Charge transport through a SET with a mechanically oscillating island

0.24 MB·English

by  N. M. Chtchelkatchev

Charge transport through a SET with a mechanically oscillating island
N. M. Chtchelkatchev,1,2,3 W. Belzig,1 and C. Bruder1
1Departement Physik und Astronomie, Universit¨at Basel, Klingelbergstr. 82, 4056 Basel, Switzerland
2L.D. Landau Institute for Theoretical Physics, Russian Academy of Sciences, 117940 Moscow, Russia
3Institute for High Pressure Physics, Russian Academy of Sciences, Troitsk 142092, Moscow Region, Russia
4
(Dated: February 2, 2008)
0
0
Weconsiderasingle-electrontransistor(SET)whosecentralislandisananomechanicaloscillator.
2
The gate capacitance of the SET depends on the mechanical displacement, thus, the vibrations of

n the island vibrations may strongly influence the current-voltage characteristics, current noise, and
a higher cumulants of the current. Harmonic oscillations of the island and oscillations with random
J amplitude (e.g., due to the thermal activation) change the transport characteristics in a different

5 way. The noise spectrum has a peak at the frequency of the island oscillations; when the island
2 oscillatesharmonically,thepeakreducestoaδ-peak. WeshowthatknowledgeoftheSETtransport
properties helpsto determine in what way theisland oscillates, to estimate theamplitude, and the

] frequency of theoscillations.
l
l
a PACSnumbers: 73.23.Hk,85.35.-p,72.70.+m
h
-
s
e The interplay of electric currents through nanostruc- tal realization of this scenario [7].
m tures with their mechanical degrees of freedom has at-
The charge of the island is coupled to the leads and
. tracted a lot of interest recently, both from the experi-
t the gate through the capacitances CL, CR, and Cg(z);
a mental and theoretical side [1, 2, 3, 4, 5, 6, 7, 8, 9]. One
z is the transverse deviation of the island from its equi-
m
ofthecentralquestionsofthisnewfieldofnanophysicsis
librium position. When the island oscillates, the gate
- howthevibrationsoftheoscillatingpartofananodevice
d capacitance changes with the position of the island cen-
influenceitstransportpropertiesandviceversa. Anum-
n ter z, and therefore the transport properties of the SET
o ber of nanomechanical devices were investigated in the change.
c last years,e.g., so-calledsingle-electronshuttles [3, 4, 5].
[ On the theoretical side, it was shown recently that elec-

1 triccurrentspassingthroughadirtynanowirecanstimu- L z=0 R
v lateitsvibrations[10]. Indicationsforthermalvibrations
6 of suspended single-wall nanotubes doubly clamped be- z
8 a)
tween two contacts were observed [11].
4 gate
1 Recently, the nanomechanical properties of single-
0 electron transistors (SETs) have started to attract at-
4 tention. SETsbuiltfromcarbonnanotubesareanatural VL CL,RL CR,RR VR
0 candidate for this type of question. For instance, it was b)
/
t shown that the equilibrium shape of a suspended nan-
a
m otube studied as a function of a gate voltage shows fea-
turesrelatedtosingle-electronelectronics,e.g.,Coulomb C(z)
- g
d “quantization” of the nanotube displacement [7]. V
n g
InthisLetter,wediscusshowvibrationsofthecentral
o
c island of the SET change the current and the noise. We FIG.1: (a)Sketchofasingle-electron transistor(SET)with
: show that the transportcharacteristicsof the SET differ an oscillating island. The island is coupled to the left (L)
v
for islands oscillating thermally or harmonically. The and right(R) leads by tunnel junctions, and its capacitance
i
X current noise spectrum has a peak at the frequency of tothegateCg(z)dependsonthecoordinatez thatmeasures
the deviation of the island from its equilibrium position. (b)
r the island oscillations that reduces to a δ-peak when the
a Equivalent circuit of the device.
island oscillates harmonically. Therefore, knowing the
transportpropertiesofthe SETcanhelptodeterminein
whatwaytheislandoscillates,andtofindthe amplitude We assume that electronic transportthroughthe SET
and frequency of the oscillations. can be described by sequential tunneling. In this case,
The system that we want to study – a SET with a it is governed by four tunneling rates [12, 13, 14]: the
mechanically oscillating island – is sketched in Fig. 1. rate for electrons to tunnel onto the central region from
We assume that the island is coupled to the left (L) and the left (ΓL ) and right (ΓR ) and the rates for
n→n+1 n→n+1
right(R) leadsby tunneljunctions butcanmechanically electrons to tunnel off the central region. The rates can
vibrate. A suspended nanotube is a possible experimen- becalculatedviaFermi’sgoldenrule. Theenergychange
2
corresponding to the first tunnel process is many electrons go through the SET and thus it is pos-
sible to neglect the time derivative in the master equa-
∆EnL→n+1 =e(Vg−VL)+εn, (1) tion (3). Then all the methods used for current calcu-
e2 1 C (V −V )+C (V −V ) lation in “usual” SETs are applicable to the case with
R R g L L g
ε = +n+ ,
n C (cid:20)2 e (cid:21) the oscillating island [14]. Averaging the current over
Σ
time can be replaced by averaging over P(z), the den-
where CΣ =CL+CR+Cg(z). Defining γ(ε)≡−ǫ/(1− sity of the probability distribution for the deviation z,
exp(βǫ)), the rates can be written as i.e., I¯ = P(z)I(z)dz. If the island oscillations are
1 thermally Ractivated, P(z) ∝ exp(−z2/2hz2iT), where
ΓLn→n+1(z)= e2R γ(∆EnL→n+1(z)), (2) hz2iT = kBT/mω02. If the island oscillates harmoni-
L cally, z(t) = z sin(ωt), then P(z) = 1/π 1−(z/z )2.
0 0
whereR istheresistanceoftheleftjunction. Theother In these expressions, the driving terms ∼pf in Eq. (4)
L
rates can be written similarly. The charge state of the that couple the current in the SET with its mechanical
SET is characterized by the probability ρ (t) to find n degrees of freedom were neglected. This term is usually
n
excess electrons on the island. The time evolution of much smaller than ω02z on the left-hand side of Eq. (4)
ρ (t) is governedby the masterequation[12, 13,14, 15]: (e.g., for the SET parameters in recent experiments, see
n
Ref.7): thesmallparameterisz max ∂ lnC . Thedriv-
0 i z i
∂ρi ingtermsmaybecomeimportant,e.g.,whenana.c. bias
= (Γi+a→iρi+a−Γi→i+aρi), (3)
∂t near the resonance frequency ω is applied to the termi-
aX=±1 0
nal(s) of the SET. Then the a.c. bias drives the island
where Γi→j = ΓRi→j +ΓLi→j. The current and all its cu- oscillations and the f-terms in Eq. (4) are important to
mulants can be expressed through Γ and ρ [16]. determine the amplitude z0 of the oscillations [19].
Equation(3)hastobesupplementedwiththeequation
In general, the current and noise in a SET cannot be
describingtheoscillationsoftheisland. Theelectrostatic
calculated analytically for arbitrary transport voltages
force that acts on the island [17] is f(z) = ∂ C (V −
z g g [20] even when the island is static. Analytical progress
ϕ)2/2, where ϕ = (C V +C V +C V +q)/C is
L L R R G G Σ can be made if we restrict ourself to the case of small
thepotentialofthe island,andq = ρ netheaverage
n n drivingvoltagesneartheonsetvoltage,andtemperatures
charge. Then the motion of the islaPnd can be described much below the charging energy e2/C , i.e., γ(ε)≈θ(ε)
Σ
by a Langevin equation,
inEq.(2). Inthecaseofastaticisland,theperformance
of the SET as a transistor and electrometer reaches an
z¨+ηz˙+ω2z =y+[f(z)−f(0)]/m, (4)
0 optimum in this regime [20]. In this region, the trans-
port characteristics of the SET are also most sensitive
where m is the island mass; η ∼ ω /Q where Q is
0
to mechanical oscillations of the island, so this regime
the quality factor; finally, y is a random force simulat-
ing the interaction with the thermal bath, hy(t)y(t′)i = is the most interesting. Only two states of the island
2ηmTδ(t − t′) [18]. Typically Q ∼ 103 − 104 [1] and haveto be takenintoaccount;the probabilityρhas only
two nonzero values ρ ,ρ [20]. If V < V an elec-
η ≪ ω0. If the island is a nanotube, Cg is a functional n n+1 L R
tron enters the island with the rate ΓL (z) from the
ofthe deviationz(x)where xis the coordinatealongthe n→n+1
leftleadandgoesawaywiththe rateΓR (z)intothe
nanotube [7]. The rates depend only on integral quan- n+1→n
right lead. The average current will be
tities like z(x)dx/L (L is the length of the nanotube).
Their dynRamics can be described by Eq. (4) if the am-
plitude of the nanotube oscillations does not exceed its eΓL (z)ΓR (z)
diameter by several orders of magnitude. I¯= dzP(z) n→n+1 n+1→n , (6)
Z ΓL (z)+ΓR (z)
The current going through the left junction [14] is n→n+1 n+1→n
I(t)=e [ΓLj→j+1(t)−ΓLj→j−1(t)]ρj(t). (5) where n = [−Cg(0)Vg/e], Cg(0) ≡ Cg(z = 0), and [...]
Xj means the integer part. Assuming that the capacitances
depend only weakly on z, Eq. (6) can be expanded with
We are interested in the current averaged over a time
respect to z. To proceed, we define
interval τ much larger than the characteristic period T
0
oftheislandoscillations: I¯= τ I(t,z(t))dt/2τ,τ →∞.
−τ
is Tωh0e∼ty1p0i0caMlHfrze.quIefneclyecotrfomnRsicrtoumnneeclhathnricoaulgohsctihlleatSioEnTs J(z)= 1eR ∆∆EELnL→n+1+∆REnR∆+1E→Rn . (7)
with a similar frequency, the current will be of the or- R n→n+1 L n+1→n
der of I ∼ eω ∼ 10−11A. However, the current going
0
through a typical SET is usually several orders of mag- Using ∆ER =e(V −V )−∆EL and defining
n+1→n R L n→n+1
nitude bigger. Hence, during the oscillation period T z ,z tobe therootsofthe equations∆EL =0and
0 1 2 n→n+1
3

0.15
harmonic island oscillations 1.8 harmonic island oscillations
mthoertimonallleys sd irsivlaennd i sland |V| 2Vosc 1.6 motionless island
0.10 a) thermally driven island
1.4
V 0.2
0.05 |V|/2 Vosc 1I.02
V /
1I. 0

0.00 0.8
b) V 0.5
00.2 |V|/2 Vosc 0.6
I
I/ 0.1 0.4
0.2

0.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
c) V 1
VC (z 0)/e
0.4 |V|/2 Vosc
FIG. 3: Coulomb staircase, RR/RL = 1000, VL = −VR =
0.2 V/2,CL=CR =Cg(0),Q0 =40.6(Cg(0) andotherparameters
are asin Fig. 2). Theaverage squareamplitude of theisland
0.0 deviations is equal to that of the harmonically oscillating is-
40.0 40.5 41.0 41.5 42.0 42.5 43.0 land. The Coulomb steps are smeared on the scale V . For
osc
Q0/e other choices of Vg thestaircase graph looks similarly.
FIG. 2: Current gate-charge characteristics for symmetric
QSE0T=(−RLCg(0=)VgR.R()a)wVith=C0g.2≫, (|VC|L/,2R,<VVLos=c),−(bV)RV==V0/.52,, In the limiting case CL,CR ≪Cg(0),
(|V|/2 ≈ V ), (c) V = 1, (|V|/2 > V ). The voltage is
mI0ea=su(reed/Cing(o0s)uc)n/(itRsLof+|eR|/R2)C.Σ(Tzh=e d0)a,shthedoescccuurrrveentcoinrreusnpiotsndosf Vosc = z0∂[CzCg(0g)|]z2→0e(n+1/2). (10)
to a static island, the solid curve to a harmonically oscil-
lating island, z = z0sin(ω0t); z0(∂zCg)/Cg = 5.6 · 10−3 Eq. (8) is valid if Vosc < e/CΣ. If the driving voltages
(this is typical for SETs where the island is a nanotube applied to the SET terminals are much larger than V ,
osc
[7]; then z0 ≈ 5r, where r is a typical nanotube diame- the integration limits in Eq. (8) can be extended to in-
ter). The dotted curve in Fig. 1a illustrates what happens
finity because they far exceed z , the scale of decay of
if V /(|e|/2C(0)) = 5 > 1 and Eq. (8) is not valid. The 0
osc g P(z). The second term in Eq. (8) vanishes and
dash-dotted curves correspond to the case of thermal mo-
tion; the thermal average hz2iT ≡ kBT/mω02 is chosen to 1 ∂2
bisetheqeunaulmh(bze0rsionf(ωel0etc)t)r2oints=onz02t/h2e.isTlahnedinintetgheer sptaartticofreQgi0m/ee I¯=I(z =0)+ 2hz2i∂z2I(z)|z→0 , (11)
whenVL=VR =0. Thecurvesareperiodicinthestaticcase, where hz2i = P(z)z2dz. The first term in Eq. (11) is
but not if the island oscillates. The areas underthepeaks in
thestatic and in thedynamical cases are thesame. the current foRr a static island. If the driving voltages
applied to SET terminals are smaller than V then the
osc
second term (linear in z) in Eq. (8) does not vanish; in
this regime the current-voltage characteristics is largely
∆ER =0, we get from Eq. (6)
n+1→n influenced by island oscillations. The small parameter
in the expansion Eq. (8) is z ∂ C . The second term
0 z g
max(z1,z2) in Eq. (11) is of second order in this parameter. If we
I¯≈ dzP(z)[J(0)+ investigate the I −V characteristics, island oscillations
Z
min(z1,z2) will smear the Coulomb gap within a voltage band of
1
z(∂zJ(z)|z→0)+z22(∂z2J(z)|z→0)]. (8) widTthheoIf−orVderchVaosrca.cteristics of a symmetric SET (R =
g L
R , C =C ≪C ) with an oscillating island is shown
R L R g
This formula is valid also for VL > VR. Using Eq. (1) in Fig. 2. The dashed curves correspond to the case of
we find z2 −z1 ≈ e(VR −VL)/∂zεn|z→0. Thus, if z0 is the static island. The solid anddash-dottedcurvesshow
a characteristicamplitude of islandoscillationsthen it is the case of a harmonically oscillating island or an island
natural to define the voltage scale subjecttothermalequilibriumfluctuations. Figures2(a-
c) also show how the I−V characteristicschange when
g
z0 thedrivingvoltageV issmaller,oftheorderof,orlarger
Vosc = (∂zεn|z→0). (9)
e thanV . Within eachpeak ofthe curves,nis constant,
osc
4

sitionoftheisland(e.g.,asitisdoneinRef.14)andthen
1.0
averagingovertimeasitwasalreadydoneforthecurrent
above. Theresultofthis procedureispresentedinFig.4
or
ct which shows the dependence of the Fano factor [21] on
a
o-f0.8 Vg. Here we assumed that the driving voltage is smaller
n
a than V , i.e., the system is in the region in which the
F osc
influence of the oscillations of the island on the trans-
0.6 port properties of the SET is maximal. The Fano factor
dips get split due to harmonic oscillations of the island;
the scale of the splitting is V . In contrast to that, the
osc
40.0 40.5 41.0 41.5 42.0 42.5 43.0
dips are washed out by thermal equilibrium oscillations
Q0/e ofthe island. Here,weassumedthatthe noise frequency
ismuchbelowthefrequencyoftheislandoscillations,ω .
0
FIG. 4: Fano factor for V < Vosc. The dashed line corre- We now consider the noise spectral density, i.e., the
sponds to the static case, the solid line to the harmonically
Fourier transform S(ω,Θ) of S(τ,Θ). If ω approaches
oscillating island, and the dash-dotted line to the thermally
ω , the correctionto the noise fromthe motionof the is-
drivenisland. Theaveragesquareamplitudeoftheislandos- 0
land is ∼ (∂ I)2hhz2ii , where hhz2ii is the irreducible
cillationsareequalfortheharmonicallyoscillatingislandand z ω ω
thecase of thermal equilibrium. correlator of the deviation z at frequency ω. It has a
δ-peak at ω if the island oscillates harmonically. In
0
contrast, if the island moves due to thermal activation,
thereforeVosc isalsoconstantwithinthepeak. Themost hhz2iiω = 2ηkBT/m((ω2 −ω02)2 +ω2η2) [18], the noise
interesting case is shown in the first panel of Fig. 2(a). peak has a width of the order of the oscillation damp-
For the harmonically oscillating island, the peaks split ing factor η, see Eq. (4). A similar result was obtained
and their width becomes larger with the characteristic numerically in Ref. 22. Thus, measuring the noise spec-
scale Vosc; when the island moves due to thermal activa- trum allows to find the frequency of island oscillations
tion the peaks do not split but their width also becomes and gives information on the nature of the oscillations.
larger with V . Thus, the type of motion of the island
osc Inconclusion,we have discussedhowvibrationsof the
leaves a characteristic trace in the I −V plot. Equa-
g island in a SET change its transport properties. The
tion (8) well describes I −V characteristics when the
g transport characteristics of the SET can be used to de-
peaks do not overlap, like in Figs. 2a and b. It follows
termine the nature of island motion, in particular, to
fromEq.(6) that the areasunder the peaks the in static
estimate the amplitude and frequency of its oscillations.
anddynamicalcasesareequal. TheI−V characteristics
g We would like to thank Ya. M. Blanter and C.
is periodic in the static case, but not periodic for an os-
Scho¨nenberger for stimulating discussions. This work
cillating island because V changes from peak to peak,
osc wasfinanciallysupportedbyRFBRandtheRussianMin-
see, e.g., Eq. (10). For this graph, we used the parame-
istry of Sciences (N.M.C.), by the Swiss NSF, and the
ters of the nanotube model of the island (see Ref. 7); we
NCCR Nanoscience.
chose,e.g,C (z)=L/2ln(2(R−z)/r),wherer =0.65nm
g
andL=500nmarethe nanotuberadiusandlength, and
R=100nmisthedistancetothegate. IfC ,C andC
L R g
areofthesameorder,theI−V characteristicsisqualita-
g
tively similar to what is shown in Fig. 2. If the junction [1] A.N.Cleland,FoundationsofNanomechanics(Springer,
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osc
(1998).
We will now discuss the currentnoise in a SET with a
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movingisland. Whentheislandoscillates,theirreducible thaus, Appl.Phys. Lett. 73, 3751 (1998).
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the dependence of S on τ is much stronger than on Θ,
cond-mat/0307528.
andthezero-frequencynoisecanbefoundas S(τ,Θ)dτ,
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5
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