Single electron tunneling with "slow" insulators

9.7 MB·English

by  S. A. Fedorov

Single electron tunneling with “slow” insulators
S. A. Fedorov,1,2 N. M. Chtchelkatchev,3,1,4,5 O. G. Udalov,3,6 and I. S. Beloborodov3
1Department of Theoretical Physics, Moscow Institute of Physics and Technology, Moscow 141700, Russia
2P.N. Lebedev Physical Institute of the Russian Academy of Sciences, Moscow 119991, Russia
3Department of Physics and Astronomy, California State University Northridge, Northridge, CA 91330, USA
4Institute for High Pressure Physics, Russian Academy of Science, Troitsk 142190, Russia
5L.D. Landau Institute for Theoretical Physics, Russian Academy of Sciences,117940 Moscow, Russia
6Institute for Physics of Microstructures, Russian Academy of Science, Nizhny Novgorod, 603950, Russia
(ΩDated: January 5, 2015)
Usual paradigm in the theory of electron transport is related to the fact that the dielectric per-
mittivity of the insulator is assumed to be constant, no time dispersion. We take into account
the “slow” polarization dynamics of the dielectric layers in the tunnel barriers in the fluctuating
5
electricfieldsinducedbysingle-electrontunnelingeventsandstudytransportinthesingleelectron
1
transistor (SET). Here “slow” dielectric implies slow compared to the characteristic time scales of
0
2 the SET charging-discharging effects. We show that for strong enough polarizability, such that the
inducedchargeontheislandiscomparablewiththeelementarycharge,thetransportpropertiesof
n
theSETsubstantiallydeviatefromtheknownresultsoftransporttheoryofSET.Inparticular,the
a
coulomb blockade is more pronounced at finite temperature, the conductance peaks change their
J
shapeandthecurrent-voltagecharacteristicsshowthememory-effect(hysteresis). However,incon-
1 trast to SETs with ferroelectric tunnel junctions,1,2 here the periodicity of the conductance in the

gate voltage is not broken, instead the period strongly depends on the polarizability of the gate-
]
l dielectric. We uncover the fine structure of the hysteresis-effect where the “large” hysteresis loop
e mayincludeanumberof“smaller”loops. Alsowepredictthememoryeffectinthecurrent-voltage
-
r characteristics I(V), with I(V)(cid:54)=−I(−V).
t
s
. PACSnumbers: 77.80.-e,72.80.Tm,77.84.Lf
t
a
m
I. INTRODUCTION to drain. The electric field in the tunnel junctions is
-
d changing in time while electrons tunnel through the is-
n land. Thedielectriclayersinthetunneljunctionsarepo-
Thesingleelectrontransistor(SET)isoneofthemost
o larizedatfiniteelectricfield. Theusualassumptioninthe
studiednanosystem,1–6 Thisisthesimplestdevicewhere
c theory ofSET isrelatedto thefact that thepolarization
[ strong electron correlations and quantum nature of elec-
of any dielectric layer in the tunnel barrier follows the
tron can be directly observed. It consists of two elec-
1 electric field in time: P(t) = αE(t), where the constant
trodes known as the drain and the source, connected
v α is the dielectric permittivity of the dielectric layer. It
throughtunneljunctionstoonecommonelectrodewitha
1 followsfromthelastexpressionthatthecapacitanceC of
6 low self-capacitance, known as the island. The electrical
anytunneljunctionintheSETisrelatedtothegeometric
2 potentialoftheislandcanbetunedbyathirdelectrode,
capacitanceC(0)asC =(cid:15)C(0),where(cid:15)=(1+4πα). And
0 knownasthegate,capacitivelycoupledtotheisland,see
this is the only place where the polarization appears in
0 Fig. 1.
. thetheoryofSET.However, theserelationshavelimited
1 Fordecadestherewasaparadigminthetheoryofelec- applicability. Ingeneral,thepolarizationofthedielectric
0
5 tron transport at the nanoscale related to the fact that is nonlocal in time: P(t) = (cid:82)t χ(t−τ)E(τ)dτ, where
thedielectricpermittivityofnanojunctionswasassumed −∞
1 χ(t) is the dynamical electric permittivity. [Here we as-
: to be constant, without any time dispersion.7–10 How- sume the linear response regime.] The time dependence
v
ever,thisparadigmisnotalwaystrue. Anumberofphys-
i offunctionχ(t)impliesthattuningofdielectricpolariza-
X icalprocessescontributetothepolarizationofdielectrics.
tion P(t) by an electric field can not be done arbitrary
Someofthemarefastandsomeareslowcomparedtothe
r fast. This is happening, for example in dielectric mate-
a time scales of electric field change in the nanojunctions.
rials with polarization being due to shift of heavy and
Recently,therewasaprogressinthedevelopmentofnew
inert ions.
types of dielectric materials with strong and at the same
timeveryslowresponsetotheexternalelectricfield.11,12
The SET is a perfect device where this physics can be The response of polarization P(t) to the external field
studied. This is related to the fact that the charging- is characterised by the time-scale τ , the decay time of
P
discharging effects in the SET are controllable and have function χ(t). The second characteristic time-scale in
well-defined time scales. theproblem: thetimeoftheelectricfieldcorrelation,τ .
E
The Coulomb blockade suppresses the electron trans- For τ (cid:28)τ the polarization has the form P(t)≈αE(t),
P E(cid:82)∞
portexceptforvaluesofthegatevoltagewhereelectrons where α = χ(τ)dτ. In the opposite case, τ (cid:28) τ ,
−∞ E P
sequentially tunnel one by one through SET from source the polarization P(t) does not follow the electric field
2
in the ferroelectric SET, however non of them have been
found before.
The paper is organized as follows. In Sec. II we dis-
cuss the general properties of SET with slow dielectric
and the methods for investigation of transport proper-
ties. In Sec. III we investigate the SET with slow di-
electriclocatedinthegateelectrodeatzerobiasvoltage,
V −V . InSec.IVweconsiderthecasewithslowdielec-
2 1
FIG. 1. (Color online) The equivalent scheme of single elec- tric in the left and right tunnel barriers of the SET and
tron transistor (SET). uncoverthememoryeffectinthecurrent-voltagecharac-
teristics, I(V). Finally, in Sec. V we discuss the validity
of our approach and the requirements for slow dielectric
E(t) instantaneously and it has the form materials which are necessary to observe the effects pre-
dicted in this paper. In the same section we show that
P(t)≈α(cid:104)E(cid:105), (1)
theCoulombblockadeinSETwithslowdielectricsisless
affected by temperature.
where(cid:104)E(cid:105)istheelectricfieldaveragedoverthetimescale
τ . It follows from Eq. (1) that the simple relation for
P
capacitance, C = (cid:15)C(0), is not valid at shorter times.
II. ELECTRON TRANSPORT THROUGH SET
Therefore the theory of single-electron tunneling in the
WITH SLOW TUNNEL BARRIERS
SET should be modified and this is the main goal of our
paper.
HereweconsiderthetheoryofSETwithslowbarriers.
ThecharacteristictimeofchargerelaxationintheSET
In the following it is convenient to distinguish between
isτ =R C ,whereR isoftheorderofthebaretunnel
E Σ Σ Σ the geometrical junction capacitances C(0) and the low-
resistance of the left and right tunnel junctions and C i
Σ frequencycapacitancesC thatincludetheslowdielectric
is the sum of all the capacitances, see Fig. 1. The time i
response. The difference between them, aside from the
scale τ is in the range of dozens of nano- to picoseconds
E unimportant geometrical factor, is
depending on the system geometry and materials. The
switchingtimeofadielectricmaterial,τ ,isintherange
P ∆C =C −C(0) =α S /d , (2)
of seconds to femto-seconds depending on the material i i i i i i
and the particular physical process behind the polariza-
where α is the dielectric polarizability of the i-th junc-
tion phenomena. i
tion (i = 1,2,g), S — the junction surface area and d
Therefore the regime of “slow” insulator, τ (cid:28) τ , i i
E P — the effective electrode-island distance.
is very important for SET-devices.However, there is
Weassumethattheelectrodesarebiasedwiththevolt-
paradigm that the existing theories with τ (cid:28) τ sat-
P E ages V = −V/2, V = V/2 and V . The grain potential
isfactory explain most experiments with SETs. What 1 2 g
φ(n)atagivennumberofexcesselectronsncanbefound
is the justification for new theory? The answer is sim-
balancing the induced charges:
ple: theeffectsdiscussedinthispaperareespeciallypro-
nounced in SETs when on average the polarization of a ne=(cid:88)C(0)(φ(n)−V )+(cid:88)∆C ((cid:104)φ(cid:105)−V ), (3)
dielectric tunnel junction in the SET is strong enough i i i i
meaningthatthechargeinducedonthegrainbythepo- i i
∞
larized dielectric is of the order of the electron charge. (cid:88)
(cid:104)φ(cid:105)= p φ(n), (4)
Thisconditioncanbereachedforlargeenoughdielectric n
n=−∞
permittivity (cid:15) only. How large we will discuss below.
Recently we have found a number of transport effects wherep istheprobabilitytofindnexcesschargesonthe
n
in the SET with slow ferroelectric in the capacitors, see grain. Twotermsoriginatein(3)becausewedistinguish
Refs.1and2. Inparticular, weinvestigatedthememory the electric field produced by the capacitance C(0) and
effect in this SET. Here we uncover new physical phe- i
the contribution due to polarized dielectric with slow re-
nomena and report about the memory-effect (hystere-
sponse. So the terms proportional to the coefficient ∆C
i
sis) where conductance periodicity in the gate voltage is
in Eq. (4) can be considered as charges induced on the
not broken. Instead, the period strongly depends on the
grain by the polarized dielectric layers that are constant
polarizability of the gate-dielectric due to the linear de-
in tunneling events.
pendence of the polarization on the external field in the
Theprobabilitydistributionp inthesteadystatecan
n
dielectric. Also, we uncover the unusual fine structure of be found using the detailed balance equation3–6
the hysteresis-effect, where “large” hysteresis loop may
includeanumberof“smaller”loops. Wepredictthatthe p Γn→n+1 =p Γn+1→n, (5)
n n+1
memory effect exists in the current-voltage characteris-
tics, meaning that I(V) (cid:54)= −I(−V) for a given memory where the rate Γn→n+1[V ,n,(cid:104)φ(cid:105)] describes the change
i
branch even at V = 0. The last two effects may exist of grain charge from n to n+1 electrons. The electric
g
3
current has the form 0.18
∞ 0.16 D C g/C (S0)=
I =e (cid:88) pn(cid:2)Γns→n−1−Γns→n+1(cid:3) . (6) /GT10.14 00 ..03
n=−∞ e, G0.12 0 .6
nc0.10
Here the lower index of Γ refers to the tunneling rate ucta0.08
correspondingtotheparticulartunneljunction, s=1or nd
o0.06
c
2 and the rate Γ in Eq. (5) is equal to Γ +Γ . Solving
1 2 0.04
Eqs. (3)-(5) self-consistently we find the current-voltage 0.02
characteristics of the SET using Eq. (6). -1.0 -0.5 0.0 0.5 1.0
We use the “orthodox” theory to calculate the Q /|e |
Coulomb-blockade peaks in the differential conductance
FIG. 3. (Color online) Conductance peaks for ∆C /C(0) =
oftheSET.ThecalculationofΓ-ratesrequirestheknowl- g Σ
0, 0.3, 0.6. The“unit”ofconductanceG istheconductance
edge of the difference in the electrostatic energies when T1
of the first tunnel junction of the SET. Parameters are: ca-
the number of excess charges on the grain differ by
pacitancesC(0) =0.3C(0),C(0) =0.5C(0)andC(0) =0.2C(0)
one elementary charge: ∆U± = U(n ± 1) − U(n). If 1 Σ 2 Σ g Σ
n andtemperatureT =0.2E(0). Theslowdielectricinthegate
the polarization in dielectric layers on electron jumps c
capacitor modifies the shape of the conductance peaks but
follow φ adiabatically, P = α (φ − V )/d , we have
i i i i preserves the periodicity in parameter Q in contrast to the
∆Un± = Ec(1 ± 2n)(cid:80), where Ec = e2/2CΣ with all the SET with ferroelectric in the gate capacitor.2
capacitances C = C being properly renormalized,
Σ i i
C = C(0)(1+4πα ). However, for slow dielectric layers
i i i
thepolarizationP =α ((cid:104)φ(cid:105)−V )/d staysconstantdur-
i i i i where the effective gate-induced charge Q(cid:48) is
ing the tunneling, and for the energy difference we find
(see App. A)
(cid:88)
∆U± =E(0)(1±2n∓2(cid:88) P S /e), (7) Q(cid:48) =−Cg(0)Vg+ i∆Ci((cid:104)φ(cid:105)−Vi). (9)
n c i i
i
where Ec(0) = e2/2CΣ(0), CΣ(0) = (cid:80)iCi(0) and PiSi = Below we use the notation Q = −CgVg for the tradi-
∆Ci((cid:104)φ(cid:105)−Vi). tional gate-induced charge. We show that although the
The work done by the leads and the gate to transfer effects of slow polarization are far from being a simple
anelectronto/fromthegrainremainsthesameasinthe renormalization of capacitances C(0) → C , the conduc-
“orthodox” theory except for the fact that only the geo- i i
tance periodicity in Q holds and maintains its period |e|
metrical capacitances C(0) should be taken into account. for any values of the parameters ∆C .
i i
This implies that for temperature T → 0 the effective
The detailed balance equation (5) can be solved an-
ground state free energy is defined as
alytically for the set of voltages V near the “degener-
g
F =E(0)min(n−Q(cid:48)/e)2, (8) acy points”, where the ground state energy of the SET
0 c
n changesfromnton±1excesscharges. Thelastcondition
requires the effective charge Q(cid:48) to be close to e(n+1/2).
1 .0 In this case the only two probabilities pn are finite while
the other probabilities are exponentially suppressed by
the factor e−Ec(0)/T. In order to illustrate the origin of
thememoryeffect,wewillfocusonthedegeneracypoint
between n=0 and n=1 at V =0. Using Eqs. (3)-(4)
1,2
we find for the average potential (cid:104)φ(cid:105)
0 .5
n [(1−2Q(cid:48)/e)E(0)]=e(cid:104)φ(cid:105)/2E(0)+Q(cid:48)/e, (10)
F c c
where n is the Fermi-function. Equation (10) has one
F
0 .0 or three solutions for a given gate voltage Q. The latter
-0 .2 -0 .1 0 .0
e < f > /2 E (0) case is shown in Fig. 2. The presence of three distinct
c
solutions for the average potential (cid:104)φ(cid:105) at a given param-
FIG. 2. (Color online) Graphical solution of Eq. (10) show- eter Q indicates the memory effect instability. Using the
ingthreepossiblesolutionsforanaveragegrainpotential(cid:104)φ(cid:105) graphicalsolutionofEq.(10)weestimatethecriteriafor
Catg(0a) g=ive0n.5CgaΣ(t0e),v∆olCtagg/eCVΣ(g0). =Pa0ra.5m,eatnerdsTare=: Q0.4=Ec(−0)0..07T|eh|e, tThheismcermiteorriyoneffceocrtreisnpsotanbdislittyo, t(cid:80)hei∆crCitii/cCalΣ(0v)al(cid:38)ue2oTf/∆Ec(C0)Σ.
three distinct solutions for (cid:104)φ(cid:105) at a given Q0 correspond to when the memory effect just appears, see Eq. (32) below
the memory effect instability. for the exact expression.
4
T10 .1 5 D C g /C (S0)= Q evolu tion ∆C is ∆Cg.
/G a ) 0 .6 For ∆C =0 the conductance is a periodic function of
ce, G 0 .1 0 12 ..00 the effectivge gate voltage Q, see the gray curve in Fig. 3.
tan 4 .0 The conductance peaks are well fitted by the orthodox
duc 0 .0 5 theory where near the peak maximum the conductance
n
co is
-1 .2 -1 .0 -0 .8 -0 .6 -0 .4 -0 .2 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 1 .2
eδQ(0)/C(0)T
T10 .1 5 b ) G(0)(δQ(0))≈ Σ . (11)
/G Q evolu tion 2(R +R )sinh(eδQ(0)/C(0)T)
, G0 .1 0 1 2 Σ
e
c
ctan0 .0 5 Here δQ(0)/e=mink[−Cg(0)Vg/e−(2k+1)/2](cid:28)1.
u
d At finite but small ∆C , when the induced charge on
n g
o
c -1 .2 -1 .0 -0 .8 -0 .6 -0 .4 -0 .2 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 1 .2 theislandduetopolarizationissmallerthantheelemen-
tary charge, the conductance peaks change their shape,
| 2
/|eg c ) but preserve their amplitude and position (see Fig. 3).
n, P 1 The opposite case, with dielectric polarization being
atio 0 strong enough to induce the charge on the island of
lariz -1 the order of the elementary charge or larger, is more
o
p interesting. In this case the conductance peaks show
-2
-1 .2 -1 .0 -0 .8 -0 .6 -0 .4 -0 .2 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 1 .2 the hysteresis and their shape depends on the direction
of Q-evolution, see Fig. 4. The hysteresis appears for
f(0)|<>/Ec00 ..12 d ) ∆effCecgt(cid:38)thCeΣc(0o)n2dTu/cEtac(n0)ce(sreeemEaqin.s2p5e).rioDdeicspiintethteheremnoermmoarly-
l, 2|e 0 .0 ized gate voltage Q = −(Cg(0) +∆Cg)Vg with the same
tia-0 .1 period |e| for any ∆C . This behavior is in striking con-
n g
ote-0 .2 trasttotheSETwithferroelectricinthegate wheredue
p -1 .2 -1 .0 -0 .8 -0 .6 -0 .4 -0 .2 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 1 .2
to the nonlinearity of polarization–electric field depen-
e ffe c tiv e g a te v o lta g e , Q /|e |
dencetheperiodicityofconductanceisbroken,seeRef.2.
FIG. 4. (Color online) Memory effect instability in the SET
with slow insulator in the gate capacitor. (a) and (b) the Now we discuss the structure of the memory effect.
conductancebranchescorrespondingtotheincreasingandde- Above the critical value of ∆Cg there are many brunch-
creasingparameterQand∆C /C(0) =0.6, 1, 2, 4,(c)polar- solutions of the self-consistency equation for the average
g Σ
ization and (d) the average grain potential (arrows show the grain potential, Eq. (4), for the given temperature, bias
directionofQevolutionforagivenbranch)for∆C /C(0) =4. and gate voltage. The question is - how to choose the
g Σ
Grey lines show stable and unstable branches of polariza- right branch? Figure 5 provides an answer to this ques-
tionandtheaveragepotential. Parametersare: capacitances tion. According to the branching theory13 the jumps oc-
C1(0) =0.3CΣ(0), C2(0) =0.5CΣ(0) and Cg(0) =0.2CΣ(0) and tem- cur at the “branching points” where the observable has
perature T =0.2E(0) as in Fig. 3. aninfinitederivativeinparameterQ. Ontheotherhand,
c
the branch should correspond to the minimum of some
effective energy functional. In our case (no bias) the role
III. SET WITH SLOW INSULATOR IN THE of the effective energy plays the free energy
GATE CAPACITOR
(cid:32) (cid:33)
(cid:88) Ec(0)(n−Q(cid:48)/e)2
F =−T lnZ, Z = exp − .
A. Numerical study of electron transport through T
n
SET
(12)
HerewestudyelectrontransportthroughSETnumer- For zero temperature it reduces to the free energy F
0
ically. We consider the SET with slow dielectric layer in discussed above.
the gate capacitor. This set-up is the most favourable Theplotsofthefreeenergyhaveasimilardependence
forexperimentsinceinthiscasethereisnoelectrontun- on the parameter Q as the zero-bias conductance G. To
nelingthroughthegateelectrodeanditcanbearbitrary illustrate this point we show in Fig. 6 the free energy for
thicktoallowawidechoiceofdielectricmaterials. More- ∆C /C(0) = 0.6, 1.3. Figure 5b shows that the conduc-
g Σ
over, as we will show in the following Sec. IV, at V = 0 tancebranchbetweenpoints“A”and“B”ismetastable:
byconsideringthegatecapacitorwestillpreserveallthe the free energy for this curve is larger than the free en-
qualitativeeffectsintroducedbyslowdielectricsinagen- ergyforbranchbelow. However,duringtheadiabatically
eral case. slow increase of parameter Q the system does not switch
Thus, for a time, we assume that the only non-zero to the lowest branch at point A, instead it may go up
5
/GT10 .1 5 a ) 0 .1 5 b ) c )
, G
ce B
n0 .1 0 0 .1 0
nducta etastable
co0 .0 5 0 .0 5 A m
-1 .0 -0 .5 0 .0 0 .5 1 .0 -1 .0 -0 .5 0 .0 0 .5 1 .0 -1 .5 -1 .0 -0 .5 0 .0 0 .5 1 .0 1 .5
Q /|e | Q /|e | Q /|e |
T10 .1 5 0 .1 6
/G
, G d )
e
nc0 .1 0 0 .1 4 f)
ta
c
u
d
n
co0 .0 5 T10 .1 2
/G
, G
e 0 .1 0
-2 -1 0 1 2 nc
ta
4 e ) uc
d
| n 0 .0 8
/|eg 2 co
, P
n
tio 0 0 .0 6
a
riz
la
po -2 0 .0 4
-4
0 .0 2
-2 -1 0 1 2 -2 -1 0 1 2
Q /|e | Q /|e |
FIG.5. (Coloronline)Memoryeffect: Plots(a)-(d)and(f)showtheconductancefor∆C /C(0) =1.3, 3.3, 5.3, 10, 20forstable
g Σ
and unstable branches of Eq.(4) for the average grain potential. Plot (e) shows the polarization for ∆C /C(0) = 10. Arrows
g Σ
indicate the position of hysteresis jumps for particular branch with increasing Q. All plots are shown at fixed temperature
T =0.2E(0).
c
to the metastable branch. The same applies to all other polarization in such a process the system would have to
plots in Fig. 6. The external perturbation can drive the passtheenergybarrierofapproximatelyE(0)/4(freeen-
c
systemtooutsideofthemetastablebranchbeforethebi- ergy maximum). Thus the higher order jumps (over the
furcation point. Usually the role of this “perturbation” average charge difference) are suppressed by the factor
plays the Langevin forces induced by the thermostat. In exp(−E(0)/4T).
c
this case the jumps occur randomly within the same re-
gionbeforethebifurcationpoint. Thisscenarioistypical
for any hysteresis.
B. The fine structure of the memory effect
Intuitively one may suppose that if conductance
“jumps” from one branch to another the final branch
should have the lowest possible free energy for the pa- Doing numerical studies of memory effect we assumed
rameterQcorrespondingtothejump. Indeed,thisisthe that parameter Q increases (or decreases) monotonically
caseinFigs.5(a)-(c). However, inFigs.5(d)and(f)this from minus to plus infinity (or vice-versa). However, for
rule is violated. The system could jump, for example, to large enough parameter ∆C , when polarization induces
g
the point marked by the red-ball in Fig. 5(d), instead of more than one electron on the grain, the hysteresis loop
finishingatthepointmarkedbythegrey-ballwhichhasa depends on the interval where the parameter Q changes.
largerfreeenergy. However,thisenergeticallyfavourable ThisisshowninFig.7withtwopossiblehysteresisloops:
transitionis“forbidden”: whilecontiniouslychangingthe The red hysteresis loop corresponds to back and forth
6
C. Analytical description of the conductance peaks
and the memory effect
0 .0 5
c D C g/C (S0)= Herewepresenttheanalyticaldescriptionoftransport
/E 0 .0 4 0 .6 properties of SET. At V = 0 and within the two-state
F
, 2 1 .3 approximation the form of the conductance peaks G(Q)
y
rg 0 .0 3 canbefoundusingEq.(11)withthepropersubstitution
ne Q(0) → Q(cid:48), where Q(cid:48) is defined in Eq. (9): with this
e e substitution we have for conductance G(Q) = G(0)(Q(cid:48)).
fre 0 .0 2 For average potential, generalizing Eq. (10), we obtain
(cid:32) (cid:32) (cid:33) (cid:33)
0 .0 1 e 1 Ec(0)δQ(cid:48) δQ(cid:48)
(cid:104)φ(cid:105)= tanh − , (13)
C(0) 2 T e e
Σ
0 .0 0
where δQ(cid:48)/e = min (Q(cid:48)/e − (k + 1/2)). Combining
k
-1 .0 -0 .5 0 .0 0 .5 1 .0 Eqs. (13) with (9) we find,
Q /|e |
(cid:32) (cid:33)
C e∆C E(0)δQ(cid:48)
FIG.6. (Coloronline)FreeenergyinEq.(12)for∆C /C(0) = δQ(cid:48) Σ − gtanh c =δQ, (14)
0.6, 1.3 and temperature T = 0.2Ec(0). The shapge ofΣfree CΣ(0) 2CΣ(0) T e
energyplotissimilartotheconductanceG(Q)plotinFig.5a.
where δQ=Q−(k+1/2)e is the deviation of parameter
Q, k is the same as for δQ(cid:48) and C = C(0) +∆C . It
Σ Σ g
should be noted that the above equations are valid for
any δQ as long as δQ(cid:48) (cid:28)1.
1. Small polarization
Here we discuss the limit of small polarization, mean-
ing that the induced charge on the island is small com-
pared to the elementary charge e. Using the small pa-
rameter, ∆C /C(0) (cid:28) 1, we expand Eq. (14) up to the
g Σ
second order
C(0)
(δQ(cid:48)) =δQ Σ , (15)
0 C
Σ
(cid:32) (cid:33)
e∆C E(0)(δQ(cid:48))
(δQ(cid:48)) =(δQ(cid:48)) + gtanh c 0 . (16)
1 0 2 C T e
Σ
The conductance now may be found by substituting
δQ(0) with (δQ(cid:48)) in Eq. 11.
0,1
FIG. 7. (Color online) Memory effect in (a) the conductance G(δQ)=G(0)(δQ(cid:48)) (17)
and (b) the polarization of the gate-insulator. The red hys-
teresis loop corresponds to back and forth change of param- The numerical calculations in Fig. 8(a) show that the
eter Q in the interval (−2,1), while the blue curve corre- first order approximation, Eq. (16), well describes the
sponds to (−2,2) interval. Parameters are: ∆C = 7.5C(0), peak shape for small parameter ∆C /C(0) ≈ 0.1, while
g Σ g Σ
T = 0.2E(0), while C(0), i = 1,2,g and R , j = 1,2 similar the zero order approximation is not sufficient. We note
c i j
to Fig. 3. that parameter ∆C /C(0) and thus the renormalization
g g
of the conductance period over V can be arbitrary in
g
this approximation.
change of parameter Q in the interval (−2,1) while the
blue curve corresponds to the interval (−2,2). In the
second case the larger hysteresis loop “includes” smaller 2. Amplitude and form of the conductance peak in the
loops. As a result, the understanding of memory effect hysteresis regime
at finite intervals of parameter Q evolution requires con-
sideration of all branches of the SET observables such as Solution of Eq. (14) becomes ambiguous for large val-
conductance and polarization. uesofparameter∆C ,whereconductanceG(Q)acquires
g
7
FIG. 8. (Color online) (a) Numerical solution for conductance peak for ∆C /C(0) = 0.05 (blue line), orange line is the zero
g Σ
order solution from Eq. (15), and the red line is the first order solution from Eq. (16). (b) Numerical and analytical solutions
of conductance for ∆C /C(0) =0.1. The first order approximates well the conductance peak at small ∆C . (c) Conductance
g Σ g
at critical ∆C , where hysteresis appears. (d) Conductance hysteresis. (e) Amplitude of conductance peak vs ∆C . Points
g g
represent the numerical solution; the red curve is G = 1/2(R +R ); and the orange curve shows Eq. (20) for G .
max 1 2 max
Parameters are: T =0.06E(0), C(0), i=1,2,g and R , j =1,2 similar to Fig. 3.
c i j
hysteresis. In this case the form of conductance peaks The predicted conductance maximum amplitude varia-
becomes nonsymmetric and the conductance G(Q) has tionisshowninFig.8. Onecanseethatthecurvebreaks
a maximum at the branching (bifurcation) point corre- atcriticalvalueofparameter∆C indicatingthestartof
g
spondingtothejumpofthepolarization. Thebifurcation the hysteresis regime.
points in Eq. (14) can be found as follows
d (cid:32) C e∆C (cid:32)E(0)δQ(cid:48)(cid:33)(cid:33) We note that since within the scope of the two-state
δQ(cid:48) Σ − gtanh c =0, (18) approximation and for ∆C above the critical value the
dQ(cid:48) CΣ(0) 2CΣ(0) T e Eq. (20) gives exact maximgum, its applicability depends
only on temperature. At finite ∆C the conductance
that reduces to g
maximum does not exactly correspond to a degeneracy
(cid:32) (cid:33)
cosh2 Ec(0)(δQ(cid:48))max = Ec(0)∆Cg. (19) point δQ(cid:48) = 0, but still δ(Q(cid:48))max (cid:28) 1 for T (cid:28) Ec(0).
T e 2T CΣ For example, for temperature T =0.06E(0) and ∆C →
c g
∞ we have δ(Q(cid:48)) /|e| ≈ 0.1 (cid:28) 1, meaning that our
The two solutions of Eq. (19) correspond to the in- max
consideration is valid (see Fig. 8).
creasing and decreasing evolution of parameter Q (solu-
tions with δQ(cid:48) <0 and δQ(cid:48) >0 respectively). These two
solutions result in mirror-reflected shapes for the peaks, Now we find the form of conductance peaks. Expand-
so we focus only on the decreasing parameter Q. For ing Eq. (14) up to the second order near δ(Q(cid:48)) we
max
conductance maximum we find obtain
(cid:18)(cid:113) (cid:19)
arccosh Ec(0)∆Cg
1 2T CΣ
G = . (20)
max 2(R +R )(cid:114) (cid:16) (cid:17)
1 2 E2c(T0)∆CCΣg E2c(T0)∆CCΣg −1 A0+A2(δQ(cid:48)−δ(Q(cid:48))max)2 =δQ, (21)
8
where where we introduce the parameter
(cid:32)(cid:114) (cid:33) (cid:88)
eT C E ∆C ∆C = ∆C . (28)
A = Σ arccosh c g − Σ i
0 Ec(0)CΣ(0) 2T CΣ i=1,2,g
(cid:115) Here we explicitly show that the functions Q(cid:48) and (cid:104)φ(cid:105)
e∆C 2T C
g 1− Σ , (22) dependonvoltageV. Ingeneral,thisdependenceresults
2CΣ(0) Ec(0)∆Cg inanadditionalcontributiontotheconductancepropor-
tional to ∂Q(cid:48)/∂V:
and
(cid:115)
E(0) C 2T C ∂I(0)(Q(cid:48),V)
A = c Σ 1− Σ . (23) G(Q,V)= =
2 eT CΣ(0) Ec(0)∆Cg ∂V ∂I(0)(Q(cid:48),V)∂Q(cid:48)
G(0)(Q(cid:48),V)+ , (29)
It follows from Eqs. (21) and (17) that the conductance
√ ∂Q(cid:48) ∂V
derivative in δQ diverges as 1/ x near its maximum
value. where I(0)(Q,V) is the current in the orthodox theory,
generally not limited by the two-state approximation.
However, the current I is zero for zero bias voltage for
3. The peak form at the bifurcation point any Q, therefore the last term can be omitted at V =0.
Thisexplainswhyintwo-stateapproximationwecancal-
To find the conductance peak at the critical value of culate the conductance by replacing Q by Q(cid:48) in Eq. (11)
parameter ∆C we expand the hyperbolic tangents in of the orthodox theory.
g
Eq. (14) up to the third order. As a result we obtain For zero voltage, V =0, Eq. (27) reduces to
(cid:32) ∆C (cid:32)E(0) (cid:33)(cid:33) e∆C (cid:32)E(0)δQ(cid:48)(cid:33)3 Q(cid:48) =Q+∆CΣ×(cid:104)φ(cid:105)(Q(cid:48)). (30)
δQ(cid:48) 1− g c −1 + g c
C(0) 2T 6C(0) T e Then
Σ Σ
(cid:32) (cid:33) (cid:32) (cid:33)
=δQ. (24) ∆C e∆C E(0)δQ(cid:48)
δQ(cid:48) 1+ Σ − Σtanh c =δQ.
The linear term equals zero at the critical point. For C(0) 2 C(0) T e
Σ Σ
critical polarizability of the gate-insulator we find (31)
As we can see, the only distinction of the Eq. 31 from
∆C(c) =C(0)(E(0)/2T −1)−1. (25) Eq. 14 is the replacement of ∆C with ∆C . It follows
g Σ c g Σ
that for V = 0 the SET with slow insulators in tunnel
Also we find that
junctionsbehavesqualitativelysimilartotheonly∆C >
g
δQ(cid:48) = eT (cid:118)(cid:117)(cid:117)(cid:116)3 6δQ(cid:32)Ec(0) −1(cid:33). (26) 0reltahtaetdwtaostchoensfiadcetretdhaptretvhieousslolyw. Tdiheeleoctnrliycdiniffetrheencgeaties
E(0) e 2T capacitorrenormalizestheperiodoftheQ-oscillationsof
c
conductance while slow dielectrics in all other capacitors
UsingEq.(17)wefindthatthepeakmaximumcanbeap- of the SET do not.
proximatedwiththefunction1/(1+x2/3)(herex∝δQ), Nowwecangeneralizeourresultsforpositive∆C >0
g
while the derivative diverges at the conductance maxi- obtained earlier. In particular, the critical polarization,
√
mum as 1/3x. As follows from Fig. 8(c) and Eq. (26) where memory effect in the conductance G(Q) first ap-
this approximation for conductance works well near its pears, becomes the integral quantity, see Eq. (28), that
maximum value only. includes properties of all the slow dielectric layers:
∆C(c) =C(0)(E(0)/2T −1)−1. (32)
Σ Σ c
IV. SINGLE ELECTRON TUNNELING
THROUGH SLOW DIELECTRIC LAYER The amplitude of conductance peaks can be found using
the substitution, ∆C →∆C in Eq. (20). The shape of
g Σ
thepeakscanbeobtainedusingthesamesubstitutionin
A. Conductance peaks with slow dielectrics in all
capacitors the equations of Sec. IIIC3 where still δQ = −(Cg(0) +
∆C )V .
g g
Here we consider the general case, with slow dielectric
layers in all capacitors with polarizabilities ∆C , ∆C
1 2
B. Memory effect in current-voltage characteristics
∆C . Using Eq. (9) we find
g
Q(cid:48) =Q+∆C ×(cid:104)φ(cid:105)(Q(cid:48),V)−(∆C −∆C )V , (27) Above we discussed the properties of SET with slow
Σ 2 1 2 dielectric barriers, related to the variation of the gate
9
FIG.9. (Coloronline)Current-voltagecharacteristics,I(V)ofSETwithzerogate-capacitance,C =0. Plot(a)showsI(V)for
g
∆C /C(0) =0.0,0.25,0.5,0.75,1.0,1.25. Smoothercurvescorrespondtosmaller∆C . Plot(b)correspondsto∆C /C(0) =1.5.
1 Σ 1 1 Σ
The jumps in I(V) in (b),(c), and (e),(f) correspond to the memory effect: the branch depends on the direction of voltage
change. Plot (c) shows I(V) for ∆C /C(0) = 3.5. The I(V)-curve can have many hysteresis loops depending on the amount
1 Σ
ofelectronchargeinducedonthegrainbythedielectricpolarization. Insertsin(b),(c)showthedetailsofthehysteresis. Plot
(d) shows I(V) for ∆C /C(0) = 1.25, ∆C /C(0) = 0 (black curve) and ∆C /C(0) = ∆C /C(0) = 1.25 (orange curve), while
1 Σ 2 Σ 1 Σ 2 Σ
plot(e)showsthegraphsfor∆C /C(0) =1.5,∆C /C(0) =0(blackcurve)and∆C /C(0) =∆C /C(0) =1.5(orangeandblue
1 Σ 2 Σ 1 Σ 2 Σ
curves). Plot(f)showsI(V)for∆C /C(0) =∆C /C(0) =3.5. Parametersare: T =0.06E(0),C(0) =0.6C(0),C(0) =0.4C(0),
1 Σ 2 Σ c 1 Σ 2 Σ
and R , i=1,2 similar to Fig. 3. The unit of voltage is E(0)/|e|, and the current is normalised to E(0)/|e|R .
i c c T1
voltage V at bias V = 0. In this subsection we instead jumpsinplots(b)correspondtotheregionsofhysteresis
g
concentrate on the current-voltage characteristic I(V) of while the arrows show the evolution of voltage. Plot (c)
SET in the case of electron tunneling through slow insu- shows the hysteresis in I(V) for ∆C /C(0) = 3.5. The
1 Σ
lator in the left and the right capacitors, see Fig. 1. We current-voltage characteristics may have many hystere-
neglect the gate to simplify the situation, putting thus sis loops, depending on the amount of electron charge
Cg = 0. Such systems have been extensively studied in that the dielectric polarization may induce on the grain.
experiments over the last two decades. They can exhibit The hysteresis in the current-voltage characteristics ap-
coulomb blockade at room temperature(11, 14–16) and pears for the first time for parameter ∆C being larger
1
their ease of fabrication makes a wide range of barrier than C(0). This is the first critical value of polarization.
materials available for experiments. Following Ref. 11 Σ
For ∆C (cid:38) 2C(0) the second hysteresis loop appears in
weconsiderthecurrent-voltagecharacteristicoftheSET 1 Σ
I(V). Therefore this is the second critical value of ∆C .
in a wide range of bias voltages. 1
For larger values of ∆C we expect further increase in
1
The typical current-voltage characteristics I(V) are the number of hysteresis loops.
shown in Fig. 9; in Fig. 9(a)-(c) the coefficients ∆C =0 Two cases of current-voltage characteristics are com-
2
and ∆C are finite. It follows that there is a memory pared in plots (e)-(d) : i) finite ∆C and zero ∆C and
1 1 2
effect in I(V) at large enough ∆C and this effect de- ii) ∆C = ∆C . In both cases the set of critical values
1 1 2
pendsonthedirectionofthebiasvoltageevolution. The of∆C isthe sameandforlargebias voltagethecurrent-
10
voltage characteristics asymptotically coincide. Interestingly,theshiftoftheI(V)curveinexperiments
Figure9showsthatthecurrent-voltagecharacteristics is a well-known effect. It is usually accounted for by
oftheSETstronglydependonthedirectionofbiasvolt- assumingthepresenceofsomeadditionalspuriouscharge
age V. Moreover, for a given hysteresis branch Q, induced on the grain (as in Ref. 11 and 18). However
the shift that we predict is notably different at least in
I(V)(cid:54)=−I(−V) (33) oneaspect—itreversesitssignwiththedirectionofthe
evolution of V.
that happens in the absence of Q, notably different from
WestressthatthedescribedrescalingandshiftofI(V)
the result for a regular SET.
takes place only under specific conditions V > |e|/C(0)
Σ
and R (cid:29) R . If R are of the same order the intro-
1 2 2
C. Influence on coulomb ladder duction of slow dielectric may change the ladder steps
in a more complex way. Such a situation is shown in
Fig. 10(b) where the ladder period do not correspond to
By coulomb ladder in this section we mean a step-like
the one we would expect from the simple capacitance-
behaviorofI(V)intheregimeofcoulombblockade. The
renormalization consideration. If R /R is even closer
coulombladderisoftenusedasanindicationofcoulomb 1 2
to unity, the slow dielectric barriers qualitatively change
blockade (Ref. 11, 14, 17–19). In the following we show
thecurrent-voltagecurveaswasdiscussedintheprevious
howtheslowpolarizationinfluencestheshapeofthelad-
section (see Fig. 9).
der. Again we take C = 0 and, consider the conditions
g
when the ladder is the most pronounced, i.e. T =0 and
strongly asymmetric barriers R (cid:29)R . At zero temper-
1 2
aturetunnelingmayoccuronlyinthedirectionofchem- V. DISCUSSION
ical potential drop, that is from the 1-st electrode to the
2-nd assuming V >0. Due to the relatively high tunnel-
A. Requirements for dielectric materials
ingratethroughthe2-ndelectrode,thenumberofexcess
electronsontheislandisalmostalwaysstaysatthemin-
Here we discuss several possible dielectric materials
imum energetically allowed number n . n can be
min min
determined as the lowest n for which ∆Fn+1→n < 0 is which can be considered as slow insulators. At finite
true, since ∆Fn→n+1 < 0 holds for any n2 < 0. For a external electric field the localized electric charges are
1 shifted and the dielectric material is polarized. There
given n the current can be calculated as
min
are several physical processes contributing to the polar-
1 ization: 1) the shift and deformation of electron-cloud,
I = ∆Fnmin→nmin+1, (34)
eR 1 2) the shift of ions in the lattice, and 3) the molecular
1
and/or macro dipole reorientation. Electrons, ions, and
where∆F isthefreeenergychangeontunnelingthrough
1 dipolescanformadifferentpolarization. Theslowestpo-
the 1-st electrode. For a conventional SET the above
larization formation corresponds to the electrocalorical
formulaleadstoaladder-shapedI(V)characteristicwith
and migration (electron, ion or dipole) mechanisms with
the step width
thecharacteristicdispersionfrequencybeingintherange
10−4−10−1 Hz and 10−3−103 Hz, respectively at tem-
∆V =|e|/C(0), (35)
step 1 perature T = 300K. The electromechanical mechanism
corresponds to frequencies 105 −108 Hz, while thermal
jumps of the current between the steps
mechanism correspond to 105−1010 Hz. The dielectrics
∆I =|e|/R C(0), (36) wherethermalmechanismisthelargestarepromisingfor
step 1 Σ applications in nanostructures and can be considered as
and the I(V) slope between the jumps “slow” dielectrics.
Dithiol self-assemble monolayers (SAMs) have a static
dI/dV =C(0)V/C(0)R (37) dielectric permittivity (cid:15)(ω = 0) ∼ 3 and the character-
2 Σ 1
istic relaxation frequency ∼ 104 Hz.12 These materials
Introducing slow dielectric into the tunnel junctions re-
are good candidates for slow dielectrics. Such dielectric
sult in some new effects (for the details of calculations layers have been used in double junction SET11. The
seeAppendixB).AtV >|e|/C(0)slowpolarizationleads hysteresis have not been observed in these experiments,
Σ
to the rescaling of the ladder that may be described by buttherewasaconsiderablediscrepancybetweenthethe
substituting the capacitances in Eqs. 35-37 with the new valuesofcapacitancesobtainedfromthefitoftheexper-
values C = C(0) +∆C , exactly as when dealing with imental data with the orthodox model and the ad-initio
i i i
a conventional fast dielectric (see Fig. 10(b)). But con- calculations.
trary to the fast dielectric, the slow one shifts the lad- Another promising materials to observe the hysteresis
der, making it asymmetric and, moreover, dependent on are polar crystal dielectrics e.g., BaTiO or KDP with
3
the direction of the evolution of V, as illustrated at the static dielectric permittivity (cid:15)(ω = 0) ∼ 103 and the
Fig. 10(c,d). typical relaxation frequency ω ∼106 Hz.
c