PHYSICAL REVIEW B 109, 035308 (2024)
Magne ic swi ching o spin-sca e ing cen e s in D esselhaus [110] ci cui s
E. J. Rod íguez ,1,*A. A. Reynoso,1,2,†J. P. Bal anás,1,‡J. Ni a ,3,4,§and D. F us aglia1,
1Depa amen o de Física Aplicada II, Uni e sidad de Se illa, E-41012 Se illa, Spain
2Ins i u o Balsei o and Cen o A ómico Ba iloche, Comisión Nacional de Ene gía A ómica, 8400 Ba iloche, A gen ina
3Depa men o Ma e ials Science, Tohoku Uni e si y, Sendai 980-8579, Japan
4NTT Basic Resea ch Labo a o ies, A sugi 243-0198, Japan
(Recei ed 22 Feb ua y 2023; e ised 23 Decembe 2023; accep ed 2 Janua y 2024; published 24 Janua y 2024)
Spin ca ie s subjec o D esselhaus [110] (D110) spin-o bi coupling (SOC) ga he null spin phases in closed
ci cui s, con a y o usual Rashba and D esselhaus [001] SOC. We show ha D110 spin phases can be ac i a ed
in squa e ci cui s by in oducing an in-plane Zeeman ield, whe e localized ield inhomogenei ies ac as e ec i e
spin-sca e ing cen e s. Ou simula ions show ich in e e ence pa e ns in he quan um conduc ance, which
wo k as maps o a geome ic classi ica ion o he p opaga ing spin s a es. We also ind ha diso de acili a es
low- ield implemen a ions.
DOI: 10.1103/PhysRe B.109.035308
I. INTRODUCTION
Spin-o bi coupling (SOC) in wo-dimensional elec on
gases (2DEG) [1] is a s a egic esou ce o quan um elec-
onics and spin-based echnologies [2–5]. The cases o
Rashba [6] and D esselhaus [001] [7] SOC in zincblende
III-V compound semiconduc o quan um wells ha e been
discussed ex ensi ely in he li e a u e o e he las decades.
Al hough hey belong o di e en symme y classes, bo h
Rashba and D esselhaus [001] SOC p esen in-plane e ec-
i e ield ex u es exploi ed in, e.g., Aha ono -Cashe (AC)
spin in e e ome y [8–20] and he manipula ion o geome ic
spin phases in elec onic anspo [11,17,21–23] by elec ical
con ol o he co esponding SOC s eng hs [18,24,25]. By
con as , D esselhaus [110] (D110) SOC [7] has ecei ed el-
a i ely li le a en ion (wi h some no able excep ions [1,4,26–
32]). Howe e , D110 o e s a unique ea u e o p ac i-
cal in e es : an e ec i e ield ex u e pe pendicula o he
2DEG’s plane. Such a obus symme y acili a es he de el-
opmen o a so-called pe sis en spin helix (PSH) [31,33,34]
and he co esponding supp ession o spin elaxa ion [27],
spin dephasing [35], spin Hall e ec [36], and weak an-
ilocaliza ion [26] o conduc ion elec ons. One addi ional
consequence o u mos impo ance is he absence o AC in-
e e ence in D110 sys ems, as in any PSH. This is ue no
only o III-V compound semiconduc o 2DEGs bu also o
g oup III and IV monochalcogenide monolaye s. Mo eo e ,
D110 has been p oposed o ealizing opological phases in
noncen osymme ic supe conduc o s ha could hos he cel-
eb a ed (and elusi e) Majo ana modes [32].
In his pape , we discuss D110-based spin in e e ome y
and con ol in mesoscopic ci cui s o squa e shape. Due o
*[email p o ec ed]
†[email p o ec ed]
‡[email p o ec ed]
§[email p o ec ed]
[email p o ec ed]
i s symme y p ope ies, D110 alone does no mani es any
sign o AC in e e ence in elec onic anspo . We show ha
he in oduc ion o an addi ional in-plane Zeeman ield ac i-
a es complex spin dynamics leading o he de elopmen o
spin phases and in e e ence e ec s ha modula e he quan-
um conduc ance. In polygonal ci cui s, he in e play be ween
D110 and Zeeman ields c ea es e ec i e- ield discon inui ies
localized a he e ices ha p oduce spin sca e ing. This
esul s in ield-dependen phase di e ences be ween coun e -
p opaga ing spin ca ie s leading o ich in e e ence pa e ns.
The e ec is ema kable, as i enables he magne ic swi ch-
ing o spin-sca e ing cen e s in D110 ci cui s (di icul o
achie e wi h o he SOC classes) and i s cha ac e iza ion by
means o anspo expe imen s. Fu he mo e, we show ha
his esul is obus agains diso de and ealizable a low
ield s eng hs hanks o he Al’ shule -A ono -Spi ak (AAS)
e ec [37].
He e, we ocus on he modeling o squa e D110 ci cui s
buil on 2DEG lodged in III-V semiconduc o he e os uc-
u es. O he geome ies may wo k as well; s ill, we choose
squa e ci cui s based on i s simple expe imen al ealiza-
ion, modeling, nume ical simula ion, and in e p e a ion.
Expe imen s ha e shown ha ich conduc ance pa e ns
a ise in Rashba squa es [38–40]. Mo eo e , one-dimensional
(1D) models ha e been applied wi h success o Rashba
squa es [41–45], esul ing in excellen ag eemen wi h he
expe imen al obse a ions. These models a e especially sui ed
o la ge loop a ays [13,17,18,22,38,40], whe e highe slow-
p opaga ing modes a e mo e likely o decohe e han lowe
as -p opaga ing ones. The su i ing in e e ence obse ed in
expe imen s wi h loop a ays is well desc ibed by he physics
o one single (quasi-1D) o bi al mode p opaga ing along a
single mesoscopic loop [17,18,22,23,40].
II. MODEL
We conside a 1D squa e ci cui con ained in he xy plane
consis ing o conduc ing segmen s o leng h Lsubjec o
2469-9950/2024/109(3)/035308(9) 035308-1 ©2024 Ame ican Physical Socie y
E. J. RODRÍGUEZ e al. PHYSICAL REVIEW B 109, 035308 (2024)
FIG. 1. (a) Squa e ci cui subjec o D110 SOC. The a ows
ep esen he SOC ield expe ienced by CCW p opaga ing spin ca -
ie s. The ield lies in he e ical di ec ion, poin ing up o down
depending on he ca ie s’ pymomen um a each ci cui ’s sec ion.
Fo CW p opaga ing ca ie s, he SOC ield in e s i s sign due
o ime- e e sal symme y. (b) The in oduc ion o a Zeeman ield
ac i a es discon inui ies in he e ec i e- ield di ec ion localized a
e ices 2 and 4.
D110 SOC and Zeeman coupling, see Fig. 1. Each segmen
connec s e ices uand (u, =1,...,4) and is o ien ed
along he di ec ion ˆ
γ=(cos γ,sin γ,0), om u owa ds .
The spin-ca ie dynamics along each segmen is gi en by he
Hamil onian [1,4,29–31]:
ˆ
H u=p2
2m∗+β
¯hpyσz+μB·σ,(1)
whe e pis he ca ie ’s linea momen um along ˆ
γ,βis he
( eno malized) linea D110 SOC s eng h, m∗ he ca ie ’s
e ec i e mass, μis he magne ic momen , and σ he ec o
o Pauli ma ices. No ice ha he D110 SOC con ibu ion—
second e m on he igh -hand side ( .h.s.) o Eq. (1)—appea s
as a momen um-dependen magne ic ield o magni ude
(β/¯hμ)pycoupled o he spin along he zdi ec ion. Impo -
an ly, since py=psin γ, he o ien a ion o each segmen
modula es he ampli ude and sign o he D110 SOC s eng h.
This e m es ablishes a PSH [31,33,34]. The e is expe imen al
e idence [46] o a domina ing linea -in-momen um con ibu-
ion o D110 SOC.1S ill, highe -o de pcon ibu ions can be
inco po a ed by in oducing a eno malized βin Eq. (1)[4].
This sugges s he possibili y o con olling he D110 SOC
s eng h by elec ical means, simila o wha was ecen ly
ound in D esselhaus [001] sys ems [18,25]. The in-plane
Zeeman ield, B=Bˆ
bα, wi h ˆ
bα=(cos α, sin α,0), is com-
mon o all segmen s. Hence, he e ec i e ield unde gone by a
spin-ca ie a eling ei he clockwise (CW) o coun e clock-
wise (CCW) a ound he squa e loop esembles a s oboscopic
Rabi d i ing, namely, a cons an Zeeman ield coexis ing wi h
a no mal D110 componen ha akes a sequence o disc e e
alues, see Fig. 1(b). No ice ha he Zeeman ield unde mines
he PSH es ablished by he D110 SOC ield.
The squa e ci cui o Fig. 1is o ien ed such ha , s a ing
om node 1, he di ec ional angles a e γ={−π
4,π
4,3π
4,5π
4}.
This choice maximizes he ampli ude o he D110 SOC ield.
The esul ing D110 ield ex u e o a spin ca ie a eling
1No ice ha he lesse highe -o de con ibu ion p opo ional o
p3sin 3ϕσ
z[4], wi h ϕ he pola angle wi h espec o he xaxis,
does no comp omize he PSH since i ac s along he zaxis as well
as he linea con ibu ion in Eq. (1).
FIG. 2. Tigh -binding simula ion o he quan um conduc ance (in
uni s o e2/h) o 1D squa e loops co esponding o (a) a symme ic
ci cui in he ballis ic egime and (b) a diso de ed ci cui .
CCW is shown in Fig. 1(a). I ep esen s a ield oscilla ing
along he zdi ec ion, ixing a global spin quan iza ion axis.
Unde his ci cums ance, he spin phase ga he ed by a ca ie
in a ound ip is ze o. Figu e 1(b) shows how he in oduc ion
o an ex e nal in-plane Zeeman ield induces sudden changes
in he di ec ion o he e ec i e ield a nodes 2 and 4, des oy-
ing he global quan iza ion axis. This a o s he de elopmen
o complex spin dynamics and phases due o he misalignmen
o he local spin quan iza ion axis a di e en segmen s. In his
way, e ices 2 and 4 ac as e ec i e spin-sca e ing cen e s.
In o de o maximize in e e ence e ec s, in he ollowing
we s udy quan um anspo be ween e ices 1 and 3. Any
o he choice would unde mine spin-dependen signa u es (in
pa icula , symme y dic a es ha la in e e ence pa e ns a e
expec ed o anspo be ween nodes 2 and 4). S ill, misalign-
men s up o 30◦be ween he leads and he c ys allog aphic
axis xa e accep able. See Appendix A.
To s udy he quan um conduc ance as a unc ion o D110
and Zeeman ields we implemen a igh -binding model o
he squa e loop and use a G een’s unc ion o malism. This
pe mi s he inclusion o con ac leads and accoun s o all
possible p opaga ion pa hs con ibu ing o quan um anspo
be ween con ac s (see Appendix B). Following he Landaue -
Bü ike o malism [47], he ze o- empe a u e conduc ance is
gi en by G=(e2/h)T, wi h T he quan um ansmission be-
ween con ac s e alua ed a he Fe mi ene gy, EF. No ice ha
a sui able desc ip ion o he expe imen al condi ions equi es
an ene gy a e age o he compu ed conduc ance o disca d
esonances due o ini e-size e ec s. We ocus on he semi-
classical limi in which he Fe mi wa eleng h o he elec on,
λF=2πk−1
F, wi h kF=√2m∗EF he Fe mi momen um, is
much smalle han he pe ime e P=4L, i.e., λFP[11,22].
III. RESULTS
In Fig. 2we show he compu ed conduc ance as a unc ion
o kDP≡(βm∗/¯h2)Pand kZP≡(μBm∗/¯h2kF)P. These pa-
ame e s a e chosen o coincide wi h he spin phases ga he ed
by a spin ca ie p opaga ing along a s aigh quan um wi e o
leng h Po ien ed along he yaxis due o he ac ion o he D110
and Zeeman ields, espec i ely, wi h a ligh ime Pm∗/¯hkF.
Figu e 2(a) shows esul s o a symme ic squa e iden i ied
wi h ballis ic anspo h ough a egula a ay o squa e loops.
No ice, howe e , ha a ypical sample consis s o hund eds o
loops, so ha a mo e ealis ic desc ip ion o he expe imen al
035308-2
MAGNETIC SWITCHING OF SPIN-SCATTERING CENTERS … PHYSICAL REVIEW B 109, 035308 (2024)
condi ions equi es a sel -a e aging o e diso de ealiza ions
(modeled he e by in oducing segmen s o luc ua ing leng hs
aken om a andom dis ibu ion). We plo he co esponding
igh -binding esul s in Fig. 2(b).
Ou nume ical simula ions show ha spin in e e ence
e ec s appea as pe iodic pa e ns in he quan um conduc-
ance o e a wide ange o ield s eng hs. Howe e , he e
we ocus ou a en ion on ela i ely low kZPand kDP al-
ues, acco ding o he limi ed ield s eng hs accessible in
expe imen s [22,40]. In bo h cases—ballis ic, Fig. 2(a), and
diso de ed, Fig. 2(b)—we obse e ha he coexis ence o
D110 and Zeeman ields is essen ial o he eme gence o
spin in e e ence e ec s: D110 o Zeeman ields alone do
no imp in any ele an spin phase con ibu ing o in e e -
ence due o he exis ence o global spin quan iza ion axes
(along ˆ
zand ˆ
bα, espec i ely), so ha he conduc ance is spin
independen .
In con as , he in e play be ween D110 and Zeeman ields
p oduces discon inui ies in he di ec ion o he local spin-
quan iza ion axis a e ices 2 and 4, Fig. 1(b), ha u n
on spin-sca e ing p ocesses and spin phases con ibu ing o
in e e ence. When bo h ields a e compa able in magni ude,
spin in e e ence leads o a pe iodic se ies o lobes co e-
sponding o minimum, Fig. 2(a), o maximum, Fig. 2(b),
conduc ance. No ice ha diso de ed ci cui s display spli
lobes wi h espec o he ballis ic case, wi h a pe iodici y
o o de 2√2/3πalong he diagonal kDP=kZP. The lobe
spli ing is a mani es a ion o he AAS e ec [37] due o
dominan ime- e e sed in e e ing pa hs in diso de ed loops,
also ela ed o weak (an i)localiza ion in diso de ed hin lay-
e s [48,49]. This has d ama ic consequences o expe imen al
e i ica ion, since i allows o de ec ion o ac i e ield discon-
inui ies o 50% lowe ield magni udes i one ocuses on he
i s spli lobe.
In-plane magne ic ields o 2.5T lead o kZP≈2πin
squa e loops wi h pe ime e P=2.8 µm buil on 2DEGs
lodged in InGaAs quan um wells (QW) [40]. Mo eo e , in
single-mode QWs o wid h wone can app oxima e β≈
γD(π/w)2[29], wi h γD he bulk-in e sion asymme y co-
e icien unning om 11 eVÅ3in GaAs and InGaAs [50]
o 490 eVÅ3in InSb [51]. The QWs can be as na ow as
5–10 nm [25,29]. Mo eo e , he ca ie s’ e ec i e mass m∗
uns om 0.014 in InSb o 0.023 in InAs, 0.041 in InGaAs,
and 0.067 in GaAs (in uni s o he elec on mass m0)[1]. To
be conse a i e, o squa e loops wi h P=2.8 µm buil on
aw=10 nm InGaAs QW we ind kDP≈π. These alues
could be uned on he same sample by building a ays o
squa e loops wi h di e en pe ime e P, o be es ed in ans-
po expe imen s. The elec ical uning o he eno malized β
( h ough cubic e ms) can also con ibu e wi h a uning ange
p opo ional o γD[18,25].
We poin ou ha he in e e ence pa e ns shown in Fig. 2
o D110 squa es a e essen ially di e en om hose epo ed
in Re . [40] o Rashba squa es. The e, he Rashba SOC lead
o AC in e e ence pa e ns subjec o addi ional modula ion
by an in-plane Zeeman ield (wi h lobes cen e ed on he
Rashba axis). He e, ins ead, no AC in e e ence a ises as a
consequence o D110 SOC: bo h D110 and Zeeman ields a e
necessa y o he eme gence o in e e ence pa e ns (wi h no
lobes on he D110 axis).
FIG. 3. (a) Di ec pa hs, V+and V−, and (b) ime- e e sed pa hs,
U+and U−, o a squa e loop. (c) Quan um conduc ance G1 om
di ec pa hs, co esponding o he ballis ic case o Fig. 2(a). (d) Quan-
um conduc ance G2 om ime- e e sed pa hs, applicable o he
diso de ed case o Fig. 2(b). The dashed lines indica e ield se ings
such ha U±=I, also de ining winding-numbe ansi ions o he
spin s a es. We show sec o s wi h ω=1,3,5, and 7. The ωis
unde ined o anishing Zeeman o D110 ields. All conduc ances
a e in uni s o e2/h.
Finally, he conduc ance shows no dependence on he angle
α, namely, he esul s o Figs. 2(a) and 2(b) a e alid o any
o ien a ion o he in-plane Zeeman ield (see Appendix C o
a demons a ion).
IV. DISCUSSION
To gain u he physical insigh on he exac nume ical
esul s p esen ed abo e, we make use o a 1D model ha cap-
u es he spin dynamics along he loop, dis ega ding o bi al
backsca e ing a he e ices [40,44]. The solu ions o he 1D
Sch ödinge equa ion along a segmen based on Eq. (1)a e
plane wa es such ha he spino wa e unc ion p opaga es
om e ex u owa ds e ex along he di ec ion ˆ
γas
|ψ()=e−ikFe−ikDsin γσ
z−ikZˆ
bα·σ|ψ(0),(2)
whe e pa ame izes he dis ance om e ex u, wi h =L
a e ex . The p e ac o kFco esponds o he kine ic phase
o he ca ie associa ed o he cha ge dynamics, while he
emaining ac o s ep esen he spin phase due o spin p e-
cession in he p esence o D110 and Zeeman ields. No ice
ha he D110 SOC ield in e s i s sign o coun e p opaga ing
ca ie s. Indeed, since ˆ
γpoin s om o u, he sign e e sal
o he D110 con ibu ion is made explici in Eq. (2) by no ing
ha sin(γ+π)=−sin γ.
Acco ding o Eq. (2), he spin e olu ion along a ull seg-
men is gi en by he spin o a ion ope a o :
R u=exp[−ikDLsin γσ
z−ikZLˆ
bα·σ].(3)
By using he labeling shown in Figs. 3(a) and 3(b),we
de ine he spin e olu ion ope a o s U+=R14R43R32R21,
035308-3
E. J. RODRÍGUEZ e al. PHYSICAL REVIEW B 109, 035308 (2024)
U−=R12R23R34R41,V+=R32R21, and V−=R34R41, wi h he
subindex +(−) indica ing CCW (CW) p opaga ion. The
uni a y U± ep esen ull ound ips wi h o igin in e ex
1, whe eas V±co espond o di ec pa hs om e ex 1 o
e ex 3. We eso again o he Landaue -Bü ike o mal-
ism and ob ain he conduc ance, G=(e2/h)T, by compu ing
he ansmission T o wo di e en domina ing in e e ence
p ocesses. In he i s case, Tis compu ed om he in e -
e ence be ween he di ec pa hs V+and V−, see Fig. 3(a).
This con ibu ion is dominan o geome ies p ese ing a
wo old e lec ion symme y along he axis connec ing he
con ac leads. I equi es symme ic and clean samples as, e.g.,
he ballis ic ones discussed in Fig. 2(a). In he second case, he
ansmission is compu ed as T=2−Ra e no ing ha he
e lec ion coe icien R ollows om he in e e ence be ween
pa hsU+andU−, see Fig. 3(b). This con ibu ion domina es in
diso de ed samples, as hose discussed in Fig. 2(b), since U+
and U−desc ibe he spin e olu ion along ime- e e sed pa hs.
Figu es 3(c) and 3(d) show he conduc ances G1and G2
calcula ed om di ec and ime- e e sed pa h con ibu ions,
espec i ely, as a unc ion o kZPand kDP. We ind ha hese
lowe -o de semiclassical models ep oduce e y well he
ully quan um nume ical esul s o Fig. 2 o he ballis ic (G1)
and diso de ed (G2) cases. In pa icula , he ag eemen be-
ween he diso de ed igh -binding simula ions, Fig. 2(b), and
he esul s om ime- e e sed pa h con ibu ions, Fig. 3(d),
demons a es ha he lobe spli ing has i s o igin in he AAS
e ec .
Mo eo e , an inspec ion o U±un eils ele an geome ic
cha ac e is ics o he spin s a es. We ind ha U± educes
o he SU(2) iden i y σ0whene e ˜
kP =2nπ, wi h ˜
k=
k2
Z+k2
D/2 and nin ege . No ice ha he ca ie s ga he no
ne spin phase in a ull ound ip o hese pa icula ield se -
ings. The condi ion de ines a se ies o lines ha spli he lobes
in Figs. 3(c) and 3(d). In e es ingly, we ind ha such lines
also de ine opological ansi ions o he ex u es displayed
by he spin modes |()in he Bloch sphe e (wi h |(0)
an eigensolu ion o U±). This is demons a ed by de ining he
winding numbe ωa ound he Zeeman- ield axis ˆ
bαas
ω=1
2πP
0
d
n×dˆ
n
d·ˆ
bα,(4)
whe e ˆ
n() is he no malized p ojec ion o he spin ex u e
ˆ
s()=()|σ|()on he plane o hogonal o ˆ
bα(see
Appendix D o u he de ails). We ind ha he spin s a es
o ganize in sec o s wi h de ini e odd ω, wi h only one single
lobe pe sec o in Fig. 3(d). This inding sugges s ha he
in e e ence pa e n in he quan um conduc ance can wo k as
a map o he geome ic cha ac e iza ion o he spin s a es.
V. CONCLUSIONS AND OUTLOOK
We show ha he combined ac ion o D110 SOC and
Zeeman ields is a esou ce ul ool o he manipula ion o
spin ca ie s in quan um ci cui s. This is demons a ed in
squa e ci cui s as a p oo o concep , hough o he geome-
ies may wo k as well. When ei he ield domina es o e he
o he , anspo is spin independen (di e ing om p e ious
esul s wi h Rashba squa es displaying AC in e e ence [40]).
By con as , he applica ion o D110 and Zeeman ields o
simila magni ude u ns on e ec i e spin-sca e ing cen e s
ha igge complex spin dynamics and he de elopmen o
spin phases modula ing quan um anspo . The esul ing in-
e e ence pa e n in he conduc ance wo ks as a guide h ough
he geome y o he p opaga ing spin s a es. Rema kably, he
p esence o diso de acili a es he expe imen al ealiza ion in
loop a ays a ela i ely low ield s eng hs due o he AAS
e ec . This also p o ides a way o de e mine he magni ude
o he D110 SOC s eng h in a 2DEG. Ou esul s p o ide
c ucial in o ma ion abou he ac ion o D110 SOC ields on
spin ca ie s and show how spin dynamics can be ac i a ed in a
con olled ashion, demons a ing a po en ial o applica ions
in spin onics and spin-based quan um echnologies.
We u he no ice some al e na i e implemen a ions. Sim-
ila e ec s could be ound in usual Rashba and D esselhaus
[001] squa e ci cui s (wi h in-plane e ec i e SOC ields)
p o ided ha a PSH is es ablished by se ing equal SOC
s eng hs [4]. This, howe e , has he disad an age o e-
qui ing a ine uning o he SOC ields (con a y o D110
ci cui s whe e he PSH is buil in). Mo eo e , g oup-III me al-
monochalcogenide monolaye s such as GaSe and GaS display
h ee old symme ic, cubic-in-momen um D110 [52], which
could be op imally exploi ed by using, e.g., iangula o
hexagonal ci cui s. Recen ly, i has been sugges ed [53] ha
g oup-IV monochalcogenide MX monolaye s (M=Sn o Ge
and X=S, Se, o Te) de elop D110 PSHs o igina ed by
in-plane e oelec ici y ha can be con olled elec ically,
appea ing as possible pla o ms. An in e es ing al e na i e
would be elec on spin esonance (ESR) expe imen s in D110
zigzag ballis ic channels ins ead o D110 squa e loops [54].
ACKNOWLEDGMENTS
We acknowledge suppo om he Spanish MICINN-AEI
h ough P ojec No. PID2021-127250NB-I00 and om he
Andalusian Go e nmen h ough PAIDI 2020 P ojec No.
P20-00548 and FEDER P ojec No. US-1380932. We hank
R. Winkle o sugges ing us he s udy o D110 ci cui s.
APPENDIX A: SPIN DYNAMICS
IN OTHER CONFIGURATIONS
Gi en he o bi al aniso opy o he D110 spin-o bi e m,
he o ien a ion o he sample wi h espec o he c ys al-
log aphic axis a ec s he esul s. He e we ocus on he
dependence wi h he angle o o ien a ion, θ, o he conduc-
ances G1and G2ob ained using he spin- o a ion model
p esen ed in he main ex . Figu es 4(a)–4(d) and 4(e)–4(h)
show esul s o G1and G2, espec i ely, co esponding o θ=
{0,π/6,π/3,π/2}as a unc ion o he Zeeman and D110
s eng hs. The s onges modula ion is ob ained o θ=0 wi h
con ac s in e ices 1 and 3 (sha ing he same ycoo dina e),
Figs. 4(a) and 4(e), which co esponds o he con igu a ion
s udied in he main ex . As θg ows, in e e ence con as
dec eases. Fo θ=π/2, Figs. 4(d) and 4(h), he in e e ence
disappea s.
The sha p di e ence be ween he θ=0 and θ=π/2 cases
is a consequence o he D110 s eng h being p opo ional o
py. Following Eq. (3) o he main ex he spin e olu ion along
a leng h Land o ien a ion γsegmen is gi en by he uni a y
035308-4
MAGNETIC SWITCHING OF SPIN-SCATTERING CENTERS … PHYSICAL REVIEW B 109, 035308 (2024)
FIG. 4. Quan um conduc ance G1[G2] o sample o ien a ion angle: (a) [(e)] θ=0, i.e., he main- ex con igu a ion, (b) [( )] θ=π/6,
(c) [(g)] θ=π/3, and (d) [(h)] θ=π/2. Inse s: Ou semiclassical model conside s spin ca ie s p opaga ing along pa hs s a ing in one ed
spo and ending in he opposi e one o G1o , o G2, s a ing and ending a he ed spo . All conduc ances a e in uni s o e2/h.
R u=exp[−i(kDLsin γˆ
z+kZLˆ
x)·σ] wi h he Zeeman ield
along he xaxis. Ha ing nonze o weigh in bo h ac o s is
essen ial o he ans o ma ions on di e en segmen s being
noncommu a i e, leading o phase di e ences o di e en
a eled pa hs. Fo he conduc ance G1 he in e e ence is
be ween he wo di ec pa hs om e ex 1 o e ex 3 [see
V±in Fig. 3(a)]: ei he ia e ex 2 o ia e ex 4. Fo sample
o ien a ion θ=0 he i s pa h (V+) is de ined by he segmen
o ien a ions γ={−π/4,π/4}, whe eas he second (V−)is
de ined by γ={π/4,−π/4}. This leads o sin γ=±1/√2
and phase di e ences depending on he pa icula alues o
kDand kZ. This con igu a ion maximizes he phase di e -
ences and he ampli ude o he in e e ence pa e n. On he
o he hand, o sample o ien a ion θ=π/2, as he con ac ed
e ex 1 and 3 become e ically aligned, he i s pa h has
γ={π/4,3π/4}and he second pa h has γ={3π/4,π/4}
wi h sin γ=1/√2 in all cases. This means ha bo h pa hs
p oduce an iden ical spin ans o ma ion, leading o cons uc-
i e in e e ence o he quan um ansmission om e ex
1 o e ex 3, esul ing in maximum conduc ance o any
alue o kDand kZ. An analogous analysis can be ex ended
o explain ha o θ=π/2 he conduc ance G2is minimized
independen ly on kDand kZ: esul ing om an iden ical se-
quence o sin γ alues o bo h in e e ing pa hs gene a ing
cons uc i e in e e ence o he e u n ansmission o e ex
1 [see ime- e e sal pa hs U±in Fig. 3(b)]. On he o he
hand, o θ=0, he wo pa hs ele an o G2ha e di e en
sequences o sin γ, namely, {−1/√2,1/√2,1/√2,−1/√2}
and {1/√2,−1/√2,−1/√2,1/√2}; as shown in Fig. 4(d),
his is he o ien a ion gene a ing he s onges in e e ence-
pa e n modula ion, induced by he coexis ence o kDand kZ.
APPENDIX B: TIGHT-BINDING APPROACH
We ocus on 1D ci cui s o polygonal shape. Each seg-
men o an N-sided egula polygon is disc e ized in Ns+1
si es labeled j={0,1,...,Ns}and sepa a ed by he la ice
dis ance a0=P/(NNs). The uni ec o ˆγde ines he angula
o ien a ion o a segmen om e ex u o e ex , hus he
coo dina e o si e jis j= u+ja0ˆγ, wi h u he coo di-
na e o e ex u. By applying he cus oma y ini e di e ence
me hod o he segmen Hamil onian o Eq. (1) one ge s he
ollowing 1D igh -binding Hamil onian:
ˆ
H u=
Ns

j=0
σσ
(2 hσ0+μB·σ)σσˆc†
jσˆcjσ
+
Ns−1

j=0
σσ
[(i Dcos γσ
z− hσ0)σσˆc†
j+1,σ ˆcjσ+H.c.],
(B1)
whe e σ0is he SU(2) iden i y ma ix, ˆcjσ(ˆc†
jσ) e e s o
he annihila ion (c ea ion) ope a o o an elec on in si e j
wi h spin σ={↑,↓} along he zdi ec ion, h=¯h2/(2m∗a2
0)
is he hopping ene gy, and D=β/(2a0) is he D110 hopping
ene gy. The Hamil onian o he ull polygon, ˆ
Hp,simply
ollows om he sum o he Hamil onians o he Nsegmen s
(a oiding double coun ing local e ms a he N e ices).
The sys em is connec ed o a sou ce con ac lead and a
d ain con ac lead, labeled η=s,d, modeled by semi-in ini e
igh -binding chains pa ame ized by he hopping ene gy η
and he si e ene gy η. The e a ded G een’s unc ion e al-
ua ed a he ene gy ε o he edge si e o each semi-in ini e
chain is ˆg
η(ε), ha ing a local and spin- i ial ma ix elemen ,
g
η(ε), which is easily ob ained nume ically o analy ically
om he con inuous ac ion equa ion 1/g
η(ε)=ε+i0+−
η− 2
ηg
η(ε). Each lead is connec ed o a pa icula e ex
o he sys em (e.g., e ices 1 o 3 o he N=4 polygon
shown in Fig. 1) by a spin-independen hopping ope a o ˆ
Tη;
which is p opo ional o he hopping ene gy η,h. The e ec
o he leads is encoded in he e a ded sel -ene gy ope a o s
ˆ

η(ε)=ˆ
T†
ηˆg
η(ε)ˆ
Tη, which allow us o ob ain he e a ded
G een’s unc ion, ˆ
G (ε), om he equa ion [ε+i0+−ˆ
Hp−
035308-5

E. J. RODRÍGUEZ e al. PHYSICAL REVIEW B 109, 035308 (2024)
ˆ

d(ε)−ˆ

s(ε)] ·ˆ
G (ε)=I. In wha ollows, o simplici y,
we do no explici ly w i e he dependence wi h εo hese op-
e a o s. The ze o- empe a u e linea conduc ance is compu ed
using he Landaue -Bü ike o mula,
G=e2
hT [ ˆ
sˆ
G ˆ
dˆ
Ga],(B2)
whe e he ad anced ope a o s, ha ing supe sc ip a,a e he
adjoin o he e a ded ope a o s ˆ
Oa=(ˆ
O )†, he lead a es
ope a o s ha e ma ix elemen s ˆ
η
jσ,jσ=i( ˆ

η−ˆ
a
η)jσ,jσ,
and all he ope a o s inside he ace a e e alua ed a he
Fe mi ene gy, i.e., aking ε=EF. Impo an ly, he Gob ained
om (B2) is exac and hus i con ains all he in e e ing pa hs
om he sou ce o he d ain.
The simula ions p esen ed in Fig. 2a e ob ained as ollows.
We se N=4 and ake EF/ h≈0.2. The la e implies ha
λF/a0∼13.75, i.e., he Fe mi wa eleng h is well esol ed
by he disc e iza ion. Besides, since we choose Ns=512, he
semiclassical egime is ensu ed as he pe ime e , P=2048a0,
is much la ge han he Fe mi wa eleng h, P/λF≈149. To
simula e con ac leads wi h a b oad ene gy bandwid h we se
η=2.5 hand choose η o se he wo king ene gy a he
cen e o he band. To minimize sca e ing he in e media e
hopping ene gies a e se o he a e age o he hopping in he
lead and in he polygon, i.e., η,h=( η+ h)/2. We pe o m a
Fe mi ene gy a e age o he conduc ance in o de o disca d
esonance-induced a ia ions due o ini e-size e ec s. Such
a e age ene gy window includes he Fe mi wa e ec o ange
kF∈[2π(n0−2)/P,2π(n0+2)/P], wi h n0=149, hus en-
su ing a e aging o e se e al o bi al esonances. The ballis ic
case p esen ed in Fig. 2(a) ollows di ec ly by conside ing all
ou segmen s being o equal leng h, i.e., L0=P/4. Ins ead,
o he case o Fig. 2(b), we also a e age he conduc ance o e
diso de ealiza ions, hus simula ing he sel -a e aging o he
diso de ha a ises when measu ing he conduc ance o e he
ull a ay o hund eds o squa es. Each diso de ealiza ion
is gene a ed by andomly modi ying he segmen leng hs o
he ou -sided polygon, i.e., L=L0(1 +δl), whe e δlis aken
om a [−0.1,0.1] uni o mly dis ibu ed p obabili y densi y.
APPENDIX C: SPIN DYNAMICS IN SQUARE LOOPS:
ANALYTICAL DERIVATION
The spin dynamics o CCW and CW p opaga ing ca ie s
in a ull ound ip a ound a squa e ci cui o pe ime e P=4L
is de e mined by he uni a y ope a o s U±in oduced in he
main ex . The conduc ance G2p esen ed in Fig. 3(b) ollows
om G2=e2/h(2 −T [0†
0]), wi h 0 he o e lapped e o-
lu ion along ime- e e sed pa hs:
U+=R14R43R32R21,U−=R12R23R34R41,
0=U++U−
2.(C1)
He e we ocus on θ=0 sample o ien a ion wi h he D es-
selhaus and Zeeman ields’ di ec ions chosen as shown in
Fig. 1.Aspy∝sin γ, i.e., each segmen angle con ols he
ampli ude and sign o he D110 e m, i can be seen ha he
same spin ans o ma ions apply o segmen s wi h γ=π/4
o γ=3π/4(γ=−π/4o γ=−3π/4) because bo h ha e
posi i e (nega i e) sin γbeing iden ical o +1/√2(−1/√2).
Taking in o accoun he la e conside a ion in Eq. (3), he spin
e olu ion ope a o s o di e en segmen s can be g ouped as
ollows:
R+=R41 =R12 =R32 =R43 =exp [−i +·σL],R−=R34 =R23 =R21 =R14 =exp [−i −·σL],(C2)
whe e he ec o s ± ep esen he wo possible e ec i e ields, which can be w i en as:
±=˜
kˆ
±,˜
k≡k2
Z+1
2k2
D,ˆ
±≡1
˜
kkZcos α,kZsin α,±1
√2kD.(C3)
By expanding he ma ix exponen ials R±=exp[−i ±·σL] one ob ains:
R±=aσ0+[xσx+yσy±zσz],(C4)
whe e
a=cos ˜
kL,x=−ikZcos α
˜
ksin ˜
kL,y=−ikZsin α
˜
ksin ˜
kL,z=− i
√2
kD
˜
ksin ˜
kL (C5)
wi h a∈Rand x,y,z∈C. The wo possible pai wise p oduc s o he R±ope a o s can be w i en as ollows:
R+R−=Aσ0+(B−iC)σx+(D+iE)σy,R−R+=Aσ0+(B+iC)σx+(D−iE)σy,
U+=R+R−R−R+=(A2+B2+C2+D2+E2)σ0+2A(Bσx+Dσy)+2(CD +EB)σz,
U−=R−R+R+R−=(A2+B2+C2+D2+E2)σ0+2A(Bσx+Dσy)−2(CD +EB)σz,(C6)
whe e we ha e in oduced he de ini ions: A=a2+x2+y2−z2,B=2ax,C=2yz,D=2ay and E=2xz. P oceeding o
calcula e 0leads o
0=1
2(U++U−)=(A2+B2+C2+D2+E2)σ0+2A(Bσx+Dσy).(C7)
035308-6
MAGNETIC SWITCHING OF SPIN-SCATTERING CENTERS … PHYSICAL REVIEW B 109, 035308 (2024)
The conduc ance G2 hen becomes:
G2=e2
h(2 −T [0†
0])
=2e2
h−2e2
h(A2+B2+C2+D2+E2)2
+2e2
h4A2(B2+D2),(C8)
which in e ms o kZand kD eads:
G2
2e2
h=1−k2
D/2+k2
Zcos ˜
kP2
˜
k4−kZk2
Dsin ˜
kP
2+k3
Zsin ˜
kP2
˜
k6.
(C9)
As discussed in he main ex , hese analy ical esul s
show ha G2is independen o he in-plane Zeeman ield’s
o ien a ion α.
APPENDIX D: GEOMETRICAL INTERPRETATION
OF RESULTS
He e, we in es iga e he ela ion be ween he conduc ance
o he squa e ci cui discussed in he main ex (θ=0) and he
opology o he spin ex u es de e mined by he p opaga ing
spin modes in a ound ip, |(), wi h ∈[0,P] a linea
pa ame iza ion o he ci cui ’s pe ime e . Such p opaga ing
spin modes a e ound a e diagonalizing he CCW e olu ion
ope a o , U+, and ob aining he eigensolu ions e alua ed a
e ex 1, which a e hen p opaga ed along he sequence o
segmen s using Eq. (2) om he main ex . The spin ex u e o
he solu ion is eadily ob ained om ˆ
s()=()|σ|(),
which desc ibes a pe iodical ajec o y on he Bloch sphe e.
The opological cha ac e iza ion o he solu ions is made in
e ms o he winding numbe o hei spin ex u e a ound
he di ec ion o he Zeeman ield.2Such winding numbe is
compu ed as [40],
ω=1
2πP
0
d
n×dˆ
n
d·ˆ
bα(D1)
whe e ˆ
n() is he no malized p ojec ion o ˆ
s() on he plane
o hogonal o he Zeeman ield di ec ion ˆ
bα.
Figu e 5(a) shows ha he opology o he spin ex u es as
a unc iono kZPand kDPde ines zones o inc easing odd
winding numbe . Figu es 5(c)–5(d) show ha he bounda ies
be ween di e en opological numbe s a e co ela ed wi h he
in e e ence pa e ns o he conduc ances. Fo he conduc ance
G2, as shown in Fig. 5(d), each lobe o maximal conduc-
ance is enclosed by a zone o cons an winding numbe .
We ind ha he bounda ies be ween egions wi h di e en
winding numbe s sa is y he condi ion U±=σ0, leading o
G2=0 due o he cons uc i e in e e ence o ime- e e sed
pa hs. Mo e explici ly, G2=e2/h(2 −T [0†
0]) =0 since
0=(U++U−)/2=σ0. This condi ion is sa is ied when-
e e (R+R−)−1=R−R+which, om Eqs. (C6), equi es ha
D=B=0. The la e imposes ˜
kP =2nπwi h nin ege ,
lying exac ly a he winding-numbe bounda ies.
2One can ake any o he wo CCW o CW solu ions o pe o m his
calcula ion because hey all sha e he same winding numbe .
FIG. 5. (a) Spin ex u es’ winding numbe , ω, along he di ec-
ion o he applied Zeeman ield. (b) Zoom on zone A, a bounda y
be ween wo di e en opological numbe s: see he co esponding
spin ex u es in Fig. 6. (c) Winding numbe o e lapped wi h he con-
duc ance G1. (d) Winding numbe o e lapped wi h he conduc ance
G2. All conduc ances a e in uni s o e2/h.
In Fig. 6we p esen a se ies o spin ex u es unde going
a opological ansi ion along he zone shown in Fig. 5(b).
The spin ex u es in his D110 squa e ci cui in ol e wo
o e lapping conelike ajec o ies, which a e seen as eigh like
ajec o ies a ound he di ec ion o he applied magne ic ield.
A simila beha io is known o appea in wo-le el sys ems
subjec o a Rabi d i e [55–57], whe e he in e play o a
cons an magne ic ield (he e he Zeeman ield con ained in
he xy plane) coexis ing wi h a no mal oscilla ing sinusoidal
componen (he e D110 along he zaxis, which in his case
akes a sequence o disc e e alues ins ead o a ying con in-
uously) gi es ise o spin ex u es wi h an analog s uc u e.
FIG. 6. (Top) Spin ex u es o p opaga ing modes in he Bloch
sphe e o di e en SO s eng hs a he zone A shown in Fig. 5(b).
(Bo om) P ojec ion o he spin ex u es on he no mal plane o he
Zeeman ield di ec ion. The colo indica es he ci cula ion o he
local spin s a es as he ca ie p opaga es h ough he pe ime e , om
ed o iole . The inse s show he di ec ion o he applied Zeeman
ield wi h espec o he Bloch sphe e’s op and bo om pe spec i es.
035308-7
E. J. RODRÍGUEZ e al. PHYSICAL REVIEW B 109, 035308 (2024)
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