Euler buckling instability and enhanced current blockade in suspended single-electron transistors

1.1 MB·English

by  Guillaume Weick

Eulerbucklinginstabilityandenhancedcurrentblockadeinsuspendedsingle-electrontransistors
Guillaume Weick,1 Felix von Oppen,2 and Fabio Pistolesi3,4
1Institut de Physique et Chimie des Mate´riaux de Strasbourg (UMR 7504),
CNRSandUniversite´ deStrasbourg, 23rueduLoess, BP43, F-67034StrasbourgCedex2, France
2Dahlem Center for Complex Quantum Systems & Fachbereich Physik,
Freie Universita¨t Berlin, Arnimallee 14, D-14195 Berlin, Germany
3Centre de Physique Mole´culaire Optique et Hertzienne (UMR 5798),
CNRSandUniversite´ deBordeauxI,351CoursdelaLibe´ration, F-33405TalenceCedex, France
4Laboratoire de Physique et Mode´lisation des Milieux Condense´s (UMR 5493),
CNRSandUniversite´ JosephFourier,25avenuedesMartyrs,BP166,F-38042GrenobleCedex,France
1
Single-electrontransistorsembeddedinasuspendednanobeamorcarbonnanotubemayexhibiteffectsorig-
1
inatingfromthecouplingoftheelectronicdegreesoffreedomtothemechanicaloscillationsofthesuspended
0
structure. Here, we investigate theoretically the consequences of a capacitive electromechanical interaction
2
when the supporting beam is brought close to the Euler buckling instability by a lateral compressive strain.

n Ourcentralresultisthatthelow-biascurrentblockade,originatingfromtheelectromechanicalcouplingforthe
a classicalresonator,isstronglyenhancedneartheEulerinstability.Wepredictthatthebiasvoltagebelowwhich
J transportisblockedincreasesbyordersofmagnitudefortypicalparameters. Thismechanismmaymakethe

4 otherwiseelusiveclassicalcurrentblockadeexperimentallyobservable.
2
PACSnumbers:73.63.-b,85.85.+j,63.22.Gh

]
l
l
a
I. INTRODUCTION
h source drain F
- µL µR
s
e Single-electron transistors (SET) are extremely sensitive
m devices, and are investigated as position detectors for nano- Fe
electromechanical systems (NEMS).1–3 But the reduced size
. gate
at of the mechanical resonator implies that the back-action of Vg
m theSETcanhavesignificanteffectsonthemechanicaldegree
of freedom, such as the generation of self-oscillations.4–6 In
-
d practice,thedetectorandtheresonatorhavetobeinvestigated FIG. 1. Sketch of the considered system: a suspended doubly-
n collectively as a single system. A prominent example of the clamped beam forming a quantum dot electrically connected to
o source and drain electrodes held at chemical potentials µ and µ
new effects displayed by this device is the current blockade L R
c bythebiasvoltageV,respectively.Thebeamiscapacitivelycoupled
that appears at low bias voltage V when the SET is coupled
[ toametallicgatekeptatavoltageV ,whichinducesaforceF that
capacitively to a classical oscillator.7 The physical idea be- g e
attracts the beam towards the gate electrode. An additional, exter-
2 hind this phenomenon is simple: The presence of an extra
nallycontrolledcompressionalforceFactsonthebeamandinduces
v
electrononthecentralislandoftheSETinducesanadditional
abucklinginstability.
0
electrostaticforceF ontheoscillator(seeFig.1). Theequi-
0 e
libriumpositionoftheoscillatoristhusshiftedbyadistance
8
0 Fe/k, where k is the oscillator spring constant. After such a andforVg nearthedegenerateregion. Thiseffectisaprecur-
0. displacement,thegatevoltageVgseenbytheSETchangesby sorofthemechanicalinstabilityandthusofthecurrentblock-
1 aquantityoftheorderof Fe Fe/ek EE/e, whereeisthe ade. But the complete observation of the latter phenomenon
× ≡
0 electroncharge. Thedimensionoftheconductingwindowin isdifficultsincethetypicalvalueof E isonlyofafewµeV,
E
1 Vg is controlled by V, since at low temperatures current can thussmallerthancryogenictemperatures. Inordertoincrease
v: flow through the device only if |Vg| (cid:46) V (when measuring EE, one can increase the electrostatic coupling between the
i Vg fromthedegeneracypoint). Thus, foreV < EE, thefluc- oscillatorandtheSETsinceEE dependsquadraticallyonFe.
X tuation of theelectronic occupation of thecentral island suf- But another way of strengthening the effect would be to re-
r fices to bring the device out of the conducting window. The ducethespringconstantkoftheoscillator. Thereasonisthat
a
current is blocked for eV < EE and a mechanical bistability softeroscillatorswilldisplacemoreundertheinfluenceofthe
appears.7 Thisphenomenonistheclassicalcounterpartofthe electrostaticforceF ,andthuswillseealargerchangeinthe
e
Franck-Condonblockadeinmoleculardevices8,9 thathasre- gate voltage when electrons tunnel in. A way of reducing k
centlybeenobservedinsuspendedcarbonnanotubesforhigh- in a controlled manner is to operate a doubly-clamped beam
energyvibrationalmodes.10 Theclassicalcasehasbeentheo- subjecttoalateralcompressionforce F. Thelattercanbring
reticallystudiedinthecaseofasingle-levelquantumdot,11,12 thebeamtothewell-knownEulerbucklinginstability18,19(see
aswellasinthemetalliccase.7,13–15 Fig.1). UndertheactionoftheforceF,thesystemexhibitsa
Recent experiments16,17 on suspended carbon nanotubes continuoustransitionfromaflattoabuckledstate,whilethe
have observed a reduction of the mechanical resonance fre- fundamentalbendingmodebecomessofterasoneapproaches
quencyofthefundamentalbendingmodeatlowbiasvoltages the mechanical instability (k 0). It is clear that this does
→
2
not imply a divergence of E , since at the transition anhar- nanobeamthustakestheLandau-Ginzburgform21,24–26,28
E
monic terms will modify the simple arguments given above.
However, a strong enhancement of EE is expected. The Eu- H = P2 + mω2X2+ αX4, (2)
ler instability has been studied both experimentally20–23 and vib 2m 2 4
theoretically24–27 inmicro-andnanomechanicalsystems. We
where P is the momentum conjugate to X. For a doubly-
have recently considered the Euler instability in NEMS for
clamped uniform nanobeam of length L, linear mass density
the case where F is negligible with respect to the intrinsic
e σ, andbendingrigidityκ, onecanshow25,26 thatclosetothe
electron-phononcoupling.28
instabilitytheeffectivemassofthebeamism = 3σL/8. The
In this paper we investigate in detail the idea of increas-
fundamentalbendingmodefrequencyreads
ing the current blockade by exploiting the Euler instability,
(cid:114)
consideringhowtheanharmonicterms, thetemperature, and F
thenon-equilibriumfluctuationsmodifythesimplifiedpicture ω=ω0 1− F , (3)
givenabove.Wefindthatnearthebucklinginstability,thecur- c
rentblockadeinducedbythemechanicalresonatorisstrongly where F isthecompressionforce, F = κ(2π/L)2 thecritical
c
enhanced,renderingthiseffectexperimentallyobservable. forceatwhichbucklingoccurs,andω = √κ/σ(2π/L)2. The
0
Thepaperisstructuredasfollows:InSec.IIthemodelused positive parameter α = F L(π/2L)4 ensures the stability of
c
to describe the system is introduced. In Sec. III, a statistical the system for F > F .30 For F < F (ω2 > 0), X = 0 is
c c
description in terms of a Fokker-Planck equation is given. It the only stable solution, and the beam remains straight. For
is then used in the remainder of the paper to determine the F > F ,itbucklesintooneofthetwometastablestatesatX =
(cid:112) c
currentandthemechanicalbehaviorofthesystem.InSec.IV, mω2/α. NoticethatinwritingEq.(2),weassumedthat
the enhancement of EE is obtained at mean-field level. We t±hen−anobeamcannotrotatearounditsaxisduetoclampingat
discuss the effects of thermal and charge fluctuations on the itstwoends.
resultsinSec.V.InSec.VI,weinvestigatetheconsequences Electronic transport is accounted for by the SET Hamilto-
of a finite average excess charge on the quantum dot for our nianconsistingofthreeparts,
results. Sec.VIIpresentssomeestimatesoftheeffectweare
predictingforrecently-realizedexperiments. Weconcludein H = H +H +H , (4)
SET dot leads tun
Sec.VIII.Werelegateseveraltechnicalissuestoappendices.
where H describesthequantumdot, H theleft(L)and
dot leads
right(R)leads,andH thetunnelingbetweenleadsanddot.
tun
Explicitly,
II. MODEL
(cid:16) (cid:17) U
H = (cid:15) eV¯ n + n (n 1), (5)
dot d g d d d
− 2 −
As a representative model for the problem outlined in
the Introduction, we consider a quantum dot embedded in a
withn = d d,andd (d)creates(annihilates)anelectronon
d † †
doubly-clampedbeamasshowninFig.1.Thepresenceofthe the dot, V¯ = C V /C , with C and C the gate and total
g g g Σ g Σ
metallicgatenearthedotisresponsibleforthecouplingofthe
capacitancesoftheSET,respectively. Theintra-dotCoulomb
bendingmodesofthebeamtothechargestateofthedot. The
repulsion is denoted by U. In the following, we set (cid:15) = 0,
d
Hamiltonianofthesystemcanthenbewrittenas
measuring V from the degeneracy point. The left and right
g
leads are assumed to be Fermi liquids at temperature T with
H = Hvib+HSET+Hc, (1) chemical potentials µL and µR (measured from (cid:15)d), respec-
tively. A(symmetric)biasvoltageV isappliedtothejunction
where H describestheoscillatingmodesofthenanobeam, suchthatµ = µ =eV/2. TheleadHamiltonianreads
vib L R
−
H theelectronicdegreesoffreedomofthesingle-electron
SET (cid:88)
transistor,and Hc thecouplingbetweentheSETandtheres- Hleads = ((cid:15)k−µa)c†kacka, (6)
onator. The model describes, for instance, transport through ka
suspendedcarbonnanotubesasconsideredintheexperiments
with c the annihilation operator for a spinless electron of
of Refs. 10, 16, 17, and 29. Notice that the model also de- ka
momentum k in lead a = L,R.31 Finally, tunneling is ac-
scribes an alternative setup that may be experimentally re-
countedforbytheHamiltonian
alized, namely a non-suspended quantum dot coupled to a
beam-likegateelectrodetowhichacompressivestrainisap- (cid:88)
plied. Htun = (tac†kad+h.c.), (7)
ka
Using standard methods of elasticity theory one can show
that, close to the buckling instability, the frequency ω of the witht thetunnelingamplitudebetweenthequantumdotand
a
fundamental bending mode of the nanobeam vanishes while leada.
thoseofthehighermodesremainfinite.19 Thisallowsoneto Two different kinds of couplings exist between the elec-
retain only the fundamental mode parametrized by the dis- tronicoccupationofthedotn andthevibrationaldegreesof
d
placement X of the center of the beam. As detailed in Ap- freedom: (i) an intrinsic one that originates from the varia-
pendixA,theHamiltonianrepresentingtheoscillationsofthe tion of the electronic energy due to the elastic deformation
3
of the beam,32 and (ii) an electrostatic one, induced by the fectscausedissipationofthemechanicalenergy,withdamp-
capacitive coupling to the gate electrode of the SET.33–36 By ingcoefficientη(X).
symmetry, theformerisquadraticintheamplitude X andits To account for the quality factor Q = mω /η of the
0 e
effectontheEulerinstabilityhasbeenconsideredinRef.28. nanobeam mode, the mechanical degree of freedom is cou-
ThelatterislinearinX andhereweareinterestedinthecase pledtoanadditionalenvironmentatequilibrium(suchas,e.g.,
wherethesecondcouplingdominatesoverthefirstone. Their a generic phonon bath within the Caldeira-Leggett model39),
relative intensity is controlled by the distance h between the implyingdissipationandfluctuationscontrolledbyanextrin-
gateelectrodeandthebeam,sincetheintrinsiccouplingdoes sicdampingconstantη enteringEq.(9).Thisextrinsicdamp-
e
not depend on h, while the electrostatic force depends log- ing comes from several mechanisms coupling the mechani-
arithmically on h.33 Assuming that the beam is sufficiently cal mode to other degrees of freedom: localized defects at
close to the gate electrode such that the capacitive coupling the surface of the sample (thought to be the main source of
dominates,37wecanwrite dissipation in semiconductor resonators40 and which can be
modeledastwo-levelsystems41,42),clampinglosses,thermo-
H = F Xn , (8)
c e d elastic losses, ohmic losses due to the gate electrode (which
havebeenpredictedtobethedominantsourceofextrinsicdis-
where F is the force exerted on the tube when one excess
− e sipation for graphene-based resonators in Ref. 43), etc. Due
electronoccupiesthequantumdot(seeFig.1). Themodelas-
tothewidevarietyofthesepossiblesourcesofextrinsicdis-
sumesthatthegatevoltageissuchthatonlychargestateswith
sipation, we here assume for simplicity that they can all be
n =0and1areaccessible.Forlargergatevoltagesovercom-
d lumpintothegeneric(Ohmic,memory-free39)dampingcon-
ingthechargingenergyofthequantumdot,thechargeonthe
stantη .Noticealsothatthephonontemperatureofthebathis
dotwillinsteadfluctuatesbetweenNandN+1. Thisinduces e
typicallylower,butofthesameorderastheelectronictemper-
anadditionalconstantforcebendingthetubefurther. Theef-
ature T.16 For simplicity, we assumed in writing Eq. (9) that
fect of such a force on the classical current blockade will be
bothtemperaturescoincide,aswedonotexpectaqualitative
discussed in Sec. VI. Notice that in the case the suspended
changeofourresultsduetothisassumption.
structure is the gate capacitance coupled to a static quantum
The explicit form of the coefficients entering into the
dot,theintrinsiccouplingisnotpresent,andonlythecapaci-
Fokker-Planck equation (9) depends on the transport regime
tiveelectromechanicalcouplinghastobetakenintoaccount.
oneconsiders(sequentialorresonanttransport),aswellason
the nature of the quantum dot (metallic or single-level quan-
tumdot).Inthispaperweconsiderthecaseofasinglelevelin
III. FOKKER-PLANCKDESCRIPTION
thesequentialtunnelingregime,butasimilaranalysiscanbe
carriedoutforthemetallic(e.g.,alongthelinesofRefs.7,14,
Weareinterestedindescribingthevicinityoftheinstability
28) and the resonant transport regime (cf. Refs. 11, 12, 38).
wheretherelevantresonatorfrequencyvanishes[seeEq.(3)]. Tobespecific,weassumethat(cid:126)Γ = (cid:80) (cid:126)Γ k T,with
Themechanicaldegreeoffreedomcanthenbetreatedclassi- Γ = 2πt 2ν/(cid:126) and ν the density of staa=teLs,Rat tahe(cid:28)FerBmi level
cally since for any reasonable temperature, (cid:126)ω (cid:28) kBT. The ofathelea|das|. Wealsoassumetheintra-dotCoulombrepulsion
softeningofthemechanicalmodeimpliesalsoanaturalsep-
U such that double occupancy of the dot is forbidden.
arationoftimescalesbetweentheslowmechanicalmodeand → ∞
Inthistransportregime,theposition-dependentratesfortun-
thefastelectronicdegreesoffreedom, controlledbythetyp-
nelingintoandoutofthedotread44
ical tunneling rate Γ. As detailed in Appendix B, it is thus
convenient to eliminate the fast modes and obtain a Fokker-
(cid:88) (cid:32)F X eV¯ µ (cid:33)
Planck equation for the probability distribution (X,P,t) of e g a
Γ (X)= Γ f − − , (10)
theslowmode:7,11,14,28,38 P 01 a F k T
a=L,R B
P η(X)+η (cid:88) (cid:34) (cid:32)F X eV¯ µ (cid:33)(cid:35)
∂tP=− m∂XP−Feff(X)∂PP+ m e∂P(PP) Γ10(X)= Γa 1− fF e −k Tg− a , (11)
(cid:32) (cid:33) a=L,R B
D(X)
+ +η k T ∂2 . (9)
2 e B PP
respectively, where f (z) = (ez +1) 1 is the Fermi function.
F −
Thus,theaverageoccupationofthedotforagivenmodeam-
The effective force F (X) = ∂ H + F (X) acting on
eff X vib c-i
− plitudeXis
the mechanical degree of freedom consists of two parts: a
force arising from the Hamiltonian (2) of the nanobeam,
−∂XHvib = −mω2X −αX3, and a current-induced conserva- n0(X)= Γ01(X), (12)
tive force F (X) = F n (X), proportional to the occupa- Γ
c-i e 0
tionofthedotaverage−doveratimelongwithrespecttoΓ 1,
−
but short with respect to the period of the mechanical mo- and,asshowninAppendixB,wehaveD(X)=2F2n (X)[1
e 0 −
tion, n (X) = n . In Eq. (9), the diffusion constant D(X) n (X)]/Γ and η(X) = F ∂ n (X)/Γ for the current-induced
0 d X 0 e X 0
accounts for th(cid:104)e fl(cid:105)uctuations of the force associated with the diffusionanddamping−termsinEq.(9).14,45 Theaveragecur-
coupling Hamiltonian (8) originating from the stochastic na- rent I throughthedevicecanbeobtainedfromthestationary
ture of the charge-transfer processes. Finally, retardation ef- solutionoftheFokker-Planckequation(9),∂ = 0,byav-
t st
P
4
x
eragingtheposition-dependentcurrent
(X)= eΓLΓR (cid:34)f (cid:32)FeX−eV¯g−eV/2(cid:33) 20
I Γ F k T
B
(cid:32)F X eV¯ +eV/2(cid:33)(cid:35)
e g
f − (13)
− F k T
B
1 0.5 0 0.5 1 δ
withthephase-spacedistribution, − −1
(cid:34) −√3α˜
20
I = dXdP (X,P) (X). (14) −
st
P I
Before we proceed, it is convenient to introduce reduced
variablesintermsoftherelevantenergyscaleoftheproblem FIG. 2. Example of a solution x(δ) of the equation for dynamic
E0 = F2/mω2, the polaronic shift (cid:96) = F /mω2, and the vi- equilibrium(21)forn =0(dashedline),n =1(solidline),andfor
E e 0 e 0 0 0
brational frequency for vanishing compression force ω0 [see n0=1andα˜ =0(dottedline).Inthefigure,α˜ =10−3.
Eq. (3)]. By denoting x = X/(cid:96), p = P/mω (cid:96), τ = ω t, the
0 0
Fokker-Planckequation(9)becomes
with a normalization constant. In order to obtain trans-
(cid:0) (cid:1) N
∂ = p∂ f (x)∂ + γ(x)+γ ∂ (p ) parentanalyticalresults,wealsoassumezerotemperature(in
τ x eff p e p
P −+(cid:32)d(2xP)−+γeT˜(cid:33)∂2pPP P (15) fdaicsttr,ib(cid:126)uωtio(cid:28)n(k1B9T)b(cid:28)ecoEmE0e),ssPusct(hx,thpa)t=thδe(pst)aδt(ioxn−arxym)p.roHbearbei,lixtmy
istheglobalminimumoftheeffectivepotential
withthescaledeffectiveforcegivenby (cid:90) x
v (x)= dx f (x) (20)
f (x)=δx α˜x3 n (x), (16) eff − (cid:48) eff (cid:48)
eff 0
− −
the reduced force δ = F/F 1, and the anharmonicity pa- corresponding to the effective force (16) and can be deter-
c
rameterα˜ = α(cid:96)4/E0. Wefurt−herintroducedascaledcurrent- minedfromthedynamicalequilibriumequation
E
induceddiffusionconstant
df (x)
d(x)= 2ω0n (x)[1 n (x)], (17) feff(x)=0, edffx <0. (21)
0 0
Γ −
Notice that the latter equation can have more than one so-
anddampingcoefficient
lution, such that the system is multi-stable. In this zero-
ω temperaturelimit,thecurrentcantheneasilybeobtainedfrom
γ(x)= 0∂xn0(x). (18) Eq.(14). Doingsoasafunctionofthegateandbiasvoltages,
− Γ
wecandeterminetheCoulombdiamondforagivencompres-
InEq.(15),γ =η /mω = Q 1,whereQisthequalityfactor sionforceδ. Atzerotemperature,onefindsthattherealways
e e 0 −
ofthemechanicalresonatorandT˜ =k T/E0. Inthesescaled exists a region at low bias voltage where the current is sup-
B E
units,theelectromechanicalcouplingappearsonlyintheco- pressed.
efficient of the quartic term, α˜ = αF2/(mω2)3. It is impor- To characterize this classical current blockade, we define
e 0
tanttonoticethatforactualexperimentsonsuspendedcarbon ∆ , the minimal value of bias voltage, for which a finite cur-
v
nanotubes,16,17,29 α˜ 1aswewilldiscussmoreextensively rentflowsthroughthedeviceatzerotemperature. Itisuseful
(cid:28)
inSec.VII. to first derive a simple estimate of the maximally obtainable
∆ . Todoso,wesolvethedynamicequilibriumequation(21)
v
forn = 0andn = 1,correspondingtoemptyandoccupied
0 0
IV. MEAN-FIELDAPPROACH:ENHANCEMENTOFTHE centralisland, respectively. Forn = 0onehasthesolutions
0
CURRENTBLOCKADE x=0foranyδandx= √δ/αforδ>0ofthepristineEuler
±
instability(seeFig.2,dashedline). Forn = 1,thesolutions
0
We begin our analysis by assuming ω /Γ 0.46 Note can easily be sketched for α˜ 1 as an interpolation of the
0 → (cid:28)
that the diffusion and dissipation coefficients d(x) and γ(x) α˜ = 0 solutions (dotted line in Fig. 2) and the solutions for
inEq.(15)areproportionaltoω /Γ[cf.Eqs.(17)and(18)], n = 0. The exact result is shown as a solid line in Fig. 2.
0 0
sothatthisimpliestoneglectcurrent-inducedfluctuations. In We are interested in the maximum shift in x that the system
this limit, the stationary solution for is given by a Boltz- undergoesinresponsetoafluctuationofn byoneunit,∆x.It
0
manndistributionattemperatureT˜, P isapparentfromthefigurethatthishappensforδ = 0where
∆x = 1/√3α˜. The corresponding change in the effective po-
(cid:32) (cid:33)
p2/2+v (x) tential (20) is ∆v 1/√3α˜. This provides an estimate of
Pst(x,p)=Nexp − T˜ eff , (19) the maximal eneregffy∼gap generated by the electromechanical
5
coupling,andthusagoodestimateof∆ . Noticethatthesim-
v
ple argument above is not specific to the transport model we 30 α˜ =10−6 1000
are considering here, as the specific form of n0 in the con- α˜ =10−3 100
ductingregiondoesnotenterourargument. Thus,weexpect
δ √3α˜ 10
frthoegartimmoeuet.ralelsictimquaatentoufmadmoatsx,iamsawleglalpas∆ivn∼th1e/r√e3sα˜onreamntatirnasnsvpaolirdt ∆v 20 δδ(cid:28)(cid:29)−√3√3α˜α˜ 110−9 10α˜−6 10−3
| |(cid:28)
WenowturntothecompletesolutionofEq.(21). Forsim- 10
plicity, we assume symmetric coupling to the leads (Γ =
L
Γ =Γ/2),suchthattheaverageoccupationofthedotatfixed (a)
R
x [enteringintotheeffectiveforce(16)]isobtainedfromthe
0
zero-temperaturelimitofEqs.(10)and(12)andgivenby
1000
1(cid:20) (cid:18) v(cid:19) (cid:18) v(cid:19)(cid:21)
n (x)= Θ x+v + +Θ x+v , (22)
0 g g
2 − 2 − − 2
100
where v = eV/E0 (assumed positive for definiteness), v =
E g
eV¯ /E0,andΘ(z)istheHeavisidestepfunction. Inthemost
g E vg 10
generalcase,wesolveEq.(21)numerically. However,trans- −
parent analytical expressions can be obtained in the limits
δ √3α˜ and δ √3α˜. Inparticular,wecanobtainexplicit 1
| | (cid:29) | | (cid:28)
expressionsforthevalueofvbeyondwhichthecurrentbegins
(b)
toflow, i.e., thegap∆ . Tofirstorderinthesmallparameter
v
√3α˜/δ (farfromtheinstability)and δ/√3α˜ (inthevicinityof 0.1
thein|s|tability),wefindthat | | 1 0.5 0 0.5 1
− −
δ
 1
∆v =−4√13δ22√3,δ−α˜, 1(cid:32)243/3 − √3δα˜(cid:33), −|δδ1|(cid:29)(cid:28)(cid:54)√3δ√3α˜(cid:28)α˜,.−√3α˜, (23) FsttdδhhuiIeeaGl=tmbsa.isoF3afyno./smdrF(a,Cpncα˜rtdoeo−ls=gtoip1acre.t1oecb0tneTvi−vlhohi3eanletlveaayir)ng)oed(erdaasss)1ci(a0nGi2r−3fcra6ue)lp,endasrc∆unetcvadiseonpad(nden2cudo4btnf)i(lviubtftehseo)leraygs,tsfaqcottwuhearaeclhvereiaeodcspslhctbeaaoxeagrmlreoeoefpwnvtrcguheo(m(essmdsoCeielporifioiadncunraellfeioloddnmrrecate)boes-,
above (dashed line), and in the vicinity (dotted line) of the critical
Far from the mechanical instability, the gap is simply given forceF . Inset: Gap∆ 1/√3α˜ fromEq.(23)asafunctionofα˜ at
by the result of a harmonic theory (see Appendix C) where themecchanicalinstabilvity∼(δ=0).
∆ = 1/2v (x¯), withv (x¯)thecurvatureofthebarepotential
v (cid:48)(cid:48) (cid:48)(cid:48)
v(x)[i.e.,theeffectivepotentialwithoutthecontributionfrom
n (x)]atitsglobalminimum x¯. Farbelowthebucklinginsta- The gaps of Eq. (23) are obtained for values of the gate
0
bility ( 1 (cid:54) δ √3α˜), x¯ = 0 and v (x¯) = δ, such that voltageapproximatelygivenby
(cid:48)(cid:48)
∆ = −1/2δ. Far(cid:28)ab−ovethemechanicalinstabili−ty(δ √3α˜), 
aax¯ppvpp=raor−e−anc√htδed/siαv˜tehraegnebdnucvc(cid:48)ek(cid:48)s(lxi¯in)ngt=hine2sfitδar,bssitluticwtyhoftrlhoinamtes∆boveflo=Ewq1.o/(r42δa3.b)(cid:29)oaAvrees,cotunhtee- v =21δ1, (cid:114)δ, δ−1(cid:54)√3δα˜(cid:28), −√3α˜, (24)
mo|δff|eT(cid:28)rbihcyea√3tlahα˜cne,aaltclchyuuetbilcmiacatialotxenriremmsouafillnttgshaexopgfin∆aEpvtqh∼i.en(1e2F/ff3i√eg3)c.α˜at3irvi(esear)cfeoofaomrccrhpeαe˜ad(r1.e=6d)1,t0oana3danfnuod-r g −−44δ√13α−˜ (cid:32)3+α˜ √23δα˜(cid:33), |δ|(cid:29)(cid:28) √3α˜,
−
α˜ = 10−6 (red dots and blue squares in the figure, respec- which are shown in Fig. 3(b) and compared to a numerical
tively). It is evident from the figure that there is a dramatic calculation. Equations (23) and (24) define the apexes of
increase of the gap close to the instability. Furthermore, the the Coulomb diamonds which are shown in Fig. 4. The ef-
smaller α˜, i.e., the smaller the electromechanical coupling, fectofthecompressionforceisthustocontinuouslydisplace
the larger is the increase of the gap at the instability relative theCoulombdiamondinthev-v planetowardsnegativegate
g
to its value for vanishing compression force [see the inset in voltages[seealsoFig.3(b)],andtoopenagapwhichismaxi-
Fig.3(a)]. However,ofcourse,themaximalvalueofthegap malclosetotheEulerinstabilityatδ=0[seeFig.4(d)]. Note
inabsolutetermsincreaseswiththestrengthoftheelectrome- thattheshiftingatevoltageisstronglyasymmetricaboutthe
chanical coupling as F4/3. It would thus be of great experi- Euler instability. While the shifts are only small below the
e
mentalinteresttoexploittheEulerinstabilitytoobtainaclear Euler instability [see Figs. 4(a)-(c) and Fig. 3(b)], the shifts
signatureoftheclassicalcurrentblockadeintransportexper- ingatevoltageareordersofmagnitudeslargeronthebuckled
imentsonsuspendedquantumdots. sideoftheEulerinstability[seeFigs.4(e)-(g)andFig.3(b)].
6
v v
(a)δ = 1 (e)δ = 0.05
−
v
g
v
(b)δ = 0.1
−
v
(f)δ = 0.1
v
(c)δ = 0.05 v
− g
v v
(d)δ = 0 (g)δ = 1
v v
g g
FIG.4.(Coloronline)Mean-fieldcurrentIatzerotemperatureandforsymmetriccouplingtotheleads(Γ =Γ =Γ/2),asafunctionofbias
L R
vandgatevoltagev (measuredinunitsoftheelasticenergyE0). The(scaled)compressionforceδincreasesfrom(a)to(g). Noticethatthe
g E
scaleofthev -axisisdifferentin(e),(f),and(g),andin(a)-(d). ThereddashedlinesindicatethepositionoftheCoulombdiamondinthe
g
absenceofelectromechanicalcoupling(F =0). Inthefigure,α˜ =10 6,anddarkblueandwhiteregionscorrespondtoI =0andI =eΓ/4,
e −
respectively.
Infact,itmaybethattheseshiftswouldbethemosteasilyde- by the latter for compression forces below the critical force
tectedconsequenceoftheEulerbucklinginstabilityinNEMS. (δ < 0),incontrasttothepresentcase. Thisisduetothefact
In Fig. 4, the bias and gate voltages are measured in units that,intheflatstate,thequadraticelectromechanicalcoupling
of the elastic energy E0 which is of the order of a few µeV merelyrepresentsarenormalizationofthefundamentalbend-
E
for typical experiments on suspended carbon nanotubes (see ingmodefrequency,anddoesnotleadtocurrentblockade.
Sec.VII).Thesmallnessofthisenergyscaleexplainswhythe
It is instructive to make the analogies with standard re-
scaled numerical values of the shifts become so large on the
sults of Landau mean field theory for continuous phase
buckledsideoftheEulerinstability.
transitions48 explicit. According to Eq. (21) governing the
dynamicalequilibrium,wecanmakethefollowingidentifica-
It is also interesting to comment on the shape of the
tions: xcorrespondstotheorderparameterinLandautheory,
Coulomb blockade diamond. In Ref. 28, we showed for the
δ to the reduced temperature, and α˜ to the coefficient of the
caseofintrinsicelectron-phononcoupling(quadraticinx)and
quartic term in the Landau free energy. Finally, n plays a
forametallicquantumdotthattheEulerbucklinginstability 0
role similar to a symmetry-breaking (magnetic) field. In the
leads to nonlinear deformations of the Coulomb diamonds,
present case, this field is in general dependent on x, which
aphenomenonthatwehavenamed“tricriticalcurrentblock-
has no correspondence in Landau theory. Nevertheless, the
ade”.Incontrast,ourpresentresultsshowthatforacapacitive
analogybetweenn andamagneticfieldishelpfulsincesome
electromechanicalcoupling(linearinx)andforasingle-level 0
of ourresults canbe understoodby comparingthe situations
quantumdot,theshapeoftheCoulombdiamondremainsun-
with zero (n = 0) and one (n = 1) electrons on the dot,
changed. Theconventionaltriangularshapeinthev-v plane 0 0
g as illustrated by the above estimate for the maximal ∆ (see
is delineated by straight lines with v 2v for any value v
of the compressive strain. As we have∼ch±eckged,47 the differ- Fig.2).
ence between the present results and those of Ref. 28 is due With these correspondences, we can now establish analo-
to the difference in the transport models considered (metal- giesbetweensomeofourresultsandstandardresultsofLan-
licvs.single-levelquantumdot), andnottothetypeofelec- dautheory. Tostartwith,thedependenceofthedisplacement
tromechanical coupling (intrinsic vs. extrinsic). Specifically, x δ1/2 in the buckled state is analogous to the result of
∼ ±
wefindthatthedifferenceisduetothefactthatinthesingle- Landautheorythattheorderparameterexponentisβ = 1/2.
level case, the average occupation of the dot abruptly jumps Giventhat∆ dependslinearlyon x,wecanalsointerpretthe
v
as a function of gate voltage, while in the metallic case, this relations in Eq. (23) in terms of Landau theory. Let us start
occupation gradually changes due to the continuous density with the case of small δ, in the immediate vicinity of the in-
of states of the dot. Notice also that for intrinsic electrome- stability. Inthiscase,wefindthat∆ α˜ 1/3. Theexponent
v −
∼
chanical coupling, the Coulomb diamond is not influenced ofα˜ correspondstothecriticalexponentδ=3ofLandauthe-
7
ory governing the dependence of the order parameter on the 0.25
symmetry-breaking field at the critical temperature. Further T˜/∆v=0
from the instability, we have ∆v ∼ 1/|δ|. This relation is re- 0.2 T˜/∆v=0.01
latedtothefamiliarCurielawfortheorderparameter(orthe T˜/∆ =0.1
v
swuistchempteibanil-ifitye)ldascrfiutniccatlioenxpoofnteenmtpγer=at1u.reinanexternalfield, Γ 0.15 T˜/∆v=1
e
/
LetusfinallyemphasizethattheanalogywithLandauthe- I 0.1
oryisrestrictedtomean-fieldlevel,sincecontrarilytocritical
phenomenawhereaninfinitenumberofmodesispresent,the
0.05
systemwedescribeisconstitutedbyasinglemode.Moreover,
beyond mean-field theory, fluctuations in Landau theory are
0
purelythermal,whileinthepresentcontext,non-equilibrium
0 1 2 3 4
fluctuations play an essential role. It is the effects of these
fluctuationswhichweturntointhenextsection. v/∆v
FIG.5. (Coloronline)CurrentIattheapexoftheCoulombdiamond
V. THERMALANDCURRENT-INDUCED asafunctionofbiasvscaledbytheenergygap∆vforvariousvalues
FLUCTUATIONS ofT˜/∆v,andforcompressionforcesinthevicinityofthebuckling
instability(δ √3α˜). Inthefigure, onlythetemperature-induced
| | (cid:28)
fluctuationsareconsidered. Inset: Sameasthemainfigureforcom-
We now go beyond the mean-field results of the previous
pressionforcesfarfromthebucklinginstability(δ √3α˜).
sectionbytakingintoaccounttheeffectsofthethermalaswell | |(cid:29)
asthecurrent-inducedfluctuations. Itisphysicallyclearthat
these fluctuations will lead to a smoothening of the current
Ournumericalandanalyticalresults(cf.AppendixD)thus
blockadeatlowbiasvoltages,asthesystemcanexploremore
confirm that tuning the system near the buckling instability
conductingstatesinphasespace.
where∆ dramaticallyincreasesallowsonetoenlargethetem-
v
peratureregionoverwhichthecurrentblockadeisobservable.
A. Temperatureeffects
B. Non-equilibriumdynamicsclosetothemechanical
Wefirstneglectthecurrent-inducedfluctuationsandfocus instability
on thermal fluctuations only. As discussed in Sec. IV, this
becomes asymptotically exact in the extreme adiabatic limit
We now consider the non-equilibrium Langevin dynamics
ofω /Γ 0, wherethetermsγ(x)andd(x)canbedropped
0 → of the nanobeam by solving the full Fokker-Planck equation
fromEq.(15). Thestationarysolutionfor isthengivenby
P (15). This is done by discretization of the Fokker-Planck
theBoltzmanndistribution(19)withtheeffectivepotential
equation and solution of the resulting linear system. We fo-
(cid:32) (cid:32) (cid:33)(cid:33) cusonthetransitionregion(δ = 0)andcalculatethecurrent
δx2 α˜x4 T˜ x v +v/2
g
v (x)= + +x+ ln f − forv attheapexoftheCoulombdiamond[seeEq.(24)and
eff − 2 4 2 F T˜ Fig.3g(b)]andtemperaturelowerthanthegapT˜/∆ =0.1.Be-
(cid:32) (cid:32) (cid:33)(cid:33) v
T˜ x v v/2 forewepresentourresults,wenoticethatfor(ω /Γ,γ ) 1,
+ ln f − g− . (25) 0 e (cid:28)
2 F T˜ we can show that the stationary distribution of the Fokker-
Planckequationapproximatelyonlydependsontheratio γe ,
The current can now be easily calculated by numerical inte- aresultwehavealsocheckednumerically(seeAppendixEω0f/oΓr
grationofEq.(14)withEq.(19).TheresultisshowninFig.5 details). Thereasonforthisbehavioristhat,for(ω /Γ,γ )
0 e
(cid:28)
asafunctionofthebiasvoltage,forgatevoltagescorrespond- 1, the stationary distribution is almost a function of the (re-
ing to the apex of the modified zero-temperature Coulomb duced)energyE = p2/2+v (x)only.
eff
diamond [cf. Eq. (24)]. Once plotted as a function of v/∆ , NumericalresultsforthecurrentareshowninFig.6forvar-
v
onefindsthatthecurrent behaviorissimilaratthetransition iousratiosoftheinversequalityfactorQ 1 =γ asquantified
− e
(Fig. 5) and far from the transition (inset of Fig. 5). In both by the damping coefficient γ and the adiabaticity parameter
e
cases,thelow-biasblockadeofthecurrentbecomeslesspro- ω /Γ. Our principal observation is that the current blockade
0
nounced as temperature increases, and vanishes completely becomessharperforlow-Qresonators.
fortemperaturesoftheorderofthegap∆ . AsshowninAp- Onecanqualitativelyunderstandthebehaviorofthecurrent
v
pendixD[cf.Eq.(D1)],thecurrenthasaFermi-function-like inFig.6bydefininganeffectivetemperatureofthesystem
behavior as a function of the bias voltage for temperatures
muchsmallerthantheenergygap∆ (seedashedanddashed- d /2+γ T˜
v T˜ = (cid:104) (cid:105) e (26)
dottedlinesinFig.5). Itisthusexponentiallysuppressedfor eff γ +γ
e
biasvoltagebelowthegap. Atlargertemperatures,Eq.(D4) (cid:104) (cid:105)
showsthatthecurrentislinearinthebiasvoltage(seedotted in close analogy with the fluctuation-dissipation theorem.49
lineinFig.5). In Eq. (26), d and γ are the averages over the phase-
(cid:104) (cid:105) (cid:104) (cid:105)
8
0.25
ωγ0/eΓ =∞ ) 0.2 (a)
0.2 ωγ0/eΓ =1 (xI0.1
γe =10 1
0.15 ω0/Γ − 0
eΓ ωγ0/eΓ =10−2 5
/
I 0.1 ωγ0/eΓ =10−3 (b)
γe =0
0.05 ω0/Γ 2.5
)
x
0 (eff 0
0 1 2 3 4 v
v/∆ =0.5
v/∆ v
v
v/∆ =1
2.5 v
FIG. 6. (Color online) Current I at the apex of the Coulomb dia- − v/∆v=1.5
mondforδ = 0asafunctionofv/∆ forα˜ = 10 6 andT˜/∆ = 0.1.
v − v
Solidlines: non-equilibriumLangevindynamicsfor ωγ0e/Γ =1,10−1, −100 −50 0 50
10 2, 10 3, and 0, from the highest to the lowest curve at large x
− −
bias. Dashed-dottedline: fullyadiabaticlimit( γe = ),i.e.,cur-
ω0/Γ ∞
rentonlyincludingthermalfluctuations(cf.dashed-dottedcurvein FIG. 7. (a) Current as a function of x [Eq. (13)] and (b) effec-
Fig.5).Inournumericalcalculations,weusedω /Γ=10 2. tivepotentialv (x)[EIq.(25)]forbiasvoltagesbelow(dashedline),
0 − eff
above(dottedline),andat(solidline)theenergygap∆.Theparam-
v
etersarethesameasinFig.6,i.e.,δ=0,α˜ =10 6,T˜/∆ =0.1,and
− v
space probability distribution of the current-induced fluctu- v = 3/4√3α˜.
g −
ations and dissipation [cf. Eqs. (17) and (18)], respectively.
Notice that the strength of these two quantities is controlled
bytheadiabaticityparameterω /Γ.
0
assumingthatthephase-spacedistributionisaBoltzmanndis-
AsonecanseefromFig.6,forv<∆ ,thecurrentisalmost
v tributionatthetemperatureT˜ . Approximatingtheeffective
insensitive to the quality factor, and is the same as without eff
potential by its zero temperature expression, and averaging
current-inducedfluctuations(seedashed-dottedlineinFigs.5
d(x) and γ(x) over the effective Boltzmann distribution, we
and6). UsingEqs.(17)and(18),wehave
findforv ∆
v
ω (cid:34) (cid:32)x v v/2(cid:33) (cid:32)x v +v/2(cid:33)(cid:35)2 (cid:29)
d(x)=2γ(x)T˜ + 2Γ0 fF − gT˜− − fF − gT˜ π∆ (cid:32) Av2 (cid:33)
(27) T˜eff = 128vAexp ∆ T˜ , (28)
for symmetric coupling to the leads. However, for v < ∆ , v eff
v
positions x for which the current (x) of Eq. (13) is sup-
pressed are most stable (see dashIed line in Fig. 7), such with A = 9(1 2−1/3)/211/3. We thus have T˜eff/∆v
thatthecurrent-induceddiffusionanddampingconstantsap- (v/∆v)2/ln(v/∆v)− T˜/∆v, which explains why for γe = ∼0
(cid:29)
proximately satisfy a “local” fluctuation-dissipation theorem thecurrentismoresuppressedthanforfiniteγe. Itisalsoin-
for all relevant positions x that are significantly populated, terestingtonotethatthisestimateoftheeffectivetemperature
d(x) 2T˜γ(x). Wethushave d 2T˜ γ ,andaccordingto ismuchlargerthanforametallicquantumdot,whereT˜eff v
thede(cid:39)finition(26),wehaveT˜ (cid:104) (cid:105)T(cid:39)˜. He(cid:104)nc(cid:105)e,forbiasvoltages (Ref.13). Thereasonforthisdifferenceisthat,inthemeta∼llic
eff
lowerthantheenergygap∆ ,the(cid:39)current-inducedfluctuations case,thefluctuationandthedissipationareofthesameorder
v
behaveasthethermalones,essentiallykeepingthemechani- insidethebiaswindow,whileinthesingle-levelcase,theav-
calsystematequilibrium. eragedissipationisexponentiallysuppressedasγ(x)onlyhas
Onthecontrary, forv > ∆ , positions xforwhichthesys- asignificantcontributionforpositions xcorrespondingtothe
v
temisconductingarethemoststableones(seedottedlinein bordersoftheCoulombdiamond[seeEq.(18)].
Fig.7), andonehas d ω0/2Γ, while γ isexponentially Ourresultsshowthatalowqualityfactorismoresuitable
(cid:104) (cid:105) (cid:39) (cid:104) (cid:105)
small. The mechanical system is then subject to strong non- for the observation of the current blockade in classical res-
equilibrium fluctuations. For γ γe, we thus have from onators. It is interesting to note that this conclusion is also
Eq. (26) T˜eff (cid:39) T˜ + ω40γ/eΓ. This(cid:104)e(cid:105)sti(cid:28)mate of the effective tem- validinthequantumcase,8,9wheretheFranck-Condonblock-
perature shows that the system becomes “hotter” as the ratio ade is more pronounced for fast equilibration of the vibron
γe decreases.50 Hence, the system can explore more states mode. Due to the scaling of our results for the classical cur-
iωn0/pΓhasespaceforwhich (x)issuppressed,and,inturn,the rent blockade with the parameter γe (see Fig. 6), we also
currentdecreasesfordecrIeasing γe forv > ∆ (seeFig.6). conclude that it is advantageous foωr0/Γthe observation of this
ω0/Γ v
Thelatterargumentbreaksdownwhenγ γ .Inthatcase, phenomenon to have a resonator which is slow compared to
e
(cid:28)(cid:104) (cid:105)
wecanestimatetheeffectivetemperatureself-consistently,by thetunnelingdynamics,i.e.,ω Γ.
0
(cid:28)
9
VI. EFFECTOFAFINITEEXCESSCHARGEONTHE
f =0
QUANTUMDOT 30 N
f =1 10
N
Within the transport model of a single resonant electronic fN =2 1
level with infinite charging energy that we have used so far, 20 fN =5
thenumberofelectronsonthedotcanonlyvarybetween0or ∆v fN =15 0.11 10 100 1000
1(seeSec.II).Moregenerally,therangeofgatevoltagescan f =50 fN
N
exceedsthechargingenergyandtheaveragenumberofexcess 10 f =200
N
electronsNonthedotcanbemuchlargerthan1.Duetothese
(a)
excess electrons, an additional force F further bends the
N
nanotube, and hence increases its vib−rational frequency.16,17 0
Wecanthusexpectthatthebiasvoltagebelowwhichthecur-
1000
rentisblockedwilldecreasewhenN increases.
Inordertoinvestigatetheeffectofanon-vanishingaverage
excess charge on the quantum dot, we assume that the gate 100
voltageissuchthatthereiseitherN orN+1electronsonthe
dot. Wemeasurethefluctuationofthedotoccupationn with
d vg 10
respecttoN,andincorporatetheresultingadditionalforcein −
Eq. (16) by writing f (x) = δx α˜x3 n (x) f , where
eff 0 N
− − −
f = F /F . We neglect thermal and current-induced fluc-
N N e 1
tuations,andworkwithinamean-fieldapproximationatzero
(b)
temperature, such that n (x) is given by Eq. (22). For finite
0
f , the bare potential v(x) (i.e., without the current-induced 0.1
N
contribution) can be approximated by a harmonic potential 1 0.5 0 0.5 1
− −
closetoitsglobalminimumx¯,suchthatthebiasvoltagebelow δ
which the current is blocked is given by ∆ = 1/2v (x¯) (see
v (cid:48)(cid:48)
AppendixC).Theenergygap(resultingfromthemoststable FIG.8. (Coloronline)(a)Gap∆ and(b)gatevoltagev attheapex
v g
solution of δx¯ α˜x¯3 = f ) is plotted in Fig. 8(a), and the of the Coulomb diamond for α˜ = 10 6 as a function of the scaled
N −
−
gatevoltageattheapexoftheCoulombdiamondinFig.8(b). compressionforceδforincreasingvaluesof f [fromtoptobottom
N
As anticipated, the increase of the energy gap close to the and bottom to top at δ = 0 in (a) and (b), respectively]. The blue
mechanical instability is reduced as fN increases. Moreover, squares(fN =0)correspondstothenumericalresultsofFig.3,while
the solid lines result from the harmonic approximation (see text).
the displacement of the Coulomb diamond in v is less pro-
g Inset: Gap ∆ 1/f2/3 from Eq. (29) as a function of f at the
nouncedforlarge f(cid:113)N. (cid:113) mechanicalinvsta∼bilityN(δ=0). N
Farfrom(δ 3 α˜f2)andinthevicinityof(δ 3 α˜f2)
| |(cid:29) N | |(cid:28) N
theEulerinstability,weanalyticallyfindforthegap
symmetry-breakingfield.48
∆v =−461δ(cid:113)231,δ1α˜,f2 1− 3(cid:113)3δα˜f2, δ|−δ1|(cid:29)(cid:28)(cid:54)(cid:113)3δ(cid:113)3α˜(cid:28)α˜fNf2−N2,,(cid:113)3 α˜fN2, (29) tEtiSchhnhiqeesnaW.tcragi(geebn2aeicfp9lcoroi)eatranyanttlhsatcteherehogesegmoetdiafimpNonptltsahe,attNteaFtee,blNgy(cid:38)istlahai(cid:39)pdtye2y,i/Fssfδca√hoeopNor3=mpcu/3eelαp2˜d−al,,rae1stvtbth.heaoaielnsvnSyieidmisnnvhcδewcaraenoeh=ninaiαs˜scsch0ehteeih.ssoatffthWtbNyetiyphfe(cid:38)ieietconhqagbceu1lartlaa/aeyptvia√inasesnmt3ergta3htahgαoia˜lneeftl.
N N
(seeSec.VII),weexpectthatasignificantincreaseofthecur-
(cid:113)
rentblockadeatthemechanicalinstabilitypersistsforawide
tofirstorderinthesmallparameter 3 α˜f2/δ (farfromthein-
N | | rangeofgatevoltages.
(cid:113)
stability)and δ/ 3 α˜f2 (inthevicinityoftheinstability). Far
| | N
belowandabovetheinstability,thegapfollowsthesamebe-
haviorasfor f =0[seeEq.(23)]. Thisisduetothefactthat VII. EXPERIMENTALREALIZATION
N
forlarge δ f ,thestablepositionofthebeamissimilarto
N
| |(cid:29)
the one for fN = 0. In the vicinity of the instability, the gap The electromechanical coupling (8) is typically weak in
is reduced as f increases as 1/f2/3 (see the inset in Fig. 8). experiment. For this reason, only a precursor of the classi-
N N
The reduction of the maximal gap close to the instability is calcurrentblockadehasbeenseenintworecentexperiments
a direct consequence of the smoothening of the mechanical onsuspendedcarbonnanotubequantumdots,16,17 butthefull
transitionbetweenflatandbuckledstatesduetothepresence current blockade has not yet been observed. Indeed, we can
of the symmetry-breaking force f , similar to the behavior obtain an estimate for the frequency shift of the fundamen-
N
oftheorderparameteratasecond-orderphasetransitionina talbendingmode,inducedbytheelectromechanicalcoupling,
10
fromtheeffectivepotentialassociatedwithF (X). Theshift sametime,theincreaseislimitedtoasmallforcerange. The
eff
arisesfromthepositiondependenceofn (X). Expandingthe increase of the gap near the transition goes as 1/F F .
0 c
| − |
current-induced force for weak electromechanical coupling, Thus when α˜ is very small, a stringent requirement will be
wefind theprecisioninF thatwedenotebyδF. Inthiscasethemax-
(cid:18) (cid:19)
F (X) F n (0) F2 ∂n0 (cid:12)(cid:12)(cid:12)(cid:12) X, (30) imalgapwillbeoftheorderof 43πδFFc ln δF√3/αF˜c ,assumingthat
c-i (cid:39)− e 0 − e ∂eV¯g(cid:12)(cid:12)Fe=0 δluFti/oFnco(cid:29)f th√e3α˜g.apTh(2is3)rewsuitlht caanLoeraesniltyziabne ochfewckidetdhbδyFc.oTnvhois-
i.e., the current-induced force generates a term in the effec- impliesthatifoneisabletocontroltheforcewithaprecision
tive potential which is quadratic in X. Far from the current- sufficient to see the buckling instability (δF/Fc 1), there
(cid:28)
inducedinstability,thistermgivesasmallrenormalizationof remainsalargeenhancementofthegap.
(cid:12)
the resonance frequency, ∆ω0/ω0 = (EE0/2)∂n0/∂eV¯g(cid:12)(cid:12)Fe=0,
from which we can extract a reliable estimate of the energy
scaleofthecurrentblockade,
EE0 = 2CCΣg∆ωω00 ∂∂enV0g(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Fe=0−1. (31) In this work, weVhIIaIv.e iCnOveNsCtigLaUteSdIOthNeSconsequences of a
FromtheexperimentsofRefs.16and17, weextractavalue capacitiveelectromechanicalcouplinginasuspendedsingle-
of ∆ω /ω of a few percents for V at the degeneracy point. electrontransistorwhenthesupportingbeamisbroughtclose
0 0 g
The derivative of n with respect to eV¯ can be estimated as to the Euler buckling instability by a lateral compression
0 g
theinverseofthewidthoftheconductancepeakintheV-V force. Our main result is that the low-bias current blockade
g
plane, divided by C /C . This last quantity can, in turn, be originatingfromthecouplingbetweentheelectronicdegrees
g Σ
estimatedfromtheslopeoftheCoulombdiamonds. Collect- of freedom and the classical resonator can be enhanced by
ing these ingredients we find that for the suspended carbon several orders of magnitude in the vicinity of the instability.
nanotubes of Ref. 16, E0 3–5 µeV which corresponds to We show that both the mechanical as well as the electronic
α˜ 10 10, while for thoEse(cid:39)or Ref. 17, we get E0 20 µeV properties of this regime can be described in an asymptoti-
(cid:39) − E (cid:39) cally exact manner based on a Langevin equation. These re-
andα˜ 10 8(seeRef.30).
(cid:39) − sultsareadirectconsequenceofthecontinuousnatureofthe
We now use these numbers to estimate the possible en-
Euler buckling instability and the associated “critical slow-
hancement of the current blockade near the Euler buckling
ing down” of the fundamental bending mode of the beam at
instability. BasedonEq.(23),theseparametersyieldapossi-
the instability. In fact, more generally our results frequently
bleincreaseofthemechanically-inducedgapbythreeorders
have close and instructive analogies with the mean-field the-
of magnitude, leading to a maximal ∆ (converted into a di-
v ory of second-order phase transitions. We focused on the
mensionful quantity using the energy scale E0) of the order
E sequential-tunnelingtransportregimeofsingle-levelquantum
of 3 to 5meV. Such large gaps would be much more easily
dots, but many of our qualitative results should remain valid
observableinexperiment. Theimplementationofsuchade-
also in the metallic case as well as for the resonant trans-
vicecouldbeperformedbythemethodroutinelyemployedto
portregime.47 Infact, ourbasicapproachshouldapplyquite
controlbreakjunctionsthroughaforcepushingthesubstrate
generallyforanycontinuousmechanicalinstabilityofanano-
ofthedevice.
electromechanicalsystem.
Wealsoemphasizehereagainthatitispreferabletooperate
thesystemnearzeroexcesschargeonthequantumdot,where Our result apply most directly to quantum dots situated
there are only a few electrons on the nanotube such that the on nanobeams or carbon nanotubes. Applying strain to the
increaseoftheenergygapclosetotheEulerinstabilityisnot nanobeam in a controlled manner could, in principle, be ex-
smeared out by the additional force exerted on the nanotube perimentallyperformedwiththehelpofabreakjunction. In
(see Sec. VI). However, for the parameters of Refs. 16 and fact, it is quite conceivable that, e.g., some carbon nanotube
17,weestimatethattheenhancementofthecurrentblockade structures happen to be close to the Euler instability due to
remains very substantial for any realistic value of the excess specifics in the fabrication of individual nanostructures. Our
charge. predictionsmaybehelpfultoidentifysuch“anomalous”(and
When the tunneling-induced width Γ becomes larger than potentiallyinteresting)samples.
or of the order of temperature, co-tunneling effects tend to
smearthecurrentblockade,9,12 asdirectelectronictransitions
betweenleftandrightleadstakeplace. However,closetothe
bucklinginstabilitythegapmayremainlargerthantempera-
ACKNOWLEDGMENTS
turesothatco-tunnelingcorrectionsshouldbesuppressedin
theimmediatevicinityoftheinstability.
Alastcommentisinorderontherequiredprecisioninthe We acknowledge stimulating discussions with Eros Mari-
control of the lateral compression force F. As mentioned ani, as well as financial support by ANR contract JCJC-036
above, the increase of ∆ is larger for smaller α˜. But at the NEMESIS,andbytheDFGthroughSfb658.
v