time,

text: and

text: anomalous

text: diffusion.

text: Physical

text: Review

text: Letters

text: Stojanova,

text: M.,

text: Santucci,

text: S.,

text: Vanel,

text: L.

text: Ramos,

text: O.

text: High

text: frequency

text: monitoring

text: reveals

text: after-

text: shocks

text: in

text: subcritical

text: crack

text: growth.

text: Physical

text: Review

text: Letters

text: Kagan,

text: Y.

text: Y.

text: Short

text: term

text: properties

text: of

text: earthquake

text: catalogs

text: and

text: models

text: of

text: earthquake

text: source.

text: Bulletin

text: of

text: the

text: Seismological

text: Society

text: of

text: America


text: 45. Snoeijer, J. H. & Andreotti, B. Moving contact lines: Scales, regimes, and dynamical transi- tions. Annual Review of Fluid Mechanics 45, 269–292 (2013). 46. Durin, G. & Zapperi, S. Scaling exponents for barkhausen avalanches in polycrystalline and amorphous ferromagnets. Physical Review Letters 84, 4705–4708 (2000). 47. Zapperi, S., Castellano, C., Colaiori, F. & Durin, G. Signature of effective mass in crackling- noise asymmetry. Nature Physics 1, 46–49 (2005). 48. Papanikolaou, S. et al. Universality beyond power laws and the average avalanche shape. Nature Physics 7, 1745–2473 (2011). 49. Cambonie, T. et al. Effect of the porosity on the fracture surface roughness of sintered mate- rials: From anisotropic to isotropic self-afﬁne scaling. Physical Review E 91 012406 (2015). 50. Barés, J., Barbier, L. & Bonamy, D. Crackling versus continuumlike dynamics in brittle failure. Physical Review Letters 111, 054301 (2013). Acknowledgements We thank Thierry Bernard for technical support, and Hervrcegol and Vadim Niko- layev for fruitful discussions. We also thank Francois Daviaud, Hugues Chatrancois Ladieu, Cindy Rountree and Julien Scheibert for the careful reading of the manuscript. Funding through ANR project MEPHYSTAR (ANR-09-SYSC-006-01) and by "Investissements d’Avenir" LabEx PALM (ANR-10-LABX-0039-PALM) is also gratefully acknowledged. 25

text: Snoeijer,

text: J.

text: H.

text: Andreotti,

text: B.

text: Moving

text: contact

text: lines:

text: Scales,

text: regimes,

text: and

text: dynamical

text: transi-

text: tions.

text: Annual

text: Review

text: of

text: Fluid

text: Mechanics

text: Durin,

text: G.

text: Zapperi,

text: S.

text: Scaling

text: exponents

text: for

text: barkhausen

text: avalanches

text: in

text: polycrystalline

text: and

text: amorphous

text: ferromagnets.

text: Physical

text: Review

text: Letters

text: Zapperi,

text: S.,

text: Castellano,

text: C.,

text: Colaiori,

text: F.

text: Durin,

text: G.

text: Signature

text: of

text: effective

text: mass

text: in

text: crackling-

text: noise

text: asymmetry.

text: Nature

text: Physics

text: Papanikolaou,

text: S.

text: et

text: al.

text: Universality

text: beyond

text: power

text: laws

text: and

text: the

text: average

text: avalanche

text: shape.

text: Nature

text: Physics

text: Cambonie,

text: T.

text: et

text: al.

text: Effect

text: of

text: the

text: porosity

text: on

text: the

text: fracture

text: surface

text: roughness

text: of

text: sintered

text: mate-

text: rials:

text: From

text: anisotropic

text: to

text: isotropic

text: self-afﬁne

text: scaling.

text: Physical

text: Review

text: E

text: Barés,

text: J.,

text: Barbier,

text: L.

text: Bonamy,

text: D.

text: Crackling

text: versus

text: continuumlike

text: dynamics

text: in

text: brittle

text: failure.

text: Physical

text: Review

text: Letters

text: Acknowledgements

text: We

text: thank

text: Thierry

text: Bernard

text: for

text: technical

text: support,

text: and

text: Hervrcegol

text: and

text: Vadim

text: Niko-

text: layev

text: for

text: fruitful

text: discussions.

text: We

text: also

text: thank

text: Francois

text: Daviaud,

text: Hugues

text: Chatrancois

text: Ladieu,

text: Cindy

text: Rountree

text: and

text: Julien

text: Scheibert

text: for

text: the

text: careful

text: reading

text: of

text: the

text: manuscript.

text: Funding

text: through

text: ANR

text: project

text: MEPHYSTAR

text: and

text: by

text: "Investissements

text: d’Avenir"

text: LabEx

text: PALM

text: is

text: also

text: gratefully

text: acknowledged.


text: Author contributions J.B., D.D. and D.B. conceived the experiment; J.B., M.L.H., D.D. and D.B. per- formed the experiments; J.B. A.D. and D.B. analyzed the data; D.B. wrote the manuscript; all authors reviewed the manuscript. Competing Interests The authors declare that they have no competing ﬁnancial or other interests. Correspondence Correspondence and requests for materials should be addressed to D.B. (email: daniel.bonamy@cea.fr). 26 ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Author

text: contributions

text: J.B.,

text: D.D.

text: and

text: D.B.

text: conceived

text: the

text: experiment;

text: J.B.,

text: M.L.H.,

text: D.D.

text: and

text: D.B.

text: per-

text: formed

text: the

text: experiments;

text: J.B.

text: A.D.

text: and

text: D.B.

text: analyzed

text: the

text: data;

text: D.B.

text: wrote

text: the

text: manuscript;

text: all

text: authors

text: reviewed

text: the

text: manuscript.

text: Competing

text: Interests

text: The

text: authors

text: declare

text: that

text: they

text: have

text: no

text: competing

text: ﬁnancial

text: or

text: other

text: interests.

text: Correspondence

text: Correspondence

text: and

text: requests

text: for

text: materials

text: should

text: be

text: addressed

text: to

text: D.B.

text: (email:

text: daniel.bonamy@cea.fr).

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text: experiment No. microstructure scale d total AE number activity rate R mean crack speed v × − − − 1 583 µm 33481 evt 1.35 10 1 evt s 1 2.7 µm s 1 × − − − 2 583 µm 5704 evt 3.06 10 2 evt s 1 0.87 µm s 1 − − 3 583 µm 18228 evt 1.90 evt s 1 33 µm s 1 × − − − 4 583 µm 6063 evt 6.78 10 1 evt s 1 13 µm s 1 − − 5 583 µm 36795 evt 2.10 evt s 1 40 µm s 1 − − 6 583 µm 31149 evt 5.41 evt s 1 100 µm s 1 − − 7 583 µm 9133 evt 22.8 evt s 1 0.96 mm s 1 − 8 233 µm 160145 evt 1.75 evt s 1 - − 9 233 µm 65436 evt 62.1 evt s 1 - − 10 24 µm 21590 evt 3.5 evt s 1 - − 11 24 µm 19442 evt 31.4 evt s 1 - Table 1: Synthesis of the samples and experiments analyzed here. Figures a, b, c, , and Supplementary Figs. 2, 3, 4, 9, 10 and 11 involve experiment No. 1. Figure d involves experiments No. 1 and 7. Figure and Supplementary Fig. 1 involve experiments No. 1 to 7. Supplementary Fig. 8 involves experiments No. 8 to 11. 30 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: experiment

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text: No.

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text: Synthesis

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text: samples

text: and

text: experiments

text: analyzed

text: here.

text: Figures

text: a,

text: b,

text: c,

text: and

text: Supplementary

text: Figs.

text: and

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text: experiment

text: No.

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text: Figure 1 Acoustic emission going along with crack propagation in a inhomogeneous solid. (a) Typical snapshot showing the time evolution of the energy of recorded AE in linear (black) and logarithmic (blue) scales. The snapshot duration is 1500 seconds while the whole experiment lasts 5 hours 30 minutes. (b) Schematic of the advancing crack front with localized AE (Supplementary movie 1). Black points show all sources localized from the starting of the experiment. Red points only show AE emitted over the last second (Supplementary movie 1). AE gather in the immediate vicinity of the propagating front. (c) This point is conﬁrmed by comparing the time evolution of the crack front as detected from a side-view imaging (blue line) with that of the AE center of mass over a moving time window of one second (open circles). Panels (a), (b) and (c) all concern the same experiment, where the microstructure length-scale is d = 583 µm and the mean crack − speed v = 2.7 µm s 1. (d), main panel: Cumulative number of AE as a function of crack { − } { length in experiments with d = 583 µm, v = 2.7 µm s 1 (plain) and d = 583 µm, v = − } { 960 µm s 1 (dash). (d), inset: Cumulative number of AE as a function of time for d = − } 583 µm, v = 2.7 µm s 1 . Figure 2 Gutenberg-Richter law and time-energy self-similarity. (a) Distribution of AE energy in one of the experiments (microstructure length-scale: d = 583 µm, crack speed: − ∝ − − v = 2.7 µm s 1). Solid magenta line is a gamma function P (E) E β exp( E/E ) for 0 ≥ − ± E E = 10 4 (vertical magenta dashed line), with ﬁtted parameters β = 0.96 0.02 min ± and E = 38 9. (b) Distribution of the time interval ∆t separating two successive AE of 0 31 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Figure

font: ZWBUJP+NimbusSanL-Bold

text: Acoustic

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text: emission

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text: with

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text: (a)

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text: snapshot

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text: scales.

text: The

text: snapshot

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text: (b)

text: Schematic

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text: from

text: the

text: starting

text: of

text: the

text: experiment.

text: Red

text: points

text: only

text: show

text: AE

text: emitted

text: over

text: the

text: last

text: second

text: (Supplementary

text: movie

text: AE

text: gather

text: in

text: the

text: immediate

text: vicinity

text: of

text: the

text: propagating

text: front.

text: (c)

text: This

text: point

text: is

text: conﬁrmed

text: by

text: comparing

text: the

text: time

text: evolution

text: of

text: the

text: crack

text: front

text: as

text: detected

text: from

text: a

text: side-view

text: imaging

text: (blue

text: line)

text: with

text: that

text: of

text: the

text: AE

text: center

text: of

text: mass

text: over

text: a

text: moving

text: time

text: window

text: of

text: one

text: second

text: (open

text: circles).

text: Panels

text: (a),

text: (b)

text: and

text: (c)

text: all

text: concern

text: the

text: same

text: experiment,

text: where

text: the

text: microstructure

text: length-scale

text: is

text: d

text: µm

text: and

text: the

text: mean

text: crack

text: speed

text: v

text: µm

text: s

text: (d),

text: main

text: panel:

text: Cumulative

text: number

text: of

text: AE

text: as

text: a

text: function

text: of

text: crack

text: length

text: in

text: experiments

text: with

text: d

text: µm,

text: v

text: µm

text: s

text: (plain)

text: and

text: d

text: µm,

text: v

text: µm

text: s

text: (dash).

text: (d),

text: inset:

text: Cumulative

text: number

text: of

text: AE

text: as

text: a

text: function

text: of

text: time

text: for

text: d

text: µm,

text: v

text: µm

text: s

text: Figure

text: Gutenberg-Richter

text: law

text: and

text: time-energy

text: self-similarity.

text: (a)

text: Distribution

text: of

text: AE

text: energy

text: in

text: one

text: of

text: the

text: experiments

text: (microstructure

text: length-scale:

text: d

text: µm,

text: crack

text: speed:

text: v

text: µm

text: s

text: Solid

text: magenta

text: line

text: is

text: a

text: gamma

text: function

text: P

text: (E)

text: E

text: β

text: exp(

text: E/E

text: for

text: E

text: E

text: (vertical

text: magenta

text: dashed

text: line),

text: with

text: ﬁtted

text: parameters

text: β

text: min

text: and

text: E

text: (b)

text: Distribution

text: of

text: the

text: time

text: interval

text: ∆t

text: separating

text: two

text: successive

text: AE

text: of

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: independent (background or MS) and dependent (offspring or AF) earthquakes29. Most of these methods are based on the spatio-temporal proximity of the events. Here, we adopted a procedure 19–21 used in compressive fracture experiments where spatial information is not available and con- sidered as MS all AE with energies in in a predeﬁned interval. The AS sequence following each of these MS is then deﬁned as all subsequent AE, till an event of energy equal or larger than that of the MS is encountered. Figure 3a shows the mean number of AS, N , triggered by a MS of energy E , in a typical AS MS fracture experiment. The productivity law is fulﬁlled, and N goes as a power-law with E as AS MS long as E is not too large (below a crossover energy value E ). The curve remains unchanged MS c after (i) having permuted randomly the energy between the events (that is having attributed to each event i the energy E of another event j chosen randomly), and (ii) having arbitrary set the time step j to unity (that is having arbitrary set the time occurrence of the event i to t = i + 1). This indicates i (cid:82) that the productivity law, here, simply emerges from the Gutenberg-Richter distribution of the AE energy, without any further information on their time organization. Calling F (E) = E P (u)du Emin the cumulative distribution of energy, we then expect (Supplementary Note 2): − N (E ) = F (E )/(1 F (E )), (3) AS MS MS MS which compares very well with the experimental curve (Fig. 3a). When β is larger than unity 8

text: independent

text: (background

text: or

text: MS)

text: and

text: dependent

text: (offspring

text: or

text: AF)

text: Most

text: of

text: these

text: methods

text: are

text: based

text: on

text: the

text: spatio-temporal

text: proximity

text: of

text: the

text: events.

text: Here,

text: we

text: adopted

text: a

text: procedure

text: used

text: in

text: compressive

text: fracture

text: experiments

text: where

text: spatial

text: information

text: is

text: not

text: available

text: and

text: con-

text: sidered

text: as

text: MS

text: all

text: AE

text: with

text: energies

text: in

text: in

text: a

text: predeﬁned

text: interval.

text: The

text: AS

text: sequence

text: following

text: each

text: of

text: these

text: MS

text: is

text: then

text: deﬁned

text: as

text: all

text: subsequent

text: AE,

text: till

text: an

text: event

text: of

text: energy

text: equal

text: or

text: larger

text: than

text: that

text: of

text: the

text: MS

text: is

text: encountered.

text: Figure

text: shows

text: the

text: mean

text: number

text: of

text: AS,

text: N

text: triggered

text: by

text: a

text: MS

text: of

text: energy

text: E

text: in

text: a

text: typical

text: AS

text: MS

text: fracture

text: experiment.

text: The

text: productivity

text: law

text: is

text: fulﬁlled,

text: and

text: N

text: goes

text: as

text: a

text: power-law

text: with

text: E

text: as

text: AS

text: MS

text: long

text: as

text: E

text: is

text: not

text: too

text: large

text: (below

text: a

text: crossover

text: energy

text: value

text: E

text: The

text: curve

text: remains

text: unchanged

text: MS

text: c

text: after

text: (i)

text: having

text: permuted

text: randomly

text: the

text: energy

text: between

text: the

text: events

text: (that

text: is

text: having

text: attributed

text: to

text: each

text: event

text: i

text: the

text: energy

text: E

text: of

text: another

text: event

text: j

text: chosen

text: randomly),

text: and

text: (ii)

text: having

text: arbitrary

text: set

text: the

text: time

text: step

text: j

text: to

text: unity

text: (that

text: is

text: having

text: arbitrary

text: set

text: the

text: time

text: occurrence

text: of

text: the

text: event

text: i

text: to

text: t

text: i

text: This

text: indicates

text: i

text: that

text: the

text: productivity

text: law,

text: here,

text: simply

text: emerges

text: from

text: the

text: Gutenberg-Richter

text: distribution

text: of

text: the

text: AE

text: energy,

text: without

text: any

text: further

text: information

text: on

text: their

text: time

text: organization.

text: Calling

text: F

text: (E)

text: E

text: P

text: (u)du

text: Emin

text: the

text: cumulative

text: distribution

text: of

text: energy,

text: we

text: then

text: expect

text: (Supplementary

text: Note

text: N

text: (E

text: F

text: (E

text: F

text: (E

text: AS

text: MS

text: MS

text: MS

text: which

text: compares

text: very

text: well

text: with

text: the

text: experimental

text: curve

text: (Fig.

text: When

text: β

text: is

text: larger

text: than

text: unity


text: We ﬁnally address Omori-Utsu law, which states that the number of AS per unit time, R , AS − decays algebraically with the elapsed time since the MS occurrence, t : R (t) = R /(1 + (t MS AS 0 t )/τ)p, where R and τ are characteristic rates and times, and p deﬁnes the Omori exponent. In MS 0 | − our experiments, R (t E ) is computed by binning the AS events over t t , and subsequently AS MS MS averaging the so-obtained curves over all MS with energy falling into the prescribed interval. Fig- ure 3c shows the so-obtained curves. The algebraic decay predicted by Omori is fulﬁlled, over almost ﬁve decades. The Omori-Utsu exponent p is independent of E . Conversely, the prefactor MS increases with E . MS (cid:82) ∞ | As N = R (t E )dt, making the Omori-Utsu law consistent with the productivity AS tMS AS MS law yields either R or τ to be proportional to N (E ). The former scaling proves to be wrong 0 AS MS while the second yields a perfect collapse of the curves (Fig. 3d). The collapse also reveals an exponential cutoff in Omori-Utsu law, which ﬁnally writes: (cid:18) (cid:19) (cid:16) (cid:17) − | R − t t R (t E ) = 0 exp MS (4) AS MS 1 + t−tMS p τ0NAS(EMS) τminNAS(EMS) The four constants, the Omori-Utsu exponent p, the lower and upper time scales τ and τ , and min 0 the characteristic activity rate R are interrelated (Supplementary Note 4 and Supplementary Eq. 0 | − 17).The very same law holds for the foreshock (FS) rate R (t E ) versus time to MS, t t FS MS MS (Supplementary Fig. 5). This symmetry along time reversal is a consequence from the fact that the 10 ##LTLine## ##LTLine## ##LTLine##

text: We

text: ﬁnally

text: address

text: Omori-Utsu

text: law,

text: which

text: states

text: that

text: the

text: number

text: of

text: AS

text: per

text: unit

text: time,

text: R

text: AS

text: decays

text: algebraically

text: with

text: the

text: elapsed

text: time

text: since

text: the

text: MS

text: occurrence,

text: t

text: R

text: (t)

text: R

text: (t

text: MS

text: AS

text: t

text: )/τ)p,

text: where

text: R

text: and

text: τ

text: are

text: characteristic

text: rates

text: and

text: times,

text: and

text: p

text: deﬁnes

text: the

text: Omori

text: exponent.

text: In

text: MS

text: our

text: experiments,

text: R

text: (t

text: E

text: is

text: computed

text: by

text: binning

text: the

text: AS

text: events

text: over

text: t

text: t

text: and

text: subsequently

text: AS

text: MS

text: MS

text: averaging

text: the

text: so-obtained

text: curves

text: over

text: all

text: MS

text: with

text: energy

text: falling

text: into

text: the

text: prescribed

text: interval.

text: Fig-

text: ure

text: shows

text: the

text: so-obtained

text: curves.

text: The

text: algebraic

text: decay

text: predicted

text: by

text: Omori

text: is

text: fulﬁlled,

text: over

text: almost

text: ﬁve

text: decades.

text: The

text: Omori-Utsu

text: exponent

text: p

text: is

text: independent

text: of

text: E

text: Conversely,

text: the

text: prefactor

text: MS

text: increases

text: with

text: E

text: MS

text: As

text: N

text: R

text: (t

text: E

text: )dt,

text: making

text: the

text: Omori-Utsu

text: law

text: consistent

text: with

text: the

text: productivity

text: AS

text: tMS

text: AS

text: MS

text: law

text: yields

text: either

text: R

text: or

text: τ

text: to

text: be

text: proportional

text: to

text: N

text: (E

text: The

text: former

text: scaling

text: proves

text: to

text: be

text: wrong

text: AS

text: MS

text: while

text: the

text: second

text: yields

text: a

text: perfect

text: collapse

text: of

text: the

text: curves

text: (Fig.

text: The

text: collapse

text: also

text: reveals

text: an

text: exponential

text: cutoff

text: in

text: Omori-Utsu

text: law,

text: which

text: ﬁnally

text: writes:

text: R

text: t

text: t

text: R

text: (t

text: E

text: exp

text: MS

text: AS

text: MS

text: t−tMS

text: p

text: τminNAS(EMS)

text: The

text: four

text: constants,

text: the

text: Omori-Utsu

text: exponent

text: p,

text: the

text: lower

text: and

text: upper

text: time

text: scales

text: τ

text: and

text: τ

text: and

text: min

text: the

text: characteristic

text: activity

text: rate

text: R

text: are

text: interrelated

text: (Supplementary

text: Note

text: and

text: Supplementary

text: Eq.

text: very

text: same

text: law

text: holds

text: for

text: the

text: foreshock

text: (FS)

text: rate

text: R

text: (t

text: E

text: versus

text: time

text: to

text: MS,

text: t

text: t

text: FS

text: MS

text: MS

text: (Supplementary

text: Fig.

text: This

text: symmetry

text: along

text: time

text: reversal

text: is

text: a

text: consequence

text: from

text: the

text: fact

text: that

text: the

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Figure 16: Isogonal Conjugates. If the ﬁrst diagram exists (with appropriate central angles), then so does the second. − Figure 17: Cyclic Quadrilateral. x¯ = τ/4 x, a + b + c + d = τ/2. The ˆ diagram here is a doubled quadrilateral, with A glued to A, etc. Front and back exist individually by EZ Holonomy. The front is inscribable in a circle. ˆ ˆ Front and back ﬁt together because B/A = B/A = sin(b)/ sin(a), etc. The back shows that an edge has the same visual angle from either of the two opposite vertices. This is another aspect of the Inscribed Angle Theorem. 14 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

text: Figure

text: Isogonal

text: Conjugates.

text: If

text: the

text: ﬁrst

text: diagram

text: exists

text: (with

text: appropriate

text: central

text: angles),

text: then

text: so

text: does

text: the

text: second.

text: Figure

text: Cyclic

text: Quadrilateral.

text: x¯

text: x,

text: a

text: b

text: c

text: d

text: The

text: ˆ

text: diagram

text: here

text: is

text: a

text: doubled

text: quadrilateral,

text: with

text: A

text: glued

text: to

text: A,

text: etc.

text: Front

text: and

text: back

text: exist

text: individually

text: by

text: EZ

text: Holonomy.

text: The

text: front

text: is

text: inscribable

text: in

text: a

text: circle.

text: ˆ

text: ˆ

text: Front

text: and

text: back

text: ﬁt

text: together

text: because

text: B/A

text: B/A

text: sin(b)/

text: sin(a),

text: etc.

text: The

text: back

text: shows

text: that

text: an

text: edge

text: has

text: the

text: same

text: visual

text: angle

text: from

text: either

text: of

text: the

text: two

text: opposite

text: vertices.

text: This

text: is

text: another

text: aspect

text: of

text: the

text: Inscribed

text: Angle

text: Theorem.

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##