- Email: [email protected]
Diamagnetism and charge transfer in graphite intercalation compounds
Volume 105A, number 4,5
PHYSICS LETTERS
15 October 1984
DIAMAGNETISM AND CHARGE TRANSFER IN GRAPHITE INTERCALATION COMPOUNDS T. TSANG and H.A. RESING Department of Physics, Howard University, Washington, DC 20059 USA and Code 6122, Naval Research Laboratory, Washington, DC 20375, USA Received 5 June 1984
From magnetic susceptibility data, the charge transfer parameters f a r e found to be 0.3 and 0.09 for graphite-Br 2 and IC1 intercalation compounds, and the charge screening length is about 2 A. These results are consistent with other measurements.
Because o f their two-dimensional characteristics and highly anisotropic electrical conductivities, there is considerable interest in graphite intercalation compounds (GIC) [ 1 - 3 ]. The charge transfer (CT) parameter f (fractional electronic charge transferred per intercalant molecule) is o f particular importance for these compounds. Nevertheless, there has been considerable amount o f disagreement on the values o f f obtained by different types of experiments [4,5]. The magnetic susceptibilities (MS) of GIC are very sensitive toward intercalation [ 6 - 1 1 ] and can be very useful as probes for CT. Studies have been made on GIC o f the donor types which are paramagnetic because o f the ring currents [ 12,13]. In the present work, we have used MS as probes for CT in GIC o f the acceptor type. In contrast to GIC donor types, these compounds are diamagnetic and are similar to pristine graphite itself where the diamagnetism comes from "interband" (IB) contributions [ 14]. We have calculated the MS of the GIC ofacceptor type and then inferred the CT parameter f from a rigid band model [ 1 5 - 1 7 ]. We will follow the standard notations o f the threedimensional graphite band structures [12,18,19] and use the parameters [1] (in eV) 3'0 = 3.16, 3'1 = 0.39, ")'2 = --0.020 and A = --0.008, while omitting the effects of the parameters 3'3, 3'4, and 3'5. Near the Brillouin zone edges, the 7r-electron energies E a ..... E d are given by:
Xc,d = e 2 +(e 2 + 72o2)1/2 ,
(lb)
where e I : ~x( E 1 - E 3 ) , e 2 = ½ ( E 2 - E 3 ) , X a = E a E 3, etc. Standard notations [18,19] have been used for E 1 = A + 3'1 r', E 2 = A -- 3'1 r', E 3 = ½,),2F2 , 1" = 2 cos(½kzco)and o = ½x/~a0K, where a 0 = 2.46 A and c o = 6.74 A are unit cell dimensions of graphite crystal, k z is the z-component of the wave vector, K is the perpendicular distance away from the BriUouin zone edges. We will denote ½kzc 0 as u. Without loss of generality, we may restrict to one-half o f the Brillouin • ~,~1 zone with 0 ~< u -.~ ~/a and e I / > e 2 • The Fermi level/a is at - 0 . 0 2 4 eV for graphite. Usually, we have -23' 1 < / a < - 0 . 0 2 4 for GIC of acceptor type. If the smaller band parameters 3'2 and A are omitted, then the E a and E c bands are empty and the E b band is filled up to/a in the low temperature limit. Let us define the "crossover" angle u c = cos -1 (-/a/23' 1) as the angle where the Fermi surface crosses E 2. The E d band is filled up to/a for u c < u < ½rr, but is Idled up only to E 2 for 0 < u < u c. The hole concentration p per carbon atom is p = [/a20r - Uc) - 2/23'1 sin Uc]/,VF37r23'2 .
(2)
We will now consider the presence of magnetic field H parallel to the c-axis (or z-axis) of graphite. The energy levels are now given by [14] X2(X - 2el)(X - 2e2) - B ( 2 n + 1)X(X - e I - e2)
~ka,b = e I +_(e2 + 3'02 02)1/2 ,
(la)
0.375-9601/84/$ 03.00 ©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
+ (n + ½)2B2 -- 1 B 2 = 0,
(3)
251
Volume 105A, number 4,5
where B = 72b, b = 3a2eH/2Bc is the dimensionless magnetic field, e is the absolute value o f the electronic charge, h is Planck's constant divided by 2rr, c is the light velocity, n = 0, 1, 2, ... is the quantum number. Let us consider the interval u to u + du. The last term in eq. (3) may be treated as a perturbation term and we may write X = t~ + fl where t~ is the zeroth order solution and fl is the perturbation correction. To calculate a, the -~-B 2 term is omitted, then eq. (3) may be readily factorized and we get the Landau levels O~a,b = e I +[e 2 + (n + ½)B] 1/2 , ac, d = e 2 + [e 2 + (n + 1 ) B ] l / 2
(4)
This is very similar to eq. (1) with 72o 2 replaced by (n + ½)B. The MS due to the formation o f Landau levels is however negligibly small [14]. For a graphite crystal o f volume V, the degeneracy o f each Landau level is Ddu, where
D = 8Vb/3rc2a02c 0 = (e/~c)(4V/lr2co)H. Since/x and 3'2 are usually small, the approximation e 2 = - e 1 may be used. Then the perturbation correction term has the very simple form fib =
-B2 /32el °( ° - e l ) ,
(5)
where o = [e 2 + (n + ½)B] 1/2. Since fib > 0, the energy o f the system increases when the external magnetic field H is applied along the c-axis. This is the IB contribution to diamagnetism. The forms for fla, tic and fld are similar. However, we note that a- and b-band contributions are diamagnetic while the c- and d-band contributions are paramagnetic, hence cancellations can often occur. Let us consider the IB contributions for/a < A at low temperatures. (a) For/a < E3, the a- and c-bands are empty. For u c < u < ½rr, both E b- and Ed-bands are filled up to/a and their IB contributions would cancel. For 0 < u < u c, the Eb-band is filled below # while the Ed-band is filled below E 2. The contributions from the two bands below E 2 would cancel, hence the net IB contribution would come from fib for the E b-band energy levels between E 2 and/a. (b) For # > E3, the Eb-band is occupied below E 3 only. On the other hand, theEc-bandis nowoccupied between levels E 3 and ~t. The IB contribution from the Ec-band is paramagnetic and would cancel the Eb-contribution from 2E 3 - / z to E 3. Hence the net IB contribution 252
15 October 1984
PHYSICS LETTERS
would again come from •b for the Eb-levels , but between E 2 and 2E 3 - #. The IB volume MS X0 in the low-temperature limit is given by
- ½ VXoH2 = f f
(Ddu)~bdn)
= ff(dflhdU)(2odv/B).
(6)
From eq. (5), integration over o in eq. (6) simply gives In Iv - e I I = In IXb I = In IEb - E 3 I. Using the proper integration limits, we get the low-temperature MS as
- ---'~ 47r
~-c
Co 71
e du In - - A + 2,yl cosu + 2T2 cos2u X0
c--~su
# - 272 cos2u
, (7)
where u c is given by the crossover condition/a = E 2 = A -- 271 cos u c. The combined factor (in front of the integral sign) has the value - 6 . 4 7 X 10 - 6 . The results for # > A are similar except that the crossover condition is now/a = E 1 = A + 271 cos u c. In fig. 1, the calculated values of X0 from eq. (7) are shown as functions of/a or p, where the relationship between p and /z is given by eq. (2). We have )CO = 0 for/a < - 23'1 or/z > 271. The MS XT at finite temperature T may be obtained by averaging with the density of states Ip'l = idp/d#l
qOOOp - oo ,o,,, ? ....
0
X -
50
-O.8
J
I
-0.4
I
I
0
0.4
0.8
F Fig. 1. Volume susceptibility x (in units o f ppm or 10 4 ) versus Fermi energy/~ and hole concentration p. Solid line and dashed line are for 0 K and 300 K respectively.
Volume 105A, number 4,5 -80
PHYSICS LETTERS
i
- 5C
i
~ o ~ .
o
o
-x_ I"
-40
15 October 1984
~
C
/. ~e
.p
O(
I
I
5
I0
15
5 ~ooopA
I000/T Fig. 2. Susceptibility x (in ppm) of pure graphite versus inverse temperature. Solid line, theoretical from eq. (8); crosses and circles, experimental data of refs. [20, 211.
and the derivative of the Fermi-Dirac distribution function F(~) = {1 + exp[(~ - la)/kT] ) - 1 (wherek is the Boltzmann constant):
XT (Id) =f F' (~)P'(~)Xo(~)d~ ( f F '
(~)p' ( ~ ) d ~ ) - l .
(8) The calculated room temperature (300 K) MS XR are shown as the dashed line in fig. 1. It differs from the solid line only when p is close to zero, thus we expect the temperature dependence of MS to disappear quickly on intercalation. This is in general agreement with the experimental observations on GIC with IC1 [10]. The calculated X may be compared with the anisotropy (Xc - ×a where Xc and Xa are MS for H parallel to the c- and a-axis) of the experimental volume MS multiplied b~;crystal density). The results for pure graphite (/a = - 0 . 0 2 4 eV) at various temperatures are shown in fig. 2, and there is general agreement with the experimental data o f Ganguli and Krishnan [20] and o f Pouquet et al. [21 ]. The agreement with the more recent data o f Maaroufi et al. [22] is also good. Only the room temperature MS (XR) are readily available for GIC. The dashed line o f XR in fig. 1 has been re-plotted versus the linear scale of p and is denoted as curve A in fig. 3. The experimental data [8] o f CBx are shown as the triangles using p = fx and choosing the CT parameter f = 0.7. The general agreement is good. For GIC with halogens or other large inorganic molecules (such as HNO3, H2SO4, AsFs, etc.), intercalation usually proceeds in stages [ 1 - 3 ] . For example, we have CsNBr 2 and C 13N(ICt) where N is
I0
Fig. 3. Susceptibility x (in ppm) of graphite intercalation compounds at 300 K versus average hole concentration OACurve A, theoretical, no screening, f = 0.7; solid triangles, CBx data, ref. [8]. Curve B, theoretical, f = 0.09, with screening; crosses, C 13/V (IC1) data, ref. [ 10]. Curve C, Theoretical, f = 0.-3 with screening; diamonds, open circles, and closed circles, CsN(Br2) data of refs. [6,7,91.
the stage number or the number of layers of carbon atoms in between two intercalant layers. In contrary to the case o f CBx, the hole concentration p would not be uniform for the various layers. Co is largest for the carbon layer adjacent to the intercalant layer while p is smallest for the "center" layer midway between two intercalant layers when N > 2.) For C(Br2)x and C(IC1)x, the average value o f p of the different layers is PA =fx. The layer number will be denoted asj and the "center" layer is chosen a s / = 0. A simple screening model [5] may be used to estimate p among the various layers:
pq) = C coshq/L).
(9)
Only a single parameter, the screening length L, h a s been introduced. The constant C is determined by the condition PA = F-'PQ)/N =fx where the summation is f o r j = - ( N - 1)/2 to (N - 1)/2 (over all integers or all half-integers). The MS o f GIC is then ~,X[pQ)]/N. The MS at 300 K for C 13NIC1 is shown as curve B in fig. 3, usingf = 0.09. The MS for C8N(Br2) is shown as curve C using f = 0.3. In both cases, we have chosen L = 0.6 (screening length o f 2 A). Again, there is general agreement with experimental data [6,7,9,10]. The IBr intercalate data [11] are also similar to the Br 2 compounds. The CT parameters.fare in general agreement with previous estimates [1,23]. The screening length of 2.0 A is in between several theoretical estimates [4,5,24] also. Quantitative comparisons are difficult because of the variations between different experi253
Volume 105A, number 4,5
PHYSICS LETTERS
mental samples, the scatterings of experimental data points, the possibility that band parameters may change on intercalation, etc. In conclusion, a simplified approach to MS appears to give good results for pure graphite and may also be applied to GIC. From the experimental MS data at 300 K, reasonable estimates of CT parameters and screening lengths may be obtained. The partial financial support by National Aeronautical and Space Administration grant NAG5-156 is gratefully acknowledged.
References [1 ] M.S. Dresselhaus and G. Dresselhaus, Adv. Phys. 30 (1981) 139. [2] N. Bartlett and B.W. McQuillan, in Intercalation chemistry, eds. M.S. Whittingham and A.J. Jacobson (Academic Press, New York, 1982) p.19. [3] S.A. Solin, Adv. Chem. Phys. 49 (1982) 455. [4] I.L. Spain and D.J. Nagel, Mater. Sci. Eng. 31 (1977) 183. [5] L. Pietronero, S. Strassler, H.R. Zeller and M.J. Rice, Phys. Rev. Lett. 41 (1978) 763. [6] G.R. Henni_gand J.D. McCleUand,J. Chem. Phys. 24 (1955) 1431. [7] L.H. Reyerson, J.E. Wertz, W. Weltner Jr. and H. Whitehurst, J. Phys. Chem. 61 (1957) 1334.
254
15 October 1984
[8] D.E. Soule, Proc. Fifth Conf. on Carbon, Vol. 1 (Pergamon, New York, 1962) p.13. [9] H. Suematsu, S. Tanuma and K. Higuchi, Physica 99B (1980) 420. [10] H. Okura, K. Kawamura and T. Tsuzuku, J. Phys. Soc. Japan 50 (1981) 1194. [11] P. Pfluger, P. Oelhafen, H.U. Kunzi, R. Jeker, E. Hauser, K.P. Ackermann, M. Muller and H.J. Gunterodt, Physica 99B (1980) 395. [12] F.J. DiSalvo, S.A. Safran, R.C. Haddon, J.V. Waszczak and J.E. Fischer, Phys. Rev. B20 (1979) 4883. [13] S.A. Safran and F.J. DiSalvo, Phys. Rev. B20 (1979) 4889. [14] J.W. McClure, Phys. Rev. 119 (1960) 606. [ 15 ] S. Loughin, R. Grayeski and J.E. Fischer, J. Chem. Phys. 69 (1978) 3740. [16] B.R. Weinberger, J. Kaufer, A.J. Heeger, J.E. Fischer, M. Moran and N.A.W. Holzwarth, Phys. Rev. Lett. 41 (1978) 1417. [17] L. Mattix, J. MuUiken,H.A. Resing, J. Mintmire and D.C. Weber, Syn. Metals 8 (1983) 117. [18] J.C. Slonczewski and P.R. Weiss, Phys. Rev. 109 (1958) 272. [19] J.W. McClure, IBM J. Res. Develop. 8 (1964) 255. [20] N. Ganguli and K.S. Krishnan, Proc. R. Soc. A177 (1941) 168. [21] E. Poquet, N. Lumbroso, J. Hoarau, A. Marchand, A. Pacault and D.E. Soule, J. Chim. Phys. (Paris) 57 (1960) 866. [22] A. Maaroufi, S. Flandrois, C. Coulon and J.C. Rouillon, J. Phys. Chem. Solids 43 (1982) 1103. [23] D.A. Young, Carbon 15 (1977) 373. [24] T. Ohno and H. Kamimura, Physica 117B (1983) 611.
PHYSICS LETTERS
15 October 1984
DIAMAGNETISM AND CHARGE TRANSFER IN GRAPHITE INTERCALATION COMPOUNDS T. TSANG and H.A. RESING Department of Physics, Howard University, Washington, DC 20059 USA and Code 6122, Naval Research Laboratory, Washington, DC 20375, USA Received 5 June 1984
From magnetic susceptibility data, the charge transfer parameters f a r e found to be 0.3 and 0.09 for graphite-Br 2 and IC1 intercalation compounds, and the charge screening length is about 2 A. These results are consistent with other measurements.
Because o f their two-dimensional characteristics and highly anisotropic electrical conductivities, there is considerable interest in graphite intercalation compounds (GIC) [ 1 - 3 ]. The charge transfer (CT) parameter f (fractional electronic charge transferred per intercalant molecule) is o f particular importance for these compounds. Nevertheless, there has been considerable amount o f disagreement on the values o f f obtained by different types of experiments [4,5]. The magnetic susceptibilities (MS) of GIC are very sensitive toward intercalation [ 6 - 1 1 ] and can be very useful as probes for CT. Studies have been made on GIC o f the donor types which are paramagnetic because o f the ring currents [ 12,13]. In the present work, we have used MS as probes for CT in GIC o f the acceptor type. In contrast to GIC donor types, these compounds are diamagnetic and are similar to pristine graphite itself where the diamagnetism comes from "interband" (IB) contributions [ 14]. We have calculated the MS of the GIC ofacceptor type and then inferred the CT parameter f from a rigid band model [ 1 5 - 1 7 ]. We will follow the standard notations o f the threedimensional graphite band structures [12,18,19] and use the parameters [1] (in eV) 3'0 = 3.16, 3'1 = 0.39, ")'2 = --0.020 and A = --0.008, while omitting the effects of the parameters 3'3, 3'4, and 3'5. Near the Brillouin zone edges, the 7r-electron energies E a ..... E d are given by:
Xc,d = e 2 +(e 2 + 72o2)1/2 ,
(lb)
where e I : ~x( E 1 - E 3 ) , e 2 = ½ ( E 2 - E 3 ) , X a = E a E 3, etc. Standard notations [18,19] have been used for E 1 = A + 3'1 r', E 2 = A -- 3'1 r', E 3 = ½,),2F2 , 1" = 2 cos(½kzco)and o = ½x/~a0K, where a 0 = 2.46 A and c o = 6.74 A are unit cell dimensions of graphite crystal, k z is the z-component of the wave vector, K is the perpendicular distance away from the BriUouin zone edges. We will denote ½kzc 0 as u. Without loss of generality, we may restrict to one-half o f the Brillouin • ~,~1 zone with 0 ~< u -.~ ~/a and e I / > e 2 • The Fermi level/a is at - 0 . 0 2 4 eV for graphite. Usually, we have -23' 1 < / a < - 0 . 0 2 4 for GIC of acceptor type. If the smaller band parameters 3'2 and A are omitted, then the E a and E c bands are empty and the E b band is filled up to/a in the low temperature limit. Let us define the "crossover" angle u c = cos -1 (-/a/23' 1) as the angle where the Fermi surface crosses E 2. The E d band is filled up to/a for u c < u < ½rr, but is Idled up only to E 2 for 0 < u < u c. The hole concentration p per carbon atom is p = [/a20r - Uc) - 2/23'1 sin Uc]/,VF37r23'2 .
(2)
We will now consider the presence of magnetic field H parallel to the c-axis (or z-axis) of graphite. The energy levels are now given by [14] X2(X - 2el)(X - 2e2) - B ( 2 n + 1)X(X - e I - e2)
~ka,b = e I +_(e2 + 3'02 02)1/2 ,
(la)
0.375-9601/84/$ 03.00 ©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
+ (n + ½)2B2 -- 1 B 2 = 0,
(3)
251
Volume 105A, number 4,5
where B = 72b, b = 3a2eH/2Bc is the dimensionless magnetic field, e is the absolute value o f the electronic charge, h is Planck's constant divided by 2rr, c is the light velocity, n = 0, 1, 2, ... is the quantum number. Let us consider the interval u to u + du. The last term in eq. (3) may be treated as a perturbation term and we may write X = t~ + fl where t~ is the zeroth order solution and fl is the perturbation correction. To calculate a, the -~-B 2 term is omitted, then eq. (3) may be readily factorized and we get the Landau levels O~a,b = e I +[e 2 + (n + ½)B] 1/2 , ac, d = e 2 + [e 2 + (n + 1 ) B ] l / 2
(4)
This is very similar to eq. (1) with 72o 2 replaced by (n + ½)B. The MS due to the formation o f Landau levels is however negligibly small [14]. For a graphite crystal o f volume V, the degeneracy o f each Landau level is Ddu, where
D = 8Vb/3rc2a02c 0 = (e/~c)(4V/lr2co)H. Since/x and 3'2 are usually small, the approximation e 2 = - e 1 may be used. Then the perturbation correction term has the very simple form fib =
-B2 /32el °( ° - e l ) ,
(5)
where o = [e 2 + (n + ½)B] 1/2. Since fib > 0, the energy o f the system increases when the external magnetic field H is applied along the c-axis. This is the IB contribution to diamagnetism. The forms for fla, tic and fld are similar. However, we note that a- and b-band contributions are diamagnetic while the c- and d-band contributions are paramagnetic, hence cancellations can often occur. Let us consider the IB contributions for/a < A at low temperatures. (a) For/a < E3, the a- and c-bands are empty. For u c < u < ½rr, both E b- and Ed-bands are filled up to/a and their IB contributions would cancel. For 0 < u < u c, the Eb-band is filled below # while the Ed-band is filled below E 2. The contributions from the two bands below E 2 would cancel, hence the net IB contribution would come from fib for the E b-band energy levels between E 2 and/a. (b) For # > E3, the Eb-band is occupied below E 3 only. On the other hand, theEc-bandis nowoccupied between levels E 3 and ~t. The IB contribution from the Ec-band is paramagnetic and would cancel the Eb-contribution from 2E 3 - / z to E 3. Hence the net IB contribution 252
15 October 1984
PHYSICS LETTERS
would again come from •b for the Eb-levels , but between E 2 and 2E 3 - #. The IB volume MS X0 in the low-temperature limit is given by
- ½ VXoH2 = f f
(Ddu)~bdn)
= ff(dflhdU)(2odv/B).
(6)
From eq. (5), integration over o in eq. (6) simply gives In Iv - e I I = In IXb I = In IEb - E 3 I. Using the proper integration limits, we get the low-temperature MS as
- ---'~ 47r
~-c
Co 71
e du In - - A + 2,yl cosu + 2T2 cos2u X0
c--~su
# - 272 cos2u
, (7)
where u c is given by the crossover condition/a = E 2 = A -- 271 cos u c. The combined factor (in front of the integral sign) has the value - 6 . 4 7 X 10 - 6 . The results for # > A are similar except that the crossover condition is now/a = E 1 = A + 271 cos u c. In fig. 1, the calculated values of X0 from eq. (7) are shown as functions of/a or p, where the relationship between p and /z is given by eq. (2). We have )CO = 0 for/a < - 23'1 or/z > 271. The MS XT at finite temperature T may be obtained by averaging with the density of states Ip'l = idp/d#l
qOOOp - oo ,o,,, ? ....
0
X -
50
-O.8
J
I
-0.4
I
I
0
0.4
0.8
F Fig. 1. Volume susceptibility x (in units o f ppm or 10 4 ) versus Fermi energy/~ and hole concentration p. Solid line and dashed line are for 0 K and 300 K respectively.
Volume 105A, number 4,5 -80
PHYSICS LETTERS
i
- 5C
i
~ o ~ .
o
o
-x_ I"
-40
15 October 1984
~
C
/. ~e
.p
O(
I
I
5
I0
15
5 ~ooopA
I000/T Fig. 2. Susceptibility x (in ppm) of pure graphite versus inverse temperature. Solid line, theoretical from eq. (8); crosses and circles, experimental data of refs. [20, 211.
and the derivative of the Fermi-Dirac distribution function F(~) = {1 + exp[(~ - la)/kT] ) - 1 (wherek is the Boltzmann constant):
XT (Id) =f F' (~)P'(~)Xo(~)d~ ( f F '
(~)p' ( ~ ) d ~ ) - l .
(8) The calculated room temperature (300 K) MS XR are shown as the dashed line in fig. 1. It differs from the solid line only when p is close to zero, thus we expect the temperature dependence of MS to disappear quickly on intercalation. This is in general agreement with the experimental observations on GIC with IC1 [10]. The calculated X may be compared with the anisotropy (Xc - ×a where Xc and Xa are MS for H parallel to the c- and a-axis) of the experimental volume MS multiplied b~;crystal density). The results for pure graphite (/a = - 0 . 0 2 4 eV) at various temperatures are shown in fig. 2, and there is general agreement with the experimental data o f Ganguli and Krishnan [20] and o f Pouquet et al. [21 ]. The agreement with the more recent data o f Maaroufi et al. [22] is also good. Only the room temperature MS (XR) are readily available for GIC. The dashed line o f XR in fig. 1 has been re-plotted versus the linear scale of p and is denoted as curve A in fig. 3. The experimental data [8] o f CBx are shown as the triangles using p = fx and choosing the CT parameter f = 0.7. The general agreement is good. For GIC with halogens or other large inorganic molecules (such as HNO3, H2SO4, AsFs, etc.), intercalation usually proceeds in stages [ 1 - 3 ] . For example, we have CsNBr 2 and C 13N(ICt) where N is
I0
Fig. 3. Susceptibility x (in ppm) of graphite intercalation compounds at 300 K versus average hole concentration OACurve A, theoretical, no screening, f = 0.7; solid triangles, CBx data, ref. [8]. Curve B, theoretical, f = 0.09, with screening; crosses, C 13/V (IC1) data, ref. [ 10]. Curve C, Theoretical, f = 0.-3 with screening; diamonds, open circles, and closed circles, CsN(Br2) data of refs. [6,7,91.
the stage number or the number of layers of carbon atoms in between two intercalant layers. In contrary to the case o f CBx, the hole concentration p would not be uniform for the various layers. Co is largest for the carbon layer adjacent to the intercalant layer while p is smallest for the "center" layer midway between two intercalant layers when N > 2.) For C(Br2)x and C(IC1)x, the average value o f p of the different layers is PA =fx. The layer number will be denoted asj and the "center" layer is chosen a s / = 0. A simple screening model [5] may be used to estimate p among the various layers:
pq) = C coshq/L).
(9)
Only a single parameter, the screening length L, h a s been introduced. The constant C is determined by the condition PA = F-'PQ)/N =fx where the summation is f o r j = - ( N - 1)/2 to (N - 1)/2 (over all integers or all half-integers). The MS o f GIC is then ~,X[pQ)]/N. The MS at 300 K for C 13NIC1 is shown as curve B in fig. 3, usingf = 0.09. The MS for C8N(Br2) is shown as curve C using f = 0.3. In both cases, we have chosen L = 0.6 (screening length o f 2 A). Again, there is general agreement with experimental data [6,7,9,10]. The IBr intercalate data [11] are also similar to the Br 2 compounds. The CT parameters.fare in general agreement with previous estimates [1,23]. The screening length of 2.0 A is in between several theoretical estimates [4,5,24] also. Quantitative comparisons are difficult because of the variations between different experi253
Volume 105A, number 4,5
PHYSICS LETTERS
mental samples, the scatterings of experimental data points, the possibility that band parameters may change on intercalation, etc. In conclusion, a simplified approach to MS appears to give good results for pure graphite and may also be applied to GIC. From the experimental MS data at 300 K, reasonable estimates of CT parameters and screening lengths may be obtained. The partial financial support by National Aeronautical and Space Administration grant NAG5-156 is gratefully acknowledged.
References [1 ] M.S. Dresselhaus and G. Dresselhaus, Adv. Phys. 30 (1981) 139. [2] N. Bartlett and B.W. McQuillan, in Intercalation chemistry, eds. M.S. Whittingham and A.J. Jacobson (Academic Press, New York, 1982) p.19. [3] S.A. Solin, Adv. Chem. Phys. 49 (1982) 455. [4] I.L. Spain and D.J. Nagel, Mater. Sci. Eng. 31 (1977) 183. [5] L. Pietronero, S. Strassler, H.R. Zeller and M.J. Rice, Phys. Rev. Lett. 41 (1978) 763. [6] G.R. Henni_gand J.D. McCleUand,J. Chem. Phys. 24 (1955) 1431. [7] L.H. Reyerson, J.E. Wertz, W. Weltner Jr. and H. Whitehurst, J. Phys. Chem. 61 (1957) 1334.
254
15 October 1984
[8] D.E. Soule, Proc. Fifth Conf. on Carbon, Vol. 1 (Pergamon, New York, 1962) p.13. [9] H. Suematsu, S. Tanuma and K. Higuchi, Physica 99B (1980) 420. [10] H. Okura, K. Kawamura and T. Tsuzuku, J. Phys. Soc. Japan 50 (1981) 1194. [11] P. Pfluger, P. Oelhafen, H.U. Kunzi, R. Jeker, E. Hauser, K.P. Ackermann, M. Muller and H.J. Gunterodt, Physica 99B (1980) 395. [12] F.J. DiSalvo, S.A. Safran, R.C. Haddon, J.V. Waszczak and J.E. Fischer, Phys. Rev. B20 (1979) 4883. [13] S.A. Safran and F.J. DiSalvo, Phys. Rev. B20 (1979) 4889. [14] J.W. McClure, Phys. Rev. 119 (1960) 606. [ 15 ] S. Loughin, R. Grayeski and J.E. Fischer, J. Chem. Phys. 69 (1978) 3740. [16] B.R. Weinberger, J. Kaufer, A.J. Heeger, J.E. Fischer, M. Moran and N.A.W. Holzwarth, Phys. Rev. Lett. 41 (1978) 1417. [17] L. Mattix, J. MuUiken,H.A. Resing, J. Mintmire and D.C. Weber, Syn. Metals 8 (1983) 117. [18] J.C. Slonczewski and P.R. Weiss, Phys. Rev. 109 (1958) 272. [19] J.W. McClure, IBM J. Res. Develop. 8 (1964) 255. [20] N. Ganguli and K.S. Krishnan, Proc. R. Soc. A177 (1941) 168. [21] E. Poquet, N. Lumbroso, J. Hoarau, A. Marchand, A. Pacault and D.E. Soule, J. Chim. Phys. (Paris) 57 (1960) 866. [22] A. Maaroufi, S. Flandrois, C. Coulon and J.C. Rouillon, J. Phys. Chem. Solids 43 (1982) 1103. [23] D.A. Young, Carbon 15 (1977) 373. [24] T. Ohno and H. Kamimura, Physica 117B (1983) 611.