Table Of ContentUnusual size effects on thermoelectricity in a strongly correlated oxide
J. Ravichandran,1,2,∗ A. K. Yadav,2,3 W. Siemons,4 M. A. McGuire,4
V. Wu,3 A. Vailionis,5 A. Majumdar,6 and R. Ramesh2,3,7,8
1Applied Science and Technology Graduate Group,
University of California, Berkeley, CA 94720, USA
2Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
3Department of Materials Science and Engineering,
University of California, Berkeley, CA 94720, USA
4Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA
5Geballe Laboratory for Advanced Materials, Stanford University, Stanford, California 94305, USA
6ARPA-E, US Department of Energy, 1000 Independence Avenue, Washington, DC 20585, USA
7Department of Physics, University of California, Berkeley, CA 94720, USA
2 8SETP, US Department of Energy, 1000 Independence Avenue, Washington, DC 20585, USA
1 (Dated: January 31, 2012)
0
2 We investigated size effects on thermoelectricity in thin films of a strongly correlated layered
cobaltate. At room temperature, the thermopower is independent of thickness down to 6 nm.
n
This unusual behavior is inconsistent with the Fuchs-Sondheimer theory, which is used to describe
a
conventionalmetalsandsemiconductors,andisattributedtothestrongelectroncorrelationsinthis
J
material. Although the resistivity increases, as expected, below a critical thickness of ∼ 30 nm.
0
The temperature dependent thermopower is similar for different thicknesses but resistivity shows
3
systematic changes with thickness. Our experiments highlight the differences in thermoelectric
behavior of strongly correlated and uncorrelated systems when subjected to finite size effects. We
]
l usetheatomiclimitHubbardmodelatthehightemperaturelimittoexplainourobservations. These
e
findings provide new insights on decoupling electrical conductivity and thermopower in correlated
-
r systems.
t
s
. PACSnumbers: 71.27.+a,79.10.-n,73.50.Lw,73.50.-h
t
a
m
I. INTRODUCTION films are the commonly used to study two dimensional
-
d transport behavior. There are several reports on thick-
n nessdependenttransportmeasurementsonthinfilmma-
o Among various energy conversion methods, thermo- terials showing band-like transport,5–11 but few studies
c electricity deals with direct inter-conversion of thermal focus on size effects on strongly correlated materials.12
[ and electrical energy. The efficiency of a thermoelectric
Several of these investigations have centered around the
1 heat engine is related to a material dependent figure of effect of transverse confinement on thermoelectric trans-
v merit,Z,givenbyS2σ/κ,whereS isthethermopoweror port. Recent investigations have shown large thermo-
4 Seebeck coefficient, σ and κ are the electrical and ther- electric responses in complex oxides.13,15,16 Particularly,
7 mal conductivities (lattice and electronic), respectively.
strongly correlated oxides such as cobaltates show en-
2
In conventional thermoelectric materials, electrical con-
6 hanced thermopower, which cannot be explained by
ductivity and thermopower are governed by the density
. band-like transport. The transport behavior in these
1 of states, chemical potential and the scattering mecha-
cobaltates has been explained by the Hubbard model,
0 nism. Due to this coupling between thermopower and with the incorporation of spin degeneracy.14,17 Size
2
electricalconductivity,achievinghighZ hasbeenachal-
1 dependent thermoelectric measurements on a strongly
lenging task. Hicks and Dresselhaus1,2 proposed quan-
: correlated cobaltate poses several interesting questions
v tum confinement as a means to enhance the thermo-
about the role of quantum confinement in thermoelec-
Xi electric power factor (S2σ) in nanostructured materials. trictransport,theeffectofthicknessonthemobilityand
Nanostructuring3,4showednosignificantenhancementin
r spin degeneracy. In this article, we report unusual size
a power factor, as the enhancement in thermopower was effectsonthermoelectricityinastronglycorrelatedther-
offsetbythedecreaseinelectricalconductivity(bothmo-
moelectric oxide, Bi Sr Co O (abbreviated as BSCO)
2 2 2 y
bilityandcarrierdensity). Nevertheless,nanostructuring
andusetheHubbardmodeltoexplainthephysicsbehind
isaneffectivemeanstoreducethelatticepartofthether-
theobservations. Weperformedthermoelectrictransport
mal conductivity without significantly affecting electri-
measurements in thin films of BSCO both as a function
cal transport. Investigations exploring quantum confine-
of thickness and temperature to elucidate size effects on
menteffectshaveprimarilycenteredaroundconventional
this system and discuss possible directions for correlated
semiconductors, which show band-like transport.
thermoelectrics.
Reduced dimensions in materials can have profound
influence on transport properties due to effects such as
quantizationandchangesinscatteringmechanism. Thin
2
(b) 5 0
(a)
N
o
105 YSZ (002) 1 0 0 4 0 m
ensity (cps)111000234 BSC(O00 (40)0(20)06)(008)(00 10) (0Intensity (cps)0 1F2(W0)0H M1 4=)0.0O(60m°0e 1g6a) (( 0)0( 01008.0 )22°0) (c) mpower (V/K) 57 05 TR he es rism tiov pitoy w e r 23 00 inal resistivity
nt o (
I101 rm 2 5 1 0 mW
e
100 Th cm
10 20 30 40 50 60 )
0 0
2θ (°)
0 5 0 1 0 0 1 5 0 2 0 0
T h ic k n e s s ( n m )
FIG. 2. The thickness dependent thermopower and resis-
FIG. 1. (a) The out-of-plane x-ray diffraction pattern for tivity of thin films of BSCO measured at room temperature.
BSCOfilmonYSZisshown. Theinsetcomparestherocking The thicknesses of the films range from 6 – 170 nm. The re-
curve for the substrate and the film. (b) Pole figure scan sistivityofthefilmswascalculatedbydividingthemeasured
for (116) plane of BSCO. The four peaks at φ = 45◦, 135◦, sheetresistancebythemeasuredthickness. Theactualresis-
225◦ and 315◦ correspond to the substrate YSZ (111). The tivity can be estimated only after considering the thickness
corresponding 2θ value for the scan is 29.75◦. (c) Pole figure of dead layer present in these films, which we show in Fig. 3,
scan for (11 14) plane of BSCO. The four peaks at φ = 0◦, but the overall trend remains the same.
90◦, 180◦ and 270◦ correspond to the substrate YSZ (2 2 0).
The corresponding 2θ value for the scan is 50.46◦.
indexed to the (00l) planes of BSCO and no secondary
phase or other orientations were observed. The inset of
II. EXPERIMENTAL METHODS
Fig.1(a)showstherockingcurveforthefilmascompared
to the substrate. The films showed rocking curves with
Thin films of BSCO (3 – 170 nm thick) were grown on full-width-at-half-maximumof∼0.05–0.2◦ (comparedto
yttria stablized zirconia (YSZ) substrates using pulsed substrate’s 0.02◦). In order to establish the in-plane epi-
laser deposition from a stoichiometric ceramic target of taxial relationship between the film and substrate, we
BSCO. The growth was carried out at 700◦C, with an performed phi-pole scans about (116) and (11 14) peaks
oxygen partial pressure of 500 mTorr. All films were ofBSCO.Thepolefiguresforthetwocasesareshownin
grown using a 248 nm KrF excimer laser with a fluence Fig.1(b)andFig.1(c)respectively. Itisevidentfromthe
of 2 J/cm2 at a repetition rate of 1 Hz. Films were char- figures that the BSCO film doesnt have any preferential
acterized by X-ray diffraction (XRD) for phase purity in-plane epitaxial relationship with the substrate. Thus,
and crystallinity, X-ray reflectivity (XRR) for thickness, we have excellent out-of-plane texturing but no in-plane
atomic force microscopy (AFM) for surface roughness relationship with the substrate, leading to a wire tex-
and Rutherford backscattering (RBS) for chemical com- ture scenario. In Fig. 1(b), four peaks at φ = 45◦, 135◦,
position. All transport measurements were carried out 225◦ and 315◦ correspond to the substrate YSZ (111).
in the van der Pauw geometry. Triangular ohmic metal The corresponding 2θ value for the scan is 29.75◦. In
contacts of side 1 mm (15 nm Ti/100 nm Pd) were de- Fig. 1(c), four peaks at φ = 0◦, 90◦, 180◦ and 270◦ cor-
posited on the corners of the films using electron beam respond to the substrate YSZ (220). The corresponding
evaporation. Hall measurements at room temperature 2θ value for the scan is 50.46◦.
werecarriedoutinairusingahome-builtapparatuswith
TheprimaryresultofthisworkissummarizedinFig.2
a1Teslaelectromagnet. Lowtemperatureresistivityand
where thickness dependent thermopower and resistivity
thermopower measurements were performed in a Quan-
for BSCO films at room temperature are shown. The
tum Design PPMS.
nominalresistivityremainedconstanttill∼30nmandfor
lowerthicknessestheresistivityincreasedwithdecreasing
thickness. Surprisingly, the thermopower remained con-
III. RESULTS AND DISCUSSION stant(∼100–110µV/K)overthestudiedthicknessrange.
In comparison, thermopower of Se doped Bi Te de-
2 3
The XRD measurements indicated that the films were creasesby65%fromthebulkvaluewhenthethicknessis
single phase and oriented with c-axis (axis perpendicu- decreased to 50 nm.7 Unlike BSCO, in a typical metal5,6
lar to the layers) along the out-of-plane direction. The orasemiconductor7,8, decreasingthicknessresultsinde-
out-of-plane x-ray diffraction pattern for the BSCO film crease in thermopower and increase in resistivity, due
grownonYSZisshowninFig.1(a). Allthepeakscanbe to surface scattering. In those systems, the observed
3
1 .5
3
1 0 0
Sheet carrier density16-2(x10 cm)01 ..50 L in e a r fit 12 ()Mobility cmVs2-1-1 mopower (V/K) 468 000 mower (V/K) 44444 33445 .....05050 TTTT hhhh iiiicccc kkkk nnnn eeeeH ssss ssss |====| 1811D1865 5 Tnnn nmmm m
m p
0 .0 0 er 2 0 ermo 4 2 .5 C a lc u la te d S p in e n tro p y
1 5 h Th 4 2 .0 c o n trib u tio n to th e rm o p o w e r
T
0 2 4 6 8
1 0 0 R 0 H (T )
/K) 8 0 S u r fa c e s c a tte r in g fits 1 2 esis 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0
mr (V 6 0 TR he es rims tiov pitoy w e r U = 0 .2 9 tivity T e m p e r a t u r e ( K )
we b ( m FIG. 4. Temperature dependent thermopower for films of
o T h e r m o p o w e r U = 0 .1 6
p 4 0 b W thickness 115, 88, 16and 15 nm. The filmsshowed very sim-
o c
m ilar temperature dependence over the measured temperature
m
er 2 0 3 ) range. Theinsetshowsthemagneticfielddependenceofther-
h mopowerfora88nmfilmat20Kwiththefieldappliedalong
T
0 0 the temperature gradient. The calculated spin entropic con-
0 5 0 1 0 0 1 5 0 tribution to thermopower is shown as a red line.
T h ic k n e s s ( n m )
ularity parameter and U is the energy dependent scat-
FIG. 3. The sheet carrier density and Hall mobility as a b
teringterm. IftheFuchs-Sondheimermodelisapplicable
function of thickness at room temperature. (Top panel) The
sheet carrier density data was an excellent fit for a straight to our system, we expect a decrease in thermopower as
line with a small offset in thickness ∼ 4 nm. (Bottom panel) thickness decreases, contradictory to the observed con-
The thermopower and the actual resistivity as a function of stant thermopower. We use in-depth transport measure-
thickness,withthesurfacescatteringsfitsclearlyshowingthe ments, to eliminate different scenarios under which this
deviation for the thermopower data. The surface scattering conventional theory can be applicable to our case and to
modelusedthemeanfreepathestimatedfromtheresistivity establish the role of strong correlations in explaining our
data and typical values for the energy scattering dependent
observations.
scattering term as 0.1 and 0.2.7 The actual resistivity of the
First, it is essential to understand the contribution of
films is calculated accounting for the dead layer.
carrier concentration and mobility in increasing resistiv-
ity with decreasing thickness. Hall measurements were
used to measure the sheet carrier density and mobility
thicknessdependentthermoelectricpropertiescanbeex-
plainedbyFuchs-Sondheimertheory18 quantitativelyfor (see Fig. 3). The sheet carrier density scaled linearly
metals5,6 andqualitativelyforsemiconductors.7–9 Fuchs- with thickness and has a small offset of ∼ 4 nm in the
thickness axis. Thus, there is no thickness dependent
Sondheimer theory uses the energy dependent surface
change in the volume carrier density but an insulating
scattering as an additional scattering mechanism which
dead layer of thickness ∼ 4 nm is present. We confirmed
becomes dominant when the thickness of the films are
this by growing a film of ∼ 3 nm thickness and found it
comparable to the bulk mean free path of the electrons.
to be insulating. The measured Hall mobility showed a
The thickness dependent thermopower and resistivity as
sharp decrease below 30 nm, as shown in Fig. 3. Hence,
predicted by this theory are given below.
we can conclude that the increase in resistivity shown in
(cid:18) 3l (cid:19) Fig. 2 is caused by the presence of a dead layer and the
ρ =ρ 1+ b(1−p) (1)
f b 8t mobility reduction caused by surface scattering below ∼
30 nm. We account for the presence of dead layer and
plot the revised resistivity in Fig. 3, which still shows
(cid:18) (cid:19)
3l U very similar thickness dependence as depicted in Fig. 2
S =S 1− b(1−p) b (2)
f b 8t 1+U and an excellent fit for the surface scattering dominated
b
resistivityshowninEqn.1. Ontheotherhand,thecalcu-
where ρ is film resistivity, ρ is bulk resistivity, S is lated thermopower values (using Eqn.2) with the mean
f b f
film thermopower, S is bulk thermopower, l is bulk free path values obtained from the resistivity fit and a
b b
electron mean free path, t is the thickness, p is the spec- typicalenergydependencescatteringtermof0.1and0.2,
4
show clear deviations from the experimentally measured
values below 50 and 30 nm respectively. 4 6 n m
1 0
The presence of a competing mechanism such as 8 .5 n m
1 2 n m
quantum confinement,1which compensates for the ther-
) 2 3 n m
mopowerdecreaseduetosurfacescattering, canvalidate m 3 3 1 n m
theapplicabilityofFuchs-Sondheimertheory. Thelackof c1 0 4 3 n m
changes in the bulk carrier concentration is inconsistent W 1 0 2 n m
with the presence of quantum confinement but is not a (m
definitiveprooftoruleoutthisscenariocompletely. Typ- ity1 0 2
ically, the surface scattering is not expected to show a iv
t
strongtemperaturedependenceasitisaboundarydom- is
inated mechanism. On the other hand, mechanisms like s
e1 0 1
quantumconfinementareexpectedtoshowastrongtem-
R
perature dependence, as the effect is stronger at lower
temperatures. Thus, if the temperature dependent ther-
mopower does not show any significant deviation from 0 1 0 0 2 0 0 3 0 0
the bulk at the lower thicknesses, we can conclude that T e m p e r a t u r e ( K )
Fuchs-Sondheimer theory is not applicable to our case
andthermopowerisinsensitivetosurfacescattering. The
FIG.5. Temperaturedependentresistivityforfilmsofdiffer-
temperature dependent thermopower as shown in Fig. 4 entthickness. Atthebulklimit,thefilmsshowthecharacter-
clearlydepictsaverysimilartemperaturedependencefor isticmetal–insulatortransitionwithatransitiontemperature
thicknesses of 115, 88, 16 and 15 nm films down to 80 K ∼100K.Asthethicknessisdecreasedthetransitiontemper-
(forthinnerfilmsthethermopowermeasurementbecame atureshiftstohighertemperatureandbelow23nm,thefilms
unreliable below this temperature due to high resistance remained insulating till 300 K.
ofthefilms). Thus, thetemperaturedependenceofther-
mopowerremainsbulk-likeeveninthinsamplesconfirm-
sary to uncover the exact origin of this shift.
ingthatthermopowerisrobustagainstsurfacescattering
AswehavealreadyestablishedthatBSCOisalsoacor-
and the non-applicability of Fuchs-Sondheimer theory in
related system similar to other cobaltates, it is essential
this system. It is important to note that the presence
toputthesefindingsinperspectivewithintheframework
ofwire-textureinthesefilms, suggeststhatgrainbound-
of the Hubbard model.17,24–26 The transport coefficients
ary scattering should also be considered for the electron
predictedforNa CoO 26 usingtheatomiclimitHubbard
scattering mechanisms. Typically, grain boundary scat- x 2
model in the high temperature limit are given as:
tering doesn’t show any thickness dependence,5,6 hence,
will not change the conclusions derived here.
The magnetic field dependence of thermopower for an k (cid:18)2(1−x)(cid:19)
S =− Blog (4)
88nmfilmat20KisshownintheinsetofFig.4. Theob- e x
servedfielddependentthermopowerisinexcellentagree-
mentwiththespinentropiccontributiontothermopower
(Eqn. 3). Vh2
ρ= (5)
8π2e2ηa2t2τβx(1−x)
Q(H,T)/Q(0,T)=ln[2cosh(u)]−utanh(u)/ln(2) (3)
where x is filling, k is the Boltzmann constant, e is the
B
where u = gµ H/2k T and g is the Land´e g-factor electronic charge, β is 1/k T, η is the lattice structure
B B B
(here, g=2). This observation is consistent with the ex- dependent constant, a is the lattice constant, t is the
periments on Na CoO (g=2.2),14 confirming the role bandwidth, τ is the relaxation time, V is the unit cell
x 2
of strong correlation and spin entropy in thermoelectric volume and h is Planck’s constant.
properties of BSCO. ThethermopowerrelationshowninEqn.4doesnotde-
Finally, we studied the size effects on resistivity at pend on the relaxation time and hence is consistent with
low temperatures by performing temperature depen- our conclusion that thermopower is robust to changes in
dent resistivity measurements (shown in Fig. 5) on films thescatteringmechanism. Besides,theresistivityclearly
with different thicknesses. Single crystals19,20 and thin shows an inverse scaling with the scattering time, hence
films21,22 of BSCO have shown a metal-insulator transi- consistent with our observations. Thus, it is evident
tion with transition temperatures ∼80–140 K. Interest- that the Fuchs-Sondheimer theory doesn’t account for
ingly, the transition temperature shifted to higher tem- the thermopower and resistivity measurements, but the
peratures as we decreased the thickness. This shift in simple Hubbard based clearly explains the thickness and
transition temperature needs further investigation for a temperature dependence of thermoelectric properties in
clear understanding. Moreover, due to the presence of this strongly correlated system. It is important to note
unconventionalHalleffect23 inthecobaltatesatlowtem- that Eqn.4 and Eqn. 5 change qualitatively, if the lim-
peratures, extensive Hall effect investigations are neces- its on the energy scales such as thermal energy(k T)
B
5
and bandwidth (t) are different from the assumed limit conventional thermoelectric materials, the insensitivity
here (t (cid:28) k T). These variations still doesn’t change ofthermopowertoscatteringmechanisminstronglycor-
B
the overall conclusion that thermopower is independent relatedsystemssimplifiesthedecouplingofthermopower
of scattering time. Our experiment elegantly establishes andelectricalconductivity. Hence,thenextsteptowards
that the thermopower, at the high temperature limit, is complete decoupling of thermopower and electrical con-
independent of scattering parameter. It is important to ductivity in a correlated system is only dependent on
commentontherelevanceoftheEqn.4,inestimatingthe understanding the limits of filling dependence of ther-
valencestateofCo. Usingtheaverageroomtemperature moelectric properties. Since nanostructuring is a proven
thermopowerof110µV/K,weestimatethextobe0.36. routetodecreasethelatticepartofthermalconductivity
Thus the estimated average cobalt valence in this com- without affecting electrical properties, designing nanos-
poundis3.36,whichisveryclosetothereportedvalueof tructured correlated materials can lead to decoupling
3.3.27 Further, the resistivity in the metallic regime can of all the three thermoelectric parameters and hence, a
be explained using Eqn. 5 but there is no clear insight pathway to high thermoelectric efficiency.
on how the metal insulator transition can be understood
using the Hubbard model.
ACKNOWLEDGMENTS
The work was supported by the Division of Materi-
IV. SUMMARY
als Sciences and Engineering, Office of Basic Energy Sci-
ences, U.SDepartmentofEnergy. JRacknowledgessup-
In summary, we have studied the size effects on ther- port from the Link foundation. The authors gratefully
moelectricity in thin films of a strongly correlated cobal- acknowledge the assistance of Dr. K. M. Yu with the
tate system. The thermopower is insensitive to surface RBS measurements, target preparation by J. Wu and
scatteringunlikeresistivity,whichincreaseswithdecreas- useful discussions with Dr. S. Mukerjee, Dr. Jay Sau,
ing thickness below ∼ 30 nm. These observations can be Dr. B. Kavaipatti, Dr. M. Trassin, Dr. C.-W Liang and
explained by the atomic limit Hubbard model. Unlike Dr. Y.-H Chu.
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