The temperature variation of electronic mobility in rutile (TiO2) and α-Nb2O5 at elevated temperatures

J. Phys. Chem. Solids

Pergamon

Press 1964. Vol. 25, pp. 881-887.

Printed in Great Britain.

THE TEMPERATURE VARIATION

OF ELECTRONIC

MOBILITY IN RUTILE (TiO,) AND a-Nb,O, AT ELEVATED TEMPERATURES J. YAHIA Department

of Physics, The Pennsylvania (Receioed

14 October

State University,

University

1963; in rewised form 3 February

Park, Pennsylvania

1964)

Abstract-It is pointed out that thermogravimetric and electrical conductivity data in combination afford a method of determining mobility at elevated temperatures in low mobility semiconductors. The temperature variation of the electronic mobility calculated using this method is presented for rutile in the range 860-1050°C and for a-NbsOs in the range 900-1400°C. The mobility for both substances is low, ~5 x 10-s ems/V-set at 1200°K for rutile and ~7 x 1OF ems/V-set for a-NbsOs at the same temperature; further, the mobility increases with an increase in temperature for both materials. An analysis of these results in terms of self-trapping of the charge carrier (giving rise to a diffusion mode of motion) is also presented. I. INTRODUCTION IN CALCULATING the (experimental) electronic mobility in solids, many methods have been used: (i) A direct method is one where the transit time of the charge carriers across the crystal is measured and the mobility thereby deduced.(l) (ii) The Hall mobility (PH) is obtained by combining Hall coefficient (RH) with the electrical conductivity (CT)according to the relation RHO = ,LLH.@)(iii) An indirect method may be employed whereby the density of charge carriers (n) is obtained from gravimetric data and combined with the electrical conductivity to yield the mobility according to the relation cs = nep (where e is the charge). Methods (i) and (ii) have been extensively used in the past but this does not seem to be the case for method (iii). This method is however well suited for the determination of mobility in oxide semiconductors where gravimetric studies are readily made and where electrical conductivity may be measured with reasonable precision. Contrasting with this situation is a difficult Hall measurement possibly involving a.c. apparatus (for reasons having to do with the low mobilities and high resistances encountered in these materials) for method (ii), or a rather complicated and not always successful direct measurement, method (i). 881

We present in this paper the results of simple calculations leading to the mobility as a function of temperature (at high temperatures) in sintered c+NbsOs and in single crystal TiOs. In the crNbs05 case, conductivity data of GREENERet al.(s) is combined with gravimetric studies by KOFSTAD and ANDERSON.@)For TiOs, the author’s electrical conductivity measurements are used in conjunction with the weight loss measurements of BUESSEM and BUTLER.(~) In obtaining the density of charge carriers from a gravimetric measurement an assignment for the defect mechanism assumed to be predominant in the formation of the charge carriers must be made. Although this assignment may sometimes be disputed, this fact does not alter the general usefulness of method (iii) for determining mobility. 2. DETERMINATION OF MOBILITY VARIATION WITH TEMPEXL4!IURE IN RUTILE (TiOs) The temperature variation of mobility in two undoped crystals of rutile has been determined in the temperature range 860 to 1050°C by measuring the electrical conductivity at various oxygen pressures and temperatures and combining these data with the weight loss measurements in the same pressure-temperature range reported by BUESSEM

882

J.

YAHIA

and BUTLER.@)The weight loss measurements give the number of charge carriers per ems (n) and this, together with the electrical conductivity (o) gives the mobility (p) according to the relation o = nep where e is the charge. Measurement of the conductivity was by a potentiometric four-probe method. The experimental procedures for producing and maintaining fixed oxygen pressures and temperatures are described in an article on the conductivity and thermoelectric power in pure and aluminum-doped rutile by the author.(s) The defects assumed to exist in the crystals are uncharged anion vacancies and quasi-free electrons, this assignment being reasonable from previous pressure-dependence data in the range investigated.(s) In addition, for both crystals the sign of the thermo-electric power was negative indicating negative charge carriers. According to this defect picture, the concentration of charge carriers [e-l may be written as [e-l = 41lsKrlsp-l/s where K is a reaction constant calculated from experimental data (and tabulated) for various values of temperature by BUESSEMand BUTLER(~) and where p is the oxygen pressure. Figures 1 and 2 represent log conductivity vs. temperature-1 for the two r-utile crystals at various oxygen pressures. These two crystals are from different sources: crystal A was

obtained from the National Bureau of Standards* and was grown by a flame fusion technique. The chemical impurities in the starting powder for this crystal are given in reference (6). Crystal B was obtained from Mr. V. PORTERof The Pennsylvania State University Materials Research Group and contained the following chemical impurities (by weight per cent) : VaO5 ( < O-01), CraOs( < 0*003), MnaO4( <0*0005),FeaOs( < O*OOl),CoO( < O-003), NiO ( < 0.003), CuO (< O*OOl), PbO ( < 0*003), SbaOs ( < 0.004), SiOa ( < 0*04), MgO ( < 0*0005) and CaO (< 0.002). The values of conductivity are seen not to be very different for the two crystals and also the variation with temperature for both of them is of the form log o = A -log B/T with A, B constants. The temperature variation of the mobility (for several oxygen pressures) obtained by combining weight loss and conductivity data is shown in Fig. 3. It is seen that the mobility of the two crystals is close for similar conditions of oxygen pressure and temperature and that at 1200°K its value -5.10-5 cma/V-sec. It should be noted that for both rutile crystals the mobility shows an increase with an increase in * Thanks are due to Mr. E. ROBERTSof the SolidState Physics Section for supplying the author with the boule from which this crystal was cut.

-3.3

-3.1

7 E

A

344mm

l

186mm

t

93mmHg

X

26mmHg

Hg Hg

-2.9

G

G

-2.7

3 -2.5

-2.3

T-’ x IO*

FIG. 1. Log electrical conductivity vs. temperature-l rutile, temperature in “K.

Crystal A, undoped

ELECTRONIC

MOBILITY

..

IN R’IITILE

(TiOs)

I

I

7.4

7.0

AND

a-NbsOs

1 8.2

AT ELEVATED

TEMPERATmES

1

I

6.6

9.0

883

T-’x IO’ Fro. 2. Log electrical conductivity

vs. temperature-l undoped rutile, temperature in “K.

temperature. This behavior is also observed in ct-Nbs05 as we will show presently. We shall return to this point.

Crystal B,

4. DISCUSSION

A great deal of work has been done on the electrical properties of rutilefr*s) whereas this is not the case for c(-NbsO5. In discussing the results obtained, therefore, emphasis will be placed on those for rutile with the expectation that similar 3. DETERMINATION OF MOBILITY VARIATION WITH TEMPERATURE IN SINTERED a-NbsOs considerations might apply to a-NbsOs. A first As has been mentioned before the mobility in question that may be raised is: how valid is the sintered a-Nbz05 has been determined by comprocedure of combining resistivity data on one set bining electrical conductivity data@) and thermoof (single) crystals with weight loss data on other gravimetric data.@) In using KOFSTADand ANDERsamples (not necessarily single crystals)? The SON’s(~) data it was assumed that equation (5) in answer to this is that it is probably not important their paper obtained and this is self-consistent with whether the weight loss samples be single crystals their determination of the reaction constant X but or not so long as they are reasonably pure since represents a different defect assignment from that what is calculated in a weight loss measurement is of GRTSNERet af.6) We have here taken KOFSTA~ not a transport property. In fact, for practical and ~ERSO~‘s~~~ defect assignment to be correct reasons (time taken to reach equilibria) it is and used GREENERet aL@) experimental values of preferable in a weight loss measurement to have the electrical conductivity. The calculation of the sample in the form of a powder. The degree mobility was performed for three oxygen pressures. of impurity is an important matter. However from Figure 4 shows the temperature variation of data on oxygen pressure vs. conductivity(s) and mobility in c+Nbs05 at different oxygen pressures. BUESSEM and BUTLER’S arguments@) the main The mobility at 1200°K is seen to be -7-10-s effects in the pressure-temperature range of the ems/V-see, considerably larger than that in rutile investigation are due to non-stoichiometry, in at the same temperature. It will be noted that here particular to anion vacancies. Further, it is unagain, as for the rutile crystals (section 2) the likely that the number of electrons will be affected mobility increases with increase in temperature. by intrinsic excitation since _Eo N 35 kT at

J.

884

YAHIA

fi0a IO -

“0 x z

;

P=26mm

A

crystal

Hg

P-93mm

-_.>“’

Hg

l/*

./

2-

,z E u P=186mm a f .Z 4

Hg

P=344mm

Hg

IO 8

“UL ii00

1200

1200

1300

1300

T, OK

TIO, P =26mm

)-

crystal

f3

Hg

P=46mm .

.

.’

. j -- - -./’

!

Hg 9/

I -r----

-

P=93mm

1100

Hg

P=lSOmm

Hg

I

I

I

I

1200

I300

1200

1300

FIG.3. Temperature variation of electronic mobility in two crystals of rutile (TiOa) at various values of oxygen vapor pressure.

T = 1000°K for rutile and YAHIA(@ observes no hole conductivity for pure rutile at these temperatures even though hole mobility is estimated to be greater than electron mobility. It is found that the Hall mobility in pure rutiie decreases with increasing temperature in the range lOO-300°K approaching an exponential of arguThe magnitude of the ment N +(O.OS/M’).@,s) Hall mobility at 300°K is -0.2 cms/V-sec.(g) If this value is extrapolated to 1200°K the mobility is found to be N 5.10-s cms,fV-sec. The question arises then of why the discrepancy exists between the extrapolated value of mobility and our results.

In the first place, it should be realized that extrapolation of Hall mobility data from 300 to 1200°K is rather dangerous for the entire picture of conduction may change in this interval of 1000°K (indeed we shall give a qualitative account of our mobility results at high temperatures using results from the FRBHLICH theory(m) and a diffusion model for the motion. This differs substantially from a band theory presumably obtaining at low temperatures). It seems, therefore, that a more reasonable approach-compared to that of extrapolation over a large interval-is to compute mobility as described in this paper.

ELECTRONIC

MOBILITY

IN RUTILE

(TiOs) AND a-NbaO5 AT ELEVATED

The model For rutile, where many of the fundamental properties are known-such as the effective mass, the static and optical dielectric constants and the rest-strahl frequency-it is possible to explain qualitatively both the magnitude and temperature variation of the mobility (as calculated from experimental values of weight loss and electrical conductivity). For a-Nbs05 little is known about the fundamental properties so we shall not discuss this case apart from noting that the same general mechanism (to be described for rutile) is probably the one giving rise to the mobility characteristics of this substance also. We start from the fact that the mobility in rutile is small at elevated temperatures; thus it is plausible that the electron is in some fashion trapped within a region of the crystal with a mean time of stay that is large in order to account for the small mobility. This model then gives a diffusion mode of motion (i.e. an activated mobility) which is in line with our results. A calculation of the binding (activation) energy on this model is problematical. We have taken the value of the total (negative) energy of a slow electron in a polar lattice (calculated by FROTHLICH)oO) for the high frequency case as an appropriate choice for this binding energy. There are two reasons for this: (a) it was felt that since w was large (N 1Or4 set-1 as will be discussed later) the static approximation in Friihlich’s theory would not obtain: some account of the dynamic character of the interaction needs to be made. (b) For the high frequency approximation, the polarization potential decreases for large distances like a Coulomb field; for distances less than V/W (v is the velocity of the electron and w is the circular frequency of the polarization modes) it remains nearly constant (this point is discussed by Frohlich in the article referred to). The expression for the total energy of the electron is the one we are identifying as a binding energy. Clearly, the polarization potential described above is one that may readily be thought of as giving rise to self-trapping. The FR~~HLICH theory(ra) gives the following expression for the energy of an electron for GC< 10 (a being the coupling constant) :

E(0) = -G&q

Here E(0) is the self-energy,

E the static dielectric

885

constant, ns the optical dielectric constant, e the electron’s charge, m, the electron’s mass, m the effective mass, cc the coupling constant and wl = (+za)l’swr where wr is the rest-strahl frequency. We have used the expression for E(0) in the case a < 10 since a calculation of a using values for basic constants reported in the literature gives a N 8.(rl)* We find, using the same constants that E(0) = 0.65 eV. We now use this number to calculate the drift mobility according to a hopping or diffusion mode of motion.(la) We find for the mobility p = 2~

exp[-

E(O)/KT].

Here a is a jump distance and Y the frequency of oscillation of the trapped particle. According to FREDERIKSE(~~)the value of the Hall mobility at ca. 400°K indicates a mean free path of the order of a lattice cell. Let us assume a -2 A. Furthermore, v N 1014 set-1. With these numbers, we calculate for the mobility at 1000°K: p( 1000°K) N 24 x IO-5cms/V-sec. This calculated value agrees fairly well with the observed mobility, p N 5 x 10-5 ems/V-sec. Moreover, the activation energy for the mobility is about the one observed experimentally (N O-6 eV). Our choice of energy for the electron is, as we have stated, based on the high frequency case which seems reasonable since a N 8. On the other hand, FREDERIK~E@) in discussing low temperature transport results in rutile treats the polarization of the lattice by the electron statically to arrive at a binding energy N ( - e%*/Asr*2) (m* is the effective mass and Z* an effective dielectric constant which Frederikse takes to equal the static dielectric constant). This is equivalent to the low frequency case (a very large) treated by PEKAR,~~) FRBHLICH(~@ and others.(l5) In view of our calculated value of a however, it seemed more reasonable to use the high frequency results (as we have done by using E(0) (cc < 10) as a binding energy).

* We use cc =

= _e2(J3?3._J’2.

TEMPERATURES

For

rutile,

f(zy”g.y’2.

at the temperatures E N

under

consideration,

70, n2m 7, wl N 1014 set-1,

886

J.

20

YAHIA

ah

P=l

P = IO’ atm

*_._-.-.A

./*’

l

I

I

I

I

1300

I400

1500

1600

P=10-%1m 20

-

./’ IO

T,

FIG. 4. Temperature

variation of electronic

where IMI is an electronic matrix element, S is a coupling constant and 7t = l/exp O/T- 1 where 0 is the Debye temperature. lo is a modified Bessel function,

A-J(z)

=

277

exp(x cos x) dx.

-

2rr

-*l

_*-•

I

I

I

I

1300

1400

1500

1600

OK

mobility in cr-NbzOs at various values of oxygen vapor pressure.

The electronic mobility temperature variation observed here in TiOs and a-Nbz05 was observed by MORIN(~@ in p-type NiO (Morin used Seebeck effect and electrical conductivity to deduce charge carrier density and mobility). He found for example that the mobility in some samples would increase by a factor of ten on doubling the temperature. YAMASHITA and &JROSAWA(~~) explained this variation by suggesting that the motion of charge carriers in this material does not take place according to a band mechanism but by a random walk from site to site. In their theory, the mobility p is determined by the transition probability:

1

-

s

0

The authors calculate the value of exp{ - S(2n + 1)) xI0{25’2/n[~+ l)]} for various values of T/O

(Table 2 in their paper). A formula for S on the continuum model is also derived. If the appropriate values for rutile are introduced into this formula(we assume that the only effect is contained in the term ((l/P) -(l/c)) where & = optical dielectric constant, E = static dielectric constant), a value of S N 5 is obtained. For the temperatures under consideration (T/ 0 N 1.4, 0 for rutile ~670”K(ls) this corresponds to a value of mobility p N 10-l ems/V-set if it is assumed that IMI N KO. This value is considerably larger than the observed mobility. This might be due to the following factors: (1) The continuum expression derived by Yamashita and Kurosawa is based on a NaCl structure with periodicity a which is taken to be -4 A; for a tetragonal structure it is not clear that such a result may be used without modification (apart from the ((l/G) -(l/e)) term already referred to). (2) The expression for IMI is quite arbitrary; this parameter could presumably be picked so that agreement resulted between the theory and the observations. Quite apart from these considerations, if the temperature variation of mobility t.~is calculated for an S corresponding to the most rapid temperature variation (S = 20) it is found that the Yamashita-Kurosawa theory gives ~(940°K) N l-4 p (SOO’K). The observed

ELECTRONIC

MOBILITY

IN RUTILE

(TiOs}

AND

a-NbeOs

AT

ELEVATED

~MPERATURES

887

temperaturevariation of mobility is much morerapid (activation energy 0.6 eV giving p (940°K) = 3.4~ (8OOOK) by extrapolation). In view of these discrepancies, it was felt more appropriate to use the FrShlich results (as indicated above) in an analysis of the data since one is led quite directly thereby to answers that agree with experiment.

4. KOFSTA~ P. and ANDERSONP. B., _T. Phys. Chem. Solids 21, 280 (1961). 5. BUE~~EMW, R. and BUTLER S. R. in Kinetics af High Tempwature Processes (ed KINGBRY E),

author is indebted to Mr. T. ~&&CORD wk measured the electrical conductivity in the crystals of rutile and to Frofs. E, Hmxa and R. G. Wmmn for discussions on this work.

fl960). 10. FRSHL~CR H,, A&rnc. P&s. 3, 325 (1954). 11. P~~~~~~~~A.,P~ys.R~.1~,1719~196l)~for~; es was obtained from the paper by CRONEMEYXR D. C., P&s+ Rm. 87, 876 (1952). WI and m/nte were obtained from FREDERIX~EH. P. R. (private communication). 12. MOTT N. F. and GURNEY R., Electronic Processes in Ionic Crystals (2nd Ed.), p. 43. Oxford University Press (1940)” H. P. R., private communication. 13. FREDERIKSE 14. PEKAR S., Gax. I%. SSSR 10, 341, 347 (1946). 15. A~~coce G. R., R&anc. Phys. 5,412 (1956). 16. MOIUN F. J., Phys. Rev. 93, 119 (1954). 17. Yz%MhSHITh J. and Kunos~w~. T., J_ Phys. Che-m.

~~~~o~~~~g~~~-T~~

REFERENCES 1. HAYNES J. R. and SHOCKLEYW., Phyf. RN. 81,835 (1951); PRINCE M. B., Phys, Rev.‘W, 271 (1953); WOESTADTERR., Nucleon& 4, No. 4, 2 (1949) ; 1, NO. 5, 29 (1949); CHOLLET L. and ROSSEL J., ReZv. Phvs. Acta 33.627 (19601. 2. SsOCKLEY *W., Electropls anh H&s in Senaiconductars, D. van Nostrand, New York (1950); Y.~IA J. and FRFS~SR~KSE H. P. R., P&s. Rm. 123, 1257 (1961); Bmrcxmm lDGE R. G, and HO~LFR w. A., Pkys, Rec. 91,793 (1953). 3. GREENERE. H., Fmm G. -4. and HrR’rHE W. M,,

J_ Chem. Phys. 38,133 (1963).

John Wiley,

New York (1958).

6. YAHIA J., P&t. Reru. 130, 1711 i1963). 7. GRANT F. A., Rew. Mod. Phys. 31,646 (1959). a. FREDZRIKSEH. P. R., J. Appl. Phys. Suppl. 32, 2211 (1961). 9. F&ERIKSS

N. P. R., Proceedings Internationu.? Cofifezence on ~~‘c~~~~ Physics. Prague

S&ids $34 (1958). H. J, and Sxr.rz H., .j. Amer. Chem. 1% McDo~&n Sot. d&2405 (1939).