Electron-phonon vertex and its influence on the superconductivity of two-dimensional metals on a piezoelectric substrate

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by  David G. González

Electron-phonon vertex and its influence on the superconductivity of two-dimensional
metals on a piezoelectric substrate
David G. González,1,2,∗ Fernando Sols,1,2 Francisco Guinea,3,4 and Ivar Zapata1
1Departamento de Física de Materiales, Universidad Complutense de Madrid, E-28040 Madrid, Spain
2Campus de Excelencia Internacional, Campus Moncloa UCM-UPM, E-28049 Madrid, Spain
3IMDEA Nanociencia, Calle de Faraday 9, E-28049 Madrid, Spain
4Department of Physics and Astronomy, University of Manchester,
Oxford Road, Manchester M13 9PL, United Kingdom
(Dated: September 16, 2016)
6
Weinvestigatetheinteractionbetweentheelectronsofatwo-dimensionalmetalandtheacoustic
1
phonons of an underlying piezoelectric substrate. Fundamental inequalities can be obtained from
0
generalenergyarguments. Asaresult,phononmediatedattractioncanbeproventoneverovercome
2
electron Coulomb repulsion, at least for long phonon wavelengths. We study the influence of these

p phonons on the possible pairing instabilities of a two-dimensional electron gas such as graphene.
e
S
5 I. INTRODUCTION the one with the largest electromechanical couplings at
1 all. Moreover,theobtainedvertexareexpressedinterms
Surface acoustic waves (SAWs) [1] have been used for of a matching constant whose physical interpretation is
l] decadesasavaluablescientificandtechnologicaltool. In ratherobscure, allowingforjustorderofmagnitudeesti-
al thecontextofelectronics,theyareoftenexcitedinpiezo- mates.
h electric materials [2–7], where the mechanical and elec- In the present work we calculate a general electron-
- trical fields are coupled. In particular, they have been phonon interaction [see Eq. (7)], which is expressed
s
e applied as experimental probes of the quantum Hall ef- solely in terms of physical quantities characterizing the
m fects in two-dimensional electron gases (2DEG) [8]. response of the substrate surface. We emphasize that
. On the other hand, there is an increased interest in all the quantities appearing in the vertex are both com-
t
a 2DEGsincetheisolationofgraphenein2004andthepro- putable from linear piezo-elasticity theory and experi-
m duction of other two-dimensional (2D) materials which mentallymeasurable. Oneofthem,theelectromechanical
- followed it. Due to their unusual character, the prop- coupling coefficient,KR,willturnouttobecentraltoall
d erties of graphene electrons have been intensively stud- computations,servingasanaturaldimensionlessparam-
n ied during the last decade [9]. Although graphene on a eterwhichprovidesthescalefortheeffectofthesubstrate
o
substrate has received considerable attention, relatively piezoelectricity on the 2D electron system. Moreover,
c
[ few studies have been devoted to the case of graphene from very general considerations explained in Appendix
3 in contact with a piezoelectric material. These include A,weareabletoprovideboundsonitssize: 0≤KR <1
the propagation of surface acoustic waves on graphene [see Eq. (16)]. It is important to note that the vertex
v
[10], some acoustoelectric effects [11] [12], and the relax- written here is derived within the framework of linear
8
0 ation induced by the surface acoustic wave quanta on piezo-elastictheory, whichmeansthatitsvalidityshould
5 graphene electrons [13], among others. Recently, it has be restricted to low amplitude, low frequency and long
4 been proposed that surface acoustic waves (SAW) can wavelength phenomena. Bulk modes are also left aside
0 provide a diffraction grating for the conversion of light in this work. On the other hand, our study is not re-
1. into graphene plasmons [14]. strictedto anyapproximationbasedon thesymmetry or
0 ThecouplingofthepiezoelectricSAWtotheelectrons the piezoelectric softness of the substrate.
6 ina2DEGoringraphenehasbeencomputed,withincer-
Equipped with the effective electron-electron inter-
1 tain simplifying assumptions and for definite substrate
v: crystal structures, in Ref. [15, 16]. The derivation of action which results from taking into account the ex-
change of these acoustic phonons between the electrons
i theelectron-surfacephononinteractionforapiezoelectric
X in graphene (or other 2D materials), the question can
material has been performed only within a purely elastic
be raised of whether these interactions might be at-
r Rayleigh wave approximation [16] or for definite propa-
a tractive and, depending on some material parameters
gatingdirections[15]. Butthesemethodsfailinstronger
and the tunable electronic density of graphene, perhaps
piezoelectric materials and for other crystal symmetries.
strong enough to generate electron Cooper pairing and
For instance, the isotropic Rayleigh wave approximation
superconductivity [18]. We can further ask whether
in the case of lithium niobate leads to surface acoustic
such a superconductivity could be observed at temper-
wave velocities about 15 % too low and lacking the cor-
atures attainable in a laboratory without the recurring
rect angular dependence [3, 17], and this material is not
to huge non gate-achievable doping levels, as predicted
forintrinsicgraphenephonons[19]orforKohn-Luttinger
or electronic superconductivity in other graphene het-
∗ email:[email protected] erostructures with repulsive interactions [20]. From the
2
z → 0+ side) and ϕ is the electric potential, which is
continuous because we do not allow for anything more
singular than surface charges. The same ratio taken be-
low the surface (z → 0− ) allows us to introduce the
Air or vacuum (relative) piezoelectric surface permittivity,
1 D−(q,ω)
ε˜(q,ω):=− 3 , (3)
ε qϕ(q,ω)
vac
Piezoelectric
whichcanalsobestraightforwardlyexpressedintermsof
the surface impedance tensor [21]. From Poisson’s equa-
tion and Eqs. (1-3) it follows
FIG. 1. A flat piezoelectric substrate with an interface to air
orvacuum,withb=[cos(θ),sin(θ)]thepropagationdirection
of the piezoelectric SAW. σ(q,ω)=D3+(q,ω)−D3−(q,ω)=qϕ(q,ω)[1+ε˜(q,ω)]εvac.
(4)
Further analysis summarized in Appendix A shows
interaction vertex derived in the present work, it can
that ε˜(q,ω) has a dependence of the form ε˜(q,ω) =
be shown that the relative size of the static phonon-
ε˜(q/ω). An immediate conclusion from Eq. (4) is
mediated electron-electron interaction with respect to
that a purely piezoelectric wave (i.e., without sources,
the original Coulomb repulsion turns out to be exactly
σ(q,ω)=0),canpropagatewithoutdampingifandonly
K2 < 1, because of the aforementioned general inequal-
R if ε˜(q/ω)+1 = 0. Thus, if the phase velocity is vs(θ),
ity. However, by applying the Eliashberg formalism to
then ε˜(b/v (θ))+1 = 0 and the dispersion relation of
s
graphene [19], we are able to assess, in terms of K , the
R the obtained, called piezoelectric Rayleigh waves (here
influence that these low-frequency and long-wavelength
referred to as SAW) is ω =v (θ)q.
s
phonons have on possible BCS type instabilities. The
For the high-frequency limit we introduce ε˜ (θ) :=
conclusion is that present piezoelectric materials are not HF
ε˜(q/ω),ω →∞. Thisshouldbetheanisotropicdielectric
able to either induce s−wave pairing by themselves or
function valid all the way up to the optical region, and
affect in a significant way any pairing instability which
should take into account all screening processes in the
couldbealreadypresentingraphene. Wenote, however,
substrate except for the slow piezoelectric ones, which
that this conclusion could be substantially changed in
areestimatedbelowforthesubstrateofthe2Delectronic
case new hard piezoelectric materials would be found.
material.
A central quantity in the evaluation of devices which
II. ELECTRON-PHONON INTERACTION use piezoelectric Rayleigh waves is the SAW electrome-
chanical coupling coefficient K (θ), introduced through
R
the relation at 1+ε˜(b/v (θ))=0:
ThesystemtobeconsideredisdepictedinFig.1. The s
z =0 interfacial surface is supposed to be free of tension
and, when acting as a substrate to a deposited graphene K2(θ)/2 (cid:20) ∂ε˜(b/v)(cid:12) (cid:21)−1
strhoedetes(aosrwaenlyl. Hotohweerv2erD,bcyhainrtgreodduscyisntgema)s,urffraeceeocfhaerlegce-, ε˜HFR(θ)+1 = vs(θ) ∂v (cid:12)(cid:12)v=vs(θ) . (5)
onecanexpressthe2Dresponseofthepiezoelectricsub-
strate, the piezoelectric surface permittivity, as the ratio In Appendix A we show that very general considera-
ofanelectricdisplacementtoanelectricfield. Tobepre- tions require
cise, let us allow for a surface charge with a harmonic
dependence along q=q[cosθ,sinθ] (here R=(x,y)) 0≤K (θ)<1, (6)
R
σ(R,t)=σ(q,ω)ei(q·R−ωt), (1)
which is one of the central results of this work. The
and, from linearity, all quantities evaluated at the sur- inequality (6) is crucial because, as we will show, it im-
face have the same 2D space-time dependence. We use pliesthatatsmallfrequenciespiezoelectricphononscan-
SI units throughout this work. Because the medium at not provide the sufficient screening to overcome the bare
z >0 has a dielectric constant εvac, then, from Poisson’s Coulomb repulsion.
equation it follows that
In Ref. [5] it is shown that there is a relation be-
D+(q,ω) tween the amplitude of electric potential at the surface,
εvac = qϕ3(q,ω) , (2) ϕ0 =ϕ(q,ω) and the total energy Hharm, see Eq. (A17).
Hence, standard quantization procedure (see the Ap-
whereD+isthenormaltothesurfaceFouriercomponent pendix, subsection A2) shows that the interaction be-
3
of the electric displacement field over the surface (on the tween the 2D electronic material sheet and the sponta-
3
neous piezoelectric Rayleigh waves can be written as electric function and effective interaction are:
e2
HeP-Aph = √1A (cid:88)γqPAa†k+q,σak,σbq+H.c. , Veff(q,ω)= 2ε(q,ω)εvacq
k,q,σ v(0)+VPA(q,ω)
K (θ)(cid:115) (cid:126)e2v (θ) (cid:115)πα (cid:126)2v v (θ) = (cid:104) q ph (cid:105) (12)
γPA = R s =K (θ) fs F s , 1− v(0)+VPA(q,ω) Π (q,ω)
q 2 ε (θ)ε R ε (θ) q ph 0
HF vac HF
(7) which can also be written as
(cid:12) (cid:12)2
v(0) (cid:12) γPA (cid:12)
where A is the area of the sample, ak,σ,a†k,σ the electron Veff(q,ω)= ε q(q,ω) +(cid:12)(cid:12)ε q(q,ω)(cid:12)(cid:12) G˜PA(q,ω) .
operators with σ = ± the electron spin, b ,b† are the RPA (cid:12) RPA (cid:12)
piezoelectric phonon operators, α :=e2/4πkε k(cid:126)v and (13)
fs vac F
εHF(θ):=(ε˜HF(θ)+1)/2. Thevalidityoftheshowninter- where εRPA(q,ω) ≡ 1−vq(0)Π0, with Π0 the irreducible
action hamiltonian Eq. (7) requires two further assump- polarization function,
tions: first, the 2DEG or multilayered graphene sample GPA(q,ω)
shouldbethinenoughsothatinEq. (A8),kd(cid:28)1,where G˜PA(q,ω)= 0 . (14)
z ≡ d is the width of the sample and k is the maximum 1− |γqPA|2GP0A(q,ω)Π0(q,ω)
εRPA(q,ω)
allowed phonon momentum. And second, this maximum
In the low frequency limit, Π (q,ω (cid:39) 0) (cid:39) −D(E ) =
allowed momentum should be sufficiently small for the 0 F
−2k /π(cid:126)v , for monolayer graphene [23], or Π (q,ω (cid:39)
classical piezo-elasticity theory, as shown in Eqs. (A2- F F 0
0) (cid:39) −D(E ) = −m/2π(cid:126)2 for a 2DEG with effective
A4), to remain valid. We assume that a maximum mo- F
mass m [24]. These two last static limits are exact for
mentum on the order of k ∼ 106 −107cm−1 does not
F q < 2k . In Eq. (13), the total interaction has been
violate this last restriction. F
rewritten as the sum of a purely electronically screened
The resulting total Hamiltonian for the combined sys-
Coulomb repulsion and a phonon-induced effective part
temof2DelectrongasandpiezoelectricRayleighphonon
in which the vertex and phonon-propagator are also
is
screened by just the conducting electrons of the 2DEG
H =(cid:88)E a† a +(cid:126)(cid:88)ω b†b or graphene [18, 25]. For frequencies small in the scale
k k,σ k,σ q q q
of the acoustic phonons (or the Bloch-Grüneisen tem-
k,σ q
perature k T := 2(cid:126)v k ), the bare electron-phonon-
1 (cid:88) B BG s F
+HPA + v(0)ρ(q)ρ(−q) , (8) electron interaction contributes to the long-range partof
e-ph 2A q
q thetotalinteractionwithaq-dependencesimilartothat
of the Coulomb repulsion:
where E is the electron energy for a 2D wave-vector
k
k, ω = v (θ)q is the dispersion relation for the acous- 2|γPA|2
q s VPA(q,ω (cid:39)0)=|γPA|2GPA(q,ω (cid:39)0)=− q ,
tic piezoelectric SAW phonon of 2D wave-vector q and ph q 0 (cid:126)v q
s
v (θ) is the SAW propagation velocity, and ρ(q) = (15)
s
(cid:80) a† a is the Fourier transform of the electron Note that it is the acoustic phonon propagator
k,σ k+q,σ k,σ
density. GPA(q,ω (cid:39)0) that introduces the coulombic long-range
0
WeusethebareCoulombelectron-electroninteraction dependence in q via the dispersion of the modes. In the
as next subsection we shall see that a similar final q depen-
dence has a different origin.
e2 In the limit of low frequencies, ω (cid:39) 0, there can be
v(0) = , (9)
q 2ε (θ)ε q no effective attraction for electrons close to the Fermi-
HF vac
surface because, as shown in Eq. (A34), the following
whichcontainsallhigh-frequencyscreeningprocessesex- inequality is satisfied
cept for piezoelectric ones.
−VPA(q,ω (cid:39)0)
The bare electron-electron interaction mediated by ph =K2(θ)<1 (16)
phonons is [18, 22] v(0) R
q
VPA(q,ω)=|γPA|2GPA(q,ω), (10) whichis,inconjunctionwiththeinteractionvertexgiven
ph q 0 by Eq. (7), a central result of this paper.
where
2ω A. Comparison with optical phonons
GPA(q,ω)= q . (11)
0 (cid:126)(cid:0)ω2−ω2 +iη(cid:1)
q
For simplicity, we focus a single branch of the longitu-
isthebarepiezoelectricacousticphononpropagator(η → dinal optical (LO) for which we assume a constant fre-
0+). The resulting RPA-type approximation to the di- quency ω . The total Hamiltonian reads as in Eq. (8)
0
4
except for the replacements: Moreover, in case such over-screening occurred, the
staticdielectricconstantfromEq. (12)wouldpredictun-
ωq →ω0 (17) physical features such as unstable phononic modes with
e2 ω˜(qc)=0forsomeqc (cid:54)=0andevenimaginaryfrequencies
v(0) →v(∞) := (18) for q < q . No matter how small the absolute difference
q q 2ε ε q c
∞ vac |1−K2(θ)| happened to be, there would always exist a
ε +1 R
ε := ∞ (19) pole for the static (12) at small enough q (what cannot
∞ 2
occurinstandardBCSmetals),signalingadifferenttype
(cid:115)
e2(cid:126)ω of instability, possibly a charge density wave.
γPA →γOP := g 0 (20)
q q 2ε q Ontheotherhand,theresult(16)forthevertexcould
vac
still lead to higher angular momentum pairing instabili-
(cid:18) (cid:19)
1 1
g := − >0, (21) ties (as in the Kohn-Luttinger mechanism [28]) provided
ε∞+1 ε0+1 that K2(θ) is sufficiently large and anisotropic, a case
R
not considered by us.
wherestandardnotationfordielectricsisused: ε thedi-
∞
electric constant coming from very high frequency inter-
bandelectronictransitionsandε wouldbestaticdielec-
0
tric constant in the absence of the piezoelectric phonons A. Eliashberg formalism [19]
at frequencies much smaller than ω .
0
Again, as shown in the discussion around Eq. (15),
Thepreviousreasoningabouttheabsenceofsupercon-
for small frequencies (ω (cid:28) ω ), the bare phonon-
0 ducting instabilities, is incomplete and somewhat over-
mediated electron-electron interaction contributes to the
simplified. Three reasons support this claim: (i) Long-
long-range part of the total interaction like the Coulomb
wave piezoelectric phonon excitations (as considered in
repulsion:
thepresentwork)canneverbetheonlysourceofeffective
electron-electron interactions; in particular, we have not
e2
VOP(q,ω (cid:39)0)=|γOP|2GOP(q,ω (cid:39)0)=−g . taken into account the short range electric fluctuations
ph q 0 εvacq of the substrate. (ii) There is definitely some dynamic
(22)
over-screening at high frequencies [see Eq. (10)]. And
However, in contrast to the piezoelectric case, here it is
(iii) Coulomb interaction has to be properly renormal-
the vertex that introduces the coulombic dependence in
ized by taking into account collisions with high momen-
q.
tum transfer, which diminishes the Coulomb repulsion
At small frequencies, ω (cid:28)ω , a single optical phonon
0 and thus comparatively strengthens the other attraction
is not enough to provide over-screening, because
mechanisms.
Leaving aside the first objection momentarily, we can
−VOP(q,ω (cid:39)0) ε −ε
ph = 0 ∞ <1 . (23) use the Eliashberg formalism, as applied to graphene in
vq(∞) ε0+1 Ref. [19], to deal with the other two objections. The ef-
fectiveinteractioncouldcausesuperconductinginstabili-
tiesifadimensionlesselectron-phononcouplingλPA hap-
III. EFFECT OF PIEZOELECTRIC PHONONS pened to be greater than an also dimensionless Coulomb
ON SUPERCONDUCTING INSTABILITIES pseudopotential µ∗ coming from high-energy renormal-
izations [19, 29]. The coupling constant λPA in the
From (16) and (12), we see that, in the static limit Eliashberg formalism is the same appearing in (the real
(ω (cid:39)0), and for q <2k , V can be written in the form part of the) self-energy calculations to renormalize the
F eff
Fermi velocity [30] and is given by:
[1−K2(θ)]v(0)
Veff(q,0)= 1+[1−KR2(Rθ)]vq(0q)D(EF) , (24) α2 F(ω)= |γPA|2 (cid:113)1−(ω2/kvFs)2 ,
wherewenotethatwehavenotassumedq (cid:28)kF, asdis- PA 2π2(cid:126)2vsvF (1+ ωk/TvFs)2
cussed inthe paragraphfollowing(7). From theinequal- ∞
(cid:90) α2 F(ω) r
ity in (16), we are led to conclude that over-screening of λPA =2 PA dω = sK2 F(2r ) , (25)
ω π R s
the Coulomb repulsion by the phonon-mediated attrac-
0
tion is not possible. Moreover, and following standard
1 √
textbook reasoning (see for example [26]), we conclude (cid:90) t 1−t2dt (1−2x2)acosh(x−1)
F(x)= =−2+xπ+ √ ,
that BCS-type instabilities must also be ruled out. A (t+x)2 1−x2
similarresultholdsforasinglebranchofopticalphonons, 0
as can be seen from Eq. (23) (see however [27] for the
effect of multiple optical phonon branches from the sub- where r (θ):=α /ε(θ), and the symbols r ,v ,γPA,K2
s fs s s R
strate on superconducting instabilities). stand for the Fermi surface angle-averaged quantities of
5
the same name. The constant µ∗ equals
1D(E )V
µ∗ = 4 F , (26)
1+ 41D(EF)V log((cid:126)EωFc)
where ω is some energy cutoff which should satisfy
c
ω (cid:28) ω (cid:28) E /(cid:126) [19] and V comes from the
Debye c F
FermisurfaceaverageoftheThomas-Fermirenormalized
Coulomb repulsion vq(0)/(1+ kTqF). We have
1 r
D(E )V = sG(2r ), (27)
4 F π s
1 √
(cid:90) 1−t2dt πx (cid:112)
G(x)= =−1+ + 1−x2acosh(x−1) ,
t+x 2
0
and therefore, provided that one takes (cid:126)ω (cid:39)k T , so
c B BG
(cid:16) (cid:17)
that log EF (cid:39)log(vF )(cid:39)5. Thus, an estimate of the
(cid:126)ωc 2vs
effective pseudo-potential is
µ∗ (cid:39) rπsG(2rs) . (28)
1+ 5πrsG(2rs)
There could be intravalley [31] superconducting instabil-
ities provided that
FIG. 2. (Color online) Critical temperature and variation
λPA F(2r )(cid:20) 5r (cid:21) from the “bare” one in [19] for graphene on a piezoelectric
1< =K2 s 1+ sG(2r ) , (29) substrate,asafunctionoftheconductionbanddensity. Three
µ∗ RG(2rs) π s pairs of plots are given for three different values of KR and
the two values of the constant C =3.5 and 5 in Eq. (5.1) for
which imposes a constraint on the value of K2 from the λ in the previous reference [19].
R
piezoelectricsubstratewithrespecttoquantitiesdepend-
ingonr . ThecouplingK shouldbeverylargeandac-
s R
tuallygreaterthan1forthischoiceofω ,althoughthere instabilities,asshowninFig. 2. Weemphasize,however,
c
could exist superconductivity in this idealized case of a thatwearenotclaimingthatapiezoelectricsubstrateper
systemconsistingjustofthegrapheneelectronsandlong se necessarily increases the critical temperature, since it
wavelength piezoelectric phonons, provided that E /ω couldbethecasethatotherpiezoelectricfluctuationsnot
F c
is larger and K2 close to 1. considered in the present study (e.g. shorter wavelength
R
In order to amend the first objection, we have to modes) could work against pairing instabilities.
consider proper phonons of the electronic system (here
we go on considering graphene), in conjunction with
the short range of the piezoelectric ones. Then, pair- IV. CONCLUSIONS
ing instabilities due to intervalley scattering have to
be considered as well, because intravalley scattering
Inconclusion,wehavederivedageneralexpressionfor
terms contribute also to the intervalley pairing gap.
the two-dimensional electron-phonon piezoelectric inter-
With the notation in Ref. [19], an estimate on the
action valid for any piezoelectric substrate covered by a
critical temperature for the intravalley pairing is [19]
two-dimensional electron system, as in the classical 2D
(cid:16) (cid:17)
Tintra = 1.13ω exp − 1+λ , and a very sim- Fröhlich hamiltonian for the optical phonons, and char-
ilacr is obtainedDfeobyrethe inteλr1v1a−llµe∗1y1 transition Tinter = acterized the magnitude of the interaction. Our results
c
(cid:16) (cid:17) showthatelectronoverscreeningcannotbeachievedjust
1.13ω exp − 1+λ , with λ = λ +λ and the
Debye λ−µ∗ 11 12 with the strongest piezoelectric phonons within our as-
12
previously computed λPA included into the intravalley sumptionsbecauseK2 <1isalwayssatisfied. Neverthe-
R
term λ (λ denotes the contribution from all interval- less, these phonons could help further in other contexts
11 12
leyterms). Herethepseudo-potentialµ∗ isonlyslightly wherethe2Dsuperconductivityisknowntoexist,forex-
12
larger than µ , and both are given by similar formulas ample in bulk few-layer MoS with most of the carriers
11 2
as in Eq. (26), but with ω →ω . confined to the first layer [32, 33]; or postulated to ex-
c Debye
The upshot of this discussion is that the long- ist but not yet observed due to experimental difficulties
wavelengthpiezoelectricphononsworkinfavorofpairing (e.g. very heavily doped graphene [19]). Other example
6
is the recent high-temperature superconductor system of motion(inthepresentappendix, itisusedimplicitsums
2D FeSe on top of the ferroelectric SrTiO , whose opti- on repeated indexes):
3
cal phonons have been analyzed arriving at conclusions
similar to ours [27], and where the strong piezoelectric ∂2ui = ∂Tij, (A2)
phonons could play a role as well. ∂t2 ∂x
j
whereT (r,t)isthesymmetricalstresstensor. Poisson’s
ij
ACKNOWLEDGMENTS equationfortheelectricdisplacementDi(r,t)(nocharges
inside the material) is:
WewanttothankFernandoCalleandJürgenSchiefele
∂D
for valuable discussions. DGG acknowledges financial i =0 . (A3)
∂x
supportfromCampusdeExcelenciaInternacional(Cam- i
pus Moncloa UCM-UPM). This work has been sup-
The coupled constitutive (linear) equations relate the
ported by the Spanish Ministry of Economy (MINECO)
stress tensor and electric displacement with the strain
through Grants No. FIS2011-23713, FIS2013-41716-P;
tensor and electric field (here written as the gradient of
the European Research Council Advanced Grant (con-
the electric potential E =−∂ϕ/∂x )
i i
tract 290846), and the European Commission under the
Graphene Flagship, contract CNECTICT- 604391. T =c ∂u /∂x +e ∂ϕ/∂x
ij ijkl k l kij k
D =−ε ε ∂ϕ/∂x +e ∂u /∂x , (A4)
i jk vac j ijk j k
Appendix A: Piezo-SAW phonon-electron where we have introduced the elastic constant tensor
interaction vertex c ≡cE measured at constant electric field, the elec-
ijkl ijkl
tric (relative) permittivity tensor ε ≡ εS measured at
ij ij
The situation is depicted in Fig. 1. The z = 0 in-
constant strain and the piezoelectric tensor e .
ijk
terfacial surface is supposed to be free of tension and,
TheSAWsaresolutionsto(A2-A4)intheformofplane
when acting as a substrate to a deposited 2D electronic
waves propagating along the surface z = 0 in the direc-
material sheet, free of electrodes as well. However, in
tion specified by b=[cosθ,sinθ]
the present section, we will allow for flat electrodes (at
z = 0+) which supply no mechanical stresses. The pur- u =α exp[ik(b x −vt)]
j j i i
pose of this section is to show that the interaction be-
ϕ=α exp[ik(b x −vt)] , (A5)
tween the propagating piezoelectric SAWs and the elec- 4 i i
trons of the 2D electronic sheet can be described with a
and we have extended here to 3D the definition of b :=
Hamiltonian of the form Eq. (8) with (we use SI units
[cos(θ),sin(θ),b ]sothatb isnowavariabletobedeter-
3 3
throughout the present section, as it is typical for piezo-
mined by the requirements of boundedness or causality
electrics):
ofnormalmodes(seebelow). Inwhatfollows,weassume
always v >0 and k >0.
ω =v (θ)|q| ,
q s The resulting linear equations for the amplitudes α ,
a
K (θ)(cid:20) v (θ)(cid:126)e2 (cid:21)1/2 (here a,b=1,2,3,4 and i,j,k,l=1,2,3) are:
γPA = √R s , (A1)
q 2 (ε˜HF(θ)+1)εvac 0=(cid:0)Γ −δ(cid:48) ρv2(cid:1)α
ab ab a
where we have written q := q[cos(θ),sin(θ)] and ε is Γ =b b c
vac jk i l ijkl
theairorvacuumelectricpermittivity. Thepiezoelectric Γ =b b e
j4 i k ijk
specific parameters are v (θ), the piezoelectric SAW ve-
s Γ =−b b ε ε , (A6)
locity; 0≤K (θ)2 <1, the SAW electromechanical cou- 44 i k ik vac
R
pling coefficient; and ε˜HF(θ) := ε˜(q/ω),ω → ∞ (in the with δ(cid:48) =δ , δ(cid:48) =δ(cid:48) =0, and ρ the constant density
acoustic frequency scale), the high-frequency (HF) limit ij ij 4a a4
of the piezoelectric solid.
of the piezoelectric surface permittivity (see [5, 6]). They
Note that k disappears, which means that there is
all depend on the propagation direction of the SAW, as
no dispersion for a given propagating direction. Hence,
the notation suggests.
given the propagation direction θ and the velocity v, the
solutions for det(cid:0)Γ −δ(cid:48) ρv2(cid:1) = 0 as a function of b
ab ab 3
is a set of no more than 8 complex values, in which,
1. Piezoelectric Surface Acoustic Waves because of the reality of the coefficients, each complex
rootcomestogetherwithitsconjugate, andamongthese
For a general introduction to piezoelectric SAWs see we have to choose the ones with Imb < 0, so that the
3
Refs. [3, 5, 6, 17]. The point displacement, u (r,t), modes are not exponentially growing deep into the solid.
i
where i = 1,2,3 for the x,y,z directions respectively in In the case of purely real solutions, usual arguments on
the piezoelectric substrate, obeys the elastic equation of causality demand that we have to take only those modes
7
with radiation (outgoing from the surface z =0) bound- i.e. the phase velocity v (θ) of the wave is given by 1+
s
ary conditions db (v)/dv <0 (see [34]). Hence, the total ε˜(b/v (θ))=0. This is the piezoelectric Rayleigh waves
3 s
number of allowed modes is 4, and the general solution condition.
we write as (we use now u := ϕ and write somehow In [35] it is shown that the energetic stability of the
4
loosely r=(R,z), with the 2D R=(x,y)): piezoelectric guarantees that Imε˜(b/v (θ)) = 0 up to
s
a v (θ) > v (θ), with ε˜(b/v (θ)) = 0. In that range,
L 0 0
ua(r,t)=Cnαa(n)eikb(3n)zexp[ik(b·R−vt)] , (A7) the four modes in Eq. (A7) are purely decaying on the
substrate side. v (θ) marks the starting point at which
L
with n=1...4 indexing the normal modes. the piezoelectric surface permittivity has an imaginary
Much simpler is the equation at vacuum/air. The so- part, which reflects the influence of bulk modes.
lution is purely electric and can be written as:
ϕ(R,z,t)=u (R,0,t)e−kz , (A8) 2. Hamiltonian and interaction vertex
4
because of continuity of the potential.
The linear equations of piezoelectricity, Eqs. (A2-A4),
The mechanical boundary condition at the in-
can be derived from a Lagrangian (see [36])
terface T (R,0,t) = 0 leads to (here b(n) :=
i3
(cid:104)cos(θ),sin(θ),b(3n)(cid:105)): L[uj,ϕ]= 21(cid:90) d3r[ρu˙iu˙i−cijklui,juk,l
C b(n)(α(n)c +α(n)e )=0 , (A9) − 2eijkϕ,iuj,k+εijεvacϕ,iϕ,i] , (A15)
n k j 3ijk 4 k3i
wherewehavewritten :=∂/∂x andu˙ :=∂u /∂t. The
hence C are proportional to C . ,j j i i
i 4
canonical momentum to ϕ is zero, so that the system is
The normal component of the electric displacement is,
constrained. The Hamiltonian is then
at the interface:
1(cid:90)
D3(R,0−,t)=ikexp[ik(b·R−vt)] H[uj,ϕ]= 2 d3r(ρu˙iu˙i+cijklui,juk,l+εijεvacϕ,iϕ,i).
×C b(n)(α(n)e −α(n)ε ε ) , (A16)
n k j 3jk 4 3k vac
(A10)
Foragivenharmonicpropagating(nosurfacecharges)
andthisallowstointroducethepiezoelectric surface per- piezoelectric SAW, i.e., a wave with the form of
mittivity as the ratio: Reua(r,z,t)fromEqs.(A7-A8)fulfillingtheequationsof
motionEqs.(A2-A4)andboundaryconditionsEqs.(A9-
D (R,0−,t) A13) with σ(0) = 0, it is straightforward to show that
ε˜(k/ω):=− 3
kϕ(R,0−,t)ε the kinetic energy (first term in Eq. (A16), coming ex-
vac
clusively from elastic vibrations in the substrate) is the
C b(n)(α(n)e −α(n)ε )
=−i n k j 3jk 4 3k , (A11) sameasthepotentialenergy(lasttwotermsinEq.(A16),
C α(m)ε containscontributionsfromelasticdeformationandelec-
m 4 vac
trostatic stored energy both in the substrate and in
which only depends on v and θ, through the relations free space). On the other hand [5], for the interval
k:=kb and ω :=kv. 0<v (θ)<v (θ), positivity of the kinetic and potential
s L
Similarly,ontheothersideoftheinterfacewehavethe energies give ∂ε˜(k/ω)/∂ω > 0. For these kind of waves
obvious relation we have that [5] (when 1+ε˜(k/ω)=0)
1= D3(R,0+,t) . (A12) H = 1Akω∂ε˜(k/ω)εvac|ϕ |2
kϕ(R,0+,t)εvac harm 4 ∂ω 0
1 (ε˜ (θ)+1)ε
Hence, the surface charge at the interface can be ex- = Ak|ϕ |2 HF vac , (A17)
2 0 K2(θ)
pressed as: R
σ(0)=D3(0+)−D3(0−)=kϕ(0)[1+ε˜(k/ω)]εvac , where A is the area of the sample, ϕ0 := Cnα4(n) is the
(A13) amplitude of the electric potential at the interface (see
wherethedependenceexp[ik(b·r−vt)]isimplicitlyas- Eq. (A7)), and we have introduced the high-frequency
sumedandtheelectrodesshouldbeplacedperpendicular limit ε˜HF(θ):=ε˜(k/ω),ω →∞ and the SAW electrome-
to the propagation direction. chanical coupling coefficient, KR(θ) through the relation
From Eq. (A13), a source free propagating wave only at 1+ε˜(k/ω)=0:
exists if
K2(θ)/2 (cid:20) ∂ε˜(k/ω)(cid:21)−1
R = ω . (A18)
1+ε˜(k/ω)=0 , (A14) ε˜ (θ)+1 ∂ω
HF
8
The electrons of the graphene sheet (or any other All this amounts to writing the general linear causal
chargedtwodimensionalstructuredepositedatthepiezo- relation [37]
electric substrate) feel the electric potential of the piezo-
(cid:90)
electric SAW. The interaction is then the total potential
ϕ(x,t)= dx(cid:48)[γ (x−x(cid:48))σ(x(cid:48),t)
at the position of the electron ∞
(cid:90) t (cid:21)
V (R)=−eϕ(R,0,t=0) . (A19) + dt(cid:48)φ(x−x(cid:48),t−t(cid:48))σ(x(cid:48),t(cid:48)) . (A24)
PA
−∞
On the other hand, the one-phonon normalization
The power delivered to the electrodes to maintain a
means that ϕ from Eq. (A17) should be chosen so that
0 given ϕ(x,t),σ(x,t) (in this subsection we assume that
H =(cid:126)ω =(cid:126)v (θ)k,andthuswefinallygetthehamil-
harm s all fields which depend on space-time are real) is:
tonian Eq. (7).
dU(t) √ (cid:90)
= A dxϕ(x,t)σ˙(x,t) , (A25)
dt
3. Response functions
√
where A is the length along the y-direction.
We now consider a 1D situation, in which flat elec-
If starting from zero fields and charges, we adiabati-
trodes parallel to the y-axis operate on top of the piezo-
callyturnonagivensurfacechargedistributionσ(x,t)=
electric substrate shown in Fig. 1. Therefore, we chose
σ(x)exp(ηt), from Eqs.(A24-A25), the total energy sup-
θ =0 and there is no y dependence. We omit to write θ
plied is:
in this subsection.
Thecharge-potentialrelation(A13)fortheamplitudes ∆U (cid:90) (cid:90) γ (x−x(cid:48))
√ad = dx dx(cid:48)σ(x) 0 σ(x(cid:48))
is written so that we define the complex admittance
A 2
χ(k,ω) as:
1 (cid:90) dk|σ(k)|2
= . (A26)
ϕ(k,ω)=γ(k,ω)σ(k,ω) 2(ε˜LF+1)εvac 2π |k|
1 1
γ(k,ω):= , (A20) Analogously, an instantaneous charging to the same
|k|(ε˜(k,ω)+1)ε
vac final charge distribution σ(x,t)=θ (t)σ(x), with θ (t) a
(cid:15) (cid:15)
differentiable approximation to the Heaviside θ-function
where we have to allow now for the possibility of nega-
such that θ (t) → θ(t),τ → 0+, requires an amount of
tive k, because we are omitting the θ dependence. From τ
work given by:
Eq. (A6), ε˜(k,ω) = f((ω/k)2) = f(v2), and its analyt-
ical extensions can be guessed from the requirements of ∆U (cid:90) (cid:90) γ (x−x(cid:48))
causality,whichforω >0meansthatthepolesandzeros √inst = dx dx(cid:48)σ(x) ∞ σ(x(cid:48))
A 2
of γ(k,ω) are placed in the lower complex ω half-plane.
We define the instantaneous part 1 (cid:90) dk|σ(k)|2
= . (A27)
2(ε˜ +1)ε 2π |k|
HF vac
1 1
γ (k):=
∞ |k|(ε˜ +1)ε The second process being non-adiabatic, it absorbs
HF vac
(cid:90) more energy from the source that exerts a work on the
= dxe−ikxγ∞(x) , (A21) system. This extra energy is employed in inducing sur-
face and bulk wave excitations. As a result, ∆U >
inst
and the retarded and static contributions ∆Uad, which implies
γret(k,ω):=γ(k,ω)−γ∞(k) ε˜HF <ε˜LF . (A28)
(cid:90) (cid:90) ∞
= dx ei(ωt−kx)φ(x,t) , (A22) After the sudden charge, i.e. at t > 0, the time
0 evolution and relaxation of the potential are, due to
1 1
Eqs. (A23,A24):
γ (k):=γ(k,0)=
0 |k|(ε˜ +1)ε
LF vac
(cid:90) (cid:90) (cid:20) (cid:90) t (cid:21)
= dxe−ikxγ (x) ϕ(x,t)= dx(cid:48)σ(x(cid:48)) γ (x−x(cid:48))+ dt(cid:48)φ(x−x(cid:48),t−t(cid:48))
0 ∞
0
(cid:90) ∞ (cid:90)
γ (x)=γ (x)+ dsφ(x,s)e−ηs , (A23) −t−→−∞→ dx(cid:48)σ(x(cid:48))γ (x−x(cid:48)) , (A29)
0 ∞ 0
0
where ε˜ := ε˜(k/ω),ω → 0 and η is to be understood this relaxed field being the same as that obtained after
LF
as η →0+. the adiabatic process to the same charge distribution.
9
The space Fourier, time Fourier-Laplace transform of Therearetwopossibilitiesforthevariationofbasv →
this potential is: ∞, either (a) b→ b <∞, (“sm” means small) or (b)
sm
b∼b →∞ (“bg” is for big).
(cid:90) (cid:90) ∞ bg
ϕ(k,ω):= dx dtei(ωt−kx)ϕ(x,t) Incase(a),Γ−ρv21willneverbesingular,sousingthe
0 determinantformulafromSchur’scomplementdet(Mˆ)=
iσ(k) det(Γ−ρv21)det(−ε−(cid:126)γ·(Γ−ρv21)−1·(cid:126)γ),itisimmediate
= γ(k,ω) , (A30)
ω+iη torealizethatε=0+O(v−2),whichleadstothedecaying
root b =−(ε +iε )/ε .
wherethechangeω →ω+iη (η ≡0+)ismadetoensure sm 31 p 33
From the modes equation (A36), we find that:
convergence.
As γ(k,ω) has poles at the Rayleigh waves condition (cid:18) (cid:126)u (cid:19) (cid:18)O(v−2)(cid:19)
(A14), we can isolate their contribution, ϕ (x,t) to sm (cid:39) , (A37)
RW ϕ 1
sm
ϕ(x,t),
wherehereandintherestofthissubsection, wenormal-
iσ(k) K2/2
ϕRW(k,ω):= |k| (ε˜ +R1)ε ize the modes amplitudes so that ϕsm,bg =1.
HF vac Fortheothercase(b),fromthemodesequation(A36)
(cid:18) (cid:19)
1 1 we find that b =O(v), hence, expanding Mˆ from:
+ , (A31) bg
ω−ω +iη ω+ω +iη
k k
Γ (cid:39)b2 c
where ω = vk and the two terms come from the two ij bg 3ij3
k
identicalSAWspropagatingtotherightandleft. Asmall γi (cid:39)e33ib2bg
0+ hasbeenaddedtoensurethatthepolesoftheadmit- ε(cid:39)ε b2 , (A38)
33 bg
tanceareinthelowercomplexω half-plane. Invertingto
get the spacetime behavior, we obtain two dispersionless but now the general form of these modes is
propagating SAWs:
(cid:18) (cid:126)u (cid:19) (cid:18)α(i) (cid:19)
ϕRW(x,t)= K2R2 (cid:2)ϕ(x−vt,0+)+ϕ(x+vt,0+)(cid:3) , ϕbbgg (cid:39) 1j , (A39)
(A32)
where ϕ(x±vt,0+) = (cid:82)(dk/2π)eik(x±vt)γ (k)σ(k) [see wherewehaveusedthenotationinEq. (A7)andchosen
Eq. (A29)]. The energy carried by these ∞two pulses is, αa(1,2,3) for the three ((cid:126)ubg,ϕbg) modes and αa(4) for the
using Eq. (A47): ((cid:126)usm,ϕsm) mode.
Choosing the constant C =1, the mechanical bound-
4
√ K2/2 (cid:90) dk|σ(k)|2 ary condition (A9) leads to:
∆U = A R , (A33)
RW (ε˜ +1)ε 2π |k|
HF vac
0(cid:39)C b(k)(α(k)c +e )+(e +b(4)e ) , (A40)
which is the energy stored in each traveling SAW, i.e. k j 3ij3 33i 13i 33i
∆U = K2 ∆U from Eq. (A27). Since we have
RW R inst and C = O(v−1), so the denominator in Eq. (A11) can
at our disposal no more than ∆U −∆U > 0, the k
inst ad be approximated as C α(m) (cid:39)1.
condition∆URW <∆Uinst−∆Uadmustbefulfilled. From m 4
On the other hand, the “big” (bg) contribution to the
Eqs.(A26-A28) we conclude that:
displacement field is, to order O(v0):
ε˜ −ε˜
K2 ≤ LF HF <1 . (A34)
R ε˜ +1 D (0−)| (cid:39)ikc b(i)(α(i)e −ε ε )(cid:39)0 , (A41)
LF 3 bg i j 33j 33 vac
the last approximate equality comes from the second
4. High frequency limit of ε˜(k/ω) Eq. (A36) together with Eq. (A38).
Collecting all these results together with the “small”
In this section we want to show that, if we take the (sm) contribution to D (0+) into Eq. (A11), we finally
3
propagating direction along x-axis, then: get [38]:
(cid:112)
ε˜ =ε := ε ε −(ε )2 . (A35) ε˜ =−ib4(ε )=ε . (A42)
HF p 11 33 13 HF k 3k p
In fact, we write the modes equation Eq. (A6) as,
(cid:18) (cid:126)u (cid:19) (cid:18)Γ−ρv21 (cid:126)γ (cid:19)(cid:18) (cid:126)u (cid:19) 5. Energy carried by the piezoelectric SAW pulse
Mˆ ≡ =0 , (A36)
ϕ (cid:126)γ(cid:62) −εε ϕ
vac
For piezoelectric phenomena, the Poynting vector is
where the form of the 3×3 matrix Γ, 3×1 vector(cid:126)γ and (see [5]):
constant ε as a function of b (where b=(1,0,b)) can be
read from Eq. (A6). P =−T u˙ +ϕD˙ , (A43)
j ij i j
10
which, after use of Eq. (A4) can be seen to be a bilinear where because of reality g (k,z)∗ = g (−k,z), we
r r
expression in the vectors (u ,u ) and (u˙ ,u˙ ) (here obtain:
a a,i b b,j
i,j = 1,2,3 and a,b = 1,2,3,4; where u = ϕ). For a
givenpulsepropagatinginthex-direction,4ua(x,y,z,t)= ∆U =√A1(cid:90) dk (cid:90) dzg (k,z)P g (k,z)∗ ,
f (x−vt,z), we are interested in the total energy which pulse v 2π r rs s
a
crosses x (is obviously independent of x) (A45)
but then [3, 5]:
∞
(cid:90)(cid:90)(cid:90)
∆U = dtdydzP (x,y,z,t) 1(cid:90) v|k| ∂ε˜(k,ω)
pulse 1 dzg (k,z)P g (k,z)∗ = ω |ϕ(k,0)|2
−∞ 2 r rs s 4 ∂ω
∞ (A46)
√ (cid:90)(cid:90)
is the time-average power per unit length crossing a yz-
= A dtdzg (x−vt,z)P g (x−vt,z) ,
r rs s
section by a harmonic piezoelectric SAW, whose electric
−∞ potentialamplitudeisϕ(k,0)attheinterface. Theresult
(A44)
is:
where gr/s are taken from the components √ (ε˜ +1)ε (cid:90) dk
u ,u ,u˙ ,u˙ with r,s = 1,2,...,16, and P is a con- ∆U = A HF vac |k||ϕ(k,0)|2 . (A47)
a a,i a a,i rs pulse K2 2π
stant matrix with elements of the tensors eˆ,cˆ,εˆ. Fourier R
analyzing g (x − vt,z) = (cid:82)(dk/2π)eik(x−vt)g (k,z),
r r
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