Supercooling and the Metal-Insulator Phase Transition of NdNiO$_{3}$

0.53 MB·English

by  Devendra Kumar

3
Super
ooling and the Metal-Insulator Phase Transition of NdNiO
∗
Devendra Kumar and K. P. Rajeev
Department of Physi
s, Indian Institute of Te
hnology, Kanpur, 208016, India
J. A. Alonso and M. J. Martínez-Lope
Instituto de Cien
ia de Materiales de Madrid, CSIC, Cantoblan
o, E-28049 Madrid, Spain
We report the temperature and time dependen
e of ele
tri
al resistivity on high temperature,
3 3
highoxygenpressure preparedpoly
rystalline samplesofNdNiO . NdNiO ismetalli
above195K
8
and below that temperature it undergoes a transition to an insulating state. We (cid:28)nd that on
0 3

ooling NdNiO below 195K it goes into a state whi
h is not in thermodynami
equilibrium and
0
slowly relaxes over several hours. As we
ool it further and go below about 110K it goes into a
2
stable insulating state. On heating the system from the insulating state towards 200K we (cid:28)nd

n that it remains stable and insulating and undergoes a rather sharp insulator to metal transition in
a the temperature range 185K to 195K. We try to make sense of these and a few other interesting
J observationsonthebasisofour
urrentunderstandingof(cid:28)rstorderphasetransitions,super
ooling,

0 and metal-insulator transitions.
1
PACSnumbers: 64.60.A-,64.60.ah,64.60.My,64.70.K-,71.30.+h,

] Keywords: Metal-Insulator Transition, Nonequilibrium, Hysteresis, Super
ooling, Phase Transition, Relax-
r ation,TimeDependen
e
e
h
ot I. INTRODUCTION aresignaturesofa(cid:28)rstorderphasetransition11,12. There
t. isalsoTahysteresisin ele
troni
transportproperties just
a MI11
below . The hysteresis in transport measurements
m Rare earth ni
kelates with the
hemi
al formula
3 points toward the
oexisten
e of metalli
and insulating
RNiO , whereRstandsforarareearthion, isoneofthe T
- phases below MI11. In this paper we report a detailed
d few families of oxides that undergoes a (cid:28)rst ordermetal- 3 TMI
study of the nature of NdNiO below through tem-
n insulator phase transition. A metal-insulator transition
o (M-I transition) is an ele
troni
phase transition whi
h perature and time dependent ele
tri
al resistivity mea-
c surements.
is usually solid to solid. The (cid:28)rst order ele
troni
phase
[
transition in many systems is asso
iated with a hystere-

2 sis, and they exhibit a phase separated (PS) state in the
v vi
inity of the transition1,2,3. The PS state has drawn II. EXPERIMENTAL DETAILS
0
a lot of attention in the
ase of manganites, mainly due
9 3
9 to its impli
ations for
olossal magnetoresistan
e, and it Poly
rystalline NdNiO samples in the form of 6mm
3 exhibitsavarietyoftimedependente(cid:27)e
tssu
hasrelax- diameterand1mmthi
kpelletswerepreparedand
har-
. ation of resistivity2,4 and magnetization5 and giant 1/f a
terized as des
ribed elsewhere13. Th◦e preparation
2 6
1 noise . method uses a high temperature of 1000 C and a high
3
7 RNiO
rystallizes in the orthorhombi
ally distorted oxygen pressure of 200bar.
bnm
0 perovskite stru
ture with spa
e group P . The de- Allthetemperatureandtimedependentmeasurements
: gree of distortion in
reases with de
rease in size of rare were done in a home made
ryostat. To avoid ther-
v
i earth ion as one moves towards the right in t6=he peri- mal gradients in the sample during measurement it was
X 7
odi
table . The ground state of ni
kelates (R La) is mounted inside a thi
k-walled
opper en
losure so that
r insulatingandantiferromagneti
. Onin
reasingthetem- during the measurement the sample temperature would
a
peraturethese
ompoundsundergoatemperaturedriven be uniform. It was found that mounting the sample like
antiferromagneti
to paramagneti
transition,and an in- this improved the reprodu
ibility of the data, espe
ially
3 3
sulator to metal transition. ForTNdNiO and PrNiO , in the time dependen
e measurements, signi(cid:28)
antly. A
MI
the M-I transition tempTerature ( ) and the maTgneti
LakeshoreCryotroni
stemperature
ontrollermodel 340
N MI
ordering tempTerature ( ) are the same, while is was used to
ontrol the temperature and the tempera-
N
greater3than for l3ater meTmMbIers of the series su
h as ture stability was found to be better than 3mK during
EuNiO and SmNiO . The of ni
kelates in
reases
onstant temperature measurements.
with de
reasing size of the rare earth ion and the later Below TMI NdNiO3 is not in thermodynami
equilib-
membersoftheseriesalsoshowa
hargeorderingtransi- rium and slowlyrelaxes. Be
auseof this the data weget
7,8,9
tion asso
iated with the M-I transition . The
harge depend on the pro
edure used for the measurement. We
orderedstatehasalsobeenobservedinepitaxiallygrown usedthefollowingpro
eduretomeasurethetemperature
310
thin (cid:28)lms of NdNiO . dependen
e of resistivity. While
ooling we start from
3
The M-I transition in NdNiO is asso
iated with a la- 300K, and then re
ord the data in steps of 1K interval
tentheatandasuddenin
reaseinunit
ellvolumewhi
h afterallowingthe temperatureto stabilizeat ea
hpoint.
2
Inbetweentwotemperaturepointsthesamplewas
ooled one of the minimum temperatures mentioned above and
at a (cid:28)xed
ooling rate of 2K/min. After the
ooling run then heat it ba
k to 220K, both operations being
ar-
isoverwewaitforonehourat82Kandthentheheating ried outat a(cid:28)xed rate of2K/min. Loopsformedin this
data was
olle
ted at every one degree interval. The in- fashionare
alledminorloopsandtheseareindi
atedby
termediate heating ratebetweentemperaturepoints was the labels a, b and
in Figure 1. In the
ooling
y
le
the same as the
ooling rate used earlier. This
y
le of all the three minor loops
oin
ide with the
ooling
urve
measurements was repeated with a di(cid:27)erent
ooling and of the full hysteresisloop. In the heating
y
le, for loops
heating rate of 0.2K/min also. a and b with lowerminimum temperatures, the resistiv-
T
MI
It was observed that the resistivity above does ity de
reases with in
reasing temperature and joins the
not show any time dependen
e, and it is also indepen- full loop at 195K. In the
ase of loop
, as we in
rease
dent of measurement history. Thus to avoid the e(cid:27)e
t the temperature, the resistivity in
reases somewhat till
of any previous measurements, all time dependent ex- about 182K and then it falls and joins the full loop at
periments in the
ooling run were done as follows: (cid:28)rst 195K.
T
MI
take the sample to 220K (above ), wait for half an The resistivity also shows a noti
eable dependen
e on
hour,then
oolat2.0K/mintothetemperatureofinter- the rate of temperature
hange in the
ooling
y
le as
est and on
e the temperature has stabilized re
ord the shown in the inset of Figure 1. The data for the lowest
resistan
e as a fun
tion of time. In the heating run the
urve was
olle
ted at 2K/min and for the middle
urve
time dependent resistivity was done in a similar fashion: at 0.2K/min. The uppermost
urve is an estimate ob-
(cid:28)rsttakethesampleto220K,waitforhalfanhour,then tainedbyextrapolatingthetimedependen
edatashown

ool at 2.0K/min to 85K, wait for one hour, and then in Figure 2 to in(cid:28)nite time. We did not see any rate de-
heatat2.0K/mintothetemperatureofinterestandon
e penden
ein theheating
y
le. Thisisan indi
ation that
the temperaturehas stabilized re
ord the resistan
e as a while
ooling, below the M-I transitiontemperature, the
fun
tion of time. system is not in equilibrium and hen
e the resistivity
The four probe van der Pauw method was used to evolveswith time. The non-ratedependen
e observed in
measure the resistivity and standard pre
autions, su
h the data while heating tells us that in the heating
y-
as
urrentreversalto take
are ofstray emfs, weretaken
during the measurement. We also took
are to ensure
that the measuring
urrent was not heating up the sam-
ple. A Keithley
urrent sour
e model 224 and a DMM
model 196 were used for the resistivity measurements. 300 200 T (K) 100
10
III. RESULTS
3
Figure 1 shows the ele
tri
al resistivity of NdNiO as 1
a
a fun
tion of temperature. We see that the fun
tion is m)
c
multiple valued, the
ooling and heating data di(cid:27)ering
(
signi(cid:28)
antlyfromea
hotherandformingalargehystere- 0.1 m)
3 c
sis loop. The resistivity plot indi
ates that NdNiO un- b 0 .1
(
dergoes a relatively sharp M-I transition at about 190K
while heating with a width of about 10K. In
ontrast, 0.01
while
ooling, the resistivity shows a rather broad M-I
c
transition
entered around 140Kwith a spread of about 0.01
0.0068 -1 0.0070
40K. Below 115K or so, the heating and
ooling data 1E-3 1/T (K )
logρ 1/T
merge and the vs plot is linear. This indi
ates 0.004 0.006 0.008 0.010 0.012
-1
that the sample is insulating at low temperatures and, 1/T (K )
∆
if the band gap is , the resistivity should follow the
relation
ρ 1/T 3
ρ(T)=ρ0exp(∆/kBT) Figure 1: (Color online) vs. plot for NdNiO . The
(1) blue
ir
les represent
ooling data and the red squares stand
forheating data (
ooling/heating rate2.0 K/min). Thesolid
Below 115K the data (cid:28)t quiteRw2ell to this moρd0el, wit∆h lineisaleastsquare(cid:28)ttothebandgapinsulatormodelbelow
a
oe(cid:30)
ient of determination, = 0.99955. Ω and 115K.The
urveslabeleda, b&
weretakenasdes
ribedin
for the insulating region turn out to be 99m
m and the text. The a

ura
y of the data points is better than 1%
Ω
42meV respe
tively, whi
h are in reasonable agreement everywhere. The error does not ex
eed 0.8 m
m anywhere.
ρ 1/T
with the values previously reported11. The inset shows vs for three di(cid:27)erent
ooling rates:
We
olle
ted more hysteresis data with di(cid:27)erent min- lower
urve(bluestars): 2K/min, middle
urve (red
ir
les):
0.2K/min, upper
urve (bla
k pluses): in(cid:28)nitely slowly (ex-
imum temperatures su
h as 140K, 146K and 160K. In
plained in text). The
onne
ting lines are to guide theeyes.
these measurements we
ool the sample from 220K to
3
ρ1/ρ0 τ 103 γ χ2/DOF R2
# T(K) ( s)
145 K 1 140.0 0.764(5) 1.52(1) 0.538(3) 9.5 0.99971
3.0
2 142.5 1.89(1) 2.02(2) 0.554(2) 5.9 0.99981
3 145.0 4.59(3) 5.05(5) 0.567(1) 0.67 0.99993
4 147.5 3.39(6) 7.9(3) 0.568(3) 0.51 0.99978
2.5
147.5 K
5 150.0 1.22(1) 4.04(6) 0.582(2) 0.0014 0.99988
0
/ 6 155.0 0.391(5) 2.6(1) 0.557(7) 0.0003 0.99845
142.5 K
2.0
Table I: Fit parameters for the time dependen
e data shown
DOF ≈1000
150 K in Figχu2r/eD2O.FThe degrees of freedomof the(cid:28)ts .
The for 150K and 155K are too small, indi
ating
1.5 140 K
that we have overestimated the erRro2r in resistivity in these

ases. Anyway, we note that, the values are
onsistently
155 K
good and indi
ate reasonably good (cid:28)ts.
1.0
Heating run, 145 K
0 1000 2000 3000
time (s)
4
4
Figure2: (Coloronline)Timedependen
eofresistivitywhile 0

ooling,atvarioustemperaturesintherange140to155K,for /1
a period of one hour. Maximum time dependen
e is seen at 3 2
145Kwhi
h isabout300%. The
urveat thebottom,whi
h
lookslikeastraightline,showsthein
reaseinresistivityina m)
c
heatingruntakenat 145Kandthe
hangeinthis
aseisless
than 0.2%. Not all thedata are shown hereto avoid
lutter. ( 2 1010 130 150
T (K)

le the system is either in, or very
lose to, equilibrium. 1
These observations are
orroborated by the data shown
in Figure 2.
A subset of the time dependent resistivity data taken
0
while
ooling is shown in Figure 2. The data are pre-
ρ(t)/ρ(t=0) 125 135 145 155
sentedas sothatthe valuesarenormalized T (K)
t = 0
to unity at for easy
omparison. We found that
that below 160K, the resistivity of the sample in
reases
ρ
with time
onsiderably. A maximum relative in
rease Figure 3: (Color online) Temperature dependen
e of on
in resistivity of about 300% for a duration of one hour
ooling with ageing by intermediate stops of one hour ea
h
(blue open
ir
les) at 155, 150, 145, 140, 135, and 130 K.
is seen at 145K, the time dependen
e being lower both
ρ(T,t) Redsolid
ir
les showtheresistivitywithout ageing. The in-
above and below this temperature. We (cid:28)tted the ρ1/ρ0
set
omparesthetemperaturedependen
eof
al
ulated

urves in (cid:28)gure 2 to the stret
hed exponential fun
tion ρ(T,t)
fromthe(cid:28)tofEquation(2)to datawith intermediate
ρ(t)=ρ0+ρ1(1−exp(−(t/τ)γ)) ageing (blue
ir
les) to the time dependen
e data shown in
(2)
Figure 2(red squares).
ρ0 ρ1 τ γ
where , , andR2are (cid:28)t parameters. The (cid:28)ts are
quite good with the value greaterthan 0.999in most
γ

ases. See Table I. We note that the exponent lies to 155K,
olle
t resistivity time dependen
e again for
0.5 < γ < 0.6 τ
in the range and has a peak around one hour and so on down to 110K for every 5K inter-
147.5K.Thevariationofresistivitywithtimeshowsthat val. We note thρa(tTw)hen
ooling is resumed after ag-
thesystemslowlyevolvestowardsaninsulatingstateata ing for an hour,
urve merges smoothly with the

onstant temperature. We
olle
ted data up to 12 hours
urve obtained without aging. A rather similar observa-
(not shown here) to
he
k whether the system rea
hes tionhasbeenreportedinthephase-separatedmanganite
0.5 0.5 0.95 0.05 34
anequilibrium state, but foundthatitwas
ontinuingto La Ca Mn Fe O .
ρ1/ρ0
relax even after su
h a long time. The redsquaresin theinsetofFigure3displays
Figure3
omparestheresistivityin a
oolingrunwith obtained from stret
hed exponential (cid:28)tting of time de-
ρ1/ρ0
and without intermediate aging. These data were taken penden
eresistivitydataofFigure2. The in
reases
as follows: we start from 220K,
ome down to 160K on de
reasing the temperature below 160 K, attains a
at 2K/min,
olle
t time dependen
e data for one hour maximum around 145 K, and then de
reases on further
and after that resume
ooling at 2K/min and go down loweringthe temperature. Wehavealreadyseen this be-
4
ρ1/ρ0
havior in Table I. The value of obtained from the fa
ts we proposethat the metastable state in the
ase of
3
(cid:28)tting of time dependen
e data with intermediate aging NdNiO
onsistsofsuper
ooled(SC)metalli
phasesand
is shown as blue
ir
les in the inset of (cid:28)gure 3, and we stable insulating phases.
ρ1/ρ0
note that the value of is relativelysmall
ompared In a (cid:28)rst order transition, a metastable SC phase
an
T
C
to what one obtains without intermediate aging. survivebelowthe(cid:28)rstordertransitiontemperature( ),
ρ T
In the heating
y
le, the vs exhibits a rather till∗a
ertaintemperature
alledthelimitofmetastabi∗lity
T T <
3,14,15
sharp insulator to metal transition. The magnitude of ( ) is rea
hed . In the temperature range
≈ 0.2% T < T
C
time dependen
e in heating runs (maximum ) there is an energy barrier separating the SC
is negligible
ompared to what one gets in
ooling runs phase from the stable ground state. The height of the
≈300%) U
(maximum ((cid:28)gure 2), whi
h suggest that in the energy barrier ( ) separating the SC metastable phase
T U ∝f(T−
MI
heating run, below , the sample is almost fully insu- fro∗mthestablegroundstate
anbewrittenas ∗
T ) f T −T
lating and stable. , where is a
∗ontinuous fun
tion of ( ), and
T ≤ T
Themajorobservationsthathavebeenmadesofarare vanishe∗s for . As the temperature is lowered, at
T =T
summarized below. , the SC metastable phase be
omes unstable and∗
T > T
15
swit
hes over to the stable ground state . At
1. The metal to insulator transition in
ooling runs the SC metastable phase
an
ross over to the stable
is rather broad and extends from about 160K to groundstatewithaprobability(p) whi
hisgovernedby
120K. the Arrhenius equation
p∝exp(−U/ T)
2. In
oolingruns a largeamount of time dependen
e kB (3)
in resistivity is seen
ompared to whi
h the time
dependen
e seen in heating runs is negligible. whi
h tells us that the barrier will be
rossed with an
τ ∝ 1/p
ensemble average time
onstant . If we imagine
3 U
3. NdNiO undergoes a rather sharp insulator to an ensemble of su
h SC phases with the same barrier ,
metal transition between 185K and 195K while then the volume of the metastable phase will exponen-
τ
heating. tially de
ay with a time
onstant .
U
The energy barrier that wetalked about in the pre-
4. Thereis alargehysteresisbetweenthe
oolingand
vious paragraph is an extensive quantity. Our system is
heating data below the MI transition temperature.
poly
rystalline, ie., it is made up of tiny
rystallites of
di(cid:27)erent sizes. Ea
h one of the
rystallites will have an
energy barrier proportional to its volume. Thus for a
IV. DISCUSSION

rystallite the energy barrier separating the metastable
state from the ground state
an be written as
3
It has been known that the M-I transition in NdNiO ∗
U ∝Vf(T −T )
is asso
iated with a latent heat and a jump in the unit (4)

ell volume whi
h are the
hara
teristi
s of a (cid:28)rst order
V
11,12
phasetransition . DSCmeasurementsofGranadoset where isthevolumeofthe
rystallite. Asalreadymen-
f
al. showthat,while
ooling,thelatentheatreleaseddur- tioned isa
ontinuousfun
tionwhi
hvanishesfornon-
ing the M-I transition extends overa broad temperature positive values of its argumen∗t. This means that various
T
11
range . This is
onsistent with the results of the resis-
rystallites with the same will have di(cid:27)erent energy
3
tivitymeasurementsthattheM-ItransitioninNdNiO is barriers depending on their size whi
h implies that the
3 τ
broadenedwhile
ooling. Thuswe
ansaythatNdNiO time
onstant , of the previous paragraph, will spread
hasabroadened(cid:28)rstordermetaltoinsulatorphasetran- outandbe
omeadistributionoftime
onstantsdepend-
sition while
ooling. ingonthedistributionofthesizeofthe
rystallites. This
T
MI
We have seen that, below , while
ooling, the sys-
angiverisetothevolumeofthemetalli
statede
aying
16,17
tem is not in equilibrium and evolves with time whi
h in a stret
hed exponential manner with time . This
suggests that it is in a metastable state. The resistivity behavior of the metalli
volume with time will give rise
ofthis metastablestateslowlyin
reaseswithtime whi
h totheresistivityalsoevolvingwithtimeinasimilarfash-
means that a slow metal to insulator transition is going ion. Weshallseelaterin thedis
ussionhowthemetalli
on in the system. On the other hand while heating up volume, the resistivity, and their time dependen
es are
fromlowtemperaturewesawthatthesystemremainsin-
onne
ted with ea
h other.
sulatingallthewayupto185Kandthenittransitionsto Imry and Wortis have argu∗ed that a weak disorder
T
the metalli
state by 195K. The resistivity was found to
an
ause a distribution of 's in di(cid:27)erent regions of
3,18
have negligible time dependen
e during the whole of the a sample . This has been experimentally observed
T
C
heating run. This result indi
ates that the low tempera- in a vortex latti
e melting experiment in a high
19
tureinsulatingstateisstableandisprobablytheground super
ondu
tor . In our
ase grain boundaries, shape
state of the system. It is well known that below a phase of the
rystallite et
.
ould b∗e sour
es of disorder whi
h
T
transitiontemperatureahightemperaturephase
ansur- would give rise to di(cid:27)erent 's for di(cid:27)erent
rystallites.
vive as a metastable super
ooled state. Based on these This e(cid:27)e
t will further broaden the distribution of time
5

onstantsinoursystemwhi
harisefromthedistribution TableIandalsointheinsetofFigure3andFigure
of
rystallite sizes. 6.
Comingba
ktotheheatingrunswenotethatthela
k
Let us take up these issues one by one.
of time dependen
e of the resistivity data indi
ates that
the system is in a stable state. Now the question arises:
what is the reason for the width of about 10K observed A. Cooling Rate Dependen
e
intheinsulatortometaltransition? Coulditbeasuper-
heating e(cid:27)e
t? It
annot be be
ause, as already noted,
In the inset of Figure 1 we saw that while
ooling a
the la
k of time dependen
e during heating runs rules
lower
ooling rate in
reases the resistivity of the sample
out the possibility of any metastable phase in the sys-
in the hysteresis region. A lower
ooling rate allows a
tem. The broadening while heating ismost likely
aused
larger number of the metastable metalli

rystallites to
by the rounding of the phase transition due to (cid:28)nite size
swit
hovertotheinsulatingstateandthustheresistivity
of the
rystallites. A similar observation on broadening
will be largerif the measurement is done slowly.
ofphase transitionsdue to (cid:28)nite size ofgrainshavebeen
20 Ifthemeasurementisdonein(cid:28)nitelyslowlyinthe
ool-
reported in a CMR manganite .
ing run we shouldget the uppermost
urveshownin the
Now we understand the reason for the hysteresis seen
inset of Figure1. It is interestingto note that this
urve
in this system. During a heating run from low tempera-
does not
ome anywhere near the heating run
urve. To
ture the system is insulating and hen
e its resistivity is
paraphrase,evenifwesitforanin(cid:28)nitetimeonthe
ool-
high. While
ooling some of the
rystallites transform
ing
urvewe
annotrea
htheheating
urve. The reason
themselves into the stable insulating state while others
for this is that only a small number super
ooled
rystal-
remain in the super
ooled metalli
state . The presen
e
liteswhi
hhavearelativelysmallenergybarrierareable
of the metalli

rystallites lowers the resistivity of the
toswit
hovertotheinsulatingstateatanygiventemper-
system. Thus we get di(cid:27)erent values of resistivity dur-
ature. The others with the largerenergybarriersremain
ing heating and
ooling runs giving rise to hysteresis in
trappedin themetalli
state. These
rystalliteswiththe
resistivity.
Thepi
turesofar: NdNiO3 ispoly
rystallineandthus larger barriers
an swit
h thTeir state only if the temper-

onsists of tiny
rystallites. Below TMI, while
ooling, aTt∗ure is lowered su
h that
omes su(cid:30)
iently
lose to
and thus the barrierbe
omes small and easy to
ross
thetiny
rystallites
anbeineitherasuper
ooledmetal-
(Equation (4)).
li
state or a stable insulating state. A
rystallite in the
super
ooled metalli
state
an make a transition to the
U
stableinsulatingstateif it
rossesthebarrier withthe B. Minor Loops
help of energy (cid:29)u
tuations whi
h in our
ase are ther-
mal in origin. Ea
h
rystallite would undergo the tran-
The minor loops a, b, and
shown in Figure 1 follow
sition independent of ea
h other. This is supported by
the full hysteresis loop in the
ooling
y
le till we start
the fa
t that the resistivity, and as we shall see later,
the heating. While heating,
urve a, with the lowest
the volume of the metastable metalli
state, behave in a
minimum temperature (140K), shows a falling resistiv-
stret
hed exponentialmanner withtime. Ataparti
ular
ity with in
reasingtemperature. Curveb with the inter-
temperature, as a fun
tion of time, the transition from
mediate minimum temperature (146K) shows a rather
the metalli
to the insulating phase will begin with the
(cid:29)at resistivity with in
reasing temperature, while
urve
smaller
rystallites,be
ausethebarrierissmallforthem,

with the highest minimum temperature (160K) shows
and then pro
eed with the larger
rystallites, in the in-
a rising resistivity with in
reasing temperature initially

reasing order of barrier size. While heating the system
before falling and joining the full hysteresis loop at the
remains insulating and stable all the way up to the M-
MI transition temperature 195K.
I transition and then it goes over to the stable metalli
The behavior of the minor loops
an be understood
state. There is no superheating during heating runs.
by re
ognizing that the nature of resistivity, when heat-
Now that we have understood the origin of time de-
ing is resumed, in the metastable region is determined
penden
e and hysteresis in the system we would like to
bythe
ompetitionbetweenthetemperaturedependen
e

onsider some more issues that needs to be understood.
and time dependen
e of resistivity. In the
ase of
urve
These are
a, with the lowest minimum temperature, most of the

rystallites will be in the insulating state as
an be in-
1. The dependen
e of resistivity on
ooling rate as
ferred from the large value of resistivity. Only few of
shown in the inset of Figure 1.
the
rystalliteswill be in the metastable state and hen
e
thetimedependen
e e(cid:27)e
ts
anbeexpe
ted tobe small.
2. The behavior of the minor loops seen in Figure 1.
Thus during heating the resistivity of the sample will be
3. Theintermediate ageingbehaviorseenin Figure3. dominated by the temperature dependen
e of resistivity
oftheinsulating
rystallites. Thisisthereasontheheat-
ρ1/ρ0 τ
4. Both and ofEquation(2)gothroughapeak ing part of the
urve looks more or less parallel to the
around145Kand 147.5Krespe
tively as shownin main heating
urve before the onset of the transition.
6
The behavior of
urve
is easier to understand and of the sample. We had already rea
hed this
on
lusion
hen
e let us
onsideritbefore dis
ussing
urveb. Inthis earlier when we dis
ussed the
ooling rate dependen
e.

asethe resistivityis lowand mostof the
rystallitesare
inthemetalli
state. Themetalli

rystallitesformaper- ρ1/ρ0 τ

olatingnetworkandtheresistivityofthe samplewillbe D. Peaks in and
governed by their behavior. The insulating
rystallites
ρ1/ρ0 τ
areessentiallyshorted out and will havelittle rolein de- Wehadnotedearlierthatboth and gothrough
termining the resistivity. The metalli

rystallites, being peaks around 145K and 147.5K respe
tively. The peak
ρ1/ρ0
metastable, will swit
h to the insulating state governed in is not surprising be
ause it shows up in a re-
by Equation (3), thus slowly in
reasing the resistivity of gion of temperature in the
ooling
urve where the re-
the sample as a fun
tion of time. When the resistiv- sistivity
hanges very fast with temperature. Thus one
ity values are
olle
ted while heating, there will be some would expe
t that a large number of metastable
rystal-
time di(cid:27)eren
e between two data points depending on lites are swit
hing over to the insulating state near this
the heating rate, allowing a rise in resistivity to register. region. Hen
e if a relaxation measurement is done here
Wewouldthen(cid:28)nd thatthe resistivityin
reaseswithin- we should see the e(cid:27)e
t of a large number of
rystallites
ρ1/ρ0

reasingtemperature. Curveb liesbetween
urvesa and transitioning to the insulating state as a peak in .
τ

and in this
ase there will be a good number of both But the peak in at about 147.5K is not that easily
insulating and metalli

rystallites. Here there would be understood. To
he
k whether this peak has anything

ompetition between the temperature dependen
e of the to with per
olation we should examine how the metalli
insulating
rystallites and the time dependen
e of the and insulating volumes depend on temperature.
metalli

rystallites and hen
e one
an expe
t that the The
ondu
tivityofaphaseseparatedsystemdepends

urve b would be more or less (cid:29)at at the onset of heat- on thevolumefra
tion ofinsulatingand metalli
phases,
ing. their geometries, distribution, and their respe
tive
on-
σ σ
I M
du
tivities ( and ). M
La
hlan has proposed an
equationbasedonagenerale(cid:27)e
tivemedium(GEM)the-
σ
E
C. Intermediate Ageing oryfor
al
ulatingthee(cid:27)e
tiveele
tri
al
ondu
tivity
21
of a binary MI mixture , whi
h is
Now let us
onsider the intermediate ageing data 1/t 1/t 1/t 1/t
(σ −σ ) (σ −σ )
shown in Figure 3. We note that after ageing for one (1−f) I E +f M E =0
hour, when the
ooling is resumed, the slope of the the (σ1/t+Aσ1/t) (σ1 /t+Aσ1/t) (5)
ρ T I E M E
vs.
urve is less in magnitude than for the
urve
f
obtained without ageing. When we stop the
ooling and where isthevolumefra
tionofthemetalli
phasesand
A=(1−f )/f f
c c c
age the sample at a (cid:28)xed temperature the super
ooled , being the volumefra
tionofmetalli
U t
metalli

rystallites with relatively small will swit
h phases at the per
olation threshold, and is a
riti
al
over to the insulating state. This de
reases the number exponent whi
h is
lose to 2 in three dimensions22,23.
f
c
of
rystallites in the metalli
state and raises the resis- The
onstant depends on the latti
e dimensionality,
tivity. So now, when the
oolingis resumed, the number and for 3D its value is 0.1624. The GEM equation has
ofmetalli

rystalliteswhi
h
answit
htotheinsulating been su

essfully applied to a wide variety of isotropi
,
state would be less and hen
e the
hange in resistivity binary, ma
ros
opi
mixtures and it works well even in
21,23,25
with temperature would be lower. the per
olation regime .
FromFigure3itis
learthatonresuming
oolingafter In order to
al
ulate the volume fra
tion of metalli
anintermediatestopofonehourthe
ooling
urvemerges and insulating phases from Equation (5), we need their
ρ (T) ρ (T) ρ (T)
M I M
withthe
urveobtainedwithoutageingwithinabout3K respe
tiveresistivities and . wasob-
ρM = ρ0 +AT A
orless. Asalreadynotedinthepreviousparagraphwhen tained using , where is the tempera-
we stop the
ooling and age the sample at a (cid:28)xed tem- ture
oe(cid:30)
ient of resistivity estimated from the resistiv-
ρ (T)
I
perature the super
ooled metalli

rystallites with rela- ity data above the M-I transition. was
al
ulated
U
tivelysmall willswit
hovertotheinsulatingstate. As using the parameters obtained by (cid:28)tting the resistivity

an be inferred from Equation (4) those
rystallites will data below 115K to Equation (1). Using the above in-
U
have a smal∗l whi
h have (i) their metastability tem- formationandtheresistivitydata,we
al
ulatedthevol-
T T
perature ( )
lose to their temperature ( ) (ii) a small ume fra
tion of metalli
and insulating phases and it is
size. The major
ontributions to resistivity
hange will shown in Figure 4. In the
ooling
y
le the volume fra
-
V =1−f

ome from the relatively larger
rystallites making the tionoftheinsulatingphases( )slowlyin
reases
T
MI
transitionfrom the metalli
stateto the insulatingstate. on de
reasing the temperature below , while in the
In the light of this the mergerof the
ooling
urves with
ase of heating
y
le it remains nearly
onstant up to
andwithoutageingwithinasmalltemperature
hangeof about185K,and thendropstozeroby200K.As
anbe
about 3K means that most of the
rystallites whi
h un- seenfromFigure4theper
olationthresholdforthe
ool-
dergo the transit∗ion from metal to insulator during age- ingrunso

ursataround144K.Belowthis temperature
T
ing havetheir within a few kelvin of the temperature therewill be no
ontinuousmetalli
paths in thesystem.
7

T (K)
50 100 150 200 147.5 K
0.070
1.0 150 K
145 K
VTH
155 K
V
0.035 142.5 K
V0.5 0.09
V/dT|0.04 0.06 V 140 K
|d0.02 0.03
0.00 0.000
0.00 0 1000 2000 3000
0.0 120 150 180 time (s)
T (K)
50 100 150 200
T (K)
Figure 5: (Color online) In
rease in volume fra
tion of insu-
∆V
lating phases ( )with time, at various temperatures.
V =1−f
Figure4: (Coloronline)Temperaturevariationof ( ),
theinsulatingvolumefra
tion. Theblue
ir
lesshowthe
ool-
ing
y
le(2K/min
oolingrate)andtheredsquaresrepresent
the heating
y
le. The green pluses represent the insulating 8000
volume fra
tion for 0.2K/min
ooling rate. The dashed hor-
izontal line at 84% insulating volume represents the per
ola-
|dV/dT|
tionthreshold. Theinsetshowsthevariationof (blue

ir
les:
ooling rate 2K/min, and green pluses:
ooling rate 6000
∆V∞
0.2K/min)and (bla
ksquares) with temperature. s)
(
4000
The rate of
hange of metalli
volume fra
tion
−dV/dT
( ) has a maximum around 147.5 K in the
ool-
ing runs (inset of Figure 4). Figure 5 displays the in-
∆V 2000

rement ( ) of insulating volume fra
tion with time,
that has been extra
ted from the time dependent re-
sistivity data of Figure 2. Just as in the
ase of time
∆V(t) 120 135 150 165
dependen
e of resistivity, the
urves of Figure 5
∆V(t) = T (K)
w∆eVr∞e((cid:28)1tt−edextpo(a−(stt/reτt)
γh)e)d exponential fun
tion∆V∞
, where the
onstant gives
τ
thein
reaseintheinsulatingvolumefra
tionwhenwait-
Figure 6: (Color online)Temperature variation of obtained
ρ(t)
ing for an in(cid:28)nitely long∆tiVm∞e. The inset of Figure 4 f∆roVm(tt)hestret
hedexponential(cid:28)ttingof (red
ir
les)and
alsoshowsthevariationo|dfV/dT|withtemperature,whi
h (bla
ksquares)
urves.
mimi
s the behavior of . Figure 6
ompares the
τ ρ(t)

onstantof relaxation obtained from the (cid:28)tting of
∆V(t) T∗
aρn(td)
uτrvesofFigure2and5respe
tively. Forthe of the
ooling
urve represents the distribution of 's
∆V(t
)urves, has a peak aτround 147.5 K while for the in the system. In
oming to the above
on
lusion we

urves, surprisingly, is nearly
onstant. have disregarded the small fra
tion of
rystallites that
|dV/dT|
During
ooling of the inset of Figure 4 rep- would be swit
hing due to the time elapsed in
overing
resents the amount of volume that will
hange from the the small temperature
hange. This is supported by the
|dV/dT|
super
ooled metalli
to the insulating state for a unit fa
tthatthe values
al
ulatedfromthe2K/min
temperature
hange. Nowasmall
hangeintemperatur∗e and 0.2K/min
ooling
urves are essentially the same.
T △V∞
will result in thosesuper
ooled
rystalliteswith their Thepeakin o∗

ursat147.5Kwhi
hisalsowhere
T
falling in that temperature range swit
hing to the insu- thedistributionof 'speak. Th∗isisunderstandablebe-
|dV/dT| T T
latingstate. Thismeansthatthevalueof at is
ause all the
rystallites with
lose to 147.5K would
agoodmeasureofthefra
tion∗ofsuper
ooledmetastable be swit
hing to the insulating state if we wait for a suf-
T T |dV/dT|

rystalliteswhi
hhavetheir
loseto . Thus (cid:28)
iently long time. Sin
e the maximum number of
rys-
8
∗
T
talliteswith are
loseto147.5K,wewillgetthemax- longer time period.
△V∞
imum temperature dependen
e for at 147.5K.
ρ1/ρ0
One would have expe
ted that the peak in and
△V∞
ρ1/wρ0ould o

ur at the same temperature. The peak V. CONCLUSION
in is at 145K whi
h is 2.5K below the peak of
△V∞
. The reason for this is that the resistivity would
Ourexperimentalresultsdemonstratethatwhile
ool-

hange faster for a given metalli
volume
hange nearer 3
ing the physi
al state of NdNiO is phase separated be-
the per
olationthreshold. We guessthat even if the vol-
low the M-I transitiontemperature; the phase separated
ume
hangeissmallerat145Kalargerresistivity
hange
state
onsists of SC metalli
and insulating
rystallites.
due to the proximity to the per
olation threshold
om-
Ametastablemetalli

rystalliteswit
hesfromthemetal-
pensates for the smaller volume
hange and shifts the
ρ1/ρ0 li
to the insulating state probabilisti
ally depending on
maximum of to that temperature.
τ the
losenessofthetemperatureofmetastabilityandthe
As
an be seen from Figure 6 the parameter of the
size of the
rystallite. At low temperature, below 115K
resistivitytimedependen
e(cid:28)thasapeakaround147.5K
τ or so, the system is insulating, all the
rystallites hav-
whilethereisnosu
hpeakinthe ofthevolumefra
tion
ing swit
hed over to the insulating state. While heating
time dependen
e. The rather (cid:29)at temperature depen-
τ the
rystallites remain in the stable insulating state till
den
eofthe ofthevolumefra
tionisprobablyanindi-
the MI transition temperature is rea
hed and then they

ation that the distribution of energy barrierslook more
swit
h over to the metalli
state.
or less the same at all temperatures. We note that the
τ
di(cid:27)eren
e in behavior of the 's from resistivity and vol-
umefra
tionismostpronoun
ed
losetothe per
olation
τ
A
knowledgments
threshold, while away from the threshold both 's be-
havein avery similarmanner. Inaresistivityrelaxation
measurement,
lose to the per
olation threshold, the re- DKthankstheUniversityGrantsCommissionofIndia
sistivity will
hange signi(cid:28)
antly even when the metalli
for(cid:28)nan
ial supportduringthis work. JAAand MJM-L
volume
hangesverylittle. Thus,onewill(cid:28)ndthat,near a
knowledgetheSpanishMinistryofEdu
ationforfund-
theper
olationthreshold,theresistivitywillrelaxovera ing the Proje
t MAT2007-60536
∗
Ele
troni
address: kprajiitk.a
.in densedMatterPhysi
s(CambridgeUniversityPress1998)
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