J. M. Honig Lincoln Laboratory* Massachusetts Institute of Technology Lexington
Electrical Properties of Metal Oxides Which Have "Hopping" Charge Carriers
For many years chemists have tended to regard oxides of metals in the solid phase as ionic insulators. While there are numerous examples to support this concept, many systems have recently been characterized which display interesting electrical conduction properties. Some oxides such as T i 0 or ReOs are metals over the entire temperature range of the investigation; some ferromagnetic metallic oxides are also known. Other oxides such as SnOl resemble Ge or Si in their semiconducting properties. Still others-V20s and VO, among theseshow interesting transitions from semiconductivity to metallic behavior over a narrow temperature range. While these properties are very interesting in their own right, this paper will deal with yet another class of metal oxides, namely those whose electrical pmperties arise from an activated transfer process in which charge carriers "hop" from site to site. Materials of this type are frequently referred to as electron transfer materials. The discussion has many points in common with a review paper by Johnston (I), though the outlook and aims are different. Thus, these two presentations are complementary. Here we shall mostly be concerned with the basic principles which make it possible to correlate the conduction characteristics with the stoichiometry or impurity content of these materials; concurrently, we also discuss certain thermal properties of this class of solids. For later use, a review of some elementary concepts pertaining to electrical phenomena will he helpful. The conductivity a (which is the inverse of the resistivity, p) may be specified as = l j p = neu
(1)
where n is the density of charge carriers, e the charge on an electron, and n the charge mobility (i.e., the drift velocity acquired on the average by a charge carrier subjected to unit electric field). Since p is related to the resistance, R, by a geometric factor involving sample dimensions, it is easy, in principle, to determine a.
I t should be noted that this quantity enters Ohm's laws as: J =
- eV$
( V T = 0)
(2)
where J is the current density, + the electrostatic potential, and E = - V+ the electric field vector.' Considerable interest attaches to the manner in which a depends on the temperature, T. We shall see that n is governed solely by the impurity content or by the sample stoichiometry, a t least in our approximation
* Operated with support from the U. S. Air Force. 76
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scheme. Consequently, any variation of o with T a t fixed sample composition is due to the dependence of u on T. Finally, we should note that eqn. (1) applies only to cases where one type of charge carrier participates in conduction. Frequently, it is convenient to regard the carriers of charge as electrons; materials for which this viewpoint is appropriate are said to he ntype. However, we shall show later that under certain conditions the entity responsible for conductivity may be considered positively charged; such quasiparticles are commonly termed holes, and materials characterized by this type of conduction are said to be P-type. We next turn to a consideration of the thermal conductivity, K. In the absence of a net current flow, the rate of transfer of energy past unit cross section is specified by Fourier's law, Js=
-*VT
(J=O)
(3)
where .Ig is the energy flux vector, and VT is the temperature gradient. The presence of the minus sign indicates that the energy flow occurs in the direction opposite to that along which T increases. Thus, - a is related to Fourier's law as a is related to Ohm's law. For later use it suffices to note that, subject to certain restrictions, K can be split into two additive parts K = xL K'. The first contribution, KL, is due to the participation of the lattice in transporting thermal energy from the hot to the cold side of the sample. The second contribution, x ' , is associated with the convective transport of thermal energy by a circulatory movement of charge carriers in the sample. We shall show later that in the types of oxides under discussion in this paper K' can be neglected compared to K L ; as a first approximation, the latter quantity is specified as
+
XL
=
'/dCaA/V)
(4)
where C , is the heat capacity measured a t constant volume, V is the volume, X is the mean free path of phonons, and v, is the velocity of sound. Another electrical property of interest is the so-called Seebeck coefficient or thermoelectric power.% The
' Equation (2) is an approximation; it does not sllow for the fact that the conductivity may vary with sample orientation and it does not allow for the possibility that the sample may he nonuniform in composition. However, for purposes of the present discussion the orientation effects will be ignored; a generalize tion of eqn. (2) is presented in eqn. (11). The use of the t,erminology "thermoelectric power" is most unfortunate, as t,he effect under discussion bas nothing whatsoever to do with r&es of energy transfer. I t is, therefore, better to follow the recent trend of using the alternate term; "Seebeck coefficient" honors -4. Seebeck, who first reported the effect in 1824.
POTENTIOMETER
Tb
From either viewpoint it is appropriate to define the Seeheck coefficient a asJ
k-1
Ar/e
To
Figure 1 . Schematic d i a g r a m showing an experimental set-up in d e termlnatim of Seebeck coefficientr.
effect may he readily visualized as follows: Consider a bar of material with sides A and B clamped between two heat reservoirs a t temperature T. > Tb,as illustrated in Figure 1. Charge carriers initially a t A are a t higher temperature than those on the other side. With their greater thermal energy they possess a greater velocity component along the negative x direction than charge carriers located at B possess along the positive x direction. Therefore, a net transfer of charge occurs away from the hot and toward the cold side of the sample. I n this process excess carriers pile up on the cold side and a deficit of these carriers occurs on the warm side; hence, an internal electric field builds up which impedes, and ultimately stops, any further net transfer of charge. Under these steady state conditions, a potentiometer, connected across the ends of the sample as shown in Figure 1, will register an emf. If the charge carriers are electrons, side B will be negatively charged relative to side A; if the carriers are holes, B will be positive relative to A . Hence, from the polarity of the cold relative to the warm side of the sample, one can deduce the sign of charge carrier. Experimentally, it is found that, nnder a large variety of operating conditions, the potentiometer readings are proportional to the temperature difference across the sample, AT = T , - Th. One might, therefore, he inclined to define the Seeheck coefficient a by the relation A+ = aAT, where A+ is the difference in potential established across the sample. However, this expression needs modification, as may be seen by two diierent types of reasoning: A potentiometer always measures differences in electrochemical potential, Ar, rather than simply differences in electrostatic potential. From the alternate viewpoint, there may he a change not only in electrostatic potential, hut also in concentration of charge carriers with position x along the sample. The latter effect may he taken into account by introducing the carrier chemical potential, where is a parameter independent of concentration, k is the Boltzmann constant, and n is the activity (concentration) of the charge carrier species under consideration. Since in the Seebeck phenomenon the effects due to fi and to always occur in conjunction, it is convenient to define an electrochemical potential by:
+
the plus or minus sign depending on whether the carriers are holes or electrons.
(7)
( J = 0)
aAT
Let us now interpret the quantity or. The chemical potential is equivalent to the partial molal Helmholtz free energy:
HEAT RESERVOIR
HEAT RESERVOIR
=
p=E-TS
(8)
where E and S are the partial molal internal energy and partial molal entropy respectively. As concerns their temperature dependence, we note that measurements of a are usually made under conditions which render AT small compared to the average sample temperature, T= (To+To). I n these circumstances one may regard E and S as temperature-independent; @ is likewise. Hence, if one substitutes eqn. (8) into eqn. (6) keeping E, 3,and constant, and if the results are used in eqn. (7), rewritten as
+
one obtains the final relation of interest,' a = -S/e
(10)
Using the concepts introduced in the preceding paragraphs, one may generalize Ohm's law, eqn. (2). From the methods of irreversible thermodynamics, one may show that the generalized version reads J = ~V(r/e)
( V T = 0)
(11)
Qualitative Arguments Pertaining to "Hopping" Processes in Metal Oxides
I n the last section we covered some basic concepts concerning resistivity [conductivity], Seeheck coefficient, and thermal conductivity. We now proceed to apply these ideas. Oxides that are not metallic, insulating or semiconducting (i.e., those mentioned in the introduction) are called, for lack of better terminology, primarily ionic. Any cation-anion interactions in these oxides produce sets of low-lying bonding energy levels which are completely occupied, and sets of very high-lying antibonding energy levels that remain unoccupied. These particular levels or energy bands do not contribute to electrical conduction, and hence may be omitted from further consideration. The origin of electrical conduction phenomena in these oxides must then be sought in processes which permit an electron to
Equation (7) is not strictly correct. To take into account the effects due to sample inh~mo~eneities, eqn. (7) as derived by the methods of irreversible thermodynamics reads where the V stands for the gradient vector (in one dimension, V, = a/bz). Equation (7a) reduces to eqn. (7)for homogeneous materiala. Also, one should recognize that in the potentiometric measurements discussed earlier, one always determines the Seeheck coefficientof the specimen relative to that of the lead wires connecting the specimen to the potentiometer. 'The result u = -S/e can he derived rigorously by the methods of irreversible thermodynamics. From the more exact treatment it also emerges that S should be regarded as the entropy carried by a charge carrier moving through the sample. Volume 43, Number 2, February 1966
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77
transfer directly from one cation to another. The intercationic spacing is usually large enough that the overlap between wave functions centered about two nearest neighbor cations is very small. I n this situation, i t is reasonable to regard charge carriers as essentially localized in the vicinity of the cationic lattice positions. Charge transfer then occurs by a "hopping" process; that is, after a certain time interval, a localized electron acquires the activation energy necessary to jump to the adjacent cationic site, where it becomes relocalized until it regains the necessary activation energy to execute another jump, and so on. It is a striking experimental feature of the primarily ionic oxides that the lattice must contain the same cation in at least two different valency states if the material is to be conducting; pure stoichiometric materials of this type are almost invariably insulators, even when the valence shell of each cation is only partially filled with electrons (3). This state of affairs can be understood in terms of a diagram shown in Figure 2 for a hypothetical oxide 1110. The unperturbed lattice in its ground state is depicted schematically along one dimension in diagram (a). I n transferring an electron from ion A to ion B, one arrives a t the situation depicted in diagram (b). As is seen, the charge transfer places an effective negative charge on ion B and a positive charge on ion A. Considerable energy must be expended for overcoming the repulsive force in placing an additional electron onto ion A: hence, the state depicted in Figure 2(b) is energetically very unfavorable relative to the state depicted in Figure 2(a). Under normal operating conditions, an ionic oxide with cations of the same valency will therefore remain an insulator.
Figure 2. Schematic diagram representing the crystal M O dong one dimension. lo) Ground state configvrdon for M O . ( b ) Excited mnfiguration for M O ; cotionr A and B ore in t r i d e n t and monovalsnt stater (4ond i d ) Oxide M O , with oltervoient cotionic states, re.pectively. stabilized in the host iatticeM0 as described in the text. Noteequivalence of configurotionr (Jand id).
It is, however, a relatively simple matter to obtain ionic oxides with the same cations in different valency states by two methods: either the material is made deliberately nonstoichiometric, which involves having one component of the binary combination present in excess over the other; or else selected impurity ions are introduced in the crystal so as to generate mixed valency states among the ions of the host lattice. These two possible modes of converting an insulator to a semiconducting material will be discussed below in further detail. Meanwhile, it is easy to see from Figures 2(c) and 2(d) that if cations of differing valency (e.g., di- and trivalent states) can somehow be stabilized in the lattice, the transfer of an electron from cation A 78
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Journal of Chemical Education
(of one valency) to cation B (of a second valency) produces an energetically equivalent configuration with the valencies of A and B interchanged. There is no energetic impediment to such an electron transfer mechanism; only the necessary activation energy for the jump is required. This activation energy may he sizable, particularly in cases where the lattice is easily polarized; for the lattice polarization will tend to follow the motion of the charge carriers. Nevertheless, the initial and final states depicted in Figures 2(c) and 2(d) are thermodynamically equivalent, in contrast to the situation depicted in Figures 2(a) and 2(b).6 Methods of Introducing Mixed Valency States into Primarily Ionic Oxides
We now discuss in some further detail the two methods of generating two cationic valency states within an oxide lattice. To deal with the deviations from stoichiometry, let us postulate the existence of an oxide AO, in which x can cover the range 1 5 x 5 2. I n this case, the principal stoichiometric compositions are AO, A203 (which is equivalent to writing A01.6) and AOz. Corresponding to these compositions, the cation A is in a divalent, trivalent, or tetravalent state, designated as A", A"', and A17 respectively. Experimentally, i t is usually found that when a cation is capable of assuming different valence states, the oxygen to metal ratio x in the oxide can be adjusted almost a t will (i.e., x is not restricted to just 1, 1.5, or 2) by equilibration of the oxide with an appropriate partial pressure of oxygen gas at the proper temperature. Consider now what happens when the material AOz is heated under conditions such that oxygen is r e moved from the lattice. Initially, all cations are in the A'" state, and, as remarked earlier, the material is an insulator in those circumstances. Since these oxides are relatively ionic, it is not unreasonable to consider 0 as being present in the ionic form 02-at lattice sites. Thus, to render the material nonstoichiometric, it is necessary to remove some anions from their lattice position to the surface, whence they can escape into the gas phase. The important point is that at some stage in this transition, two electrons are left behind in the crystal for every 0 which has been detached from the solid. I n broad band semiconductors, electrons left behind in this manner normally would he accommodated in the lowest available energy states of the band structure. However, since AO, has been postulated to be an electron transfer (mixed valence) material, it is now more appropriate to consider each such electron to be placed on a cation, thereby converting two A17 ions to the A111 state for every 0 removed from the solid. Thus for x < 2, but close to this upper limit, a sprinkling of A"' ions is present in a background of AIV. As discussed earlier, a t sufficiently high temperatures, it is possible for an electron so introduced to move from one cation A1"I to a second initially in the A21" state, thereby givingrise to the two new states A p and Aprwhich accounts for the fact that the nonstoichiometric
The model described above leads to interesting magnetic properties which are beyond the purview of the present paper; certain aspects of thin problem are described elsewhere (I).
oxide has become an *type semiconductor. With further removal of 0, more and more cations are converted to the tripositive state; a t x = 1.75, there is an equal concentration of A"' and A'v. As still more oxygen leaves the crystal and the composition x = 1.5 is approached from ahove, one ultimately reaches the stage where the majority of cations is in the form A"', with a small quantity of AIv still present. This situation is analogous to the presence of positively charged centers A'V in a relative background of negatively charged entities A"'; thus, in this composition region AO, should be a p t y p e semiconductor. At x = 1.5, all cations are in the A111 state; the material is once more an insulator. The argument can now be repeated for the range 1 5 x 5 1.5, the only difference being that when x -t 1, the tripositive cations are almost totally converted to A". I n particular, for 1.25 < x < 1.5, the material is expected to be n-type, whereas for 1 < x < 1.25, i t should he p-type. According to this very crude argument, the B sistivity would be expected to pass through broad minima near x = 1.75 and x = 1.25, since for these two values, both the number of charge carriers and the number of available sites to which these can jump are maximal; away from these mid-composition points either the number of carriers or the number of available sites to which the electrons can transfer diminishes. Hence, p should increase as the stoichiometric compositions are approached.
Figure 3. Expected dependence of rerisUvih/ of electron transfer semiconductors on oxide composition, x in A O , a t wnstmt Rmperoture.
Concerning the Seebeck coefficient, ru, one should recall that as x is diminished from 2.0 to 1.5 or again from 1.5 to 1.0, the charge carrier changes sign-a fact that is reflected in a sign change for a. Figure 4 is drawn so that a is negative in the ranges 2 > x > 1.75, 1.5 > x > 1.25 and positive otherwise. It will later be seen that the shape of the curves in Figures 3 and 4 is dictated by the appropriate theoretical analysis. While the above discussion has referred specifically to the compound AO,, it can obviously be extended to cover more generally the class of all nonstoichiometrio electron transfer materials.' We now consider the second method for altering the electrical properties of electror transfer materials; here, ions of different (generally baed) valence are substituted in a host lattice comprisi~~g cations of variable This discussion is obviously oversimplified. Few, if any, materials are known that retain the same crystal structure over a wide range of z valulues. In general, changes in crystal structure are encountered when z is appreciably altered. This gives rise to concomitant changes in p and a, causing deviations from the symmetrical variation of a and p with z depicted in the diagrams. Nevertheless, the trends discussed above should remain in evidence.
valency. As a simple example, consider the substitution of Li+ for Ni2+ ions in the NiO lattice. Generally, the material is prepared by admixing the oxides Li20 and NiO (or the corresponding carbonates) and heating to high temperature in the presence of oxygen. The subsequent mixing process may then be represented by the quasi-chemical reaction L40
+ l/~O?(g)
2 Li-(Ni)
+ 2@ + 2NiO(s)
which shows that (a) the oxide LizO enters the host lattice together with (b) an atom of 0 abstracted from the gas phase. The following sequence of events now occurs: (c) The 2Li+ ions in Li20 take the place of two Ni2+ ions a t their normal lattice position, giving rise to the presence of a net negative charge at each of these locations [this is the interpretation of the symbol 2Li- (Ni)]. (d) The 0 atom originally derived from the gas phase acquires two electrons to form an 02-ion, leaving a deficit of two electrons (holes), @, in the lattice. (e) The 02-thus generated and the 0%ion originally present in LipO combine with the two displaced NiZ+ ions mentioned in (c) to form two additional Ni2+ 02-units [designated as 2NiO(s)]. Although not shown in the equation, the two holes @may associate with two Ni2+ions to form two Ni3+units. The reader should recognize that the actual sequence of events may be far more complicated than outlined above. However, the important features of the process are brought out in this discussion. These are that Li+in a iked valence state enters the NiO lattice substitutionally in the presence of atmospheric oxygen, to convert an equivalent number of Ni2+ to the trivalent state. One now encounters the situation described earlier: an admixture of two cations of the same species in different valence states. Thus, the final material Liz+ NiZ3+NilX2+01-,2-should be an electron transfer semiconductor. Indeed, it is found that the mixed valence material exhibits a very enhanced electrical conductivity relative to pure NiO (3). The incorporation of a trivalent impurity cation (Gaa+) into a divalent host lattice (ZnO) may be illustrated schematically with the quasichemical reaction Ga,O.
2 Ga+(Zn)
+ 2 0 + 2 ZnO(s) + '/~Odg)
Here the sequence of events can be depicted along the following lines: (a) The oxide G&Oa comes in intimate contact with the host lattice; eventually (b) the gallium ions displace two ZnZ+ions from their normal positions, giving rise to an excess positive charge at those locations. (c) Two of the three 0%-ions originally derived from Ga203and the two displaced Zn2+ ions form additional Zn2+02- units designated as ZnO(s); (d) the third 0 2 - ion originally associated with
p-TYPE MATERIAL
h>OI "-TYPE MATERIAL
l. 0 for 1.50 5 x 5 1.72 (7). Thus, it appears that the conceptually simple theory here presented does provide a reasonable account of experimental data in solids that can be regarded as electron-transfer materials. Acknowledgment
The author acknowledges his indebtedness to Dr. Thomas B. Reed, of Lincoln Laboratory, for his searching criticism of various drafts of this manuscript. Literature Cited (1) JOHNSTON, W. D., J. CHEH.EDUC.,38. 605 (1959). (2) JONKER, G . H., AND VANHOUTEN, S., "Halbleiterprobleme," F. SAUTER, Ed., Vieweg, Braunschweig, 1960-61, Vol. 5, p. ..118.
(3) VAN HOUTEN, S., J. Phys Chem. S o l i b , 17, 7 (1960). (4) HEIKES,R. R., AND URE, R. W., JR., "Thermoelectricity; Science and Engineering," Interscience ( a division of John Wiley & Sons, Inc.), New York, 1961, p. 40. (5) Ibid., Chap. 4. G. H., J. Phys Chem. Solids, 9, 165 (1959). (6) JONKER, (7) H?NIQ,J. M., CELLA,A. A., AND CORNWELL, J. C., Proceedlngs of the Third Rare Earth Conference, Clearwater, Florida, 1963, p. 555.
