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Electronic properties and band model of carbons
ELECTRONIC
Carbon Research
PROPERTIES AND BAND MODEL OF CARBONS*
S. MROZOWSKI Laboratory and Department of Physics, State University of New York at Buffalo, Buffalo, New York, 14214 U.S.A. (Received
12June
1970)
Abstract-The early electronic energy band model of carbon, as proposed in 1950, and its later modifications such as faster collapse of energy gap and the presence of band overlap in graphites are recalled to introduce the main topic of the paper. The problem of the band structure in the range between the collapse of the energy gap and the development of the overlap, that is for fine crystalline turbostratic and for disordered structures, was in recent years attacked in our laboratory by four different experimental techniques. Two of these turned out to furnish interesting results, which unfortunately, cannot yet be related to the band structure. However, the two other techniques, the electron spin resonance and the Hall coefficient studies, have yielded definite and concordant information: starting from well graphitized material as the disorder increases the overlap decreases at first, but after passing through a minimum it starts to increase again. The possible sources of such overlap and the general requirements for a band theory of disordered structures are discussed. Two additional new results, namely, the absence of the Hall minimum at HTT 1650” for the soft Kendall coke, and the reversal in the effect of boronation and of H,SO., doping below HTT 185O”C, are mentioned.
I‘wenty years ago, when I was just starting to organize carbon research at the University of Buffalo, the carbon field was wide open to the physicists. The pioneering researches of Warren and Riley on X-ray structure, of Ganguli and Krishnan on diamagmetic susceptibility and of Coulson and Wallace on electronic band structure of a two dimensional graphite lattice having demonstrated 11ow interesting and fruitful such work can be, we, that is myself and my collaborators fanned out at first in many directions of exploration. As the carbon field started to attract more attention and the competition rose, and as obtaining meaningful results began to require more precise and sophisticated apparatus and better standardization *George Skakel Memorial Award Lecture, 9th Biennial Conference on Carbon, Boston Massachusetts, June 19, 1969.
experimental have of techniques, we gradually narrowed our efforts to one, still broad field of carbon physics, that is to electronic properties. In this paper, I would like to discuss some of our more recent experimental findings and describe the way they have modified our thinking about the electronic processes and the band structure of carbons. Today’s physics of the solid state is basically still physics of crystals. This was true even more so 20 years ago. But carbons are disortlered structuresL: In the first stages of formation from organic compounds they rescmhle polymers, in some cases with preferential organization and quasi-crystallisation occurring already very early in their evolution (raw cokes). In later stages, a polycrystalline structure develops changing gradually from turbostratic into graphitic order. It is only at the final stages of growth of crystallites that $17
CARVd.!lNo.%A
S. MROZOWSKI
98
carbons can be considered as agglomeration of graphite crystals, and band consideration can be safely applied. At the other end of the evolution one can visualize the process of carbonization starting from an organic crystal via increasing aromatic ring condensation. Akamatu and Inokuchi have shown that such organic crystals are intrinsic semiconductors with an energy gap rapidly decreasing with increase of the molecular size. Thus at the two ends of the process, one had reasonably well established and theoretically supported band models of the solids, with a wide variety of disordered materials in various stages of organisation linking the two ends for which an electronic model based on theoretical considerations was (and still is) not available. As it frequently happens in the course of events, when a theoretical solution to a problem is unattainable, an experimental physicist must step in with a temporary substitute. This will permit him to pick out questions to be asked of nature and to inter-
\
Aromotit
pret the answers obtained in terms of a model so that further questions can be formulated. Twenty years ago, I tried to do just that, and as the reader will see later, I am still continuing these efforts. To link the two ends of the carbonization-graphitization series, I have assumed that a band model is applicable throughout and that the transition is gradual [ 1,2]. The assumption of the applicability of the band model to disordered systems was not so unjustified at the time as it might have sounded: changes in electric conductivity of metals and semiconductors upon melting were known to be not very great, in spite of the complete collapse of periodic order. Today, there are many more indications available, that for disordered systems something of a kind of “energy bands” is present. Some modifications necessary to be introduced into the classical energy band picture to fit it to disordered solids will be suggested at the end of this paper. Figure 1, shows the original band model as proposed for carbons in 1950. On the left
crystal Chors Raw
-
Corbons
Calcined
coke
-
coke
Polycrystolline graphite
Single crystal
n(E)\
HTT
I 700
I
1000
I 1500
do0
25hO
30bo
OC
Fig. 1. The electronic energy band model for carbons covering the transition from simple organic crystals through disordered structures of chars and carbons up to single crystal graphite, as proposed in 1950-2 [ 1,2]. The depression of the Fermi level is caused by intrinsic traps of unknown origin, which become eliminated in graphitization. The corresponding heattreatment temperatures for a soft type carbon are indicated.
ELECTRONIC
PROPEKI‘IES
hand side the IT energy levels of aromatic molecules broadened into bands are indicated. The interband distances (gaps) decrease as the molecular size increases. Since in chars, mixtures of various sizes are present, it was assumed that the bands coalesce into two wide r-bands: an electron occupied and an empty one, which are separated by an energy gap, rapidly decreasing as the size of the aromatic systems in the carbon increases with advancing carbonization and graphitization, ending in the touching band model of a single graphite crystal. To obtain a qualitative description of the electronic properties of carbons introduction of a second assumption was necessary: that in the process of carbonization (driving out peripheral volatiles) traps are formed which remove some of the electrons from the band, thus depressing the Fermi level, and making the carbons essentially hole conductors. These traps would be gradually eliminated as a result of growth of crystals in graphitization, the Fermi level returning to the midposition, as required by the Wallace’s model for the single graphite crystal. It was pointed out at the end of the early paper[2] that this band model for carbons should be applicable not only to the carbonization-graphitization transition but also to the reverse case when an order is being destroyed by atomic displacements. As well known, the subsequent work of ours and of others in a broad way supported these two assumptions. The gradual decrease of the energy gap and appearance of continuous absorption due to hole conduction were checked in infrared for chars[3]. Studies of electrical resistivity, Hall and thermoelectric effects in graphitization and in reverse processes such as neutron irradiation amply demonstrated the existence of traps. Still many years later, not only the nature of the traps is unknown but even their relation to the defects in the lattice and to the presence of the localized spin centers in carbons remain
OF (:ARBONS
!)!I
obscure. The progress seems disappointingl) slow. Very soon after having proposed the band model Fig. 1, I realized that it requires modification. At first I had mistakenly believed that the presence of a tiny energ gap is responsible for the low temperature dependence of the electric resistance in polycrystalline graphite [2 I. Sonle colleagrles pointed out to me however that the gap for aromatic systems decreases at such fast rate with increase in molecular si;le. that it should practically close on calcination of soft cokes (HTT - 1300°C). Furthermore, the actual decrease in the observed average gap should be even faster than that, the aromatic structures with various trap concentrations shifting relatively in energy scale val-ious amounts so to adjust their individual Fernli levels to the common Fermi level of the solid. Accordingly in the modified model Fig. 2. the gal) is indicated to disappear at full carbonization (after driving out of most of hydrogens). The second modification came about from a careful consideration of all known properties of graphites. This led me to conclude that the energy bands in real graphite do not just touch, as required by the Wallace’s m&lel. but must slightly overlap [4]. ‘Illis conclusion has been shown two years later to be correct by theoretical calculations of Slonczewski and Weiss and others. But at what stage of heat treatment does the overlap develop? Studies of the temperature and HTT dependence of the Hall coefficient in particular the IOU temperature studies by Chaberski 1.51, shop definitely that a partial overlap is alreacl~, present at HTl“s corresponditlg to 111~. positive maximum of the Hall coefficient (about HTT 21OOY: for sof’t carbons) thr* Fermi level moving at that stage into thca overlap region. On the other hanc-I, the overlap can not develop much 5ooncr, sincts as X-ray diffraction studies show, it is only in this heattreatment region that turbostratit crystals begin to transform slowly into graphiric order (N, 6. U, b). Since the hnd
S. MROZOWSKI
100
\ n (El
HTT
II
700
1000
I
I
1500
2000
I
2500
I 3000
-C
Fig. 2. The improved electronic band model for carbon, incorporating some early modifications, namely a faster collapse of the gap and the development of a band overlap in graphitization. The four boxes indicate the four .types of experimental research aimed at clarification of the band structure in thx transition region. overlap is the result of tridimensional order, the proper band model for turbostratic crystallites seemed to be the two dimensional touching band scheme. So it was then to be expected that the overlap starting from zero will increase gradually up to the graphite limit as more and more pairs of neighboring planes rotate into proper graphitic order inside of a turbostratic crystal. Our recent studies of the ESR and Hall effects seem to support this expectation, as will be discussed later. Correspondingly in Fig. 2 an increasing overlap is indicated, with the Fermi level entering the overlap at an early stage. What is the band structure of carbon like in the region between the closure of the gap and the development of the overlap? As experimental data piled up throughout the years, I had more and more doubts if a conventional band model could be used to describe the behavior of carbons. I do believe that Boy and Marchand[6] finally delivered the “coup de grace” to the convent.ional model, by taking various models with and without an energy gap and with various depressions of the Fermi level and by trying to fit the parameters so as to obtain a reasonable interpretation of electronic properties. They found that no proper values of para-
meters can be selected which would give simultaneously a correct description of three electronic properties considered: para, dia-magnetism and Hall coefficient. Thus the conclusion was clear: a band model for disordered systems if it does exist must be an unconventional one. During the sixties, myself and my collaborators have continued to attack vigorously the problem of electronic properties of carbons. The work was aided by acquisition of some good equipment in particular for ESR and for low temperature studies, but much more by the developed experimental technique for a rapid study of the band structure, consisting of introduction of acceptors and donors in steps and then lowering of the position of the Fermi level by gradually oxidizing out the donor. We have attacked the problem from four the magnetoresistsides; by investigating ante, the specific heat at low temperature, the electron spin resonance (ESR) and the Hall coefficient (Fig. 2). As well known, nature always repays hard work, but frequently not in the type of currency one expects. This happened to us in two out of four cases: in our work we have stumbled upon some interesting effects, but
ELECTRONIC
PROPERTIES
it turned out that no direct information about the band structure could be obtained from these experiments. The first case was the negative magnetoresistance. Since the time we have shown with Chaberski [7] that the negative magnetoresistance is an effect characteristic to carbons in heattreatment range just before the incipient graphitization, I have hoped that important information about the band structure of carbons could be extracted from such investigations. Detailed low temperature studies revealed a number of interesting facts, such as the low temperature turnaround point, the negative magnetoresistance turning positive at sufficiently low temperature[8,9], the saturation of the negative magnetoresistance higher magnetic at at low tields, its square field dependence fields, etc., the assembled data supporting the original supposition of the additivity of the two magnetoresistance effects of opposite signs 171. However, the experiments with doping have shown that while the positive magnetoresistance is readily destroyed by doping (which disturbs the graphitic order by separating the layers and shifts the Fermi level), the negative effect almost does not depend on the position of the Fermi level iI1 the valence as well as in the conduction band [lo, 1t]. Thus, the liegative magnetortGstance is an effect caused by scattering and is ;I measure of disorder (no negati\.e eftect iir graphite crystal [12]. It does not furnish XII? direct information concerning tile band structure. The second case is the specific heat. When \‘an der Hoeven and Keesom [ IS] have shown that the linear component of specific heat of graphite can be obtained with a reasonably good precision if data are obtained in the HeZl temperature range, we started preparations for such a project in the Ilope that measurements carried out on carbons would give us directly information about the density of energy states n(E) at the position of the Fermi level. Doping of the specimens
OF CARBONS
101
would widen the range of information obtained. However, even before our experiments were started the skies became clouded by the publication of a subsequent paper by Van der Hoeven, et al. [14] in which it was reported that whereas boron doping causes a theoretically expected change in the linear term the difference observed between the polycrystalline graphite and Madagascar single crystal graphite is too large to be band explainable in terms of electronic model, Our work on carbons has not only confirmed but amplified this discrepancy 115 I. As the heat-treatment (and degree of graphitization) of the polycrystalline material decreases the linear term increases much faster than any electronic band model would reasonably allow. More recent measurements of‘samples of raw carbon (H’l‘~1‘ 7OO”C),which do not exhibit electron conductivity have shown the presence of a very large linear component [16]. Consequentlv. most of the effect must be due to lattice \ihrations of’ ;I special kind, not yet theoreticallv understood. the situation becoming inverted relative to our original expectations: I;nowledge of the density of states at the Fermi level rl(E,,) from other source becoming necessary for the e\.aluation of the new and unexpected linear lattice contribution. So we are left fi)r the time being with onI>, two avenues of approach: the electroil slain resonance (ESR) and the Hall effect. I,et mc discuss the information furnished bv the E:SR investigations first. Essentially the I&R is giving very strai~ghtfor\~ard answers: the integrated absorption intensity gives the spin concentration in case of‘ localized spins antI roughly the density of‘ energy states at tI~(k Fermi level r@,.,) in case of electron and hf)Ic carriers. The fil3t approach IY;I~used tn. man\ investigators to e\.aluate the spin concentration in chars, the second In Wagoner and Singer [17, 181 to demon&te the car-ricr nature of the ESR in single crystals ant1 highly graphitized polycrystallirie materi;tls. In the late fif‘ties and early sixties. ;i
102
S. MROZOWSKI
number of people began to suspect that the ESR in carbons is due to both localized and carrier spin centers, and as soon as more precise temperature dependence data for the absorption became available, it was noticed that the total absorption intensity can be represented as a sum of a pure Curie term for localized spins and an approximately temperature independent Pauli term for conduction carriers*. However, not. having an explanation why only a single ESR line with a g-value varying from experiment to experiment is always observed, made such an assumption of the composite nature of the line and of the additivity of intensities of questionable validity. It is only after I was able to show that the variations in the g-value are resulting from the changes in relative contributions of carrier and localized spins to total paramagnetic susceptibility, the g-value and its temperature dependence being predictable on basis of a simple mixing formula presupposing an exchange interaction between the localized and carrier spin systems that this formal separation could be applied with confidence [19]. Subsequent work by Kester and by others, including the recent work by Marchand and Amiell[20] fully supported the belief in correctness of such separation procedure. Having a reliable separation technique on hand, I have first shown that introduction of alkali donors does not lead to creation of any additional localized spin centers [19]. Taking after that a well-graphitized material with a depressed Fermi level by boronation, I have swept across the band overlap region back and *In terms of classical statistics the ~ramag~letic susceptibility is x = CNefflT but N,, is composed of two parts: a constant number of localized centers NL and a variable number N, of conduction spins which can be flipped by thermal energy. The effective number NC is equal to the density of states n(E) at the Fermi level times the depth
akT from which the spins on the average can be raised. Thus N eff = IV,,+ akTu(E) and x = ( CNL/?‘) +c&CE(E), the first being a Cune type and the second a Pauli type of paramaglletism.
forth by introducing and driving out sodium and checked that the minimum value of n(E) is well reproduced in these experiments 1211. Having this information we were ready to investigate pregraphitic materials. Since most of this work has been published [22,23] let me here review it briefly without going into details and limiting myself only to those results which concern directly the problem of the band structure of carbons. Fig. 3 gives the variation of the total absorption intensity as observed in the process of heattreatment and its separation into localized and carrier components. On the same figure, the minimum values of n(E) as found in sweeping back and forth across the band overlap by sodium doping are shown. The minimum values were determined directly by intensity measurements in combination with temperature dependence separation technique. In the graphitization range where the Fermi level is located already in the overlap region an increasing overlap with increasing graphitization is indicated by the increasing ESR absorption intensity (a result supporting the Fig. 2). On the other hand in the pregraphitic range an increasing coalescence of the bands is observed as lower HTT’s are taken. Certainly no touching band scheme nor band gaps are brought into evidence. It might be interesting to note that in graphitization range for soft carbons very little if any increasing overlap is observed (almost no intensity minimLIm; unpublished results) in distinction to the case of the P33 carbon black given in Fig. 3. It will be shown below how this observation ties in nicely with the Hall effect observations. There is one more important information which can be drawn from these ESR experiments. By observing the amount of introduced sodium needed to reach the minimum value of n(E) it was established in a qualitative way that the total concentration of holes Nh(traps) increases as lower HTT’s are taken, fast in the range HTT
ELECTRONIC
200
PROPERTIES
OF CARBOSS
-
100 z!? h x E e m
50-
i 6 = g 5
zo-
: uo c :: IO -
600
1000
14cxJ
Heat
1800
treatment
temperature
2200
2600
3coo
-
,“C
Fig. 3. A graph presenting schematically the results of the work on electron spm resonance in carbons and on the separation of the spin resonatlce intensity into localized and carrier contributions. The variation with heattreatment is given along the horizontal axis [221. The variation with the position of the Fermi level in doping as obtained for a given heattreated carbon is indicated by vertical down and up lines with arrows giving the direction of increasin donor concentrations[23]. Note that in the process of sodium doping, t7le concentration of localized qi>in centers remams unchanged. An increasing density of states with increasing disorder below H7‘T 2.500” is obser\ed. and slower beyond (HT’T 20002400-2000 1600). In view of the increasing coalescence of bands (and of the lack of knowledge of the ionization efficiency of the donor) the increase Emin.
in the integral
J1;$E) dE might or might not fi mean a progressing depression of the Fermi level on the energy scale. Whereas t,he ESR technique furnishes the two quantities TL(E~) and (qualitativelyiN,, = Emill S,;(E)
dE, other closely related quantities are
obtained from studies of the Hall coefficient. We have recently been carrying out a series of experiments on Hall coefficient for variously doped soft carbons at their various stages of heattreatment [24]. When doping
are introduced and the Fermi level sweeps across the band overlap. the Hall coefficient A goes throtlgh a maximum alld a minimum (Fig. 4). By taking a look a~. the elementary formula for two carrier conduction* one can see that at 0°K the Hall coefficient extremes will occur slightly inside of l.he overlap limits, whereas they will decrease and move out to the outside of the overlap region as the temperature is increased. The limiting values of ‘4 increase rapidly; as the overlap decreases, the 0°K values tending to 2~0 for a touching band model (or for a model with an energy gap where 22 is attained at the band edges). Even in cases where quantitative agents
104
S. MROZOWSKI
Fig. 4. Graphical comparison of the variation in the Wall coefficient A for two cases of band overlap as the Fermi level sweeps across the overlap. A large change in total variation of A resulting from a small change in overlap illustrates the sensitivity of the Hall coefficient to such changes. The curves were drawn for equal mobility of holes and electrons (crossing point of curves for 0°K and T > 0°K on the zero axis).
information concerning the exact position of the Fermi level at each stage is not obtainable (for that the exact shape of the n(E) dependence on E and the ionization efficiency of the doping agent must be known: a quite heavy order when it is n(E) what we want to find something about) the total amplitude in variation of A (A\A= A,,, - Ami,,) as the Fermi level sweeps across the overlap gives some information about the extent of the band overlap (see Fig. 4). The AA is an extremely sensitive measure of the overlap a decrease in overlap by one-half causes an increase of AA by a factor of four! The very large maximum values of Hall constant measured for P33 and Thermax carbon blacks[5] must be due to a small overlap indicated by the presence of minimal values in n(E) on the heattreatment curve (see Fig. 3 and Fig. 10 of Ref. [23]). In the heattreatment process one does not sweep across the overlap however - the Fermi level is moving up with advancing graphitiza-
tion, but only less than half of the way across of the complete A curve of Fig. 4 is covered. Unfortunately, the solid binderless carbon black samples are very fragile and no doping experiments could be performed to investigate the A curves, in particular around the n(E) minimum. We had therefore to limit our work to carbons made of soft filler and soft binder which materials do not show such a distinct minimum on the ESR intensity vs heattreatment curve. Figure 5 condenses the main results of our work [24]. Before discussing them in their entirety, I should like to focus attention on the behavior of well-graphitized polycrystalline materials (2600” c HTT C 3200°C) shown in the last graph on Fig. 5. In our experiments, we have introduced boron by a relatively low temperature technique (heating at 1700°C) in which no new vacancies to be filled with boron were created-some boron atoms fill the available vacancies without changing the Fermi level, but the greatest part of the doping agent lodges itself between the microcrystals [21,23]. No evidence was found of spreading the graphitic layers (interstitial atoms). Similarly in our sodium dopings, although sodiums go in-between the layers at first, they do migrate out very quickly on cooling, the doping ions being located preferentially at defects or in spaces between the microcrystals [23]. In other obtained were in words, the compounds both cases mostly of residual type. On the other hand, potassium is held firmly between the layers and what is investigated are certainly to a great extent lamellar compounds. The striking difference in the magnitude of variat.ion of the Hall coefficient for residual and lamellar compounds as seen in Fig. 5 was already noted by Hennig[25]. Evidently the introduction of a doping agent can modify the band structure, that very thing which one wants to investigate. Introductions of interstitials increases the interlayer distance c and should therefore decrease the band overlap. Consequently the
ELECTRONIC
PROPERTIES
K-Kendall T-Texas
L.N. R.T
---
OF (:ARRONS
coke coke
Kl650-Tl710 An= z6.f3
I )
k /I
KT 2600 -3200 Lamellar
T2200
-\
\
\
No+
B-
K+
K+
\ ,/ \‘., \_I
TK
1
Fig. 5. Changes in Hall coefficient A as obsel-ved ar 2OKC; for v;lriowl\ hr;lttreated c;lrtwns with variation of the position of the Fermi level caused by doping [24]. ‘1‘11~amplit~~de AA ih definitely smaller when ;I ta1denc.y of’ forming lamcllar compo~lrlds is III-CSYII~.A maxim11n1 valr~e liar- A.4 is obtait~etl fi,r carbons jrlst lvhen turbostratic strlcctllrcs Iw@n to tr;~11s1orn~ into grapllitic fol-111.At lower heattkatrncnts an increzlsing overlap ij illtllcxtcd as t tic disorder increases.
lower
value
must
he
for
of’ AA
due
to
lamellar
another
a series of experiments
compounds
more
influetltial
opposite
direction.
HTT
2400°C
The
on the
carbon
was
in
doping
in a number of steps up to a high of level, and then a large amount
distribution of a doping agent. leads to a decrease in the total amplitude A,4. Similarly,
sodium
introduced
any
microscopic
out
tion
of
f>lctor
working
(Iearly,
in
the
any macroscopic
create
ions
nonuniformity
(periodic
variations
cause
a
band
system. as
c-oncerned
‘I’he far
or
in local
fluctuating-
geneity
non-uniformity
as
result the
is equivalent
iii distribu-
nonperiodic) potential
depression of Hall
boronated
was performed
material.
and
rvill
turned
sodium,
was closely when sodium
coefficient
is
in the
type doping
leads to reliable results, as far as the original tindisturbed band structure is concerned,
original mum
the of
In gradual minimum
A were
out that the minimum
this
inhomo-
so as to pass beyond
minimum.
values
of the energy
to an increase
if residual
the
maximum
ing this effect were discussed in previous publications (Fig. 26 in [l l] and Fig. 9 in [23]). to check
of
will
temperature or (better) to an increase in the effective band overlap. Two pictures illustrat-
In order
Hall coefficient
equal
to the
nonboronated value
maximum
value
was introduced only
slightly
value observed
and
tile
measured.
It
value
reached
of
,4 found
directly
material.
the
driving
into the
a~lc-ithe maxi-
IOIVCI. ttiall in boronation
the (Fig.
5). Evidently. the presence of’ a large amount of‘ boron and still larger amount of sodium does Ilot modify the band structure from what it is when only a small anlount of sodium is introduced resull
gives
into us
the
original
confidence
material. that
as
This far
;IS
106
S. MROZOWSKI
residual type doping is concerned, the original band structure is not greatly modified and meaningful results are obtained from such doping experiments. Looking at Hall coefficient curves for lower heattreatment the decreasing difference between Na and K dopings is noticeable. This undoubtedly is caused by the fact that as material becomes less crystallized and more disordered, potassium atoms also become lodged all throughout the material and not preferentially between the layers. In general, it seems that as the disorder in the carbon increases, the introduction of doping agents should have progressively less disturbing effect on the band structure. Figure 5 shows three important results: (1) whereas AA remains about constant in the range of heattreatments 2400-3200°C (this is the range where the plateau of diamagnetic susceptibility is observed), it increases by more than 40 per cent in the vicinity of the Hall effect maximum (HTT 2150°C). This seems to be a direct evidence of slightly decreasing band overlap, a result which could not be obtained by ESR technique for a soft carbon (no definite intensity minimum on the HTT curve observed). (2) A rapid decrease of AA is observed for decreasing heattreatments below the Hall effect maximum. This indicates in complete agreement with the mentioned earlier ESR results (Fig. 3) that the two rr-energy bands must gradually fuse, the overlap reversing its previous trend and beginning to increase with increasing disorder and decreasing crystallinity. (3) Since the position above or below the zero axis of the crossing point of Hall curves measured at different temperatures tells us roughly which carriers have a higher mobility, one can see from Fig. 5, that whereas electrons are more mobile in wellgraphitized materials, in carbons heattreated to below the Hall effect maximum, the holes are the more mobile of the two. Before discussing the implications of these results in relation to the band structure of
poorly ordered and disordered systems, I should like to mention two other recent results of our work closely connected with this problem [24]. The first concerns the presence of a temperature dependent Hall coefficient minimum at about HTT 1650” and the negative values of the Hall coefficient in the range 1000” < HTT < 1720”, both believed in the past to be characteristic of the soft carbons. Figure 6 shows the dependence of the Hall coefficient on HTT for two soft carbons, one made using a Texas coke filler, the other using a Kendall coke filler. Since the much purer Kendall coke (which was produced by distillation of pitch) does not show negative values nor the negative minimum at HTT 1650, these features of the Texas petroleum coke and of the French coke GFEC must be due to impurities. The fact that by introduction of large quantities of doping agents (boron into HTT 2400”, see Fig. 5) or by strong neutron irradiation, no negative Hall values can be obtained, seems to support this contention. The interpretation of the negative values and of the minimum in terms of a band model remain however as mysterious as ever, and maybe even more so in view of the similar behavior shown in Fig. 5 for the two cokes K1650 and T1710. The second result is the surprising behavior of boron when introduced into carbons of HTT s 1900°C. Already in ESR work an anomalous behavior of boron was noted when boron was introduced into P33 carbon black samples of HTT G 2OOO”C[23]*. At least some of the difficulties were found to originate in the boron atoms forming localized spin centers of variable concentration from experiment to experiment. But it was clear at the time that this cannot be the whole story. Now we have found that whereas *The particularly investigated samples in [23] were HTT 2000” and 1700”; the reference that they were HTT 2200 and 1700”, was a printing mistake.
ELECTRONIC:
PROPERTIES
for HTT > 1900” the boronation still changes the Hall coefficient in the direction of lower HTT’s, the effectiveness decreases almost to a stop at HTT = 1900”, where the values of the Hall coefficient for successive boronations oscillate around the nondoped value. For lower HTT’s the trend becomes reversed: boronation changes the Hall coefficient in the direction of higher HTT’s. These results are indicated by arrows in Fig. 6. As we have ascertained, the reversal in behavior is not specific to boron, but also is observed for bisulfate doping*. Thus when the Fermi level is greatly depressed by the formation of a large number of negatively charged traps, the power of acceptors becomes insufficient to pick up electrons 1rom the depleted band-the acceptors are forced to become donors-possibly by forming some new molecular groupings. Returning to the main subject of this paper. one can say that in the two pronged advance concordant results were obtained concerning the electronic band structure of pregraphitic carbons. A minimum band overlap is found ~just when the process of order f’ormation of three-dimensional begins. Above the overlap increases from two dimensional towards the graphitic limit. Kelow. an increase in the overlap with increase of‘ disorder and decrease in crystallinity is found. ‘These results seem to fall in line with the more recent discoveries of the French investigators in the X-ray field (see for instance 126 I). At low heattreatments the tused v-bands would correspond to the weakly diamagnetic form of layers F,(acu). which in further heattreatment would transt&m into strongly diamagnetic but still turbostratic f’orm F,( WY), giving a minimum *At the time this papel- wag given, it looked to us as if the donors also were Inverting their effect about in the same region of HT’I‘. Later more refined experiments have shown however. that we have been missing the Hall coefficient maxima and that consequently this conclusion was not correct (see Fig. .5.)
OF CARBONS
107
band overlap. Finally, the process of graphitization does change the structure into F3’(~P) and F,“(a/3) the first being partly and second f~llly graphitized material. One of the ways to try to justify the band overlaps in pregraphitic carbons would be to take a two-dimensional touching band model for individual benzene ring planes, and assume variable corlcentration of traps on the microscopic scale leading to adjustment to an average Ferrni level-some k&d of smearing out of the zero position, as described previously ((23 1, Fig. 9.) McClure has indicated1271 that in case of‘ two;1 pure turbostratic arrangement. dimensional model might not be applicable due to interlayer interactions. and has proposed a mixed distribution of energy surfaces. This would explain the increase in band overlap without the need of introduction of variable concentration of traps. However, no matter how attractive. such conil)letely do not seem considerations satisfying since not all the carbon is in f’cjrm of benzene ring planes; a morfl general viewpoint seems to be called for which would be extendable to even less ordered but electronically well conducting systenls. It was pointed out by a number of people [28], that when some disorder is introduced into a lattice localized energy levels will appear in the otherwise empty energ-y gap. (;ubano\ 1291 has developed arguments for stich a model of “amorphization” and Young [30] has used these ideas to interpret the properties of a glassy carboll. .4cc_ording to these views for a turbostratic type of‘ carbon. the region of the low values ot r/(e) in the neighborhood and across the contact region of the r-bands woultl w4th advancing disorder become filled with localized nonconducting eneqq states. .l’hese energy states would be observable in the ESK, but being nonconducting could Ilot in itiny way aff‘ect the Hall coefficients dependence on the position of the Fermi level. contrary to our experimental results which show definitely that below HTT 2100” the region of’
S. MROZOWSKI
108
2
Texas coke Kendall coke
-
----
Heat treatment
temperature,
‘C
Fig. 6. Detailed study of the Hall coefficient A (20KG) in the heattreatment range 1450-2000°C for Texas and Kendall coke carbons at room (RT) and liquid nitrogen (LN) temperature[24]. The Kendall coke which is less impure than Texas coke does not show the HTT 1650” minimum. The wiggles on the curves are probably connected with temperature ranges where impurities are being driven out. The changes caused by doping with boron and with H,SO, are indicated by arrows. Wiggles in the arrows indicate variable direction in consecutive boronations. It is evident that the effect of doping changes sign somewhere between HTT 1800, and HTT 1900”.
the
minimum
of
conduction levels. To understand
n(E)
becomes
filled
what
happens
one
with might
have to look between the
more closely at the interactions carriers and the lattice. When
an
moves
solid
electron its free
path
have an energy the whole solid,
through is very short-
state extending and in moving
continuously energy with the point of view of the
a
disordered it does
not
throughout it exchanges
the lattice. From electron alone it
moves up and down along a bumpy road. If the potential field of the lattice be stationary, the depressions in these surfaces would represent localized states, such as were discussed by Gubanov. But the electronic coupling to the lattice being so strong, the separation of the electronic energy from the lattice vibrations under these conditions is a
half measure and might have little or no meaning. Neither the electronic energy nor the lattice vibrations can separately even roughly be quantitized - the situation being similar to the case of polyatomic molecules, where a combination of both in terms of vibronic levels has to be considered. Should it be so, the twisted levels would be in constant motion; Gubanov’s localized depressions would travel all throughout the solid. It would be the task of the theoreticians to analyse the effect of the electric field on these entities (electrophonons), to clarify the meaning of mass and acceleration, and to derive selection rules for transitions at energy level crossings. * How much will be achiev*Note added in proof: It may be noted that the phonon drag effect would in terms of this model receive a new and interesting interpretation.
ELECTRONIC
able by analytical
methods
PROPERTIES
and how much the
will have to be left to giant computers future
only might tell.
REFERENCES 1. Mrozowski S.. Phys. Rev. 77,838 (1950). 2. Mrozowski S., Phys. Rev. 85, 609 (1952); Errata Phys. Rev. 86, 1056 (1952). 3. Kmetko E. A., Phy~. Rev. 82,456 (1951). 4. Mrozowski S.,J. Chem. Phys. 21,492 (1953). ii. Chaberski A., Ph.D. Thesis, State Univ. of New York at Buffalo (1965). 6. Boy F. and Marchand A. Carbon 5,227 (1967). 7. Mrozowski S. and Chaberski A., Phys. Rev. 104, 74 (1956).
8. Inagaki
M., Komatsu Y. and Zanchetta J. V.,
Carbon 7, 163 (1969). 9. Komatsu Y., Carbon 7,229
(1969).
10. Zanchetta J. and Mrozowski S., C.R. Acad. Sci. II. 12. 13. 14.
15.
264, 1621 (1967). Toyoda S. and Mrozowski S., Carbon 7, 239 (1969). Hishiyama Y.. Carbon 8,259 (1970). van der Hoeven B. J. C. and Keesom P. H., Plzys. Re71. 130, 1318 (1963). van der Hoeven B. J. C., Keesom P. H., McClure J. W. and Wagoner G., Phy~. RP~J. 152, 796 (1966). Delhaes P. and Hishiyama Y., Carbon 8, 31 (1970).
OF CARBONS
109
16. Kamiya K., Mrozowski S. and Vagh A. S., Carbon. To be published. 17. Wagoner G., Proc. Fourth Carbon Cons, p. 197. Pergamon Press, Oxford (1960). 18. Singer L. S. and Wagoner G., Proc. Fifth Carbon ConJ, Vol 2, p. 65. Pergamon Press, Oxford (1963). 19. Mrozowski S., Carbon 3,305 (1965). A. and Amiell J.. Carbon. To be 20. Marchand published. 21. Mrozowski S., Carbon 4,227 (1966). 22. Arnold G. and Mrozowski S., Carbon 6, 243 (1968). 23. Mrozowski S., Carbon 6,841 (1968). 24. Bulawa J., Mrozowski S. and Vagh A. S., Carbon. To be published. 25. Hennig G.,J. Chem. Phys. 19,922 (1951). 26. Maire J., Thesis, University of Paris (1967). 27. McClure J. W., Sympo.tium on Carbon, 11. 14. Tokyo (1964). 28. See the review by N. F. Molt, Advances in Physics 16, 49 (1967). Surprisinglv in spite of the all-inclusive title Electrons 1,~ IIiAorderrd Structures not a word is said about the most important example of disordered structure, namely, carbons. 29. Gubanov A. I., Quantum electron theory of amorphous conductors. Consultants Bureau, New York (1965). 30. Young D. A., Carbon 6, 135 (1968).
Carbon Research
PROPERTIES AND BAND MODEL OF CARBONS*
S. MROZOWSKI Laboratory and Department of Physics, State University of New York at Buffalo, Buffalo, New York, 14214 U.S.A. (Received
12June
1970)
Abstract-The early electronic energy band model of carbon, as proposed in 1950, and its later modifications such as faster collapse of energy gap and the presence of band overlap in graphites are recalled to introduce the main topic of the paper. The problem of the band structure in the range between the collapse of the energy gap and the development of the overlap, that is for fine crystalline turbostratic and for disordered structures, was in recent years attacked in our laboratory by four different experimental techniques. Two of these turned out to furnish interesting results, which unfortunately, cannot yet be related to the band structure. However, the two other techniques, the electron spin resonance and the Hall coefficient studies, have yielded definite and concordant information: starting from well graphitized material as the disorder increases the overlap decreases at first, but after passing through a minimum it starts to increase again. The possible sources of such overlap and the general requirements for a band theory of disordered structures are discussed. Two additional new results, namely, the absence of the Hall minimum at HTT 1650” for the soft Kendall coke, and the reversal in the effect of boronation and of H,SO., doping below HTT 185O”C, are mentioned.
I‘wenty years ago, when I was just starting to organize carbon research at the University of Buffalo, the carbon field was wide open to the physicists. The pioneering researches of Warren and Riley on X-ray structure, of Ganguli and Krishnan on diamagmetic susceptibility and of Coulson and Wallace on electronic band structure of a two dimensional graphite lattice having demonstrated 11ow interesting and fruitful such work can be, we, that is myself and my collaborators fanned out at first in many directions of exploration. As the carbon field started to attract more attention and the competition rose, and as obtaining meaningful results began to require more precise and sophisticated apparatus and better standardization *George Skakel Memorial Award Lecture, 9th Biennial Conference on Carbon, Boston Massachusetts, June 19, 1969.
experimental have of techniques, we gradually narrowed our efforts to one, still broad field of carbon physics, that is to electronic properties. In this paper, I would like to discuss some of our more recent experimental findings and describe the way they have modified our thinking about the electronic processes and the band structure of carbons. Today’s physics of the solid state is basically still physics of crystals. This was true even more so 20 years ago. But carbons are disortlered structuresL: In the first stages of formation from organic compounds they rescmhle polymers, in some cases with preferential organization and quasi-crystallisation occurring already very early in their evolution (raw cokes). In later stages, a polycrystalline structure develops changing gradually from turbostratic into graphitic order. It is only at the final stages of growth of crystallites that $17
CARVd.!lNo.%A
S. MROZOWSKI
98
carbons can be considered as agglomeration of graphite crystals, and band consideration can be safely applied. At the other end of the evolution one can visualize the process of carbonization starting from an organic crystal via increasing aromatic ring condensation. Akamatu and Inokuchi have shown that such organic crystals are intrinsic semiconductors with an energy gap rapidly decreasing with increase of the molecular size. Thus at the two ends of the process, one had reasonably well established and theoretically supported band models of the solids, with a wide variety of disordered materials in various stages of organisation linking the two ends for which an electronic model based on theoretical considerations was (and still is) not available. As it frequently happens in the course of events, when a theoretical solution to a problem is unattainable, an experimental physicist must step in with a temporary substitute. This will permit him to pick out questions to be asked of nature and to inter-
\
Aromotit
pret the answers obtained in terms of a model so that further questions can be formulated. Twenty years ago, I tried to do just that, and as the reader will see later, I am still continuing these efforts. To link the two ends of the carbonization-graphitization series, I have assumed that a band model is applicable throughout and that the transition is gradual [ 1,2]. The assumption of the applicability of the band model to disordered systems was not so unjustified at the time as it might have sounded: changes in electric conductivity of metals and semiconductors upon melting were known to be not very great, in spite of the complete collapse of periodic order. Today, there are many more indications available, that for disordered systems something of a kind of “energy bands” is present. Some modifications necessary to be introduced into the classical energy band picture to fit it to disordered solids will be suggested at the end of this paper. Figure 1, shows the original band model as proposed for carbons in 1950. On the left
crystal Chors Raw
-
Corbons
Calcined
coke
-
coke
Polycrystolline graphite
Single crystal
n(E)\
HTT
I 700
I
1000
I 1500
do0
25hO
30bo
OC
Fig. 1. The electronic energy band model for carbons covering the transition from simple organic crystals through disordered structures of chars and carbons up to single crystal graphite, as proposed in 1950-2 [ 1,2]. The depression of the Fermi level is caused by intrinsic traps of unknown origin, which become eliminated in graphitization. The corresponding heattreatment temperatures for a soft type carbon are indicated.
ELECTRONIC
PROPEKI‘IES
hand side the IT energy levels of aromatic molecules broadened into bands are indicated. The interband distances (gaps) decrease as the molecular size increases. Since in chars, mixtures of various sizes are present, it was assumed that the bands coalesce into two wide r-bands: an electron occupied and an empty one, which are separated by an energy gap, rapidly decreasing as the size of the aromatic systems in the carbon increases with advancing carbonization and graphitization, ending in the touching band model of a single graphite crystal. To obtain a qualitative description of the electronic properties of carbons introduction of a second assumption was necessary: that in the process of carbonization (driving out peripheral volatiles) traps are formed which remove some of the electrons from the band, thus depressing the Fermi level, and making the carbons essentially hole conductors. These traps would be gradually eliminated as a result of growth of crystals in graphitization, the Fermi level returning to the midposition, as required by the Wallace’s model for the single graphite crystal. It was pointed out at the end of the early paper[2] that this band model for carbons should be applicable not only to the carbonization-graphitization transition but also to the reverse case when an order is being destroyed by atomic displacements. As well known, the subsequent work of ours and of others in a broad way supported these two assumptions. The gradual decrease of the energy gap and appearance of continuous absorption due to hole conduction were checked in infrared for chars[3]. Studies of electrical resistivity, Hall and thermoelectric effects in graphitization and in reverse processes such as neutron irradiation amply demonstrated the existence of traps. Still many years later, not only the nature of the traps is unknown but even their relation to the defects in the lattice and to the presence of the localized spin centers in carbons remain
OF (:ARBONS
!)!I
obscure. The progress seems disappointingl) slow. Very soon after having proposed the band model Fig. 1, I realized that it requires modification. At first I had mistakenly believed that the presence of a tiny energ gap is responsible for the low temperature dependence of the electric resistance in polycrystalline graphite [2 I. Sonle colleagrles pointed out to me however that the gap for aromatic systems decreases at such fast rate with increase in molecular si;le. that it should practically close on calcination of soft cokes (HTT - 1300°C). Furthermore, the actual decrease in the observed average gap should be even faster than that, the aromatic structures with various trap concentrations shifting relatively in energy scale val-ious amounts so to adjust their individual Fernli levels to the common Fermi level of the solid. Accordingly in the modified model Fig. 2. the gal) is indicated to disappear at full carbonization (after driving out of most of hydrogens). The second modification came about from a careful consideration of all known properties of graphites. This led me to conclude that the energy bands in real graphite do not just touch, as required by the Wallace’s m&lel. but must slightly overlap [4]. ‘Illis conclusion has been shown two years later to be correct by theoretical calculations of Slonczewski and Weiss and others. But at what stage of heat treatment does the overlap develop? Studies of the temperature and HTT dependence of the Hall coefficient in particular the IOU temperature studies by Chaberski 1.51, shop definitely that a partial overlap is alreacl~, present at HTl“s corresponditlg to 111~. positive maximum of the Hall coefficient (about HTT 21OOY: for sof’t carbons) thr* Fermi level moving at that stage into thca overlap region. On the other hanc-I, the overlap can not develop much 5ooncr, sincts as X-ray diffraction studies show, it is only in this heattreatment region that turbostratit crystals begin to transform slowly into graphiric order (N, 6. U, b). Since the hnd
S. MROZOWSKI
100
\ n (El
HTT
II
700
1000
I
I
1500
2000
I
2500
I 3000
-C
Fig. 2. The improved electronic band model for carbon, incorporating some early modifications, namely a faster collapse of the gap and the development of a band overlap in graphitization. The four boxes indicate the four .types of experimental research aimed at clarification of the band structure in thx transition region. overlap is the result of tridimensional order, the proper band model for turbostratic crystallites seemed to be the two dimensional touching band scheme. So it was then to be expected that the overlap starting from zero will increase gradually up to the graphite limit as more and more pairs of neighboring planes rotate into proper graphitic order inside of a turbostratic crystal. Our recent studies of the ESR and Hall effects seem to support this expectation, as will be discussed later. Correspondingly in Fig. 2 an increasing overlap is indicated, with the Fermi level entering the overlap at an early stage. What is the band structure of carbon like in the region between the closure of the gap and the development of the overlap? As experimental data piled up throughout the years, I had more and more doubts if a conventional band model could be used to describe the behavior of carbons. I do believe that Boy and Marchand[6] finally delivered the “coup de grace” to the convent.ional model, by taking various models with and without an energy gap and with various depressions of the Fermi level and by trying to fit the parameters so as to obtain a reasonable interpretation of electronic properties. They found that no proper values of para-
meters can be selected which would give simultaneously a correct description of three electronic properties considered: para, dia-magnetism and Hall coefficient. Thus the conclusion was clear: a band model for disordered systems if it does exist must be an unconventional one. During the sixties, myself and my collaborators have continued to attack vigorously the problem of electronic properties of carbons. The work was aided by acquisition of some good equipment in particular for ESR and for low temperature studies, but much more by the developed experimental technique for a rapid study of the band structure, consisting of introduction of acceptors and donors in steps and then lowering of the position of the Fermi level by gradually oxidizing out the donor. We have attacked the problem from four the magnetoresistsides; by investigating ante, the specific heat at low temperature, the electron spin resonance (ESR) and the Hall coefficient (Fig. 2). As well known, nature always repays hard work, but frequently not in the type of currency one expects. This happened to us in two out of four cases: in our work we have stumbled upon some interesting effects, but
ELECTRONIC
PROPERTIES
it turned out that no direct information about the band structure could be obtained from these experiments. The first case was the negative magnetoresistance. Since the time we have shown with Chaberski [7] that the negative magnetoresistance is an effect characteristic to carbons in heattreatment range just before the incipient graphitization, I have hoped that important information about the band structure of carbons could be extracted from such investigations. Detailed low temperature studies revealed a number of interesting facts, such as the low temperature turnaround point, the negative magnetoresistance turning positive at sufficiently low temperature[8,9], the saturation of the negative magnetoresistance higher magnetic at at low tields, its square field dependence fields, etc., the assembled data supporting the original supposition of the additivity of the two magnetoresistance effects of opposite signs 171. However, the experiments with doping have shown that while the positive magnetoresistance is readily destroyed by doping (which disturbs the graphitic order by separating the layers and shifts the Fermi level), the negative effect almost does not depend on the position of the Fermi level iI1 the valence as well as in the conduction band [lo, 1t]. Thus, the liegative magnetortGstance is an effect caused by scattering and is ;I measure of disorder (no negati\.e eftect iir graphite crystal [12]. It does not furnish XII? direct information concerning tile band structure. The second case is the specific heat. When \‘an der Hoeven and Keesom [ IS] have shown that the linear component of specific heat of graphite can be obtained with a reasonably good precision if data are obtained in the HeZl temperature range, we started preparations for such a project in the Ilope that measurements carried out on carbons would give us directly information about the density of energy states n(E) at the position of the Fermi level. Doping of the specimens
OF CARBONS
101
would widen the range of information obtained. However, even before our experiments were started the skies became clouded by the publication of a subsequent paper by Van der Hoeven, et al. [14] in which it was reported that whereas boron doping causes a theoretically expected change in the linear term the difference observed between the polycrystalline graphite and Madagascar single crystal graphite is too large to be band explainable in terms of electronic model, Our work on carbons has not only confirmed but amplified this discrepancy 115 I. As the heat-treatment (and degree of graphitization) of the polycrystalline material decreases the linear term increases much faster than any electronic band model would reasonably allow. More recent measurements of‘samples of raw carbon (H’l‘~1‘ 7OO”C),which do not exhibit electron conductivity have shown the presence of a very large linear component [16]. Consequentlv. most of the effect must be due to lattice \ihrations of’ ;I special kind, not yet theoreticallv understood. the situation becoming inverted relative to our original expectations: I;nowledge of the density of states at the Fermi level rl(E,,) from other source becoming necessary for the e\.aluation of the new and unexpected linear lattice contribution. So we are left fi)r the time being with onI>, two avenues of approach: the electroil slain resonance (ESR) and the Hall effect. I,et mc discuss the information furnished bv the E:SR investigations first. Essentially the I&R is giving very strai~ghtfor\~ard answers: the integrated absorption intensity gives the spin concentration in case of‘ localized spins antI roughly the density of‘ energy states at tI~(k Fermi level r@,.,) in case of electron and hf)Ic carriers. The fil3t approach IY;I~used tn. man\ investigators to e\.aluate the spin concentration in chars, the second In Wagoner and Singer [17, 181 to demon&te the car-ricr nature of the ESR in single crystals ant1 highly graphitized polycrystallirie materi;tls. In the late fif‘ties and early sixties. ;i
102
S. MROZOWSKI
number of people began to suspect that the ESR in carbons is due to both localized and carrier spin centers, and as soon as more precise temperature dependence data for the absorption became available, it was noticed that the total absorption intensity can be represented as a sum of a pure Curie term for localized spins and an approximately temperature independent Pauli term for conduction carriers*. However, not. having an explanation why only a single ESR line with a g-value varying from experiment to experiment is always observed, made such an assumption of the composite nature of the line and of the additivity of intensities of questionable validity. It is only after I was able to show that the variations in the g-value are resulting from the changes in relative contributions of carrier and localized spins to total paramagnetic susceptibility, the g-value and its temperature dependence being predictable on basis of a simple mixing formula presupposing an exchange interaction between the localized and carrier spin systems that this formal separation could be applied with confidence [19]. Subsequent work by Kester and by others, including the recent work by Marchand and Amiell[20] fully supported the belief in correctness of such separation procedure. Having a reliable separation technique on hand, I have first shown that introduction of alkali donors does not lead to creation of any additional localized spin centers [19]. Taking after that a well-graphitized material with a depressed Fermi level by boronation, I have swept across the band overlap region back and *In terms of classical statistics the ~ramag~letic susceptibility is x = CNefflT but N,, is composed of two parts: a constant number of localized centers NL and a variable number N, of conduction spins which can be flipped by thermal energy. The effective number NC is equal to the density of states n(E) at the Fermi level times the depth
akT from which the spins on the average can be raised. Thus N eff = IV,,+ akTu(E) and x = ( CNL/?‘) +c&CE(E), the first being a Cune type and the second a Pauli type of paramaglletism.
forth by introducing and driving out sodium and checked that the minimum value of n(E) is well reproduced in these experiments 1211. Having this information we were ready to investigate pregraphitic materials. Since most of this work has been published [22,23] let me here review it briefly without going into details and limiting myself only to those results which concern directly the problem of the band structure of carbons. Fig. 3 gives the variation of the total absorption intensity as observed in the process of heattreatment and its separation into localized and carrier components. On the same figure, the minimum values of n(E) as found in sweeping back and forth across the band overlap by sodium doping are shown. The minimum values were determined directly by intensity measurements in combination with temperature dependence separation technique. In the graphitization range where the Fermi level is located already in the overlap region an increasing overlap with increasing graphitization is indicated by the increasing ESR absorption intensity (a result supporting the Fig. 2). On the other hand in the pregraphitic range an increasing coalescence of the bands is observed as lower HTT’s are taken. Certainly no touching band scheme nor band gaps are brought into evidence. It might be interesting to note that in graphitization range for soft carbons very little if any increasing overlap is observed (almost no intensity minimLIm; unpublished results) in distinction to the case of the P33 carbon black given in Fig. 3. It will be shown below how this observation ties in nicely with the Hall effect observations. There is one more important information which can be drawn from these ESR experiments. By observing the amount of introduced sodium needed to reach the minimum value of n(E) it was established in a qualitative way that the total concentration of holes Nh(traps) increases as lower HTT’s are taken, fast in the range HTT
ELECTRONIC
200
PROPERTIES
OF CARBOSS
-
100 z!? h x E e m
50-
i 6 = g 5
zo-
: uo c :: IO -
600
1000
14cxJ
Heat
1800
treatment
temperature
2200
2600
3coo
-
,“C
Fig. 3. A graph presenting schematically the results of the work on electron spm resonance in carbons and on the separation of the spin resonatlce intensity into localized and carrier contributions. The variation with heattreatment is given along the horizontal axis [221. The variation with the position of the Fermi level in doping as obtained for a given heattreated carbon is indicated by vertical down and up lines with arrows giving the direction of increasin donor concentrations[23]. Note that in the process of sodium doping, t7le concentration of localized qi>in centers remams unchanged. An increasing density of states with increasing disorder below H7‘T 2.500” is obser\ed. and slower beyond (HT’T 20002400-2000 1600). In view of the increasing coalescence of bands (and of the lack of knowledge of the ionization efficiency of the donor) the increase Emin.
in the integral
J1;$E) dE might or might not fi mean a progressing depression of the Fermi level on the energy scale. Whereas t,he ESR technique furnishes the two quantities TL(E~) and (qualitativelyiN,, = Emill S,;(E)
dE, other closely related quantities are
obtained from studies of the Hall coefficient. We have recently been carrying out a series of experiments on Hall coefficient for variously doped soft carbons at their various stages of heattreatment [24]. When doping
are introduced and the Fermi level sweeps across the band overlap. the Hall coefficient A goes throtlgh a maximum alld a minimum (Fig. 4). By taking a look a~. the elementary formula for two carrier conduction* one can see that at 0°K the Hall coefficient extremes will occur slightly inside of l.he overlap limits, whereas they will decrease and move out to the outside of the overlap region as the temperature is increased. The limiting values of ‘4 increase rapidly; as the overlap decreases, the 0°K values tending to 2~0 for a touching band model (or for a model with an energy gap where 22 is attained at the band edges). Even in cases where quantitative agents
104
S. MROZOWSKI
Fig. 4. Graphical comparison of the variation in the Wall coefficient A for two cases of band overlap as the Fermi level sweeps across the overlap. A large change in total variation of A resulting from a small change in overlap illustrates the sensitivity of the Hall coefficient to such changes. The curves were drawn for equal mobility of holes and electrons (crossing point of curves for 0°K and T > 0°K on the zero axis).
information concerning the exact position of the Fermi level at each stage is not obtainable (for that the exact shape of the n(E) dependence on E and the ionization efficiency of the doping agent must be known: a quite heavy order when it is n(E) what we want to find something about) the total amplitude in variation of A (A\A= A,,, - Ami,,) as the Fermi level sweeps across the overlap gives some information about the extent of the band overlap (see Fig. 4). The AA is an extremely sensitive measure of the overlap a decrease in overlap by one-half causes an increase of AA by a factor of four! The very large maximum values of Hall constant measured for P33 and Thermax carbon blacks[5] must be due to a small overlap indicated by the presence of minimal values in n(E) on the heattreatment curve (see Fig. 3 and Fig. 10 of Ref. [23]). In the heattreatment process one does not sweep across the overlap however - the Fermi level is moving up with advancing graphitiza-
tion, but only less than half of the way across of the complete A curve of Fig. 4 is covered. Unfortunately, the solid binderless carbon black samples are very fragile and no doping experiments could be performed to investigate the A curves, in particular around the n(E) minimum. We had therefore to limit our work to carbons made of soft filler and soft binder which materials do not show such a distinct minimum on the ESR intensity vs heattreatment curve. Figure 5 condenses the main results of our work [24]. Before discussing them in their entirety, I should like to focus attention on the behavior of well-graphitized polycrystalline materials (2600” c HTT C 3200°C) shown in the last graph on Fig. 5. In our experiments, we have introduced boron by a relatively low temperature technique (heating at 1700°C) in which no new vacancies to be filled with boron were created-some boron atoms fill the available vacancies without changing the Fermi level, but the greatest part of the doping agent lodges itself between the microcrystals [21,23]. No evidence was found of spreading the graphitic layers (interstitial atoms). Similarly in our sodium dopings, although sodiums go in-between the layers at first, they do migrate out very quickly on cooling, the doping ions being located preferentially at defects or in spaces between the microcrystals [23]. In other obtained were in words, the compounds both cases mostly of residual type. On the other hand, potassium is held firmly between the layers and what is investigated are certainly to a great extent lamellar compounds. The striking difference in the magnitude of variat.ion of the Hall coefficient for residual and lamellar compounds as seen in Fig. 5 was already noted by Hennig[25]. Evidently the introduction of a doping agent can modify the band structure, that very thing which one wants to investigate. Introductions of interstitials increases the interlayer distance c and should therefore decrease the band overlap. Consequently the
ELECTRONIC
PROPERTIES
K-Kendall T-Texas
L.N. R.T
---
OF (:ARRONS
coke coke
Kl650-Tl710 An= z6.f3
I )
k /I
KT 2600 -3200 Lamellar
T2200
-\
\
\
No+
B-
K+
K+
\ ,/ \‘., \_I
TK
1
Fig. 5. Changes in Hall coefficient A as obsel-ved ar 2OKC; for v;lriowl\ hr;lttreated c;lrtwns with variation of the position of the Fermi level caused by doping [24]. ‘1‘11~amplit~~de AA ih definitely smaller when ;I ta1denc.y of’ forming lamcllar compo~lrlds is III-CSYII~.A maxim11n1 valr~e liar- A.4 is obtait~etl fi,r carbons jrlst lvhen turbostratic strlcctllrcs Iw@n to tr;~11s1orn~ into grapllitic fol-111.At lower heattkatrncnts an increzlsing overlap ij illtllcxtcd as t tic disorder increases.
lower
value
must
he
for
of’ AA
due
to
lamellar
another
a series of experiments
compounds
more
influetltial
opposite
direction.
HTT
2400°C
The
on the
carbon
was
in
doping
in a number of steps up to a high of level, and then a large amount
distribution of a doping agent. leads to a decrease in the total amplitude A,4. Similarly,
sodium
introduced
any
microscopic
out
tion
of
f>lctor
working
(Iearly,
in
the
any macroscopic
create
ions
nonuniformity
(periodic
variations
cause
a
band
system. as
c-oncerned
‘I’he far
or
in local
fluctuating-
geneity
non-uniformity
as
result the
is equivalent
iii distribu-
nonperiodic) potential
depression of Hall
boronated
was performed
material.
and
rvill
turned
sodium,
was closely when sodium
coefficient
is
in the
type doping
leads to reliable results, as far as the original tindisturbed band structure is concerned,
original mum
the of
In gradual minimum
A were
out that the minimum
this
inhomo-
so as to pass beyond
minimum.
values
of the energy
to an increase
if residual
the
maximum
ing this effect were discussed in previous publications (Fig. 26 in [l l] and Fig. 9 in [23]). to check
of
will
temperature or (better) to an increase in the effective band overlap. Two pictures illustrat-
In order
Hall coefficient
equal
to the
nonboronated value
maximum
value
was introduced only
slightly
value observed
and
tile
measured.
It
value
reached
of
,4 found
directly
material.
the
driving
into the
a~lc-ithe maxi-
IOIVCI. ttiall in boronation
the (Fig.
5). Evidently. the presence of’ a large amount of‘ boron and still larger amount of sodium does Ilot modify the band structure from what it is when only a small anlount of sodium is introduced resull
gives
into us
the
original
confidence
material. that
as
This far
;IS
106
S. MROZOWSKI
residual type doping is concerned, the original band structure is not greatly modified and meaningful results are obtained from such doping experiments. Looking at Hall coefficient curves for lower heattreatment the decreasing difference between Na and K dopings is noticeable. This undoubtedly is caused by the fact that as material becomes less crystallized and more disordered, potassium atoms also become lodged all throughout the material and not preferentially between the layers. In general, it seems that as the disorder in the carbon increases, the introduction of doping agents should have progressively less disturbing effect on the band structure. Figure 5 shows three important results: (1) whereas AA remains about constant in the range of heattreatments 2400-3200°C (this is the range where the plateau of diamagnetic susceptibility is observed), it increases by more than 40 per cent in the vicinity of the Hall effect maximum (HTT 2150°C). This seems to be a direct evidence of slightly decreasing band overlap, a result which could not be obtained by ESR technique for a soft carbon (no definite intensity minimum on the HTT curve observed). (2) A rapid decrease of AA is observed for decreasing heattreatments below the Hall effect maximum. This indicates in complete agreement with the mentioned earlier ESR results (Fig. 3) that the two rr-energy bands must gradually fuse, the overlap reversing its previous trend and beginning to increase with increasing disorder and decreasing crystallinity. (3) Since the position above or below the zero axis of the crossing point of Hall curves measured at different temperatures tells us roughly which carriers have a higher mobility, one can see from Fig. 5, that whereas electrons are more mobile in wellgraphitized materials, in carbons heattreated to below the Hall effect maximum, the holes are the more mobile of the two. Before discussing the implications of these results in relation to the band structure of
poorly ordered and disordered systems, I should like to mention two other recent results of our work closely connected with this problem [24]. The first concerns the presence of a temperature dependent Hall coefficient minimum at about HTT 1650” and the negative values of the Hall coefficient in the range 1000” < HTT < 1720”, both believed in the past to be characteristic of the soft carbons. Figure 6 shows the dependence of the Hall coefficient on HTT for two soft carbons, one made using a Texas coke filler, the other using a Kendall coke filler. Since the much purer Kendall coke (which was produced by distillation of pitch) does not show negative values nor the negative minimum at HTT 1650, these features of the Texas petroleum coke and of the French coke GFEC must be due to impurities. The fact that by introduction of large quantities of doping agents (boron into HTT 2400”, see Fig. 5) or by strong neutron irradiation, no negative Hall values can be obtained, seems to support this contention. The interpretation of the negative values and of the minimum in terms of a band model remain however as mysterious as ever, and maybe even more so in view of the similar behavior shown in Fig. 5 for the two cokes K1650 and T1710. The second result is the surprising behavior of boron when introduced into carbons of HTT s 1900°C. Already in ESR work an anomalous behavior of boron was noted when boron was introduced into P33 carbon black samples of HTT G 2OOO”C[23]*. At least some of the difficulties were found to originate in the boron atoms forming localized spin centers of variable concentration from experiment to experiment. But it was clear at the time that this cannot be the whole story. Now we have found that whereas *The particularly investigated samples in [23] were HTT 2000” and 1700”; the reference that they were HTT 2200 and 1700”, was a printing mistake.
ELECTRONIC:
PROPERTIES
for HTT > 1900” the boronation still changes the Hall coefficient in the direction of lower HTT’s, the effectiveness decreases almost to a stop at HTT = 1900”, where the values of the Hall coefficient for successive boronations oscillate around the nondoped value. For lower HTT’s the trend becomes reversed: boronation changes the Hall coefficient in the direction of higher HTT’s. These results are indicated by arrows in Fig. 6. As we have ascertained, the reversal in behavior is not specific to boron, but also is observed for bisulfate doping*. Thus when the Fermi level is greatly depressed by the formation of a large number of negatively charged traps, the power of acceptors becomes insufficient to pick up electrons 1rom the depleted band-the acceptors are forced to become donors-possibly by forming some new molecular groupings. Returning to the main subject of this paper. one can say that in the two pronged advance concordant results were obtained concerning the electronic band structure of pregraphitic carbons. A minimum band overlap is found ~just when the process of order f’ormation of three-dimensional begins. Above the overlap increases from two dimensional towards the graphitic limit. Kelow. an increase in the overlap with increase of‘ disorder and decrease in crystallinity is found. ‘These results seem to fall in line with the more recent discoveries of the French investigators in the X-ray field (see for instance 126 I). At low heattreatments the tused v-bands would correspond to the weakly diamagnetic form of layers F,(acu). which in further heattreatment would transt&m into strongly diamagnetic but still turbostratic f’orm F,( WY), giving a minimum *At the time this papel- wag given, it looked to us as if the donors also were Inverting their effect about in the same region of HT’I‘. Later more refined experiments have shown however. that we have been missing the Hall coefficient maxima and that consequently this conclusion was not correct (see Fig. .5.)
OF CARBONS
107
band overlap. Finally, the process of graphitization does change the structure into F3’(~P) and F,“(a/3) the first being partly and second f~llly graphitized material. One of the ways to try to justify the band overlaps in pregraphitic carbons would be to take a two-dimensional touching band model for individual benzene ring planes, and assume variable corlcentration of traps on the microscopic scale leading to adjustment to an average Ferrni level-some k&d of smearing out of the zero position, as described previously ((23 1, Fig. 9.) McClure has indicated1271 that in case of‘ two;1 pure turbostratic arrangement. dimensional model might not be applicable due to interlayer interactions. and has proposed a mixed distribution of energy surfaces. This would explain the increase in band overlap without the need of introduction of variable concentration of traps. However, no matter how attractive. such conil)letely do not seem considerations satisfying since not all the carbon is in f’cjrm of benzene ring planes; a morfl general viewpoint seems to be called for which would be extendable to even less ordered but electronically well conducting systenls. It was pointed out by a number of people [28], that when some disorder is introduced into a lattice localized energy levels will appear in the otherwise empty energ-y gap. (;ubano\ 1291 has developed arguments for stich a model of “amorphization” and Young [30] has used these ideas to interpret the properties of a glassy carboll. .4cc_ording to these views for a turbostratic type of‘ carbon. the region of the low values ot r/(e) in the neighborhood and across the contact region of the r-bands woultl w4th advancing disorder become filled with localized nonconducting eneqq states. .l’hese energy states would be observable in the ESK, but being nonconducting could Ilot in itiny way aff‘ect the Hall coefficients dependence on the position of the Fermi level. contrary to our experimental results which show definitely that below HTT 2100” the region of’
S. MROZOWSKI
108
2
Texas coke Kendall coke
-
----
Heat treatment
temperature,
‘C
Fig. 6. Detailed study of the Hall coefficient A (20KG) in the heattreatment range 1450-2000°C for Texas and Kendall coke carbons at room (RT) and liquid nitrogen (LN) temperature[24]. The Kendall coke which is less impure than Texas coke does not show the HTT 1650” minimum. The wiggles on the curves are probably connected with temperature ranges where impurities are being driven out. The changes caused by doping with boron and with H,SO, are indicated by arrows. Wiggles in the arrows indicate variable direction in consecutive boronations. It is evident that the effect of doping changes sign somewhere between HTT 1800, and HTT 1900”.
the
minimum
of
conduction levels. To understand
n(E)
becomes
filled
what
happens
one
with might
have to look between the
more closely at the interactions carriers and the lattice. When
an
moves
solid
electron its free
path
have an energy the whole solid,
through is very short-
state extending and in moving
continuously energy with the point of view of the
a
disordered it does
not
throughout it exchanges
the lattice. From electron alone it
moves up and down along a bumpy road. If the potential field of the lattice be stationary, the depressions in these surfaces would represent localized states, such as were discussed by Gubanov. But the electronic coupling to the lattice being so strong, the separation of the electronic energy from the lattice vibrations under these conditions is a
half measure and might have little or no meaning. Neither the electronic energy nor the lattice vibrations can separately even roughly be quantitized - the situation being similar to the case of polyatomic molecules, where a combination of both in terms of vibronic levels has to be considered. Should it be so, the twisted levels would be in constant motion; Gubanov’s localized depressions would travel all throughout the solid. It would be the task of the theoreticians to analyse the effect of the electric field on these entities (electrophonons), to clarify the meaning of mass and acceleration, and to derive selection rules for transitions at energy level crossings. * How much will be achiev*Note added in proof: It may be noted that the phonon drag effect would in terms of this model receive a new and interesting interpretation.
ELECTRONIC
able by analytical
methods
PROPERTIES
and how much the
will have to be left to giant computers future
only might tell.
REFERENCES 1. Mrozowski S.. Phys. Rev. 77,838 (1950). 2. Mrozowski S., Phys. Rev. 85, 609 (1952); Errata Phys. Rev. 86, 1056 (1952). 3. Kmetko E. A., Phy~. Rev. 82,456 (1951). 4. Mrozowski S.,J. Chem. Phys. 21,492 (1953). ii. Chaberski A., Ph.D. Thesis, State Univ. of New York at Buffalo (1965). 6. Boy F. and Marchand A. Carbon 5,227 (1967). 7. Mrozowski S. and Chaberski A., Phys. Rev. 104, 74 (1956).
8. Inagaki
M., Komatsu Y. and Zanchetta J. V.,
Carbon 7, 163 (1969). 9. Komatsu Y., Carbon 7,229
(1969).
10. Zanchetta J. and Mrozowski S., C.R. Acad. Sci. II. 12. 13. 14.
15.
264, 1621 (1967). Toyoda S. and Mrozowski S., Carbon 7, 239 (1969). Hishiyama Y.. Carbon 8,259 (1970). van der Hoeven B. J. C. and Keesom P. H., Plzys. Re71. 130, 1318 (1963). van der Hoeven B. J. C., Keesom P. H., McClure J. W. and Wagoner G., Phy~. RP~J. 152, 796 (1966). Delhaes P. and Hishiyama Y., Carbon 8, 31 (1970).
OF CARBONS
109
16. Kamiya K., Mrozowski S. and Vagh A. S., Carbon. To be published. 17. Wagoner G., Proc. Fourth Carbon Cons, p. 197. Pergamon Press, Oxford (1960). 18. Singer L. S. and Wagoner G., Proc. Fifth Carbon ConJ, Vol 2, p. 65. Pergamon Press, Oxford (1963). 19. Mrozowski S., Carbon 3,305 (1965). A. and Amiell J.. Carbon. To be 20. Marchand published. 21. Mrozowski S., Carbon 4,227 (1966). 22. Arnold G. and Mrozowski S., Carbon 6, 243 (1968). 23. Mrozowski S., Carbon 6,841 (1968). 24. Bulawa J., Mrozowski S. and Vagh A. S., Carbon. To be published. 25. Hennig G.,J. Chem. Phys. 19,922 (1951). 26. Maire J., Thesis, University of Paris (1967). 27. McClure J. W., Sympo.tium on Carbon, 11. 14. Tokyo (1964). 28. See the review by N. F. Molt, Advances in Physics 16, 49 (1967). Surprisinglv in spite of the all-inclusive title Electrons 1,~ IIiAorderrd Structures not a word is said about the most important example of disordered structure, namely, carbons. 29. Gubanov A. I., Quantum electron theory of amorphous conductors. Consultants Bureau, New York (1965). 30. Young D. A., Carbon 6, 135 (1968).