Electrical and seebeck effect measurements in Nb doped VO2

1 Phys. Chem. Solids Vol. 43. No. 3. pp. 205-21 Printed in Great Britain.

I, 1982

W22_3697/82/03lXOM7SO3@3/0 Pcrgamon Press Ltd.

ELECTRICAL AND SEEBECK EFFECT MEASUREMENTS IN Nb DOPED VO, B. FISHER? Technion-Israel Institute of Technology, Haifa, Israel (Received 28 July 1980;accepted in revisedform 8 July 1981) Abstract-The resistance of pure and Nb doped VO* and the Seebeck coefficient of Nb doped VOz have been measured in the temperature range of 78-360K. A simple analysis of the results shows that above 140K and below the transition temperature the effective density of states in the conduction band of VO*is of the order of (but larger than) one state per vanadium atom. This high effective density of states is consistent with the large effective mass (and low mobility) of electrons in this material. It is shown also that in this range, the temperature dependence of the electronic mobility in V02 is F wherey 2 2. Additional resultsare discussedin the text. 1.INTRODUCTION

Vanadiumdioxide received considerableattention in the past two decades due to its semiconductor to metal transition at 340K. In spite of the large experimental effort invested in it, some of its properties in the semiconducting state such as the value of the effective density of states in the conduction band, NC,or the temperature factor of the mobility are yet uncertain due to conflictingresults in the literature. The low mobility of VO, (p < 1cm’lvoltset) can be a result of either a polaronic mechanism of conduction or of a very high effective mass of a very narrow band (a high effective density of states). Previous measurements showed no indicationof the firstpossibilitybut also could not rule it out due to the limited temperature interval over which they were performed. On the other hand, as will be discussed later, the values of the effective mass of the electrons and the impliedeffective density of states used in the literature are contrary to a narrow band model. Some reasons for this situation are due to technical problems which VOz raises: the difficulty in growing single crystals with good quality, the tendency of the crystals to break when cycled throughthe transition,this tendency being enhanced by purity, and the low mobility which complicatesthe Hall measurements.It willalso be shown that the unusual temperature dependence of the gap in VO, may lead to misinterpretationof the temperature dependence of the conductivity. Therefore it was felt that additionalmeasurements,over a wide range of temperatures could be of use in findinga more consistent model for the transport properties of VO,. The approach in this work was to measure the transport properties in single crystals on n-type VOz.Doped samplesare more robust than pure samplesbecause their phase transition is less abrupt and the drdp in their conductivitywith decreasingtemperatureis smaller,thus enablingthe increase of the temperature range. Also the Hall effect measurements have been replaced by Seebeck effect measurements. tPresent address: Department of Physics, Purdue University, West Lafayette, IN 47907,U.S.A.

The work consists of conductivity and Seebeck effect measurementsin V1_,Nb,02 for x = 0 and 2.5x lop3c x ~4 x lo-* in the temperature range 78-360K. An extensive study of the electrical and thermoelectrical properties of the system V,_,Nb,02 for x < 0.5 was done by Willeneuveet al.[l]. Their measurementswere done on sintered hors and the emphasis was on understanding the evolution of the system in the various phases and on constructinga phase diagram.The present work, which is done on single crystals, is restricted to small enough values of x so that the Nb atoms may be regarded as donors in the native VO, lattice[l]. In principle the study of the variation of rr(T) as a function of x should provide enough information by which it is possible to separate between the carrier concentrationand their mobilityin the conductivity.The condition required to obtain this information is small enough x (but well determined)so that the mobilityand the energy bands and levels are yet unaffected by x. If this can be achieved then the obtained results may be superior to Hall effect measurements because they produce the drift mobility, a notion which is better understood than the Hall mobility. For some unknown reason the crystal growth technique used for x 2 2.5x 10e3and x = 0 was unsuccessful for 0 < x < 2.5x 10e3.It was immediatelyobserved that the activation energy of the conductivity varied with x, therefore a straightforwardinterpretation of the results was impossible without complementary results. Additional information was obtained by use of Seebeck measurements. The small dimensionsof the samples on which four probes were applied, the diameters of which were comparable with their separation, introduced a large uncertainty in the absolute value of the conductivity. However, the phase transition to the metallic state provides the calibration of u in the semiconductingstate because the metallicconductivityis insensitiveto a small concentrationof impurities.The mobilityof the electrons is definedso that the carrier concentrationin the metallic state is 1 per Vanadiumatom. Finally the variationof the transitiontemperaturewith 205

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x found by several techniques in Ref. (11is used in this work as a check of the Nb concentration vs the concentration introduced before the growth of the crystals. Although VO, is not a standard semiconductor, an attempt is made in this work to interpret the experimental results in terms of standard formulas from the theory of semiconductors[2,3]. It will be shown that according to such analysis the effective density of states in the conduction band is larger than 1 state per Vanadium atom at T,(N, =2 if the donor level is an s state). This value, which is about two orders of magnitude higher than that implied in Ref. [4], is consistent with a high effective mass and a low mobility.It will be shown that the electron mobility in the semiconductingstate at T, is smallerbut comparablewith that in the metallicstate and in a wide range of temperatures, it drops with increasing temperature like T-’ - T-‘. 2. THEORETICAL BACKGROUND

For convenience the formulas used for the carrier concentration and the Seebeck coefficient are summarized below. The units used for all the concentrations are l/Vanadium atom, same units as for the donor concentration x. The electron concentration n in the intrinsic regime of a semiconductoris121:

scattering mechanism. In a standard semi-conductorin which the dependence of the relaxation time &.L = e (7)/m*) on the thermal energy E of the electrons can be

expressed by 7 = E” is given by: 6=5/2+s.

(4)

For a given s, the averagingof 7 over energy results in a temperaturedependence of the mobilitypucuT’where y is comparable with s. From eqns (2’) and (3) it is seen that in the extrinsic regime the plot of a vs l/T is a straightline with slope EJ2e (if the smallpossiblevariation of NCwith T is neglected).The extrapolationof this plot to l/T = 0 gives: a, = (K/e)[W(N,I/?x) + 61.

(5)

The temperature dependence of NC has various effects on the apparent activation energies found in the conductivity and in the Seebeck coefficient. A temperature dependence of N, of the type 7’” increases the apparent conductivityactivationenergy (in the extrinsic regime)to 1/2(E, t +T) while it reduces the a activationenergy to (l/2) (Ed - aKT). On the other hand any temperature dependence of Ed has similar effects on both measured activation energies.

3. EXPERIMENTAL.METHOD The starting materialsfor the crystal growth were high where N, and NC are the effective density of states in purity VzO, and Nb,Os. The powders were thoroughly the valence and the conduction band respectively and mixed in the appropriate concentrations and then heat E, is the energy gap. In wide parabolic bands the tem- treated in an inert atmosphere of flowing nitrogen for perature dependence of each of the effective densitiesof several hours at 600°Cand later for about two weeks at states is given by the factor T3” but it may be weaker 900°C.The result of a successful growth was a batch of when the bulk of the states is concentrated in a width shiny needles. All the growths run with 0 < x < 0.0025 were unsuccessful. Though some needles were fairly comparablewith kT. In a nondegenerate, noncompensated, n-type semi- large the best samples had typical dimensions of conductor the carrier concentration n is given by the 5x.1x.1mm3. For the resistance measurements the needles were attached with indiumamalgamdots to four formula? potential probes printed on the basis of the sample holder. The minimalseparation between adjacent probes n’/(x - n) = @NC exp (- EdlZkT) (2) was 1mm. The temperature was varied between liquid where x is the donor concentration, &-the ionization nitrogen and lOOC In the first run the sample was energy of electrons from the ground state donor level cycled through the transition by slowly increasing and and j? is the spin degeneracy factor of this level (/3= l/2 decreasingthe temperature.Only those samplesin which if this level is Is). Equation (2) does not consider occu- the reproducibilityof the resistance in both phases was pation of excited levels of the donors. In the regime better than 10%were used for further measurementsat n < x it can be approximatedby: low temperatures. The others were suspected of having microcrackscaused by the transition. For the thermoelectric power measurementsthe samn = ~@NJ) exp ( - E,,/ZkT). (2’) ples bridged two separated bricks of BN, each brick The regime in which n becomes comparablewith x, the having a small heater and a Chromel-Alumelthermoexhaustion regime, is characterized by a decrease couple. The thermocouple tips were immersed with the ends of the sample in amalgamdots. The thermoelectric towards zero of the slope of the curve In n vs l/T. The Seebeck coefficient3- Q in an n-type semicon- voltagewas measuredbetween the Alumelwires, and the ductor is given by small contribution of the Seebeck coefficient of these wires was taken into account. The temperature drop AT (I = (K/e)[ln(NC/n)t 61 (3) on the sample varied between 0 and several degrees in both polarities. The initial rise of the thermoelectric where e is negativeand 6, the so called heat of transport voltage AV with increasingAT was for some unknown term, is normallya smallconstant which depends on the reason nonlinear and nonsymmetric about AT = 0 but n = v(N,N,) exp (- Egl2kT)

(1)

Electricaland Seebeckeffectmeasurementsin Nb dopedV02 above a few tens of a degree it continued into a wide

linear and symmetric regime. The Seebeck coefficient reported in this work is the slope of the linear regime of AV vs AT. The quality of the results of these measurements is much poorer than those of the resistance probably due to the poor geometry. The samplesbroke very often at low temperatures and therefore an effect of thermal stress cannot be ruled out. The measurementsof the Seebeck coefficientin pure VO, close to the transitionwere very irreproducibleeven at high temperatures and therefore only results for Nb,V,-,02 are shown here. Both the conductivity and the Seebeck measurementswere done along the natural axis of the crystals (tetragonal c axis) and therefore no effects of anisotropy could be measured. 4. -AL RESULTS AND DISCUSSIONS 4.a The transition temperature and the resistance jump Figure l(a) is a plot of the metal-semiconductortransition temperature T, as a function of the nominal Nb concentration labelled x. In order to emphasize the logarithmicscale of the chosen concentrationsthe scale of x is logarithmic.The temperatures T, are plotted on a scale in which T,( VOz)- T1(V,_,Nb,0J is also logarithmic.The broken line in this plot represents the initial drop in T, of 11 K per cent Niobium found by

Villeneuveet al. [ 11from X-ray data in this system. The triangles for x = 0.005, 0.01 and 0.02 represent data a

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Fii. 1. Semiconductorto metal transition temperature(a) and resistancejump Rzj& (b) in VI_,NbX02 plotted vs X.The bars represent results of the present work. The triangles represent

results from Ref. [l].

207

obtained by the same group by D.T.A. measurements and the triangle for 5% Niobium was taken from their

phase diagram.The vertical bars represent the range of transition temperatures found in this work in each nominal concentration. They are based in each concentration on at least three samples.The hysteresis (of less than 1 K) is includedin the bars. In sampleswith x s 0.02 the smearing of the transition is negligiblebut in the samplewith x = 0.04it may reach 10K and therefore T, is not well determined.Althoughthe range of transition temperatures found for samples with nominal concentration x = 0.02 is included in that of the samples with x = 0.04the two types of samplesbehave very differently at the transition and below it. It is seen that if the transition temperatureis used for the calibrationof concentration for x ~0.02 then the concentrations in this work are determined within an error of less than 50%. Chemicalanalysis on individualneedles could not reach this accuracy. Figure l(b) represents the dependence on concentration of the resistancejump at the transition,expressed by the ratio R.JRM where RS and RM are the semiconductor and metal resistance respectively. Here again the ratios R.JR, are plotted on a logarithmic scale. The vertical bars represent the range of values of R$R, found in each nominal concentration. The maximal resistance jump found in this work in a sample of pure VO2is 0.53 x lo’, about two orders of magnitude above the highest RJR, found in x = 2.5x 1OW.The bars for 0.0025IX I 0.02 lie in a region between two lines of slopes - 1 and -l/2. The lower line (slope - 1) would be obtained if all the donors were ionized at the transition. For a metalliccarrier concentrationof 1 electron per Vanadiumatom this line corresponds to a mobility ration ~Jlc, 0.54.The upper line (slope - l/2) would be obtained if the majority of donors were non-ionizedand Ed, N, and pLswould be x independent(see eqn 2’). In such a case ~,I~,,,> l/5. A ratio 0.1~ ~JP,,, < 1 was claimedin the past in pure VOZ4e5. In principle one can calculate pm from the metallic conductivityif it were not for the large uncertaintyin the geometry of the samples mentioned above. The estimated metallicconductivity at the transition in ten samples with x I 0.02rangedbetween 1.3x lo3 to 10”n cm-’ with an average of 4 x l@a-’ cm-‘. This average agrees well with the conductivityof a large sampleof pure VOz measuredin this work and with data from the literature.’ This value yields CL,,, ~0.7 cm*v-’ set-‘. The value found for pcL, by Rosevear and Paul6 in semiconducting V02 is 0.4cm2V-’ sec.-‘. The ratio ~JP,,, = 417would correspond to a straightline of slope - 1 slightlyshifted downwardsin Fig. l(b). This means that all the samples are not yet in the exhaustionregime but not far from it. The feeling is that the resistance jump would be an even more effective tool in determining H/P,,, (at TI) for x < 0.0025and that an additionaleffort in this directionis worthwhile. 4.b The temperature dependence of the resistance in pure vo2

Following the discussion above it is seen that in order

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to avoid the dimensions uncertainty the best representation of the temperaturedependenceof the conductivity (or resistivity) is by the ratio r(T) = R,( T)/&( T,). The upper curve (solid line) in Fig. 2 is a semilogplot of r(T) vs 1000/T of a sample of pure V02 with the highest resistance jump found in this work. In an intrinsic sample the major temperature dependence of its carrier concentration is governed by the term exp [ E,I2kT]. The temperature dependence of the optical gap of VO, is plotted in Fig. IV-14 of Ref. [5]. In the range of temperaturesfor which r(T) is plotted the optical gap found in Ref. [5] decreases nonlinearly with increasing temperaturesfrom 0.7eV (at 223K) to 0.62eV (at 333K). The function exp [E,I2kT] calculated point by point on the basis of that figure was pinned to experimental r( 7’) at T = 333K and plotted vs 1000/T. The result is a perfect fit between the two curves over two orders of magnitudeof the resistance. At lower temperatures the experimentalcurve lies below the theoretical curve (dotted lines). This fit leaves no doubt that the sample is intrinsic with a conduction gap equal to the optical gap. This also leaves room for only a small variation of CL, with temperature. It should be noted that if one expands the upper curve over the first 60 degrees below the transition and takes the slope of this portion as a representative of E,/2, then the values of E, so obtained

r

will be almost twice the correct value. This is because in this narrow range of temperatures E,(T) can be approximated by 0.9(1-9 x 10m4T)eV. The nonlinear terms reduce the value of E, at lower temperatures. Now if the current is carried only by electrons [S] then from eqn (1) we obtain r(Z) = (~,,,/~,)(N,N,))“* exp (E$KT). From the values of r and Ep at 333K we obtain (~J~,,,)~\I(N,N,)= 0.7. This value ISvery close to the value of pS/p,,,estimatedin the previous Section 4.a. This impliesa value of the product N,N, around 1. Energy band calculationsby Caruthers and Kleinman[7] show a high density of states at the bottom of the conduction band in VO, (quantitative results are not specified)and a lower density of states at the top of the valence band. Therefore if N, > N, then N, > 1. The classical expression]21for N, (for a wide parabolic band) yields m*/m,,100 for N, = 1 at T,. Paul]81noticed that if the mean free paths of the electrons is minimal,that is, one interatomic distance, then their effective mass must be about 100m,, in order to obtain pL,= 0.4cm*/Vsec. The discrepancy between the two curves at lower temperatures may be due either to impurities(less than IO-’ per V atom) or to a small net decrease of the product pcLsd(NcNO) with temperature. Rosevear and Paul6 mentioned that the semiconductingHall mobility

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1000/T Fig. 2. Semiconducting lines). The resistance

resistance

vs 1000/T

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solid line) and in Nb doped VO, (lower solid in the semiconducting state is represented in units of the resistance in the metallic state. The doted line represents the plot of the function exp EJZRT pinned to the experimental line at T = 333 K (E,.(r) is taken from Ref. [.(I). The dashed lines represent

in pure VO,

(upper

the product of the experimental text).

lines with the function

(T,/r)*

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Electricaland Seebeckeffectmeasurementsin Nb doped VOz behaves like the metallic conductivity, that is, it decreases with increasingtemperature. Accordingto the above analysis the electron concentration at the transition is about 1Ol9cm-‘. Therefore samples with x > lo-’ will have carrier concentrations above the intrinsic concentration. The large value of NC ensures that the carrier concentrations in the doped samplesobey nondegeneratestatistics.

4.c The temperature dependence of the resistance and of the Seebeck coejicient

in Ir,_xNb 0,.

The lower (solid) curves in Fig. 2 are typical plots of r(T) vs 1000/T for 2.5X 10m3 5 x S4 X lo-*. Though curves of r(T) for different samples having the same nominal concentration x may be slightly shifted vertically, one with respect to another, the shapes of the curves are very reproducible.This fact indicatesthat the concentration uncertainty is probably smaller than that observed in Fig. 1 and that a large part of the shift factor is a geometrical effect. The semilog plot of r(T) for x = 2.5x lo-’ has a wide linear range at low temperatures followed by a range in which the slope decreases monotonically.In the high concentrations the slope first increases and later decreases monotonically. The apparent break in slope which occurs between 100K and 14OKis barely noticeable in the plots for x3.01 but is very clear in the plot for x = 0.04. The main features of these plots can be recognizedin similarplots for higher concentrations(see Fig. 3 or Ref. [l]). Like in Ref. [l] the linear portions of each plot are formally interpreted as ra exp [E,/~KT]. The values of E,+ so obtained are plotted vs x in Fig. 4(a) (open circles for T> 140K and open squares for T< 100K). These values match well the corresponding plot for x ~0.05 and T = 125K in Fig. 5 of Ref. [l]. There, only a rise of the activation energy with x is observed. Here, for lower

x, the rise of Ed with x (at high temperatures) is preceded by a drop of Ed with x. Assuminga negligibledependence on temperature of CL,,NC and Ed one could interpret the monotonic decrease of the slope before the transition as representing the exhaustion regime. On the other hand a temperature factor which has a negligible effect on the function exp EJ~KT (over a narrow range of temperatures) may have a drastic effect on a function with a much lower activation energy (and over a wide range of temperatures).Therefore, no further conclusionscan be drawn at this stage before comparing with the complementaryresults of the Seebeck coefficient. Figure 3 represents typical plots of the Seebeck coefficient vs 1000/T for the various concentrations x. The coefficients are negative and their absolute value increases with l/T. The large experimentalerror and the low density of points at low temperatures (caused by experimental difficulties) preclude drawing continuous curves through the points. The values of (Yfor x = 0.04 close to the transition match well with the curve of (I vs l/T (in a narrow range of temperature) for x = 0.05 given in Ref. [l]. This fact excludes the possibility of a systematic error in the present results, caused by geometry, because in the cited work[l] the measurements were done on large

sintered samples. No evidence of exhaustionlike in Fig. 2 is observed in Fig. 3 except, perhaps for x = 2.5x lo-‘. Moreover the drop of (Ywith l/T is faster than the rise of In r(T). Therefore it must be concluded that the decreasing slopes of r(T) towards T, are caused by a temperature factor of the pre-exponentialterm of r(T). Therefore a function of the type T’ was tried for the term ~S~(/3NC) (see eqn 2’).The best results were obtained for y = - 2 (but as seen below, y can be decreased to about - 3 for x = 2.5x 10m3).This means that, while d(N,) can

1000/T

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Fig. 3. Seebeck coefficientvs 1000/Tin Nb doped V02. The dashed lines have slopes consistent with the high temperatureactivationenergiesfound from the modifiedcurves of Fig. 2. FT.9 Vol. 43, No. 3A

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Fig. 4. Ionization energy Ed of electronsfrom the donor levels (4a)and the parameter(~,/j~~)~l(fiN~,)(4b)vs Nb concentration, in two temperatureregimes,found from the modifiedand the unmodifiedcurves of Fig. 2. Ed vs ‘d(x) (the inverse of the averagedistancebetweendonors)is shownin the insertedplot at the bottomof the figure.

only increase with T (as most as T3“‘)the mobility must drop faster than T-*. Such a temperature factor of the

mobility is indicative of an acoustic scattering mechanism[2,3]. The plots in Fig. 2 have been multiplied by (T,/ T)’ where lOOO/ T, = 3 was chosen for convenience.The new curves which should now represent (~L,JPJ (NC,x)-“‘exp(EJ2~7’) are depicted in Fig. 2 by the dashed lines. It can be easily checked that if this procedure is also used for the experimental curve of pure VO, the agreement with the theoretical curve increases over one more order of magnitudeof the resistance. It is seen that the apparent carrier exhaustion is wiped out completely for all concentrations except for x = 0.0025in which a weak drop of the slope is retained above 200K. The break in the slope at lower temperatures is more visible for x > 5 x 10e3simply because the high temperature linear portion became wider. The new, higher values of Ed for the two regimesare plotted in Fig. 4 (see dark circles for T > 140K and dark squares for T < 100K). The straight dashed lines in Fig. 3 have slopes E,,/2e correspondingto the new values of Ed for T > 140K. These lines seem to be a fairly good continuation of the behavior of a in the high temperature range. Tbe values of a, (the extrapolations of the dashed lines to l/T = 0) range between - 200 to + 50V/K. This corresponds to - 0.6< In (NC/n)t 6 < 2.3 (see eqn. 3;

k/e = 86 @V/K).Since In NC/n> 1, then s = S -2.5 (see eqn. 4) must be negative. This is consistent with the negative temperature factor of the mobility. Inspection of the plots of a vs l/T given in Ref. [ 11shows that much more positive a, is obtained for x > 0.05but there, the plot is over a very narrow range of temperatures. Assumingthat NCand 6 do not depend on x we obtain from eqn (5) that AaJAx = - (K/2f_?)(A In x). The observed spread of about 250pV/K is twice the expected spread of a,, mainly due to ~~(0.0025).Increasing y to about - 3 for this concentration alone (thus increasing Ed from 0.18eV to 0.21eV) can increase the agreement. However, the quality of the Seebeck coefficient measurements does not call for the introduction of an additionalfittingparameter. Figure 4(b) represents the values of (~J~,,,,)~(/3Ncl) found from the extrapolation to l/T = 0 of the modified r( T,/T)* in the two regimes (dark circles for T > 140K and dark squares for T < lOOK).For comparison, the values of this factor obtained for y = 0 are also plotted (open circles for T > 140K and open squares for T < 100K). It is immediately seen that the effect of the modificationon the preexponentialterm for T > 140K is at most a factor of two. Its main effect consists in makingthis term less dependent on X.Its average value is 0.67+ 0.10, practically equal to the value of pL,,/~,,,, (see Section 4.a), which leaves fiNC,= l(N,, 52 for p = (l/2)). This value is consistent with the prediction

ElectricalandSeebeckeffectmeasurements in NbdopedV02

211

high effective mass of the electrons. The measurements in pure VO, produced a low limit N,(X) > 1 per Vanadiumatom while the simple analysis of the combined resistance and Seebeck measurements in V,_,Nb,02 produced BN,(T,) = 1 where B is the spin degeneracy factor of the ground state donor level. The (~.l/~,,)&?Ncr) found from the modified and the unpresent measurementscould not separate /3 from N, but since ,S is smaller than one (/3= l/2 for an s type donor modified curves drop with increasing x. The results of the level) this result is consistentwith the previous low limit. Seebeck coefficient are of little help in this regime due to Thus it is believed that the uncertainty in NC has been their poor quality. One possible reason for the drop of the reduced in this work from about two orders of magnitude mobility with x is the contribution of the (mainly nonionized) impurities to scattering. This contribution is to a factor of 2.Thisis regardedtherefore as the mainresult of this work. Such high effective density of states may expected to become noticeable when the average distance between the donors becomes comparablewith the mean represent a spikein the density of states of the conduction band of V02 which contains many more states. free path determinedby the lattice scattering. Additionalresults obtained are the dependence of the The plots of the activation energies in both temperature regimes (before and after modificationby the ionizationenergy of the electrons from the donors E,, on function (Z/T)* show an initial drop of Ed with 1. This Nb concentration x. At low temperatures (T< 140K) effect is common for high doping levels. It is interesting highdopingcaused an additionaleffect whichis probably to observe this drop when Ed is plotted vs 3~(x) (the due to the competitionof impurity scatteringwith lattice inverse of the average distance between the donors). scattering.Althoughthese last results may be of interest This is done in the inserted plots at the bottom of Fig. 4. by themselvesthey were outside the scope of the present When the initial slopes of the plots (-0.49 < work. The variation of Ed with x complicatedthe analydEJd(x)“3 < - 0.34)are interpreted in terms of a repul- sis and reduced the accuracy of the results. It is hoped sive coulomb energy of the form kc*/(r) the value that the same type of measurements in samples with obtained for the dielectric constant is 10c,,< E< 1460. x -=c 2.5x 10m3will produce additionaland more accurate These limits of 6 are at least of the right order of results. For low enough concentrations the ionization energy and the mobility should not be dependent on x. magnitude[91. Moreover, the lower is x the wider is the exhaustion regime which by its nature produces additional infor5. CONCLUSIONSANDREMARKS mation. Independent determination of Ed by optical The main results of this work are: (a) The temperature dependence of the carrier con- measurementsare also very desirable. centration in semiconductingVO, is governed by the REFERENCES conductivitygap which is equal to the optical gap. (b) The electron mobilityin a wide temperatureregime 1. ViIleneuveG., BordetA., CasalotA., PougetJ. P., LaunoisH. and LedererP., 1. Phys. C/tern.Solids 33, 1953(1972). below the transition drops with increasingtemperature. 2. BlakemoreJ. S., Semiconductor Stutistics. PergamonPress, The temperature factor of the mobility has not been Oxford(1%2). separated from that of the effective density of states. 3. Heikes R. R. and Ure R. W., Jr., Thermoelectricity.Interscience,New York(l%l). However, the joint temperature factor T-* of the 4. BerglundC. N. and GuggenheimH. J., Phys. Reu. 185, 1022 product p,d(I(BN,)represents the least temperaturevari(1%9)and Ref. therein. ation of y because NC can only increase with tem- 5. LaddL. A., Officeof Naval ResearchTechnicalReport Nos. perature. Such a temperaturedependenceof c~sis typical HP-26and ARPA-41(1971)(unnublished). for lattice scattering.At T, the mobility in the semicon- 6. RosevearW. and Paul W.; ‘Bull.Am. Pby~. Sot. Ser. II, 15, (1970). ducting phase is smaller but comparablewith that in the 7. 316 CaruthersE. and KleinmanL., Phys. Rev. B7,3760(1973). metallicphase. 8. Paul W., Mat. Res. Bull. 5,691 (1970). (c) The effective density of states in the conduction 9. Zylbersztejn A., Pannetier B. and Merenda P., Phys. L&t. AMA, 145(1975). band is unusually high and consistent with an unusually NC> 1 made on the basis of the results from pure VOz (4b).This consistency is the mainargumentin support of the model. The estimated uncertainty in NC due to the experimental error and to a possible variation of y are not more than a factor of 2. In the low temperature regime, the values of