Table Of ContentInverse slow relaxation in granular hopping systems
N. Kurzweil and A. Frydman
The Department of Physics, Bar Ilan University, Ramat Gan 52900, Israel
(Dated: February 5, 2008)
7 Wepresentexperimentalresultsthatdemonstrateaglassybehaviorintheconductanceofquench
0 condensed insulating granular metals that is different from that observed in continuous disordered
0 systems. Exciting the granular system by biasing the sample with a high electric field results in a
2 slow conductance change both during the excitation and during the relaxation back to its steady
state. The time scales for these processes are many orders of magnitudes longer than the typical
n
hoppingtime. Wefindthat,initially,thisconductancechangehasanoppositesigntothatobserved
a
J in similar experiments performed on continuous films. Only after relatively long times this trend
is reversed and the samples exhibit conventional behavior. We suggest that the granular systems
6
exhibitanewglassyprocessrelatedtochargeredistributionamongthegrains. Thisprocesscombines
1
with the relaxation processes that are characteristic of disordered systems to give rise to a unique
relaxation profile.
]
l
l
a PACSnumbers: 73.23.-b;73.40.Rw;75.50.Cc
h
-
s
Glassy behavior of the conductivity of strongly disor- rium.
e
m dered systems has attracted attention for over a decade. In this letter we report on an experiment in which
Excitingahoppingsystemoutofitsthermalequilibrium we apply a similar technique to that of Orlyanchik and
.
at leads to an increase in conductivity. The relaxation pro- Ovadyahu to excite a granular metal. Surprisingly (un-
m cessestowardequilibriumarecharacterizedbyextremely like the case of applying a gate voltage), we find that
long times, memory phenomena and aging effects. The applying a bias voltage gives rise to an opposite trend
-
d majority of experiments were performed on Indium Ox- to that seen in the continuous disordered systems. We
n
ide thin films which were excited by applying a gate attribute this difference to the unique electric field dis-
o
voltage,thusabruptlychangingthecarrierconcentration tribution in granular metals.
c
[ [1, 2, 3, 4, 5]. Following the sharp initial increase, the Thesamplesstudiedinthisworkwerevarious(Ag,Au
conductivityexhibitedalogarithmictimedecaytoanew and Ni) thin granular films fabricated using the quench
1
equilibrium state with time scales of the order of many condensation technique. This method enables growing
v
2 hours or days. This slow conductivity relaxationshowed andmeasuringa2Dgranularsample atlow temperature
6 manyphenomenacharacteristicofglassessuchasscaling and high vacuum without exposing it to the atmosphere
3 behavior in aging experiments and memory effects. orthermallycyclingit[9,10,11]. Priortothecool-down,
1
Very similar glassy behavior (e.g., slow relaxation, twoGoldleadswereevaporatedonainsulatingSi/SiOor
0
memory and aging) was reported also in granular Au glasssubstrate. ThesubstratewasthenplacedinaUHV
7
0 or Al [6, 7] demonstrating that this phenomena is not system and immersed in liquid He. An ultrathin metal
/ unique to Anderson insulators. This is not surprising film was then evaporated on the cold substrate, form-
t
a since, in principle, the electrical transport in a granular ing a discontinuous, granular structure with grain sizes
m metal in the insulating phase is very similar to that of a of 100-200˚A and intergraindistances of a few tens of ˚A.
- disorderedcontinuousmetal. Themaindifferenceisthat Sample resistance was monitored during the growthand
d
in a granular system the electrons are localized on iso- theprocesswasstoppedatadesiredresistance. Westud-
n
o lated grains instead of localized electronic states created ied samples with resistances in the range of 50 MΩ−5
c bytherandompotential. Eachgrainisasmallmetaland GΩ.Afterterminatingthefilmdepositionthesamplewas
v: cancontainalargenumberofelectrons,unlikethesingle allowed to relax for a few hours until the conductance
i localized state. change was smaller than 0.1 %/hour and a base conduc-
X
Recently,OrlyanchikandOvadyahu[8]usedadifferent tance line was determined. Then, a large electric field,
r technique to excite Indium oxide samples in the hopping F =2−600 V/cm, was applied abruptly and the time-
a n
regime. Theybiasedthesamplebyapplyingalargeelec- dependent-conductance, G(t), was monitored. After a
tricfield,thusdrivingthesystemoutofitsohmicregime, desired waiting time, t , the field was abruptly reduced
w
and studied the time-dependent conductance both dur- to F0 ≈ 0.5 V/cm and the low voltage relaxation G(t)
ingexcitation(whileapplyingthevoltage)andrelaxation was measured. All conductance measurements were per-
(after the field was removed). They found that the ap- formed by measuring the current through a small series
plication of the field caused a fast conductance increase resistor (at least two orders of magnitude smaller than
followedbya slowrisetowardsanasymptotic value. Re- the sample resistance), similar to the method described
moving the field caused a fast conductance decrease fol- in ref [8].
lowed by a very slow drop toward the thermal equilib- Typicalexcitation and relaxationG(t) profiles for two
2
2
1.0 (a)
(a) 0
0.5
-2
0.0
-4
-0.5
-1.0 A -6 100sec
-1.5 τ -8 1000sec
10000sec
-2.0 -10
-2.5 -12
)
%-3.0 )-14
( %
G 10 100 1000 10000 500 1000 1500
G/1.6 G( 2 t(s )
∆1.4 (b) G/ 0 (b)
1.2 ∆ -2
1.0
-4
0.8
-6
0.6
-8
0.4
0.2 -10
0.0 -12
-0.2 -14
1 10 100 1000 10000 100000
t (s) 0.5 1.0 1.5 2.0 2.5 3.0
t/t
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FIG.1: RelativeconductancechangeversustimefortwoAg
granular films excited with a bias field Fn=200 V/cm. (a) FIG. 2: (a) Conductance during relaxation for different val-
relaxation curveunder F0=0.5 V/cm, R=360 MΩ, T=4.2 K. ues of tw (b) The same data as in (a) plotted as a function
(b)therelativeconductanceduringtheexcitation,R=80MΩ. of t/tw. R=5.4 GΩ, Fn=100 V/cm. The conductance os-
cillations which appear in thegraphs are rather reproducible
andprobablyresultfromthemesoscopicnatureofthesample
different samples are shown in fig. 1. It is seen that the [12].
initialtrendwasoppositetothatobservedbyOrlyanchik
and Ovadyahu [8]. During the excitation the conductiv-
ity decreased with time instead of the increase observed of ∼ 100 MΩ and the measured circuit capacitance was
in continuous films, and during the relaxation process ∼ 100pF yielding RC ∼10ms,whichis many ordersof
the conductivity increased slowly contrary to decrease magnitude smaller than the typical measured relaxation
observedincontinuousfilms. After a periodoftime that time τ.
depended on t (typically, a few hundred to a few thou- Inordertostudytheglassynatureofthe”oppositeef-
w
sand seconds), the trend reversed and the conventional fect”weperformedagingexperimentsusingthefollowing
behavior typical of Anderson insulators was observed. scheme: We stabilized the system at low applied electric
This behavior was observed in all investigated granular field, F0, for several hours, thus reaching a steady state
metals without significant dependence on the material. of the system. Then we excited the system by applying
Since the behavior during excitation and relaxation high bias field, Fn, for a waiting time, tw after which
is similar (with opposite signs) for the reminder of this the fieldwasreducedbackto F0. Conductivity measure-
paper we will focus on the processes of relaxation af- ments were performed for a time of the order of 3×tw
ter switching off the electric field. In figure 1 we plot after removing the high field. The relaxation curves of
the change in conductivity versus time, ∆G(t)/G = the conductivity versus time for three different tw are
(G(t)−G0)/G0, where G0 is the extremum conductance plottedinfig. 2(a). Thesamedataisplottedinfig. 2(b)
atwhichthe changeinthe trendoccurredandt=0when as a function of t/tw. It is seen that all curves collapse
theexcitationwasintroducedorremoved. Wedefinetwo onto a single curve as a function of t/tw giving rise to
parameters, (as seen in fig.1a): τ, the time it takes the simple scaling behavior [4, 13].
conductivity to reach G=G0 and A, the change in con- Varying the strength of the exciting field also had a
ductivity during τ. In all measurements we identify two significant effect on the G(t) profiles. Fig. 3a which
regimes of the relaxation G(t) profiles: the initial con- depicts ∆G/G for different F s shows that the relative
n
ductanceincreaseatt<τ wenamethe”oppositeeffect” conductancechange,A,growslinearlywithF . Forhigh
n
and the conductance decrease at t>τ which is the con- enough F , switching off the field gave rise to a short
n
ventionalbehaviorsimilartotheresultsincontinuousin- periodof”negativeconductance”(seefig. 3b),indicating
sulators. We note that a typical sample had a resistance that, for short times, the current flow was opposite to
3
byelectronhoppingbetweenthemetalgrains[14]. When
an external electric field F is applied to the system the
n
0.0 charge on each grain reorganizes itself in order to reach
50 electrostatic equilibrium. If the metallic grains are far
40 from each other the charge on the grains is fixed and
%) %)30 is redistributed on the grain so that the internal field
G/G(∆-2.0x102 100V/cmA(12000 cchanarcaeclsteoruisttitchme eetxatlelricnasclrFeenn.inTghtisimpero(ctyespsicoacllcyurlessosvtehrana
250V/cm 0 20 40 60 80
F(v/cm) 1 ps). However, for closely packed grains, strong elec-
400V/cm n
(a) 600V/cm trons interactions cause the rearrangementof the charge
-4.0x102
0 50 100 150 2 00 250 300 350 400 distributionineachgraintoaffectthearrangementofits
5.0x10-9
10V/cm 25V/cm 50V/cm 100V/cm neighbors. Therefore both the magnitude and the direc-
tionofthescreeninginternalfieldineachgrainchangein
0.0
time, mostly by hopping events,until it reachesa steady
state. This process is interactive - hierarchic, hence is
G(S)-5.0x10-9 characterized by very long times leading to a slow de-
crease of the electric current in the system. After the
-1.0x10-8 excitation is removed, the grains discharge in order to
adapt to the new electrostatic potential. This too is a
(b)
-1.5x10-8 slow process leading to a long-time current decrease, in
8000 12000 16000 20000
t(s) which the direction of the current is opposite to that
FIG. 3: (a) Relaxation curves after excitations by different of the original excitation. This leads to a new glassy
Fn. tw=2000 sec and R=150 MΩ. The inset shows the am- processwhichcompeteswiththe conventionalrelaxation
plitude,A,asafunctionofappliedbiasvoltagefora320MΩ processes of a hopping system relaxing to equilibrium.
sample . (b) Relaxation processes where ”negative conduc- In order to test this model we simulated the time de-
tance” was obtained, R=250 MΩ.
pendence conductivity in a one dimension granular sys-
tem. We considerachainofmetallic grainsconnectedto
0 two large leads which are biased by a voltage, V. Each
grainis connectedto its neighborby junctions ofcapaci-
)
% tancec andtunnelingresistanceR ,asillustratedinthe
-1 i i
(
G insetoffig 5. The potential, ϕi, ofeachgrainis givenby
G/
∆-2
c ϕ +n e
ϕ = P j j i , (1)
i
cΣ
-3
where j is the index of the neighboring capacitors and
1 10 100 1000 10000
t(s) grains, n is the number of the electrons added to the
i
grainifromtheinitiallyneutralconditionandcΣisgiven
FIG. 4: The excess conductance versus time after switching by Pcj. The potentials of the two leads are 0 and V.
off a 2 HzAC field of 25 V/cm. The total electrostatic energy of the system is given by:
1
U =XUi = Xci(ϕi+1−ϕi)2. (2)
2
the voltage direction. We found that τ does not depend
on the value of the exciting field and no correlation was By subtracting the total coulomb energy of two dif-
found between A and tw. ferent combinations of ni and V (state 1 and state 2),
All the above results were obtained employing a DC we find ∆E = U2 −U1, the total coulomb energy cost
voltage excitation. We performed similar measurements of the system transferring from one state to the other.
using an AC voltage excitation and measured the AC However, the charge distribution in the system changes
current through the sample. Fig.4 shows that exciting inordertokeepaconstantbiasvoltage,thereforeweadd
the samples with an AC voltage, even with frequencies to ∆E the energy supplied by the voltage source when
aslowas2Hz,causedthe”oppositeeffect”todisappear, an electron tunnels through any capacitor.
−→
leavingonly the conventional,OrlyanchikandOvadyahu Wedefineavector ρ(V,n1,....,ni,t)whichistheprob-
relaxation trend. ability to have n1....ni electrons on grains 1....i respec-
In the following paragraphs we propose a model for tively, under bias voltage V at time t.
the observedresults,basedonthe granularnature ofthe Atnonzerotemperature,thereisaprobabilityoftun-
samples. The conductivity in these systems is governed neling processes, for which the rate of change from one
4
0.0 currentprofile similar to those seenin figure3b inwhich
thetotalconductivitymaybeginnegativebuteventually
8.25x102
approachesasteadystatevalueofG0. Thechangeinthe
A)-1.0x10-15 /tτhop67..7550xx110022 cfioenldd.uWctahnecnetihseodbitraeicnteidonasofatrheespfioelndsechtoantgheesDfaCstaerppthliaedn
(Ι 6.00x102 the redistributionprocess(i.e. as a resultofanAC bias)
-2.0x10-15 3 4 5 the opposite effect does not show up.
number of dots
Ri‘Ci φi
v=0 v=V
0.0 4.0x102 8.0x102 1.2x103 In summary, we presented results of the time depen-
t/t
hop dentconductivityofinsulatinggranularmetalsafterexci-
tationwithahighelectricfield. Wefindthatadisordered
FIG. 5: Simulation of the current versus time, normalized granular system behaves differently than its continuous
to the hopping time, after an excitation with bias voltage counterpart. Weinterprettheresultsasoriginatingfrom
V=500 V of a 5 grains chain. Ci = 10−17 F,Ri = 108 MΩ. a competition between two different trends. The first
The inserts show a schematic description of the model and process is the relaxation of the systems toward equilib-
the dependence of the characteristic relaxation time on the
rium and is typical of all hopping systems. The second
numberof dots in thechain.
is a consequence of charge rearrangement on the metal-
lic grains and is expected only in granular systems. In
distribution to another is: a sense, the granular metal excited by an electric field
1 −∆E is similar to spin glass subject to a sudden magnetic
Γ(n11...n1i,n21...n2i)= Re2(1−exp(∆E/k T)), (3) field change. Despite the fact that a spin flip occurs on
B
timesofafewns,themagneticinteractionsinthesystem
wherenk1...nki isthechargedistributioninthestatekand cause the arrangement of the whole system to occur on
R is the resistance of the junction. timescales which are many orders of magnitude larger.
The master equation for the system is thus given by: We are grateful for helpful discussions with O. Agam,
V. Orlyanchik,Z. Ovadyahu, R. Berkovitsand M. Gold-
−→
∂ρ(V,n1,....,ni,t) =Γ·−→ρ(V,n1,....,ni,t), (4) sfoteuinnd.aTtihoins(rgersaeanrtcnhuwmabsesru2p4p9o/r0t5ed).by the Israel Science
∂t
−→
where Γ is the rate matrix and ρ is the state vec-
tor. Because the system is not in equilibrium, the cur-
rent through each junction is not the same. The mea-
sured current is the sum of all the possibilities to tun- [1] M. Ben Chorin, D. Kowal and Z. Ovadyahu,Phys. Rev.
nel through the last junction. Our initial conditions are B44, 3420 (1991).
probability1forzeroelectronsoneachgrain. Wethenin- [2] M. Ben Chorin, Z. OvadyahuandM. Pollak, Phys.Rev.
B48, 15025 (1993).
troduce a high bias voltage V for a time of t . We solve
w
[3] A.Vaknin,Z.OvadyahuandM.Pollak,Phys.Rev.Lett.
eq. 4 and calculate the current for every step. Fig. 5
81, 669 (1998).
shows the relaxationcurve of the currentafter switching
[4] A.Vaknin,Z.OvadyahuandM.Pollak,Phys.Rev.Lett.
off the highvoltage excitationso that V=0. The current 84, 3402 (2000).
isnegativeandhasanamplitude whichdecreasestozero [5] A.Vaknin,Z.OvadyahuandM.Pollak,Phys.Rev.B65,
over a time which is much longer (∼3 orders of magni- 134208 (2002).
tude)thanatypicalhoppingprocessevenforasystemof [6] C.J.Adkins,J.D.Benjamin,J.M.D.Thomas,J.W.Gard-
5 grains. The inset of fig 5 shows that the characteristic ner, A.J.Mc Geown, . J. Phys. C 17, 4633 (1984).
[7] T. Grenet, Eur. Phys.J. B 32, 275 (2003).
relaxation times increase monotonically as a function of
[8] V. Orlyanchik, Z. OvadyahuPhys. Rev. Lett. 92, 66801
the number of grains, hence the times can be expected
(2004).
to be much longer in a macroscopic sample. We note [9] M. Strongin, R. Thompson, O. Kammerer and J. Crow,
that this is clearly a simplistic model which takes into Phys. Rev.B1, 1078 (1970).
account only nearest-neighbor hops and doesn’t include [10] R.C.Dynes,J.P.GarnoandJ.M.Rowell,Phys.Rev.Lett.
the disorderpropertiesofthe systembut webelieve that 40, 479 (1978).
it conveys the essential physics of the slow, glassy dy- [11] H.M. Jaeger, D.B. Haviland, B.G. Orr and A.M. Gold-
man, Phys. Rev.B40, 182 (1989).
namics responsible for the time dependent conductivity.
[12] A. Cohen, N. Kurzweil, L. Bitton and A. Frydman, in
In the actual experiment the electric field is not re-
Preparation .
duced to zero but rather to a small dc field for conduc-
[13] G.F.Rodrigues,G.G.KenningandR.Orbach,Phys.Rev.
tivitymeasurements. Sothenegativecurrentobservedin Lett. 91, 037203 (2003).
the simulations has to be added to a finite positive cur- [14] A.Frydman,T.L.KirkandR.C.Dynes,SolidStateCom-
rent. The sum of those two can be expected to lead to a mun. 114, 481 (2000).
