Magnetic effects on nonlinear mechanical properties of a suspended carbon nanotube

1.1 MB·English

by  A. Nocera

Magnetic effects on nonlinear mechanical properties of a suspended carbon nanotube
A. Nocera1,2, C.A. Perroni3,4, V. Marigliano Ramaglia1,4, G. Cantele3, and V. Cataudella3,4
1CNISM, UdR Napoli Universita’, Monte SantAngelo, Via Cintia, I-80126 Napoli, Italy
2Dipartimento di Fisica E. Amaldi, Universita’ di Roma Tre, Via della Vasca Navale 84, I-00146 Roma,Italy
3CNR−SPIN, Monte SantAngelo, Via Cintia, I-80126 Napoli, Italy
4Universita’ degli Studi di Napoli Federico II, Complesso Universitario Monte Sant’Angelo, Via Cintia, I-80126 Napoli, Italy
We propose a microscopic model for a nanoelectromechanical system made by a radio-frequency
driven suspended carbon nanotube (CNT) in the presence of an external magnetic field perpendic-
ulartothecurrent. Asamainresult,weshowthat,whenthedeviceisdrivenfarfromequilibrium,
one can tune the CNT mechanical properties by varying the external magnetic field. Indeed, the
3 magnetic field affects the CNT bending mode dynamics inducing an enhanced damping as well as
1 a noise term due to the electronic phase fluctuations. The quality factor, as observed experimen-
0 tally, exhibits a quadratic dependence on external magnetic field strength. Finally, CNT resonance
2 frequencies as a function of gate voltage acquire, increasing the magnetic field strength, a peculiar
dip-peak structure that should be experimentally observed.
n
a
J
I. INTRODUCTION ple effective model15. In that model, the CNT descrip-
2
tion is reduced to a single electronic level15,20 coupled to

two metallic leads and interacting by means of a charge-
]
l NanoElectroMechanical Systems (NEMS) made of displacement term with a single vibrational degree of
l
a suspended CNTs have received increasing attention freedom describing the bending mode. Due to the low
h recently1–8. These devices are ideal for NEMS applica- frequency associated to the bending mode21, the vibra-
- tions due to the extreme mechanical properties of CNTs
s tionalCNTdynamicsisdescribedbyaclassicalLangevin
e (lowmassdensityandahighYoung’smodulus),resulting equation20,22–28. In this simple approach, we were able
m in a wide range of resonance frequencies for the funda-
to reproduce, in the absence of magnetic field, the main
. mental bending mode vibration (from MHz up to GHz9 results reported in Ref.[2] and [3] on a similar CNT-
t
a range). In particular, CNT-based electromechanical de- resonator device and even to predict features not yet
m vices working in the semiclassical regime (resonator fre-
observed. In particular, we have shown that when bias
- quenciesinMHzrangecomparedtoanelectronichopping voltages are smaller than the broadening due to tunnel
d frequencyfromtheleadsoftheorderoftensofGHz)have
coupling, the resonance frequency shows a single dip as
n
attracted great interest due to the extremely large qual-
a function of gate voltage while, at bias voltages exceed-
o
ity factors (Q > 105) attainable2,3,10. Furthemore, at
c ingthebroadeningduetotunnelcoupling,theresonance
[ cryogenictemperatures,suchdevicesbehaveasquantum frequency shows a double dip structure. The successive
dots11–13 with a strong interplay between single-electron experimental observation of the predicted effects14 has
3 tunneling and bending mode mechanical motion2,14–18. confirmedthevalidityofthemodeladoptedbyus15,22 in
v
This means that the electronic current is very sensitive
7 describing these devices.
to the CNT bending mode dynamics and can be used as
6 The main result of the present paper is to prove that
2 a quantum (due to the intrinsic quantum nature of the theexternalmagneticfieldmodifiesthemechanicalprop-
0 chargecarriers)measurementdevice. Ontheotherhand,
erties of the CNT-resonator. In particular, the magnetic
. largequalityfactorsallowtotuneCNTmechanicalprop-
2 field provides an additional damping mechanism for the
1 erties,e.g. theresonancefrequencyofthebendingmode, resonator mechanical motion. Interestingly, a quadratic
2 by adjusting electronic parameters such as gate and bias decrease of the quality factor Q as a function of the ex-
1 voltages2,14. As a main consequence, one expects that
ternalmagneticfieldstrength, inquantitativeagreement
: the application of a static magnetic field perpendicular
v with the experiment performed in Ref.[1], emerges.
i to the CNT device modifies the above picture since the When the device is driven far from the equilibrium
X
electronic current flow is affected by the Lorentz force.
we find that, increasing the magnetic field, the peculiar
r This,inturn,inducesachangeinthemechanicalproper-
a features (single or double dip) observed in the resonance
tiesofthe CNTresonatorthatcan be measuredthrough
frequency against gate voltage curves get distorted and
the current itself.
acquireadip-peakstructurethatcouldbeexperimentally
Motivated by recent experiments1,14, we study a gen- observed. In particular, this effect should be more easily
eral model describing the mechanical properties of a detectable at bias voltages that exceed the broadening
radio-frequency driven suspended CNT-based NEMS in due to tunnel coupling.
the presence of a transverse magnetic field. Indeed, we Weshowthat,atafixedgate,biasvoltageandtemper-
have recently shown that the effects of damping, spring ature,ifchargeandcurrentvariationsoftheoppositesign
stiffening and softening, and nonlinearity observed in occur, damping increases and Q is reduced. Vice-versa,
similar devices in the absence of a magnetic field2,3,19 charge and current variations of the same sign reduce
can be successfully described in terms of a very sim- damping with a consequent increase of quality factors.
2
is
Hˆ =U dˆ†dˆ+(cid:88)VH (x)cˆ† dˆ+h.c.+(cid:88)ε cˆ† cˆ ,
el gate k,α k,α k,α k,α k,α
k,α k,α
(1)
where CNT’s electronic level has energy U with cre-
gate
ation (annihilation) operators dˆ†(dˆ). The operators
cˆ† (cˆ ) create (annihilate) electrons with momentum
k,α k,α
FIG. 1. A carbon nanotube (CNT) subject to an external k and energy εk,α = Ek,α −µα in the left (α = L) or
magnetic field H suspended between two normal metal leads right (α = R) free metallic leads. The chemical poten-
biased by a voltage eVbias. tialsintheleadsµL andµR areassumedtobebiasedby
an external voltage eV =µ −µ .
bias L R
In the presence of a magnetic field, the phases of the
As concerns the physical mechanism triggered by the tunneling amplitudes with the leads depend on the CNT
magnetic field, we show that the application of a field displacement x,30,31
perpendicular to the current flux modifies all the terms
describing the CNT-resonator dynamics. Actually, the VkH,L(x)=Vk,Le−ıpx,
coupling with a transverse magnetic field introduces an VH (x)=V eıpx, (2)
electronic tunneling phase which depends on the me- k,R k,R
chanical displacement of the CNT-resonator itself. This where p = δeHL/2¯h gives the CNT momentum change
modifies the effective force acting on the resonator by induced by the Lorentz force when an electron tunnels
a pure non-equilibrium correction term proportional to from the CNT to a lead, and δ (cid:39) 1 is a numerical fac-
the magnetic field as well as to the electronic current29. tor determined by the spacial profile of the fundamental
Moreover, even at zero bias voltage, damping and dif- mode32. Above, e is the modulus of the electron charge,
fusive terms are both modified by quantum electronic ¯h is the Plank constant and L is the CNT length. Slow
current-current fluctuations corrections whose strength time varying tunneling amplitudes V (t) are also rele-
k
is quadratic in magnetic field. Finally, we further show vant in the case of adiabatic quantum pumping through
that,atzerobias,charge-displacementandmagneticfield quantumdots23,33. Furthermore,whentheexternalmag-
mediated electron-oscillator couplings cooperate behav- netic field values are sufficiently small, such as the Zee-
ing as a whole as a standard thermal bath at leads’ tem- man splitting is negligible compared to broadening due
perature. to tunnel coupling, we can neglect the effect of the elec-
The paper ends with a study of the device response tronic spin degrees of freedom (this issue will be consid-
when the CNT-resonator motion is actuated by an ex- ered elsewhere34).
ternal antenna at fixed frequency and amplitude. In For the sake of simplicity, we will suppose symmet-
this case, the current-gate voltage characteristic exhibits ric coupling V = V and a flat density of states for
k,L k,R
specific structures corresponding to the mechanical res- the leads ρ (cid:55)→ ρ , considered as thermostats at fi-
k,α α
onances (antenna frequency equal to the bending mode nitetemperatureT,withinthewide-bandapproximation
frequency). (V (cid:55)→ V , α = L,R)20,35. Definitely, the total tunnel-
k,α α
The paper is organized as follows: In Sec. II we dis- ing rate is ¯hΓ=(cid:80) ¯hΓ , with Γ =2πρ |V |2/¯h.
α=L,R α α α k,α
cussthemodel,InSec. IIIweconstruct,bymeansofthe The Hamiltonian of the mechanical degree of freedom
adiabatic approximation, the stochastic Langevin equa- is given by
tionforthedynamicsoftheoscillatorincludingmagnetic
field and external antenna effects. In Sec. IV we present pˆ2 1
Hˆ = + mω2xˆ2, (3)
numerical results. osc 2m 2 0
characterized by the frequency ω and the effective mass
0
m (k = mω2). The charge-displacement interaction is
II. MODEL 0
provided by35,36
We consider the system sketched in Fig.(1), which Hˆ =λxˆnˆ, (4)
int
shows a single wall CNT suspended between two nor-
malmetalleads. AnexternalmagneticfieldH isapplied whereλisthecharge-displacementcouplingstrengthand
perpendiculartotheCNT.WealsorestricttheCNTme- nˆ = dˆ†d represents the electronic charge-density on the
chanicaldegreesoffreedomtothelowfrequencybending CNT37. Definitely, the overall Hamiltonian is
mode and model it as a harmonic oscillator with fre-
quency ω0. Hˆ =Hˆel+Hˆosc+Hˆint. (5)
The electronic part of the CNT device is modeled as a
single electronic level coupled to the leads through stan- For the experiment discussed in Ref.[1], one has a
dard tunneling terms15,22. The electronic Hamiltonian strong separation between vibrational (ω (cid:39) 500 MHz(cid:39)
0
3
2µeV)andelectronictimescales(Γ(cid:39)50Ghz)sothatwe atfirstorder,alinearcorrectionintheoscillatorvelocity.
can solve our model in the adiabatic limit, ω /Γ << 1. Asaresultoftheadiabaticapproximation,thetruncated
0
The experimental values of bias voltages and tempera- CNT level Green functions will depend on the instanta-
tures allow also a semi-classical treatment of the oscil- neous value of the position and velocity of the resonator
lator dynamics22,27,38. In this paper, we will measure Gr,a,<,>(ω,x,v).
lengths in units of x = r, where r is a small frac- The adiabatic expansion of the Fourier transformed
0
tion of CNT radius (r = 60pm) appropriate to resolve retarded CNT level Green function is
the CNT bending dynamics at relatively small temper-
atures (T (cid:39) 25mK). For the sake of simplicity, we Gr(ω,x,v)=Gr (ω,x)+Gr (ω,x,v), (6)
(0) (1)
will indicate dimensionless displacement variable with
x. Energies are measured in units of ¯hΓ = 200µeV, where the expression of Gr (ω,x) is
(0)
and times in units of t = 1/ω . In terms of these
0 0
units, the dimensionless spring constant is k/mω2 (cid:39) 1, 1
0 Gr (ω,x)= , (7)
since, following Ref.[1], the effective mass of the nan- (0) ¯hω−U (x)+ı¯hΓ/2
gate
otube is m = 1.3×10−21kg. Definitely, the adiabatic
ratio is ω /Γ = 0.01, while the dimensionless temper- and that of Gr (ω,x,v) is
0 (1)
ature k T = 0.01. Magnetic fields are measured in
B
terms of the quantity B = H where our magnetic field ∂Gr (ω,x)
H0 Gr (ω,x,v)=ı¯hU˙ (x)Gr (ω,x) (0) . (8)
unit is H0 = 2¯h/eLr (cid:39) 16.6T, since the CNT length is (1) gate (0) ∂¯hω
L (cid:39) 700nm. Throughout this paper, we keep fixed the
dimensionless charge-displacement coupling strength to Above, U (x) = U +λx and the dot indicates the
gate gate
λ=0.1(ourforceunitish¯Γ/r),correspondingtoanesti- time derivative U˙ =λ∂x =λv.
gate ∂t
mateEp =λ2/2k (cid:39)1µeV,implyingamoderatecoupling Using the adiabatic approximation31 x(t1)−x(t2) (cid:39)
betweentheelectronicandvibrationaldegreesoffreedom x˙(t )(t −t ), we obtain for the lesser and greater com-
0 1 2
(Ep/¯hω0 =0.5). Summarizing,theregimeoftherelevant ponents in Fourier space
parameters is h¯ω (cid:39)E (cid:39)k T <<eVeff ≤¯hΓ.
0 p B bias
In the next section, we show how adiabatic approxi- Σ< (ω,v)(cid:39)Σ< (ω)+Σ (ω,v) (9)
leads leads,(0) leads,(1)
mation works on the coupled electron-oscillator problem
in the presence of a transverse magnetic field. where the expression of Σ< (ω) is
leads,(0)
Σ< (ω)=ı[¯hΓ f (ω)+¯hΓ f (ω)], (10)
leads,(0) L L R R
III. ADIABATIC APPROXIMATION
and that of Σ (ω,v) is
leads,(1)
As analyzed in the previous section, we work in the
(cid:32) (cid:33)
physical regime where the vibrational motion of the ∂[h¯Γ f (ω)+¯hΓ f (ω)]
Σ (ω,v)=−ıeH˜ L L R R v.
CNT-resonator is ’slow’ with respect to all electronic en- leads,(1) ∂[eV ]
bias
ergy scales and can be considered ”classical”: ω << Γ.
0 (11)
This regime of the parameters leads to the adiabatic ap-
Above, we have defined H˜ = 2p¯h/e. The adiabatic
proximation for the electronic problem. In contrast to
correction to the lesser component of the leads self-
previousworksthattreatedtheadiabaticapproximation
energy in Eq.(11) is entirely due to the transverse mag-
in the absence of a magnetic field22–24, we here investi-
netic field and represent one of the main results of the
gate the effect of a transverse magnetic field on the elec-
present paper, in contrast to previous works where mag-
tronicproblemdescribedbytheHamiltonianEq.(5). We
netic field effects in the adiabatic expansion were not
remark that the adiabatic approximation has been used
considered15,22,27,28,38.
todescribelargersystems39 forthestudyofspectraland
Definitely, for the CNT level occupation we get
transport properties of organic semiconductors39–41.
(cid:104)nˆ(cid:105)(x,v)(cid:39)(cid:104)nˆ(cid:105) (x)+(cid:104)nˆ(cid:105) (x,v), (12)
(0) (1)
A. Adiabatic approximation for the electron
where at zero order in the adiabatic expansion we get
problem in the presence of a magnetic field
(cid:90) d¯hω
(cid:104)nˆ(cid:105) (x)= (f (ω)+f (ω))C(ω,x), (13)
Inthissubsection,weshowhowtheadiabaticapproxi- (0) 4π L R
mationontheelectronicCNTlevelGreenfunctionworks
in the presence of a transverse magnetic field. with the spectral function C(ω,x) = −2(cid:61)Gr (ω,x) of
(0)
Assuming a slow time dependence of electronic Green the CNT level given by
functionsontheresonatordisplacementx,weareableto
calculate truncated expressions for the CNT level Green ¯hΓ
C(ω,x)= . (14)
functions which acquire a `slow´ time dependence and, (¯hω−U (x))2+[¯hΓ]2/4
gate
4
Thefirstordercorrectionsintheadiabaticexpansionare As already discussed referring to adiabatic corrections
linear in the oscillator velocity to the average charge density of the CNT level, U (x)
(1)
is the the adiabatic correction to the electronic current
(cid:104)nˆ(cid:105) (x,v)=v[R (x)+R (x)], (15)
(1) (1) (2) related to the charge-displacement coupling described in
the interaction Hamiltonian Eq.(4), already described in
with
Ref.[24] and [25]. U (x) is the adiabatic correction to
λ (cid:90) d¯hω (2)
R (x)= g(1)(ω)C(ω,x)T(ω,x), (16) the current exclusively due to magnetic-coupling effects.
(1) Γ 2π +
As we show below, Eq.(26) is one of the main result of
eH˜ (cid:90) d¯hω the present paper, describing the increase of damping
R (x)= g(2)(ω)C(ω,x), (17)
(2) 2 2π + acting on the CNT resonator given by the application of
a transverse magnetic field.
where we have defined the transmission function T(ω,x)
In next subsection, we show that, even in the pres-
¯hΓ ¯hΓ ence of a transverse magnetic field, the dynamics of the
T(ω,x)= L R , (18) CNT-resonatorcanbeaccuratelydescribedbyastochas-
[(h¯ω−U (x))2+[¯hΓ]2/4]
gate
tic Langevin equation.
and
∂[f (ω)+f (ω)]
g(1)(ω)=− L R , (19)
+ ∂¯hω B. Langevin equation for the oscillator
∂[f (ω)+f (ω)]
g(2)(ω)=− L R . (20)
+ ∂[eV ] Intheabsenceofamagneticfield,theeffectoftheelec-
bias
tron bath and the charge-displacement coupling on the
Above, R (x) is the the adiabatic correction to the
(1) oscillator dynamics gives rise to a stochastic Langevin
density related to the charge-displacement coupling de-
equationwithapositiondependentdissipationtermand
scribedintheinteractionHamiltonianEq.(4),alreadyde-
white noise force22. As in Ref.[15], even in the present
scribed in many papers in the literature22,27,28. R (x)
(2) casetheequationfortheoscillatordynamicscanbewrit-
is the adiabatic correction exclusively due to magnetic-
ten as follows
coupling effects modifying the electronic phase of elec-
trons flowing from the leads to the CNT. mx¨+A(x)x˙ =F (x)+(cid:112)D(x)ξ(t)+A cos(ω t),
(0) ext ext
Finally,inthehypothesisofsymmetriccouplingtothe
(cid:104)ξ(t)(cid:105)=0, (cid:104)ξ(t)ξ(t(cid:48))(cid:105)=δ(t−t(cid:48)), (29)
leadsΓ =Γ ,onecancalculatetheadiabaticexpansion
L R
for the symmetrized current (cid:104)Iˆ(cid:105)=[(cid:104)Iˆ (cid:105)−(cid:104)Iˆ (cid:105)]/2
L R where ξ(t) is a standard white noise term. We have in-
e (cid:90) d¯hω cluded in our schematization the effect of an external
(cid:104)Iˆ(cid:105)(x,v)= |Gr(ω,x)|2(ΣR,>(ω,v)ΣL,<(ω,v)
¯h 2π antenna exciting the motion of the CNT, where Aext,
ω represent the antenna amplitude and frequency, re-
−ΣL,>(ω,v)ΣR,<(ω,v)). (21) ext
spectively. In this section, we describe how all the terms
Using Eqs.(6),(9), we get appearinginaboveequationmodifyinthepresenceofan
external transverse magnetic field.
(cid:104)Iˆ(cid:105)(x,v)(cid:39)(cid:104)Iˆ(cid:105) (x)+(cid:104)Iˆ(cid:105) (x,v), (22)
(0) (1) The total force acting on the CNT-resonator is
where
F =−kx−λ(cid:104)nˆ(cid:105)(x,v)+H˜(cid:104)Iˆ(cid:105)(x,v). (30)
(cid:90) d¯hω
(cid:104)Iˆ(cid:105) (x)=eΓ (f (ω)−f (ω))C(ω,x), (23)
(0) 8π L R The linear elastic force exerted on the oscillator is mod-
with linear corrections in the oscillator velocity ified by two relevant nonlinear correction terms: the
former is proportional to the electronic charge density
(cid:104)Iˆ(cid:105)(1)(x,v)=v[U(1)(x)+U(2)(x)], (24) on the CNT level Eq.(13), while the latter to the elec-
troniccurrentEq.(23). Thefirstterm,duetothecharge-
with
displacement interaction on CNT-resonator and propor-
U (x)=−eλ(cid:90) d¯hωg(1)(ω)C(ω,x)T(ω,x), (25) tional to λ was already discussed in Refs.[2] and [15].
(1) 2 2π − Far from equilibrium and in the presence of a mag-
neticfield, amagnetomotivecouplingbetweentheCNT-
e2 (cid:90) d¯hω resonator displacement and the electronic flow through
U (x)=− H˜ g(2)(ω)T(ω,x), (26)
(2) ¯h 2π − thedevicecomesintoplay. Actually,thetransversemag-
netic field introduces a phase in the electronic tunneling
where we have defined
that is proportional to the displacement of the CNT res-
g(1)(ω)= ∂[fL(ω)−fR(ω)], (27) onator as well as on the field strength. This originates
− ∂¯hω a Lorentz-like additive correction, linear in the magnetic
∂[f (ω)−f (ω)] fieldstrengthandintheelectroniccurrent,totheaverage
g(2)(ω)= L R . (28)
− ∂[eV ] force acting on the resonator.
bias
5
In the limit of the adiabatic approximation, the force Intheadiabaticlimit,exploitingtheeffectofthe’fast’
Eq.(30)canbedecomposedindifferentexpansionterms. electronic environment on the oscillator motion, one de-
It explicitly depends on the oscillator position x through rives
U (x)=U +λx and velocity v. The force is
gate gate
(cid:104)δFˆ(t)δFˆ(t(cid:48))(cid:105)=D(x)δ(t−t(cid:48)), (39)
F(x,v)=F (x)+F (x,v), (31)
(0) (1)
where in the presence of a magnetic field we have
where
D(x)=Dλ(x)+DH(x)+DH,λ(x), (40)
F (x)=−kx−λ(cid:104)nˆ(cid:105) (x)+H˜(cid:104)Iˆ(cid:105) (x), (32)
(0) (0) (0)
with
and (cid:90) d¯hω
Dλ(x)=λ2¯h G< (ω,x)G> (ω,x)=
F (x,v)=−λ(cid:104)nˆ(cid:105) (x,v)+H˜(cid:104)Iˆ(cid:105) (x,v) 2π (0) (0)
(1) (1) (1) (cid:90) d¯hω ¯hΓ f (ω)+¯hΓ f (ω)
=−A(x)v. (33) =λ2¯h L L R R ×
2π ((¯hω−U (x))2+[¯hΓ]2/4)2
gate
ThetotaldampingtermA(x)isgivenbythreecontribu- ×(¯hΓ (1−f (ω))+¯hΓ (1−f (ω))) (41)
L L R R
tions
and
A(x)=Aλ(x)+AH(x)+AH,λ(x), (34)
eH˜ (cid:90) d¯hω
DHλ(x)= λ |Gr (ω,x)|2C(ω,x)×
where both 2 2π (0)
×(ΣL,>(ω)ΣL,<(ω)−ΣR,<(ω)ΣR,>(ω))=
Aλ(x)=λR (x), (35) (0) (0) (0) (0)
(1) (cid:90) dω ¯hΓ +¯hΓ
=eλH˜ L R ×
coming from the charge-displacement coupling, and 2π [(h¯ω−Ugate(x))2+[¯hΓ]2/4]2
(cid:110) (cid:111)
× [h¯Γ ]2f (ω)(1−f (ω))−[¯hΓ ]2f (ω)(1−f (ω))
AH(x)=−H˜U (x), (36) L L L R R R
(2)
(42)
due to magnetic field coupling, are positive definite.
The function AH,λ(x) is proportional to both charge- whereC(ω,x)=−2(cid:61)Gr (ω,x)istheelectronicspectral
(0)
displacement coupling strength λ and magnetic field H function of the electronic level defined in Eq.(14). The
noise strength contribution coming from current-current
AH,λ(x)=λR (x)−H˜U (x), (37) fluctuations is
(2) (1)
(cid:90) d¯hω(cid:104) (cid:105)2
andisnotpositivedefinite. Remarkably,wehaveverified DH(x)=H˜2¯h f (ω)−f (ω) T(ω,x)×
that the whole sum appearing in Eq.(34) is positive def- 2π L R
(cid:110)
inite in all parameters regime of our model. This shows ×(1−T(ω,x))+ f (ω)(1−f (ω))+
L L
that, using a spinless fermionic model in the presence of
(cid:111)
normal (not ferromagnetic) electronic leads, the CNT- +f (ω)(1−f (ω)) T(ω,x), (43)
R R
resonatorexperiencesnonegativedampingregions. This
is in contrast to results of Ref.[31], where the authors where T(ω,x) = h¯ΓC(ω,x). In the absence of electron
use a normal and a ferromagnetic lead and observe neg- 4
bias voltage, one has D(x) = 2k TA(x), that is the
B
ative damping and consequent nano-electromechanical
fluctuation-dissipationconditionisverifiedforeachfixed
self-excitations of the CNT-resonator system.
position x. Moreover, it is possible to show that in the
A fluctuating term has to be be included to take cor-
chosen units, the dimensionless damping A(x) (Eq.(34))
rectly into account the effect of the bath degrees of free-
and diffusive term D(x) (Eq.(40)) result proportional to
dom. When a magnetic field is present, the force-force
the adiabatic ratio ω /Γ.
0
fluctuations are given by three contributions (see ap-
It is important to point out that, when there is no
pendix A)
intrinsic charge-displacement coupling (λ = 0), in the
absence of the antenna effects and at zero bias (V =
(cid:104)δFˆ(t)δFˆ(t(cid:48))(cid:105)=λ2(cid:104)δnˆ(t)δnˆ(t(cid:48))(cid:105)+ bias
0),theoscillatorisstillgovernedbyaLangevinequation
−H˜λ[(cid:104)δnˆ(t)δIˆ(t(cid:48))(cid:105)+(cid:104)δIˆ(t)δnˆ(t(cid:48))(cid:105)]+
(cid:113)
+H˜2(cid:104)δIˆ(t)δIˆ(t(cid:48))(cid:105), (38) mx¨+AH(x)x˙ =kx+ DH(x)ξ(t), (44)
where we get a mixed current-density fluctuation con- with a harmonic force F(0)(x) = −kx, an intrinsic
tribution [(cid:104)δnˆ(t)δIˆ(t(cid:48))(cid:105) + (cid:104)δIˆ(t)δnˆ(t(cid:48))(cid:105)], and a current- positive-definite dissipative term AH(x), and a diffusive
current fluctuation contribution (cid:104)δI (t)δI(t(cid:48))(cid:105) to the term DH(x) proportional to the thermal current-current
α
noise. noise. Looking at Eqs.(36) and (43), one can clearly see
6
0 eVbias =0.1 Aλ(x) eVbias =1.5
AλH(x)
(a) 5 12 (a) (d) 6
-1 -0 AH(x)
pm] -2 (x)/1 48 B=0.1λ Atot(x) 24
[ A
min -3 U =0µeV 0 0
x --54 Vbias=20µV UUgggaaattteee==9200 0µ eµVeV -4/1011..05 (b) (e) 0.6
(b) (x)0.5 B=λ 0.3
0 A
] 0.0
m 0.0
3
p 4 (c) (f)
[n -5 U =0µeV -10 4 2
xmi Uggaattee=150 µeV x)/ 2 B=5λ 1
-10 Vbias=300µV Ugate=300 µeV A( 0 0
-10 -5 0 5 10 -20 -10 0 10 20 -10 0 10 20
B(T) x-2U /λ x-2U /λ
gate gate
FIG. 2. (Color online) Panel(a): minimum of the effective FIG. 3. (Color online) Spatial dependence of the dimension-
potential(comingfromtheforceEq.(32))affectingtheCNT- less damping coefficient A(x) at low bias (Panels(a-b-c)) and
resonator as a function of the magnetic field at low bias at large bias voltage applied (Panels(d-e-f)). See main text
eV = 0.1¯hΓ (V = 20µV in our units). Solid (black) for discussion.
bias bias
line indicates U = 0, dashed (red) line U = 0.45
gate gate
(V = 90µV in our units), dotted (green) line U = 1.0
bias gate
(Vbias = 200µV in our units). Panel(b): same as above at resonator effective potential minimum is displaced to-
largebiaseV =1.5¯hΓ(V =300µV inourunits). Solid
bias bias wards positive displacements x with respect to the min-
(black)lineindicatesU =0,dashed(red)lineU =0.75
gate gate imum set by the charge-displacement interaction (see
(V =150µV in our units), dotted (green) line U =1.5
bias gate Panel (a-b) of Fig.(2)). In Fig.(2), one can observe that,
(V =300µV in our units).
bias the minimum of the effective potential acting on the res-
onator depends linearly on the magnetic field strength.
This comes from the linear dependence on the mag-
thatanaturalquadraticdependenceofdampinganddif-
neticfieldoftheLorentz-likecorrectiontermtotheforce
fusive strength on the magnetic field emerges. This can
Eq.(32). In particular, as shown in Panel(a) in Fig.(2),
be explained observing that, even at zero bias voltage,
in the low bias regime for the device (small compared
the electronic tunneling events, whose phase is depen-
to the broadening of the CNT level), the larger is the
dent linearly on the CNT displacements as well as on
gate voltage, the smaller is the displacement of the po-
the magnetic field strength, perturb the CNT mechani-
tential minimum as a function of the external magnetic
calmotionwithaforcewithzeroaverage(dueto(cid:104)Iˆ(cid:105)=0,
field with respect to the shift produced by the charge-
H˜ can be also different from zero) and square mean pro- displacement interaction on the CNT (whose position is
portional to the magnetic field square. Definitely, even indicated by a (black) square for U = 0, a (red) cir-
gate
in the absence of external bias voltage Vbias, the mag- cle for Ugate = 0.45 (Vbias = 90µV in our units), and
netic field applied perpendicular to the CNT couples to a (green) triangle U = 1.0 (V = 200µV in our
gate bias
the bending mode dynamics behaving as a surrounding units)). This can be explained observing that in the low
thermal bath at leads temperature kBT. conductingregimeofthedevicetheresonatorislesseffec-
We end this section with a systematic study of the tivelycoupledwiththeelectronicsubsystem. Inthelarge
spatial dependence of the total force, the damping (see biasregime(Panel(b)ofFig.(2)),asmallermagneticfield
Fig.(3)) and diffusive terms (see Fig.(4)) as a function of is sufficient to displace the potential minimum of the
the bias voltage as well as on the magnetic field. same quantity produced by the sole charge-displacement
As concerns the total force acting on the CNT res- interaction on the CNT. Again, the larger is the gate
onator,wepointoutthat,forthemagneticfieldstrengths voltage, the smaller is the displacement of the potential
investigated in this paper, the effective potential pre- minimumasafunctionoftheexternalmagneticfieldwith
servesitsparabolicshapewithadisplacedminimumand respecttotheshiftproducedbythecharge-displacement
renormalized curvature. For instance, when a left-to- interaction on the CNT.
right current flows through the device (see the sketch The renormalization of the effective potential curva-
in Fig.(1)) in the presence of a positive magnetic field ture, that is of the resonance frequency of the resonator,
(outgoingfromthesketchreportedinFig.(1)),theCNT- will be discussed in subsection B of next section.
7
In this section, we limit ourself to discuss the damp-
ing term A(x), since for the diffusive term D(x), un- eVbia s=0.1 Dλ(x) eVbias =1.5
lsehsoswenxpalbicoitvley(ssteaeteEdq,.a(3s4i)m),ilawreacnaanlydsiisstcinangubisehdboentew.eAens -500.6 (a) DDλHH((xx)) (d) 4
tlahtroere:coantpruibreutciohnasrgteo-dtihspelafrciecmtioenntacffoencttreidbubtyiotnheAoλs(cxi)l-, (x)/10.3 B=0.1λ Dtot(x) 2
D
depicted in Fig.(3) with a solid (black) line, already
0.0 0
discussed in Refs.[22] and [28]; a damping contribution 5 0.6 (b) (e)
due to current-current fluctuations AH(x), depicted in -0 4
1
Fig.(3) with a dotted (blue) line; a mixed damping term x)/0.3 B=λ 2
due to current-density fluctuations (not positive defi- (
D
nite), indicated by AλH(x) and depicted in Fig.(3) with 0.0 0
15
a short-dashed (red) line. The total damping A(x) is re- -501.0 (c) (f)
ported with a dashed (pink) line. As one can observe 1 B=5λ 10
in Panel(a), at low bias voltage, when the external mag- x)/0.5 5
netic field strength is smaller than charge-displacement D(
coupling λ, the damping contributions coming from the 0.0 0
-20 -10 0 10 20 -10 0 10 20
current-current AH(x) and current-density AλH(x) fluc-
x-2U /λ x-2U /λ
tuations are negligible with respect to that generated
gate gate
by the pure charge-displacement contribution Aλ(x). In
Panel (a), Aλ(x) and AH(x) have a single peak struc-
ture centered at x−2U /λ (cid:39) 0, while AλH(x) is an FIG. 4. (Color online) Spatial dependence of the dimension-
gate
less diffusive coefficient D(x) at large bias voltage applied
odd symmetric function with respect to this point. We
(Panels(a-b-c)). See main text for discussion.
point out that these peculiar structures emerge only at
larger values of magnetic field (Panels(b-c) of Fig.(3)).
The total damping affecting the resonator is peaked at
tions of the electronic current flowing through the CNT
configurations where large density variations take place
as a function of the gate voltage (see Eq.(32)). These
|x−2U /λ| < ¯hΓ/λ. Indeed, the density of the CNT
gate current variations are positive for x−2U /λ < 0 and
level goes from a region x−2U /λ < −¯hΓ/λ corre- gate
gate negative otherwise. At x−2U /λ (cid:39) −eV /2, large
sponding to almost completely filled states ((cid:104)n(cid:105) ∼ 1) to gate bias
negative variations of the electronic density and positive
a region x − 2U /λ > ¯hΓ/λ corresponding to com-
gate variations of the electronic current cooperate giving an
pletely empty states ((cid:104)nˆ(cid:105) ∼ 0). Definitely, the CNT
enhanced damping.
level experiences an unit charge variation across the
We intend now to study the regime realized when the
|x−2U /λ|<¯hΓ/λ region14,15.
gate external magnetic field strength is larger than charge-
At large bias voltages applied (Panel (b)), Aλ(x) has
displacement coupling strength λ. In this case, the
two peaks centered at x − 2U /λ (cid:39) eV /2λ and
gate bias contribution to the damping coming from the current-
x−2Ugate/λ (cid:39) −eVbias/2λ, respectively. AH(x) shows currentfluctuationsAH(x)aredominantwithrespectto
the same behavior, while AλH(x) is an odd symmet- those corresponding to charge-displacement Aλ(x) and
ric function with respect to these two points. As in current-density AλH(x) fluctuations. In the low bias
Panel (a), AH(x) and AλH(x) are negligible with re-
regime, the total damping term preserves its single peak
spect to Aλ(x). The total damping affecting the res-
structure which, due to the intrinsic asymmetry of the
onator is peaked at configurations where the CNT level current-density term AλH(x), is slightly distorted. For
experiences a half-unit charge variation across the |x−
the same reason, in the large bias regime, the double dip
2U /λ±eV /2λ|<¯hΓ/λ regions14,15.
gate bias structure of the total damping term is preserved with
When the external magnetic field is turned on, an en- an enhanced asymmetry. In the large magnetic field
hanceddampingaswellasasnoisestrengthemergeswith regime,itisimportanttopointouttheparticularspatial
a quadratic dependence on the magnetic field intensity dependence of the noise strength D(x) (Panels (c-f) of
Eqs.(34)-(40). In Panels(b-e) of Fig.(3), one can observe Fig.(4)). Here,thenoisecontributionduetothecurrent-
that, as the dimensionless ratio B/λ is equal to one, the current fluctuations (DH(x)) emerges with the charac-
totaldampingaffectingtheresonatorisonlyslightlyper- teristic double peak structure even at low bias regime
turbed by the application of the magnetic field. At low (dotted(blue)curveinPanel(c)ofFig.(4)). Comparing
bias, A(x) preserves its single peak structure with an thedashed (pink)curves inPanel (c)of Figs.(3)-(4), one
enhanced strength (dashed (pink) curve in Panel (b) of can observe that, even at low bias voltage, the applica-
Fig.(3)). At large bias, the strength of the two peaks tionofalargemagneticfielddrivetheCNT-resonatorfar
becomes asymmetric, with an enhanced damping of the out of equilibrium, breaking the validity of the Einstein
peak at x−2U /λ (cid:39) −eV /2. This effect can be relation D(x)=2k T A(x) with an effective tempera-
gate bias B eff
explained as follows: when a magnetic field is applied to ture. Far from equilibrium, this relation is strictly valid
the device, the resonator starts to feel even the varia- only at very low bias voltages22,38.
8
In the next section, we study numerical results of
our model concerning mechanical properties of CNT- eVbias= 0.1 eVbias =1.0
3 5
resonator(resonancefrequencyandqualityfactor)aswell
as the electronic observables inherent to the transport 4
problem (I-V characteristic). 2
40 3
1
Q/ 1 2
IV. MECHANICAL AND ELECTRONIC 1
CHARACTERISTICS OF THE DEVICE 0 (a) (b)
0
5
4 (c) (d)
Given the assumption about the separation between -0 4 2
the slow vibrational and fast electronic (tunneling) 1
timescales, the problem of evaluating a generic observ- >/ 3
able(electronicornot)ofthesystemreducestotheeval- x) 2 1
(
A
uationofthatquantityforafixedpositionxandvelocity 1
<
voftheoscillator,withtheconsequentaveragingoverthe
0 0
stationary probability distribution P(x,v). From the so- -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0
lution of the Langevin equation (29), one can determine U U
the distribution P(x,v) which allows to calculate all the gate gate
electronic observables O:
(cid:90) FIG.5. (Coloronline)Devicequalityfactorasafunctionthe
(cid:104)O(cid:105)= dxdvP(x,v)O(x,v). (45) gate voltage U for different magnetic field strengths at
gate
low (Panel (a)) and large (Panel (b)) bias voltage. Panels(c-
d)Sameasabovefortheaveragetotaldamping(cid:104)A(x)(cid:105)ofthe
Weanalyzeinthenextsectiontheeffectsofthemagnetic
system. Panel(a-c): solid thick line B = 0.0, solid normal-
fieldonthemechanicalaswellaselectronicpropertiesof
thickness line B =1.5, and solid thin line B =3.0. Panel(b-
the device.
d): solid thick line B =0.0, solid normal-thickness line B =
0.2, and solid thin line B =0.4.
A. Device Quality factors
voltage regime, respectively. In the absence of a trans-
One of the main findings of Ref.[1] is the observation versemagneticfield,wereproducethequalitativebehav-
of a quadratic dependence of the device quality factor Q ior obtained in the experiment of Ref.[14]. When bias
on external magnetic field strength. Within our model, voltages are smaller than the broadening due to tunnel
asalsostressedintheprevioussections,suchaquadratic coupling, the quality factor shows a single dip feature
dependence on B emerges naturally. In order to include (solid (black) thick line in Panel (a) of Fig.(5)). At bias
back-actions effects of the out of equilibrium electronic voltages that exceed (or are equal to) the broadening
bath on the resonator, we have calculated the average due to tunnel coupling, the quality factor shows a dou-
device quality factor as ble dip structure (solid (black) thick line in Panel (b) of
Fig.(5)). This behavior, already addressed in Refs.[14]
(cid:90) ∞ 1 and [15], can be easily explained looking at the average
Q= dx P(x), (46)
A(x) charge and dissipation of the CNT-resonator. As also
−∞
discussed referring to total damping affecting the CNT-
whereA(x)isthetotaldampingataparticularresonator resonatorintheprevioussection,atlowbiasvoltageand
displacementxandP(x)isthereduceddisplacementdis- intheabsenceofmagneticfield, thetotalaveragedamp-
tributionprobabilityoftheCNT-resonator. Wehavever- ing affecting the resonator is peaked at electronic config-
ified that this particular way of extracting quality fac- urationswheretheCNTlevelexperiencesanunitcharge
tors is completely equivalent to measure the width at variation across the region where the small conduction
half-highin the current-frequencycurves obtained inthe window is placed |U |<¯hΓ (solid (black) thick line in
gate
linear response to an external antenna exciting the nan- Panel (c) of Fig.(5)). At large bias voltages, the conduc-
otube motion15. tion window, whose extension is proportional to eV ,
bias
Motivated by the experiment performed in Ref.[1] and becomeslarger thanthe broadening ofthe CNTlevel, so
byrecentexperimentalstudyonasimilarCNTdevice14, that the total average damping affecting the resonator
we here performed a systematic study of the quality cal- is peaked at electronic configurations where the CNT
culated from our model as a function of the bias, gate level experiences half-unit charge variations, that is at
voltage as well as on the magnetic field. In Panels (a- |U − eV /2| < ¯hΓ and |U + eV /2| < ¯hΓ.
gate bias gate bias
b) of Fig.(5), we investigate the device quality factor Q When the transverse magnetic field is turned on, the
as a function of gate voltage in the low and large bias above scenario modifies as follows. At low bias volt-
9
ages, the total damping affected by the CNT-resonator
increasesquadraticallywiththefieldateverypointinthe Ugate= 0.45 Ugate =0.0
8
configurationspaceoftheoscillator. Moreover,theCNT-
resonator distribution probabilities P(x) depend slightly 8 (a) (b)
6
on the magnetic field as well as on the gate voltages and 6
4
0
are actually centered at configurations close to the har- 1 4
4
monicpotentialminimumx(cid:39)0intheabsenceofcharge- Q/
displacement interaction λ. The overall result is an en- 2 2
hanced average total damping as one increases the mag-
0 0
netic field (solid normal-thickness (B = 1.5) and thin
-0.8 -0.4 0.0 0.4 0.8-0.8 -0.4 0.0 0.4 0.8
(cBorr=esp3o.n0d)in(bgladcekc)relainseesoifntPheanqeula(lcit)yoffacFtiogr.(5in))aallndthae 3 (c) B (d) B 2
4
gate voltage range investigated (solid normal-thickness 0
1
(B = 1.5) and thin (B = 3.0) (black) lines in Panel / 2 1
Q
(a) of Fig.(5)). At large bias voltage, the P(x) still
depends only slightly on the magnetic field but is very 1
spread on the configuration space. Therefore, the aver- 0
-1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0
age in Eq.(46) reproduces the spatial dependence struc-
eV eV
ture of the total damping coefficient reciprocal 1/A(x). bias bias
The double peak structure of the average total damping
term (solid thick (black) line in Panel (d) of Fig.(5)) is
canceled by the magnetic field, giving a single peak at FIG. 6. (Color online) Device quality factor as a func-
tion the magnetic field strength B for different bias volt-
U = eV /2 where a cooperation between negative
gate bias ages (Solid thick line eV = 0.1, solid normal-thickness
charge-density and positive current variations take place bias
line eV = 0.75, and solid thin line eV = 1.5) at low
(solid normal-thickness (B = 0.2) and thin (B = 0.4) bias bias
(Panel(a))andhigh(Panel(b))conductingstates. Panels(c-
(black) lines in Panel (d) of Fig.(5)). As a consequence,
d) Device quality factor as a function the bias voltage eV
bias
the quality factor loses its double dip structure getting a for different magnetic field strengths at low (Panel (c)) and
single dip at Ugate = −eVbias/2 (solid normal-thickness high (Panel (d)) conducting states. Panel(c): solid thick line
(B = 0.2) and thin (B = 0.4) (black) lines in Panel (b) B =0.0, solid normal-thickness line B =0.05, and solid thin
of Fig.(5)). lineB =0.1. Panel(d): solidthicklineB =0.0,solidnormal-
thickness line B =0.25, and solid thin line B =0.5.
We intend now to study the device quality factors as
a function of the transverse magnetic field B comparing
different conducting states of the device. In Panel (a) of
Fig.(6), onecanobservecalculateddevicequalityfactors
as a function of the magnetic field at a low conducting
state of the device (U =0.45). Different curves, from
gate
thicker to thinner, refer to increasing bias voltages ap- 25000
plied eV = 0.1−0.75−1.5. At every fixed bias volt-
bias
age, a clear quadratic dependence of the total average
damping on the magnetic field strength is observed (not
showninFig.(6)),withaLorentzianshapeofthequality
Q
factor curves (see Panel (a) of Fig.(6)). It is important 20000
to point out that the range of magnetic field strengths
experimentally investigated in Ref.[1], B = 0−3T, cor-
Exp. Ref. [1]
responds to small magnetic fields in our units (we recall
Theory
that H = 16.6T). Remarkably, at low bias and small
0
magnetic fields, a quadratic decrease of the Q against 15000
magnetic field is observed (see solid thick (black) line in
Panel(a)ofFig.(6)). InFig.(7),weshowthequantitative 0 1 2 3
agreement between experimental and calculated quality B(T)
factorsagainstmagneticfieldwhenthedeviceisinalow
conducting state, with eV = 0.1 and U = −0.45.
bias gate
The slight increase of the quality factor Q as a function FIG. 7. (Color online) Device quality factor as a function
ofthefieldforsmallmagneticfields,isduetoasymmetry the magnetic field strength. Squares represent experimental
introduced by the gate voltage Ugate = 0.45 applied to values obtained in Ref.[1] at T = 25mK, Vbias = 0.3mV
thedevice(seealsoPanel(a)ofFig.(6)). Forgatevoltage and distance from the current peak Ugate = −90mV. Solid
(red) line is calculated Q at k T = 0.01, eV = 0.1 and
equaltozero,thatinthehighconductingstateofthede- B bias
U =0.45.
vice,thecalculatedQagainstB curveisaparabolawith gate
10
a maximum at zero magnetic field applied.
Static Dyn.+Antenna
ComingbacktoPanel(a)ofFig.(6),onecanobservean
interesting increase of the quality factor peak as a func-
1.000
tion of the bias voltage. In particular, for eV = 1.5 1.000 (3)
bias (2)
(thinnerlineinPanel(a)ofFig.(6))aqualityfactorpeak 0) (1) 0.998
at B (cid:39) U /2 = 0.225 occurs. This can be directly re- ω(0.996
lated to tghateeaverage total damping dip, not shown in /s 0.996
e
Fig.(6). This effect can be explained noting that, when ωr0.992 λ=0.1 0.994
the bias voltage applied to the electronic device is in- (a) eV =0.1 (b)
bias
0.988 0.992
creased, a transition from a single peak to a double peak

structure in the spatial dependence total damping af- 1.000 (3) 1.000
fected by the CNT-resonator con be observed (compare 0) (2)
Panel(a) and (d) of Fig.(3)), while at the same time, the ω( (1) 0.998
/ 0.996
tdhisepclaocnefimguenrattidoinstrsipbaucteiornemparionbinabgilcietnietserPed(xa)tscpornefiagduroan- ωres λ=0.1 0.996
tions close to the harmonic potential minimum x (cid:39) 0 0.992 (c) eV =1.5 (d) 0.994
bias
characteristic of the low bias regime. The overall result
-2 -1 0 1 2 -1 0 1 2
is a reduction of the average total damping affecting the
U U
CNT-resonator whose minimum is translated by a quan- gate gate
tity proportional to the gate voltage applied to the de-
vice. This argument becomes even more clear when no
FIG.8. (Coloronline)Panel(a): Resonatorfrequencyagainst
gate voltage is applied to the device which is therefore
effective gate voltage calculated as minimum of the effective
placed in a high conducting state. In this case a perfect
potential in the static approximation at small bias eV =
bias
symmetry of Q-factor curves with respect to zero mag-
0.1¯hΓfordifferentmagneticfieldvalues: SolidthicklineB =
netic field is obtained (see Panel (b) of Fig.(6)). 0.0, solid normal-thickness line B = 1.5, solid thin line B =
We end this section with a study of the device qual- 3.0. Panel(c): sameasPanel(a)atlargebiaseV =1.5¯hΓ
bias
ity factors as function of the bias voltages and magnetic for different magnetic field values: From thicker to thinner
fieldscomparinglowandhighconductingstatesofthede- line B =0.0−0.1−0.2−0.3−0.4. Panel (b-d): Resonator
frequency calculated using an external antenna (with A =
vice. In Panel (c) of Fig.(6), one can observe calculated ext
10−3) at mechanical resonance against effective gate voltage
devicequalityfactorsasafunctionofthebiasvoltagesat
forsameparametersofPanel(a-c),respectively. Dashed(red
a low conducting state of the device (U =0.45). Dif-
gate online)andsolid(blueonline)portionsofeachcurveindicate
ferent curves, from thicker to thinner, refer to increasing
resonancefrequencyvalueswithpositiveandnegativecurrent
magneticfieldappliedtothedeviceB =0.0−0.05−0.1.
change ∆I, respectively. In Panel (d), only curves referring
At zero magnetic field, a clear double dip feature in the to magnetic field strengths B =0.0−0.2−0.4 are reported.
quality factor Q, as experimentally observed in Ref.[14],
is visible. This can be explained looking at the aver-
age total damping and in terms of the average charge ity factors as a function of the bias voltages at a high
present on the CNT level. The total average damp- conducting state of the device (U = 0.0). Different
gate
ing, in the absence of magnetic field, has two peaks at curves, from thicker to thinner, refer to increasing mag-
eV = −2U = −0.9 and at eV = 2U = 0.9. netic field applied to the device B = 0.0−0.25−0.5.
bias gate bias gate
Indeed, as also discussed previously, the total average As above, at zero magnetic field, a single dip feature
dampingispeakedatelectronicconfigurationswherethe in the quality factor Q, as experimentally observed in
CNT level experiences half-unit charge variations, that Ref.[14], is visible. This behavior can be discussed with
isat|U −eV /2|<¯hΓand|U +eV /2|<¯hΓ. the same argument given for discussing the Panels(a-b)
gate bias gate bias
Therefore, when the edges of the conduction window of Fig.(6), where a reduction of the total average damp-
(whose width is proportional to eV ) meet the CNT ingasafunctionofthebiasvoltageappliedtothedevice
bias
level energy (given by U ), a maximum total average was observed. Again, a decrease of the quality factor in
gate
damping (minimum quality factor) is observed. As also all the gate voltage range investigated as a function of
discussed in reference of Fig.(5), the double peak struc- the magnetic field is observed (see Panel (d) of Fig.(6)).
tureoftheaveragetotaldampingtermiscanceledbythe
magnetic field, giving a single peak at eV = −2U
bias gate
whereacooperationbetweennegativecharge-densityand B. Resonance frequency renormalization and
positive current variations take place. As a consequence, current-voltage curves
the quality factor loses its double dip structure getting
a single dip at eVbias = −2Ugate (solid normal-thickness Inthissection,weaddressthemagneticfieldeffectson
(B =0.05) and thin (B =0.1) (black) lines in Panel (c) the renormalization of the CNT-resonator resonance fre-
of Fig.(6)). quencies and its back-action effects on the current volt-
InPanel(d)ofFig.(6),weshowcalculateddevicequal- ages curves of the device. In order to study the CNT