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Electronic and ionic conductivities and point defects in ytterbium sesquioxide at high temperature
00~3697/82/101003-lOSO3.OCMO PergamonPms Ltd.
I. Phys. Chem Solids Vol.43,No.IO.pp.IlW-1012,I982 PrinkdinGreatBritain.
ELECTRONIC AND IONIC CONDUCTIVITIES AND POINT DEFECTS IN YTTERBIUM SESQUIOXIDE AT HIGH TEMPERATURE JEAN-LOUISCARPENTIER, AND& LEBRUN,FRANCISPERDUand PIERRETEL,LIER Laboratoiredes MatCriauxsemi-conducteurs,U.E.R. Sciences Exactes et Naturelles,33, rue Saint-Leu80039
Amiens Cedex, France (Receioed12 June 1981;acceptedin revisedform
16 March 1982)
Ahs&aet-From the study of complex impedancediagramsapplied to a symmetriccell Pt-Yb,O,-Pt, the authors have shown the mixed characterof electricalconductionwithin the ytterbiumsesquioxide.The measurementswere performedat thermodynamicequilibriumin the temperaturerangefrom 1423to 1623K and the partialpressureof oxygen range from lO-‘2 to 1 atm. The variationsof ionic and electronic conductivityas a function of Pal were interpretedin terms of point defects e’, h’, U;, and ybr in the generalcase of a Frenkeldisorder.The relative contributionsand the activationenergies of conductionof these differentdefects were determined. 1. INTRODUCTION
The previous studies [ l] to [S] concerning the rare earth
sesquioxides all conclude that these oxides are both ionic and electronic mixed conductors. Two observations support this conclusion. Fist the d.c. conductivity is notably different from the a.c. conductivity and second, a continued electrolysis above 873 K reduces slightly the conductivity. However, the results obtained until now vary on the relative importance of the ionic contribution characterized by the ionic transport number ti* Noddack and Walch[l] and[2] have found from experiments of electrolysis in vacuum (PO2= lo-* atm) at 873 K and from developing current-voltage curves in air and in vacuum in the temperature range 873-1573 K that the ionic transport number is less than 0.05. Tare and Schmalzried[3] have shown using the e.m.f. method that rare earth sesquioxides are essentially complete ionic conductors around 1073K (ti > 0.9) for Pot between 10m6and 10atm. Rao and CoU[4] primarily have studied electronic conduction using the a.c. method and have found a p-type electronic conduction (for PO2= 150mmHg). They have determined a Pai dependence of conductivity which they have interpreted in terms of cationic vacancies. Moreover, they have evaluated the ionic contribution as approximately 10% in the temperature range 673-l 173K. Wilbert and Coll[5] established that the oxides of general formula Ln203 are practically stoichiometric in the temperature range 1073-1473K and PO2ranging from lo-” to 1 atm. In addition, they have considered them as amphoteric and mixed semiconductors characterized by a p+n transition in the decreasing range of oxygen pressure with the ionic and electronic conductivities of the same order. In this paper, we try to specify the relative contribution of ionic and electronic conduction within a sesquioxide of the lanthanides family, using electrical impedance measurements in a large range of frequencies.
We have chosen the ytterbium sesquioxide under quasistoichiometric conditions. The complex impedance method widely used in electrochemistry of solutions[l4] allow to separate the different kinetic steps governing the current transport. The use of semi-blocking electrodes to only the ions permit us to adapt this method to case of oxide with mixed conduction. Thus, from the impedance of the sample, we can separate the diBerent kinetic steps and accurately isolate the electronic and ionic ohmic resistances from electrode phenomena. 2. EXPERIMENTAL The investigated cell can be schematically represented by a symmetrical chain Pt-Yb203-Pt. The polycristalline samples of Ytterbium sesquioxide were prepared from a 99.99% pure powder (Knock-Light L.T.D). The electronic microsonde analysis did not detect any impurities beyond the detection threshold (50 p.p.m) before or after experiments. Samples consist of small cylinders (0= 3.8 mm, 1 = 14 mm) pressed under 2 000 bars for 2 hr in an isostatic press. Two platinum wires were introduced at the samples extremities before pressing and served to assure good electrical contacts. The impedance measurements were carried out in the frequency range 10-10s Hz with a classical Sauty bridge (with 1% accuracy) and with a Hewlett-Packard 4342 A Q-meter in the frequency range Id-lo’ Hz (with 10% accuracy). High stable temperatures were achieved in a carborundum furnace by means of proportional regulator in the temperature ranging from 1423 to 1623K with 0.1% accuracy. The gas mixtures used to fix the oxygen pressure around the sample were obtained by a generator of oxygenated atmospheres fabricated in the laboratory[6]. This apparatus allowed the continuous production of PO2 ranging from 10-l’ to 1 atm. The partial pressures of oxygen were measured while the sample was in the thermodynamic equilibrium at high temperature using a pure rutile sensor, sensitive to the Po,[7]. Thus, using @is experimental device, we have been
1004
JEAN-LOUISCARPENTIER et al.
able to study the electrical impedance of a ytterbium sesquioxide sample under isothermal conditions at the temperatures 1423, 1473, 1523, 1573 and 1623K over the frequency interval 10-10’ Hz and the PO2interval 10-‘21 atm. The most reductive Pe was deliberately limited to lo-l2 atm in order to avoid the formation of platinum hydrides at the junctions at the highest temperatures. 3. DETERMINATIONOF TOTAL IONIC AND ELECTRONICCONDUCTMTIES
The measured impedances Z(o) = Z’(o) + jZ”(o) were represented in the complex plane for each Ps and temperature. The various diagrams obtained resembled those plotted in Figs. 1 and 2. The analysis of the diagrams in relation to the frequency f = (0/21~) of the sinusoidal voltage (< 10mV) applied to the terminals of the measuring cell can be interpreted from an equivalent electrical circuit which takes into account the mixed character of conduction within the sample and the metallic nature of the junctions. Such a circuit comprised of three parallel branches is represented in Fig. 3. Co= lim (- oZ”)-’ represents the capacitance of the measurYing cell. It is independent of the frequency of measurement, of the Ps and the temperature. R, = lim Z’ is the o+o ohmic electronic resistance of the sample, since the metallic nature of the platinum contacts allows the free
1
passage of the electronic charge carriers which implies a negligible transfer resistance. Finally, the complex impedance Zi(o) represents both the ionic conduction within the sample as well as the process of polarization at the electrodes. We have plotted, in Fig. 4, some characteristic impedance diagrams Zi(0) = ZXo)+ jZ”&). By comparing with the proposed model by Sluyters 181,in the case of an electrochemical cell, we can describe the electrical behaviour of Zi using a circuit yielding a partial blocking to the ionic charge carriers at one (Fig. 5a) or both junctions (Fig. Sb). The measuring chain is symmetrical in our case, thus the appropriate circuit is that in Fig. 5b in which R, = lim Z, is the ohmic O-LIO ionic resistance of the non disturbed homogeneous sample and Z, is the impedance linked to an inhomogeneous zone existing next to each oxide-platinum junction. The real and imaginary parts of complex impedance Z,(o) = ZXo) t jZ@) both approach zero as the frequency approaches inlinity. This electrical behaviour at the interfaces is characteristic of kinetic phenomena occurring in an ionic conduction. In fact, each junction introduces a partial blocking of ions next to each interface. Blocking creates on one hand a space charge and therefore the double layer capacitance C,, = lim (- oa-’ IUand on the other hand prevents the ions from freely discharging at the interfaces. This kinetic brake is
I
I
1
_Z"(lOW) T=
0.5 -
0
0.5
1
ls23k
1.5
2
Z’(lOW)
Fig. 1. Complex impedance diagrams of a Pt -Yb,Ol-Pt cell at T = 1523K. (a) Pti = 10-O.”atm; (b) Pti = 10-2.55 atm; (c) Ps = 10-7Aoatm; (d) Pb = 10-9~70 atm. Frequencies are reported in Hz.
0,21
P0.p
atm
60k
-\ I
2.5
0
Fig.
2. Complex
5
1.5
-.
I
lo
\
Z’(lO4R)
impedance diagrams of a Pt-YhO,-Pt cell at various T for Ph = 0.21atm. (a) T = 1423K; (b) T = 1523K; (c) T = 1573K; (d) T = 1623K. Frequencies are reported in Hz.
1005
Electronicand ionic conductivities
Fig. 3. Electrical equivalent circuit for Pt-YhzOS-Ptcell in the frequencyrange 1040’ Hz.
characterized at the oxide-platinum junction by the transfer resistance R, = lim (C2&2ZX’. Furthermore, the concentration gradient”zposed by the kinetics of the charge transfer implies the existence of a difbrsional impedance which is revealed here as the Warburg type Zd = CMJ-“*(~ - j)[9] with up = lim (- Zb”*). Thus, the spectral analysis
of the successive
impedance
diagrams 2, Zi and 2, allows not only the determination by extrapolation at zero frequency or inlXty of the six parameters of the electrical equivalent circuit but above ail to isolate the electronic conductance G, = R;’ on one hand, and the ionic conductance Gi = Rf’ on the other hand, independently of the phenomena occur& at the interfaces. Figures 6 and 7 show isothermal variations of G, and Gi respectively, for T between 1423 and 1623K as a function of Fs in the lo-‘* - 1 atm range. From these, we can deduce the isothermal variations of the ionic transport number ti = (GJG. t Gi)represented in Fig. 8. Thus the mixed conduction is confirmed. This is explained by the presence in the centered cubic structure of the T,,’ group of this sesquioxide, of unoccupied sites which allow the migration of ions within the lattice. Towards the higher PO2 electronic conduction predominates while ionic conduction predominates in the zone of intermediate Pti where ti reaches its maximum. The maximum value of ti increases as the temperature decreases and is obtained for a higher Ps as the temperature decreases. These results contirm quantitatively
T=1623k
Fig. 4. 4 partial complex impedance diagrams at T= 1523K. (a) Ps = lo-‘.“atm; (b) PO,= 10-2.5satm;
(c)
PO2= lo-‘.” atm; (d) PO,= 10-9.70 atm. Frequenciesare reportedin Hz.
.
-y+=~+
=
++z
a 6
b
b
E
Fig. 5. Electricalequivalentcircuitrepresentingthe.partialimpedanceZi (a) for one junctionYh20,Pt
two symmetricaljunctions Pt-YbZO,-Pt.
only;
(b)
for
JEAN-LOUISCARPENTIER et a/.
loo6
\
\
\ I
_
\
Y -6
-12
-10
-6
-;o$J PO2
Fig. 6. Plots of isothermal electronic conductance G, against peratures.
the qualitative results [3, N-121 which indicated that the ionic conduction predominates at lower temperatures and at intermediate oxygen pressures. It must be emphasized that the method used here for separating ionic and electronic conductances take electrode polarization into account contrary to the standard e.f.m. method which is only accurate for a predominant ionic conductivity.
4. DETFXUMINATION OF PARTIAL IONIC AND ELECTRONIC CONDUCTIONS
The curves (log G,, log Paz>and (log Gi. log Pe) exhibit a minimum which conlirms the amphoteric character of the electronic conductance G, and ionic conductance Gi. Each curve representing the electronic conductance presents a symmetry axis as the equation Pg,$ =constant. The last value corresponds to the PO2of the minimum isothermal conductivity and decreases as the temperature does. However, the curves representing ionic conduction do not exhibit this symmetry. Furthermore, towards high Paz, the slope of the curves (log G., log P%) approaches (3/16). This result coincides
the
(*ii%)
partial pressure of oxygen at various tem-
with that of RaO and Co11[4] and implies the existence of V’& vacancies. The amphoteric behaviour of the conduction and the presence of V”, defects at high Ps, can be interpreted from point defect theory in the disorder cases of either Frenkel or Schottky. A theoretical study indicates that electronic conductivity should vary at low oxygen pressure with a Pz” dependence for the Schottky disorder whereas the Frenkel disorder involves a p Z”” dependence in the same Ps range. The choice between these two models depends on the accuracy obtained for the slope-of-conductance curves in the n type range. With a slope equal to 0.195 0.01, the best agreement with experimental data is obtained in whole temperature and whole Ps range (for each isothermal curve-conductance, the value of chi squared is less than 3.96 and the correlation factor is greater than 0.99). Therefore, the Frenkel disorder is the more probable mechanism and it alone will be studied here. (a) Study of Fynkel disorder within Yb~0, We assume the complete ionisation of defects (very probably at the temperatures used) and we use the Krager and Vink notation[l3]. Thus, the equ&ria of the
Electronic and ionic conductivities
Fig. 7. Plots of isothermal ionic conductance Gi agaiust the partial pressure of oxygen at various temperatures.
I
-12
10
I
I
-8
-6
I
Fig. 8. Isothermal variations of the ionic transport number ti of Yb20, as a function of Pe at various temperatures.
formation of defects are the foIlowing:
(c) Experimental determination of partial electronic conductances G. passes through its minimum
Ge.min = 2qWKf”= 2q&VcN,)“2exp -3 (
) (14)
with the associated eq~ib~~
constants
for
K$ = [ Vc]2p6P&3f2
w
K2 = [ Ybr]n’Pg
(2)
Ki = np
(3)
KF = [ Yby][ V’b] = (K,K#“Kf3.
(4)
I/3
Pe,,=
(
2
1)
Further, the electroneutrality condition is expressed by
which corresponds to the apparent stoichiometric composition definedby n = p = K,‘“. With the help of the eqns (6)(8)(9)(10)and (14) for all Ps we can obtain
n + 3[V”h]= p + 3[Y&j.
- G&)‘+ G’,miJ”2. Ge= [tGe>~
(5)
(b) Ionic aud electronic conduction induced by the ~igrution of post defects
We assign JL as the mobility of electrons and holes which is assumedto be equal due to the symmetry of the curve (log G, log Ps) and fiUand pr as the mobility of vacancies V%, and interstitials yb? respectively. If q and g denote the protonic charge and geometric factor, the electronic and ionic conductances are given by
In the high P* range, the holes and vacancies of extrinsic origin are pr~o~~t (1) and (5) lead to pp %(9K,)*‘nP~6
Thus, G, = G,, since nr, co, GeSand GcV3 in this case are negligeable. In the low Ps range, the electrons and interstitialsof extrinsic origin are predominant(2) and (5) lead to
G, = gqlil(n+ P)
n, = (9K2)“8P;“6. Thus, G, c=G,, since pn co, Q, and Gc,3in this case are
with
negligible.From this, we deduce that Gel P “_&=&= G e.2 n,
II,, pe, c,and cr are the concentrationsof charge carriers of eXt&iC origin, and co and cF of intrinsic origin (native disorder). The conductances G. and Gi are thus the sum of three partial conductances G,.i and G,,
3C&L F==i Gr = ’ G&k F
G,,P r
Ge=
-1
(7)
with G,J = gqcLp. Go.2 = ima
Ge,3= 2gq& = gqpIb
rK (
k
‘i’p~=p~~~~.
(17)
1
Therefore, from the eqns (16), (17) and (6) and the experimental values Pg,$, Gl.mh and G, we can determine at all P% the three partial electronic conductances Gc.k,And, further, experimental results show that the products G,,P;‘==
G,r = 3gqpucv
(16)
= a,,(T)
(18)
= gq~(9K2)“’ = a&‘?
(19)
gqp(%)“*
are constant for a given tern~~~e at ali P+ The following table shows the values of G&mint PQ, a,i and 5,2 as a function of temperature.
(8) (9) - n.)’ + 4kil”2- be + ndl (10) 01)
‘J’(K) 1% Jk,, log Gr,min fog ad log %2
1423
1473
1523
1573
1623
;8.20 -6.09 -4.95 -8.03
-7.49 -5.73 -4.7f -7.52
-6.84 -5.39 -4.48 -7.04
-6.22 -5.07 -4.26 -6.60
-5.65 -4.78 -4.06 -6.18
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Electronicand ionic conductivities
We can determine the variation of the three partial electronic conductances Ge.kas a function of Ps. Figure 9 shows a typical example at T = 1473K. We can also calculate the different ratios of electronic conduction T& = (GJG,) of extrinsic origin (k = 1,2) and of intrinsic origin (k = 3) in the experimental range studied (Fig. 10). The slope of the curves in Fig. 6 is related to by the relationship (aG./&J = (3/16) the %k (~5~- 72,~). It becomes zero for 7,, = T~.~,i.e. at the apparent stoichiometry. The influence of the native electronic disorder characterized by 7c.3is thus at its maximum. It is weak at high temperature (8% at 1623K) but increases as T decreases (20% at 1423K).
of the diKerent electronic charge carriers (holes and electrons) are of the form:
(d) Activation energies of electronic conduction The variations of log a,, et log a,2 as a function of T-’ represented in Fig. 11 allow the determination of activation energies of conduction W,, and WQ of extrinsic defects associated with partial conductances
Therefore, it depends on T and on Ps and in the experimental range studied, it remains between 2.05 and 4.24 eV.
Ge.l and G.2. The observed linear variations allow the following:
ac2 = A,, exp a,.1 = A,, exp (- ( W,JRT)) and ( W,2/kTj where &.r and A,3 are constants independent of T and of Ps and W,, and W,, are activation energies which include both the enthalpy of creation and the enthalpy of migration of the charge carriers. On the other hand, eqn (14) shows that the slope of the curve log G,h as a function of T’ plotted in Fig. 11, is linked to the bandgap Ep Since this latter slope is constant, the mobility
p=bexp
(
-9.
)
We find W,,, = 2.05eV, W,, =4.24eV and E, + 2AH,,, = 6.02 eV. The apparent activation energy of the
total electronic conduction is thus given by aG
we =-kcatl/T)=Tf,,we,,+d.2we.,
(e) Experimental determination of partial ionic conductance As pr[ YIJY]= p.[ V’&,],the ionic conductance passes through a minimum Gi,mio= 6gq(~+~)“*K$*
observed at Ps = Pg.,. In Fig. 7, we notice this minimum does not correspond to the apparent stoichiometry defined by [ Yb y] = [ V$] = Kg*, since Pg., 3 P,,,.
Fig. 9. Isothermalvariationsof partial electronic conductances Cc* as a function of Ps at T = 1473K. Tbe total electronicconductance G. is tbe sum of tbe three partial conductancesGet (k 1.2.3). PCS Vd. 43, No. 10-E
(20)
1010
JEAN-LOUIS CAWENTIER et al.
0,4 _
0,2
-
-12
-10
-3
-6
-4
-2
0
lo@02(atm)
Fig. 10. Isothermal variations of rations T.,~= (G,dG,) as a function of PO2at the hvo temperatures 1423 and 1623K. 423
1'
!3
b
Tlkr\
-a- .
L'\ '.
A \
‘A<
\
‘( k-‘1 Fig. 11. Plots of a.1 (0). arJ (O), Gemi”(Cl) and ail (a), aiz temperature.
(+)
and G,,
0
vs the reciprocal of absolute
1011
Electronicand ionic conductivities I ‘ON+ (fr-‘1
1
I
I
I
Fig. 12.Isothermalvariationsof partial ionic conductsaces GLkas a function of P%at T = 1473K.
At stoichiometric composition, G, takes the value of Gi.1~ 3g&
+ /JI)KFI” = $$
Gi,min
(21)
with .,k.
(22)
clr
Similarly to paragraph (c), using eqns (11), (12), (13). (7) and (21) and the experimental values Gi,, G,_min,Ps., and Gi, we can determine a and the three partial ionic conductances GLk at ah PO,. Similarly, we observe that the products Gi.,p&‘1’6= gqcL$‘K,)“* = at,(T)
Figure 12 shows the variations of GLkas a function of P* at T = 1473K. Fiie 13 represents the variations of the different ratios of ionic conduction TLk= (GiJGi) for T = 1473and 1673K. The Frenkel disorder characterized by 713 is never negligeable in the experimental range studied. It even surpasses 50% near stroichiometry. (f) Activation energies of ionic conduction The apparent activation energy of the total ionic conduction W, = -2.3 k (a log GJa(l/T’)) depends both on T and on P* and varies between 1.85 and 3.11 eV. Nevertheless, we can, as in the case of electronic conduction, obtain from quasilinear variations of log ai1 and log uL2 as a function of (l/Z’), the activation energies of partial ionic conduction W,., and Wi.2given by:
GiT2Pg6 = gqpl(9K2)“* = uO( 7’) are constants at all Paz for a given temperature. The table below gives the values of Gimin, Q, ui.1 and fri2 as a function of temperature. T(K)
1423
logGmin -5.73
1473
-5.39 0.56 io_x=!n,, Xl -4.79 -7173 -7.35 iug 42
1523
1573
1623
-5.08 -4.78 -4.51 0.72 0.90 1.11 -4.58 -4.39 -4.20 -7.00 -6.68 -6.37
%.,=-2,3
k v=
1.85eV
air0 W.l.2 =-23&%?-Qlop313eV . 1 * aT0
.
Furthermore, the quasi-linear variations of log Gi,minvs (l/T’) (Fig. 11) show that the mobility of the defects Yh:’ et V”, respectively are of the form.
JEANLouts CARPENTER et
1012
----
a&
T=w=k
Fig. 13.Isothermalvariationsof ratios 7i.t= (GidGi) as a function ofP, at the two temperatures1423and 1623K.
wL;exp(-$g) and
REFERENCES
~.=&exp(-s)
where POIand PL”,are constants and AH,,,’ and AH,,,” are the migration enthalpies of interstitials and vacancies. Since KF = K$exp (-EdkT), we can also calculate the activation energy of intrinsic conduction W, = EF + AH,,,’+ AH,,,”= 5.60 eV, where IIF is the creation energy of a pair of Frenkel defects. CONCLUSION
Thus, by studying impedance diagrams, we have been able to precisely determine the electrical conductivity in the homogeneous phase of the material studied. The equivalent electrical circuit we have established allows the qualitative and quantitative differentiation of ionic and electronic conduction. These were shown to be of comparable importance in the case of I’!& and were interpreted by the model of point defects. This study could be extended to other oxides of rare earths. In fact, it appears that under certain conditions of PO2and temperature most of them exhibit electrical conduction whose mixed character makes a classical study difficult.
1. NoddackW. and WalchH., Z. Elektrochem.63,269 (1959). 2. Noddack W. and Walch H., Z. Phys. Chem. (Leipzig) 211, 194(1959). 3. Tare V. B. and ScbmalzriedH., 2. Phys. Chem. (Frankfurt am Main) 43,30 (1964). 4. Subba Rao G. V., RamdasS., MebrotraP. N. and Rao C. N. R., J. Solid St. Chem. 2,377 (1970). 5. WilbertY., DherbomezN. and BreuilH., Compt. Rend SC C 280,465(1975). 6. OebligJ. J., Jamet A. and DuquesnoyA., Compt. Rend. sir. C, 274, 1021(1972). 7. DuquesnoyA. and MarionF., Compt. Rend.256,2862(1%3). 8. SluytersJ. H., Rec. Tmu. Chim. 79, 1092(1960). 9. WarburgE., Ann. Phys.67,493 (1899);6, 125(1901). 10. Neuimin A. D. and Pal’guev S. F., Chem.Abstr. 59, 9417 (1%3). 11. SchmalzriedH., Z. Phys. Chem. (Frankfurt am Main) 38.87 (1%3). 12. Rapp R. A., U.S. At. Energy Comm. 33, 1440(1%7). 13. KrQex F. A. and Vink H. T., Solid State Physics (Editedby F. Seitz and D. Turnbull).AcademicPress, New York(1966). 14. Delabay P., New Instmmental Methods in Electrochemistry. Interscience,New York (1954).
I. Phys. Chem Solids Vol.43,No.IO.pp.IlW-1012,I982 PrinkdinGreatBritain.
ELECTRONIC AND IONIC CONDUCTIVITIES AND POINT DEFECTS IN YTTERBIUM SESQUIOXIDE AT HIGH TEMPERATURE JEAN-LOUISCARPENTIER, AND& LEBRUN,FRANCISPERDUand PIERRETEL,LIER Laboratoiredes MatCriauxsemi-conducteurs,U.E.R. Sciences Exactes et Naturelles,33, rue Saint-Leu80039
Amiens Cedex, France (Receioed12 June 1981;acceptedin revisedform
16 March 1982)
Ahs&aet-From the study of complex impedancediagramsapplied to a symmetriccell Pt-Yb,O,-Pt, the authors have shown the mixed characterof electricalconductionwithin the ytterbiumsesquioxide.The measurementswere performedat thermodynamicequilibriumin the temperaturerangefrom 1423to 1623K and the partialpressureof oxygen range from lO-‘2 to 1 atm. The variationsof ionic and electronic conductivityas a function of Pal were interpretedin terms of point defects e’, h’, U;, and ybr in the generalcase of a Frenkeldisorder.The relative contributionsand the activationenergies of conductionof these differentdefects were determined. 1. INTRODUCTION
The previous studies [ l] to [S] concerning the rare earth
sesquioxides all conclude that these oxides are both ionic and electronic mixed conductors. Two observations support this conclusion. Fist the d.c. conductivity is notably different from the a.c. conductivity and second, a continued electrolysis above 873 K reduces slightly the conductivity. However, the results obtained until now vary on the relative importance of the ionic contribution characterized by the ionic transport number ti* Noddack and Walch[l] and[2] have found from experiments of electrolysis in vacuum (PO2= lo-* atm) at 873 K and from developing current-voltage curves in air and in vacuum in the temperature range 873-1573 K that the ionic transport number is less than 0.05. Tare and Schmalzried[3] have shown using the e.m.f. method that rare earth sesquioxides are essentially complete ionic conductors around 1073K (ti > 0.9) for Pot between 10m6and 10atm. Rao and CoU[4] primarily have studied electronic conduction using the a.c. method and have found a p-type electronic conduction (for PO2= 150mmHg). They have determined a Pai dependence of conductivity which they have interpreted in terms of cationic vacancies. Moreover, they have evaluated the ionic contribution as approximately 10% in the temperature range 673-l 173K. Wilbert and Coll[5] established that the oxides of general formula Ln203 are practically stoichiometric in the temperature range 1073-1473K and PO2ranging from lo-” to 1 atm. In addition, they have considered them as amphoteric and mixed semiconductors characterized by a p+n transition in the decreasing range of oxygen pressure with the ionic and electronic conductivities of the same order. In this paper, we try to specify the relative contribution of ionic and electronic conduction within a sesquioxide of the lanthanides family, using electrical impedance measurements in a large range of frequencies.
We have chosen the ytterbium sesquioxide under quasistoichiometric conditions. The complex impedance method widely used in electrochemistry of solutions[l4] allow to separate the different kinetic steps governing the current transport. The use of semi-blocking electrodes to only the ions permit us to adapt this method to case of oxide with mixed conduction. Thus, from the impedance of the sample, we can separate the diBerent kinetic steps and accurately isolate the electronic and ionic ohmic resistances from electrode phenomena. 2. EXPERIMENTAL The investigated cell can be schematically represented by a symmetrical chain Pt-Yb203-Pt. The polycristalline samples of Ytterbium sesquioxide were prepared from a 99.99% pure powder (Knock-Light L.T.D). The electronic microsonde analysis did not detect any impurities beyond the detection threshold (50 p.p.m) before or after experiments. Samples consist of small cylinders (0= 3.8 mm, 1 = 14 mm) pressed under 2 000 bars for 2 hr in an isostatic press. Two platinum wires were introduced at the samples extremities before pressing and served to assure good electrical contacts. The impedance measurements were carried out in the frequency range 10-10s Hz with a classical Sauty bridge (with 1% accuracy) and with a Hewlett-Packard 4342 A Q-meter in the frequency range Id-lo’ Hz (with 10% accuracy). High stable temperatures were achieved in a carborundum furnace by means of proportional regulator in the temperature ranging from 1423 to 1623K with 0.1% accuracy. The gas mixtures used to fix the oxygen pressure around the sample were obtained by a generator of oxygenated atmospheres fabricated in the laboratory[6]. This apparatus allowed the continuous production of PO2 ranging from 10-l’ to 1 atm. The partial pressures of oxygen were measured while the sample was in the thermodynamic equilibrium at high temperature using a pure rutile sensor, sensitive to the Po,[7]. Thus, using @is experimental device, we have been
1004
JEAN-LOUISCARPENTIER et al.
able to study the electrical impedance of a ytterbium sesquioxide sample under isothermal conditions at the temperatures 1423, 1473, 1523, 1573 and 1623K over the frequency interval 10-10’ Hz and the PO2interval 10-‘21 atm. The most reductive Pe was deliberately limited to lo-l2 atm in order to avoid the formation of platinum hydrides at the junctions at the highest temperatures. 3. DETERMINATIONOF TOTAL IONIC AND ELECTRONICCONDUCTMTIES
The measured impedances Z(o) = Z’(o) + jZ”(o) were represented in the complex plane for each Ps and temperature. The various diagrams obtained resembled those plotted in Figs. 1 and 2. The analysis of the diagrams in relation to the frequency f = (0/21~) of the sinusoidal voltage (< 10mV) applied to the terminals of the measuring cell can be interpreted from an equivalent electrical circuit which takes into account the mixed character of conduction within the sample and the metallic nature of the junctions. Such a circuit comprised of three parallel branches is represented in Fig. 3. Co= lim (- oZ”)-’ represents the capacitance of the measurYing cell. It is independent of the frequency of measurement, of the Ps and the temperature. R, = lim Z’ is the o+o ohmic electronic resistance of the sample, since the metallic nature of the platinum contacts allows the free
1
passage of the electronic charge carriers which implies a negligible transfer resistance. Finally, the complex impedance Zi(o) represents both the ionic conduction within the sample as well as the process of polarization at the electrodes. We have plotted, in Fig. 4, some characteristic impedance diagrams Zi(0) = ZXo)+ jZ”&). By comparing with the proposed model by Sluyters 181,in the case of an electrochemical cell, we can describe the electrical behaviour of Zi using a circuit yielding a partial blocking to the ionic charge carriers at one (Fig. 5a) or both junctions (Fig. Sb). The measuring chain is symmetrical in our case, thus the appropriate circuit is that in Fig. 5b in which R, = lim Z, is the ohmic O-LIO ionic resistance of the non disturbed homogeneous sample and Z, is the impedance linked to an inhomogeneous zone existing next to each oxide-platinum junction. The real and imaginary parts of complex impedance Z,(o) = ZXo) t jZ@) both approach zero as the frequency approaches inlinity. This electrical behaviour at the interfaces is characteristic of kinetic phenomena occurring in an ionic conduction. In fact, each junction introduces a partial blocking of ions next to each interface. Blocking creates on one hand a space charge and therefore the double layer capacitance C,, = lim (- oa-’ IUand on the other hand prevents the ions from freely discharging at the interfaces. This kinetic brake is
I
I
1
_Z"(lOW) T=
0.5 -
0
0.5
1
ls23k
1.5
2
Z’(lOW)
Fig. 1. Complex impedance diagrams of a Pt -Yb,Ol-Pt cell at T = 1523K. (a) Pti = 10-O.”atm; (b) Pti = 10-2.55 atm; (c) Ps = 10-7Aoatm; (d) Pb = 10-9~70 atm. Frequencies are reported in Hz.
0,21
P0.p
atm
60k
-\ I
2.5
0
Fig.
2. Complex
5
1.5
-.
I
lo
\
Z’(lO4R)
impedance diagrams of a Pt-YhO,-Pt cell at various T for Ph = 0.21atm. (a) T = 1423K; (b) T = 1523K; (c) T = 1573K; (d) T = 1623K. Frequencies are reported in Hz.
1005
Electronicand ionic conductivities
Fig. 3. Electrical equivalent circuit for Pt-YhzOS-Ptcell in the frequencyrange 1040’ Hz.
characterized at the oxide-platinum junction by the transfer resistance R, = lim (C2&2ZX’. Furthermore, the concentration gradient”zposed by the kinetics of the charge transfer implies the existence of a difbrsional impedance which is revealed here as the Warburg type Zd = CMJ-“*(~ - j)[9] with up = lim (- Zb”*). Thus, the spectral analysis
of the successive
impedance
diagrams 2, Zi and 2, allows not only the determination by extrapolation at zero frequency or inlXty of the six parameters of the electrical equivalent circuit but above ail to isolate the electronic conductance G, = R;’ on one hand, and the ionic conductance Gi = Rf’ on the other hand, independently of the phenomena occur& at the interfaces. Figures 6 and 7 show isothermal variations of G, and Gi respectively, for T between 1423 and 1623K as a function of Fs in the lo-‘* - 1 atm range. From these, we can deduce the isothermal variations of the ionic transport number ti = (GJG. t Gi)represented in Fig. 8. Thus the mixed conduction is confirmed. This is explained by the presence in the centered cubic structure of the T,,’ group of this sesquioxide, of unoccupied sites which allow the migration of ions within the lattice. Towards the higher PO2 electronic conduction predominates while ionic conduction predominates in the zone of intermediate Pti where ti reaches its maximum. The maximum value of ti increases as the temperature decreases and is obtained for a higher Ps as the temperature decreases. These results contirm quantitatively
T=1623k
Fig. 4. 4 partial complex impedance diagrams at T= 1523K. (a) Ps = lo-‘.“atm; (b) PO,= 10-2.5satm;
(c)
PO2= lo-‘.” atm; (d) PO,= 10-9.70 atm. Frequenciesare reportedin Hz.
.
-y+=~+
=
++z
a 6
b
b
E
Fig. 5. Electricalequivalentcircuitrepresentingthe.partialimpedanceZi (a) for one junctionYh20,Pt
two symmetricaljunctions Pt-YbZO,-Pt.
only;
(b)
for
JEAN-LOUISCARPENTIER et a/.
loo6
\
\
\ I
_
\
Y -6
-12
-10
-6
-;o$J PO2
Fig. 6. Plots of isothermal electronic conductance G, against peratures.
the qualitative results [3, N-121 which indicated that the ionic conduction predominates at lower temperatures and at intermediate oxygen pressures. It must be emphasized that the method used here for separating ionic and electronic conductances take electrode polarization into account contrary to the standard e.f.m. method which is only accurate for a predominant ionic conductivity.
4. DETFXUMINATION OF PARTIAL IONIC AND ELECTRONIC CONDUCTIONS
The curves (log G,, log Paz>and (log Gi. log Pe) exhibit a minimum which conlirms the amphoteric character of the electronic conductance G, and ionic conductance Gi. Each curve representing the electronic conductance presents a symmetry axis as the equation Pg,$ =constant. The last value corresponds to the PO2of the minimum isothermal conductivity and decreases as the temperature does. However, the curves representing ionic conduction do not exhibit this symmetry. Furthermore, towards high Paz, the slope of the curves (log G., log P%) approaches (3/16). This result coincides
the
(*ii%)
partial pressure of oxygen at various tem-
with that of RaO and Co11[4] and implies the existence of V’& vacancies. The amphoteric behaviour of the conduction and the presence of V”, defects at high Ps, can be interpreted from point defect theory in the disorder cases of either Frenkel or Schottky. A theoretical study indicates that electronic conductivity should vary at low oxygen pressure with a Pz” dependence for the Schottky disorder whereas the Frenkel disorder involves a p Z”” dependence in the same Ps range. The choice between these two models depends on the accuracy obtained for the slope-of-conductance curves in the n type range. With a slope equal to 0.195 0.01, the best agreement with experimental data is obtained in whole temperature and whole Ps range (for each isothermal curve-conductance, the value of chi squared is less than 3.96 and the correlation factor is greater than 0.99). Therefore, the Frenkel disorder is the more probable mechanism and it alone will be studied here. (a) Study of Fynkel disorder within Yb~0, We assume the complete ionisation of defects (very probably at the temperatures used) and we use the Krager and Vink notation[l3]. Thus, the equ&ria of the
Electronic and ionic conductivities
Fig. 7. Plots of isothermal ionic conductance Gi agaiust the partial pressure of oxygen at various temperatures.
I
-12
10
I
I
-8
-6
I
Fig. 8. Isothermal variations of the ionic transport number ti of Yb20, as a function of Pe at various temperatures.
formation of defects are the foIlowing:
(c) Experimental determination of partial electronic conductances G. passes through its minimum
Ge.min = 2qWKf”= 2q&VcN,)“2exp -3 (
) (14)
with the associated eq~ib~~
constants
for
K$ = [ Vc]2p6P&3f2
w
K2 = [ Ybr]n’Pg
(2)
Ki = np
(3)
KF = [ Yby][ V’b] = (K,K#“Kf3.
(4)
I/3
Pe,,=
(
2
1)
Further, the electroneutrality condition is expressed by
which corresponds to the apparent stoichiometric composition definedby n = p = K,‘“. With the help of the eqns (6)(8)(9)(10)and (14) for all Ps we can obtain
n + 3[V”h]= p + 3[Y&j.
- G&)‘+ G’,miJ”2. Ge= [tGe>~
(5)
(b) Ionic aud electronic conduction induced by the ~igrution of post defects
We assign JL as the mobility of electrons and holes which is assumedto be equal due to the symmetry of the curve (log G, log Ps) and fiUand pr as the mobility of vacancies V%, and interstitials yb? respectively. If q and g denote the protonic charge and geometric factor, the electronic and ionic conductances are given by
In the high P* range, the holes and vacancies of extrinsic origin are pr~o~~t (1) and (5) lead to pp %(9K,)*‘nP~6
Thus, G, = G,, since nr, co, GeSand GcV3 in this case are negligeable. In the low Ps range, the electrons and interstitialsof extrinsic origin are predominant(2) and (5) lead to
G, = gqlil(n+ P)
n, = (9K2)“8P;“6. Thus, G, c=G,, since pn co, Q, and Gc,3in this case are
with
negligible.From this, we deduce that Gel P “_&=&= G e.2 n,
II,, pe, c,and cr are the concentrationsof charge carriers of eXt&iC origin, and co and cF of intrinsic origin (native disorder). The conductances G. and Gi are thus the sum of three partial conductances G,.i and G,,
3C&L F==i Gr = ’ G&k F
G,,P r
Ge=
-1
(7)
with G,J = gqcLp. Go.2 = ima
Ge,3= 2gq& = gqpIb
rK (
k
‘i’p~=p~~~~.
(17)
1
Therefore, from the eqns (16), (17) and (6) and the experimental values Pg,$, Gl.mh and G, we can determine at all P% the three partial electronic conductances Gc.k,And, further, experimental results show that the products G,,P;‘==
G,r = 3gqpucv
(16)
= a,,(T)
(18)
= gq~(9K2)“’ = a&‘?
(19)
gqp(%)“*
are constant for a given tern~~~e at ali P+ The following table shows the values of G&mint PQ, a,i and 5,2 as a function of temperature.
(8) (9) - n.)’ + 4kil”2- be + ndl (10) 01)
‘J’(K) 1% Jk,, log Gr,min fog ad log %2
1423
1473
1523
1573
1623
;8.20 -6.09 -4.95 -8.03
-7.49 -5.73 -4.7f -7.52
-6.84 -5.39 -4.48 -7.04
-6.22 -5.07 -4.26 -6.60
-5.65 -4.78 -4.06 -6.18
1009
Electronicand ionic conductivities
We can determine the variation of the three partial electronic conductances Ge.kas a function of Ps. Figure 9 shows a typical example at T = 1473K. We can also calculate the different ratios of electronic conduction T& = (GJG,) of extrinsic origin (k = 1,2) and of intrinsic origin (k = 3) in the experimental range studied (Fig. 10). The slope of the curves in Fig. 6 is related to by the relationship (aG./&J = (3/16) the %k (~5~- 72,~). It becomes zero for 7,, = T~.~,i.e. at the apparent stoichiometry. The influence of the native electronic disorder characterized by 7c.3is thus at its maximum. It is weak at high temperature (8% at 1623K) but increases as T decreases (20% at 1423K).
of the diKerent electronic charge carriers (holes and electrons) are of the form:
(d) Activation energies of electronic conduction The variations of log a,, et log a,2 as a function of T-’ represented in Fig. 11 allow the determination of activation energies of conduction W,, and WQ of extrinsic defects associated with partial conductances
Therefore, it depends on T and on Ps and in the experimental range studied, it remains between 2.05 and 4.24 eV.
Ge.l and G.2. The observed linear variations allow the following:
ac2 = A,, exp a,.1 = A,, exp (- ( W,JRT)) and ( W,2/kTj where &.r and A,3 are constants independent of T and of Ps and W,, and W,, are activation energies which include both the enthalpy of creation and the enthalpy of migration of the charge carriers. On the other hand, eqn (14) shows that the slope of the curve log G,h as a function of T’ plotted in Fig. 11, is linked to the bandgap Ep Since this latter slope is constant, the mobility
p=bexp
(
-9.
)
We find W,,, = 2.05eV, W,, =4.24eV and E, + 2AH,,, = 6.02 eV. The apparent activation energy of the
total electronic conduction is thus given by aG
we =-kcatl/T)=Tf,,we,,+d.2we.,
(e) Experimental determination of partial ionic conductance As pr[ YIJY]= p.[ V’&,],the ionic conductance passes through a minimum Gi,mio= 6gq(~+~)“*K$*
observed at Ps = Pg.,. In Fig. 7, we notice this minimum does not correspond to the apparent stoichiometry defined by [ Yb y] = [ V$] = Kg*, since Pg., 3 P,,,.
Fig. 9. Isothermalvariationsof partial electronic conductances Cc* as a function of Ps at T = 1473K. Tbe total electronicconductance G. is tbe sum of tbe three partial conductancesGet (k 1.2.3). PCS Vd. 43, No. 10-E
(20)
1010
JEAN-LOUIS CAWENTIER et al.
0,4 _
0,2
-
-12
-10
-3
-6
-4
-2
0
lo@02(atm)
Fig. 10. Isothermal variations of rations T.,~= (G,dG,) as a function of PO2at the hvo temperatures 1423 and 1623K. 423
1'
!3
b
Tlkr\
-a- .
L'\ '.
A \
‘A<
\
‘( k-‘1 Fig. 11. Plots of a.1 (0). arJ (O), Gemi”(Cl) and ail (a), aiz temperature.
(+)
and G,,
0
vs the reciprocal of absolute
1011
Electronicand ionic conductivities I ‘ON+ (fr-‘1
1
I
I
I
Fig. 12.Isothermalvariationsof partial ionic conductsaces GLkas a function of P%at T = 1473K.
At stoichiometric composition, G, takes the value of Gi.1~ 3g&
+ /JI)KFI” = $$
Gi,min
(21)
with .,k.
(22)
clr
Similarly to paragraph (c), using eqns (11), (12), (13). (7) and (21) and the experimental values Gi,, G,_min,Ps., and Gi, we can determine a and the three partial ionic conductances GLk at ah PO,. Similarly, we observe that the products Gi.,p&‘1’6= gqcL$‘K,)“* = at,(T)
Figure 12 shows the variations of GLkas a function of P* at T = 1473K. Fiie 13 represents the variations of the different ratios of ionic conduction TLk= (GiJGi) for T = 1473and 1673K. The Frenkel disorder characterized by 713 is never negligeable in the experimental range studied. It even surpasses 50% near stroichiometry. (f) Activation energies of ionic conduction The apparent activation energy of the total ionic conduction W, = -2.3 k (a log GJa(l/T’)) depends both on T and on P* and varies between 1.85 and 3.11 eV. Nevertheless, we can, as in the case of electronic conduction, obtain from quasilinear variations of log ai1 and log uL2 as a function of (l/Z’), the activation energies of partial ionic conduction W,., and Wi.2given by:
GiT2Pg6 = gqpl(9K2)“* = uO( 7’) are constants at all Paz for a given temperature. The table below gives the values of Gimin, Q, ui.1 and fri2 as a function of temperature. T(K)
1423
logGmin -5.73
1473
-5.39 0.56 io_x=!n,, Xl -4.79 -7173 -7.35 iug 42
1523
1573
1623
-5.08 -4.78 -4.51 0.72 0.90 1.11 -4.58 -4.39 -4.20 -7.00 -6.68 -6.37
%.,=-2,3
k v=
1.85eV
air0 W.l.2 =-23&%?-Qlop313eV . 1 * aT0
.
Furthermore, the quasi-linear variations of log Gi,minvs (l/T’) (Fig. 11) show that the mobility of the defects Yh:’ et V”, respectively are of the form.
JEANLouts CARPENTER et
1012
----
a&
T=w=k
Fig. 13.Isothermalvariationsof ratios 7i.t= (GidGi) as a function ofP, at the two temperatures1423and 1623K.
wL;exp(-$g) and
REFERENCES
~.=&exp(-s)
where POIand PL”,are constants and AH,,,’ and AH,,,” are the migration enthalpies of interstitials and vacancies. Since KF = K$exp (-EdkT), we can also calculate the activation energy of intrinsic conduction W, = EF + AH,,,’+ AH,,,”= 5.60 eV, where IIF is the creation energy of a pair of Frenkel defects. CONCLUSION
Thus, by studying impedance diagrams, we have been able to precisely determine the electrical conductivity in the homogeneous phase of the material studied. The equivalent electrical circuit we have established allows the qualitative and quantitative differentiation of ionic and electronic conduction. These were shown to be of comparable importance in the case of I’!& and were interpreted by the model of point defects. This study could be extended to other oxides of rare earths. In fact, it appears that under certain conditions of PO2and temperature most of them exhibit electrical conduction whose mixed character makes a classical study difficult.
1. NoddackW. and WalchH., Z. Elektrochem.63,269 (1959). 2. Noddack W. and Walch H., Z. Phys. Chem. (Leipzig) 211, 194(1959). 3. Tare V. B. and ScbmalzriedH., 2. Phys. Chem. (Frankfurt am Main) 43,30 (1964). 4. Subba Rao G. V., RamdasS., MebrotraP. N. and Rao C. N. R., J. Solid St. Chem. 2,377 (1970). 5. WilbertY., DherbomezN. and BreuilH., Compt. Rend SC C 280,465(1975). 6. OebligJ. J., Jamet A. and DuquesnoyA., Compt. Rend. sir. C, 274, 1021(1972). 7. DuquesnoyA. and MarionF., Compt. Rend.256,2862(1%3). 8. SluytersJ. H., Rec. Tmu. Chim. 79, 1092(1960). 9. WarburgE., Ann. Phys.67,493 (1899);6, 125(1901). 10. Neuimin A. D. and Pal’guev S. F., Chem.Abstr. 59, 9417 (1%3). 11. SchmalzriedH., Z. Phys. Chem. (Frankfurt am Main) 38.87 (1%3). 12. Rapp R. A., U.S. At. Energy Comm. 33, 1440(1%7). 13. KrQex F. A. and Vink H. T., Solid State Physics (Editedby F. Seitz and D. Turnbull).AcademicPress, New York(1966). 14. Delabay P., New Instmmental Methods in Electrochemistry. Interscience,New York (1954).