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Effect of structural misalignments on the temperature variation of basal plane electrical conductivity of graphite
1. Phys. Chem. Solids Vol. Printed in Great Britain.
42. pp. 433-437,
OWZ-3697/8l/05043~05$02.00/0 Pergdmon Press Ltd.
1981
EFFECT OF STRUCTURAL MISALIGNMENTS ON THE TEMPERATURE VARIATION OF BASAL PLANE ELECTRICAL CONDUCTIVITY OF GRAPHITE R. BHAITACHARYAand A. K. DUITA Department of Magnetism, Indian Association for the Cultivation of Science, Jadavpur, Calcutta 700 032, India (Received 10 March 1980;accepted in reuised form 18 August
1980)
Abstract-The amount and extent of structural misaligments in natural graphite crystals have been determined, and the temperature variation of the basal plane electrical conductivity ((rl) of naturally occurring graphite has also been studied from 300 to WK. The conductivity (uL) has been found to obey a law uL~(l/T) down to a certain temperature 0 (0 varying from sample to sample), below which the variation deviates from linearity towards lower values of oI. This behaviour, which was earlier thought to be a characteristic of graphite and whose origin could not be traced. has been shown to be an effect of the structural misalignments usually present in natural samples of graphite.
peratures. Some details of these measurements and the results of observation have already been reportedIl31, and it has been found that while in the case of magnetic ductivity. Its basal plane conductivity (rl has been reported by different workers[l-71 to vary from 10’ to susceptibility an attempt could be made to find a quantitative correlation between susceptibility and misalign10”a-’ cm-‘, and (~11, the conductivity in a perpendicular ment, no such attempt was possible in the case of direction, from 1 to 10’ R-’ cm-‘. (TVand its temperature electrical conductivity. However, the observation that (TV variation have been the subject of many experimental and theoretical investigations. At high temperature (TV is appreciably affected by these misalignments may have follows a linear variation with reciprocal temperature, serious consequences on the characteristics of free but at low temperatures there are deviations from charge carriers in graphite as deduced from their depenlinearity and a knee is observed in the curve at the point dence on uL and its temperature variation. The situation of deviation. Although this knee can be detected in the therefore needs careful analysis so that some light may observations of practically all workers (although at be thrown on the possible mechanism by which electrical different temperatures)[6-91, none has given any special conductivity in graphite is affected by these misalignattention to this point except Soule and ments. The present communication gives an account of McClure [7,9, IO]. The former considered this knee to be an attempt towards this end with the relevent obsera characteristic of graphite, and from its position the vations taken with some fresh samples of Ceylon value of y2, in the overlap band model was calculated graphite. and found to be in good agreement with that obtained EXPERIMRNTAL PROCEDURES ANDRESULTS from DeHaas-VanAlphen measurements. Simultaneously with these investigations it has been shown The experimental procedure consisted of the deterthat natural graphite crystals, with which all in- mination of the amount of misalignment and the basal vestigations have been carried out, possess appreciable plane electrical conductivity gI. The degree and extent amounts of misalignment? among a considerable of misalignment present in different samples of Ceylon number of crystallites present[l, 11-131. It has been graphite has been estimated by observing the spread of shown further than these misalignments appreciably the basal (0002) plane X-ray reflection with the help of a affect the electrical and magnetic properties of decoupled X-ray diffractometer. Three samples with graphite [8,131. Taking different specimens Tsuzuku [ 141 widely varying amounts of misalignment were chosen as has found that the value of PII moves towards the in- our specimens. The details and results of X-ray studies trinsic value as misalignment and other stacking faults have already been published[l3], and are summerised in decrease more and more. In order to investigate Table 1 below. The basal plane electrical conductivity systematically the relationship between the misalign- (Us) has been measured over the temperature range ments and the abovementioned properties the present 90-300 K by the usual d.c. potentiometric technique, and authors have undertaken an extensive study for quan- the results are shown in Fig. 1, where clobs has been titative estimation of these misalignments and simul- plotted against l/T (T is the temperature). taneous determination of the principal electrical conducDISCUSSION tivities and magnetic susceptibilities at different temIt is observed from Fig. 1 that the values of (TVare tBy misalignmentit is understoodthat small crystilites inside different for different samples, being larger for specimens with lesser amounts of misalignment, and that oL the natural sin& crystal are oriented with their c-axes at slightly differentangles to the c-axis of the natural crystal. increases with lowering of temperature. Further, the INTRODUCTION Graphite is well known for its anistropic electrical con-
PCS Vol. 42, No.
5-G
433
R. BHATTACHARYA and A. K. DUITA
fore be written as u~,~~=u~~(T)-Au,(T)
for
T<8
(la)
ulobr = u,“(T)-AU,(~)
for
Ta 8
(lb)
where Au,(O) is the value of Au,(T) at 7’ = 0 and is a constant for T 2 8. If the linear portion of the curve in Fig. 1 is extrapolated to the low temperature region it becomes evident that in spite of an overall increase, the value of ulobs decreases more and more from the extrapolated straight line as the temperature is lowered. Representing this change (difference between observed u, and extrapolated uI in the region T < 0) by Su, we find from relations (la) and (lb) that
,,i
0
2.3
5.0
,7.5
1.0
12.5
lo,
T Fig. I. Temperature variation of basal plane electrical conductivity (oJ for different specimens.
curves in Fig. I are in general linear at higher temperatures but deviate from linearity at lower temperatures. The temperature 0 at which this deviation starts (the position of the “knee” in the curve) is not the same for all samples but is higher, the larger the misalignment in the sample.7 The observed conductivity may therefore be expressed by the simple relation
(1)
UlObS = UL p-Aa,
where gIp varies linearly with 1/T and Aa, is the change in oLP. It is obvious from Fig. 1 that the nature of A(T~is such that (i) Aa, varies with temperature in the region 7’~ 0 and causes the curve ulobs vs l/T to deviate from linearity for T < 0 (ii) it is independent of temperature for T > 0 and is different in different crystals. In the two regions of temperature, eqn (1) may theretin XI, the most misaligned sample, no linear portion could be observed (see Fig. 1).
ISuLIT
(2a)
J8uIITaB = 0.
(2b)
In Fig. 2 a plot of log )6u,l against T (for T < 19)shows that the curves for all the samples are straight lines. Considering these facts and that the sign of Su, is negative in the region under consideration, So, can be expressed as &, = A em”(@-T)- A, i.e. AuL( T) - Au,( 0) = A e-0(e-r) - A :.
Au,(T)=Ae-“(8-r) Au,(O) = A
for
for T> 0 .
SamDIe X6
5”
c*t X,$
8 14”
Half width of intensity distribution
0
n
2 5” 9
121K 208K 300K
0.384x IO-’ 0.121x10-’ 0.076x IO-’
tin addition to misalignment, twinning is also present.
iFor X, no linear portion has been observed in the crlobsvs l/T curve in the region of temperature studied. To find SgI, the deviation from a straight line parallel to the linear portion of X6 (Fig. 1) was measured.
T
Equation (4) indicates (in combination with eqn 1) that above the temperature 0, Au, lowers the value of ulobS by an amount A without affecting its temperature variation, but for T< 0, it also affects the temperature variation of ulobs. The quantity “a” determines the rapidity with which the curve approaches linearity and is found to be greater the lesser the amount of misalignment present in the sample (Table 1). From the above it appears that an association exists between the structural misalignments present in the samples and the observed results. The observed deviation from linearity in the uloba vs l/T curve may also then be associated with the effect of these misalignments on the temperature variation of uI. In what follows we
Table 1. Extent of the distribution of misalignment
(3)
435
Structureal misaligments in natural graphite crystals
II 50
I
I
100
150
2(
0
Pressure + kg wt/cm2 Fig. 3. Dependence of oI/on pressure.
low so that the reproducibility of results was not affected. We observed that the pressure variation of conductivity follows an exponential law. Such a dependence of ~11on pressure may naturally be ascribed to the Tpresence of microgaps between the crystallites in the Fig. 2. Temperature variation of the deviations from linearity. specimen.t The potential inside such a crystal is different from that of a perfect single crystal and at the boundaries of the crystallites having microgaps between them shall show that the temperature variation of (TVcan be the carriers will also encounter potential barriers. satisfactorily explained if we consider the following The magnitude of these potential barriers will depend model for the crystals of graphite. on the dimensions of the microgaps and the angle of (1) A crystal of graphite contains a considerable orientation of the crystallites. Transport of carriers will number of misaligned crystallites, and therefore be affected in such a specimen mainly due to: (2) Such misalignment will be accompanied by the (1) lattice vibrations (2) impurities and (3) the potential presence of microgaps between the crystallites which are barrier between the crystalites..$ randomly distributed in the sample (we give below an Of the above three factors affecting conductivity the experimental observation made here which supports this second one is temperature independent and may be view). The first of these is well known[8,12-151. The presence neglected in discussing the temperature variation of globs. Since graphite is a semimetal, the effect of lattice of microgaps as envisaged above can be easily realised by studying the effect of pressure on the electrical convibrations is to decrease the conductivity linearly with ductivity of graphite. l/T as the temperature is raised. The effect of pressure on the conductivity of graphite We now consider the effect of barriers in the paths of along the c-axis has been studied up to a pressure flow of electrons mentioned under factor (3) above. Let - lo* Kg/cm*, and the results for two specimens are us now examine the effect of temperature on the ranshown in Fig. 3. The maximum pressure was sufficiently domly distributed microgaps inside the crystal. The thermal expansion coefficient of graphite along the c-axis (all) is much larger than that in a perpendicular direction tThe effect of pressure on the properties of graphite has been (a,), the ratio being all/a1 -20[21]. When a naturally studied by many workers[l6-191 upto 20 kb. They aimed at measuring the pressure coefficients of the properties of graphite occurring specimen of a graphite crystal is heated, the and therefore applied such high pressures to decrease the misalmisaligned crystallites expand randomly. Due to the ignment and other stacking faults to a minimum. They have different coefficients of expansion along different directherefore not performed studies under low pressures. It is tions of these crystallites the dimensions of the microevident that their studies will not give any idea about how the misalignment changes with pressure and the consequent changes gaps will continue to decrease with rise of temperature in the conductivity. As our aim is to find the effect of misalignuntil at a particular temperature depending on the ment on the conductivity of graphite, studies under low pressure amount and extent of misalignment, i.e. history of forare necessary. We therefore have not been able to compare our mation in nature, the microgaps assume a minimum results with those of the workers mentioned above. SThe contribution of the tensor component of or of the misalpossible value and do not change any further.8 The width igned crystallites along the basal plane will also affect the oiob,, of the barrier encountered by the moving carriers in such but this will cause only small changes in the numerical value of natural specimens of graphite therefore decreases with oLobsand leave the temperature variation practically unaffected, rise of temperature, and above a particular temperature because alI=+cl. §A similar phenomenon has been observed in naturally occurring remains constant. This gives rise to the A(T~of eqn (1). coal[l5,20]. The electrons and holes in the system encountering such
436
R. BHATTACHARYA and A. K. DLJITA
barriers have to tunnel through them for transport. Now in the case of tunneling the transmission coefficient of the carriers varies exponentially with the width of the barrier[22] and is given by pl = eecw
(5)
where w is the width of the barrier and c a constant. Since w decreases with rise of temperature, transmission of free carriers is more and more favoured, thereby tending to increase the value of ulobs with rise of temperature. This will, however, be opposed by the scattering due to phonons, which will decreases the value of ulobs with rise of temperature. From our experimental observations it is obvious that the second effect is larger, the observed resultant effect being a decrease of (Tlobs with rise of temperature. Above the particular temperature where w becomes a constant, the effect of barriers will become temperature independent, leaving only the phonon scattering to account for the temperature dependence, and so (Tl&s varies linearly with l/T. This is the nature of the temperature variation which has been observed experimentally. The particular temperature mentioned above corresponds to the experimentally observe temperature 8. Thus the presence of microgaps explains the general nature of the variation of (Tl&s with l/T. In view of the above proposition it is obvious that at constant temperature the electrical conductivity should depend on the voltage applied for measuring the conductivity. This effect was studied and it has been found that conductivity increases with the applied voltage initially and then becomes constant (Fig. 4). The observation therefore favours the explanation given above. It is now appropriate to examine the consequences of our suggestion that the presence of microgaps is responsible for the anomalous behaviour in the temperature variation of (Tlobsin graphite.
CONSEQUENCEs OF THE PRFSENCEOF MICROGAPS VARYINGWITHTEMPERATURF,
Let us take the case of one representative microgap in between two crystallites in a specimen of graphite and let (Ybe the coefficient of thermal expansion of the crystallites along the direction of the gap. Let the dimensions of the crystallites at the temperature 0 be lo and that of the gap oo. Then at any temperature T < 8 p* = e-cw, = e-ctwO+lOn(B-T) =p
e-b(B-T)
(64
where P =eCwo
and
b = cloa
In the region of temperature 7’3 0 where the dimension of the gap becomes constant (= wo) (6b) This implies that for T 2 0, the effect of misalignment is to lower the value of ml&s by a constant amount proportional to (1 - P) without affecting its temperature variation, and for T < 8, (Tlobsis lowered by an at’UOUt’tt proportional to (1 - P e- b(e--T)) which increases as the temperature is lowered. This is what has been observed experimentally, and eqns (6a) and (6b) resemble eqns (4) where (1 - P), (1 - Pt) and b correspond to Acial(0), Au~( T), and a, respectively. It should be noted here that large values of P or P, means smaller values of the decrease in (Tl&,s, i.e. smaller ValUeS of Aul(0) and Aul( T) and vice versa. Again from eqns (6a) and (6b), P, depends on w. and b, which are constants for the specimen under consideration. We can reasonably believe that in crystals
9
i Current
pA
Fig. 4. Dependence of current through the sample on applied voltage.
Structuralmisalignmentsin natural graphite crystals with lesser amount of misalignment, w. will be smaller and lo larger than those in a crystal with a larger amount of misalignment. Hence the expression P = eeFYOsuggests that for crystals with less imperfections, P is larger and so Aul(0) in eqn (lb) is smaller than in the case of crystal with more imperfections. Therefore the value of globs will be larger for the former type of crystals than for the latter, as has been observed experimentally. Similarly, if we plot log Pt vs (19- T) as per eqn (6a) the slope of the curve is given by - b = - doa, showing that the slope is larger in less imperfect crystals, as has also been observed experimentally (Table 1). Further, as the microgap decreases due to the thermal expansion of the crystallites constituting the specimen, those with larger L and smaller w. (i.e. crystals with lesser misalignment) will attain the minimum allowable dimension for the microgaps at a comparatively lower temperature than crystals with smaller &and larger wo.This explains the experimental observation that 0 is lower for crystals with lesser misalignment (Table 1). SUMMARY ANDCONCLUSIONS
It is seen from the above discussion that consideration of the presence of microgaps in natural crystals of graphite containing misaligned crystallites can successfully explain the experimental features of the basal plane electrical conductivity. (1) The conductivity is larger for crystals containing lesser amounts of misalignment. (2) The value of “a” in eqn (6) which given the rapidity with which the temperature variation of ulobs approaches a linear relation with l/T, is larger for crystals with lesser amounts of misalignment. (3) The temperature 0, the position of the knee in the curve ulobs vs l/T, is dependent on the amount and extent of misalignment, and is lower for crystals with lesser amounts of misalignment. (4) For temperatures above 0, (Tlobsobeys a linear relation with l/T. In view of the above findings and explanations, the
431
suggestion that the deviation from linearity of the curve uLObSvs l/T is a characteristic property of graphite, does not appear to be properly justified. Instead it appears to be due to the presence of microgaps and hence misalignment in the naturally occurring samples. It may be mentioned here that a more direct proof of the presence of microgaps might be possible by a study of the specimens under an electron microscope, where one could perhaps “see” the microgaps. Unfortunately, such facilities are not available to us. REFERENCES 1.
Krishnan K. S. and Ganguli N., Proc. Roy. Sot. A177, 168 (1941).
2. Dutta A. K., Phys. Rev. 90, 187(1953). Dutta A. K. and Chowdhury A., Ind. J. Phys. 28,312 (1954). Kinchin G. H., Proc. Roy. Sot. A217,9 (1953). Primak W., Phys. Rev. 103.541 (1%9). Primak W. andFuchs L. H., Phis. Rkv. 95,22 (1954). Soule D. E., Phys. Rev. 112,698 (1958). Bhattacharya R., Thesis, University of Calcutta (1974). Haerring R. R. and Mrozowski S., Prowess of Semiconducfors (Edited by A. F. Gibson), Vol. 5,~. 27j. Heywood & Company Ltd., London (l%O). 10. McClure J. W., Phys. Rei. lli, 715 (1958). 11. Ray S., Ind. J. Phys. 33, 282 (1959). 12. Bhattacharya R. and Ray S., Ind. J. Phys. 39,300 (1%5). 13. Bhattacharya R. and Dutta A. K., Ind. J. Phys. 50, 495 (1976). 14. Tsuzuku T., Carbon 17,293 (1979). 15. Mrozowski S., Proc. 1st and 2nd Carbon Conference, p. 33. University of Buffalo (1956). 16. Noto K. and Tsuzuku T., J. Phys. Sot. Japan 35,1649 (1973). 17. Dillon R. O., Spain I. L. and McClure J. W., J. Phys. Chem. Solids 38,635 (1977). 18. Anderson J. R., O’Sullivan W. J., Schirber J. E. and Soul D. E., Phys. Rev. 164, 1038(1%7). 19. Itskevitch E. S. and Fisher L. M., Sov. Phys. JETP Lett. 5, 114(1%7). 20. Collins F. M., Proc. 3rd Carbon Conference, p. 659. University of Buffalo (1957). 21. Ubbelohde A. R. and Lewis F. A., Graphite and Its Crystal Compounds, p. 50. Clarendon Press, Oxford (l%O). 22. Bloch F., Phys. Rev. B2, 109(1970). 3. 4. 5. 6. 7. 8. 9.
42. pp. 433-437,
OWZ-3697/8l/05043~05$02.00/0 Pergdmon Press Ltd.
1981
EFFECT OF STRUCTURAL MISALIGNMENTS ON THE TEMPERATURE VARIATION OF BASAL PLANE ELECTRICAL CONDUCTIVITY OF GRAPHITE R. BHAITACHARYAand A. K. DUITA Department of Magnetism, Indian Association for the Cultivation of Science, Jadavpur, Calcutta 700 032, India (Received 10 March 1980;accepted in reuised form 18 August
1980)
Abstract-The amount and extent of structural misaligments in natural graphite crystals have been determined, and the temperature variation of the basal plane electrical conductivity ((rl) of naturally occurring graphite has also been studied from 300 to WK. The conductivity (uL) has been found to obey a law uL~(l/T) down to a certain temperature 0 (0 varying from sample to sample), below which the variation deviates from linearity towards lower values of oI. This behaviour, which was earlier thought to be a characteristic of graphite and whose origin could not be traced. has been shown to be an effect of the structural misalignments usually present in natural samples of graphite.
peratures. Some details of these measurements and the results of observation have already been reportedIl31, and it has been found that while in the case of magnetic ductivity. Its basal plane conductivity (rl has been reported by different workers[l-71 to vary from 10’ to susceptibility an attempt could be made to find a quantitative correlation between susceptibility and misalign10”a-’ cm-‘, and (~11, the conductivity in a perpendicular ment, no such attempt was possible in the case of direction, from 1 to 10’ R-’ cm-‘. (TVand its temperature electrical conductivity. However, the observation that (TV variation have been the subject of many experimental and theoretical investigations. At high temperature (TV is appreciably affected by these misalignments may have follows a linear variation with reciprocal temperature, serious consequences on the characteristics of free but at low temperatures there are deviations from charge carriers in graphite as deduced from their depenlinearity and a knee is observed in the curve at the point dence on uL and its temperature variation. The situation of deviation. Although this knee can be detected in the therefore needs careful analysis so that some light may observations of practically all workers (although at be thrown on the possible mechanism by which electrical different temperatures)[6-91, none has given any special conductivity in graphite is affected by these misalignattention to this point except Soule and ments. The present communication gives an account of McClure [7,9, IO]. The former considered this knee to be an attempt towards this end with the relevent obsera characteristic of graphite, and from its position the vations taken with some fresh samples of Ceylon value of y2, in the overlap band model was calculated graphite. and found to be in good agreement with that obtained EXPERIMRNTAL PROCEDURES ANDRESULTS from DeHaas-VanAlphen measurements. Simultaneously with these investigations it has been shown The experimental procedure consisted of the deterthat natural graphite crystals, with which all in- mination of the amount of misalignment and the basal vestigations have been carried out, possess appreciable plane electrical conductivity gI. The degree and extent amounts of misalignment? among a considerable of misalignment present in different samples of Ceylon number of crystallites present[l, 11-131. It has been graphite has been estimated by observing the spread of shown further than these misalignments appreciably the basal (0002) plane X-ray reflection with the help of a affect the electrical and magnetic properties of decoupled X-ray diffractometer. Three samples with graphite [8,131. Taking different specimens Tsuzuku [ 141 widely varying amounts of misalignment were chosen as has found that the value of PII moves towards the in- our specimens. The details and results of X-ray studies trinsic value as misalignment and other stacking faults have already been published[l3], and are summerised in decrease more and more. In order to investigate Table 1 below. The basal plane electrical conductivity systematically the relationship between the misalign- (Us) has been measured over the temperature range ments and the abovementioned properties the present 90-300 K by the usual d.c. potentiometric technique, and authors have undertaken an extensive study for quan- the results are shown in Fig. 1, where clobs has been titative estimation of these misalignments and simul- plotted against l/T (T is the temperature). taneous determination of the principal electrical conducDISCUSSION tivities and magnetic susceptibilities at different temIt is observed from Fig. 1 that the values of (TVare tBy misalignmentit is understoodthat small crystilites inside different for different samples, being larger for specimens with lesser amounts of misalignment, and that oL the natural sin& crystal are oriented with their c-axes at slightly differentangles to the c-axis of the natural crystal. increases with lowering of temperature. Further, the INTRODUCTION Graphite is well known for its anistropic electrical con-
PCS Vol. 42, No.
5-G
433
R. BHATTACHARYA and A. K. DUITA
fore be written as u~,~~=u~~(T)-Au,(T)
for
T<8
(la)
ulobr = u,“(T)-AU,(~)
for
Ta 8
(lb)
where Au,(O) is the value of Au,(T) at 7’ = 0 and is a constant for T 2 8. If the linear portion of the curve in Fig. 1 is extrapolated to the low temperature region it becomes evident that in spite of an overall increase, the value of ulobs decreases more and more from the extrapolated straight line as the temperature is lowered. Representing this change (difference between observed u, and extrapolated uI in the region T < 0) by Su, we find from relations (la) and (lb) that
,,i
0
2.3
5.0
,7.5
1.0
12.5
lo,
T Fig. I. Temperature variation of basal plane electrical conductivity (oJ for different specimens.
curves in Fig. I are in general linear at higher temperatures but deviate from linearity at lower temperatures. The temperature 0 at which this deviation starts (the position of the “knee” in the curve) is not the same for all samples but is higher, the larger the misalignment in the sample.7 The observed conductivity may therefore be expressed by the simple relation
(1)
UlObS = UL p-Aa,
where gIp varies linearly with 1/T and Aa, is the change in oLP. It is obvious from Fig. 1 that the nature of A(T~is such that (i) Aa, varies with temperature in the region 7’~ 0 and causes the curve ulobs vs l/T to deviate from linearity for T < 0 (ii) it is independent of temperature for T > 0 and is different in different crystals. In the two regions of temperature, eqn (1) may theretin XI, the most misaligned sample, no linear portion could be observed (see Fig. 1).
ISuLIT
(2a)
J8uIITaB = 0.
(2b)
In Fig. 2 a plot of log )6u,l against T (for T < 19)shows that the curves for all the samples are straight lines. Considering these facts and that the sign of Su, is negative in the region under consideration, So, can be expressed as &, = A em”(@-T)- A, i.e. AuL( T) - Au,( 0) = A e-0(e-r) - A :.
Au,(T)=Ae-“(8-r) Au,(O) = A
for
for T> 0 .
SamDIe X6
5”
c*t X,$
8 14”
Half width of intensity distribution
0
n
2 5” 9
121K 208K 300K
0.384x IO-’ 0.121x10-’ 0.076x IO-’
tin addition to misalignment, twinning is also present.
iFor X, no linear portion has been observed in the crlobsvs l/T curve in the region of temperature studied. To find SgI, the deviation from a straight line parallel to the linear portion of X6 (Fig. 1) was measured.
T
Equation (4) indicates (in combination with eqn 1) that above the temperature 0, Au, lowers the value of ulobS by an amount A without affecting its temperature variation, but for T< 0, it also affects the temperature variation of ulobs. The quantity “a” determines the rapidity with which the curve approaches linearity and is found to be greater the lesser the amount of misalignment present in the sample (Table 1). From the above it appears that an association exists between the structural misalignments present in the samples and the observed results. The observed deviation from linearity in the uloba vs l/T curve may also then be associated with the effect of these misalignments on the temperature variation of uI. In what follows we
Table 1. Extent of the distribution of misalignment
(3)
435
Structureal misaligments in natural graphite crystals
II 50
I
I
100
150
2(
0
Pressure + kg wt/cm2 Fig. 3. Dependence of oI/on pressure.
low so that the reproducibility of results was not affected. We observed that the pressure variation of conductivity follows an exponential law. Such a dependence of ~11on pressure may naturally be ascribed to the Tpresence of microgaps between the crystallites in the Fig. 2. Temperature variation of the deviations from linearity. specimen.t The potential inside such a crystal is different from that of a perfect single crystal and at the boundaries of the crystallites having microgaps between them shall show that the temperature variation of (TVcan be the carriers will also encounter potential barriers. satisfactorily explained if we consider the following The magnitude of these potential barriers will depend model for the crystals of graphite. on the dimensions of the microgaps and the angle of (1) A crystal of graphite contains a considerable orientation of the crystallites. Transport of carriers will number of misaligned crystallites, and therefore be affected in such a specimen mainly due to: (2) Such misalignment will be accompanied by the (1) lattice vibrations (2) impurities and (3) the potential presence of microgaps between the crystallites which are barrier between the crystalites..$ randomly distributed in the sample (we give below an Of the above three factors affecting conductivity the experimental observation made here which supports this second one is temperature independent and may be view). The first of these is well known[8,12-151. The presence neglected in discussing the temperature variation of globs. Since graphite is a semimetal, the effect of lattice of microgaps as envisaged above can be easily realised by studying the effect of pressure on the electrical convibrations is to decrease the conductivity linearly with ductivity of graphite. l/T as the temperature is raised. The effect of pressure on the conductivity of graphite We now consider the effect of barriers in the paths of along the c-axis has been studied up to a pressure flow of electrons mentioned under factor (3) above. Let - lo* Kg/cm*, and the results for two specimens are us now examine the effect of temperature on the ranshown in Fig. 3. The maximum pressure was sufficiently domly distributed microgaps inside the crystal. The thermal expansion coefficient of graphite along the c-axis (all) is much larger than that in a perpendicular direction tThe effect of pressure on the properties of graphite has been (a,), the ratio being all/a1 -20[21]. When a naturally studied by many workers[l6-191 upto 20 kb. They aimed at measuring the pressure coefficients of the properties of graphite occurring specimen of a graphite crystal is heated, the and therefore applied such high pressures to decrease the misalmisaligned crystallites expand randomly. Due to the ignment and other stacking faults to a minimum. They have different coefficients of expansion along different directherefore not performed studies under low pressures. It is tions of these crystallites the dimensions of the microevident that their studies will not give any idea about how the misalignment changes with pressure and the consequent changes gaps will continue to decrease with rise of temperature in the conductivity. As our aim is to find the effect of misalignuntil at a particular temperature depending on the ment on the conductivity of graphite, studies under low pressure amount and extent of misalignment, i.e. history of forare necessary. We therefore have not been able to compare our mation in nature, the microgaps assume a minimum results with those of the workers mentioned above. SThe contribution of the tensor component of or of the misalpossible value and do not change any further.8 The width igned crystallites along the basal plane will also affect the oiob,, of the barrier encountered by the moving carriers in such but this will cause only small changes in the numerical value of natural specimens of graphite therefore decreases with oLobsand leave the temperature variation practically unaffected, rise of temperature, and above a particular temperature because alI=+cl. §A similar phenomenon has been observed in naturally occurring remains constant. This gives rise to the A(T~of eqn (1). coal[l5,20]. The electrons and holes in the system encountering such
436
R. BHATTACHARYA and A. K. DLJITA
barriers have to tunnel through them for transport. Now in the case of tunneling the transmission coefficient of the carriers varies exponentially with the width of the barrier[22] and is given by pl = eecw
(5)
where w is the width of the barrier and c a constant. Since w decreases with rise of temperature, transmission of free carriers is more and more favoured, thereby tending to increase the value of ulobs with rise of temperature. This will, however, be opposed by the scattering due to phonons, which will decreases the value of ulobs with rise of temperature. From our experimental observations it is obvious that the second effect is larger, the observed resultant effect being a decrease of (Tlobs with rise of temperature. Above the particular temperature where w becomes a constant, the effect of barriers will become temperature independent, leaving only the phonon scattering to account for the temperature dependence, and so (Tl&s varies linearly with l/T. This is the nature of the temperature variation which has been observed experimentally. The particular temperature mentioned above corresponds to the experimentally observe temperature 8. Thus the presence of microgaps explains the general nature of the variation of (Tl&s with l/T. In view of the above proposition it is obvious that at constant temperature the electrical conductivity should depend on the voltage applied for measuring the conductivity. This effect was studied and it has been found that conductivity increases with the applied voltage initially and then becomes constant (Fig. 4). The observation therefore favours the explanation given above. It is now appropriate to examine the consequences of our suggestion that the presence of microgaps is responsible for the anomalous behaviour in the temperature variation of (Tlobsin graphite.
CONSEQUENCEs OF THE PRFSENCEOF MICROGAPS VARYINGWITHTEMPERATURF,
Let us take the case of one representative microgap in between two crystallites in a specimen of graphite and let (Ybe the coefficient of thermal expansion of the crystallites along the direction of the gap. Let the dimensions of the crystallites at the temperature 0 be lo and that of the gap oo. Then at any temperature T < 8 p* = e-cw, = e-ctwO+lOn(B-T) =p
e-b(B-T)
(64
where P =eCwo
and
b = cloa
In the region of temperature 7’3 0 where the dimension of the gap becomes constant (= wo) (6b) This implies that for T 2 0, the effect of misalignment is to lower the value of ml&s by a constant amount proportional to (1 - P) without affecting its temperature variation, and for T < 8, (Tlobsis lowered by an at’UOUt’tt proportional to (1 - P e- b(e--T)) which increases as the temperature is lowered. This is what has been observed experimentally, and eqns (6a) and (6b) resemble eqns (4) where (1 - P), (1 - Pt) and b correspond to Acial(0), Au~( T), and a, respectively. It should be noted here that large values of P or P, means smaller values of the decrease in (Tl&,s, i.e. smaller ValUeS of Aul(0) and Aul( T) and vice versa. Again from eqns (6a) and (6b), P, depends on w. and b, which are constants for the specimen under consideration. We can reasonably believe that in crystals
9
i Current
pA
Fig. 4. Dependence of current through the sample on applied voltage.
Structuralmisalignmentsin natural graphite crystals with lesser amount of misalignment, w. will be smaller and lo larger than those in a crystal with a larger amount of misalignment. Hence the expression P = eeFYOsuggests that for crystals with less imperfections, P is larger and so Aul(0) in eqn (lb) is smaller than in the case of crystal with more imperfections. Therefore the value of globs will be larger for the former type of crystals than for the latter, as has been observed experimentally. Similarly, if we plot log Pt vs (19- T) as per eqn (6a) the slope of the curve is given by - b = - doa, showing that the slope is larger in less imperfect crystals, as has also been observed experimentally (Table 1). Further, as the microgap decreases due to the thermal expansion of the crystallites constituting the specimen, those with larger L and smaller w. (i.e. crystals with lesser misalignment) will attain the minimum allowable dimension for the microgaps at a comparatively lower temperature than crystals with smaller &and larger wo.This explains the experimental observation that 0 is lower for crystals with lesser misalignment (Table 1). SUMMARY ANDCONCLUSIONS
It is seen from the above discussion that consideration of the presence of microgaps in natural crystals of graphite containing misaligned crystallites can successfully explain the experimental features of the basal plane electrical conductivity. (1) The conductivity is larger for crystals containing lesser amounts of misalignment. (2) The value of “a” in eqn (6) which given the rapidity with which the temperature variation of ulobs approaches a linear relation with l/T, is larger for crystals with lesser amounts of misalignment. (3) The temperature 0, the position of the knee in the curve ulobs vs l/T, is dependent on the amount and extent of misalignment, and is lower for crystals with lesser amounts of misalignment. (4) For temperatures above 0, (Tlobsobeys a linear relation with l/T. In view of the above findings and explanations, the
431
suggestion that the deviation from linearity of the curve uLObSvs l/T is a characteristic property of graphite, does not appear to be properly justified. Instead it appears to be due to the presence of microgaps and hence misalignment in the naturally occurring samples. It may be mentioned here that a more direct proof of the presence of microgaps might be possible by a study of the specimens under an electron microscope, where one could perhaps “see” the microgaps. Unfortunately, such facilities are not available to us. REFERENCES 1.
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