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Comple e unneling o acous ic wa es be ween piezoelec ic c ys als
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Published e sion
Geng, Zhuo an; Maasil a, Ila i J.
Geng, Z., & Maasil a, I. J. (2023). Comple e unneling o acous ic wa es be ween piezoelec ic
c ys als. Communica ions Physics, 6, A icle 178. h ps://doi.o g/10.1038/s42005-023-01293-y
2023
ARTICLE
Comple e unneling o acous ic wa es be ween
piezoelec ic c ys als
Zhuo an Geng 1✉& Ila i J. Maasil a 1✉
The mechanical displacemen s in piezoelec ic ma e ials ca y along mac oscopic elec ic
fields, allowing unneling o acous ic wa es ac oss a acuum gap beyond he cha ge-cha ge
in e ac ion dis ance. Howe e , no igo ous p oo o comple e acous ic wa e unneling has
been p esen ed, and he condi ions o achie e comple e unneling ha e no been iden ified.
He e, we demons a e analy ically he condi ion o such phenomenon o a bi a y aniso-
opic c ys al symme ies and o ien a ions, and ha comple e ansmission o he incoming
wa e occu s a he exci a ion equency o leaky su ace wa es. We also show ha he
comple e ansmission condi ion can be ela ed o he su ace elec ic impedance and
he e ec i e su ace pe mi i i y o he piezoelec ic ma e ial, ele an o ealize he comple e
unneling expe imen ally. We suppo ou findings wi h nume ical esul s o he maximum
powe ansmi ance o a slow ans e se wa e unneling be ween iden ical ZnO c ys als.
The esul s show ha comple e unneling can be achie ed o a la ge ange o o ien a ions.
h ps://doi.o g/10.1038/s42005-023-01293-y OPEN
1Nanoscience Cen e , Depa men o Physics, Uni e si y o Jy askyla, P. O. Box 35, FI-40014 Jy äskylä, Finland. ✉email: zhgeng@jyu.fi;maasil a@jyu.fi
COMMUNICATIONS PHYSICS | (2023) 6:178 | h ps://doi.o g/10.1038/s42005-023-01293-y | www.na u e.com/commsphys 1
1234567890():,;
Acous ic wa es (acous ic phonons) a e de o ma ions o
ib a ions p opaga ing h ough a ma e ial medium. As
such, hey do no exis in acuum, leading o he ini ial
conclusion ha i is impossible o he acuum o ansmi he
ene gy o an acous ic wa e be ween wo sepa a ed media. How-
e e , a he a omic scale he ib a ions o he nuclei can p opaga e
ia hei elec ical in e ac ions h ough acuum. Thus, a ques ion
can be aised, whe he acous ic phonons can also be ansmi ed
ac oss la ge han a omic scale acuum gaps h ough some
elec omagne ic mechanism. This is a ele an ques ion, as wi h
he ad ances in expe imen al echniques, nanome e o sub-
nanome e scale acuum gaps can be achie ed1–4. The possibili y
o such acous ic phonon unneling, as i is o en called in he
li e a u e, has a ac ed a conside able amoun o heo e ical wo k
in ecen yea s o in es iga e possible mechanisms o he e ec
such as Casimi and an de Waals o ces, pa icula ly in he
con ex o nea -field hea ans e 5–18.
One possible mechanism o acous ic wa e unneling is pie-
zoelec ici y, as in piezoelec ic ma e ials mechanical displace-
men s ca y along mac oscopic elec ic fields. When an acous ic
wa e in a piezoelec ic solid impinges on a ee su ace, i ex ends
a decaying, e anescen elec ic field in o he acuum19. The leng h
scale o his decay is de e mined by he wa eleng h o he acous ic
wa e, so by b inging ano he piezoelec ic solid wi hin a wa e-
leng h, acous ic powe can be ansmi ed in o he second pie-
zoelec ic solid ac oss he acuum gap. Wha makes his
piezoelec ically media ed acous ic wa e unneling pa icula ly
a ac i e is i s leng h scale: i is no fixed o be in he nanoscale,
bu ope a es on he ypically much la ge wa eleng h scale
defined by he equency (1 GHz would co espond o ~5 μm).
The e ec was in oduced20,21 and obse ed22 long ago ( o mo e
de ailed backg ound, see Geng and Maasil a23), bu de eloped
u he mo e ecen ly5,23,24. In pa icula , a gene al o malism
was in oduced23 ha is applicable o any inciden bulk wa e
mode o any aniso opic c ys allog aphic o ien a ion. One o
he mos in e es ing sugges ions5,21,24 is he possibili y o uni y
ansmission o some pa icula condi ions, meaning ha
he inciden wa e could pe haps be comple ely ansmi ed
in o he adjacen solid. Howe e , he discussions in p e ious
li e a u e5,21,24 a e limi ed ei he by he simplified models used,
o only show nume ical esul s o he highes symme y c ys al
o ien a ions. Un il now, no igo ous p oo o comple e acous ic
wa e unneling has been p esen ed, no ha e gene ally alid
comple e unneling condi ions been pu o wa d.
In his wo k, we ocus on he powe ansmi ance o acous ic
wa e unneling. We use he gene al o malism de eloped o
piezoelec ic acous ic wa e unneling23 o analy ically p o e he
exis ence o he comple e unneling phenomenon be ween wo
acuum sepa a ed iden ical solids. In addi ion, a esonan un-
neling condi ion is also de i ed, co esponding o he exci a ion
o leaky su ace wa es. We also p opose ha his condi ion could
be checked expe imen ally. Fu he discussion o he esul s a e
p esen ed wi h a ew nume ical examples o ZnO c ys als. In
pa icula , we find ou esul s di e om hose ob ained be o e5.
Resul s and discussion
Tunneling o acous ic wa es. We s udy a sys em o wo aniso-
opic, semi-infini e piezoelec ic solids sepa a ed by a acuum
gap o wid h d, as shown in Fig. 1. Two coo dina e sys ems
desc ibe he ela ion be ween he c ys al in insic o ien a ion,
deno ed by XYZ, and he ex e nal labo a o y space, deno ed by
xyz. The su aces o he solids a e assumed o be mechanically
and elec ically ee23, wi h su ace no mals aligned wi h he
z-axis. We conside an incoming homogeneous acous ic plane
(bulk) wa e expðik þiω Þ, whe e kand ωa e he wa e
ec o and angula equency, p opaga ing inside he xz-plane
(sagi al plane) om he posi i e z-axis di ec ion owa d he
su ace a z=0, wi h a posi i e x-componen o wa e ec o
(k
x
> 0). In addi ion, we only conside low equency acous ic
wa es wi h linea dispe sion and assume he usual quasis a ic
app oxima ion o piezoelec ic acous ic wa es19 sa is ying
E=−∇Φ, whe e Eand Φa e he elec ic field and he elec ic
po en ial, espec i ely.
An inciden bulk wa e sca e s in o a linea combina ion o
pa ial wa es a an in e ace. These pa ial wa es a e ei he eflec ed
o ansmi ed, and can ei he be homogeneous (bulk) wa es o
inhomogeneous (e anescen ) wa es bound on he su ace o he
solid23. The single su ace eflec ion and ansmission coe ficien s,
which desc ibe he ampli udes o hese sca e ed wa es, can be
calcula ed ollowing he mul iple eflec ion me hod p esen ed in
Sec ion III.B in Geng and Maasil a23. We deno e hese coe ficien s
wi h an o e head ba , as ollows: 
ð1Þ
in!Vis he coe ficien o an
incoming wa e om solid 1 ansmi ed in o a acuum elec ic
wa e, 
ð2Þ
V!αis he coe ficien o an acuum wa e ansmi ed in o
mode αin solid 2, and 
ðiÞ
Vis he coe ficien o an acuum wa e
eflec ed on he acuum side o he in e ace o solid i=1, 2. In
hese coe ficien s, α=1, ..., 4 co espond o he ou physically
allowed elec oacous ic pa ial wa e modes in he co esponding
solids 1 o 2. I should be no ed ha hese coe ficien s a e no he
di ec analogs o he F esnel coe ficien s25 om op ics.
A o al ansmission coe ficien
α
, which desc ibes he
ampli ude a io o a ansmi ed pa ial wa e αin solid 2 o an
incoming bulk wa e om solid 1, akes a o m23 ( o de ails, see
“Me hods”sec ion):
α¼

ð1Þ
in!V
ð2Þ
V!α
ekxd
ð1Þ
V
ð2Þ
Vekxd¼
ð1Þ
in!V
ð2Þ
V!α mðdÞ;ð1Þ
wi h k
x
he wa e ec o componen along he su aces,
which is conse ed in he unneling p ocess. This exp ession
can be in e p e ed as wo single su ace ansmission
Fig. 1 Schema ic o he sys em unde s udy. Two piezoelec ic solids 1, 2
a e sepa a ed by a acuum gap o wid h d. An incoming acous ic wa e om
solid 1 (posi i e z-axis o a labo a o y coo dina es xyz) wi h an inciden
angle θ
i
unnels ac oss he acuum gap in o solid 2 inside he xz-plane. XYZ
desc ibe he in insic c ys al coo dina es, which can be o a ed w. . . he xyz
coo dina es.
ARTICLE COMMUNICATIONS PHYSICS | h ps://doi.o g/10.1038/s42005-023-01293-y
2COMMUNICATIONS PHYSICS | (2023) 6:178 | h ps://d oi.o g/10.1038/s42005-023-01293-y | www.na u e.com/commsphys
coe ficien s 
ð1Þ
in!Vand 
ð2Þ
V!αcoupled by a geome ical mul iple
eflec ion ac o o e anescen elec ical wa es in he
gap mðdÞ¼½expðkxdÞ
ð1Þ
V
ð2Þ
VexpðkxdÞ123. I implici ly
depends on he inciden angle θ
i
no only ia kx¼ksin θi,
bu also ia he coe ficien s 
and 
, which a e unc ions o
x¼ω=ðksin θiÞ(“Me hods”).
De i a ion o he condi ion o comple e unneling. To ully
desc ibe he unneling o he acous ic wa e, we also look a he
ene gy ans e be ween he solids. The ime-a e aged powe flow
densi y (ene gy flux, uni s [Wm−2]) o a ansmi ed pa ial wa e
in he di ec ion no mal o he su aces (deno ed as P
α
) can be
ob ained om he eal pa o he no mal componen o piezo-
elec ic Poyn ing ec o (“Me hods”). Fo he unneled bulk
pa ial wa es, he ansmi ed powe ela es o he no mal com-
ponen o he inciden powe by P
α
=∣
α
∣2P
in
, in which he inpu
powe P
in
can be om a cohe en bulk wa e o om a he mal
phonon, whe eas he eflec ed o ansmi ed e anescen pa ial
wa es in solids 1,2 a e bound on o he su ace and ca y no powe
in he no mal di ec ion (P
α
=0i αis an e anescen mode).
As he e is no dissipa ion inside he acuum gap, he no mal
di ec ion powe flow densi y inside he acuum (deno ed by P
V
)
is equal o he o al no mal di ec ion ansmi ed powe densi y
(deno ed by P
Σ
). I is clea ha P
Σ
is he sum o P
α
o e all he
ansmi ed bulk wa es in solid 2, and we can w i e i using
Eq. (1)asPΣ¼∑αj
ð1Þ
in!V
ð2Þ
V!α mðdÞj2Pin, whe e α uns only o e
he bulk modes. The numbe o ansmi ed bulk modes can be
om ze o o h ee (in some cases ou 26), and i he e is no bulk
mode a ailable, he powe flow in bo h he acuum and solid 2
a e ze o. On he o he hand, he no mal powe flow inside he
acuum gap can be exp essed using he Poyn ing’s heo em unde
he quasis a ic app oxima ion as PV¼2j
ð1Þ
in!V mðdÞj2Re½
ð2Þ
VPin
(see Supplemen a y No e 1 o he de i a ion). As a esul , om
P
V
=P
Σ
we find a ela ion:
2Re 
ð2Þ
V
hi
¼∑
α¼bulk
j
ð2Þ
V!αj2:ð2Þ
Fu he mo e, i we assume ha he wo solids consis o he
same ma e ial wi h iden ical c ys al o ien a ions, wo addi ional
ela ions ha link he single su ace coe ficien s o he wo solids
can be ound by exploi ing he comple eness o he eigensolu ions
o he sca e ing p oblem (see Supplemen a y No e 2 o he
de i a ions). The fi s one ela es he eflec ion coe ficien s 
ðiÞ
Vo
he wo solids as:

V
ð2Þ
V¼

ð1Þ
V:ð3Þ
The second one s a es ha i he ansmi ed bulk wa e mode γin
he solid 2 is he same mode as he inciden wa e in solid 1, he e
exis s a ela ion:

ð1Þ
γ!V¼
ð2Þ
V!γ:ð4Þ
In addi ion, by compa ing he ela ion (4) wi h Eq. (2), we find
he condi ion:
2Reð
VÞ≥j
ð1Þ
in!Vj2;ð5Þ
whe e he equali y is sa isfied when he e exis s only one
ansmi ed bulk wa e mode in solid 2 and he mode is he
same as he inciden wa e in solid 1. By applying he ela ions (2)
and (3), P
Σ
can hen be simplified o:
PΣ
Pin
¼2Reð
VÞj
ð1Þ
in!Vj2
4Reð
VÞ2þe2kxdj

Vj2

2e2kxd;ð6Þ
which explici ly depends only on wo single su ace coe ficien s:

ð1Þ
in!Vand 
V.
Equa ion (6) shows ha he o al ansmi ed powe P
Σ
is
always less han he inciden powe P
in
i mo e han one
ansmi ed bulk wa e modes exis , since in ha case he
inequali y Eq. (5) akes he g ea e - han sign. This esul has
he implica ion ha comple e unneling, i.e., he ull ansmission
o he inciden powe , can’ be achie ed i he ansmi ed wa e
consis s o mul iple pa ial bulk wa es, in con adic ion o
p e ious wo k5.
In con as , i he e is only one ansmi ed homogeneous bulk
mode and i is he same mode as he inciden wa e, hen he equal
sign o Eq. (5) is alid, and Eq. (6) simplifies o:
PΣ
Pin
¼4Reð
VÞ2
4Reð
VÞ2þe2kxdj

Vj2

2e2kxd;ð7Þ
which e y much esembles he Fab y-Pe o -like o m o
ansmission coe ficien s o he nea -field adia i e hea
ans e 27. F om Eq. (7), i is clea ha he maximum ansmi ed
powe is exac ly equal o he inciden powe (P
Σ
=P
in
) when he
esonance condi ion:
j
Vj¼ekxd;ð8Þ
is sa isfied, simila o he co esponding condi ion o pe ec
pho on unneling in nea -field hea ans e 28,29. This p o es ha
(1) uni y ansmission (comple e unneling) o an acous ic wa e
ac oss a acuum gap is possible, and (2) he condi ion o i
depends explici ly only on he single su ace eflec ion coe ficien

V, he wa e ec o componen k
x
and he gap wid h d. The
physical explana ion o such comple e unneling is he exci a ion
o esonan coupled leaky su ace wa es on bo h in e aces
(mo e de ails below and in Supplemen a y No e 4), which is
undamen ally di e en om he p inciple o an i eflec ion in
op ics25.
In pa icula , wi h a gi en ma e ial and c ys al o ien a ion, 
V
is only a unc ion o he inciden angle and is independen o he
gap wid h o he exis ence o he adjacen solid. We p opose ha ,
as a ma e ial pa ame e , 
Vcould be de e mined expe imen ally
by measu ing he e ec i e su ace pe mi i i y ϵ
e
(
x
)24,30,31 o
he TM-wa e su ace impedance Z
p
(ω,
x
)32,33 o he piezoelec ic
solid. They a e ound o be ela ed by exp essions (see
Supplemen a y No e 3):

V¼iϵe ϵ0
ϵe þϵ0
;
V¼i1þi xϵ0Zp
1i xϵ0Zp
;ð9Þ
whe e ϵ
0
is he acuum pe mi i i y. The e ec i e su ace
pe mi i i y concep is use ul in he s udy o piezoelec ic
ma e ials, o example o he gene a ion and de ec ion o
acous ic wa es by ansduce s31 o o de e mining he gap wa e
modes be ween piezoelec ic solids24.Wefind he symme ic and
an isymme ic gap wa e condi ions can be simply exp essed by

V¼±iexpðkxdÞ, which a e he poles o he ansmission
coe ficien o Eq. (1) (Supplemen a y No e 3).
Nume ical examples and physical in e p e a ion o comple e
unneling be ween iden ical ZnO c ys als. We now u n o
demons a e he comple e unneling e ec wi h nume ical
examples o wo iden ical ZnO c ys als, using he o malism
de eloped be o e23. The fi s example is shown in Fig. 2, whe e
he wo c ys als a e sepa a ed wi h a scaled gap wid h o kd =
0.01, and a e bo h o a ed fi s wi h espec o he x-axis by
ϑ=46.89∘and hen o he z-axis by φ=88° (see Geng and
Maasil a23 o de ails on he c ys al o a ion p ocedu e). The
mode o he inciden wa e in his example is chosen o be he
slowes quasi- ans e sal wa e (ST), so ha he e exis s a c i ical
COMMUNICATIONS PHYSICS | h ps://doi.o g/10.1038/s42005-023-01293-y ARTICLE
COMMUNICATIONS PHYSICS | (2023) 6:178 | h ps://doi.o g/10.1038/s42005-023-01293-y | www.na u e.com/commsphys 3
inciden angle beyond which only one bulk ansmi ed wa e can
be ound, hus sa is ying he gene al condi ion o comple e
unneling.
In Fig. 2a, we plo he ansmi ance in o each bulk mode
P
α
/P
in
as a unc ion o he inciden angle θ
i
, whe e αcan be he
quasi-longi udinal (L), he as quasi- ans e sal (FT) o he slow
quasi- ans e sal (ST) mode, ca ego ized based on hei phase
eloci ies. We see ha o mos angles, ansmi ance is low,
excep o he wo sha p ansmission peaks o he ST mode
gi ing exac ly uni y ansmission a angles be ween 75° and 80°.
Ab up cu -o s a e isible o he ansmi ed L and FT modes,
co esponding o he c i ical inciden angles θLc28and
θFTc63:5. Beyond hese c i ical angles, he co esponding
modes become e anescen , bound on he su ace o he solid wi h
no di ec ene gy ansmission in o he bulk.
Figu e 2b p o ides a zoomed iew on he esonan ansmis-
sion peaks, now wi h wo di e en scaled gap alues kd =1 (blue
solid line) and kd =0.01 (o ange solid line), wi h an o e lay o
he j
Vjcu e (black dashed line), helping us also o unde s and
he double s uc u e. The wo addi ional ho izon al dashed lines
ep esen he alues o he RHS o Eq. (8) o he wo kd alues,
whe eas he dashed black cu e ep esen s he LHS o Eq. (8). I is
clea ha he uni y ansmission occu s whe e he esonance
condi ion is alid, p o ing consis ency be ween he analy ical
heo y and he nume ical app oach. In addi ion, we see ha wi h
he inc ease o he scaled gap wid h om 0.01 o 1, he sepa a ion
o he peaks is educed, and wi h a u he inc ease he wo
solu ions would me ge in o one a he maximum o j
Vj. Wi h
his pa icula ZnO c ys al o ien a ion, his maximum is abou 4
as shown in he plo , which leads o a maximum gap wid h o
kd ≈1.4 o obse e comple e unneling (me ged uni y ansmis-
sion peak). Fo ZnO (ST wa e eloci y =2780 m/s), and a
2 GHz equency ele an o de ice applica ions, his co e-
sponds o a qui e long physical dis ance o d=300 nm wi h he
pa ame e s and he o ien a ion used in he example.
In addi ion, i is also use ul o b iefly discuss how sensi i e he
powe ansmi ance is o de ia ions om he esonance
condi ion, based on he abo e nume ical example. Fo a fixed
equency o 2 GHz, he ansmi ance as a unc ion o bo h he
inciden angle θ
i
and he gap wid h dis p esen ed in Fig. 2c. We
see ha when he acuum gap is small, e.g., d< 100 nm, he wo
esonances a e well sepa a ed and a e sensi i e o de ia ions o
bo h dand θ
i
. Fo example, a 50% d op in ansmi ance (whi e
dashed lines) occu s wi hin a 10 nm change in do a 0.1 deg ee
change in θ
i
o he igh b anch. Howe e , he me ging o he
wo esonances leads o a highe de ia ion ole ance. θ
i
=76. 4° is
an example, whe e he ansmi ance emains highe han 50% o
a wide ange 150 nm < d< 500 nm, significan ly elaxing he
cons ain o he gap wid h con ol in measu ing he unneling.
Simila discussion can also be applied o a fixed gap wid h, o
which he ansmi ance becomes a unc ion o he angula
equency ωand he in-plane wa e ec o k
x
, as illus a ed in he
inse o panel (c).
Fig. 2 Angula and gap wid h dependence o he powe ansmi ance o an incoming ST wa e. a Powe ansmi ance P
α
/P
in
o he longi udinal α=L
(g een), he as ans e se α=FT (o ange) and he slow ans e se α=ST (blue) wa es, o an incoming ST wa e as unc ion o he inciden angle θ
i
,
o wo iden ical ZnO c ys als sepa a ed by a scaled gap kd =0.01ando ien edwi hazeni hangleϑ=46.89°andanazimu hangleφ=88° (inse ).
We used he aniso opic c ys al pa ame e s c
11
=20.97 × 1010 Nm−2,c
33
=21.09 × 1010 Nm−2,c
44
=4.247 × 1010 Nm−2,c
12
=12.11 × 1010 Nm−2,
c
13
=10.51 × 1010 Nm−2,c
66
=(c
11
−c
12
)/2, ϵ
xx
=8.55ϵ
0
,ϵ
zz
=10.2ϵ
0
,e
x5
=−0.48 Cm−2,e
z1
=−0.573 Cm−2,e
z3
=1.32 Cm−2, and he densi y
ρ=5680 kgm−3, aken omAuld
19.bZoomed iew on he peaks o he ansmi ance (le axis) wi h wo alues kd =1 (blue solid line) and kd =0.01
(o ange solid line). The single su ace eflec ion coe ficien j
Vjcu e (black dashed line) is o e layed ( igh axis) oge he wi h expðkxdÞ¼expðsin θiÞ
(blue ho izon al dashed line) and expðkxdÞ¼expð0:01 sin θiÞ(o ange ho izon al dashed line) o demons a e he esonance condi ion, Eq. (8), o he
woscaledgaps, espec i ely.cPowe ansmi ance (colo scale) as a unc ion o inciden angle θ
i
and gap wid h da a fixed equency o 2 GHz
(main panel), and as a unc ion o ω/k
x
and k
x
,wi hafixed d=300 nm (inse panel). The black solid line ep esen s he esonance condi ion (Eq. (8)),
and he whi e dashed lines indica e 50% powe ansmi ance. dF equencies o he symme ic (blue) and an isymme ic (o ange) esonances as
unc ions o gap wid h d o a fixed k
x
o 2π/kx =1μm, whe e he symme y e e s o he shape o he elec ical po en ial unc ion in he gap. The g een
dashed line shows he equency di e ence be ween he wo esonances.
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Fu he mo e, i is possible o ake ad an age o he esonances
in expe imen s and po en ial applica ions, such as he p ecise
con ol o a gap dis ance. By exci ing a bulk wa e wi h a known
in-plane wa e ec o k
x
( o example wi h an in e digi al
ansduce o finge spacing 2π/k
x
31), he equencies o he wo
unneling esonances become unc ions o he gap dis ance, as
demons a ed o ou nume ical example case in Fig. 2d. We see
ha while he highe - equency esonance
sym
depends weakly
on d, he lowe - equency esonance
an i
is highly sensi i e o d,
wi h ∂ /∂d≈0.3 MHz/nm a d< 20 nm. In addi ion, he equency
di e ence Δ =
sym
−
an i
(g een dashed line, igh axis scale) is
also sensi i e o d, eaching a esponsi i y ~0.5 MHz/nm a
d< 20 nm. Such ela ions can be en isioned o be used no only o
expe imen ally demons a e acous ic wa e unneling, bu also o
con ol a bu ied nanoscale gap dis ance wi h nanome e accu acy.
In gene al, he comple e esonan unneling can ake place o a
ange o c ys al o ien a ions. In Fig. 3, we show he nume ically
calcula ed maximal powe ansmi ance P
ST
/P
in
(o e all θ
i
)o an
inciden ST mode o a ansmi ed ST mode, as a unc ion o all
possible c ys al o a ions (ZnO has a c ys allog aphic 6 mm
sys em wi h uniaxial symme y19, hence all he unique o ien a-
ions o he c ys al can be ep esen ed by he di ec ion o he
c ys al c-axis (Z-axis) using a zeni h angle ϑ∈(0°, 180°) and an
azimu h angle φ∈(−90°, 90°)), using again pa ame e s o
aniso opic ZnO19 and a fixed scaled gap k
x
d=0.01. We find a
significan pa ame e space o o ien a ions, wi h mul iple
sepa a e egions, whe e comple e unneling is possible (da k ed
egions). To alida e he consis ency o he nume ics wi h he
analy ical condi ion, Eq. (8), we also plo a se o do ed con ou
lines in Fig. 3a o enci cle he o ien a ions sa is ying j
Vj>1, whe e
uni y ansmission is possible, finding excellen ag eemen .
Ano he obse a ion is ha he inciden angle θ
i
sa is ying
comple e unneling a ies o di e en c ys al o ien a ions
(Fig. 3b, c), eaching as low alues as 60° in some cases (Fig. 3c).
To unde s and he physics, we fi s conside he h ee
ellipsoidal uni y ansmission a eas a ound ϑ=90° (Fig. 3).
Inside hese a eas, he inciden ST wa es a e no pu e shea wa es
and he e o e couple o he o he pa ial wa es (L, FT) a he
su ace. As a esul , when he incoming ST wa e has an inciden
angle beyond he c i ical angle o he FT mode, he eflec ed FT
wa e becomes e anescen , wi h i s ene gy concen a ed on he
su ace. Fo hose o ien a ions he FT-mode wa es a e p edomi-
na ely pola ized in he di ec ion o he c-axis, he di ec ion o he
piezoelec ic dipole, c ea ing a s ong piezoelec ic esponse. Tha
exci es la ge elec ic po en ial di e ences on he su ace and
hence gi es ise o a s ong elec ic coupling ac oss he gap, which
finally enables he esonan ansmission. On he o he hand,
when he azimu h o a ions app oach φ=±90° wi h ϑ=90°, he
c-axis aligns wi h he x-axis and he ST mode becomes a pu e
shea mode, pola ized pe pendicula o he sagi al plane. Then
he inciden ST wa es a e e y weakly piezoelec ic, and also
decouple om all o he pa ial modes.
O he ea u es can also be obse ed in Fig. 3a. Nodes ha ing
low ansmission a a ound φ=±25° and ϑ=90° appea . This is
because he elec ic po en ial exci ed by he eflec ed FT wa e
mode change pola i y a ound hese nodes, leading o minimized
po en ial di e ences and weak coupling be ween he wo su aces.
In addi ion, uni y ansmission is also obse ed in ou small
a eas a ound φ=±90°, whe e he single su ace eflec ion
coe ficien 
ð1Þ
ST!FT o he eflec ed FT pa ial wa es inc eases
significan ly. This indica es an enhanced mode con e sion
be ween he ST and FT pa ial wa e modes a hese o ien a ions,
p o iding la ge elec ic po en ial di e ences on he solid- acuum
in e ace again ia he e anescen FT wa e, leading o s ong
unneling signal. A mo e de ailed discussion o he physical
in e p e a ion o he esonance can be ound in Supplemen a y
No e 4.
Ou nume ical o malism can also be applied o he pa icula
case s udied wi h a simplified model be o e5, he de ails o which
can be ound in Supplemen a y No e 5. We do no find comple e
unneling o he incoming modes and he c ys al o ien a ion in
ques ion, in con adic ion o p e ious wo k5.
Conclusions
In conclusion, we ha e analy ically and nume ically p o en i is
possible o acous ic wa es o comple ely unnel ac oss a acuum
gap be ween wo piezoelec ic solids, up o gap sizes o abou a
wa eleng h. We showed ha such comple e unneling, wi h uni y
powe ansmi ance, is possible only i one ansmi ed pa ial
bulk mode is exci ed, i being he same mode as he inciden
wa e. We de i ed a simple esonance unneling condi ion o he
comple e unneling e ec , Eq. (8), and p o ed i s alidi y and
ange o applicabili y wi h nume ical examples o a bi a ily
o a ed ZnO c ys als. As his is a s ong and no a a e e ec , i
could ha e an impac in u u e acous ic wa e de ices, as well as in
o he applica ion a eas conce ning phonons, such as con olling
hea anspo , op omechanics and quan um in o ma ion science.
Me hods
Ex ended S oh o malism and mul iple eflec ion app oach. In piezoelec ic
solids, he dynamics o a p opaga ing plane (bulk) wa e expðik þiω Þa e
go e ned by he elas ic equa ion o mo ion ∇⋅σ=ρ∂2u/∂ 2and Gauss’s law ∇⋅
D=0, oge he wi h he piezoelec ic cons i u i e ela ions19:
σ¼cE:SeE
D¼e:SþϵSEð10Þ
whe e S,σ,u,Da e he elas ic s ain, elas ic s ess, mechanical displacemen and
elec ic displacemen fields, ρ,cE,e,ϵSa e he mass densi y, elas ic s i ness enso
a cons an elec ic field, piezoelec ic s ess enso and elec ic pe mi i i y enso
a cons an s ain, espec i ely. The double do p oduc indica es summa ion o e
pai ed indices be ween second- ank and highe - ank enso s, and he s ain-
displacemen ela ion eads as S
ij
=(∂u
i
/∂
j
+∂u
j
/∂
i
)/2.
An inciden plane wa e is sca e ed in o a linea combina ion o pa ial wa es a
an in e ace, which a e ei he eflec ed o ansmi ed. The gene al solu ions o such
Fig. 3 C ys al o a ion map o comple e unneling be ween ZnO c ys als.
aColo scale o he ST- o-ST mode maximum powe ansmi ance P
ST
/P
in
o e all inciden angles, plo ed as unc ion o c ys al o a ion angles ϑand
φ o aniso opic ZnO. The do ed lines enci cle he egions whe e j
Vj>1.
b,cshow he ange o θ
i
whe e j
Vj>1 o wo fixed φ(b), o ϑ(c). He e, we
fixkxd¼kd sin θiins ead o kd, as comple e unneling can be achie ed by
uning k
x
ei he by changing he inciden angle θ
i
o by he angula
equency ω.
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pa ial wa es ha sa is y he go e ning equa ions ake he exp essions34–36:
u¼∑
αbαAαeiðkxxþkyyþpαkxzω Þ
Φ¼∑
αbαϕαeiðkxxþkyyþpαkxzω Þ
nσ¼ikx∑
αbαLαeiðkxxþkyyþpαkxzω Þ
nD¼ikx∑
αbαDαeiðkxxþkyyþpαkxzω Þ;
ð11Þ
in which nis he uni ec o o he z-axis. A
α
,ϕ
α
,L
α
,D
α
a e he no malized
cons an s desc ibing he pola iza ion ec o , he elec ic po en ial, he ac ion o ce
and he no mal p ojec ion o he elec ic displacemen o a pa ial wa e mode α,
espec i ely. b
α
a e dimensionless ampli udes o he pa ial wa es, and p≡k
z
/k
x
.To
a oid edundan w i ing in he ollowing exp essions, we omi he common phase
ac o expðikxxikyyþiω Þsha ed by all solu ions.
In his s udy, we sol ed hese go e ning equa ions unde he amewo k o
ex ended S oh o malism23, in which Eq. (10) is combined and ea anged in o an
eigh -dimensional eigen alue p oblem37,38 in he o m o :
Nð xÞξα¼pαξα;ð12Þ
whe e Nis 8 × 8 eal ma ix and
x
≡ω/k
x
is he x-componen o he phase eloci y.
Gene ally, eigh linea ly independen eigen ec o s ξα¼½Aα;ϕα;Lα;DαTand
co esponding eigen alues p
α
can be ob ained o pa ial wa e modes α=1, ..., 8.
These eigen ec o s ollow he o hono maliza ion and comple eness condi ions:
ξT
α
^
Tξβ¼δαβ ð13Þ
∑ξα^
Tξα¼^
I8´8;ð14Þ
whe e he ope a o ⊗deno es he ou e p oduc o wo ma ices, δ
αβ
is he
K onecke del a, ^
I8´8is 8 × 8 uni ma ix, and ^
T akes he o m:
^
T¼O4´4^
I4´4
^
I4´4O4´4
"# ð15Þ
whe e O
4×4
and ^
I4´4a e 4 × 4 ze o and uni y ma ices.
The con inui y o he elec ic po en ial (Φ(i)=Φ
V
, whe e he subsc ip i=1, 2
indica es he medium index and he subsc ip V indica es he acuum) and he
no mal componen o elec ic displacemen (n⋅D(i)=n⋅D
V
), as well as he
condi ion o a mechanically ee su ace (n⋅σ(i)=0) en o ce he bounda y
condi ions o he wo solid- acuum in e aces:
bð1Þ
in Uð1Þ
in þ∑
4
α¼1bð1Þ
αUð1Þ
α¼bVþUVþþbVUV;
∑
4
α¼1
~
bð2Þ
αUð2Þ
α¼bVþUVþekxdþbVUVekxd;
ð16Þ
in which we in oduce 5 × 1 column ec o s UðiÞ
γ¼½ϕðiÞ
γ;DðiÞ
γ;LðiÞ
γT o wa e modes
γ=in, α, whe e he subsc ip in indica es he inciden wa e mode, α=1, ..., 4
co esponds o ou physically allowed wa e modes in hei co esponding medium
i¼1;2;UV±¼½ϕV±;DV±;0;0;0T, and ~
bð2Þ
αbð2Þ
αexpðipð2Þ
αkxdÞ o simplici y. In
he acuum egion, he elec ic po en ial and displacemen fields ake he o m:
ΦVðzÞ¼bVþϕVþekxzþbVϕVekxz
nDVðzÞ¼ϵ0kxbVþϕVþekxzþϵ0kxbVϕVekxz;ð17Þ
whe e ϕV±¼1=ffiffiffiffiffiffiffiffiffiffiffiffi
±2iϵ0
p, no ing he no maliza ion condi ion o he acuum
mode 2ϕV±DV±¼1.
The ampli ude ac o s bðiÞ
αcan be sol ed om he bounda y condi ions o Eq.
(16), ollowing he mul iple eflec ion me hod in oduced in Geng and Maasil a23:
he single su ace eflec ion (
ðiÞ
in!α;
ðiÞ
V) and ansmission (
ðiÞ
V!α;
ðiÞ
in!V) coe ficien s
a e calcula ed fi s ia sca e ing ma ices SðiÞ o solid i=1, 2, and he o al
ansmission coe ficien
α
o a pa ial mode αis hen ob ained by coupling he
single su ace coe ficien s wi h a mul iple eflec ion ac o
m
(d), which explici ly
depends on he gap dis ance d. We no e he e ha he o e lined single su ace
coe ficien s desc ibe he sca e ing o he elec oacous ic wa e as i he e is no
second adjacen solid.
The 5 × 2 sca e ing ma ices Sð1Þand Sð2Þ ake he ollowing o m:
Sð1Þ¼
ð1Þ
ð1Þ

ð1Þ
in!V
ð1Þ
V
"#
¼Uð1Þ
1; :::; Uð1Þ
4;UVþ
hi
1Uð1Þ
in ;UV
hi
;
Sð2Þ¼
ð2Þ
ð2Þ

ð2Þ
in!V
ð2Þ
V
"#
¼Uð2Þ
1; :::; Uð2Þ
4;UV
hi
1Uð2Þ
in ;UVþ
hi
ð18Þ
whe e he exp ession 
ðiÞ¼½

ðiÞ
in!1; :::;
ðiÞ
in!4Tand 
ðiÞ¼½

ðiÞ
V!1; :::;
ðiÞ
V!4Ta e he
single su ace eflec ion and ansmission coe ficien s o modes α=1, ..., 4.
The o al ansmission coe ficien
α
om an incoming bulk wa e in solid 1 in o
a pa ial wa e o mode αin solid 2 can be ob ained as:
α
~
bð2Þ
α
bð1Þ
in
¼
ð1Þ
in!V
ð2Þ
V!α mðdÞð19Þ
whe e he mul iple eflec ion ac o is mðdÞ¼½expðkxdÞ
ð1Þ
V
ð2Þ
VexpðkxdÞÞ1.
Equa ion (19) is iden ical o Eq. (1) in he main ex .
The ime-a e aged ansmi ed powe flow densi y in he di ec ion no mal o
he su aces om solid 1 o 2 can be exp essed by he eal pa o he piezoelec ic
Poyn ing ec o in he no mal di ec ion34:
Pα¼ωkx
4jbαj2ξT
α
^
Tξ
α:ð20Þ
Fo ansmi ed homogeneous (bulk) wa es, ξT
α
^
Tξ
α¼±ξT
α
^
Tξα¼±1, due o
he S oh-no maliza ion condi ion (Eq. (13)). The e o e ∣
α
∣2can be in e p e ed as
he powe flow a io ( he ansmi ance) o he ansmi ed bulk pa ial wa e o e
he inciden wa e in he no mal di ec ion:
Pα¼j αj2Pin:ð21Þ
Da a a ailabili y
All ele an da a a e a ailable om he au ho s upon eques .
Code a ailabili y
All ele an code o simula ions a e a ailable om he au ho s upon eques .
Recei ed: 26 Janua y 2023; Accep ed: 30 June 2023;
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